• Algebraic Geometry
• Workshop: Brill-Noether methods in the study of hyper-Kähler and Calabi-Yau manifolds

• Schedule
• Participants

We will describe recent results on totally geodesic submanifolds and Shimura subvarieties of A_g contained in the Torelli locus T_g. Using the second fundamental form of the Torelli map we give an upper bound on the dimension of totally geodesic submanifolds contained in T_g, which depends on the gonality of the curve. We will also describe some new examples of Shimura subvarieties in T_g obtained as non-abelian Galois coverings of P¹. These are results in collaboration with E. Colombo, A. Ghigi and M. Penegini.

I will discuss joint work with Verra concerning a complete birational classification of the moduli space of odd spin curves of genus g. In particular, for g<12, we find explicit unirational parametrizations of the moduli space, by constructing new models of the spin moduli space mirroring Mukai's well-known work on the structure of canonical curves of genus at most 9.

In this talk we will study the moduli space Z_g of smooth genus g curves admitting a singular model on a K3 surface. Using the Mori-Mukai approach of rank two, non-Abelian Brill-Noether loci we will work out the dimension of Z_g, and further we will work out the Brill-Noether theory of curves C in Z_g via Lazarsfeld-Mukai bundles. If time permits, we will give a Wahl-type obstruction for a smooth curve to have a singular model on a K3 surface.

Two of the four known types of compact hyperkahler manifolds arise from K3 surfaces and two from abelian surfaces. The moduli spaces of polarised varieties of the K3 types have been extensively studied, but less attention has been paid to the abelian types. I shall describe some work in progress, part of it due to my student Matthew Dawes, in which we examine the moduli of polarised varieties that are deformations of generalised Kummer varieties. They share many features with the moduli of deformations of Hilbert schemes on K3 surfaces, but there are also significant differences, particularly concerning the singularities.

Given a subvariety of a holomorphic symplectic manifold, the symplectic form on the ambient space induces a natural foliation on the subvariety. In this talk we will consider situations where this 'characteristic foliation' is integrable, and look at applications of these ideas. Many of the examples can be interpreted as Brill-Noether loci in Mukai moduli spaces of stable sheaves.

We study the geometry of some moduli spaces of stable ACM bundles on cubic fourfolds and show that they provide birational models of some hyperkaehler manifolds associated to the cubic. More precisely, we will show that the twisted K3 surface associated to a cubic 4-fold containig a plane and the hyperkaehler manifold Z constructed by Lehn-Lehn-Sorger-van Straten are birational to (a component) of two moduli spaces of Gieseker stable ACM bundles. This is a joint work, partly in progess, with M. Lahoz and E. Macri.

We show that for certain nodal quartic double solids X, there does not exist a family of curves for which the Abel-Jacobi map is surjective onto the intermediate Jacobian J(X), with rationally connected fibers. In particular, there is no universal codimension 2 cycle on J(X) x X.

In the talk the universal family U_g, over the moduli space A_g of complex p.p. abelian varieties of dimension g, is considered. The unirationality of U₅ is proven. The construction is used to study the perfect cone compactification of A₆ and bound its slope. Joint work with G. Farkas.

I will report on a joint work with François Charles, in which we study families of rational curves on certain irreducible holomorphic symplectic varieties. In particular, we prove that projective holomorphic symplectic fourfolds of -type contain uniruled divisors and rationally connected lagrangian surfaces.