Fiber Cone of Codimension 2 Lattice Ideals

Abstract

Let I ⊂ ℛ: = 𝒦[x 1, x 2, …, x r ] be a codimension two lattice ideal. In this article we study the arithmetic properties of the blow-up of the ideal I in ℛ. Let ℱ(I) = ⨁ n≥0 I n /𝔪 I n be the Fiber cone of I, we prove that In addition, if 𝒦 is infinite and I is radical, noncomplete intersection, then: ℱ(I) has dimension 3, is reduced, arithmetically Cohen–Macaulay, of minimal degree. Moreover, a presentation of ℱ(I) is effective from the minimal system of generators of I. An explicit minimal reduction of ℱ(I) is given. The blow-up ring, or Rees ring ℛ(I) = ⨁ n≥0 I n , is arithmetically Cohen–Macaulay and has a presentation by linear and quadratic forms. This article completes and extends to the general case of codimension 2 lattice ideals previous results for the simplicial toric case by Morales and Simis (1992), Gimenez et al. (1999), and Barile and Morales (1998).