Simultaneous categorical resolutions

TL;DR

This paper introduces the concept of simultaneous categorical resolutions of singularities, providing a new categorical framework for resolving surface singularities and applying it to families of higher-dimensional varieties.

Contribution

It proposes a novel notion of simultaneous categorical resolutions and constructs such resolutions for families of varieties with ordinary double points, linking geometric and categorical properties.

Findings

Verified fiberwise the relative smoothness and properness of geometric triangulated categories.

Constructed a smooth family of K3 categories with specific geometric fibers.

Demonstrated the applicability to families of high-dimensional varieties with singularities.

Abstract

We introduce the notion of a simultaneous categorical resolution of singularities, a categorical version of simultaneous resolutions of rational double points of surface degenerations. Furthermore, we suggest a construction of simultaneous categorical resolutions which, in particular, applies to the case of a flat projective 1-dimensional family of varieties of arbitrarily high even dimension with ordinary double points in the total space and central fiber. As an ingredient of independent interest, we check that the property of a geometric triangulated category linear over a base to be relatively smooth and proper can be verified fiberwise. As an application we construct a smooth and proper family of K3 categories with general fiber the K3 category of a smooth cubic fourfold and special fiber the derived category of the K3 surface of degree 6 associated with a singular cubic fourfold.