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Given a morphismf \colon X \to Yof schemes over a field, we prove several finiteness results about the fibers of the induced mapf_{\infty} \colon X_{\infty} \to Y_{\infty}on arc spaces. Assuming thatfis quasi-finite andXis separated and quasi-compact, our theorem states thatf_{\infty}has topologically finite fibers of bounded cardinality and its restriction toX_{\infty} \setminus R_{\infty}, whereRis the ramification locus off, has scheme-theoretically finite reduced fibers. We also provide an effective bound on the cardinality of the fibers off_{\infty}whenfis a finite morphism of varieties over an algebraically closed field, describe the ramification locus off_{\infty}, and prove a general criterion forf_{\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 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. 

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.