On the existence of rational curves on projective hyperk\"ahler fourfolds

TL;DR

This paper proves that certain projective hyperk"ahler fourfolds contain rational curves if they have specific boundary divisors, supporting a special case of Oguiso's conjecture.

Contribution

It establishes the existence of rational curves on hyperk"ahler fourfolds under new conditions related to boundary divisors, advancing understanding of their geometric structure.

Findings

Hyperk"ahler fourfolds with specific boundary divisors contain rational curves.

Supports a special case of Oguiso's conjecture.

Identifies conditions under which rational curves must exist.

Abstract

We show that if $X$ is a projective hyperk\"ahler fourfold and there exists a nonzero effective divisor $D$ which is not of bi-elliptic type and contained in the boundary of the nef cone of $X$, then $X$ contains a rational curve. This is a very special case of Oguiso's conjecture for projective hyperk\"ahler fourfolds.