The minimal resolution conjecture for points on del Pezzo surfaces

Mustaţǎ (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 IZ. 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 ⊆ ℙ d - up to three sporadic cases - whose cardinality {pipe}Z{pipe} sits into the interval [P X(r - 1),m(r)] or [n(r),P X(r)], r ≥ 4, where P X(r) is the Hilbert polynomial of X, m(r):= dr 2 + r(2 - d) and n(r):= dr 2 + r(d - 2). As a corollary we prove: (1) Mustaţǎ's conjecture for a general set of s ≥ 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 ≥ 6d + 1 on a del Pezzo surface X ⊆ ℙ d. © 2012 by Mathematical Sciences Publishers