Registration

BMS Summer School registration opens on Friday 21 August in room 3037 from 11:00 - 13:00

The registration office will also be open during further lecture breaks for late arrivals.

Schedule

Course schedule as a PDF-file 

Below please find the abstracts of the respective lectures

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.