Wonderful resolutions and categorical crepant resolutions of singularities

TL;DR

This paper introduces the concept of wonderful resolutions for Gorenstein singularities and demonstrates that such resolutions lead to categorical crepant resolutions, with applications to determinantal varieties.

Contribution

It defines wonderful resolutions for singularities and proves their link to categorical crepant resolutions, extending to determinantal varieties and exploring minimality notions.

Findings

Wonderful resolutions imply categorical crepant resolutions for certain singularities.

All determinantal varieties from generic matrices admit categorical crepant resolutions.

Links between minimality and crepancy in categorical resolutions are discussed.

Abstract

Let $X$ be an algebraic variety with Gorenstein singularities. We define the notion of a wonderful resolution of singularities of $X$ by analogy with the theory of wonderful compactifications of semi-simple linear algebraic groups. We prove that if $X$ has rational singularities and has a wonderful resolution of singularities, then $X$ admits a categorical crepant resolution of singularities. As an immediate corollary, we get that all determinantal varieties defined by the minors of a generic square/symmetric/skew-symmetric matrix admit categorical crepant resolution of singularities. We also discuss notions of minimality for a categorical resolution of singularities and we explore some links between minimality and crepancy for such resolutions.