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Rational points on del Pezzo surfaces of low degree

Jakob Glas, Leonhard Hochfilzer

Abstract

We give upper bounds for the number of rational points of bounded anti-canonical height on del Pezzo surfaces of degree at most five over any global field whose characteristic is not equal to two or three. For number fields these results are conditional on a conjecture relating the rank of an elliptic curve to its conductor, while they are unconditional in positive characteristic. For quartic or quintic del Pezzo surfaces with a conic bundle structure, we establish even stronger estimates unconditionally as long as the characteristic is not two.

Rational points on del Pezzo surfaces of low degree

Abstract

We give upper bounds for the number of rational points of bounded anti-canonical height on del Pezzo surfaces of degree at most five over any global field whose characteristic is not equal to two or three. For number fields these results are conditional on a conjecture relating the rank of an elliptic curve to its conductor, while they are unconditional in positive characteristic. For quartic or quintic del Pezzo surfaces with a conic bundle structure, we establish even stronger estimates unconditionally as long as the characteristic is not two.
Paper Structure (20 sections, 34 theorems, 274 equations, 1 table)

This paper contains 20 sections, 34 theorems, 274 equations, 1 table.

Table of Contents

  1. Introduction
  2. Outline.
  3. Conventions
  4. Background
  5. Geometry
  6. del Pezzo surfaces
  7. Dual varieties
  8. Algebraic number theory
  9. Height functions
  10. Rational points on varieties
  11. Geometry of numbers and rational points on conics
  12. Lattices
  13. Rational points on conics
  14. Rational points on smooth genus 1 curves
  15. Conic bundles
  16. ...and 5 more sections

Key Result

Theorem 1.2

Let $X$ be a del Pezzo surface of degree $1\leq d\leq 5$ over a global field $K$ with $\mathop{\mathrm{char}}\nolimits(K)\neq 2,3$. Then unconditionally when $\mathop{\mathrm{char}}\nolimits(K)>3$ and assuming Conjecture Conj: RGH when $\mathop{\mathrm{char}}\nolimits(K)=0$. Moreover, when $d=1$ the implied constant is independent of $X$.

Theorems & Definitions (65)

  • Conjecture 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 1.4
  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • Remark
  • Proposition 2.4
  • proof
  • ...and 55 more