Abstract

Midwest Algebra, Geometry and their Interactions Conference
MAGIC05

University of Notre Dame, Notre Dame
October 7-11, 2005

Contact loci and valuations
by Lawrence EIN, University of Illinois at Chicago
Abstract: This is joint work with R. Lazarsfeld and M. Mustata. Let X be a smooth complex variety. Given m ³ 0, we denote by X_m = Hom (Spec C[t]/(t^m+1), X) the space of m^th order arcs on X. Similarly we define the space of formal arcs on X as X_¥ = Hom (Spec C[[t]], X) Consider now a non-zero ideal sheaf a Í O_X defining a subscheme Y Í X. Given a finite or infinite arc g on X, the order of vanishing of a --- or the order of contact of the corresponding scheme Y --- along g is defined in the natural way.¹ For a fixed integer p ³ 0, we define the contact loci Cont^p(Y) = Cont^p(a) = { g Î X_¥ | ord_g(a) = p }. These are locally closed cylinders, i.e. they arise as the common pull-back of the locally closed sets (1) Cont^p(Y)_m = Cont^p (a)_m =_def { g Î X_m | ord_g(a) = p } defined for any m ³ p. Let W be the closure of an irreducible component of Cont^p(a). We can naturally associate a valuation of the function field of X to W in the following manner. Let f be a nonzero rational function of X. We define val_W(f) = ord_g(f) for a general g Î W. Such a valuation is called a contact valuation. Suppose m: X' → X be a proper birational morphism. Assume that E is an irreducible divisor in X'. We can define the valuation associated to E by val_E(f) = the vanishing order of f along E. A valuation on the function field of X is called a divisorial valuation if it is of the form m × val_E for some positive integer m. A basic invariant of val_E from higher dimensional birational geometry is the discrepancy along E which is defined as k_E = val_E(det(J(m ))) where J(m ) is the Jacobian matrix of m. k_E is just the coefficient of the relative canonical divisor K_X'/X along E.

Theorem A. Every contact valuation is a divisorial valuation. Conversely, every divisorial valuation can be realized uniquely as a contact valuation.

In the above correspondence, suppose that a contact valuation val_W is equal to a divisorial valuation m × Val_E. The following theorem relates the geometry between the two valuations.

Theorem B. codim(W,X_¥) = m × (kE + 1).

The above two theorems also hold for singular varieties after some minor modifications using Nash's blow-up and Mather's canonical class.

These results can be used to study singularities of pairs and inversion of adjunction.

References
[1] L. Ein, R. Lazarsfeld, M. Mustata, Contact loci in arc spaces, Compositio Math. 140 (2004) 1229-1244.
[2] L. Ein, M. Mustata, and T. Yasuda, Jet schemes, log discrepancies and inversion of adjunction, Invent. Math. 153 (2003) 519-535.
[3] L. Ein, M. Mustata, Inversion of adjunction for local complete intersection variety, Amer. J. Math. 126 (2004)1355-1365.
[4] M. Mustata, Singularities of pairs via jet schemes, J. Amer. Math. Soc. 15 (2002), 599-615.
[5] M. Mustata, Jet schemes of locally complete intersection canonical singularities, with an appendix by D. Eisenbud and E. Frenkel, Invent. Math. 145 (2001), 397-424.

¹ Specifically, pulling a back via g yields an ideal (t^e) in C[t]/(t^m+1) or C[[t]], and one sets ord_g(a) = ord_g (Y) = e. (Take ord_g(a) = m+1 when a pulls back to the zero ideal in C[t]/(t^m+1) and ord_g(a) = ¥ when it pulls back to the zero ideal in C[[t]].)