Quartic del Pezzo surfaces without quadratic points

Brendan Creutz and Bianca Viray

Vol. 337 (2025), No. 2, 215–224

DOI: 10.2140/pjm.2025.337.215

Abstract

We previously showed that every quartic del Pezzo surface over a number field has index dividing $2$ (i.e., has a closed point of degree $2$ modulo $4$), and asked whether such surfaces always have a closed point of degree $2$. We resolve this by constructing infinitely many quartic del Pezzo surfaces over $\mathbb{Q}$ without degree-$2$ points. These are the first examples of smooth intersections of two quadrics with index strictly less than the minimal degree of a closed point.

Keywords

rational points, index, del Pezzo surfaces

Mathematical Subject Classification

Primary: 14G05, 14G12

Milestones

Received: 26 August 2024

Revised: 8 April 2025

Accepted: 14 May 2025

Published: 3 June 2025