# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1907.01208