The Universal Severi Variety of Rational Curves on K3 Surfaces

TL;DR

This paper studies the universal Severi variety of rational curves on K3 surfaces, exploring its irreducibility using deformation theory and providing new proofs for high genus cases.

Contribution

It advances understanding of the irreducibility of universal Severi varieties on K3 surfaces, especially for primitive classes and high genus cases.

Findings

Partial results on irreducibility for primitive classes

A new proof of irreducibility for high genus curves

Insights into deformation theory of stable maps on K3 surfaces

Abstract

We investigate the universal Severi variety of rational curves on K3 surfaces, which parametrises irreducible rational curves in a fixed class on varying K3 surfaces of fixed genus. We investigate the conjecuted irreducibility of this space in the case of primitive classes using the deformation theory of stable maps and obtain partial results. In an appendix we give a short proof of the irreducibility of universal Severi varieties of high genus curves on K3 surfaces.