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Linear growth and moduli spaces of rational curves

Jakob Glas

TL;DR

The paper develops a framework to bound the number of $bF_q(t)$-points of bounded height on smooth Fano varieties by leveraging the geometry of moduli spaces of rational curves. By proving a general Bézout-based bound on the number of $bF_q$-points of any variety and combining it with new results on the dimensions of Mor$\_e(\bbP^1,X)$ from positive characteristic, the authors obtain explicit, near-sharp upper bounds for $N_U(q^e)$ in several important cases: del Pezzo surfaces of degree $d\le5$ (with $N_U(q^e)=O(C_d^e q^e)$), smooth cubic hypersurfaces, and smooth intersections of two quadrics with dimension at least $3$. A key technical thread is showing that the moduli spaces of rational curves of fixed degree are of the expected dimension in these cases, aided by detailed analysis of the Fano scheme of lines and the Kontsevich spaces of stable maps. The results approach the linear growth predicted by the Batyrev--Manin conjecture when $q$ is large and highlight a flexible, geometry-driven method for function-field Manin-type questions in positive characteristic.

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.

Linear growth and moduli spaces of rational curves

TL;DR

The paper develops a framework to bound the number of -points of bounded height on smooth Fano varieties by leveraging the geometry of moduli spaces of rational curves. By proving a general Bézout-based bound on the number of -points of any variety and combining it with new results on the dimensions of Mor from positive characteristic, the authors obtain explicit, near-sharp upper bounds for in several important cases: del Pezzo surfaces of degree (with ), smooth cubic hypersurfaces, and smooth intersections of two quadrics with dimension at least . A key technical thread is showing that the moduli spaces of rational curves of fixed degree are of the expected dimension in these cases, aided by detailed analysis of the Fano scheme of lines and the Kontsevich spaces of stable maps. The results approach the linear growth predicted by the Batyrev--Manin conjecture when is large and highlight a flexible, geometry-driven method for function-field Manin-type questions in positive characteristic.

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 over to obtain strong estimates for the number of -points of bounded height on . 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 , the bounds obtained come arbitrarily close to the linear growth predicted by the Batyrev--Manin conjecture.
Paper Structure (5 sections, 20 theorems, 41 equations)

This paper contains 5 sections, 20 theorems, 41 equations.

Table of Contents

  1. Introduction
  2. Uniformly counting rational points over finite fields
  3. Counting $\mathbb{F}_q$-points on moduli spaces of rational curves
  4. Fano scheme of lines
  5. Kontsevich spaces of stable maps

Key Result

Theorem 1.1

Let $X$ be a smooth del Pezzo surface of degree $d$ over $\mathbb{F}_q$ with $1\leq d\leq 5$. If we take $U$ to be the complement of the union of all rational curves $C\subset X$ such that $-K_X\cdot C\leq 1$, then where and the implied constant depends on $q$, but not on $e$.

Theorems & Definitions (40)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Proposition 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • proof : Proof of Proposition \ref{['Prop: UniformPointCount']}
  • Proposition 3.1
  • ...and 30 more