Right unimodal and bimodal singularities in positive characteristic

Abstract

The problem of classification of real and complex singularities was initiated by Arnol'd in the sixties who classified simple, unimodal and bimodal singularities w.r.t. right equivalence. The classification of simple singularities 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. It is surprising that in positive characteristic, there are no infinite series of unimodal and bimodal singularities. Moreover, the Milnor number of simple, unimodal and bimodal singularity satisfies $\mu(f)\leq 4p$. 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.