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Program

All lectures and talks are in Weierstrass Hörsaal, Room 3038, at Unter den Linden 6.


Summer School's Schedule

Conference's Schedule

Summer School's Outlines

Jean-Benoit Bost Formal, analytic, and algebraic curves in algebraic varieties over number fields
Diverse classical transcendence results may be reformulated as theorems asserting that formal curves in algebraic varieties over number fields (for instance, formal curves defined by integrating algebraic vector fields) are algebraizable when they satisfy suitable arithmetic or analytic conditions. In my lectures, I will discuss this geometric point of view on transcendence theorems and some of their applications to elliptic curves and their moduli spaces.

Carel Faber Cohomology of moduli spaces of curves
The first lecture will be a survey lecture concerning tautological classes on the moduli spaces of smooth or stable pointed curves and the subalgebras of the Chow and cohomology rings generated by them. Both results and open problems will be discussed. In the second lecture, I will first discuss some results concerning the natural action on the moduli spaces of curves with marked points of the symmetric groups permuting the points. Then I will talk about how this action, in combination with other results, can be used (at least in principle) to detect non-algebraic cohomology classes, which typically are related to various kinds of modular forms.

Gerard van der Geer Modular forms and moduli spaces
After introducing elliptic modular forms we will explain their geometrical and cohomological interpretation using moduli spaces of elliptic curves. We then will try to extend this to the case of higher genus by introducing Siegel modular forms and treat their role in the geometry of moduli spaces of curves of higher genus and the moduli of abelian varieties.

Hisanori Ohashi Enriques surfaces covered by a fixed K3 surface
It can happen that more than one Enriques surfaces are covered by a single K3 surface. Such an example was first stated explicitly by S. Kondo. Using lattice theory, period maps and the Torelli theorems for K3 & Enriques, I will present several results around this phenomena.
1. The number of Enriques quotients (modulo isomorphisms) of one K3 surface is always finite.
2. The K3-cover of generic Enriques surfaces have exactly one Enriques quotient, but in general the number can tend to infinity.
3. We can classify explicitly the Enriques quotients of a generic Kummer surface of product type.
4. We can classify explicitly the Enriques quotients of a generic Jacobian Kummer surface. Application to the generators of Aut(X).

David Smyth Birational geometry of the moduli space of curves
LECTURE 1: Explicit presentations of M_{0}, M_{1}, M_{2}, M_{3} and the questions raised (e.g. unirationality, connectedness). Quick definition of DM-compactification. Definition of basic invariants of birational geometry: nef cone, effective cone, chamber decomposition of effective cone, with some quick examples. Explain how the classical problems have become subsumed into the project of computing these for M_{g}.
LECTURE 2: Explanation of how to do intersection theory on M_{g}, i.e. how to compute degrees of natural line-bundles on some easy families. Computations on M_{2}, M_{3}, where we will see that the relevant cones are all "determined by geometry". Description of what's known in general (e.g. discussion of slope conjecture, F-conjecture).
LECTURE 3: Raise the problem of alternate compactifications and MMP for M_{g}. Discuss known results on M_{g}, M_{0,n}, M_{1,n}.

Fillippo Viviani Torelli theorem for stable curves
The classical Torelli theorem asserts that a smooth projective curve is determined by its Jacobian together with the principal polarization induced by the theta divisor. In modular terms, it asserts that the natural (Torelli) map from the moduli space of smooth projective curves of genus g into the moduli space of principally polarized abelian varieties of dimension g is injective on geometric points. Quite recently, Alexeev has extended the Torelli map, in a geometrically meaningful way, to modular compactifications of the above moduli spaces, namely the moduli space of Deligne-Mumford stable curves and the moduli space of principally polarized stable semi-abelic pairs. In a joint work with L. Caporaso, we study the geometric fibers of the above compactified Torelli map.

Conference's Abstracts

Arnaud Beauville Enriques surfaces and Brauer groups
The Brauer group is an invariant of algebraic varieties, which plays an important role in arithmetic geometry. For an Enriques surface S (quotient of a K3 surface X by an involution), Br(S) has only one non trivial element. Does this element become trivial on X ? I'll show that the surfaces for which this happens form a countable union of hypersurfaces in the moduli space of Enriques surfaces.

Jean-Benoit Bost Integral curves in algebraic varieties over number fields
In this talk, I will discuss some recent joint work with Hugues Randriam, concerning the transcendence of points on some integral curves of algebraic vector fields on algebraic varieties defined over a number field. Our results extend a famous theorem of Nesterenko concerning the transcendence of values of classical modular forms, and are expected to admit consequences concerning Diophantine properties of moduli spaces of Calabi-Yau manifolds.

Izzet Coskun The birational geometry of Kontsevich moduli spaces
I will discuss the birational geometry of Kontsevich moduli spaces of genus-zero stable maps, focusing on cones of ample and effective divisors, stable base locus decompositions and the corresponding birational models.

Carel Faber On M_{3, 17} and M_{24}
I will discuss some recent results concerning the two moduli spaces in the slightly exotic title. The results may be said to be of geometric nature in one case and of arithmetic nature in the other case.

Gerard van der Geer Siegel modular forms of low genera
The Eichler-Shimura theorem expresses elliptic modular forms in terms of cohomology of local systems. This theorem generalizes to higher genera. The cohomology of local systems can be explored by counting points over finite fields. We show how this can be succesfully applied to obtain a lot of information about Siegel modular forms of genus 2 and 3. This is joint work with Jonas Bergstroem and Carel Faber.

Bert van Geemen Calabi-Yau threefolds parametrised by Shimura varieties
We recall the basic properties of the period map for Calabi-Yau threefolds. The associated Picard-Fuchs equations are introduced and some new results on these equations for certain quintic threefolds are given. Next we explicitly describe a one parameter family of CY threefolds, first constructed by Rohde, and we show how to determine its Picard-Fuchs equation. We also discuss some generalisations to higher dimensional families. An important feature is that these familes do not have maximal unipotent monodromy, which might have consequences for Mirror Symmetry.

Valery Gritsenko Reflective modular forms and Kodaira dimension of modular varieties.
In this talk I present two different applications of reflective modular forms. The first one is the series of results obtained together with K. Hulek and G. Sankaran about the general type of moduli spaces of polarised symplectic varieties. The second application gives us new examples of modular varieties of Kodaira dimension minus infinity or zero.

Sam Grushevsky Differentials with real periods and the geometry of the moduli space of curves M_g
We use meromorphic differentials on Riemann surfaces with prescribed singular parts and real periods to define a foliation of M_g. This foliation allows us to give a quick proof of the Diaz' bound on the dimension of complete subvarieties of M_g, and of some vanishing results for the cohomology of M_g. We will also discuss the motivation for this construction, from the Whitham perturbation theory of integral systems, and potential further applications to the study of cohomology and subvarieties of M_g. Joint work with Igor Krichever.

Klaus Hulek Characters of orthogonal groups and the fundamental group of modular varieties
We consider the commutator subgroup of integral orthogonal groups belonging to indefinite quadratic forms and show that the index of this subgroup is 2 for many groups that occur in the construction of moduli spaces in algebraic geometry, in particular the moduli of K3 surfaces. We give applications to modular forms and to computing the fundamental groups of some moduli spaces. This is joint work with V. Gritsenko and G. K. Sankaran.

Ulf Kühn Scattering constants and Neron-Tate heights
Neron-Tate heights of degree zero divisors on algebraic curves are closely related to scattering constants coming from particular hyperbolic uniformisations of that curves. We explain this relation and illustrate it in an example.

Ben Moonen Chern classes of automorphic bundles
In my talk I shall discuss tautological classes on compactified Shimura varieties. The picture we have is that for the Shimura variety associated to a pair (G,X) the tautological ring should be isomorphic to the cohomology ring of the compact dual of X, which is a completely explicit ring. We give a simple geometric explanation for this. In cohomology this works. However, on more refined levels (Deligne cohomology, Chow rings) we know this only in special cases and there is a fundamental obstacle that remains. I shall describe where the difficulty lies and will discuss what is known about it.

Shigeru Mukai Non-abelian Brill-Noether loci of curves and application to the unirationality of moduli of K3 surfaces
Two types of K3 surfaces appears in the moduli space of 2- bundles on curves as certain Brill-Noether loci. I would like to explain how they were found and discuss their application to the uniratonality problem.

Riccardo Salvati Manni Two-point functions for three-loop superstring scattering amplitudes
In this talk I will report about some joint results obtained with Sam Grushevsky. I will consider the two-point function for the three-loop chiral superstring measure ansatz proposed by Cacciatori, Dalla Piazza, van Geemen and Grushevsky vanishes. The proof uses specifically the theory of the Gamma_{00} linear system on Jacobians introduced by van Geemen and van der Geer. I also discuss the possible approaches to proving the vanishing of the two-point function for the proposed ansatz in higher genera.

Gregory K. Sankaran Moduli problems for holomorphic symplectic manifolds
I shall describe part of some joint work with Gritsenko and Hulek in which we show how the moduli spaces of polarised holomorphic symplectic manifolds are related to locally symmetric varieties. In particular I shall examine the failure of the global Torelli theorem in this case and suggest possible alternatives.

Joachim Schwermer Geometric cycles, arithmetic groups and related automorphic forms

David Smyth Moduli spaces of curves with A_{k}-singularities
I will describe work-in-progress relating to the construction of compactified moduli spaces of curves with A_{k}-singularities (y^2=x^{k+1}). These spaces are expected to arise in Hassett's log minimal model program for the moduli space for curves. This is joint work with Jarod Alper and Fred van der Wyck.

Emmanuel Ullmo Generalised Tate, Mumford-Tate and Shafarevich conjectures
The aim of this talk is to formulate some natural generalisations of the conjectures of Tate and Shafarevich proved by Faltings and to show that these generalisations are both equivalent to the Mumford-Tate conjecture.

Ravi Vakil The ring of invariants of n points on the projective line
The GIT quotient of a small number of points on the projective line has long been known to have beautiful geometry. For example, the case of six points is intimately connected to the outer automorphism of S_6. We extend this picture to an arbitrary number of points, completely describing the equations of the moduli space. (In some sense there is only one equation.) The case of eight points is particularly entertaining. This is joint work with Ben Howard, John Millson, and Andrew Snowden.

Alessandro Verra Some geometry of the moduli of roots of line bundles in low genus
The talk deals with the moduli spaces of spin curves, and with the moduli spaces R(g,n) of roots of the trivial line bundle, in the case of curves of low genus g. After a brief survey, the case n = 3 is considered and a geometric description of R(4,3) is given, together with a proof of its rationality. For spin curves some work in progress is presented, about the values of the Kodaira dimension and rational parametrizations of the moduli of odd spin curves of genus g < 10.

Claire Voisin Infinitesimal invariants for cycles modulo algebraic equivalence and 1-cycles on Jacobians
We define infinitesimal invariants for families of cycles modulo algebraic equivalence, in an arbitrarily high level of the Bloch-Beilinson filtration. We apply this invariant to the study of the Beauville decomposition of the cycle of a general curve in its Jacobian. The goal is to prove that the Colombo-Van Geemen upper-bound for the last non vanishing component is optimal. We achieved this for the second component, (the case next to Ceresa), showing that it is non zero starting from genus 5. The general computation is still ``work in progress''.

Rainer Weissauer The supersymmetric square of the theta divisor in dimension 4