Kernels of categorical resolutions of nodal singularities

TL;DR

This paper investigates categorical resolutions of nodal singularities, showing that their kernels are generated by spherical objects, and applies these results to nodal cubic fourfolds linked to K3 surfaces.

Contribution

It introduces a new description of kernels of categorical resolutions for nodal singularities, identifying spherical objects as generators, and connects these to geometric objects like K3 surfaces.

Findings

Kernels are generated by 2- or 3-spherical objects depending on dimension.

Explicit description of the kernel for nodal cubic fourfolds.

Links between categorical resolutions and geometric objects like K3 surfaces.

Abstract

In this paper we study derived categories of nodal singularities. We show that for all nodal singularities there is a categorical resolution whose kernel is generated by a $2$ or $3$-spherical object, depending on the dimension. We apply this result to the case of nodal cubic fourfolds, where we describe the kernel generator of the categorical resolution as an object in the bounded derived category of the associated degree six K3 surface. This paper originated from one of the problem sessions at the Interactive Workshop and Hausdorff School "Hyperk\"ahler Geometry", Bonn, September 6-10, 2021.