Weak weak approximation and the Hilbert property for degree 2 del Pezzo surfaces

Julian Demeio; Sam Streeter; Rosa Winter

doi:10.1112/plms.12601

Weak weak approximation and the Hilbert property for degree 2 del Pezzo surfaces

Julian Demeio, Sam Streeter, Rosa Winter

Department of Mathematical Sciences

Research output: Contribution to journal › Article › peer-review

Abstract

We prove that del Pezzo surfaces of degree 2 over a field (Formula presented.) satisfy weak weak approximation if (Formula presented.) is a number field and the Hilbert property if (Formula presented.) is Hilbertian of characteristic zero, provided that they contain a (Formula presented.) -rational point lying neither on any 4 of the 56 exceptional curves nor on the ramification divisor of the anticanonical morphism. This builds upon results of Manin, Salgado–Testa–Várilly-Alvarado, and Festi–van Luijk on the unirationality of such surfaces, and upon work of the first two authors verifying weak weak approximation under the assumption of a conic fibration.

Original language	English
Article number	e12601
Journal	Proceedings of the London Mathematical Society
Volume	128
Issue number	5
Early online date	15 May 2024
DOIs	https://doi.org/10.1112/plms.12601
Publication status	Published - 31 May 2024

Funding

We thank Jean-Louis Colliot-Th\u00E9l\u00E8ne, Daniel Loughran, Cec\u00EDlia Salgado, and Alexei Skorobogatov for useful discussions and feedback. We thank the anonymous referee for useful comments that improved the quality of the paper. Meetings to complete this work were made possible by funding from ICMS Edinburgh and Pierre Le Boudec's SNSF Professorship grant. The first author started this project while being a guest at the Max Planck Institute of Bonn, which he thanks for its wonderful hospitality and optimal working conditions, and continued while being supported by Pierre Le Boudec's SNSF Professorship grant. The second author was supported by the University of Bristol and the Heilbronn Institute for Mathematical Research. The third author was supported by UKRI Fellowship MR/T041609/2.

Funders	Funder number
International Centre for Mathematical Sciences (ICMS)
Max Planck Instituut voor Psycholinguïstiek
Pierre Le Boudec's SNSF
Heilbronn Institute for Mathematical Research
University of Bristol
UK Research and Innovation	MR/T041609/2
UK Research and Innovation

ASJC Scopus subject areas

General Mathematics

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Weak weak approximation and the Hilbert property for degree 2 del Pezzo surfaces

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