The minimal resolution conjecture for points on del Pezzo surfaces

Miró-Roig, Rosa M.; Pons-Llopis, Joan

doi:10.2140/ant.2012.6.27

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The minimal resolution conjecture for points on del Pezzo surfaces

Rosa M. Miró-Roig and Joan Pons-Llopis

Vol. 6 (2012), No. 1, 27–46

DOI: 10.2140/ant.2012.6.27

Abstract

Mustaţă (1997) stated a generalized version of the minimal resolution conjecture for a set $Z$ of general points in arbitrary projective varieties and he predicted the graded Betti numbers of the minimal free resolution of $I_{Z}$ . In this paper, we address this conjecture and we prove that it holds for a general set $Z$ of points on any (not necessarily normal) del Pezzo surface $X \subseteq ℙ^{d}$ — up to three sporadic cases — whose cardinality $| Z |$ sits into the interval $[P_{X} (r - 1), m (r)]$ or $[n (r), P_{X} (r)]$ , $r \geq 4$ , where $P_{X} (r)$ is the Hilbert polynomial of $X$ , $m (r) : = \frac{1}{2} d r^{2} + \frac{1}{2} r (2 - d)$ and $n (r) : = \frac{1}{2} d r^{2} + \frac{1}{2} r (d - 2)$ . As a corollary we prove: (1) Mustaţă’s conjecture for a general set of $s \geq 19$ points on any integral cubic surface in $ℙ^{3}$ ; and (2) the ideal generation conjecture and the Cohen–Macaulay type conjecture for a general set of cardinality $s \geq 6 d + 1$ on a del Pezzo surface $X \subseteq ℙ^{d}$ .

Keywords

minimal free resolutions, del Pezzo surfaces, $G$-liaison

Mathematical Subject Classification 2000

Primary: 13D02

Secondary: 13D40, 14M05

Milestones

Received: 2 July 2010

Revised: 20 January 2011

Accepted: 19 February 2011

Published: 15 June 2012

Authors

Rosa M. Miró-Roig
Facultat de Matemàtiques Department d’Algebra i Geometria University of Barcelona Gran Via des les Corts Catalanes 585 08007 Barcelona Spain

Joan Pons-Llopis
Facultat de Matemàtiques Department d’Algebra i Geometria University of Barcelona Gran Via des les Corts Catalanes 585 08007 Barcelona Spain

Vol. 6, No. 1, 2012