The genericity theorem for the essential dimension of tame stacks

Abstract: 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 $\operatorname{ed}{k}X{k} \le \operatorname{ed}{K}X{K}$ if $X\to \operatorname{Spec} R$ is smooth and $\operatorname{ed}{k}X{k} \le \operatorname{ed}{K}X{K}+1$ in general.