Positivity of the tangent bundle of rational surfaces with nef anticanonical divisor

TL;DR

This paper investigates the conditions under which the tangent bundle of rational surfaces with nef anticanonical divisor is big, providing complete and partial classifications based on surface degree and configurations of (-2)-curves.

Contribution

It offers a complete characterization of the bigness of tangent bundles for degree 4 weak del Pezzo surfaces and partial results for degrees 1-3, using geometric divisor constructions.

Findings

Tangent bundle not big for rational elliptic surfaces.

Complete classification for degree 4 weak del Pezzo surfaces.

Partial results for degrees 1-3 based on (-2)-curves configurations.

Abstract

In this paper, we study the property of bigness of the tangent bundle of a smooth projective rational surface with nef anticanonical divisor. We first show that the tangent bundle $T_S$ of $S$ is not big if $S$ is a rational elliptic surface. We then study the property of bigness of the tangent bundle $T_S$ of a weak del Pezzo surface $S$. When the degree of $S$ is $4$, we completely determine the bigness of the tangent bundle through the configuration of $(-2)$-curves. When the degree $d$ of $S$ is less than or equal to $3$, we get a partial answer. In particular, we show that $T_S$ is not big when the number of $(-2)$-curves is less than or equal to $7-d$, and $T_S$ is big when $d=3$ and $S$ has the maximum number of $(-2)$-curves. The main ingredient of the proof is to produce irreducible effective divisors on $\mathbb{P}(T_S)$, using Serrano's work on the relative tangent bundle when $S$ has a fibration, or the total dual VMRT associated to a conic fibration on $S$.