On the number of certain Del Pezzo surfaces of degree four violating the Hasse principle

TL;DR

This paper studies the frequency of degree four Del Pezzo surfaces that violate the Hasse principle due to Brauer-Manin obstructions within a specific family, providing an asymptotic density under certain hypotheses.

Contribution

It offers the first asymptotic formula for counting such Del Pezzo surfaces violating the Hasse principle in a particular family, assuming Schinzel's hypothesis and Tate-Shafarevich finiteness.

Findings

Asymptotic expansion for the density of violating surfaces

Conditional asymptotic formula for the total count

Identification of the role of Brauer-Manin obstructions

Abstract

We give an asymptotic expansion for the density of del Pezzo surfaces of degree four in a certain Birch Swinnerton-Dyer family violating the Hasse principle due to a Brauer-Manin obstruction. Under the assumption of Schinzel's hypothesis and the finiteness of Tate-Shafarevich groups for elliptic curves, we obtain an asymptotic formula for the number of all del Pezzo surfaces in the family, which violate the Hasse principle.