Weighted projective lines and Hochschild cohomology

TL;DR

This paper computes the Hochschild (co)homology dimensions of weighted projective curves, revealing they depend mainly on the genus and exceptional points, using classical and modern representation theory techniques.

Contribution

It provides a new description of Hochschild (co)homology for weighted projective lines, connecting classical representation theory with modern algebraic geometry.

Findings

Hochschild (co)homology dimensions depend primarily on genus and exceptional points

Concrete realizations of weighted projective lines as quotient stacks

Extension of classical representation-theoretic methods to weighted settings

Abstract

We describe the dimensions of Hochschild (co)homology groups of weighted projective curves over complex numbers. Surprisingly, all but one of those numbers depend only on the genus of the underlying non-weighted curve and the number of exceptional points. Our proof involves revising a classical representation-theoretic argument of Happel together with more recent results of Lenzing and Arinkin, C\u{a}ld\u{a}raru and Hablicsek. We give concrete realizations of a large class of weighted projective lines as quotient stacks. This paper conicides with the author's master's thesis submitted to the University of Bonn in 2019.