Fiber Cone of Codimension 2 Lattice Ideals

Abstract

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 . Let I = n≥0 I n / I n be the Fiber cone of I, we prove thatIn addition, if is infinite and I is radical, noncomplete intersection, then:• I has dimension 3, is reduced, arithmetically Cohen-Macaulay, of minimal degree. Moreover, a presentation of I is effective from the minimal system of generators of I. • An explicit minimal reduction of I is given. • The blow-up ring, or Rees ring 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).

Simplicial ideals, 2-linear ideals and arithmetical rank

“…A particular class of varieties, of important interest in classical Geometry are Cohen-Macaulay varieties of minimal degree, they were classified geometrically by the successive contribution of Del Pezzo (1885) [DP], Bertini (1907) [B], and Xambo (1981) [X] and algebraically in Barile and Morales (2000) [BM2]. They appear naturally studying the fiber cone of a codimension two toric ideals Morales (1995) [M], Gimenez et al (1993Gimenez et al ( , 1999 [GMS1,GMS2], Barile and Morales (1998) [BM1], Ha (2006) [H], Ha and Morales (2009) [HM].…”