Cubic surfaces violating the Hasse principle are Zariski dense in the moduli scheme

TL;DR

This paper constructs new examples of cubic surfaces that violate the Hasse principle, demonstrating that such counterexamples are densely distributed in the moduli space over any number field.

Contribution

It provides the first explicit construction of Zariski dense sets of cubic surfaces violating the Hasse principle over all number fields.

Findings

Counterexamples are Zariski dense in the moduli scheme

Violations occur over every number field

New explicit examples of such cubic surfaces

Abstract

We construct new examples of cubic surfaces, for which the Hasse principle fails. Thereby, we show that, over every number field, the counterexamples to the Hasse principle are Zariski dense in the moduli scheme of non-singular cubic surfaces.