On arc fibers of morphisms of schemes
Publisher DOI
Abstract
Given a morphism f:X→Y of schemes over a field, we prove several finiteness results about the fibers of the induced map f∞:X∞→Y∞ on arc spaces. Assuming that f is quasi-finite and X is separated and quasi-compact, our theorem states that f∞ has topologically finite fibers of bounded cardinality and its restriction to X∞∖R∞, 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∞ when f is a finite morphism of varieties over an algebraically closed field, describe the ramification locus of f∞, and prove a general criterion for f∞ 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 on 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 also discussed.
Date Issued
2024-07-01
Publication Type
Article
Subject(s)
Subjects
arc space
•
generically finite morphism
•
embedding dimension
•
stable point
Language(s)
en
Author(s)
de Fernex, Tommaso | |
Docampo, Roi |
Additional Credits
Journal
Journal of the European Mathematical Society
Publisher
EMS Press
ISSN
1435-9855
1435-9863
Access(Rights)
open.access