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## Program

Thursday, 15 May 2014:

14:00–15:00 Geom H6: Oliver Fabert (Universität Hamburg)
Coffee break
15:30–16:30 Geom H6: Fabio Tonini (HU Berlin)
Coffee break
17:00–18:00 Geom H6: Klaus Hulek (Leibniz Universität Hannover)

18:30 Conference dinner: Terzetto

Friday, 16 May 2014:

10:00–11:00 Geom H4: Lei Zhang (FU Berlin)
Coffee break
11:30–12:30 Geom H4: Jaap Top (Groningen)
Lunch
14:00–15:00 Geom H4: Franziska Schroeter (Universität Hamburg)
15:15–16:15 Geom H4: Eduardo Esteves (HU Berlin)

Abstracts:

• Limit linear series: Degenerating points.
Eduardo Esteves (HU Berlin)
Abstract:
The theory of limit linear series was devised by Eisenbud and Harris in the early 80’s to systematize certain arguments using degenerations of linear series to singular curves. In view of its many applications, it was unfortunate that the theory worked only for curves of compact type, and thus Eisenbud and Harris posed the problem of extending it to any stable curve. In the past 30 years, there have been certain breakthroughs in the work on this problem, especially Caporaso's construction of a compactification of the relative Jacobian over the moduli of stable curves, and Osserman's construction of a different variety of limit linear series, encoding all the data in a degeneration, for the case of two-component curves of compact type. However, all these approaches failed to produce a solution to the original problem. In this talk, I will review the theory and present a new approach to the problem, one that takes into account the works by Caporaso and Osserman, but introduces a new idea, that of degeneration of points.
• Classical mirror symmetry for open Calabi-Yau manifolds
Oliver Fabert (Universität Hamburg)
Abstract:
The classical mirror symmetry conjecture for closed Calabi-Yau manifolds X and Y relates the rational Gromov-Witten theory of X with the (extended) deformation theory of complex structures on Y (and vice versa). In my talk I will illustrate how this correspondence is supposed to generalize from closed to open Calabi-Yau manifolds.
• Extending the Prym map to toroidal compactifications of A_g
Klaus Hulek (Leibniz Universität Hannover)
Abstract:
In this talk we discuss the question whether the Prym map $R_{g+1} \to A_g$ extends as a regular map $\overline{R}_{g+1} \to A_g^{\operatorname{tor}}$ from the space of admissible double covers to toroidal compactifictations of $A_g$. It was already shown by Friedman and Smith that this is not the case for "meaningful" toroidal compactifications. By work of Alexeev, Birkenhake and Hulek as well as Vologodsky it is known that the indeterminacy locus of the Prym map to the second Voronoi compactification is the closure of the so called Friedman-Smith loci. Motivated by work of Alexeev and Brunyate on the Torelli map we investigate the Prym map to other torioidal compactifications of $A_g$, in particular the perfect cone compactifiation. We develop a systematic approach which separates the geometric aspects from combinatorial issues and reduces the problem to the computation of certain monodromy cones. This is joint work with Sebastian Casaleina-Martin, Sam Grushevsky and Radu Laza, with a contribution from Mathieu Dutour.
• Refined invariants in tropical geometry
Franziska Schroeter (Universität Hamburg)
Abstract:
Mikhalkin changed the way of considering enumerative problems in algebraic geometry when he considered tropical curves counted with a numerical multiplicity and proved that the invariance of classical enumerative numbers can be proven tropically. Block and Göttsche recently introduced polynomial multiplicities (in the variable y) for plane tropical curves which yield an invariant number when counting curves passing through a generic point configuration. In addition, they reveal deep relations to classical enumerative problems: specializing y=1 we obtain the corresponding Gromov Witten invariant and for y=-1 we retrieve the corresponding Welschinger invariant. After a recap of basic tropical notions, I will present a similar approach for broccoli invariants which have been introduced to prove the invariance of tropical Welschinger numbers for certain real curves.
• Stacks of ramified Galois covers
Fabio Tonini (HU Berlin)
Abstract:
We introduce the notion of Galois cover for a finite group G and discuss the problems of constructing them and of the geometry of the stack G-Cov they form. When G is abelian, we describe certain families of G-covers in terms of combinatorial data associated with G. In the general case, we present a correspondence between G-covers and particular monoidal functors and study the problem of Galois covers of normal varieties whose total space is normal.
• The sections of a certain elliptic K3 surface
Jaap Top (Groningen)
Abstract:
In his PhD thesis finished in 2011, Bas Heijne describes the elliptic surfaces that are obtained as quotients of Fermat surfaces by a specific kind of abelian subgroup of automorphisms. Tetsuji Shioda coined the name "Delsarte surfaces" for such quotients. Heijne computed the ranks of the group of sections of the elliptic Delsarte surfaces, and in particular showed that up to possibly finite index, all sections can be explained in terms of base changes from either rational or K3 elliptic surfaces. Moreover, he found the right number of independent sections for all rational or K3 elliptic surfaces occurring here, with one exception: the surface corresponding to y^2=x^3+t^7x+1. We will explain how to tackle this remaining case, and how this relates to a recent preprint by Masamichi Kuroda. This is joint work with my master's student Luuk Disselhorst.
• Galois theory for schemes
Lei Zhang (FU Berlin)
Abstract:
In this talk we will firstly briefly recall Grothendieck's etale fundamental group constructed in SGA1, then we introduce Nori's generalization of the etale fundamental group for proper varieties using the language of essentially finite vector bundles. On the way we will also introduce some recent developments on this notion. Next, we introduce Nori's second definition on the fundamental group scheme. But we will not completely follow Nori's second idea, instead, we will slightly generalize it and discuss some kind of Galois theory coming out of it.

 Impress 2014-05-13, wwwmath (MJ)