IJPAM: Volume 60, No. 4 (2010)

AN ASYMPTOTIC VANISHING THEOREM
FOR GENERIC FAT POINTS IN
POSITIVE CHARACTERISTIC

E. Ballico
Department of Mathematics
University of Trento
38 123 Povo (Trento) - Via Sommarive, 14, ITALY
e-mail: ballico@science.unitn.it

Abstract.Let be an integral projective variety defined over an algebraically closed field $\mathbb {K}$ such that $p:= \mbox{char}(\mathbb {K})>0$ . Fix an integer and $M, L\in \mbox{Pic}(X)$ with ample. Here we prove (following a paper by Alexander and Hirschowitz) the existence of an integer such that for all integers $d \ge d(M,X,L,M)$ either $h^0(X,\mathcal {I}_Z\otimes M\otimes L^{\otimes d}) =0$ or $h^1(X,\mathcal {I}_Z\otimes M\otimes L^{\otimes d}) =0$ , where is a general union of i-points of with $i \le m$ .