Right simple singularities in positive characteristic

TL;DR

This paper classifies right simple hypersurface singularities over algebraically closed fields of positive characteristic, revealing only finitely many such singularities exist, and explores the algebraic notion of modality and its properties.

Contribution

It extends the classification of simple singularities to positive characteristic and generalizes the concept of modality within algebraic geometry.

Findings

Only finitely many right simple singularities exist in positive characteristic.

Modality is semicontinuous in any characteristic.

The classification parallels the ADE-series in characteristic zero.

Abstract

We classify isolated hypersurface singularities $f\in K[[x_1,..., x_n]]$, $K$ an algebraically closed field of characteristic $p>0$, which are simple w.r.t. right equivalence, that is, which have no moduli up to analytic coordinate change. For $K=\mathbb R$ or $\mathbb C$ this classification was initiated by Arnol'd, resulting in the famous ADE-series. The classification w.r.t. contact equivalence for $p>0$ was done by Greuel and Kr\"oning with a result similar to Arnol'd's. It is surprising that w.r.t. right equivalence and for any given $p>0$ we have only finitely many simple singularities, i.e. there are only finitely many $k$ such that $A_k$ and $D_k$ are right simple, all the others have moduli. A major point of this paper is the generalization of the notion of modality to the algebraic setting, its behaviour under morphisms, and its relations to formal deformation theory. As an application we show that the modality is semicontinuous in any characteristic.