Search for: All records

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.