The genericity theorem for the essential dimension of tame stacks (Q2093655)

In the paper under review, the authors prove several results comparing the essential and generic essential dimension of tame stacks. Given a stack \(X\) over a field \(k\) the \textit{essential dimension} of an object \(\xi \ni X(\ell)\) for an extenison \(\ell\) of \(k\). Is the smallest transcendence degree of an extension of \(k\) to which \(\xi\) descends. The essential dimension \(ed_k(X)\) of \(X\) itself is the supremum of the essential dimensions of all points of \(X\) over all field extensions of \(k\). Thus it gives some bound to how complicated objects of \(X\) can be to define. By restricting to the essential dimenson of objects corresponding to dominant morphisms \(\xi: \mathrm{Spec}(\ell) \to X\) one may define the \textit{generic essential dimension} \(ged_k(X)\). The authors now show close relations between the two notions. If \(X\) is a regular integral weakly tame stack locally of finite type over a field \(k\) then essential and generic essential dimensions agree. If \(X\) is an integral weakly tame stack with a smooth morphism \(X \rightarrow S\) to a regular integral scheme and \(s \in S\) is a point then \(ed_{k(s)}X_{k(s)} \leq ged_{k(S)} X_{k(S)}\). If \(X\) is a regular integral weakly tame stack locally of finite type over a noetherian 1-dimensional local domain \(R\) with fraction field \(K\) and residue field \(k\) then \(ed_k X_K \leq ged_K X_K + 1\). The proofs rely on a new valuative criterion of properness for stacks using root stacks of a discrete valuation ring. This result became available as a pre-print after this article was published [\textit{G. Bresciani} and \textit{A. Vistoli}, ``An arithmetic valuative criterion for proper maps of tame algebraic stacks'', Preprint, \url{arXiv:2210.03406}].