Density of rational points on a quadric bundle in ${\mathbb{P}}^3\times {\mathbb{P}}^3$

An asymptotic formula is established 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.