On arc fibers of morphisms of schemes

TL;DR

This paper investigates the fibers of morphisms between arc spaces of schemes over a field, establishing finiteness, bounds, and structural properties, and explores implications for local geometry and invariants of arc spaces.

Contribution

It provides new finiteness results, bounds, and structural insights for arc space fibers of morphisms, extending classical algebraic geometry concepts to arc spaces.

Findings

Fibers of the induced map on arc spaces are topologically finite with bounded cardinality.

The restriction of the arc space map outside the ramification locus has scheme-theoretically finite reduced fibers.

Local rings at stable points of arc spaces have finitely generated maximal ideals and topologically Noetherian spectra.

Abstract

Given a morphism $f \colon X \to Y$ of schemes over a field, we prove several finiteness results about the fibers of the induced map on arc spaces $f_\infty \colon X_\infty \to Y_\infty$. Assuming that $f$ is quasi-finite and $X$ is separated and quasi-compact, our theorem states that $f_\infty$ has topologically finite fibers of bounded cardinality and its restriction to $X_\infty \setminus R_\infty$, where $R$ is the ramification locus of $f$, has scheme-theoretically finite reduced fibers. We also provide an effective bound on the cardinality of the fibers of $f_\infty$ when $f$ is a finite morphism of varieties over an algebraically closed field, describe the ramification locus of $f_\infty$, and prove a general criterion for $f_\infty$ to be a morphism of finite type. We apply these results to further explore the local structure of arc spaces. One application is that the local ring at a stable point of the arc space of a variety has finitely generated maximal ideal and topologically Noetherian spectrum, something that should be contrasted with the fact that these rings are not Noetherian in general; a lower-bound to the dimension of these rings is also obtained. Another application gives a semicontinuity property for the embedding dimension and embedding codimension of arc spaces which extends to this setting a theorem of Lech on Noetherian local rings and translates into a semicontinuity property for Mather log discrepancies. Other applications are discussed in the paper.