Invertible Sheaves on Generic Rational Surfaces and a Conjecture of Hirschowitz ' s by Giuseppe

We study the cohomology of invertible sheaves C on surfaces X, blowings-up of P2 at points pl,...,p, in general position (generic rational surfaces). The main theme is when such sheaves have the natural cohomology, i.e. at most one cohomology group is non zero. Our approach is geometrical. On one hand we are lead to deform the configuration of points to a special position, notably surfaces with a reduced and irreducible anticanonical divisor, for which, thanks to the extensive work of Harbourne, the cohomology is known or computable. Semicontinuity theorems give then vanishing of the cohomology for line sheaves on generic surfaces satisfying a kind of positivity condition. On the other hand we fiber our surfaces over the projective line, r : Xr -+ P1, and reduce the problem to a cohomological estimate for locally free sheaves on the base. Indeed, under mild numerical assumptions on £ = Ox,(dH E=rmiE&) (where H denotes the divisor on Xr for which Ox,(H) = a*(OpV(1)), a : Xr -+ P2 being the blowing-up map and Ei the (-1)-curves a1 (ps)) the 7rC.£'s turn out to be locally free, hence, thanks to the Birkhoff-Grothendiek theorem, sums of invertible sheaves on P . We can thereby reduce a conjecture of Andr6 Hirschowitz -to the effect that the invertible sheaves C have the natural cohomology provided c1((£).E > -1 for any exceptional curve of the first kind E such that 2H.E < H.cl(C(£)to the thesis that certain locally free sheaves are direct sums of Ori(-1)'s, or, equivalently, are semistable. Thesis Supervisor: Victor G. Kae Title: Professor of Mathematics Thesis Supervisor: Fedor A. Bogomolov Title: Professor of Mathematics