ACM tilting bundles on a Geigle-Lenzing projective plane of type $(2,2,2,p)$

TL;DR

This paper investigates ACM tilting bundles on a specific Geigle-Lenzing projective plane, classifying and constructing them, and revealing their connection to certain infinite-dimensional algebras.

Contribution

It classifies ACM tilting bundles on the Geigle-Lenzing plane of type (2,2,2,p) and provides a construction method linking them to (almost) 2-representation infinite algebras.

Findings

A tilting bundle of line bundles is the 2-canonical tilting bundle up to shift.

A construction program for ACM tilting bundles is established.

Classification of ACM tilting bundles on the given Geigle-Lenzing plane is achieved.

Abstract

Let $\mathbb{X}$ be a Geigle-Lenzing projective plane of type $(2,2,2,p)$ and $\mathsf{coh} \mathbb{X}$ the category of coherent sheaves on $\mathbb{X}$. This paper is devoted to study ACM tilting bundles over $\mathbb{X}$, that is, tilting objects in the derived category $\mathsf{D}^{\rm b}(\mathsf{coh} \, \mathbb{X})$ that are also ACM bundles. We show that a tilting bundle consisting of line bundles is the $2$-canonical tilting bundle up to degree shift. We also provide a program to construct ACM tilting bundles, which give a rich source of (almost) $2$-representation infinite algebras. As an application, we give a classification result of ACM tilting bundles.