** The Tangent Cone of a Local Ring of Codimension 2 **

* Mousumi Mandal, Maria Evelina Rossi *

**Abstract**

Let *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.