Pointwise surjective presentations of stacks

TL;DR

This paper demonstrates that stacks of finite type over Noetherian schemes can be presented by schemes with surjective pointwise maps over various fields, with applications in counting classes and real algebraic stacks.

Contribution

It introduces pointwise surjective presentations of stacks, extending their applicability over different fields and rings, with new applications in algebraic and real geometry.

Findings

Existence of pointwise surjective presentations over finite and real closed fields.

Extension of results to certain Henselian rings.

Applications in counting isomorphism classes and real algebraic stacks.

Abstract

We show that any stack $\mathfrak{X}$ of finite type over a Noetherian scheme has a presentation $X \rightarrow \mathfrak{X}$ by a scheme of finite type such that $X(F) \rightarrow \mathfrak{X}(F)$ is onto, for every finite or real closed field $F$. Under some additional conditions on $\mathfrak{X}$, we show the same for all perfect fields. We prove similar results for (some) Henselian rings. We give two applications of the main result. One is to counting isomorphism classes of stacks over the rings $\mathbb{Z}/p^n$; the other is about the relation between real algebraic and Nash stacks.