Helices on del Pezzo surfaces and tilting Calabi-Yau algebras

TL;DR

This paper explores the relationship between helices on del Pezzo surfaces, tilting Calabi-Yau algebras, and quiver mutations, providing a comprehensive description of the tilting process in this geometric context.

Contribution

It establishes a connection between tilting operations on Calabi-Yau algebras and mutations of exceptional collections on del Pezzo surfaces, extending understanding of algebraic and geometric structures.

Findings

Complete description of tilting via quiver mutations

Relation between helices and Calabi-Yau algebra tilting

Application of Herzog, Kuleshov, and Orlov's results

Abstract

We study tilting for a class of Calabi-Yau algebras associated to helices on Fano varieties. We do this by relating the tilting operation to mutations of exceptional collections. For helices on del Pezzo surfaces the algebras are of dimension three, and using an argument of Herzog, together with results of Kuleshov and Orlov, we obtain a complete description of the tilting process in terms of quiver mutations.