Rational curves on lattice-polarised K3 surfaces

TL;DR

This paper proves that generic lattice-polarized K3 surfaces contain integral nodal rational curves in certain linear systems, using degeneration techniques, and extends previous results to higher rank lattices.

Contribution

It establishes the existence of rational curves on generic lattice-polarized K3 surfaces, strengthening prior work and applying degeneration methods to higher rank lattices.

Findings

Existence of integral nodal rational curves on generic K3 surfaces with lattice polarization.

Method extends to many higher rank lattices.

Strengthens previous results on rational curves on K3 surfaces.

Abstract

Fix a K3 lattice $\Lambda$ of rank two and $L\in\Lambda$ a big and nef divisor that is positive enough. We prove that the generic $\Lambda$-polarised K3 surface has an integral nodal rational curve in the linear system $|L|$, in particular strengthening previous work of the first named author. The technique is by degeneration, and also works for many lattices of higher rank.