On the conductor of a surface at a point whose projectivized tangent cone is a generic union of lines (Q2717183)

Let \(x\) be a singular point of a projective surface \(S\subset \mathbb{P}^r\) and let \((A,m)\) be the local ring of \(S\) at \(x\). Call \(G\) the corresponding tangent cone. The author assumes that the normalization \(\overline A \) of \(A\) is regular and \(G\) splits in a union of planes, in such a way that its projectification is a union of lines in generic position (i.e. with generic Hilbert function). With this setting, the author proves that the conductor of \(A\) in \(\overline A\) is a power of \(m\). This result extends to higher dimension the known fact that a curve singularity, whose projectivized tangent cone is a set of points in generic position, has conductor equal to a power of the maximal ideal.NEWLINENEWLINEFor the entire collection see [Zbl 0960.00042].