Orateur
Daniel Naie
Description
(joint work Igor Reider)
Let $X$ be a projective surface. A twisted Kodaira-Spencer class is
an element of the cohomology group $H^1(T_X(-D))$, with $D$
``sufficiently positive''. We study the connection between the
existence of a non-trivial twisted class and the geometry of $X$. In
particular, we show that, for a minimal general type surface
satisfying $c_2/c_1^2<5/6$, the non-vanishing of $H^1(T_X(-K_X))$
imposes the existence of configurations of rational curves on the
surface.
The techniques used to obtain this result are based on the
interpretation of a non-trivial twisted class as an extension ---a
short exact sequence of locally free sheaves on $X$---, and on the
detailed study of this sequence.
The above point of view and techniques are applied to the study of
surfaces in $\mathbb{P}^4$. Indeed, a surface of non-negative Kodaira
dimension contained in a hypersurface of degree $\leq 5$ displays a
natural non-trivial twisted class, allowing us to address the
Hartshorne-Lichtenbaum problem for, and to slightly control the
irregularity of these surfaces.
