# Curves on K3 surfaces

## TL;DR

This paper advances the understanding of rational and integral curves on K3 surfaces by completing conjectural cases, introducing new deformation techniques, and proving the existence of such curves across various characteristics.

## Contribution

It introduces two novel deformation methods, 'regeneration' and the 'marked point trick', to establish the existence of rational and integral curves on K3 surfaces in all characteristics.

## Key findings

- Complete remaining cases of the rational curves conjecture in characteristic zero.
- Prove almost all cases of the conjecture in positive characteristic.
- Establish existence of integral curves of arbitrary degree and genus on K3 surfaces.

## Abstract

We complete the remaining cases of the conjecture predicting existence of infinitely many rational curves on K3 surfaces in characteristic zero, prove almost all cases in positive characteristic and improve the proofs of the previously known cases. To achieve this, we introduce two new techniques in the deformation theory of curves on K3 surfaces. Regeneration, a process opposite to specialisation, which preserves the geometric genus and does not require the class of the curve to extend, and the marked point trick, which allows a controlled degeneration of rational curves to integral ones in certain situations. Combining the two proves existence of integral curves of unbounded degree of any geometric genus g for any projective K3 surface in characteristic zero.