Counting Rational Points on a Certain Bihomogeneous Threefold

Let $V\subset {\mathbb{P}}^2\times {\mathbb{P}}^2$ be the threefold $x_0{y}_0^2+x_1{y}_1^2+x_2{y}_2^2=0$, and let $U$ be the complement of its coordinate hyperplane sections. For the anticanonical height $H([\mathbf{x}],[\mathbf{y}])=\mathrm{\mid }\mathbf{x}{\mathrm{\mid }}^2\mathrm{\mid }\mathbf{y}\mathrm{\mid }$, evaluated on primitive integral representatives, we prove $$N_U(B)=\frac{12\pi -14}{\zeta (2)\zeta (3)}B\log B+o(B\log B)\mathrm{.}$$ The two projections combine lattice counting with uniform conic estimates. The main arithmetic step is an average of the fiber singular series, whose two main terms are related by Hilbert reciprocity. A moment estimate controls the full bad-prime part, and shape estimates permit the errors to be summed over the height region. We also check the leading constant with Peyre’s constant.