Rational curves and strictly nef divisors on Calabi--Yau threefolds

TL;DR

This paper establishes criteria for semi-ampleness of nef divisors on Calabi--Yau threefolds and explores conditions under which strictly nef divisors are ample, linking divisor properties to the existence of rational curves.

Contribution

It provides a new criterion for semi-ampleness of nef divisors on Calabi--Yau threefolds under specific conditions and relates divisor nefness to the presence of rational curves.

Findings

Nef divisor D with D^3=0 and c_2(X)·D=0 is semiample.

Strictly nef divisors with ν(D)≠1 are ample.

Existence of a nef non-ample divisor implies rational curves if the Euler characteristic is non-zero.

Abstract

We give a criterion for a nef divisor $D$ to be semiample on a Calabi--Yau threefold $X$ when $D^3=0=c_2(X)\cdot D$ and $c_3(X)\neq 0$. As a direct consequence, we show that on such a variety $X$, if $D$ is strictly nef and $\nu(D)\neq 1$, then $D$ is ample; we also show that if there exists a nef non-ample divisor $D$ with $D\not\equiv 0$, then $X$ contains a rational curve when its topological Euler characteristic is not $0$.