Partial local resolution by characteristic zero methods

TL;DR

This paper explores how characteristic zero techniques for resolving singularities can be partially applied to improve singularities in various classes, including generic determinantal varieties, with implications for positive or mixed characteristic cases.

Contribution

It demonstrates the partial applicability of characteristic zero methods to specific singularities, providing a foundation for constructing resolutions in broader contexts.

Findings

Resolution via blowing ups in regular centers for certain singularities

Application to generic determinantal varieties

Guidance for non-experts on extending resolution techniques

Abstract

We discuss to what extent the local techniques of resolution of singularities over fields of characteristic zero can be applied to improve singularities in general. For certain interesting classes of singularities, this leads to an embedded resolution via blowing ups in regular centers. We illustrate this for generic determinantal varieties. The article is partially expository and is addressed to non-experts who aim to construct resolutions for other special classes of singularities in positive or mixed characteristic.