# Stability conditions and inequalities for surfaces

## Thursday 21 October 2010 - Lidia Stoppino

ARGOMENTI: Seminari

Lidia Stoppino (Università dell'Insubria)
"Stability conditions and inequalities for surfaces"

Thursday, October 21, room 2AB45 h. 14:00

-Abstract
Let $S$ be a complex surface with a fibration $f$ over a curve $B$ . Consider a divisor $L$ on $S$ , and a subsheaf $mathcal G$ of the pushforward $f_*mathcal{O}_S(L)$ . Note that the fibre of $mathcal G$ on $bin B$ represents a linear system $mathcal G_bsubseteq H0(L_{|f^*(b)})$ on the fibre $f^*(b)$ . I will describe three different stability properties for these objects, and three methods that exploit these properties for deriving bounds on the ratio $L2/deg mathcal G$ . There is a yet another stability property which seems to be the key concept that unifies the three approaches: the $textit{linear stability}$ of the linear system $mathcal G_b$ on the general fibre $f^*(b)$ . I will then describe an application of these methods to the geography of fibred surfaces. The geographical problem for fibred surfaces is to find the range of variation of the basic invariants $K_f2$ and $chi_f$ , and constrains imposed by geometrical properties of $f$ . Linear stability can be proved for suitable canonical projections on a curve of genus $geq 2$ , and this allows us to obtain a lower bound for $K2_f/chi_f$ , that depends on the relative irregularity $q_f$ and on the Clifford index of the general fibre. These are results obtained in collaboration with Miguel Àngel Barja.

Rif. int. A. Bertapelle