An inverse mapping theorem for blow-Nash maps on singular spaces
aa r X i v : . [ m a t h . AG ] M a y An inverse mapping theorem for blow-Nash maps onsingular spaces
Jean-Baptiste Campesato ⋆ May 11, 2015
Abstract
A semialgebraic map f : X → Y between two real algebraic sets is called blow-Nash if itcan be made Nash (i.e. semialgebraic and real analytic) by composing with finitely manyblowings-up with non-singular centers.We prove that if a blow-Nash self-homeomorphism f : X → X satisfies a lower boundof the Jacobian determinant condition then f − is also blow-Nash and satisfies the samecondition.The proof relies on motivic integration arguments and on the virtual Poincaré polyno-mial of McCrory–Parusiński and Fichou. In particular, we need to generalize Denef–Loeserchange of variables key lemma to maps that are generically one-to-one and not merely bira-tional. Contents
References 19
Blow-analytic maps were introduced by T.-C. Kuo in order to classify real singularities [26, 27, 28].A map f : X → Y between real algebraic sets is called blow-analytic if there exists σ : M → X afinite sequence of blowings-up with non-singular centers such that f ◦ σ is analytic. In the samevein a semialgebraic map between real algebraic sets is called blow-Nash if the compositionwith some finite sequence of blowings-up with non-singular centers is Nash (i.e. semialgebraicand analytic). Arc-analytic maps were introduced by K. Kurdyka [29]. A map f : X → Y betweentwo real algebraic sets is called arc-analytic if every real analytic arc on X is mapped by f to a real ⋆ Univ. Nice Sophia Antipolis, CNRS, LJAD, UMR 7351, 06100 Nice, France.E-mail address:
A blow-Nash inverse mapping theoremanalytic arc on Y . By a result of E. Bierstone and P. D. Milman [5] in response to a question of K.Kurdyka, if f : X → Y is semialgebraic (i.e. its graph is semialgebraic) and if X is non-singularthen f is arc-analytic if and only if it is blow-Nash. When X is non-singular, the set of pointswhere such a map is analytic is dense [29, 5.2] and thus the Jacobian determinant of f is definedeverywhere except on a nowhere dense subset of X .The following Inverse Function Theorem is known for X non-singular [13]: if the Jacobiandeterminant of a blow-Nash self-homeomorphism h : X → X is locally bounded from below by a non-zero constant, on the set it is defined, then h − is blow-Nash and its Jacobian determinant is also locallybounded from below by a non-zero constant on the set it is defined. In this paper, we generalize this theorem for singular algebraic sets.We first introduce, in subsection 2.3, the notion of generically arc-analytic maps which aremaps f : X → Y between real algebraic sets such that there exists a nowhere dense subset S of X with the property that every arc on X not entirely included in S is mapped by f to a real analyticarc on Y . When dim Sing ( X ) ≥ , we see that this condition is strictly weaker than being arc-analytic, otherwise a continuous generically arc-analytic map is an arc-analytic map. Then weshow that the semialgebraic generically arc-analytic maps are exactly the blow-Nash ones.Given f : X → X a blow-Nash self-map on a real algebraic set X , we have the followingdiagram M σ ~ ~ ⑦⑦⑦⑦⑦⑦⑦⑦ ˜ σ ❅❅❅❅❅❅❅❅ X f / / X with σ given by a sequence of blowings-up with non-singular centers and ˜ σ a Nash map.We may now give an analogue of the lower bound of the Jacobian determinant condition: wesay that f satisfies the Jacobian hypothesis if the Jacobian ideal of σ is included in the Jacobianideal of ˜ σ . This condition doesn’t depend on the choice of σ .We are now able to state the main theorem of this paper: let f : X → X be a semialgebraic self-homeomorphism with X an algebraic subset then f is blow-Nash and satisfies the Jacobian hypothesis ifand only if f − satisfies the same conditions. Heuristically, the main idea of the proof consists in comparing the “motivic volume” of theset of arcs on X and the “motivic volume” of the set of arcs on X coming from arcs on M by˜ σ . This allows us to prove that we can uniquely lift by ˜ σ an arc not entirely included in somenowhere dense subset of X . Thereby, such an arc is mapped to an analytic arc by f − . Thus f − is generically arc-analytic and so blow-Nash.Therefore, we first define the arc space on an algebraic subset X of R N as the set of germs ofanalytic arcs on R N which lie in X , i.e. γ : ( R , 0 ) → X such that ∀ f ∈ I ( X ) , f ( γ ) = . For n ∈ N ,we define the space of n -jets on X as the set of n -jets γ on R N such that ∀ f ∈ I ( X ) , f ( γ ( t )) ≡ mod t n + . The subsection 2.4 contains some general properties of these objects and some usefulresults for the proof of the main theorem.The additive invariant used in order to apply motivic integration arguments is the virtualPoincaré polynomial which associates to a set of a certain class, denoted AS , a polynomial withinteger coefficients. We recall the main properties of the collection AS in subsection 2.1. Thevirtual Poincaré polynomial was constructed by C. McCrory, A. Parusiński [36] and G. Fichou[11]. The subsection 2.2 contains the main properties of this invariant and motivates its use.In order to compute the above-cited “motivic volumes” , we first prove a version of Denef–Loeserkey lemma for the motivic change of variables formula which fulfills our requirements and witha weaker hypothesis: we don’t assume the map to be birational but only generically one-to-one.Based on these results, we may finally prove there exists a subset on X such that every ana-lytic arc on X not entirely included in this subset may be uniquely lifted by ˜ σ . This part relieson real analysis arguments and on the fact that an arc not entirely included in the center of ablowing-up may be lifted by this blowing-up.ean-Baptiste Campesato I am very grateful to my thesis advisor Adam Parusiński for his help andsupport during the preparation of this work.
Arc-symmetric sets have been first defined and studied by K. Kurdyka in [29]. A subset ofan analytic manifold M is arc-symmetric if all analytic arcs on M meet it at isolated points orare entirely included in it. Semialgebraic arc-symmetric sets are exactly the closed sets of anoetherian topology AR on R N . We work with a slightly different framework defined by A.Parusiński in [41] and consider the collection of sets AS defined as the boolean algebra gener-ated by semialgebraic arc-symmetric subsets of P n R . The advantages of AS over AR are that weget a constructible category and a better control of the behavior at infinity. We refer the readerto [31] for a survey. Definition 2.1 ([41, 2.4]) . Let C be a collection of semialgebraic sets. A map between two C -sets is a C -map if its graph is a C -set. We say that C is a constructible category if it satisfies thefollowing axioms:A1. C contains the algebraic sets.A2. C is stable by boolean operations ∩ , ∪ and \ .A3. a. The inverse image of a C -set by a C -map is a C -set.b. The image of a C -set by an injective C -map is a C -set.A4. Each locally compact X ∈ C is Euler in codimension 1, i.e. there is a semialgebraic subset Y ⊂ X with dim Y ≤ dim X − such that X \ Y is Euler ⋆ . Remark 2.2.
A locally compact semialgebraic set X is Euler in codimension if and only if itadmits a fundamental class for the homology with coefficient in Z . For instance, this propertyis crucial in the construction of the virtual Poincaré polynomial in order to use the Poincaréduality.Given a constructible category C , we have a notion of C -closure. Theorem 2.3 ([41, 2.5]) . Let C be a constructible category and let X ∈ C be a locally closed set. Then forany subset A ⊂ X there is a smallest closed subset of X which belongs to C and contains A . It is denotedby A C . Any other closed subset of X that is in C and contains A must contain A C . Remark 2.4 ([41, 2.7]) . If A is semialgebraic then dim A C = dim A . In particular, if A ∈ C then A C = A ∪ A \ A C and hence dim (cid:16) A C \ A (cid:17) < dim A . Definition 2.5 ([41, §4.2]) . A semialgebraic subset A ⊂ P n R is an AS -set if for every real analyticarc γ : (−
1, 1 ) → P n R such that γ ((−
1, 0 )) ⊂ A there exists ε > 0 such that γ ((
0, ε )) ⊂ A .Using the proof of [41, Theorem 2.5], we get the following proposition. Proposition 2.6.
There exists a unique noetherian topology on P n R whose closed sets are exactly the closed AS -subsets. Theorem 2.7 ([41]) . • The algebraically constructible sets form a constructible category denoted by AC . • AS is a constructible category. • Every constructible category contains AC and is contained in AS . This implies that each locallycompact set in a constructible category is Euler. ⋆ A locally compact semialgebraic set X is Euler if for every x ∈ X the Euler-Poincaré characteristic of X at xχ ( X, X \ x ) = P (− ) i dim H i ( X, X \ x ; Z ) is odd. A blow-Nash inverse mapping theorem • AS is the only constructible category which contains the connected components of compact realalgebraic sets.
In what follows, constructible subset stands for AS -subset, constructible map stands for mapwith constructible graph and constructible isomorphism stands for AS -homeomorphism.In our proof of Lemma 4.5 we need the following result which is, in some sense, a replace-ment of Chevalley’s theorem for Zariski-constructible sets over an algebraically closed field. Theorem 2.8 ([41, 4.3]) . Let A be a semialgebraic subset of a real algebraic subset X of P n R . Then A ∈ AS if and only if there exist a regular morphism of real algebraic varieties f : Z → X and Z ′ the union ofsome connected components of Z such that x ∈ A ⇔ χ (cid:16) f − ( x ) ∩ Z ′ (cid:17) ≡ mod
2x / ∈ A ⇔ χ (cid:16) f − ( x ) ∩ Z ′ (cid:17) ≡ mod where χ is the Euler characteristic with compact support.In particular the image of an AS -subset by a regular map whose Euler characteristics with compactsupport of all the fibers are odd is an AS -subset. In this paper, we need to work with AS -sets in order to use the virtual Poincaré polynomialdiscussed below.In our settings, the noetherianity of the AS topology will also allow us to prove a versionof J. Denef and F. Loeser key lemma for the motivic change of variables formula with a weakerhypothesis. Indeed, we won’t assume that the map is birational but only Nash, proper andgenerically one-to-one. C. McCrory and A. Parusiński proved in [36] there exists a unique additive invariant of realalgebraic varieties which coincides with the Poincaré polynomial for (co)homology with Z co-efficients for compact and non-singular real algebraic varieties. Moreover, this invariant behaveswell since its degree is exactly the dimension and the leading coefficient is positive. This vir-tual Poincaré polynomial has been generalized to AS -subsets by G. Fichou in [11]. FurthermoreNash-equivalent AS -subsets have the same virtual Poincaré polynomial. These proofs use theweak factorization theorem [49, 1] in a way similar of what has been done by Bittner in [7] togive a new description of the Grothendieck ring in terms of blowings-up. Theorem 2.9 ([11]) . There is an additive invariant β : AS → Z [ u ] , called the virtual Poincaré polyno-mial , which associates to an AS -subset a polynomial with integer coefficients β ( X ) = P β i ( X ) u i ∈ Z [ u ] and satisfies the following properties: • β k G i = X i ! = k X i = β ( X i ) • β ( X × Y ) = β ( X ) β ( Y ) • For X = ∅ , deg β ( X ) = dim X and the leading coefficient of β ( X ) is positive ⋆ . • If X is non-singular and compact then β i ( X ) = dim H i ( X, Z ) . • If X and Y are Nash-equivalent then β ( X ) = β ( Y ) . The virtual Poincaré polynomial is a more interesting additive invariant than the Euler char-acteristic with compact support since it stores more information, like the dimension. Notice thatit is well known that if we forget the arc-symmetric hypothesis and work with all semialgebraicsets, the Euler characteristic with compact support is the only additive invariant [45]. ⋆ β ( ∅ ) = ean-Baptiste Campesato For the sake of convenience, we recall some basics of Nash geometry and arc-analytic mapsbefore introducing generically arc-analytic maps.A Nash function on an open semialgebraic subset of R N is an analytic function which satisfiesa non-trivial polynomial equation. This notion coincides with C ∞ semialgebraic functions. Wecan therefore define the notion of Nash submanifold in an obvious way. This notion is powerfulsince we can use tools from both algebraic and analytic geometries, for example we have a Nashimplicit function theorem. For more details on Nash geometry, we refer the reader to [8] and[47].Arc-analytic maps were first introduced by K. Kurdyka in relation with arc-symmetric setsin [29]. These are maps that send analytic arcs to analytic arcs by composition and hence itis suitable to work with arc-analytic maps between arc-symmetric sets. A semialgebraic map f : M → N is blow-Nash if there is a finite sequence of blowings-up with non-singular centers σ : ˜ M → M such that f ◦ σ : ˜ M → N is Nash. Let M be an analytic manifold and f : M → R ablow-analytic map, since we can lift an analytic arc by a blowing-up with non-singular centerof a non-singular variety, f is clearly arc-analytic. Kurdyka conjectured the converse with anadditional semialgebraicity ⋆ hypothesis and E. Bierstone and P. D. Milman brought us the proofin [5]. A. Parusiński gave another proof in [40]. We refer the reader to [31] for a survey on arc-symmetric sets and arc-analytic maps. Definition 2.10.
Let U be a semialgebraic open subset of R N . Then an analytic function f : U → R is said to be Nash if there are polynomials a , . . . , a d with a d = such that a d ( x ) ( f ( x )) d + · · · + a ( x ) = Theorem 2.11 ([8, Proposition 8.1.8]) . Let U be a semialgebraic open subset of R N . Then f : U → R is a Nash function if and only if f is semialgebraic and of class C ∞ . Definition 2.12. A Nash submanifold of dimension d is a semialgebraic subset M of R p suchthat every x ∈ M admits a Nash chart ( U, ϕ ) , i.e. there are U an open semialgebraic neighbor-hood of ∈ R n , V an open semialgebraic neighborhood of x in R p and ϕ : U → V a Nash-diffeomorphism satisfying ϕ ( ) = x and ϕ (( R d × { } ) ∩ U ) = M ∩ V . Remark 2.13.
A non-singular algebraic subset M of R p has a natural structure of Nash subman-ifold given by the Jacobian criterion and the Nash implicit function theorem. Definition 2.14 ([29]) . Let X and Y be arc-symmetric subsets of two analytic manifolds. Then f : X → Y is arc-analytic if for all analytic arcs γ : (− ε, ε ) → X the composition f ◦ γ : (− ε, ε ) → Y is again an analytic arc. Theorem 2.15 ([5]) . Let f be a semialgebraic map defined on a non-singular algebraic subset. Then f isarc-analytic if and only if f is blow-Nash. Remark 2.16.
Let f : X → Y be a semialgebraic arc-analytic map between algebraic sets. Then f is blow-Nash even if X is singular. Indeed we may first use a resolution of singularities ρ : U → X given by a sequence of blowings-up with non-singular centers [19] and apply Theorem 2.15 to f ◦ ρ : U → Y . Remark 2.17. If M is a non-singular algebraic set and ρ : ˜ M → M the blowing-up of M with anon-singular center, it is well known that we can lift an arc on M by ρ to an arc on ˜ M . This resultis obviously false for a singular algebraic set as shown in the following examples. However, if X is a singular algebraic set and ρ : ˜ X → X the blowing-up of X with a non-singular center wecan lift an arc on X not entirely included in the center † and this lifting is unique. ⋆ The question is still open for the general case: is a map blow-analytic if and only if it is subanalytic and arc-analytic? † Such an arc meets the center only at isolated points since it is algebraic and hence arc-symmetric.
A blow-Nash inverse mapping theorem
Example 2.18.
Consider the Whitney umbrella X = V ( x − zy ) and ρ : ˜ X → X the blowing-upalong the singular locus I ( X sing ) = ( x, y ) . Then we can’t lift by ρ an arc included in the handle { x =
0, y =
0, z < 0 } ( ρ is not even surjective). Example 2.19.
This phenomenon still remains in the pure dimensional case. Let X = V ( x − zy ) .Then X is of pure dimension and the blowing-up ρ : ˜ X → X along the singular locus I ( X sing ) =( x, y ) is surjective. However we can’t lift the (germ of) analytic arc γ ( t ) = (
0, 0, t ) to an analyticarc. In the y -chart, ˜ X = { ( X, Y, Z ) ∈ R , X = Z } and ρ ( X, Y, Z ) = (
XY, Y, Z ) . Then the lifting of γ should have the form ˜ γ ( t ) = ( t , 0, t ) . Remark 2.20.
A continuous subanalytic map f : U → V is locally Hölder, i.e. for each compactsubset K ⊂ U , there exist α > 0 and C > 0 such that for all x, y ∈ K , k f ( x ) − f ( y ) k ≤ C k x − y k α .See for instance [17], it’s a consequence of [20, §9, Inequality III]. See also [4, Corollary 6.7].Or we can directly use Łojasiewicz inequality [4, Theorem 6.4] with ( x, y ) → | f ( x ) − f ( y ) | and ( x, y ) → | x − y | .The following result will be useful. Proposition 2.21.
Let f : X → Y be a surjective proper subanalytic map (resp. proper semialgebraicmap) and γ : [
0, ε ) → Y a real analytic (resp. Nash) arc. Then there exist m ∈ N >0 and ˜ γ : [
0, δ ) → X analytic (resp. Nash) with δ m ≤ ε such that f ◦ ˜ γ ( t ) = γ ( t m ) .Proof. The proof is divided into two parts. First we use the properness of f to lift γ to an arc on X and then we conclude thanks to Puiseux theorem.Consider the following diagram X f (cid:15) (cid:15) ˜ X = X × Y [
0, ε ) pr X o o ˜ f (cid:15) (cid:15) Y [
0, ε ) γ o o Let X = ˜ f − ((
0, ε )) . Since f is proper, X \ X ⊂ ˜ X . Let x ∈ X \ X , then by the curve selectionlemma ([8, Proposition 8.1.13] for the semialgebraic case) there exists γ : [
0, η ) → ˜ X analytic(resp. Nash) such that γ ( ) = x and γ ((
0, η )) ⊂ X . We have the following diagram˜ X ˜ f (cid:15) (cid:15) [
0, η ) γ o o h { { ①①①①①①①①① [
0, ε ) Then, h ( ) = and h ((
0, η )) ⊂ (
0, ε ) . Hence there exists α ∈ (
0, η ) such that h : [
0, α ) → [
0, β ) is a subanalytic (resp. semialgebraic) homeomorphism.By Puiseux theorem ([8, Proposition 8.1.12] for the semialgebraic case; see also [42]), thereexist m ∈ N >0 and δ ≤ β such that h − ( t m ) is analytic (resp. Nash) for t ∈ [
0, δ ) .Finally, ˜ γ : [
0, δ ) → X defined by ˜ γ ( t ) = pr X γ h − ( t m ) satisfies f ◦ ˜ γ ( t ) = γ ( t m ) . (cid:4) In the singular case we will work with a slightly different framework.
Definition 2.22.
Let X and Y be two algebraic sets. A map f : X → Y is said to be genericallyarc-analytic in dimension d = dim X if there exists an algebraic subset S of X with dim S < dim X such that for all analytic arc γ : (− ε, ε ) → X not entirely included ⋆ in S , f ◦ γ : (− ε, ε ) → Y isanalytic.If X is non-singular, these maps are exactly the arc-analytic ones. ⋆ γ − ( S ) = (− ε, ε ) ean-Baptiste Campesato Let X be a non-singular algebraic set of dimension d ⋆ and Y an algebraic set. Let f : X → Y be a continuous semialgebraic map. If f is generically arc-analytic in dimension d then f is arc-analytic.Proof. Let S be as in Definition 2.22. By the Jacobian criterion and the Nash implicit functiontheorem we may assume that S is locally a Nash subset of R d . Taking the Zariski closure wemay moreover assume that S is an algebraic subset of R d since it doesn’t change the dimension.Let γ : (− ε, ε ) → R d be an analytic arc entirely included in S .As in [30, Corollaire 2.7], by Puiseux theorem, we may assume that f ( γ ( t )) = X i ≥ b i t ip , t ≥ ( γ ( t )) = X i ≥ c i (− t ) ir , t ≤ By [30, Corollaire 2.8 & Corollaire 2.9], two phenomena may prevent f ( γ ( t )) from being analytic:either one of these expansions has a non-integer exponent or these expansions don’t coincide.To handle the first case, we assume that one of these expansions, for instance for t ≥ , hasa non-integer exponent, i.e. f ( γ ( t )) = m X i = b i t i + bt pq + · · · , b =
0, m < pq < m +
1, t ≥ It follows from Remark 2.20 there exists N ∈ N such that for every analytic arc δ we have f ( γ ( t )+ t N δ ( t )) ≡ f ( γ ( t )) mod t m + . We are going to prove that for η ∈ R d generic, the arc ˜ γ ( t ) = γ ( t ) + t N η is not entirely included in S in order to get a contradiction since f ( ˜ γ ( t )) ≡ f ( γ ( t )) mod t m + .Let t ∈ (− ε, ε ) \ { } . Since dim S < d , there is ˜ η ∈ R d \ C γ ( t ) S where C γ ( t ) S is the tangentcone of S at γ ( t ) . Thus there exists f ∈ I ( S ) with f ( γ ( t ) + x ) = f m ( x ) + · · · + f m + r ( x ) wheredeg f i = i and such that f m ( ˜ η ) = . Then f (cid:0) γ ( t ) + st N0 ˜ η (cid:1) = (cid:0) st N0 (cid:1) m f m ( ˜ η ) + (cid:0) st N0 (cid:1) m + g ( s, t ) and hence for s small enough the arc γ ( t ) + t N s ˜ η isn’t entirely included in S .Then we prove that the expansions coincide in a similar way. Assume that the expansionsare different, i.e. f ( γ ( t )) = m − X i = a i t i + bt m + · · · , t ≥ ( γ ( t )) = m − X i = a i t i + ct m + · · · , t ≤ with b = c . As in the previous case, we may construct an arc ˜ γ not entirely included in S suchthat fγ ( t ) and f ˜ γ ( t ) coincide up to order m + . That leads to a contradiction. (cid:4) Remark 2.24.
If dim Sing ( X ) = then a generically arc-analytic map X → Y is also arc-analyticsince the analytic arcs contained in the singular locus are constant. Remark 2.25.
The previous proof fails when X isn’t assumed to be non-singular. Let X = V ( x − zy ) and S = X sing = O z . Consider (germ of) analytic arc γ ( t ) = (
0, 0, t ) entirely included in S . Given any N ∈ N we can’t find η ( t ) such that ˜ γ ( t ) = γ ( t ) + t N η ( t ) isn’t entirely includedin S . Indeed, if we inject the coordinates of ˜ γ in the equation x = zy we get a contradictionconsidering the orders of vanishing. Remark 2.26.
A continuous semialgebraic generically arc-analytic in dimension d = dim X map f : X → Y may not be arc-analytic if dim Sing ( X ) ≥ . Indeed, let X = V ( x − zy ) and f : X → R be defined by f ( x, y, z ) = xy . Then f (
0, 0, t ) = t is not analytic. ⋆ We mean that every point of X is non-singular of dimension d . A blow-Nash inverse mapping theoremIn the non-singular case, by Theorem 2.15, the blow-Nash maps are exactly the semialgebraicarc-analytic ones. With the following proposition, we notice that more generally the blow-Nashmaps are exactly the semialgebraic generically arc-analytic ones.
Proposition 2.27.
Let X be an algebraic set of dimension d . Let f : X → Y be a semialgebraic map whichis continuous on Reg d X . Then f is generically arc-analytic in dimension d if and only if it is blow-Nash.Proof. Assume that f is generically arc-analytic. Let ρ : U → X be a resolution of singularitiesgiven by a sequence of blowings-up with non-singular centers, then f ◦ ρ : U → Y is semialge-braic and generically arc-analytic with U non-singular. Thus f ◦ ρ is arc-analytic by Lemma 2.23.By Theorem 2.15, there exists a sequence of blowings-up with non-singular centers η : M → U such that f ◦ ρ ◦ η is Nash. Finally f ◦ σ is Nash where σ = ρ ◦ η : M → X is a sequence ofblowings-up with non-singular centers.Assume that f is blow-Nash. Then there is σ : M → X a sequence of blowings-up withnon-singular centers such that f ◦ σ : M → Y is Nash. Let γ be an arc on X not entirely includedin the singular locus of X and the center of σ , then there is ˜ γ an arc on M such that γ = σ ( ˜ γ ) .Thus f ( γ ( t )) = f ◦ σ ( ˜ γ ( t )) is analytic. (cid:4) Arc spaces and truncations of arcs were first introduced by J. F. Nash in 1964 [38] and their studyhas gained new momentum with the works of M. Kontsevich [24], J. Denef and F. Loeser [9] onmotivic integration. We can notice that K. Kurdyka [29], A. Nobile [39], M. Lejeune-Jalabert [34][15], M. Hickel [18] and others studied arc space and jet spaces before the advent of motivicintegration. Most of these works concern the relationship between the singularities of a varietyand its jet spaces.In this section, we define the arc space and the jet spaces of a real algebraic set. We first workwith the whole ambient Euclidean space and then use the equations of the algebraic set to definearcs and jets on it. Finally we will give and prove a collection of results concerning these objects.The arc space on R N is defined by L (cid:16) R N (cid:17) = (cid:10) γ : ( R , 0 ) → R N , γ analytic (cid:11) and, for n ∈ N , the set of n -jets on R N is defined by L n (cid:16) R N (cid:17) = L (cid:0) R N (cid:1). ∼ n where γ ∼ n γ if and only if γ ≡ γ mod t n + . Obviously, L n (cid:0) R N (cid:1) ≃ (cid:0) R { t } (cid:14) t n + (cid:1) N . We alsoconsider the truncation maps π n : L (cid:0) R N (cid:1) → L n (cid:0) R N (cid:1) and π mn : L m (cid:0) R N (cid:1) → L n (cid:0) R N (cid:1) , where m > n . These maps are clearly surjective.Next, assume that X ⊂ R N is an algebraic subset. The set of analytic arcs on X is L ( X ) = (cid:10) γ ∈ L (cid:16) R N (cid:17) , ∀ f ∈ I ( X ) , f ( γ ( t )) = (cid:11) and, for n ∈ N , the set of n -jets on X is L n ( X ) = (cid:10) γ ∈ L n (cid:16) R N (cid:17) , ∀ f ∈ I ( X ) , f ( γ ( t )) ≡ mod t n + (cid:11) When X is singular, we will see that the truncation maps may not be surjective. Example 2.28.
Let X ⊂ R N be an algebraic subset, then L ( X ) ≃ X and L ( X ) ≃ T Zar X = F T Zar x X .Indeed, we just apply Taylor expansion to f ( a + bt ) where f ∈ I ( X ) (or we may directly use thatthe Zariski tangent space at a point is given by the linear parts of the polynomials f ∈ I ( X ) aftera translation).ean-Baptiste Campesato The following lemma is useful to find examples which are hypersurfaces since the construc-tions of arc space and jet spaces on an algebraic set are algebraic. See [8, Theorem 4.5.1] for amore general result with another proof. We may find similar results for non-principal ideals in[8, Proposition 3.3.16, Theorem 4.1.4]. See also [33, §6].
Lemma 2.29.
Let f ∈ R [ x , . . . , x N ] be an irreducible polynomial which changes sign, then I ( V ( f )) =( f ) .Proof. The following proof comes from [33, Lemma 6.14]. After an affine change of coordinates,we may assume that f ( a, b ) < 0 < f ( a, b ) with a = ( a , . . . , a N − ) . Let g ∈ I ( V ( f )) and assumethat f ∤ g in R [ x , . . . , x N ] . In the PID (and hence UFD) R ( x , . . . , x N − )[ x N ] , f is also irreducibleand f ∤ g too. In this PID, we may find ϕ and γ such that ϕf + γg = . Let ϕ = ϕ /h and γ = γ /h with = h ∈ R [ x , . . . , x N − ] and ϕ , γ ∈ R [ x , . . . , x N − ][ x N ] . Then ϕ f + γ g = h .Let V be a neighborhood of a in R N − such that ∀ v ∈ V, f ( v, b ) < 0 < f ( v, b ) . By the IVT, forall v ∈ V , there is b ≤ b v ≤ b such that f ( v, b v ) = , and so g ( v, b v ) = . Then ∀ v ∈ V, h ( v ) = and hence h ≡ which is a contradiction. (cid:4) Example 2.30.
Let X = V (cid:0) y − x (cid:1) . Since y − x is irreducible and changes sign, we have I ( X ) = (cid:0) y − x (cid:1) by Lemma 2.29. Hence we get, L ( X ) = (cid:10) ( a + a t, b + b t ) ∈ (cid:0) R { t } (cid:14) t (cid:1) , ( b + b t ) − ( a + a t ) ≡ mod t (cid:11) = (cid:10) ( a + a t, b + b t ) ∈ (cid:0) R { t } (cid:14) t (cid:1) , a = b , 3a a = b (cid:11) L ( X ) = (cid:14) ( a + a t + a t , b + b t + b t ) ∈ (cid:0) R { t } (cid:14) t (cid:1) , ( b + b t + b t ) − ( a + a t + a t ) ≡ mod t (cid:15) = ( a + a t + a t , b + b t + b t ) ∈ (cid:0) R { t } (cid:14) t (cid:1) , a = b ,3a a = b ,3a a + a = b + b Then the preimage of (
0, t ) ∈ L ( X ) by π is obviously empty.We therefore take care not to confuse the set L n ( X ) of n -jets on X and the set π n ( L ( X )) of n -jets on X which can be lifted to analytic arcs. Thanks to Hensel’s lemma and Artin approxi-mation theorem [2], this phenomenon disappears in the non-singular case. Proposition 2.31.
Let X be an algebraic subset of R N . The following are equivalent:(i) For all n , π n + : L n + ( X ) → L n ( X ) is surjective.(ii) For all n , π n : L ( X ) → L n ( X ) is surjective.(iii) X is non-singular.Proof. (iii) ⇒ (ii) is obvious using Hensel’s lemma and Artin approximation theorem [2].(ii) ⇒ (i) is obvious since π n = π n + ◦ π n + .(i) ⇒ (iii): Assume that is a singular point of X . We can find γ = αt ∈ L ( X ) which doesn’t liein the tangent cone of X at , i.e. such that f ( αt ) mod t m + for some f ∈ I ( X ) of order m .Such a -jet can’t be lifted to L m ( X ) . (cid:4) The set L n ( X ) of n -jets on X ⊂ R N can be seen as a algebraic subset of R ( n + ) N . By a theoremof M. J. Greenberg [16], given an algebraic subset X ⊂ R N , there exists c ∈ N >0 such that forall n ∈ N , π n ( L ( X )) = π cnn ( L cn ( X )) . Then if we work over C the sets π n ( L ( X )) are Zariski-constructible by Chevalley theorem. See for instance [34] ⋆ , [15] or [9].In our framework, the following example shows that the π n ( L ( X )) may not even be AS . ⋆ She uses a generalization of [3, Theorem 6.1] instead of Greenberg theorem. A blow-Nash inverse mapping theorem
Example 2.32.
Let X = V (cid:0) x − zy (cid:1) . Then for every a ∈ R , γ a ( t ) = (
0, t , at ) ∈ L ( X ) . Let η ( t ) = ( bt + t η ( t ) , t + t η ( t ) , at + t η ( t )) ∈ L ( R ) . Let f ( x, y, z ) = x − zy , then f ( η ( t )) =( b − a ) t + t ˜ η ( t ) . So if a < 0 , γ a ( t ) / ∈ π ( L ( X )) . However if a ≥ , γ a ( t ) = π (cid:0) √ at , t , at (cid:1) ∈ π ( L ( X )) . Proposition 2.33.
Let X ⊂ R N be an algebraic subset of dimension d . Then:(i) dim ( π n ( L ( X ))) = ( n + ) d (ii) dim ( L n ( X )) ≥ ( n + ) d (iii) The fibers of ˜ π mn = π mn | π m ( L ( X )) : π m ( L ( X )) → π n ( L ( X )) are of dimension smaller than or equalto ( m − n ) d where m ≥ n .(iv) A fiber (cid:0) π n + (cid:1) − ( γ ) of π n + : L n + ( X ) → L n ( X ) is either empty or isomorphic to T Zar γ ( ) X .If moreover we assume that X is non-singular, we get the following statement since L n ( X ) = π n ( L ( X )) :(v) dim ( L n ( X )) = ( n + ) d Proof.
We first notice that (i) is a direct consequence of (iii).(ii) ( π n0 ) − ( X \ X sing ) is of dimension ( n + ) d since the fiber of π n0 over a non-singular pointis of dimension nd .(iii) We may assume that m = n + . Let γ ∈ π n ( L ( X )) . We may assume that γ ∈ ( R n [ t ]) N .We consider the following diagram R N × R p z z ✈✈✈✈✈✈✈✈✈ p ❋❋❋❋❋❋❋❋❋❋ R N R with p ( x, t ) = γ ( t )+ t n + x and p ( x, t ) = t . Let X = p − ( X ) ∩ { t = } Zar . For c = , X ∩ p − ( c ) ≃ X and dim X ∩ p − ( c ) = dim X − . Hence dim X ∩ p − ( ) ≤ dim X − = dim X .We are looking for objects of the form π n + ( γ ( t ) + t n + α ( t )) with γ ( t ) + t n + α ( t ) ∈ L ( X ) . Suchan α is equivalent to a section of p | X i.e. R → X t → ( α ( t ) , t ) . Since we want an arc modulo t n + ,we are looking for the constant term of α , therefore ( ˜ π n + ) − ( γ ) ⊂ X ∩ p − ( ) .(iv) Let γ ∈ L n ( X ) . Let η ∈ R N . Assume that I ( X ) = ( f , . . . , f r ) . By Taylor expansion we get f i ( γ + t n + η ) ≡ f i ( γ ( t )) + t n + (cid:0) ∇ γ ( t ) f i (cid:1) ( η ) mod t n + Assume that f i ( γ ( t )) ≡ t n + α i mod t n + . Since t n + (cid:0) ∇ γ ( t ) f i (cid:1) ( η ) ≡ t n + (cid:0) ∇ γ ( ) f i (cid:1) ( η ) mod t n + ,we have f i ( γ + t n + η ) ≡ t n + (cid:0) α i + (cid:0) ∇ γ ( ) f i (cid:1) ( η ) (cid:1) mod t n + Hence, γ ( t ) + t n + η is in the fiber (cid:0) π n + (cid:1) − ( γ ) if and only if α i + (cid:0) ∇ γ ( ) f i (cid:1) ( η ) =
0, i =
1, . . . , r . (cid:4) An arc-analytic map f : X → Y induces a map f ∗ : L ( X ) → L ( Y ) . Moreover, if f : X → Y is analytic, then we also have maps at the level of n -jets f ∗ n : L n ( X ) → L n ( Y ) such that thefollowing diagram commutes L ( X ) f ∗ / / π n (cid:15) (cid:15) L ( Y ) π n (cid:15) (cid:15) L n ( X ) f ∗ n / / L n ( Y ) In particular, if X is non-singular, Im f ∗ n ⊂ π n ( L ( Y )) since π n : L ( X ) → L n ( X ) is surjective.For M a non-singular algebraic set and σ : M → X ⊂ R N analytic, we define Jac σ ( x ) theJacobian matrix of σ at x with respect to a coordinate system at x in M . For γ an arc on M with origin γ ( ) = x , we define the order of vanishing of γ along Jac σ by ord t Jac σ ( γ ( t )) = ean-Baptiste Campesato min { ord t δ ( γ ( t )) , ∀ δ m -minor of Jac σ } where m = min ( d, N ) and γ is expressed in the local co-ordinate system. This order of vanishing is independent of the choice of the coordinate system.The critical locus of σ is C σ = { x ∈ M, δ ( x ) = ∀ δ m -minor of Jac σ } . If E ⊂ M is lo-cally described by an equation f = around x and if γ is an arc with origin γ ( ) = x thenord γ E = ord t f ( γ ( t )) . Lemma 3.1.
Let X be an algebraic subset of R N and f : X → X a blow-Nash map. Let σ : M → X be asequence of blowings-up with non-singular centers such that ˜ σ = f ◦ σ : M → X is Nash. M σ ~ ~ ⑦⑦⑦⑦⑦⑦⑦⑦ ˜ σ ❅❅❅❅❅❅❅❅ X f / / X After adding more blowings-up, we may assume that the critical loci of σ and ˜ σ are simultaneously normalcrossing and denote them by P i ∈ I ν i E i and P i ∈ I ˜ ν i E i .Then the property (1) ∀ i ∈ I, ν i ≥ ˜ ν i doesn’t depend on the choice of σ .Proof. Given σ and σ as in the statement and using Hironaka flattening theorem lemma [21](which works as it is in the real algebraic case), there exist π and π regular such that thefollowing diagram commutes: f M π ~ ~ ⑤ ⑤ ⑤ ⑤ π ❇❇❇❇ M ˜ σ (cid:29) (cid:29) σ ! ! ❈❈❈❈❈❈❈❈ M ˜ σ (cid:1) (cid:1) σ } } ④④④④④④④④ X f (cid:15) (cid:15) X The relation 1 means exactly that the Jacobian ideal of σ i is included in the Jacobian ideal of ˜ σ i .By the chain rule, the relations at the level M i are preserved in f M . Again by the chain rule andsince the previous diagram commutes, the relations in M and M must coincide. (cid:4) Definition 3.2.
We say that a map f : X → X as in Lemma 3.1 verifying the relation (1) satisfiesthe Jacobian hypothesis . Question 3.3.
May we find a geometric interpretation of this hypothesis?The following example is a direct consequence of the chain rule.
Example 3.4.
Let X be a non-singular algebraic set and f : X → X a regular map satisfying | det d f | > c for a constant c > 0 , then f satisfies the Jacobian hypothesis . Theorem 3.5 (Main theorem) . Let X be an algebraic subset of R N and f : X → X a semialgebraichomeomorphism (for the Euclidean topology). If f is blow-Nash and satisfies the Jacobian hypothesis then f − is blow-Nash and satisfies the Jacobian hypothesis too. A blow-Nash inverse mapping theoremBy Lemma 2.23 and Proposition 2.27, if X is a non-singular algebraic subset we get the fol-lowing corollary. Corollary 3.6 ([13]) . Let X be a non-singular algebraic subset and f : X → X a semialgebraic homeomor-phism (for the Euclidean topology). If f is arc-analytic and if there exists c > 0 satisfying | det d f | > c then f − is arc-analytic and there exists ˜ c > 0 satisfying | det d f − | > ˜ c . Remark 3.7.
We recover [13, Theorem 1.1] using the last corollary and [13, Corollary 2.2 &Corollary 2.3].
An algebraic version of the following lemma was already known in [10], [43] or [44, §2] with aproof in [48, 4.1]. The statement given below is more geometric and the proof is quite elemen-tary.
Lemma 4.1.
Let X be a d -dimensional algebraic subset of R N . We consider the following ideal of R [ x , . . . , x N ] H = X f ,...,f N − d ∈ I ( X ) ∆ ( f , . . . , f N − d ) (( f , . . . , f N − d ) : I ( X )) where ∆ ( f , . . . , f N − d ) is the ideal generated by the ( N − d ) -minors of the matrix (cid:16) ∂f i ∂x j (cid:17) i = − dj = . Then V ( H ) is the singular locus ⋆ X sing of X .Proof. Let x / ∈ V ( H ) then there exist f , . . . , f N − d ∈ I ( X ) , δ a ( N − d ) -minor of (cid:16) ∂f i ∂x j (cid:17) i = − dj = and h ∈ R [ x , . . . , x N ] with hI ( X ) ⊂ ( f , . . . , f N − d ) and hδ ( x ) = . Since δ ( x ) = , x is a non-singularpoint of V ( f , . . . , f N − d ) . Furthermore we have X = V ( I ( X ))) ⊂ V ( f , . . . , f N − d ) ⊂ V ( hI ( X )) and,since h ( x ) = , in an open neighborhood U of x in R N we have V ( hI ( X )) ∩ U = X ∩ U . Hence V ( f , . . . , f N − d ) ∩ U = X ∩ U . So x is a non-singular point of X by [8, Proposition 3.3.10]. Weproved that X sing ⊂ V ( H ) .Now, assume that x ∈ X \ X sing . With the notation of [8, §3], the local ring R X,x = R R N ,x (cid:14) I ( X ) R R N ,x is regular, so we may find a regular system of parameters ( f , . . . , f N ) of R X,x such that I ( X ) R R N ,x =( f , . . . , f N − d ) R R N ,x by [25, VI.1.8&VI.1.10] † (see also [8, Proposition 3.3.7]). Moreover, we mayassume that the f , . . . , f N − d are polynomials. We may use the following classical argument. θ : R [ x , . . . , x N ] → R N defined by f → f ( x ) induces an isomorphism θ ′ : m x / m → R N . Thenrk (cid:16) ∂f i ∂x j ( x ) (cid:17) = dim θ (( f , . . . , f N − d )) which is, by θ ′ , the dimension of (( f , . . . , f N − d ) + m ) . m as a subspace of m x (cid:14) m . If we denote by m the maximal ideal of R X,x = (cid:0) R [ x , . . . , x N ] (cid:14) ( f , . . . , f N − d ) (cid:1) m x ,we have m (cid:14) m ≃ m x (cid:14) (( f , . . . , f N − d ) + m ) . So we have dim (cid:0) m (cid:14) m (cid:1) + rk (cid:16) ∂f i ∂x j ( x ) (cid:17) = N . Fur-thermore, since R X,x is a d -dimensional regular local ring, dim (cid:0) m (cid:14) m (cid:1) = d . Hence (cid:16) ∂f i ∂x j ( x ) (cid:17) i = − dj = is of rank N − d and so there exists δ a ( N − d ) -minor of (cid:16) ∂f i ∂x j (cid:17) i = − dj = such that δ ( x ) = . Assumethat I ( X ) = ( g , . . . , g r ) in R [ x , . . . , x N ] . Then g i = P f j q j with q j ( x ) = , so g i h i ⊂ ( f , . . . , f N − d ) with h i = Q q j . Then h = Q h i satisfies h ( x ) = and hI ( X ) ⊂ ( f , . . . , f N − d ) . So x / ∈ V ( H ) .Hence V ( H ) ⊂ X sing ∪ ( R N \ X ) . ⋆ By singular locus we mean the complement of the set of non-singular points in dimension d as in [8, 3.3.13] (andnot the complement of non-singular points in every dimension). We may avoid this precision with the sup-plementary hypothesis that every irreducible component of X is of dimension d or in the pure dimensionalcase. † Since R R N ,x = R [ x , . . . , x N ] m x ean-Baptiste Campesato To complete the proof, it remains to prove that V ( H ) ⊂ X . Let x / ∈ X . There exist f , . . . , f N − d ∈ I ( X ) such that f i ( x ) = . We construct by induction N − d polynomials of the form g i = a i f i with g i ( x ) = and ( d g ∧ · · · ∧ d g N − d ) x = . Suppose that g , . . . , g j − are constructed, if ( d g ∧ · · · ∧ d g j − ∧ d f j ) x = , we can take a j = , so we may assume that ( d g ∧ · · · ∧ d g j − ∧ d f j ) x = . Then we just have to take some a j satisfying ( d g ∧ · · · ∧ d g j − ∧ d a j ) x = and a j ( x ) = since ( d g ∧ · · · ∧ d g j − ∧ d ( a j f j )) x = f j ( x )( d g ∧ · · · ∧ d g j − ∧ d a j ) x . Then we have g , . . . , g N − d ∈ I ( X ) whose a ( N − d ) -minor δ satisfies δ ( x ) = . Moreover we have g i ( x ) = and g i I ⊂ ( g , . . . , g N − d ) . So x / ∈ V ( H ) . (cid:4) Definition 4.2.
Let X be an algebraic subset of R N . For e ∈ N , we set L ( e ) ( X ) = (cid:10) γ ∈ L ( X ) , ∃ g ∈ H, g ( γ ( t )) mod t e + (cid:11) where H is defined in Lemma 4.1. Remark 4.3. L ( X ) = [ e ∈ N L ( e ) ( X ) ! G L ( X sing ) Remark 4.4.
In [9], Denef–Loeser set L ( e ) ( X ) = L ( X ) \ π − (cid:0) L e ( X sing ) (cid:1) and used the Nullstellen-satz to get that I ( X sing ) c ⊂ H for some c since X sing = V ( H ) . Since we can’t do that in our case,we defined differently L ( e ) ( X ) .The following lemma is an adaptation of Denef–Loeser key lemma [9, Lemma 3.4] to fulfillour settings. The aim of the above-mentioned lemma is to allow the proof of a generalization ofKontsevich’s birational transformation rule (change of variables) of [24] to handle singularities.We can find a first adaption to our settings in the non-singular case in [23, Lemma 4.2]. Lemma 4.5.
Let σ : M → X be a proper generically ⋆ one-to-one Nash map where M is a non-singularalgebraic subset of R p of dimension d and X an algebraic subset of R N of dimension d . For e, e ′ ∈ N , weset ∆ e,e ′ = (cid:10) γ ∈ L ( M ) , ord t ( Jac σ ( γ ( t ))) = e, σ ∗ ( γ ) ∈ L ( e ′ ) ( X ) (cid:11) For n ∈ N , let ∆ e,e ′ ,n be the image of ∆ e,e ′ by π n . Let e, e ′ , n ∈ N with n ≥ max ( ′ ) , then:(i) Given γ ∈ ∆ e,e ′ and δ ∈ L ( X ) with σ ∗ ( γ ) ≡ δ mod t n + there exists a unique η ∈ L ( M ) suchthat σ ∗ ( η ) = δ and η ≡ γ mod t n − e + .(ii) Let γ, η ∈ L ( M ) . If γ ∈ ∆ e,e ′ and σ ( γ ) ≡ σ ( η ) mod t n + then γ ≡ η mod t n − e + and η ∈ ∆ e,e ′ .(iii) The set ∆ e,e ′ ,n is a union of fibers of σ ∗ n .(iv) σ ∗ n ( ∆ e,e ′ ,n ) is constructible and σ ∗ n | ∆ e,e ′ ,n : ∆ e,e ′ ,n → σ ∗ n ( ∆ e,e ′ ,n ) is a piecewise trivial fibra-tion † with fiber R e . Remark 4.6.
It is natural to use Taylor expansion to prove some approximation theorems con-cerning power series as we are going to do for 4.5.(i). For instance, we may find similar argumentin [16], [3], or [10]. For 4.5.(i), we will follow the proof of [9, Lemma 3.4] with some differencesto match our framework. Concerning 4.5.(iv), we can’t use anymore the section argument of [9]since σ is not assumed to be birational. Lemma 4.7 (Reduction to complete intersection) . Let X be an algebraic subset of R N of dimension d . For each e ∈ N , L ( e ) ( X ) is covered by a finite number of sets of the form A h,δ = (cid:10) γ ∈ L ( R N ) , ( hδ )( γ ) mod t e + (cid:11) ⋆ i.e. σ is a Nash map which is one-to-one away from a subset S of X with dim S < dim X . † By a trivial piecewise fibration, we mean there exist a finite partition of σ ∗ n ( ∆ e,e ′ ,n ) with constructible parts and atrivial fibration given by a constructible isomorphism over each part. A blow-Nash inverse mapping theorem with δ a N − d -minor of the matrix (cid:16) ∂f i ∂x j (cid:17) i = − dj = and h ∈ (( f , . . . , f N − d ) : I ( X )) for some f , . . . , f N − d ∈ I ( X ) .Moreover, L ( X ) ∩ A h,δ = (cid:10) γ ∈ L (cid:16) R N (cid:17) , f ( γ ) = · · · = f N − d ( γ ) =
0, hδ ( γ ) mod t e + (cid:11) Remark 4.8.
We may have different polynomials f , . . . , f N − d for two different A h,δ . Proof.
By noetherianity, we may assume that H = ( h δ , . . . , h r δ r ) with h i , δ i as desired. There-fore, L ( e ) ( X ) ⊂ ∪ A h i ,δ i .Finally, L ( X ) ∩ A h,δ = (cid:10) γ ∈ L (cid:16) R N (cid:17) , ∀ f ∈ I ( X ) , f ( γ ) =
0, hδ ( γ ) mod t e + (cid:11) = (cid:10) γ ∈ L (cid:16) R N (cid:17) , f ( γ ) = · · · = f N − d ( γ ) =
0, hδ ( γ ) mod t e + (cid:11) Indeed, for the second equality, if f ∈ I ( X ) then hf ∈ ( f , . . . , f N − d ) , hence if γ vanishes the f i ,then hf ( γ ) = , and so f ( γ ) = since h ( γ ) = . (cid:4) Proof of Lemma 4.5.
We first notice that 4.5.(iii) is a consequence of 4.5.(ii): ∀ π n ( γ ) ∈ ∆ e,e ′ ,n wehave π n ( γ ) ∈ σ − ∗ n ( σ ∗ n ( π n ( γ ))) = (cid:10) π n ( η ) , η ∈ L ( M ) , σ ( η ) ≡ σ ( γ ) mod t n + (cid:11) using that L ( M ) → L n ( M ) is surjective since M is smooth and that π n ◦ σ ∗ = σ ∗ n ◦ π n . ⊂ (cid:10) η ∈ ∆ e,e ′ ,n , γ ≡ η mod t n − e + (cid:11) ⊂ ∆ e,e ′ ,n by 4.5.(ii)Next 4.5.(ii) is a direct consequence of 4.5.(i). We apply 4.5.(i) to γ with δ = σ ∗ ( η ) , hence thereexists a unique ˜ η such that ˜ η ≡ γ mod t n − e + and σ ∗ ( ˜ η ) = σ ∗ ( η ) . By the assumptions on σ andthe definition of ∆ e,e ′ , for ϕ ∈ L ( M ) and ϕ ∈ ∆ e,e ′ with ϕ = ϕ we have σ ( ϕ ) = σ ( ϕ ) .Hence η = ˜ η and η ≡ γ mod t n − e + . Since σ ( γ ) ≡ σ ( η ) mod t n + and n ≥ e ′ , σ ( η ) ∈ L ( e ′ ) ( X ) .We may write η ( t ) = γ ( t )+ t n + − e u ( t ) and applying Taylor expansion to Jac σ ( γ ( t )+ t n + − e u ( t )) we get that Jac σ ( η ( t )) ≡ Jac σ ( γ ( t )) mod t e + since n + − e ≥ e + . So η ∈ ∆ e,e ′ .So we just have to prove 4.5.(i) and 4.5.(iv).We begin to refine the cover of Lemma 4.7: for e ′′ ≤ e ′ , we set A h,δ,e ′′ = γ ∈ A h,δ , ord t δ ( γ ) = e ′′ and ord t δ ′ ( γ ) ≥ e ′′ for all ( N − d ) -minor δ ′ of (cid:18) ∂f i ∂x j (cid:19) i = − dj = Fix some A = A h,δ,e ′′ , then it suffices to prove the lemma for ∆ e,e ′ ∩ σ − ( A ) .Up to renumbering the coordinates, we may also assume that δ is the determinant of the first N − d columns of ∆ = (cid:16) ∂f i ∂x j (cid:17) i = − dj = .We choose a local coordinate system of M at γ ( ) in order to define Jac σ and express arcs of M as elements of R { t } d .Now, a crucial observation is that the first N − d rows of Jac σ ( γ ) are R { t } -linear combinationsof the last d rows: the application M − → X − → R N − d y → σ ( y ) → ( f i ( σ ( y ))) i = − d is identically zero, so its Jacobian matrix is identically zero too and thus ∆ ( σ ( γ )) Jac σ ( γ ) = .Let P be the transpose of the comatrix of the submatrix of ∆ given by the first N − d columnsof ∆ , then P∆ = ( δI N − d , W ) . Moreover, we have W ( σ ( γ )) ≡ mod t e ′′ . Indeed, if we denoteean-Baptiste Campesato ∆ , . . . , ∆ N − d the N − d first columns of ∆ and W , . . . , W d the columns of W , then W j ( σ ( γ )) issolution of ( ∆ ( σ ( γ )) , . . . , ∆ N − d ( σ ( γ ))) X = δ ( σ ( γ )) ∆ N − d + j ( σ ( γ )) since δ ( σ ( γ )) ∆ ( σ ( γ )) = ( ∆ ( σ ( γ )) , . . . , ∆ N − d ( σ ( γ ))) P ( σ ( γ )) ∆ ( σ ( γ ))= ( ∆ ( σ ( γ )) , . . . , ∆ N − d ( σ ( γ ))) ( δ ( σ ( γ )) I N − d , W ( σ ( γ ))) So, by Cramer’s rule, ( W j ( σ ( γ ))) i = det ( ∆ ( σ ( γ )) , . . . , ∆ i − ( σ ( γ )) , ∆ N − d + j ( σ ( γ )) , ∆ i + ( σ ( γ )) , . . . , ∆ N − d ( σ ( γ ))) Finally, the congruence arises because the minor formed by the N − d first columns is of minimalorder by definition of A .Now the columns of Jac σ ( γ ) are solutions of(2) (cid:16) t − e ′′ · P ( σ ( γ ))) · ∆ ( σ ( γ )) (cid:17) X = but since t − e ′′ · P ( σ ( γ ))) · ∆ ( σ ( γ )) = (cid:16) t − e ′′ δ ( σ ( γ )) I N − d , t − e ′′ W ( σ ( γ )) (cid:17) we may express the first N − d coordinates of each solution in terms of the last d coordinates. This completes the proofof the observation.For 4.5.(i), it suffices to prove that for all v ∈ R { t } N satisfying σ ( γ ) + t n + v ∈ L ( X ) there existsa unique u ∈ R { t } d such that(3) σ ( γ + t n + − e u ) = σ ( γ ) + t n + v By Taylor expansion, we have(4) σ ( γ ( t ) + t n + − e u ) = σ ( γ ( t )) + t n + − e Jac σ ( γ ( t )) u + t ( n + − e ) R ( γ ( t ) , u ) with R ( γ ( t ) , u ) analytic in t and u . By (4), (3) is equivalent to(5) t − e Jac σ ( γ ( t )) u + t n + − R ( γ ( t ) , u ) = v with n + − ≥ by hypothesis.Since σ ( γ ( t )) + t n + v ∈ L ( X ) and using Taylor expansion, we get = f i ( σ ( γ ( t )) + t n + v ) = t n + ∆ ( σ ( γ ( t ))) v + t ( n + ) S ( γ ( t ) , v ) with S ( γ ( t ) , v ) analytic in t and v . So v is a solution of (2) and hence the first N − d coefficients of v are R { t } -linear combinations of the last d coefficients with the same relations that for Jac σ ( γ ) .This allows us to reduce (5) to(6) t − e Jac p ◦ σ ( γ ( t )) u + t n + − p ( R ( γ ( t ) , u )) = p ( v ) where p : R N → R d is the projection on the last d coordinates. The observation ensures thatord t Jac p ◦ σ ( γ ( t )) = ord t Jac σ ( γ ( t )) = e and thus (6) is equivalent to(7) u = (cid:0) t − e Jac p ◦ σ ( γ ( t )) (cid:1) − p ( v ) − t n + − (cid:0) t − e Jac p ◦ σ ( γ ( t ) (cid:1) − p ( R ( γ ( t ) , u )) Applying the implicit function theorem to u ( t, v ) ensures that given an analytic arc v ( t ) thereexists a solution u v ( t ) = u ( t, v ( t )) . Using the same argument as in the proof of 4.5.(ii), thesolution u v ( t ) is unique. This proves 4.5.(i).Let us prove 4.5.(iv). Let γ ∈ ∆ e,e ′ ∩ σ − ( A ) then σ − ∗ n ( π n ( σ ∗ ( γ ))) = { η ∈ L n ( M ) , σ ∗ n ( η ) = π n ( σ ∗ ( γ ) } = (cid:10) π n ( η ) , η ∈ L ( M ) , σ ( η ) ≡ σ ( γ ) mod t n + (cid:11) using that L ( M ) → L n ( M ) issurjective since M is smooth and that π n ◦ σ ∗ = σ ∗ n ◦ π n . = (cid:10) γ ( t ) + t n + − e u ( t ) mod t n + , u ∈ R { t } d , Jac p ◦ σ ( γ ( t )) u ( t ) ≡ mod t e (cid:11) by 4.5.(ii) and (6) A blow-Nash inverse mapping theoremThus, the fiber is an affine subspace of R de . There are invertible matrices A and B with co-ordinates in R { t } such that A Jac p ◦ σ ( γ ( t )) B is diagonal with entries t e , . . . , t e d such that e = e + · · · + e d . Therefore the fiber is of dimension e .Since σ is not assumed to be birational, we can’t use the section argument of [9, 3.4] or [23,4.2], instead we use a topological noetherianity argument to prove that σ ∗ n | ∆ e,e ′ ,n is a piecewisetrivial fibration.We may assume that M is semialgebraically connected, then by Artin-Mazur theorem [8,8.4.4], there exist Y ⊂ R p + q a non-singular irreducible algebraic set of dimension dim M , M ′ ⊂ Y an open semialgebraic subset of Y , s : M → M ′ a Nash-diffeomorphism and g : Y → R N apolynomial map such that the following diagram commutes R p + qΠ (cid:15) (cid:15) Y ? _ o o g ! ! ❈❈❈❈❈❈❈❈ M ′ ?(cid:31) O O R N R p M s ≃ O O σ = = ③③③③③③③③ ? _ o o Thus, we have σ − ∗ n ( π n ( σ ∗ ( γ ))) = (cid:10) γ ( t ) + t n + − e u ( t ) mod t n + , u ∈ R { t } d , Jac g ◦ s ( γ ( t )) u ( t ) ≡ mod t e (cid:11) So ∆ e,e ′ ,n is constructible and we may assume that σ ∗ n : ∆ e,e ′ ,n → σ ∗ n ( ∆ e,e ′ ,n ) is polynomialup to working with arcs over M ′ via s . The fibers (i.e. R e ) have odd Euler characteristic withcompact support, so by Theorem 2.8 the image σ ∗ n ( ∆ e,e ′ ,n ) is constructible.Let V = { u + u t + · · · + u n t n , u i ∈ R d } and fix Λ : V → V a linear projection on asubspace of dimension e . The set Ω = { π n ( γ ( t )) ∈ ∆ e,e ′ ,n , dim Λ ( σ − ∗ n ( π n ( σ ∗ ( γ )))) < e } isclosed, constructible and union of fibers of σ ∗ n . Therefore ( σ ∗ n , Λ ) : ∆ e,e ′ ,n \ Ω → σ ∗ n ( ∆ e,e ′ ,n \ Ω ) × V is a constructible isomorphism. We now repeat the argument to the closed constructiblesubset σ ∗ n ( Ω ) and so on. Indeed, assume that ∆ e,e ′ ,n ) Ω ) Ω ) · · · ) Ω i − are constructedas previously and that Ω i − = ∅ , then we may choose Λ i such that Ω i ( Ω i − . So on the onehand the process continues until one Ω i is empty, on the other hand it must stop because of thenoetherianity of the AS -topology. Therefore after a finite number of steps, one Ω i is necessarilyempty. (cid:4) By our hypothesis, there exists a sequence of blowings-up σ : M → X with non-singular centerssuch that ˜ σ = f ◦ σ : M → X is Nash. M σ ~ ~ ⑦⑦⑦⑦⑦⑦⑦⑦ ˜ σ ❅❅❅❅❅❅❅❅ X f / / X After adding more blowings-up, we may assume that the critical loci of σ and ˜ σ are simulta-neously normal crossing and denote them by P ν i E i and P ˜ ν i E i . Our hypothesis ensures that ν i ≥ ˜ ν i .In the same way, we may ensure that the inverse images of H (defined in Lemma 4.1) by σ and ˜ σ are also simultaneously normal crossing and denote them σ − ( H ) = P i ∈ I λ i E i (resp.˜ σ − ( H ) = P i ∈ I ˜ λ i E i ).We recall the usual notation ⋆ . For j = ( j i ) i ∈ I ∈ N I , we set J = J ( j ) = { i, j i = } ⊂ I , E J = ∩ i ∈ J E i and E • J = E J \ ∪ i ∈ I \ J E i . ⋆ This notation is natural and classical. See [22, Chapter II, §1] for some properties of this stratification. ean-Baptiste Campesato We also define: B j = { γ ∈ L ( M ) , ∀ i ∈ J, ord γ E i = j i , γ ( ) ∈ E • J } and for all n ∈ N , B j ,n = π n ( B j ) and X j ,n ( σ ) = π n ( σ ∗ B j ) = σ ∗ n ( B j ,n ) . Lemma 4.9.
We have B j ⊂ ∆ e ( j ) ,e ′ ( j ) ( σ ) where e ( j ) = X i ∈ I ν i j i and e ′ ( j ) = X i ∈ I λ i j i .Proof. Let γ ∈ B j and choose a local coordinate system of M at γ ( ) such that the critical locusof σ is locally described by the equation Q i ∈ J x ν i i = and E i by the equation x i = . Sinceord γ E i = j i , we have γ i ( t ) = c j i t j i + · · · and c j i = . Then Q i ∈ J γ ν i i = ct e ( j ) + · · · with c = .So we have ord t ( Jac σ ( γ ( t ))) = e ( j ) .In the same way, ord γ σ − ( H ) = e ′ ( j ) thus ord σ ( γ ) ( H ) = e ′ ( j ) . (cid:4) Therefore we set A n ( σ ) = (cid:8) j , P i ∈ I ν i j i ≤ n2 , P i ∈ I λ i j i ≤ n (cid:9) . Indeed, for each j ∈ A n ( σ ) , B j ⊂ ∆ e ( j ) ,e ′ ( j ) ( σ ) and we may apply Lemma 4.5 at the level of n -jets.The argument of the following lemma is essentially the same as [13, §4.2]. Lemma 4.10 (A decomposition of jet spaces) . For all j ∈ A n ( σ ) , the sets X j ,n ( σ ) are constructiblesubsets of L n ( X ) and dim X j ,n ( σ ) = d ( n + ) − s j − P i ∈ I ν i j i where s j = P i ∈ I j i . Moreover Im ( σ ∗ n ) = Z n ( σ ) ⊔ G j ∈ A n ( σ ) X j ,n ( σ ) and the set Z n ( σ ) satisfies dim Z n ( σ ) < d ( n + ) − nc where c = max ( max , λ max ) .Proof. Consider j such that E • J = ∅ and ∀ i ∈ I, 0 ≤ j i ≤ n . The fiber of B j ,n → E • J is Y i ∈ J ( R ∗ × R n − j i ) × ( R n ) d − | J | ≃ ( R ∗ ) | J | × R dn − s j since truncating the coordinates of γ ∈ B j to degree n produces d − | J | polynomials of degree n with fixed constant terms and for i ∈ J a polynomial of the form c j i t j i + c j i + t j i + + · · · + c n t n with c j i ∈ R ∗ and other c k ∈ R . We conclude that dim B j ,n = d ( n + ) − s j .We first assume that j ∈ A n ( σ ) . By Lemma 4.9, B j ⊂ ∆ e ( j ) ,e ′ ( j ) ( σ ) . Hence by 4.5.(iv), X j ,n ( σ ) is constructible since it is the image of the constructible set B j ,n by the map σ ∗ n | ∆ e ( j ) ,e ′ ( j ) ,n withfibers of odd Euler characteristic with compact support. Let γ ∈ B j ,n and γ ∈ ∆ e ( j ) ,e ′ ( j ) ,n with σ ∗ n ( γ ) = σ ∗ n ( γ ) , then, by 4.5.(ii), γ ≡ γ mod t n − e ( j )+ with n − e ( j ) ≥ e ( j ) and hence γ ∈ B j ,n . Thus by 4.5.(iv) the map B j ,n → X j ,n ( σ ) is a piecewise trivial fibration with fiber R e ( j ) .So we have dim X j ,n ( σ ) = d ( n + ) − s j − e ( j ) as claimed.Otherwise j / ∈ A n ( σ ) and then dim X j ,n ≤ dim B j ,n = d ( n + ) − s j < d ( n + ) − nc (since n2 < e ( j ) ≤ ν max s j or n < e ′ ( j ) ≤ λ max s j ). (cid:4) Remark 4.11.
The two previous lemmas work as they are if we replace σ by ˜ σ , ν i by ˜ ν i , λ i by ˜ λ i and c by ˜ c . Remark 4.12.
Remember that Im σ ∗ n ⊂ π n ( L ( X )) (resp. Im ˜ σ ∗ n ⊂ π n ( L ( X )) ). Moreover, sincewe may lift by σ an arc not entirely included in the singular locus, π n ( L ( X )) \ Im σ ∗ n ⊂ π n ( L ( X sing )) .The second part only works for σ and doesn’t stand for ˜ σ .In order to apply the virtual Poincaré polynomial, we are going to modify the objects of thepartitions of Lemma 4.10. Notation 4.13.
We set ^ π n ( L ( X )) := Z n ( σ ) ⊔ ( π n ( L ( X )) \ Im σ ∗ n ) A S ⊔ G j ∈ A n ( σ ) X j ,n ( σ ) resp. ^ Im ˜ σ ∗ n := Z n ( ˜ σ ) A S ⊔ G j ∈ A n ( ˜ σ ) X j ,n ( ˜ σ ) A blow-Nash inverse mapping theoremwhere the closure is taken in the complement of G j ∈ A n ( σ ) X j ,n ( σ ) resp. in ^ π n ( L ( X )) \ G j ∈ A n ( ˜ σ ) X j ,n ( ˜ σ ) .Hence we still have the inclusion ^ Im ˜ σ ∗ n ⊂ ^ π n ( L ( X )) , the unions are still disjoint and the dimen-sions remain the same. Lemma 4.14.
For j ∈ A n ( σ ) we have β (cid:0) X j ,n ( σ ) (cid:1) = β (cid:0) E • J (cid:1) ( u − ) | J | u nd − P ( ν i + ) j i .(resp. for j ∈ A n ( ˜ σ ) we have β (cid:0) X j ,n ( ˜ σ ) (cid:1) = β (cid:0) E • J (cid:1) ( u − ) | J | u nd − P ( ˜ ν i + ) j i )Proof. We have β (cid:0) X j ,n ( σ ) (cid:1) = β (cid:0) B j ,n (cid:1) u − P ν i j i by Lemma 4.5 and Lemma 4.9 = β (cid:16) E • J × ( R ∗ ) | J | × R dn − s j (cid:17) u − P ν i j i by the beginning of the proof of Lemma 4.10 = β (cid:0) E • J (cid:1) ( u − ) | J | u nd − s j − P ν i j i The same argument works for ˜ σ too. (cid:4) Lemma 4.15. ∀ i ∈ I, ν i = ˜ ν i Proof.
Applying the virtual Poincaré polynomial to the partitions of Notation 4.13, we get β (cid:16) ^ π n ( L ( X )) (cid:17) − β (cid:16) ^ Im ˜ σ ∗ n (cid:17) − X j ∈ A n ( σ ) ∩ A n ( ˜ σ ) (cid:0) β ( X j ,n ( σ )) − β ( X j ,n ( ˜ σ )) (cid:1) = X j ∈ A n ( σ ) \ A n ( ˜ σ ) β ( X j ,n ( σ ))− X j ∈ A n ( ˜ σ ) \ A n ( σ ) β ( X j ,n ( ˜ σ ))+ β (cid:16) Z n ( σ ) ⊔ ( π n ( L ( X )) \ Im σ ∗ n ) AS (cid:17) − β (cid:16) Z n ( ˜ σ ) AS (cid:17) We set P n = β (cid:16) ^ π n ( L ( X ) (cid:17) − β (cid:16) ^ Im ˜ σ ∗ n (cid:17) , Q n = − X j ∈ A n ( σ ) ∩ A n ( ˜ σ ) (cid:0) β ( X j ,n ( σ )) − β ( X j ,n ( ˜ σ )) (cid:1) ,R n = X j ∈ A n ( σ ) \ A n ( ˜ σ ) β ( X j ,n ( σ )) , S n = − X j ∈ A n ( ˜ σ ) \ A n ( σ ) β ( X j ,n ( ˜ σ )) ,T n = β (cid:16) Z n ( σ ) ⊔ ( π n ( L ( X )) \ Im σ ∗ n ) AS (cid:17) , U n = − β (cid:16) Z n ( ˜ σ ) AS (cid:17) . Assume there exists i ∈ I such that ν i > ˜ ν i .Then for n big enough, K n = (cid:14) s j + X i ∈ I ˜ ν i j i , j ∈ A n ( σ ) ∩ A n ( ˜ σ ) , X i ∈ I ( ν i − ˜ ν i ) j i > 0 (cid:15) is notempty. The minimum k n = min K n stabilizes for n greater than some rank n . Let k = k n .Then, for n ≥ n , the degree of Q n is max (cid:8) d ( n + ) − s j − P i ∈ I ˜ ν i j i (cid:9) = d ( n + ) − k using thecomputation at the beginning of the proof of Lemma 4.10.The leading coefficients of P n is positive since P n = β (cid:16) ^ π n ( L ( X )) \ ^ Im ˜ σ ∗ n (cid:17) . The leadingcoefficient of Q n is also positive. Hence the degree of the LHS is at least d ( n + ) − k .Moreover, we have deg R n < d ( n + ) − n ˜ c , deg S n < d ( n + ) − nc , deg T n < d ( n + ) − n max ( c,1 ) and deg U n < d ( n + ) − n ˜ c . Indeed, for T n , π n ( L ( X )) \ Im σ ∗ n ⊂ π n ( L ( X sing )) anddim (cid:0) π n ( L ( X sing )) (cid:1) ≤ ( n + )( d − ) < d ( n + ) − n by 2.33.(i). So the degree of the RHS is lessthan d ( n + ) − n max ( c, ˜ c,1 ) .We get a contradiction for n big enough. (cid:4) Corollary 4.16. Q n = Since ˜ σ : M → X is a proper Nash map generically one-to-one, there exists a closed semial-gebraic subsets S ⊂ X with dim S < d such that for every p ∈ X \ S , ˜ σ − ( p ) is a singleton. Corollary 4.17.
Every arc on X not entirely included in S ∪ X sing may be uniquely lifted by ˜ σ . ean-Baptiste Campesato Proof.
Let γ be an analytic arc on X not entirely in S and not entirely in the singular locus of X .Assume that γ / ∈ Im ˜ σ ∗ . Then, by Proposition 2.21, we have˜ σ − ( γ ( t )) = m X i = b i t i + bt pq + · · · , b =
0, m < pq < m +
1, t ≥ Since ˜ σ − is locally Hölder by Remark 2.20, there is N ∈ N such that for every analytic arc η on X with γ ≡ η mod t N we have ˜ σ − ( η ( t )) ≡ ˜ σ − ( γ ( t )) mod t m + . Hence such an analytic arc η isn’t in the image of ˜ σ ∗ and for n ≥ N , π n ( η ) isn’t in the image of ˜ σ ∗ n : L n ( M ) → π n ( L ( X )) .Hence (cid:16) π nN | π n ( L ( X )) (cid:17) − ( π N ( γ )) ⊂ π n ( L ( X )) \ Im ( ˜ σ ∗ n ) .The first step consists in computing the dimension of the fiber (cid:16) π nN | π n ( L ( X )) (cid:17) − ( π N ( γ )) where n ≥ N . For that, we will work with a resolution ρ : ˜ X → X (for instance σ ) instead of ˜ σ sinceevery analytic arc on X not entirely included in X sing may be lifted to ˜ X by ρ . Let θ be the uniqueanalytic arc on ˜ X such that ρ ( θ ) = γ . Let e = ord t (cid:0) Jac ρ ( θ ( t )) (cid:1) and e ′ be such that γ ∈ L ( e ′ ) ( X ) .We may assume that N ≥ max ( ′ ) in order to apply Lemma 4.5 to ρ for γ .We consider the following diagram L ( ˜ X ) ρ ∗ / / π n (cid:15) (cid:15) (cid:15) (cid:15) L ( X ) π n (cid:15) (cid:15) (cid:15) (cid:15) L n ( ˜ X ) ρ ∗ n / / π nN (cid:15) (cid:15) (cid:15) (cid:15) π n ( L ( X )) π nN (cid:15) (cid:15) (cid:15) (cid:15) L N ( ˜ X ) ρ ∗ N / / π N ( L ( X )) Since the fibers of ρ ∗ n | ∆ e,e ′ ,n and ρ ∗ N | ∆ e,e ′ ,N are of dimension e , and since the fibers of π nN : L n ( ˜ X ) → L N ( ˜ X ) are of dimension ( n − N ) d , we have dim (cid:18)(cid:16) π nN | π n ( L ( X )) (cid:17) − ( π N ( γ )) (cid:19) = ( n − N ) d .Hence dim ( π n ( L ( X )) \ Im ( ˜ σ ∗ n )) ≥ ( n − N ) d . And so, with the notation of Lemma 4.15, we have P n + = R n + S n + T n + U n with deg P n ≥ ( n − N ) d = ( n + ) d − ( N + ) d and deg ( R n + S n + T n + U n ) < ( n + ) d − n max ( c, ˜ c,1 ) .We get a contradiction for n big enough. (cid:4) End of the proof of Theorem 3.5.
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