# Pragmatic 2004

Scientific activities

Olivier Debarre:
An introduction to the theory of algebraic and analytic hyperbolicity. Ample line bundles. Ample vector bundles. Analytic hyperbolicity. Construction of varieties with ample cotangent bundle. Construction of analytically hyperbolic hypersurfaces. Algebraic hyperbolicity of hypersurfaces in the projective space. Algebraic and entire curves on surfaces of general type. Entire curves in complex tori.

Lucia Caporaso: Aritmetic hyperbolicity and moduli of curves. Overview and open problems. Generalities about moduli heory. Rational points of varieties and moduli schemes. Moduli of nonsingular curves: existence and coarseness. Examples of (non-trivial) isotrivial families and non-modular rational points in the moduli space of smooth curves. The Shafarevich problem and Mordell problem. Uniformity of rational points of curves. Uniform versions of Shafarevich problem. The moduli space of stable curves. The Picard moduli problem. Combinatorial aspects of the moduli space of stable curves.

Problems

1. (Meng CHEN and Yann SEPULCRE)
Prove the following Bertini type result: let X be an integral subscheme of  P^n of dimension at least 2 and let V_e  be the projective space of hypersurfaces of degree e in  P^n. The codimension of the complement of  {F \in V_{e} |  X\cap F is integral of codimension 1 in X} in V_e is at least e-1.

2. (Abel CASTORENA and Erwan ROUSSEAU)
Present a detailed account of Siu's results on the hyperbolicity of very general hypersurfaces of sufficiently high degree in P^n.

3. (Denis CONDUCHE' and Eleonora PALMIERI)
Describe the set of all ratios c^2_1(X)/ c_2(X)}, for X smooth complex projective surface with ample cotangent bundle. Is it equal to (1,3]\cap Q?

4. (Roi DO CAMPO ALVAREZ and Ethan COTTERILL)
Construct new examples of analytically hyperbolic hypersurfaces or complete intersections of low degree in P}^n.

5. (Dajano TOSSICI and Francesca VETRO)
Modular curves in M_g: estimates on their numerical invariants.

6. (Alessio DEL PADRONE  and  Ernesto MISTRETTA)
Hyperbolicity of the functor of moduli of smooth curves (or: Non existence theorems for families of smooth curves, after Beauville).

7. (Sergei GORCHINSKY and Filippo VIVIANI)
Uniformity results for rational points of curves over function fields having maximal variation of moduli.

8. (Simone BUSONERO, Margarida MELO and Lidia STOPPINO)
Combinatiorial properties of stable curves: use of the Degree class group.