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

TL;DR

This paper establishes that, in positive characteristic with $p>5$ and cardinality $q=p^e>19$, a geometrically integral smooth 3-fold $X$ over a finite field $\mathbb{F}_q$ with $-K_X$ nef and $\kappa(X)=-\infty$ has a rational point, and likewise a $K_X\sim_{\mathbb{Q}}0$ 3-fold with $b_1(X)\neq0$ has a rational point. The authors develop an MMP-based framework that reduces to Mori fiber spaces and extends the arithmetic conclusions to generalized log Calabi–Yau 3-fold pairs, leveraging the EP23 result on Albanese surjectivity in positive characteristic. A key technical innovation is treating singular cases via generalized pairs and constructing rational points by analyzing bases and fibers of Mori fiber spaces, as well as demonstrating Albanese surjectivity in this generalized setting. These results provide a significant advance toward a positive-characteristic analog of the Green–Griffiths–Lang perspective for varieties with nef anti-canonical class over finite fields and connect birational geometry techniques with arithmetic point existence results.

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.

Theorems & Definitions (7)

• Conjecture 1
• Theorem 1.1: Smooth case of \ref{['thm: intro_rat_pts_anti-nef']}
• Theorem 1.2: Smooth case of \ref{['thm: intro_alb_non_trivial']}
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Definition 1