Orbifold Riemann–Roch for Threefolds and Applications to Calabi–Yaus

Anita Buckley

A thesis submitted for the degree of Doctor of Philosophy

University of Warwick, Mathematics Institute May 2003 Contents

Acknowledgements v

Declaration vi

Introduction vii

1 Preliminaries 1 1.1 Basic definitions and notation . . . . . . . . . . . . . . . . . . 1 1.2 Embedding in a weighted projective space . . . . . . . . . . . 2 1.3 Quotient singularities . . . . . . . . . . . . . . . . . . . . . . . 3 1.4 The Riemann–Roch formula . . . . . . . . . . . . . . . . . . . 7

2 Proof of the Riemann–Roch formula 10 2.1 Existence of the Riemann–Roch formula . . . . . . . . . . . . 10 2.2 Resolution of singularities . . . . . . . . . . . . . . . . . . . . 11 2.3 Contributions from A_{r−1} curves . . . . . . . . . . . . . . . . . 25

2.3.1 Quotients by finite groups . . . . . . . . . . . . . . . . 25 2.3.2 The Atiyah–Singer equivariant Riemann–Roch formula 26 2.3.3 The μ_{r} ⊕ μ_{r} group action . . . . . . . . . . . . . . . . . 31 2.3.4 Construction of an auxiliary cover . . . . . . . . . . . . 39 2.3.5 RR does not depend on the auxiliary cover . . . . . . . 47

2.4 Contributions from singular points . . . . . . . . . . . . . . . 53

3 Calabi–Yau threefolds 59 3.1 The definition of a Calabi–Yau and its Hilbert series . . . . . . 59 3.2 CYs in weighted projective spaces . . . . . . . . . . . . . . . . 61

3.2.1 Resolution of the graded ring . . . . . . . . . . . . . . 61 3.2.2 The computer algorithm . . . . . . . . . . . . . . . . . 63

4 Construction of examples 66

4.1 Complete intersections . . . . . . . . . . . . . . . . . . . . . . 66 4.1.1 Classification of singularities . . . . . . . . . . . . . . . 66 4.1.2 The degree of curves . . . . . . . . . . . . . . . . . . . 68 4.1.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.2 Pfaffians in codimension 3 and 4 . . . . . . . . . . . . . . . . . 73 4.2.1 The Type I unprojection . . . . . . . . . . . . . . . . . 73 4.2.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.3 The weighted Grassmannian wGr(2, 5) . . . . . . . . . . . . . 82 4.3.1 The definition of wGr(2, 5) . . . . . . . . . . . . . . . . 82 4.3.2 Quasilinear sections of wGr(2, 5) . . . . . . . . . . . . . 83 4.3.3 The Hodge numbers . . . . . . . . . . . . . . . . . . . 85

4.4 Weighted orthogonal Grassmannians . . . . . . . . . . . . . . 101 4.4.1 Definitions and the spinor embedding . . . . . . . . . . 101 4.4.2 Examples in codimension 5 . . . . . . . . . . . . . . . 103

5 Outstanding questions 111

Appendix 113 A.1 The Riemann–Hurwitz formula . . . . . . . . . . . . . . . . . 113 A.2 Orbifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 A.3 The Hodge numbers and mirror pairs . . . . . . . . . . . . . . 115

References 117

List of Figures

I Resolution of an A_{r−1} point . . . . . . . . . . . . . . . . . . . 14 II Resolution of an A_{r−1} curve . . . . . . . . . . . . . . . . . . . 15 III Resolution of an A_{3} curve . . . . . . . . . . . . . . . . . . . . 70 IV The Type I unprojection Y → X . . . . . . . . . . . . . . . . 91 V Resolution of the dissident curve . . . . . . . . . . . . . . . . . 92 VI Fans of the exceptional surfaces . . . . . . . . . . . . . . . . . 93 VII Configuration of the exceptional surfaces . . . . . . . . . . . . 94 VIII Type I unprojection Y → X . . . . . . . . . . . . . . . . . . . 99

Acknowledgements

I am indebted to Balázs Szendrői for his constant help, encouragement and advice. His bright ideas and sharp eyes make him as excellent a supervisor as one could hope for. I thank Balázs for his time and friendship.

I was also privileged to have Miles Reid as my supervisor. I am grateful to Miles for inviting me to the University of Warwick, for many stimulating conversations and countless priceless comments.

I also benefitted immensely from Gavin Brown’s skills in Magma. I thank him for his interest in my work and long hours spent on implementing my ideas.

My research was supported by a Special Research Studentship in the Department of Mathematics, University of Warwick. I was also awarded the position of Mladi Raziskovalec (Young Researcher) by the Slovenian Ministry of Science.

Tomaž Košir from the University of Ljubljana enthusiastically followed my progress and gave of his time generously in sorting out the necessary administration.

Finally I thank my husband Dave for believing in me and being my tower of strength.

Declaration

This thesis is the result of my own work except when explicitly stated otherwise. Chapter 1 (except 1.4) and the Appendix recall material from literature. To the best of my knowledge, the formula in 1.4 and the proofs in Chapter 2 are original. Chapters 3 and 4 apply the formula to the construction and study of new examples.

Introduction

The aim of this thesis is to state and prove the Riemann–Roch formula for threefolds with quotient singularities. Let X be a normal projective threefold and D a Weil divisor which is ℚ–Cartier and Cartier outside Sing X. We prove

χ(X, O_{X}(D)) = RR type expression in D + ∑_{P} c_{P} (D) + ∑_{C} s_{C}(D),

where ∑_{P} c_{P} (D) are the contributions from the singular points of O_{X}(D) and ∑_{C} s_{C}(D) are the contributions from the curves of singularities. The contributions depend on the type of quotient singularities and on the embedding of Sing X in X.

Chapter 1 recalls notations and results from literature. It also states Theorem 1.4.2, which is the main result of the thesis.

In Chapter 2 we prove the stated formula. The contributions from the A_{r−1} curves of singularities are computed in two ways. First we resolve the singularities

f: Y ⟶ X,

so that f_{∗}O_{Y} (D′) = O_{X}(D), and then use the Riemann–Roch formula for D′ on the smooth Y. The resolution also determines the shape of the formula. In the second proof we use the Atiyah–Singer equivariant Riemann–Roch formula on the auxiliary cover

π: Z ⟶ X,

where π is a (local) quotient morphism for μ_{r} ⊕μ_{r} group action. The formula follows from comparing the singularities of O_{X}(D) and L_{i,j}, which is the (i, j)th eigensheaf of the μ_{r} ⊕ μ_{r} action on π_{∗}O_{Z}. The contributions from the singular points

\frac{1}{s}(a_{1}, a_{2}, a_{3})

are local analytic invariants and depend only on the type of singularity. Thus, they can be computed on a suitable model by equating O_{X}(D) with L_{i}, the eigensheaf of the μ_{s} action with weights a_{1}, a_{2}, a_{3}.

The Riemann–Roch formula is proved with the purpose to construct new examples of Calabi–Yau threefolds. In Chapter 3 we state the formulae for h^{0}(X, O_{X}(D)) and the Hilbert series. We describe the algorithm for the computer search of Calabi–Yaus in weighted projective spaces, by using the properties of the following graded ring

R = ⨁_{n≥0} H^{0}(X, O_{X}(n)).

In Chapter 4 we find examples of Calabi–Yaus in codimensions 1, …, 5. We give a thorough analysis of Calabi–Yau 3–folds that are complete intersections in the weighted Grassmannian wGr(2, 5) or orthogonal Grassmannian wOGr(5, 10) of Corti and Reid [8]. A method for computing their Hodge numbers by using the Type I unprojection is described in Section 4.3. The technique of unprojections is also used to construct varieties in codimension 4. In Section 4.1 we classify the quotient singularities of complete intersections in weighted projective spaces.

Chapter 5 explores interesting questions, new ideas and possibilities that arise throughout the thesis. A number of these should serve as suggestions for further research. We suggest using our computer algorithm to generate new examples; and then applying the technique of unprojection to analyse these examples, their Hodge numbers and mirror pairs.

The Appendix recalls some additional material from literature. We prove the Riemann–Hurwitz formula for smooth 3–folds and define the mirror pairs for smooth Calabi–Yaus. We include the definition of orbifolds and the dualising sheaf ω_{X}.

1 Preliminaries

1.1 Basic definitions and notation

Let X be a normal projective threefold. We define:

Div (X) = the group of Weil divisors

PDiv (X) = the group of principal divisors

Cl (X) = Div (X) / PDiv (X)

Pic (X) = Cartier divisors / PDiv (X) = the group of line bundles.

A ℚ–divisor is a formal linear combination of prime divisors with coefficients in ℚ. A divisor D is ℚ–Cartier if mD is Cartier for some positive integer m ≠ 0. The canonical divisor K_{X} is defined so that its restriction to the nonsingular part of X is a divisor of a rational 3–form. The corresponding sheaf ω_{X} = O_{X}(K_{X}) is called the canonical sheaf.

Definition 1.1.1 Let f: Y → X be a resolution of singularities of a normal variety X. We say that X has canonical singularities if

(i) mK_{X} is Cartier for some positive integer m,

(ii) mK_{Y} = f^{∗}(mK_{X}) + ∑_{i} α_{i}E_{i} with α_{i} ≥ 0,

where E_{i} are prime exceptional divisors.

If all α_{i} > 0, then X has terminal singularities.

It is known [31] that canonical 3–fold singularities are Cohen–Macaulay, which means that the local rings are Cohen–Macaulay. A variety with canonical singularities is Gorenstein if K_{X} is Cartier. X is ℚ–factorial if every Weil divisor is ℚ–Cartier.

We sometimes add the condition that rank Pic_{ℚ} X = Pic X ⊗ ℚ is 1, or even stronger Cl X = ℤ · D for some Weil divisor D. For example, if X is embedded in a weighted projective space as a complete intersection, then Pic_{ℚ} X has rank 1 by [7, Lemma 3.5].

1.2 Embedding in a weighted projective space

A weighted projective space ℙ = ℙ(a_{0}, …, a_{n}) is defined by

ℙ = Proj ℂ[x_{0}, …, x_{n}],

where deg x_{i} = a_{i}. Note that ℙ is a quotient of ℂ^{n+1} − {0} by the ℂ^{∗} action

x_{0}, …, x_{n} ↦ λ^{a_{0}}x_{0}, …, λ^{a_{n}}x_{n}.

Every weighted projective space is a normal projective variety with quotient singularities.

Let X be a normal projective threefold over ℂ with canonical singularities. On X choose a Weil divisor D which is ℚ–Cartier and Cartier outside Sing X. By [27, Appendix to §1], there exists a bijective correspondence

\{ \begin{matrix}Weil divisors \\ on X \end{matrix} \} ↔ \{ \begin{matrix}divisorial subsheaves \\ L ⊂ k(X) \end{matrix} \} /Γ(X, O_{X})^{∗}
D  ↔  O_{X}(D).

Definition 1.2.1 Let r be the index of a ℚ−Cartier divisor D. Locally identify O_{X}(rD) with O_{X}. Define an O_{X} algebra structure on

A = O_{X} ⊕ O_{X}(D) ⊕ ⋯ ⊕ O_{X}((r − 1)D),

by the multiplication of sections O_{X}(aD) ⊗ O_{X}(bD) → O_{X}((a + b)D).

The pair (X, D) is well formed if the above multiplication is an isomorphism in codimension zero and codimension one, and is quasismooth if the local ℤ_{r} covers Spec_{X}A → X are nonsingular at each point.

From here on we will always assume that (X, D) is quasismooth and well formed.

On X there exists an ample Cartier divisor H such that D′ = D + mH is also ample for some positive integer m. Note that the sheaves O_{X}(D) and O_{X}(D′) have the same type of singularities. The same way as in Definition 1.2.1 define locally

B = O_{X} ⊕ O_{X}(D′) ⊕ ⋯ ⊕ O_{X}((r − 1)D′).

Since Spec_{X}A and Spec_{X}B are isomorphic, (X, D′) is well defined and quasismooth.

The finite dimensional vector spaces

H^{0}(X, nD′) = H^{0}(X, O_{X}(nD′)) = {f ∈ ℂ(X) | div f + nD′ ≥ 0}

define the graded ring

R(X, D′) = ⨁_{n≥0} H^{0}(X, nD′).

Since D′ is ample and ℚ–Cartier, R(X, D′) is finitely generated.

In all the cases we consider in this thesis, R(X, D′) is a Gorenstein ring. This happens if and only if K_{X} is ℚ–Cartier and K_{X} is linearly equivalent to tD′ for some t ∈ ℤ. We say that such (X, D′) is projectively Gorenstein.

In other words, Proj R(X, D′) is an embedding i: X ↪ ℙ into a weighted projective space, where O_{X}(D′) corresponds to O_{X}(1) = i^{∗}O_{ℙ}(1). Here O_{X}(1) denotes the twisting sheaf of Serre.

Remark 1.2.2 Our assumptions imply that the embedded variety X ⊂ ℙ(a_{1}, …, a_{n}) is quasismooth and well formed (check [11] for definitions).

When X is quasismooth, its singularities are cyclic quotient singularities induced by the weights a_{1}, …, a_{n}. Another important fact is that a quasismooth variety is an orbifold, that is, locally a complex manifold divided by a finite group of automorphisms. This will be used in Section 2.3.

When X is also well formed, the adjunction formula holds [11, Formula 6.14].

1.3 Quotient singularities

Definition 1.3.1 The quotient π: 𝔸^{n} → 𝔸^{n}/μ_{r}, where μ_{r} acts on 𝔸^{n} by

μ_{r} ∋ ε: (x_{1}, …, x_{n}) ↦ (ε^{b_{1}}x_{1}, …, ε^{b_{n}}x_{n})

is a cyclic quotient singularity of type \frac{1}{r}(b_{1}, …, b_{n}). For the sake of clearer notation we always assume that all the b_{i} and r have no common divisor, which is equivalent to the action μ_{r} being effective. The sheaf π_{∗}O_{𝔸^{n}} decomposes into r eigensheaves

L_{i} = {f | ε(f) = ε^{i} · f for all ε ∈ μ_{r}},

for i = 1, …, r − 1. A singularity Q ∈ X polarised by a Weil divisor D is a cyclic quotient singularity of type _{i}(\frac{1}{r}(b_{1}, …, b_{n})), if Q is locally isomorphic to a point of type \frac{1}{r}(b_{1}, …, b_{n}) and O_{X}(D) ≅ L_{i} near the singular point.

The cyclic quotient singularities are special cases of quotient singularities. We say that x ∈ X has a quotient singularity if locally

(x ∈ X) ≅ (0 ∈ 𝔸^{n})/G

for a finite group G.

In Section 1.2 we restricted our study to a quasismooth, well formed threefold X embedded in a weighted projective space ℙ. The singularities of X are induced by the singularities of ℙ, which are cyclic quotient singularities. Since X is normal its singular locus has codimension 2. Thus Sing X consists of a union of intersecting curves and isolated points.

Examples 1.3.2

(1) ℙ(1, 4, 5, 15) with variables x, y, z_{1}, z_{2} has an isolated singular point P_{Y} of type \frac{1}{4}(1, 1, 3) and a curve of singularities C, which is the (z_{1}, z_{2}) line. A general point on C is of transverse type \frac{1}{5}(1, 4), but there is also a dissident point P_{z_{2}} ∈ C which is of type \frac{1}{15}(1, 4, 5).

(2) X_{13} ⊂ ℙ(1, 1, 2, 3, 6) with variables x_{1}, x_{2}, y, z, t is a Calabi–Yau threefold with two curves of singularities. The first C_{1} = {x_{1} = x_{2} = y = 0} is of type \frac{1}{3}(1, 2) and the second C_{2} = \{x_{1} = x_{2} = z = 0\} is of type \frac{1}{2}(1, 1). They intersect in (0, 0, 0, 0, 1) of type \frac{1}{6}(1, 2, 3).

Through the entire thesis we will assume that all curves of singularities are compound Du Val singularities. It is well known that the only cyclic Du Val singularities are of type

A_{r−1} = _{k}(\frac{1}{r}(1, −1)).

The precise definition is the following:

Definition 1.3.3 Let X be a normal projective threefold with a curve C of singularities and D a divisor which is ℚ–Cartier and Cartier away from C.

Take a generic hyperplane S which intersects C transversely in a finite number of points. Assume that every point in the intersection is a singular point of type _{k}(\frac{1}{r}(1, −1)) on the polarised surface (S, D|_{S}). A curve with this property will be named a curve of singularities of transverse type _{k}(\frac{1}{r}(1, −1)), or shorter a _{k}(\frac{1}{r}(1, −1)) curve. When the index k is not important we will name it an A_{r−1} curve.

Every A_{r−1} curve can contain a finite number of points of different type. We call them dissident points. In order to emphasise that C contains dissident points, we call it a dissident A_{r−1} curve.

Note that r and k in the above definition are not necessarily coprime.

Remark 1.3.4 Let P be a point of type \frac{1}{r}(a, b, c). The numbers a, b, c, r determine whether P is an isolated singular point, or it lies on a number of dissident curves. There are 4 possibilities:

• If a, b, c are all coprime to r, then P is an isolated point singularity. Consider for example (0, 0, 0, 1) ∈ ℙ(1, 1, 1, 3).

• If a, b are coprime to r, and hcf(c, r) = γ, then P is a dissident point on a curve of generically transverse type \frac{1}{γ}(a, b). A model for such singularity is ℙ(1, 4, 20, 25) with coordinates x, y, z, t. In this example, P_{t} is of type \frac{1}{25}(1, 4, 20) and lies on a dissident A_{4} = \frac{1}{5}(1, 4) curve (z, t).

• The same example demonstrates the case where a is coprime to r and hcf(b, r) = β and hcf(c, r) = γ. Note that if b, c, r had a common divisor, this would contradict the well formedness. Thus P is an intersection of two curves of different types \frac{1}{β} (a, c) and \frac{1}{γ}(a, b). In our example P_{z} ∈ ℙ(1, 4, 20, 25) is of type \frac{1}{20}(1, 4, 5) and lies on the intersection of the line (y, z) of type \frac{1}{4}(1, 1) with the line (z, t) of type \frac{1}{5}(1, 4).

• If hcf(a, r) = α, hcf(b, r) = β and hcf(c, r) = γ, then P is an intersection of three curves. As in the previous case, all the curves are of different types \frac{1}{α}(b, c), \frac{1}{β} (a, c) and \frac{1}{γ}(a, b). Consider for example P_{t} ∈ ℙ(3, 4, 35, 42) which is the intersection of a \frac{1}{3}(1, 2) line (x, t), a \frac{1}{2}(1, 1) line (y, t) and a \frac{1}{7}(3, 4) line (z, t).

Therefore, the positive integers a, b, c, r corresponding to P do not only determine the type of a point singularity, but also determine the transverse type of curve singularities for all curves that pass through P.

Often there is some confusion with the notion of the types of singularities in a weighted projective space, which we will now try to clarify.

Lemma 1.3.5 Consider X in a weighted projective space ℙ(b_{1}, …, b_{m}). Let P ∈ X be a singular point of type \frac{1}{r}(a_{1}, …, a_{n}), which is induced by the singularities of ℙ. This means that a_{i} = b_{p_{i}} mod r, for i = 1, …, n and some weights b_{p_{i}} in ℙ. Then locally at P

O_{X}(i) is of type \overline{−i}(\frac{1}{r}(a_{1}, …, a_{n})) for all i ∈ ℕ,

where denotes the smallest residue mod r.

Proof A divisor D on a projective variety by definition contains a singular point Q of type _{j}(\frac{1}{r}(a_{1}, …, a_{n})) if Q is locally isomorphic to a point of type \frac{1}{r}(a_{1}, …, a_{n}) and

O_{X}(D) ≅ L_{j} = {f | ε(f) = ε^{j} · f, ∀ε ∈ μ_{r}}

near the singular point.

Now, consider X in a weighted projective space ℙ(b_{1}, …, b_{m}) which has a singular point P of type \frac{1}{r}(a_{1}, …, a_{n}). The singularities of X are induced by the weights of ℙ, thus we can write the following relations (after permuting the weights in ℙ)

b_{1} = a_{1} mod r
⋮
b_{n} = a_{n} mod r
b_{n+1} = 0 mod r.

Without loss of generality we can assume that P = (\underbrace{0, …, 0}_{n}, 1, …) are the

coordinates of P in ℙ and that b_{n+1} = r. Moreover, if x_{1}, …, x_{m} denote the variables of ℙ, then x_{1}, …, x_{n} define local parameters of P. On the affine piece x_{n+1} ≠ 0 the group ε ∈ μ_{r} acts by

x_{1}, …, x_{n} ↦ ε^{a_{1}}x_{1}, …, ε^{a_{n}}x_{n}.

Recall that

O_{X}(k) = \{\frac{p}{q} | p, q homogeneous of deg p = deg q + k\} .

Observe that O_{X}(b_{1}) has a section x_{1} = 0. It defines a divisor D_{1} = {x_{1} = 0} and O_{X}(b_{1}) ≅ O_{X}(D_{1}). By definition O_{X}(D_{1}) = { functions with poles at most at x_{1}}. It is generated by

f = \frac{\sqrt[r]{x_{n+1}}^{b_{1}}}{x_{1}}

in the neighborhood of P. It is obvious that f ∈ L_{−a_{1}}, since ε(f) = ε^{−a_{1}} · f. Thus we proved that locally O_{X}(b_{1}) ≅ O_{X}(D_{1}) ≅ L_{−a_{1}}.

For the same reason

O_{X}(i) ≅ L\overline{−i} for all i ∈ ℤ,

where denotes the smallest residue mod r. □

For varieties with quotient singularities the following holds:

Proposition 1.3.6 ( [21], Prop. 5.15) Let X be a variety over ℂ with quotient singularities only. Then X is ℚ–factorial and has rational singularities.

1.4 The Riemann–Roch formula

Definition 1.4.1 Let an A_{r−1} curve C contain a finite number of dissident points \{P_{λ} : λ ∈ Λ\} of types \frac{1}{rτ_{λ}} (a_{1λ}, a_{2λ}, a_{3λ}). We define the index of C to be τ = lcm{τ_{λ}}.

Note that for A_{r−1} curves with no dissident points τ = 1.

Consider a threefold X and a ℚ–Cartier divisor D. Then tD is Cartier for some t ∈ ℕ. For a curve C on X we define the degree of D on C by:

deg D|_{C} = \frac{1}{t} · deg_{C} O_{X}(tD) = \frac{1}{t}C · tD.

We also define the intersection D·c_{2}(X) either as \frac{1}{t}f^{∗}(tD)·c_{2}(Y) computed on a resolution f: Y → X, or we compute \frac{1}{t} tD · c_{2}(X) on the nonsingular part of X.

At this stage we can state the main theorem of this thesis:

Theorem 1.4.2 Let X be a normal projective threefold with canonical singularities and let D be a ℚ–Cartier divisor. Assume that (X, D) is quasismooth and well formed. Let the singularities of O_{X}(D) consist of

• points P of type _{n}(\frac{1}{s}(a_{1}, a_{2}, a_{3})),

• curves C of generic type _{k}(\frac{1}{r}(1, −1)) with index τ_{C}.

Then for all positive integers m,

\begin{aligned}χ(X, O_{X}(mD)) & = & χ(O_{X}) + \frac{1}{12}mD(mD − K_{X})(2mD − K_{X}) + m\frac{D·c_{2}(X)}{12} \\ & & + ∑_{P} c_{P} (mD) + ∑_{C} s_{C}(mD), \end{aligned}

where

c_{P} (mD) = \frac{1}{s} ∑_{\begin{matrix}ε∈μs, \\ ε^{a_{i}} ≠1 ∀i=1,2,3 \end{matrix}} (ε^{−nm} − 1)\frac{1}{(1 − ε^{−a_{1}})(1 − ε^{−a_{2}})(1 − ε^{−a_{3}})}

and

\begin{aligned}s_{C}(mD) & = & −m\frac{\overline{mk}(r−\overline{mk})}{2r} deg D|_{C} + \frac{\overline{mk}(r−\overline{mk})}{4r} deg K_{X}|_{C} \\ & & +\frac{\overline{mk}(r−\overline{mk})(r−2·\overline{mk})}{12r^{2}τ_{C}}N_{C}. \end{aligned}

Here denotes the smallest residue mod r and the integer N_{C} is an invariant of X in the neighbourhood of C.

Remark 1.4.3 Note that P is either (i.) an isolated point singularity if hcf(a_{i}, s) = 1 for all i = 1, 2, 3, or (ii.) a dissident point on some curve if hcf(a_{i}, s) = α_{i} ≠ 1 for some i.

In the first case c_{P} (mD) equals to the so called basket contribution to the Riemann–Roch formula.

In the second case we always assume that a_{j} +a_{k} = 0 mod α_{i} for different indexes i, j, k. Thus, the singularities along the curve are of A_{α_{i}−1} transverse type. Since X is well formed a_{i}, a_{j}, s have no common divisor.

This formula is a version of the orbifold Riemann–Roch (RR) theorem for the singular O_{X}(mD). It is of the type

χ(X, O_{X}(mD)) = RR–type expression in mD +

contributions from singular points + contributions from curves of singularities.

It only deals with the ℚ–divisor class D ∈ Pic X ⊗ ℚ.

These types of formulae were first studied by Miles Reid in [30]. Reid considers the plurigenus formula of 3–folds with canonical singularities for which K_{X} generates Cl X. The orbifold RR for isolated cyclic singularities can be found in [23, Chapter 4]. In Chapters 3 and 4 of this thesis we apply Theorem 1.4.2 to Calabi–Yau threefolds. The Riemann–Roch formula for Calabi–Yaus with only A_{1} curves is particularly nice and was proved in [32].

2 Proof of the Riemann–Roch formula

In this chapter we prove Theorem 1.4.2. The contributions from curves will be computed in two ways: first by resolving the singularities and second by the method of cyclic group coverings. The contributions from singular points will be computed on suitable models.

2.1 Existence of the Riemann–Roch formula

We will use resolution of singularities on many occasions. We need the following version of Hironaka’s resolution of singularities.

Theorem 2.1.1 ( [21], Thm 0.2) Let X be an irreducible and reduced projective threefold over ℂ with canonical singularities. On X pick a ℚ–Cartier Weil divisor D. There exists a smooth threefold Y with a Cartier divisor D′ and a projective morphism f: Y → X such that f^{∗}O_{X}(D) → O_{Y} (D′) is surjective and f_{∗}O_{Y} (D′) ≅ O_{X}(D).

Consider the surjection f^{∗}O_{X}(D) → O_{Y} (D′). Since X is normal f_{∗}O_{Y} = O_{X}. Hence the corresponding short exact sequence

0 ⟶ K ⟶ f^{∗}O_{X}(D) ⟶ O_{Y} (D′) ⟶ 0

gives rise to a long exact sequence

0 ⟶ f_{∗}K ⟶  O_{X}(D)  ⟶^{≅} f_{∗}O_{Y} (D′)  ⟶
R^{1}f_{∗}K ⟶ R^{1}f_{∗}f^{∗}O_{X}(D) ⟶ R^{1}f_{∗}O_{Y} (D′) ⟶
R^{2}f_{∗}K ⟶ R^{2}f_{∗}f^{∗}O_{X}(D) ⟶ R^{2}f_{∗}O_{Y} (D′) ⟶ …

By Theorem (3.8) in [30] the canonical singularities in characteristic zero are rational. Thus R^{i}f_{∗}O_{Y} = 0 for i > 0. Using the projection formula we get

R^{i}f_{∗}f^{∗}O_{X}(D) = R^{i}f_{∗}(O_{Y} ⊗ f^{∗}O_{X}(D)) = R^{i}f_{∗}O_{Y} ⊗ O_{X}(D) = 0

for every i > 0. Hence the long exact sequence becomes

0 ⟶ f_{∗}K ⟶ O_{X}(D) ⟶^{≅} f_{∗}O_{Y} (D′) ⟶
R^{1}f_{∗}K ⟶  0  ⟶ R^{1}f_{∗}O_{Y} (D′) ⟶^{≅}
R^{2}f_{∗}K ⟶  0  ⟶ R^{2}f_{∗}O_{Y} (D′) ⟶^{≅}
R^{3}f_{∗}K …

Note that K is supported on the fibres of f. The fibre dimension of f is ≤ 2, thus

R^{i}f_{∗}O_{Y} (D′) ≅ R^{i+1}f_{∗}K = 0

for i ≥ 2. Finally, the Leray spectral sequence yields

χ(Y, O_{Y} (D′)) = ∑_{i}(−1)^{i}χ(X, R^{i}f_{∗}O_{Y} (D′))  (E. i)
= χ(X, O_{X}(D)) − χ(X, R^{2}f_{∗}K).

Note that R^{2}f_{∗}K is supported only over the isolated and dissident singularities.

On the smooth Y we can use the Riemann–Roch formula

χ(Y, O_{Y} (D′)) = ch(D′) · td(T_{Y}).

In the next section we will express D′ and K_{Y} with D, K_{X} and the exceptional divisors of the resolution. This will prove

χ(X, O_{X}(D)) = Riemann–Roch type expression in D +

contributions from singularities.

2.2 Resolution of singularities

We start by a technical lemma.

Lemma 2.2.1 Let X be a quasismooth and well formed threefold embedded in a weighted projective space. Assume that (X, O_{X}(1)) contains a _{k}(\frac{1}{r}(1, −1)) curve C. Then r and k are coprime.

Proof A generic point P on the curve is of transverse type A_{r−1}. For P on C choose local transverse parameters x and y. Since X is quasismooth, all singularities of X come from the singularities of the weighted projective space ℙ. In other words, ℙ has at least two variables of degrees r∗ (since it contains a singular curve), where ∗ is an arbitrary integer. Moreover, there exists an 0 < a < r such that variables x, y are of degrees a + r∗, −a + r ∗ . The cyclic group ε ∈ μ_{r} then acts by

(x, y) ∼ (ε^{a}x, ε^{r−a}y)

.

We will first prove that in the above set-up r and a have to be coprime. Assume that there exists s > 1 such that r = sr′ and a = sa′ and argue by contradiction. The quasismooth X is locally a complete intersection in the neighbourhood of P. It is given by independent equations

f_{1}, f_{2}, …, f_{n+m+l−4} ⊂ ℙ(b_{1}, …, b_{n}, rc_{1}, …, rc_{m}, sd_{1}, …, sd_{l}),

where f_{i} have degrees a_{i}, entries sd_{i} are not divisible by r, and b_{i} are not divisible by s for all i. Since X contains a curve of transverse type A_{r−1}, m − 2 of degrees a_{i} must be divisible by r, say the last m − 2 of them a_{n+l−1}, …, a_{n+m+l−4}. Without loss of generality we can assume that the type of singularity is induced by the last two variables of degrees sd_{l−1}, sd_{l}. More precisely, sd_{l−1} = a + r∗ and sd_{l} = −a + r ∗ .

Threefold X is quasismooth along the curve, so we can use the implicit function theorem in the neighbourhood of P to expand out the variables in the remaining degrees b_{1}, …, b_{n}, sd_{1}, …, sd_{l−2}. This gives us

a_{1} = b_{1} + r∗
⋮
a_{n} = b_{n} + r∗
a_{n+1} = sd_{1} + r∗
⋮
a_{n+l−2} = sd_{l−2} + r∗

Thus, (l − 2) + (m − 2) = m + l − 4 of equations have degrees divisible by s. They define a threefold

f_{n+1}, …, f_{n+l−2}, f_{n+l−1}, …, f_{n+m+l−4} ⊂ ℙ(0, …, 0, rc_{1}, …, rc_{m}, sd_{1}, …, sd_{l}),

which means that X contains a threefold fixed by μ_{s} action. This contradicts the fact that X is well formed.

Using the notation in Lemma 1.3.5, we proved that the singularities of O_{X}(1) are of type

r−1(\frac{1}{r}(a, −a)) = _{1}(\frac{1}{r}(a, −a)),

where a and r are coprime. This is the same as _{k}(\frac{1}{r}(1, −1)) where ak = 1 mod r. Clearly such k and r are coprime. □

We will prove Theorem 1.4.2 by resolving the singularities. By [29, Main Theorem] we can choose a resolution f: Y → X which factors as Y →^{h} X̄ →^{g} X, where g is a partial crepant resolution, X̄ only has terminal singularities and h is an isomorphism outside terminal points.

Before we proceed, we describe the precise configuration of exceptional surfaces: Let ^{C}E_{i} and ^{P}G_{j} be the exceptional surfaces of the chosen f.

Every singular point P (dissident or isolated) is locally analytically isomorphic to 𝔸^{3}/μ_{s}, thus toric. Using the terminology of [12], the resolution of toric singularities depends only on the quotient type of the singularity. It corresponds to subdividing the defining fan in a lattice. Every divisor corresponds to a ray in the fan and is determined by its Star. Therefore ^{P}G_{j} depend only on the singularity type of P. They are contracted to P by f.

A generic point of a _{k}(\frac{1}{r}(1, −1)) curve C allows a crepant resolution, which is a composite of ⌊\frac{r}{2}⌋ blow–ups. Thus Y has no discrepant divisors over the generic point of C. A dissident point on C has a toric resolution.

In more detail we can write

K_{Y} = f^{∗}K_{X} + S, where S = ∑_{P} ∑_{j} ^{P}γ_{j} ^{P}G_{j},

and

f^{∗}D = D′ + ∑_{C} R_{C} + M, where R_{C} = ∑_{i} ^{C}α_{i} ^{C}E_{i} and M = ∑_{P} ∑_{j} ^{P}β_{j} ^{P}G_{j},

with γ_{j} and α_{i}, β_{j} ∈ ℚ. Here f^{∗}D is by definition \frac{1}{t}f^{∗}(tD) for a Cartier divisor tD.

Lemma 2.2.2 R_{C} = \frac{k}{r} ^{C}E_{1} + \frac{\overline{2k}}{r} ^{C}E_{2} + … + \frac{\overline{(r−1)k}}{r} ^{C}E_{r−1}, where denotes the smallest residue mod r.

Proof The type of singularities determines the geometry of ^{C}E_{i} and the coefficients ^{C}α_{i} in

R_{C} = ∑ ^{C}α_{i} ^{C}E_{i}.

Choose a generic point Q on C and a transverse hypersurface S that contains P. On the polarised (S, D|_{S}) the type of Q is _{k}(\frac{1}{r}(1, −1)). Such a point has unique crepant resolution f|S̃: S̃ → S and coefficients in

f^{∗}D|_{S} = D′|S̃ + \frac{k}{r} F_{1} + \frac{\overline{2k}}{r} F_{2} + … + \frac{\overline{(r − 1)k}}{r} F_{r−1},

where F_{i} are exceptional lines that form the configuration on Figure I.

Figure I: Resolution of an A_{r−1} point

Each F_{i} is a fibre in one of the surfaces ^{C}E_{j}. By Lemma 2.2.1 k and r are coprime, thus all numbers k, \overline{2k}, …, \overline{(r − 1)k} are different. Hence F_{1}, …, F_{r−1} must be fibres in different surfaces ^{C}E_{1}, …,^{C}E_{r−1} and ^{C}α_{i} = \frac{\overline{ik}}{r}.

□

In particular, every ^{C}E_{i} is isomorphic to a blow–up of a few points (over the dissident points) of a smooth ruled surface. The base of the ruling is isomorphic to C. Moreover, ^{C}E_{i} and ^{C}E_{j} intersect along a curve isomorphic to C if j = i + 1 and are disjoint otherwise. Define ^{C}γ_{i} = ^{C}E_{i} ∩ ^{C}E_{i+1}. The situation is shown on Figure II. For two different curves C and C̄, the surfaces ^{C}E_{i} and C̄E_{j} intersect only over the dissident points.

Figure II: Resolution of an A_{r−1} curve

The Riemann-Roch formula for the smooth threefold Y gives

χ(Y, O_{Y} (D′)) =

χ(O_{Y}) + \frac{1}{12}D′(D′ − K_{Y})(2D′ − K_{Y}) + \frac{1}{12}D′c_{2}(Y) =

χ(O_{Y}) + \frac{1}{12} (f^{∗}D − ∑_{C} R_{C} − M) (f^{∗}D − ∑_{C} R_{C} − M − f^{∗}K_{X} − S) (2f^{∗}D − 2 ∑_{C} R_{C} − 2M − f^{∗}K_{X} − S) +

\frac{1}{12} (f^{∗}D − ∑_{C} R_{C} − M) · c_{2}(Y).

Take t big, so that the zero locus of D̄ = tD defines a smooth surface. The restriction tD|_{C} is a finite number of nondissident points on C. Then f^{∗}D̄·^{C}E_{i} is a finite union of fibres in ^{C}E_{i}, since f^{∗}D̄|_{^{C}E_{i}} is a pull back of tD|_{C}. Now, using the projection formula

f^{∗}D̄ · f^{∗}D̄ ·^{C} E_{i} = f^{∗}D̄ · (fibres) = tD · f_{∗}(fibres) = 0.

This holds for all i, thus f^{∗}D · f^{∗}D · R_{C} = 0 and the same way

f^{∗}D · f^{∗}K_{X} · R_{C} = 0.

The projection formula also gives

(f^{∗}D)^{3} = D^{3}, (f^{∗}D)^{2}f^{∗}K_{X} = D^{2}K_{X}, f^{∗}D(f^{∗}K_{X})^{2} = DK_{X}^{2}.

The intersection numbers involving M or S depend only on the isolated or dissident singular points of X. Moreover, for different curves of singularities, the intersections

f^{∗}D · R_{C}RC̄, f^{∗}K_{X} · R_{C}RC̄ and (R_{C})^{2}RC̄

can only be nonzero because of intersections over dissident points. Recall Formula (E. i) from Section 2.1

χ(X, O_{X}(D)) = χ(Y, O_{Y} (D′)) + χ(X, R^{2}f_{∗}K).

The fibre dimension of f over a generic point on a curve of singularities is 1. Thus R^{1}f_{∗}O_{Y} (D′) ≅ R^{2}f_{∗}K is supported on the isolated or dissident singular points of X.

The above considerations and the fact that χ(O_{X}) is a birational invariant simplify the RR formula to

χ(X, O_{X}(D)) = χ(O_{X}) + \frac{1}{12}D(D − K_{X})(2D − K_{X}) + \frac{1}{12}f^{∗}D · c_{2}(Y) +
\frac{1}{2}f^{∗}D · ∑_{C} R_{C}^{2} − \frac{1}{4}f^{∗}K_{X} · ∑_{C} R_{C}^{2} −  (E. ii)

\frac{1}{6} ∑_{C} R_{C}^{3} − \frac{1}{12} ∑_{C} R_{C} · c_{2}(Y) +

contributions from singular points of X.

The contributions from the singular points of X only depend on the analytic neighbourhood of these points. We will compute them on suitable models in Section 2.4. In this section we focus our attention to the contributions from the curves of singularities. For this reason we will denote any expression which depends only on the singular points by P and will not worry about its exact value.

The next two lemmas will analyse the intersection numbers in (E. ii).

Lemma 2.2.3 For a _{k}(\frac{1}{r}(1, −1)) curve C in (X, D) the following holds

\frac{1}{2}f^{∗}D · R_{C}^{2} = − deg D|_{C} · \frac{\overline{k}(r − \overline{k})}{2r}

and

\frac{1}{4}f^{∗}K_{X} · R_{C}^{2} = − deg K_{X}|_{C} · \frac{\overline{k}(r − \overline{k})}{4r}.

Proof Choose a surface S ⊂ X which intersects C transversely. By Bertini’s theorem the only singularities on the polarised (S, D|_{S}) are points C ∩ S which are all of type _{k}(\frac{1}{r}(1, −1)). Write down Riemann–Roch for a resolution f|S̃: S̃ → S of these points

χ(S, O_{S}(D)) = χ(O_{S}) + \frac{1}{2}D|_{S}(D|_{S} − K_{S}) + \frac{1}{2}B2 ; \overline{k},

as in the proof of Theorem 9.1 in [30]. The term \frac{1}{2}B2 ; \overline{k} equals to the so called basket contribution ∑_{P} c_{P} (D|_{S}), where each point P ∈ C ∩ S contributes

c_{P} (D|_{S}) = −\frac{\overline{k}(r − \overline{k})}{2r}.

In our notation R_{C}|S̃ = B\overline{k}.

Note that S can be defined by a section of tD in X for t big enough. Finally we can compute

\begin{matrix}\frac{1}{2}f^{∗}D · R_{C}^{2} = \frac{1}{2t}f^{∗}S · R_{C}^{2} = \frac{1}{2t} · B2 ; \overline{k} = \\ \frac{1}{t} ∑_{P ∈C∩S} c_{P} (D|_{S}) = − deg D|_{C} · \frac{\overline{k}(r−\overline{k})}{2r}. \end{matrix}

Similarly, if we define S by a section of a multiple of K_{X}, we get

\frac{1}{4}f^{∗}K_{X} · R_{C}^{2} = − deg K_{X}|_{C} · \frac{\overline{k}(r − \overline{k})}{4r}

which ends the proof of Lemma 2.2.3.

□

Lemma 2.2.4 Let C be a _{k}(\frac{1}{r}(1, −1)) curve in (X, D). Then

\begin{matrix}R_{C}^{3} + \frac{1}{2}R_{C} · c_{2}(Y) = \\ −\frac{1}{r^{2}} \overline{k}(r − \overline{k})(r − 2\overline{k}) [(r − 2) (1 − g + \frac{1}{2}K_{Y} ^{C}E_{1} ^{C}E_{2}) + ^{C}E_{r−2}^{2} ^{C}E_{r−1}] + P, \end{matrix}

where g is the genus of C and P depends only on the singular points of X.

Proof By Lemma 2.2.2

R_{C} = \frac{\overline{k}}{r} E_{1} + \frac{\overline{2k}}{r} E_{2} + … + \frac{\overline{(r − 1)k}}{r} E_{r−1}.

We drop the upper index in ^{C}E_{i} for the sake of clearer notation. Then

\begin{aligned}R_{C}^{3} & = & ∑_{i=1}^{r−1} (\frac{\overline{ik}}{r})^{3} E_{i}^{3}+ \\ & & 3 ∑_{i=1}^{r−2} (\frac{\overline{ik}}{r})^{2} \frac{\overline{(i+1)k}}{r}E_{i}^{2}E_{i+1} + \frac{\overline{ik}}{r} (\frac{\overline{(i+1)k}}{r})^{2} E_{i}E_{i+1}^{2} \end{aligned}

and

R_{C} · c_{2}(Y) = (\frac{\overline{k}}{r} E_{1} + \frac{\overline{2k}}{r} E_{2} + … + \frac{\overline{(r − 1)k}}{r} E_{r−1}) · c_{2}(Y).

These expressions will simplify, using the properties of ruled surfaces. Recall that E_{i} equals to a smooth ruled surface Ê_{i} blown–up in a number of points for i = 1, …, r − 1. The blow–ups E_{i} ⟶^{π_{i}} Ê_{i} happen only over the dissident points. In other words, the resolution f: Y → X contracts Exc(π_{i}) ⊂ E_{i} to the dissident points on C.

Denote γ_{i} = E_{i} ∩ E_{i+1} which is isomorphic to C of genus g. Note that π_{i}(γ_{i}) = π_{i+1}(γ_{i}) is a section in both ruled surfaces Ê_{i} and Ê_{i+1}. Then

K_{E_{i}} = π_{i}^{∗}KÊ_{i} + P

is numerically equivalent to

−2γ_{i} + (2g − 2 + γ_{i}^{2})f_{i} + P,

where f_{i} is a generic fibre on E_{i}. In particular K_{E_{i}}^{2} = 8(1 − g) + P. Similarly

K_{E_{i+1}} ∼ −2γ_{i} + (2g − 2 + γ_{i}^{2})f_{i+1} + P.

By the adjunction formula we can compute

γ_{i}^{2} in E_{i+1} = E_{i}|_{E_{i+1}}E_{i}|_{E_{i+1}} = E_{i}^{2}E_{i+1} =
E_{i}|_{E_{i}}E_{i+1}|_{E_{i}} = (K_{E_{i}} − K_{Y} |_{E_{i}})E_{i+1}|_{E_{i}} = K_{E_{i}}γ_{i} − K_{Y} E_{i}E_{i+1} =
−γ_{i}^{2} in E_{i} + 2(g − 1) + P − K_{Y} E_{i}E_{i+1}.

Similarly

γ_{i}^{2} in E_{i} = E_{i}E_{i+1}^{2} = −γ_{i}^{2} in E_{i+1} + 2(g − 1) + P − K_{Y} E_{i}E_{i+1}.

Both equalities together give

E_{i}^{2}E_{i+1} + E_{i}E_{i+1}^{2} = 2(g − 1) − K_{Y} E_{i}E_{i+1} + P for i = 1, …, r − 2.

Next,

E_{i}^{2}E_{i+1} + E_{i+1}E_{i+2}^{2} = γ_{i}^{2} in E_{i+1} + γ_{i+1}^{2} in E_{i+1} = P for i = 1, …, r − 3

since π_{i+1}(γ_{i}) and π_{i+1}(γ_{i+1}) are disjoint sections in Ê_{i+1}. And the sum of the self–intersections of any two disjoint sections in a ruled surface is 0.

From expanding the equalities

\begin{aligned}0 & = & f^{∗}K_{X} · f^{∗}D · E_{i} = \\ & & (K_{Y} + P) (D′ + \frac{\overline{k}}{r}E_{1} + \frac{\overline{2k}}{r}E_{2} + … + \frac{\overline{(r−1)k}}{r}E_{r−1} + P) E_{i} \end{aligned}

for various i it follows

K_{Y} E_{i−1}E_{i} − K_{Y} E_{i}E_{i+1} = P and K_{Y} E_{i}^{2} + 2K_{Y} E_{i}E_{i+1} = P.

Finally, putting together all of the above yields

\begin{aligned}E_{i}^{2}E_{i+1} & = & −(r − 2 − i)2(g − 1) + (r − 2 − i)K_{Y} E_{1}E_{2} + E_{r−2}^{2}E_{r−1} + P, \\ E_{i}E_{i+1}^{2} & = & (r − 1 − i)2(g − 1) − (r − 1 − i)K_{Y} E_{1}E_{2} − E_{r−2}^{2}E_{r−1} + P, \end{aligned}

for all i = 1, …, r − 2.

The second Chern class c_{2}(Y) can also be computed without difficulty. Let N_{i} be the normal bundle of E_{i} in the smooth threefold Y, and T_{E_{i}}, T_{Y} the tangent sheaves. There is an exact sequence

0 ⟶ T_{E_{i}} ⟶ j_{i}^{∗}T_{Y} ⟶ N_{i} ⟶ 0,

where j_{i}: E_{i} → Y is the inclusion. The Chern polynomial for E_{i} is

c_{t}(T_{E_{i}}) = 1 − K_{E_{i}}t + c_{2}(E_{i})t^{2}.

By the self–intersection formula

c_{t}(N_{i}) = 1 + E_{i}|_{E_{i}}t.

By comparing the coefficients of t^{2} and then using the adjunction formula, we find

c_{2}(Y) · E_{i} = c_{2}(E_{i}) − K_{E_{i}}E_{i}|_{E_{i}} =  (E. iii)
c_{2}(E_{i}) − (K_{Y} + E_{i})|_{E_{i}}E_{i}|_{E_{i}} = c_{2}(E_{i}) − (K_{Y} + E_{i})E_{i}^{2}.

Recall that f: Y → X is crepant over a generic point on C, which implies K_{Y}^{2} E_{i} = (f^{∗}K_{X} + P)^{2}E_{i} = P. This yields another relation

8(1 − g) + P = K_{E_{i}}^{2} = (K_{Y} + E_{i})^{2}E_{i} = E_{i}^{3} + 2K_{Y} E_{i}^{2} + P.

Also c_{2}(E_{i}) can be expressed with other invariants

\frac{1}{12} (K_{E_{i}}^{2} + c_{2}(E_{i})) = 1 + p_{a}(E_{i}) = 1 − g,

where the last equality holds since Ê_{i} is a ruled surface and p_{a} is a birational invariant. This in particular implies c_{2}(E_{i}) = 4(1 − g) + P.

Pluging all this in (E. iii) we get,

c_{2}(Y) · E_{i} = −\frac{1}{2}E_{i}^{3} + P,

where E_{i}^{3} = 8(1 − g) + 4K_{Y} E_{1}E_{2} + P.

This reduces the expression for R_{C}^{3} to

\frac{8(1−g)+4K_{Y} E_{1}E_{2}}{r^{3}} ∑_{i=1}^{r−1} (\overline{ki})^{3} +
\frac{3}{r^{3}} ∑_{i=1}^{r−2}(\overline{ik})^{2}\overline{(i + 1)k} [−(r − 2 − i)2(g − 1)+
(r − 2 − i)K_{Y} E_{1}E_{2} + E_{r−2}^{2}E_{r−1}]+
\frac{3}{r^{3}} ∑_{i=1}^{r−2} \overline{ik}(\overline{(i + 1)k})^{2} [(r − 1 − i)2(g − 1)−
(r − 1 − i)K_{Y} E_{1}E_{2} − E_{r−2}^{2}E_{r−1}]+
P

and R_{C} · c_{2}(Y) reduces to

−\frac{4(1 − g) + 2K_{Y} E_{1}E_{2}}{r} ∑_{i=1}^{r−1} \overline{ki} + P.

Together R_{C}^{3} + \frac{1}{2}c_{2}(Y)R_{C} equals

(1 − g + \frac{1}{2}K_{Y} E_{1}E_{2}) [\frac{8}{r^{3}} ∑_{i=1}^{r−1} \overline{ki}^{3}−

\frac{2}{r} ∑_{i=1}^{r−1} \overline{ki} + \frac{6}{r^{3}} ∑_{i=1}^{r−2} \overline{ki}^{2}\overline{k(i + 1)}(r − 2 − i) − \overline{ki}\overline{k(i + 1)}^{2}(r − 1 − i)] +

\frac{3}{r^{3}} E_{r−2}^{2}E_{r−1} ∑_{i=1}^{r−2} \overline{ki}^{2}\overline{k(i + 1)} − \overline{ki}\overline{k(i + 1)}^{2} + P.

We will rewrite this into a more compact form. First observe that −\overline{ki} + \overline{k(i + 1)} is either k or k − r. More precisely, it has value k exactly r − k − 1 times and value k − r exactly k − 1 times. This can be easily seen, since the permutation

( \begin{matrix}1 & 2 & 3 & … & r − 1 \\ k & \overline{2k} & \overline{3k} & … & r − k \end{matrix} )

changes the sign exactly k − 1 times. This and

∑_{i=1}^{r−2} (−\overline{ki} + \overline{k(i + 1)})^{3} = ∑_{i=1}^{r−2} −\overline{ki}^{3} + 3\overline{ki}^{2}\overline{k(i + 1)} − 3\overline{ki}\overline{k(i + 1)}^{2} + \overline{k(i + 1)}^{3}

yields

3 ∑_{i=1}^{r−2} \overline{ki}^{2}\overline{k(i + 1)} − \overline{ki}\overline{k(i + 1)}^{2} = −rk(r − k)(r − 2k).  (E. iv)

In the next step we prove that

\frac{8}{r^{3}} ∑_{i=1}^{r−1} \overline{ki}^{3} − \frac{2}{r} ∑_{i=1}^{r−1} \overline{ki} + \frac{3}{r^{3}} ∑_{i=1}^{r−2} \overline{ki}^{2}\overline{k(i + 1)}(r − 2 − 2i) −  (E. v)
\overline{ki}\overline{k(i + 1)}^{2}(r − 2i) = 0.

By Formula 8.30. in [19] define

a_{j} = \frac{\overline{kj}}{r} − \frac{\overline{k(j − 1)}}{r} for j = 1, …, r,

which in particular means a_{1} = \frac{k}{r} and a_{r} = \frac{k−r}{r}. Then

\begin{aligned}∑_{1≤i<j≤r}(a_{i} − a_{j})^{3} & = & \frac{8}{r^{3}} ∑_{i=1}^{r−1} \overline{ki}^{3} − \frac{3}{r^{3}} ∑_{i=1}^{r−1} \overline{ki}^{2}\overline{k(i + 1)} + \overline{ki}\overline{k(i + 1)}^{2}+ \\ & & \frac{3}{r^{3}} ∑_{i=1}^{r−1}(r − 2i − 1)(\overline{ki}^{2}\overline{k(i + 1)} + \overline{ki}\overline{k(i + 1)}^{2}). \end{aligned}

This rewrites (E. v) into

−\frac{2}{r} ∑_{i=1}^{r−1} \overline{ki} + ∑_{1≤i<j≤r}(a_{i} − a_{j})^{3}.

By Lemma 2.2.1 r and k are coprime, thus \overline{ki} ≠ 0 for all i. The sum

∑_{i=1}^{r−1} \overline{ki} = \frac{r(r − 1)}{2}

does not depend on k. We can write the terms a_{i}−a_{j} into an upper–triangular matrix

a_{1} − a_{2} a_{1} − a_{3} … a_{1} − a_{r−1} a_{1} − a_{r}
a_{2} − a_{3} … a_{2} − a_{r−1} a_{2} − a_{r}
...  ⋮
a_{r−2} − a_{r−1} a_{r−2} − a_{r}
a_{r−1} − a_{r}

Observe that

a_{1} − a_{r} = 1  and
a_{1} − a_{j} = −(a_{r−j+1} − a_{r}) + 1,
a_{i} − a_{j} = −(a_{r−j+1} − a_{r−i+1}),
a_{i} − a_{r−i+1} = 0  for i, j = 2, …, r − 1.

This means that the (1r)th minor is always antisymmetric along the anti– diagonal and has zeros along the anti–diagonal. Thus,

∑_{1≤i<j≤r}(a_{i} − a_{j})^{3} = ∑_{i=2}^{r−1}(a_{1} − a_{i})^{3} + ∑_{i=2}^{r−1}(a_{i} − a_{r})^{3} + (a_{1} − a_{r})^{3}.

The same as in the proof of (E. iv), a_{i} has value \frac{k−r}{r} exactly k − 1 times and value \frac{k}{r} exactly r − k − 1 times. We get

∑_{1≤i<j≤r}(a_{i} − a_{j})^{3} = (k − 1) + (r − k − 1) + 1 = r − 1,

which finishes the proof of (E. v).

Equations (E. iv) and (E. v) simplify R_{C}^{3} + \frac{1}{2}R_{C} · c_{2}(Y) into the beautiful form

−\frac{1}{r^{2}} \overline{k}(r − \overline{k})(r − 2\overline{k}) [(r − 2) (1 − g + \frac{1}{2}K_{Y} E_{1}E_{2}) + E_{r−2}^{2}E_{r−1}] + P.

This ends the proof of Lemma 2.2.4. □

Corollary 2.2.5 Let (X, D) satisfy the conditions of Theorem 1.4.2. Then for all positive integers m

\begin{aligned}χ(X, O_{X}(mD)) & = & χ(O_{X}) + \frac{1}{12}mD(mD − K_{X})(2mD − K_{X}) + m\frac{D·c_{2}(X)}{12} \\ & & + ∑_{C} s_{C}(mD) + P, \end{aligned}

where

\begin{aligned}s_{C}(mD) & = −m\frac{\overline{mk}(r−\overline{mk})}{2r} deg D|_{C} + \frac{\overline{mk}(r−\overline{mk})}{4r} deg K_{X}|_{C} \\ & +\frac{\overline{mk}(r−\overline{mk})(r−2·\overline{mk})}{6r^{2}} ((r − 2)(1 − g) + \frac{r−2}{2}K_{Y} ^{C}E_{1} ^{C}E_{2} + ^{C}E_{r−2}^{2} ^{C}E_{r−1}) . \end{aligned}

and P only depends on the singular points of X.

Proof For m = 1 we only need to put the results of Lemma 2.2.3 and Lemma 2.2.4 into Formula (E. ii) on page 16. We define D ·c_{2}(X) to be f^{∗}D · c_{2}(Y). The existence of the formula proves that K_{Y} ^{C}E_{1} ^{C}E_{2}, ^{C}E_{r−2}^{2} ^{C}E_{r−1} and f^{∗}D · c_{2}(Y) are independent of the resolution and are thus invariants of the threefold X.

For m > 1 consider O_{X}(mD) which is of transverse type \overline{mk}(\frac{1}{r}(1, −1)) on C. We can write

f^{∗}(mD) = D^{(m)} + ∑_{C} R_{C}^{(m)} + P,

where D^{(m)} is a Cartier divisor on Y and

R_{C}^{(m)} = \frac{\overline{mk}}{r} ^{C}E_{1} + \frac{\overline{2mk}}{r} ^{C}E_{2} + … + \frac{\overline{(r − 1)mk}}{r} ^{C}E_{r−1}

by Lemma 2.2.2. Repeat all arguments in Lemma 2.2.3 and Lemma 2.2.4 for R_{C}^{(m)}. The only assumption that might fail is that \overline{mk} and r are coprime. This fact was only used in the proof of (E. v).

We will show that (E. v) also holds when k and r are not coprime. Write h = hcf(k, r) and k′ = \frac{k}{h}, r′ = \frac{r}{h}. Then

k, \overline{2k}, …, \overline{(r′ − 1)k}, \overline{r′k}, \overline{(r′ + 1)k}, …, \overline{(r − 1)k}, \overline{rk}

equal to

k, \overline{2k}, …, \overline{(r′ − 1)k}, 0,  k,  …, \overline{(r′ − 1)k}, 0,

where denotes mod r. The sum \frac{2}{r} ∑_{i=1}^{r−1} \overline{ki} consists of h equal parts, each of them being r′ − 1.

The matrix of (a_{i} − a_{j})_{1≤i<j≤r} has the form

a_{1}−a_{2} a_{1}−a_{3} ... a_{1}−a_{r′−1}  a_{1}−a_{r′}  a_{1}−a_{1}  a_{1}−a_{2} ...  a_{1}−a_{r′}  a_{1}−a_{1}  ...
a_{2}−a_{3} ... a_{2}−a_{r′−1}  a_{2}−a_{r′}  a_{2}−a_{1}  ...  ...  a_{2}−a_{r′}  a_{2}−a_{1}  ...
...  ⋮  ⋮
a_{r′−1}−a_{r′} a_{r′−1}−a_{1}  ...  ... a_{r′−1}−a_{r′} a_{r′−1}−a_{1} ...
a_{r′}−a_{1} a_{r′}−a_{2} ...  a_{r′}−a_{r′}  a_{r′}−a_{1} ...
a_{1}−a_{2} ...  a_{1}−a_{r′}  a_{1}−a_{1}  ...
...  ⋮

Observe that by symmetry most of the terms cancel out. Only the triangular blocks contribute to the sum ∑_{1≤i<j≤r}(a_{i} − a_{j})^{3}. More precisely, the same way as in the coprime case each triangle contributes r′ − 1. There is h of the triangles. Thus we proved that also when r and k are not coprime

−\frac{2}{r} ∑_{i=1}^{r−1} \overline{ki} + ∑_{1≤i<j≤r}(a_{i} − a_{j})^{3} = 0.

□

2.3 Contributions from A_{r−1} curves

Until now we studied contributions to the Riemann–Roch formula via the resolution f: Y → X. The aim of this section is to describe another approach, using cyclic quotients. We will construct a smooth threefold Z such that X = Z/G for a finite group G. We will give a new proof of Corollary 2.2.5 for curves of singularities with no dissident points.

2.3.1 Quotients by finite groups

Theorem 2.3.1 ([16], Ex. 10.25) The quotient of a projective variety Z ⊂ ℙ^{n} by a finite group G always exists and is a projective variety.

Not every projective threefold with an A_{r−1} curve can be realised as a global quotient by μ_{r}. We begin with an example:

Example 2.3.2 Consider ℙ(1, 2, 3, 3) with the variables x, y, t_{1}, t_{2}. It contains an A_{2} curve C = {x = y = 0} and an isolated singular point (0, 1, 0, 0). In this section we are only interested in the contributions to RR induced by C.

Lemma 2.3.3 ℙ(1, 2, 3, 3) can not be realised as a global quotient of a smooth threefold under μ_{3} action.

Proof We prove this by contradiction. Assume that there exists Z, which is smooth in the neighbourhood of the curve, such that

Z/μ_{3} = ℙ(1, 2, 3, 3).

On the affine piece U_{1} = {t_{1} ≠ 0} of ℙ(1, 2, 3, 3) choose coordinates a_{1}, b_{1}, c_{1}, which are identified by a_{1}, b_{1}, c_{1} ∼ εa_{1}, ε^{2}b_{1}, c_{1}, for all ε ∈ μ_{3}. This defines a cyclic μ_{3} cover

\begin{matrix}ℂ^{3} & ⟶ & U_{1} \\ a_{1}, b_{1}, c_{1} & ⟼ & a_{1}^{3}, a_{1}b_{1}, b_{1}^{3}, c_{1}. \end{matrix}

The same way denote by a_{2}, b_{2}, c_{2} the coordinates of U_{2} = {t_{2} ≠ 0}, which are identified by a_{2}, b_{2}, c_{2} ∼ εa_{2}, ε^{2}b_{2}, c_{2}. This defines another cyclic cover ℂ^{3} → U_{2}. Since ℙ(1, 2, 3, 3) = Z/μ_{3}, we can obtain Z by gluing together the affine pieces of the covers. The transition function is induced by identification between U_{1} and U_{2} :

a_{1}, b_{1}, c_{1} ↔ a_{1}c_{1}^{− \frac{1}{3}} , b_{1}c_{1}^{− \frac{2}{3}} , c_{1}^{−1}.

But these are not rational functions, which gives the contradiction. □

On the other hand, ℙ(1, 2, 3, 3) is isomorphic to

ℙ(1, 2, 1, 1)/μ_{3} ⊕ μ_{3}.

The elements (ε, ε′) ∈ μ_{3} ⊕ μ_{3} act on ℙ(1, 2, 1, 1) via 1, 1, ε, ε′ and fix either points, curve or surfaces. Note that ℙ(1, 2, 1, 1) is smooth in the neighbourhood of C.

The above example demonstrates what happens in a general case. In Subsection 2.3.4 we will prove that a projective threefold with an A_{r−1} curve can be realised as a global quotient by the group μ_{r} ⊕ μ_{r}. Various elements of μ_{r} ⊕ μ_{r} fix different loci in the cover Z, i.e. points, curves or surfaces.

2.3.2 The Atiyah–Singer equivariant Riemann–Roch formula

Through the entire section we will use the Atiyah-Singer equivariant RR formula. The version that we need can be found on page 565 in [3]: Let Z be a compact complex manifold, V a holomorphic vector bundle over Z and O(V) the sheaf of germs of holomorphic sections of V. Moreover, let G be a finite group of automorphisms of the pair (Z, V). Let Z^{g} denote the fixed point set of an element g ∈ G and N^{g} is the normal bundle of Z^{g} in Z. Under the G–action, the complex vector bundle N^{g} has a decomposition

N^{g} = ∑ N^{g}(θ),

where N^{g}(θ) is the subbundle on which g acts as e^{iθ}. Then

∑_{p}(−1)^{p}Trace(g|H^{p}(Z, O(V))) = \{\frac{ch(V|Z^{g})(g)T (Z^{g})}{∏_{θ}(∏_{j_{θ}}(1 − e^{−x_{j_{θ}} −iθ})(N^{g}(θ)))}\} [Z^{g}],

where x_{j_{θ}} are Chern classes of N^{g}(θ) and T (Z^{g}) is the Todd class of the tangent bundle on Z^{g}.

Note that the formula is written in the Grothendieck group K(Z^{g}). The Grothendieck group is connected to the Chow ring A(Z^{g}) in the following way: If Z^{g} is nonsingular, we can compute K(Z^{g}) using only locally free sheaves ([17], Ex. II 6.10). This extends the definition of Chern classes from A(Z^{g}) to K(Z^{g}) and induces maps

\begin{matrix}c_{t} : K(Z^{g}) ⟶ A(Z^{g})[t] \\ ch : K(Z^{g}) ⟶ A(Z^{g}) ⊗ ℚ. \end{matrix}

We will use the Atiyah-Singer formula in a simpler form. For V we take the trivial vector bundle, whose total Chern class is 1.

Theorem 2.3.4 Let Z be a smooth projective threefold with a cyclic group action μ_{r} whose fixed locus is a curve C of genus g. Assume also that the group μ_{r} ∋ ε acts in the normal direction of the curve by ε^{a}, ε^{r−a} for some a coprime to r.

If r ≠ 2, the normal sheaf decomposes into N_{C/Z} = N_{1} ⊕ N_{2} with the corresponding Chern classes x_{1} = c_{1}(N_{1}), x_{2} = c_{1}(N_{2}) in the Chow ring A(C) ⊗ ℚ. Then

∑_{p}(−1)^{p}Trace(ε|H^{p}(Z, O_{Z})) =
\frac{1−g}{(1−ε^{−a})(1−ε^{a})} − \frac{ε^{−a}}{(1−ε^{−a})^{2}(1−ε^{a})} deg_{C} x_{1} − \frac{ε^{a}}{(1−ε^{−a})(1−ε^{a})^{2}} deg_{C} x_{2}.

If r = 2,

∑_{p}(−1)^{p}Trace(−1|H^{p}(Z, O_{Z})) = \frac{1−g}{4} − \frac{1}{8} deg_{C} c_{1}(N_{C/Z}).

Proof Every element ε ∈ μ_{r} fixes exactly a curve C, in other words Z^{ε} = C for all ε. On a curve C with the canonical divisor K_{C} and genus g the following always holds

tdT_{C} = 1 − \frac{1}{2}K_{C} and deg K_{C} = 2(g − 1).

The normal bundle N_{C/Z} of C in Z is of rank two, and is also independent on ε.

If r ≠ 2, a and r − a are different. Thus N_{C/Z} decomposes into two subbundles of rank one

N_{C/Z} = N_{1} ⊕ N_{2}.

The group ε ∈ μ_{r} acts on each component of the normal bundle via ε^{a}, ε^{r−a}.

These considerations simplify the above formula. In the language of algebraic geometry it rewrites to

∑_{p}(−1)^{p}Trace(ε|H^{p}(Z, O_{Z})) = deg \{\frac{1 · td(T_{C})}{(1 − ε^{−a}e^{−x_{1}})(1 − ε^{−(r−a)}e^{−x_{2}})}\}_{1} ,

where x_{i} is the first Chern class of N_{i} for i = 1, 2; and { }_{1} denotes the component of degree 1 in the Chow ring A(C) ⊗ ℚ.

In the Taylor series for \frac{1}{1−ε^{−a}e^{−x_{1}}} we only need to consider the terms of degree ≤ 1. Thus,

\frac{1}{1−ε^{−a}e^{−x_{1}}} = 1 + ε^{−a}e^{−x_{1}} + ε^{−2a}e^{−2x_{1}} + …
= 1 + ε^{−a}(1 − x_{1}) + ε^{−2a}(1 − 2x_{1}) + … + terms of degree ≥ 2
= \frac{1}{1−ε^{−a}} − \frac{ε^{−a}}{(1−ε^{−a})^{2}} x_{1} + terms of degree ≥ 2.

The same way we compute

\frac{1}{1 − ε^{a}e^{−x_{2}}} = \frac{1}{1 − ε^{a}} − \frac{ε^{a}}{(1 − ε^{a})^{2}} x_{2} + terms of degree ≥ 2.

From comparing the factors in degree 1 we get

∑_{p}(−1)^{p}Trace(ε|H^{p}(Z, O_{Z})) = \frac{1−g}{(1−ε^{−a})(1−ε^{a})}−
\frac{ε^{−a}}{(1−ε^{−a})^{2}(1−ε^{a})} deg_{C} x_{1}−
\frac{ε^{a}}{(1−ε^{−a})(1−ε^{a})^{2}} deg_{C} x_{2}

This expression will play a crucial role in the following paragraphs.

For r = 2 the normal sheaf N_{C/Z} might not decompose. If it does, we have already proved that

∑_{p}(−1)^{p}Trace(−1|H^{p}(Z, O_{Z})) = \frac{1 − g}{4} + \frac{1}{8}(deg_{C} x_{1} + deg_{C} x_{2}).

When N_{C/Z} does not decompose, the Atiyah–Singer equivariant RR has the form

∑_{p}(−1)^{p}Trace(−1|H^{p}(Z, O_{Z})) = deg \{\frac{1 · td(T_{C})}{(1 + e^{−y_{1}})(1 + e^{−y_{2}})}\}_{1} ,

where y_{i} = c_{i}(N_{C/Z}) for i = 1, 2. By comparing the factors in degree 1 we get

∑_{p}(−1)^{p}Trace(−1|H^{p}(Z, O_{Z})) = \frac{1 − g}{4} − \frac{1}{8} deg_{C} y_{1}.

□

Remark 2.3.5 Note that we get the same formula for r = 2 whether N_{C/Z} decomposes or not. Therefore we can in all further computations assume that N_{C/Z} = N_{1} ⊕ N_{2}.

We will also prove the analogy to Theorem 2.3.4 for surfaces and points.

Lemma 2.3.6 Let Z be a smooth projective threefold with a finite group G action that fixes exactly a surface S. Let the group element g ∈ G act in the normal direction of the surface by ε^{−1} for some ε ∈ μ_{r}.

Then,

\begin{matrix}∑_{p}(−1)^{p}Trace(g|H^{p}(Z, O_{Z})) = \\ [\frac{ε}{(1−ε)^{3}} − \frac{ε}{2(1−ε)^{2}} ] S^{3} + \frac{1}{2} \frac{ε}{(1−ε)^{2}} K_{S} · S^{2} + \frac{1}{12(1−ε)}(K_{S}^{2} + c_{2}), \end{matrix}

where S has the canonical divisor K_{S} and the second Chern class c_{2}.

Proof The Atiyah–Singer equivariant RR formula for this case rewrites to

∑_{p}(−1)^{p}Trace(g|H^{p}(Z, O_{Z})) = deg \{\frac{1 · td(T_{S})}{(1 − εe^{−c_{1}})}\}_{2} ,

where c_{1} is the first Chern class of the rank 1 normal bundle N; and { }_{2} denotes the component of degree 2 in the Chow ring A(S) ⊗ ℚ.

In the Taylor series for \frac{1}{1−εe^{−c_{1}}} we only need to consider the terms of degree ≤ 2. Thus,

\frac{1}{1−εe^{−c_{1}}} = 1 + εe^{−c_{1}} + ε^{2}e^{−2c_{1}} + …
= \frac{1}{1−ε} − \frac{ε}{(1−ε)^{2}} c_{1} + [\frac{ε}{(1−ε)^{3}} − \frac{ε}{2(1−ε)^{2}} ] c_{1}^{2} + terms of degree ≥ 3.

On a surface S with the canonical divisor K_{S} and the second Chern class c_{2} the following always holds

tdT_{S} = 1 − \frac{1}{2}K_{S} + \frac{1}{12}(K_{S}^{2} + c_{2}).

Comparing the factors in degree 2 yields

∑_{p}(−1)^{p}Trace(g|H^{p}(Z, O_{Z})) = [\frac{ε}{(1 − ε)^{3}} − \frac{ε}{2(1 − ε)^{2}} ] c_{1}^{2} +

\frac{1}{2} \frac{ε}{(1 − ε)^{2}} K_{S} · c_{1} +

\frac{1}{12(1 − ε)}(K_{S}^{2} + c_{2}).

By the self–intersection formula c_{1}(N) = S · S = S|_{S}, thus

∑_{p}(−1)^{p}Trace(g|H^{p}(Z, O_{Z})) = [\frac{ε}{(1 − ε)^{3}} − \frac{ε}{2(1 − ε)^{2}} ] S^{3} +

\frac{1}{2} \frac{ε}{(1 − ε)^{2}} K_{S} · S^{2} +

\frac{1}{12(1 − ε)}(K_{S}^{2} + c_{2}).

□

Lemma 2.3.7 Let Z be a smooth projective threefold with a finite group G action that fixes exactly a point P. Let g ∈ G act on the point by ε^{a_{1}}, ε^{a_{2}}, ε^{a_{3}} for some a_{1}, a_{2}, a_{3} coprime to r and ε ∈ μ_{r}.

Then

∑_{p}(−1)^{p}Trace(g|H^{p}(Z, O_{Z})) = \frac{1}{(1−ε^{−a_{1}})(1−ε^{−a_{2}})(1−ε^{−a_{3}})}.

Proof Write dg_{P} : T_{Z,P} → T_{Z,P} for the differential at point P fixed by the action of g ≡ ε ∈ μ_{r}. Atiyah-Singer equivariant RR formula for points takes the simple form

∑_{p}(−1)^{p}Trace(g | H^{p}(Z, O_{Z})) =

\frac{1}{det(1 − dg_{P})} = \frac{1}{(1 − ε^{−a_{1}})(1 − ε^{−a_{2}})(1 − ε^{−a_{3}})}.

□

2.3.3 The μ_{r} ⊕ μ_{r} group action

Definition 2.3.8 Let Z be a smooth projective threefold with the group μ_{r} ⊕ μ_{r} acting on it in a way that we now describe. For a chosen generator ε ∈ μ_{r} write

μ_{r} ⊕ μ_{r} = {(ε^{α}, ε^{β}) | α, β = 0, …, r − 1}.

Let a, d_{1} and d_{2} be integers that are all coprime to r. Then the group element (ε^{α}, ε^{β})

• fixes a curve C, if α = β. It acts in the normal direction of the curve by ε^{aα}, ε^{(r−a)α}.

• fixes a number of points P_{λ} on C, if α ≠ 0 and α ≠ β. It acts on each point P_{λ} by ε^{aα}, ε^{(r−a)α}, ε^{d_{2}(α−β)}.

• fixes a number of points Q_{ξ} on C, if β ≠ 0 and α ≠ β. It acts on each point Q_{ξ} by ε^{aβ}, ε^{(r−a)β}, ε^{d_{1}(β−α)}.

• fixes a number of surfaces S_{λ} such that S_{λ} ∩ C = P_{λ}, if α = 0. It acts in the normal direction of each S_{λ} by ε^{−βd_{2}}.

• fixes a number of surfaces T_{ξ} such that T_{ξ} ∩ C = Q_{ξ}, if β = 0.

It acts in the normal direction of each T_{ξ} by ε^{−αd_{1}}. From now on call such an action Action ♠.

Now write π: Z → X for the quotient map and let L_{i,j} be the (i, j)th eigensheaf of the μ_{r} ⊕ μ_{r} action on π_{∗}O_{Z}. Then

π_{∗}O_{Z} = ⨁_{i,j=0}^{r−1} L_{i,j}

implies

H^{p}(Z, O_{Z}) = ⨁_{i,j=0}^{r−1} H^{p}(X, L_{i,j}).

The group action on any f ∈ L_{i,j} is given by (ε^{α}, ε^{β})(f) = ε^{αi+βj} · f, thus

Tr((ε^{α}, ε^{β}) : H^{p}(Z, O_{Z})) = ∑_{i,j=0}^{r−1} h^{p}(X, L_{i,j}) · ε^{αi+βj}.

Moreover,

∑_{p}(−1)^{p}Tr((ε^{α}, ε^{β}) : H^{p}(Z, O_{Z})) = ∑_{i,j=0}^{r−1} χ(X, L_{i,j}) · ε^{αi+βj}.

Observe that the Action ♠ on the curve C is a μ_{r} action, since

μ_{r} ≅ {(ε^{α}, ε^{α}) | α = 0, …, r − 1} ⊂ μ_{r} ⊕ μ_{r}.

The normal sheaf to C of genus g decomposes into N_{C/Z} = N_{1} ⊕ N_{2} with the corresponding Chern classes x_{1} = c_{1}(N_{1}), x_{2} = c_{1}(N_{2}). Thus by Theorem 2.3.4

∑_{p}(−1)^{p}Trace((ε^{α}, ε^{α}) | H^{p}(Z, O_{Z})) =

\frac{1−g}{(1−ε^{−aα})(1−ε^{aα})} − \frac{ε^{−aα}}{(1−ε^{−aα})^{2}(1−ε^{aα})} deg_{C} x_{1} − \frac{ε^{aα}}{(1−ε^{−aα})(1−ε^{aα})^{2}} deg_{C} x_{2},

for every 1 ≠ (ε^{α}, ε^{α}) ∈ μ_{r} ⊕ μ_{r}. Denote the right hand side of this equation by A_{α,α}.

If α ≠ β are both nonzero, the element (ε^{α}, ε^{β}) ∈ μ_{r} ⊕ μ_{r} fixes the points P_{λ} and Q_{ξ}. Then Lemma 2.3.7 implies

\begin{matrix}∑_{p}(−1)^{p}Trace((ε^{α}, ε^{β}) | H^{p}(Z, O_{Z})) = \\ \frac{Λ}{(1−ε^{−aα})(1−ε^{aα})(1−ε^{−d_{2}(α−β)})} + \frac{Ξ}{(1−ε^{−aβ})(1−ε^{aβ})(1−ε^{−d_{1}(β−α)})}, \end{matrix}

where Λ and Ξ denote the number of points P_{λ} and Q_{ξ} respectively. Denote the right hand side of this equation by A_{α,β}.

In the case α = 0 and β ≠ 0 the element (1, ε^{β}) ∈ μ_{r} ⊕ μ_{r} fixes the surfaces S_{λ} and the points Q_{ξ}. By Lemma 2.3.6

∑_{p}(−1)^{p}Trace((1, ε^{β}) | H^{p}(Z, O_{Z})) =

∑_{λ} [\frac{ε^{d_{2}β}}{(1−ε^{d_{2}β})^{3}} − \frac{ε^{d_{2}β}}{2(1−ε^{d_{2}β})^{2}} ] S_{λ}^{3} + \frac{1}{2} \frac{ε^{d_{2}β}}{(1−ε^{d_{2}β})^{2}} K_{S_{λ}} · S_{λ}^{2} + \frac{1}{12(1−ε^{d_{2}β})}(K_{S_{λ}}^{2} + c_{2}(S_{λ}))

+\frac{Ξ}{(1−ε^{−aβ})(1−ε^{aβ})(1−ε^{−d_{1}β})}.

Denote this equation by A_{0,β}.

Similarly for α ≠ 0 and β = 0 the element (ε^{α}, 1) ∈ μ_{r} ⊕ μ_{r} fixes the points P_{λ} and the surfaces T_{ξ}. We get

∑_{p}(−1)^{p}Trace((ε^{α}, 1) | H^{p}(Z, O_{Z})) =

∑_{ξ} [\frac{ε^{d_{1}α}}{(1−ε^{d_{1}α})^{3}} − \frac{ε^{d_{1}α}}{2(1−ε^{d_{1}α})^{2}} ] T_{ξ}^{3} + \frac{1}{2} \frac{ε^{d_{1}α}}{(1−ε^{d_{1}α})^{2}} K_{T_{ξ}} · T_{ξ}^{2} + \frac{1}{12(1−ε^{d_{1}α})}(K_{T_{ξ}}^{2} + c_{2}(T_{ξ}))

+\frac{Λ}{(1−ε^{−aα})(1−ε^{aα})(1−ε^{−d_{2}α})}.

Denote this equation by A_{α,0}.

Clearly, in the introduced notation A_{0,0} = χ(O_{Z}).

Therefore, for every 1 ≠ (ε^{α}, ε^{β}) ∈ μ_{r} ⊕ μ_{r} we get

∑_{i,j=0}^{r−1} χ(X, L_{i,j}) · ε^{αi+βj} = A_{α,β}.

In the case (ε^{α}, ε^{β}) = 1 we get

∑_{i,j=0}^{r−1} χ(X, L_{i,j}) = χ(O_{Z}).

The last two formulas can be considered as a linear system of r^{2} equations in χ(X, L_{i,j}) and variable ε, given by Vandermonde matrix

( \begin{matrix}1 & 1 & … & 1 \\ 1 & & … \\ … & & ε^{αi+βj} \\ 1 & … & & … \end{matrix} ) ( χ(X, L_{0,0}) ; ⋮ ; χ(X, L_{i,j}) ; ⋮ ) = ( χ(O_{Z}) ; ⋮ ; A_{α,β} ; ⋮ ) .

Since a Vandermonde matrix is invertible, we can solve the system

\frac{1}{r^{2}} ( \begin{matrix}1 & 1 & … & 1 \\ 1 & & … \\ … & & ε^{−αi−βj} \\ 1 & … & & … \end{matrix} ) ( χ(O_{Z}) ; ⋮ ; A_{α,β} ; ⋮ ) = ( χ(X, L_{0,0}) ; ⋮ ; χ(X, L_{i,j}) ; ⋮ ) .

Write χ(X, L_{0,0}) = χ(O_{X}) and eliminate χ(O_{Z}) from the solution. We end up with

χ(X, L_{i,j}) = χ(O_{X}) + \frac{1}{r^{2}} ∑_{α,β=0}^{r−1}(ε^{−αi−βj} − 1)A_{α,β}.  (E. vi)

Insert A_{α,β} in this formula. Finally we get χ(X, L_{i,j}) = χ(O_{X})

+\frac{1}{r^{2}} ∑_{α=1}^{r−1}(ε^{−αi−αj} − 1) (\frac{1−g}{(1−ε^{−aα})(1−ε^{aα})} − \frac{ε^{−aα}}{(1−ε^{−aα})^{2}(1−ε^{aα})} deg_{C} x_{1}−

\frac{ε^{aα}}{(1−ε^{−aα})(1−ε^{aα})^{2}} deg_{C} x_{2})

+\frac{1}{r^{2}} ∑_{α=1,β=0; and α≠β}^{r−1}(ε^{−αi−βj} − 1)\frac{Λ}{(1−ε^{−aα})(1−ε^{aα})(1−ε^{−d_{2}(α−β)})} +\frac{1}{r^{2}} ∑_{α=0,β=1; and α≠β}^{r−1}(ε^{−αi−βj} − 1)\frac{Ξ}{(1−ε^{−aβ})(1−ε^{aβ})(1−ε^{−d_{1}(β−α)})}

+\frac{1}{r^{2}} ∑_{β=1}^{r−1}(ε^{−βj} − 1) ∑_{λ} ([\frac{ε^{d_{2}β}}{(1−ε^{d_{2}β})^{3}} − \frac{ε^{d_{2}β}}{2(1−ε^{d_{2}β})^{2}} ] S_{λ}^{3} + \frac{1}{2} \frac{ε^{d_{2}β}}{(1−ε^{d_{2}β})^{2}} K_{S_{λ}} · S_{λ}^{2}+

\frac{1}{12(1−ε^{d_{2}β})}(K_{S_{λ}}^{2} + c_{2}(S_{λ})))

+\frac{1}{r^{2}} ∑_{α=1}^{r−1}(ε^{−αi} − 1) ∑_{ξ} ([\frac{ε^{d_{1}α}}{(1−ε^{d_{1}α})^{3}} − \frac{ε^{d_{1}α}}{2(1−ε^{d_{1}α})^{2}} ] T_{ξ}^{3} + \frac{1}{2} \frac{ε^{d_{1}α}}{(1−ε^{d_{1}α})^{2}} K_{T_{ξ}} · T_{ξ}^{2}+

\frac{1}{12(1−ε^{d_{1}α})}(K_{T_{ξ}}^{2} + c_{2}(T_{ξ}))) .

We can think of χ(O_{X}) as RR–type expression in L_{ij}, where L_{ij} = 0 ∈ DivX ⊗ ℚ is the corresponding ℚ−divisor. The rest of the formula is a sum of contributions coming from

- the curve C

- the points {P_{λ}} and {Q_{ξ}}

- the surfaces {S_{λ}} and {T_{ξ}} which are the singularities of L_{i,j} on X.

In the sequel we prove two technical lemmas, which will simplify the above expressions.

Lemma 2.3.9 Let C be a smooth curve in a smooth threefold Z. Then the first Chern class

c_{1}(N_{C/Z}) = K_{C} − K_{Z}|_{C},

where K_{Z} is the canonical divisor of Z, K_{C} is the canonical divisor of the curve and N_{C/Z} its normal bundle.

Proof There exists a short exact sequence

0 → T_{C} → i^{∗}T_{Z} → N_{C/Z} → 0,

where i: C → Z is the inclusion. The corresponding Chern polynomials are

c_{t}(T_{C}) = 1 − K_{C}t
c_{t}(N_{C/Z}) = 1 + c_{1}(N_{C/Z})t + …
c_{t}(T_{Z}) = 1 − K_{Z}t + …

By comparing the coefficients of t in c_{t}(i^{∗}T_{Z}) = c_{t}(T_{C}) · c_{t}(N_{C/Z}), we find

c_{1}(N_{C/Z}) = K_{C} − K_{Z}|_{C}.

This in particular implies that

deg c_{1}(N_{C/Z}) = deg K_{C} − deg K_{Z}|_{C} = 2(g − 1) − deg K_{Z}|_{C}.
□

Lemma 2.3.10 Let C be a smooth curve of genus g in a smooth threefold Z and ε ∈ μ_{r}. Assume that the normal sheaf to C decomposes as N_{C/Z} = N_{1} ⊕ N_{2} with x_{1} = c_{1}(N_{1}), x_{2} = c_{1}(N_{2}). Then

(1.) ∑_{α=1}^{r−1}(ε^{−αi} − 1) (\frac{1−g}{(1−ε^{−aα})(1−ε^{aα})} − \frac{ε^{−aα}}{(1−ε^{−aα})^{2}(1−ε^{aα})} deg_{C} x_{1}−
\frac{ε^{aα}}{(1−ε^{−aα})(1−ε^{aα})^{2}} deg_{C} x_{2})

= \frac{\overline{ik}(r−\overline{ik})}{4} deg K_{Z}|_{C} + \frac{1}{12} \overline{ki}(r − \overline{ki})(r − 2 · \overline{ki})(deg_{C} x_{1} − deg_{C} x_{2})

and

(2.) ∑_{α=1,β=0 and α≠β}^{r−1}(ε^{−αi} − 1)\frac{1}{(1−ε^{−aα})(1−ε^{aα})(1−ε^{−d(α−β)})} =
−(r − 1)\frac{\overline{ki}(r−\overline{ki})}{4},

where a, r, d, i, k are integers with a and d coprime to r, denotes the smallest residue mod r, and \overline{ak} = 1.

Proof (1.) First observe that

\frac{1−g}{(1−ε^{−a})(1−ε^{a})} − \frac{ε^{−a}}{(1−ε^{−a})^{2}(1−ε^{a})} deg_{C} x_{1} − \frac{ε^{a}}{(1−ε^{−a})(1−ε^{a})^{2}} deg_{C} x_{2} =

\frac{1}{(1−ε^{−a})(1−ε^{a})} (1 − g + \frac{deg_{C} x_{1}}{2} + \frac{deg_{C} x_{1}}{2} ) −

\frac{1+ε^{−a}}{2(1−ε^{−a})^{2}(1−ε^{a})} deg_{C} x_{1} − \frac{1+ε^{a}}{2(1−ε^{−a})(1−ε^{a})^{2}} deg_{C} x_{2} =

\frac{1}{(1−ε^{−a})(1−ε^{a})} (1 − g + \frac{deg_{C} x_{1}}{2} + \frac{deg_{C} x_{1}}{2} ) − \frac{1+ε^{a}}{2(1−ε^{−a})(1−ε^{a})^{2}} (deg_{C} x_{2} − deg_{C} x_{1})

for all ε ∈ μ_{r} − {1}.

In the notation of Section 8 in [30]

∑_{ε≠1} \frac{ε^{−i} − 1}{(1 − ε^{a})(1 − ε^{r−a})}

equals to σ_{i}(\frac{1}{r}(r − a, a)) − σ_{0}(\frac{1}{r}(r − a, a)). This is by definition the same as σ\overline{ki}(\frac{1}{r}(−1, 1))−σ_{0}(\frac{1}{r}(−1, 1)), where ak ≡ 1 mod r and denotes the smallest residue mod r. Using Proposition (8.9) in [30] we get

−\frac{\overline{ik}(r − \overline{ik})}{2r}.

The same way

\begin{aligned}& & ∑_{ε≠1} \frac{(ε^{−i}−1)(1+ε^{a})}{2(1−ε^{−a})(1−ε^{a})^{2}} = \\ \frac{1}{2} & ∑ & ε≠1 & [−\frac{ε^{−i}−1}{(1−ε^{−a})(1−ε^{a})} + \frac{2(ε^{−i}−1)}{(1−ε^{−a})(1−ε^{a})^{2}} ] \end{aligned}

equals

−\frac{1}{2}σ_{i}(\frac{1}{r}(a, r−a))+\frac{1}{2}σ_{0}(\frac{1}{r}(a, r−a))+σ_{i}(\frac{1}{r}(a, r−a, r−a))−σ_{0}(\frac{1}{r}(a, r−a, r−a)).

Using Proposition (8.10) in [30] we get

\begin{matrix}−\frac{1}{2}σ\overline{ki}(\frac{1}{r}(1, −1)) + \frac{1}{2}σ_{0}(\frac{1}{r}(1, −1)) + σ_{r−\overline{ki}}(\frac{1}{r}(−1, 1, 1)) − σ_{0}(\frac{1}{r}(−1, 1, 1)) = \\ \frac{\overline{ki}(r−\overline{ki})}{4} − (r − \overline{ki})\frac{r^{2}−1}{12} + ∑_{j=1}^{r−\overline{ki}−1} \frac{j(r−j)}{2} = \frac{1}{12} \overline{ki}(r − \overline{ki})(r − 2 · \overline{ki}). \end{matrix}

These observations simplify the expression ∑_{α=1}^{r−1}(ε^{−αi} − 1) (\frac{1−g}{(1−ε^{−aα})(1−ε^{aα})} − \frac{ε^{−aα}}{(1−ε^{−aα})^{2}(1−ε^{aα})} deg_{C} x_{1}−

\frac{ε^{aα}}{(1−ε^{−aα})(1−ε^{aα})^{2}} deg_{C} x_{2}) =

∑_{μ_{r}∋ε≠1}(ε^{−i} − 1) (\frac{1−g}{(1−ε^{−a})(1−ε^{a})} − \frac{ε^{−a}}{(1−ε^{−a})^{2}(1−ε^{a})} deg_{C} x_{1} − \frac{ε^{a}}{(1−ε^{−a})(1−ε^{a})^{2}} deg_{C} x_{2}) into

−\frac{\overline{ik}(r−\overline{ik})}{2}(1 − g + \frac{deg_{C} x_{1}}{2} + \frac{deg_{C} x_{1}}{2} )+
\frac{1}{12r} \overline{ki}(r − \overline{ki})(r − 2 · \overline{ki})(deg_{C} x_{1} − deg_{C} x_{2}).

By Lemma 2.3.9, the term 1 − g + \frac{deg_{C} x_{1}}{2} + \frac{deg_{C} x_{1}}{2} = −\frac{1}{2} deg K_{Z}|C which finishes the proof of (1.). (2.) Since d and a are coprime to r

∑_{α=1,β=0 andα≠β}^{r−1}(ε^{−αi} − 1)\frac{1}{(1−ε^{−aα})(1−ε^{aα})(1−ε^{−d(α−β)})} =

∑_{j=1}^{r−1} \frac{1}{1−ε^{j}} ∑_{α=1}^{r−1}(ε^{−αi} − 1)\frac{1}{(1−ε^{−aα})(1−ε^{aα})} =
∑_{j=1}^{r−1} \frac{1}{1−ε^{j}} (σ_{i}(\frac{1}{r}(a, r − a)) − σ_{0}(\frac{1}{r}(a, r − a))) =
−\frac{\overline{ik}(r−\overline{ik})}{2} · ∑_{j=1}^{r−1} \frac{1}{1−ε^{j}} =
−\frac{r−1}{2} · \frac{\overline{ik}(r−\overline{ik})}{2}
□

2.3.4 Construction of an auxiliary cover

By Section 1.2 we can restrict ourselves to a threefold X embedded in a weighted projective space. In the neighbourhood of an A_{r−1} curve we will find two parameters t_{1}, t_{2} such that \sqrt[r]{t_{1}}, \sqrt[r]{t_{2}} define parameters in a cover Z. Then

X = Z/μ_{r} ⊕ μ_{r},

where μ_{r} ⊕ μ_{r} acts on \sqrt[r]{t_{1}}, \sqrt[r]{t_{2}}.

In the construction of the μ_{r} ⊕μ_{r} cover we will need the following lemma.

Lemma 2.3.11 Let

X ⊂ ℙ(a_{1}, …, a_{k}, b_{1}, …, b_{l})

with variables x_{1}, …, x_{k}, z_{1}, …, z_{l} be a quasismooth well formed threefold with singularities of transverse type A_{r−1} along a curve C. Assume that a_{1}, …, a_{k} are not divisible by r and b_{1}, …, b_{l} are. The singularities along C are induced by the weights b_{1}, …, b_{l}.

After reembedding C ⊂ X in a larger weighted projective space if necessary, we can always assume that

hcf (\frac{b_{1}}{r} , r) = hcf (\frac{b_{2}}{r} , r) = 1

and

C ∩ {z_{1} = 0} ∩ {z_{2} = 0} = ∅.

Proof Denote by D the ℚ−Cartier divisor given by O_{X}(1).

Since b_{1}, …, b_{l} induce the singularities along C that are all of type A_{r−1}, observe that hcf(b_{1}, …, b_{l}) = r. Let d be a positive number, coprime to r. By the Chinese Remainder Theorem there exist integers γ_{1}, …, γ_{l} such that

∑_{i=1}^{l} γ_{i}b_{i} = dr.

By adding ∑_{i=1}^{l} r^{2} ∗_{i} b_{i} for suitable nonnegative integers ∗_{i} to both sides of the equation, we get

∑_{i=1}^{l} γ′_{i}b_{i} = dr + r^{2} ∑_{i=1}^{l} ∗_{i}b_{i},

where all γ′_{i} are nonnegative. This proves that |(dr + r^{2} ∑_{i=1}^{l} ∗_{i}b_{i})D| has a section. Note that d + r ∑_{i=1}^{l} ∗_{i}b_{i} is coprime to r.

By the above construction, we can always find d_{1} and d_{2} that are both coprime to r and |d_{1}rD| and |d_{2}rD| have sections. Sections f_{1} ∈ |d_{1}rD| and f_{2} ∈ |d_{2}rD| can be expressed as relations among x_{1}, …, x_{k}, z_{1}, …, z_{l}, namely f_{1} = p_{1}(x_{i}, z_{j}) and f_{2} = p_{2}(x_{i}, z_{j}).

Define

w_{1} = p_{1}(x_{i}, z_{j}) and w_{2} = p_{2}(x_{i}, z_{j}).

Then

C ⊂ X ⊂ ℙ(a_{1}, …, a_{k}, b_{1}, …, b_{l})

with the variables x_{1}, …, x_{k}, z_{1}, …, z_{l} is isomorphic to

C′ ⊂ X′ ⊂ ℙ(a_{1}, …, a_{k}, b_{1}, …, b_{l}, d_{1}r, d_{2}r),

with the variables x_{1}, …, x_{k}, z_{1}, …, z_{l}, w_{1}, w_{2}, where the corresponding ideals are

I_{C′} = I_{C}+ < w_{1} − p_{1}, w_{2} − p_{2} > and I_{X′} = I_{X}+ < w_{1} − p_{1}, w_{2} − p_{2} > .

Each general section in |d_{1}rD| and |d_{2}rD| cuts C in a number of points. Pick two disjoint sections which fulfill the condition C ∩ {w_{1} = 0} ∩ {w_{2} = 0} = ∅. This finishes the proof. □

Remark 2.3.12 Note that the reembedding in the proof really is necessary. Consider for example

X_{900} ⊂ ℙ(1, 29, 60, 90, 150).

Proposition 2.3.13 Let

X ⊂ ℙ(a_{1}, …, a_{k}, b_{1}, …, b_{l})

be a quasismooth well formed threefold where a_{1}, …, a_{k} are not divisible by r and b_{1}, …, b_{l} are. Assume that O_{X}(1) contains a _{k}(\frac{1}{r}(1, −1)) curve C of genus g, where the singularities along the curve are induced by the weights b_{1}, …, b_{l} and (X, O_{X}(1)) contains no other singularities.

Then for all m > 0

χ(X, O_{X}(m)) = RR–type expression in O_{X}(m)+
\frac{\overline{mk}(r−\overline{mk})}{4r^{2}} deg K_{Z}|_{C_{Z}}+
\frac{1}{12r^{2}} \overline{km}(r − \overline{km})(r − 2 · \overline{km})(deg_{C_{Z}} x_{2} − deg_{C_{Z}} x_{1})−
Λ(r − 1)\frac{\overline{km}(r−\overline{km})}{4r^{2}} − Ξ(r − 1)\frac{\overline{km}(r−\overline{km})}{4r^{2}},

where

\begin{aligned}C_{Z} & ⊂ & Z \\ & & ↓ \\ C & ⊂ & X \end{aligned}

is a smooth auxiliary cover of X and the normal sheaf to C_{Z} decomposes into N_{C_{Z}/Z} = N_{1} ⊕N_{2} with x_{1} = c_{1}(N_{1}), x_{2} = c_{1}(N_{2}). The integers Λ and Ξ also depend on Z.

Proof Fix a generator ε ∈ μ_{r}. We construct an auxiliary cover of X in the following way. Consider the global quotient of

ℙ (a_{1}, …, a_{k}, \frac{b_{1}}{r} , \frac{b_{2}}{r} , b_{3}, …, b_{l})

with the variables x_{1}, …, x_{k}, z_{1}, …, z_{l}, by the group μ_{r}⊕μ_{r} ∋ (ε^{α}, ε^{β}) acting via

\underbrace{1, …, 1}_{k}, ε^{−α \frac{b_{1}}{r}} , ε^{−β \frac{b_{2}}{r}} , \underbrace{1, …, 1}_{l−2} .

By Lemma 2.3.11 we can assume that \frac{b_{1}}{r} and \frac{b_{2}}{r} are both coprime to r. Thus, the invariants x_{1}, …, x_{k}, z_{3}, …, z_{l} and z_{1}^{r}, z_{2}^{r} generate the coordinate ring of

ℙ (a_{1}, …, a_{k}, \frac{b_{1}}{r} , \frac{b_{2}}{r} , b_{3}, …, b_{l}) /μ_{r} ⊕ μ_{r}

which is exactly

ℙ(a_{1}, …, a_{k}, b_{1}, b_{2}, b_{3} …, b_{l}).

Write

\begin{aligned}π: & ℙ (a_{1}, …, a_{k}, \frac{b_{1}}{r} , \frac{b_{2}}{r} , b_{3}, …, b_{l}) & ⟶ & ℙ (a_{1}, …, a_{k}, b_{1}, b_{2}, b_{3}, …, b_{l}) \\ & x_{1}, …, x_{k}, z_{1}, z_{2}, z_{3}, …, z_{l} & ⟼ & x_{1}, …, x_{k}, z_{1}^{r}, z_{2}^{r}, z_{3}, …, z_{l} \end{aligned}

for the quotient map.

The threefold X ⊂ ℙ (a_{1}, …, a_{k}, b_{1}, …, b_{l}) is defined by a set of equations and relations in variables

x_{1}, …, x_{k}, t_{1} = z_{1}^{r}, t_{2} = z_{2}^{r}, z_{3}, …, z_{l}.

The singularities along the curve C are induced by weights b_{1}, …, b_{l}, thus

C = X ∩ {x_{1} = … = x_{k} = 0}.

Now define

C_{Z} ⊂ Z ⊂ ℙ (a_{1}, …, a_{k}, \frac{b_{1}}{r} , \frac{b_{2}}{r} , b_{3}, …, b_{l})

by the same equations and relations as

C ⊂ X ⊂ ℙ (a_{1}, …, a_{k}, b_{1}, …, b_{l}) .

In other words, replace t_{1} by z_{1}^{r} and t_{2} by z_{2}^{r} in all equations and relations. By Lemma 2.3.11 C ∩ {t_{1} = t_{2} = 0} = ∅. Thus, by the definition of C_{Z} also C_{Z} ∩{z_{1} = z_{2} = 0} = ∅, which implies that Z is smooth in the neighbourhood of C_{Z}.

The threefold Z inherits μ_{r} ⊕ μ_{r} = {(ε^{α}, ε^{β}) | α, β = 0, …, r − 1} action which we now describe:

• If α = β the action is

\underbrace{1, …, 1}_{k}, ε^{−α \frac{b_{1}}{r}} , ε^{−α \frac{b_{2}}{r}} , \underbrace{1, …, 1}_{l−2},

which is the same as

ε^{αa_{1}}, …, ε^{αa_{k}}, \underbrace{1, …, 1}_{l} .

It fixes C_{Z} and acts in the normal direction of the curve by ε^{aα}, ε^{(r−a)α} for some a coprime to r.

• If α ≠ 0 and α ≠ β it fixes a number of points given by {z_{2} = 0} on C_{Z}. The action is

ε^{αa_{1}}, …, ε^{αa_{k}}, 1, ε^{(α−β) \frac{b_{2}}{r}} , \underbrace{1, …, 1}_{l−2}

which acts on each point in Z by ε^{aα}, ε^{(r−a)α}, ε^{(α−β) \frac{b_{2}}{r}} . Denote the number of these points by Λ.

• If β ≠ 0 and α ≠ β it fixes a number of points given by {z_{1} = 0} on C_{Z}. The action is

ε^{βa_{1}}, …, ε^{βa_{k}}, ε^{(β−α) \frac{b_{1}}{r}} , 1, \underbrace{1, …, 1}_{l−2}

which acts on each point in Z by ε^{aβ}, ε^{(r−a)β}, ε^{(β−α) \frac{b_{1}}{r}} . Denote the number of these points by Ξ.

• If α = 0 it fixes surfaces given by {z_{2} = 0} on Z. The action is

\underbrace{1, …, 1}_{k}, 1, ε^{−β \frac{b_{2}}{r}} , \underbrace{1, …, 1}_{l−2},

which acts in the normal direction by ε^{−β \frac{b_{2}}{r}}.

• If β = 0 it fixes surfaces given by {z_{1} = 0} on Z. The action is

\underbrace{1, …, 1}_{k}, ε^{−α \frac{b_{1}}{r}} , 1, \underbrace{1, …, 1}_{l−2},

which acts in the normal direction by ε^{−α \frac{b_{1}}{r}}.

This is again an example of Action ♠ introduced in Definition 2.3.8.

Now write π: Z → X for the quotient map and let L_{i,j} be the (i, j)th eigensheaf of the action of μ_{r} ⊕ μ_{r} on π_{∗}O_{Z}.

Without loss of generality we can assume that the variables x_{1} and x_{2} in ℙ induce the action on the normal direction of C. In other words, the corresponding weights are a_{1} = a mod r and a_{2} = −a mod r. Observe that O_{X}(a_{1}) has a global section x_{1}. The divisor D_{a} = {x_{1} = 0} defines a divisorial subsheaf O_{X}(D_{a}) ⊂ k(X). On the open set U_{1} = {t_{1} ≠ 0} we can write

\frac{\sqrt[r]{t_{1}}^{a_{1}}}{x1^{\frac{b_{1}}{r}}} ∈ Γ (U_{1}, O_{X} (\frac{b_{1}}{r} D_{a})) ,

and the same way on U_{2} = {t_{2} ≠ 0} we have

\frac{\sqrt[r]{t_{2}}^{a_{1}}}{x1^{\frac{b_{2}}{r}}} ∈ Γ (U_{2}, O_{X} (\frac{b_{2}}{r} D_{a})) .

This means that the sheaf O_{X}(\frac{b_{1}}{r} D_{a}) is generated by \frac{\sqrt[r]{t_{1}}^{a_{1}}}{x \frac{b_{1}}{r} ; 1} on U_{1}, and

O_{X}(\frac{b_{2}}{r} D_{a}) is generated by \frac{\sqrt[r]{t_{2}}^{a_{1}}}{x \frac{b_{2}}{r} ; 1} on U_{2}. In order to simplify the notation

substitute t_{1} = z_{1}^{r} and t_{2} = z_{2}^{r}. On U_{1}, the function

\frac{z_{1}^{a_{1}}}{x1^{\frac{b_{1}}{r}}} ∈ L_{\overline{−a \frac{b_{1}}{r}} ,0},

since

(ε^{α}, ε^{β}) (\frac{z_{1}^{a_{1}}}{x1^{\frac{b_{1}}{r}}} ) = ε^{−αa \frac{b_{1}}{r}} · \frac{z_{1}^{a_{1}}}{x1^{\frac{b_{1}}{r}}} .

The same way on U_{2}, the function

\frac{z_{2}^{a_{1}}}{x1^{\frac{b_{2}}{r}}} ∈ L_{0,\overline{−a \frac{b_{2}}{r}}} ,

since

(ε^{α}, ε^{β}) (\frac{z_{2}^{a_{1}}}{x1^{\frac{b_{2}}{r}}} ) = ε^{−βa \frac{b_{2}}{r}} · \frac{z_{2}^{a_{1}}}{x1^{\frac{b_{2}}{r}}} .

Thus we proved

O_{X} (\frac{b_{1}}{r} D_{a})|_{U_{1}} has the same singularities as L_{\overline{−a \frac{b_{1}}{r}} ,0}|_{U_{1}}

and

O_{X} (\frac{b_{2}}{r} D_{a})|_{U_{2}} has the same singularities as L_{0,\overline{−a \frac{b_{2}}{r}}} |_{U_{2}},

where denotes the smallest residue mod r.

Let m be an arbitrary positive integer. Since integers a, a_{1} and \frac{b_{1}}{r} , \frac{b_{2}}{r} are all coprime to r, we can find n_{1}, n_{2} such that

n_{1} · a_{1} \frac{b_{1}}{r} = m mod r and n_{2} · a_{1} \frac{b_{2}}{r} = m mod r.

This implies:

O_{X}(m)|_{U_{1}}

has the same singularities as

O_{X}(n_{1} · a_{1} \frac{b_{1}}{r})|_{U_{1}} ≅ O_{X}(n_{1} \frac{b_{1}}{r} D_{a})|_{U_{1}},

which has the same singularities as

L_{\overline{−m},0}|_{U_{1}}.

The first part of the statement holds, since the corresponding divisorial sheaves differ only by tensoring with an invertible sheaf. If n_{1} ·a_{1} \frac{b_{1}}{r} = m+∗r for some positive ∗, then

O_{X}(m + ∗r) ≅ O_{X}(m) ⊗ O_{X}(Cartier divisor).

The second part of the statement is true since the section (\frac{z_{1}^{a_{1}}}{x \frac{b_{1}}{r} ; 1} )^{n_{1}} of

O_{X}(n_{1} \frac{b_{1}}{r} D_{a})|_{U_{1}} lies in L_{\overline{−m},0}|_{U_{1}}.

The same way we prove

O_{X}(m)|_{U_{2}}

has the same singularities as

O_{X}(n_{2} · a_{1} \frac{b_{2}}{r})|_{U_{2}} ≅ O_{X}(n_{2} \frac{b_{2}}{r} D_{a})|_{U_{2}},

which has the same singularities as

L_{0,\overline{−m}}|_{U_{2}}.

Note also that L_{\overline{−m},0}|_{U_{1}∩U_{2}} has the same singularities as L_{0,\overline{−m}}|_{U_{1}∩U_{2}}.

In (E. vi) on page 34 we proved

χ(X, L_{i,j}) = χ(O_{X}) + \frac{1}{r^{2}} ∑_{α,β=0}^{r−1}(ε^{−αi−βj} − 1)A_{α,β}

for any Action ♠. Here we think of χ(O_{X}) as RR–type expression in L_{ij}, where L_{ij} = 0 ∈ DivX ⊗ ℚ is the corresponding ℚ−divisor. The rest of the formula is a sum of contributions coming from the singularities contained in L_{i,j}.

This proves

χ(X, O_{X}(m)) = RR–type expression in O_{X}(m) +
contributions from singularities of L_{\overline{−m},0}|_{U_{1}}+
contributions from singularities of L_{0,\overline{−m}}|_{U_{2}}−
contributions from singularities of L_{\overline{−m},0}|_{U_{1}∩U_{2}}.

The following lemma will finish the proof of Proposition 2.3.13

Lemma 2.3.14 The contributions from the singularities of L_{i,0} on {t_{1} ≠ 0} are

\frac{\overline{ik}(r−\overline{ik})}{4r^{2}} deg K_{Z}|_{C_{Z}} + \frac{1}{12r^{2}} \overline{ki}(r − \overline{ki})(r − 2 · \overline{ki})(deg_{C_{Z}} x_{1} − deg_{C_{Z}} x_{2})−
Λ(r − 1)\frac{\overline{ki}(r−\overline{ki})}{4r^{2}}.

The contributions from the singularities of L_{0,j} on {t_{2} ≠ 0} are

\frac{\overline{jk}(r−\overline{jk})}{4r^{2}} deg K_{Z}|_{C_{Z}} + \frac{1}{12r^{2}} \overline{kj}(r − \overline{kj})(r − 2 · \overline{kj})(deg_{C_{Z}} x_{1} − deg_{C_{Z}} x_{2})−

Ξ(r − 1)\frac{\overline{kj}(r−\overline{kj})}{4r^{2}}.

The contributions from the singularities of L_{i,0} on {t_{1} ≠ 0} ∩ {t_{2} ≠ 0} are

\frac{\overline{ik}(r−\overline{ik})}{4r^{2}} deg K_{Z}|_{C_{Z}} + \frac{1}{12r^{2}} \overline{ki}(r − \overline{ki})(r − 2 · \overline{ki})(deg_{C_{Z}} x_{1} − deg_{C_{Z}} x_{2}).

Proof By (E. vi) the contributions from the singularities of L_{i,0} are

\frac{1}{r^{2}} ∑_{α,β=0}^{r−1}(ε^{−αi} − 1)A_{α,β}.

On {t_{1} ≠ 0} the contributions come from the curve C_{Z}, points C_{Z} ∩ {z_{2} = 0} and surfaces {z_{2} = 0}. By the definition of A_{α,β} this simplifies the above sum into

\frac{1}{r^{2}} ∑_{α=1}^{r−1}(ε^{−αi} − 1) (\frac{1−g}{(1−ε^{−aα})(1−ε^{aα})} − \frac{ε^{−aα}}{(1−ε^{−aα})^{2}(1−ε^{aα})} deg_{C} x_{1}−
\frac{ε^{aα}}{(1−ε^{−aα})(1−ε^{aα})^{2}} deg_{C} x_{2})
+\frac{1}{r^{2}} ∑_{α=1,β=0; and α≠β}^{r−1}(ε^{−αi} − 1)\frac{Λ}{(1−ε^{−aα})(1−ε^{aα})(1−ε^{−d_{2}(α−β)})}

By Lemma 2.3.10 the sum can be written in a more compact form

\frac{\overline{ik}(r−\overline{ik})}{4r^{2}} deg K_{Z}|_{C_{Z}} + \frac{1}{12r^{2}} \overline{ki}(r − \overline{ki})(r − 2 · \overline{ki})(deg_{C_{Z}} x_{1} − deg_{C_{Z}} x_{2})−
Λ(r − 1)\frac{\overline{ki}(r−\overline{ki})}{4r^{2}}.

Exactly the same way we compute the contributions of L_{0,j} on {t_{2} ≠ 0}.

On {t_{1} ≠ 0} ∩ {t_{2} ≠ 0} the only contributions come from the curve C_{Z} and equal

\frac{\overline{ik}(r − \overline{ik})}{4r^{2}} deg K_{Z}|_{C_{Z}} + \frac{1}{12r^{2}} \overline{ki}(r − \overline{ki})(r − 2 · \overline{ki})(deg_{C_{Z}} x_{1} − deg_{C_{Z}} x_{2}).

□

2.3.5 RR does not depend on the auxiliary cover

We would like to express the quantities deg K_{Z}|_{C_{Z}}, Λ and Ξ in Proposition 2.3.13 (depending on the auxiliary cover Z) with parameters of X.

In this construction we will need the following two lemmas.

Lemma 2.3.15 Under the conditions of Proposition 2.3.13,

deg K_{Z}|_{C_{Z}} = r · deg K_{X}|_{C} + deg R,

where R is the ramification divisor of a finite r to 1 morphism of curves π|_{C_{Z}}: C_{Z} → C.

In particular

deg K_{Z}|_{C_{Z}} − (r − 1)(Λ + Ξ) = r · deg K_{X}|_{C}

Proof Consider the auxiliary cover

C_{Z} ⊂ Z ⊂ ℙ (a_{1}, …, a_{k}, \frac{b_{1}}{r} , \frac{b_{2}}{r} , b_{3}, …, b_{l})
↓^{π|_{C_{Z}}} ↓^{π|_{Z}} ↓^{π}
C ⊂ X ⊂ ℙ (a_{1}, …, a_{k}, b_{1}, b_{2}, b_{3}, …, b_{l})

with the variables

x_{1}, …, x_{k}, z_{1}, z_{2}, z_{3}, …, z_{l} ⟶ x_{1}, …, x_{k}, t_{1} = z_{1}^{r}, t_{2} = z_{2}^{r}, z_{3}, …, z_{l},

constructed in the proof of Proposition 2.3.13.

The variety X is an orbifold, as defined in Appendix A.2. As usual, restrict ourselves to a neighbourhood of C, which can be covered by local charts U_{λ}/G_{λ}. Every G_{λ} is isomorphic to μ_{r} ∋ ε with action

x_{λ}, y_{λ}, z_{λ} ↦ ε^{a}x_{λ}, ε^{r−a}y_{λ}, z_{λ}

on the local parameters of U_{λ}. Then,

\begin{matrix}U_{λ} ⊂ ℂ^{3} & ⟶^{q_{λ}} & U_{λ}/μ_{r} \\ x_{λ}, y_{λ}, z_{λ} & ↦ & x_{λ}^{r}, x_{λ}y_{λ}, y_{λ}^{r}, z_{λ}. \end{matrix}

The corresponding ring of invariants is

ℂ[x_{λ}^{r}, x_{λ}y_{λ}, y_{λ}^{r}, z_{λ}] ≅ ℂ[u_{λ}, v_{λ}, w_{λ}, z_{λ}]/(u_{λ}w_{λ} = v_{λ}^{r}).

Observe that dx_{λ} ∧ dy_{λ} ∧ dz_{λ} is a basis for ω_{U_{λ}} which is invariant under the group action. Now set s = \frac{du_{λ}∧dv_{λ}∧dz_{λ}}{u_{λ}} . We claim that s is a basis for ω_{U_{λ}/μ_{r}}. Indeed, by differentiating u_{λ} = x_{λ}^{r}, v_{λ} = x_{λ}y_{λ} we get

q_{λ}^{∗}s = r · dx_{λ} ∧ dy_{λ} ∧ dz_{λ}.

Recall from Appendix A.2 that the sheaves ω_{U_{λ}/μ_{r}} glue together to define ω_{X} = O_{X}(K_{X}).

There are two different types of points on C:

• points that have one of the coordinates t_{1} or t_{2} zero. Recall that there are Λ + Ξ such points.

• points for which t_{1} and t_{2} are nonzero.

Let P be a point of the first type with a local chart U/μ_{r} ⊂ X. Note that π^{−1}(P) is only one point. Choose V ⊂ π^{−1}(U/μ_{r}) ⊂ Z, a neighborhood of π^{−1}(P) with local parameters x, y, t. Then

\begin{aligned}π|_{V} : & V & ⟶ & U/μ_{r} \\ & x, y, t & ↦ & x^{r}, xy, y^{r}, t^{r}. \end{aligned}

Obviously the diagram

U  x, y, z
↓  ↓
V ⟶ U/μ_{r} x^{r}, xy, y^{r}, z
x, y, t ↦ x^{r}, xy, y^{r}, t^{r}

factors. The induced map

\begin{aligned}f: & V & ⟶ & U \\ & x, y, t & ↦ & x, y, t^{r} \end{aligned}

is a r to 1 morphism with the ramification divisor R = {t = 0}. Its ramification index is r. By Lemma A.1.1

ω_{V} ≅ f^{∗}ω_{U} ⊗ O_{R}.

Let Q be a point of the second type with a local chart U′/μ_{r} ⊂ X. Note that π^{−1}(Q) contains r points. Denote by {V′_{i}}_{i=1,... ,r} the corresponding neighborhoods in π^{−1}(U′/μ_{r}) ⊂ Z. In this case the induced map f|_{V′_{i}}: V′_{i} ⟶ U′ is an isomorphism, and

ω_{V′_{i}} ≅ ω_{U′} or eqivalently f^{∗}ω_{U′} ≅ ω_{∪_{i}V′_{i}}.

Recall from the previous page that ω_{U_{λ}/μ_{r}} are defined by ω_{U_{λ}}. Gluing together different charts V_{λ} in Z and U_{λ}/μ_{r} in X yields

ω_{Z} ≅ π^{∗}ω_{X} ⊗ O_{R},

where R is the ramification divisor of π supported on Λ + Ξ surfaces given by t_{1} = 0 or t_{2} = 0. For the corresponding divisors on Z this implies

K_{Z} ∼ π^{∗}K_{X} + R.

Restrict these divisors to the curve C_{Z}. Then deg R|_{C_{Z}} = (r − 1)(Λ + Ξ), since each ramification surface intersects C_{Z} in exactly one point with the ramification index r.

Next consider deg(π^{∗}K_{X})|_{C_{Z}}. Since X is ℚ–Gorenstein, a multiple of K_{X} can be moved away from the branched points on C. By definition tK_{X} intersects C in t deg K_{X}|_{C} points. Since π|_{C_{Z}}: C_{Z} → C is r to 1 and etale away from R|_{C_{Z}}, the divisor π^{∗}tK_{X} intersects C_{Z} in r · t deg K_{X}|_{C} points. (Think of K_{X} set theoretically as a Weil divisor. Then P ∈ π^{∗}K_{X} ∩ C_{Z} ⇒ π(P) ∈ K_{X} ∩ C. Conversely, for a generic Q ∈ K_{X} ∩ C ⇒ π^{−1}(Q) contains r points in π^{∗}K_{X} ∩ C_{Z}, since π^{−1}(C) = C_{Z}.) This finishes the proof of Lemma 2.3.15. □

Lemma 2.3.16 Under the conditions of Proposition 2.3.13

deg_{C_{Z}} x_{2} − deg_{C_{Z}} x_{1} = 2r(genus(C) − 1) − r deg K_{X}|_{C} − 2 deg_{C_{Z}} x_{1}.

Proof Recall that the auxiliary cover Z is smooth in the neighborhood of C_{Z} with c_{1}(N_{C_{Z}/Z}) = x_{1} + x_{2}. By Lemma 2.3.9

deg c_{1}(N_{C_{Z}/Z}) = deg K_{C_{Z}} − deg K_{Z}|_{C_{Z}} = 2(genus(C_{Z}) − 1) − deg K_{Z}|_{C_{Z}}.

Now use Hurwitz theorem and Lemma 2.3.15

deg c_{1}(N_{C_{Z}/Z}) = deg_{C_{Z}} x_{2} + deg_{C_{Z}} x_{1} =
2r(genus(C) − 1) + deg R − r deg K_{X}|_{C} − deg R.

From here we get

deg_{C_{Z}} x_{2} − deg_{C_{Z}} x_{1} = 2r(genus(C) − 1) − r deg K_{X}|_{C} − 2 deg_{C_{Z}} x_{1}.
□

Theorem 2.3.17 Let X be a quasismooth and well formed threefold embedded in a weighted projective space and let D be a divisor on X given by O_{X}(1). Assume that the singularities of (X, D) consist of

• curves C of type _{k}(\frac{1}{r}(1, −1)) and genus g with no dissident points,

• isolated singular points P.

The Riemann–Roch formula for the divisorial sheaf O_{X}(mD) is

\begin{aligned}χ(X, O_{X}(mD)) & = & χ(O_{X}) + \frac{1}{12}mD(mD − K_{X})(2mD − K_{X}) + m\frac{D·c_{2}(X)}{12} \\ & & + ∑_{C} s_{C}(mD) + P, \end{aligned}

where

s_{C}(mD) = −m\frac{\overline{mk}(r−\overline{mk})}{2r} deg D|_{C} + \frac{\overline{mk}(r−\overline{mk})}{4r} deg K_{X}|_{C}
+\frac{\overline{mk}(r−\overline{mk})(r−2·\overline{mk})}{6r} · (g − 1 − \frac{1}{2} deg K_{X}|_{C} − \frac{ι_{C}}{r} )

and P only depends on the isolated points of X. The integer ι_{C} is an invariant of X in the neighbourhood of C.

Proof Every component of the singular locus gives a contribution to the Riemann–Roch formula which is a local analytic invariant. This means that the contributions can be computed separately on the local neighbourhood of each singularity.

Proposition 2.3.13, Lemma 2.3.15 and Lemma 2.3.16 together imply

χ(X, O_{X}(mD)) = (RR–type expression in mD)+
∑_{C} s′_{C}(mD) + P,

where

s′_{C}(mD) = \frac{\overline{mk}(r−\overline{mk})}{4r} deg K_{X}|_{C}
+ \frac{\overline{mk}(r−\overline{mk})(r−2·\overline{mk})}{6r} · (g − 1 − \frac{1}{2} deg K_{X}|_{C} − \frac{1}{r} deg_{C_{Z}} x_{1})

The existence of the RR formula proves that the integers ι_{C} := deg_{C_{Z}} x_{1} do not depend on the auxiliary covers and are thus invariants of X.

The Riemann–Roch type expression in mD is just ch(mD) · td(T_{X}). This contains the terms

\frac{1}{6}m^{3}D^{3}, \frac{1}{4}m^{2}D^{2}K_{X}, \frac{1}{12}mDK_{X}^{2}, 1 − p_{a} and \frac{1}{12}mD · c_{2}(X).

Some multiple of D is a Cartier divisor, i.e. |tD| defines a surface S for some t. Therefore D^{3} can be computed away from the singularities. The same way D^{2}K_{X} and DK_{X}^{2} can be computed away from the singularities. The arithmetic genus p_{a} is a birational invariant of variety and 1 − p_{a} = χ(O_{X}).

On the other hand the Cartier divisor tD always intersects the singular locus C in a finite number of points. These singular points will induce additional contributions to our RR formula, which we now describe. Consider the short exact sequence

0 → T_{S} → i^{∗}T_{X} → N_{S/X} → 0,

where i: S → X is the inclusion. The corresponding Chern polynomials are

\begin{aligned}c_{t}(T_{S}) & = & 1 − K_{S}t + c_{2}(S)t^{2}, \\ c_{t}(N_{S/Z}) & = & 1 + c_{1}(N_{S/X})t, \\ c_{t}(T_{X}) & = & 1 − K_{X}t + c_{2}(X)t^{2} + c_{3}(X)t^{3}. \end{aligned}

Use the self–intersection formula c_{1}(N_{S/X}) = S·S and the adjunction formula K_{S} = S|_{S} + K_{X}|_{S} and then compare the coefficients of t^{2}. We get c_{2}(S) − (S|_{S})^{2} − S|_{S}K_{X}|_{S} = c_{2}(X)|_{S}, which is the same as

c_{2}(S) − S^{3} − K_{X}S^{2} = S · c_{2}(X).

Recall that S was defined by tD and that S^{3} and K_{X}S^{2} can be computed away from the singular locus. The contributions to Riemann–Roch coming from the term \frac{1}{12}mD ·c_{2}(X) = \frac{m}{t} \frac{1}{12}S ·c_{2}(X) will thus equal the contributions induced by \frac{m}{t} \frac{1}{12}c_{2}(S) on S. Riemann–Roch for a surface with isolated singular points was computed by Miles Reid in [30]. Given the ℚ−Cartier divisor mD|_{S}, the term \frac{1}{12}c_{2}(S) contributes exactly the so called basket −\frac{\overline{mk}(r−\overline{mk})}{2r} for each singular point. There is t · deg D|_{C} points in the intersection S ∩ C.

By these considerations, we can interpret the term \frac{1}{12}mD·c_{2}(X) appearing in ch(mD) · td(T_{X}) as

\frac{1}{12}mD · c_{2}(X) computed on the nonsingular part of X−
m · \frac{\overline{mk}(r−\overline{mk})}{2r} deg D|_{C}.

Finally we get

RR–type expression in mD =

χ(O_{X}) + \frac{1}{12}mD(mD − K_{X})(2mD − K_{X}) + m · \frac{D·c_{2}(X)}{12}−

m · \frac{\overline{mk}(r−\overline{mk})}{2r} deg D|_{C}

where D^{3}, K_{X}D^{2}, K_{X}^{2}D and D · c_{2}(X) are computed away from the singularities. □

Observe that Theorem 2.3.17 and Corollary 2.2.5 indeed are identical for X whose curves of singularities contain no dissident points.

Now all that remains is to calculate the contributions from the singular points.

2.4 Contributions from singular points

We proved that it is enough to compute the contributions to RR from singular points on any model that contains such singularities. First we will show the existence of such projective varieties.

Proposition 2.4.1 Fix positive integers s and a_{1}, a_{2}, a_{3}, and assume that a_{i}, a_{j}, s have no common factor for all different i, j ∈ {1, 2, 3}. There exists a smooth projective 3–fold Z together with μ_{s} = ℤ/sℤ action that we now describe: Choose a generator ε ∈ μ_{s}. The action fixes a number of points on which ε acts by

ε: z_{1}, z_{2}, z_{3} ↦ ε^{a_{1}}z_{1}, ε^{a_{2}}z_{2}, ε^{a_{3}}z_{3}.

If hcf(a_{i}, s) = α_{i} ≠ 1, these points lie on curves which are fixed by ε^{\frac{s}{α_{i}}} . Here ε^{\frac{s}{α_{i}}} ∈ μ_{α_{i}} acts in the normal direction of each curve by

ε^{\frac{s}{α_{i}}} : z_{j}, z_{k} ↦ ε^{\frac{s}{α_{i}} a_{j}}z_{j}, ε^{\frac{s}{α_{i}} a_{k}}z_{k}, for j, k ∈ \{1, 2, 3\} − \{i\}

and freely away from the curve.

Proof The assumption hcf(a_{i}, a_{j}, s) = 1 implies that the action fixes no two dimensional stratum, thus the quotient variety is well formed. We imitate the proof of (8.4) in [30]. Choose an integer l ≥ 3 and consider action of μ_{s} on ℙ^{l+3}(1, 1, …, 1) given by

x_{1}, x_{2}, x_{3}, x_{4}, …, x_{l+4} ↦ ε^{a_{1}}x_{1}, ε^{a_{2}}x_{2}, ε^{a_{3}}x_{3}, x_{4}, …, x_{l+4}.

This action fixes ℙ^{l} = {x_{1} = x_{2} = x_{3} = 0} and acts in the normal direction by

x_{1}, x_{2}, x_{3} ↦ ε^{a_{1}}x_{1}, ε^{a_{2}}x_{2}, ε^{a_{3}}x_{3}.

If hcf(a_{i}, s) = α_{i} ≠ 1, the action is not free on ℙ^{l+1} = {x_{j} = x_{k} = 0} (for j, k ∈ \{1, 2, 3\} − \{i\}), which is fixed by ε^{\frac{s}{α_{i}}} . It acts in the normal direction by

ε^{\frac{s}{α_{i}}} : x_{j}, x_{k} ↦ ε^{\frac{s}{α_{i}} a_{j}}x_{j}, ε^{\frac{s}{α_{i}} a_{k}}x_{k}.

Another locus on which the action might not be free, is {x_{4}, …, x_{l+4} = 0}. We will avoid this locus by defining

X ⊂ ℙ^{l+3}/μ_{s},

as a complete intersection of l general very ample divisors. Let Z be the inverse image of X under the quotient ℙ^{l+3} → ℙ^{l+3}/μ_{s}. Such Z clearly satisfies the conditions in the proposition. □

Using the above notation, we additionally assume that

a_{j} + a_{k} = 0 mod α_{i}.

This implies that the singular locus of X equals to a number of dissident

A_{α_{i}−1} = \frac{1}{α_{i}} (a_{j}, a_{k})

curves, which intersect in dissident points of type \frac{1}{s}(a_{1}, a_{2}, a_{3}).

Let now X be a projective threefold with a singularity of type \frac{1}{s}(a_{1}, a_{2}, a_{3}), which is a quotient of a smooth projective variety Z under μ_{s} action as described in Proposition 2.4.1.

Write π: Z → X for the quotient map and let L_{n} be the nth eigensheaf of the action of μ_{s} on π_{∗}O_{Z}. Then

π_{∗}O_{Z} = ⨁_{n=0}^{s−1} L_{n}

implies

H^{p}(Z, O_{Z}) = ⨁_{n=0}^{s−1} H^{p}(X, L_{n}).

The group action on any f ∈ L_{n} is given by ε(f) = ε^{n} · f, thus

Tr(ε : H^{p}(Z, O_{Z})) = ∑_{n=0}^{s−1} h^{p}(X, L_{n}) · ε^{n}.

Moreover,

∑_{p}(−1)^{p}Tr(ε : H^{p}(Z, O_{Z})) = ∑_{n=0}^{s−1} χ(X, L_{n}) · ε^{n}.

In order to simplify the notation, denote ∑_{p}(−1)^{p}Tr(ε : H^{p}(Z, O_{Z})) by A_{ε}. Then

∑_{n=0}^{s−1} χ(X, L_{n}) · ε^{n} = A_{ε}.

In the case ε = 1, we get

∑_{n=0}^{s−1} χ(X, L_{n}) = χ(O_{Z}).

From now on fix a generator ε ∈ μ_{s}. The last two formulas can be considered as a linear system of s equations in χ(X, L_{n}) and variable ε, given by Vandermonde matrix,

( \begin{matrix}1 & 1 & 1 & … & 1 \\ 1 & ε & ε^{2} & … & ε^{s−1} \\ 1 & ε^{2} & ε^{4} & … & ε^{2(s−1)} \\ … & … & … & … \\ 1 & ε^{s−1} & … & … & ε \end{matrix} ) ( χ(X, L_{0}) ; χ(X, L_{1}) ; χ(X, L_{2}) ; ⋮ ; χ(X, L_{s−1}) ) = ( χ(O_{Z}) ; A_{ε} ; A_{ε^{2}} ; ⋮ ; A_{ε^{s−1}} ) .

Since the Vandermonde matrix is invertible, we can solve the system

\frac{1}{s} ( \begin{matrix}1 & 1 & 1 & … & 1 \\ 1 & ε^{−1} & ε^{−2} & … & ε \\ 1 & ε^{−2} & ε^{−4} & … & ε^{2} \\ … & … & … & … \\ 1 & ε & … & … & ε^{s−1} \end{matrix} ) ( χ(O_{Z}) ; A_{ε} ; A_{ε^{2}} ; ⋮ ; A_{ε^{s−1}} ) = ( χ(X, L_{0}) ; χ(X, L_{1}) ; χ(X, L_{2}) ; ⋮ ; χ(X, L_{s−1}) ) .

Write χ(X, L_{0}) = χ(O_{X}) and eliminate χ(O_{Z}) from the solution. We end up with formula

χ(X, L_{n}) = χ(O_{X}) + \frac{1}{s} ∑_{j=1}^{s−1}(ε^{−jn} − 1)A_{ε^{j}}  (E. vii)
= χ(O_{X}) + \frac{1}{s} ∑_{ε∈μ_{s}}(ε^{−n} − 1)A_{ε}.

We have already computed terms A_{ε} for all ε ∈ μ_{s} using the Atiyah-Singer formula. If ε^{a_{i}} ≠ 1 for all i = 1, 2, 3, then the fixed locus of ε is a number of points. By Lemma 2.3.7 we get

A_{ε} = ∑_{p}(−1)^{p}Trace(ε|H^{p}(Z, O_{Z})) = \frac{1}{(1−ε^{−a_{1}})(1−ε^{−a_{2}})(1−ε^{−a_{3}})}.

If, on the other hand ε^{a_{i}} = 1 for one of the i = 1, 2, 3, then the fixed locus of ε is a curve C_{i}. In this case hcf(a_{i}, s) = α_{i} ≠ 1 and ε is an element of μ_{α_{i}}. The action in the normal direction equals

ε^{a_{j}}, ε^{a_{k}} = ε^{a_{j}}, ε^{−a_{j}}.

Thus, we can use Theorem 2.3.4 for the cyclic group μ_{α_{i}}. Let C_{i} have genus g_{i}, and let x_{1}, x_{2} denote the first Chern classes of N_{C_{i}/Z} = N_{1} ⊕ N_{2}. We get

A_{ε} = ∑_{p}(−1)^{p}Trace(ε|H^{p}(Z, O_{Z})) =

\frac{1−g_{i}}{(1−ε^{−a_{j}})(1−ε^{a_{j}})} − \frac{ε^{−a_{j}}}{(1−ε^{−a_{j}})^{2}(1−ε^{a_{j}})} deg_{C_{i}} x_{1} − \frac{ε^{a_{j}}}{(1−ε^{−a_{j}})(1−ε^{a_{j}})^{2}} deg_{C_{i}} x_{2}.

Since we assumed that a_{i}, a_{j}, s have no common divisor for all i, j = 1, 2, 3, we can split the sum over {ε ∈ μ_{s}} into four subsumes over (possibly empty) disjoint sets

{ε ∈ μ_{s} s.t. ε^{a_{i}} ≠ 1 ∀i = 1, 2, 3},
{ε ∈ μ_{s} s.t. ε^{a_{1}} = 1} = {ε ∈ μ_{α_{1}}},
{ε ∈ μ_{s} s.t. ε^{a_{2}} = 1} = {ε ∈ μ_{α_{2}}} and
{ε ∈ μ_{s} s.t. ε^{a_{3}} = 1} = {ε ∈ μ_{α_{3}}}.

This rewrites Formula (E. vii) into

χ(X, L_{n}) = χ(O_{X})
+\frac{1}{s} ∑_{ε∈μs, ε^{a_{i}} ≠1 ∀i=1,2,3}(ε^{−n} − 1)\frac{1}{(1−ε^{−a_{1}})(1−ε^{−a_{2}})(1−ε^{−a_{3}})}
+\frac{1}{s} ∑_{ε∈μα_{1}}(ε^{−n} − 1) (\frac{1−g_{1}}{(1−ε^{−a_{2}})(1−ε^{a_{2}})} − \frac{ε^{−a_{2}}}{(1−ε^{−a_{2}})^{2}(1−ε^{a_{2}})} deg_{C_{1}} x_{1}−
\frac{ε^{a_{2}}}{(1−ε^{−a_{2}})(1−ε^{a_{2}})^{2}} deg_{C_{1}} x_{2})
+\frac{1}{s} ∑_{ε∈μα_{2}}(ε^{−n} − 1) (\frac{1−g_{2}}{(1−ε^{−a_{3}})(1−ε^{a_{3}})} − \frac{ε^{−a_{3}}}{(1−ε^{−a_{3}})^{2}(1−ε^{a_{3}})} deg_{C_{2}} x_{1}−
\frac{ε^{a_{3}}}{(1−ε^{−a_{3}})(1−ε^{a_{3}})^{2}} deg_{C_{2}} x_{2})
+\frac{1}{s} ∑_{ε∈μα_{3}}(ε^{−n} − 1) (\frac{1−g_{3}}{(1−ε^{−a_{1}})(1−ε^{a_{1}})} − \frac{ε^{−a_{1}}}{(1−ε^{−a_{1}})^{2}(1−ε^{a_{1}})} deg_{C_{3}} x_{1}−
\frac{ε^{a_{1}}}{(1−ε^{−a_{1}})(1−ε^{a_{1}})^{2}} deg_{C_{3}} x_{2}) .

By Lemma 2.3.10 (1) we can put the last formula in a more compact form

χ(X, L_{n}) = χ(O_{X})

+\frac{1}{s} ∑_{ε∈μs, ε^{a_{i}} ≠1 ∀i=1,2,3}(ε^{−n} − 1)\frac{1}{(1−ε^{−a_{1}})(1−ε^{−a_{2}})(1−ε^{−a_{3}})}

+\frac{\overline{nk_{1}}(α_{1}−\overline{nk_{1}})}{4s} deg K_{Z}|_{C_{1}} + \frac{1}{12s} \overline{nk_{1}}(α_{1} − \overline{nk_{1}})(α_{1} − 2 · \overline{nk_{1}})(deg_{C_{1}} x_{1} − deg_{C_{1}} x_{2})

+\frac{\overline{nk_{2}}(α_{2}−\overline{nk_{2}})}{4s} deg K_{Z}|_{C_{2}} + \frac{1}{12s} \overline{nk_{2}}(α_{2} − \overline{nk_{2}})(α_{2} − 2 · \overline{nk_{2}})(deg_{C_{2}} x_{1} − deg_{C_{2}} x_{2})

+\frac{\overline{nk_{3}}(α_{3}−\overline{nk_{3}})}{4s} deg K_{Z}|_{C_{3}} + \frac{1}{12s} \overline{nk_{3}}(α_{3} − \overline{nk_{3}})(α_{3} − 2 · \overline{nk_{3}})(deg_{C_{3}} x_{1} − deg_{C_{3}} x_{2}),

where a_{j}k_{i} = 1 mod α_{i} and denotes the smallest residue mod α_{i} for pairs (j, i) = (2, 1), (3, 2), (1, 3) respectively.

We can think of χ(O_{X}) as RR-type expression for L_{n}, where L_{n} = 0 ∈ DivX ⊗ ℚ is the corresponding ℚ−divisor. The rest of the formula is a sum of contributions coming from

- dissident points of type \frac{1}{s}(a_{1}, a_{2}, a_{3}),

- dissident curve of type \frac{1}{α_{1}} (a_{2}, a_{3}),

- dissident curve of type \frac{1}{α_{2}} (a_{3}, a_{1}),

- dissident curve of type \frac{1}{α_{3}} (a_{1}, a_{2}),

which are singularities of L_{n} on X.

Now we have everything needed to conclude the proof of the main theorem. Proof of Theorem 1.4.2 By Corollary 2.2.5 we only need to add the contributions P from the singular points to the RR formula. These were computed above.

Let C be a \overline{mk}(\frac{1}{r}(1, −1)) curve for O_{X}(mD) with dissident points

\{P_{λ} | λ ∈ Λ\} of types \overline{mn_{λ}}(\frac{1}{s_{λ}} (a_{1λ}, a_{2λ}, a_{3λ})).

Note that for every λ there exists i such that hcf(a_{iλ}, s_{λ}) = r, since P_{λ} ∈ C. Moreover, mn_{λ}k_{iλ} = mk mod r, where k_{iλ} is defined by a_{jλ}k_{iλ} = 1 mod r for i ≠ j.

The contribution to RR from this locus is a sum of

\frac{\overline{mk}(r − \overline{mk})(r − 2 · \overline{mk})}{6r^{2}} ((r − 2)(1 − g) + \frac{r − 2}{2}K_{Y} ^{C}E_{1} ^{C}E_{2} + ^{C}E_{r−2}^{2} ^{C}E_{r−1})

by Corollary 2.2.5, and of

\begin{matrix}\frac{1}{12s_{λ}} \overline{mk}(r − \overline{mk})(r − 2 · \overline{mk})(deg_{C_{iλ}} x_{1} − deg_{C_{iλ}} x_{2}) \\ + \frac{1}{s_{λ}} ∑_{ε∈μs_{λ} , ε^{a_{iλ}} ≠1 ∀i=1,2,3}(ε^{−mn_{λ}} − 1)\frac{1}{(1−ε^{−a_{1λ}})(1−ε^{−a_{2λ}})(1−ε^{−a_{3λ}})}. \end{matrix}

for every point P_{λ}. The first two rows of the sum can be written together as

\frac{\overline{mk}(r − \overline{mk})(r − 2 · \overline{mk})}{12r^{2}τ_{C}}N_{C},

where τ_{C} = \frac{1}{r} lcm\{s_{λ}\} is the index of C and the integer N_{C} is an invariant of X in the neighbourhood of C.

□

3 Calabi–Yau threefolds

3.1 The definition of a Calabi–Yau and its Hilbert series

A Calabi–Yau threefold is a normal projective threefold X with canonical Gorenstein singularities, satisfying K_{X} ∼ 0 and H^{1}(X, O_{X}) = 0. Calabi– Yaus with only terminal singularities are minimal models with Kodaira dimension 0. There is a huge, possibly infinite number of deformation types. A lot of research on Calabi–Yaus was done because of their connection with string theory (check Appendix A.3).

Corollary 3.1.1 Let X be a Calabi–Yau threefold and let D be a ℚ–Cartier divisor. Assume that (X, D) is quasismooth and well formed with the following singularities

• points Q of type _{n}(\frac{1}{s}(a_{1}, a_{2}, a_{3})),

• curves C of generic type _{k}(\frac{1}{r}(1, −1)) with index τ_{C}.

Then for all positive integers m,

\begin{aligned}h^{0}(X, O_{X}(mD)) & = & \frac{1}{6}m^{3}D^{3} + m\frac{D·c_{2}(X)}{12} \\ & & + ∑_{Q} c_{Q}(mD) + ∑_{C} s_{C}(mD), \end{aligned}

where

c_{Q}(mD) = \frac{1}{s} ∑_{\begin{matrix}ε∈μs, \\ ε^{a_{i}} ≠1 ∀i=1,2,3 \end{matrix}} (ε^{−nm} − 1)\frac{1}{(1 − ε^{−a_{1}})(1 − ε^{−a_{2}})(1 − ε^{−a_{3}})}

and

s_{C}(mD) = −m\frac{\overline{mk}(r − \overline{mk})}{2r} deg D|_{C} + \frac{\overline{mk}(r − \overline{mk})(r − 2 · \overline{mk})}{12r^{2}τ_{C}}N_{C}.

Here denotes the smallest residue mod r and the integer N_{C} is an invariant of X in the neighbourhood of C.

Proof This is a direct corollary of Theorem 1.4.2. By Kodaira vanishing h^{p}(X, mD) = 0 for all p > 0 and by Serre duality χ(O_{X}) = 0. Moreover, K_{X} ∼ 0 by the definition of Calabi–Yau . □

Remark 3.1.2 Observe that for a \frac{1}{2}(1, 1) curve of singularities the term involving N_{C} is 0. The Riemann–Roch formula for this special case was already proved in [32].

Theorem 3.1.3 Let X be the same as in Corollary 3.1.1. The Hilbert series for X is

\begin{aligned}P_{X}(t) & = & 1 + \frac{1}{6}D^{3} · \frac{t^{3}+4t^{2}+t}{(1−t)^{4}} + \frac{c_{2}(X)·D}{12} · \frac{t}{(1−t)^{2}} \\ & & + ∑_{Q} P̃_{Q}(t) + ∑_{C} P̃_{C}(t), \end{aligned}

where every curve C contributes

\begin{aligned}P̃_{C}(t) & = & − deg D|_{C} (\frac{1}{1−t^{r}} ∑_{i=1}^{r−1} i\frac{\overline{ik}(r−\overline{ik})}{2r}t^{i} + \frac{rt^{r}}{(1−t^{r})^{2}} ∑_{i=1}^{r−1} \frac{\overline{ik}(r−\overline{ik})}{2r}t^{i}) \\ & & + \frac{N_{C}}{12r^{2}τ_{c}} · \frac{1}{1−t^{r}} ∑_{i=1}^{r−1} \overline{ik}(r − \overline{ik})(r − 2 · \overline{ik})t^{i} \end{aligned}

and every point Q contributes

P̃_{Q}(t) = \frac{1}{1 − t^{s}} ∑_{i=1}^{s−1} c_{Q}(iD)t^{i}.

Here

c_{Q}(iD) = \frac{1}{s} ∑_{\begin{matrix}ε∈μs, \\ ε^{a_{j}} ≠1 ∀j=1,2,3 \end{matrix}} \frac{ε^{−ni} − 1}{(1 − ε^{−a_{1}})(1 − ε^{−a_{2}})(1 − ε^{−a_{3}})}.

Proof The Hilbert series is defined by

P_{X}(t) = 1 + ∑_{m=1}^{∞} h^{0}(X, O_{X}(mD))t^{m}.

Use the following trivial expansions:

• \frac{t^{3}+4t^{2}+t}{(1−t)^{4}} = t + 2^{3}t^{2} + 3^{3}t^{3} + … + m^{3}t^{m} + …, • \frac{t}{(1−t)^{2}} = t + 2t^{2} + … + mt^{m} + …,

• \frac{1}{1−t^{r}} ∑_{i=1}^{r−1} ia_{i}t^{i} + \frac{rt^{r}}{(1−t^{r})^{2}} ∑_{i=1}^{r−1} a_{i}t^{i} = a_{1}t + 2a_{2}t^{2} + … + ma\overline{m}t^{m} + …,

• \frac{1}{1−t^{r}} ∑_{i=1}^{r−1} a_{i}t^{i} = a_{1}t + … + a_{r−1}t^{r−1} + a_{1}t^{r+1} + … + a\overline{m}t^{m} + …,

where a_{1}, …, a_{r−1} are arbitrary rational numbers, a_{0} = 0 and denotes the smallest residue mod r. The formula for P_{X}(t) follows directly from Corollary 3.1.1, as for m ∈ ℤ the factors

\overline{mk}(r − \overline{mk}) and \overline{mk}(r − \overline{mk})(r − 2 · \overline{mk})

are r periodic and c_{Q}(mD) are s periodic. □

3.2 CYs in weighted projective spaces

The formula for the Hilbert series in Theorem 3.1.3 was constructed with the aim to generate new examples of Calabi–Yau threefolds. In this section we will describe a computer search for Calabi–Yaus embedded in weighted projective spaces.

3.2.1 Resolution of the graded ring

Many cases of projective algebraic varieties can be studied in terms of graded rings.

Consider a projectively normal threefold X ⊂ ℙ(w_{0}, …, w_{n}). The graded ring

R = R(X, O_{X}(1))

over the polynomial ring A = ℂ[x_{0}, …, x_{n}] is a finite module over A. By the Hilbert syzygies theorem, there exists a finite free resolution

0 → L_{c} → ⋯ → L_{1} → A → R → 0,

where each L_{i} = ⨁ A(−b_{i,j}) is a free graded module. We assume that R is Gorenstein, thus the length of the resolution equals to the codimension of X in ℙ.

Then the Hilbert series of R is

P (t) = \frac{Q(t)}{∏_{k=0}^{n}(1 − t^{w_{k}})},

where Q(t) is the multiplied out Hilbert polynomial

Q(t) = ∑_{i,j}(−1)^{i}t^{b_{i,j}}.

The terms b_{1,j} are the degrees of relations of R, the terms b_{2,j} are the degrees of first syzygies and so on. Note that the polynomial Q(t) does not entirely determine the resolution of R.

Lemma 3.2.1 The degree of the c codimensional X in ℙ equals

\frac{∑_{i,j}(−1)^{i}\binom{b_{i,j}}{c}}{∏_{k=0}^{n} w_{k}}.

Proof The degree of a variety is defined in the following way

deg X = lim_{m→∞} \frac{H^{0}(X, O_{X}(m)) · d!}{m^{d}},

where d = n − c is the dimension of X. We can compute this limit from the Hilbert series

P_{X}(t) = \frac{Q(t)}{∏_{k=0}^{n}(1 − t^{w_{k}})} = \frac{(1 − t)^{n−d}R_{1}(t)}{(1 − t)^{n+1}R_{2}(t)},

where the polynomials R_{i}(t) only have positive terms. This implies

deg X = lim_{m→∞} \binom{d + m}{d} · \frac{R_{1}(1)}{R_{2}(1)} · \frac{d!}{m^{d}} = \frac{R_{1}(1)}{R_{2}(1)}.

Obviously R_{2}(1) = ∏_{k=0}^{n} w_{k}, and R_{1}(1) can be computed from Q(t) by differentiating it c times and then inserting t = 1. □

We can conclude more from the free resolution of R. The shape of R suggests the shape of the embedding.

In codimension 1, X can only be a hypersurface. If X has codimension 2, it is a complete intersection. In codimension 3, X is defined by the submaximal Pfaffians of some skew-symmetric matrix of odd size. This was proved by Buchsbaum and Eisenbud in [6]. Note that a 3 × 3 skew–symmetric matrix defines a complete intersection. For higher codimensions there are no similar theorems about the defining equations and relations of X. Thus, it is much harder to prove the existence of X. In Chapter 4 we construct some examples of Corti–Reid [8] varieties in high cosimensions. The technique that works well on concrete examples (check Section 4.2), is the theory of unprojections developed by Miles Reid. More on this can be found in [26] and [25].

3.2.2 The computer algorithm

Gavin Brown wrote a computer program in MAGMA which we now describe. The input data are:

• rational numbers D^{3}, D · c_{2}(X),

• integers a_{1}, a_{2}, a_{3}, s, n for each singular point

• integers r, k, τ_{C}, rτ_{C} · deg D|_{C} and N_{C} for each curve of singularities,

which is equivalent to giving the data:

• integers h^{0}(X, D) and h^{0}(X, 2D),

• Points: \{_{n}(\frac{1}{s}(a_{1}, a_{2}, a_{3})) \} ,

• Curves: \{ type = _{k}(\frac{1}{r}(1, −1)), deg D|_{C}, τ_{C} and N_{C} \} .

In the first step the computer generates the Hilbert series P (t) from Theorem 3.1.3 by using the input data. In the next step it finds a multiplied out Hilbert polynomial Q(t). This is equivalent to finding weights w_{0}, …, w_{n} such that

∏_{k=0}^{n}(1 − t^{w_{k}})P (t)

is a polynomial. As described in Subection 3.2.1, Q(t) corresponds to the resolution of a graded ring R. This gives a candidate for X embedded in ℙ(w_{0}, …, w_{n}). Of course we need to check the existence of X in every separate case. Since (X, O_{X}(1)) satisfies the Riemann–Roch formula it is likely to have the prescribed singularities.

Remarks 3.2.2

(I) Different types of singularities can give the same contributions to the Riemann-Roch formula and thus define the same Hilbert series.

(II) In practice it is difficult to ”guess” the suitable rational numbers D^{3}, D · c_{2}(X) such that all coefficients in the generated series are integers. Instead we choose two integers for h^{0}(X, D) and h^{0}(X, 2D) as part of the input data. The terms D^{3} and D · c_{2}(X) can then be evaluated from the RR formula.

(III) Note that rτ_{C} · deg D|_{C} is an integer, since rτ_{C}D is Cartier in the neighbourhood of C.

(IV) The contribution from a singular point of type \frac{1}{s}(a_{1}, a_{2}, a_{3}) involves summing over the complex numbers ε ∈ μ_{s}, ε^{a_{j}} ≠ 1 ∀j = 1, 2, 3. In order to avoid introducing complex numbers in the computer algorithm, we read off the contributions from a suitable model. We will briefly describe the method which was suggested by Miles Reid and is also part of Brown’s program in Magma.

Consider ℙ(a_{1}, a_{2}, a_{3}, s). As always assume that hcf(a_{i}, a_{j}, s) = 1 for different i, j = 1, 2, 3 and denote hcf(s, a_{i}) = α_{i}. Its Hilbert series equals

P_{ℙ}(t) = \frac{1}{(1 − t^{a_{1}})(1 − t^{a_{2}})(1 − t^{a_{3}})(1 − t^{s})}.

The coefficient in front of t^{m} in the Taylor extension of P_{ℙ}(t) is h^{0}(ℙ, O_{ℙ}(m)). By Theorem 1.4.2,

h^{0}(ℙ, O_{ℙ}(m)) = RR–type expression in m +

contributions from the singularities.

The singularities of (ℙ, O_{ℙ}(1)) are the point (0, 0, 0, 1) of type

s−1(\frac{1}{s}(a_{1}, a_{2}, a_{3}))

and for i = 1, 2, 3 and j, k ≠ i

a curve of type \frac{1}{α_{i}} (a_{j}, a_{k})  if α_{i} ≠ 1,
an isolated point \frac{1}{a_{i}} (a_{j}, a_{k}, s) if α_{i} = 1.

For

m = \frac{a_{1}a_{2}a_{3}}{α_{1}α_{2}α_{3}} , 2\frac{a_{1}a_{2}a_{3}}{α_{1}α_{2}α_{3}} , …, (s − 1)\frac{a_{1}a_{2}a_{3}}{α_{1}α_{2}α_{3}}

the contributions to RR will come only from the singularity (0, 0, 0, 1). Indeed, O_{ℙ}(\frac{a_{1}a_{2}a_{3}}{α_{1}α_{2}α_{3}} ) can be moved away from the other singular points of ℙ. Since \frac{a_{1}}{α_{1}} , \frac{a_{2}}{α_{2}} , \frac{a_{3}}{α_{3}} and s are coprime, we get all s − 1 periodic contributions, which is what we wanted to compute.

(IV) We can put no bounds on the invariants N_{C}, which are determined by the type of singularities along C and its embedding in X.

When an A_{r−1} curve C contains no dissident points (τ_{C} = 1), we have computed N_{C} in two different ways. First, for the resolution f: Y → X which is crepant over the generic point of C

N_{C} = 2 ((r − 2)(1 − g) + ^{C}E_{r−2}^{2} ^{C}E_{r−1})

by Corollary 2.2.5. Second, for the auxiliary cover π: Z → X

N_{C} = 2 (r(g − 1) − deg_{C_{Z}} x_{1})

by Theorem 2.3.17. Recall that g is the genus of C, ^{C}E_{i} are the exceptional surfaces of f over C and x_{1} = c_{1}(N_{1}) where N_{C/Z} = N_{1} ⊕ N_{2}.

We will prove an interesting relation between the above invariants. Write e for the intersection ^{C}E_{r−2}^{2} ^{C}E_{r−1} and ē for ^{C}E_{1} ^{C}E_{2}^{2}. By the formula on page 19

e + ē = 2(r − 2)(g − 1).

In Section 2.3 we defined ι = deg c_{1}(N_{1}). Write ῑ for deg_{C_{Z}} x_{2} = deg c_{1}(N_{2}). In Lemma 2.3.16 we showed that

ι + ῑ = deg c_{1}(N_{C_{Z}/Z}) = 2r(g − 1).

By comparing the formulas for N_{C} we get

\frac{1}{r}((r − 2)(1 − g) + e) = − ((1 − g) + \frac{ι}{r}) .

Substituting e by −2(r − 2)(1 − g) − ē, we see that

ē = ι + 2(1 − g).

Similarly, substitute ι by −2(1 − g)r − ῑ to get

e = ῑ+ 2(1 − g).

4 Construction of examples

4.1 Complete intersections

The simplest examples of Calabi–Yau threefolds are complete intersections

X_{d_{1},... ,d_{l}} ⊂ ℙ(b_{1}, …, b_{l+4}),

where d_{1} + … + d_{l} = b_{1} + … + b_{l+4}. In this section we use the notation of [11].

4.1.1 Classification of singularities

Proposition 4.1.1 Let X be a quasismooth and well formed Calabi–Yau defined as a complete intersection in a weighted projective space. Then Sing(X) consists of:

• points of type \frac{1}{r}(a, b, c), where a + b + c = 0 mod r.

• curves of transverse type \frac{1}{r′} (a′, b′), where a′ + b′ = 0 mod r′.

Proof Since X is normal, Sing X consists of a number of isolated points and a number of (possibly intersecting) curves. The singularities of X are induced by the singularities of ℙ.

Consider first a point P ∈ X of type \frac{1}{r}(a, b, c). If P is an isolated point or a dissident point, it can only appear in a complete intersection of the form

X_{d_{1},... ,d_{l},e_{1}r,... ,e_{k}r} ⊂ ℙ(a_{1}, …, a_{l+3}, b_{1}r, …, b_{k+1}r),

where d_{i} and a_{j} are not divisible by r. After a suitable permutation of the variables, we can assume

a_{l+1} = a mod r,
a_{l+2} = b mod r,  (E. viii)
a_{l+3} = c mod r.

Let f_{1}, …, f_{l}, g_{1}, …, g_{k} be the defining equations of X, and let x_{1}, …, x_{l+3}, y_{1}, …, y_{k+1} be the coordinates of ℙ. Since X is quasismooth, we can use the implicit function theorem to express x_{1}, …, x_{l} out of f_{1}, …, f_{l}. This means that after a suitable permutation of equations

f_{i} = x_{i} · ∏_{j} y_{j}^{α_{j}} + other monomials of deg d_{i},

for i = 1, …, l. The corresponding degrees are

d_{i} = a_{i} + r∗_{i},  (E. ix)

where ∗_{i} denote positive integers.

By the adjunction formula for Calabi–Yau threefolds we get

0 = d_{1} + … + d_{l} + e_{1}r + … + e_{k}r −  (E. x)
a_{1} − … − a_{l+3} − b_{1}r − … − b_{k+1}r.

Then (E. viii), (E. ix) and (E. x) imply

a + b + c = 0 mod r.

Next consider a curve C of type \frac{1}{r′} (a′, b′). Such a singularity can only appear in a complete intersection of the form

X_{d_{1},... ,d_{l},e_{1}r′,... ,e_{k}r′} ⊂ ℙ(a_{1}, …, a_{l+2}, b_{1}r′, …, b_{k+2}r′),

where d_{i} and a_{j} are not divisible by r′. We can assume

a_{l+1} = a′ mod r′,  (E. xi)
a_{l+2} = b′ mod r′.

Let f_{1}, …, f_{l}, g_{1}, …, g_{k} be the defining equations of X, and let x_{1}, …, x_{l+2}, y_{1}, …, y_{k+2} be the coordinates of ℙ. Since X is quasismooth, we can use the implicit function theorem on every point of C. This eliminates x_{1}, …, x_{l} out of f_{1}, …, f_{l}. The remaining variables x_{l+1}, x_{l+2} are local parameters to the normal direction of C. This means that after a suitable permutation of equations

f_{i} = x_{i} · ∏_{j} y_{j}^{α_{j}} + other monomials of deg d_{i},

for all i = 1, …, l. The corresponding degrees are

d_{i} = a_{i} + r′∗_{i},  (E. xii)

where ∗_{i} denote positive integers.

Use the adjunction formula for Calabi–Yaus together with (E. xi) and (E. xii) to get

a′ + b′ = 0 mod r′.
□

4.1.2 The degree of curves

For a threefold X in a weighted projective space the sheaf O_{X}(1) defines a ℚ–Cartier divisor D. By slight abuse of notation we call deg D|_{C} defined on page 8 the degree of C. In the case of complete intersections deg C can be computed by the following lemma.

Lemma 4.1.2 Let X be a quasismooth complete intersection in a weighted projective space ℙ containing an A_{r−1} curve C. Choose the defining equations of X that have degrees d_{1} … d_{l} divisible by r. Similarly, let a_{1} … a_{l+2} be the weights of ℙ divisible by r. Then

deg C = \frac{d_{1} … d_{l}r}{a_{1} … a_{l+2}} .

Proof The singularities along C are induced by the weights of ℙ that are divisible by r. Thus C can be expressed as a complete intersection X_{d_{1},... ,d_{l}} ⊂ ℙ(a_{1}, …, a_{l+2}), where d_{1}, …, d_{l} and a_{1}, …, a_{l+2} are divisible by r. Choose t big enough and containing the factor

∏_{i=1}^{l+2} a_{i}.

Then tD = O_{X}(t) is Cartier and defines a surface H. The intersection C ∩H equals to X_{t,d_{1},... ,d_{l}} ⊂ ℙ(a_{1}, …, a_{l+2}) and is a finite number of points. It is isomorphic to a quasismooth

X_{\frac{t}{r} , \frac{d_{1}}{r} ,... , \frac{d_{l}}{r}} ⊂ ℙ(\frac{a_{1}}{r} , …, \frac{a_{l+2}}{r}).

By quasismoothness all intersection points lie outside the fundamental simplex of ℙ and by the generalised Bezout theorem there is \frac{td_{1}...d_{l}r}{a_{1}...a_{l+2}} of them. Finally the degree equals to

deg C = \frac{d_{1} … d_{l}r}{a_{1} … a_{l+2}} .

□

4.1.3 Examples

Example 4.1.3 Using Fletcher’s notation [11], a general hypersurface X_{13} ⊂ ℙ(1, 1, 3, 4, 4) with coordinates x_{1}, x_{2}, y, z_{1}, z_{2} is given by an equation

f = x_{1}^{13} + x_{2}^{13} + y^{3}z_{1} + yx_{1}^{10} + z_{1}^{3}x_{1} + z_{2}^{3}x_{2} + ...

It is easy to check that X_{13} is quasismooth. Thus the adjunction formula holds and X_{13} is Calabi–Yau . It contains a singular point P = (0, 0, 1, 0, 0). Since \frac{∂f}{∂z_{1}} (P) ≠ 0, we can use the implicit function theorem to express z_{1} out of f. This means that x_{1}, x_{2}, z_{2} are the local parameters for X_{13} around P. The group μ_{3} ∋ ε acts by

x_{1}, x_{2}, z_{2} ∼ εx_{1}, εx_{2}, ε^{4}z_{2}.

Thus we proved that (X_{13}, O_{X}(1)) contains P of type _{2}(\frac{1}{3}(1, 1, 1)).

The threefold X_{13} also contains a curve C defined by {x_{1} = x_{2} = y = 0}, which is a copy of ℙ^{1}. Choose a point on C, e.g. (0, 0, 0, 0, 1). Using the implicit function theorem again, we can choose x_{1}, y to be the local parameters on a hyperplane that contains the chosen point. The elements of ε ∈ μ_{4} act via x_{1}, y ∼ εx_{1}, ε^{3}y. Thus every point along C is of transverse type \frac{1}{4}(1, 3).

Define D by O_{X}(1). Then

D^{3} = \frac{13}{48}, deg C = \frac{1}{4} and 1 − g = 1.

We can also compute E_{2}^{2}E_{3}, where E_{i} are the exceptional surfaces of the resolution along C : In the toric variety ℙ(1, 1, 3, 4, 4), the blow up along the A_{3} curve corresponds

Figure III: Resolution of an A_{3} curve

to Picture III. We add rays E_{1}, E_{2}, E_{3} to a three dimensional facet. On the hypersurface f^{∗}(4D), the restrictions of E_{2} and E_{3} intersect once, thus f^{∗}(4D)E_{2}E_{3} = 1. This yields

\begin{aligned}3 & = & 4f^{∗}(3D)E_{2}E_{3} = 4f^{∗}(D_{3})E_{2}E_{3} & = \\ & & (4D′_{3} + 3E_{1} + 2E_{2} + E_{3})E_{2}E_{3} & = & 2E_{2}^{2}E_{3} + E_{2}E_{3}^{2} \end{aligned}

On the other hand, as in the proof on page 19

E_{2}^{2}E_{3} + E_{2}E_{3}^{2} = 2(g − 1) = −2.

Finally we get E_{2}^{2}E_{3} = 5. Thus we computed N_{C} = 14.

The only missing input data is c_{2}(X)·D. We can compute it directly from the RR formula for m = 12, since then there are no contributions. We get \frac{1}{12}c_{2}(X) · D = \frac{3}{2}.

By Corollary 3.1.1 we get

h^{0}(X, mD) = \frac{13}{288}m^{3}+\frac{3}{2}m+c_{P} (mD)−\frac{1}{32}m(4−\overline{m})\overline{m}+\frac{7}{6 · 16}\overline{m}(4−\overline{m})(4−2\overline{m}),

where

c_{P} (mD) = \{ \begin{matrix}\frac{1}{9} & if m = 1mod3, \\ −\frac{1}{9} & if m = 2mod3, \\ 0 & if m = 0mod3. \end{matrix}

It is easy to check that this yields the Hilbert series for X_{13}

P_{X}(t) = \frac{1 − t^{13}}{(1 − t)^{2}(1 − t^{3})(1 − t^{4})^{2}} .

Example 4.1.4 Consider the complete intersection X_{10,15} ⊂ ℙ(2, 3, 5, 5, 5, 5) with variables x, y, z_{1}, z_{2}, z_{3}, z_{4}. It is defined by two equations f, g of degrees 10 and 15 respectively. It is easy to verify that this is a quasismooth Calabi–Yau. We can avoid (1, 0, 0, 0, 0, 0) with f and (0, 1, 0, 0, 0, 0) with g. Therefore, the singular locus of X_{10,15} contains only a curve C given by X_{10,15} ∩ {x = y = 0}.

Choose an arbitrary point on C. The variables x, y are local parameters of a hypersurface around the chosen point. The group ε ∈ μ_{5} act on them by x, y ∼ ε^{2}x, ε^{3}y. This means that (X, O_{X}(1)) contains a singular curve of transverse type

4(\frac{1}{5}(2, 3)) = _{2}(\frac{1}{5}(1, −1)).

The degree of C is \frac{6}{5} and the other invariants of X_{10,15} are

D^{3} = \frac{1}{25} and \frac{1}{12}c_{2}(X) · D = \frac{5}{6}.

For N_{C} = −6 we get

h^{0}(X, mD) = \frac{1}{150}m^{3} + \frac{5}{6}m − \frac{3}{25}m · \overline{2m}(5 − \overline{2m}) − \frac{1}{50}\overline{2m}(5 − \overline{2m})(5 − 2 · \overline{2m}),

where is the smallest residue mod 5. This again yields the Hilbert series

P_{X}(t) = \frac{(1 − t^{10})(1 − t^{15})}{(1 − t^{2})(1 − t^{3})(1 − t^{5})^{4}} .

We continue by two examples generated by the computer search.

Example 4.1.5 For the input data we choose

• h^{0}(X, D) = 2 and h^{0}(X, 2D) = 4 • Points: _{4}(\frac{1}{5}(1, 1, 3))

• Curves: type = \frac{1}{3}(1, 2), deg = \frac{1}{3}, index = 1, N_{C} = 4. The output we get is

• Codimension: 2

• Weights: 1,1,2,3,3,5

• Multiplied out Hilbert polynomial: t^{15} − t^{8} − t^{7} + 1 This data defines a complete intersection

X_{7,8} ⊂ ℙ(1, 1, 2, 3, 3, 5).

The threefold X_{7,8} is a quasismooth Calabi–Yau by Theorem 8.7 in [11]. It is easy to check that the only singularities are (0,0,0,0,0,1) of type _{4}(\frac{1}{5}(1, 1, 3)) and a line of _{1}(\frac{1}{3}(1, 2)) transverse type. Example 4.1.6 For the input data we choose

• h^{0}(X, D) = 1 and h^{0}(X, 2D) = 1

• Points:

1. _{5}(\frac{1}{6}(1, 2, 3)),

2. _{13}(\frac{1}{14}(1, 6, 7)),

3. _{20}(\frac{1}{21}(1, 6, 14)),

• Curves:

1. type = \frac{1}{2}(1, 1), deg = \frac{1}{42}, index = 21,

2. type = \frac{1}{3}(1, 2), deg = \frac{1}{42}, index = 14, N_{C} = 55,

3. type = \frac{1}{7}(1, 6), deg = \frac{1}{6}, index = 6, N_{C} = 35, The output we get is

• Codimension: 1

• Weights: 1,6,7,14,21

• Multiplied out Hilbert polynomial: t^{49} − 1

This data defines a Calabi–Yau hypersurface X_{49} ⊂ ℙ(1, 6, 7, 14, 21) with variables x_{1}, x_{6}, x_{7}, x_{14}, x_{21}. It is easy to check that (X, O_{X}(1)) is quasismooth and well formed and has the following singularities

• dissident curve C_{2} = \{x_{1} = x_{7} = 0\} of type \frac{1}{2}(1, 1),

• dissident curve C_{3} = \{x_{1} = x_{7} = x_{14} = 0\} of type \frac{1}{3}(1, 2),

• dissident curve C_{7} = \{x_{1} = x_{6} = 0\} of type \frac{1}{7}(1, 6),

• point P_{2,3} of type _{5}(\frac{1}{6}(1, 2, 3)) on the intersection of C_{2} and C_{3},

• point P_{2,7} of type _{13}(\frac{1}{14}(1, 6, 7)) on the intersection of C_{2} and C_{7},

• point P_{3,7} of type _{20}(\frac{1}{21}(1, 6, 14)) on the intersection of C_{3} and C_{7}.

4.2 Pfaffians in codimension 3 and 4

4.2.1 The Type I unprojection

As discussed in Subsection 3.2.1, it is difficult to analyse the varieties in high codimension. Therefore we project them to lower codimension by the so called unprojection method. The same method can be used to construct new varieties. We will now briefly describe the Type I unprojection.

Let Y be a projective Gorenstein variety of codimension c in ℙ(a_{0}, …, a_{n}), containing a projective Gorenstein variety Π of codimension c + 1. Let the homogenous polynomials h_{1}, …, h_{l} generate the ideal of Π and f_{1}, …, f_{k} the ideal of Y. Then there exists a rational function t on Y with pole along Π. In other words, there exist homogeneous polynomials m_{1}, …, m_{l} such that

t = \frac{m_{1}}{h_{1}} = ⋯ = \frac{m_{l}}{h_{l}} .

Then, X defined by

f_{1} = ⋯ = f_{k} = th_{1} − m_{1} = ⋯ = th_{l} − m_{l} = 0

is a projective Gorenstein variety of codimension c + 1 in ℙ(a_{0}, …, a_{n}, a_{n+1}). Here a_{n+1} = wt t. The construction φ: Y ⟶ X is called the Type I unprojection.

Denote the point (\underbrace{0, …, 0}_{n+1}, 1) ∈ X by P and the locus

X ∩ {h_{1} = … = h_{l} = 0}

by Γ. The inverse of φ is the projection from P. Thus the restriction

φ: Y \ Π ⟶ X \ Γ

is an isomorphism.

4.2.2 Examples

In this subsection we use the standard notion of Pfaffians as in [1], [25] and [28].

Example 4.2.1 For the input data we choose

• h^{0}(X, D) = 2 and h^{0}(X, 2D) = 4

• Points: _{4}(\frac{1}{5}(1, 1, 3))

• Curves: type = \frac{1}{3}(1, 2), deg = \frac{2}{3}, τ_{C} = 1, N_{C} = 8.

The output we get is

• Codimension: 3

• Weights: 1,1,2,3,3,3,5

• Multiplied out Hilbert polynomial: −t^{18} + t^{12} + 2t^{11} + 2t^{10} − 2t^{8} − 2t^{7} − t^{6} + 1

This output data suggests that X is a threefold in ℙ(1, 1, 2, 3, 3, 3, 5) defined by submaximal Pfaffians of a 5×5 skew-symmetric matrix M. There are five relations of degrees 6,7,7,8,8 and five syzygies of degrees 10,10,11,11,12. Denote the variables of ℙ(1, 1, 2, 3, 3, 3, 5) by x_{1}, x_{2}, y, z_{1}, z_{2}, z_{3}, t. We fill in the entries of M as follows:

M = ( \begin{matrix}t & a_{4} & b_{3} & c_{3} \\ & a_{5} & b_{4} & c_{4} \\ & & z_{2} & −z_{1} \\ & & & y \end{matrix} ) ,

where the polynomials a_{i}, b_{i}, c_{i} have degrees

( \begin{matrix}5 & 4 & 3 & 3 \\ & 5 & 4 & 4 \\ & & 3 & 3 \\ & & & 2 \end{matrix} ) .

The corresponding Pfaffians are

Pf_{1} = a_{5}y + b_{4}z_{1} + c_{4}z_{2},
Pf_{2} = a_{4}y + b_{3}z_{1} + c_{3}z_{2},
Pf_{3} = ty − b_{3}c_{4} + c_{3}b_{4},
Pf_{4} = −tz_{1} − a_{4}c_{4} + c_{3}a_{5},
Pf_{5} = tz_{2} − a_{4}b_{4} + b_{3}a_{5}.

When X is quasismooth and well formed the adjunction formula holds and has the form

ω_{X} ≅ O_{X}((10+10+11+11+12)−(6+7+7+8+8)−(1+1+2+3+3+3+5)) = O_{X},

where the sum runs over the degrees of syzygies, relations and weights respectively. In this case X is a Calabi–Yau .

The last thing we need to check is whether X is quasismooth. It is enough to examine whether Cone(X) is smooth on each singular stratum of ℙ(1, 1, 2, 3, 3, 3, 5). point (0,0,1,0,0,0,0): Since some power of y appears in Pf_{2}, Pf_{4} and Pf_{5}, this point is not contained in X.

point (0,0,0,0,0,0,1): The Pfaffians contain no monomial t^{n} for some n, thus this point lies in X. The rank of M at this point is 2 and by the implicit function theorem we can express y, z_{1} and z_{2} out of the equations. Indeed, on the affine piece t = 1 there are locally three linearly independent Pfaffians

Pf_{3} = y − b_{3}c_{4} + c_{3}b_{4},
Pf_{4} = −z_{1} − a_{4}c_{4} + c_{3}a_{5},
Pf_{5} = z_{2} − a_{4}b_{4} + b_{3}a_{5}.

This means that x_{1}, x_{2} and z_{3} are the local parameters of X around the point. The group ε ∈ μ_{5} acts by

x_{1}, x_{2}, z_{3} ∼ εx_{1}, εx_{2}, ε^{3}z_{3},

which implies that the point is a quotient singularity of type \frac{1}{5}(1, 1, 3).

stratum (0, 0, 0, z_{1}, z_{2}, z_{3}, 0): is defined by {x_{1} = x_{2} = y = t = 0} and M restricted to this stratum equals

( \begin{matrix}0 & 0 & b_{3} & c_{3} \\ & 0 & 0 & 0 \\ & & z_{2} & −z_{1} \\ & & & 0 \end{matrix} ) .

A point of the form (0, 0, 0, z_{1}, z_{2}, z_{3}, 0) will lie in X iff rk M|_{{x_{1}=x_{2}=y=t=0}} ≤ 2. But,

rk M|_{\{x_{1}=x_{2}=y=t=0\}} ≤ 2 ⟺ rk ( \begin{matrix}b_{3} & c_{3} \\ z_{2} & −z_{1} \end{matrix} ) ≤ 1.

This condition gives exactly the Pfaffian

Pf_{2} = b_{3}z_{1} + c_{3}z_{2},

which defines a curve C = X ∩ {x_{1} = x_{2} = y = t = 0}. Choose for example b_{3} = z_{3} and c_{3} = z_{2}. At every point on the curve rk M = 2, thus Cone(X) is smooth along C. This finishes the proof that X is quasismooth.

Next we determine the type of singularities along C. For every point on C one of z_{i} is nonzero. First, let z_{1} ≠ 0. Use the implicit function theorem on

Pf_{4} = −tz_{1} − a_{4}c_{4} + c_{3}a_{5}

to eliminate t. Assume that b_{4} contains the monomial z_{1}x_{1}. Then we can express x_{1} out of

Pf_{1} = a_{5}y + b_{4}z_{1} + c_{4}z_{2}.

This means that x_{2} and y are the local transverse parameters of X around the points on C for which z_{1} ≠ 0. The group ε ∈ μ_{3} acts by

x_{2}, y ∼ εx_{2}, ε^{2}y.

The same way we will choose the local parameters for z_{2} ≠ 0. Assume that c_{4} contains x_{1}z_{2}, and then eliminate x_{1} out of Pf_{1} and t out of

Pf_{5} = tz_{2} − a_{4}b_{4} + z_{3}a_{5}.

In the case z_{3} ≠ 0 we also assume that a_{5} contains yz_{3} and c_{4} contains x_{1}z_{3}. Then we express y out of Pf_{5} and x_{1} out of

Pf_{3} = ty − z_{3}c_{4} + c_{3}b_{4}.

Thus x_{2} and t are the local transverse parameters in the neighbourhood of z_{3} ≠ 0 with the group ε ∈ μ_{3} acting as before by

x_{2}, t ∼ εx_{2}, ε^{2}t.

This implies that C contains quotient singularities of transverse type \frac{1}{3}(1, 2).

It is obvious that X is well formed. Indeed, ℙ(1, 1, 2, 3, 3, 3, 5) is well formed and X contains no codimension 4 singular stratum of ℙ.

Example 4.2.2 For the input data we choose

• h^{0}(X, D) = 2 and h^{0}(X, 2D) = 4

• Points: _{4}(\frac{1}{5}(1, 1, 3))

• Curves: type = \frac{1}{3}(1, 2), deg = 1, τ_{C} = 1, N_{C} = 12.

The output is

• Codimension: 4

• Weights: 1,1,2,3,3,3,3,5

• Multiplied out Hilbert polynomial: t^{21} −3t^{15} −3t^{14} −3t^{13} +2t^{12} +6t^{11} + 6t^{10} + 2t^{9} − 3t^{8} − 3t^{7} − 3t^{6} + 1

This data suggests that X is a threefold in ℙ(1, 1, 2, 3, 3, 3, 3, 5) and there are nine relations of degrees 6, 6, 6, 7, 7, 7, 8, 8, 8 respectively. The resolution of the graded ring R(X, O_{X}(1)) is

0 → A(−21) → ⨁_{3} A(−15) ⨁_{3} A(−14) ⨁_{3} A(−13) →

⨁_{2} A(−12) ⨁_{6} A(−11) ⨁_{6} A(−10) ⨁_{2} A(−9) →
⨁_{3} A(−8) ⨁_{3} A(−7) ⨁_{3} A(−6) → A → R → 0,

where ⨁_{i} A denotes the direct sum of i copies of the same module.

We will prove the existence of X by the Type I unprojection. Consider

Π = {y = z_{1} = z_{2} = z_{3} = 0} ⊂ Y ⊂ ℙ(1, 1, 2, 3, 3, 3, 3),

where x_{1}, x_{2}, y, z_{1}, z_{2}, z_{3}, z_{4} are the coordinates in ℙ(1, 1, 2, 3, 3, 3, 3) and Y is defined by a set of Pfaffians

Pf_{1} = z_{2}q_{3} − z_{3}q_{2} + z_{4}q_{1} of degree 7,
Pf_{2} = z_{1}q_{3} − z_{2}q_{2} + z_{3}q_{1} of degree 7,
Pf_{3} = yq_{3} − z_{2}z_{4} + z_{3}^{2}  of degree 6,
Pf_{4} = yq_{2} − z_{1}z_{4} + z_{2}z_{3} of degree 6,
Pf_{5} = yq_{1} − z_{1}z_{3} + z_{2}^{2}  of degree 6.

These Pfaffians come from a 5 × 5 skew–symmetric matrix

M = ( \begin{matrix}y & z_{1} & z_{2} & z_{3} \\ & z_{2} & z_{3} & z_{4} \\ & & q_{1} & q_{2} \\ & & & q_{3} \end{matrix} ) ,

where all polynomials q_{i} have degree 4. The existence of the codimension 3 Calabi–Yau Y can be checked the same way as in Example 4.2.1.

By the unprojection there exists a rational function t on Y with pole along Π. In other words, there exist homogeneous polynomials m_{1}, …, m_{4} such that

t = \frac{m_{1}}{z_{1}} = \frac{m_{2}}{z_{2}} = \frac{m_{3}}{z_{3}} = \frac{m_{4}}{y} .

By [25] t has degree wt x_{1} + wt x_{2} + wt z_{4} = 5. We define X by

Pf_{1}, …, Pf_{5}

of degrees 7, 7, 6, 6, 6 respectively, and by

ty − m_{4}, tz_{1} − m_{1}, tz_{2} − m_{2}, tz_{3} − m_{3}

of degrees 7, 8, 8, 8.

In the next step we check that X has the expected quotient singularities. point (0,0,1,0,0,0,0,0): Since some power of y appears in Pf_{3}, Pf_{4} and Pf_{5}, this point is not contained in X.

point (0,0,0,0,0,0,0,1): The equations contain no monomial of t^{n} for some n, thus this point lies in X. We will express y, z_{1}, z_{2} and z_{3} with the other parameters using the implicit function theorem. On the affine piece t = 1 we have linearly independent equations

y = m_{4}, z_{1} = m_{1}, z_{2} = m_{2}, z_{3} = m_{3}.

This means that x_{1}, x_{2} and z_{4} are the local parameters of X around the point. The group ε ∈ μ_{5} acts by

x_{1}, x_{2}, z_{4} ∼ εx_{1}, εx_{2}, ε^{3}z_{4},

which implies that the point is a quotient singularity of type \frac{1}{5}(1, 1, 3).

stratum (0, 0, 0, z_{1}, z_{2}, z_{3}, z_{4}, 0): is defined by {x_{1} = x_{2} = y = t = 0}. Its intersection with X is given by

Pf_{3} = −z_{2}z_{4} + z_{3}^{2},
Pf_{4} = −z_{1}z_{4} + z_{2}z_{3},
Pf_{5} = −z_{1}z_{3} + z_{2}^{2},

which define a twisted cubic C. Choose an arbitrary point on C, e.g. one with z_{1} ≠ 0. Assume also that q_{3} contains the monomial z_{1}x_{1}. On the affine piece z_{1} = 1 we can eliminate t from the equation t = m_{1}, and x_{1} from Pf_{2} . Thus we get the local transverse parameters x_{2}, y for the neighbourhood of the chosen point. The group ε ∈ μ_{3} acts by

x_{2}, y ∼ εx_{2}, ε^{2}y,

which implies that C contains quotient singularities of transverse type \frac{1}{3}(1, 2).

The fact that X is quasismooth can be checked by the computer that generates the Jacobian matrix J(X) and then checks if Sing(X) = ∅. The singular locus of X is defined by the suitable minors of J(X) and the ideal I(X) of X.

It is obvious that X is well formed.

Example 4.2.3 For the input data we choose • h^{0}(X, D) = 3 and h^{0}(X, 2D) = 6,

• Points: _{2}(\frac{1}{3}(1, 1, 1)),

• Curves: type = \frac{1}{4}(1, 3), deg = \frac{1}{4}, τ_{C} = 1, N_{C} = 14. The output we get is

• Codimension: 3,

• Weights: 1,1,1,3,3,4,4,

• Multiplied out Hilbert polynomial: −t^{17} + t^{12} + t^{11} + 2t^{10} + t^{8} − t^{9} − 2t^{7} − t^{6} − t^{5} + 1.

The output data suggests that X is a threefold in ℙ(1, 1, 1, 3, 3, 4, 4). Let x_{1}, x_{2}, x_{3}, y_{1}, y_{2}, z_{1}, z_{2} be the variables of ℙ. Then X is defined by the submaximal Pfaffians of

M = ( \begin{matrix}y_{2} & a_{2} & x_{1} & x_{2} \\ & a_{6} & b_{5} & c_{5} \\ & & z_{2} & −z_{1} \\ & & & y_{1} \end{matrix} ) with degrees ( \begin{matrix}3 & 2 & 1 & 1 \\ & 6 & 5 & 5 \\ & & 4 & 4 \\ & & & 3 \end{matrix} ) ,

where a_{2}, a_{6}, b_{5} and c_{5} are general polynomials of degrees 2, 6, 5, 5 respectively. The corresponding Pfaffians are

Pf_{1} = a_{6}y_{1} + b_{5}z_{1} + c_{5}z_{2},

Pf_{2} = a_{2}y_{1} + x_{1}z_{1} + x_{2}z_{2},
Pf_{3} = y_{2}y_{1} − c_{5}x_{1} + b_{5}x_{2},
Pf_{4} = −y_{2}z_{1} − a_{2}c_{5} + a_{6}x_{2},
Pf_{5} = y_{2}z_{2} − a_{2}b_{5} + a_{6}x_{1}.

It is easy to check that X is a quasismooth well formed Calabi–Yau . Its singularities consist of the projective line {x_{1} = x_{2} = x_{3} = y_{1} = y_{2} = 0} of transverse type A_{3} and the isolated point {x_{1} = x_{2} = x_{3} = y_{1} = z_{1} = z_{2} = 0} of type \frac{1}{3}(1, 1, 1). Example 4.2.4 For the input data we choose • h^{0}(X, D) = 3 and h^{0}(X, 2D) = 6

• Points:

1. _{2}(\frac{1}{3}(1, 1, 1)),

2. _{8}(\frac{1}{9}(1, 3, 5)),

• Curves: type = \frac{1}{3}(1, 2), deg = \frac{1}{9}, τ_{C} = 3, N_{C} = 22. The output we get is

• Codimension: 3

• Weights: 1,1,1,3,3,5,9

• Multiplied out Hilbert polynomial: −t^{23} + t^{17} + t^{15} + 2t^{13} + t^{11} − t^{12} − 2t^{10} − t^{8} − t^{6} + 1

This data suggests that X is a threefold in ℙ(1, 1, 1, 3, 3, 5, 9) with the variables x_{1}, x_{2}, x_{3}, y_{1}, y_{2}, z, v. It is defined by the submaximal Pfaffians of a 5 × 5 skew-symmetric matrix

M = ( \begin{matrix}y_{2} & a_{7} & a_{5} & x_{1} \\ & b_{7} & b_{5} & x_{2} \\ & & v & −z \\ & & & y_{1} \end{matrix} ) with degrees ( \begin{matrix}3 & 7 & 5 & 1 \\ & 7 & 5 & 1 \\ & & 9 & 5 \\ & & & 3 \end{matrix} ) ,

were a_{i}, b_{i} are general polynomials. The corresponding Pfaffians are

Pf_{1} = b_{7}y_{1} + b_{5}z + x_{2}v,
Pf_{2} = a_{7}y_{1} + a_{5}z + x_{1}v,
Pf_{3} = y_{2}y_{1} − a_{5}x_{2} + b_{5}x_{1},
Pf_{4} = −y_{2}z − a_{7}x_{2} + b_{7}x_{1},
Pf_{5} = y_{2}v − a_{7}b_{5} + a_{5}b_{7}.

The same way as in Example 4.2.1 we check that X is quasismooth and well formed. The only singular stratum is

{x_{1} = x_{2} = x_{3} = z = 0} ∩ X.

It is given by the equations y_{1}y_{2} = 0, vy_{2} = 0 and defines the isolated point (0, 0, 0, 0, 1, 0, 0) and the line C = {y_{2} = 0}. Note that C is an example of a dissident curve. It is easy to check that the type of singularities is the same as in the input data.

4.3 The weighted Grassmannian wGr(2, 5)

4.3.1 The definition of wGr(2, 5)

Let Pf_{1}, …, Pf_{5} be the 4 × 4 Pfaffians of the generic skew matrix

( \begin{matrix}0 & x_{12} & x_{13} & x_{14} & x_{15} \\ −x_{12} & 0 & x_{23} & x_{24} & x_{25} \\ −x_{13} & −x_{23} & 0 & x_{34} & x_{35} \\ −x_{14} & −x_{24} & −x_{34} & 0 & x_{45} \\ −x_{15} & −x_{25} & −x_{35} & −x_{45} & 0 \end{matrix} ) .

The affine variety aGr(2, 5) is by definition

aGr(2, 5) = Spec ℂ[x_{i,j}]/I,

where ideal I is generated by Pf_{1}, …, Pf_{5}.

Using the notation of [8], let w = (w_{1}, …, w_{5}) denote a vector of halfintegers such that w_{i} + w_{j} ∈ ℤ is positive for all i, j. Impose a ℂ^{∗} action on aGr(2, 5) by

x_{ij} ⟼ λ^{w_{i}+w_{j}}x_{ij}

and define the weighted Grassmannian by

wGr(2, 5) = (aGr(2, 5) − {0}) /ℂ^{∗}.

Equivalently, we can define

wGr(2, 5) = Proj ℂ[x_{ij}]/I,

where the ring ℂ[x_{ij}] is graded by wt x_{ij} = w_{i} + w_{j}.

The Hilbert series of a weighted Grassmannian is

P (t) = \frac{1 − ∑_{i=1}^{5} t^{s−w_{i}} + ∑_{i=1}^{5} t^{s+w_{i}} − t^{2s}}{∏_{i,j}(1 − t^{w_{i}+w_{j}})} ,

where s = w_{1} +⋯+w_{5}. If wGr(2, 5) is well formed, its canonical class equals

K_{wGr} = O(−2s).

Moreover, by Lemma 3.2.1

deg wGr = O(1)^{6} = \frac{∑ \binom{s−w_{i}}{3} − ∑ \binom{s+w_{i}}{3} + \binom{2s}{3}}{∏(w_{i} + w_{j})}.

4.3.2 Quasilinear sections of wGr(2, 5)

Many codimension 3 Calabi-Yau threefolds can be represented as quasilinear sections of a weighted Grassmannian wGr(2, 5) or a cone over wGr(2, 5).

Definition 4.3.1 Write CwGr(2, 5) for the projective cone over wGr(2, 5). This means that we add one extra variable to the homogeneous coordinate ring, which is not involved in any relation.

Let w = (w_{1}, …, w_{5}) be the defining vector of wGr(2, 5). The section

X = wGr(2, 5) ∩ (f_{1}) ∩ … ∩ (f_{l})

defined by general forms is a quasilinear section of wGr(2, 5) if for every k = 1, …, l exist i, j such that deg f_{k} = w_{i} + w_{j}. The same way we define the quasilinear section of CwGr(2, 5).

Lemma 4.3.2 The codimension 3 threefolds from Examples 4.2.1, 4.2.3 and 4.2.4 can be represented as quasilinear sections of a cone over wGr(2, 5).

Proof We claim that X from Example 4.2.3 can be represented as a quasilinear section in

CwGr(2, 5) ⊂ ℙ(1^{3}, 2, 3^{2}, 4^{2}, 5^{2}, 6).

Choose

w = (−\frac{1}{2}, \frac{7}{2}, \frac{5}{2}, \frac{3}{2}, \frac{3}{2}) ,

which defines as skew-symmetric matrix with weights

w_{i} + w_{j} = ( \begin{matrix}3 & 2 & 1 & 1 \\ & 6 & 5 & 5 \\ & & 4 & 4 \\ & & & 3 \end{matrix} ) .

These are exactly the degrees of the elements in M, the defining matrix for X. Note that h^{0}(X, D) = 3 but M contains only two entries of degree 1. Thus we must add an extra variable of degree 1 to the homogeneous coordinate ring of wGr(2, 5). This variable is not involved in any relation and thus defines the projective cone CwGr(2, 5).

We will prove that X is a general quasilinear section

(2) ∩ (5)^{2} ∩ (6) ∩ CwGr(2, 5) ⊂ ℙ(1^{3}, 2, 3^{2}, 4^{2}, 5^{2}, 6).

At the beginning of this section we wrote

K_{CwGr} = O(−2(−1/2 + 7/2 + 5/2 + 2 × 3/2) − 1) = O(−18).

By the adjunction formula the canonical class of the quasilinear section equals K_{X} = O(−18 + 2 + 2 × 5 + 6), which is trivial.

The Hilbert series of CwGr(2, 5)

P_{CwGr}(t) = \frac{−t^{17} + t^{12} + t^{11} + 2t^{10} + t^{8} − t^{9} − 2t^{7} − t^{6} − t^{5} + 1}{(1 − t)^{3}(1 − t^{2})(1 − t^{3})^{2}(1 − t^{4})^{2}(1 − t^{5})^{2}(1 − t^{6})}

yields the Hilbert series of X, which is the same as the computer output.

Also X from Example 4.2.4 can be represented as a quasilinear section in the 7 dimensional

CwGr(2, 5) ⊂ ℙ(1^{3}, 3^{2}, 5^{3}, 7^{2}, 9).

Let

w = (\frac{3}{2}, \frac{3}{2}, \frac{11}{2}, \frac{7}{2}, −\frac{1}{2}) .

Then w_{i} + w_{j} correspond to the degrees of the elements in M, which are

( \begin{matrix}3 & 7 & 5 & 1 \\ & 7 & 5 & 1 \\ & & 9 & 5 \\ & & & 3 \end{matrix} ) .

Since h^{0}(X, D) = 3 and M contains only two entries of degree 1, we must add an extra variable of degree 1 to the homogeneous coordinate ring of wGr(2, 5). This defines the projective cone CwGr(2, 5).

We claim that X is a general quasilinear section

(5)^{2} ∩ (7)^{2} ∩ CwGr(2, 5) ⊂ ℙ(1^{3}, 3^{2}, 5^{3}, 7^{2}, 9).

Since K_{CwGr} = O(−2(2 × 3/2 + 11/2 + 7/2 − 1/2) − 1) = O(−24), the adjunction formula implies that O_{X}(−24 + 2 × 5 + 2 × 7) is trivial.

The Hilbert series of CwGr(2, 5) is

P_{CwGr}(t) = \frac{−t^{23} + t^{17} + t^{15} + 2t^{13} + t^{11} − t^{12} − 2t^{10} − t^{8} − t^{6} + 1}{(1 − t)^{3}(1 − t^{3})^{2}(1 − t^{5})^{3}(1 − t^{7})^{2}(1 − t^{9})}.

Thus the multiplied out Hilbert polynomial of X is consistent with the computer output.

In Example 4.2.1 take w = (2, 3, 2, 1, 1). This gives a wGr(2, 5) ⊂ ℙ(2, 3^{4}, 4^{3}, 5^{2}) defined by a skew-symmetric matrix with the same weights w_{i} + w_{j} as

M = ( \begin{matrix}5 & 4 & 3 & 3 \\ & 5 & 4 & 4 \\ & & 3 & 3 \\ & & & 2 \end{matrix} ) .

Its canonical class equals K_{wGr} = O(−18). The threefold X can not be realised directly as a general quasilinear section in wGr(2, 5). The point is that h^{0}(X, D) = 2, so the graded ring R(X, D) has 2 generators of degree 1. On the other hand, the matrix M has no elements of weight 1. Therefore X is defined as a quasilinear section in

CCwGr(2, 5) ⊂ ℙ(1, 1, 2, 3^{4}, 4^{3}, 5^{2}).

This means that we add two extra variables of degree 1 to the homogeneous coordinate ring, which are not involved in any relation.

Thus we proved that X ⊂ ℙ(1, 1, 2, 3, 3, 3, 5) is isomorphic to

(3) ∩ (4)^{3} ∩ (5) ∩ CCwGr(2, 5) ⊂ ℙ(1, 1, 2, 3^{4}, 4^{3}, 5^{2}).

Since K_{CCwGr} = O(−20), the adjunction formula implies K_{X} = O_{X}(3 + 3 × 4 + 5 − 20). This shows that X is indeed a Calabi–Yau . □

4.3.3 The Hodge numbers

In this subsection we compute the Hodge numbers for Examples 4.2.3 and 4.2.4. The method is based on the Type I unprojection and was suggested by Balázs Szendrői. We apply the same method to Example 4.2.1 but we encounter worse singularities.

We start by proving two technical lemmas.

Lemma 4.3.3 Let X be a quasilinear section in wGr(2, 5) or CwGr(2, 5), which is quasismooth. Then dim Pic_{ℚ} X = 1.

Proof Consider Gr(2, 5) embedded in an ordinary projective space ℙ^{9} by the Plűcker embedding. Denote by aGr(2, 5) the affine cone over Gr(2, 5) in 𝔸^{10}. By [17, Ex. II 6.3] there is an exact sequence

0 ⟶ ℤ ⟶ Cl (Gr) ⟶ Cl (aGr) ⟶ 0,

where the first arrow sends 1 ↦ O(1) and the second is the restriction to aGr −{0} followed by the inclusion in aGr. By [13, page 274] the sheaf O(1) generates Pic (Gr(2, 5)). This means that the first map is surjective and

Cl (aGr) = Cl (aGr −{0}) = 0.

In other words, aGr is factorial.

Let X be a quasilinear section in wGr(2, 5). Write aX for the affine cone over X. This is the complete intersection singularity given by the same (quasilinear) equations as X. Since X is quasismooth, 0 ∈ aX is an isolated singularity. By Grothendieck theorem and Lefschetz principle (check [7, Lemma 3.5]), aX is factorial. Thus every Weil divisor on X can be represented as the quotient by ℂ^{∗} of a principal divisor on aX, which is invariant under the ℂ^{∗} action.

The same proof works for quasilinear sections in CwGr(2, 5), since for every factorial ring R the ring R[x] is also factorial. □

Lemma 4.3.4 Let α, α′, α′′, β, β′, β′′ be homogenous polynomials on ℙ^{2} of degrees a, a′, a′′, b, b′, b′′ respectively, such that

a − b = a′ − b′ = a′′ − b′′.

Consider a 0-dim scheme defined by

rank ( \begin{matrix}α & α′ & α′′ \\ β & β′ & β′′ \end{matrix} ) ≤ 1.

For a generic choice of polynomials, this scheme will contain

aa′ + b′b′′ + ab′′

points.

Proof Consider the matrix

( \begin{matrix}α & α′ & α′′ \\ β & β′ & β′′ \end{matrix} )

from the lemma. Its minors define a set S in ℙ^{2} whose ring has resolution

\begin{matrix}O_{S} ← O_{ℙ^{2}} ← O(−a − b′) ⊕ O(−a − b′′) ⊕ O(−a′ − b′′) ← \\ ← O(−a − a′ − b′′) ⊕ O(−a − b′ − b′′) ← 0. \end{matrix}

Thus, the Hilbert function for S is

P_{S}(t) = \frac{1 − t^{a+b′} − t^{a+b′′} − t^{a′+b′′} + t^{a+a′+b} + t^{a+b′+b′′}}{(1 − t)^{3}} .

Since S is of codimension 2 in ℙ^{2}, by Lemma 3.2.1 its degree equals

\begin{matrix}−\binom{a+b′}{2} − \binom{a+b′′}{2} − \binom{a′+b′′}{2} + \binom{a+a′+b′′}{2} + \binom{a+b′+b′′}{2} \\ = aa′ + b′b′′ + ab′′. \end{matrix}

□

Let h^{1,1} and h^{2,1} be the Hodge numbers and e the Euler characteristic with compact support of a smooth Calabi–Yau as described in Appendix A.3.

Proposition 4.3.5 Let X be the Calabi–Yau from Example 4.2.3 and X̂ its desingularisation. Then

h^{1,1}(X̂) = 5, h^{2,1}(X̂) = 73 and e(X̂) = −136.

Proof We will divide the proof into 5 steps.

∘ Step 1: Project X to a codimension 2 complete intersection Y.

By the Type I unprojection, X can be constructed from a complete intersection Y ⊂ ℙ(1, 1, 1, 3, 4, 4), which we now describe. Denote

J = ( \begin{matrix}a_{2} & x_{1} & x_{2} \\ a_{6} & b_{5} & c_{5} \end{matrix} ) ,

where a_{2}, a_{6}, b_{5}, c_{5} are general polynomials in x_{1}, x_{2}, x_{3}, y_{1}, z_{1}, z_{2} of degrees 2, 6, 5, 5 respectively. There exists an isomorphism

φ: Y \ Π ⟶ X \ Γ,

where

Π = \{y_{1} = z_{1} = z_{2} = 0\} ⊂ Y = \{J ( \begin{matrix}y_{1} \\ z_{1} \\ z_{2} \end{matrix} ) = 0\} ⊂ ℙ(1, 1, 1, 3, 4, 4)

and Γ ⊂ X is defined by

rank J|_{{y_{1}=z_{1}=z_{2}=0}} ≤ 1.

In other words, Y can be viewed as a codimension 2 complete intersection of degrees 5 and 9 in ℙ(1, 1, 1, 3, 4, 4) on which we impose the plane Π.

∘ Step 2: The Euler number of a general c.i. Y_{5,9}.

Denote by Y_{5,9} a general quasismooth complete intersection in ℙ(1, 1, 1, 3, 4, 4), and by Ŷ_{5,9} its MPCP–desingularisation. We use the notation of Appendix A.3. By [22] or A. Klemm’s list of the codimension 2 complete intersections we find that

e(Ŷ_{5,9}) = 2(h^{1,1} − h^{2,1}) = 2(4 − 109) = −210.

The only singularities of Y_{5,9} lie along the projective line C = {x_{1} = x_{2} = x_{3} = y_{1} = 0}. Then Ŷ_{5,9} is obtained from Y_{5,9} by resolving along C. The exceptional locus consists of 3 ruled surfaces E_{1}, E_{2}, E_{3}. All E_{i} are ruled over C ≅ ℙ^{1}, thus e(E_{i}) = e(ℙ^{1}) × e(ℙ^{1}). The intersections E_{1} ∩ E_{2} and E_{2} ∩ E_{3} both meet in a copy of ℙ^{1}, whereas E_{1} ∩ E_{3} = ∅. Therefore

e(E_{1} ∪ E_{2} ∪ E_{3}) = 3e(ℙ^{1})^{2} − 2e(ℙ^{1}) = 3 · 4 − 2 · 2 = 8.

By the definition of blow up

Y_{5,9} − C ≅ Ŷ_{5,9} − {E_{1} ∪ E_{2} ∪ E_{3}}

and by additivity of the Euler number

e(Y_{5,9}) = e(Ŷ_{5,9}) − e(E_{1} ∪ E_{2} ∪ E_{3}) + e(C) = −210 − 8 + 2 = −216.

∘ Step 3: The difference between Sing Y_{5,9} and Sing Y.

By Bertini’s theorem Y is quasismooth away from the base locus Π. Recall the defining equations of Y

f_{5} = a_{2}y_{1} + x_{1}z_{1} + x_{2}z_{2},  (E. xiii)
f_{9} = a_{6}y_{1} + b_{5}z_{1} + c_{5}z_{2}.

Since we can assume that a_{6} contains y_{1}^{2}, the point (0, 0, 0, 1, 0, 0) does not lie in Y. On the other hand z_{1} and z_{2} induce a line of singularities of transverse type \frac{1}{4}(1, 3). Consider the affine piece z_{1} ≠ 0 and assume that b_{5} contains x_{2}z_{1}. We can eliminate x_{1}, x_{2} out of f_{5} and f_{9}. Thus x_{3} and y_{1} are the local transverse parameters around the curve with the ε ∈ μ_{4} group action

x_{3}, y_{1} ∼ εx_{3}, εy_{1}.

The same argument holds for z_{2} ≠ 0.

We proved that Y and Y_{5,9} have the same singularities away from Π.

Next we determine the singularities of Y that lie on Π. The Jacobian of (E. xiii) restricted to Π equals

( \begin{matrix}0 & 0 & 0 & a_{2} & x_{1} & x_{2} \\ 0 & 0 & 0 & a_{6} & b_{5} & c_{5} \end{matrix} )|_{Π} .

Therefore,

Sing Y |_{Π} = rank J|_{{y_{1}=z_{1}=z_{2}=0}} ≤ 1.

Thus Sing Y consists of 37 points on Π by Lemma 4.3.4.

Now we will determine the type of these singularities. For every point of Π one of x_{1}, x_{2}, x_{3} is not zero. We will use the implicit function theorem on (E. xiii). On the affine piece x_{1} = 1, we can express z_{1} out of the first equation f_{5}. The quadratic monomials in the second equation f_{9} come from a_{6}y_{1} + c_{5}z_{2}. They give a quadratic

y_{1}(x_{2} + x_{3} + y_{1} + z_{2}) + z_{2}(x_{2} + x_{3} + y_{1} + z_{2})

of rank 4. This tells that all singularities for which x_{1} ≠ 0 are simple nodes. Exactly the same argument holds for x_{2} ≠ 0.

We can also assume that a_{6} contains monomial x_{3}^{6}. Then, on the affine piece x_{3} = 1 we can express y_{1} out of f_{9}. The quadratic monomials in the first equation f_{5} come from x_{1}z_{1} + x_{2}z_{2}, which is clearly a quadratic of rank

4. Therefore also the singularities for which x_{3} ≠ 0 are simple nodes.

We proved that Y is a complete intersection of degrees 5 and 9 in ℙ(1, 1, 1, 3, 4, 4) with 37 simple nodes. It is well known [14], that every node adds 1 to the Euler characteristic. This is because nodes have very simple topology and each node has the sphere S^{3} as its Milnor fibre. Therefore

e(Y) = e(Y_{5,9}) + 37 = −216 + 37 = −179.

∘ Step 4: Unproject from Y to compute e(X).

By the Type I unprojection there exists an isomorphism

φ: Y \ Π ⟶ X \ Γ,

where X is defined by Y and an extra variable

y_{2} = \frac{x_{1}c_{5} − x_{2}b_{5}}{y_{1}} = \frac{x_{2}a_{6} − a_{2}c_{5}}{z_{1}} = \frac{a_{2}b_{5} − x_{1}a_{6}}{z_{2}} .

Note that these equations are indeed the defining Pfaffians of X in Example 4.2.3. Note also that Γ in X ⊂ ℙ(1, 1, 1, 3, 3, 4, 4) is defined by

rank J|_{{y_{1}=z_{1}=z_{2}=0}} ≤ 1.

This inequality restricted to {y_{2} = 0} ⊂ X satisfies the conditions of Lemma 4.3.4. Thus Γ is a union of 37 lines that all meet in (0, 0, 0, 0, 1, 0, 0) as shown on Figure IV.

Finally we can compute

e(X) = e(Y) − e(Π) + e(Γ) = −179 − 3 + 37 + 1 = −144.

∘ Step 5: The Hodge numbers of X̂.

Let X̂ be the MPCP-desingularisation of X. In Example 4.2.3 we prove that X is quasismooth and has a \frac{1}{4}(1, 3) transverse line C and a \frac{1}{3}(1, 1, 1) singular point P.

Figure IV: The Type I unprojection Y → X

The resolution of C ≅ ℙ^{1} induces three exceptional surfaces E_{1}, E_{2}, E_{3}. In Step 2 we show that e(E_{1} ∪ E_{2} ∪ E_{3}) = 8. Note that P allows a crepant toric resolution. Its exceptional divisor E is a copy of ℙ^{2}. This yields

\begin{aligned}e(X̂) & = & e(X) − e(P) − e(C) + e(ℙ^{2}) + e(E_{1} ∪ E_{2} ∪ E_{3}) \\ & = & −144 − 1 − 2 + 3 + 8 = −136. \end{aligned}

The desingularisation adds the divisors E_{1}, E_{2}, E_{3} and E to the Picard group. Thus, by Lemma 4.3.3

h^{1,1}(X̂) = ρ(X̂) = 1 + 1 + 3 = 5.

From here we get

h^{2,1}(X̂) = h^{1,1}(X̂) − \frac{1}{2}e(X̂) = 73.

□

In the next example we will need the following lemma:

Lemma 4.3.6 Let ℙ(1, 3, 5, 9) have the coordinates t_{1}, t_{3}, t_{5}, t_{9}. Denote by E the exceptional locus of the toric resolution of C = {t_{1} = t_{5} = 0} and (0, 0, 0, 1) ∈ C. Then e(E) = 12.

Proof The toric threefold ℙ(1, 3, 5, 9) can be described by the fan in ℤ^{3} spanned on the vectors

v_{1} = (−9, −5, −3),
v_{3} = (0, 0, 1),
v_{5} = (0, 1, 0),
v_{9} = (1, 0, 0).

Note that v_{1} + 3v_{3} + 5v_{5} + 9v_{9} = 0.

Figure V: Resolution of the dissident curve

We resolve C by adding rays

\begin{matrix}w = \frac{2}{3}v_{1} + \frac{1}{3}v_{5} = (−6, −3, −2), \\ u = \frac{1}{3}v_{1} + \frac{2}{3}v_{5} = (−3, −1, −1) \end{matrix}

to the fan. These two vectors correspond to the exceptional divisors E_{w} and E_{u}. Next we add

a = \frac{1}{9}(v_{1} + 3v_{3} + 5v_{5}) = \frac{1}{3}(v_{3} + v_{5} + u) = (−1, 0, 0),
b = \frac{1}{9}(2v_{1} + 6v_{3} + v_{5}) = \frac{2}{3}v_{3} + \frac{1}{3}w = (−2, −1, 0),
c = \frac{1}{9}(4v_{1} + 3v_{3} + 2v_{5}) = \frac{1}{3}v_{3} + \frac{2}{3}w = (−4, −2, −1),

which correspond to the divisors E_{a}, E_{b}, E_{c}. Note that this is a crepant resolution.

In order to compute the Euler number e(E_{w} ∪ E_{u} ∪ E_{a} ∪ E_{b} ∪ E_{c}), we need to understand what are these divisors topologically and how do they intersect. Using the terminology of [12], every divisor is determined by its Star. In ℤ^{2}, this corresponds to the fans on Figure VI.

Figure VI: Fans of the exceptional surfaces

The exceptional surfaces form the configuration on Figure VII Every nontrivial intersection of two exceptional surfaces is a copy of ℙ^{1}

with the Euler number 2. Observe also that all exceptional divisors are toric surfaces. By [12, Section 3.2.], the Euler number of a compact toric variety defined by a fan F in ℤ^{n} equals to the number of n-dimensional cones in F.

Figure VII: Configuration of the exceptional surfaces

Thus e(E_{w} ∪ E_{u} ∪ E_{a} ∪ E_{b} ∪ E_{c}) = e(E_{w}) + e(E_{u}) + e(E_{a}) + e(E_{b}) + e(E_{c})− e(E_{u} ∩ E_{a}) − e(E_{u} ∩ E_{b}) − e(E_{u} ∩ E_{c})− e(E_{u} ∩ E_{w}) − e(E_{b} ∩ E_{c}) − e(E_{c} ∩ E_{w})+ e(E_{u} ∩ E_{b} ∩ E_{c}) + e(E_{u} ∩ E_{c} ∩ E_{w}) = 4 + 7 + 3 + 4 + 4 − (2 + 2 + 2 + 2 + 2 + 2) + 2 = 12.

□

Proposition 4.3.7 Let X be the Calabi–Yau from Example 4.2.4 and X̂ its desingularisation. Then

h^{1,1}(X̂) = 7, h^{2,1}(X̂) = 92 and e(X̂) = −170.

Proof We will use the same method as in Proposition 4.3.5 ∘ Step 1: Project X to a codimension 2 complete intersection Y.

By the Type I projection we project X to Y ⊂ ℙ(1, 1, 1, 3, 5, 9). Write

J = ( \begin{matrix}a_{7} & a_{5} & x_{1} \\ b_{7} & b_{5} & x_{2} \end{matrix} ) ,

where a_{i}, a_{6}, b_{j} are general polynomials in x_{1}, x_{2}, x_{3}, y_{1}, z, v and define

Y = \{J ( \begin{matrix}y_{1} \\ z \\ v \end{matrix} ) = 0\} .

There exists an isomorphism

φ: Y \ Π ⟶ X \ Γ,

where

Π = {y_{1} = z = v = 0} ⊂ Y ⊂ ℙ(1, 1, 1, 3, 4, 4),

and Γ ⊂ X is defined by

rank J|_{{y_{1}=z=v=0}} ≤ 1.

The singular threefold Y is a codimension 2 complete intersection in ℙ(1, 1, 1, 3, 4, 4). It is defined by two equations of degrees 10 on which we impose Π.

∘ Step 2: The Euler number of a general c.i. Y_{10,10}.

Denote by Y_{10,10} a generic quasismooth complete intersection in ℙ(1, 1, 1, 3, 5, 9), and by Ŷ_{10,10} its MPCP-desingularisation. The only singularities of Y_{10,10} lie along the projective line C = {x_{1} = x_{2} = x_{3} = z = 0}. It is easy to see that C is a curve of transverse type \frac{1}{3}(1, 2) with a \frac{1}{9}(1, 3, 5) dissident point on it. From [22] we read

e(Ŷ_{10,10}) = 2(h^{1,1} − h^{2,1}) = 2(6 − 138) = −264.

By the definition of desingularisation

Ŷ_{10,10} − E ≅ Y_{10,10} − C,

where E denotes the exceptional locus. Lemma 4.3.6 implies

e(Y_{10,10}) = e(Ŷ_{10,10}) − e(E) + e(C) = −264 − 12 + 2 = −274.

∘ Step 3: The difference between Sing Y_{10,10} and Sing Y.

The defining equations of Y are

f_{10} = a_{7}y_{1} + a_{5}z + x_{1}v  (E. xiv)
g_{10} = b_{7}y_{1} + b_{5}z + x_{2}v.

By Bertini’s theorem Y is quasismooth away from the base locus Π. We use the implicit function theorem on (E. xiv) to find the local transverse parameters in the neighbourhood of the curve {x_{1} = x_{2} = x_{3} = z = 0} ⊂ Y. This shows that Y has a curve of transverse type \frac{1}{3}(1, 2) with a dissident point \frac{1}{9}(1, 3, 5). These singularities are the same as the singularities of the generic Y_{10,10}.

Additionally, Y has singularities on Π. They are given by

Sing Y |_{Π} = rank J|_{{y_{1}=z=v=0}} ≤ 1.

By Lemma 4.3.4 we get 47 points on Π.

Next we will prove that all 47 points are nodes. For every point on Π one of the x_{1}, x_{2}, x_{3} is not zero. We will use the implicit function theorem on the defining equations (E. xiv). On the affine piece x_{1} = 1, we can express v out of f_{10}. This gives hypersurface singularities, which are determined by the quadratic monomials in g_{10}. They define a quadric

y_{1}(x_{2} + x_{3} + y_{1} + z) + z(x_{2} + x_{3} + y_{1} + z)

of rank 4.

Exactly the same argument holds for x_{2} ≠ 0.

For x_{3} ≠ 0 we can assume that a_{7} contains x_{3}^{7}. On the affine piece x_{3} = 1 we can express y_{1} out of f_{10}. The quadratic monomials in g_{10} come from z(x_{1} + x_{2} + z) + x_{2}v, which is again a quadric of rank 4.

We proved that Y is a complete intersection in ℙ(1, 1, 1, 3, 5, 9) with 47 simple nodes. Therefore

e(Y) = e(Y_{10,10}) + 47 = −274 + 47 = −227.

∘ Step 4: Unproject from Y to compute e(X).

Recall the Type I unprojection and the isomorphism

φ: Y \ Π ⟶ X \ Γ,

where X is constructed from Y by adding

y_{2} = \frac{x_{2}a_{5} − x_{1}b_{5}}{y_{1}} = \frac{x_{1}b_{7} − x_{2}a_{7}}{z} = \frac{a_{7}b_{5} − a_{5}b_{7}}{v}

and Γ ⊂ X ⊂ ℙ(1, 1, 1, 3, 3, 5, 9) is defined by

rank J|_{{y_{1}=z=v=0}} ≤ 1.

For {y_{2} = 0} in X this inequality satisfies the conditions of Lemma 4.3.4. Thus Γ is a union of 47 lines that all meet in (0, 0, 0, 0, 1, 0, 0).

Finally we get

e(X) = e(Y) − e(Π) + e(Γ) = −227 − 3 + 47 + 1 = −182.

∘ Step 5: The Hodge numbers of X̂.

In Example 4.2.3 we proved that Sing(X) contains a \frac{1}{3}(1, 1, 1) singular point P and a \frac{1}{3}(1, 2) curve C with a dissident point Q.

The line C has a resolution with the exceptional locus E that was described in Lemma 4.3.6. The point P also allows a crepant toric resolution whose exceptional divisor E_{P} is a copy of ℙ^{2}. For the resolved X̂ this yields

\begin{aligned}e(X̂) & = & e(X) − e(P) − e(C) + e(ℙ^{2}) + e(E) \\ & = & −182 − 1 − 2 + 3 + 12 = −170. \end{aligned}

The desingularisation introduces the divisors E_{w}, E_{u}, E_{a}, E_{b}, E_{c} and E_{P}. Thus

h^{1,1}(X̂) = 1 + 1 + 5 = 7.

From here we get

h^{2,1}(X̂) = h^{1,1}(X̂) − \frac{1}{2}e(X̂) = 92.

□

Lemma 4.3.8 In Example 4.2.1 the complete intersection Y obtained by the Type I unprojection contains worse singularities than simple nodes.

Proof Consider X from Example 4.2.1. It is quasismooth and has a \frac{1}{5}(1, 1, 3) isolated point Q = (0, 0, 0, 0, 0, 0, 1) and a \frac{1}{3}(1, 2) transverse curve C. We can obtain X by the Type I unprojection from a complete intersection Y. Define Y ⊂ ℙ(1, 1, 2, 3, 3, 3) with variables x_{1}, x_{2}, y, z_{1}, z_{2}, z_{3} by

Π = \{y = z_{1} = z_{2} = 0\} ⊂ Y = \{( \begin{matrix}a_{4} & b_{3} & c_{3} \\ a_{5} & b_{4} & c_{4} \end{matrix} ) ( \begin{matrix}y \\ z_{1} \\ z_{2} \end{matrix} ) = 0\} ,

where a_{i}, b_{j}, c_{k} are generic homogeneous polynomials. The singular threefold Y can be viewed as a codimension 2 complete intersection of degrees 6 and 7 in ℙ(1, 1, 2, 3, 3, 3) on which we impose Π ≅ ℙ(1, 1, 3). Then X is the unprojection of Y by adding the parameter

t = \frac{b_{3}c_{4} − b_{4}c_{3}}{y} = \frac{c_{3}a_{5} − a_{4}c_{4}}{z_{1}} = \frac{a_{4}b_{4} − b_{3}a_{5}}{z_{2}}

of degree 5. The defining equations of X are exactly the Pfaffians in Example 4.2.1.

Denote by Y_{6,7} a general quasismooth complete intersection in ℙ(1, 1, 2, 3, 3, 3). The only singularities of Y_{6,7} lie on the quadric C = {x_{1} = x_{2} = y = 0}. Indeed, by an equation of degree 6 we can avoid the \frac{1}{2} singular point of ℙ. Moreover, the singularities along C are of the transverse type \frac{1}{3}(1, 2).

By Bertini’s theorem it is easy to see that Y and Y_{6,7} have the same singularities away from the base locus Π.

In the next step we determine the singularities of Y that lie on Π. Denote

( \begin{matrix}a_{4} & b_{3} & c_{3} \\ a_{5} & b_{4} & c_{4} \end{matrix} )

by J. Then

Sing Y |_{Π} = rank J|_{{y=z_{1}=z_{2}=0}} ≤ 1.

We can not use Lemma 4.3.4 in this case since Π ≅ ℙ(1, 1, 3). Instead we use the computer to count the number of solutions for the equations defined by the minors of J. If we choose a_{i}, b_{j}, c_{k} in the variables x_{1}, x_{2}, z_{3} general enough, we get 15 points on Π. Observe that P = (0, 0, 1) is always one of them. Note also also that

P = Π ∩ C.

The situation is shown on Figure VIII.

Figure VIII: Type I unprojection Y → X

We will prove that 14 of these points are nodes on Y, whereas P is a worse singularity. For every point on Π one of x_{1}, x_{2}, z_{3} is not zero. We can use the implicit function theorem on

a_{4}y + b_{3}z_{1} + c_{3}z_{2} = 0  (E. xv)
a_{5}y + b_{4}z_{1} + c_{4}z_{2} = 0.

Assume that a_{4} contains the monomial x_{1}^{4}. Then, use the implicit function theorem on the affine piece x_{1} = 1, to express y out of the first equation. The quadratic monomials in the second equation come from a_{6}y_{1}+c_{5}z_{2}. They give a quadric

z_{1}(x_{2} + z_{1} + z_{2} + z_{3}) + z_{2}(x_{2} + z_{1} + z_{2} + z_{3})

of rank 4. This tells that all singularities for which x_{1} ≠ 0 are simple nodes. Exactly the same argument holds for x_{2} ≠ 0.

We can also assume that b_{3} = z_{3}. At P we set z_{3} = 1 and eliminate z_{1} out of the first equation in (E. xv). The quadratic part of the second equation is y^{2} + (x_{1} + x_{2})z_{2}, which is a quadric of rank 3. Therefore we also need to consider the cubic monomials

y^{2} + (x_{1} + x_{2})z_{2} + x_{1}^{2}y + x_{1}x_{2}y + x_{2}^{2}y + y^{2}z_{2} + x_{1}z_{2}^{2} + x_{2}z_{2}^{2}.

We proved that P is a singularity of type

{p(x_{1}, x_{2}, y, z_{2}) = 0} ⊂ 𝔸^{4}/μ_{3},

where p denotes the above cubic equation and ε ∈ μ_{3} acts by

x_{1}, x_{2}, y, z_{2} ↦ εx_{1}, εx_{2}, ε^{2}y, z_{2}.
□

4.4 Weighted orthogonal Grassmannians

In this section we discuss some examples of codimension 5 Calabi–Yau threefolds. The ambient spaces of these varieties will be weighted orthogonal Grassmannians, which are weighted projective analogs of the homogeneous orthogonal Grassmannian OGr(5, 10).

4.4.1 Definitions and the spinor embedding

The construction we recall in this section was introduced in [8].

Consider V = ℂ^{10} equipped with a nondegenerate quadratic form q. Choose a basis

e_{1}, …, e_{5}, f_{1}, …, f_{5}

for V such that

q = ( \begin{matrix}0 & I \\ I & 0 \end{matrix} ) and V = U ⊕ U^{∨},

where U = ⟨e_{1}, …, e_{5}⟩ and U^{∨} = ⟨f_{1}, …, f_{5}⟩. A vector space F ⊂ V is isotropic if q is identically zero on F. Define the orthogonal Grassmannian OGr(5, 10) by

OGr(5, 10) = \{F ∈ Gr(5, V) | \begin{matrix}F is isotropic and \\ dimF ∩ U is odd \end{matrix} \} .

A 5-plane F in the neighbourhood of U has a basis of five vectors in V. Thus it can be written as a matrix (I, M), where M and identity I are 5 × 5 matrices. Observe that F is isotropic for q if and only if M is a skew symmetric matrix. Therefore, an affine neighbourhood of U in OGr(5, 10) is given by (I, M), where M is a skew 5 × 5 matrix.

OGr(5, 10) can be covered by 16 affine pieces. Each one is obtained from the affine piece (I, M), by swapping evenly many of the columns in I with the corresponding columns in M.

Define the space of spinor

S^{+} = ℂ ⊕ ⋀^{2} U ⊕ ⋀^{4} U.

For an element (x, M, P) in S^{+}, where x ∈ ℂ we introduce notation

M = ( \begin{matrix}0 & x_{12} & x_{13} & x_{14} & x_{15} \\ −x_{12} & 0 & x_{23} & x_{24} & x_{25} \\ −x_{13} & −x_{23} & 0 & x_{34} & x_{35} \\ −x_{14} & −x_{24} & −x_{34} & 0 & x_{45} \\ −x_{15} & −x_{25} & −x_{35} & −x_{45} & 0 \end{matrix} ) and P = ( \begin{matrix}x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \\ x_{5} \end{matrix} ) .

This defines the spinor embedding

OGr(5, 10) ↪ ℙ(S^{+}),

where (x, M, P) satisfy the 10 equations

xP = Pf M, and MP = 0.

More precisely, these equations are:

N_{1} = xx_{1} − x_{23}x_{45} + x_{24}x_{35} − x_{25}x_{34},
N_{2} = xx_{2} + x_{13}x_{45} − x_{14}x_{35} + x_{15}x_{34},
N_{3} = xx_{3} − x_{12}x_{45} + x_{14}x_{25} − x_{15}x_{24},
N_{4} = xx_{4} + x_{12}x_{35} − x_{13}x_{25} + x_{15}x_{23},
N_{5} = xx_{5} − x_{12}x_{34} + x_{13}x_{24} − x_{14}x_{23},
N_{−1} = x_{2}x_{12} + x_{3}x_{13} + x_{4}x_{14} + x_{5}x_{15},
N_{−2} = −x_{1}x_{12} + x_{3}x_{23} + x_{4}x_{24} + x_{5}x_{25},
N_{−3} = −x_{1}x_{13} − x_{2}x_{23} + x_{4}x_{34} + x_{5}x_{35},
N_{−4} = −x_{1}x_{14} − x_{2}x_{24} − x_{3}x_{34} + x_{5}x_{45},
N_{−5} = −x_{1}x_{15} − x_{2}x_{25} − x_{3}x_{35} − x_{4}x_{45}.

The same as for a weighted projective space, we can assign weights to x, x_{ij}, x_{k}. Using representation theory (check Section 4 in [8] for the proof), we can introduce the following numerology. Let w_{1}, …, w_{5} be weights of U = ℂ^{5} and u a further overall weight on S^{+}. Define s = w_{1} + … + w_{5}. Then the generators have weights

wt x = u
wt x_{ij} = u + w_{i} + w_{j}
wt x_{i} = u + s − w_{i}.

The weighted orthogonal Grassmannian wOGr(5, 10) has the Hilbert numerator

Q(t) = 1 − t^{s+2u}Q_{V} + t^{2s+3u}Q_{S^{−}} − t^{2s+5u}Q_{S^{+}} + t^{3s+6u}Q_{V} − t^{4s+8u}, where

\begin{aligned}Q_{V} & = & ∑_{i=1}^{5} t^{w_{i}} + t^{−w_{i}}, \\ Q_{S^{+}} & = & 1 + ∑_{i,j} t^{w_{i}+w_{j}} + ∑_{i} t^{s−w_{i}}, \\ Q_{S^{−}} & = & 1 + ∑_{i,j} t^{−w_{i}−w_{j}} + ∑_{i} t^{−s+w_{i}}. \end{aligned}

The corresponding Hilbert series is

P_{wOGr}(t) = \frac{Q(t)}{(1 − t^{u}) · ∏_{i}(1 − t^{u+s−w_{i}}) · ∏_{ij}(1 − t^{u+w_{i}+w_{j}})}.

If wOGr(5, 10) is well formed, its canonical class equals

K_{wOGr} = O(−4s − 8u).

4.4.2 Examples in codimension 5 In this subsection we construct new examples of Calabi–Yau threefolds as quasilinear sections of a fixed weighted orthogonal Grassmannian.

Choose the weights

(w_{1}, w_{2}, w_{3}, w_{4}, w_{5}) = (0, 0, 1, 1, 1) and u = 1,

defining wOGr(5, 10). Using the notation of Subsection 4.4.1, we get • s = 3,

• the coordinates x and x_{12} have weight 1,

• the coordinates x_{13}, x_{14}, x_{15} and x_{23}, x_{24}, x_{25} have weight 2,

• the coordinates x_{34}, x_{35}, x_{45} and x_{3}, x_{4}, x_{5} have weight 3,

• the coordinates x_{1} and x_{2} have weight 4.

The spinor embedding takes wOGr(5, 10) into ℙ(1^{2}, 2^{6}, 3^{6}, 4^{2}), with the defining equations N_{1}, …, N_{5}, N_{−1}, …, N_{−5} of degrees 5, 5, 4, 4, 4, 5, 5, 6, 6, 6 respectively.

Its Hilbert function equals

P_{wOGr}(t) =_{\frac{1−3t^{4}−4t^{5}−t^{6}+6t^{7}+6t^{8}+2t^{9}−2t^{11}−6t^{12}−6t^{13}+t^{14}+4t^{15}+3t^{16}−t^{20}}{(1−t)^{2}(1−t^{2})^{6}(1−t^{3})^{6}(1−t^{4})^{2}}}

and the canonical class is K_{wOGr} = O(−20).

Example 4.4.1 For the input data we choose

• h^{0}(X, D) = 2 and h^{0}(X, 2D) = 6

• Points: /

• Curves: type = \frac{1}{3}(1, 2), deg = 1, index = 1, N_{C} = −6.

The output we get is

• Codimension: 5

• Weights: 1,1,2,2,2,3,3,3,3

• Multiplied out Hilbert polynomial: 1 − 3t^{4} − 4t^{5} − t^{6} + 6t^{7} + 6t^{8} + 2t^{9} − 2t^{11} − 6t^{12} − 6t^{13} + t^{14} + 4t^{15} + 3t^{16} − t^{20}

We will prove that the output defines a general quasilinear section

X = wOGr(5, 10) ∩ (2)^{3} ∩ (3)^{2} ∩ (4)^{2}

in the chosen wOGr(5, 10).

The fact that X is a Calabi–Yau can be checked in two ways. Either X ⊂ ℙ(1^{2}, 2^{3}, 3^{4}) with the adjunction number 20 yields −2 × 1 − 3 × 2 − 4 × 3 + 20 = 0, or the adjunction formula for (2)^{3} ∩ (3)^{2} ∩ (4)^{2} ⊂ wOGr(5, 10) gives K_{X} = O_{X}(−20 + 3 × 2 + 2 × 3 + 2 × 4) ∼ 0.

In the next paragraph we study the cyclic quotient singularities of X, induced by the singularities of ℙ. The Calabi–Yau X contains no \frac{1}{2} or \frac{1}{4} singularities, since there are enough equations of even degrees. Next consider C defined by the weights of degree 3 in ℙ. More precisely, C is given by

\begin{aligned}N_{−3} : & x_{4}x_{34} + x_{5}x_{35} & = & 0, \\ N_{−4} : & −x_{3}x_{34} + x_{5}x_{45} & = & 0, \\ N_{−5} : & −x_{3}x_{35} − x_{4}x_{45} & = & 0, \end{aligned} and (3)^{2} : l_{1} = l_{2} = 0,

where l_{1} and l_{2} are two linear equations in x_{3}, x_{4}, x_{5}, x_{34}, x_{35}, x_{45}. Rewrite the above as

rank ( \begin{matrix}x_{3} & x_{4} & x_{5} \\ x_{45} & x_{35} & x_{34} \end{matrix} ) ≤ 1 and l_{1} = l_{2} = 0,

which proves that C is a twisted cubic.

Choose a ℚ−Cartier divisor D such that O_{X}(D) ≅ O_{X}(1). Then deg D|_{C} = 1.

We claim that C is of transverse type \frac{1}{3}(1, 2). For example, a point on C at which x_{3} ≠ 0 has the local transverse parameters x_{12} and x_{24}. Indeed, by the implicit function theorem we can express x out of N_{3}, x_{13}, x_{23} out of N_{−1}, N_{−2} respectively; and the variables x_{14}, x_{15}, x_{25} and x_{1}, x_{2} out of the linear equations corresponding to (2)^{3} and (4)^{2}. The same way we find the local parameters for any other point on C.

Recall the formula in Corollary 3.1.1. The data h^{0}(X, D) = 2, h^{0}(X, 2D) = 6 are equivalent to

D^{3} = \frac{5}{3} and D · c_{2}(X) = 26.

Then

h^{0}(X, mD) = \frac{5}{18}m^{3} + \frac{13}{6}m + α_{m},

where

α_{m} = \{ \begin{matrix}−\frac{1}{3}m − \frac{1}{9} & if m = 1 mod 3, \\ −\frac{1}{3}m + \frac{1}{9} & if m = 2 mod 3, \\ 0 & if m = 0 mod 3. \end{matrix}

From this we calculate the Hilbert function of X

\begin{matrix}P_{X}(t) = \\ \frac{1−3t^{4}−4t^{5}−t^{6}+6t^{7}+6t^{8}+2t^{9}−2t^{11}−6t^{12}−6t^{13}+t^{14}+4t^{15}+3t^{16}−t^{20}}{(1−t)^{2}(1−t^{2})^{3}(1−t^{3})^{4}} , \end{matrix}

which is what we predicted on page 104.

Example 4.4.2 For the input data we choose

• h^{0}(X, D) = 1 and h^{0}(X, 2D) = 6

• Points: 3 × _{1}(\frac{1}{3}(1, 1, 1))

• Curves: type = \frac{1}{2}(1, 1), deg = \frac{7}{2}, index = 1.

The output we get is

• Codimension: 5

• Weights: 1,2,2,2,2,2,3,3,3

• Multiplied out Hilbert polynomial: 1 − 3t^{4} − 4t^{5} − t^{6} + 6t^{7} + 6t^{8} + 2t^{9} − 2t^{11} − 6t^{12} − 6t^{13} + t^{14} + 4t^{15} + 3t^{16} − t^{20}

This output suggests a general quasilinear section in the chosen wOGr(5, 10)

X = wOGr(5, 10) ∩ (1) ∩ (2) ∩ (3)^{3} ∩ (4)^{2}.

Use the adjunction formula for (1) ∩ (2) ∩ (3)^{3} ∩ (4)^{2} ⊂ wOGr(5, 10) to compute K_{X} = O(−20 + 1 + 2 + 3 × 3 + 2 × 4) ∼ 0. Thus X is a Calabi–Yau.

We will study the cyclic quotient singularities of X, induced by the weights of ℙ. We can avoid the weights of degree 4 by two equations (4)^{2}, thus there are no \frac{1}{4} singularities.

On the other hand, the weights of degree 2 determine a singular locus C. More precisely, C is defined by the equations involving the variables of even degrees:

\begin{aligned}N_{3} : & x_{14}x_{25} − x_{15}x_{24} & = & 0, \\ N_{4} : & −x_{13}x_{25} + x_{15}x_{23} & = & 0, \\ N_{5} : & x_{13}x_{24} − x_{14}x_{23} & = & 0, \\ N_{−3} : & −x_{1}x_{13} − x_{2}x_{23} & = & 0, \\ N_{−4} : & −x_{1}x_{14} − x_{2}x_{24} & = & 0, \\ N_{−5} : & −x_{1}x_{15} − x_{2}x_{25} & = & 0, \end{aligned} and (2) ∩ (4)^{2} : l = q_{1} = q_{2} = 0,

where l is a linear equation of degree 2 and q_{1}, q_{2} are equations of degree 4 which are linear in x_{1}, x_{2} and quadratic in x_{13}, x_{14}, x_{15}, x_{23}, x_{24}, x_{25}. The above system can be written into

rank ( \begin{matrix}x_{13} & x_{14} & x_{15} \\ x_{23} & x_{24} & x_{25} \end{matrix} ) ≤ 1,

(x_{1}, x_{2}) · ( \begin{matrix}x_{13} & x_{14} & x_{15} \\ x_{23} & x_{24} & x_{25} \end{matrix} ) = 0,

l = q_{1} = q_{2} = 0.

This proves that C is a curve of transverse type \frac{1}{2}(1, 1).

We will also compute the degree of C. Let a divisor D be given by O_{X}(1). The divisor 2D is Cartier in the neighbourhood of C and has a section, defined by an equation l′ = 0 of degree 2. Observe that

rank ( \begin{matrix}x_{13} & x_{14} & x_{15} \\ x_{23} & x_{24} & x_{25} \end{matrix} ) ≤ 1 ⋂ \{l = l′ = 0\}  (E. xvi)

is a twisted cubic which can be parametrically given by:

rank ( \begin{matrix}1 & a & a^{2} \\ a & a^{2} & a^{3} \end{matrix} ) ≤ 1.

Now eliminate the variables x_{1}, x_{2} of weight 4 out of q_{1}, q_{2} and represent them by a. For i = 1, 2 we get x_{i} = r_{i}(a), where r_{i} is a polynomial of degree 6 in a. The last equation we need to consider is (x_{1}, x_{2}) · (x_{13}, x_{23}) = 0. This equals to

(r_{1}(a), r_{2}(a)) · (1, a) = 0,

which has degree 7 and thus has 7 solutions. Thus we proved that deg D|_{C} = \frac{7}{2}.

The weights of degree 3 define a locus given by the equations

\begin{aligned}N_{−3} : & x_{4}x_{34} + x_{5}x_{35} & = & 0, \\ N_{−4} : & −x_{3}x_{34} + x_{5}x_{45} & = & 0, \\ N_{−5} : & −x_{3}x_{35} − x_{4}x_{45} & = & 0, \end{aligned} and (3)^{3} : l_{1} = l_{2} = l_{3} = 0,

where l_{1}, l_{2} and l_{3} are linear in x_{3}, x_{4}, x_{5}, x_{34}, x_{35}, x_{45}. The same way as in Example 4.4.1 this defines a twisted cubic intersected by a hyperplane. Denote the intersecting points by P_{1}, P_{2}, P_{3}.

Next we determine the type of singularities at these points. Assume that x_{4} ≠ 0 at P_{1}. By the implicit function theorem we can express x out of N_{4} and x_{14}, x_{24} out of N_{−1}, N_{−2}. We can also eliminate x_{12}, x_{13}, x_{1}, x_{2} from the equations (1), (2), (4)^{2}. Thus, the variables x_{15}, x_{23}, x_{25} of weights 2 are the local parameters in the neighbourhood of P_{1}. Exactly the same computation determines the type of P_{2} and P_{3}. We proved that P_{1}, P_{2}, P_{3} are of type

2(\frac{1}{3}(2, 2, 2)) = _{1}(\frac{1}{3}(1, 1, 1)).

From h^{0}(X, D) = 1 and h^{0}(X, 2D) = 6 we calculate

D^{3} = \frac{5}{4} and D · c_{2}(X) = 24.

Then by Corollary 3.1.1

h^{0}(X, mD) = \frac{5}{24}m^{3} + 2m + α_{m} + 3 · β_{m},

where

α_{m} = \{ \begin{matrix}−\frac{1}{4} · m · \frac{7}{2} & if m = 1 mod 2, \\ 0 & if m = 0 mod 2, \end{matrix}

β_{m} = \{ \begin{matrix}\frac{1}{3} ∑_{ε∈μ_{3}} \frac{ε^{−1}−1}{(1−ε^{−1})^{3}} = −\frac{1}{9} & if m = 1 mod 3, \\ \frac{1}{3} ∑_{ε∈μ_{3}} \frac{ε^{−2}−1}{(1−ε^{−1})^{3}} = \frac{1}{9} & if m = 1 mod 3, \\ 0 & if m = 0 mod 3. \end{matrix}

Here α_{m} are the contributions from the singularities along C and β_{m} are the basket contributions from P_{1}, P_{2}, P_{3}.

It is easy to check that this yields the Hilbert function P_{X}(t).

Example 4.4.3 For the input data we choose

• h^{0}(X, D) = 2 and h^{0}(X, 2D) = 7

• Points: _{3}(\frac{1}{4}(2, 3, 3))

• Curves: type = \frac{1}{2}(1, 1), deg = \frac{7}{4}, index = 2.

The output we get is

• Codimension: 5 • Weights: 1,1,2,2,2,2,3,3,4

• Multiplied out Hilbert polynomial: 1 − 3t^{4} − 4t^{5} − t^{6} + 6t^{7} + 6t^{8} + 2t^{9} − 2t^{11} − 6t^{12} − 6t^{13} + t^{14} + 4t^{15} + 3t^{16} − t^{20}

The output suggests the following interesting example. Consider a general quasilinear section

X = wOGr(5, 10) ∩ (2)^{2} ∩ (3)^{4} ∩ (4).

By the adjunction formula for (2)^{2} ∩ (3)^{4} ∩ (4) ⊂ wOGr(5, 10) we get K_{X} = O(−20 + 2 × 2 + 4 × 3 + 4) ∼ 0. Thus X is a Calabi–Yau .

Next we will analyse the cyclic quotient singularities of X which are induced by the weights of ℙ. Consider the equations involving weights of degree 3. They define a cubic intersected with two hyperplanes.This defines an empty set, thus there are no \frac{1}{3} singularities.

There is only one equation of degree 4, which is linear in x_{1} and x_{2}. Therefore X contains a singular point P of index \frac{1}{4}. Note that P lies on a curve C determined by the equations

\begin{aligned}N_{3} : & x_{14}x_{25} − x_{15}x_{24} & = & 0, \\ N_{4} : & −x_{13}x_{25} + x_{15}x_{23} & = & 0, \\ N_{5} : & x_{13}x_{24} − x_{14}x_{23} & = & 0, \\ N_{−3} : & −x_{1}x_{13} − x_{2}x_{23} & = & 0, \\ N_{−4} : & −x_{1}x_{14} − x_{2}x_{24} & = & 0, \\ N_{−5} : & −x_{1}x_{15} − x_{2}x_{25} & = & 0, \end{aligned} and (2)^{2} ∩ (4) : l_{1} = l_{2} = q = 0,

where l_{1}, l_{2} are linear equations of degree 2, and q is a degree 4 equation that is linear in x_{1}, x_{2} and quadratic in x_{13}, x_{14}, x_{15}, x_{23}, x_{24}, x_{25}.

Next we determine the type of P. Assume that x_{1} ≠ 0 at P. By the implicit function theorem we can express x, x_{12} out of equations N_{1}, N_{−2} and x_{13}, x_{14}, x_{15} out of N_{−3}, N_{−4}, N_{−5}. Moreover, we can eliminate x_{23}, x_{24} from the equations (2)^{2} : l_{1} = l_{2} = 0 and x_{3}, x_{4}, x_{5}, x_{34} from the four equations (3)^{4}. The remaining variables are x_{25}, x_{35}, x_{45} of degrees 2, 3, 3 respectively, which are the local parameters for P. This proves that P is of type

3(\frac{1}{4}(2, 3, 3)) = _{1}(\frac{1}{4}(2, 1, 1)).

Thus C is an example of a dissident \frac{1}{2}(1, 1) curve.

In the last step we compute deg D|_{C}. As before, let D = O_{X}(1). Then 4D is Cartier along C. The intersection number C ∩ |4D| is determined by (E. xvi) on page 107 and therefore equals 7. Thus

deg D|_{C} = \frac{7}{4}.

From h^{0}(X, D) = 2, h^{0}(X, 2D) = 7 and Corollary 3.1.1 we get

h^{0}(X, mD) = \frac{5}{16}m^{3} + \frac{9}{4}m + α_{m} + β_{m},

where

α_{m} = \{ \begin{matrix}−\frac{1}{4} · m · \frac{7}{4} & if m = 1 mod 2, \\ 0 & if m = 0 mod 2, \end{matrix}

β_{m} = \{ \begin{matrix}\frac{1}{4} ∑_{ε∈μ_{4}−\{1\}, ε^{2}≠1} \frac{ε^{−1}−1}{(1−ε^{−1})^{2}(1−ε^{−2})} = −\frac{1}{8} & if m = 1 mod 4, \\ \frac{1}{4} ∑_{ε∈μ_{4}−\{1\}, ε^{2}≠1} \frac{ε^{−2}−1}{(1−ε^{−1})^{2}(1−ε^{−2})} = 0 & if m = 2 mod 4, \\ \frac{1}{4} ∑_{ε∈μ_{4}−\{1\}, ε^{2}≠1} \frac{ε^{−3}−1}{(1−ε^{−1})^{2}(1−ε^{−2})} = \frac{1}{8} & if m = 3 mod 4, \\ 0 & if m = 0 mod 4. \end{matrix}

Here α_{m} are contributions from the transverse singularities along C and β_{m} are the basket contributions from the dissident point P ∈ C.

This yields the expected Hilbert function.

5 Outstanding questions

Question 1 How precisely does the input data for the Riemann–Roch formula determine the output model?

The basket contributions to RR coming from different singular points can cancel out. Consider for example _{1}(\frac{1}{3}(1, 1, 1)) and _{2}(\frac{1}{3}(1, 1, 1)). On the other hand, a _{k}(\frac{1}{r}(1, −1)) curve C always contributes the term

m\frac{\overline{mk}(r − \overline{mk})}{2r} deg D|_{C}

which is linear in m. Since deg D|_{C} is positive, the contributions from curves can not cancel. Note also that

f(m) := \overline{mk}(r − \overline{mk}) and g(m) := \overline{mk}(r − \overline{mk})(r − 2 · \overline{mk})

are both periodic mod r and f(m) = f(−m), g(m) = −g(−m).

Question 2 How can we compute the Hodge numbers for the Calabi–Yaus in the thesis?

There exist many papers on the Hodge numbers of Calabi–Yau complete intersections in toric varieties. The essence of all proofs are [10] and [15], which compute the Hodge numbers of hypersurfaces in a torus. Concrete formulas and algorithms can be found in [9] and [18, Formula 4.21.], and some interesting examples are in [22], [20].

In Section 4.3 we compute the Hodge numbers of some Calabi–Yau quasilinear sections in wGr(2, 5) by using the Type I unprojection. This method can not be used in general as we encounter worse singularities (Lemma 4.3.8).

Can we compute the Milnor fibre of the singularities that are worse than nodes? Can we use different types of unprojections to avoid the bad singularities?

Question 3 An interesting problem would be to find the mirror pairs for the Calabi–Yaus constructed in this thesis.

It is often possible to construct new Calabi–Yau threefolds from old ones by degenerating the complex structure and then resolving the singularities. In Section 4.2 we explain how this can be done by using the method of Type I unprojection. As suggested in [24], the method of projection–unprojection should also be used to produce new mirrors from old. The mirrors are known for complete intersections in weighted projective spaces [18] or straight Grassmannians [5].

Question 4 Do there exist birational maps between Examples 4.2.3 and 4.2.4, and the Calabi–Yau complete intersections in toric ambient varieties with the same Hodge numbers?

Question 5 In Proposition 4.1.1 we classified all possible singularities of quasismooth well formed Calabi–Yau complete intersections. The same result holds for a codimension 3 Calabi–Yau, which is constructed from a codimension 2 c.i. by the Type I unprojection.

Can we determine all possible quotient singularities of Calabi–Yaus in weighted projective spaces?

Question 6 For an A_{r−1} curve C in a quasismooth and well formed threefold X there exists a unique crepant resolution (Lemma 2.2.2). It is a sequence of ⌊\frac{r}{2}⌋ blow–ups and induces r − 1 different exceptional surfaces.

If X is not quasismooth and well formed the exceptional surfaces can form a different configuration. It can be proved that another configuration exists only when r is even. Moreover, it consists of \frac{r}{2} exceptional surfaces; one is ruled over C and the rest are ruled over a double cover of C.

The interesting fact is that the Riemann-Roch formula remains the same. Can we use RR to construct smooth 3–folds with more complicated con-

figurations of surfaces [33]?

Question 7 What is the general orbifold Riemann–Roch formula? Is it possible to state and prove the formula without using the Atiyah–Singer equivariant RR?

Appendix

A.1 The Riemann–Hurwitz formula

Lemma A.1.1 Let f: S ⟶ T be a finite separable morphism of smooth threefolds. Then

K_{S} ∼ f^{∗}K_{T} + R,

where R is the ramification divisor on S.

Proof There exists an exact sequence

f^{∗}Ω_{T} ⟶ Ω_{S} ⟶ Ω_{S/T} ⟶ 0.  (E. xvii)

Since S and T are smooth, Ω_{S} and f^{∗}Ω_{T} are locally free sheaves of rank 3. Ω_{S/T} is a torsion sheaf that measures the difference between Ω_{S} and Ω_{T}. It can be locally defined as div(J), where

J = \frac{∂(s_{1}, s_{2}, s_{3})}{∂(t_{1}, t_{2}, t_{3})}

is determinant of the Jacobian matrix. Therefore the support of Ω_{S/T} equals to the ramification divisor R on S. In other words, considering R as a closed subscheme of S, its structure sheaf O_{R} is isomorphic to Ω_{S/T} .

We claim that the sequence (E. xvii) is short exact. Since the sheaves f^{∗}Ω_{T} and Ω_{S} are torsion free and have the same rank, we only need to prove that the map f^{∗}Ω_{T} ⟶ Ω_{S} is surjective at the generic point of S. But since k(S) is separable over k(T), the sheaf Ω_{S/T} is zero at the generic point of S by II, 8.6.A in [17].

Take the highest exterior powers in the exact sequence

0 ⟶ f^{∗}Ω_{T} ⟶ Ω_{S} ⟶ O_{R} ⟶ 0

and since f^{∗} commutes with ⋀, we get

ω_{S} ≅ f^{∗}ω_{T} ⊗ O_{R}.

Moreover, K_{S} ∼ f^{∗}K_{T} + R, which ends the proof.

A.2 Orbifolds

Definition A.2.1 A variety X of dimension n is an orbifold if every point has a neighborhood analytically equivalent to 0 ∈ U/G, where G ⊂GL(n, ℂ) is a finite subgroup with no complex reflections other than identity and U ⊂ ℂ^{n} is a G−stable neighborhood of the origin.

The condition that G has no complex reflections is equivalent to X being quasismooth.

Definition A.2.2 Let

X = ( \begin{matrix}U_{λ} \\ ↓ \\ U_{λ}/G_{λ} \end{matrix} ) and Y = ( \begin{matrix}V_{ξ} \\ ↓ \\ V_{ξ}/H_{ξ} \end{matrix} )

be orbifolds. An orbifold morphism is a morphism of varieties f: X → Y that (after refining U_{λ}/G_{λ} if necessary) lifts to

f_{#} : G_{λ} ⟶ H_{ξ} a homomorphism of groups

and

f_{λξ}: U_{λ} ⟶ V_{ξ}  a map of covers

so that for all g ∈ G_{λ} and u ∈ U_{λ}

f_{λξ}(gu) = f_{#}(g)(f_{λξ}(u)).

The main ”rule” in the theory of orbifolds is that over ℚ they behave like manifolds. The above definitions show that the theories which only depend on the local analytic nature of varieties can be developed on the smooth cover U, and then suitably adjusted under the finite map U → U/G. In all cases that we consider, G will be isomorphic to μ_{r} for some r.

For example, we can define intersection theory, Hodge theory etc.

There are two equivalent ways of defining ω_{X} on an orbifold X :

1. ω_{X} = j_{∗}(Ω_{X^{0}}^{n}), where j: X^{0} ↪ X is the inclusion of the smooth locus of X.

2. The sections on a local chart U/G are defined to be the G−invariant sections on U, which means Γ(U/G, ω_{X}) = Γ(U, ω_{U})^{G}. There is a natural way to define the sheaf by glueing together different charts.

Since orbifolds are Cohen-Macaulay, ω_{X} is the dualising sheaf of X.

We restrict ourselves to Gorenstein orbifolds which have at worst canonical singularities.

Remark A.2.3 A morphism of varieties f: X → Y, where X is smooth and Y an orbifold, does not automatically lift to a morphism of orbifolds. For example consider the inclusion

ℂ^{2} = Spec ℂ[x, y]
↓
ℂ = Spec ℂ[z] ⟶ ℂ^{2}/μ_{2} = Spec ℂ[u, v, w]/(uw = v^{2})

defined by

\begin{matrix}ℂ[u, v, w]/(uw = v^{2}) & ⟶ & ℂ[z] \\ u & ↦ & z \\ v & ↦ & 0 \\ w & ↦ & 0. \end{matrix}

The quotient by μ_{2} is given by u = x^{2}, v = xy, w = y^{2}. The diagram lifts to

Spec ℂ[z] ⊗_{ℂ[u,v,w]/(uw=v^{2})} ℂ[x, y] ≅ Spec ℂ[x, y]/(xy, y^{2})

which is not smooth.

A.3 The Hodge numbers and mirror pairs

Topological field theories for Calabi–Yau threefolds predict the following property of mirror manifolds: If two smooth Calabi–Yau threefolds V and V^{∗} form a mirror pair, then their Hodge numbers must satisfy the relations

h^{p,q}(V) = h^{3−p,q}(V^{∗}), for 0 ≤ p, q ≤ 3.

In particular, for the Euler numbers we get

e(V) = −e(V^{∗}).

Since V is a Calabi–Yau threefold,

h^{1,0} = h^{1}(V, O_{V}) = 0,
h^{2,0} = h^{2}(V, O_{V}) = 0,
h^{3,0} = h^{0,3} = h^{0}(V, Ω_{V}^{3} ) = h^{0}(V, O_{V}) = 1.

Thus we only need to consider h^{1,1}, h^{1,2} and the Euler number

e = 2(h^{1,1} − h^{1,2}).

It is well known that dim Pic_{ℚ} V = dim H^{2}(V, ℚ) = h^{1,1}(V).

By [10], the cohomology with compact support H_{c}^{k}(X, ℚ) of a quasismooth projective variety X carries a natural Hodge structure which coincides with the classical one. This implies that

• e(X) = ∑_{i∈I} e(X_{i}), if X is a finite disjoint union of locally closed subvarieties {X_{i}}_{i∈I};

• e(X) = e(Y) × e(F) for a locally trivial bundle f: X → Y with fibre F.

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