Right unimodal and bimodal singularities in positive characteristic

TL;DR

This paper classifies right unimodal and bimodal singularities in positive characteristic fields, providing explicit normal forms, analyzing their adjacencies, and exploring properties like the smoothness of the $$-constant stratum and bounds on the Milnor number.

Contribution

It offers a complete classification of right unimodal and bimodal singularities in positive characteristic, including explicit normal forms and adjacency relations.

Findings

The $$-constant stratum is smooth with dimension equal to the right modality for singularities of modality at most 2.

Finitely many 1- and 2-dimensional families of unimodal and bimodal singularities exist in positive characteristic.

The Milnor number of such singularities is bounded above by 4 times the characteristic p.

Abstract

The problem of classification of real and complex singularities was initiated by Arnol'd in the sixties who classified simple, unimodal and bimodal w.r.t. right equivalence. The classification of right simple singularities in positive characteristic was achieved by Greuel and the author in 2014. In the present paper we classify right unimodal and bimodal singularities in positive characteristic by giving explicit normal forms. Moreover we completely determine all possible adjacencies of simple, unimodal and bimodal singularities. As an application we prove that, for singularities of right modality at most 2, the $\mu$-constant stratum is smooth and its dimension is equal to the right modality. In contrast to the complex analytic case, there are, for any positive characteristic, only finitely many 1-dimensional (resp. 2-dimensional) families of right class of unimodal (resp. bimodal) singularities. We show that for fixed characteristic $p>0$ of the ground field, the Milnor number of $f$ satisfies $\mu(f)\leq 4p$, if the right modality of $f$ is at most 2.