Simplicial ideals, 2-linear ideals and arithmetical rank

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 .•We prove that , 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].