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

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 $S$.