Complete Intersections in Binomial and Lattice Ideals

Abstract

For the family of graded lattice ideals of dimension 1, we establish a complete intersection criterion in algebraic and geometric terms. In positive characteristic, it is shown that all ideals of this family are binomial set theoretic complete intersections. In characteristic zero, we show that an arbitrary lattice ideal which is a binomial set theoretic complete intersection is a complete intersection.2010 Mathematics Subject Classification. Primary 13F20; Secondary 14H45, 13P25, 11T71.

Minimal generating sets of lattice ideals

“…Otherwise, we say that L is a non-positive lattice. Almost all results in the literature deal with positive lattices, with few exceptions like in [5,13,15,17,21,22,27]. We survey below several well known facts for positive lattices, as they pertain to our study.…”

Computing the degree of a lattice ideal of dimension one

“…Lattice ideals have been studied extensively, see [13,16,24,27,28] The next lemma gives the general form of the binomials in a lattice ideal.…”

“…Lattice ideals have been studied extensively, see [13,16,24,27,28] and the references therein. The concept of a lattice ideal is a natural generalization of a toric ideal [38, Given a binomial g = t a − t b , we set g = a− b.…”

“…The complete intersection property of I(X) was recently characterized in [24] in algebraic and geometric terms (see also [34]). If X is parameterized by the edges of a clutter, then I(X) is a complete intersection if and only if X is a projective torus [33].…”

Vanishing ideals over complete multipartite graphs

“…[17,18,19,20]) and also focusing on the vanishing ideal of parameterized algebraic toric sets (cf. [15,16]).…”