Alessandra Bernardi -Damiano

Fulghesu

ON GENERALIZED KUMMER OF RANK 3 VECTOR BUNDLES OVER A GENUS 2 CURVE

Introduction.

Let X be a smooth projective complex curve and let U X (r, d) be the moduli space of semi-stable vector bundles of rank r and degree d on X (see [START_REF] Seshadri | Fibrés vectoriels sur les courbes algébriques[END_REF]). It contains an open Zariski subset U X (r, d) s which is the coarse moduli space of stable bundles, i.e. vector bundles satisfying inequality

d F r F < d E r E .
The complement U X (r, d) \ U X (r, d) s parametrizes certain equivalence classes of strictly semi-stable vector bundles which satisfy the equality

d F r F = d E r E .
Each equivalence class contains a unique representative isomorphic to the direct sum of stable bundles. Furthermore one considers subvarieties SU X (r, L) ⊂ U X (r, d) of vector bundle of rank r with determinant isomorphic to a �xed line bundle L of degree d . In this work we study the variety of strictly semistable bundles in SU X (3, O X ), where X is a genus 2 curve. We call this variety the generalized Kummer variety of X and denote it by Kum 3 (X ). Recall that the classical Kummer variety of X is de�ned as the quotient of the Jacobian variety Jac(X ) = U X (1, 0) by the involution L � → L -1 . It turns out that our Kum 3 (X ) has a similar description as a quotient of Jac(X ) × Jac(X ) which justi�es the name. We will see that the �rst de�nition allows one to de�ne a natural embedding of Kum 3 (X ) in a projective space (see section 4). The second approach is useful in order to give local description of Kum 3 (X ) by following the theory developed in [START_REF] Beauville | Variété Käheleriennes dont la première classe de Chern est nulle[END_REF] (section 3). We point out the use of [START_REF] Capani -G. Niesi -L | CoCoA, A system for doing Computations in Commutative Algebra[END_REF] for local computations. We want to tanks Professor Dolgachev for his patient guidance and his generous suggestions and also Professor Ragusa for a good organization of the Pragmatic.

Generalized Kummer variety.

Let A be an s-dimensional abelian variety, A r the r -Cartesian product of A, and A (r) := A r /� r be the r -symmetric power of A. We can de�ne the usual map a r : A (r) → A such that a r ({x 1 , . . . , x r }) = x 1 + • • • + x r 1 . This surjective map is just a morphism of varieties since there is no group structure on A (r) . However, all �bers of a r are naturally isomorphic.

De�nition 2.1. The generalized K ummer r variety associated to an abelian variety A is

Kum r (A) := a -1 r (0).

It is easy to see that dim(Kum r (a)) = s(r -1).

When dim A > 1, A (r) is singular. If dim A = 2, A (r) admits a natural desingularization isomorphic to the Hilbert scheme A [r] := Hilb(A) [r] of 0dimensional subschemes of A of length r (see [START_REF] Göttsche | Hilbert schemes of points on surfaces[END_REF]). Let pr : A [r] → A (r) be the usual projection. It is known that the restriction of pr over Kum r (A) is a resolution of singularities. Also � Kum r (A) admits a structure a holomorphic symplectic manifold (see [START_REF] Beauville | Variété Käheleriennes dont la première classe de Chern est nulle[END_REF]).

The Kummer variety of Jacobians.

Let X be a smooth connected projective curve of genus g and SU X (r, L) be the set of semi-stable vector bundles on X of rank r and determinant which is 1 here {x 1 , . . . , x r } mean an unordered set of r elements. isomorphic to a �xed line bundle L. Let Jac(X ) be the Jacobian variety of X which parametrizes isomorphism classes of line bundles on X of degree 0, or, equivalently the divisor classes of degree 0. We have a natural embedding:

Kum r (Jac(X )) �→ SU X (r, O X ) {a 1 , . . . , a r } � → (L a 1 ⊕ . . . ⊕ L a r )
where L a i := O X (a i ). Obviously, the condition a 1 + . . . + a r = 0 means that det(L a 1 ⊕ . . . ⊕ L a r ) = 0 and deg(L A i ) = 0 for all i = 1, . . . , r . Consequently the Kummer variety Kum r (Jac(X )) describes exactly the completely decomposable bundles in SU X (r) (from now on well write only SU X (r) instead of SU X (r, O X )).

In this paper we restrict ourselves with the case g = 2 and rank r = 3. In this case Kum 3 (Jac(X )) is a 4-fold.

Singular locus of Kum 3 (Jac(X )).

From now we let A denote Jac(X ). Let us de�ne the following map:

π : A (2) → Kum 3 (A) {a, b} � → L a ⊕ L b ⊕ L -a-b .
This map is well de�ned and it is a (3 : 1) -covering of Kum 3 (A). Let now ρ : A 2 → A (2) be the (2 : 1)-map which sends (x , y) ∈ A 2 to {x , y} ∈ A (2) . If we consider the map: [START_REF] Beauville | Variété Käheleriennes dont la première classe de Chern est nulle[END_REF] p := (π • ρ) : A 2 → A (2) → Kum 3 (A) ⊂ A (3) we get a (6 : 1)-covering of Kum 3 (A).

Notations: Let X and Y be two varieties and f : X → Y be a �nite morphism. We let Sing(X ) denote the singular locus of X , B f ⊆ Y the branch locus of f and R f ⊆ X the rami�cation locus of f .

Observation: B π = π (B ρ ).
Proof. Since B ρ = {{x , y} ∈ A (2) |x = y} and π ({x

, x }) = {x , x , -2x } ∈ B π we obviously get that π (B ρ ) ⊂ B π .
Conversely, for any point {x , y, z} of B π , at least two of the three elements x , y, z are equal to some t . Therefore π ({t, t}) = {x , y, z}, and hence B π ⊂ π (B ρ ). � Since A 2 is smooth, we have Sing(A (2) ) ⊂ B ρ . Obviously B ρ ⊂ R π , hence Sing(Kum 3 (A)) ⊂ B π . As a consequence we obtain that Sing(Kum 3 (A)) ⊆ B π . Therefore we have to study the (3 : 1)-covering π : A (2) → Kum 3 (A).

Since π is not a Galois covering, in order to give the local description at every point Q ∈ Kum 3 (A), we have to consider the following three cases separately:

1. Q ∈ Kum 3 (A) s.t. π -1 (Q) is just a point; 2. Q ∈ Kum 3 (A) s.t. π -1 (Q) is a set of two different points; 3. Q ∈ Kum 3 (A) s.t. π -1 (Q) is a set of exactly three points.
Let us begin studying these cases. (2) . Then Q is a smooth point of the Kummer variety.

Case 3. When Q ∈ Kum 3 (A) s.t. �(π -1 (Q)) = 3 we have that Q / ∈ B π . Since π (B ρ ) = B π any point of π -1 (Q) is smooth in A

Case 2. When

Q ∈ Kum 3 (A) s.t. �(π -1 (Q)) = 2 we �x the two points P 1 , P 2 ∈ A (2) s.t π (P 1 ) = π (P 2 ) = Q. In this case Q = {x , x , -2x } with x � = -2x ; let us �x P 1 = {x , x }, P 2 = {x , -2x }. Let U ⊂ Kum 3 (A) be a suf�ciently small analytic neighborhood of Q such that π -1 (U ) = U 1 � U 2
where U 1 and U 2 are respectively analytic neighborhoods of P 1 and P 2 and also

U 1 ∩ U 2 = ∅. Let � Q a generic point of U , so � Q = {x + �, x + δ, -2x -� -δ}; the preimage of � Q by π is π -1 ( � Q) = {{x + �, x + δ}, {x + �, -2x -� -δ}, {x + δ, -2x -� -δ}}, but {x + �, x + δ} ∈ U 1 and {x + �, -2x -� -δ}, {x + δ, -2x -� -δ} ∈ U 2 ,
it means that P 1 has rami�cation order equal to 1 and P 2 has rami�cation order equal to 2. Therefore there is an analytic neighborhood of P 1 which is isomorphic by π to an analytic neighborhood of Q. This allows us to study a generic point of B ρ instead of a generic point of B π .

Case 1. When Q ∈ Kum 3 (A) s.t. �(π -1 (Q)) = 3 we consider a point P ∈ A (2) s.t. π -1 (Q) = P ⇒ Q = {x , x , x } s.t. 3x = 0 ⇒ x is a 3-torsion point of A.
Now our abelian variety is a complex torus of dimension 2, so we have exactly 3 2g = 3 4 = 81 such points.

Proposition 3.1. The singular locus of Kum 3 (A) is a surface which coincides with the branch locus B π of the projection π :

A (2) → Kum 3 (A) and it is locally isomorphic at a generic point to (C 2 × Q, C × o)
where Q is a cone over a rational normal curve and o is the vertex of such a cone (see [START_REF] Beauville | Variété Käheleriennes dont la première classe de Chern est nulle[END_REF]).

Moreover there are exactly 81 points of Sing(Kum 3 (X )) whose local tangent cone is isomorphic to the spectrum of:

C[[u 1 , . . . , u 7 ]] I
where I is the ideal generated by the following polynomials :

u 2 5 -u 4 u 6 u 4 u 7 -u 5 u 6 u 2 6 -u 5 u 7 u 3 u 4 + u 2 u 5 + u 1 u 6 u 3 u 5 + u 2 u 6 + u 1 u 7 . Proof. According to what we saw in Case 2, an analytic neighborhood of Q ∈ Kum 3 (A) such that �(π -1 (Q)) = 2 is isomorphic to a generic element of B ρ .
We have to study the (2 : 1)-covering A 2 → A (2) . Since A = Jac(X ), A is a smooth abelian variety, this means that A is a complex torus (C g /Z 2g ) where g is the genus of X ; in our case X is a genus 2 curve, A � (C 2 /Z 4 ). Thus, in local coordinates at

P ∈ A, � O P � C[[z 1 , z 2 ]
], so we consider U P (a neighborhood of P ∈ A) isomorphic to C 2 . Therefore we obtain that locally at

Q ∈ A 2 , � O Q � � O P ⊗ � O P � C[[z 1 , z 2 ; z 3 , z 4 ]].
We �x a coordinate system (z 1 , z 2 ; z 3 , z 4 ) in A 2 such that A 2 ⊃ U P � P = (0, 0; 0, 0). Let Q be a point in U P , in the �xed coordinate system Q = (z 1 , z 2 ; z 3 , z 4 ). Since P is such that ρ(P) ∈ B ρ , by de�nition of ρ, we have:

A (2) = A 2 / < i >,
where i is the following involution of U P :

(2) i :

U P → U P i : (z 1 , z 2 ; z 3 , z 4 ) � → (z 3 , z 4 ; z 1 , z 2 ).
The involution i is obviously linear and its associated matrix is M = e 1,3 +e 3,1 + e 2,4 + e 4,2 (where e i, j is the matrix with 1 in the i, j position and 0 otherwhere). Its eigenvalues λ 1 = -1 and λ 2 = 1 have both multiplicity 2, so its diagonal form is:

� M = (1, 1, -1, -1)
which in a new coordinate system:

       x 1 = z 1 +z 3 2 x 2 = z 2 +z 4 2 x 3 = z 1 -z 3 2 x 4 = z 2 -z 4 2 .
corresponds to the linear transformation:

(x 1 , x 2 , x 3 , x 4 ) � → (x 1 , x 2 , -x 3 , -x 4 ).
The algebra of invariant polynomials with respect to this actions is generated by the homogeneous forms (x 1 , x 2 , x 2 3 , x 2 4 , x 3 x 4 ). Let us now consider these forms as local coordinates (s 1 , s 2 , s 3 , s 4 , s 5 ) around ρ(P), here we have that the completion of the local ring is isomorphic to the following one:

� C[[s 1 , . . . s 5 ]] (s 2 1 -s 2 s 3 ) � . Therefore B ρ at a generic point is locally isomorphic to (C 2 × Q, C × o)
where Q is a cone over a rational normal curve (we can see this rational normal curve as the image of P 1 in P 3 by the Veronese map ν 2 :

(P 1 ) * → (P 3 ) * , ν 2 (L) = L 2 )
and o the vertex of this cone. (What we have just proved in our particular case of Kum 3 (A) can be found in a more general form in [START_REF] Beauville | Variété Käheleriennes dont la première classe de Chern est nulle[END_REF].) Therefore we have the same local description of singularity of Kum 3 (A) out of the correspondent points of the 81 three-torsion points of A. Now we have to study what happens at those 3-torsion. Let Q 0 be one of them, we already know that p -1 (Q 0 ) = (x , x ) := P 0 is such that 3x = 0. Let us �x (z 1 , z 2 ; z 3 , z 4 ) ∈ C 2 × C 2 a local coordinate system around P 0 in order to describe locally the (6 : 1)-covering p : A 2 → Kum 3 (A). We observe that for a generic P in that neighborhood, the pre-image of p(P) is the set of the following 6 points: P 1 := (z 1 , z 2 ; z 3 , z 4 ),

P 6 := (z 1 , z 2 ; (-z 1 -z 3 ), (-z 2 -z 4 )).
Observe that i(P 1 ) = P 2 , i(P 3 ) = P 4 , i(P 5 ) = P 6 where i is the involution de�ned in [START_REF] Beauville -M | Spectral curves and the generalised theta divisor[END_REF]. We now de�ne a trivolution τ of C 2 × C 2 as follows:

(3)

τ : C 2 × C 2 → C 2 × C 2 (z 1 , z 2 ; z 3 , z 4 ) � → (z 3 , z 4 ; (-z 1 -z 3 ), (-z 2 -z 4 )).
It is easy to see that:

P 1 τ → P 3 τ → P 5 τ → P 1 , P 2 τ → P 6 τ → P 4 τ → P 2
The matrices that represent i and τ are respectively:

i =    0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0    τ =    0 0 1 0 0 0 0 1 -1 0 -1 0 0 -1 0 -1    furthermore < τ, i >� � 3 , then the local description of Kum 3 (X ) around Q 0 is isomorphic to A 2 /� 3 .
In what follows we have used [START_REF] Capani -G. Niesi -L | CoCoA, A system for doing Computations in Commutative Algebra[END_REF] program in order to do computations. First we recall Noethers theorem ( [START_REF] Cox -J. Little -Oshea | Ideals, Varieties, and Algorithms[END_REF] pag. 331) Theorem 3.2. Let G ⊂ G L(n, C) be a given �nite matrix group, we have:

C[z 1 , . . . , z n ] G = C[R G (z β ) : |β| ≤ |G|].
where R G is the Reynolds operator.

In other words, the algebra of invariant polynomials with respect to the action of G is generated by the invariant polynomials whose degree is at most the order of the group. In our case the order of G is 6, so it is not hard to compute C[z 1 , z 2 , z 3 , z 4 ] G . Then, after reducing the generators, we obtain that

C[z 1 , z 2 , z 3 , z 4 ] G is generated by: f 1 := z 2 2 + z 2 z 4 + z 2 4 , f 2 := 2z 1 z 2 + z 2 z 3 + z 1 z 4 + 2z 3 z 4 , f 3 := z 2 1 + z 1 z 3 + z 2 3 , f 4 := -3z 2 2 z 4 -3z 2 z 2 4 , f 5 := z 2 2 z 3 + 2z 1 z 2 z 4 + 2z 2 z 3 z 4 + z 1 z 2 4 , f 6 := -2z 1 z 2 z 3 -z 2 z 2 3 -z 2 1 z 4 -2z 1 z 3 z 4 , f 7 := 3z 2 1 z 3 + 3z 1 z 2 3 . Let us now write C[z 1 , . . . , z 4 ] G = C[ f 1 , . . . , f 7 ] as: C[u 1 , . . . , u 7 ]/I G ,
where I G is the syzygy ideal. It is easy to obtain that I G is generated by the following polynomials:

u 1 (u 2 2 -4u 1 u 3 ) + 3(u 2 5 -u 4 u 6 ) u 2 (u 2 2 -4u 1 u 3 ) + 3(u 4 u 7 -u 5 u 6 ) u 3 (u 2 2 -4u 1 u 3 ) + 3(u 2 6 -u 5 u 7 ) u 3 u 4 + u 2 u 5 + u 1 u 6 u 3 u 5 + u 2 u 6 + u 1 u 7 cr
and so we have the completion of the local ring at P:

� O P � C[[u 1 , . . . , u 7 ]] I G .
Let now calculate the tangent cone in Q 0 in order to understand which kind of singularity occurs in Q 0 . With [START_REF] Capani -G. Niesi -L | CoCoA, A system for doing Computations in Commutative Algebra[END_REF] aid we �nd that this local cone is:

Spec � C[[u 1 , . . . , u 7 ]] I �
where I is the ideal generated by the following polynomials:

u 2 5 -u 4 u 6 u 4 u 7 -u 5 u 6 u 2 6 -u 5 u 7 u 3 u 4 + u 2 u 5 + u 1 u 6 u 3 u 5 + u 2 u 6 + u 1 u 7 .
The degree of the variety V (I ) ⊂ P 6 is 5, this means that Q 0 is a singular point of multiplicity 5.

What we want to do now is to describe the singular locus of the local description. Let us start to calculate the Jacobian of V (I G ), what we �nd is the following 5 × 7 matrix:

J G :=        u 2 2 -8u 1 u 3 2u 1 u 2 -4u 2 1 -3u 6 6u 5 -3u 4 0 -4u 2 u 3 3u 2 2 -4u 1 u 3 -4u 1 u 2 3u 7 -3u 6 -3u 5 3u 4 -4u 2 3 2u 2 u 3 u 2 2 -8u 1 u 3 0 -3u 7 6u 6 -3u 5 u 6 u 5 u 4 u 3 u 2 u 1 0 u 7 u 6 u 5 0 u 3 u 2 u 1       
Local equations de�ne a fourfold, so we have to �nd the locus where the dimension of K er(J G ) is at least 5. In order to do it we calculate the minimal system of generators of all 3 × 3 minors of J G , we intersect the corresponding variety with V (I G ), we �nd a minimal base of generators of the ideal corresponding to this intersection and we compute its radical; the polynomials we �nd de-�ne, after suitable change of coordinates, the (reduced) variety of singular locus Let X be a curve of genus g and � Jac(X ) is the Riemann theta divisor. It is known that it is an ample divisor and dim|r� Jac(X ) | = r g -1

(see [START_REF] Grif�ths -Harris | Principles of algebraic geometry[END_REF] Theorem p. 317). Recall that for any �xed point q 0 ∈ X there exists an isomorphism:

ψ g-1,0 : Pic g-1 (X ) → Jac(X ) = Pic 0 (X ).
The set W g-1 of effective line bundles of degree g -1 is a divisor in Pic g-1 (X ) denoted by � Pic g-1 (X ) . By Riemanns Theorem there exists a divisor k of degree 0 such that:

ψ g-1,0 (� Pic g-1 (X ) ) = � Jac(X ) -k.
In a similar way we can de�ne the generalized theta divisor as follows:

� gen SU X (r,L) = {E ∈ Pic g-1 (X ) : h 0 (E ⊗ L) > 0}.
It is known that

Pic(SU X (r, L)) = Z� gen SU X (r,L) ,
and there exists a canonical isomorphism:

|r� Pic g-1 (X ) | � |� gen SU X (r) | *
(see [START_REF] Beauville -M | Spectral curves and the generalised theta divisor[END_REF]).

Degree of Kum 3 ( A)

Let us consider the (2 : 1)-map

φ 3 : SU 3 (X ) -→ |3� Pic 1 (X ) | � |� gen SU X (3) | * E � -→ D E = {L ∈ Pic 1 (X ) : h 0 (E ⊗ L) > 0}. De�nition 4.1. � η := {E ∈ SU X (3) : h 0 (E ⊗ η) > 0} ⊂ SU X (3)
where η is a �xed divisor in Pic 1 (X ).

Observation: φ 3 (� η ) = H η ⊂ |3� Pic 1 (X )
| and H η is a hyperplane. Since

φ 3 � � Kum 3 ( A) : Kum 3 (A) → φ 3 (Kum 3 (A)
) is a (1 : 1)-map (it is a well known fact but we will see it in the next section), we have that � η ∩ Kum 3 (X ) � H η ∩ φ 3 (Kum 3 (X )). In order to study the degree of Kum 3 (A) we have to take four generic divisors η 1 , . . . , η 4 ∈ Pic 1 (X ) and consider the respective

� η 1 , . . . , � η 4 ⊂ SU X (3). The intersection � η i ∩ Kum 3 (A) is equal to {L a ⊕ L b ⊕L -a-b ∈ Kum 3 (X ) : h 0 (L a ⊕L b ⊕L -a-b ⊗η i ) > 0} = {L a ⊕L b ⊕L -a-b ∈ Kum 3 (A) : h 0 (L a ⊗ η i ) > 0} ∪ {L a ⊕ L b ⊕ L -a-b ∈ Kum 3 (A) : h 0 (L b ⊗ η i ) > 0} ∪ {L a ⊕ L b ⊕ L -a-b ∈ Kum 3 (A) : h 0 (L -a-b ⊗ η i ) > 0} for all i = 1, . . . , 4. If L a ⊕ L b ⊕ L -a-b
is a generic element of Kum 3 (A) and p is the (6 : 1)covering of Kum 3 (A) de�ned as in [START_REF] Beauville | Variété Käheleriennes dont la première classe de Chern est nulle[END_REF]

, then p -1 (L a ⊕ L b ⊕ L -a-b ) ⊂ A 2 is a set of 6 points. Its easy to see that p((a, b)) ∈ � η i ∩ Kum 3 (X ) if and only if or h 0 (L a ⊗ η i ) > 0 or h 0 (L b ⊗ η i ) > 0 or h 0 (L -a-b ⊗ η i ) > 0 where (a, b) ∈ A 2 and L a , L b , L -a-b ∈ Pic 0 (X ) are three line bundles respectively associated to a, b, -a -b ∈ A.
Let us recall Jacobis Theorem ( [START_REF] Grif�ths -Harris | Principles of algebraic geometry[END_REF] page: 235): Jacobis Theorem: Let X be a curve of genus g, q 0 ∈ X and ω 1 , . . . , ω g a basis for H 0 (X, � 1 ). For any λ ∈ Jac(X ) there exist g points p 1 , . . . , p g ∈ X such that

µ( g � i=1 ( p i -q 0 )) = λ, where µ : Div 0 (X ) → Jac(X ) � i ( p i -q i ) � → � � i � p i q i ω 1 , . . . , � i � p i q i ω g � .
Since Jac(X ) is isomorphic to Pic 0 (X ), this theorem has the following two corollaries:

1. if q 0 is a �xed point of C, then for all L a ∈ Pic 0 (X ), there are two points

P 1 , P 2 in X such that L a � O X (P 1 + P 2 -2q 0 ); 2. Consider the isomorphism ψ 1,0 : Pic 1 (X ) ∼ -→ Pic 0 (X ) η � → η ⊗ O X (-q 0 ).
For every i = 1, . . . , 4 there are

q i 1 , q i 2 ∈ C such that η i � O X (q i 1 + q i 2 - q 0 ).
Now these two facts imply that h 0 (L a ⊗ η i ) > 0 if and only if h 0 (O X (P 1 + P 2 -2q 0 ) ⊗ O X (q i,1 + q i,2 -q 0 )) > 0, and this happens if and only if h 0 (O X (P 1 + P 2 + q i,1 + q i,2 -3q 0 )) > 0.

Notations: � -k is a translate of theta divisor by k ∈ Pic 0 (X ).

By Riemanns Singularity Theorem (see [START_REF] Grif�ths -Harris | Principles of algebraic geometry[END_REF], p. 348) the dimension h 0 (O X (P 1 + P 2 + q i,1 + q i,2 -3q 0 )) is equal to the multiplicity of ψ 1,0 (P 1 + P 2 + q i,1 + q i,2 -3q 0 ) in � -k (by a suitable k ∈ Pic 0 (X )), i.e. it is equal to the multiplicity of (P 1 + P 2 + q i,1 + q i,2 -4q 0 ) in � -k . It follows from this fact that h 0 (O X (P 1 + P 2 + q i,1 + q i,2 -3q 0 )) is greater than zero if and only if

(P 1 + P 2 + q i,1 + q i,2 -4q 0 ) ∈ � -k .
Notations:

� i := � -k-η i +q 0 ; R i := {(a, b) ∈ A 2 : (a + b) ∈ {-� i }}; � i := (� i × A) ∪ (A × � i ) ∪ R i . Now (P 1 + P 2 + q i,1 + q i,2 -4q 0 ) ∈ � -k iff P 1 + P 2 -2q 0 ∈ � i which is equivalent to say that L a belongs to � i , but this implies that p((a, b)) ∈ � η i ∩ Kum 3 (A) if and only if L a ∈ � i or L b ∈ � i or L -a-b ∈ � i (or equivalently L a+b belongs to {-� i }), i.e. (a, b) ∈ � i .
Therefore we can conclude:

(a, b) ∈ A 2 is such that p((a, b)) ∈ Kum 3 (A) ∩ � η i , i = 1, . . . , 4 if and only if (a, b) ∈ � i .
The last conclusion together with the observation that �( pr -1 (L a ⊕ L b ⊕ L -a-b )) = 6 gives the following proposition:

Proposition 4.2. deg(Kum 3 (A)) = 1 6 (�(� 1 ∩ � 2 ∩ � 3 ∩ � 4 )). Proof. �(� 1 ∩ � 2 ∩ � 3 ∩ � 4 ) = 6 • �(Kum 3 (A) ∩ � η 1 ∩ � η 2 ∩ � η 3 ∩ � η 4 ) = 6 • deg(Kum 3 (A)). � Notations: R a,i j = {(a, b) ∈ A 2 : a ∈ � i and (a + b) ∈ {-� j }}, R b,i j = {(a, b) ∈ A 2 : b ∈ � i and (a + b) ∈ {-� j }} and R 1,2 = {(a, b) ∈ A 2 : (a + b) ∈ {-� 1 } ∩ {-� 2 }}. Instead of computing directly � 1 ∩ � 2 ∩ � 3 ∩ � 4 , we will compute (� 1 ∩ � 2 ) ∩

On action of the hyperelliptic involution and Kum 3 ( A).

Let X be a curve of genus 2. Consider the degree 2 map:

φ 3 : SU X (3) 2:1 -→ P 8 = |3� Pic 1 (X ) | E � -→ D E = {L ∈ Pic 1 (X )/ h 0 (E ⊗ L) > 0}
(see [START_REF] Lazlo | Local structure of the moduli space of vector bundles over curves[END_REF]). Let τ � be the involution on SU X (3) acting by the duality:

τ � (E) = E *
and τ the hyperelliptic involution on Pic 1 (X ):

τ (L) = ω X ⊗ L -1 .
We will use the following well known relation:

τ • φ 3 (E) = φ 3 • τ � (E).
On SU X (3) there is also the hyperelliptic involution h * :

E � → h * (E)
induced by the hyperelliptic involution h of the curve X . We de�ne σ := τ � •h * . It is the involution which gives the double covering of SU X (3) on P 8 . The �xed locus of σ is obviously contained in SU X (3) and we recall: [START_REF] Göttsche | Hilbert schemes of points on surfaces[END_REF] φ 3 (Fix(σ )) = Coble sextic hypersurface (see [START_REF] Lazlo | Local structure of the moduli space of vector bundles over curves[END_REF]). By de�nition, the strictly semi-stable locus SU X (3) ss of SU X (3) consists of isomorphism classes of split rank 3 semi-stable vector bundles of determinant O X . Its points can be represented by the vector bundles of the form F ⊕ L or L a ⊕ L b ⊕ L c with trivial determinant where L, L a , L b , L c are line bundles and F is a rank 2 vector bundle. We want to consider the elements of the form L a ⊕ L b ⊕ L c (those belonging to Kum 3 (A)) and actions of previous involutions on them:

• τ � (L a ⊕ L b ⊕ L c ) = (L a ⊕ L b ⊕ L c ) * = L -a ⊕ L -b ⊕ L -c ; • τ � (h * (L a ⊕ L b ⊕ L c )) = L a ⊕ L b ⊕ L c .
This implies that σ (Kum 3 (A)) = Kum 3 (A) ⊂ SU X (3) which means that Kum 3 (A) ⊂ Fix(σ ) and then φ 3 (Kum 3 (X )) ⊂ Coble sextic (see 5).

Let us now consider rank 2 semistable vector bundles of trivial determinant: SU X (2). If we take its symmetric square, we obtain a semisable rank three vector bundle with trivial determinant:

SU X (2) → SU X (3); E � → Sym 2 (E).
We want to study the action of involutions de�ned on the beginning of this paragraph on Sym 2 (E) with E ∈ SU X [START_REF] Beauville -M | Spectral curves and the generalised theta divisor[END_REF]. Since Sym 2 (E) * = Sym 2 (E) = h * (Sym 2 (E)), then σ (Sym 2 (E)) = Sym 2 (E) ⊂ SU X (3), so Sym 2 (SU X (2)) ⊂ Fix(σ ), and, again by ( 5), φ 3 (Sym 2 (SU X (2))) ⊂ Coble sextic. Now we want to see the action of τ on |3� Pic 1 (X ) |. It is known that Fix(τ ) = P 4 � P 3 .

Notations:

We denote by P 3 τ and P 4 τ , respectively, the P 3 and the P 4 which are �xed by action of τ .

Since the image of Sym 2 (SU X (2)) by φ 3 in P 8 has dimension 3 and also φ 3 (Sym 2 (SU X (2))) ⊂ Fix(τ ), we obtain �

φ 3 (Sym 2 (SU X (2))) ⊂ P 4 τ . Let L a ⊕ L -a be an element of Kum 2 (X ) ⊂ SU X (2), then Sym 2 (L a ⊕ L -a ) = L 2a ⊕ L -2a ⊕ O ∈ Kum 3 (A) ⊂ SU X (3). It means that Sym 2 (Kum 2 (A)) ⊂ Kum 3 (A). Observation: Since {L 2a ⊕ L -2a ⊕ O ∈ SU X (3)} is isomorphic to S 2 ({L a ⊕ L -a }), we can view {L 2a ⊕ L -2a ⊕ O ∈ SU X ( 3

P 2 :

 2 = (z 3 , z 4 ; z 1 , z 2 ), P 3 := (z 3 , z 4 ; (-z 1 -z 3 ), (-z 2 -z 4 )), P 4 := ((-z 1 -z 3 ), (-z 2 -z 4 ); z 3 , z 4 ), P 5 := ((-z 1 -z 3 ), (-z 2 -z 4 ); z 1 , z 2 ),

  )} as the image of Kum 2 (A) inside SU X (3) under the symmetric square map. Moreover it follows from the surjectivity of the multiplication by 2 map[START_REF] Beauville -M | Spectral curves and the generalised theta divisor[END_REF] : A → A that the image of Kum 2 (A) in SU X (3) is isomorphic to Kum 2 (A).We have already observed that φ 3 � � Kum 3( A) is a (1 : 1)-map on the image; this fact allows us to view φ 3 (Kum 3 (A)) as the Kum 3 (A) in |3� Pic 1 (X ) |. For the same reason we can view φ 3 (Sym 2 (SU X (2))) as Kum 2 (A) ⊂ |3� Pic1(X ) |. Using this language we can say that Kum 2 (A) is left �xed by the action ofτ in Kum 3 (A) ⊂ |3� Pic1(X ) | because |3� Pic 1 (X ) | ⊃ φ 3 (Kum 3 (A)) ⊃ φ 3 (Sym 2 (SU X (2)) = Kum 2 (A) ⊂ P 4 ⊂ Fix(τ ) ⊂ |3� 1 Pic (X )|. Proposition 5.1. Fix(τ ) ∩ φ 3 (Kum 3 (A)) = φ 3 (Sym 2 (Kum 2 (A))). Proof. By de�nition τ (L a ⊕ L b ⊕ L c ) = L -a ⊕ L -b ⊕ L -c then L a ⊕ L b ⊕ L cbelongs to Fix(τ ) if and only if {a, b, c} = {-a, -b, -c}. Let P belong to {-a, -b, -c} and a = P .• If P is different from -a, suppose that P = -c, then {-a, -b, -c} = {-a, -b, a}; moreover a + b + c = 0 because L a ⊕ L b ⊕ L c ∈ Kum 3 (A), then b = 0.• Now, if P = -a or, equivalently a = -a, then a = 0 and b = -c.In both cases L a ⊕L b ⊕L c ∈ Kum 3 (A) such that τ (L a ⊕L b ⊕L c ) = L a ⊕L b ⊕L c are of the form L a ⊕ L -a ⊕ L 0 . This means that they belong to Kum 2 (A) ⊂ |3� Pic 1 (X ) |. �The previous proposition tells us also thatP 3 τ ∩ Kum 3 A = ∅. So the projection of Kum 3 (A) ⊂ |3� Pic 1 (X ) | from P 3 τ to P 4τ is a morphism. It would be interesting to �nd its degree.Our �nal observation is the following. Proposition 5.2. Sing(Kum 3 (A))∩ Kum 2 (A) = Sing(Kum 2 (A)) Proof. Points of Kum 2 (A) ⊂ Kum 3 (A) are of the form (P, -P, 0). Singular points of Kum 3 (A) are those which have at least two equal components, then Sing(Kum 3 (A))∩ Kum 2 (A) = {(P, -P, 0)} where 2P = 0 that are exactly the 15 points of 2-torsion and one more point (O X , O X , O X ) which are singularities of the usual Kum 2 (A). This implies that �(Sing(Kum 3 (A))∩ Kum 2 (A)) = 16 and Sing(Kum 3 (A))∩ Kum 2 (A) = Sing(Kum 2 (A)).

V (I S ), where I S = (u 2 6 -u 5 u 7 , u 5 u 6 -u 4 u 7 , u 2 5 -u 4 u 6 , u 3 u 6 -u 2 u 7 , u 3 u 5u 1 u 7 , u 2 u 6 -u 1 u 7 , u 3 u 4 -u 1 u 6 , u 2 u 5 -u 1 u 6 , u 2 u 4 -u 1 u 5 , u 2 2 -u 1 u 3 , u 3 3u 2 7 , u 2 u 2 3 -u 6 u 7 , u 1 u 2 3 -u 5 u 7 , u 1 u 2 u 3 -u 4 u 7 , u 2 1 u 3 -u 4 u 6 , u 2 1 u 2 -u 4 u 5 , u 3 1 -u 2 4 ). We veri�ed that the only one singular point of V (I S ) is the origin. Now, let us consider the map from C 2 to C 7 such that:

(4) (t, s) � → (t 2 , ts, s 2 , t 3 , t 2 s, ts 2 , s 3 ). This is the parametrization of V (I S ); as we have already done we can �nd relations between these polynomials and verify that the ideal we get is equal to I S . Now we can consider the following smooth parametrization from C 2 to C 9 : (t, s) � → (t, s, t 2 , ts, s 2 , t 3 , t 2 s, ts 2 , s 3 ) (which is nothing but the graph of ( 4)) whose projective closure is the Veronese surface ν 3 (P 2 ) = V 2,3 where ν 3 : (P 2 ) * → (P 9 ) * , ν 3 (L) = L 3 . What we want to �nd now is the tangent cone in Q 0 seen inside the singular locus. Using [START_REF] Capani -G. Niesi -L | CoCoA, A system for doing Computations in Commutative Algebra[END_REF] we �nd that its corresponding ideal � I C is generated by following polynomials:

The ideal � I C has multiplicity 4 (the corresponding variety has degree four) and its radical is the following ideal:

Then V (I C ) is a cone and V ( � I C ) is a double cone. This gives the description of the singularity at one of the 81 3-torsion points. �

Degree of Kum 3 ( A).

To �nd the degree of Kum 3 (A), we have to recall some general facts about theta divisors.

The Riemann theta divisor.

(� 3 ∩ � 4 ):

At the end we will obtain that �(� 1 ∩ � 2 ∩ � 3 ∩ � 4 ) = 216 (see also tables 1. and 2.) and so:

Proof. In the following two tables we write at place (i, j ) the cardinality of intersection of the subset of � 1 ∩ � 2 which we write at the place (0, j ), with the subset of � 3 ∩ � 4 which we write at the place (i, 0). Table 1.

In order to be more clear we show some cases:

�xed a+b the couple (a, b) has to belong to {(h 1 , -h 1 ), (h 2 , -h 2 )} where

Then, as in the previous case, we have

, but since � i are generic curves on a surface, their intersection two by two is the empty set, then �(R a, 1 2 )∩((� 3 ∩� 4 )× A) = 0. �

The degree of Sing(Kum 3 ( A))

As we have already seen, the singular locus of Kum 3 (A) is a surface. What we want to do now is to compute its degree. We use the notation from the previous section.

Let us �x two divisors � 1 and � 2 in A 2 . We denote by � the diagonal of Proof. The following table is used in the same way as we used Table 1 and Table 2 in the previous section:

Table 3.

The following list describes Table 3:

� ∩ R b,1 2 : the previous argument can be used for � ∩ R b,1 2 and � ∩ R b,2 1 : we have already counted these intersection points respectively in R a, 1 2 and in R a, 2 1 . R a,1 2 ∩�: we have now to show that �(R a,1

} which is equal to {(a, a) ∈ A × A : 2a ∈ ((-� 2 ) ∩ (2 • � 1 )) and a ∈ � 1 }. Let now L 1 be the line bundle on A associated to � 1 . The line bundle L 2 1 is associated to (2 • � 1 ) and its divisor is linearly equivalent to 2� 1 . As a consequence of this fact we have that 2a ∈ (2� 1 ∩ (-� 2 )) then �{2� 1 ∩ (-� 2 )} = 4. Now, since the map from � 1 to (2 • � 1 ) is 1 : 1 we get the conclusion. R 1,2 ∩�: �nally we have that (R 1,2 ∩ �) is equivalent to the set {a ∈ A|2a ∈ ((-� 1 ) ∩ (-� 2 ))} whose cardinality is 32. �


A Bernardi

M V Catalisano

Some defective secant varieties to osculating varieties of Veronese surfaces

We consider the k-osculating varieties O k,d to the Veronese d-uple embeddings of P 2 . By studying the Hilbert function of certain zero-dimensional schemes Y ⊂ P 2 , we find the dimension of O s k,d , the (s -1) th secant varieties of O k,d , for 3 ≤ s ≤ 6 and s = 9, and we determine whether those secant varieties are defective or not.

Introduction.

The problem of determining the dimension of the higher secant varieties of a projective variety is a classical subject of study. In the present paper we are concerned with the (s -1) th higher secant varieties of O k,V n,d , where O k,V n,d is the k-osculating variety to the Veronese embedding V n,d of P n into P N (N = d+n n -1) via the complete linear system R d , where R = K[x 0 , . . . , x n ], and K is an algebraically closed field of characteristic zero.

This matter has been dealt with by several authors in the last few years (see [START_REF] Ballico | On the secant varieties to the tangent developable of a Veronese variety[END_REF], [START_REF] Ballico | On the secant varieties to the osculating variety of Veronese surfaces[END_REF], [START_REF] Ballico | A Terracini Lemma for osculating spaces with applications to Veronese surfaces[END_REF], [START_REF] Bernardi | Osculating varieties of Veronesean and their higher secant varieties[END_REF], [START_REF] Catalisano | On the Secant Varieties to the Tangential Varieties of a Veronesean[END_REF]). We wish to mention E.Ballico and C.Fontanari. In [START_REF] Ballico | On the secant varieties to the tangent developable of a Veronese variety[END_REF] and [START_REF] Ballico | On the secant varieties to the osculating variety of Veronese surfaces[END_REF] they study the higher secant varieties of O k,V n,d for n = 2 and k = 1, 2, and they prove the following results: 0.1. Proposition. For k = 1, the (s -1) th higher secant variety of the tangential variety to V 2,d has the expected dimension, unless s = 2 and d = 3. 0.2. Proposition. For k = 2, the (s -1) th higher secant variety of 2-osculating variety to V 2,d has the expected dimension, unless s = 2 and d = 4.

In this note, for n = 2, for 3 ≤ s ≤ 6 and s = 9, and for all k, we will determine the dimension of O s k,V 2,d . The methods for proving our results are similar to the ones used by Ballico and Fontanari. The basic idea is to use Terracini's Lemma (see [START_REF] Terracini | Sulle V k per cui la varietà degli S h (h + 1)-seganti ha dimensione minore dell'ordinario[END_REF]), then, via apolarity, the calculation of dim O s k,V 2,d is related to the evaluation of the Hilbert function of certain 0-dimensional schemes Y ⊂ P 2 supported at s generic points (see [START_REF] Bernardi | Osculating varieties of Veronesean and their higher secant varieties[END_REF]), and this is done using geometric constructions, Bezout's theorem, and the Horace Method ( [START_REF] Hirschowitz | La méthode d'Horace pour l'interpolation à plusieurs variables[END_REF]).

In the first section we fix some notation, and describe the relationship between the higher secant varieties we want to study, and the 0-dimensional schemes Y .

In the second section we relate the Hilbert function of Y to the Hilbert function H(X, d) of a scheme X of s generic (k + 1)-fat points (Lemma 2.2, Proposition 2.6 to 2.9, and Proposition 2.12), and in Theorem 2. [START_REF] Terracini | Sulle V k per cui la varietà degli S h (h + 1)-seganti ha dimensione minore dell'ordinario[END_REF] we prove the main result of this paper, i.e., for s ≤ 6 and s = 9:

dim O s k,V 2,d = min{H(X, d) + 2s, N + 1} -1, except when s = 2, d = k + 2. In this case dim O 2 k,V 2,k+2 = H(T, d) -1 = N -1,
where H(T, d) is the Hilbert function of a scheme T of s generic (k + 2)-fat points.

We wish to warmly thank Monica Idà and Sandro Gimigliano for many interesting conversation about the questions considered in this paper, and the referee for his several helpful comments.

1. Preliminaries and notation.

1.1. Definition. If V ⊂ P N is an irreducible projective variety, an m-fat point on V is the (m -1) th infinitesimal neighborhood of a smooth point P in V , and it will be denoted by mP (i.e., the scheme mP is defined by the ideal sheaf I m P,V ⊂ O V ). If dim V = n, then, mP is a 0-dimensional scheme of length m-1+n n

. If X is the union of the (m -1) th infinitesimal neighborhoods in V of s generic smooth points of V , we will say for short that X is union of s generic m-fat points on V .

1.2. Remark. In general it is a hard problem to determine the postulation for a union of m-fat points. If V = P 2 , there is a conjecture for the postulation of a generic union X ⊂ P 2 of s m-fat points (e.g. see [START_REF] Harbourne | Problems and progress: A survey on fat points in P 2 . Queen's Pap[END_REF]): for s ≥ 10 the conjecture says that X is regular in any degree d. This has been proved for m ≤ 20 in [START_REF] Ciliberto | Bivariate Hermite interpolation and linear systems of plane curves with base fat points[END_REF], and, when s is a square, by L.Evain in [START_REF] Evain | Computing limit linear series with infinitesimal methods[END_REF]. For s ≤ 9 all the defective cases are known (e.g., see [START_REF] Ciliberto | Bivariate Hermite interpolation and linear systems of plane curves with base fat points[END_REF] or [START_REF] Harbourne | Problems and progress: A survey on fat points in P 2 . Queen's Pap[END_REF]), more precisely, for any m and s ≤ 9 the cases in which X ⊂ P 2 is not regular are: i) s = 2, and m ≤ d ≤ 2m -2;

ii) s = 3, and 3m 2 ≤ d ≤ 2m -2; iii) s = 5, and 2m ≤ d ≤ 5m-2 2 ; iv) s = 6, and 12m 5 ≤ d ≤ 5m-2 2 ; v) s = 7, and 21m 8 ≤ d ≤ 8m-2 3 ; vi) s = 8, and 48m 17 ≤ d ≤ 17m-2

6

. Now we recall the notions of higher secant variety and k th osculating variety.

1.3. Definition. Let V ⊂ P N be a closed irreducible projective variety; the (s -1) th higher secant variety of V is the closure of the union of all linear spaces spanned by s points of V , and it will be denoted by

V s . Let dim V = n; the expected dimension for V s is ( †) expdimV s = min {sn + s -1, N }
where the number sn + s -1 corresponds to ∞ sn choices of s points on V , plus ∞ s-1 choices of a point on the P s-1 spanned by the s points. When dim V s < min{sn + s -1, N }, the variety V s is said to be defective, with defect δ = min{sn + s -1, N } -dim V s .

1.4. Definition. Let V ⊂ P N be a variety, and let P ∈ V be a smooth point; we define the k th osculating space to V at P, and we denote it by O k,V,P , as the linear space defined by the vanishing of all linear forms L such that L| V vanishes to order k + 1 on V at P . Let V 0 ⊂ V be the dense set of the smooth points where O k,V,P has maximal dimension. The k th osculating variety to V is defined as:

O k,V = P ∈V0
O k,V,P .

1.5. Notation. Set R = K[x, y, z] = R d . Let V d ⊂ P N , N = d+2 2
-1, denote the d-ple Veronese embedding of P 2 , defined by the linear system R d of all forms of a given degree d. Set O k,d = O k,V d , so that the (s -1) th higher secant variety to the k th osculating variety to the Veronese surface V d will be denoted by O s k,d .

1.6. Remark. We have (see [START_REF] Bernardi | Osculating varieties of Veronesean and their higher secant varieties[END_REF], Lemma 2.3) that the dimension of O k,d is always the expected one, that is

dimO k,d = min k + 2 2 + 1, d + 2 2 -1 .
For d ≤ k we immediately get O k,d = P N , hence for d ≤ k and for all s, we have O s k,d = P N . Now we briefly recall how to associate to O s k,d , a zero dimensional scheme Y ⊂ P 2 (see [START_REF] Bernardi | Osculating varieties of Veronesean and their higher secant varieties[END_REF], Remark 2.2).

Remark.

Let P N = P(R d ), and let d ≥ k + 1. A form M ∈ R d will denote, depending on the situation, a vector in R d or a point in P N . We can view V d as the image's closure of the map (P 2 ) * → P N , where

L → L d , L ∈ R 1 . Hence V d = {L d , L ∈ R 1 }. At the point Q = L d we have O k,V d ,Q = {L d-k F, F ∈ R k } and O k,d = Q∈V d O k,V d ,Q . So we have: O k,d = {L d-k F, L ∈ R 1 , F ∈ R k } hence O s k,d = {L d-k 1 F 1 + • • • + L d-k s F s , L i ∈ R 1 , F i ∈ R k , i = 1, . . . , s}. Let P i = L d-k i F i be a generic point in O k,d , and let T O k,d ,Pi be the tangent space of O k,d at P i . The affine cone over T O k,d ,Pi is W i =< L d-k i R k , L d-k-1 i F i R 1 >.
Terracini's Lemma (see [START_REF] Terracini | Sulle V k per cui la varietà degli S h (h + 1)-seganti ha dimensione minore dell'ordinario[END_REF]) says that the tangent space of O s k,d at a generic point of < P 1 , . . . , P s >,

(P 1 , . . . , P s ∈ O k,d ), is the span of the tangent spaces of O k,d at P i (1 ≤ i ≤ s); if T O k,d ,Pi = P(W i ), then dim O s k,d = dim < T O k,d ,P1 , . . . , T O k,d ,Ps >= dim < W 1 , . . . , W s > -1
Now consider the ortogonal space W ⊥ i ⊂ R d , (1 ≤ i ≤ s) via the apolarity action (for the definition of W ⊥ i see [START_REF] Bernardi | Osculating varieties of Veronesean and their higher secant varieties[END_REF], Remark 2.5). It generates an ideal in R defining a scheme

Z i (k, d) ⊂ P 2 . Let Y be a generic union of s schemes Z i (k, d) in P 2 , (1 ≤ i ≤ s). Since dim < W 1 , . . . , W s > -1 = N -dim[< W 1 , . . . , W s >] ⊥ = N -dim(W ⊥ 1 ∩ • • • ∩ W ⊥ s ) = N -h 0 (P 2 , I Y (d)),
we have (see also [START_REF] Bernardi | Osculating varieties of Veronesean and their higher secant varieties[END_REF], Remark 2.5):

( ‡) dim O s k,d = N -h 0 (P 2 , I Y (d)) = H(Y, d) -1
where [START_REF] Bernardi | Osculating varieties of Veronesean and their higher secant varieties[END_REF], Proposition 3.4 C)), then for s ≥ 2 we immediately get:

H(Y, d) is the Hilbert function of Y in degree d. Since for d = k + 1, O 2 k,d = P N (see
1.8. Proposition. For d = k + 1 and s ≥ 2, we have O s k,d = P N .
For d ≥ k + 2, the schemes Z i (k, d) are zero-dimensional, and do not depend on d, in fact we have the following lemma (see [START_REF] Bernardi | Osculating varieties of Veronesean and their higher secant varieties[END_REF], Lemmata 2.6, 2.7, 2.8):

1.9. Lemma. Let Z(k, d) = Z i (k, d) be one such scheme with support at P . For d ≥ k + 2, we have:

i) (k + 1)P ⊂ Z(k, d) ⊂ (k + 2)P ; ii) the length of Z(k, d) is l(Z) = k+2 2 + 2; iii) Z(k, d) = Z(k, k + 2).
Henceforth for d ≥ k + 2 we will denote Z(k, d) by Z(k), or Z, if k is obvious by the context. From ( ‡) and the lemma above it follows that for d ≥ k + 2 in order to study the dimension of O s k,d , we only need to study the postulation of unions of generic schemes Z(k).

W =< L d-k R k , L d-k-1 F R 1 >, with L ∈ R 1 and F ∈ R k . Now
we want to give a specialization of the scheme Z(k): put L = x and F = y k ; we get

W =< x d-k R k , x d-k-1 y k R 1 > hence W ⊥ =< x d-k-1 y k-1 z 2 , . . . , x d-k-1 yz k , x d-k-1 z k+1 , x d-k-2 y k+2 , x d-k-2 y k+1 z, . . . , x d-k-2 yz k+1 , x d-k-2 z k+2 , x d-k-3 y k+1 , x d-k-3 y k z, . . . , x d-k-3 yz k , x d-k-3 z k+1 , . . . , xy d-1 , xy d-2 z, . . . , xyz d-2 , xz d-1 , y d , y d-1 z, . . . , yz d-1 , z d > .
Let I be the ideal generated by W ⊥ . By a direct computation, it is easy to show that the saturation of I is the ideal

(I) sat = (y, z) k+1 ∩ ((y, z) k+2 + (z 2 ))
that defines a scheme supported at a point of P 2 , whose structure is given by the union of its k th infinitesimal neighbourhood, with the intersection of its (k + 1) th infinitesimal neighbourhood with a double line.

1.11. Notation. We fix the following notation: i) let P 1 , . . . , P s be s generic points in P 2 ; ii) let X be the union of s generic (k + 1)-fat points in P 2 , with support in P 1 , . . . , P s ;

iii) let T be the union of s generic (k + 2)-fat points in P 2 , with support in P 1 , . . . , P s ; iv) let Z i be a 0-dimensional scheme in P 2 , as defined in Remark 1.7, with support in

P i ; v) let Y = Z 1 + • • • + Z s ;
vi) denote by (k + 1, k + 2)P a 0-dimensional scheme whose defining ideal is ℘ k+1 ∩ (℘ k+2 + l 2 ) where ℘ is the homogeneous ideal in R = K[x, y, z] of a point P ∈ P 2 , and l is the ideal of a generic line through P ; we call (k + 1, k + 2)P a (k + 1, k + 2) point; vii) let Z i be a (k +1, k +2) point with support in P i . By Remark 1.10, the scheme Z i is a specialization

of Z i ; viii) let Y = Z 1 + • • • + Z s (so Y is a specialization of the scheme Y ). We have deg Y = deg Y = s k + 2 2 + 2 = deg X + 2s; ix) if C ⊂ P 2 is
a curve, and Z is a zero-dimensional scheme, the scheme Z defined by the ideal (I Z :

I C )
is called the residual of Z with respect to C, and denoted by Res C Z.

In the following lemma we determine the subscheme of a (k + 1, k + 2) point with support in P , residual to a curve C. 1.12. Lemma. Let Z be a (k + 1, k + 2) point, with support in P with defining ideal

℘ k+1 ∩ (℘ k+2 + l 2 ),
where ℘ is the ideal of P , and l = (L) is the ideal of a generic line through P . Let C ⊂ P 2 be a curve having at P a singularity of multiplicity m, and having L as tangent direction with multiplicity t. Then Res C (Z) is defined by the ideal

I Res C (Z) = ℘ max{k+1-m;0} ∩ (℘ max{k+2-m;0} + l max{2-t;0} ). Res C (Z) is a fat point, or a (k + 1 -m, k + 2 -m)
point, except for m < k + 1 and t = 1, more precisely:

Res C (Z) =          0P f or m ≥ k + 2, or m = k + 1 and t ≥ 2 1P
f or m = k + 1 and t ≤ 1 (k + 1 -m)P f or m < k + 1 and t ≥ 2 2P

f or m = k and t = 0 (k + 1 -m, k + 2 -m)P f or m < k and t = 0 . Proof. Without loss of generality, we assume that ℘ = (x, y), L = x, and, by abuse of notation, that x, y are affine coordinates. Let x t f 1 + f 2 = 0 be an equation defining the curve C, where f 1 is a homogeneous polynomial of degree m -t, f 1 / ∈ (x) , and f 2 ∈ (x, y) m+1 . We have to prove that

((x, y) k+1 ∩ ((x, y) k+2 + (x 2 ))) : (x t f 1 + f 2 ) = (x, y) max{k+1-m;0} ∩ ((x, y) max{k+2-m;0} + (x max{2-t;0} )).
This is obvious for m ≥ k + 2, and for

m = k + 1, t ≥ 2, since in these cases Res C (Z) is the emptyset. Let m = k + 1, t ≤ 1.
The equality above becomes

((x, y) k+1 ∩ ((x, y) k+2 + (x 2 ))) : (x t f 1 + f 2 ) = (x, y). "⊆" : To prove this inclusion, let g = a + h, a ∈ K, h ∈ (x, y). If g (x t f 1 + f 2 ) = (a + h) (x t f 1 + f 2 ) ∈ ((x, y) k+2 + (x 2 )), since f 2 ∈ (x, y) m+1 , hx t f 1 ∈ (x, y) m+1 , and m + 1 = k + 2, it follows that ax t f 1 ∈ ((x, y) k+2 + (x 2 )). But f 1 is a homogeneous polynomial of degree m -t, f 1 / ∈ (x), t ≤ 1,
so it easily follows that a = 0, hence g ∈ (x, y). The reverse inclusion is obvious.

Since

I Res C (Z) = (x, y), we have Res C (Z) = 1P . Now, let m < k + 1, t ≥ 2.
In this case we have to prove that

((x, y) k+1 ∩ ((x, y) k+2 + (x 2 ))) : (x t f 1 + f 2 ) = (x, y) k+1-m . If g (x t f 1 + f 2 ) ∈ (x, y) k+1
, it immediately follows that g ∈ (x, y) k+1-m , and the reverse inclusion is obvious. Moreover, since

I Res C (Z) = (x, y) k+1-m , we have that Res C (Z) = (k + 1 -m)P . Let m ≤ k, t ≤ 1. Now we have to prove that ((x, y) k+1 ∩ ((x, y) k+2 + (x 2 ))) : (x t f 1 + f 2 ) = (x, y) k+1-m ∩ ((x, y) k+2-m + (x 2-t )). "⊆" : As in the previous case, if g (x t f 1 + f 2 ) ∈ (x, y) k+1 , it follows that g ∈ (x, y) k+1-m , so we can write g = xg 1 + ay k+1-m + g 2 ,
where

g 1 ∈ (x, y) k-m is homogeneous of degree k -m, g 2 ∈ (x, y) k+2-m , a ∈ K.
In order to prove that

g (x t f 1 + f 2 ) = (xg 1 + ay k+1-m + g 2 )(x t f 1 + f 2 ) ∈ ((x, y) k+2 + (x 2 )))
since g 2 x t f 1 , and f 2 ∈ (x, y) k+2 , it suffices to prove that

x t+1 g 1 f 1 + ax t y k+1-m f 1 ∈ ((x, y) k+2 + (x 2 )).
Since x t+1 g 1 f 1 +ax t y k+1-m f 1 is homogeneous of degree k+1, and f 1 / ∈ (x), we get that x t+1 g 1 +ax t y k+1-m ∈ (x 2 ). For t = 1, this implies a = 0, so g ∈ ((x, y) k+2-m + (x)). For t = 0 this implies a = 0, and g 1 ∈ (x), so g ∈ ((x, y) k+2-m + (x 2 )).

"⊇" : This inclusion is obvious. So we have proved that, for m ≤ k and t ≤ 1:

I Res C (Z) =    (x, y) k+1-m ∩ ((x, y) k+2-m + (x)) = (x, y) k+1-m ∩ (x, y k+2-m ) f or m ≤ k and t = 1 (x, y) ∩ ((x, y) 2 + (x 2 )) = (x, y) 2 f or m = k and t = 0 (x, y) k+1-m ∩ ((x, y) k+2-m + (x 2 )) f or m < k

Osculating varieties to

Veronese surface and some of their higher secant varieties.

In this section we will compute the dimension of O s k,d for 3 ≤ s ≤ 6 and s = 9.

2.1. Remark. We recall that for d ≤ k + 1 and s ≥ 2 (see Remark 1.6 and Proposition 1.8):

dim O s k,d = N.
So we have to study the dimension of O s k,d only for

d ≥ k + 2. Since, for d ≥ k + 2 (see( ‡)) dim O s k,d = H(Y, d) -1,
then, if we know the postulation of Y , we are done. We wish to notice that, by ( †), the expected dimension for

O s k,d is expdim O s k,d = min{sn + s -1, N }, where n = dimO k,d = min k+2 2 + 1, d+2 2 -1 = min k+2 2 + 1, N = min deg Y s -1, N (see Re- mark 1.6 and Lemma 1.9 ii)). Hence it easily follows that expdim O s k,d = min{deg Y, N + 1} -1 = exp H(Y, d) -1
where exp H(Y, d) is the expected value for H(Y, d).

In the next lemmata we show that the postulation of Y is strictly related with the postulation of the specialized scheme Y, and of the scheme of fat points X.

Lemma. If the Hilbert function of the specialized scheme

Y in degree d is H(Y, d) = min{H(X, d) + 2s, N + 1}, then H(Y, d) = min{H(X, d) + 2s, N + 1}.
Proof. It follows from the obvious inequalities: ii) follows from [START_REF] Ballico | On the secant varieties to the osculating variety of Veronese surfaces[END_REF] Theorems 1 and 2.

H(Y, d) ≤ H(Y, d) ≤ min {H(X, d) + 2s, N + 1}.
2.4. Lemma. i) If H(Y, d 0 ) = H(X, d 0 ) + 2s, then for every d ≥ d 0 we have H(Y, d) = H(X, d) + 2s; ii) if (I Y ) d0 = (0), then for every d ≤ d 0 we have (I Y ) d = (0). Proof. i) Since X ⊂ Y and H(Y, d 0 ) = H(X, d 0 ) + 2s, then it easily follows that dim (I X /I Y ) d0 = 2s.
Therefore there are 2s forms f 1 , . . . , f 2s ∈ (I X ) d0 linearly independent modulo (I Y ) d0 . Let {l = 0} be a line not through any of the points P 1 , . . . , P s . The forms

f 1 l d-d0 , . . . , f 2s l d-d0 ∈ (I X ) d are linearly independent modulo (I Y ) d , hence dim (I X /I Y ) d ≥ 2s, so we have H(Y, d) ≥ H(X, d) + 2s. Since obviously H(Y, d) ≤ H(X, d) + 2s
, then the conclusion follows.

ii) Obvious. Now we will study the postulation of Y for each s separately (s = 3, 4, 5, 6, 9), but first we wish to mention the case s = 2.

2.5. Proposition. For s = 2 we have:

H(Y, d) =                                for k = 1 :    N + 1 if d ≤ 2 H(T, d) = 9 < expH(Y, d) if d = 3 H(X, d) + 4 = deg Y if d ≥ 4
for k = 2 :

   N + 1 if d ≤ 3 H(T, d) = 14 < expH(Y, d) if d = 4 H(X, d) + 4 = deg Y if d ≥ 5
for k ≥ 3 :

     N + 1 if d ≤ k + 1 H(T, d) = N < expH(Y, d) if d = k + 2 H(X, d) + 4 < expH(Y, d) if k + 3 ≤ d ≤ 2k H(X, d) + 4 = deg Y if d ≥ 2k + 1 .
Proof. The case d ≤ k + 1 follows from Lemma 2.4 ii), and [START_REF] Bernardi | Osculating varieties of Veronesean and their higher secant varieties[END_REF], Proposition 3.4, C).

For d = k + 2, observe that the line L through P 1 and P 2 is a component of multiplicity at least 2(k + 1) -d = k for the curves defined by the forms both of (I Y ) d and of (I [START_REF] Bernardi | Osculating varieties of Veronesean and their higher secant varieties[END_REF], Corollary 3.5).

T ) d . Since Res kL Y = Res kL T = 2P 1 + 2P 2 (see Lemma 1.12), we get dim(I Y ) k+2 = dim(I T ) k+2 = dim(I 2P1+2P2 ) 2 = 1 (the only curve is the (k + 2)-uple line through the two points). Thus H(Y, d) = H(T, d). Moreover, since T is not regular in degree k + 2 (see Remark 1.2), we get H(Y, d) < expH(Y, d) (see
For k = 1, 2 and d ≥ k + 3, see [START_REF] Bernardi | Osculating varieties of Veronesean and their higher secant varieties[END_REF], Corollary 3.

(I Y ) k+3 = dim(I Y ) k+3-(k-1) = dim(I Y ) 4 = 15 -10 = 5, dim(I X ) k+3 = dim(I X ) 4 = 9,
where Y = Res νL Y = (2, 3)P 1 + (2, 3)P 2 (see Lemma 1.12), and

X = Res νL X = 2P 1 + 2P 2 . It follows that H(Y, k + 3) = H(X, k + 3) + 4. Hence by Lemma 2.4 i), for every d ≥ k + 3 we have H(Y, d) = H(X, d) + 4.
Since two (k + 1)-fat points impose independent conditions to curves of degree d if and only if

d ≥ 2k + 1 (see Remark 1.2), then, for k + 3 ≤ d ≤ 2k, we have H(X, d) < deg X, thus H(Y, d) = H(X, d) + 4 < deg X + 4 = deg Y. Moreover, since for d = k + 3, dim(I Y ) k+3 = 5, then for d ≥ k + 3, dim(I Y ) d is positive, that is H(Y, d) < d+2 2 . It follows that if k + 3 ≤ d ≤ 2k, then H(Y, d) < min deg Y, d+2 2
= expH(Y, d). (For k ≥ 3, and k + 3 ≤ d ≤ 2k, see also [START_REF] Bernardi | Osculating varieties of Veronesean and their higher secant varieties[END_REF], Proposition 3.10).

2.6. Proposition. For s = 3 we have:

i) H(Y, d) =      N + 1 if d ≤ 3(k+1) 2 H(X, d) + 6 < deg Y if 3(k+1) 2 + 1 ≤ d ≤ 2k H(X, d) + 6 = deg Y if d ≥ max{ 3(k+1) 2 + 1; 2k + 1} . ii) H(Y, d) < exp H(Y, d) iff 3(k+1) 2 + 2 ≤ d ≤ 2k if k + 1 is even 3(k+1) 2 + 1 ≤ d ≤ 2k if k + 1 is odd . Proof. i) In case d ≤ 3(k+1) 2 , it suffices to prove that (I Y ) d = (0) for d = 3(k+1) 2 .
Let C be the curve formed by the three lines P 1 P 2 , P 1 P 3 ,

P 2 P 3 . For d = 3(k+1) 2
, the curve C is a fixed component, of multiplicity at least

ν = 2(k + 1) -d = k+1 2 if k + 1 is even k 2 if k + 1 is odd
for the curves defined by the forms of (I Y ) d , so we have (see Lemma 1.12)

dim(I Y ) d = dim(I Y ) d-3ν
where

Y = Res νC Y = P 1 + P 2 + P 3 if k + 1 is even 2P 1 + 2P 2 + 2P 3 if k + 1 is odd , d -3ν = 0 if k + 1 is even 2 if k + 1 is odd .
ν = 2(k + 1) -d = k-1 2 if k + 1 is even k-2 2 if k + 1 is odd
for the curves defined by the forms of both (I Y ) d and (I X ) d , then we have

dim(I Y ) d = dim(I Y ) d-3ν , dim(I X ) d = dim(I X ) d-3ν
where (see Lemma 1.12)

d -3ν = 4 if k + 1 is even 6 if k + 1 is odd , Y = Res νC Y = (2, 3)P 1 + (2, 3)P 2 + (2, 3)P 3 if k + 1 is even (3, 4)P 1 + (3, 4)P 2 + (3, 4)P 3 if k + 1 is odd , X = 2P 1 + 2P 2 + 2P 3 if k + 1 is even 3P 1 + 3P 2 + 3P 3 if k + 1 is odd .
Since it is well known that dim(I 2P1+2P2+2P3 ) 4 = 6 and dim(I 3P1+3P2+3P3 ) 6 = 10, we have

dim(I X ) d-3ν = 6 if k + 1 is even 10 if k + 1 is odd , moreover, by Lemma 2.3 we get that dim(I Y ) d-3ν = 0 if k + 1 is even 4 if k + 1 is odd . It follows that dim(I X ) d -dim(I Y ) d = 6, hence H(Y, d) -H(X, d) = 6.
Since three (k + 1)-fat points impose independent conditions to curves of degree d if and only if

d ≥ 2k + 1 (see Remark 1.2), then for 3(k+1) 2 + 1 ≤ d ≤ 2k we have H(X, d) < deg X, while if d ≥ max 3(k+1) 2 + 1; 2k + 1 , then H(X, d) = deg X. Since deg Y = deg X + 6 we get: H(Y, d) = H(X, d) + 6 < deg Y if 3(k+1) 2 + 1 ≤ d ≤ 2k H(X, d) + 6 = deg Y if d ≥ max 3(k+1) 2 + 1; 2k + 1 . ii) For d ≤ 3(k+1) 2 , or d ≥ max 3(k+1) 2 + 1; 2k + 1 , from i) we have H(Y, d) = expH(Y, d).
If k + 1 is even, and

d = 3(k+1) 2 + 1, then dim(I Y ) d = 0, hence H(Y, d) = d+2
2 , the expected one. If k + 1 is even, and

d = 3(k+1) 2 + 2, from i), since dim(I X ) d-1 = 6 implies dim(I X ) d > 6, we have: dim(I Y ) d = d + 2 2 -H(Y, d) = d + 2 2 -H(X, d) -6 = dim(I X ) d -6 > 0. Hence, if k + 1 is even, for d = 3(k+1) 2
+ 2, and so also for

d ≥ 3(k+1) 2 + 2, we have dim(I Y ) d > 0, that is H(Y, d) < d+2 2 . Since, by i), if 3(k+1) 2 + 1 ≤ d ≤ 2k, then H(Y, d) < deg Y , it
H(Y, d) =        for k ≤ 6 : N + 1 if d ≤ 2k + 2 H(X, d) + 8 = deg Y if d ≥ 2k + 3 for k ≥ 6 : N + 1 if d ≤ 2k + 1 H(X, d) + 8 = deg Y if d ≥ 2k + 2 .
Proof. If d ≤ 2k + 1, by Bezout Theorem, each element of (I Y ) d is divisible by every form defining an irreducible conic through P 1 , . . . , P 4 , hence (I Y ) d = (0).

Let d = 2k + 2. Recall that the ideal of the scheme

Z i is ℘ k+1 i ∩ (℘ k+2 i + l 2 i )
, where l i defines a generic line

L i through P i (1 ≤ i ≤ 4) such that deg(Y ∩ L i ) = k + 2.
Let C i be the conic through P 1 , . . . , P 4 , tangent in P i to L i . For the genericity of the L i 's, the conics C 1 , . . . , C 4 are irreducible and distinct. Bezout's Theorem implies that each conic C i is a component of any curve defined by the forms of (I Y ) d . By Lemma 1.12 we can determine I Res C 1 +•••+C 4 Y , and it is an easy computation (wich will be omitted) that the intersection multiplicities of the curves defined by the forms of (I

Res C 1 +•••+C 4 Y ) d-8 with a conic C i , is bigger than 2(d -
H( Y, 2k + 3) = 2k + 5 2 - 2k + 3 2 + 4 k + 1 2 + 8 = deg Y + 3 = deg Y.
Hence Y imposes independent conditions to curves of degree 2k + 3. Since Y ⊂ Y, then also Y imposes independent conditions to curves of degree 2k + 3, that is H(Y, 2k + 3) = deg Y = deg Y .

Proposition.

For s = 5 we have:

H(Y, d) =      N + 1 if d ≤ 2k + 3 H(X, d) + 10 < expH(Y, d) if 2k + 4 ≤ d ≤ 5(k+1) 2 -1 H(X, d) + 10 = deg Y if d ≥ max 2k + 4; 5(k+1)
dim(I Y ) 2k+4 = dim(I Y ) 2k+4-2(k-3) = dim(I Y ) 10 , dim(I X ) 2k+4 = dim(I X ) 10 ,
where (see Lemma 1.12)

Y = Res (k-3)C Y = (4, 5)P 1 + • • • + (4, 5)P 5 , X = Res (k-3)C 4P 1 + • • • + 4P 5 .
From the case k = 3 it follows that

dim(I Y ) 2k+4 = 6, dim(I X ) 2k+4 = 16, hence H(Y, d) = H(X,
If d ≥ max 2k + 4; 5(k+1) 2 , then H(X, d) = deg X, so H(Y, d) = deg Y .
2.9. Proposition. For s = 6 we have:

H(Y, d) =                                        for k = 1 : N + 1 if d ≤ 6 H(X, d) + 12 = deg Y if d ≥ 7 for k = 2 : N + 1 if d ≤ 8 H(X, d) + 12 = deg Y if d ≥ 9 for k ≥ 3 k ≡ 2 (mod 5) :      N + 1 if d ≤ 12(k+1) 5 -1 H(X, d) + 12 < exp H(Y, d) if 12(k+1) 5 ≤ d ≤ 5(k+1) 2 -1 H(X, d) + 12 = deg Y if d ≥ max 12(k+1) 5 ; 5(k+1) 2 for k ≥ 3 k ≡ 2 (mod 5) :      N + 1 if d ≤ 12(k+1) 5 H(X, d) + 12 < exp H(Y, d) if 12(k+1) 5 + 1 ≤ d ≤ 5(k+1) 2 -1 H(X, d) + 12 = deg Y if d ≥ max 12(k+1) 5 + 1; 5(k+1) 2 .
Proof. We start by proving four particular cases, that we need later in the proof. Proof. Though all of the above equalities can be checked using CoCoA (see [START_REF] Capani | CoCoA, A system for doing computations in Commutative Algebra[END_REF]), we prove i) by specializing the scheme Y. The proofs of ii), iii), and iv) by using a specialization of Y are similar, and we left them to the reader.

i) Let Q ∈ P 2 be a generic point, and let F = {F = 0} be a rational integral curve of degree 5 passing through Q, and having at each P i , 1 ≤ i ≤ 6, an ordinary singularity of multiplicity 2, (so F ∈ (I 2P1+•••+2P6 ) 5 ), and let { l i = 0} be one of the two distinct lines contained in the tangent space T F ,Pi to F at the point P i . Recall that the defining ideal of Y = (8, 9)P 1 + • • • + (8, 9)P 6 is

I Y = (℘ 8 1 ∩ (℘ 9 1 + l 2 1 )) ∩ • • • ∩ (℘ 8 6 ∩ (℘ 9 6 + l 2 6 )).
Specialize the scheme Y putting l i = l i for i

Now let

k + 1 = 5q + r, (0 ≤ r ≤ 4). Thus k ≡ 2 (mod 5) iff r = 3. For k = 1, 2 see Lemma 2.3.
Let k ≥ 3. Let C i be the conic through P 1 , . . . , P i , . . . , P 6 , ( i = 1, . . . , 6), and let C = 6 i=1 C i . Observe that if 2d < 5(k + 1), then the curves defined by the forms of (I Y ) d , and by the forms of (I X ) d have the six conics C i as fixed components with multiplicity at least ν = 5(k + 1) -2d.

Then dim(I

Y ) d = dim(I Y ) d-12ν , dim(I X ) d = dim(I X ) d-12ν ,
where

Y = Res νC Y = (k + 1 -5ν, k + 2 -5ν)P 1 + • • • + (k + 1 -5ν, k + 2 -5ν)P 6 , X = Res νC X = (k + 1 -5ν)P 1 + • • • + (k + 1 -5ν)P 6 .
We split the proof in four cases.

Case 1): k ≡ 2 (mod 5), and d ≤ 12(k+1)

5

-1 = 12q + 7. In this case it suffices to prove that (I Y ) d = (0) for d = 12q + 7. Since 2d = 2(12q + 7) < 5(k + 1) = 5(5q + 3), then the curves defined by the forms of (I Y ) d should have a fixed locus of degree 12ν = 12q + 12, and this is impossible, since d = 12q + 7. It follows that (I Y ) d = (0). Since for ν = 0, we have Y = Y, X = X, and d = d, then for every ν ≥ 0 we have:

k + 1 -5ν = 5q + 3 -5(q -1) = 8. So dim(I Y ) d = dim(I Y ) 20 = 3, dim(I X ) d = dim(I X ) 20 = 15, where Y = Res νC Y = (8, 9)P 1 + • • • + (8, 9)P 6 , X = Res νC X = 8P 1 + • • • + 8P 6 . So we have proved that H(Y, d) = H(X, d) + 12. Now, since for d = 12q + 8, dim(I Y ) d is positive (and in fact it is equal to dim(I Y ) 20 = 3), then H(Y, d) < d+2
r k + 1 d ν Y d 0 5q 12q q > 0 P 1 + • • • + P 6 0 1 5q + 1 12q + 3 q -1 ≥ 0 (6, 7)P 1 + • • • + (6,
r k + 1 d ν k + 1 -5ν Y X d 0 5q 12q + 1 q -2 10 6 i=1 (
dim(I Y ) d = dim(I Y ) d , dim(I X ) d = dim(I X ) d . It follows that H(Y, d) -H(X, d) = H(Y , d ) -H(X , d ).
Hence in case ν ≥ 0 we have only to prove that: (a) H( Proof. i) Let F be (as above) a rational curve of degree 5 having at each P i (1 ≤ i ≤ 6), an ordinary singularity of multiplicity 2. Let Q 1 , . . . , Q η ∈ F be generic points. Since 5(d + 5) < 6(2(m + 2)) + η, by Bezout Theorem F is a fixed component for the curves defined by the forms of (

I Y+Q1+•••+Qη ) d+5 . It follows that dim(I Y+Q1+•••+Qη ) d+5 = dim(I Y ) d . Since d+5+2 2 -deg( Y + Q 1 + • • • + Q η ) = 1 2 (d+7)(d+6)-(deg Y +6(m+2)+6(m+1)+η) = d+2 2 -deg Y = d+2 2 -H(Y, d) = dim(I Y ) d , we have dim(I Y+Q1+•••+Qη ) d+5 = d + 5 + 2 2 -deg( Y + Q 1 + • • • + Q η ) hence H( Y + Q 1 + • • • + Q η , d + 5) = deg( Y + Q 1 + • • • + Q η ).
Since obviously Proof. For k = 1, 2 the statement is known by [START_REF] Ballico | On the secant varieties to the tangent developable of a Veronese variety[END_REF] and [START_REF] Ballico | On the secant varieties to the osculating variety of Veronese surfaces[END_REF].

X ⊂ Y ⊂ Y + Q 1 + • • • + Q η ,
Let k = 3, so Y = (4, 5)P 1 + • • • + (4, 5)P 9 .

For d = 13, by CoCoA (see [START_REF] Capani | CoCoA, A system for doing computations in Commutative Algebra[END_REF]), or by specializing the scheme Y it is easy to check that dim(I Y ) 13 = 0, hence for d ≤ 13 the conclusion follows from Lemma 2.4. Now let C be the unique (smooth) cubic curve passing through the support of Y, i.e., through P 1 , . . . , P 9 . Consider the following exact sequence, where Y = Res C Y: In the other cases by Lemma 2.2, and from Propositions 2.5 to 2.9, and Proposition 2.12, we have From what we have seen above, and by Remark 1.6, Propositions 2.5 to 2.9 and 2.12, we immediately get that O s k,d is defective only in the cases i) to vii). In cases iii) and iv), computing the dimension of (I X ) d by cutting off the three lines P 1 P 2 , P 1 P 3 , P 2 P 3 , 2(k + 1) -d times each, we have:

0 → I Y (d -3) → I Y (d) → I Y∩C,C (d) → 0 We have that I Y∩C,C (d) = O C (dH -Y ∩ C),
Y = (k + 1, k + 2)P 1 + • • • + (k + 1, k + 2)P 9 Y = (k, k + 1)P 1 + • • • + (k, k +
h 0 (I Y (d -3)) = 0 for d -3 ≤ 3(k -1) + 3, h 1 (I Y (d -3)) = 0 for d -3 ≥ 3(k -1) + 4.
dim(I X ) d = dim(I X ) d-3(2k+2-d) = dim(I X ) 2(2d-3k-3) = 2(2d -3k -3) + 2 2 -3 2d -3k -2 2 = 2d -3k -1 2 ,
where X =

1. 10 .

 10 Remark. Let d ≥ k + 2. Recall that Z(k) is defined by the ideal generated by W ⊥ ⊂ R d , where

and t = 0 ,

 0 hence for m = k and t = 0 we have Res C (Z) = 2P , for m < k and t = 0 we have Res C (Z) = (k + 1 -m, k + 2 -m)P , while for m ≤ k and t = 1, Res C (Z) is the union of the fat point (k + 1 -m)P with the intersection of the line {x = 0} with the fat point (k + 2 -m)P .

2. 3 .

 3 Lemma. Let s > 2. Then: i) for k = 1, Y = Y = (2, 3)P 1 + • • • + (2, 3)P s , and H(Y, d) = min{deg Y, N + 1}; ii) for k = 2, Y = (3, 4)P 1 + • • • + (3, 4)P s , and H(Y, d) = min{deg Y, N + 1}. Proof. i) If d = 2 see [7], Proposition 3.3; for d = 3 see [7], Proposition 4.5; for d ≥ 4 see [2], Theorem 1.

8 .

 8 For k ≥ 3, and d ≥ 2k + 1 see [5], Proposition 3.9. Now let k ≥ 3, and k + 3 ≤ d ≤ 2k. For d = k + 3 the line L through P 1 and P 2 is a component of multiplicity at least ν = 2(k + 1) -d = k -1 for the curves defined by the forms of both (I Y ) d , and (I X ) d , hence from the case k = 1, d = 4, we get dim

2 + 1 . 2 + 1 . 2 + 1 .

 212121 It immediately follows that (I Y ) d = (0). Now let d ≥ 3(k+1) In order to prove that H(Y, d) = H(X, d) + 6, by Lemma 2.4 it suffices to prove that H(Y, d) = H(X, d) + 6 for d = 3(k+1) Let d = 3(k+1) The curve C is a fixed component, with multiplicity at least

follows that for 3 (k+1) 2 + 2 ≤ 2 = 2 + 1 , 2 + 1

 32222121 d ≤ 2k we have H(Y, d) < min deg Y, d+2 exp H(Y, d). If k + 1 is odd and d ≥ 3(k+1) from the proof of i) we get dim(I Y ) d > 0, that is H(Y, d) < d+2 2 . Moreover, by i), if 3(k+1) ≤ d ≤ 2k, then H(Y, d) < deg Y , and the conclusion immediately follows. 2.7. Proposition. For s = 4 we have:

2 - 2 - 2 - 2 +

 2222 8).Hence by Bezout's Theorem we get that each conic C i is a component with multiplicity at least 2 of any curve defined by the forms of (I Y ) d . So these curves have a component of degree 16. It follows that, if (I Y ) d = (0), then d ≥ 16, that is k ≥ 7. Thus, for k ≤ 6, we have (I Y ) d = (0), that is H(Y, d) = N + 1. Observe that for k = 6, we have N + 1 = H(X, d) + 8 = deg Y , in fact in this case d = 2k + 2 = 14, N + 1 = 16 2 = 120, and, since four 7-fat points impose independent conditions to curves of degree 14 (see Remark 1.2), thenH(X, d) = 112. If k ≥ 7 we have dim(I Y ) 2k+2 = dim(I Y ) 2k+2-16 , where Y = Res 2C1+•••+2C4 Y = (k -7)P 1 + • • • + (k -7)P 4 is a scheme of four (k -7)-fat points (see Lemma 1.12). Since Y imposes independent conditions to curves of degree 2k -14 (see Remark 1.2), then H(Y, 2k + 2) = 2k+4 dim(I Y ) 2k+2 = 2k+4 dim(I Y ) 2k-14 = 2k+4 2k + 2) + 8 = deg Y . Now let d ≥ 2k + 3. It suffices to prove that H(Y, 2k + 3) = H(X, 2k + 3) + 8 = deg Y (see Lemma 2.4 i)), so let d = 2k + 3. By induction on k. For k = 1 see Lemma 2.3. Let k ≥ 2. Let C be an irreducible conic through P 1 , . . . , P 4 , and let Q 1 , Q 2 , Q 3 be three points on C. Let Y = Y+Q 1 + Q 2 + Q 3 . By Bezout's Theorem, the conic C is a fixed component for the curves of degree 2k + 3 through Y, then dim(I Y ) 2k+3 = dim(I Y ) 2k+1 = 2k + 3 2 -H( Y , 2k + 1), where Y = Res C Y = Res C Y = 4 i=1 (k, k + 1)P i (see Lemma 1.12). By the inductive hypothesis we have that H( Y , 2k + 1) = deg Y = 4 k+1

2 . 2 -

 22 Proof. Let d ≤ 2k + 3. If we prove that (I Y ) d = (0) for d = 2k + 3 we are done. So let d = 2k + 3. For k = 1 see Lemma 2.3. Let k ≥ 2. Any curve defined by a nonzero element of (I X ) d has the conic C through P 1 , . . . , P 5 as a component of multiplicity at least 5(k + 1) -2d = k -1, where X is the fat point subscheme of 5 points of multiplicity k + 1, hence the same is true for Y in place of X, since X ⊂ Y, so we have:dim(I Y ) 2k+3 = dim(I Y ) 2k+3-2(k-1) = dim(I Y ) 5 ,where, by Lemma 1.12, Y = Res (k-1)C Y = (2, 3)P 1 + • • • + (2, 3)P 5 . Since, by Lemma 2.3 i), dim(I Y ) 5 = 0, then the conclusion follows. Now let d ≥ 2k + 4. We have to prove that H(Y, d) = H(X, d) + 10. By Lemma 2.4, it suffices to prove that H(Y, d) = H(X, d) + 10 for d = 2k + 4, so let d = 2k + 4. For k = 1, 2 see Lemma 2.3. If k = 3 (hence d = 10), let Q be a point on the conic C through P 1 , . . . , P 5 . The scheme Y + Q imposes independent conditions to curves of degree 10. In fact, since the conic C is a fixed locus for (I Y+Q ) 10 , from the case k = 2 we get: dim(I Y+Q ) 10 = dim(I Y ) deg(Y + Q), where Y = Res C (Y + Q) = (3, 4)P 1 + • • • + (3, 4)P 5 (see Lemma 1.12). Since Y + Q imposes independent conditions to curves of degree 10, then Y and X also do the same. It follows that H(Y, 10) = deg Y = deg X + 10 = H(X, 10) + 10. For k ≥ 4, since C is a fixed component with multiplicity at least (k -3) for curves defined both by (I Y ) 2k+4 and by (I X ) 2k+4 , it follows that

2 - 1 ,

 21 d) + 10. So we have proved that for d ≥ 2k + 4 H(Y, d) = H(X, d) + 10. Now, since dim(I Y ) 2k+4 is positive, then H(Y, d) < d+2 2 for any d ≥ 2k + 4. Moreover, since five generic (k +1)-fat points impose independent conditions to curves of degree d if and only if d ≥ 5(k+1) 2 (see Remark 1.2), then for 2k + 4 ≤ d ≤ 5(k+1) we have H(X, d) < deg X, hence H(Y, d) = H(X, d) + 10 < min deg X + 10,

2. 10 .

 10 Lemma. We have: i) dim(I (8,9)P1+•••+(8,9)P6 ) 20 = 3; ii) dim(I (6,7)P1+•••+(6,7)P6 ) 15 = 0; iii) dim(I (5,6)P1+•••+(5,6)P6 ) 13 = 3; iv) dim(I (4,5)P1+•••+(4,5)P6 ) 11 = 6.

2 - 2 -

 22 = 1, 2, 3, 4, and let Y * be such specialization of Y. Since the expected dimension of (I Y ) 20 is 20+2 deg Y = 231 -228 = 3, then if we prove that dim(I Y * ) 20 = 3, we are done. It is easy to see that the curves defined by the forms of (I Y * +Q ) 20 have the quintic F as fixed component with multiplicity at least 2, hence dim(I Y * +Q ) 20 = dim(I W ) 10 where W = Res 2F (Y * + Q) = 4P 1 + 4P 2 + 4P 3 + 4P 4 + (4, 5)P 5 + (4, 5)P 6 . Now let W * be a specialization of W obtained by putting l i = l i for i = 5, 6. Since the quintic F is as fixed component with multiplicity at least 2 for (I W * +Q ) 10 , and since Res 2F (W * + Q) = ∅ (see Lemma 1.12) we have dim(I W * +Q ) 10 = dim(I Res 2F (W * +Q) ) 0 = 1. Thus for the specialized scheme W * we have dim(I W * ) 10 = 2 = 10+2 deg W * . Then W * , and so W also, imposes independent conditions to curves of degree 10. It follows that dim(I W ) 10 = 2. So dim(I Y * +Q ) 20 = 2, hence dim(I Y * ) 20 = 3, and we are done.

Case 2 ) 5 = 12q + 8 .

 258 : k ≡ 2 (mod 5), and d ≥ 12(k+1) First we will prove that H(Y, d) = H(X, d) + 12. By Lemma 2.4, it suffices to prove that H(Y, d) = H(X, d) + 12, for d = 12q + 8. Since k ≥ 3, and k + 1 = 5q + 3, then we have q ≥ 1. Let q = 1, so d = 20, k + 1 = 8, Y = (8, 9)P 1 + • • • + (8, 9)P 6 , and X = 8P 1 + • • • + 8P 6 . Since dim(I (8,9)P1+•••+(8,9)P6 ) 20 = 3 (see Lemma 2.10 i)), and six 8-fat points impose independent conditions to curves of degree 20 (see Remark 1.2), we have dim(I X ) 20 = 15. It follows that H(Y, d) = H(X, d) + 12. If q > 1, then νC = 6 i=1 νC i is a fixed locus for (I Y ) d and (I X ) d . Since ν = 5(k + 1) -2d = 5(5q + 3) -2(12q + 8) = q -1, we have d -12ν = 12q + 8 -12(q -1) = 20, and

2 2 ( 2 - 1 , 2 , 5 . 5 =

 221255 for any d ≥ 12q + 8. Since six generic (k + 1)-fat points impose independent conditions to curves of degree d if and only if d ≥5(k+1) see Remark 1.2), then for 12q + 8 ≤ d ≤5(k+1) we have H(X, d) < deg X, henceH(Y, d) = H(X, d) + 12 < min deg X + 12,While for d ≥ max 12q + 8;5(k+1) we haveH(X, d) = deg X, so H(Y, d) = H(X, d)+12 = deg X+12 = deg Y .That is enough to finish the proof of this case.Case 3): k ≡ 2 (mod 5), and d ≤ 12(k+1) By Lemma 2.4 we have only to prove that H(Y, d) = N + 1 for d =12(k+1) 12q + 12r 5 . Since k ≥ 3, we have k + 1 = 5q + r ≥ 4, hence q ≥ 4-r 5 . As above, let ν = 5(k + 1) -2d, Y = Res νC Y, and let d = d -12ν. We have:

1 + 5 + 1 . 5 + 1 = 12q + 12r 5 + 1 .

 1515151 • • • + (4, 5)P 6 10 Since for ν = 0, we have Y = Y and d = d, then for every ν ≥ 0 we have: dim(I Y ) d = dim(I Y ) d . Now we will prove that dim(I Y ) d = 0. For r = 0 it is obvious. For r = 2 see Lemma 2.3. For r = 1 by Lemma 2.10 ii), we have dim(I (6,7)P1+•••+(6,7)P6 ) 15 = 0. For r = 4, let F = {F = 0} be a rational integral curve of degree 5 having at each P i (1 ≤ i ≤ 6) an ordinary singularity of multiplicity 2, (F ∈ (I 2P1+•••+2P6 ) 5 ). If there exists a form G = 0, G ∈ (I (4,5)P1+•••+(4,5)P6 ) 10 , then F G = 0 and F G ∈ (I (6,7)P1+•••+(6,7)P6 ) 15 , but this is impossible by the previous case r = 1. Case 4): k ≡ 2 (mod 5), and d ≥ 12(k+1) First we will prove that H(Y, d) = H(X, d) + 12. By Lemma 2.4, it suffices to prove that H(Y, d) = H(X, d) + 12 for d = 12(k+1) As usual, let ν = 5(k + 1) -2d, Y = Res νC Y, X = Res νC X, and d = d -12ν. We have:

2. 11 . 6 .

 116 Lemma. Let:Y = (m, m + 1)P 1 + • • • + (m, m + 1)P 6 , Y = (m + 2, m + 3)P 1 + • • • + (m + 2, m + 3)P 6 , X = (m + 2)P 1 + • • • + (m +2)P If the integer η = 5(d + 5) -12(m + 2) + 1 ≥ 0, and H(Y, d) = deg Y, then i) H( Y, d + 5) = deg( Y), H( X, d + 5) = deg( X);ii) H( Y, d + 5) = H( X, d + 5) + 12.

2 = 9 •

 29 where H is a line section of C, and deg(OC (dH -Y ∩ C)) = 3d -9(k + 1). Let d = 14. Since k = 3, we have deg(O C (dH -Y ∩ C)) = 14 • 3 -4 • 9 = 6. It follows that h 1 (O C (dH -Y ∩ C)) = 0. Since Y = (3, 4)P 1 + • • • + (3, 4)P 9 , from the case k = 2 we get h 1 (I Y (d -3)) = h 1 (I Y (11)) = 0.So by the exact sequence above it follows that h 1 (I (4,5)P1+•••+(4,5)P9 (14)) = 0, which implies H(Y, 14) = deg Y . For d > 14 the conclusion follows from Lemma 2.4. Let k ≥ 4. Now we proceed by induction on k. For k = 4, we have Y = (5, 6)P 1 + • • • + (5, 6)P 9 , and 3k + 4 = 16. By CoCoA (see [6]), or by specializing the scheme Y it is easy to check that dim(I Y ) 16 = 0. So, since N + 1 = 16+2 17 = deg Y, it follows that H(Y, 16) = N + 1 = deg Y. Hence by Lemma 2.4 it follows that for d ≤ 16 we have H(Y, d) = N + 1, while for d ≥ 16 we have H(Y, d) = deg Y. Now let k > 4. We have:

1 )P 9 .

 19 Since obviously if d ≤ 3k + 3, then d -3 ≤ 3(k -1) + 3, and if d ≥ 3k + 4, then d -3 ≥ 3(k -1) + 4, by the induction hypothesis we have H(Y , d -3) = N + 1 for d -3 ≤ 3(k -1) + 3, (N = d-3+2 2 ), and H(Y , d -3) = deg Y for d -3 ≥ 3(k -1) + 4. That is:

1 except when s = 2 , 2 - 2 ., d+2 2 ,

 12222 Moreover, since deg(O C (dH -Y ∩ C)) = 3d -9(k + 1) ≤ 0 for d ≤ 3k + 3, and deg(O C (dH -Y ∩ C)) = 3d -9(k + 1) ≥ 3 for d ≥ 3k + 4, we have: h 0 (I Y∩C,C (d)) = 0 for d ≤ 3k + 3, h 1 (I Y∩C,C (d)) = 0 for d ≥ 3k + 4. So whenever d ≤ 3k + 3, we get h 0 (I Y (d -3)) = h 0 (I Y∩C,C (d)) = 0, which by the exact sequence above implies h 0 (I Y (d)) = 0. When d ≥ 3k + 4, we get h 1 (I Y (d -3)) = h 1 (I Y∩C,C (d)) = 0, so by the exact sequence above we have h 1 (I Y (d)) = 0, and we are done.With all these partial results we have actually proved the main theorem of this paper: 2.13. Theorem. For s ≤ 6, or s = 9, then dim O s k,d = min H(X, d) + 2s, d = k + 2 where dim O 2 k,k+2 = H(T, d) -1 = d+2 Proof. For s = 1, since H(X, d) = min k+2 2 then the result follows from Remark 1.6. For s = 2 and d = k + 2, since H(Y, d) = H(T, d) (see Propositions 2.5), by the obvious inequalities H(Y, d) ≤ H(Y, d) ≤ H(T, d) we get H(Y, d) = H(Y, d) = H(T, d) and the conclusion follows from Remark 1.7 ( ‡).

2 ; 2 + 2 ≤ 2 ; 2 - 6 . 2 + 1 ≤ 2 ; 2 - 1 , 2 ; 5 d-2k-1 2 - 9 . 5 ≤ d ≤ 5 (k+1) 2 - 1 , 2 ; 2 - 5 + 1 2 - 1 , 2 ; 5d-12k-10 2 - 12 . 2 +2 2 + 1 ≤ 2 - 1 5 ≤ 5 (k+1) 2 - 1 5 + 1 ≤ 5 (k+1) 2 - 1 ,

 2222262122125295521225121221222121552151521 H(Y, d) = H(Y, d) = min{H(X, d) + 2s, N + 1}, hence from Remark 1.7 ( ‡) we get the conclusion.2.14. Corollary. Let δ = min {deg Y -1, N } -dim O s k,d be the defect of O s k,d . If s ≤ 6, or s = 9, then O s k,d is defective only in the following cases: i) s = 2, d = k + 2, with defect: δ = 1. ii) s = 2, k ≥ 3, k + 3 ≤ d ≤ 2k, with defect: δ = min 2(k+1)-d (d -k) 2 -4 .iii) s = 3, k ≥ 7, k odd ,3(k+1) d ≤ 2k, with defect: δ = min 3 2(k+1)-d 2d-3k-1 iv) s = 3, k ≥ 6, k even ,3(k+1) d ≤ 2k, with defect: δ = min 3 2(k+1)-d k ≥ 5, 2k + 4 ≤ d ≤5(k+1) with defect δ = min 5(k+1)-2dvi) s = 6, k ≡ 2 (mod 5), k ≥ 17,12(k+1) with defect: δ = min 6 5(k+1)-2d 5d-12k-10 12 .vii) s = 6, k ≡ 2 (mod 5), k ≥ 19 if k odd 24 if k even , 12(k+1) ≤ d ≤5(k+1) with defect:δ = min 6 5(k+1)-2dProof. First observe that: k +3 ≤ 2k implies k ≥ 3; if k is odd and 3(k+1) ≤ 2k, then 3(k +1)+4 ≤ 4k, that is k ≥ 7, while if k is even and3(k+1) 2k, then k ≥ 6; from 2k + 4 ≤ 5(k+1) we get k ≥ 5; finally, for k ≡ 2 (mod 5), it is easy to compute that12(k+1) implies k ≥ 17, while for k ≡ 2 (mod 5), if12(k+1) then k ≥ 19 if k odd 24 if k even .

For s = 2

 2 and d = k + 2, since dim O 2 k,k+2 = N -1, while the expected dimension is N , we have δ = 1. In the other cases we know that H(Y, d) = H(X, d) + 2s, so we haveδ = min{deg Y -1, N } -dim O s k,d = min{deg Y -1, N } -H(Y, d) + 1 = min{deg Y -H(X, d) -2s, N + 1 -H(X, d) -2s} = min{deg X -H(X, d), dim(I X ) d -2s}.For s = 2, k ≥ 3 and k + 3 ≤ d ≤ 2k, computing the dimension of (I X ) d by removing the line P 1 P 2 (2(k + 1) -d) times, we get:dim(I X ) d = dim(I X ) 2(d-k-1) = 2(d -k --k) 2 , where X = (d -k -1)P 1 + (d -k -1)P 2 , hence deg X -H(X, d) + 1) -d 2 ; (d -k) 2 -4 .

3 i=1 (k + 1 - 2 ( 6 . 5 i=1 (k + 1 - 5 i=1

 3126515 2k + 2 -d))P i = 3 i=1 (2d -3k -3)P i , and from here we easily get: deg X -H(X, d) For s=5, computing the dimension of (I X ) d (by cutting off the fixed conics), we get:dim(I X ) d = dim(I X ) d-2(5k+5-2d) = dim(I X ) 5(d-2k-2) (5k + 5 -2d))P i = (2d -4k-4)P i , and from here we get: deg X -H(X, d)

  it follows that H( Y, d + 5) = deg( Y), and H( X, d + 5) = deg( X). , 30) + 12. Analogously, by Lemma 2.11, we have that (b) ⇒ (c) ⇒ (d), so, for ν ≥ 0, we have proved that H(Y, d) = H(X, d) + 12. Now let ν < 0. In this case, since k + 1 = 5q + r ≥ 3, we are left with the folloving cases:

		ii) Obvious.
		By Case 2) we know that H(	6 i=1 (8, 9)P i , 20) = H(	6 i=1 8P i , 20) + 12 = deg(	6 i=1 (8, 9)P i ), so by
	Lemma 2.11 ii) we have (a) : H(	6 i=1 (10, 11)P i , 25) = H(	6 i=1 10P i , 25) + 12. Moreover, by Lemma 2.11 i),
	H(	6 i=1 (10, 11)P i , 25) = deg(	6 i=1 (10, 11)P i ), hence by Lemma 2.11 ii) we get (b) : H(	6 i=1 (12, 13)P i , 30)
	= H(	6 i=1 12P i r	q	k + 1		Y	X	d
			0		1	5		6 i=1 (5, 6)P i	6 i=1 5P i	13
			1		1	6		6 i=1 (6, 7)P i	6 i=1 6P i	16
			1		2	11		6 i=1 (11, 12)P i	6 i=1 11P i	28
			2		1	7		6 i=1 (7, 8)P i	6 i=1 7P i	18
			4		0	4		6 i=1 (4, 5)P i	6 i=1 4P i	11
			4		1	9		6 i=1 (9, 10)P i	6 i=1 9P i	23
		hence we have to prove that:
		(e) H(	6 i=1 (5, 6)P i , 13) = H(	6 i=1 5P i , 13) + 12;
		(f ) H(	6 i=1 (6, 7)P i , 16) = H(	6 i=1 6P i , 16) + 12;
		(g) H(	6 i=1 (11, 12)P i , 28) = H(	6 i=1 11P i , 28) + 12;
		(h) H(	6 i=1 (7, 8)P i , 18) = H(	6 i=1 7P i , 18) + 12;
		(i) H(	6 i=1 (4, 5)P i , 11) = H(	6 i=1 4P i , 11) + 12;

* All authors supported by MIUR funds.

(l) H( 6 i=1 (9, 10)P i , 23) = H( 6 i=1 9P i , 23) + 12. By Remark 1.2 and by Lemma 2.10 iii), and iv), it easily follows that (e) and (i) hold, moreover by Lemma 2.11 we have that (e) ⇒ (h) ⇒ (l) ⇒ (g), and (i) ⇒ (f ), so we have proved that H(Y, d) = H(X, d) + 12 also for ν < 0. Now observe that for d = 12(k+1) 5 + 1, dim(I Y ) d is positive. In fact, as shown above, we have if ν ≥ 0: 2.12. Proposition. For s = 9 we have:

Finally, for s=6, calculating the dimension of (I X ) d by removing every conic C i (see the proof of Proposition 2.9) (5(k + 1) -2d) times, we get

where X = 6 i=1 (k + 1 -5(5k + 5 -2d))P i = 6 i=1 (10d -24k -24)P i , and from here we get:

2.15. Remark. Some examples, some computations, and a lack of geometric reasons, lead us to conjecture that also for s = 7, and s = 8 we have

Unfortunately, by methods similar to the ones utilized for s ≤ 6, the proof splits into many cases, and becomes too long and tedious to justify including. E.Ballico and C.Fontanari in [START_REF] Ballico | A Terracini Lemma for osculating spaces with applications to Veronese surfaces[END_REF] give partial results about the regularity of O s k,d for 2 ≤ s ≤ 8. Our Corollary 2.14, for s ≤ 6 or s = 9, improves the results of [START_REF] Ballico | A Terracini Lemma for osculating spaces with applications to Veronese surfaces[END_REF] and gives a complete classification of all the defective cases.

2.16. Remark. We wish to notice that there are no defective cases for s = 4 or s = 9.

In case s = 2, d = k + 2 defectivity is forced by the defectivity of T , in fact, since Y ⊂ T implies that H(Y, k + 2) ≤ H(T, k + 2), and since H(T, k + 2) = N < expH(Y, k + 2) = N + 1, it follows that H(Y, k + 2) < expH(Y, k + 2) . In the other cases defectivity of O s k,d is forced by the defectivity of X.

k,d is as expected. In fact if s is a square, by [START_REF] Evain | Computing limit linear series with infinitesimal methods[END_REF]