  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.
