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Geometric Manin's conjecture in characteristic $p$

Brian Lehmann, Sho Tanimoto

TL;DR

Geometric Manin's conjecture in characteristic $p$ proposes a structural description of the moduli spaces of curves on Fano varieties in positive characteristic and connects to the Batyrev--Manin--Peyre--Tschinkel conjecture over global function fields. The paper develops a framework built from numerical spaces $N^1(X)$ and $N_1(X)$, Fujita invariants $a(X,L)$ and $b(k,X,L)$, and the vanishing properties of globally $F$-regular varieties to control deformations and exceptional loci. It introduces exceptional maps and Manin components to separate breaking phenomena from genuine families of sections of good Fano fibrations, and formulates standard and all-height versions of Manin's conjecture over $K(B)$ with Tamagawa measures. Key constants such as $ obreakoldsymbol{ extalpha}(oldsymbol{ ext{X}}_oldsymbol{oldsymbol{ exteta}})$, $oldsymbol{eta}(oldsymbol{ ext{X}}_oldsymbol{oldsymbol{ exteta}})$, and the Tamagawa number $ au_{oldsymbol{ ext{X}}}(-oldsymbol{ ext{K}}_{oldsymbol{ ext{X}}_oldsymbol{oldsymbol{ exteta}}})$ are described, and several examples and methods, including the homological sieve, are reviewed. The survey gathers conjectures, evidence, and strategies for transferring results from characteristic $0$ and global fields to characteristic $p$, highlighting pathologies and recent progress.

Abstract

Geometric Manin's conjecture for complex Fano varieties describes the structure of the moduli space of curves. We propose a version of this conjecture in characteristic $p$ and describe its connection to the Batyrev--Manin--Peyre--Tschinkel conjecture over global fields. This is a survey paper written for a volume of the Summer Research Institute in Algebraic Geometry held at Colorado State University in 2025.

Geometric Manin's conjecture in characteristic $p$

TL;DR

Geometric Manin's conjecture in characteristic proposes a structural description of the moduli spaces of curves on Fano varieties in positive characteristic and connects to the Batyrev--Manin--Peyre--Tschinkel conjecture over global function fields. The paper develops a framework built from numerical spaces and , Fujita invariants and , and the vanishing properties of globally -regular varieties to control deformations and exceptional loci. It introduces exceptional maps and Manin components to separate breaking phenomena from genuine families of sections of good Fano fibrations, and formulates standard and all-height versions of Manin's conjecture over with Tamagawa measures. Key constants such as , , and the Tamagawa number are described, and several examples and methods, including the homological sieve, are reviewed. The survey gathers conjectures, evidence, and strategies for transferring results from characteristic and global fields to characteristic , highlighting pathologies and recent progress.

Abstract

Geometric Manin's conjecture for complex Fano varieties describes the structure of the moduli space of curves. We propose a version of this conjecture in characteristic and describe its connection to the Batyrev--Manin--Peyre--Tschinkel conjecture over global fields. This is a survey paper written for a volume of the Summer Research Institute in Algebraic Geometry held at Colorado State University in 2025.
Paper Structure (26 sections, 7 theorems, 61 equations)

This paper contains 26 sections, 7 theorems, 61 equations.

Table of Contents

  1. Introduction
  2. Historical background: curves on Fano varieties
  3. Historical background: Manin's conjecture on rational points
  4. Batyrev's heuristic and extensions
  5. Preliminaries
  6. Numerical spaces
  7. Curves
  8. Brauer groups
  9. Thin sets
  10. Fujita invariants
  11. Globally $F$-regular varieties
  12. Rational curves on Fano varieties in characteristic p
  13. Counterexamples
  14. Positive examples
  15. Sections of Fano fibrations in characteristic p
  16. ...and 11 more sections

Key Result

Theorem 2.9

Let $k$ be a field of characteristic $p$. Suppose $X$ is a smooth projective geometrically integral $k$-variety that is geometrically globally F-regular. Then for any big and nef Cartier divisor $L$ on $X$ we have $H^{i}(X,\mathcal{O}_{X}(K_{X}+L)) = 0$ for every $i>0$.

Theorems & Definitions (57)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Example 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Theorem 2.9: SS10
  • Corollary 2.10
  • ...and 47 more