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 TS of S is not big if S is a rational elliptic surface. We then study the property of bigness of the tangent bundle TS 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 TS is not big when the number of $(-2)$-curves is less than or equal to $7-d$, and TS 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 P(TS), 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.