The Tangent Cone of a local ring of codimension 2

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 find 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, S. Goto ${\it{et al.}}$ 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.

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