CONTACT LOCI AND VALUATIONSLAWRENCE EINThis is joint work with R. Lazarsfeld and M. Mustaţǎ. Let X be a smooth complexvariety. Given m ≥ 0, we denote by¡¢
Xm = Hom Spec C[t]/(tm+1 ) , X
the space of mth 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 ⊆ OX defining a subscheme Y ⊆ X. Given afinite or infinite arc γ on X, the order of vanishing of a — or the order of contact of thecorresponding scheme Y — along γ is defined in the natural way.1 For a fixed integerp ≥ 0, we define the contact loci©
ªContp (Y ) = Contp (a) = γ ∈ X∞ | ordγ (a) = p .
These are locally closed cylinders, i.e. they arise as the common pull-back of the locallyclosed sets©
ª(1)Contp (Y )m = Contp (a)m =def γ ∈ Xm | ordγ (a) = p
defined for any m ≥ p. Let W be the closure of an irreducible component of Contp (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 definevalW (f ) = ordγ (f ) for a general γ ∈ W .
Such a valuation is called a contact valuation. Suppose µ : X 0 −→ X be a properbirational morphism. Assume that E is an irreducible divisor in X 0 . We can define thevaluation associated to E by valE (f ) = the vanishing order of f along E. A valuationon the function field of X is called a divisoriral valuation if it is of the form m · valE forsome positive integer m. A basic invariant of valE from higher dimensional birational
geometry is the discrepancy along E which is defined as
kE = valE (det(J(µ)) where J(µ) is the Jacobian matrix of µ.
kE is just the coefficient of the relative canonical divisor KX 0 /X along E.
Theorem A. Every contact valuation is a divisorial valuation. Conversely, every divisorial valuation can be realized uniquely as a contact valuation.
1Specifically, pulling a back via γ yields an ideal (te ) in C[t]/(tm+1 ) or C[[t]], and one setsordγ (a) = ordγ (Y ) = e.
(Take ordγ (a) = m + 1 when a pulls back to the zero ideal in C[t]/(tm+1 ) and ordγ (a) = ∞ when it pulls
back to the zero ideal in C[[t]].)
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In the above correspondence, suppose that a contact valuation valW is equal to adivisorial valuation m · V alE . The following theorem relates the geometry between thetwo 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. Let X be a normal QGorenstein complex variety and Y be closed subscheme of X. We also fix a positive
number λ. One can study the singularities of the pair (X, λ · Y ) using log-resolution.
Consider a divisorial valuation of the form valE with center cX (E) in X. We consider
the log discrepancy of (X, λ · Y ) along E,a(E, X, λ · Y ) = kE + 1 − λ · valE (IY ),where IY is the ideal of Y in X. Suppose that B is a closed subset of X. We can measure
the singularities of the pair (X, λ·Y ) along B using the invariant minimal log-discrepancy.
Definition 0.1. Let B ⊆ X be a nonempty closed subset. The minimal log discrepancyof (X, λ · Y ) on W is defined by(2)mld(B; X, λ · Y ) :=infcX (E)⊆W{a(E; X, λ · Y )}.

Suppose D is a normal effective Cartier divisor in X and B be a closed subset ofD. The following theorem is a joint result with Mustaţǎ. It allows us to compute theminimal log-discrepancy mld(B; X, D + λ · Y ) using mld(B; D, λ · Y |D ).
Theorem C. Let X be a normal, local complete intersection variety, and Y be a proper
closed subscheme of X. Let λ is a positive number. Assume D ⊂ X is a normal effective
Cartier divisor such that D 6⊆ Y , then for every proper closed subset B ⊂ D, we havemld(B; X, D + λ · Y ) = mld(B; D, λ · Y |D ).
The theorem is first proved in the case that X is smooth in a joint paper of Ein,
Mustaţǎ and Yasuda. In general, Kollár, and Shokurov have conjectured that the result
is true when X is just normal and Q-Gorenstein (Inversion of Adjunction). See Kollár’s
article for a discussion of this conjecture and related topics. The following are somegeometric applications of the Theorem.
Theorem D. If X is a normal, local complete intersection variety, and Y is a closed
subscheme. Suppose that λ is a positive number then the function x −→ mld(x; X, λ · Y ),x ∈ X, is lower semicontinuous.
Theorem E. Let X be a normal, local complete intersection variety. X has log canonical (canonical, terminal) singularities if and only if Xm is equidimensional (respectivelyirreducible, normal) for every m.

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3References[1] F. Ambro, Inversion of Adjunction for non-degenerate hypersurfaces, arXiv: math.AG/0108168.
[2] L. Ein, R. Lazarsfeld, M. Mustaţǎ, Contact loci in arc spaces, Compositio Math. 140 (2004)
1229-1244 math.AG/0303002.
[3] L. Ein, M. Mustaţǎ, and T. Yasuda Jet schemes, log discrepancies and inversion of adjunction
Invent. Math. 153 (2003) 519-535.
[4] L. Ein, M. Mustaţǎ, Inversion of adjunction for local complete intersection variety, Am. J. Math.
126 (2004)1355-1365.
[5] J. Kollár, Singularities of pairs, in Algebraic Geometry, Santa Cruz 1995, volume 62 of Proc. Symp.
Pure Math Amer. Math. Soc. 1997, 221–286.
[6] M. Mustaţǎ, Singularities of pairs via jet schemes, J. Amer. Math. Soc. 15 (2002), 599–615.
[7] M. Mustaţǎ, Jet schemes of locally complete intersection canonical singularities, with an appendix
by D. Eisenbud and E. Frenkel, Invent. Math. 145 (2001), 397–424.