Strictly nef divisors on singular threefolds

TL;DR

This paper investigates the ampleness of divisors on singular threefolds with klt singularities, proving that under certain conditions, the sum of the canonical divisor and a large multiple of a strictly nef divisor is ample, except in specific Calabi-Yau cases.

Contribution

It extends Serrano's conjecture to singular threefolds, establishing ampleness results for $K_X + tL_X$ under assumptions like $Q$-factoriality and Gorenstein terminal singularities, with exceptions for certain Calabi-Yau varieties.

Findings

$K_X + tL_X$ is ample for large $t$ on $Q$-factorial Gorenstein terminal threefolds.

The ampleness fails only when $X$ is a weak Calabi-Yau with $L_X eq 0$ and $L_X ullet c_2(X)=0$.

The results generalize known conjectures to singular threefolds with mild singularities.

Abstract

Let $X$ be a normal projective variety with only klt singularities, and $L_X$ a strictly nef $\mathbb{Q}$-divisor on $X$. In this paper, we study the singular version of Serrano's conjecture, i.e., the ampleness of $K_X+t L_X$ for sufficiently large $t\gg 1$. We show that, if $X$ is assumed to be a $\mathbb{Q}$-factorial Gorenstein terminal threefold, then $K_X+tL_X$ is ample for $t\gg 1$ unless $X$ is a weak Calabi-Yau variety (i.e., the canonical divisor $K_X\sim_\mathbb{Q}0$ and the augmented irregularity $q^\circ(X)=0$) with $L_X\cdot c_2(X)=0$.