The Tangent Cone of a Local Ring of Codimension 2

Abstract

Let $(S, \mathfrak {n})$ be a regular local ring and let $I \subseteq \mathfrak {n}^{2}$ be a perfect ideal of S. Sharp upper bounds on the minimal number of generators of I are known in terms of the Hilbert function of R = S/I. Starting from information on the ideal I, for instance the minimal number of generators, a difficult task is to determine good bounds on the minimal number of generators of the leading ideal I ^∗ which defines the tangent cone of R or to give information on its graded structure. Motivated by papers of S. C. Kothari and S. Goto et al. concerning the leading ideal of a complete intersection I = (f, g) in a regular local ring, we present results provided ht (I) = 2. If I is a complete intersection, we prove that the Hilbert function of R determines the graded Betti numbers of the leading ideal and, as a consequence, we recover most of the results of the previously quoted papers. The description is more complicated if ν(I) > 2 and a careful investigation can be provided when ν(I) = 3. Several examples illustrating our results are given.