nLab Jouanolou cover

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

• Aravind Asok, The Jouanolou-Thomason homotopy lemma, pdf

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.

• M. Kontsevich, A. Rosenberg, Noncommutative smooth spaces, The Gelfand Mathematical Seminars, 1996–1999, 85–108, Gelfand Math. Sem., Birkhäuser Boston, Boston, MA, 2000; (arXiv:math/9812158)