Rational curves on elliptic K3 surfaces

TL;DR

This paper proves that non-isotrivial elliptic K3 surfaces over algebraically closed fields contain infinitely many rational curves, extending known results to arbitrary characteristic and providing new insights into their geometric structure.

Contribution

It establishes the existence of infinitely many rational curves on non-isotrivial elliptic K3 surfaces over fields of any characteristic, generalizing previous characteristic-zero results.

Findings

Infinitely many rational curves exist on non-isotrivial elliptic K3 surfaces.

The result holds in arbitrary characteristic, including cases where characteristic is 2 or 3.

Extension of known results from characteristic zero to positive characteristic.

Abstract

We prove that any non-isotrivial elliptic K3 surface over an algebraically closed field $k$ of arbitrary characteristic contains infinitely many rational curves. In the case when $\mathrm{char}(k)\neq 2,3$, we prove this result for any elliptic K3 surface. When the characteristic of $k$ is zero, this result is due to the work of Bogomolov-Tschinkel and Hassett.