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

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 + ⋯ + x 4 y 4 2 = 0 in P 3 × P 3 . This confirms the modified Manin conjecture for this variety, in which the removal of a “thin” set of rational points is allowed.