Minimal generation of ideals of algebraic varieties at points with multilinear tangent cones (Q480394)

Let \(V\subset \mathbb A^n\) be an affine variety defined over an algebraically closed field of characteristic \(0\). let \(P\in V\) be an isolated singularity of \(V\). The authors assume that the normalization of \(V\) is regular at \(P\) and that the tangent cone \(W\) of \(V\) at \(P\) consists of a union of linear spaces, all of the same dimension. With the assumptions, the authors prove that the minimal number of generators for the local ideal at \(P\) of \(V\) equals the minimal number of generators for the ideal of \(W\). The result extends to varieties \(V\) of arbitrary dimension a previous theorem of the authors, limited to the case \(\dim(V)=1,2\).