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Generalized Eckardt points on del Pezzo surfaces of degree 1

Julie Desjardins, Yu Fu, Kelly Isham, Rosa Winter

TL;DR

This work initiates a systematic study of generalized Eckardt points on del Pezzo surfaces of degree $1$, focusing on points where ten exceptional curves meet outside char $2$ and $3$. It provides a complete combinatorial classification of ten-line concurrency configurations via the $E_8$ root system and Weyl-group action, and constructs a new ramification-curve family yielding such a concurrency, extending prior examples. The authors develop computational strategies (Magma) to test realizability over finite fields, obtaining nonexistence results for several prime fields and explicit realizations in $oldsymbol{ m F}_{19}$, while also describing ramification-curve constructions that produce infinite families in characteristic $0$. By linking geometric concurrency with arithmetic questions about rational points and weak approximation, the paper lays groundwork for broader classifications and practical search methods for generalized Eckardt points on degree-$1$ del Pezzo surfaces.

Abstract

We study intersections of exceptional curves on del Pezzo surfaces of degree 1, motivated by questions in arithmetic geometry. Outside characteristics 2 and 3, at most 10 exceptional curves can intersect in a point. We classify the different ways in which 10 exceptional curves can intersect, construct a new family of surfaces with 10 exceptional curves intersecting in a point, and discuss strategies for finding more such examples.

Generalized Eckardt points on del Pezzo surfaces of degree 1

TL;DR

This work initiates a systematic study of generalized Eckardt points on del Pezzo surfaces of degree , focusing on points where ten exceptional curves meet outside char and . It provides a complete combinatorial classification of ten-line concurrency configurations via the root system and Weyl-group action, and constructs a new ramification-curve family yielding such a concurrency, extending prior examples. The authors develop computational strategies (Magma) to test realizability over finite fields, obtaining nonexistence results for several prime fields and explicit realizations in , while also describing ramification-curve constructions that produce infinite families in characteristic . By linking geometric concurrency with arithmetic questions about rational points and weak approximation, the paper lays groundwork for broader classifications and practical search methods for generalized Eckardt points on degree- del Pezzo surfaces.

Abstract

We study intersections of exceptional curves on del Pezzo surfaces of degree 1, motivated by questions in arithmetic geometry. Outside characteristics 2 and 3, at most 10 exceptional curves can intersect in a point. We classify the different ways in which 10 exceptional curves can intersect, construct a new family of surfaces with 10 exceptional curves intersecting in a point, and discuss strategies for finding more such examples.
Paper Structure (13 sections, 15 theorems, 34 equations, 3 figures, 1 table)

This paper contains 13 sections, 15 theorems, 34 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background
  3. Del Pezzo surfaces of degree one and exceptional curves
  4. The weighted graph on exceptional classes
  5. Configurations of ten concurrent lines
  6. Strategies - generalized Eckardt points outside the ramification curve
  7. Clique Representatives
  8. Strategy for Computing Examples
  9. Surfaces with generalized Eckardt points on the ramification curve
  10. Finite Fields
  11. Results for $\mathbb{F}_p$, $p$ prime
  12. General $\mathbb{F}_q$
  13. Further questions

Key Result

Theorem 1.3

Outside characteristic 2, there are 9 possible configurations for 10 exceptional curves on a del Pezzo surface of degree 1 to intersect in one point, and they are listed in Figures table cliques and table clique 9. In characteristic 0, at most 7 of the configurations (1, 2, 3, 5, 6, 8 in Figure tabl

Figures (3)

  • Figure 1: Maximal clique $T$ of size twelve in $G$.
  • Figure 2: The isomorphism types of 8 cliques in $G$ with only edges of weights 1 and 2. Each clique is a fully connected subgraphs of $G$ with the edges drawn corresponding to edges of weight 2. The edges not drawn correspond to edges of weight 1 in the clique.
  • Figure 3: The isomorphism type of a clique of size 10 in $G$ with only edges of weights 1 and 3. The clique is a fully connected subgraph of $G$ with the edges drawn corresponding to edges of weight 3. The edges not drawn correspond to edges of weight 1 in the clique.

Theorems & Definitions (41)

  • Theorem 1.3
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Theorem 2.4: Manin
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Lemma 2.9: vLW
  • ...and 31 more