IJPAM: Volume 61, No. 4 (2010)

ON THE POSTULATION OF A GENERAL UNION OF
TRIPLE POINTS AND DOUBLE POINTS IN $\mathbb {P}^4$

E. Ballico
Department of Mathematics
University of Trento
38 123 Povo (Trento) - Via Sommarive, 14, ITALY
e-mail: [email protected]

Abstract.Fix integers such that $d \ge 8$ , $x \ge 0$ , $y \ge 0$ . Let $Z \subset \mathbb {P}^4$ be a general union of triple points and double points. Here we prove that either $h^1(\mathbb {P}^4,\mathcal {I}_Z(d)) = 0$ (case $15x+5y \le \binom{d+4}{4}$ ) or $h^0(\mathbb {P}^4,\mathcal {I}_Z(d)) = 0$ (case $15x+5y \ge \binom{d+4}{4}$ ). When we are only able to prove the corresponding result assuming $(x,y) \notin \{ (22,0),(21,3)\}$ .