# which deformation of a matrix lead to flat deformations of determinantal varieties (fitting ideals)?

Let $(R,m)$ be a complete local ring over a field (of char=0). Consider a (not necessarily square) matrix $A$ over $R$. Consider its fitting ideal, $I_j(A)$. In general, a deformation of the matrix, $A\to A+\epsilon B$, does not lead to a flat family of fitting ideals $I_j(A+\epsilon B)$. (A notable nice case: the ideal $I_0(A)$ in the case of square matrices :) )

Are there some known sufficient conditions on $A$, $B$ to ensure flatness? At least for a given $I_j(A)$? An immediate necessary condition is that the codimension of $I_j(A)$ is the "expected one". I guess this is far from being sufficient.

What about: $I_j(A)$ has the expected codimension and all the entries of $B$ belong to a high enough power of $m\subset R$?

Well, at least the case: the ideal of maximal minors $I_0(A)$, of expected codimension, such that the ring $R/I_0(A)$ is an isolated curve singularity (though not necessarily reduced). If we perturb the entries of $A$ by elements of $m^N$, for $N\gg0$, will we get a flat deformation of this curve?

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