Local uniformization and arc spaces.

Let X be a variety over a perfect field k and let X ∞ be its space of arcs. Given a closed subset Z of X , let X ∞ Z denote the subscheme of X ∞ consisting of all arcs centered at some point of Z . We prove that Local Uniformization implies that X ∞ Z has a finite number of irreducible components for each closed subset Z of X . In particular, Local Uniformization implies that X ∞ Sing X has a finite number of irreducible components. [ABSTRACT FROM AUTHOR]