Classification of simple 0-dimensional isolated complete intersection singularities

TL;DR

This paper classifies simple 0-dimensional isolated complete intersection singularities in positive characteristic, extending known complex analytic classifications with new methods suitable for positive characteristic fields.

Contribution

It provides the first classification of simple ICIS in positive characteristic, introducing new normal forms and a framework for formal deformation theory in this setting.

Findings

Classification of simple ICIS in positive characteristic completed

New normal forms identified for low characteristic cases

Semicontinuity of modality established in formal deformation context

Abstract

The aim of this article is the classification of simple 0-dimensional isolated complete intersection singularities in positive characteristic. As usual, a singularity is called simple or 0-modal if there are only finitely many isomorphism classes of singularities into which the given singularity can deform. The notion of simpleness was introduced by V. I. Arnold and the classification of low modality singularities has become a fundamental task in singularity theory. Simple complex analytic isolated complete intersection singularities (ICIS) were classified by M. Giusti. However, the classification in positive characteristic requires different methods and is much more involved. The final result is nevertheless similar to the classification in characteristic 0 with some additional normal forms in low characteristic. The theoretical results in this paper mainly concern families of ICIS that are formal in the fiber and algebraic in the base (formal deformation theory is not sufficient). In particular, we give a definition of modality in this situation and prove its semicontinuity.