Small irreducible components of arc spaces in positive characteristic

Abstract

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