Singular curves and quasi-hereditary algebras

TL;DR

This paper develops a categorical resolution for singular algebraic curves using sheaves of orders, linking the derived categories of the curve, its normalization, and certain quasi-hereditary algebras, and explores implications for Rouquier dimension.

Contribution

It introduces a new categorical resolution of singular curves via sheaves of orders and establishes connections with quasi-hereditary algebras and derived category embeddings.

Findings

Constructed a categorical resolution using sheaves of orders.

Proved results on Rouquier dimension of the derived category.

Embedded the derived category of perfect complexes into a module category over a quasi-hereditary algebra.

Abstract

In this article we construct a categorical resolution of singularities of an excellent reduced curve $X$, introducing a certain sheaf of orders on $X$. This categorical resolution is shown to be a recollement of the derived category of coherent sheaves on the normalization of $X$ and the derived category of finite length modules over a certain artinian quasi-hereditary ring $Q$ depending purely on the local singularity types of $X$. Using this technique, we prove several statements on the Rouquier dimension of the derived category of coherent sheaves on $X$. Moreover, in the case $X$ is rational and projective we construct a finite dimensional quasi-hereditary algebra $\Lambda$ such that the triangulated category of perfect complexes on $X$ embeds into $D^b(\Lambda-\mathsf{mod})$ as a full subcategory.