Jouanolou cover of a projective $n$-space by quadrics has been introduced by Jean-Pierre Jouanolou for the purposes of algebraic K-theory
J.-P. Jouanolou, Une suite exacte de Mayer-Vietoris en K-th´eorie alg´ebrique, Algebraic K-theory, I (Seattle, 1972), 293–316, Lecture Notes in Mathematics 341, Springer (1973).
MathOverflow: The Jouanolou trick
This is also used in Michel Van den Bergh’s proof that every projective variety over complexes is a quiver variety (see blog)
There is a noncommutative version of a Jouanolou cover for noncommutative projective space in a paper of Rosenberg and Konstevich.