Embedding pointed curves in K3 surfaces

TL;DR

This paper studies how pointed algebraic curves can be embedded into K3 surfaces with rational curves, exploring related enumerative problems and applications to del Pezzo fibrations.

Contribution

It extends the understanding of curve embeddings into K3 surfaces by analyzing morphisms with marked points and solving enumerative problems with applications to fibrations.

Findings

Solved specific enumerative problems involving curve embeddings.

Established conditions for morphisms from pointed curves to K3 surfaces.

Applied results to the existence of sections in del Pezzo fibrations.

Abstract

We analyze morphisms from pointed curves to K3 surfaces with a distinguished rational curve, such that the marked points are taken to the rational curve, perhaps with specified cross ratios. This builds on work of Mukai and others characterizing embeddings of curves into K3 surfaces via non-abelian Brill-Noether theory. Our study leads naturally to enumerative problems, which we solve in several specific cases. These have applications to the existence of sections of del Pezzo fibrations with prescribed invariants.