IJPAM: Volume 80, No. 3 (2012)


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

Abstract. A typical statement proved here. Fix integer $r\ge 4$ and $k\ge 2$. Let $S\subset \mathbb {P}^r$ be a finite set such that $d:= \sharp (S)\le rk+1$ and $\sharp (S\cap V)\le r-1$ for each $(r-2)$-dimensional linear subspace $V\subset \mathbb {P}^r$. We have $h^1(\mathcal {I}_S(k)) >0$ if and only if there is a hyperplane $H\subset \mathbb {P}^r$ such that $h^1(H,\mathcal {I}_{S\cap H,H}(k)) >0$. If there is such a hyperplane $H$, then it is unique and $H$ is the unique hyperplane containing the maximal number of points of $S$. If $d \le (r-1)k+1$, then $h^1(\mathcal {I}_S(k)) =0$.

Received: July 14, 2012

AMS Subject Classification: 14N05, 15A69, 15A21

Key Words and Phrases: finite sets, linear subspace

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Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2012
Volume: 80
Issue: 3