Right Simple Singularities in Positive Characteristic

View/ Open
Date
2013MFO Scientific Program
OWLF 2013Series
Oberwolfach Preprints;2013,28Author
Greuel, Gert-Martin
Nguyen, Hong Duc
Metadata
Show full item recordOWP-2013-28
Abstract
We classify isolated singularities $f \in K[[x_1,...,x_n]]$, which are simple, i.e. have no moduli, w.r.t. right equivalence, where $K$ is an algebraically closed field of characteristic $p>0$. 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 Kröning with a result similiar to Arnol'd's. It is surprising that w.r.t. right equivalence and 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.