Complete intersection lattice ideals

Set-theoretic complete intersection lattice ideals in monoid rings

“…Further, if the characteristic of k is zero, and if the lattice ideal I is set-theoretic complete intersection on binomials on a fixed lattice ideal J (i.e., I/J is set-theoretic complete intersection on binomials), then I is a complete intersection on J (i.e., I/J is a complete intersection). This is a generalization of results in [2,9,16]. In Proposition 2.3, we prove that, if a lattice ideal of height r is generated by r − 1 binomials and one polynomial G up to radical, then G is congruent to a power of a binomial modulo the lattice ideal generated by the given binomials.…”

“…, F (v s ) define a lattice ideal I (W ) and that V /W is torsion free, if the characteristic of k is zero. Further, Theorem 2.1(3) is a generalization of [2, Theorem 2], [9, Theorem 2.1] and [16,Corollary 3.10], in which it was proved that a lattice ideal is a complete intersection, if it is set-theoretic complete intersection on binomials, and if the characteristic of k is zero. It says that the same result holds on a fixed lattice ideal.…”

On the geometry of complete intersection toric varieties

“…of the affine semigroup INA) is a complete intersection cone. In this description, it turns out that the key concept needed is that of the s-gluing of semigroups, a generalization of the gluing and p-gluing of semigroups, see relatively [1,5]. Definition 4.1 Let E 1 , E 2 be two nonempty subsets of {1, .…”

“…M. Morales and A. Thoma [5] have characterized the subsemigroups INA with no invertible elements that are complete intersection, A is a finite set of elements of Z Z n ⊕ T with T a torsion group. Semigroups of this kind correspond to lattice ideals.…”

“…To any such semigroup, one can associate the cone generated by the non torsion parts of the elements of A. It follows from the results in [5] and Theorem 4.3 that complete intersection lattice ideals have also complete intersection cones.…”