Algorithmic equiresolution of possibly non-reduced families of singularities

TL;DR

This paper explores the concept of algorithmic equiresolution for families of singularities, including non-reduced parameter spaces, establishing conditions under which various definitions are equivalent.

Contribution

It introduces a new definition of algorithmic equiresolution applicable to non-reduced parameter spaces and proves equivalence of definitions when the parameter space is regular.

Findings

Proposes a new definition of algorithmic equiresolution for non-reduced families.

Shows equivalence of different definitions when the parameter space is regular.

Extends the theory of equiresolution to more general families of singularities.

Abstract

This paper studies the concept of algorithmic equiresolution of a family of embedded varieties or ideals, which means a simultaneous resolution of such a family compatible with a given (suitable) algorithm of resolution in characteristic zero. The paper's approach is more indirect: it primarily considers the more general case of families of basic objects (or marked ideals). A definition of algorithmic equiresolution is proposed, which applies to families whose parameter space T may be non-reduced, e.g., the spectrum of a suitable artinian ring. Other definitions of algorithmic equiresolution are also discussed. These are geometrically very natural, but the parameter space T of the family must be assumed regular. It is proven that when T is regular all the proposed definitions are equivalent.