TULS IV

(A meeting in Algebraic Geometry)

Ljubljana, Saturday, May 13, 2006

Abstracts

Michela Brundu: Geometrical aspects of surfaces ruled by conics

Rational projective surfaces ruled by conics arise in a natural way in studying four-gonal curves. In this talk we give an overview of the subject and classify all the possible surfaces ruled by conics and their singularities. We give also a hint of a procedure which allows us to associate to any surface ruled by conic a ''canonical'' birational model geometrically ruled.

Anita Buckley and Tomaž Košir: Self-adjoint determinantal representations of cubic surfaces

We show that a real nonsingular cubic surface can have 0, 2, 4, 6 or 24 nonequivalent self-adjoint determinantal representations. The number depends on the Segre type F_j, j=1,...,5. Only a surface of type F_5 has defnite determinantal representations.

Etienne Mann: Orbifold cohomology of weighted projective spaces

In 1991, Dubrovin defined the Frobenius structure on a complex manifold. These Frobenius manifolds can be constructed in different domain of mathematics : in quantum cohomology (A side) and on singularity theory (B side). Mirror symmetry can be formulated in terms of an isomorphism between Frobenius manifolds constructed from the A side and those constructed from the B side. The first example of such a correspondence was done by Barannikov (2000) who proved that the complex projective space of dimension n and the Laurent polynomial x_0+ ...+x+n+ (x_0 ... x_n) are mirror partners. We can generalize this correspondence to weighted projective spaces and some Laurent polynomial. For the A side, we use the construstions of W.Chen and Y.Ruan on orbifolds and for the B side, we use the results of Antoine Douai and Claude Sabbah.

Emilia Mezzetti: Congruences of lines in the projective space and  systems of conservation laws

S.Agafonov and E.Ferapontov have introduced a construction that allows one to associate naturally to every system of partial differential equations of conservation laws a congruence of lines in an appropriate projective space. In particular, to hyperbolic systems of Temple type, there correspond congruences of lines that form planar pencils of lines. The language of Algebraic Geometry turns out to be very natural in the study of these systems. In the talk, after recalling the definition and the basic facts on congruences of lines, I will illustrate the Agafonov-Ferapontov construction and some recent results related to Temple systems.

Miles Reid: Orbifold Riemann-Roch and Plurigenera

Orbifold Riemann--Roch concerns the contributions made by cyclic quotient singularities to the dimension of cohomology groups. Simple cases include many "cyclic" phenomena of elementary mathematics, such as the number of integer points in a lattice simplex. The colloquium lecture will present an elementary but powerful formula for the Hilbert series of an orbifold with isolated cyclic quotient singularities.

Jean-Pierre Rosay: Extension of holomorphic bundles and Serre's problem on Stein bundles

We give an example of a holomorphic fiber bundle over the unit disc, with fiber C^2 and with polynomial transition automorphisms of the fiber, whose total space fails to be Stein. Earlier examples of this type, but with a non-simply connected base or with non-polynomial fiber transition maps, were given by H. Skoda and J.-P. Demailly.