The minimal resolution conjecture for points on del Pezzo surfaces

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 $X\subseteq {\mathbb{P}}^d$ — up to three sporadic cases — whose cardinality $\left|Z\right|$ 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 ${\mathbb{P}}^3$; 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 $X\subseteq {\mathbb{P}}^d$.