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.