# Geometry of stable ruled surface over an elliptic curve

## TL;DR

This paper investigates the geometric structure of a stable ruled surface over an elliptic curve, revealing a locally parallelizable web structure formed by the fibration, foliation, and 2-web.

## Contribution

It demonstrates that the 4-web composed of the fibration, foliation, and 2-web on the stable ruled surface is locally parallelizable, providing new geometric insights.

## Key findings

- The stable ruled surface admits a unique transverse foliation.
- The minimal self-intersection sections define a 2-web.
- The combined 4-web is locally parallelizable.

## Abstract

We consider the stable ruled surface $S_1$ over an elliptic curve. There is a unique foliation on $S_1$ transverse to the fibration. The minimal self-intersection sections also define a 2-web. We prove that the 4-web defined by the fibration, the foliation and the 2-web is locally parallelizable.