IJPAM: Volume 57, No. 4 (2009)

$\mathbb {P}^n$, $n \ge 4$, OF A RATIONAL CURVE AND A FEW PLANES

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

Abstract.Here we prove the existence of an integer $\alpha \ge 0$ with the following property. Let $X \subset \mathbb {P}^4$ be a general union of $x$ planes and a degree $y$ smooth rational cuve. Let $k\ge 1$ be the minimal integer such that $x\binom{k+2}{2} -x(x-1)/2 + ky+1 \le \binom{k+4}{4}$. Assume $x \le k-\alpha$. Then $X$ has the expected postulation. We extend the result to $\mathbb {P}^n$, $n\ge 5$, when the planes are either disjoint or contained in a $4$-dimensional linear subspace.

Received: November 3, 2009

AMS Subject Classification: 14N05

Key Words and Phrases: postulation, unions of planes, planes in $\mathbb {P}^4$

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2009
Volume: 57
Issue: 4