Severi Varieties on Ruled Surfaces over Elliptic Curves

TL;DR

This paper investigates Severi varieties on Atiyah ruled surfaces over elliptic curves, demonstrating that their general members typically have nodes and triple points as singularities.

Contribution

It establishes the singularity types of general members of Severi varieties on these specific ruled surfaces, a new result in algebraic geometry.

Findings

General members have nodes as singularities.

They also have ordinary triple points.

Singularity types are characterized for these varieties.

Abstract

We proved that the general members of Severi varieties on an Atiyah ruled surface over a general elliptic curve have nodes and ordinary triple points as singularities.