B Pasquier

N Perrin

Elliptic curves on some homogeneous spaces

Keywords: Elliptic curves, minuscule homogeneous spaces, adjoint homogeneous spaces. Mathematics Subject Classification: 14M15, 14N35

Let X be a minuscule homogeneous space, an odd quadric, or an adjoint homogenous space of type different from A and G 2 . Le C be an elliptic curve. In this paper, we prove that for d large enough, the scheme of degree d morphisms from C to X is irreducible, giving an explicit lower bound for d which is optimal in many cases.

Introduction

In this paper we study the scheme Hom d (C, X) of degree d morphisms from an elliptic curve C to a rational homogeneous space X. Specifically we will assume that X is either a minuscule homogeneous space or an adjoint homogeneous space. Minuscule homogeneous spaces are natural generalisation of Grassmann varieties (see Table 1). Adjoint homogeneous space are also called quasi-minuscule and are obtained as the unique closed orbits in Pg under the adjoint action. These varieties are also called minimal nilpotent orbit (see Table 2).

We prove that the scheme Hom d (C, X) is irreducible as soon as d is large enough and we give an explicit bound d(X) for d (see Tables 1 and2). Theorem 0.1 Let C be a smooth elliptic curve and let X be a minuscule or an adjoint homogeneous space. If X is of adjoint type, assume furthermore that the group G is not of type A or G 2 . For d ≥ d(X), the scheme of morphisms Hom d (C, X) is irreducible of dimension c 1 (X)d.

In many cases, we also prove that the bound d(X) is optimal in the following sense: for d < d(X), the scheme Hom d (C, X) has dimension strictly bigger than the expected dimension c 1 (X)d. Note that it may happen that for d < d(X) the scheme Hom d (C, X) is irreducible. This is the case for all d when X is isomorphic to the maximal isotropic Grassmann variety G Q (n, 2n) (see Table 1 and Corollary 3.9). Remark 0.2 (ı) Note that B. Kim and R. Pandharipande proved the connectedness of the moduli space of stable maps M d,g (X) to any homogeneous space and for any genus. However, this space is almost never irreducible since there will be irreducible components of unexpected dimension (see Proposition 4.2).

(ıı) Note also that for d = 2, the scheme of morphisms Hom d (C, X) to any homogeneous space with Picard rank 1 is irreducible of dimension c 1 (X) + dim X + 1. Indeed, any degree 2 morphism factors through a line.

1 Minuscule and adjoint homogeneous spaces are of the form G/P for G a semisimple algebraic group. Except for the adjoint homogeneous space in type A, the parabolic subgroup P is a maximal parabolic subgroup. This is the reason why we will assume that if X is of adjoint type the group G is not of type A. Recall that the Dynkin diagram of the group G has vertices indexed by simple roots. In the following tables, we give the list of all minuscule homogeneous spaces. Note that we also include odd dimensional quadrics which are not minuscule but cominuscule. The above statement is still true for odd dimensional quadrics.

T ype

V ariety Diagram Dimension Index d(X)

A n-1 G(k, n) • • • • • • • k(n -k) n n B n Q 2n-1 • • • • • • • 2n -1 2n -1 3 D n Q 2n-2 • • • • • • • • b b " " 2n -2 2n -2 3 D n G Q (n, 2n) • • • • • • • b b " " n(n-1) 2 2n -2 n -1 E 6 OP 2 • • • • • • • 16 12 3 E 7 E 7 /P 7 • • • • • • • • 27 18 8
Table 1. Minuscule homogeneous spaces and odd quadrics.

In the following table, we give the list of all adjoint homogeneous spaces for a group of type different from A. In this table we also include the adjoint variety of type G 2 even if we have no result for this variety.

T ype

V ariety Diagram Dimension Index d(X)

B n G Q (2, 2n + 1) • • • • • • • 4n -5 2n -2 2n
C n P 2n-1 • • • • • • • 2n -1 2n 2 D n G Q (2, 2n) • • • • • • • • b b " " 4n -7 2n -3 2n -1 E 6 E 6 /P 2 • • • • • • • 21 11 9 E 7 E 7 /P 7 • • • • • • • • 33 17 11 E 8 E 8 /P 8 • • • • • • • • 57 29 15
F 4 F 4 /P 1 • • • • • 15 8 8 G 2 G 2 /P 1 • • • 5 3
Table 2. Adjoint homogeneous spaces of Picard number one.

2

In these tables, we followed the notation of [Bou54, Tables] and we depicted the set Σ(P ) of simple roots not in P with plain vertices. The minuscule and adjoint varieties X = G/P are described in the second column. By convention we denote by G(k, n) (resp. G Q (k, n)) the Grassmann variety of k-dimensional subspaces in C n (resp. isotropic k-dimensional subspaces in C n for a non-degenerate quadratic form Q in C n ). For G Q (n, 2n) we only consider one of the two connected components of the above Grassmann variety. We denoted by Q m any smooth m-dimensional quadric. The varieties OP 2 = E 6 /P 1 and E 7 /P 7 are the Cayley plane and the Freudenthal variety. Recall that the index of X = G/P is the integer c 1 (X) such that the anticanonical divisor -K X equals c 1 (X)H where H is an ample generator of the Picard group. The last column gives the bound d(X) of our main theorem.

Remark 0.3 Note that the above statement was already known for Grassmann varieties by results of A. Bruguières [START_REF] Bruguières | The scheme of morphisms from an elliptic curve to a Grassmannian[END_REF] and for quadrics by results of E. Ballico [START_REF] Ballico | On the Hilbert scheme of curves in a smooth quadric[END_REF]. Note also that for orthogonal Grassmann varieties, the above result was obtained by another technique by the second author in the unpublished paper [START_REF] Perrin | Courbes elliptiques sur la variété spinorielle[END_REF]. This case was the very first motivation for our study. It answered a question of D. Markushevitch and has been used in [START_REF] Iliev | Parametrization of sing Θ for a Fano 3-fold of genus 7 by moduli of vector bundles[END_REF].

Let us briefly describe the content of the paper. The strategy of the proof relies on a description given in [START_REF] Perrin | Courbes rationnelles sur les variétés homogènes[END_REF] of a big open cell U in any homogeneous space X as a tower of affine bundles φ : U → Y where Y is again homogeneous under a smaller group. In section 1 we explain how to restrict the study of the irreducibility of Hom d (C, X) to the one of Hom d (C, U ) and we explain, under some conditions on the fibration φ : U → Y , how to deduce the irreducibility of Hom d (C, U ) from the irreducibility of Hom d (C, Y ). In section 1.3, we recall the construction of U and of the the fibration φ : U → Y and describe the tower of affine fibrations. In section 2, we prove some results on the cohomology of the restriction to elliptic curves of equivariant locally free sheaves on projective spaces quadrics. In section 3, we gather all these results to prove Theorem 0.1. In section 4, we include some remarks on the moduli space of stable maps to minuscule or adjoint homogeneous spaces with source an elliptic curve. Finally, in an appendix we deal with of the quadric of dimension 3, which we cannot consider with the same method as all other homogeneous spaces.

Acknowledgments. The second author would like to thank Christian Peskine for enlightening discussions on the paper [START_REF] Gruson | On a theorem of Castelnuovo, and the equations defining space curves[END_REF] and Piotr Achinger for useful comments on Spinor bundles. A The quadric of dimension 3 18

1 Reduction of the theorem

Restriction to an open subset

Let C be a smooth curve, let X be a homogeneous space with Picard number 1 and let α ∈ A 1 (X).

The strategy of the proof will be similar to the one in [START_REF] Perrin | Courbes rationnelles sur les variétés homogènes[END_REF]. We prove that there exists an open subset U of X whose complement is of codimension at least 2 and such that U can be realised as a tower of affine bundles φ : U → Y over a homogeneous space Y of smaller dimension. The next result from [Per02, Proposition 2] proves that the irreducibility of Hom α (C, X) is equivalent to the irreducibility of Hom i * α (C, U ) (where i : U → X is the inclusion morphism). This result is an easy application of Kleiman-Bertini Theorem [START_REF] Kleiman | The transversality of a general translate[END_REF].

Proposition 1.1 Let C be a smooth curve and let X be a homogeneous space under a group G. Assume that U be an open subset of X such that Codim X (X \ U ) ≥ 2 and let i : U → X be the inclusion morphism. For α ∈ A 1 (X), if Hom i * α (C, U ) is irreducible, then so is Hom α (C, X).

We then need to study the open subset U . It will be the open orbit of the action of some parabolic subgroup of G on X. The second author proved in [START_REF] Perrin | Courbes rationnelles sur les variétés homogènes[END_REF] that U can be realised as a tower φ : U → Y of affine bundles over a homogeneous space Y of smaller dimension. However, if this was enough to study rational curves, we need to be more precise if we want to study higher genus curves. If φ : U → Y is an affine bundle associated to a vector bundles E, we explain in the next subsection how a good understanding of the restriction of E to higher genus curves in Y enables to proceed by induction on the dimension.

Passing through affine bundles

Let us fix some notation. In this section X is a homogeneous space of Picard number 1, we denote by U an open subset whose complement Z is of codimension 2 and we assume that U can be realised as a sequence of affine bundles φ : U → Y over a homogeneous space Y of smaller dimension.

Note that because of the inequality Codim X Z ≥ 2, we have Pic(U ) = Pic(X). Furthermore since φ : U → Y is a sequence of affine bundles, then we also have Pic(U ) = Pic(Y ). For f : C → U a morphism, we may define a linear form [f ] on Pic(X) by L → deg(f * L). We shall say that for α ∈ Pic(X) ∨ a curve f : C → U is of class α if [f ] = α. We shall write α, L for the evaluation of α at L ∈ Pic(X). For α such a class in Pic(X) ∨ , we denote by Hom α (C, X) resp. Hom α (C, U ),

Hom α (C, Y ) the scheme of morphisms from f : C → X resp. f : C → U , f : C → Y such that [f ] = α.
Note that for X (or for Y ) we have identifications A 1 (X) ≃ Pic(X) ∨ and A 1 (Y ) ≃ Pic(Y ) ∨ and that these schemes of morphisms are the classical schemes of morphisms as defined in [START_REF] Grothendieck | Techniques de construction et théorèmes d'existence en géométrie algébrique. IV. Les schémas de Hilbert[END_REF].

We now prove a general result on the scheme of morphisms to an affine bundle. For this we introduce the following notation. Let φ : U → Y be an affine bundle whose direction vector bundle is associated to a locally free sheaf E on Y and let α ∈ Pic(U ) ∨ . We denote by H b the locally closed subset of Hom φ * α (C, Y ) defined by

H b = {f ∈ Hom φ * α (C, Y ) / dim H 1 (C, f * E) = b}.
Proposition 1.2 Let φ : U → Y be an affine bundle over a variety Y whose direction vector bundle is associated to a locally free sheaf E. Let α ∈ Pic(U ) ∨ and let C be a smooth curve such that the scheme

Hom φ * α (C, Y ) is irreductible of dimension φ * α, c 1 (T Y ) .
(ı) Assume that there exists f ∈ Hom φ * α (C, Y ) satisfying the vanishing

H 1 (C, f * E) = 0. If for all integer b > 0 we have the inequality Codim H b > b, then Hom α (C, U ) is irreducible of dimension α, c 1 (T U ) .
(ıı) Assume that φ is a vector bundle and let a be the smallest integer such that there exists an element f ∈ Hom φ * α (C, Y ) with dim H 1 (C, f * E) = a. If there exists an integer b > a satisfying the inequality Codim H b ≤ ba, then the scheme Hom α (C, U ) is reductible.

Proof. (ı) Let us consider the natural morphism Hom α (C, U ) → Hom φ * α (C, Y ). As explained in [START_REF] Perrin | Courbes rationnelles sur les variétés homogènes[END_REF] for rational curves, the fiber of this map over an element f ∈ Hom φ * α (C, Y ) is given by the sections s ∈ H 0 (C, f * E). In particular, such a section exists if and only if the affine bundle C × Y U → C obtained by pull-back to C is a vector bundle. For a general element f ∈ Hom φ * α (C, Y ) this is the case thanks to the vanishing

H 1 (C, f * E) = 0.
For b > 0 the fibers over the locally closed subset H b are either empty (if the pull-back is not a vector bundle) or of dimension dim

H 0 (C, f * E) = χ(C, f * E) + dim H 1 (C, f * E) = φ * α, c 1 (E) + b.
The dimension of the inverse image of H b in Hom α (C, U ) is therefore of dimension strictly less than φ * α, c 1 (T Y ) + c 1 (E) = α, c 1 (T U ) which is the expected dimension of Hom α (C, U ). As any irreducible component is at least of this expected dimension (cf. [START_REF] Mori | Projective manifolds with ample tangent bundles[END_REF]) these locally closed subsets do not form irreducible components. Therefore the inverse image of the open subset H 0 is dense in Hom α (C, U ). As explained in [START_REF] Perrin | Courbes rationnelles sur les variétés homogènes[END_REF] for rational curves, this inverse image is an affine bundle over the base and is thus irreducible of dimension φ * α, c 1 (T Y ) + c 1 (E) = α, c 1 (T U ) .

(ıı) In this case, the inverse image of the locally closed subset H a is of dimension α, c 1 (T U ) + a while the inverse image of the locally closed subset H b is of dimension at least α, c 1 (T U ) + a. This second locally closed subset is therefore not contained in the closure of the first one thus they have to be contained in two different irreducible components.

Decomposition of X

The aim of this subsection is to describe a big open cell U in the homogeneous space X and to realise it as a fibration φ : U → Y over a space Y homogeneous under a group of smaller rank. This fibration can be decomposed in general as a sequence of affine bundles (see [START_REF] Perrin | Courbes rationnelles sur les variétés homogènes[END_REF]Proposition 5]). Let us recall this result.

Let T be a maximal torus in G and let B be a Borel subgroup of G containing T . Denote by w 0 the longest element of the Weyl group W of (G, T ). For any subgroup H of G, we set H w 0 := w 0 Hw -1 0 and for any parabolic subgroup P of G containing B, we denote by L P the Levi subgroup of P containing T and by U P the unipotent radical of P . We also denote by U r P ⊂ • • • ⊂ U 1 P the central filtration of U P . For any parabolic subgroup P containing B, we denote by Σ(P ) the set of simple roots which are not roots of P . Finally, for α a simple root, we denote by ι(α) its image via the Weyl involution i.e. ι(α) = -w 0 (α).

Lemma 1.3 (Proposition 5 and 6, [START_REF] Perrin | Courbes rationnelles sur les variétés homogènes[END_REF]) Let P and Q be two parabolic subgroup of G containing B and assume that ι(Σ(P )) ∩ Σ(Q) = ∅. Set U := Q w 0 P/P . Then U is an open subset of G/P with complement in codimension at least 2 and is an

L Q w 0 equivariant fibration φ : U → Y over the flag variety Y = L Q w 0 /(L Q w 0 ∩ P ).
Moreover this fibration decomposes in a sequence of affine bundles whose associated vector bundles are defined over Y and are the L Q w 0 -equivariant vector bundles associated to the

L Q w 0 ∩ P representations U i Q w 0 /U i+1 Q w 0 • (U i Q w 0 ∩ P ).
Remark 1.4 Note that it is easy to determine the weights of T on the representation

U i Q w 0 /U i+1 Q w 0 • (U i
Q w 0 ∩ P ) defining the vector bundle. For example for P and Q maximal parabolic subgroups with Σ(P ) = {α P } and Σ(Q) = {α Q }, then the weights of the above representation are the negative roots α such that the coefficient of α on α P is negative and the coefficient of α on α Q is exactly -i.

Keeping the previous notation we deduce the existence of the following fibrations. In the next result we will use the description of simple roots given in [START_REF] Bourbaki | Groupes et algèbres de Lie[END_REF]Tables] Corollary 1.5 We have the following fibrations.

1. If G is of type A n , if Σ(P ) = {α i } with 2 ≤ i ≤ n and if Σ(Q) = {ι(α i-1 )}, then Y is isomorphic to P n-i+1
and φ : U → Y is the vector bundle associated to the locally free sheaf

(T P n-i+1 (-1)) ⊕i-1 ≃ (Ω n-i P n-i+1 (n -i + 1)) ⊕i-1 . 2. If G is of type B n , if Σ(P ) = {α 1 } and if Σ(Q) = {α n }, then Y is isomorphic to P n-1 and φ : U → Y is
a sequence of affine fibrations whose direction vector bundles are associated to the locally free sheaves O P n-1 (1) and

Ω 1 P n-1 (2). 3. If G is of type B n , if Σ(P ) = {α 2 } and if Σ(Q) = {α 1 }, then Y is isomorphic to Q 2n-3
and φ : U → Y is a vector bundle associated to the locally free sheaf (O Q 2n-3 (1)) ⊥ i.e. to the restriction of the tautological quotient bundle on

P 2n-2 to Q 2n-3 . 4. If G is of type D n , if Σ(P ) = {α 1 } and if Σ(Q) = {α n }, then Y is isomorphic to P n-2
and φ : U → Y is the vector bundle associated to the locally free sheaf

Ω 1 P n-2 (2). 5. If G is of type D n , if Σ(P ) = {α 2 } and if Σ(Q) = {α 1 }, then Y is isomorphic to Q 2n-4
and φ : U → Y is a vector bundle associated to the locally free sheaf (O Q 2n-4 (1)) ⊥ i.e. to the restriction of the tautological quotient bundle on

P 2n-3 to Q 2n-4 . 6. If G is of type D n , if Σ(P ) = {α n } and if Σ(Q) = {ι(α n-1 )}, then Y is isomorphic to P n-1
and φ : U → Y is the vector bundle associated to the locally free sheaf

Ω n-3 P n-1 (n -2). 7. If G is of type E 6 , if Σ(P ) = {α 1 } and if Σ(Q) = {ι(α 6 )} = {α 1 }, then Y is isomorphic to Q 8 and φ : U → Y is the vector bundle associated to the locally free sheaf E = S one of the two spinor bundles on Q 8 . 8. If G is of type E 6 , if Σ(P ) = {α 2 } and if Σ(Q) = {α 1 , α 6 }, then Y is isomorphic to Q 6 and φ : U → Y
is a sequence of affine bundles whose direction vector bundles are associated to the locally free sheaves E = (O Q 6 (1)) ⊥ and E ′ = S ⊕ S ′ where E is the restriction of the tautological quotient bundle of P 7 to Q 6 and S and S ′ are the two spinor bundles on Q

6 . 9. If G is of type E 7 , if Σ(P ) = {α 7 } and if Σ(Q) = {α 2 },
then Y is isomorphic to P 6 and φ : U → Y is a sequence of affine bundles whose direction vector bundles are associated to the locally free sheaves E = Ω 5 P 6 (6) and 12. If G is of type

E ′ = Ω 2 P 6 (3). 10. If G is of type E 7 , if Σ(P ) = {α 2 } and if Σ(Q) = {α 6 }, then Y is isomorphic to Q 8 and φ : U → Y
F 4 , if Σ(P ) = {α 1 } and if Σ(Q) = {α 4 }, then Y is isomorphic to Q 5 and φ : U → Y
is a sequence of affine bundles whose direction vector bundles are associated to the locally free sheaves E = (O Q 5 (1)) ⊥ and E ′ = S where E is the restriction of the tautological quotient bundle of P 6 to Q 5 and S is the spinor bundle on Q 5 .

Proof. According to the previous Lemma, we only need to identify in each case the direction vector bundles of the affine bundles and to notice (see [START_REF] Perrin | Courbes rationnelles sur les variétés homogènes[END_REF] proof of Proposition 5) that the first affine bundle over Y is always a vector bundle.

To identify the vector bundles, we only have to identify the corresponding representations. Indeed, recall from [Per02, Proposition 5] that the direction vector bundles are equivariant and correspond to the

L Q w 0 ∩ P -representations U i Q w 0 /U i+1 Q w 0 • (U i Q w 0 ∩ P ).
We therefore only have to compute their highest weight which is an easy check using [START_REF] Bourbaki | Groupes et algèbres de Lie[END_REF]Tables]. Note that the representations corresponding to the sheaf (O Qn (1)) ⊥ on an n-dimensional quadric is not irreducible as a representation of the semisimple part of L Q w 0 ∩P but is an irreducible non simple representation of L Q w 0 ∩ P . Note also that the above description of vector bundles for the adjoint varieties were given in [START_REF] Chaput | Rationality of some Gromov-Witten varieties and application to quantum K-theory[END_REF].

Rectriction of some homogeneous vector bundles to curves

In this section we gather some known results and some new ones on the geometry of curves in the projective space and in smooth quadrics. As explained in the previous sections, all the affine bundles φ : U → Y we have to consider are homogeneous vector bundles over these varieties. Therefore the main aim of the section is to control the cohomology of the pull-back to curves of some equivariant vector bundles. Even if we only need these results for elliptic curves, we state them for any curve of genus g.

Projective spaces

In this subsection we recall some results on secants to a curve. The following result can be found in [Har77, Proposition IV.3.8 and Theorem IV.3.9].

Proposition 2.1 Let C be a reduced and irreducible curve in P r such that any secant line to C is a multisecant, then C is a line.

We will use this statement via the following corollary.

Corollary 2.2 Let C be a reduced and irreducible curve in P r which is not a line, then there exists a point x in C such that the projection from x is birational from C onto its image.

Let C be a smooth curve of genus g and let f : C → P r be a morphism such that f * O P r (1) is of degree d. We want to prove that the pull back of the homogeneous bundles Ω k P r (k) through f admits nice filtrations. The following lemma can be found in [GLP83, Remark (2) page 498].

Lemma 2.3 Assume that f is locally injective and that f (C) is non degenerate i.e. not contained in an hyperplane. Then, for almost all choices of r -1 points

(x i ) i∈[1,r-1] in C, there exists a filtration 0 = F r+1 ⊂ F r ⊂ • • • ⊂ F 1 = f * Ω 1 P r (1) such that F i /F i+1 = O C (-x i ) for i ∈ [1, r -1] and F r /F r+1 = f * O P r (-1) ⊗ O C ( i x i ). Proof. Let us set M = f * Ω 1 P r (1) and L = f * O P r (1)
. Let us also denote by V ∨ the vector space H 0 (P r , O P r (1)). Because f (C) is non degenerate, we may consider V ∨ as a subspace of H 0 (C, L).

We have an exact sequence 0

→ M → V ∨ ⊗ O C → L → 0.
We proceed by induction on r. Let us choose a point x 1 on C. Let us denote by

W ∨ the intersection V ∨ ∩ H 0 (C, L(-x 1 )) in the vector space H 0 (C, L). If f (x 1 ) is smooth, the map W ∨ ⊗ O C → L(-x 1
) is surjective and we have the commutative diagram:

W ∨ ⊗ O C / / L(-x 1 ) 0 / / M / / V ∨ ⊗ O C / / L / / 0 0 / / O C (-x 1 ) / / O C / / O x 1 / / 0.
The snakes lemma implies that the left most vertical map is surjective and we get the term F 2 of the filtration by taking the kernel of this map which is also the kernel of the map W ∨ ⊗ O C → L(-x 1 ). Replacing V by W and L by L(-x 1 ) we get a morphism f 2 from C to P r-1 and F 2 = f * 2 Ω P r-1 (1). The morphism f 2 is the composition of f with the projection of center x 1 and because, in characteristic zero, general secants are not multisecant (see Corollary 2.2) we have that f 2 is also generically injective for x 1 general. We may apply then the induction hypothesis. For r = 1, the result is true.

Remark 2.4 Note that in the above filtration, all the vector bunndles F i are contructed as tautological subbundles therefore their dual are globally generated.

Corollary 2.5 Assume that f (C) is non degenerate, then for almost all choices of r -1 points

(x i ) i∈[1,r-1] in C, there exists a filtration 0 = G s+1 ⊂ G s ⊂ • • • ⊂ G 1 = f * Ω k P r (k) such that G i /G i+1 = O C (-(x i 1 + • • • + x i k )) with {i 1 , • • • , i k } ⊂ [1, r -1] or G i /G i+1 = f * O P r (-1)⊗ O C (x i 1 + • • • + x i r-k ) with {i 1 , • • • , i r-k } ⊂ [1, r -1].
Proof. We simply take the k-th exterior product of the previous filtration.

Corollary 2.6 Assume that f (C) is non degenerate. Let g be the genus of C and d be the degree of f * O P r (1). If g ≤ rk, then we have the vanishing

H 1 (C, f * Ω k P r (k + 1)) = 0.
Proof. Indeed, for g ≤ rk and for general points (x i ) i∈[1,r-1] on C, we have the vanishing

H 1 (C, O C (x i 1 + • • • + x i r-k )) = 0.
Note that because f (C) is non degenerate, we must have the inequality g ≤ dr ≤ dk and this condition implies the following vanishing

H 1 (C, f * O P r (1) ⊗ O C (-(x i 1 + • • • + x i k ))) = 0.
Let us assume that C is elliptic. Then even if f (C) is degenerate we may obtain results. Indeed, if f (C) is degenerate, there is a minimal linear space H in P r of codimension say a containing f (C) and such that f (C) is non degenerate in H. Furthermore we have the exact sequence 0

→ O a C → f * Ω 1 P r (1) → f * Ω 1 H (1)
→ 0 from which we deduce a filtration of f * Ω 1 P r (1). In particular we get the following result.

Proposition 2.7 Assume that C is elliptic. If f (C) is non degenerate in a linear subspace of codimension a, then we have the equality dim

H 1 (C, f * Ω k P r (k + 1)) = a r-k (with a r-k = 0 if a < r -k).
In particular, if we have the inequality dim H 1 (C, f * Ω k P r (k +1)) > 0, then there exists an integer a ≥ r-k such that dim H 1 (C, f * Ω k P r (k+1)) = a r-k and f (C) in non degenerate in a linear subspace of codimension a.

Proof. Taking the k-th exterior power of the exact sequence 0

→ O a C → f * Ω 1 P r (1) → f * Ω 1 H (1) → 0 and tensoring it by f * O P r (1), we get a filtration of f * Ω k P r (k + 1) by the vector bundles Λ u (O a C ) ⊗ f * Ω v H (v + 1) for u + v = k
and u, v non negative. By Corollary 2.6, the first cohomology groups of these bundles vanish except maybe for

v = 0 or v = r-a. For v = 0 we have Λ u (O a C )⊗f * Ω v H (v+1) = Λ k (O a C ) ⊗ f * O P r (1) and for v = r -a we have Λ u (O a C ) ⊗ f * Ω v H (v + 1) = Λ k+a-r (O a C
). The first cohomology group in the first case vanishes while the first cohomology group in the second case has dimension a r-k . We thus have an inequality dim H 1 (C, f * Ω k P r (k + 1)) ≤ a r-k . For a < rk, we are done. If a ≥ rk, let us consider the identification Ω k P r (k + 1) = (Ω r-k P r (rk)) ∨ . By the above exact sequence, we have a trivial subbundle of rank a of f * Ω 1 P r (1) therefore we have a trivial subbundle of rank a r-k in f * Ω r-k P r (rk) (take the rk exterior power). By duality, there is a trivial quotient bundle of rank a r-k of f * Ω k P r (k + 1).

Lemma 2.8 Let E be a globally generated vector bundle on C such that E has a trivial quotient of rank n, then dim H 1 (C, E) ≥ n.

Proof. By Serre duality we have an isomorphism

H 1 (C, E) ≃ H 0 (C, E ∨ ) ∨ . We therefore need to prove that dim H 0 (C, E ∨ ) ≥ n. But we have a surjection E → O n C giving rise to an injection O n
C → E ∨ and the result follows.

We conclude using the previous lemma and because Ω k P r (k + 1) is globally generated.

Spinor bundles on quadrics

In this section we consider the spinor bundles on quadrics and their restriction to elliptic curves.

Proposition 2.9 Let Q n be a smooth n-dimensional quadric and let E be a spinor bundle on Q n . Let f : C → Q n be a morphism from a smooth irreducible curve to Q n . Then the pull-back f * E is isomorphic to a direct sum O C 2 β ⊕ F where F has no trivial factor if and only if f factors through an isotropic subspace of (linear) dimension [n/2] + 1α with α =    β for n odd, β + 1 for n even and β > 0, 0 for n even and β = 0 and in the last case, the maximal isotropic subspace has to be an element of a fixed component of the two connected components of maximal isotropic subspaces.

Proof. We shall here use the results of G. Ottaviani [START_REF] Ottaviani | Spinor bundles on quadrics[END_REF] on the spinor bundles. Let us write h = ⌈n/2⌉ -1 and p = [n/2] + 1. There is an embedding ϕ :

Q n → G(2 h , 2 h+1 ) such that E = ϕ * K with K the tautological subbundle in the Grassmann variety G(2 h , 2 h+1 ) of 2 h dimensional subspaces in C 2 h+1 .
Lemma 2.10 Let x and y be points on Q n , then we have the equality C 2 h+1 = K ϕ(x) ⊕ K ϕ(y) if and only if x and y are not orthogonal.

Proof. We shall consider the dual assertion: the map

C 2 h+1 → K ∨ ϕ(x) ⊕ K ∨ ϕ(y)
is an isomorphism if and only if x and y are not orthogonal. To test this assertion, we can restrict the bundle K to some subvariety containing both x and y. If x and y are not orthogonal, then we may choose a smooth conic c : P 1 → Q n → G(2 h , 2 h+1 ) passing through x and y. By [Ott88, Theorem 1.4], the pull back c * K ∨ is isomorphic to O P 1 (1) 2 h . The surjectivity condition holds.

Conversely, if x and y are orthogonal, then there exists a maximal isotropic subspace V p (if n is even we can choose V p to be in any of the two conneced components of maximal isotropic subspaces) such that x and y are contained in P(V p ). By [Ott88, Theorems 2.5 and 2.6] the restriction of K to P(V p ) contains a trivial factor (if n is even this is true only for V p in one of the two connected components). Therefore the surjectivity condition does not hold.

Let us return to the proof of the proposition. Remark that f * E has a trivial factor if and only if for all x ∈ C the subspaces K ϕ(x) have a common intersection in C 2 h+1 . This occurs only if for all x and y in C, the points f (x) and f (y) are orthogonal. In particular the linear subspace spanned by f (C) has to be isotropic. Therefore if the curve f (C) is not contained in an isotropic subspaces then f * E has no trivial factor.

Let us assume that f * E has a trivial factor. Then f factors through an isotropic subspace V p . Furthermore by [Ott88, Theorems 2.5 and 2.6], we know that the restriction E P(Vp) is isomorphic to one of the following direct sums (the first case occurs for n odd while the other two cases occur for n even):

p k=0 Ω k P(Vp) (k), p k=0, k odd Ω k P(Vp) (k) or p k=0, k even Ω k P(Vp) (k).
By the previous results on the projective space, we conclude that if f (C) factors through a linear subspace of linear dimension pα and if there is a unique such factor then V p is in the connected component of G Q (p, 2p) corresponding to the last decomposition above. The number of trivial factors in f * E is then 2 β with β and α as above.

For elliptic curves, applying the same techniques as in the case of the projective space, we obtain the following result.

Corollary 2.11 Let Q n be a smooth n-dimensional quadric and let E be a spinor bundle on Q n . Let f : C → Q n be a morphism from a smooth irreducible elliptic curve to Q n . Then we have the equality dim H 1 (C, f * E) = 2 β if and only if f factors through an isotropic subspace of (linear) dimension [n/2] + 1α with α =    β for n odd, β + 1 for n even and β > 0, 0 for n even and β = 0 and in the last case, the maximal isotropic subspace has to be an element of a fixed component of the two connected components of maximal isotropic subspaces.

Irreducibility

We are now in position to prove our main result. Let C be a smooth elliptic curve. We proceed via a case-by-case analysis to finish the proof of Theorem 0.1.

Projective space

Our proof is based on the fact that the irreducibility of the scheme of morphisms to a projective space is well known. Let us first recall this fact. Let X be the projective space P(V ) with V of dimension n.

Fact 3.1 Let d ≥ 2 be an integer, the scheme Hom d (C, X) is irreducible of dimension dn.

Proof. The scheme of morphism is described by simple Brill-Noether data: a line bundle L on C of degree d and a linear map from V ∨ to H 0 (C, L). But for d positive, we have H 1 (C, L) = 0 therefore H 0 (C, L) is of constant dimension d for all L and the result follows from the fact that Jac d (C), the Jacobian of degree d line bundles on C, is irreducible: the variety Hom d (C, P(V )) is a locally trivial bundle over Jac d (C) with fiber P(V ⊗ H 0 (C, L)) over L.

Grassmann varieties

We now deal with Grassmann varieties. Note that in this case the result was proved by A. Bruguières [START_REF] Bruguières | The scheme of morphisms from an elliptic curve to a Grassmannian[END_REF]. We include here a proof that will serve as a model for the other cases. Let X be the Grassmann variety G(p, n) of p-dimensional vector spaces in a vector space of dimension n.

Proposition 3.2 The scheme Hom d (C, X) is irreducible of dimension nd for d ≥ n.

Proof.

Consider the open subset U given by Theorem 1.5. We have a morphism φ : U → Y with Y ≃ P n-p which is the vector bundle associated to the locally free sheaf

E = T P n-p (-1) ⊗ O p-1 P n-p = (Ω n-p-1 P n-p (n -p)) p-1 .
By Proposition 1.1, we only need to prove the irreducibility of the scheme Hom d (C, U ). For this we study the map φ. Let us denote by H a the locally closed subset of Hom d (C, Y ) of maps whose image is contained in a linear subspace of codimension a and not in a smaller linear subspace. The dimension of this locally closed subset is a(np + 1a) + d(np + 1a). Indeed, there is a morphism H a → G(np + 1a, np + 1) sending to a morphism the unique linear subspace of codimension a containing its image. The fibers are isomorphic to Hom d (C, P n-p-a ). Now the inverse image under the map Φ :

Hom d (C, U ) → Hom d (C, Y ) induced by φ of H a is irreducible of dimension dim H a + dim H 0 (C, f * E) for f ∈ H a . By Proposition 2.7 we have dim H 0 (C, f * E) = (d deg(E) + a)(p -1) = d(p -1) + a(p -1). In particular for d ≥ n, the dimension of Φ -1 (H a ) is (d + a)(n -a) = dn -a(d -n) -a 2 <
dn for a > 0. The locally closed subsets Φ -1 (H a ) for a > 0 are therefore of dimension smaller than the dimensions (at least dn) of irreducible components of Hom d (C, U ) thus the irreducible components of Hom d (P 1 , U ) are those of Φ -1 (H 0 ) which is irreducible of dimension dn.

Remark 3.3 For smaller degrees, the scheme Hom d (C, X) is not irreducible (except in degree 2). Its irreducible components were described in [START_REF] Bruguières | The scheme of morphisms from an elliptic curve to a Grassmannian[END_REF].

Quadrics

We shall now deal with smooth quadrics. Note that in this case the irreducibility of the scheme of morphisms was proved by see E. Ballico [START_REF] Ballico | On the Hilbert scheme of curves in a smooth quadric[END_REF] for quadrics of dimension n ≥ 6 for curves of genus g with g ≤ ⌊ n-1 2 ⌋ and for the degree d ≥ 2g -1. Let X be a smooth quadric of dimension n.

Proposition 3.4 Assume that dim X = 3, then the scheme Hom d (C, X) is irreducible of dimension nd for d ≥ 3.

Proof. We proceed as for the Grassmann variety and consider the open subset U given by Theorem 1.5. We have a sequence of affine bundles φ : U → Y where Y ≃ P r with r = ⌊n/2⌋ and the direction vector bundles are associated to the locally free sheaves E = Ω 1 P r (2) for n even (here φ is a vector bundle) and to E = Ω 1 P r (2) and E ′ = O P r (1) for n odd. By Proposition 2.7 the locally free sheaf E satisfies H 1 (C, f * E) = 0 for any map f : C → Y of degree d ≥ 3 as soon as f does not factor through a line. In this latter case we have dim H 1 (C, f * E) = 1 and we conclude by Proposition 1.2. Remark 3.5 (ı) Note that if dim X ≤ 2, then X is a product of projective lines and the result follows from that case. However, if dim X = 3, then in the above proof we get r = 1 and Ω 1 P r (2) = O P r and we do not get the vanishing condition.

(ıı) Note that for d = 2 the scheme Hom d (C, X) is also irreducible but not of the expected dimension since any degree 2 morphism factors through a line. The dimension is 2n + 1.

Even if we cannot prove, with our method, Proposition 3.4 when X is 3-dimensional, the result is still true. Proposition 3.6 Let X = Q 3 be a smooth quadric of dimension 3, then the scheme Hom d (C, X) is irreducible of dimension 3d for d ≥ 3.

We give a proof of this result for d ≥ 4 in the appendix, using a different method. The case where d = 3 is given by the following remark.

Remark 3.7 Any morphism of degree 2 or 3 from an elliptic curve to Q 3 has to factor through a line. This is clear for degree 2, and for degree 3 the morphism factors through a plane. But as Q 3 contains no plane, the image is contained in the intersection of Q 3 with a plane. It has to be a line for degree reasons. In particular the scheme of morphisms Hom d (C, X) is irreducible of dimension 3 + 2d for d = 2 or d = 3.

Maximal orthogonal Grassmann varieties

Let X be a maximal orthogonal Grassmann variety G Q (n, 2n). The following result already appeared in the unpublished work of the second author [START_REF] Perrin | Courbes elliptiques sur la variété spinorielle[END_REF] with a different method.

Proposition 3.8 The scheme Hom

d (C, X) is irreducible of dimension 2(n -1)d for d ≥ n -1.
Proof. The map φ : U → Y is the vector bundle over Y ≃ P n-1 associated to the locally free sheaf E = Λ 2 (T P n-1 (-1)) = Ω n-3 P n-1 (n -2). By Proposition 2.7, the fibers of Φ : Hom d (C, U ) → Hom d (C, Y ) over the locally closed subset H a of morphisms whose image is non-degenerate in a linear subspace of codimension a are of dimension d(n -2) + a 2 . Therefore we have dim p -1 (H a ) = 2d(n -1) + a(na) + a 2ad. This dimension is strictly smaller than the expected dimension 2d(n -1) for a > 0 and d ≥ n -1. The result follows.

Corollary 3.9 For d ∈ [2, n], the scheme Hom d (C, X) is irreducible of dimension 2(n -1)d + 1 2 (n -d)(n -d -1).
Proof. Let f : C → X be a morphism of degree d, then the kernel ker(f ) of f i.e. the intersection of all subspaces f (x) for x ∈ C is of dimension at least nd (see [START_REF] Buch | Quantum cohomology of Grassmannians[END_REF] for other applications of this definition). Indeed, consider the tautological exact sequence

0 → K → O 2n X → Q → 0
on X where K and Q are the tautological subbundle and quotient bundle. On X we have the identification Q ≃ K ∨ . Pulling back this sequence to C via f we get an exact sequence on the curve. Taking cohomology we obtain:

0 → H 0 (C, K) → C 2n → H 1 (C, K) → H 1 (C, K) → C 2n → H 0 (C, K) → 0.
Let us denote by h 0 and h 1 the dimensions of H 0 (C, K) and H 1 (C, K). Since the map f is of degree d in X, the degree of K is -2d and we have the equality h 0 = h 1 -2d. We deduce the inequality h 0 ≥ nd. This implies that ker(f ) is of dimension at least nd. Note also that for a general curve the kenel is of dimension exactly nd. Let us consider the following incidence variety

I = {(f, V n-d ) ∈ Hom d (C, X) × G Q (n -d, 2n) / for all x ∈ C we have V n-d ⊂ f (x) ⊂ V ⊥ n-d }.
Here

G Q (n -d, 2n) is the Grassmann variety of isotropic subspaces of C 2n of dimension n -d.
The above condition on f simply translates in V n-d ⊂ ker(f ). Note that the second inclusion is implied by the first one since we have f (x) = f (x) ⊥ . The projection p : I → Hom d (C, X) is surjective and birational (because for a general curve the kernel is of dimension nd) while the map q :

I → G Q (n -d, 2n
) is an equivariant locally trivial fibration with fibers

Hom d (C, Y ) with Y ≃ G Q (d, 2d
). By the previous proposition these fibers are irreducible of dimension 2d(d -1). The result follows.

Adjoint varieties in type B and D

Let us consider the variety X = G Q (2, n) of isotropic planes in a vector space of dimension n endowed with a non-degenerate bilinear form.

Proposition 3.10 The scheme Hom

d (C, X) is irreducible of dimension (n -3)d for d ≥ n -1.
Proof. The map φ : U → Y is a vector bundle over Y ≃ Q n-4 associated to the locally free sheaf

E = (O Q n-4 (1)) ⊥ where O Q n-4 (-1) is the tautological subbundle on Q n-4 .
Here we have to compute the dimension of the scheme of morphisms from an elliptic curve to a cone (the intersection of Q n-4 with a linear subspace of P n-3 in our case). Let C be a cone in P k+r-1 associated to a quadratic form of rank r (and therefore of kernel K of dimension k). Remark that, if k = 0, the cone C is a smooth quadric of dimension (k + r -2) and we already know the result with equality.

Proof. Consider the incidence variety

I = {(V 1 , V k+1 ) ∈ C × G Q (k + 1, k + r) / V 1 ⊂ V k+1 ⊃ K}.
There is a diagram

I p / / q Q r-2 C.
The map q is simply the blow-up of the vertex of the cone. If f : C → C has a non degenerate image then its image is not contained in the vertex P(K) of the cone and we may consider the composition of f with the projection from K. This defines a morphism f ′ : C → Q r-2 which can be obtained by lifting f to f I : C → I and composing with p. The degree of this morphism is dx where x is the multiplicity of the intersection of f (C) with the vertex of the cone.

To compute the dimension of our open set, we consider the map defined by f → f ′ . Its image is contained in Hom d-x (C, Q r-2 ) which is of dimension (dx)(r -2). The fiber is given by the morphisms L → f ′ * V k+1 where L is of degree d and V k+1 is the tautological subbundle in

Q r-2 ≃ {V k+1 ∈ G Q (k + 1, k + r) / K ⊂ V k+1 }.
Because the projection by K gives a curve of degree dx > 0 we have an extension 0

→ K ⊗ O C → f ′ * V k+1 → L ′ → 0 with L ′ of degree d -x.
This extension has to split, thus the fibres of the map are given by PHom(L, K ⊗ O C ⊕ L ′ )) which is of dimension kd + x. The dimension is therefore at most d(k + r -2)x(r -3) proving the result for r ≥ 3. But if r ≤ 2, then the cone is a union of hyperplane thus the curve is degenerate. By Proposition 2.7, since E is the restriction to Q n-4 of Ω n-4 P n-3 (n -3), the fibers of Φ : Hom d (C, U ) → Hom d (C, Y ) over the locally closed subset H a of morphisms whose image is nondegenerate in a linear subspace of codimension a are of dimension d + a. Therefore the dimension of Φ -1 (H a ) is at most the sum of d + a, the dimension of the Grassmann variety G(n -2a, n -2) and the dimension of the scheme of degree d morphisms from an elliptic curve C to a cone C in P n-3-a , so that dim Φ -1 (H a ) ≤ d(na -3) + a(n -2a) + a. This dimension is strictly smaller than the expected dimension (n -4)d for a > 0 and d ≥ n -1. The result follows.

Remark 3.12 The above argument shows that for d = n -2, the scheme of morphisms has 2 irreducible components of the expected dimension.

Type E 6

The Cayley plane

Let X be isomorphic to E 6 /P 1 . Proposition 3.13 The scheme Hom d (C, X) is irreducible of dimension 12d for d ≥ 3.

Proof. The map φ : U → Y is the vector bundle over Y ≃ Q 8 associated to the locally free sheaf E = S with S one of the two spinor bundles on Q 8 . By Corollary 2.11, the fibers of p : Hom d (C, U ) → Hom d (C, Y ) over the locally closed subset H a of morphisms whose image is non degenerate in an isotropic linear subspace of codimension a are of dimension 4d + 2 max(0,a-6) with a ≥ 5. Therefore we have the equality dim Φ -1 (H a ) = 12d+a(10-a)-(10-a)(11-a) 2 +2 max(0,a-6) -ad. This dimension is strictly smaller than the expected dimension 12d for a > 0 and d ≥ 3. The result follows.

The adjoint variety

Let X be isomorphic to E 6 /P 2 .

Proposition 3.14 The scheme Hom d (C, X) is irreducible of dimension 11d for d ≥ 9.

Proof. The map φ : U → Y is a sequence of affine bundles over Y ≃ Q 6 associated to the locally free sheaves E = (O Q 6 (1)) ⊥ (where O Q 6 (-1) is the tautological subbundle on Q 6 ) and E ′ = S ⊕ S ′ where S and S ′ are the two spinor bundles on Q 6 . Note that the bundle E is the restriction of the tautological quotient bundle of P 7 to Q 6 . By Proposition 2.7 and Corollary 2.11, the fibers of Φ : Hom d (C, U ) → Hom d (C, Y ) over the locally closed subset H a of morphisms whose image is non degenerate in a linear subspace of codimension a are of dimension at most 5d + a for a ≤ 3 or a ≥ 4 and the subspace containing the curve is non isotropic and 5d + a + 2 max(0,a-4) for a ≥ 4 and the subspace containing the curve is isotropic. Therefore we have the equality dim Φ -1 (H a ) = d(11a) + a(8a) + a for a ≤ 3 or a ≥ 4 and the subspace containing the curve is non isotropic and d(13a) + a(8a) -(8-a)(9-a) 2 + a + 2 max(0,a-4) for a ≥ 4 and the subspace containing the curve is isotropic. This dimension is strictly smaller than the expected dimension 11d for a > 0 and d ≥ 9. The result follows.

Type E 7

The Freudenthal variety

Let X be isomorphic to E 7 /P 7 .

Proposition 3.15 The scheme Hom d (C, X) is irreducible of dimension 18d for d ≥ 8.

Proof. The map φ : U → Y is a sequence of affine bundles associated to the locally free sheaves E = Ω 5 P 6 (6) and E ′ = Ω 2 P 6 (3). By Proposition 2.7, the fibers of Φ : Hom d (C, U ) → Hom d (C, Y ) over the locally closed subset H a of morphisms whose image is non degenerate in a linear subspace of codimension a are of dimension at most 11d + a + a 4 . Therefore we have dim Φ -1 (H a ) = 18d + a(7a) + a + a 4ad. This dimension is strictly smaller than the expected dimension 18d for a > 0 and d ≥ 8. The result follows.

The adjoint variety

Let X be isomorphic to E 7 /P 1 . Proposition 3.16 The scheme Hom d (C, X) is irreducible of dimension 17d for d ≥ 11.

Proof. The map φ : U → Y is a sequence of affine bundles over Y ≃ Q 8 associated to the locally free sheaves E = (O Q 8 (1)) ⊥ where O Q 8 (-1) is the tautological subbundle on Q 8 and E ′ = S 2 where S is one of the two spinor bundles on Q 8 . By Proposition 2.7 and Corollary 2.11, the fibers of Φ : Hom d (C, U ) → Hom d (C, Y ) over the locally closed subset H a of morphisms whose image is non degenerate in a linear subspace of codimension a are of dimension at most 9d + a for a ≤ 4 or a ≥ 5 and the subspace containing the curve is non isotropic and 9d + a + 2 max(0,a-6) for a ≥ 5 and the subspace containing the curve is isotropic. Therefore we have the equality dim Φ -1 (H a ) = d(17a) + a(10a) + a for a ≤ 4 or a ≥ 5 and the subspace containing the curve is non isotropic and d(19a) + a(10a) -(10-a)(11-a) 2 + a + 2 max(0,a-6) for a ≥ 5 and the subspace containing the curve is isotropic. This dimension is strictly smaller than the expected dimension 17d for a > 0 and d ≥ 11. The result follows.

Adjoint variety of type E 8

Let X be isomorphic to E 8 /P 8 . Proposition 3.17 The scheme Hom d (C, X) is irreducible of dimension 29d for d ≥ 15.

Proof. The map φ : U → Y is a sequence of affine bundles over Y ≃ Q 12 associated to the locally free sheaves E = (O Q 12 (1)) ⊥ where O Q 12 (-1) is the tautological subbundle on Q 12 and E ′ = S where S is one of the two spinor bundles on Q 12 . By Proposition 2.7 and Corollary 2.11, the fibers of Φ : Hom d (C, U ) → Hom d (C, Y ) over the locally closed subset H a of morphisms whose image is non degenerate in a linear subspace of codimension a are of dimension at most 17d + a for a ≤ 6 or a ≥ 7 and the subspace containing the curve is non isotropic and 17d + a + 2 max(0,a-8) for a ≥ 7 and the subspace containing the curve is isotropic. Therefore we have the equality dim Φ -1 (H a ) = d(29a) + a(14a) + a for a ≤ 6 or a ≥ 7 and the subspace containing the curve is non isotropic and d(31a) + a(14a) -(14-a)(15-a) 2 + a + 2 max(0,a-8) for a ≥ 7 and the subspace containing the curve is isotropic. This dimension is strictly smaller than the expected dimension 29d for a > 0 and d ≥ 15. The result follows.

Adjoint variety of type F 4

Let X be isomorphic to F 4 /P 1 .

Proposition 3.18 The scheme Hom d (C, X) is irreducible of dimension 11d for d ≥ 8.

Proof. The map φ : U → Y is a sequence of affine bundles over Y ≃ Q 5 associated to the locally free sheaves E = (O Q 5 (1)) ⊥ where O Q 5 (-1) is the tautological subbundle on Q 5 and E ′ = S where S is the spinor bundle on Q 5 . By Proposition 2.7 and Corollary 2.11, the fibers of Φ : Hom d (C, U ) → Hom d (C, Y ) over the locally closed subset H a of morphisms whose image is non degenerate in a linear subspace of codimension a are of dimension at most 6d + a for a ≤ 3 or a ≥ 4 and the subspace containing the curve is non isotropic and 6d + a + 2 max(0,a-5) for a ≥ 4 and the subspace containing the curve is isotropic. Therefore we have the equality dim Φ -1 (H a ) = d(11a) + a(7a) + a for a ≤ 3 or a ≥ 4 and the subspace containing the curve is non isotropic and d(13a) + a(7a) -(7-a)(8-a) 2 + a + 2 max(0,a-5) for a ≥ 4 and the subspace containing the curve is isotropic. This dimension is strictly smaller than the expected dimension 1d for a > 0 and d ≥ 8. The result follows.

Remark 3.19 Note that, in Propositions 3.14, 3.15, 3.16, 3.17 and 3.18, it is possible that one can improve the bound on the dimension of Φ -1 (H a ). Indeed if the affine bundle restricted to curve in H a is not a vector bundle then this bound will be strictly smaller. We therefore expect a better bound d(X) in these situations.

Stable maps

Let us consider the moduli space M 1,d (X) of stable maps with source an elliptic curve. Recall the following general result obtained in [START_REF] Kim | The connectedness of the moduli space of maps to homogeneous spaces[END_REF].

Theorem 4.1 ([KP01])

The moduli space M 1,d (X) is connected for any rational homogeneous space X and any degree d.

Note, in contrast, that for any X in the tables of the introduction, we have the proposition. Proposition 4.2 For d ≥ d(X), the moduli space M 1,d (X) is never irreducible.

Proof. Indeed, consider the locally closed subset of M 1,d (X) where the degree on the unique elliptic irreducible component of the source of the map is 2. The dimension of this locally closed subset is dc 1 (X) + dim X which is always strictly bigger than the expected dimension dc 1 (X) which is the dimension of the component containing irreducible curves for d ≥ d(X).

However, there is a natural decomposition of this space into locally closed subsets M τ 1,dτ (X) parametrised by their combinatorial graph τ and combinatorial degree d τ (see [START_REF] Kontsevich | Gromov-Witten classes, quantum cohomology, and enumerative geometry[END_REF] or also [START_REF] Kim | The connectedness of the moduli space of maps to homogeneous spaces[END_REF]). Note that for stable maps of genus one, at most one irreducible component of the source map is an elliptic curve. We call this component (if it exists) the elliptic component of τ . As a consequence of our results we obtain the following proposition.

Proposition 4.3 If the degree d τ is bigger that d(X) on the elliptic component of τ , then M τ 1,dτ (X) is irreducible.

Proof. If there is no elliptic component, then the result follows from the fact (see [START_REF] Buch | Finiteness of cominuscule quantum K-theory[END_REF]) that the moduli space of stable maps of genus 0 passing through at most 2 fixed point is irreducible for any homogeneous space X.

If there is an elliptic component, then claim that the natural forgetful map to the moduli space of elliptic curves is flat with irreducible fibers. Indeed, the fiber over C is isomorphic to the product of Hom d (C, X) with some moduli spaces of rational stable maps passing through at most 2 fixed points. These last moduli spaces are irreducible of the expected dimension by [BCMP10, Corollary • α • ξ 1 > 0, α • ξ 2 > 0 and α • ξ 3 ≥ 0, or α • ξ 1 > 0, α • ξ 2 ≥ 0 and α • ξ 3 > 0;

• d = α • (ξ 1 + 2ξ 2 + ξ 3 ).

Remark that, under conditions ( * ) d , we have α • (T 1 + T 2 + T 3 ) = 3dα • (2ξ 2 + ξ 3 ) and then the dimension of the scheme Hom α(C, X) is at most 3d -1 with equality if an only if α • ξ 1 = d -1, α • ξ 2 = 0 and α • ξ 3 = 1. We deduce the first fact from above results.

To prove the second fact, fix a curve f ∈ Hom d (C, Q 3 ) such that f (C) is not contained in an isotropic projective line of P(V ). We begin by proving the following lemma. If E is a vector subspace of V , we denote by E ⊥ the orthogonal subspace to E in V (with respect to Q). For any point x in P(V ), we denote by x the corresponding line in V .

Lemma A.5 There exist 3 points x 0 , x 1 , x 2 of f (C) and an isotropic flag W • = (W 1 , W 2 ) such that x 0 is not contained in W ⊥ 1 , x 1 is contained in W ⊥ 1 but not in W 2 , and x 2 is contained in W 2 .

Proof. First remark that, because f (C) ⊂ Q 3 , the condition saying that f (C) is not contained in an isotropic projective line of P(V ) means that f (C) is not contained in a projective line of P(V ).

Then there exist 3 non-collinear points x 0 , x 1 , x 2 in f (C). Let E := x 0 ⊕x 1 ⊕x 2 and F := x 1 ⊕x 2 . We can assume that Q| F is non degenerate, in particular V = F ⊕ F ⊥ and Q is non-degenerate on F ⊥ . Then there exists an isotropic line W 1 in F ⊥ \E ⊥ . And we can define

W 2 = W 1 ⊕ x 2 , because x 2 ⊂ F ⊥ .
It is obvious that x 2 is contained in W 2 . And we can also easily check that x 0 is not contained in W ⊥ 1 and that x 1 is contained in W ⊥ 1 but not in W 2 . x such that x is not contained in W ⊥ 1 . Recall that, over Ω, the morphism π : X W• → Q 3 is an isomorphism. Thus, since C is smooth and f (C) intersects Ω (at least in x 0 ), f lifts into a unique f : C → X W• . Moreover, we can compute that π(ξ 1 ) = {x ∈ Q 3 | x ⊂ W ⊥ 1 } and π(ξ 2 ) = π(ξ 3 ) = {x ∈ Q 3 | x ⊂ W 2 }. So, for any i ∈ {1, 2, 3}, π -1 (Ω) does not intersect ξ i so that [ f * (C)] • ξ i ≥ 0. Also, the existence of x 1 implies that [ f * (C)] • ξ 1 > 0 and the existence of x 2 implies that [ f * (C)] • ξ 2 > 0 or [ f * (C)] • ξ 3 > 0.

Let us now explain why the set of flags that we can choose is of dimension 3. In the proof of Lemma A.5, we can note that, when x 0 , x 1 , x 2 are fixed in f (C) 3 , we choose W 1 in a quadric of dimension 1 and W 2 is uniquely determined. Moreover, if W • is fixed, then we have finitely many choice for x 1 and x 2 whereas the set of possible x 0 is one-dimensional. At the end, the dimension we are looking for is 3 + 1 -1 = 3.

Finally, let us prove the third fact. Let J be the scheme of morphisms from C to X of degree d such that f (C) is contained in an isotropic projective line of P(V ). We have a natural projection from J to the variety G Q (2, 5) of isotropic projective lines in P(V ), which is of dimension 3. The fibers are isomorphic to the scheme

Contents12

  Reduction of the theorem 4 1.1 Restriction to an open subset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 Passing through affine bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Decomposition of X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Rectriction of some homogeneous vector bundles to curves 7 2.1 Projective spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 Spinor bundles on quadrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3 Irreducibility 11 3.1 Projective space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.2 Grassmann varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.3 Quadrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.4 Maximal orthogonal Grassmann varieties . . . . . . . . . . . . . . . . . . . . . . . . 13 3.5 Adjoint varieties in type B and D . . . . . . . . . . . . . . . . . . . . . . .

Lemma 3 .

 3 11 Consider the open subset in the scheme of degree d morphisms from an elliptic curve C to C of morphisms whose image is non-degenerate. Then this open subset is of dimension at most d(k + r -2).

  3.3]. Now by dimension count, any irreducible component of M τ dτ (X) has to dominate the moduli space of elliptic curves (the fibers have constant dimension 1 less than the expected dimension) thus the map is flat. The result follows by [Gro65, Corollaire 2.3.5.(ıı)]. Definition A.4 We say that α satisfies conditions ( * ) d if

Corollary A. 6

 6 There exists a flag W • such that f lifts into a unique f : C → X W• , and [ f * (C)] satisfies the conditions ( * ) d . Proof. Let W • be an isotropic flag and (x 0 , x 1 , x 2 ) ∈ f (C) 3 as in Lemma A.5. Denote by B the Borel subgroup of SO(5) corresponding to W • . Then the open B-orbit Ω in Q 3 is the set of points

  . . . . . 14 3.6 Type E 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.6.1 The Cayley plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.6.2 The adjoint variety . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.7 Type E 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.7.1 The Freudenthal variety . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.7.2 The adjoint variety . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.8 Adjoint variety of type E 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.9 Adjoint variety of type F 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

	4 Stable maps	17

  is a sequence of affine bundles whose direction vector bundles are associated to the locally free sheaves E = (O Q 8 (1)) ⊥ and E ′ = S 2 where E is the restriction of the tautological quotient bundle of P 9 to Q 8 and S is one of the two spinor bundles on Q8 . 11. If G is of type E 8 , if Σ(P ) = {α 8 } and if Σ(Q) = {α 1 }, then Y is isomorphic to Q 12 andφ : U → Y is a sequence of affine bundles whose direction vector bundles are associated to the locally free sheaves E = (O Q 12 (1)) ⊥ and E ′ = S where E is the restriction of the tautological quotient bundle of P 13 to Q 12 and S is one of the two spinor bundles on Q 12 .

A The quadric of dimension 3

In this section, we prove Proposition 3.6 for d ≥ 4 using Bott-Samelson resolutions (the reader could see [START_REF] Demazure | Désingularisation des variétés de Schubert généralisées. Collection of articles dedicated to Henri Cartan on the occasion of his 70th birthday[END_REF] for more details on Bott-Samelson resolutions). This method was already used by the second author in [START_REF] Perrin | Courbes elliptiques sur la variété spinorielle[END_REF] to study elliptic curves in spinorial varieties.

First we introduce some notation. Let V be a 5-dimensional vector space endowed with a non degenerate quadratic form Q. Then X is the subvariety of P(V ) consisting of isotropic lines in V .

For any isotropic flag W • = (W 1 , W 2 ) of V , we define a Bott-Sameslon resolution π : X → X by

The proof of Proposition 3.6 uses the 3 following facts.

Fact 1 Under some conditions ( * ) d on the class α of 1-cycles in X (see Definition A.4), the scheme Hom α(C, X) of morphisms from C to X of class α is irreducible of dimension at most 3d-1 with equality for a unique class α).

Fact 2 For all f ∈ Hom d (C, X) such that f (C) is not contained in an isotropic projective line of P(V ), we can choose a flag W • such that f lifts into a unique f : C → X W• , such that α := [ f * (C)] satisfies the conditions ( * ) d . And, for each such f , the set of flags that we can choose is a subvariety (not closed) of SO(5)/B of dimension 3.

Fact 3 For d ≥ 4 we prove that the scheme of morphisms from C to X of degree d such that f (C) is contained in an isotropic projective line of P(V ) cannot be an irreducible component of Hom d (C, X).

Assuming these facts for the moment, let us prove Proposition 3.6. Let I be the set of couples

Call p and q the natural projections from I to Hom d (C, X) and to SO(5)/B respectively. Then, by the second and third facts, p is dominant as soon as d ≥ 4 and general fibers of p have dimension 3. Now, for any flag W • , the fiber q -1 (W • ) is the disjoint union, over the α satisfying conditions ( * ) d , of the schemes Hom α(C, X W• ) of morphisms f from C to X W• such that [ f * (C)] = α. By the first fact, exactly one of these latter schemes has maximal dimension 3d -1 and moreover this scheme is irreducible, so that the fiber q -1 (W • ) is irreducible. Thus I is a union of irreducible connected components of dimension at most 3d -1+ = 3d + 3 with equality for a unique component. The image in Hom d (C, X) of these components are irreducible of dimension at most 3d + 3 -3 = 3d which is the expected dimension. Therefore all these images are contained in the closure of the image of the maximal one proving the irreducibility. Moreover the dimension of Hom d (C, X) therefore 4 + (3d -1) -3 = 3d.

Let us prove the three facts one by one. We begin with a general lemma.

Lemma A.1 Let φ : X → Y be a P 1 -fibration with a section σ. Let α be a class of 1-cycles in X.

Denote by T φ the relative tangent bundle and by ξ the divisor σ(Y ). Suppose that α • ξ ≥ 0 and that α • (T φξ) > 0.

Then, if Hom φ * α(C, Y ) is irreducible, Hom α(C, X) is also irreducible, and

Proof. Let E a rank two vector bundle on Y such that X = P Y (E). The section σ is given by a surjection E → L where L is an invertible sheaf on Y . Let N be the kernel of this map. We first study the fiber of the map Hom α (C, X) → Hom φ * α (C, Y ) given by composition of morphisms over a morphism f : C → Y . An element in this fiber is given by a lift of f i.e. a surjective map f * E → M of vector bundles on C where M is an invertible sheaf with 2 deg(M )deg(f * E) = α • T φ . An element in the fiber is therefore given by an invertible sheaf M on C with degree d = If

does not depend on f . The fiber is therefore a non empty open subset of PHom(f * E, M ). We therefore again get a smooth fibration of relative dimension α • T φ over Hom φ * α (C, Y ).

Applying this lemma to the P 1 -fibrations φ i : X i → X i-1 for i ∈ {1, 2, 3} given above by a Bott-Samelson resolution X of Q 3 , we obtain the following corollary. We denote by T i the relative tangent space of φ i , we define three divisors ξ 1 , ξ 2 , ξ 3 of X, which are given by natural sections of the φ i , by

To obtain the first fact, we use the following result that can be deduced from [Per05, Corollary 3.8 and Fact 3.7] by a short computation.

Proposition A.3 (ı) The relative tangent bundles T i are expressed in terms of the ξ i as follows:

In particular, we have α • ξ i ≥ 0 and α • (T iξ i ) > 0 for all i ∈ {1, 2, 3} as soon as α • ξ 1 > 0, α • ξ 2 > 0 and α • ξ 3 ≥ 0, or, α • ξ 1 > 0, α • ξ 2 ≥ 0 and α • ξ 3 > 0 (ıı) The pull-back π * O Q 3 (1) of the ample generator of the Picard group of Q 3 equals ξ 1 +2ξ 2 +ξ 3 . In particular, if α = π * (α) where α is the class of 1-cycles of degree d in Q 3 , we have d = α • (ξ 1 + 2ξ 2 + ξ 3 ). Hom d (C, P 1 ) of morphisms of degree d from C to P 1 and are of dimension 2d. The dimension of J is therefore 2d + 3. If d ≥ 4, we have 2d + 3 < 3d and then J cannot be an irreducible component of Hom d (C, X).


Michel Cristofol

Jimmy Garnier

François Hamel

Lionel Roques

Uniqueness from pointwise observations in a multi-parameter inverse problem

Keywords: reaction-diffusion, inverse problem, multi-parameter, heterogeneous media, uniqueness

In this paper, we prove a uniqueness result in the inverse problem of determining several nonconstant coefficients of one-dimensional reaction-diffusion equations. Such reaction-diffusion equations include the classical model of Kolmogorov, Petrovsky and Piskunov as well as more sophisticated models from biology. When the reaction term contains an unknown polynomial part of degree N, with non-constant coefficients µ k (x), our result gives a sufficient condition for the uniqueness of the determination of this polynomial part. This sufficient condition only involves pointwise measurements of the solution u of the reaction-diffusion equation and of its spatial derivative ∂u/∂x at a single point x 0 , during a time interval (0, ε). In addition to this uniqueness result, we give several counter-examples to uniqueness, which emphasize the optimality of our assumptions. Finally, in the particular cases N = 2 and N = 3, we show that such pointwise measurements can allow an efficient numerical determination of the unknown polynomial reaction term.

Introduction

Reaction-diffusion equations arise as models in many fields of mathematical biology [START_REF] Murray | Mathematical Biology[END_REF]. From morphogenesis [START_REF] Turing | The chemical basis of morphogenesis[END_REF] to population genetics [START_REF] Fisher | The wave of advance of advantageous genes[END_REF][START_REF] Kolmogorov | Étude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique[END_REF] and spatial ecology [START_REF] Shigesada | Biological invasions: theory and practice[END_REF][START_REF] Skellam | Random dispersal in theoretical populations[END_REF][START_REF] Turchin | Quantitative analysis of movement: measuring and modeling population redistribution in animals and plants[END_REF], these partial differential equations benefit from a well-developed mathematical theory.

In the context of spatial ecology, single-species reaction-diffusion models generally deal with polynomial reaction terms. In a one-dimensional case, and if the environment is supposed to be homogeneous they take the form: ∂u ∂t -D ∂ 2 u ∂x 2 = P (u), (1.1) where u = u(t, x) is the population density at time t and space position x and D > 0 is the diffusion coefficient. The function P, which stands for the growth of the population, is a polynomial of order N ≥ 1. The Fisher-Kolmogorov, Petrovsky, Piskunov (F-KPP) equation is the archetype of such models. In this model, we have P (u) = µ u -γu 2 . The constant parameters µ and γ respectively 1 correspond to the intrinsic growth rate and intraspecific competition coefficients. In this model, the lower the population density u, the higher the per capita growth rate P (u)/u. More complex models can involve polynomial nonlinearities of higher order. Examples are those taking account of an Allee effect. This effect occurs when the per capita growth rate P (u)/u reaches its maximum at a strictly positive population density and is known in many species [START_REF] W C Allee | The social life of animals[END_REF][START_REF] Dennis | Allee effects: population growth, critical density, and the chance of extinction[END_REF][START_REF] Veit | Dispersal, population growth, and the Allee effect: dynamics of the house finch invasion of eastern North America[END_REF]. A typical example of reaction term involving an Allee effect is [START_REF] T H Keitt | Allee effects, invasion pinning, and species' borders[END_REF][START_REF] Lewis | Allee dynamics and the speed of invading organisms[END_REF][START_REF] Roques | A population facing climate change: joint influences of Allee effects and environmental boundary geometry[END_REF]:

P (u) = ru (1 -u) (u -ρ) ,
with r > 0 and ρ ∈ (0, 1). The parameter ρ corresponds in that case to the "Allee threshold" below which the growth rate becomes negative. In the previous examples, the reaction terms were assumed to be independent of the space variable. However, real world is far from begin homogeneous. In order to take the heterogeneities into account, models have been adapted and constant coefficients have been replaced by space or time dependant functions. In his pioneering work, Skellam [START_REF] Skellam | Random dispersal in theoretical populations[END_REF] (and later, Shigesada, Kawasaki and Teramoto [START_REF] Shigesada | Traveling periodic-waves in heterogeneous environments[END_REF]) mentioned the following extension of the F-KPP model to heterogeneous environments:

∂u ∂t -D ∂ 2 u ∂x 2 = µ(x) u -γ(x)u 2 . (1.2)
Here, the values of µ(x) and γ(x) depend on the position x. For instance, regions of the space associated with high values of µ(x) correspond to favorable regions, whereas those associated with low or negative values of µ(x) correspond to unfavorable regions. As emphasized by recent works, the precise spatial arrangement of these regions plays a crucial role in this model, since it controls persistence and spreading of the population [START_REF] Berestycki | Analysis of the periodically fragmented environment model: I -Species persistence[END_REF][START_REF] Cantrell | Spatial ecology via reaction-diffusion equations[END_REF][START_REF] El Smaily | Homogenization and influence of fragmentation in a biological invasion model, Discrete and Continuous Dynamical Systems[END_REF][START_REF] Roques | On population resilience to external perturbations[END_REF][START_REF] Roques | Mathematical analysis of the optimal habitat configurations for species persistence[END_REF][START_REF] Roques | Species persistence decreases with habitat fragmentation: an analysis in periodic stochastic environments[END_REF][START_REF] Shigesada | Biological invasions: theory and practice[END_REF]. Models involving an Allee effect can be extended as well to heterogeneous environments, as in [START_REF] Hamel | Spreading speeds in slowly oscillating environments[END_REF][START_REF] Roques | A population facing climate change: joint influences of Allee effects and environmental boundary geometry[END_REF], where the effects of spatial heterogeneities are discussed for models of the type:

∂u ∂t = D ∂ 2 u ∂x 2 + r(x) u [(1 -u)(u -ρ(x)) + ν(x)] . (1.3)
We also refer to [START_REF] Matano | Periodic traveling waves in a two-dimensional cylinder with saw-toothed boundary and their homogenization limit[END_REF] for an analysis of propagation phenomena related to a reaction-diffusion model with an Allee effect in infinite cylinders having undulating boundaries.

In this paper, we focus on reaction-diffusion models with more general heterogeneous nonlinearities:

∂u ∂t = D ∂ 2 u ∂x 2 + N k=1 µ k (x) u k + g(x, u), for t > 0, x ∈ (a, b), (1.4)
for some interval (a, b) in R.

Since the behavior of such models depends on the precise form of the coefficients, their empirical use requires an accurate knowledge of the coefficients. Unfortunately, in applications, the coefficients cannot be directly measured since they generally result from intertwined effects of several factors. Thus, the coefficients are generally measured through the density u(t, x) [START_REF] Soubeyrand | Modelling the spread in space and time of an airborne plant disease[END_REF]. From a theoretical viewpoint, if u(t, x) is measured at any time t ≥ 0 and at all points x in the considered domain, all the coefficients in the model can generally be determined. However, in most cases, u(t, x) can only be measured in some -possibly small -subregions of the domain (a, b) [START_REF] Wikle | Hierarchical models in environmental science[END_REF]. For reaction-diffusion models as well as for many other types of models, the determination of the coefficients in the whole domain (a, b) bears on inference methods which consist in comparing the solution of the model with hypothetical values of coefficients μk , with the measurements on the subregions [START_REF] Soubeyrand | Mechanical-statistical modeling in ecology: from outbreak detections to pest dynamics[END_REF]. The underlying assumption behind this inference process is that there is a one-toone and onto relationship between the value of the solutions of the model over the subregions and the space of coefficients. This assumption is of course not true in general.

In this paper, we obtain uniqueness results for the coefficients µ k (x), k = 1, . . . , N , based on localized measurements of the solution u(t, x) of (1.4). The major differences with previous works dealing with comparable uniqueness results are (1) the size of the regions where u(t, x) has to be known in order to prove uniqueness, (2) the number of parameters we are able to determine, and (3) the general type of nonlinearity we deal with.

Uniqueness of the parameters, given some values of the solution, corresponds to an inverse coefficient problem, which is generally dealt with -for such reaction-diffusion equations -using the method of Carleman estimates [START_REF] Bukhgeim | Uniqueness in the large of a class of multidimensional inverse problems[END_REF][START_REF] Klibanov | Carleman estimates for coefficient inverse problems and numerical applications[END_REF]. This method provides Lipschitz stability, in addition to the uniqueness of the coefficients. However, this method requires, among other measurements, the knowledge of the density u(θ, x) at some time θ and for all x in the domain (a, b) (see [START_REF] Belassoued | Inverse source problem for a transmission problem for a parabolic equation[END_REF][START_REF] Benabdallah | Inverse problem for a parabolic system with two components by measurements of one component[END_REF][START_REF] Cristofol | Inverse problems for a two by two reaction-diffusion system using a carleman estimate with one observation[END_REF][START_REF] O Y Immanuvilov | Lipschitz stability in inverse parabolic problems by the Carleman estimate[END_REF][START_REF] Yamamoto | Simultaneous reconstruction of the initial temperature and heat radiative coefficient[END_REF]). The uniqueness of the couple (u, µ(x)) satisfying the equation (1.2) given such measurements has been investigated in a previous work [START_REF] Cristofol | Biological invasions: Deriving the regions at risk from partial measurements[END_REF], in any space dimension.

In a recent work, Roques and Cristofol [START_REF] Roques | On the determination of the nonlinearity from localized measurements in a reaction-diffusion equation[END_REF] have proved the uniqueness of the coefficient µ(x) in (1.2) when γ(x) is known under the weaker assumption that the density u(t, x 0 ) and its spatial derivative ∂u ∂x (t, x 0 ) are known at a point x 0 in (a, b) for all t ∈ (0, ε) and that the initial density u(0, x) is known over (a, b). This result shows that the coefficient µ(x) is uniquely determined in the whole domain (a, b) by the value of the solution u(t, x) and of its spatial derivative at a single point x 0 . The present work extends this result to the case of several coefficients µ k (x), k = 1, . . . , N : given any point x 0 in (a, b) we establish a uniqueness result for the N -uple (µ 1 (x), . . . , µ N (x)) given measurements of the N solutions u(t, x) of (1.4) and of their first spatial derivatives in (0, ε) × {x 0 }, starting with N nonintersecting initial conditions.

Hypotheses and main result

Let (a, b) be a bounded interval in R. We consider, for some T > 0, the problem:

                       ∂u ∂t -D ∂ 2 u ∂x 2 = N k=1 µ k (x)u k + g(x, u), t ∈ (0, T ), x ∈ (a, b), α 1 u(t, a) -β 1 ∂u ∂x (t, a) = 0, t > 0, α 2 u(t, b) + β 2 ∂u ∂x (t, b) = 0, t > 0, u(0, x) = u 0 (x), x ∈ (a, b), (P u 0 (µ k ) )
for some N ∈ N * , and for -unknown -functions µ k which belong to the following space M:

M := {ψ ∈ C 0,η ([a, b]) s. t. ψ is piecewise analytic on (a, b)}, (2.1)
for some η ∈ (0, 1]. The space C 0,η corresponds to Hölder continuous functions with exponent η (see e.g. [START_REF] Friedman | Partial differential equations of parabolic type[END_REF]). A function ψ ∈ C 0,η ([a, b]) is called piecewise analytic if their exist n > 0 and an increasing sequence (κ j ) 1≤j≤n such that κ 1 = a, κ n = b, and

for all x ∈ (a, b), ψ(x) = n-1 j=1 χ [κ j ,κ j+1 ) (x)ϕ j (x),
for some analytic functions ϕ j , defined on the intervals [κ j , κ j+1 ], and where χ [κ j ,κ j+1 ) are the characteristic functions of the intervals [κ j , κ j+1 ) for j = 1, . . . , n -1. In particular, if ψ ∈ M,

then, for each x ∈ [a, b) (resp. x ∈ (a, b]), there exists r = r x > 0 such that ψ is analytic on [x, x+ r] (resp. [x -r, x]).
The assumptions on the function g are:

g(•, u) ∈ C 0,η ([a, b]) for all u ∈ R, g(x, •) ∈ C 1 (R) for all x ∈ [a, b] and g(•, 0) = 0 in [a, b]. (2.2)
We also assume that the diffusion coefficient D is positive and that the boundary coefficients satisfy:

α 1 , α 2 , β 1 , β 2 ≥ 0 with α 1 + β 1 > 0 and α 2 + β 2 > 0. (2.3)
We furthermore make the following hypotheses on the initial condition:

u 0 > 0 in (a, b) and u 0 ∈ C 2,η ([a, b]), (2.4)
that is u 0 is a C 2 function such that (u 0 ) ′′ is Hölder continuous. In addition to that, we assume the following compatibility conditions:

α 1 u 0 (a) -β 1 (u 0 ) ′ (a) = 0 and -D (u 0 ) ′′ (a) = g(a, 0) if β 1 = 0, α 2 u 0 (b) + β 2 (u 0 ) ′ (b) = 0 and -D (u 0 ) ′′ (b) = g(b, 0) if β 2 = 0. (2.5)
Under the assumptions (2.1)-(2.5), for each sequence (µ k ) 1≤k≤N ∈ M N , there exists a time

T u 0 (µ k ) ∈ (0, +∞] such that the problem (P u 0 (µ k ) ) has a unique solution u ∈ C 2,η 1,η/2 [0, T u 0 (µ k ) ) × [a, b] (i.
e. the derivatives up to order two in x and order one in t are Hölder continuous). In the sequel, even if it means decreasing T u 0 (µ k ) in some cases and dropping the indices (µ k ) and u 0 , we only deal with values of t smaller than T so that the problem (P u 0 (µ k ) ) is well posed. Existence, uniqueness and regularity of the solution u are classical (see e.g. [START_REF] Pao | Nonlinear Parabolic and Elliptic Equations[END_REF]).

Our main result is a uniqueness result for the sequence of coefficients (µ k ) 1≤k≤N associated with observations of the solution and of its spatial derivative at a single point x 0 in [a, b]. Consider N ordered initial conditions u 0 i and, for each sequence (µ k ) 1≤k≤N , let u i be the solution of (P

u 0 i (µ k )
). Our result shows that for any ε > 0 the function

G : M N → C 1 ((0, ε)) 2 N (µ k ) 1≤k≤N → (u i (•, x 0 ), ∂u i /∂x(•, x 0 )) 1≤i≤N
, is one-to-one. In other words, we have the following theorem:

Theorem 2.1. Let N ∈ N * , (µ k ) 1≤k≤N and (μ k ) 1≤k≤N be two families of coefficients in M.
Let (u 0 i ) 1≤i≤N be N positive functions fulfilling (2.4) and (2.5) and such that u 0 i (x) = u 0 j (x) for all x ∈ (a, b) and all i = j. Let u i and ũi be the solutions of (P

u 0 i (µ k ) ) and (P u 0 i (μ k ) ), respectively, on [0, T ) × [a, b].
We assume that u i and ũi satisfy at some x 0 ∈ [a, b], and for some ε ∈ (0, T ]:

   u i (t, x 0 ) = ũi (t, x 0 ), ∂u i ∂x (t, x 0 ) = ∂ ũi ∂x (t, x 0 ), for all t ∈ (0, ε) and all i ∈ {1, • • • , N }. (2.6) Then µ k ≡ μk on [a, b] for all k ∈ {1, • • • , N }, and consequently u i ≡ ũi in [0, T ) × [a, b] for all i.
The main result in [START_REF] Roques | On the determination of the nonlinearity from localized measurements in a reaction-diffusion equation[END_REF] was a particular case of Theorem 2.1. A similar conclusion was indeed proved in the case N = 1 and for g(x, u) = -γ u 2 . In such case, the determination of one coefficient µ 1 (x) only requires the knowledge of the initial condition u 0 and of (u(t, x 0 ), ∂u/∂x(t, x 0 )) for t ∈ (0, ε). When N ≥ 2, the above theorem requires more than the knowledge of the initial condition for the determination of the coefficients: we need a control on the initial condition, which enables to obtain N measurements of the solution of (P u 0 (µ k ) ), starting from N different initial conditions. A natural question is whether the result of Theorem 2.1 remains true when the number of measurements is smaller than N. In Section 4, we prove that the answer is negative in general.

Proof of Theorem 2.1

For the sake of clarity, we begin with proving Theorem 2.1 in the particular case N = 2 (the proof in the case N = 1 would be similar to that of [START_REF] Roques | On the determination of the nonlinearity from localized measurements in a reaction-diffusion equation[END_REF], which was concerned with g(x, u) = -γ u 2 ). We then deal with the general case of problems (P

u 0 i (µ k ) ) and (P u 0 i (μ k ) ) with N ≥ 1.
3.1 Proof of Theorem 2.1, case N = 2.

Let (µ 1 , μ1 ) and (µ 2 , μ2 ) be two pairs of coefficients in M. Let u 0 1 (x), u 0 2 (x) be two functions verifying (2.4) and (2.5) and such that u 0 1 (x) = u 0 2 (x) in (a, b). Let u 1 and ũ1 be respectively the solutions of (P

u 0 1 µ 1 ,µ 2 ) and (P u 0 1 μ1 ,μ 2
) and u 2 and ũ2 be the solutions of (P

u 0 2 µ 1 ,µ 2 ) and (P u 0 2 μ1 ,μ 2 ).
We set, for i = 1, 2,

U i := u i -ũi , m 1 := µ 1 -μ1 and m 2 := µ 2 -μ2 .
The functions U i satisfy:

∂U i ∂t -D ∂ 2 U i ∂x 2 = b i (t, x)U i + h(x, u i (t, x)), (3.1)
for t ∈ [0, T ) and x ∈ [a, b], where b i (t, x) = μ1 (x) + μ2 (x) (u i (t, x) + ũi (t, x)) + c i (t, x), c i (t, x) =        g(x, u i (t, x)) -g(x, ũi (t, x)) u i (t, x) -ũi (t, x) if u i (t, x) = ũi (t, x), ∂g ∂u (x, u i (t, x)) if u i (t, x) = ũi (t, x), h(x, s) = s (m 1 (x) + m 2 (x)s) , (3.2)
and the boundary and initial conditions:

           α 1 U i (t, a) -β 1 ∂U i ∂x (t, a) = 0, t > 0, α 2 U i (t, b) + β 2 ∂U i ∂x (t, b) = 0, t > 0, U i (0, x) = 0, x ∈ (a, b). (3.3)
Let us first assume that x 0 < b, and set:

A + = x ≥ x 0 s.t. m 1 (y) ≡ m 2 (y) ≡ 0 for all y ∈ [x 0 , x] ,
and

x 1 := sup (A + ) if A + is not empty, x 0 if A + is empty. If x 1 = b, then m 1 (x) ≡ m 2 (x) ≡ 0 on [x 0 , b].
Let us assume on the contrary that x 1 < b.

Step 1: We show that there exist θ > 0, x 2 ∈ (x 1 , b) and j ∈ {1, 2} such that the function

(t, x) → h(x, u j (t, x)) has a constant strict sign on [0, θ] × (x 1 , x 2 ], i.e. h(x, u j (t, x)) > 0 or h(x, u j (t, x)) < 0 in [0, θ] × (x 1 , x 2 ].
To do so, let us define, for all x ∈ [x 1 , b): (3.4) Whenever m 2 (x) = 0, z(x) is a root of the polynomial h(x, •). Notice also that the limit lim y→x + -m 1 (y) m 2 (y) in the second case of the definition of z(x) is well defined since m 1 and m 2 are analytic on [x, y] for y -x > 0 small enough.

z(x) =              - m 1 (x) m 2 (x) if m 2 (x) = 0, lim y→x + - m 1 (y) m 2 (y) if m 1 (x) = m 2 (x) =
Since, u 0 1 (x 1 ) = u 0 2 (x 1 ), we have |u 0 j (x 1 ) -z(x 1 )| > 0 for some j ∈ {1, 2}. Moreover, there exists δ > 0 such that x 1 + δ < b and

|u 0 j (x) -z(x)| ≥ r > 0 on [x 1 , x 1 + δ],
for some r > 0.

(3.5) Indeed, if z(x 1 ) = ±∞, we clearly have (3.5). If z(x 1 ) = ±∞, from (3.4), z is continuous in a right neighborhood of x 1 . As the function u 0 j is also continuous in this neighborhood, we get (3.5). Moreover,

u j (t, x) ∈ C 2,η 1,η/2 ([0, T ) × [a, b]
) . This implies that u j is continuous at (t, x) = (0, x 1 ). As a consequence, there exists θ > 0 small enough so that |u j (t, x) -

u 0 j (x 1 )| ≤ r/4 in [0, θ] × [x 1 , x 1 + θ], whence |u j (t, x) -u 0 j (x)| ≤ r 2 in (t, x) ∈ [0, θ] × [x 1 , x 1 + θ]. (3.6)
Finally, setting δ ′ = min{θ, δ}, we have, from (3.5) and (3.6):

|u j (t, x) -z(x)| ≥ |u j (t, x) -u 0 j (x)| -|u 0 j (x) -z(x)| ≥ r 2 > 0 (3.7) for all (t, x) ∈ [0, θ] × [x 1 , x 1 + δ ′ ].
Now, the definition of x 1 and the piecewise analyticity of m 1 and m 2 imply that there exists δ ′′ ∈ (0, δ ′ ) such that:

for all x ∈ (x 1 , x 1 + δ ′′ ], the polynomial function h(x, •) verifies h(x, •) ≡ 0 in R. (3.8)
Indeed, assume on the contrary that there is a decreasing sequence y n → x 1 such that h(y n , •) ≡ 0 in R. Since the functions h(y n , •) are polynomial, we get m 1 (y n ) = m 2 (y n ) = 0 for all n ∈ N.

Besides, as m 1 and m 2 belong to M, for n large enough, m 1 and m 2 are analytic on [x 1 , y n ], which implies that m 1 ≡ m 2 ≡ 0 on [x 1 , x n ] for n large enough. This contradicts the definition of x 1 and we get (3.8).

From the expression (3.2) of h and using (3.8), we observe that for each x ∈ (x 1 , x 1 + δ ′′ ], either s = 0 is the unique solution of h(x, s) = 0 or m 2 (x) = 0 and the equation h(x, s) = 0 admits exactly two solutions s = 0 and s = z(x). Let us set x 2 = x 1 + δ ′′ ∈ (x 1 , b). From the strong parabolic maximum principle u j (t, x) > 0 in [0, θ] × (a, b). Thus, using (3.7) we finally get:

h(x, u j (t, x)) = 0 in [0, θ] × (x 1 , x 2 ].
This concludes the proof of step 1.

Step 2: We prove that x 1 = b.

From Step 1, let us assume in the sequel -without loss of generality -that (t, x) → h(x, u j (t, x)) is positive on [0, θ] × (x 1 , x 2 ] (the case h(x, u j (t, x)) < 0 could be treated similarly). Then, from the definition of x 1 , we deduce that h(x, u j (t, x)) is nonnegative for all (t, x)

∈ [0, θ] × [x 0 , x 2 ].
Since h(x 2 , u j (0, x 2 )) > 0 and U j (0, •) ≡ 0, it follows from (3.1) that

∂U j ∂t (0, x 2 ) = h(x 2 , u j (0, x 2 )) > 0.
Thus, for ε ′ ∈ (0, θ) small enough, U j (t, x 2 ) > 0 for t ∈ (0, ε ′ ). As a consequence, and from the assumption of Theorem 2.1, U j satisfies:

       ∂U j ∂t -D ∂ 2 U j ∂x 2 -b j (t, x)U j ≥ 0, t ∈ (0, ε ′ ), x ∈ [x 0 , x 2 ], U j (t, x 0 ) = 0 and U j (t, x 2 ) > 0, t ∈ (0, ε ′ ), U j (0, x) = 0, x ∈ (x 0 , x 2 ).
Moreover, the weak and strong parabolic maximum principles give that U j (t, x) > 0 in (0, ε ′ ) × (x 0 , x 2 ). Since U j (t, x 0 ) = 0, the Hopf's lemma also implies that

∂U j ∂x (t, x 0 ) > 0 for all t ∈ (0, ε ′ ).
This contradicts the assumption (2.6) of Theorem 2.1. Finally, we necessarily have

x 1 = b and therefore m 1 ≡ m 2 ≡ 0 on [x 0 , b].
Step 3: We prove that

m 1 ≡ m 2 ≡ 0 on [a, b].
Assuming that x 0 > a, and setting:

A -= x ≤ x 0 s.t. m 1 (y) = m 2 (y) = 0 for all y ∈ [x, x 0 ] ,
and 3.2 Proof of Theorem 2.1, general case N ≥ 1.

y 1 := inf (A -) if A -is not empty, x 0 if A -is empty,
We set for all i, k ∈ {1, • • • , N },

U i := u i -ũi , m k := µ k -μk .
The functions U i satisfy:

∂U i ∂t -D ∂ 2 U i ∂x 2 = b i (t, x)U i + h(x, u i (t, x)), (3.9)
for t ∈ [0, T ) and x ∈ [a, b], where

b i (t, x) =        ( f + g)(x, u i (t, x)) -( f + g)(x, ũi (t, x)) u i (t, x) -ũi (t, x) if u i (t, x) = ũi (t, x), ∂( f + g) ∂u (x, u i (t, x)) if u i (t, x) = ũi (t, x), h(x, s) = N k=1 m k (x)s k , and f (x, u) = N k=1 μk (x)u k .
Moreover, the functions U i satisfy the following boundary and initial conditions:

           α 1 U i (t, a) -β 1 ∂U i ∂x (t, a) = 0, t > 0, α 2 U i (t, b) + β 2 ∂U i ∂x (t, b) = 0, t > 0, U i (0, x) = 0, x ∈ (a, b).
(3.10)

Let us set

A + = x ≥ x 0 s.t. m k ≡ 0 on [x 0 , x] for all k ∈ {1, • • • , N } , and
x 1 := sup (A + ) if A + is not empty, x 0 if A + is empty. If x 1 = b, then for all k, m k (x) ≡ 0 on [x 0 , b].
Let us assume by contradiction that x 1 < b. As in the case N = 2, we prove, as a first step, that there exist θ ∈ (0, T ),

x 2 ∈ (x 1 , b) and j ∈ {1, . . . , N } such that the function (t, x) → h(x, u j (t, x)) has a constant strict sign on [0, θ] × (x 1 , x 2 ], i.e. h(x, u j (t, x)) > 0 or h(x, u j (t, x)) < 0 in [0, θ] × (x 1 , x 2 ].
To do so, observe that, from the definitions of x 1 and M, there exists δ > 0 such that x 1 + δ < b and all functions m k are analytic on [x 1 , x 1 + δ] and not all identically zero. Therefore, the integer

ρ = max ρ ′ ∈ N, m k (x) = O (x -x 1 ) ρ ′ as x → x + 1 for all k ∈ {1, . . . , N }
is well-defined. Furthermore, the function h can then be written as

h(x, s) = (x -x 1 ) ρ × H(x, s) for all (x, s) ∈ [x 1 , x 1 + δ] × R, where H(x, s) = M 1 (x) s + • • • + M N (x) s N
and the functions M 1 , . . . , M N are analytic on [x 1 , x 1 + δ] and not all zero at the point x 1 (namely, there exists i ∈ {1, . . . , N } such that M i (x 1 ) = 0). In other words, the polynomial H(x 1 , •) is not identically zero. Since its degree is not larger than N and since H(x 1 , 0) = 0 and the real numbers u 0 1 (x 1 ), . . . , u 0 N (x 1 ) are all positive and pairwise different, there exists j ∈ {1, . . . , N } such that

H(x 1 , u 0 j (x 1 )) = 0. By continuity of H in [x 1 , x 1 + δ] × R and of u j on [0, T ) × [a, b],
it follows that there exist θ ∈ (0, T ) and

x 2 ∈ (x 1 , b) such that H(x, u j (t, x)) = 0 for all (t, x) ∈ [0, θ] × [x 1 , x 2 ]. Consequently, h(x, u j (t, x)) = 0 for all (t, x) ∈ [0, θ] × (x 1 , x 2 ].
The remaining part of the proof of Theorem 2.1 in the general case N ≥ 1 is then similar to Steps 2 and 3 of the proof in the particular case N = 2. Namely, we eventually get a contradiction with the assumption that ∂U j ∂x (t, x 0 ) = 0 for all t ∈ (0, ε), yielding x 1 = b and m k = 0 on [x 0 , b] for all k ∈ {1, . . . , N }. Similarly, m k = 0 on [a, x 0 ] for all k ∈ {1, . . . , N }.

Non-uniqueness results

This section deals with non-uniqueness results for the coefficients (µ k ) in (P u 0 (µ k ) ) under assumptions weaker than those of Theorem 2.1. These results emphasize the optimality of the assumptions of Theorem 2.1.

1-Number of measurements smaller than number of unknown coefficients.

We give a counter-example to the uniqueness result of Theorem 2.1 in the case where the number of measurements is smaller than the number of unknown coefficients N .

Assume that the coefficients µ 1 , . . . , µ N are constant, not all zero, and such that the polynomial

f (x, u) = f (u) = N k=1 µ k u k
admits exactly N -1 positive and distinct roots z 1 , . . . , z N -1 . Assume furthermore that α 1 = α 2 = 0 (Neumann boundary conditions) and that g ≡ 0. Then for each i = 1, . . . , N -1, z i is a (stationary) solution of (P z i (µ k ) ). Consider a similar problem with the coefficients μk = τ µ k for τ = 1 and k = 1, . . . , N. Then, again, for each i = 1, . . . , N -1, z i is a solution of (P z i (μ k ) ). In particular, assumption (2.6) is fulfilled at any point

x 0 ∈ [a, b] for k ∈ {1, . . . , N -1}. However (µ 1 (x), . . . , µ N (x)) = (μ 1 (x), . . . , μN (x)).
This shows that the determination of N coefficients (µ k ) 1≤k≤N requires in general N observations of the solution of (P u 0 (µ k ) ), starting from N different initial conditions.

2-Lack of measurement of the spatial derivatives.

We show that if hypothesis (2.6) in Theorem 2.1 is replaced with the weaker assumption:

u i (t, x 0 ) = ũi (t, x 0 ), for all t ∈ (0, ε) and all i ∈ {1, • • • , N }, (4.1)
then the conclusion of the theorem is false in general. Let (µ k ) 1≤k≤N ∈ M N and assume that α 1 = α 2 = 0 (Neumann boundary conditions). Let (u 0 i ) 1≤i≤N satisfy the assumptions of Theorem 2.1 and assume furthermore that the functions u 0 i and g(•, u) are symmetric with respect to x = (a + b)/2, i.e.

(u 0 i ) 1≤i≤N (x) = (u 0 i ) 1≤i≤N (b -(x -a)) g(x, •) = g(b -(x -a), •) for all x ∈ [a, b]. Let μk := µ k (b -(x -a)) for all x ∈ [a, b] and k ∈ {1, • • • , N }.
Then, we claim that the solutions u i and ũi of (P

u 0 i (µ k )
) and (P

u 0 i (μ k )
) satisfy (4.1) at x 0 = a+b 2 and for ε small enough. Indeed, we observe that, for each

i ∈ {1, • • • , N }, ũi (t, b -(x -a)) is a solution of (P u 0 i (µ k ) )
. By uniqueness, we have

u i (t, x) = ũi (t, b -(x -a)), for all t ∈ (0, T ), all x ∈ [a, b] and all i ∈ {1, • • • , N }.
In particular, u i (t, a+b 2 ) = ũi (t, a+b 2 ) for t ∈ (0, T ) and i ∈ {1, • • • , N }. This shows that the assumption (4.1) alone is not sufficient to determine the coefficients (µ k ) 1≤k≤N .

The above result is an adaptation of Proposition 2.3 in [START_REF] Roques | On the determination of the nonlinearity from localized measurements in a reaction-diffusion equation[END_REF] to the general case N ≥ 1.

3-Time-dependent coefficients.

We show here that the result of Theorem 2.1 is not true in general when the coefficients (µ k ) are allowed to depend on the variable t.

We place ourselves in the simple case N = 1 and g ≡ 0, and we assume that α 1 = α 2 = 0 (Neumann boundary conditions). We assume that (a, b) = (0, π), and we set u(t, x) = 1 + t cos 2 (x) and ũ(t, x) = 1 + t sin 2 (2x).

Let us set

µ 1 (t, x) = 1 u ∂u ∂t -D ∂ 2 u ∂x 2 (t, x) and μ1 (t, x) = 1 ũ ∂ ũ ∂t -D ∂ 2 ũ ∂x 2 (t, x)),
i.e., µ 1 (t, x) = (4Dt + 1) cos 2 (x) -2Dt 1 + t cos 2 (x) and μ1 (t, x) = (16Dt + 1) sin 2 (2x) -8Dt 1 + t sin 2 (2x) .

Then, for each t ∈ [0, +∞), µ 1 (t, •) and μ1 (t, •) belong to M. The functions u and ũ are solutions of (P 1 (µ 1 ) ) and (P 1 (μ 1 ) ), respectively, and they satisfy the assumption (2.6) of Theorem 2.1 at x 0 = π/2. However, the conclusion of Theorem 2.1 does not hold since µ 1 ≡ μ1 .

4-Unknown initial data.

We show here that the result of Theorem 2.1 is not true in general if the functions u i and ũi are solutions of (P

u 0 i (µ k ) ) and (P ũ0 i (μ k ) ), with u 0 i ≡ ũ0 i .
This means that the coefficients (µ k ) cannot be determined, given only the measurements (u i (t, x 0 ), ∂u i /∂x(t, x 0 )) 1≤i≤N for t ∈ (0, ε), if the initial conditions u 0 i are unknown. We build an explicit counter-example in the simple case N = 1 and g ≡ 0, with α 1 = α 2 = 0. Assume that (a, b) = (0, π), and let us set u(t, x) = (1 + cos 2 (x)) e ρ t for some ρ > 0, and ũ(t, x) = (1 + sin 2 (2x)) e ρ t . We furthermore set

µ 1 (x) = 1 u ∂u ∂t -D ∂ 2 u ∂x 2 and μ1 (x) = 1 ũ ∂ ũ ∂t -D ∂ 2 ũ ∂x 2 ,
i.e.,

µ 1 (x) = (4D + ρ) cos 2 (x) + ρ -2 D 1 + cos 2 (x) and μ1 (x) = (16D + ρ) sin 2 (2x) + ρ -8 D 1 + sin 2 (2x) ,
thus µ 1 and μ1 belong to M. Besides, u and ũ are solutions of (P

1+cos 2 (x) µ 1
) and (P

1+sin 2 (2x) μ1
) respectively, and they satisfy the assumption (2.6) of Theorem 2.1 at x 0 = π/2. However, we obviously have µ 1 ≡ μ1 .

Numerical determination of several coefficients

In the particular case N = 1, it was shown in [START_REF] Roques | On the determination of the nonlinearity from localized measurements in a reaction-diffusion equation[END_REF] that the measurements (2.6) of Theorem 2.1 are sufficient to obtain a good numerical approximation of a coefficient µ 1 (x). In this section, we check whether the measurements (2.6) of Theorem 2.1 also allow for an accurate reconstruction of N coefficients (µ k ) 1≤k≤N in the cases N = 2 and N = 3.

G (µ 1 ,µ 2 ) . In this case µ 1 -µ * 1 L 2 (0,1) + µ 2 -µ * 2 L 2 (0,1) = 0.
-µ * 1 L 2 (0,1) + µ 2 -µ * 2 L 2 (0,1) + µ 3 -µ * 3 L 2 (0,1) = 0.38 and G (µ 1 ,µ 2 ,µ 3 ) [(µ * 1 , µ * 2 , µ * 3 )] = 3 • 10 -5 .
Given the initial data u 0 i and the measurements u i (t, x 0 ) and ∂u i ∂x (t, x 0 ), for t ∈ (0, ε) and i ∈ {1, • • • , N }, we can look for the sequence (µ k ) 1≤k≤N as a minimizer of some functional G (µ k ) . Indeed, for any sequence (μ k ) 1≤k≤N in M N , the distance between the measurements of the solutions u i of (P u 0 i (µ k ) ) and ũi of (P

u 0 i (μ k ) ), i ∈ {1, • • • , N } can be evaluated through the function: G (µ k ) [(μ k )] = N i=1 u i (•, x 0 ) -ũi (•, x 0 ) L 2 (0,ε) + ∂u i ∂x (•, x 0 ) - ∂ ũi ∂x (•, x 0 ) L 2 (0,ε) .
Then, G (µ k ) [(µ k )] = 0 and from Theorem 2.1 this is the unique global minimum of G (µ k ) in M N . In our numerical computations, we fixed (a, b) = (0, 1), D = 0.1, α 1 = α 2 = 0 and β 1 = β 2 = 1 (Neumann boundary conditions), x 0 = 2/3 and ε = 0.3. Besides, we assumed the coefficients µ k to belong to a finite-dimensional subspace E ⊂ M :

E :=    ρ ∈ C 0,η ([0, 1]) | ∃ (h j ) 0≤j≤n ∈ R n+1 , ρ(x) = n j=0 h j • J ((n -2) (x -c j )) on [0, 1]    , with c j = j-1 n-2 and J(x) = (x -2) 4 (x + 2) 4 2 8 if x ∈ (-2, 2
), and J(x) = 0 otherwise. In our computations, the integer n was set to 10.

Case N = 2 : 25 couples of functions (µ 1 , µ 2 ) have been randomly sampled in E 2 : for k = 1 and k = 2, the components h k j , in the expression

µ k (x) = n j=0 h k j • J ((n -2) (x -c j )) ,
were randomly drawn from a uniform distribution in (-5, 5).

Starting from the initial data u 0 1 ≡ 0.1, and u 0 2 ≡ 0.2, the corresponding values of u 1 (t, x 0 ), ∂u 1 ∂x (t, x 0 ), u 2 (t, x 0 ), ∂u2 ∂x (t, x 0 ) were recorded1 , which enabled us to compute G (µ 1 ,µ 2 ) [(μ 1 , μ2 )] for any couple (μ 1 , μ2 ) in E 2 . The minimizations 2 of the functions G (µ 1 ,µ 2 ) lead to 25 couples (µ * 1 , µ * 2 ), each one corresponding to a computed minimizer of the function G (µ 1 ,µ 2 ) .

The average value of the quantity µ 1 -µ * 1 L 2 (0,1) + µ 2 -µ * 2 L 2 (0,1) , over the 25 samples of couples (µ 1 , µ 2 ) is 0.25. The corresponding average value of G (µ 1 ,µ 2 ) (µ * 1 , µ * 2 ) is 1.5 • 10 -5 . Fig. 1 (a) depicts an example of a couple (µ 1 , µ 2 ) in E, together with the couple (µ * 1 , µ * 2 ) which was obtained by minimizing G (µ 1 ,µ 2 ) .

Case N = 3 : in this case, the minimization of the function G (µ 1 ,µ 2 ,µ 3 ) is more time-consuming. We therefore focused on a unique example of a triple (µ 1 , µ 2 , µ 3 ) in E3 . The initial data were chosen as follows: u 0 1 ≡ 0.1, u 0 2 ≡ 0.2, and u 0 3 ≡ 0.3. Fig. 1 (b) depicts the triple (µ 1 , µ 2 , µ 3 ) in E, together with the triple (µ * 1 , µ * 2 , µ * 3 ) obtained by minimizing G (µ 1 ,µ 2 ,µ 3 ) .

Discussion

We have obtained a uniqueness result in the inverse problem of determining several non-constant coefficients of reaction-diffusion equations. With a reaction term containing an unknown polynomial part of the form N k=1 µ k (x) u k , our result provides a sufficient condition for the uniqueness of the determination of this nonlinear polynomial part.

This sufficient condition, which is detailed in Theorem 2.1, involves pointwise measurements of the solution u(t, x 0 ) and of its spatial derivative ∂u/∂x(t, x 0 ) at a single point x 0 , during a time interval (0, ε), and starting with N nonintersecting initial conditions.

The results of Section 4 show that most conditions of Theorem 2.1 are in fact necessary. In particular, the first counter-example of Section 4 shows that, for the result of Theorem 2.1 to hold in general, the number of measurements of the couple (u, ∂u/∂x)(t, x 0 ) needs to be at least equal to the degree (N ) of the unknown polynomial term.

From a practical point of view, such measurements can be obtained if one has a control on the initial condition. Nevertheless, since our result does not provide a stability inequality, the possibility to do a numerical reconstruction of the unknown coefficients µ k , on the basis of pointwise measurements, was uncertain. In Section 5, we have shown in the cases N = 2 and N = 3 -which include the classical models (1.2) and (1.3) -that such measurements can indeed lead to good numerical approximations of the unknown coefficients, at least if they are assumed to belong to a known finite-dimensional space.

0 and m 2

 2 (y) = 0 in a right neighborhod of x,+∞otherwise.

  we can prove, by applying the same arguments as above, that y 1 = a and consequently m 1 ≡ m 2 ≡ 0 on [a, x 0 ].Finally, m 1 ≡ m 2 ≡ 0 on [a, b] which concludes the proof of Theorem 2.1 in the case N = 2.

Figure 1 :

 1 Figure 1: (a) Plain lines: examples of functions µ 1 (red line) and µ 2 (blue line) in E. Dashed lines: the functions µ * 1 (in red) and µ * 2 (in blue) obtained by minimizingG (µ 1 ,µ 2 ) . In this case µ 1 -µ * 1 L 2 (0,1) + µ 2 -µ * 2 L 2 (0,1) = 0.15, and G (µ 1 ,µ 2 ) [(µ * 1 , µ * 2 )] = 9 • 10 -6 . (b) Plain lines: functions µ 1 (red line), µ 2 (blue line) and µ 3 (black line) in E. Dashed lines: the functions µ * 1 (in red), µ * 2 (in blue), µ * 3 (in black) obtained by minimizingG (µ 1 ,µ 2 ,µ 2 ) . Here, µ 1 -µ * 1 L 2 (0,1) + µ 2 -µ * 2 L 2 (0,1) + µ 3 -µ * 3 L 2 (0,1) = 0.38 and G (µ 1 ,µ 2 ,µ 3 ) [(µ * 1 , µ * 2 , µ * 3 )] = 3 • 10 -5 .

  15, and G (µ 1 ,µ 2 ) [(µ * 1 , µ * 2 )] = 9 • 10 -6 . (b) Plain lines: functions µ 1 (red line), µ 2 (blue line) and µ 3 (black line) in E. Dashed lines: the functions µ * 1 (in red), µ * 2 (in blue), µ * 3 (in black) obtained by minimizing G (µ 1 ,µ 2 ,µ 2 ) . Here, µ 1

Numerical computation of u and ũ were carried out with Comsol Multiphysics time-dependent solver. We used a second order finite element method (FEM) with 960 elements. This solver uses a method of lines approach incorporating variable order and variable stepsize backward differentiation formulas.

The minimizations of the functions G (µ k ) were performed using MATLAB's fminunc solver. This optimization algorithm uses a Quasi-Newton method with a mixed quadratic and cubic line search procedure. The stopping criterion was based on the number of evaluations of the function G (µ 1 ,µ 2 ) which was set at 4 • 10

.

Acknowledgements

The authors are supported by the French "Agence Nationale de la Recherche" within the projects ColonSGS (third and fourth authors), PREFERED (second, third and fourth authors) and UR-TICLIM (second and fourth authors). The third author is also indebted to the Alexander von Humboldt Foundation for its support.