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## Program

Thursday, 19 January 2017:

14:30–15:30 Geom H2: Marta Pieropan (FU Berlin)
Coffee break
16:00–17:00 Geom H6: Radu Laza (Stony Brook University)
Coffee break
17:30–18:30 Geom H6: Quentin Gendron (Leibniz Universität Hannover)

19:00 Conference dinner: Rucola e Parma

Friday, 20 January 2017:

09:30–10:30 Geom H6: Carsten Liese (Leibniz Universität Hannover)
Coffee break
11:00–12:00 Geom H6: Charles Vial (University of Cambridge)
Lunch
13:30–14:30 Geom H1: Ángel Muñoz (FU Berlin)
Coffee break
15:00–16:00 Geom H1: Jérémy Guéré (HU Berlin)

Abstracts:

• On rationally connected varieties over large C_1 fields of characteristic 0
Marta Pieropan (FU Berlin)
Abstract:
In the 1950s Lang studied the properties of C_1 fields, that is, fields over which every hypersurface of degree at most n in an n-dimensional projective space has a rational point. Later he conjectured that every smooth proper rationally connected variety over a C_1 field has a rational point. The conjecture is proven for finite fields (Esnault) and function fields of curves over algebraically closed fields (Graber-Harris-de Jong-Starr). I use birational geometry to address the open case of Henselian fields of mixed characteristic with algebraically closed residue field.
• GIT vs Baily-Borel compactification for the moduli space of Quartic Surfaces
Radu Laza (Stony Brook University)
Abstract:
This is a report on joint work with Kieran O'Grady. The period map from the GIT moduli space of quartic surfaces to the Baily-Borel compactification of the period space is birational but far from regular. New birational models of locally symmetric varieties of Type IV have been introduced by Looijenga, in order to study similar problems. Looijenga's construction does not succeed in "explaining" the period map for quartic surfaces. We discovered that one can (conjecturally) reconcile Looijenga's philosophy with the phenomenology of quartic surfaces, provided one takes into account suitable Borcherds relations between divisor classes on relevant locally symmetric varieties. We work with a tower of locally symmetric varieties, in particular our results should also "explain" the period map for double EPW sextics.
• On differentials with given local invariants
Quentin Gendron (Leibniz Universität Hannover)
Abstract:
A meromorphic differential on a Riemann surface of genus g has three types of local invariants: the order of a zero, the order of a pole and the residue at a pole. It is well known that these invariants satisfy three conditions: the sum of the orders is 2g-2, the sum of the residues is zero and the poles of order -1 have a non zero residue. In this talk, I want to discuss the sufficiency of these conditions. This is part of a joint work with Guillaume Tahar.
• The KSBA compactification of the moduli space of degree 2 K3 pairs: a toroidal interpretation
Carsten Liese (Leibniz Universität Hannover)
Abstract:
Work of Gross,Hacking,Keel and Siebert shows that the Gross-Siebert reconstruction algorithm provides a partial toroidal compactification of the moduli space of polarized K3 surfaces for any genus. The construction comes with a family $\mathfrak{X}\to \bar{\mathbb{P}}^g$ over a partial toroidal compactification $\bar{\mathbb{P}}^g$ of a subset $\mathbb{P}^g$ of the Kollár-Shepherd-Barron moduli space of stable K3 pairs $M_{SP}$. A conjecture of Keel says that $\mathfrak{X}\to \bar{\mathbb{P}}^g$ extends to a toroidal compactification of ${\mathbb{P}^g}$ and in particular, all surfaces in the boundary of $M_{SP} appear as fibres of$\mathfrak{X}\to \bar{\mathbb{P}}^g$. In the genus$2$case,$M_{SP}$is known by work of Laza. We check the prediction of Keel's conjecture and show that all degenerate type III$K3$surfaces in the boundary of$M_{SP}$appear as fibres of$\mathfrak{X}\to \mathbb{P}^g\$.
• On the Chow ring of holomorphic symplectic varieties
Charles Vial (University of Cambridge)
Abstract:
Since the seminal work of Beauville and Voisin on the Chow ring of K3 surfaces, it is believed that the Chow ring of smooth projective holomorphic symplectic manifolds should have properties similar to that of the Chow ring of abelian varieties. I will explain which properties are expected, and I will give unconditional results in the case of Hilbert schemes of points on K3 surfaces or abelian surfaces, and in the case of generalised Kummer varieties. Joint work with Lie Fu and Zhiyu Tian.
• Principal G-bundles on stable curves
Ángel Muñoz (FU Berlin)
Abstract:
Alexander Schmitt has found a compactification for the moduli space of principal G-bundles on a irreducible curve with one node. In this talk we will discuss a generalization of this compactification for an arbitrary stable curve and its behaviour under degenerations.
• Enumerative theory of complex curves from singularities
Jérémy Guéré (HU Berlin)
Abstract:
In 2007, Fan, Jarvis, and Ruan developed a new kind of intersection theory on a moduli space of curves associated to a quasi-homogeneous polynomial singularity. Based on insights from Witten, it is viewed as a GIT variation of Gromov-Witten theory of hypersurfaces in weighted projective spaces. It is called the quantum singularity theory (or FJRW theory). More precisely, the moduli problem deals with a generalization of spin curves and the intersection theory on it is defined via an algebraic cycle embodying the geometry of the moduli problem. The approach I will present is due to Polishchuk and Vaintrob and relies on the concept of matrix factorizations. I will show how the same ideas lead us to a K-theoretic version as well, and how classical results on Koszul cohomology yield explicit computations.

 Impress 2017-01-18, wwwmath