  We prove that spin groups act generically freely on various spinor modules,
in the sense of group schemes and in a way that does not depend on the
characteristic of the base field. As a consequence, we extend the surprising
calculation of the essential dimension of spin groups and half-spin groups in
characteristic zero by Brosnan--Reichstein--Vistoli (Annals of Math., 2010) and
Chernousov--Merkurjev (Algebra & Number Theory, 2014) to fields of
characteristic different from 2.
