Andr É Belotto
email: andre.belottodasilva@math.univ-toulouse.fr

D A Silva

Daniel Panazzolo Dedicado

Felipe Cano

GENERALIZED FLOW-BOX PROPERTY FOR SINGULAR FOLIATIONS

Keywords: 2010 Mathematics Subject Classification. Primary 34C99, 34M35, 37F75; Secondary 14B05, 14E15, 34C08 Flow-Box theorem, vector-field, foliations, resolution of singularities

We introduce a notion of generalized Flow-Box property valid for general singular distributions and sub-varieties (based on a dynamical interpretation). Just as in the usual Flow-Box Theorem, we characterize geometrical and algebraic conditions of (quasi) transversality in order for an analytic sub-variety X (not necessarily regular) to be a section of a line foliation. We also discuss the case of more general foliations.

This study is originally motivated by a question of Jean-François Mattei in [7, Theorem 3.2.1] about the existence of local slices for a (non-compact) Lie group action.

Introduction

The Flow-Box Theorem (also called Straightening Theorem) stands as an important classical tool for the study of vector-fields. The Theorem states that the dynamic near a non-singular point is as simple as possible, that is, it is conjugated to a translation (see e.g. [START_REF] Ilyashenko | Lectures on analytic differential equations[END_REF]Theorem 1.14]). The Frobenius Theorem can be seen as a generalization for general foliations. In this work, we introduce a notion of generalized Flow-Box based on a dynamical interpretation and which is valid for singular foliations. We show, furthermore, that under "quasi-transversality" conditions the generalized Flow-Box property holds. We were originally motivated by the study of local slices for (non-compact) Lie group actions and a question of J.-F. Mattei which we briefly discuss on subsection 1.4 below.

Let us denote by M a real or complex analytic manifold and by X an analytic (not necessarily regular) sub-variety of M . For simplicity, we specialize our presentation for vector-fields before considering general involutive singular distributions in subsection 1.2.

1.1. Vector-fields. Let ∂ be an analytic vector-field over M and denote by Sing(∂) the set of singular points of ∂. We say that the triple (M, ∂, X) is transverse (or, more commonly, that X is a transverse section of ∂) if the vector-field ∂ is transverse to X, that is, at every point p ∈ X there exists an analytic germ of function f such that X ⊂ (f = 0) and ∂(f )(p) = 0. We consider the following version of the Flow-Box Theorem: Theorem 1.1 (Flow-Box Theorem). If a triple (M, ∂, X) is transverse then, at each point p ∈ X, there exists a neighbourhood U of p and a coordinate system (x, y y y) = (x, y 2 , . . . , y m ) over U centered at p such that ∂ = ∂ x and X ∩ U ⊂ {x = 0}. Remark 1.2.

(1) Sometimes the Flow-Box Theorem (see e.g. [6, Theorem 1.14]) is stated independently of a transverse section X, but the above version is an equivalent statement (which follows from the Implicit Function Theorem).

(2) Let p ∈ X and denote by ϕ p (t) the orbit of X such that ϕ p (0) = p. If (M, ∂, X) is transverse, then for every compact set K ⊂ X, there exists δ > 0 such that ϕ p (t) / ∈ X for all 0 < |t| < δ and for all p ∈ K. (3) If M is a complex manifold and X is a codimension one smooth sub-variety, then the reciprocal of statement in (2) also holds.

Based on the interpretation given in remark 1.2(2), we define:

Definition 1.3 (Generalized Flow-Box property). We say that the triple (M, ∂, X) satisfies the generalized Flow-Box property if for every compact set K ⊂ X, there exists a real number δ > 0 such that for every point p ∈ K \ Sing(∂) we have that ϕ p (t) / ∈ X for all 0 < |t| < δ (where ϕ p (t) is the orbit of ∂ such that ϕ p (0) = p).

Note that the generalized Flow-Box property is defined for singular sub-varieties X and vector-fields ∂. In order to illustrate the property, we present two examples:

(1) Consider the triple R 2 , ∂ = x∂ y -y∂ x , (y = 0) (see figure 1). In this case (M, ∂, X) satisfies the generalized Flow-Box property with, for example, δ = π/4; (2) Consider the triple R 2 , ∂ = ∂ x + x∂ y , (y = 0) (see figure 1). In this case (M, ∂, X) does not satisfy the generalized Flow-Box property;

Now, just as in the usual Flow-Box Theorem, we characterize this condition in terms of a transversality condition between ∂ and X. We start by a geometrical condition:

Definition 1.4. A triple (M, ∂, X) is geometrically quasi-transverse if the triple (M, ∂, X \ Sing(∂)) is transverse, that is, if at every point p ∈ X \ Sing(∂), there exists an analytic germ f such that X ⊂ (f = 0) and ∂(f ) is a unit (at p).

Under this condition, we prove the following result:

Theorem 1.5. Let (M, ∂, X) be a geometrically quasi-transverse triple and assume that X is non-singular. (where X is non-singular) that does not satisfy the generalized Flow-Box property.

Condition (a) of the above result is a consequence of Theorem 3.3 below and the part (b) is proved in subsection 3.1. Next, we introduce a stronger notion of quasi-transversality in order to get a sufficient condition:

Definition 1.6. A triple (M, ∂, X) is analytically quasi-transverse if it is geomet- rically quasi-transverse and ∂ 2 (I X ) ⊂ ∂(I X ) + I X
, where I X is the reduced ideal sheaf whose support is X.

The above condition avoids tangent contacts between X and ∂ which are "hidden" in the singular locus. Indeed: Theorem 1.7. If a triple (M, ∂, X) is analytically quasi-transverse, then it satisfies the generalized Flow-Box property.

The above result is a consequence of Theorem 3.3 below. 1.2. Involutive Singular Distributions. Following [1, p. 281], an analytic singular foliation is a coherent sub-sheaf θ of Der M (where Der M is the sheaf of all derivations over M ) which is closed under the Lie-Bracket operation. We say that a triple (M, θ, X) satisfies the Flow-Box property at a point p in X if there exists a neighbourhood U of p and a coordinate system (x x x, y y y) = (x 1 , . . . , x t , y t+1 , . . . , y m ) over U centered at p such that θ is generated by {∂ x1 , . . . , ∂ xt } and X ∩ U is a subset of V (x 1 , . . . , x t ):= {x 1 = . . . = x t = 0}. We say that the triple (M, θ, X) is transverse (or, more commonly, X is a transverse section of θ) if X is a regular sub-variety and at every point p in X, the leaf L of θ passing through p has maximal dimension and its tangent space has trivial intersection with the tangent space of X. The classical Frobenius theorem states that any triple (M, θ, X) which is transverse, satisfies the Flow-Box property. Following the same considerations which lead to the definition 1.3, we define: Definition 1.8. We say that a triple (M, θ, X) satisfies the generalized Flow-Box property at a point p in X, if there exists a triple (U, δ, g), where U is an open neighbourhood of p, δ > 0 is a real number and g is a sub-Riemannian metric (whose definition is recalled in subsection 4.1) over M generated by a set {∂ 1 , . . . , ∂ d } of local generators of θ such that: for every point q ∈ U ∩ X, every g-ball with radius η < δ: B g η (q) = {a ∈ U ; d g (a, q) < η} intercepts X only at q and is homeomorphic to a k q -euclidean ball, where k q is the dimension of the subspace L q ⊂ T p M generated by θ.

Remark 1.9. We demand that B g η (q) is homeomorphic to a ball in order to avoid "infinitesimal" self-returns which would not be consistent with definition 1.3. More precisely, suppose that θ is one dimensional and that there exists a sequence of periodic orbits converging to p with period converging to zero, that is, there exists sequences η i → 0 and q i → p such that B g ηi (q i ) is isomorphic to a circle. This would contradict definition 1.3, but would not contradict definition 1.8 if we did not demand that B g η (q) is homeomorphic to a ball. Such a definition coincides with the Flow-Box property whenever (M, θ, X) is transverse, but also extends to the singular locus of θ and X. Just as for line foliations, we consider the following property: Definition 1.10. A triple (M, θ, X) is geometrically quasi-transverse if at every point p in X, the Zariski tangent space of X at p has trivial intersection with the subspace of T p M generated by θ.

Under this condition, we can prove the following result: Theorem 1.11. Let (M, θ, X) be a geometrically quasi-transverse triple and assume that X is a non-singular curve, then (M, θ, X) satisfies the generalized Flow-Box property. Nevertheless, whenever dimM ≥ 4, there exists a geometrically quasitransverse triple (M, θ, X) (where X is non-singular) that does not satisfy the generalized Flow-Box property.

We prove this result in section 4.5 as a consequence of theorem 4.13. In subsection 4.4, we introduce a notion of quasi-transversality based on the stability under certain operations, including blowings-up. We give a precise definition (definition 4.10) and statement of the result (Theorem 4.13) on section 4. 1.3. Structure and notations of the manuscript. The main technique used in the manuscript is resolution of singularities subordinated to the singular distribution θ. We introduce the main technical tool (resolution by θ-invariant centers) in section 2 and we rely on [START_REF] Belotto | Global resolution of singularities subordinated to a 1-dimensional foliation[END_REF][START_REF] Belotto | Resolution of singularities in foliated spaces[END_REF] for the proofs. In section 3 we prove the main results for vector-fields, and in section 4 we provide the proofs for general foliations.

Since we use resolution of singularities arguments, it is convenient to work with ideal sheaves and to keep track of divisors. We consider a quadruple (M, θ, I, E), which we call foliated ideal sheaf, where

• I is a reduced ideal sheaf (whose support V (I) plays the role of the subvariety X); • E is a simple normal crossing divisor on M (i.e. E is the union of smooth connected hypersurfaces and, at each point p ∈ E, there exists a coordinate system x x x = (x 1 , . . . , x m ) centered at p such that E is locally given by {Π r i=1 x i = 0} for some r ≤ m); • θ is coherent sub-sheaf of Der M (-log E) closed under the Lie-bracket operation. In particular, θ is tangent to E (i.e. its local sections ∂ are everywhere tangent to E). If θ is a 1-dimensional free singular distribution (i.e. at each point p ∈ M , θ is locally generated by a vector-field ∂), we say that (M, θ, I, E) is a 1-foliated ideal sheaf. In what follows, we re-introduce the definitions of quasi-transversality and generalized Flow-Box property for foliated ideal sheaves.

1.4. Origin of the Problem. Originally, this work aimed to prove that the semiuniversal equisingular unfolding of one-forms constructed by J.-F. Mattei in [7, Theorem (3.2.1)], is actually universal. This is an important problem in the theory of families of one-forms and foliations. Mattei's construction leads to a local problem involving the action of an algebraic group over an analytic sub-variety. This problem was brought to our attention by J.-F. Mattei in a private communication and stands as the main technical obstacle to prove that the semi-universal unfolding is actually universal. More precisely, the original question can be stated as follows:

Problem. Consider an (not necessarily compact) algebraic group G acting over a regular manifold M and an analytic and regular sub-variety X of M . Suppose that at every point p in X, the orbit of G passing through p is "quasi-transverse" to X, i.e. the intersection of the tangent space of the orbit with the tangent space of X is trivial. Is it possible to find a neighbourhood U ⊂ G of the neutral element and a neighbourhood W of p such that, for every q in W and every g in U , g(q) ∈ X =⇒ g(q) = q?

Stating this problem for the Lie algebra instead of the Lie group yields the definitions of quasi-transversality and generalized Flow-Box introduced here and to the question: does geometrical quasi-transversality implies the Flow-Box property? Although we give a negative answer in Theorem 1.5, the original question of J.-F.

Mattei is still open, since it concerns an specific algebraic Lie group. We do show, nevertheless, that an approach based on the Lie Algebra is insufficient since, in this context, the answer is negative.

Based on these considerations, we could also introduce a notion of local generalized slices for (not necessarily compact) Lie groups. We recall that the existence of slices for compact Lie group actions is an important tool for the global study of Lie group actions. Under some additional conditions, it is also possible to obtain global slices over non-compact Lie groups (see [START_REF] Palais | On the existence of slices for actions of non-compact Lie groups[END_REF]). In any case, up to our knowledge, no local notion of section is given for general non-compact groups; the generalized Flow-Box property might be a natural way to do so.

Background: Resolution of singularities

We briefly recall some basic notations and ideas of resolution of singularities. Let E ⊂ M be a simple normal crossing divisor (that is, E is the union of smooth hypersurfaces and, at each point p ∈ E, there exists a coordinate system

x = (x 1 , . . . , x n ) such that E = {Π m i=1 x i i = 0}
, where i ∈ {0, 1}). Given a blowing-up σ : M → M , denote by F its exceptional divisor. We say that the blowing-up is admissible in respect to a divisor E ⊂ M if the center C is a closed and regular sub-manifold of M which has SNC with E (that is, there exists a coordinate system x x x = (x 1 , . . . , x m ) where the reduced ideal sheaf whose support is C, which we denote by I C , is locally given by I C = (x 1 , . . . , x t ) and E = {Π m i=1 x i i = 0}, where i ∈ {0, 1}). If the blowing-up is admissible, we denote it by σ :

( M , E) → (M, E), where E = σ -1 (E) ∪ F .
Consider an ideal sheaf I. If the center of blowing-up C is contained in the support of I, we say that σ is a blowing-up with order one in respect to I. In this case we denote the blowing-up by σ : ( M , I, E) → (M, I, E), where I is the weak transform of I, that is I F I = σ * I, where I F is the reduced ideal sheaf whose support is the exceptional divisor F of σ (c.f. [5, Definition 2.2]). We now consider an involutive singular distribution θ which is tangent to E. Remark 2.1. Since θ is a coherent sub-sheaf of the derivations over M , it naturally acts on the structure sheaf O M and, thus, over ideal sheaves I. More precisely, given a point p ∈ M , the ideal θ[I] • O p is defined as the ideal generated by

{∂(f ); f ∈ I • O p , ∂ ∈ θ • O p }
where ∂ is a local generator of θ. We denote by θ[I] the ideal sheaf whose stalks are defined as above. Since θ[I] is again an ideal sheaf, we can iterate the process a finite number of times, say k, and obtain an ideal which we denote by

θ k [I]. Definition 2.2 (θ-invariant blowing-up). We say that a blowing-up σ is θ-invariant, if the center of blowing-up is θ-invariant, that is, if θ[I C ] ⊂ I C , where I C is the reduced ideal sheaf whose support is C. Remark 2.3.
(1) Given a θ-invariant blowing-up of order one σ : ( M , I, E) → (M, I, E), the total transform θ = σ * (θ) is well-defined, analytic and tangent to E. Indeed, following the notation introduced in Remark 2.1, we note that:

θ

* [I F ] = σ * (θ[I C ]) ⊂ σ * (I C ) = I F
where F is the exceptional divisor of σ. Therefore, we can denote the blowing-up by

σ : ( M , θ, I, E) → (M, θ, I, E)
(2) Furthermore, by using [3, Lemma 3.2.i] and part (1), we get that:

I F • ν i=0 θ i [ I] = σ * ν i=0 θ i [I] , ∀ ν ∈ N Proposition 2.4.
Let (M, θ, I, E) be a foliated ideal sheaf (the leaf-dimension of θ is not necessarily one). Consider a relatively compact open set M 0 of M and denote by I 0 = I| M0 and E 0 = E| M0 . Let ν be a natural number such that θ ν+1 (I) ⊂ ν i=0 θ i (I) in M 0 . Then, there exists a sequence of θ-invariant blowingsup of order one:

( M , θ, I, E) = (M r , θ r ,I r , E r ) • • • (M 0 , θ 0 ,I 0 , E 0 ) σ r σ 1
such that: i) the sequence of blowings-up is an isomorphism outside of the support of ν i=0 θ i (I); ii) the pulled back foliation θ is (well-defined,) analytic and tangent to E. Furthermore ν i=0 θ i ( I) has empty support (that is,

ν i=0 θ i ( I) = O M ).
Proof. This result follows from the same argument as [3, Proposition 5.2] and relies on the functorial property of resolution of singularities [START_REF] Bierstone | Functoriality in resolution of singularities[END_REF]; a complete proof can be found in [4, Proposition 5.4.1].

Remark 2.5. If (M, θ, I, E) is geometrically quasi-transverse (see definitions 3.1 and 4.2), then condition (i) of Proposition 2.4 implies that σ is an isomorphism outside of Sing(θ) ∪ E. Indeed, the quasi-transversality hypothesis guarantees that all θ-invariant centres must be contained in Sing(θ) ∪ E and its transform.

Generalized Flow-Box property for vector-fields

Following subsection 1.3, we work with 1-foliated ideal sheaves (M, θ, I, E) instead of triples (M, ∂, X) (where Supp I = X and θ is generated by ∂). We start by extending the definitions given in subsection 1.1: Definition 3.1. Consider a 1-foliated ideal sheaf (M, θ, I, E), the variety X = Supp I and vector-fields ∂ U , defined in open sets U (which cover M ), that generate θ on U . We say that (M, θ, I, E):

• is geometrically quasi-transverse if all the triples (U, ∂ U , X \ E) are geometrically quasi-transverse. • is analytically quasi-transverse if it is geometrically quasi-transverse and

θ 2 [I] ⊂ I + θ[I].
• satisfies the generalized Flow-Box property if for every compact set K ⊂ X, there exists a real number δ > 0 such that for every point p ∈ K \ (Sing(∂) ∪ E) we have that ϕ p (t) / ∈ X for all 0 < |t| < δ, where ϕ p (t) is the orbit of ∂ U , for some U , such that ϕ p (0) = p. Remark 3.2. Given a geometrically quasi-transverse 1-foliated ideal sheaf (M, θ, I, E), note that θ may be tangent to X ∩ E.

We are ready to state the main result of this section: Theorem 3.3. Let (M, θ, I, E) be a geometrically quasi-transverse 1-foliated ideal sheaf. If one of the following conditions is verified:

i) The 1-foliated ideal sheaf (M, θ, I, E) is analytically quasi-transverse;

ii) The ideal sheaf I is regular and the variety X := Supp I: has dimension one or co-dimension one; has normal crossings with E (that is, if p ∈ X ∩E, there exists a local coordinate system x = (x 1 , . . . , x n ) such that X = V (x 1 , . . . , x t ) and E = {x i i = 0} where i ∈ {0, 1}); and is not contained in E. Then (M, θ, I, E) satisfies the generalized Flow-Box property.

Proof. Fix a point p in X = V (I) and a vector-field ∂ which generates θ • O p . By Proposition 2.4 there exists a relatively compact neighbourhood M 0 of p and a sequence of blowings-up of order one:

( M , I, E) = (M r , I r , E r ) • • • (M 0 , I 0 , E 0 ) σ r σ 1
such that: i) the sequence of blowings-up is an isomorphism outside of E ∪ Sing(∂) (see Remark 2.5; ii) the pulled back vector-field ∂ is analytic and

ν i=0 ∂ i ( I) = O M .
In particular, by (i), the inverse image of all regular orbits of ∂ outside of E are regular orbits of ∂ outside of E (i.e there is no change on the time of the orbits). Thus, since blowings-up are proper morphisms, we just need to verify that ( M , θ, I, E) satisfies the generalized Flow-Box property in a neighborhood of every point in V ( I). In what follows, we denote by X the sub-variety V ( I) and we divide in four cases depending on the initial hypothesis:

Case I -(M, θ, I, E) is analytically quasi-transverse: In this case, ν ≤ 1, which implies that ∂( I p ) = O M ,p for any point p ∈ X. Thus, the Flow-box theorem is valid at each point of the sub-variety X and we conclude by Remark 1.2 [START_REF]Sub-Riemannian geometry[END_REF].

Case II -The ideal sheaf I is regular and the variety X = V (I) has dimension one: Note that X ∩ E has dimension zero. Since E is invariant by θ (which is regular over X ∩ E), we conclude that θ must be transverse to X also over X ∩ E. Thus, the Flow-box theorem is valid at each point of the sub-variety X and we conclude as above.

Case III -The ideal sheaf I is regular, the variety V (I) has co-dimension one and K = C: Let q be a point of X, the function f be a local generator of I and consider the sub-variety W := V (f, ∂(f )). Assume, by contradiction, that ∂(f ) is not a unit, i.e that W = ∅. Then, since W has codimension two and is empty outside of E (by geometrically quasi-transversality), W is a non-empty co-dimension one subvariety of E which must be equal to X ∩ E. Since E is invariant by θ we conclude that W is also invariant by θ. This contradicts the fact that ∂ k (f ) is a unit for some k ≤ ν. Thus, ∂(f ) must be a unit itself and θ must be transverse to X at every point. Thus, the Flow-box theorem is valid at each point of the sub-variety X and we conclude as above.

Case IV -The ideal sheaf I is regular, the variety V (I) has co-dimension one and K = R: Let q be a point of X, the function f be a generator of I and consider the sub-variety W := V (f, ∂(f )). The same arguments of Case III are valid if W has codimension one in X or in E, so we may assume that W is a sub-variety of co-dimension at least two of X and E. Now, since X is locally orientable, let g be an analytic Riemannian metric in a neighborhood U of q and set N = ∇ g (f ) in U . The function:

φ : U ∩ X -→ R a → N , ∂ (a)
is continuous and equal to zero, if and only if, a is point in W . Now, since W has codimension at least two in X, without loss of generality, we can assume that φ ≥ 0. This implies that the order of tangency between ∂ and X is odd. Therefore the regular orbits of ∂ always cross X from the negative to the positive side (with respect to the orientation defined by N ). We conclude the generalized Flow-Box property in U from this observation.

3.1. Proof of Theorem 1.5(b). We present an example of a geometrically quasitransverse triple (M, ∂, X) with dimM = 4 that does not satisfy the generalized Flox-Box condition. We remark that this example is valid for a four dimension real or complex manifold M , but it can be generalized to any higher dimension. Its construction relies in the following preliminary result:

Lemma 3.4. The equation:

(3.1) (1 + s 2 ) cos(α) sin(α) -α = 0
has an analytic solution (α, s) = (h(s), s) = (sU (s), s) defined in a neighbourhood of the origin, where

U (0) = √ 6/2.
Proof. Note that the equation:

(3.2) (1 + s 2 )cos(α) sin(α) α -1 = 0
has the same solutions of equation (3.1) apart from (α, s) = (0, s). Now, taking the Taylor expansion with respect to the variable α of equation (3.2) at (α, s) = (0, 0), we get

s 2 - 2 3 (1 + s 2 )α 2 + o(α 3 ) = 0
so, by the Weierstrass Preparation Theorem and the symmetry of equation (3.2) with respect to the transform α → -α, the equation (3.2) can be written as:

(f (s) 2 -α 2 )u(α, s) = 0
where u(α, s) is an unit and f (s) is an analytic function. Furthermore, we have that f (s) 2 u(0, s) = s 2 , which implies that f (s) = ±sU (s), where U (s) is an analytic unit defined in a neighbourhood of zero. Taking h(s) = sU (s) gives the desired result.

To finish, we just need to use the Taylor expansion of equation (3.1) in respect to s to conclude that U (0) = √ 6/2. Now, let (x, y, z, w) be a globally defined coordinate system of K 4 (where K = R or C). We consider the triple (M, ∂, X) where the manifold M is a (sufficiently) small open neighbourhood of the origin of K 4 (where h(x 2 + y 2 ) is well-defined); the vector-field ∂ is given by:

∂ = y 2 ∂ ∂w + x ∂ ∂y -y ∂ ∂x
and the (non-singular) variety X is given by z = f (x, y) and w = g(x, y) where:

f (x, y) = y 2 cos(h(x 2 + y 2 )) -xy sin(h(x 2 + y 2 )) g(x, y) = 1 2 xy(x 2 + y 2 ) 2
Theorem 1.5(b) follows from the two Claims below:

Claim 1: In this example, (M, ∂, X) is geometrically quasi-transverse.

Proof. In order to prove the Claim, we just need to show that the set W of points where ∂ and X are tangent is contained in Sing(∂) ∩ X = {0, 0, 0, 0}. Indeed, W is a variety defined by z = f (x, y), w = g(x, y), and

∂[z -f (x, y)] = -2xy cos(h(x 2 + y 2 )) + (x 2 -y 2 ) sin(h(x 2 + y 2 )) = 0 2 ∂[w -g(x, y)] = 2y 2 -(x 2 -y 2 )(x 2 + y 2 ) 2 = 0
Now, from the forth equation we get two solutions in y (defined in a sufficiently small neighbourhood of the origin):

y 1 = x 3 V (x) and y 2 = -x 3 V (x)
where V (x) is an analytic function such that V (0) = 1 2 √ 2. Now, substituting in the third equation and taking its Taylor expansion in x, we get:

[±2V (0) + U (0)] x 4 + O(x 5 ) = 0
where we recall that h(s) = sU (s). Since ±2V (0) + U (0) = 0, we conclude that W = {0, 0, 0, 0} as we wanted to prove.

Claim 2: In this example, (M, ∂, X) does not satisfy the generalized Flow-Box property. In the picture r0 > 0 is fixed. The orbit ϕ(r0, t) is contained in the plane {z = 0} and the variety X is a curve that cuts the orbit ϕ(r0, t) two times.

Proof. We prove this claim when K = R since the complex case trivially follows from the real case. Since the variety C = {x = 0, y = 0} is invariant by ∂, we consider the (real) blowing-up σ : M → M with center C and exceptional divisor E, where M = R + × S 1 × R 2 and there exists a globally defined coordinate system (r, α, z, w) on M where σ is given by: x = r cos(α), y = r sin(α).

The pulled-back vector-field ∂ is equal to:

∂ = r 2 sin(α) 2 ∂ ∂w + ∂ ∂α
and the pulled-back variety X is defined by z = f (r, α) and w = g(r, α)), where:

f (r, α) = r 2 sin(α)[sin(α) cos(h(r 2 )) -cos(α) sin(h(r 2 ))] g(r, α) = 1 2 r 6 sin(α) cos(α)
For a fixed r 0 > 0, the orbit of ∂ passing by (r, α, z, w) = (r 0 , 0, 0, 0) at t = 0 is given by:

ϕ(r 0 , t) = r 0 , t, 0, r 2 0 4 [2t -sin(2t)]
Now, at one hand:

z -f (r, α) (r 0 , 0, 0, 0) = r 2 0 sin(0)[sin(0 -h(r 2 0 ))] = 0
[wg(r, α)] (r 0 , 0, 0, 0) = 1 2 r 6 0 sin(0) cos(0) = 0 and (r 0 , 0, 0, 0) ∈ X. At the other hand:

z -f (r, α) (ϕ(r 0 , h(r 2 0 ))) = r 2 0 sin(h(r 2 0 ))[sin(h(r 2 0 ) -h(r 2 ))] = 0 2 [w -g(r, α)] (ϕ(r 0 , h(r 2 0 ))) = r 2 0 2 2h(r 2 0 ) -sin(2h(r 2 0 )) -r 6 0 sin(h(r 2 0 )) cos(h(r 2 0 ))
and, re-ordering the terms of the second equation, we get:

r 2 0 h(r 2 0 ) -sin(h(r 2 0 )) cos(h(r 2 0 ))(1 + r 4 0
) and, since h(s) is a solution of equation (3.1), we conclude that [wg(r, α)] (ϕ(r 0 , h(r 2 0 ))) = 0, which implies that ϕ(r 0 , h(r 2 0 )) ∈ X. So, for each r 0 > 0 fixed, there exists an orbit of ∂ that cuts X two times. Furthermore, the time between each cut is equal to h(r 2 0 ), which goes to zero when r 0 goes to zero. Now, the center of blowing-up C was invariant by the vector field ∂ (thus, the time of the orbits is not affected) and σ is an isomorphism outside of the exceptional divisor, we conclude that (M, ∂, X) does not satisfy the generalized Flow-Box property.

Generalized Flow-Box property for singular foliations

4.1. Background on sub-Riemannian Geometry. We follow [START_REF]Sub-Riemannian geometry[END_REF]. Consider K = R, a regular analytic manifold W and globally defined analytic vector fields {∂ 1 , ..., ∂ m }. For each point p in W , we denote by L p the subspace of T p W generated by {∂ 1 (p), . . . , ∂ m (p)}. Given any vector v of L p , there always exists (u 1 , ..., u m ) ∈ K (not necessarily unique) such that v = m i=1 u i ∂ i (p). So, for each point p of W , consider the mapping:

Φ p : R m -→ T p W (u 1 , ..., u m ) → m i=1 u i ∂ i (p)
Note that Φ p restricted to the linear subspace (kerΦ p ) ⊥ is a linear isomorphism onto L p . Let Ψ p : L p → (kerΦ p ) ⊥ be the inverse mapping. Then, if v and w are vectors contained in L p , we define the sub-Riemannian metric g p (v, w) associated to {∂ 1 , ..., ∂ m } by: g p (v, w) := Ψ p (v), Ψ p (w)

Based on this metric, we define the notion of sub-Riemannian norms . g,p = |g p (•, •)| and . g,p,∞ = max i {|u i |} (where u = Ψ p (•)) generated by {∂ 1 , . . . , ∂ m }. We extend both norms to every vector v of T p W by setting v g,p = v g,p,∞ = ∞ if v is not contained in L p . Finally, given two point p and q, we defined their sub-Riemannian distance by

d g (p, q) = min 1 0 γ(t) g,γ(t) dt; γ abs. continuous curve s.t. γ(0) = p, γ(1) = q
Remark 4.1. The above definitions can be extended to the complex case in a standard way: if {∂ 1 , ..., ∂ m } are globally defined holomorphic vector fields, let { ∂1 , . . . , ∂2m } be globally defined real-analytic vector fields such that { ∂i , ∂m+i } generates ∂ i . We then extend the definitions of sub-Riemmanian metric and norm by using { ∂1 , . . . , ∂2m }.

Finally, an analytic sub-Riemannian metric on M is a function g : T M ⊗ T M -→ R ∪ {∞} which, locally may be defined as the metric associated to some system of analytic vector fields. In other words, for each point p in M , there exists a neighbourhood U p of p and a set of vector fields {∂ 1 , ..., ∂ m } defined over U p such that g| Up = g {∂1,...,∂m} where g {∂1,...,∂m} is the sub-Riemmanin metric on U p generated by {∂ 1 , ..., ∂ m }.

Geometric quasi-transversality.

Definition 4.2. A foliated ideal sheaf (M, θ, I, E) is said to be geometrically quasi-transverse if for any point p in X \ E we have

dim K L p + dim K T p X = dim K (L p + T p X)
where L p is the linear sub-space of T p M generated by θ and T p X is the Zariski tangent space of X at p. Proposition 4.3. Let (M, θ, I, E) be a geometrically quasi-transverse foliated ideal sheaf and suppose that θ[I] = O M . Then, at each point p ∈ V (I) there exists two involutive singular distributions θ Y and θ Z (defined in some open neighborhood U of p), where: θ Y is generated by a vector-field ∂ Y , and θ Z is generated by vectorfields {∂ Z1 , . . . , ∂ Zr } and has leaf dimension strictly smaller than θ, such that: i)

θ = θ Y + θ Z ; ii) ∂ Y (I • O p ) = O p ; iii) [∂ Y , ∂ Zi ] ≡ 0 for all i ≤ r.
Proof. In what follows, we use a well-known argument based on a linear ODE; we learned about this argument through [START_REF] Mitera | Junya The local analytical triviality of a complex analytic singular foliation[END_REF] (where θ is assumed to be torsion free).

Fix a point p ∈ M a let (y, z) = (y, z 1 , ..., z n-1 ) be a coordinate system centered at p and defined in a neighbourhood U of p, such that the vector field

Y = ∂ y is contained in θ • O U and the analytic function y is contained in the ideal I • O U . Let {W 1 , ..., W s } be vector fields such that {Y, W 1 , ..., W s } generate θ • O U . We can suppose that W i (x) ≡ 0, which implies that: [Y, W j ] = s j=1 A i,j (y, z)W j
Now, consider a vector field of the form W = s i=1 µ i W i , where µ i ∈ O U . Then:

[Y, W ] = s j=1 Y (µ j ) + s i=1 µ i A i,j (y, z) W j
Since Y is a regular vector field, the system of equations:

Y (µ j ) + s i=1 µ i A i,j (y, z) = 0, j = 1, . . . , s
is an analytic system of ODE's. Therefore, up to shrinking U , there exists s locally defined analytic solutions µ i = (µ i,1 , ..., µ i,s ) such that µ i (0) = e i = (0, ..., 0, 1, 0, ..., 0) (where the 1 is on the i position). Without loss of generality, we suppose that these solutions are defined on U and we denote by Z i := s j=1 µ i,j W j the vector field associated to it. Note that:

• The vector fields {Y, Z 1 , ..., Z s } generates θ • O U ;

• The Lie bracket [Y, Z i ] ≡ 0 for i = 1, . . . , s;

• The vector fields {Z 1 , ..., Z s } generate an involutive d -1-singular distribution θ Z (because of the two points above).

Lemma 4.4. Let (M, θ, I, E) be a geometrically quasi-transverse foliated ideal sheaf and consider a foliated ideal sheaf ( M , θ, I, E) defined by one of the following three operations:

• A θ-invariant blowing-up of order one, that is, σ : ( M , θ, I, E) → (M, θ, I, E) is a θ-invariant blowing-up of order one; • An open restriction, that is, given an open set U ⊂ M , we consider ( M , θ, I, E) = (U, θ| U , I| U , E| U ). • A leaf-reduction operation, that is, suppose that θ[I] = O M . In the notation of Proposition 4.3, we consider ( M , θ, I, E) = (U, θ Z , I| U , E U ).
Then ( M , θ, I, E) is also geometrically quasi-transverse.

Proof. The proof follows directly from the definition of quasi-transversality and the fact that a blowing-up is an isomorphism outside its center.

Remark 4. [START_REF] Bierstone | Functoriality in resolution of singularities[END_REF]. In what follows, we consider the category, which we denote by GQT , whose objects are geometrically quasi-transverse foliated ideal sheaves and the morphisms are (composition of) θ-invariant blowings-up of order one, restrictions and leaf-reduction operation (see Lemma 4.4).

4.3.

The generalized Flow-Box property.

Definition 4.6. We say that a foliated ideal sheaf (M, θ, I, E) satisfies the generalized Flow-Box property at a point p ∈ X = V (I), if there exists a triple (U, δ, g), where U is an open neighbourhood of p, δ > 0 is a real number and g is a sub-Riemannian metric over U generated by a set {∂ 1 , . . . , ∂ d } of local generators of θ such that: for every point q ∈ U ∩ (X \ E), every g-ball of radius η < δ B g η (q) = {a ∈ U ; d g (a, q) < η} intercepts X only at q and is homeomorphic to a k q -euclidean ball, where k q is the dimension of the leaf of θ passing through q. Finally, we say that (M, θ, I, E). satisfies the generalized Flow-Box property if it is so at every point p ∈ X. Proof. Suppose that (M, θ, I, E) satisfies the generalized Flox-Box property at a point p with the sub-Riemannian metric g generated by {∂ 1 , . . . , ∂ r } and let g be a sub-Riemmanian metric generated by another set { ∂1 , . . . , ∂s } of generators of θ • O p . Since these are two sets of generators of θ • O p , there exists a matrix A = [a i,j ] r×s of analytic functions such that (∂ 1 , . . . , ∂ r ) = A( ∂1 , . . . , ∂s ). Let A p be the ∞-norm of the matrix A at p, i.e. A p = max{ a i,j (p) }. Now, given a vector v in L p , we can write it as v = r i=1 u i ∂ i (p) which implies that:

v = r i u i s j=1 a i,j ∂j (p) = s j=1 ∂j (p) r i=1 a i,j u i .
This implies that max{ r i=1 a i,j u i } ≤ A p max{ u i }. Thus:

v g,p,∞ ≤ A p v g,p,∞ .
So, up to taking a smaller open set U p , there exists C > 0 such that v g,q ≤ C v g,q for every v in L q and every q ∈ U p . Furthermore, by the symmetric argument, we conclude that there exists c > 0 such that:

c v g,q ≤ v g,q ≤ C v g,q
Finally, since the metrics are equivalent, up to shrinking the radios η, the balls B g η (q) are also homeomorphic to an Euclidean balls for every q ∈ U p ∩ (X \ E), which proves the Lemma. Proof. The result is clear for any point outside of the center of blowing-up. So, let p be a point in the center C, {∂ 1 , . . . , ∂ d } be a system of generators of θ • O p and g be the sub-Riemannian metric generated by {∂ 1 , . . . , ∂ d }. We can assume that g is well-defined in a sufficiently small relatively compact open neighbourhood U of p.

Since ( M , θ, I, E) satisfies the generalized Flow-Box property, at each point q in σ -1 (p) there exists a triple (U q , δ q , g q ) that satisfies the conditions of the generalized Flow-Box property. So, there exists a finite number N of points q i on M such that σ (∪ i≤N U qi ) = U . We can assume, by lemma 4.7, that g qi = σ * g, where σ * g is generated by the pull-back vector-fields {σ * ∂ 1 , . . . , σ * ∂ d }. Set δ = min{δ qi ; i = 1, . . . , N }. We claim that (U, δ, g) satisfies the conditions of the generalized Flow-Box property at p. Indeed, let q ∈ (V (I) ∩ U ) \ E. If q ∈ C then, by the quasi-transversality hypothesis, we have that dim(L q ) = 0. Otherwise, q / ∈ C and σ is local isomorphism at q. We conclude easily from these two observations. Proposition 4.9. Following the hypothesis and notation of Proposition 4.3, if (U, θ Z , I| U , E| U ) satisfies the generalized Flow-Box property, so does (U, θ, I| U , E| U ).

Proof. Assume that (U, θ Z , I, E) satisfies the generalized Flow-Box property and let X = V (I). Without loss of generality we can suppose that there exists a coordinate system (y, z z z) = (y, z 1 , ..., z n-1 ) centered at p such that ∂ Y = ∂ y and that y ∈ I. Now, let g be the sub-Riemannian metric generated by {∂ Y , ∂ Z1 , . . . , ∂ Zs }, g Y be the sub-Riemannian metric generated by {∂ Y } and g Z be the sub-Riemannian metric generated by {∂ Z1 , . . . , ∂ Zs }. By the usual Flow-Box Theorem, we know that (U, θ Y , I, E) also satisfies the generalized Flow-Box property and, thus, there exists δ > 0 such that:

d g Y (q, X \ {q}) > δ, d g Z (q, X \ {q}) > δ,
for all point q ∈ X \ E. We claim that d g (q, X \ {q}) > δ. Indeed, suppose by contradiction that there exists a point q in X \ E such that d g (q, V (I) \ {q}) < δ. In this case, there exists an absolutely continuous curve c : [a, b] -→ U such that c(a) = q, c(b) ∈ X \ {q} and length g (c(t)) < δ. We write c(t) = (y(t), z(t)) and we define the absolutely continuous curve ϕ(t) = (0, z(t)). Since y ∈ I, we conclude that c(a) = (0, z(a)) and c(b) = (0, z(b)) and, thus, ϕ(a) = q and ϕ(b

) ∈ X \ {q}. Now, since g Y = dy 2 ċ(t) g,c(t) ≥ ż(t) g,c(t) = ż(t) g Z ,c(t) .
Moreover, since (0, ż(t)) = φ(t) and g Z is independent of the y coordinate

ż(t) g Z ,c(t) = φ(t) g Z ,ϕ(t) =⇒ ċ(t) g,c(t) ≥ φ(t) g Z ,ϕ(t) .
Which implies that length g Z (ϕ(t)) ≤ length g (c(t)) < δ, which is a contradiction. Thus, it only rests to prove that the g-ball B g η (q), for η < δ is homeomorphic to a k q -euclidean ball, where k q is the dimension of the leaf of θ passing through q. This is clear in the ∞-norm . g,∞ since in this norm

B g,∞ η (q) = B g Y ,∞ η (q) × B g Z ,∞
η (q). So, the claim for the norm • g also follows and we are done. 4.4. Analytic quasi-transversality. Following remark 4.5, we consider the category GQT whose objects are geometrically quasi-transverse foliated ideal sheaves (M, θ, I, E) and the morphisms are composition of θ-invariant blowings-up of order one, restrictions and leaf-reduction operation (see Lemma 4.4). In this section, we define a sub-category of GQT , which we denote by AQT (and whose objects are said to be analytically quasi-transverse), which satisfies the following condition: an object (M, θ, I, E) whose leaf dimension is one belongs to AQT if, and only if, it is analytically quasi-transverse according to definition 3.1 (c.f. Remark 4.11(2) below). More precisely: Definition 4.10. We denote by AQT the maximal sub-category of GQT whose objects (M, θ, I, E) satisfy the following condition: (1) Note that, if F 1 and F 2 are two sub-categories of GQT whose objects satisfy condition (4.1), then their union F = F 1 ∪ F 2 is also a sub-category whose objects satisfy condition (4.1). This remark, together with Zorn's Lemma, guarantee that AQT is well-defined (that is, that there exists an unique maximal sub-category which is closed by condition (4.1)).

(2) Note that 1-foliated ideal sheaves (M, θ, I, E) satisfying the conditions of definition 3.1 belong to AQT . Indeed, denote by aqt the subset of foliated ideal sheaves satisfying definition 3.1, and by AQT (1) the set of objects in AQT whose leaf dimension is 1. It follows from the definition that AQT (1) ⊂ aqt. Next, it is not difficult to see that the union of aqt with all objects whose leaf dimension is zero, forms a subcategory of GQT (that is, closed under the admissible morphisms, c.f Remark 2.3(2)) and it satisfies (4.1). Therefore aqt ⊂ AQT (1) by maximality of AQT , which implies that these sets are equal. (3) A geometrically quasi-transverse foliated ideal sheaf (M, θ, I, E) may satisfy condition (4.1) and not satisfy the generalized Flow-Box property. Indeed, we consider a modification of the example given in subsection 3.1. Let M = U × K ⊂ K 5 (with coordinate system (x, y, z, w, v)), where U is a sufficiently small open neighborhood of the origin of K 4 where h(x 2 + y 2 ) is well-defined (see Lemma 3.4); we consider θ generated by

∂ 1 = y 2 ∂ ∂w + x ∂ ∂y -y ∂ ∂x , ∂ 2 = ∂ ∂v
and I = (zf (x, y), wg(x, y), v) (see definition of f and g on subsection 3.1). It is easy to see that (M, θ, I, E = ∅) satisfies condition (4.1). Furthermore, following subsection 3.1, we can verify that this example is geometrically quasi-transverse, but does not satisfy the generalized Flow-Box property. Note, nevertheless, that (M, θ, I, E = ∅) is not algebraically quasitransverse, since after a leaf-reduction operation, we recover the example of subsection 3.1.

Although the definition can not be verified without considering blowings-up and leaf-reduction operations (c.f. Remark 4.11(3)) there is at least one interesting class of examples which satisfies these conditions:

Lemma 4.12. Let (M, θ, I, E) be a geometrically quasi-transverse foliated ideal sheaf and suppose that X = V (I) is a regular analytic curve transverse to E. Then (M, θ, I, E) is analytically quasi-transverse.

Proof. Denote by F denote the subset of geometrically quasi-transverse foliated ideal sheaves such that X = V (I) is a regular curve transverse to E. We note that F is a sub-category of GQT since it is closed by θ-invariant blowings-up of order one and leaf-reduction operation. By the maximality of AQT it is enough, therefore, to prove that (4.1) is satisfied. Fix a foliated ideal sheaf (M, I, θ, E) in F and a point p in X = V (I). Without loss of generality, there exists a coordinate system (x x x, y) = (x 1 , . . . , x m-1 , y) centered at p such that I = (x 1 , . . . , x m-1 ) (furthermore, if p ∈ E, we can assume that E = (y = 0)). By the geometric quasi-transversality hypothesis and the fact that θ is tangent to E, we conclude that J := (θ[I] + I) • O p is either equal to O p (and we are done), or E = (y = 0) and J = (x 1 , . . . , x m-1 , y r ) for some r ∈ N. Now, from the fact that θ is tangent to E, we conclude that θ[y r ] ⊂ (y r ), which implies that θ[J ] ⊂ J . Thus

Figure 1 .

 1 Figure 1. In the left, the phase space of Example 1. In the right the phase space of Example 2.

  (a) If X has dimension or codimenstion one, then (M, ∂, X) satisfies the generalized Flow-Box property. (b) If dim M ≥ 4, there exists a geometrically quasi-transverse triple (M, ∂, X)

Figure 2 .

 2 Figure 2. In the picture r0 > 0 is fixed. The orbit ϕ(r0, t) is contained

Lemma 4 . 7 .

 47 The generalized Flox-Box property is independent of the choice of local generators {∂ 1 , . . . , ∂ d } of θ • O p (and, thus, it is independent of the local sub-Riemannian metric).

Lemma 4 . 8 (

 48 Blowing-up reduction). Let σ : ( M , θ, I, E) → (M, θ, I, E) be a θinvariant blowing-up of order one and suppose that (M, θ, I, E) is geometrically quasi-transverse and ( M , θ, I, E) satisfies the generalized Flow-Box property. Then (M, θ, I, E) also satisfies the generalized Flow-Box property.

(4. 1 )

 1 θ 2 [I] ⊂ θ[I] + I Foliated ideal sheaves (M, θ, I, E) which belong to AQT are said to be analytically quasi-transverse. Remark 4.11.

4 . 5 .

 45 θ 2 [I] ⊂ θ[I] + I, which proves that all elements in F satisfy (4.1). Main result. Theorem 4.13. Let (M, θ, I, E) be an analytically quasi-transverse foliated ideal sheaf, then (M, θ, I, E) satisfies the generalized Flow-Box property.Proof of Theorem 1.11. If (M, θ, I, E) is a geometrically quasi-transverse foliated ideal sheaf such that X = V (I) is a regular analytic curve transverse to E, by Lemma 4.12 is is analytically quasi-transverse and by Theorem 4.13 it satisfies the generalized Flow-Box property.Proof of Theorem 4.13: We prove by induction on the dimension of a generic leaf d; the case that d = 0 is trivial. So, assume by induction that the Theorem holds whenever θ has generic leaf dimension smaller or equal to d -1 and fix a singular distribution θ with generic leaf dimension d. Fix a point p in X = V (I) and a relatively compact open neighbourhood M 0 of p. By Proposition 2.4 there exists a sequence of θ-invariant blowings-up of order one:( M , θ, I, E) = (M r , θ r , I r , E r ) • • • (M 0 , θ 0 , I 0 , E 0 ) σ r σ 1 such that (since AQT is closed by θ-admissible blowings-up) θ[ I] + I = O M . SinceAQT is a sub-category, ( M , θ, I, E) is analytically quasi-transverse. By Proposition 4.3, for each point q in the pre-image of p, there exists a decomposition θ = θ Y + θ Z valid in a sufficiently small open neighbourhood U of q and (since AQT is closed by leaf-reduction operation) ( U , θ Z , I| U , E| U ) is analytically quasi-transverse. By the

Acknowledgments

We would like to thank Jean-François Mattei for bringing the problem to our attention and for the very useful discussions on the topic. The second author was supported by ANR-16-CE40-0008. induction hypothesis, ( U , θ Z , I| U , E| U ) satisfies the generalized Flow-Box property. By Proposition 4.9, ( U , θ, I| U , E| U ) also satisfies the generalized Flow-Box property. Since q was arbitrary, ( M , θ, I, E) satisfies the generalized Flow-Box property and, finally, by Lemma 4.8, the foliated ideal sheaf (M 0 , θ 0 , I 0 , E 0 ) satisfies the generalized Flow-Box property. Since the choice of p ∈ X was arbitrary, we have proved the result.


P Vrignat
email: pascal.vrignat@univ-orleans.fr

D Delouche
email: david.delouche@yncrea.fr

M Avila

F Kratz

Industry 4.0: introduction to the continuous improvement of an industrial process for Master's students

Keywords: industry 4.0, Edunet network, Skkynet, project pedagogy, IoT, engineering students. 

Among the many pedagogical strategies for improving students' motivation, life skills, and know-how, project-based learning is often cited as an example. It is in this context that a challenge entitled "New digital processes for the plant of the future" was launched in 2019 for Master's students learning in the field of "mechatronics". Seven teams formed randomly with last year students of the HEI graduate school of engineering, therefore, they took part in this challenge. The technical issues to be addressed were vast because they had to meet certain needs and services dedicated to issues related to the digital industry or industry 4.0. This paper details the pedagogical strategy that has been adopted, the technical specifications and the planning to be respected. It also proposes some significant results. A measurement tool concerning the efficiency of the method has also been implemented. According to these results, we refine our feedback on the challenge and thus refine our educational policy.

INTRODUCTION

Among the many pedagogical strategies for improving students' motivation, life skills, and know-how, project-based learning has often been cited as an example for several decades [START_REF] Bavota | Teaching software engineering and software project management: An integrated and practical approach[END_REF], [START_REF] Blumenfeld | Motivating project-based learning: Sustaining the doing, supporting the learning[END_REF], [START_REF] Reverdy | Des projets pour mieux apprendre ?[END_REF]. One of the essential characteristics of projects or challenges dedicated to students is their relatively long duration, which is in all cases longer than the usual pedagogical activities. This duration requires consistency in action and reflection, with organizational management of the group or working groups. However, it is necessary to be vigilant and anticipate certain actions in order not to fall into certain excesses [START_REF] Blumenfeld | Motivating project-based learning: Sustaining the doing, supporting the learning[END_REF], Fig. 1). In our department at the University of Orléans, combined with others (HEI campus Centre, INSA Centre Val de Loire), we are active in pedagogic innovation according to [START_REF] Bégot | Une possible réponse ludique pour les processus industriels communicants[END_REF], [START_REF] Vrignat | A technology shift for a fireworks controller, 2 nd International Congress in Education, Innovation and Learning Technologies[END_REF], [START_REF] Vrignat | Suivi d'une campagne d'irrigation par un enrouleur d'arrosage via un SmartPhone[END_REF]. A project is characterized by a duration whose end is known in advance. Between the beginning and the end of the project, there is a learning process in Midler (2012) described as "an irreversible dynamic in which we move from a situation where we know nothing or little but everything is possible, to another where, on the contrary, the level of knowledge has reached its maximum but where all the room for maneuver has been used". Fig. 2 shows a possible representation of this situation. We used a modern measurement tool to try to reproduce this feedback on pedagogical experience (Kahoot1 ). Section 5 of this paper is devoted to a Benchmark on this subject. Fig. 2. Adapted representation of the dynamics of the socalled "project" situation in [START_REF] Midler | L'auto qui n'existait pas, Management des projets et transformation de l'entreprise[END_REF].

As teachers, we have noticed that student enthusiasm depends on the study subject. This subject needs to be at the same time motivating, attractive, topical, playful and demonstrative. The theme chosen for this experiment was de facto carried out following industrial challenges according to, [START_REF] Bartodziej | The Concept Industry 4.0: An Empirical Analysis of Technologies and Applications in Production Logistics[END_REF], [START_REF] Champaney | Pas d'industrie du Futur sans coopération entre industriels et enseignement supérieur[END_REF], [START_REF] Lluansi | Croire en l'Industrie du futur et au futur de l'industrie[END_REF]. Higher technical, technological and scientific education already provides some answers by offering appropriate training on many topics related to the industry 4.0. This paper is composed of six sections. After this introduction, section 2 presents the technical context of the challenge associated with 2 industrial partners. Section 3 describes the pedagogical approach to work with students. Section 4 presents a range of significant results obtained by different groups. Section 5 proposes an educational Benchmark obtained at the end of the challenge. We end with a conclusion.

TECHNICAL CONTEXT OF THE CHALLENGE

The proposed work consisted of setting up an operational structure showing the foundations of a communicating industrial architecture in a process of monitoring and controlling production. It provides many numbers of additional information compared to the papers in [START_REF] Chemla | Programmation objet pour une domotique réalisée avec un API, 10 ème Colloque sur l'Enseignement des Technologies et des Sciences de l'Information et des Systèmes[END_REF], [START_REF] Endi | Three-layer PLC/SCADA system architecture in process automation and data monitoring[END_REF] and [START_REF] Vrignat | Développement et utilisation d'applications complémentaires dans une approche SCADA: retour sur expérience avec des étudiants Bac+ 2 et Bac+ 5, CETSIS[END_REF]. Based on the information from the process studied, the different groups had to develop and use several clients in a SCADA2 approach in [START_REF] Riera | L'utilisation pédagogique et l'enseignement des TIC dans les automatismes[END_REF]. It was necessary to show in this work several solutions to be implemented to carry out a wide range of tests. To do this, the data had to be centralized in an OPC DA3 server and made available to different OPC clients (Fig. 3). The entire solution was also intended to make it possible to track a certain number of variable contents (Items4 ) associated with the quality of the products manufactured "online". The states of proper functioning and dysfunction of the system also needed to be the subject of a specific work. This work was to be based on the work presented in [START_REF] Vrignat | Des exemples de brique technologiques dans le vadre d'une application pour l'industrie du furur[END_REF]. Sensitive productionrelated information had to be sent to a recipient through two transmission strategies:

 SMS with an associated message on a mobile phone (maintenance or production manager).  Mail with an associated message on a mailbox (maintenance or production manager). Fig. 7 presents in detail the different technical constraints that the students had to manage. The latter had to extract relevant information about the process (sensor information, button information, pre-actuator information, safety element information, production information, etc.) without disrupting the operation of the system. The process is described in Fig. 3.

Fig. 3. The production system

This part integrates many primary functions that can be found on an industrial process:

 temperature regulation with different manufacturing recipes,  hydraulic injection press, supplying polymer balls to mold (4 cavities),  controlling the quality of the tokens thus produced via a camera (Fig. 3). All this is automated with different Siemens brand products PLC (Industrial Programmable Logic Controller), operator panel, pre-actuators, etc.). Fig. 4 presents in detail the functional set for the Ecolinject part. This will be the core of the work support for the challenge. The PLC used is a PLC from the Phoenix Contact manufacturer (ILC 151 GSM/GPRS). The proposed platform was to be able to process on-demand the information desired by the various customers (Fig. 5). Two software tools have been installed on the students' personal computers for this purpose. These software platforms were:

 AutomationWorx 2018,  Cogent Data-Hub V9, an office automation suite (Office or other (> 2010)).

The hardware solutions used had to fit well into a context where digital wireless communications are extending their influence according to [START_REF] Loubert | Wireless communication system, Google Patents[END_REF]. With the help of the international university network EduNet 5 , Phoenix Contact encourages exchange and collaboration between various universities working with skills in the field of process automation as well as in Research and Development.

Nowaday, this international network of contacts and skills includes 110 high schools and universities in 30 countries (Fig. 5). It is within this framework that the partner training structures design their courses using Phoenix Contact equipment offered by this convention. In exchange, they must make their educational materials and experiences available to the community registered on the EduNet Internet platform. Also, "Phoenix Contact" laboratories are collaboratively initiated within Universities, thus facilitating the entry of students into the professional world. In July 2017, the Orléans University and the IUT de l'Indre signed the first agreement for France with this network in Phoenix Contact, (2017). This collaboration since then has enabled us to integrate new technologies for the challenge with negotiated hardware and software purchase rates (Fig. 8). A software platform (PC Worx, Fig. 6) is available to configure, program and diagnose the PLC (ILC 151 GSM/GPRS). The 2 nd collaboration allowed us to integrate an international software publisher: Skkynet (https://skkynet.com). We have been working for 3 years with this software publisher and an environment called Cogent DataHub. Cogent Data-Hub (Fig. 7) is a unique tool that allows you to connect, concentrate, integrate and redistribute data in real-time between different devices (PLCs...) and users (supervision applications, database...). The publisher agreed to provide the necessary licenses for the challenge so that students can use this platform on their personal computers or school computers (Fig. 8). The proposed challenge was submitted to apprentice engineering students in the 5 th year (HEI campus Centre). This challenge was compulsory for the entire student's promotion. In recent years, we have all seen at the University that pedagogical modalities and strategies must evolve in response to new learners. It is in this context that we have radically modified the pedagogical strategy adopted to date for this type of teaching, which concerns a module called "Supervision of industrial processes" (Courses, Tutorials).

The objective was then to work from a strategy called "project-based learning", validating skills and high levels of individual and collective involvement. It was natural to activate this challenge with 7 groups of students (4 people). These seven groups formed random and competed at the same time, over a limited and supervised period of 35 hours. The distribution of the different tasks to be carried out was the responsibility of a leader. 7 hours and 30 minutes were devoted upstream to provide the theoretical elements. All the necessary resources were available on the learning platform: technical documents, examples of programs to start, video presentation of the challenge... Both material resources made available were also identical for all groups. At the end of the challenge, each group had to present their work in English in a "My project in 180 seconds" format. A video game presenting the material and the different services to be reached was also the subject of available resources. Fig. 9 provides access to this resource called "La cagnotte de Pascal". At the end of each half-day of work, a quiz with the Kahoot environment was proposed to all students to measure their level of motivation, involvement, coordination, and skills. Finally, a complementary pedagogical strategy has also been implemented based on the so-called "active" pedagogy in [START_REF] Lin | Fostering creativity through education-a conceptual framework of creative pedagogy[END_REF]. The purpose of this strategy was to validate the ability of some students to provide information to others (leader to other participants in the group, Fig. 10,(a) and (b)).

Given the many tasks to be handled, this strategy also made it possible not to mobilize all human resources at the same time with the teacher. The specific knowledge contributions on the different software environments were disseminated only to leaders (Fig. 10,(a)). The leaders then had to give back, according to the needs, the knowledge they had previously acquired (Fig. 10,(b)). Very often, when faced with a problem that has arisen and needs to be solved, our reasoning process proposes the solution(s) too quickly before intellectually contextualizing the subject or exercise to be addressed. This reflex is often present to reassure us, an intellectual subterfuge not to leave our circle of trust. We have all experienced this. Sometimes, the exercise to be treated is ready for a very quick reaction and action. It is necessary to react as well as possible at the moment... Nevertheless, and very often, many problems can and must be contextualized with the use of formalization tools (functional blocks, Boolean equation, flowchart, structured analysis and design technics, algorithm...). Before any operational intervention, the various groups had to consult each other to bring out the essential data issued by the system (without modifying either the program stored in the PLC and the operator panel, or the wiring of the electrical cabinet). The majority of the groups wanted to be able to process the following information:

 Management of on and off modes: system on or off, emergency stop, electrical protection of the conveyor belt disconnection, opening/closing of the protective cover, jamming, de-stemming, thermal protection of the heating rings present before the mold.  Management of the manufacturing recipe: opening and closing the mold, position of the hydraulic cylinder, information on the quality of the tokens, the number of tokens produced. To carry out this subject, Fig. 11 describe the concatenation work that has been carried out in global concertation between the different groups. The technical documents of the system manufacturer had to be studied with the utmost seriousness. As we have mentioned, each group had all the resources to carry out the challenge. Fig. 13 shows a board being assembled: (a), the PLC and its SIM card (subscription to be chosen from a telephone access provider); (b), the beginning of wiring and programming for a group. Fig. 13. Electrical connection constraints Fig. 14 (a) presents the result for all the necessary interconnections. The students were particularly aware of the constraints that can be associated with electrical wiring. Each group has also managed the obligation to check the first conformity checks that must be carried out before starting such an installation (Fig. 14 (b)). These conformities concern electrical potentials, electrical safety, input states (detectors, sensors), the initial state of the operating part, the initial state of the program in the PLC...

Fig. 14. Global interconnections and test phase

The results obtained were of very good quality. The result for a group is presented in Fig. 15 and Fig. 16. The video had to be made with a single shot including the various demonstrations. The students could carry out as many tests as they wanted beforehand. Fig.

15

. An example of results https://youtu.be/RTw6IUp57JY Fig. 16. A 180-second presentation of the project for a group https://youtu.be/9gwX7VarkfM

EDUCATIONAL BENCHMARK

The first very positive feedback from this challenge concerns the technical results obtained by the 7 groups. Not specialists in the field and discovering during this challenge all the hardware and software, the students were particularly efficient and organized. Kahoot was used in our case to (five surveys were proposed over the challenge period):

 Revitalize groups in the restart... or sleep phase!  Establish a diagnosis of the students' work and motivation.  Evaluate the challenge.  Evaluate our work (teachers). These statistical results from the various surveys correspond to response criteria between 1 and 4. The ability to act on the project was important at the start of the challenge (3.4); mobilization of working groups, appropriation of software, hardware ... This mobilization has declined over time. The technical locks were opened step by step. The results established by the tool are presented in Fig 17 . We can indeed see an increase in the degree of motivation between the beginning and end of the challenge. This degree was at 2.8 at the beginning and ends at 3.6 (out of 4 (maximum score)). When asked about their level of prior knowledge, the results obtained are significant (1.14 at the beginning of the challenge and 3.8 at the end). The students greatly increased their knowledge and skills after the challenge. Fig. 17. Evolution of skills and motivations 6. CONCLUSION Engineering schools have been adept at challenges for many years (internal, inter-school, national and even international challenges). This challenge initiated for the first time in this training is a complete success in several respects: student involvement, the involvement of external professionals, results obtained, student motivation, a significant increase of individual and collective skills... We have shown that socalled "project-based" and "active" pedagogies can be a very good teaching strategy. But as we have shown, much preliminary work must be done by the teacher. Institutional partners: Orléans University, HEI campus Centre, IUT de l'Indre. Industrial partners: Phoenix Contact, EduNet, Skkynet.

Fig. 1 .

 1 Fig. 1. Possible deviations from the pedagogy by project.

Fig. 4 .

 4 Fig. 4. The token making systemhttps://youtu.be/NCw1b8vUljQ

Fig. 5 .

 5 Fig. 5. Edunet network

Fig. 9 .

 9 Fig. 9. A fun demonstrator to start the challenge (https://youtu.be/2Q6VYkoxn8I)

Fig

  Fig. 10. Reverse classroom 4. SIGNIFICANT RESULTS OBTAINED BY DIFFERENT GROUPS

Fig. 11 .

 11 Fig. 11. Electrical connection constraints

https://kahoot.com/. Kahoot is an online quiz system that is easy to use and challenging for participants.

SCADA: Supervisory Control and Data Acquisition. The objective of a SCADA environment is to conduct reactive process management. A SCADA environment consists of 4 subsets: the acquisition function, the control function, the monitoring function, and the supervision function.

The OPC Data Access Specification is the first of a group of specifications known as conventional OPC specifications. OPC Data Access is a group of client-server standards that provide specifications for real-time data communication from data acquisition devices such as Industrial Programmable Logic Controllers (PLCs) to display and interface devices such as Human Machine Interfaces (HMIs), SCADA systems and ERP/MES systems.

Item: designation of an object associated with a variable that can be processed by the OPC server and potential clients.