Singular Rational Curves on Elliptic K3 Surfaces

TL;DR

The paper proves the existence of infinitely many rational curves with unbounded genus on elliptic K3 surfaces and shows their lifts are Zariski dense, providing a new proof of Kobayashi's theorem in this context.

Contribution

It demonstrates the existence of unbounded rational curves on elliptic K3 surfaces and their density, offering a novel proof of Kobayashi's theorem for these surfaces.

Findings

Existence of rational curves with unbounded genus on elliptic K3 surfaces.

Lifts of these curves are dense in the Zariski topology.

Provides a new proof of Kobayashi's theorem in the elliptic case.

Abstract

We show that on every elliptic K3 surface $X$ there are rational curves $(R_i)_{i\in \mathbb{N}}$ such that $R_i^2 \to \infty$, i.e., of unbounded arithmetic genus. Moreover, we show that the union of the lifts of these curves to $\mathbb{P}(\Omega_X)$ is dense in the Zariski topology. As an application we give a simple proof of a theorem of Kobayashi in the elliptic case, i.e., there are no globally defined symmetric differential forms.