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Primitive arcs on curves.

1 Introduction

1.1 Let k be a field. A test-ring A (or (A,[m.sub.A])) is a local k-algebra, whose maximal ideal [m.sub.A] is nilpotent with residue field A/[m.sub.A] [congruent to] k. A primitive arc [gamma] of a k-curve X at x [member of] X(k) is a primitive k-parametrization [O.sub.X,x] [right arrow] k[[T]] (see definition 3.1), which satisfies the following property: For every test-ring (A,[m.sub.A]), for every commutative diagram of morphisms of local k-algebras

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

where [r.sub.A] : A[[T]] [right arrow] k[[T]] is the continuous morphism of complete local k-algebras defined by T [??] T with kernel [m.sub.A], there exists a unique power series [p.sub.A] [member of] [m.sub.A][[T]] which induces a continuous morphism of complete local k-algebras [p.sup.#.sub.A] : k[[T]] [right arrow] A[[T]] that verifies the formula [[gamma].sub.A] = [p.sup.#.sub.A] [omicron] [gamma]. If it exists, a primitive arc is unique (up to isomorphism).

1.2 The basic subject of this article could be summarized by the following question:

Question 1.1. Which class of pointed k-curves admits primitive arcs?

This article provides a complete answer to question 1.1 for analytically irreducible curves. Precisely, we establish various criterions for the existence of primitive arcs on k-curves. In this way, the existence of primitive arcs can be interpreted as an original criterion of local smoothness (for k-curves) in terms of the associated arc schemes, or as a criterion for determining when the normalization morphism induces an isomorphism at the level of the involved arc schemes. If X is a k-curve and x [member of] X(k), recall that the point x can be viewed as a constant arc of the associated arc scheme [L.sub.[infinity]](X), and we denote by [L.sub.[infinity]][(X).sub.x] the formal neighborhood of the arc x in [L.sub.[infinity]](X), i.e., the formal k-scheme Spf([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]).

Theorem 1.1. Let k be an algebraically closed field. Let X be a k-curve which is unibranch at x [member of] X(k). Then the following assertions are equivalent:

1. The k-curve X is smooth at x;

2. There exists a primitive arc [gamma] at x on X;

3. The formal k-scheme [L.sub.[infinity]][(X).sub.x] is isomorphic to Spf(k[[[([T.sub.i]).sub.i[member of]N]]]);

4. The normalization [pi] : [bar.X] [right arrow] X induces, at the level of formal neighborhoods of the associated arc schemes, an isomorphism of formal k-schemes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [bar.x] [member of] [bar.X] (k) is the lifting of x;

5. The morphism [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is surjective.

1.3 The point of view of formal neighborhoods of arc schemes has been introduced in [8] (see also [6]). If V is a variety, the formal neighborhood [L.sub.[infinity]][(V).sub.[gamma]] parametrizes the formal deformations of the arc [gamma] in [L.sub.[infinity]](V). In [8, 6] (see also [3] for an analog statement in the context of formal geometry), the authors prove a structure theorem for formal neighborhoods of arc schemes at non-constant arcs, which are not contained in the singular locus of the involved variety. The interpretation of such a result in terms of singularity theory remains a challenging problem, and works [3, 4, 5] are, to the best of our knowledge, the first steps in this direction. Let us also mention [7, 12] where some properties of formal neighborhoods of arc schemes are also studied in other frameworks.

Contrary to these works, the involved arcs in our statement are constant; hence, our result provides information for arcs contained in the singular locus. (Let us note that in general the main theorem of [8, 6] does not hold for singular constant arcs, see [2] for counter-examples). Roughly speaking, the present work (see assertion (3) of theorem 1.1) investigates the study of the smoothness of an analytically irreducible k-curve X at a point x from the point of view of the "deformations" of the constant arc x in the associated arc scheme [L.sub.[infinity]](X). In this context, the notion of rigidity (i.e., situation where there is no non-trivial deformation) corresponds to the existence of a primitive arc.

2 Preliminaries

2.1 Let k be a field. A k-variety is a k-scheme of finite type. A k-curve is a reduced k-variety of dimension 1. We say that a pointed curve (X, x), with x [member of] X(k), is unibranch (or analytically irreducible) at x if the ring [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is a domain.

2.2 Let k be a field. Let V be a k-variety. The functor S [??] Homk(S[[??].sub.k]k[[T]],V) defined from the category of k-schemes to that of sets is representable by a k-scheme [L.sub.[infinity]](V). (Let us note that this presentation uses a recent non-trivial result due to B. Bhatt, see [1, Theorem 1.1]). If V is an affine k-variety, for every k-algebra A, every element [[gamma].sub.A] [member of] [L.sub.[infinity]](V)(A) coincides with the datum of a morphism of k-algebras O(V) [right arrow] A[[T]].

2.3 Let V be a k-variety and [gamma] [member of] [L.sub.[infinity]](V)(k). Yoneda's lemma [9, 8.1.4] and the properties of completion formally imply that the formal neighborhood [L.sub.[infinity]][(V).sub.[gamma]] of the k-scheme [L.sub.[infinity]](V) at g is completely determined by the functor of points

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

when A runs over the category of test-rings, and where the considered morphisms are the continuous morphisms of complete local k-algebras from [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] to A. See [8, 6] or also [3].

3 The proof of theorem 1.1

Definition 3.1. Let k be a field. Let X be a k-curve with x [member of] X(k). A primitive k-parametrization of X at x is a morphism of local k-algebras [gamma]: [O.sub.X,x] [right arrow] k[[T]], which satisfies the following property: For every morphism [gamma]': [O.sub.X,x] [right arrow] k[[T]] of local k-algebras, there exists a power series [p.sub.k] [member of] Tk[[T]] such that we have [gamma]' = [p.sup.#.sub.k] [omicron] [gamma].

If k is algebraically closed and X is unibranch at x, the normalization [pi]: [bar.X] [right arrow] X of X provides a primitive k-parametrization of X at x by considering the induced morphism of local k-algebras [[pi].sub.x] : [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Remark 3.1. A primitive k-parametrization may not be a primitive arc. Let X be the affine plane k-curve defined by the datum of the polynomial F = [T.sup.3.sub.1] - [T.sup.2.sub.2] [member of] C[[T.sub.1], [T.sub.2]]. Let us consider the primitive k-parametrization [gamma] at the origin o in X = Spec(C[[T.sub.1], [T.sub.2]]/<[T.sup.3.sub.1] - [T.sup.2.sub.2]>) defined by the element ([T.sup.2], [T.sup.3]) [member of] C[[T]]. Let A := C[S]/<[S.sup.2]>. We observe that the element [[gamma].sub.A] [member of] [L.sub.[infinity]][(X).sub.o](A) given by [T.sub.1] [??] S, [T.sub.2] [??] S, can not be written under the form [gamma] [omicron] [p.sub.A]. So, [gamma] is not a primitive arc.

Let us mention that implication 4 [??] 5 is obvious, and that implication 4 [??] 1 also is obvious since we have [L.sub.[infinity]]([bar.X]) [bar.x] [congruent to] Spf(k[[[([T.sub.i]).sub.i[member of]N]]]). Let us prove the other implications.

1 [??] 2 Since X is smooth at x, there exists an affine open subscheme U of X, which contains x, endowed with an etale morphism of k-schemes U [right arrow] [A.sup.1.sub.k] = Spec(k[t]), corresponding to the choice of a local parameter in O(U) (i.e., a generator t of the maximal ideal [m.sub.x] in the ring [O.sub.X,x]). Up to shrinking X, we may assume that X = U. Then, let [gamma] be the arc corresponding to the following morphism of k-schemes:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

obtained by composition via the completionmorphism. It gives rise to a primitive k-parametrization of X at x. Then, it is easy to check that the arc [gamma] is primitive, since, in this case, for every test-ring A, and every [[gamma].sub.A] [member of] [L.sub.[infinity]][(X).sub.x](A), we take [p.sub.A] = [[gamma].sub.A].

2 [??] 4 Let g be a primitive arc at x on the curve X. Let A be a test-ring. By [section]2.3, we only have to prove that the map:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is a bijection. Let [[gamma].sub.A] [member of] [L.sub.[infinity]](X)x(A). By assumption, there exists a unique power series [p.sub.A] [member of] [m.sub.A][[T]] such that [[gamma].sub.A] = [gamma] [omicron] [p.sub.A] (where we identify [p.sub.A] and the induced morphisms of k-schemes). Since the morphism [pi] is proper and birational, the valuative criterion of properness implies the existence of a unique non-constant arc [bar.[gamma]] [member of] [L.sub.[infinity]]([bar.X])(k) such that [pi] [omicron] [bar.[gamma]] = [gamma]. Then, we easily observe that [bar.[gamma]] [omicron] [p.sub.A] is the unique preimage by [p.sub.A] of [[gamma].sub.A].

5[??] 1We assume that the morphism [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is surjective. Then, for every test-ring A, the induced map:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is injective. We are going to show that this property implies the smoothness of X at x. Let us denote by [mult.sub.x](X) the integer defined as follows. If [gamma] is a primitive k-parametrization at x, let us consider the ideal [gamma]([m.sub.x]) in the ring [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (t is here a generator of the maximal ideal [m.sub.[bar.x]] of the ring O[bar.X] , [bar.x]). There exists an integer n such that [gamma]([m.sub.x]) = <[t.sup.n]>. We then set n =: [mult.sub.x](X). This definition does not depend on the choice of g. If the k-curve X is singular at x, we have [mult.sub.x](X) [greater than or equal to] 2.

Let us assume that x is a singular point of X. Up to shrinking X, we may assume that X is affine, embedded in [A.sup.N.sub.k] = Spec(k[[T.sub.1] ,..., [T.sub.N]]), and, up to a translation, we may assume that x is the origin o of [A.sup.N.sub.k]. Let A := k[U]/<[U.sup.2]>. The power series [[phi].sub.1] = 0 [member of] A[[T]] and [[phi].sub.2] = UT [member of] A[[T]] define two elements of [L.sub.[infinity]]([bar.X]) [bar.x](A). It follows from the definition that

[L.sub.[infinity]](p)([[phi].sub.1]) = [L.sub.[infinity]]([pi])([[phi].sub.2]). (3)

since [mult.sub.x](X) > 1. Indeed, every variable [T.sub.i] (seen in the ring [O.sub.X,x]) is sent by the morphism of local k-algebras [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] to an element in the ideal [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where t is a generator of the ideal m[bar.x]. So, we obtain formula (3) since [L.sub.[infinity]]([pi])([[phi].sub.i]) corresponds, for every integer i [member of] {1, 2}, to the following composition of morphisms of local k-algebras:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The injectivity of the map [[pi].sub.A] then implies that [[phi].sub.1] = [[phi].sub.2]. That is a contradiction, which concludes the proof.

3 [??] 1 It is sufficient to prove that the ring [O.sub.X,x] is formally smooth for the mx-adic topology thanks to [11, 17.5.3]. By [10, 19.3.3,19.3.6],we observe that, due to our assumption, the ring [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is formally smooth, and we conclude by the existence of the following diagram of continuous morphisms of local k-algebras:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Example 3.1. Keep the notation of remark 3.1. In this case, the normalization morphism [[pi].sup.#] : O(X) [right arrow] k[T] is defined by [T.sub.1] [??] [T.sup.2], [T.sub.2] [??] [T.sup.3]; hence, every A-deformation of the origin in [bar.X] is sent to the origin in X. We easily conclude that the deformation (S, S) of the origin in X does not lift to the normalization [bar.X].

Remark 3.2. By base change, we observe that theorem 1.1 can be generalized to every geometrically unibranch integral curve X and any closed point x [member of] X.

Remark 3.3. Keep the notation and assumptions of theorem 1.1. It is not hard to prove that the morphism [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is a formal invariant of the curve singularity (X, x). By this way, formal neighborhoods at constant arcs in arc scheme provide new formal invariants of curve singularities. It would be interesting to study these invariants with respect to the classical theory of singularities.

Acknowledgement. We would like to thank the referee for his comments and David Bourqui for pointing out the current argument used in the proof of implication 3 [??] 1 to us.

References

[1] B. Bhatt, Algebraization and tannaka duality, Preprint.

[2] D. Bourqui and J. Sebag, The Drinfeld-Grinberg-Kazhdan theorem is false for singular arcs, To appear in Journal of IMJ (DOI: http://dx.doi.org/10.1017/S1474748015000341).

[3] --, Drinfeld-Grinberg-Kazhdan's theorem for formal schemes and singularity theory, Preprint (submitted, 2015).

[4] --, Formalminimalmodels of plane curve singularities, Preprint (submitted, 2016).

[5] --, Smooth arcs on algebraic varieties, Preprint (submitted, 2016).

[6] V. Drinfled, On the Grinberg-Kazhdan formal arc theorem, Preprint.

[7] L. Ein and M. Mustata, Generically finite morphisms and formal neighborhoods of arcs, Geom. Dedicata 139 (2009), 331-335.

[8] M. Grinberg and D. Kazhdan, Versal deformations of formal arcs, Geom. Funct. Anal. 10 (2000), no. 3, 543-555.

[9] A. Grothendieck, Elements de geometrie algebrique. III. Etude cohomologique des faisceaux coherents. I, Inst. Hautes Etudes Sci. Publ. Math. (1961), no. 11, 167.

[10] --, Elements de geometrie algebrique. IV. Etude locale des schemas et des morphismes de schemas. I, Inst. Hautes Etudes Sci. Publ. Math. (1964), no. 20, 259.

[11] --, Elements de geometrie algebrique. IV. Etude locale des schemas et des morphismes de schemas IV, Inst. Hautes Etudes Sci. Publ.Math. (1967), no. 32, 361.

[12] A. J. Reguera, Towards the singular locus of the space of arcs, Amer. J. Math. 131 (2009), no. 2, 313-350.

Received by the editors in July 2015--In revised form in March 2016.

Communicated by S. Caenepeel.

2010 Mathematics Subject Classification : 14E18,14B05.

Institut de recherche mathematique de Rennes

UMR 6625 du CNRS

Universite de Rennes 1

Campus de Beaulieu

35042 Rennes cedex (France)

email: julien.sebag@univ-rennes1.fr
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Author:Sebag, Julien
Publication:Bulletin of the Belgian Mathematical Society - Simon Stevin
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Date:Oct 1, 2016
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