Spinors and essential dimension

TL;DR

This paper demonstrates that spin groups act freely on spinor modules across different characteristics, extending known calculations of their essential dimension from characteristic zero to other fields.

Contribution

It proves the generic freeness of spin group actions on spinor modules and extends essential dimension results to fields of characteristic not equal to 2.

Findings

Spin groups act generically freely on spinor modules in all characteristics.

Essential dimension of spin and half-spin groups is extended to non-zero characteristic fields.

The results are characteristic-independent, broadening previous characteristic-zero findings.

Abstract

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.