HU Arithmetic Geometry Seminar

Humboldt Arithmetic Geometry Seminar

Humboldt University Berlin, Winter 2025/26

Time: Tuesdays 13:15 - 14:45

Venue: Room 3.007, Rudower Chaussee 25, 12489 Berlin

— See also the Algebraic Geometry Seminars at HU and FU —


04.11.2025	Giacomo Mezzedimi (Bonn)
Title:	On automorphism groups of K3 and Enriques surfaces
Abstract:	According to the Tits alternative, the automorphism group of a smooth projective surface is either virtually abelian (i.e. it contains an abelian subgroup of finite index) or it contains a free, non-abelian subgroup. By classification of surfaces, the latter can only occur on rational, K3, Enriques or abelian surfaces. I will present a classification of K3 and Enriques surfaces with virtually abelian automorphism groups, focusing on the dynamical properties of their automorphisms. Furthermore, I will present some geometric examples and talk about possible applications.
25.11.2025	Franco Giovenzana (Orsay)
Title:	Projective models for some generalised Kummer fourfolds
Abstract:	Generalised Kummer varieties, introduced by Beauville, are a fundamental example of hyperkähler manifolds. Nevertheless, very little is known about their projective models. In this seminar, I will present a concrete construction of projective models for four-dimensional Kummer varieties associated with Jacobian abelian surfaces, highlighting the role of the Coble cubic. I will show how this cubic makes it possible to construct a duality between two birational projective models of the Kummer variety, extending a duality result conjectured by Dolgachev and proved by Ortega (2005) and Nguyen (2007) concerning moduli spaces of sheaves on genus 2 curves. Time permitting, I will discuss some partial results in progress for generalised Kummer fourfolds associated with abelian surfaces polarised of type (1,4). This is joint work with D. Agostini, P. Beri, and Á.D. Ríos Ortiz.
09.12.2025	Haitao Zou (Bielefeld)
Title:	Pointed Shafarevich conjecture of primitive symplectic varieties
Abstract:	The Shafarevich conjecture predicts that families of varieties over number fields with “good reduction” outside finitely many places are severely restricted—in classical cases like curves or polarized abelian varieties, only finitely many isomorphism classes occur. In previous work with Lie Fu, Zhiyuan Li and Teppei Takamatsu, we proved an unpolarized version of this conjecture for hyper-Kähler varieties of a fixed deformation type: that is, even without bounding the polarization, only finitely many such varieties exist over a given number field with good reduction outside a fixed finite set. Based on that, I will introduce a pointed Shafarevich conjecture for primitive symplectic varieties (possibly singular), and sketch our new results showing that fixing a basepoint in a family forces finiteness of the (generic) fibers under mild hypotheses.

04.11.2025

Giacomo Mezzedimi (Bonn)

Title:

On automorphism groups of K3 and Enriques surfaces

Abstract:

According to the Tits alternative, the automorphism group of a smooth projective surface is either virtually abelian (i.e. it contains an abelian subgroup of finite index) or it contains a free, non-abelian subgroup. By classification of surfaces, the latter can only occur on rational, K3, Enriques or abelian surfaces. I will present a classification of K3 and Enriques surfaces with virtually abelian automorphism groups, focusing on the dynamical properties of their automorphisms. Furthermore, I will present some geometric examples and talk about possible applications.

25.11.2025

Franco Giovenzana (Orsay)

Title:

Projective models for some generalised Kummer fourfolds

Abstract:

Generalised Kummer varieties, introduced by Beauville, are a fundamental example of hyperkähler manifolds. Nevertheless, very little is known about their projective models. In this seminar, I will present a concrete construction of projective models for four-dimensional Kummer varieties associated with Jacobian abelian surfaces, highlighting the role of the Coble cubic. I will show how this cubic makes it possible to construct a duality between two birational projective models of the Kummer variety, extending a duality result conjectured by Dolgachev and proved by Ortega (2005) and Nguyen (2007) concerning moduli spaces of sheaves on genus 2 curves. Time permitting, I will discuss some partial results in progress for generalised Kummer fourfolds associated with abelian surfaces polarised of type (1,4). This is joint work with D. Agostini, P. Beri, and Á.D. Ríos Ortiz.

09.12.2025

Haitao Zou (Bielefeld)

Title:

Pointed Shafarevich conjecture of primitive symplectic varieties

Abstract:

The Shafarevich conjecture predicts that families of varieties over number fields with “good reduction” outside finitely many places are severely restricted—in classical cases like curves or polarized abelian varieties, only finitely many isomorphism classes occur. In previous work with Lie Fu, Zhiyuan Li and Teppei Takamatsu, we proved an unpolarized version of this conjecture for hyper-Kähler varieties of a fixed deformation type: that is, even without bounding the polarization, only finitely many such varieties exist over a given number field with good reduction outside a fixed finite set. Based on that, I will introduce a pointed Shafarevich conjecture for primitive symplectic varieties (possibly singular), and sketch our new results showing that fixing a basepoint in a family forces finiteness of the (generic) fibers under mild hypotheses.