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: [email protected]


Abstract.Let $X$ be an integral projective variety defined over an algebraically closed field $\mathbb {K}$ such that $p:= \mbox{char}(\mathbb {K})>0$. Fix an integer $0 < m <p$ and $M, L\in \mbox{Pic}(X)$ with $L$ ample. Here we prove (following a paper by Alexander and Hirschowitz) the existence of an integer $d(m,X,L,M)$ 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 $Z$ is a general union of i-points of $X$ with $i \le m$.

Received: March 18, 2010

AMS Subject Classification: 14N05

Key Words and Phrases: fat point, zero-dimensional scheme

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2010
Volume: 60
Issue: 4