Density of rational points on a certain biprojective hypersurface

By the circle method, 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 $$\sum _{i=1}^nx_iy_i^2=0$$ in $\mathbb {P}^{n-1}\times \mathbb {P}^{n-1}$, where $n\geq 5$. This confirms the Manin conjecture for this variety. Moreover, we get an upper bound for the number of integer solutions to diagonal quadratic forms, which refines a previous result.