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Motivic counting of curves on split quintic del Pezzo surfaces

Christian Bernert, Loïs Faisant, Jakob Glas

Abstract

We prove the "all-the-heights'' version of the Batyrev--Manin--Peyre conjecture for split quintic del Pezzo surfaces, both for counting rational points over global function fields in positive characteristic and for the motivic version over a general base field.

Motivic counting of curves on split quintic del Pezzo surfaces

Abstract

We prove the "all-the-heights'' version of the Batyrev--Manin--Peyre conjecture for split quintic del Pezzo surfaces, both for counting rational points over global function fields in positive characteristic and for the motivic version over a general base field.

Paper Structure

This paper contains 29 sections, 30 theorems, 269 equations.

Table of Contents

  1. Geometric preliminaries
  2. Decomposition of the nef cone
  3. Vector bundles on curves
  4. Semistable vector bundles
  5. Algebraic families of vector bundles
  6. Vector bundles from the adelic point of view
  7. Motivic preliminaries
  8. Rings of varieties and motivic measures
  9. Definitions
  10. Point-counting and motivic measures
  11. Toolbox from motivic arithmetics
  12. Motivic Euler products
  13. Classical and motivic Möbius functions
  14. From coefficients to residues
  15. Lifting to the universal torsor and a Möbius inversion
  16. ...and 14 more sections

Key Result

Theorem A

Assume that $k = \mathbf F_q$. Let $\omega_X^\vee$ be the anticanonical sheaf of $X$. Then, there exists $\eta > 0$ such that for any $\bm d \in \mathop{\mathrm{\mathrm{Pic}}}\nolimits ( X )^\vee$ lying in the dual of the effective cone $\mathop{\mathrm{\mathrm{Eff}}}\nolimits ( X )$, where $d=\bm{d}\cdot \omega_X^\vee$ is the anticanonical degree of $\bm{d}\in \mathop{\mathrm{\mathrm{Pic}}}\noli

Theorems & Definitions (60)

  • Theorem A
  • Theorem B
  • Lemma 1.1
  • proof
  • Proposition 1.2
  • proof
  • Corollary 1.3
  • proof
  • Definition 1.4
  • Remark 1.5
  • ...and 50 more