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

TL;DR

This paper proves the existence of rational points on certain smooth 3-folds over finite fields with specific geometric properties, using the Minimal Model Program to analyze their structure.

Contribution

It establishes the existence of rational points on 3-folds with nef anti-canonical class or trivial canonical class and non-zero first Betti number over finite fields, under certain conditions.

Findings

3-folds with nef anti-canonical class have rational points

3-folds with trivial canonical class and non-zero b_1 have rational points

structure results for log Calabi–Yau 3-fold pairs over perfect fields

Abstract

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 $b_1(X) \neq 0$ 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.