On the geometry of complete intersection toric varieties

Michelacakis, Nickolas; Thoma, Apostolos

doi:10.1007/s00013-005-1652-z

Cited by 3 publications

(1 citation statement)

References 9 publications

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“…Therefore it is interesting to find better criteria for establishing that a lattice ideal is or is not a complete intersection. In this direction such criteria were given in [17] which can be read off from the geometry of the cone σ L . Our next Theorem provides a criterion depending on the lattices associated to the faces of the cone σ L .…”

Section: Applicationsmentioning

confidence: 99%

Binomial generation of the radical of a lattice ideal

2010

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Section: Applicationsmentioning

confidence: 99%

Binomial generation of the radical of a lattice ideal

2010

Self Cite

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CIO and ring graphs: Deficiency and testing

Gitler

Reyes

Vega

2017

Journal of Symbolic Computation

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Frobenius vectors, Hilbert series and gluings of affine semigroups

Assi¹,

García-Sánchez²,

Ojeda³

2015

J. Commut. Algebra

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Let S1 and S2 be two affine semigroups and let S be the gluing of S1 and S2. Several invariants of S are then related to those of S1 and S2; we review some of the most important properties preserved under gluings. The aim of this paper is to prove that this is the case for the Frobenius vector and the Hilbert series. Applications to complete intersection affine semigroups are also given. On gluins of affine semigroupsIn this section we take a quick tour summarizing some of the more relevant results on the gluing of affine semigroups. We also introduce concepts and notations that will be used later on in the paper.An affine semigroup S is finitely generated submonoid of Z m for some positive integer m. If S ∩ (−S) = 0, that is to say S is reduced, it can be shown that it has a unique minimal system of generators (see for instance [24, Chapter 3]). The cardinality of the minimal generating system of S is known as the embedding dimension of S. Recall that each reduced affine semigroup can be embedded into N m for some m. In the following we will assume that our affine semigroups are submonoids of N m .Given an affine semigroup S ⊆ N m , denote by G(S) the group spanned by S, that is,Let A be the minimal generating system of S, and A = A 1 ∪ A 2 be a nontrivial partition of A. Let S i = A i (the monoid generated by A i ), i ∈ {1, 2}. Then S = S 1 + S 2 . We say that S is the gluing of S 1 and S 2 by d if • d ∈ S 1 ∩ S 2 and, • G(S 1 ) ∩ G(S 2 ) = dZ. We will denote this fact by S = S 1 + d S 2 .There are several properties that are preserved under gluings, and also some invariants of a gluing S 1 + d S 2 can be computed by knowing their values in S 1 and S 2 . We summarize some of them next.Assume that A = {a 1 , . . . , a k }. The monoid homomorphism ϕ : N k → S induced by e i → a i , i ∈ {1, . . . , k} is an epimorphism (where e i is the ith row of the k × k identity matrix). Thus S is isomorphic as a monoid to N k / ker ϕ, where ker ϕ is the kernel congruence of ϕ, that is, the set of pairs (a, b) ∈ N k ×N k with ϕ(a) = ϕ(b). A presentation of S is a system of generators of ker ϕ. A minimal presentation is a presentation such that none of its proper subsets is a presentation. All minimal presentations have the same (finite) cardinality (see for instance [24, Corollary 9.5]). Suppose that S = S 1 + d S 2 , with S i = A i , i ∈ {1, 2} and A = A 1 ∪ A 2 a nontrivial partition of A. We may assume without loss of generality that A 1 = {a 1 , . . . , a l } and A 2 = {a l+1 , . . . , a k }. According to [21, Theorem 1.4], if we know minimal presentations ρ 1 and ρ 2 of S 1 and S 2 , respectively, then ρ = ρ 1 ∪ ρ 2 ∪ {(a, b)}

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On the geometry of complete intersection toric varieties

Cited by 3 publications

References 9 publications

Binomial generation of the radical of a lattice ideal

Binomial generation of the radical of a lattice ideal

CIO and ring graphs: Deficiency and testing

Frobenius vectors, Hilbert series and gluings of affine semigroups

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