IJPAM: Volume 23, No. 2 (2005)

ON THE MINIMAL FREE RESOLUTION
OF GENERAL UNIONS IN $P^2$ OF
A FINITE FIX SUBSET AND
MANY GENERAL POINTS

E. Ballico
Department of Mathematics
University of Trento
380 50 Povo (Trento) - Via Sommarive, 14, ITALY
e-mail: ballico@science.unitn.it


Abstract.Let $A\subset {\bf {P}}^2$ be a finite subset such that no three points of $A$ are collinear. Set $d:= \sharp (A)$ and $m_d:= 2\lceil d/4\rceil +1$. Fix an integer $x$ such that $d+x > m_d(m_d+1)/2$, i.e. assume that the first integer $t$ such that $d+x \le (t+2)(t+1)/2$ satisfies $t \ge m_d$. Let $S \subset {\bf {P}}^2$ be a general subset such that $\sharp (S)=x$. Here we prove that the minimal free resolution of $Z:= A\cup S$ is the expected one, i.e. $h^1({\bf {P}}^2,\mathcal {I}_Z(y))
= 0$ for all $y \ge t$, $h^0({\bf {P}}^2,\mathcal {I}_Z(x)) = 0$ for all $x<t$ and the homogeneous ideal ${\bf {I}}(Z)$ of $Z$ is minimally generated by $(t+2)(t+1)/2 -d-x$ forms of degree $t$ and $\max \{0,2d+2x-t^2-2t\}$ forms of degree $t+1$.

Received: August 3, 2005

AMS Subject Classification: 14N05

Key Words and Phrases: minimal free resolution, zero-dimensional scheme, postulation, fat point, double point

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2005
Volume: 23
Issue: 2