On automorphisms of $p$-torsion $\mathbf{G}_m$-gerbes

TL;DR

This paper investigates automorphisms of $p$-torsion $ extbf{G}_m$-gerbes over smooth projective varieties, showing surjectivity of automorphism maps in certain cases and providing counterexamples in positive characteristic.

Contribution

It extends Olsson's results by analyzing automorphism groups of $ extbf{G}_m$-gerbes in positive characteristic and offers new conditions for surjectivity using deformation theory and cohomological methods.

Findings

Counterexample when Brauer class has order equal to characteristic

Sufficient conditions for automorphism surjectivity

Exposition of deformation theory and flat cohomology in positive characteristic

Abstract

Olsson showed in [Ols25] that if $\mathcal{X} \to X$ is a $\mathbf{G}_m$-gerbe over a smooth projective variety over an algebraically closed field $k$ such that the Brauer class of $\mathcal{X}$ has order prime to the characteristic of $k$, then the homomorphism of $k$-group algebraic spaces $\operatorname{Aut}^0_{\mathcal{X}} \to \operatorname{Aut}^0_X$ is surjective. We provide an example to show that this need not be the case when the Brauer class of $\mathcal{X}$ has order equal to the characteristic. Our main tools are deformation theory of the fppf sheafified Artin--Mazur formal groups and nice properties of the flat cohomology of ordinary varieties in positive characteristic which are presumably well-known, but which we collect and give an exposition of here. We additionally prove some sufficient conditions for surjectivity of $\operatorname{Aut}^0_{\mathcal{X}} \to \operatorname{Aut}^0_X$ using representability results of Bragg and Olsson.