Quartic del Pezzo surfaces without quadratic points

TL;DR

This paper constructs infinitely many quartic del Pezzo surfaces over the rationals that lack degree 2 points, answering a longstanding question and providing new examples of surfaces with index less than the minimal degree of a closed point.

Contribution

The authors explicitly construct the first known examples of smooth intersections of two quadrics with index less than the minimal degree of a closed point.

Findings

Existence of infinitely many such surfaces over 

These surfaces have no degree 2 points

They are the first known examples with index less than the minimal degree

Abstract

Previous work of the authors 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.