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Frobenius actions on Del Pezzo surfaces of degree 2

Olof Bergvall

Vol. 21 (2024), No. 1, 1–10
DOI: 10.2140/iig.2024.21.1
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 and Loughran and Loughran and Trepalin.

Keywords
Del Pezzo surfaces, moduli spaces, Frobenius endomorphism, inverse Galois problem, point counts, cohomology
Mathematical Subject Classification
Primary: 14J10, 14J26
Secondary: 05E18, 14F20
Milestones
Received: 23 November 2022
Revised: 24 February 2023
Accepted: 19 April 2023
Published: 9 January 2024
Authors
Olof Bergvall
Department of Mathematics and Physics
Mälardalen University
Västerås
Sweden