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 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 d — up to three sporadic cases — whose cardinality |Z| sits into the interval [PX(r 1),m(r)] or [n(r),PX(r)], r 4, where PX(r) is the Hilbert polynomial of X, m(r) := 1 2dr2 + 1 2r(2 d) and n(r) := 1 2dr2 + 1 2r(d 2). As a corollary we prove: (1) Mustaţă’s conjecture for a general set of s 19 points on any integral cubic surface in 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 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