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
2
1
+
⋯
+
x
4
y
2
4
=
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.