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Frobenius actions on Del Pezzo surfaces of degree 2

Olof Bergvall

Abstract

We determine the number of Del Pezzo surfaces of degree 2 over finite fields of odd characteristic with specified action of the Frobenius endomorphism, i.e. we solve the "quantitative inverse Galois problem". As applications we determine the number of Del Pezzo surfaces of degree 2 with a given number of points and recover results of Banwait-Fité-Loughran and Loughran-Trepalin.

Frobenius actions on Del Pezzo surfaces of degree 2

Abstract

We determine the number of Del Pezzo surfaces of degree 2 over finite fields of odd characteristic with specified action of the Frobenius endomorphism, i.e. we solve the "quantitative inverse Galois problem". As applications we determine the number of Del Pezzo surfaces of degree 2 with a given number of points and recover results of Banwait-Fité-Loughran and Loughran-Trepalin.
Paper Structure (9 sections, 4 theorems, 9 equations, 2 tables)

This paper contains 9 sections, 4 theorems, 9 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Del Pezzo surfaces of degree 4
  3. Del Pezzo surfaces of degree 3
  4. Del Pezzo surfaces of degree 2
  5. Del Pezzo surfaces of degree 1
  6. The present paper
  7. Del Pezzo surfaces of degree $2$
  8. Lefschetz trace formula
  9. Counts of surfaces

Key Result

Theorem 4.1

Let $\mathbb{F}_q$ be a finite field of odd characteristic and let $w$ be an element of $W(E_7)$. The number geometrically marked Del Pezzo surfaces of degree $2$ over $\mathbb{F}_q$ such that $w$ acts on $\mathrm{Pic}(X_{\overline{\mathbb{F}}_q})$ as $w$ is given in Table frob_action_tab.

Theorems & Definitions (5)

  • Remark 1.1
  • Theorem 4.1
  • Corollary 4.2: Loughran-Trepalin, loughrantrepalin Theorem 1.2
  • Corollary 4.3
  • Corollary 4.4: Banwait-Fité-Loughran, banwaitetal Theorem 1.4