Frobenius morphism and vector bundles on cycles of projective lines

TL;DR

This paper investigates how the Frobenius morphism acts on indecomposable vector bundles over cycles of projective lines, providing insights relevant to Hilbert--Kunz theory for plane cubic curves.

Contribution

It offers a detailed description of the Frobenius action on vector bundles in this geometric setting, answering a question posed by Paul Monsky.

Findings

Explicit description of Frobenius action on vector bundles

Resolution of Monsky's question in the context of cycles of projective lines

Implications for Hilbert--Kunz theory for plane cubic curves

Abstract

In this paper we describe the action of the Frobenius morphism on the indecomposable vector bundles on cycles of projective lines. This gives an answer on a question of Paul Monsky, which appeared in his study of the Hilbert--Kunz theory for plane cubic curves.