Humboldt Universität zu Berlin
Mathem.-Naturwissenschaftliche Fakultät II
Institut für Mathematik

Forschungsseminar "Algebraische Geometrie"

Sommersemester 2013


Das Forschungsseminar findet mittwochs in der Zeit von 15.00 - 17.00 Uhr in der Rudower Chaussee 25, 12489 Berlin-Adlershof, Raum 2.009 (Haus 2, Erdgeschoss), statt.

Seminar: Algebraic Geometry an der FU



10.04.2013 Timo Schürg (HU Berlin)
Title: Derived algebraic geometry and reduced obstruction theories

Abstract: Obstruction theories are essential ingredients in defining enumerative invariants as in Gromov-Witten and Donaldson-Thomas theory.
After reviewing this notion, I will give an overview of the added functoriality of obstruction theories available from derived algebraic geometry.
As an application, the reduced obstruction theory necessary for counting curves on K3 surfaces will be constructed.
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17.04.2013 15:15 - 16:15: Ingrid Bauer (Universität Bayreuth)
Title: Faithful actions of the absolute Galois group on moduli spaces and change of fundamental group

Abstract: In the 60's J.P. Serre showed that there exists a field automorphism σ ∈ Gal (Q / Q), and a variety X defined over
Q such that X and the Galois conjugate variety Xσ have no isomorphic fundamental groups, in particular they are not homeomorphic.
In a joint paper with F. Catanese and F. Grunewald we give a strong sharpening of this phenomenon discovered by Serre:

Theorem. If σ ∈ Gal (Q / Q) is not in the conjugacy class of the complex conjugation then there exists a surface
(isogenous to a product) X such that X and the Galois conjugate variety Xσ have non isomorphic fundamental groups.


Moreover, we give some faithful actions of the absolute Galois group Gal (Q / Q ), related among them, in particular:

Theorem. The absolute Galois group Gal(Q / Q) acts faithfully on the set of connected components of the (coarse) moduli spaces of surfaces of general type

We will explain the idea of the proof of the second theorem, as well as the technical difficulties. Finally we will show that the first
theorem can be deduced from the second.
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16:30 - 17:30: Fabrizio Catanese (Universität Bayreuth )
Title: Moduli spaces of automorphism marked varieties: the case of curves and surfaces

Abstract: For several reasons it is interesting to consider moduli spaces of triples (X,G,a) where X is a projective variety, G is a finite group,
and a is an effective action G on X. If X is the canonical model of a variety of general type, then G is acting linearly on some pluricanonical model,
and we have a moduli space which is a finite covering of a closed subspace MG of the moduli space.

In the case of curves this investigation is related to the description of the singular locus of the moduli space Mg, for instance of its
irreducible components (due to Cornalba), and of its compactification Mg.
In the case of surfaces there is another ocurrence of Murphy┬┤s law, as shown in my joint work with Ingrid Bauer: the deformation equivalence for
minimal models S and for canonical models differs drastically (nodal Burniat surfaces being the easiest example).
In the case of curves, there are interesting relations with topology. Moduli spaces of curves with a group G of automorphisms of a fixed topological
type have a description by Teichmüller theory, which naturally leads to conjecture genus stabilization for rational homology groups.

I will then show two equivalent descriptions of its irreducible components, surveying known irreducibility results for some special groups.
A new fine homological invariant was introduced in my joint work with Lönne and Perroni: it allows to prove genus stabilization in the ramified case,
extending a theorem of Dunfield and Thurston in the easier unramified case.
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24.04.2013 Fabio Tonini (HU Berlin)
Title: Stacks of ramified Galois covers

Abstract: We introduce the notion of Galois cover for a finite group G and discuss the problems of constructing them and the geometry
of the stack G-Cov they form. When G is abelian, we show that these problems are related to the theory of equivariant hilbert schemes
and we describe certain families of G-covers in terms of combinatorial data associated to 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.
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08.05.2013 Alessandro Verra (Università Roma Tre )
Title: The universal Prym variety in genus < 7 via nodal conic bundles

Abstract: In the talk various examples of families of nodal conic bundles are considered and some of them are used to dominate
the universal Prym variety over the Prym moduli space in genus g < 7. Possible further applications are described. Work in progress with
G. Farkas and S. Grushevsky.
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15.05.2013 Kein Seminar (NoGAGS in Hannover)
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22.05.2013 François Charles (Université de Rennes)
Title: Arithmetic aspects of the Noether-Lefschetz locus for K3 surfaces

Abstract: Given a family of smooth projective varieties, the Noether-Lefschetz locus is the set of points of the base
corresponding to varieties containing strictly more divisors than the generic fiber. In the setting of complex algebraic
geometry, Hodge theory has provided many tools leading to a thorough understanding of it.
In this talk, we will describe some arithmetic aspects of the Noether-Lefschetz locus for families of K3 surfaces, first
over finite fields, leading to a proof of the Tate conjecture for K3 surfaces, and then over number fields, where we expect
it to shed light on the geometry of rational curves on these surfaces.
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29.05.2013 Kein Seminar (Syzygies in Berlin May 27-31 )
05.06.2013
12.06.2013
19.06.2013
26.06.2013 Orsola Tommasi (Leibniz Universität Hannover)
Title: A counterexample to the Gorenstein conjecture in genus 2

Abstract: A main theme in the study of the cohomology of moduli spaces of curves is the study of the tautological ring, a subring generated
by certain geometrically natural classes. One of the open questions is whether the tautological ring is a Gorenstein ring, as conjectured
by Carel Faber in the case of smooth curves without marked points. In this talk we discuss an approach that allows to detect the existence
of non-tautological classes in the cohomology ring of the moduli space of stable curves of genus 2 with sufficiently many marked points,
such as those constructed by Graber and Pandharipande for the compact moduli space of 20-pointed curves of genus 2. We use this to prove
that the Gorenstein conjecture does not hold for these spaces. This is joint work with Dan Petersen (KTH, Stockholm).
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03.07.2013 Sean Keel (University of Texas at Austin)
Title: Canonical Coordinates

Abstract: I will give more details on my proof, joint with Gross, Hacking and Kontsevich, of the (corrected) Fock-Goncharov
dual basis conjecture, and the (positivity statement in the) Fomin-Zelevinski Laurent phenomenon conjecture, for cluster algebras.
I will aim the talk at algebraic geometers, but will not assume any familiarity with cluster algebras.
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15.07.2013 (Montag) Algebraic Geometry afternoon

The event takes place in room 3.008.

14:00 - 15:00: Balázs Szendrői (Oxford)
Title: Purity and quantum cluster positivity

Abstract: To a Calabi-Yau threefold or CY3 category, Donaldson-Thomas theory associates numerical "sheaf counting" invariants
in the first instance, which can however be refined to cohomological invariants which also carry Hodge-theoretic information.
I will explain a purity result and an application to the quantum cluster positivity conjecture in the theroy of cluster algebras.
This is joint work with Davison, Maulik, Schuermann.

15:30 - 16:30: Marian Aprodu (Bucharest)
Title: Chow forms of K3 surfaces and Ulrich bundles

Abstract: An Ulrich bundle on a projective variety is a vector bundle that admits a completely linear resolution over the polynomial algebra.
We prove the existence of rank-2 Ulrich bundles on polarised K3 surfaces with a mild Brill-Noether condition.
As a consequence, we obtain explicit Pfaffian representations of the associated Chow forms.
This talk is based on a joint work with Gavril Farkas and Angela Ortega.

17:00 - 18:00: Maksym Fedorchuk (Boston College)
Title: GIT stability of Hilbert and Syzygy points of canonical curves

Abstract: I will discuss a Geometric Invariant Theory problem concerning stability of Hilbert and Syzygy points
of canonical curves, explain its solution for the Hilbert points in the generic case (joint with Alper and Smyth)
and explain how it is extended to obtain GIT stability of the first Syzygy point of a general canonical curve
(current work-in-progress, joint with Deopurkar and Swinarski). I will also give motivation and explain applications
of these results coming from the minimal model program for Mg.

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