Hélène Esnault
Deligne's moduli of l-adic representations with bounded ramification
Abstract: Deligne showed a year ago that there are finitely many isomorphism classes of irreducible l-adic
representations with bounded ramification on X normal over a finite field. With Moritz Kerz, we slightly
simplified his proof. In the talk, we give an account of it.
Carel Faber
On the cohomology of the moduli spaces of curves
of genus three with marked points
Abstract: I will first discuss a formula for the Euler characteristic
of the compactly supported cohomology of Mg,n
of weight zero. With the help of the formula, Galois representations
in the cohomology of M3,n can be detected
that aren't associated to Siegel modular forms. These Galois representations
are rather mysterious, but appear
to be related to Teichmüller modular forms. I will also try to discuss the relation with
the recent work of
Chenevier and Renard on level one algebraic cusp forms of classical groups.
Maksym
Fedorchuk
Log canonical models
of moduli spaces via GIT
Abstract: We will
introduce the log minimal model program for the moduli space
of curves and discuss
a particular approach to it
via Geometric Invariant Theory of finite Hilbert points of
(bi)canonical curves.
We will illustrate how GIT
approach works in the case of moduli spaces of low genus
curves, particularly
curves of genus 5, where
the completion of the program is within reach.
This talk is based on joint
work with Jarod Alper and David Smyth.
Tamás
Hausel
Positivity for Kac polynomials and DT-invariants of
quivers
Abstract:
In this talk I will introduce Kac's conjecture from 1982 on the
non-negativity of coefficients of
A-polynomials counting absolutely indecomposable representations
of quivers over finite fields.
The proof is accomplished by a cohomological interpretation of
these polynomials as a certain isotypical
component on the cohomology of some associated moduli spaces of
quiver representations under the
action of a Weyl group. In the same setup we find a proof of the
positivity of refined Donaldson-Thomas
invariants associated to the same quiver, conjectured by
Kontsevich-Soibelman and first proved by Efimov.
This is joint work with Emmanuel Letellier and Fernando
Rodrigues Villegas (arXiv:1204.2375).
Klaus
Hulek
Uniruledness
of modular varieties and reflective automorphic forms
Abstract: It has long
been known that the existence of automorphic forms with special
properties has
strong consequences for the geometry of modular varieties.
Originally this has been used to prove general
type results, for example for moduli spaces of K3 surfaces. On the
other hand automorphic forms can also
be used to prove that a moduli space has negative Kodaira
dimension or is uniruled. Here we present a criterion
for uniruledness of modular varieties and illustrate this with
examples of moduli spaces of lattice-polarized K3 surfaces.
Jun-Muk
Hwang
Buser-Sarnak invariants of Prym varieties
Abstract: The
Buser-Sarnak invariant of a principally polarized abelian
variety measures the square of the
minimal length of periods. Buser and Sarnak showed that the
Buser-Sarnak invariant of a Jacobian variety J
is bounded by \frac{3}{\pi} log(4 dim J +3). Using Lazarsfeld's
work on the relation between Buser-Sarnak
invariant and Seshadri number, Bauer showed that the
Buser-Sarnak invariant of a Prym variety P is bounded
by \frac{4}{\pi} \sqrt{2 \dim P}. He raised the question whether
a bound of logarithmic order in \dim P exists,
in analogy with Buser-Sarnak's bound for Jacobians. Using a
recent work of Balacheff-Parlier-Sabourau, we give
an affirmative answer: the Buser-Sarnak invariant of a Prym
variety P is bounded by 220 log(2 dim P)$.
Alexis
Kouvidakis
Divisors
on Hurwitz spaces and moduli spaces of curves
Abstract:
We ll discuss the
geometry of the natural map from the Hurwitz space H2k,k+1
of simple
covers of the projective line of
degree d=k+1 and genus g=2k to the moduli space of stable
curves of genus g=2k.
We calculate the cycle class of the
Hurwitz divisor D2 consisting of degree k+1 covers
of the projective
line with simple ramification points, two of which lie in
the same fibre. This has applications to bounds on the slope
of the moving cone of the moduli space of curves, the
calculation of other divisor classes and led to an algebraic
proof for the formula of the Hodge bundle
of the Hurwitz space. This is joint work with G. van der Geer.
Eduard Looijenga
Cohomological amplitude of moduli spaces of curves
Abstract: We show that the cohomological amplitude of the universal curve of
genus g is at most g-1.
This implies theorems of Harer and Diaz.
Martin Möller
Connected components of strata of
the cotangent bundle of the moduli space of curves
Abstract: The cotangent bundle to the
moduli space of curves can be identified with the space of
quadratic differentials.
As such it is stratified by the number and type of zeros of
the quadratic differential. We classify connected components
of
these strata, completing
work of Lanneau. We also give applications to proving
non-varying slope and sum of Lyapunov
exponents for Teichmueller curves.
Rahul Pandharipande
Curves and sheaves on 3-folds
Abstract:
I will discuss joint
work with A. Pixton on descendent integrals on certain moduli
spaces of sheaves on 3-folds.
Descendents are simply the
Chern characters of the universal sheaf. Their study has a
rich structure which is parallel to
descendents (obtained from
cotangent line classes on the moduli spaces of stable maps).
An outcome of this work is the
proof of the GW/DT/ stable
pairs correspondences for the quintic 3-fold.
Aaron
Pixton
Tautological relations of Faber-Zagier type
Abstract: The tautological ring of the moduli
space of smooth curves of genus g is the subring of its Chow ring generated
by the kappa
classes. The Faber-Zagier relations are an explicit algebraic
description of a large number of relations in this
ring,
possibly giving all the relations. I will discuss three families of
tautological relations defined in a similar fashion
to the
Faber-Zagier relations.
The geometry of A_5 arising from
singularities of theta divisors
Abstract: Let H be
the codimension two locus in Ag parametrizing
principally polarized abelian varieties of dimension g
whose
theta divisor has a non ordinary double
point. The talk describes a series of results on H
obtained in a joint work with Farkas,
Grushevsky and Salvati Manni. The case of
dimension g = 5 is specially considered. For this case
the components of H are
described in detail and the slope of the
perfect cone compactification of A5 is
computed.
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