TULSF VIII

(A meeting in Algebraic Geometry)

Ljubljana, Thursday, September 19, 2013

Abstracts

Janez Bernik: Semitransitivity and prehomogeneity

We shall recall some recent results on semitransitive actions of Lie algebras and cast them in terms of prehomogeneous vector spaces with a very particular poset of orbit closures. Joint work with M. Mastnak

Alberto Calabri: Plane Cremona transformations of fixed degree

A report on joint work with Cinzia Bisi and Massimiliano Mella.

Matej Filip: Rank 2 ACM bundles on complete intersection Calabi-Yau threefolds

We consider the problem of classifying all the rank 2 ACM bundles on smooth Calabi-Yau threefolds which are also complete intersections. We prove the existence

of such vector bundles in some cases. Previously were known only partial results by the works of Chiantinni, Madonna, Rao and Mohan Kumar (see [1], [2] and [3]). We obtain new geometric properties of the curves corresponding to rank 2 ACM bundles (by Serre correspondence). These follow from minimal free resolutions of curves in suitably chosen fourfolds (containing Calabi-Yau threefolds as hypersurfaces). We give the idea leading to the existence of some bundles on quintic threefold conjectured in [1] and [3].

References:

[1] Chiantini, L., Madonna, C.: ACM bundles on a general quintic threefold. Le Matematiche 55, vol. 2, 239-258 (2000).

[2] Madonna, C.: ACM vector bundles on prime Fano threefolds and complete intersection Calabi-Yau threefolds. Rev. Roumaine Math. Pures Appl. 47, vol.2, 211-222 (2002).

[3] Mohan Kumar, N.: Rao A.P.: ACM bundles, quintic threefolds and counting problems. Central European Journal of Mathematics 10(4), 1380-1392 (2012)

Alessandro Logar: Groebner bases of submodules of Zⁿ

We introduce the notion of Groebner bases of submodules of Zⁿ, we study some of their properties and in particular we see how to describe them via a collection of finite abelian groups. As an application, we see how this theory can be applied to pure binomial ideals of K[x₁ ,... , x_n].

Fabio Perroni: On the components of moduli spaces of curves with symmetry

We study the moduli spaceof smooth G-curves of genus g, where G is a finite group. In particular we are interested in the classification of the connected components of . To this aim we introduce a new homological invariant, the e-invariant, of G-curves (C; a), where a: G ® Aut(C) is the homomorphism given by the G-action. The e-invariant can be seen as an extension to more general group actions of the usual invariant in Z) associated to (C; a) when the action is free. Moreover, the e-invariant of (C; a) determines the numerical type (also called the branching data or the Nielsen class) of the branched cover C ® C/G. We prove that, when G = D_n, the dihedral group of order 2n, the e-invariant distinguishes the different connected components of , for any g. In general, for any group G, we prove that there exists an integer s(d) depending on the number d of branch points of C ® C/G, such that the e-invariant distinguishes the components of , if the genus of C/G is greater than s(d). In particular, this implies that the number of connected components of stabilises for big enough g. By the Nielsen realisation problem, this also gives a complete classification of conjugacy classes of finite subgroups of the mapping class group Map_g, when g ® ∞. This is a joint work with F. Catanese and M. Lönne.

Amos Turchet: On algebraic hyperbolicity for complements of degree 4 divisors in

The function field version of Vojta-Lang Conjecture in asks for algebraic degeneracy for complements of plane curves of degree at least four. When the curve has at least four components this follows from Stothers-Mason abc Theorem. After briefly recalling recent results for the three components case obtained by Corvaja and Zannier, we will show how to extend these in two directions. Firstly we are going to describe the case in which the divisor itself is defined over the function field which leads us to prove algebraic degeneracy for sections of fibered threefolds whose fibers are complements of degree four and three components curves in . Secondly we will focus on a completely new idea to solve the conjecture for complements of two components (and very generic) curves using Log-Gromov-Witten Invariants.