Small irreducible components of arc spaces in positive characteristic

We study the Irreducibility Problem for arc space X∞ of an irreducible singular algebraic variety X defined over a perfect field k characteristic p>0. The existence reducible such is related to fact that Kolchin's Theorem fails hold in positive characteristic. obtain complete description components when surface. Section 5 introduces two main new problems arbitrary dimension: (1) line O. Zariski and H. Hironaka, blowing up any get Y→X with Y∞ irreducible; (2) J. Nash's work on arcs, characterizing irreducibility terms Resolution Singularities.