The full exceptional collections of categorical resolutions of curves

TL;DR

This paper characterizes which singular projective curves admit categorical resolutions with full exceptional collections, showing it only occurs for genus 0 curves, and that higher genus curves cannot have tilting objects in their resolutions.

Contribution

It provides a complete classification of singular curves with categorical resolutions admitting full exceptional collections, based on their geometric genus.

Findings

Full exceptional collections exist only for genus 0 curves.

Curves with genus ≥ 1 cannot have tilting objects in their categorical resolutions.

The proofs involve detailed analysis of Grothendieck and Picard groups.

Abstract

This paper gives a complete answer of the following question: which (singular, projective) curves have a categorical resolution of singularities which admits a full exceptional collection? We prove that such full exceptional collection exists if and only if the geometric genus of the curve equals to 0. Moreover we can also prove that a curve with geometric genus equal or greater than 1 cannot have a categorical resolution of singularities which has a tilting object. The proofs of both results are given by a careful study of the Grothendieck group and the Picard group of that curve.