Vol. 6, No. 1, 2012

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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
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 $\left[{P}_{X}\left(r-1\right),m\left(r\right)\right]$ or $\left[n\left(r\right),{P}_{X}\left(r\right)\right]$, $r\ge 4$, where ${P}_{X}\left(r\right)$ is the Hilbert polynomial of $X$, $m\left(r\right):=\frac{1}{2}d{r}^{2}+\frac{1}{2}r\left(2-d\right)$ and $n\left(r\right):=\frac{1}{2}d{r}^{2}+\frac{1}{2}r\left(d-2\right)$. As a corollary we prove: (1) Mustaţă’s conjecture for a general set of $s\ge 19$ 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 $s\ge 6d+1$ 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
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