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 Fq of characteristic p>5 and cardinality q=pe>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 b1(X)≠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.