Contact loci in arc spaces

Abstract

We study loci of arcs on a smooth variety defined by order of contact with a fixed subscheme. Specifically, we establish a Nash-type correspondence showing that the irreducible components of these loci arise from (intersections of) exceptional divisors in a resolution of singularities. We show also that these loci account for all the valuations determined by irreducible cylinders in the arc space. Along the way, we recover in an elementary fashion-without using motivic integration-results of the third author relating singularities to arc spaces. Moreover, we extend these results to give a jet-theoretic interpretation of multiplier ideals.

18 pages; v.2: minor changes; final version, to appear in Compositio Math

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• Bounds for log canonical thresholds with applications to birational rigidity
• The motivic zeta function and its smallest poles
• The arc space of a toric variety