Local solubility of a family of quadric surfaces over a biprojective base

Abstract

We prove an asymptotic formula for the number of everywhere locally soluble diagonal quadric surfaces 𝑦₀𝑥²₀+𝑦₁𝑥²₁+𝑦₂𝑥²₂+𝑦₃𝑥²₃=0 parametrised by points 𝑦 ∈ ℙ³ (ℚ) lying on the split quadric surface 𝑦₀𝑦₁ = 𝑦₂𝑦₃ which do not satisfy −𝑦₀𝑦₂ = □ nor −𝑦₀𝑦₃ = □. Our methods involve proving asymptotic formulae for character sums with a hyperbolic height condition and proving variations of large sieve inequalities for quadratic characters