Simplicial ideals, 2-linear ideals and arithmetical rank

Abstract

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) [BM1], Ha (2006) [H], Ha and Morales (2009) [HM]. Let S be a polynomial ring and I ⊂ S a homogeneous ideal defining a sequence of linearly joined varieties. • We compute depth S / I . • We prove that c ( V ) = depth S / I − 1 , 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].