Categorical resolutions, poset schemes and Du Bois singularities

TL;DR

This paper introduces poset schemes and demonstrates their use in providing categorical resolutions of singularities, establishing a link with Du Bois singularities and defining the de Rham-Du Bois complex via these schemes.

Contribution

It defines poset schemes, shows their role in categorical resolutions of singularities, and connects Du Bois singularities with the existence of such resolutions.

Findings

Smooth poset schemes can resolve singularities categorically.

A variety has a categorical resolution iff it has Du Bois singularities.

The de Rham-Du Bois complex can be constructed using suitable poset schemes.

Abstract

We introduce the notion of a poset scheme and study the categories of quasi-coherent sheaves on such spaces. We then show that smooth poset schemes may be used to obtain categorical resolutions of singularities for usual singular schemes. We prove that a singular variety $X$ possesses such a resolution if and only if $X$ has Du Bois singularities. Finally we show that the de Rham-Du Bois complex for an algebraic variety $Y$ may be defined using any smooth poset scheme which satisfies the descent over $Y$ in the classical topology.