Page 104 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 104
ARITHMETIC AND GEOMETRY OF ALGEBRAIC SURFACES (MS-45)
The Mumford–Tate conjecture for surfaces — state of the art
Johan Commelin, jmc@math.uni-freiburg.de
Universität Freiburg, Germany
The Mumford–Tate conjecture (MTC) provides a bridge between the Hodge structure on the
singular cohomology of an algebraic variety (over the complex numbers) and the Galois repre-
sentation on the étale cohomology of a model of that same variety over a number field. Recently,
there have been several new results on MTC for algebraic surfaces. In this talk I will give an
overview of what is currently known, and explain some of the techniques used in the new re-
sults.
Moduli space of stable (1,2)-surfaces
Stephen Coughlan, stephen.coughlan@uni-bayreuth.de
Universitaet Bayreuth, Germany
A (1,2)-surface is a surface of general type with geometric genus 2 and canonical degree 1. I
describe ongoing work with several collaborators, aiming to understand the so-called KSBA
compactification of the moduli space of (1,2)-surfaces. In particular, we are able to describe
some irreducible components of the moduli space and some boundary strata too.
Irregular covers of K3 surfaces
Alice Garbagnati, alice.garbagnati@unimi.it
Universita’ Statale di Milano, Italy
Coauthor: Matteo Penegini
The K3 surfaces are regular surfaces. They can be covered by irregular surfaces in several ways
and the easiest example is given by the abelian surfaces: these often appear as the minimal
model of Galois covers of K3 surfaces for several Galois group (e.g. all the Abelian surfaces
are birational to the double cover of their Kummer surfaces).
Nevertheless is considerably more difficult to obtain an irregular cover of a K3 surface which
is a surface of general type.
During the talk we discuss the construction of irregular covers of K3 surfaces, obtaining
both surfaces with Kodaira dimension 1 and 2. In particular we focus our attention to the covers
of order 2 and 3 (the latter not necessarly Galois).
102
The Mumford–Tate conjecture for surfaces — state of the art
Johan Commelin, jmc@math.uni-freiburg.de
Universität Freiburg, Germany
The Mumford–Tate conjecture (MTC) provides a bridge between the Hodge structure on the
singular cohomology of an algebraic variety (over the complex numbers) and the Galois repre-
sentation on the étale cohomology of a model of that same variety over a number field. Recently,
there have been several new results on MTC for algebraic surfaces. In this talk I will give an
overview of what is currently known, and explain some of the techniques used in the new re-
sults.
Moduli space of stable (1,2)-surfaces
Stephen Coughlan, stephen.coughlan@uni-bayreuth.de
Universitaet Bayreuth, Germany
A (1,2)-surface is a surface of general type with geometric genus 2 and canonical degree 1. I
describe ongoing work with several collaborators, aiming to understand the so-called KSBA
compactification of the moduli space of (1,2)-surfaces. In particular, we are able to describe
some irreducible components of the moduli space and some boundary strata too.
Irregular covers of K3 surfaces
Alice Garbagnati, alice.garbagnati@unimi.it
Universita’ Statale di Milano, Italy
Coauthor: Matteo Penegini
The K3 surfaces are regular surfaces. They can be covered by irregular surfaces in several ways
and the easiest example is given by the abelian surfaces: these often appear as the minimal
model of Galois covers of K3 surfaces for several Galois group (e.g. all the Abelian surfaces
are birational to the double cover of their Kummer surfaces).
Nevertheless is considerably more difficult to obtain an irregular cover of a K3 surface which
is a surface of general type.
During the talk we discuss the construction of irregular covers of K3 surfaces, obtaining
both surfaces with Kodaira dimension 1 and 2. In particular we focus our attention to the covers
of order 2 and 3 (the latter not necessarly Galois).
102
