IJPAM: Volume 28, No. 2 (2006)

ON THE MINIMAL FREE RESOLUTION OF
GENERAL UNIONS OF FAT POINTS IN ${\bf {P}}^2$

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

Abstract.Here we prove the following result. Fix integers , $m_1 > \cdots > m_s>0$ , , $1 \le i \le s$ . Take $\sum _{i=1}^{s} a_i$ general points $P_{i,j}\in {\bf {P}}^2$ , $1 \le i \le s$ , $1 \le j \le a_i$ and set $Z:= \cup _{i=1}^{s} \cup _{j=1}^{a_i} m_iP_{i,j} \subset {\bf {P}}^2$ and $\delta:= \sum _{i=1}^{s} a_i\binom{m_i}{2}$ . Let be the minimal integer such that $\delta \le \binom{k+2}{2}$ . Assume and either $a_s \ge km(m_1+2)(m_1+1)/2 +m_1k$ or $m_{s-1} = 2$ , $m_s \ge k$ and $2a_{s-1}+a_s \ge km(m_1+2)(m_1+1)/2 +m_1k$ or $\mbox{char}(\mathbb {K}) \ne 2$ , $m_{s-1} = 2$ , $2a_{s-1}+a_s \ge km(m_1+2)(m_1+1)/2 +m_1k$ and $m_s\ge 2$ . Then $h^0({\bf {P}}^2,\mathcal {I}_Z(t)) = 0$ for all , $h^1({\bf {P}}^2,\mathcal {I}_Z(t)) = 0$ for all $t \ge k$ and the homogeneous ideal of is minimally generated by $\binom{k+2}{2} - \delta$ forms of degree and by $\max \{0,2\delta -k(k+2)\}$ forms of degree .