Contact loci in arc spaces

Details Contact loci in arc spaces Contact loci in arc spaces

2003-03-21

arxiv.org/abs/math/0303268v4

Compositio Math. 140 (2004), 1229-1244.

Mathematics

Algebraic Geometry

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

Scientific paper

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.

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