Embedding codimension of the space of arcs

Chiu, Christopher; de Fernex, Tommaso; Docampo, Roi

Embedding codimension of the space of arcs

dc.contributor.author	Chiu, Christopher
dc.contributor.author	de Fernex, Tommaso
dc.contributor.author	Docampo, Roi
dc.date.accessioned	2022-03-11T21:02:06Z
dc.date.available	2022-03-11T21:02:06Z
dc.date.issued	2022-02-21
dc.description.abstract	We introduce a notion of embedding codimension of an arbitrary local ring, establish some general properties and study in detail the case of arc spaces of schemes of finite type over a field. Viewing the embedding codimension as a measure of singularities, our main result can be interpreted as saying that the singularities of the arc space are maximal at the arcs that are fully embedded in the singular locus of the underlying scheme, and progressively improve as we move away from said locus. As an application, we complement a theorem of Drinfeld, Grinberg and Kazhdan on formal neighbourhoods in arc spaces by providing a converse to their theorem, an optimal bound for the embedding codimension of the formal model appearing in the statement, a precise formula for the embedding dimension of the model constructed in Drinfeld’s proof and a geometric meaningful way of realising the decomposition stated in the theorem.	en_US
dc.description.peerreview	Yes	en_US
dc.identifier.citation	Forum of Mathematics, Pi, Volume 10, 2022, e4 DOI: https://doi.org/10.1017/fmp.2021.19	en_US
dc.identifier.doi	https://doi.org/10.1017/fmp.2021.19	en_US
dc.identifier.uri	https://hdl.handle.net/11244/334965
dc.language	en	en_US
dc.rights	Attribution 4.0 International	*
dc.rights.uri	http://creativecommons.org/licenses/by/4.0/	*
dc.subject	Birational geometry	en_US
dc.subject	Ring extension and related topics	en_US
dc.subject	Arithmetic rings and other special rings	en_US
dc.subject	Theory of modules and ideals	en_US
dc.subject	Local theory	en_US
dc.title	Embedding codimension of the space of arcs	en_US
dc.type	Article	en_US