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Events
A day on K3 surfaces

MPI für Mathematik in den Naturwissenschaften Leipzig

E1 05 (Leibniz-Saal)

conference

06.05.22

A day on K3 surfaces

Confirmed speakers include

Ignacio Barros (University of Paderborn)
Mario Kummer (TU Dresden)
Emanuele Macrì (Université Paris-Saclay)
Marta Panizzut (TU Berlin)

Program

10:30 - 11:00	Welcome Coffee
11:00 - 12:00	Emanuele Macrì (Université Paris-Saclay) Lagrangian fibrations on hyper-Kähler fourfolds I will present joint work with Olivier Debarre, Daniel Huybrechts and Claire Voisin on the SYZ hyper-Kähler conjecture for fourfolds under certain topological assumptions. As application, this proves a conjecture by O'Grady that a hyper-Kähler fourfold whose cohomology ring is isomorphic to the one of the Hilbert square of a K3 surface is a deformation of a Hilbert square.
12:00 - 13:00	Lunch Break
13:00 - 14:00	Ignacio Barros (Universität Paderborn) Kodaira classification of the moduli of hyperelliptic curves We study the birational geometry of the moduli spaces of hyperelliptic curves with marked points. We show that these moduli spaces have non Q-factorial singularities forcing us to work with a better birational model provided by Hurwitz spaces. We finish the Kodaira classification by proving that these spaces are of intermediate type when the number of markings is 4g + 6 and of general type when the number of markings is n ≥ 4g+7. Similarly, we consider the natural finite cover given by ordering the Weierstrass points. In this case, we show that the Kodaira dimension is one when n = 4 and of general type when n ≥ 5. For this, we carry out a singularity analysis of ordered and unordered pointed Hurwitz spaces. We show that the ordered space has canonical singularities and the unordered space has non-canonical singularities. We describe all non-canonical points and show that pluricanonical forms defined on the full regular locus extend to any resolution. Further, we provide a full classification of the structure of the pseudo-effective cone of Cartier divisors for the moduli space of hyperelliptic curves with marked points. We show the cone is non-polyhedral when the number of markings is at least two and polyhedral in the remaining cases. This is joint work with Scott Mullane.
14:00 - 14:15	Coffee Break
14:15 - 15:15	Marta Panizzut (TU Berlin) K3 polytopes and their quartic surfaces Tropical geometry is a recent area of mathematics at the interface between algebraic and polyhedral geometry. Tropical varieties, the tropical counterpart of algebraic varieties, are polyhedral complexes satisfying certain combinatorial properties. The closure of the connected components of the complement of a tropical hypersurface are called regions. They have the structure of convex polyhedra. A $3$-dimensional polytope is a K3 polytope if it is the closure of the bounded region of a smooth tropical quartic surface. In this talk we begin by studying properties of K3 polytopes. In particular we exploit their duality to regular unimodular central triangulations of reflexive polytopes in the fourth dilation of the standard tetrahedron. Then we focus on quartic surfaces that tropicalize to K3 polytopes, and we look at them through the lenses of Geometric Invariant Theory. These are first steps on the computational study of tropical quartic surfaces. In the presentation we will highlight computational aspects and further questions related to the project. The talk is based on joint work with Gabriele Balletti and Bernd Sturmfels.
15:15 - 15:45	Coffee Break
15:45 - 16:30	Mario Kummer (TU Dresden) A signed count of torsion points on real abelian varieties While the number of m-torsion points on an abelian variety of dimension g over the complex numbers is always equal to m^2g, the number of real m-torsion points varies between m^g and m^2g when m is even. I will assign a sign ±1 to each real m-torsion point on a real principally polarized abelian variety such that the sum over all signs is always m^g. I will give an interpretation of this count for Jacobians of curves and for Kummer surfaces. If time permits, I will speculate about generalizations to arbitrary ground fields.

Participants

Daniele Agostini

Max Planck Institute for Mathematics in the Sciences

Ignacio Barros

Universität Paderborn

Alessio Bottini

Université Paris-Saclay

Henry Dakin

TU Chemnitz

Elena Derunova

Max Planck Institute of Microstructure Physics

Mario Kummer

TU Dresden

Christian Lehn

Technische Universität Chemnitz

Emanuele Macrì

Université Paris-Saclay

Nima Manjila

TU Chemnitz

Emeryck Marie

Technische Universität Chemnitz

Leonid Monin

MPI MiS

Chamir Ngandjia

TU Chemnitz

Marta Panizzut

TU Berlin

Angel Rios

Max Planck Institute for Mathematics in the Sciences

Pierpaola Santarsiero

Max Planck Institute for Mathematics in the Sciences

Javier Sendra Arranz

Max Planck Institute for Mathematics in the Sciences

Christian Sevenheck

TU Chemnitz

Rainer Sinn

Universität Leipzig

Peter Spacek

TU Chemnitz

Bernd Sturmfels

Max Planck Institute for Mathematics in the Sciences

Scientific Organizers

Daniele Agostini

Max Planck Institute for Mathematics in the Sciences

Christian Lehn

Technische Universität Chemnitz

Rainer Sinn

Universität Leipzig

Administrative Contact

Saskia Gutzschebauch

Max Planck Institute for Mathematics in the Sciences Contact via Mail