Spinors and essential dimension

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 et al. [Essential dimension, spinor groups, and quadratic forms, Ann. of Math. (2) 171 ( 2010), 533-544], and Chernousov and Merkurjev [Essential dimension of spinor and Clifford groups, Algebra Number Theory 8 (2014), 457-472] to fields of characteristic different from two. We also complete the determination of generic stabilizers in spin and half-spin groups of low rank.