Del Pezzo surfaces of degree four violating the Hasse principle are Zariski dense in the moduli scheme

TL;DR

This paper proves that over any number field, del Pezzo surfaces of degree four that violate the Hasse principle are densely distributed in the moduli space, highlighting their abundance.

Contribution

It establishes the Zariski density of Hasse principle-violating degree four del Pezzo surfaces in the moduli scheme over all number fields.

Findings

Violations of the Hasse principle are Zariski dense in the moduli scheme.

Density holds over every number field.

Provides new insights into the distribution of such surfaces.

Abstract

We show that, over every number field, the degree four del Pezzo surfaces that violate the Hasse principle are Zariski dense in the moduli scheme.