Simple vector bundles on plane degenerations of an elliptic curve

TL;DR

This paper extends Atiyah's classification of simple vector bundles from elliptic curves to all reduced plane cubic curves, providing a comprehensive description based on rank, multidegree, and determinant.

Contribution

It generalizes Atiyah's classification by describing simple vector bundles on all reduced plane cubic curves using representation theory of boxes.

Findings

Simple vector bundles are determined by rank, multidegree, and determinant.

Explicit universal families of simple vector bundles are constructed.

The approach employs the representation theory of boxes for classification.

Abstract

In 1957 Atiyah classified simple and indecomposable vector bundles on an elliptic curve. In this article we generalize his classification by describing the simple vector bundles on all reduced plane cubic curves. Our main result states that a simple vector bundle on such a curve is completely determined by its rank, multidegree and determinant. Our approach, based on the representation theory of boxes, also yields an explicit description of the corresponding universal families of simple vector bundles.