In this paper we completely characterize lattice ideals that are complete intersections or equivalently complete intersections finitely generated semigroups of ZZn ⊕ T with no invertible elements, where T is a finite abelian group. We also characterize the lattice ideals that are set-theoretic complete intersections on binomials.

We present examples which show that in dimension higher than one or codimension higher than two, there exist toric ideals IA such that no binomial ideal contained in IA and of the same dimension is a complete intersection. This result has important implications in sparse elimination theory and in the study of the Horn system of partial differential equations.

This is an expository account based mainly on an article by Jack Ohm titled “Space curves as ideal-theoretic intersections”. It also gives a proof of the fact that smooth space curves can be realized as set-theoretic complete intersections. The penultimate section proves the theorem of Cowsik and Nori : Curves in affine n-space of characteristic p are set-theoretic complete intersection.

This paper proves the formulae reg(IJ) ≤ reg(I) + reg(J), reg(I ∩ J) ≤ reg(I) + reg(J) for arbitrary monomial complete intersections I and J , and provides examples showing that these inequalities do not hold for general complete intersections.