Une conjecture pour la cohomologie des diviseurs sur les surfaces rationnelles générique. (A conjecture for the cohomology of divisors on generic rational surfaces). (French) Zbl 0686.14013

Hirschowitz, André

J. Reine Angew. Math. 397, 208-213 (1989).

Let \(S_ r\) be the blowing-up of \({\mathbb{P}}^ 2\) at r general points. Here it is given a very precise conjecture about the cohomology of line bundles on \(S_ r\). It would be nice to have similar (or even much weaker) conjectures about other surfaces and/or to know more about the blowing-up at not too general points. The paper contains also a proof of an interesting special case of the conjecture.

Reviewer: E.Ballico

MSC:

14F99	(Co)homology theory in algebraic geometry
14M20	Rational and unirational varieties
14C22	Picard groups

Keywords:

rational surfaces; divisor; cohomology of line bundles; blowing-up