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2012 The minimal resolution conjecture for points on del Pezzo surfaces
Rosa M. Miró-Roig, Joan Pons-Llopis
Algebra Number Theory 6(1): 27-46 (2012). DOI: 10.2140/ant.2012.6.27


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 Xd — up to three sporadic cases — whose cardinality |Z| sits into the interval [PX(r1),m(r)] or [n(r),PX(r)], r4, where PX(r) is the Hilbert polynomial of X, m(r):=12dr2+12r(2d) and n(r):=12dr2+12r(d2). As a corollary we prove: (1) Mustaţă’s conjecture for a general set of s19 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 s6d+1 on a del Pezzo surface Xd.


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Rosa M. Miró-Roig. Joan Pons-Llopis. "The minimal resolution conjecture for points on del Pezzo surfaces." Algebra Number Theory 6 (1) 27 - 46, 2012.


Received: 2 July 2010; Revised: 20 January 2011; Accepted: 19 February 2011; Published: 2012
First available in Project Euclid: 20 December 2017

zbMATH: 1262.13019
MathSciNet: MR2950160
Digital Object Identifier: 10.2140/ant.2012.6.27

Primary: 13D02
Secondary: 13D40 , 14M05

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

Rights: Copyright © 2012 Mathematical Sciences Publishers


Vol.6 • No. 1 • 2012
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