Minitwistor spaces, Severi varieties, and Einstein-Weyl structure

TL;DR

This paper demonstrates that 3-dimensional Severi varieties of nodal rational curves on non-singular projective surfaces naturally possess Einstein-Weyl structures, extending Hitchin's work on smooth rational curves and exploring associated geometric features.

Contribution

It generalizes Einstein-Weyl structures to Severi varieties of nodal rational curves and investigates their geometric properties and explicit examples.

Findings

Severi varieties of rational curves have natural Einstein-Weyl structures.

Null surfaces and geodesics are characterized on these Severi varieties.

Real structures induce positive definite Einstein-Weyl manifolds.

Abstract

In this paper we show that the space of nodal rational curves, which is so called a Severi variety (of rational curves), on any non-singular projective surface is always equipped with a natural Einstein-Weyl structure, if the space is 3-dimensional. This is a generalization of the Einstein-Weyl structure on the space of smooth rational curves on a complex surface, given by N.Hitchin. As geometric objects naturally associated to Einstein-Weyl structure, we investigate null surfaces and geodesics on the Severi varieties. Also we see that if the projective surface has an appropriate real structure, then the real locus of the Severi variety becomes a positive definite Einstein-Weyl manifold. Moreover we construct various explicit examples of rational surfaces having 3-dimensional Severi varieties of rational curves.