The Milnor number of a hypersurface singularity in arbitrary characteristic

TL;DR

This paper investigates the behavior of the Milnor number for hypersurface singularities over fields of arbitrary characteristic, establishing conditions for its invariance and relating it to algebraic properties of the singularity.

Contribution

It provides necessary and sufficient conditions for the invariance of the Milnor number under contact equivalence in arbitrary characteristic and links it to the smoothness of the generic fiber.

Findings

Milnor number is not invariant under contact equivalence in positive characteristic.

Conditions are established for the invariance of the Milnor number.

The Milnor number equals the conductor of a plane branch under certain conditions.

Abstract

The Milnor number of an isolated hypersurface singularity, defined as the codimension $\mu(f)$ of the ideal generated by the partial derivatives of a power series $f$ whose zeros represent locally the hypersurface, is an important topological invariant of the singularity over the complex numbers, but its meaning changes dramatically when the base field is arbitrary. It turns out that if the ground field is of positive characteristic, this number is not even invariant under contact equivalence of the local equation $f$. In this paper we study the variation of the Milnor number in the contact class of $f$, giving necessary and sufficient conditions for its invariance. We also relate, for an isolated singularity, the finiteness of $\mu(f)$ to the smoothness of the generic fiber $f=s$. Finally, we prove that the Milnor number coincides with the conductor of a plane branch when the characteristic does not divide any of the minimal generators of its semigroup of values, showing in particular that this is a sufficient (but not necessary) condition for the invariance of the Milnor number in the whole equisingularity class of $f$.