Rational curves on lattice-polarised 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.

37 pages. Fixed mistake in Proposition 3.10. Thanks to A. Knutsen and an anonymous referee for pointing this out