# Mini Courses

## Shimura varieties

Dates: January 13 - 17, 10:00 - 12:00 (exception: 14:00 - 16:00 on Tuesday and Thursday)

Lecturer: Marc-Hubert Nicole

## p-adic geometry

Dates: January 27 - 31, 10:00 - 12:00

Lecturer: Jérôme Poineau

Introduction: At the end of the eighties, Vladimir Berkovich introduced a new way to define p-adic analytic spaces. A surprising feature is that, although p-adic fields are totally discontinuous, the resulting spaces enjoy many nice topological properties: local compactness, local path- connectedness, etc. On the whole, those spaces are very similar to complex analytic spaces. They already have found numerous applications in several domains: arithmetic geometry, dynamics, motivic integration, etc.

Abstract: In this course, we will introduce Berkovich spaces and study their basic properties. The program will cover the following topics:

- Tate algebras, rigid spaces and affinoid algebras
- affinoid spaces
- analytification of algebraic varieties (and thorough description of the affine line)
- analytic curves
- étale cohomology and vanishing cycles
- potential theory

## Mini-courses organized by the group "Brill-Noether methods in the study of hyper-Kähler and Calabi-Yau manifolds"

Dates: January 27 - 31, 14:15 - 15:45 and 16:30 - 18:00

Lecturers: Enrico Arbarello, Andreas Leopold Knutsen, Edoardo Sernesi

Enrico Arbarello: Curves on K3 surfaces

Monday 14:15 - 15:45, Tuesday 16:30 - 18:00, Wednesday 16:30 - 18:00

Abstract: We will illustrate the basic results of the classical Brill-Noether theory and in particular Lazarsfeld's point of view. We will discuss conditions imposed on a curve lying on a K3 surface explaining the ideas of Wahl, Beauville-Mérindol, and Voisin. We then illustrate Mukai's program to reconstruct a K3 surface S (of odd genus greater or equal than 13) from a curve C lying on it, by exhibiting S as a suitable Fourier-Mukai transform of an exceptional Brill-Noether locus of rank 2 vector bundles on C. The implementation of this program was carried out recently as a joint work with Andrea Bruno and Edoardo Sernesi and it involves the geometry of elliptic K3 surfaces with Picard number equal to two.

Andreas Leopold Knutsen: Linear series on singular curves on K3 surfaces: Vector bundle methods and degenerations

Monday 16:30 - 18:00, Thursday 14:15 - 15:45, Friday 14:15 - 15:45

Abstract: Linear series on K3 surfaces have been intensively studied the last couple of decades and vector bundle methods have played an important role in this study, starting from Lazarsfeld's proof of the Petri condition for the general curve. At the same time, also singular (especially nodal) curves on K3s have been extensively studied, for instance in connection with the famous Yau-Zaslow counting formula for rational curves obtained with methods from theoretical physics, and for this study degeneration methods (of the surface) have proved to be useful. In the lectures I will talk about the study (joint with Ciro Ciliberto) of linear series of normalizations of curves on K3 surfaces, and the role the vector bundle methods and degeneration methods play here, as well as some applications to the study of the Mori cone of Hilberts schemes of points on K3 surfaces.

Edoardo Sernesi: Syzygies of special line bundles on curves

Tuesday 14:15 - 15:45, Wednesday 14:15 - 15:45, Thursday 16:30 - 18:00

Abstract: In the last decades the study of equations and syzygies of projective curves has been at the origin of many important results, both in the case of canonical curves and of non-special line bundles of high degree. On the other hand, in the case of special line bundles different from the canonical there has been virtually no progress. In my lectures I will discuss this case, explaining some results in this direction obtained in collaboration with M. Aprodu. They are based on new geometric ideas which also provide a better understanding of some of the known results.