Linear growth and moduli spaces of rational curves

TL;DR

This paper investigates the dimension of moduli spaces of rational curves on Fano varieties over finite fields, deriving estimates for rational points and confirming predictions of the Batyrev–Manin conjecture in specific cases.

Contribution

It introduces a method to use moduli space dimensions to estimate rational points over function fields, applying it to del Pezzo surfaces, cubic hypersurfaces, and intersections of quadrics.

Findings

Bounds for rational points close to linear growth predicted by Batyrev–Manin conjecture.

Moduli spaces of rational curves have the expected dimension in studied cases.

Application to specific Fano varieties over finite fields.

Abstract

Working in positive characteristic, we show how one can use information about the dimension of moduli spaces of rational curves on a Fano variety $X$ over $\mathbb{F}_q$ to obtain strong estimates for the number of $\mathbb{F}_q(t)$-points of bounded height on $X$. Building on work of Beheshti, Lehmann, Riedl and Tanimoto~\cite{BeheshtiLehmannRiedlTanimoto.dP}, we apply our strategy to del Pezzo surfaces of degree at most 5. In addition, we also treat the case of smooth cubic hypersurfaces and smooth intersections of two quadrics of dimension at least 3 by showing that the moduli spaces of rational curves of fixed degree are of the expected dimension. For large but fixed $q$, the bounds obtained come arbitrarily close to the linear growth predicted by the Batyrev--Manin conjecture.