Solubility of a family of conics with polynomial coefficients in many variables

We study the proportion of conics given by (C_F, y): F₀(y)x₀² + F₁(y)x₁² = F₂( y)x₂² ( C F , y ) : F 0 ( y ) x 0 2 + F 1 ( y ) x 1 2 = F 2 ( y ) x 2 2 which have a rational point x = (x₀:x₁:x₂) ∈ P²(Q) x = ( x 0 : x 1 : x 2 ) ∈ P 2 ( Q ) , where y = (y₀: : yₙ)∈ Pⁿ(Q) y = ( y 0 : ⋯ : y n ) ∈ P n ( Q ) and F₀,F₁,F₂ ∈ Z[X₀,… , Xₙ] F 0 , F 1 , F 2 ∈ Z [ X 0 , … , X n ] are homogeneous polynomials in many variables of the same degree d . We provide an asymptotic formula for the number of y <mml:math xmlns:mml="http: