Degeneration and curves on K3 surfac

TL;DR

This paper introduces a new method for constructing curves on K3 surfaces, proving that a dense subset of quartic K3 surfaces contains infinitely many rational curves, supporting a longstanding conjecture.

Contribution

It presents a novel approach for constructing curves on K3 surfaces with degenerations and proves the conjecture for a dense subset of quartic K3 surfaces.

Findings

A Zariski open dense subset of quartic K3 surfaces satisfies the rational curves conjecture.

The new technique allows construction of various positive genus curves.

Supports the conjecture that all K3 surfaces contain infinitely many rational curves.

Abstract

This paper studies curves on quartic K3 surfaces, or more generally K3 surfaces which are complete intersection in weighted projective spaces. A folklore conjecture concerning rational curves on K3 surfaces states that all K3 surfaces contain infinite number of irreducible rational curves. It is known that all K3 surfaces, except those contained in the countable union of hypersurfaces in the moduli space of K3 surfaces satisfy this property. In this paper we present a new approach for constructing curves on varieties which admit nice degenerations. We apply this technique to the above problem and prove that there is a Zariski open dense subset in the moduli space of quartic K3 surfaces whose members satisfy the conjecture. Various other curves of positive genus can be also constructed.