Invariants of Hypersurface Singularities in Positive Characteristic

TL;DR

This paper investigates hypersurface singularities over fields of positive characteristic, establishing links between finiteness conditions and determinacy, and analyzing non-degeneracy conditions and their implications for singularity invariants.

Contribution

It provides new bounds for determinacy degrees and explores the behavior of non-degenerate singularities in positive characteristic, extending classical results.

Findings

Finiteness of Milnor and Tjurina numbers is equivalent to finite determinacy.

Planar Newton non-degenerate singularities satisfy Milnor's formula in positive characteristic.

Absence of wild vanishing cycles in certain non-degenerate singularities.

Abstract

We study singularities f in K[[x_1,...,x_n]] over an algebraically closed field K of arbitrary characteristic with respect to right respectively contact equivalence, and we establish that the finiteness of the Milnor respectively the Tjurina number is equivalent to finite determinacy. We give improved bounds for the degree of determinacy in positive characteristic. Moreover, we consider different non-degeneracy conditions of Kouchnirenko, Wall and Beelen-Pellikaan in positive characteristic, and we show that planar Newton non-degenerate singularities satisfy Milnor's formula mu=2 delta-r+1. This implies the absence of wild vanishing cycles in the sense of Deligne.