North German Algebraic Geometry Seminar

Schedule

All talks are in the Schrödinger Zentrum, Rudower Chaussee 26, Raum 0.311

Talk Titles and Abstracts

Lucia Caporaso, Brill-Noether theory of stable curves
Abstract: For a Riemann surface, the classical Brill-Noether theory concerns certain remarkable subschemes of the Picard scheme, parametrizing line bundles with unexpected global sections.

We shall present some work in progress extending Brill-Noether theory to stable curves, equipped with their compactified Picard scheme. As an application, classical results about smooth curves admit a new, simple, proof.

Hélène Esnault, Some aspects of the motivic and etale fundamental groups
Abstract: We'll report on recent work with Marc Levine on the motivic fundamental group and, if time permits, on work in progress with Olivier Wittenberg on the étale fundamental group.

Ulrich Görtz, Siegel modular varieties with Iwahori level structure
In this talk, I will discuss the geometry of moduli spaces of abelian varieties in positive characteristic p. The main focus will be on such spaces with Iwahori (in other words, G₀(p)-) level structure at p.

In the case of elliptic curves, we obtain the modular curves Y₀(p), which have been studied for a long time. They have been put to good use for instance in Deligne's proof that the Weil conjectures imply the Ramanujan conjecture, about 40 years ago. For abelian varieties of higher dimension, the situation is much more complicated. I will present some structural results about the supersingular locus, which is of particular interest, for instance from the point of view of the Langlands program. These results were obtained in joint work with Chia-Fu Yu.

Remke Kloosterman, The Mordell-Weil group of elliptic threefolds
Abstract: In this talk we discuss a method (obtained together with K. Hulek) to compute the rank of E(C(s,t)) for a class of elliptic curves E defined over C(s,t). This method relies on an explicit method to compute H ⁴(Y,C) for a class of singular threefolds Y.

We use this method to classify all elliptic 3-folds with constant j-invariant, and discriminant curve of degree at most 12.

Damiano Testa, Cox rings of del Pezzo surfaces
Cox rings have been introduced by D. Cox for toric varieties as an analogue of the homogeneous coordinate ring of projective space, and have later been generalized to broader classes of varieties. Moreover, Cox rings of del Pezzo surfaces have been used to establish the validity of the Hasse principle or weak approximation for certain del Pezzo surfaces over number fields.

For del Pezzo surfaces, a conjecture of Batyrev and Popov gives a presentation of the Cox rings in terms of generators satisfying only quadratic relations. For the case of del Pezzo surfaces of degree at least two, various proofs have been given for this conjecture by different methods. I will talk about joint work with A. Várilly-Alvarado and M. Velasco where we prove the conjecture for all del Pezzo surfaces. The proof relies on a tight link between a multi-graded resolution of the Cox ring and a combinatorial game on the set of exceptional curves.

Yuri Tschinkel, A Torelli theorem for curves over finite fields
Abstract: I will explain some constructions with hyperbolic curves and their Jacobians over finite fields.

Alessandro Verra,On the finiteness of the theta map for SU(r,0)
Abstract: Let SU(r,0) be the moduli spaces of semistable vector bundle with trivial determinant and even rank r on a curve C of genus g. A geometric description of the theta map t, i.e. the map associated to the ample generator of Pic(SU(r,0)), is provided: in terms of a suitable linear system of cubic hypersurfaces through a projective embedding of C of degree r(g+2). As an application the finiteness of t onto its image follows for g >> r.

Eckart Viehweg, Curves in moduli spaces
Abstract: We will present some results concerning curves (and partly higher dimensional subvarieties) in moduli spaces of canonically polarized complex manifolds, of complex abelian varieties, or more generally of complex minimal models of Kodaira dimension zero.