Elliptic ruled surfaces over arbitrary characteristic fields

TL;DR

This paper investigates elliptic ruled surfaces over fields of any characteristic, analyzing their elliptic fibrations and singular fibers based on Atiyah's classification of vector bundles on elliptic curves.

Contribution

It extends the understanding of elliptic ruled surfaces by characterizing when they admit elliptic fibrations and describing the types of singular fibers that occur, in arbitrary characteristic.

Findings

Classification of elliptic ruled surfaces with elliptic fibrations

Identification of possible singular fiber types

Extension of Atiyah's vector bundle classification to surface structures

Abstract

Atiyah classifies vector bundles on elliptic curves $E$ over an algebraically closed field of any characteristic. On the other hand, a rank $2$ vector bundle on $E$ defines a surface $S$ with a $\mathbb{P}^1$-bundle structure on $E$. We study when $S$ has an elliptic fibration according to the Atiyah's classification, and what kinds of singular fibers appear.