Strictly nef divisors on singular threefolds

TL;DR

This paper investigates the properties of strictly nef divisors on singular threefolds, establishing conditions for ampleness and rational connectedness, thereby advancing understanding of the geometry of such varieties with mild singularities.

Contribution

It proves ampleness of certain divisors under specific conditions and shows rational connectedness for threefold pairs with strictly nef anti-log canonical divisors.

Findings

Ampleness of $K_X+tL_X$ when $ exists ext{Kodaira dimension} eq 0$ and $q^{ullet}(X) eq 0$

Strictly nef anti-log canonical divisors imply rational connectedness

Results extend understanding of nef divisors on singular threefolds

Abstract

Let $X$ be a normal projective threefold with mild singularities, and $L_X$ a strictly nef $\mathbb{Q}$-divisor on $X$. First, we show the ampleness of $K_X+tL_X$ with sufficiently large $t$ if either the Kodaira dimension $\kappa(X)\neq 0$ or the augmented irregularity $q^{\circ}(X)\neq 0$. Second, we show that, if $(X,\Delta)$ is a projective klt threefold pair with the anti-log canonical divisor $-(K_X+\Delta)$ being strictly nef, then $X$ is rationally connected.