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Torsion points and concurrent exceptional curves on del Pezzo surfaces of degree one

Julie Desjardins, Rosa Winter

TL;DR

This work investigates when a point on a degree-$1$ del Pezzo surface $S$ that lies on many exceptional curves induces a torsion point on its fiber in the associated rational elliptic surface $\mathscr{E}$. By mapping exceptional curves to sections of $\mathscr{E}$ and exploiting the $E_8$-lattice structure of the Mordell–Weil group along with height pairings, the authors reduce the problem to linear relations among sections drawn from concurrent curves. They prove a sharp result: if at least $9$ exceptional curves pass through a common point, the corresponding fiber point is torsion; they also construct explicit counterexamples showing that $7$ lines do not guarantee torsion, and provide partial results for the open case of $8$ lines, including a detailed classification of possible configurations in characteristic $0$. These findings advance understanding of the distribution of rational points on degree-$1$ del Pezzo surfaces and illuminate the interplay between line configurations and torsion phenomena in the associated elliptic fibrations.

Abstract

The blow-up of the anticanonical base point on a del Pezzo surface $S$ of degree 1 gives rise to a rational elliptic surface $\mathscr{E}$ with only irreducible fibers. The sections of minimal height of $\mathscr{E}$ are in correspondence with the $240$ exceptional curves on $S$. A natural question arises when studying the configuration of these curves: if a point on $S$ is contained in 'many' exceptional curves, it is torsion on its fiber on $\mathscr{E}$? In 2005, Kuwata proved for the analogous question on del Pezzo surfaces of degree $2$, where there are 56 exceptional curves, that if 'many' equals $4$ or more, the answer is yes. In this paper, we prove that for del Pezzo surfaces of degree 1, the answer is yes if 'many' equals $9$ or more. Moreover, we give counterexamples where a \textsl{non}-torsion point lies in the intersection of $7$ exceptional curves. We give partial results for the still open case of 8 intersecting exceptional curves.

Torsion points and concurrent exceptional curves on del Pezzo surfaces of degree one

TL;DR

This work investigates when a point on a degree- del Pezzo surface that lies on many exceptional curves induces a torsion point on its fiber in the associated rational elliptic surface . By mapping exceptional curves to sections of and exploiting the -lattice structure of the Mordell–Weil group along with height pairings, the authors reduce the problem to linear relations among sections drawn from concurrent curves. They prove a sharp result: if at least exceptional curves pass through a common point, the corresponding fiber point is torsion; they also construct explicit counterexamples showing that lines do not guarantee torsion, and provide partial results for the open case of lines, including a detailed classification of possible configurations in characteristic . These findings advance understanding of the distribution of rational points on degree- del Pezzo surfaces and illuminate the interplay between line configurations and torsion phenomena in the associated elliptic fibrations.

Abstract

The blow-up of the anticanonical base point on a del Pezzo surface of degree 1 gives rise to a rational elliptic surface with only irreducible fibers. The sections of minimal height of are in correspondence with the exceptional curves on . A natural question arises when studying the configuration of these curves: if a point on is contained in 'many' exceptional curves, it is torsion on its fiber on ? In 2005, Kuwata proved for the analogous question on del Pezzo surfaces of degree , where there are 56 exceptional curves, that if 'many' equals or more, the answer is yes. In this paper, we prove that for del Pezzo surfaces of degree 1, the answer is yes if 'many' equals or more. Moreover, we give counterexamples where a \textsl{non}-torsion point lies in the intersection of exceptional curves. We give partial results for the still open case of 8 intersecting exceptional curves.
Paper Structure (8 sections, 8 theorems, 32 equations, 1 figure, 1 table)

This paper contains 8 sections, 8 theorems, 32 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Background
  4. Exceptional curves and the $E_8$ root system
  5. The elliptic surface
  6. The case of 7 lines: two examples
  7. The case of 9 or more lines: proof of Theorem \ref{['thmtorsion']}
  8. The case of $8$ lines

Key Result

Theorem 1.4

If at least 9 exceptional curves on $S$ are concurrent in a point, then the corresponding point on $\mathscr{E}$ is torsion on its fiber.

Figures (1)

  • Figure 1: Isomorphism types of the cliques in 47 orbits of the set of cliques of size 8 with only edges of weights 1 and 2. All graphs are fully connected subgraphs with edges of weights 2 (the ones that are drawn) and 1 (all other edges).

Theorems & Definitions (29)

  • Theorem 1.4
  • Remark 1.5
  • Theorem 1.6
  • Remark 1.7
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3: Manin
  • Definition 2.5
  • Definition 2.6
  • Remark 2.7
  • ...and 19 more