  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 $\nu(I) >2$ and a careful
investigation can be provided when $\nu(I)=3. $ Several examples illustrating
our results are given.
