The Hasse principle for lines on del Pezzo surfaces

TL;DR

This paper investigates whether the existence of lines on cubic and del Pezzo surfaces over number fields can violate the Hasse principle, exploring cases where local solutions exist but no global solution does.

Contribution

It introduces the problem of the Hasse principle failure for lines on del Pezzo surfaces and analyzes this phenomenon across various types of surfaces and number fields.

Findings

Identifies conditions under which the Hasse principle fails for lines on cubic surfaces.

Provides examples of del Pezzo surfaces where local solutions do not lift to global solutions.

Abstract

In this paper, we consider the following problem: Does there exist a cubic surface over $\mathbb{Q}$ which contains no line over $\mathbb{Q}$, yet contains a line over every completion of $\mathbb{Q}$? This question may be interpreted as asking whether the Hilbert scheme of lines on a cubic surface can fail the Hasse principle. We also consider analogous problems, over arbitrary number fields, for other del Pezzo surfaces and complete intersections of two quadrics.