SCHEDULE AND ABSTRACTS

Ignacio Barros: Finite generation of Noether–Lefschetz divisors and the slope of the moduli space of cubic fourfolds

I will discus aspects of the divisor geometry of moduli spaces of cubic fourfolds with simple singularities and quasi-polarized K3 surfaces of degree 2d. For the moduli space of cubic fourfolds, we introduce a slope quantity to characterize the effective cone and prove an explicit bound for it. For the K3 moduli spaces, we give an explicit finite presentation of the rational Picard group by showing that it is generated by Noether–Lefschetz divisors of discriminant less than or equal to 4d. As a byproduct, we obtain two explicit expressions for the Hodge class in terms of Noether–Lefschetz divisors, and we indicate analogous results for higher-codimension Noether–Lefschetz cycles. This is all joint work with Shi He and Paul Kiefer.

Jose Ignacio Burgos: Chern-Weil theory and Hilbert-Samuel theorem for semi-positive singular toroidal metrics on line bundles

In this talk I will report on joint work with A. Botero, D. Holmes and R. de Jong. Using the theory of b-divisors and non-pluripolar products we show that Chen-Weil theory and a Hilbert Samuel theorem can be extended to a wide class of singular semi-positive metrics. We apply the techniques relating semipositive metrics on line bundles with b-divisors to study the line bundle of Siegel-Jacobi forms with the Peterson metric. On the one hand we prove that the ring of Siegel-Jacobi forms of constant positive ratio between index and weight is never finitely generated, and we recover a formula of Tai giving the asymptotic growth of the dimension of the spaces of Siegel-Jacobi modular forms.

Sebastian Casalaina-Martin: Hodge theory and the geometry of moduli spaces

Hodge theory and moduli spaces have a long a fruitful relationship. I will discuss some of this in the context of some recent results. This will include some joint work with Samuel Grushevsky, Klaus Hulek, and Radu Laza on moduli spaces of cubic surfaces, as well as some joint work with Shend Zhjeqi on moduli spaces of varieties of general type.

Francesc Fité: Elliptic curves attached to abelian threefolds with imaginary multiplication

Let A be an abelian threefold defined over the rational numbers. Via Tate twisting, the Weil pairing exhibits its first étale cohomology group H^1(A) as a subrepresentation of its third étale cohomology group H^3(A). If the geometric endomorphism ring of A consists solely of the integer multiplications, then Serre's open image theorem ensures that no further piece of H^3 comes from the H^1 of an abelian variety. Suppose that the geometric endomorphism algebra of A is a quadratic imaginary field M. I will explain a result that attaches to A an elliptic curve E defined over the rationals with potential complex multiplication by M such that H^1(E) occurs as a subrepresentation of H^3(A)(1). I will also report on ongoing work determining the isogeny class of E when A runs over certain families of abelian varieties with imaginary multiplication by the square root of -1, -2, and -3, respectively. This combines joint works with S. Chidambaram, P. Goodman, and F. Pedret.

Olivier de Gaay Fortman: Curve classes on very general abelian sixfolds

We prove that on a very general principally polarized abelian sixfold, the smallest multiple of the minimal curve class which can be represented by an integral linear combination of curves is exactly six. This is joint work with Philip Engel and Stefan Schreieder.

Gerard van der Geer: The cycle class of the supersingular locus

Deuring gave a now classical formula for the number of supersingular elliptic curves in characteristic p. We generalize this to a formula for the cycle class of the supersingular locus in the moduli space of principally polarized abelian varieties of given dimension g in characteristic p. The formula determines the class up to a multiple and shows that it lies in the tautological ring. We also give the multiple for g up to 4. This is joint work with S. Harashita.

Sam Grushevsky: Moduli of abelian varieties near diagonal period matrices

By investigating the lowest order terms in Taylor expansions of theta functions near the locus of diagonal period matrices in the Siegel upper half-space, we obtain local characterizations of various geometrically defined loci (decomposable abelian varieties, hyperelliptic Jacobians, Jacobians, ...). Based on joint work with Riccardo Salvati Manni.

Aitor Iribar López: Intersection theory on degenerating abelian schemes

The Fourier transforms and the weight decomposition are fundamental objects to understand intersection theory on a family of abelian varieties. A remarkable consequence of this machinery is the expression of the zero section as the self-intersection of the symmetric theta divisor. Abelian varieties can degenerate into singular objects, as it was studied by Nakamura, Mumford, Alexeev and others.

The expression of the zero section in terms of simpler "tautological" classes is still a fundamental question on the intersection theory of degenerating abelian schemes. I will explain how this problem can be tackled on the simplest degenerations, those of torus rank 1. Along the way, we construct a Fourier transform and a weight decomposition in the Chow ring of these degenerating objects, verifying conjectures of Corti-Hanamura on relative Chow motives.

This is joint work (in progress) with Younghan Bae, Jeremy Feusi and Sam Molcho.

Thomas Krämer: Cubic threefolds and geometric local systems with monodromy E6

For families of subvarieties inside a fixed abelian variety, the monodromy of their abelian covers is controlled by the Tannaka group attached to sheaf convolution on a single fiber of the family. Does such a relation exist more generally when the ambient abelian variety is not fixed? In the talk I will provide evidence by discussing a recent construction of local systems with monodromy E6, related to Fano surfaces of smooth cubic threefolds (work in progress with Daniel Litt and Marco Maculan).

Eyal Markman: Cycles on abelian 2n-folds of Weil type from secant sheaves on abelian n-folds

In 1977 Weil identified a 2-dimensional space of rational classes of Hodge type (n,n) in the middle cohomology of every 2n-dimensional abelian variety with a suitable complex multiplication by an imaginary quadratic number field. These abelian varieties are said to be of Weil type and these Hodge classes are known as Weil classes.

The connected components of the moduli space of polarized abelian varieties A of Weil type have three discrete invariants, dim(A), the imaginary quadratic number field K, and the discriminant. The latter is the coset in ℚ^*/Nm(K^*) of the determinant of a natural Hermitian form.

We prove that the Weil classes are algebraic for all abelian sixfold of Weil type of discriminant -1, for all imaginary quadratic number fields. The algebraicity of the Weil classes follows for all abelian fourfolds of Weil type (for all discriminants and all imaginary quadratic number fields). The Hodge conjecture for abelian fourfolds is known to follow from the above result.

Ben Moonen: Bézout's theorem for abelian varieties

I'll report on joint work with Olivier Debarre. The main result is that if A is an absolutely simple abelian variety over some field and X1,..., Xr are subvarieties of A, then the dimension of their sum X1 + ... + Xr equals the minimum of dim(A) and Σ dim(Xi). In characteristic 0, there is a simple geometric proof for this, but that argument breaks down in characteristic p. Instead, we prove this result as a consequence of a theorem on perverse sheaves, building upon work of Krämer and Weissauer.

Angela Ortega: A family of simple Jacobians with many automorphisms and applications to the Prym map

We consider a (2g-1)-dimensional family of Jacobians of dimension (d-1)(g-1)/2 arising as quotient of curves of unramified cyclic coverings of prime degree d > 2 over hyperelliptic curves of genus g. We prove that the generic element in this family is simple by means of a deformation argument and describe completely its endomorphism algebra.

This result is used to show that the Prym map corresponding to the cyclic covering of prime degree d is generically injective under some mild numerical restrictions on d and g. This is a joint work with J.C. Naranjo, G.P. Pirola and I. Spelta.

Finally, I will mention recent results on the injectivity of the Prym map for the non-prime cyclic coverings over hyperelliptic curves obtained in a joint work with P. Borówka, J.C. Naranjo and A. Shatsila.

Irene Spelta: Families of G-curves and maximal monodromy

Maximality results for the monodromy of families of algebraic varieties are widely studied in algebraic geometry. In this talk, we study the monodromy of families of Galois coverings of curves. The natural action of the Galois group G on cohomology induces a decomposition of the associated variation of Hodge structures, and under a certain assumption on this decomposition, we prove that the algebraic monodromy group is maximal. As an application, we analyse positive-dimensional totally geodesic subvarieties of Ag arising from such families.

Alessandro Verra: On nodal cubic threefolds and the degree of a modular map

A beautiful description of complex, nodal cubic threefolds goes back to Corrado Segre. Among these, a unique one, up to projective equivalence, has the maximal number of ten nodes and it is named the Segre primal. This seems ubiquitous in Algebraic Geometry. In this talk we describe the solution of the following enumerative problem, where the Segre primal once more appears. Let V be a smooth complex cubic threefold and x a general point of it, then the six lines of V through x define a quadric cone and six points of the projective line. This defines a rational map f: V ----> M, where M is the moduli space of genus two curves. What is the degree of f? Joint work with Ciro Ciliberto.