Algorithmic equiresolution of deformations of embedded varieties

TL;DR

This paper develops a theory for simultaneous resolution of singularities in families of embedded varieties over characteristic zero fields, ensuring compatibility with existing resolution algorithms.

Contribution

It introduces a new framework for resolving singularities uniformly across families of embedded varieties parametrized by artinian rings.

Findings

Established a theory for simultaneous resolution of singularities in families

Proved compatibility with existing resolution algorithms

Extended the resolution framework to families over artinian rings

Abstract

A theory of simultaneous resolution of singularities for families of embedded varieties (over a field of characteristic zero) parametrized by the spectrum of a suitable artinian ring, and compatible with a given algorithm of resolution, is presented. As usually, this a simple consequence of a similar theory for analogous families of basic objects, to which the main portion of this article is devoted.