The genericity theorem for the essential dimension of tame stacks

Let $X$ be a regular tame stack. If $X$ is locally of finite type over a field, we prove that the essential dimension of $X$ is equal to its generic essential dimension; this generalizes a previous result of P. Brosnan, Z. Reichstein and the second author. Now suppose that $X$ is locally of finite type over a $1$-dimensional noetherian local domain $R$ with fraction field $K$ and residue field $k$. We prove that $\operatorname{ed}_k X_k \leq \operatorname{ed}_K X_K$ if $X \to \operatorname{Spec} R$ is smooth and $\operatorname{ed}_k X_k \leq \operatorname{ed}_K X_K + 1$ in general.

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The first author was partially supported by the DFG Priority Program “Homotopy Theory and Algebraic Geometry” SPP 1786.

The second author was supported by research funds from the Scuola Normale Superiore, Project SNS19_B_VISTOLI.

The paper is based upon work partially supported by the Swedish Research Council under grant no. 2016-06596 while the second author was in residence at Institut Mittag-Leffler in Djursholm.

Received 1 November 2021

Received revised 8 February 2022

Accepted 19 February 2022

Published 25 October 2022