Rational points on 3-folds with nef anti-canonical class over finite fields

We prove that a geometrically integral smooth 3-fold $X$ with nef anti-canonical class and negative Kodaira dimension over a finite field ${\mathbb{F}}_q$ of characteristic $p>5$ and cardinality $q=p^e>19$ has a rational point. Additionally, under the same assumptions on $p$ and $q$, we show that a 3-fold $X$ with trivial canonical class and non-zero first Betti number $b_1(X)\mathrm{\ne }0$ has a rational point. Our techniques rely on the Minimal Model Program to establish several structure results for generalized log Calabi–Yau 3-fold pairs over perfect fields.