Geometry of arc spaces

Transcription

1 Geometry of arc spaces Tommaso de Fernex University of Utah Differential, Algebraic and Topological Methods in Complex Algebraic Geometry Cetraro, September 2018 Tommaso de Fernex (University of Utah) Geometry of arc spaces Cetraro, September / 24

2 Introduction Introduction Tommaso de Fernex (University of Utah) Geometry of arc spaces Cetraro, September / 24

3 Introduction First definition and motivations Let X be a complex algebraic variety C the formal disk The arc space of X is X := { α X } Maps α: X are called arcs. Usage: Motivic integration (Kontsevich, Denef Loeser, Batyrev,...) Singularities in the MMP (Mustaţă, Ein Mustaţă Yasuda,...) Kobayashi hyperbolicity (Siu, D ly,...) Object of study (Greenberg, Nash, Kolchin,...) Tommaso de Fernex (University of Utah) Geometry of arc spaces Cetraro, September / 24

4 Introduction Formal definition Functor of points: As schemes, = Spec C[[t]] and for every K/C X (K) = Hom C (Spec K[[t]], X ) Inverse limit: Define the m-th jet scheme to be X m = { m X } where m C is the m-th neighborhood of the origin. That is, X m (K) = Hom C (Spec K[t]/(t m+1 ), X ) Note: X 0 = X and X 1 = TX. Inclusions m m+1 induce affine maps X m+1 X m, and X = limx m. Tommaso de Fernex (University of Utah) Geometry of arc spaces Cetraro, September / 24

5 Introduction Equations For A n = Spec C[x 1,..., x n ], we have A n = ( C[[t]]) n = Spec C[x i, x i, x i,... ] α = (a i + a i t + a i t ) (a i, a i, a i,... ) If X = {f j = 0} A n, then α X f j (a i + a it + a i t ) 0 Expanding with respect to t, we get equations X = {f j = f j = f j = = 0} A n where f (m) (x i, x i,..., x (m) i ) is the m-th derivation of f (x i ). Note: Infinitely many equations in infinitely many variables. Tommaso de Fernex (University of Utah) Geometry of arc spaces Cetraro, September / 24

6 Introduction Finiteness properties Even if X is a variety, X is not Noetherian (is infinite dimensional). Yet: Theorem (Greenberg) The images of the truncation maps X n X m stabilize for n m. In particular, the image of X X m is constructible. Theorem (Kolchin) X is irreducible (in characteristic zero). Theorem (Nash) The are finitely many families of arcs through the singularities of X. That is, π 1 (Sing(X )) has finitely many components, where π : X X. The Nash problem asks to characterize these components in terms of resolutions of singularities (and divisorial valuations). Tommaso de Fernex (University of Utah) Geometry of arc spaces Cetraro, September / 24

7 Main Theorem Main Theorem Tommaso de Fernex (University of Utah) Geometry of arc spaces Cetraro, September / 24

8 Main Theorem Our focus Points: α X α: Spec K[[t]] X. If α, as a morphism Spec K[[t]] X, is dominant, then the order of contact along α defines a valuation ord α : C(X ) This brings us to birational geometry. Local rings: O X,α α k((t)) ord t N Some questions: When is O X,α (or is completion) Noetherian? Can we compute the dimension of O X,α? Can we compute the embedding dimension of O X,α? Why do we care about these questions? Tommaso de Fernex (University of Utah) Geometry of arc spaces Cetraro, September / 24

9 Main Theorem Our approach Focusing on points and local rings of X is not new (e.g., see Lejeune-Jalabert, Denef Loeser, Ishii, Reguera,...) What is new is the approach. In joint work with Roi Docampo, we use the sheaf of Kähler differentials to address these questions. Our main result is a formula for Ω X. We deduce properties regarding the points and local rings of X. As application, we recover various known properties of arc spaces while providing a unified point of view. Tommaso de Fernex (University of Utah) Geometry of arc spaces Cetraro, September / 24

10 Main Theorem Sheaf of differentials The universal arc U = X Spf C[[t]] gives the universal family U γ X ρ X where ρ is the natural projection and γ is the evaluation map. Theorem (de Fernex Docampo) There exists a sheaf P on U such that Ω X = ρ (γ Ω X P ) The sheaf P is characterized by ρ O U = Hom X (ρ P, O X ). Tommaso de Fernex (University of Utah) Geometry of arc spaces Cetraro, September / 24

11 Main Theorem Algebraic formulation Assume X = Spec A, and write X = Spec A. The universal arc is A [[t]] ρ γ A A where γ (f ) = f + f t + f t Then and A [[t]] = A t i i=0 P = A ((t)) ta [[t]] = A t j Ω A /C = Ω A/C A P. j=0 Tommaso de Fernex (University of Utah) Geometry of arc spaces Cetraro, September / 24

12 Main Theorem The formula for jet schemes Assume X = Spec A, and write X m = Spec A m. The universal m-jet is A m [t]/(t m+1 ) ρ m γ m A A m where γ (f ) = f + f t + f t f (m) t m. m A m [t]/(t m+1 ) = A t i P m = t m A m [t] = ta m [t] and Hence i=0 Ω Am/C = Ω A/C A P m m A m t j j=0 Ω t m A/C A A m [t]/(t m+1 ) Ω Xm = ρ m (γ mω X P m ) = ρ m γ mω X Tommaso de Fernex (University of Utah) Geometry of arc spaces Cetraro, September / 24

13 Main Theorem Idea of the proof Need to show Ω A /C = Ω A/C A P where A [[t]] = Hom A (P, A ) By duality from the next formula, where M is any A -module: Der C (A, M) = Der C (A, M A A [[t]]) Geometrically, if M = K/C, this says that for any α: Spec K[[t]] X { tangent vectors } { vector fields on } = to X at α X along Im(α) Analogue to (f : Y X ) Mor(Y, X ) T f Mor(Y, X ) = H 0 (Y, f TX ) Tommaso de Fernex (University of Utah) Geometry of arc spaces Cetraro, September / 24

14 Applications Applications Tommaso de Fernex (University of Utah) Geometry of arc spaces Cetraro, September / 24

15 Applications Embedding dimension Let α X. The embedding dimension at α is emb.dim(o X,α) := dim κα (m α /m 2 α) If α m X m is the image of α under the truncation map X X m, then the jet codimension of α is ( jet.codim(α, X ) := lim (m + 1) dim(x ) dim(αm ) ) m Theorem (de Fernex Docampo) If X is a variety, then emb.dim(o X,α) = jet.codim(α, X ) Tommaso de Fernex (University of Utah) Geometry of arc spaces Cetraro, September / 24

16 Applications Birational transformation rule Let f : Y X be a resolution of singularities of a variety X β Theorem (de Fernex Docampo) Y Y f f α X X emb.dim(o X,α) = emb.dim(o Y,β) + ord β (Jac f ) This relates to the change-of-variables formula in motivic integration: Theorem (Kontsevich, Denef Loeser) L ord(d) = X Y L ord(k Y /X +f D) (X smooth, D a divisor) Tommaso de Fernex (University of Utah) Geometry of arc spaces Cetraro, September / 24

17 Applications Log discrepancies A maximal divisorial arc α E X is given by generic point β E Theorem (de Fernex Docampo) π 1 Y (E) Y π Y Y f f π X α E X X emb.dim(o X,α E ) = ord E (Jac f ) + 1 E divisor The analogous formula for O X,α E is due to Mourtada Reguera. Corollary (Ein Lazarsfeld Mustaţă, de Fernex Ein Ishii) jet.codim(α E, X ) = ord E (Jac f ) + 1 This relates to inversion of adjunction results in the MMP program. Tommaso de Fernex (University of Utah) Geometry of arc spaces Cetraro, September / 24

18 Applications Stable points α X is a stable point if it is the generic point of an irreducible component of a constructible subset (a.k.a. cylinder) of X. Theorem (de Fernex Docampo) α X stable, α Sing(X ) emb.dim(o X,α) < Note: The latter condition is equivalent to Ô X,α being Noetherian. Corollary (Reguera) For any maximal divisorial arc α E X, O X,α E is Noetherian and hence the curve selection lemma holds in a neighborhood of α E. This implies the existence of wedges for Nash-type problems. Tommaso de Fernex (University of Utah) Geometry of arc spaces Cetraro, September / 24

19 Applications Embedding codimension For a Noetherian scheme S and a point p S, one defines emb.codim(o S,p ) := emb.dim(o S,p ) dim(o S,p ) If S is not Noetherian, then we need a different definition. Fix elements x i O S,p giving a basis in m p /m 2 p, and let φ: K[[y i ]] ÔS,p, y i x i where K ÔS,p is a coefficient field. In joint work with Christopher Chiu and Roi Docampo, we define the embedding codimension: emb.codim(o S,p ) := height(ker(φ)) Tommaso de Fernex (University of Utah) Geometry of arc spaces Cetraro, September / 24

20 Applications Singular factors of formal neighborhoods Let α X (C). Theorem (Grinberg Kazhdan, Drinfeld) If α Sing(X ), then there exists a scheme Z of finite type, called singular factor, and a point q Z(C) such that We prove the converse: Spec ( ) ) Ô X,α = Spec (ÔZ,q Theorem (Chiu de Fernex Docampo) If α Sing(X ), then emb.codim(o X,α) =. In particular, there is no singular factor of finite type in Spec ( Ô X,α). Tommaso de Fernex (University of Utah) Geometry of arc spaces Cetraro, September / 24

21 A Simple Computation A Simple Computation Tommaso de Fernex (University of Utah) Geometry of arc spaces Cetraro, September / 24

22 A Simple Computation Fitting invariants Let α X and α m X m its image. Let K and K m be the residue fields. Theorem (de Fernex Docampo) Ω Xm K m = ( Km [t]/(t m+1 ) ) d m i d m K m [t]/(t ei ) Here, d m is the Betti number and e i are the invariant factors of α mω X : m + 1 > e dm e dm+1 e dm+2... These numbers stabilize for m 1. (eventually zero) Geometrically, d := d m (for m 1) is the dimension of the fiber of Ω X at the generic point of the image of α: Spec K[[t]] X. Setting e i = for i < d, we recover the Fitting invariants ord α (Fitt i (Ω X )) = e i + e i+1 + e i Tommaso de Fernex (University of Utah) Geometry of arc spaces Cetraro, September / 24

23 A Simple Computation Rank of the differential map For n > m 1, the differential map Ω Xm K n Ω Xn K n corresponds to the map ( Kn [t]/(t m+1 ) ) d K n [t]/(t e i ) ( K n [t]/(t n+1 ) ) d K n [t]/(t e i ) i d given by multiplication by t n m. For n m > ord α (Fitt d (Ω X )), this map kills the invariant factors part. Corollary Denoting π m : X X m, for m 1 we have i d rank ( dπ m : T α X T αm X m ) = (m + 1)d dim(αm ), where d dim X with equality holding if and only if α Sing(X ). Formulas for embedding dimension and codimension follow from this. Tommaso de Fernex (University of Utah) Geometry of arc spaces Cetraro, September / 24

24 Thank You Thank You Tommaso de Fernex (University of Utah) Geometry of arc spaces Cetraro, September / 24