When does a perturbation of the equations preserve the normal cone?

Abstract:

Let $(R,\frak {m})$ be a local ring and $I, J$ two arbitrary ideals of $R$. Let $gr_J(R/I)$ denote the associated graded ring of $R/I$ with respect to $J$, which corresponds to the normal cone in algebraic geometry. With regards to the finite determinacy of singularity with respect to the Jacobian ideal we study the problem for which ideal $I = (f_1,\dots ,f_r)$ does there exist a number $N$ such that if $f_i’ \equiv f_i \mod J^N$, $i = 1,\dots ,r$, and $I’ = (f_1’,\dots ,f_r’)$, then $gr_J(R/I) \cong gr_J(R/I’)$. This problem arises from a recent result of Ma, Quy and Smirnov in the case $J$ is an $\frak {m}$-primary ideal, which solves a long standing conjecture of Srinivas and Trivedi on the invariance of Hilbert functions under small perturbations. Their approach involves Hilbert functions and cannot be used to study the above general problem. Our main result shows that such a number $N$ exists if $f_1,\dots ,f_r$ is locally a regular sequence outside the locus of $J$. It has interesting applications to a range of related problems.