Density of rational points on a quadric bundle in P 3 × P 3

Abstract

We establish an asymptotic formula for the number of rational points of bounded anticanonical height which lie on a certain Zariski-dense subset of the biprojective hypersurface $$x_1{y}_1^2+\cdots +x_4{y}_4^2=0$$ in ${\mathbb{P}}^3\times {\mathbb{P}}^3$. This confirms the modified Manin conjecture for this variety, in which the removal of a “thin” set of rational points is allowed.