Resolution of Singularities of Germs in Characteristic Positive Associated with Valuation Rings of Iterated Divisor Type

TL;DR

This paper demonstrates that hypersurface singularities in positive characteristic can be resolved using iterated monoidal transformations linked to valuation rings of iterated divisor type, introducing new concepts for singularity resolution.

Contribution

It introduces a novel approach to resolving singularities in positive characteristic via valuation rings of iterated divisor type and develops foundational concepts for this method.

Findings

Resolution achieved through iterated monoidal transformations

Introduces new concepts like space germs and reduction sequences

Applicable to hypersurface singularities in positive characteristic

Abstract

In this paper we show that any hypersurface singularities of germs of varieties in positive characteristic can be resolved by iterated monoidal transformations in centers in smooth subvarieties, if we have a valuation ring of iterated divisor type associated with the germ. Besides, we introduce fundamental concepts for the study of resolution of singularities of germs such as space germs, iterated analytic monoidal transformations with a normal crossing, Weierstrass representations, reduction sequences, and so forth.