Simplicial ideals, 2-linear ideals and arithmetical rank

[摘要] In the first part of this paper we study scrollers and linearly Joined varieties Scrollers were introduced in Barile and Morales (2004) [BM4] linearly joined varieties are an extension of scrollers and were defined in Eisenbud et al (2005) [EGHP] there they proved that scrollers are defined by homogeneous ideals having a 2-linear resolution A particular class of varieties of important interest in classical Geometry are Cohen-Macaulay varieties of minimal degree they were classified geometrically by the successive contribution of Del Pezzo (1885) [DP] Bertini (1907) [B] and Xambo (1981) [X] and algebraically in Barile and Morales (2000) [BM2] They appear naturally studying the fiber cone of a codimension two toric Ideals Morales (1995) [M] Gimenez et al (1993 1999) [GMS1 GMS2] Barile and Morales (1998) [BM-1] Ha (2006) [H] Ha and Morales (2009) [HM] Let S be a polynomial ring and I subset of S a homogeneous ideal defining a sequence of linearly joined varieties We compute depth S/I. We prove that c(V) = depth S/I where c(V) is the connectedness dimension of the algebraic set defined by I We characterize sets of generators of I and give an effective algorithm to find equations as an application we prove that ara(I) = projdim(S/I) in the case where V is a union of linear spaces in particular this applies to any square free monomial ideal having a 2 linear resolution In the case where V is a union of linear spaces the ideal I can be characterized by a tableau which is an extension of a Ferrer (or Young) tableau All these results are new and extend results in Barile and Morales (2004) [BM4] Eisenbud et al (2005) [EGHP] (C) 2010 Elsevier Inc All rights reserved

[发布日期] 2010-12-15

[关键词] 2 linear resolution;Syzygies;Depth;Cohomological dimension;Arithmetical rank;Scroll