Alessandra Bernardi

Ideals of varieties parameterized by certain symmetric tensors

The ideal of a Segre variety

 and [Gr]). We extend this result to the case of Segre-Veronese varieties. The main tool is the concept of "weak generic hypermatrix" which allows us to treat also the case of projection of Veronese surfaces from a set of general points and of Veronese varieties from a Cohen-Macaulay subvariety of codimension 2.

Introduction

In this paper we study the generators of the ideal of Segre-Veronese varieties and the ideal of projections of Veronese surfaces from a set of general points and, more generally, of Veronese varieties from a Cohen-Macaulay subvariety of codimension 2.

A Segre variety parameterizes completely decomposable tensors (Definition 2.1).

The problem of tensor decomposition has been studied studied for many years and by researchers in many scientific areas as Algebraic Geometry (see for example [START_REF] Catalisano | Gimigliano Ranks of tensors, secant varieties of Segre varieties and fat points[END_REF], [LM], [LW], [AOP], [Za]), Algebraic Statistic (see [HR], [GSS], [PS]), Phylogenetic ( [AR], [Bo], [Lak], [SS]), Telecommunications ( [Com]), Complexity Theory ( [BCS], [Lan], [Li], [St]), Quantum Computing ( [BZ]), Psychometrics ( [CKP]), Chemometrics ( [Br]).

In [START_REF] Hà | Box-shaped matrices and the defining ideal of certain blowup surface[END_REF] (Theorem 1.5) it is proved that the ideal of a Segre variety is generated by all the 2-minors of a generic hypermatrix of indeterminates.

Here we prove an analogous statement for Segre-Veronese varieties (see [START_REF] Catalisano | Gimigliano Higher Secant Varieties of Segre-Veronese varieties[END_REF]). Segre-Veronese varieties parameterize certain symmetric decomposable tensors, and are the embedding of P n1 × • • • × P nt into P Π t i=1 ( n i +d i -1 d i

)-1 given by the sections of the sheaf O(d 1 , . . . , d t ) with d 1 , . . . , d t ∈ N (see Section 3). We prove (in Theorem 3.11) that their ideal is generated by the 2-minors of a generic symmetric hypermatrix (Definition 3.5).

The idea we use is the following; generalizing ideas in [START_REF] Hà | Box-shaped matrices and the defining ideal of certain blowup surface[END_REF] we define "weak generic hypermatrices" (see Definition 3.8) and we prove that the ideal generated by 2-minors of a weak generic hypermatrix is a prime ideal (Proposition 3.10). Then we show that a symmetric hypermatrix of indeterminates is weak generic and we can conclude, since the ideal generated by its 2-minors defines, set-theoretically, a Segre-Veronese variety.

An analogous idea is used in Sections 4 and 5 in order to find the generators of projections of Veronese varieties from a subvariety of codimension 2. This is a problem which has been studied classically in Algebraic Geometry (starting with the projection of Veronese surface, see [Sh]); for a quite general analysis of subalgebras of the Rees Algebra associated to embeddings of blow ups of P n along subvarieties, see [CHTV] and [MU].

Denote with Y n,d the Veronese variety obtained as the d-uple embedding of P n into P ( n+d d )-1 and consider the surface Y ⊂ P ( 2+d 2 )-s-1 which is the projection of Y 2,d from s general points on it. The defining ideal of Y has been studied in [START_REF] Hà | Box-shaped matrices and the defining ideal of certain blowup surface[END_REF] when s is a binomial and s ≤ d 2 and in [GL] and [START_REF] Hà | On the Rees algebra of certain codimension two perfect ideals[END_REF] for s > d 2 (in the second paper also the case of any set of s points is treated, when d ≥ max{4, s + 1}). Here we complete the picture for s < d 2 general points on Y 2,d ; our method follows the framework of [GG] and [GL], but uses the "hypermatrix" point of view of [START_REF] Hà | Box-shaped matrices and the defining ideal of certain blowup surface[END_REF]. We construct a hypermatrix in such a way that its 2-minors together with some linear equations generate an ideal I that defines Y set-theoretically; then we prove that such hypermatrix is weak generic and in Theorem 4.7 we prove that I is actually the ideal of the projected surface.

This construction can be generalized to projections of Veronese varieties Y n,d , for all n, d > 0, from a subvariety of codimension 2 and of degree s = t+1 2 + k ≤ d 2 for some non negative integers t, k, d such that 0 < t < d -1 and 0 ≤ k ≤ t (see Section 5). I want to thank A. Gimigliano for many useful talks and suggestions. I want to thank also J. M. Landsberg for pointing out to me that by using representation theory techniques (as for example in [LM]) it is possible to see that the equations coming from the vanishing of 2-minors of a symmetric hypermatrix of indeterminates are the generators of the ideal of a Segre-Veronese variety. By an unpublished theorem of Kostant, the ideal of any homogeneously embedded rational homogeneous variety is generated in degree two by the annihilator of a certain vector space (for the experts: the homogeneously embedded rational homogeneous variety G/P ⊂ P(V λ ), is generated in degree two by the annihilator of V 2λ in S 2 (V * λ )). While the representation-theoretic techniques identify the modules generating the ideal, they do not provide an explicit method for writing down a set of generators, which is the subject of this paper. Last but not least thanks to the anonymous referee for his careful work and suggestions.

Preliminaries

Let K = K be an algebraically closed field of characteristic zero, and let V 1 , . . . , V t be vector spaces over K of dimensions n 1 , . . . , n t respectively. We will call en element

T ∈ V 1 ⊗ • • • ⊗ V t a tensor of size n 1 × • • • × n t .
Let E j = {e j,1 , . . . , e j,nj } be a basis for the vector space V j , j = 1, . . . , t. We define a basis

E for V 1 ⊗ • • • ⊗ V t as follows: E := {e i1,...,it = e 1,i1 ⊗ • • • ⊗ e t,it | 1 ≤ i j ≤ n j , ∀j = 1, . . . , t}. (1) A tensor T ∈ V 1 ⊗ • • • ⊗ V t can
be represented via a so called "hypermatrix" (or "array")

A = (a i1,...,it ) 1≤ij ≤nj , j=1,...,t
with respect to the basis E defined in (1), i.e.:

T = 1≤ij ≤nj , j=1,...,t a i1,...,it e i1,...,it .

Definition 2.1.

A tensor T ∈ V 1 ⊗ • • • ⊗ V t is called "decomposable" if, for all j = 1, . . . , t, there exist v j ∈ V j such that T = v 1 ⊗ • • • ⊗ v t .
Definition 2.2. Let E j = {e j,1 , . . . , e j,nj } be a basis for the vector space V j for j = 1, . . . , t. Let also v j = nj i=1 a j,i e j,i ∈ V j for j = 1, . . . , t. The image of the following embedding

P(V 1 ) × • • • × P(V t ) ֒→ P(V 1 ⊗ • • • ⊗ V t ) ([v 1 ] , . . . , [v t ]) → [v 1 ⊗ • • • ⊗ v t ] = = 1≤ij ≤nj , j=1,...,t [(a 1,i1 • • • a t,it )e i1,...,it ]
is well defined and it is known as "Segre Variety". We denote it by

Seg(V 1 ⊗ • • • ⊗ V t ). Remark: A Segre variety Seg(V 1 ⊗ • • • ⊗ V t ) parameterizes the decomposable tensors of V 1 ⊗ • • • ⊗ V t . A set of equations defining Seg(V 1 ⊗ • • • ⊗ V t )
is well known (one of the first reference for a set-theoretical description of the equations of Segre varieties is [Gr]). Before introducing that result we need the notion of d-minor of a hypermatrix.

Notation:

• The hypermatrix A = (x i1,...,it ) 1≤ij ≤nj , j=1,...,t is said to be a generic hypermatrix of indeterminates (or more simply generic hypermatrix ) of S := K[x i1,...,it ] 1≤ij ≤nj , j=1,...,t , if the entries of A are the independent variables of S.

• We denote by S t the homogeneous degree t part of the polynomial ring S.

• We will always suppose that we have fixed a basis E i for each V i and the basis

E for V 1 ⊗ • • • ⊗ V t as in (1).
• When we will write "A is the hypermatrix associated to the tensor T " (or vice versa) we will always assume that the association is via the fixed basis E. Moreover if the size of

T is n 1 × • • • × n t , then A is of the same size.
It is possible to extend the notion of "d-minor of a matrix" to that of "d-minor of a hypermatrix".

Definition 2.3. Let V 1 , . . . , V t be vector spaces of dimensions n 1 , . . . , n t , respectively, and let (J 1 , J 2 ) be a partition of the set {1, . . . , t}.

If J 1 = {h 1 , . . . , h s } and J 2 = {1, . . . , t}\J 1 = {k 1 , . . . , k t-s }, the (J 1 , J 2 )- Flattening of V 1 ⊗ • • • ⊗ V t is the following: V J1 ⊗ V J2 = (V h1 ⊗ • • • ⊗ V hs ) ⊗ (V k1 ⊗ • • • ⊗ V kt-s ). Definition 2.4. Let V J1 ⊗ V J2 be any flattening of V 1 ⊗ • • • ⊗ V t and let f J1,J2 : P(V 1 ⊗ • • • ⊗ V t ) ∼ → P(V J1 ⊗ V J2 ) be the obvious isomorphism. Let A be a hypermatrix associated to a tensor T ∈ V 1 ⊗ • • • ⊗ V t ; let [T ′ ] = f J1,J2 ([T ]) ∈ P(V J1 ⊗ V J2
) and let A J1,J2 be the matrix associated to T ′ . Then the d-minors of the matrix A J1,J2 are said to be "d-minors of A". Sometimes we will improperly write "a d-minor of a tensor T ", meaning that it is a d-minor of the hypermatrix associated to such a tensor via the fixed basis

E of V 1 ⊗ • • • ⊗ V t .
Example: d-minors of a decomposable tensor.

Let V 1 , . . . , V t and (J 1 , J 2 ) = ({h 1 , . . . , h s }, {k 1 , . . . , k t-s }) as before. Consider the following composition of maps:

P(V 1 ) × • • • × P(V t ) s1×s2 → P(V J1 ) × P(V J2 ) s → P(V J1 ⊗ V J2 )
where

Im(s 1 × s 2 ) = Seg(V J1 ) × Seg(V J2
) and Im(s) is the Segre variety of two factors.

Consider the basis (made as E above) E J1 for V J1 and E J2 for V J2 . In terms of coordinates, the composition s • (s 1 × s 2 ) is described as follows.

Let v i = (a i,1 , . . . , a i,ni ) ∈ V i for each i = 1, . . . , t and

T = v 1 ⊗ • • • ⊗ v t ∈ V 1 ⊗ • • • ⊗ V t ; then: s 1 × s 2 ([(a
,...,n k t-s ) = (z 1 , . . . , z N2 ), with N 2 = n k1 • • • n kt-s , then: s([(y 1 , . . . , y N1 )], [(z 1 , . . . , z N2 )]) = [(q 1,1 , q 1,2 , . . . , q N1,N2 )] = s • (s 1 × s 2 )([T ]),
where q i,j = y i z j for i = 1, . . . , N 1 and j = 1, . . . , N 2 . We can easily rearrange coordinates and write s • (s 1 × s 2 )([T ]) as a matrix:

((s 1 × s 2 ) • s)([T ]) =    q 1,1 • • • q 1,N2 . . . . . . q N1,1 • • • q N1,N2    . (2)
A d-minor of the matrix s Example: The ideal of the 2-minors of a generic hypermatrix A = (x i1,...,it ) 1≤ij ≤nj , j=1,...,t is I 2 (A) := (x i1,...,i l ,...,it x j1,...,j l ,...,jt -x i1,...,j l ,...,it x j1,...,i l ,...,jt ) l=1,...,t; 1≤i k ,j k ≤nj , k=1,...,t .

• (s 1 × s 2 )([T ]) defined in (2) is called a d-
It is a classical result (see [Gr]) that a set of equations for a Segre Variety is given by all the 2-minors of a generic hypermatrix. In fact, as previously obseved, a Segre variety parameterizes decomposable tensors, i.e. all the "rank one" tensors.

In [START_REF] Hà | Box-shaped matrices and the defining ideal of certain blowup surface[END_REF] (Theorem 1.5) it is proved that, if A is a generic hypermatrix of a polynomial ring S of size n 1 × • • • × n t , then I 2 (A) is a prime ideal in S, therefore:

I(Seg(V 1 ⊗ • • • ⊗ V t )) = I 2 (A) ⊂ S.
Now we generalize this result to another class of decomposable tensors: those defining "Segre-Veronese varieties".

3 Segre-Veronese varieties With an abuse of notation we will say that a tensor T ∈ V ⊗d is supersymmetric if it can be represented by a supersymmetric hypermatrix. Definition 3.2. Let H ⊂ V ⊗d be the n+d-1 d -dimensional subspace of the supersymmetric tensors of V ⊗d , i.e.

H is isomorphic to the symmetric algebra Sym d (V ). Let S be a ring of coordinates on P ( n+d-1 d )-1 = P(H) obtained as the quotient S = S/I where S = K[x i1,...,i d ] 1≤ij ≤n, j=1,...,d and I is the ideal generated by all x i1,...,i d -x i σ(1) ,...,i σ(d) , ∀ σ ∈ S d . The hypermatrix (x i1,...,i d ) 1≤ij ≤n, j=1,...,d whose entries are the indeterminates of S, is said to be a "generic supersymmetric hypermatrix".

Remark:

The Veronese variety Y n-1,d ⊂ P ( n+d-1 d )-1 can be viewed as Seg(V ⊗d ) ∩ P(H) ⊂ P(H).

Let A = (x i1,...,i d ) 1≤ij ≤n, j=1,...,d be a generic supersymmetric hypermatrix, then it is a known result that:

I(Y n-1,d ) = I 2 (A) ⊂ S.
(3)

See [Wa] for set theoretical point of view. In [Pu] the author proved that I(Y n-1,d ) is generated by the 2-minors of a particular catalecticant matrix (for a definition of "Catalecticant matrices" see e.g. either [Pu] or [Ge]). A. Parolin, in his PhD thesis ( [Pa]), proved that the ideal generated by the 2-minors of that catalecticant matrix is actually I 2 (A), where A is a generic supersymmetric hypermatrix.

In this way we have recalled two very related facts:

• if A is a generic n 1 × • • • × n t hypermatrix, then the ideal of the 2-minors of A is the ideal of the Segre variety Seg(V 1 ⊗ • • • ⊗ V t ); • if A is a generic supersymmetric n × • • • × n d hypermatrix, then the ideal of the 2-minors of A is the ideal of the Veronese variety Y n-1,d , with dim(V ) = n.
Now we want to prove that a similar result holds also for other kinds of hypermatrices strictly related with those representing tensors parameterized by Segre varieties and Veronese varieties. Definition 3.3. Let V 1 , . . . , V t be vector spaces of dimensions n 1 , . . . , n t respectively. The Segre-Veronese variety S d1,...,dt

(V 1 ⊗ • • • ⊗ V t ) is the embedding of P(V 1 ) ⊗ • • • ⊗ P(V t ) into P N -1 , where N = Π t i=1 ni+di-1 di
, given by sections of the sheaf O(d 1 , . . . , d t ).

I.e. S d1,...,dt (V 1 ⊗ • • • ⊗ V t )
is the image of the composition of the following two maps:

P(V 1 ) × • • • × P(V t ) ν d 1 ו••×ν d t -→ P ( n 1 +d 1 -1 d 1 )-1 × • • • × P ( n t +d t -1 d t
)-1

and P ( n 1 +d 1 -1 d 1 )-1 × • • • × P ( n t +d t -1 d t )-1 s -→ P N -1
where

Im(ν 1 × • • • × ν t ) = Y n1-1,d1 × • • • × Y nt-1,dt and Im(s)
is the Segre variety with t factors.

Example:

If (d 1 , . . . , d t ) = (1, . . . , 1) then S 1,...,1 (V 1 ⊗ • • • ⊗ V t ) = Seg(V 1 ⊗ • • • ⊗ V t ).
Example:

If t = 1 and dim(V ) = n, then S d (V ) is the Veronese variety Y n-1,d .
Below we describe how to associate to each element of S d1,...,dt

(V 1 ⊗ • • • ⊗ V t ) a decomposable tensor T ∈ V ⊗d1 1 ⊗ • • • ⊗ V ⊗dt t . Definition 3.4. Let n = (n 1 , . . . , n t ) and d = (d 1 , . . . , d t ). If V i are vector spaces of dimension n i for i = 1, . . . , t, an "(n, d)-tensor" is defined to be a tensor T belonging to V ⊗d1 1 ⊗ • • • ⊗ V ⊗dt t .
Definition 3.5. Let n and d as above. A hypermatrix A = (a i1,1,...,i 1,d 1 ;...;it,1,...,i t,d t ) 1≤i j,k ≤nj , k=1,...,dj , j=1,...,t is said to be "(n, d)-symmetric" if a i1,1,...,i 1,d 1 ;...;it,1,...,i t,d t = a i σ 1 (1,1) ,...,i σ 1 (1,d 1 ) ;...;i σ t (t,1) ,...,i σ t (t,d t ) for all permutations σ j ∈ S(j, d j ) where S(j, d j ) ≃ S dj is the permutation group on {(j, 1), . . . , (j, d j )} for all j = 1, . . . , t.

An (n, d)-tensor T ∈ V ⊗d1 1 ⊗ • • • ⊗ V ⊗dt t
is said to be an "(n, d)-symmetric tensor" if it can be represented by an (n, d)-symmetric hypermatrix.

Definition 3.6. Let H i ⊂ V ⊗di i be the subspace of supersymmetric tensors of V ⊗di i for each i = 1, . . . , t, then H 1 ⊗ • • • ⊗ H t ⊂ V ⊗d1 1 ⊗ • • • ⊗ V ⊗dt t is the subspace of the (n, d)-symmetric tensors of V ⊗d1 1 ⊗ • • • ⊗ V ⊗dt t . Let n = (n 1 , . . . , n t ) and d = (d 1 , . . . , d t ) and let R [n,d] be the ring of coordinates on P N -1 = P(H 1 ⊗ • • • ⊗ H t ), with N = Π t i=1 ni+di-1 di
, obtained from S = K[x i1,1,...,i 1,d 1 ;...;it,1,...,i t,d t ] 1≤i j,k ≤nj , k=1,...,dj , j=1,...,t via the quotient modulo x i1,1,...,i 1,d 1 ;...;it,1,...,i t,d t -x i σ 1 (1,1) ,...,i σ 1 (1,d 1 ) ;...;i σ t (t,1) ,...,i σ t (t,d t ) , for all σ j ∈ S(j, d j ) and j = 1, . . . , t. The hypermatrix (x i1,1,...,i 1,d 1 ;...;it,1,...,i t,d t ) 1≤i j,k ≤nj , k=1,...,dj , j=1,...,t of indeterminates of R [n,d] , is said to be a "generic (n, d)-symmetric hypermatrix".

Remark: It is not difficult to check that, as sets:

P(H 1 ⊗ • • • ⊗ H t ) ∩ Seg(V ⊗d1 1 ⊗ • • • ⊗ V ⊗dt t ) = S d1,...,dt (V 1 ⊗ • • • ⊗ V t ); (4) i.e. S d1,...,dt (V 1 ⊗ • • • ⊗ V t ) parameterizes the (n, d)-symmetric decomposable (n, d)-tensors of V ⊗d1 1 ⊗ • • • ⊗ V ⊗dt t .
Since Segre variety is given by the vanishing of 2-minors of a hypermatrix of indeterminates and

H 1 ⊗ • • • ⊗ H i is a linear subspace of V 1 ⊗ • • • ⊗ V t ,
it follows that a Segre-Veronese variety is set-theoretically given by the 2-minors of an (n, d)-symmetric hypermatrix of indeterminates .

In Section 3.3 we will prove that the ideal of the 2-minors of the generic (n, d)-symmetric hypermatrix in R [n,d] is the ideal of a Segre-Veronese variety. We will need the notion of "weak generic hypermatrices" that we are going to introduce.

Weak Generic Hypermatrices

The aim of this section is Proposition 3.10 which asserts that the ideal generated by 2-minors of a weak generic hypermatrix (Definition 3.8) is prime.

Definition 3.7. A k-th section of a hypermatrix A = (x i1,...,it ) 1≤ij ≤nj ,j=1,...,t is a hypermatrix of the form

A (l) i k = (x i1,...,it ) 1≤ij ≤nj ,j=1,..., k,...,t,i k =l . Remark: If a hypermatrix A represents a tensor T ∈ V 1 ⊗ • • • ⊗ V t , then a k-th section of A is a hypermatrix representing a tensor T ′ ∈ V 1 ⊗ • • • ⊗ Vk ⊗ • • • ⊗ V t .
We introduce now the notion of "weak generic hypermatrices"; this is a generalization of "weak generic box" in [START_REF] Hà | Box-shaped matrices and the defining ideal of certain blowup surface[END_REF].

Definition 3.8. Let K[u 1 , . . . , u r ] be a ring of polynomials. A hypermatrix A = (f i1,...,it ) 1≤ij ≤nj , j=1,...,t , where all f i1,...,it ∈ K[u 1 , . . . , u r ] 1 ,

is called a "weak generic hypermatrix of indeterminates" (or briefly "weak generic hypermatrix") if:

1. all the entries of A belong to {u 1 , . . . , u r };

2. there exists an entry f i1,...,it such that f i1,...,it = f k1,...,kt for all (k 1 , . . . , k t ) = (i 1 , . . . , i t ), 1 ≤ k j ≤ n j , j = 1, . . . , t;

3. the ideals of 2-minors of all sections of A are prime ideals.

Lemma 3.9. Let I, J ⊂ R = K[u 1 , . . . , u r ] be ideals such that J = (I, u 1 , . . . , u q ) with q < r. Let f ∈ R be a polynomial independent of u 1 , . . . , u q and such that I : f = I. Then J : f = J.

Proof. We need to prove that if g ∈ R is such that f g ∈ J, then g ∈ J.

Any polynomial g ∈ R can be written as g = g 1 + g 2 where g 1 ∈ (u 1 , . . . , u q ) and g 2 is independent of u 1 , . . . , u q . Clearly g 1 ∈ J. Now f g 2 = f g -f g 1 ∈ J and f g 2 is independent of u 1 , . . . , u q . This implies that f g 2 ∈ I, then g 2 ∈ I ⊂ J because I : f = I by hypothesis. Therefore g = g 1 + g 2 ∈ J. Now we can state the main proposition of this section. The proof that we are going to exhibit follows the ideas the proof of Theorem 1.5 in [START_REF] Hà | Box-shaped matrices and the defining ideal of certain blowup surface[END_REF], where the author proves that the ideal generated by 2-minors of a generic hypermatrix of indeterminates is prime. In the same proposition (Proposition 1.12) it is proved that also the ideal generated by 2-minors of a "weak generic box" is prime. We give here an independent proof for weak generic hypermatrix, since it is a more general result; moreover we do not follow exactly the same lines as in [START_REF] Hà | Box-shaped matrices and the defining ideal of certain blowup surface[END_REF].

Proposition 3.10. Let R = K[u 1 , . . . , u r ] be a ring of polynomials and let A = (f i1,...,it ) 1≤ij ≤nj , j=1,...,t be a weak generic hypermatrix as defined in 3.8. Then the ideal I 2 (A) is a prime ideal in R.

Proof. Since A = (f i1,...,it ) 1≤ij ≤nj , j=1,...,t is a weak generic hypermatrix, there exists an entry f i1,...,it that verifies the item 2. in Definition 3.8. It is not restrictive to assume that such

f i1,...,it is f 1,...,1 . Let F, G ∈ R s.t. F G ∈ I 2 (A). We want to prove that either F ∈ I 2 (A) or G ∈ I 2 (A). Let Z = {f k 1,...,1 | k ≥ 0} ⊂ R and let R Z be the localization of R at Z. Let also ϕ : R → R Z such that ϕ(f j1,...,jt ) = f j1,1,...,1 • • • f 1,..., 1,jt f t-1 1,...,1 , ϕ(K) = K and ϕ(u i ) = u i for u i ∈ {u 1 , . . . , u r }\{f i1,...,it | 1 ≤ i j ≤ n j , j = 1, . . . , t}. Clearly ϕ(m) = 0 for all 2-minors m of A. Hence ϕ(I 2 (A)) = 0. Since F (. . . , f j1,...,jt , . . .)G(. . . , f j1,...,jt , . . .) ∈ I 2 (A) then F (. . . , ϕ(f j1,...,jt ), . . .)•G(. . . , ϕ(f j1,...,jt ), . . .) = 0 R Z . The localization R Z is a domain because R is a domain, thus either F (. . . , ϕ(f j1,...,jt ), . . .) = 0 R Z , or G(. . . , ϕ(f j1,...,jt ), . . .) = 0 R Z . Suppose that F . . . , fj 1 ,1,...,1•••f1,..., 1,j t f t-1 1,...,1 , . . . = 0 R Z . We have F (. . . , f j1,...,fj t , . . .) = F . . . , f j1,1,...,1 • • • f 1,..., 1,jt f t-1 1,...,1 , . . . + H, (5)
where H belongs to the ideal (f j1,...,jt f

t-1 1,...,1 -f j1,1...,1 • • • f 1,...,1,jt ) 1≤j k ≤nj , k=1,...,t ⊂ R Z . Now let H t-1 = f j1,...,jt f t-1 1,...,1 -f j1,1...,1 • • • f 1,...,1,jt . Then H t-1 = f 11,j2,...,jt f j1,1,...,1 f t-2 j1,...,jt + (f 1,...,1 f j1,...,jt -f 1,j2,...,jt f j1,1...,1 )f t-2 j1,...,jt - -f 1,j2,...,jt f j1,1,j3,...,jt • • • f j1,...,jt-1,1 ≡ I2(A) f 1,j2,...,jt f j1,1,...,1 f t-2 1,...,1 -f 1,j2,...,jt f j1,1,j3,...,jt • • • f j1,...,jt-1,1 = H t-2 .
Proceeding analogously for H t-2 , . . . , H 1 , it is easy to verify that H t-1 ∈ I 2 (A). Hence H belongs to the ideal of R Z generated by I 2 (A). This fact, together with (5), implies that also F belongs to the ideal of R Z generated by I 2 (A). Therefore we obtained that if ϕ(F ) = 0 R Z , then there exists ν > 0 such that

f ν 1,...,1 F (. . . , f j1,...,jt , . . .) ∈ I 2 (A) ⊂ R. (6)
Now we want to prove that if there exists ν > 0 such that f ν 1,...,1 F (. . . , f j1,...,jt , . . .) ∈ I 2 (A), then F ∈ I 2 (A). Analogously as it is done in the proof of Lemma 1.4 in [START_REF] Hà | Box-shaped matrices and the defining ideal of certain blowup surface[END_REF], we will use a triple induction: first on the dimension t of the hypermatrix A, then on t j=1 n j , and finally on deg(F ). Induction on t. For t = 2 our goal is proved in Lemma 3 of [Sh]. Assume that t > 2 and that the induction hypothesis holds for any weak generic hypermatix of size lower than t.

Induction on t j=1 n j . If n j = 1 for at least one j ∈ {1, . . . , t}, then A is a hypermatrix of order (t -1), so the result is true for the induction hypothesis on t. Assume that n j ≥ 2 for all j = 1, . . . , t and that the induction hypothesis holds for smaller values of

t j=1 n j . Induction on deg(F). If deg(F ) = 0, since ϕ(F ) = 0 R Z , we have F = 0 ∈ I 2 (A). Then let deg(F ) > 0 and
assume that the induction hypothesis holds for polynomials of degree lower than deg(F ).

In [START_REF] Hà | Box-shaped matrices and the defining ideal of certain blowup surface[END_REF], Corollary 1.1.1, it is proved that (I 2 (A), f n1,...,nt ) = ∩ t l=1 I l where A l is the hypermatrix (f i1,...,it ) i l <n l , and I l := (I 2 (A l ), {f i1,...,it | i l = n l }) . Clearly I 2 (A) ⊆ (I 2 (A), f n1,...,nt ). By (6), we have that f ν 1,...,1 F ∈ I 2 (A). Hence, by Corollary 1.1.1 in [START_REF] Hà | Box-shaped matrices and the defining ideal of certain blowup surface[END_REF], f ν 1,...,1 F ∈ I l for all l = 1, . . . , t. We can apply here the induction hypotheses on t and on t j=1 n j , hence I 2 (A l ) : f ν 1,...,1 = I 2 (A l ). Now, by Lemma 3.9, I l : f ν 1...,,1 = I l , i.e. F ∈ ∩ t l=1 I l = (I 2 (A), f n1,...,nt ). Hence we can write

F = F 1 + F 2 where F 1 ∈ I 2 (A) and F 2 ∈ (f n1,...,nt ), that is to say F = F 1 + f n1,...,nt F2 with deg( F2 ) < deg(F ). Obviously f ν 1,...,1 f n1,...,nt F2 = f ν 1,...,1 F -f ν 1,...,1 F 1 ∈ I 2 (A).
Let's notice that we checked that, since ϕ(f n1,...,nt ) = 0 R Z , for any form K for which f n1,...,nt K ∈ I 2 (A) there exists µ > 0 such that f µ 1,...,1 K ∈ I 2 (A); if we apply this to K = f ν 1,...,1 F2 , we get that f ν+µ 1,...,1 F2 ∈ I 2 (A) for some µ > 0. Now we deduce that there exists µ > 0 s. t. f ν+µ 1,...,1 F2 ∈ I 2 (A). Now, by induction hypothesis on the degree of F , we have that F2 ∈ I 2 (A). Therefore F ∈ I 2 (A).

Ideals of Segre -Veronese varieties

Since a Segre-Veronese variety is given set-theoretically by the 2-minors of an (n, d)-symmetric hypermatrix of indeterminates (see ( 4)), if we prove that any (n, d)-symmetric hypermatrix of indeterminates is weak generic, we will have, as a consequence of Proposition 3.10, that its 2-minors are a set of generators for the ideals of Segre-Veronese varieties.

Remark: If A = (a i1,...,i d ) 1≤ij ≤n; j=1,...,d is a supersimmetric hypermatrix of size n × • • • × n d , then also a k-th section A (l) i k of A is a supersymmetric hypermatrix of size n × • • • × n d-1 .
In fact, since A is supersymmetric, then a i1,...,i d = a i σ(1) ,...,i σ(d) for all σ ∈ S d . The section A (l)

i k is obtained from A by imposing i k = l. Therefore A (l) i k = (a i1,...,i k =l,...i d ) is such that a i1,...,i k =l,...i d = a i σ(1) ,...,i σ(k) =l,...,i σ(d) ,
for all σ ∈ S d such that σ(k) = l, hence such σ's can be viewed as elements of the permutation group of the set {1, . . . , l -1, l + 1, . . . , d} that is precisely S d-1 .

Remark: If [T ] ∈ Y n-1,d
, then a hypermatrix obtained as a section of the hypermatrix representing T , can be associated to a tensor

T ′ such that [T ′ ] ∈ Y n-1,d-1 .
Theorem 3.11. Let n = (n 1 , . . . , n t ) and d = (d 1 , . . . , d t ). Let H i ⊂ V ⊗di i be the subspace of supersymmetric tensors of V ⊗di i for i = 1, . . . , t and let R [n,d] be the ring of coordinates of

P(H 1 ⊗• • •⊗H t ) ⊂ P(V ⊗d1 1 ⊗• • •⊗V ⊗dt t ) defined in Definition 3.6. If A is a generic (n, d)-symmetric hypermatrix of R [n,d] ,
then A is a weak generic hypermatrix and the ideal of the Segre-Veronese variety S d1,...,dt

(V 1 ⊗ • • • ⊗ V t ) is I(S d1,...,dt (V 1 ⊗ • • • ⊗ V t )) = I 2 (A) ⊂ R [n,d] with d i > 0 for i = 1, . . . , t.
Proof. The proof is by induction on t i=1 d i . The case

t i=1 d i = 1 is not very significant because if dim(V 1 ) = n 1 , so S 1 (V 1 ) = Y n1-1,1 = P(V 1 ), then I(S 1 (V 1 )) = I(P(V )) i.
2 (V 1 ) or S 1,1 (V 1 , V 2 ). Clearly, if dim(V 1 ) = n 1 , then S 2 (V 1 ) = Y n1-1,2
is Veronese variety and the theorem holds because of (3). Analogously

S 1,1 (V 1 , V 2 ) = Seg(V 1 ⊗ V 2 )
and again the theorem is known to be true [START_REF] Hà | Box-shaped matrices and the defining ideal of certain blowup surface[END_REF]).

Assume that the theorem holds for every (n, d)-symmetric hypermatrix with t i=1 d i ≤ r -1. Then, by Proposition 3.10, the ideal generated by the 2-minors of such an (n, d)-symmetric hypermatrix is a prime ideal. Now, let A be an (n, d)-symmetric hypermatrix with

t i=1 d i = r.
The first two properties that characterize a weak generic hypermatrix (see Definition 3.8) are immediately verified for A. For the third one we have to check that the ideals of the 2-minors of all sections A (l) ip,q of A are prime ideals. If we prove that A (l) ip,q represents an (n, d ′ )-symmetric hypermatrix (with d ′ = (d 1 , . . . , d p -1, . . . , d t ))) we will have, by induction hypothesis, that A (l) ip,q is a weak generic hypermatrix and hence its 2-minors generate a prime ideal. The hypermatrix A = (a i1,1,...,i 1,d 1 ;...;it,1,...i t,d t ) 1≤i j,k ≤nj , k=1,...,dj , j=1,...,t is (n, d)-symmetric, hence, by definition, a i1,1,...,i 1,d 1 ;...;it,1,...,i t,d t = a i σ 1 (1,1) ,...,i σ 1 (1,d 1 ) ;...;i σ t (t,1) ,...,i σ t (t,d t ) for all permutations σ j ∈ S(j, d j ) where S(j, d j ) is the permutation group on {(j, 1), . . . , (j, d j )} for all j = 1, . . . , t. The hypermatrix A (l) ip,q = (a i1,1,...,i 1,d 1 ;...,ip,q=l,...;it,1,...,i t,d t ), obtained from A by imposing i p,q = l, is (n, d ′ )symmetric because a i1,1,...,i 1,d 1 ;...,ip,q=l,...;it,1,...,i t,d t = a i σ 1 (1,1) ,...,i σ 1 (1,d 1 ) ;...,i σp(p,1) ,...,ip,q=l,...i σp(p,dp) ;...;i σ t (t,1) ,...,i σ t (t,d t ) for all σ j ∈ S(j, d j ), j = 1, . . . , p, . . . , t, and for σ p ∈ S(p, d p -1), where S(p, d p -1) is the permutation group on the set of indices {(p, 1), . . . , (p, q), . . . , (p, d p )} (this is a consequence of the first Remark of this section). Hence I 2 (A (l) ip,q ) is prime by induction, and A is weak generic, so also

I 2 (A) is prime. Since by definition S d1,...,dt (V 1 ⊗ • • • ⊗ V t ) = P(H 1 ⊗ • • • ⊗ H t ) ∩ Seg(V 1 ⊗ • • • ⊗ V t ), we have that I 2 (A) is a set of equations for S d1,...,dt (V 1 ⊗ • • • ⊗ V t ) (see (4)), hence, because of the primeness of I 2 (A) that we have just proved, I 2 (A) ⊂ R [n,d] is the ideal of S d1,...,dt (V 1 ⊗ • • • ⊗ V t ).

Projections of Veronese surfaces

In this section we want to use the tool of weak generic hypermatrices in order to prove that the ideal of a projection of a Veronese surface Y 2,d ⊂ P ( d+2 d )-1 from a finite number s ≤ d 2 of general points on it is the prime ideal defined by the order 2-minors of some particular tensor.

In [START_REF] Hà | Box-shaped matrices and the defining ideal of certain blowup surface[END_REF] the case in which s is a binomial number (i.e. s = t+1 2 for some positive integer t ≤ d -1) is done. In this section we try to extend that result to a projection of a Veronese surface from any number s ≤ d 2 of general points.

Notice that in [Gi] and in [GL] the authors study the projection of Veronese surfaces Y 2,d from s = d 2 + k general points, 0 ≤ k ≤ d, for some non negative integer k, (this corresponds to the case of a number of points between the two consecutive binomial numbers d 2 and d+1 2 ).

Let Z = {P 1 , . . . , P s } ⊂ P 2 be a set of general points in P 2 , where s = t+1 2 + k ≤ d 2 with 0 < t ≤ d -1 and 0 ≤ k ≤ t (actually we may assume t ≤ d -2 because the case t = d -1 and k = 0 corresponds to the known case of the "Room Surfaces" -see [GG]). Let J ⊂ S = K[w 1 , w 2 , w 3 ] be the ideal J = I(Z), i.e. J = ℘ 1 ∩ • • • ∩ ℘ s with ℘ i = I(P i ) ⊂ S prime ideals for i = 1, . . . , s.

Let J d be the degree d part of the ideal J and let Bl Z (P 2 ) be the blow up of P 2 at Z. Since d ≥ t + 1, the linear system of the strict transforms of the curves defined by J d , that we indicate with Jd , is very ample. If ϕ J d : P 2 P ( d+2 2 )-s-1 is the rational morphism associated to J d and if ϕ Jd : Bl Z (P 2 ) → P ( d+2 2 )-s-1 is the morphism associated to Jd , the variety X Z,d we want to study is Im(ϕ J d ) = Im(ϕ Jd ). This variety can also be viewed as the projection of the Veronese surface Y 2,d ⊂ P ( d+2

2 )-1 from s general points on it. The first thing to do is to describe J d as vector space.

The ideal of general points in the projective plane

There is a classical result, Hilbert-Burch Theorem (see, for instance, [CGO]), that gives a description of the generators of J. I.e. the ideal

J ⊂ S = K[w 1 , w 2 , w 3 ] is generated by t -k + 1 forms F 1 , . . . , F t-k+1 ∈ S t and by h forms G 1 , . . . , G h ∈ S t+1 where h = 0 if 0 ≤ k < t/2 and h = 2k -d if t/2 ≤ k ≤ t.
What follows now is the constructions of the F j 's and the G i 's (the same description is presented in [GL]).

If t/2 ≤ k ≤ t, for a general choice of points P 1 , . . . , P s , the generators of J can be chosen to be the maximal minors of:

L :=    L 1,1 • • • L 1,2k-t Q 11 • • • Q 1,t-k+1 . . . . . . . . . . . . L k,1 • • • L k,2k-t Q k,1 • • • Q k,t-k+1    ∈ M k,k+1 (S) (7)
where L i,j ∈ S 1 and Q h,l ∈ S 2 for all i, h = 1, . . . , k, j = 1, . . . , 2k -t and l = 1, . . . , t -k + 1. The forms F j ∈ S t are the minors of L obtained by deleting the 2k -t+j-th column, for j = 1, . . . , t-k +1; the forms G i ∈ S t+1 are the minors of L obtained by deleting the i-th column, for i = 1, . . . , 2k -t.

The degree (t + 1) part of the ideal J is clearly J t+1 =< w 1 F 1 , . . . , w 3 F t-k+1 , G 1 , . . . , G 2k-t >. If we set Gi,j = w i F j for i = 1, 2, 3, j = 1, . . . , t -k + 1 we can write:

J t+1 =< G1,1 , . . . , G3,t-k+1 , G 1 , . . . , G 2k-t > .
Notice that w 1 F 1 = G1,1 , . . . , w 3 F t-k+1 = G3,t-k+1 are linearly independent (see, for example, [CGO]).

If 0 ≤ k < t/2, then J is generated by maximal minors of:

L :=           Q 1,1 • • • • • • • • • Q 1,t-k+1 . . . . . . Q k,1 • • • • • • • • • Q k,t-k+1 L 11 • • • • • • • • • L 1,t-k+1 . . . . . . L t-2k,1 • • • • • • • • • L t-2k,t-k+1           ∈ M t-k,t-k+1 (S) (8)
where L i,j ∈ S 1 and Q h,l ∈ S 2 for all i = 1, . . . , t -2k, j, l = 1, . . . , t -k + 1 and h = 1, . . . , k.

The forms F j ∈ S t are the minors of L obtained by deleting the j-th column for j = 1, . . . , t -k + 1. Again J t+1 =< w 1 F 1 , . . . , w 3 F t-k+1 > but now those generators are not necessarily linearly independent.

Using the same notation of the previous case one can write:

J t+1 =< G1,1 , . . . , G3,t-k+1 > .
Clearly if t/2 ≤ k ≤ t then:

J d =< w d-t-1 Gi,j , w d-t-1 G l > (9) for i = 1, 2, 3, j = 1, . . . t -k + 1, l = 1, . . . , 2k -t and w d-t-1 G = {w d-t-1 1 G, w t-d-2 1 w 2 G, . . . , w d-t-1 3 G}. If 0 ≤ k < t/2 then: J d =< w d-t-1 Gi,j > (10) for i = 1, 2, 3 and j = 1, . . . , t -k + 1. Denote          z 1 := w d-t-1 1 , z 2 := w t-d-2 1 w 2 , . . . z u := w t-d-1 3 where u = d-t+1 2 ; or z α for w α = w α1 1 w α2 2 w α3 3 , if α = (α 1 , α 2 , α 3 ) ∈ N 3 , |α| = d -t
-1 and we assume that the α's are ordered by the lexicographic order.

Let N be the number of generators of J d , and let K[x h;i,j , x h,l ] be a ring of coordinates on P N -1 with l = 1, . . . , 2k -t only if t/2 ≤ k ≤ t (in the other case the variables x h,l do not exist at all) and h = 1, . . . , u; i = 1, 2, 3; j = 1, . . . , t -k + 1 in any case. The morphism ϕ :

P 2 \ Z → P N -1 such that ϕ([w 1 , w 2 , w 3 ]) = [z 1 G1,1 , . . . , z u G3,t-k+1 , z 1 G 1 , . . . , z u G 2k-t ], if t/2 ≤ k ≤ t, or ϕ([w 1 , w 2 , w 3 ]) = [z 1 G1,1 , . . . , z u G3,t-k+1 ], if 0 ≤ k < t/2, gives a parameterization of X Z,d into P N -1 . Observe that X Z,d = ϕ J d (P 2 \ Z) is naturally embedded into P ( d+2 2 )-s-1 , because dim K (J d ) = d+2 2 -s.
In terms of the xh;i,j 's and the x h,l 's, since the parameterization of X Z,d is:

xh;i,j = z h Gi,j , x h,l = z h G l , (11)
the independent linear relations between the generators of J d will give the subspace P(< Im(ϕ Jd ) >) = P ( d+2 2 )-s-1 of P N -1 . The number of such relations has to be N -d+2 2 + s. If t/2 ≤ k ≤ t, the number of generators of J d given by ( 9) is d-t+2 2 (t -k + 1) + d-t-1+2 2 (2k -t); hence there must be d-t 2 k independent relations between those generators of J d . If 0 ≤ k < t/2, the number of generators of J d in ( 10) is d-t+2 2 (t -k + 1), hence there must be d-t+1

2 (t - k) -k(d -t) independent relations between those generators of J d .
There is a very intuitive way of finding exactly those numbers of relations between the generators of J d and this is what we are going to describe (then we will prove that such relations are also independent).

If t/2 ≤ k ≤ t, assume that β = (β 1 , β 2 , β 3 ) with |β| = d -t -2.
The determinant obtained by adding to the matrix L defined in (7) a row

w β L i,1 • • • w β L i,2k-t w β Q i,1 • • • w β Q i,t-k+1
clearly vanish for all i = 1, . . . , k:

det w β L i,1 • • • w β L i,2k-t w β Q i,1 • • • w β Q i,t-k+1 L = 0.
Computing those determinants, for i = 1, . . . , k, one gets:

2k-t r=1 w β L i,r G r + t-k+1 p=1 w β Q i,p F p = 0 ( 12
)
where the G r 's and the F p 's are defined as minors of ( 7). Since L i,r ∈ S 1 , there exist some λ i,r,l ∈ K, for i = 1, . . . , k, r = 1, . . . , 2k -t and l = 1, 2, 3, such that

L i,r = 3 l=1 λ i,r,l w l ;
analogously, since Q i,p ∈ S 2 , there exist some γ i,p,l,h ∈ K, for i = 1, . . . , k, p = 1, . . . , t -k + 1 and l, h = 1, 2, 3, such that

Q i,p = 3 l,h=1
γ i,p,l,h w l w h .

Before rewriting the equations ( 12), observe that

Q i,p F p =   3 l,h=1 γ i,p,l,h w l w h   F p = 3 l,h=1
γ i,p,l,h w l Gh,p , and set:

• µ i,α,r = λ i,r,l , if α = β + e l , 0
otherwise, for i = 1, . . . , k; |α| = t -d -1 and l = 1, 2, 3 and where e 1 = (1, 0, 0), e 2 = (0, 1, 0) and e 3 = (0, 0, 1);

• μi,α,p,h = γ i,p,l,h , if α = β + e l , 0 otherwise, for i = 1, . . . , k; p = 1, . . . , t -k + 1; l, h = 1, 2, 3 and |α| = d -t -2.
Therefore the equations ( 12), for i = 1, . . . , k, can be rewritten as follows:

|α| = d -t -1 1 ≤ r ≤ 2k -t µ i,α,r w α G r + |α| = d -t -1 1 ≤ p ≤ t -k + 1 h = 1, 2, 3 μi,α,p,h w α Gh,p = 0, (13)
which, for i = 1, . . . , k, in terms of x α,r and xα,h,p defined in (11) becomes:

|α| = d -t -1 1 ≤ r ≤ 2k -t µ i,α,r x α,r + |α| = d -t -1 1 ≤ p ≤ t -k + 1 h = 1, 2, 3 μi,α,p,h xα,h,p = 0. (E 1 )
There are exactly k of such relations for each β and the number of β's is d-t 2 . Hence in (13) we have found precisely the number of relations between the generators of J d that we were looking for; we need to prove that they are independent.

If 0 ≤ k < t/2, the way of finding the relations between the generators of J d is completely analogous to the previous one. The only difference is that in this case they come from the vanishing of two different kinds of determinants:

det w β L i,1 • • • w β L i,t-k+1 L = 0 (14)
for i = 1, . . . , t -2k, |β| = d -t -1 and L defined as in (8); and

det w β ′ Q j,1 • • • w β ′ Q j,t-k+1 L = 0 (15) for j = 1, . . . , k, |β ′ | = d -t -2 and L defined as in (8).
Proceeding as in the previous case one finds that the relations coming from ( 14) are of the form

|α| = d -t -1 1 ≤ r ≤ t -k -1 l, h = 1, 2, 3 λi,α,r,l z α Gh,r = 0 (E)
for some λi,α,r,l ∈ K and the number of them is d-t+1 2 (t -2k).

The relations coming from (15) are of the form

|α| = d -t -1 1 ≤ r ≤ t -k + 1 l, h = 1, 2, 3 μi,α,r,l z α Gh,r = 0 (EE)
for some μi,α,r,l ∈ K and the number of them is d-t 2 k. The equations (E) and (EE) allow to observe that X Z,d is contained in the projective subspace of P N -1 defined by the following linear equations in the variables xα,h,r :

             |α| = d -t -1 1 ≤ r ≤ t -k -1 l, h = 1, 2, 3 λi,α,r,l xα;h,r = 0 |α| = d -t -1 1 ≤ r ≤ t -k + 1 l, h = 1, 2, 3 μi,α,r,l xα;h,r = 0 (E 2 )
The number of relations

(E 2 ) is d-t+1 2 (t-2k)+ d-t 2 k
, that is exactly the number of independent relations we expect in the case 0 ≤ k < t/2. Now we have to prove that the relations (E 1 ), respectively (E 2 ), are independent.

Notation: Let M be the matrix of order d-t 2 k × d-t+1 2 (2t -k + 3)
given by the µ i,α,r and the μi,α,p,h appearing in all the equations (E 1 ). We have already observed that there exists an equation of type (E 1 ) for each multi-index over three variables β of weight |β| = d -t -2, and for each i = 1, . . . , k. We construct the matrix M by blocks M β,α (the triple multi-index α is such that |α| = d -t -1):

M = (M β,α ) |β|=d-t-2,|α|=d-t-1 (16)
and the orders on the β's and the α's are the respective decreasing lexicographic orders. For each fixed β and α, the block M β,α is the following matrix:

M β,α =    µ 1,α,1 • • • µ 1,α,2k-t μ1,α,1,1 • • • μ1,α,t-k+1,3 . . . . . . . . . . . . µ k,α,1 • • • µ k,α,2k-t μk,α,1,1 • • • μk,α,t-k+1,3    .
Analogously we construct the matrix N of order d-t+1

2 (t -2k) + d-t 2 k × 3 t-d+1 2 (t -k + 1) : N := N β,α N β ′ ,α |α|=|β|=d-t-1, |β ′ |=d-t-2 (17)
where

N β,α :=    λ1,α,1,1 • • • λ1,α,t-k-1,3 . . . . . . λt-2k,α,1,1 • • • λt-2k,α,t-k-1,3    and N β ′ ,α :=    μ1,α1,1 • • • μ1,αt-k+1,3 . . . . . . μk,α1,1 • • • μk,αt-k+1,3   
where the λi,α,r,l 's and the μi,α,r,l 's are those appearing in (E) and in (EE) respectively.

Proposition 4.1. The matrices M and N defined in ( 16) and ( 17), respectively, are of maximal rank.

Proof. Without loss of generality we may assume that P = [0, 0, 1] / ∈ Z and that F 1 (i.e. the first minor of the matrix L defined either in (7) or in (8)) does not vanish at P .

For the M case, one can observe that every time α = β + e l , l = 1, 2, 3, the block M β,α is identically zero, and we denote M β,β+e l with A l for l = 1, 2, 3.

Consider M the maximal square submatrix of M obtained by deleting the last columns of M (recall that we have ordered both the columns and the rows of M with the respective decreasing lexicographic orders).

All the blocks M β,α on the diagonal of M are such that the position of β is the same position of α in their respective decreasing lexicographic orders. Since |β| = |α| -1, then the blocks appearing on the diagonal of M are M β,β+e 1 = A 1 for all β's. If β = (β 1 , β 2 , β 3 ) and α = (α 1 , α 2 , α 3 ), the blocks M β,α under the diagonal are all such that β 1 < α 1 -2, hence they are all equal to zero. This is clearly sufficient to prove that M has maximal rank; then M has maximal rank too.

The N case is completely analogous.

With this discussion we have proved the following:

Proposition 4.2. The coordinates of the points in X Z,d ⊂ P N -1 = P((K[x h;i,j , x h,l ] 1 ) * ) satisfy either the equations (E 1 ) if t/2 ≤ k ≤ t, or (E 2 ) if 0 ≤ k < t/2. Moreover the relations (E 1 ), respectively (E 2 ), are linearly independent.

Remark: There exist other linear relations between the xα;i,j 's and the x α,l coming from the fact that w i Gh,j = w h Gi,j for i, h = 1, 2, 3 and all j's. If we denote z β+e i = w β w i (with |β| = d -t -2), we have that z β+e i Gh,j = z β+e h Gi,j , that is equivalent to: xβ+e i ;h,j = xβ+e h ;i,j .

The proposition just proved and the fact that the span < Im(ϕ Jd ) > has the same dimension of the subspaces of P N defined by either (E 1 ) or by (E 2 ), imply that those relations are linear combinations of either the (E 1 ), or the (E 2 ). Now the study moves from the linear dependence among generators of J d to the dependence in higher degrees.

Quadratic relations

Remark:

1. Let X := (x h;i,j , x h,l ) h;i,j,l be the matrix whose entries are the variables of the coordinate ring K[x h;i,j , x h,l ] 1 where the index h = 1, . . . , d-t+1 2 indicates the rows of X, and the indicies (i, j, l) indicate the columns and are ordered via the lexicographic order, i = 1, 2, 3, j = 1 . . . , t -k + 1, l = 1, . . . , 2k -t (when it occurs). The 2-minors of X are annihilated by points of X Z,d . Denote this set of equations with (XM).

2. The z i 's satisfy the equations of the Veronese surface Y 2,d-t-1 , i.e. the 2-minors of the following catalecticant matrix:

C :=   z 1 z 2 z 3 • • • z u-2 z 2 z 4 z 5 • • • z u-1 z 3 z 5 z 6 • • • z u   (18)
with u = d-t+1

2 . Multiplying C either by Gi,j , or by G l , for each i = 1, 2, 3; j = 1, . . . , t -k + 1 and l = 1, . . . , 2k -t, one obtains either

  x1;i,j • • • xu-2;i,j x2;i,j • • • xu-1;i,j x3;i,j • • • xu;i,j   , or   x 1,l • • • x u-2,l x 2,l • • • x u-1,l x 3,l • • • x u,l   .
Therefore on X Z,d ⊂ P N -1 , the coordinates x1;i,j , . . . , xu;i,j , for all i = 1, 2, 3 and j = 1, . . . , t -k + 1, or x 1,l , . . . , x u,l , for all l = 1, . . . , 2k -t, annihilate the 2-minors of those catalecticant matrices, respectively. Denote the set of all these equations with (Cat).

3. For all h = 1, . . . , d-t+1

2

, on X Z,d we have that Gi,j = xh,i,j /z h and G l = x h,l /z h therefore on X Z,d × Y 2,d-t-1 the following system of equations is satisfied for all h's:

                   xh;i,j z 1 = x1;i,j z h . . . xh;i,j z u = xu;i,j z h x h,l z 1 = x 1,l z h . . . x h,l z u = x u,l z h . (S h ) Proposition 4.3. Let Q : [x h;i,j , x h,l ], h = 1, . . . , d-t+1
2 , i = 1, 2, 3, j = 1 . . . , t -k + 1 and l = 1, . . . , 2k -t, such that the equations (XM) are zero if evaluated in Q. Then there exists a point P : [z 1 , . . . , z u ] ∈ P u-1 such that P and Q satisfy the equations (S h ) for all h's.

Proof. Since Q : [x 1;1,1 , . . . , x ( d-t+1
2 ),2k-t ] annihilates all the equations (XM), the rank of X at Q is 1, i.e., if we assume that the first row of X is not zero, there exist a h ∈ K, h = 1, . . . , u, such that the coordinates of Q verify the following conditions:

xh;i,j = a h x1;i,j and x h,l = a h x 1,l for h = 1, . . . , d-t+1

2 , i = 1, 2, 3, j = 1 . . . , t -k + 1 and l = 1, . . . , 2k -t. We are looking for a point P : [z 1 , . . . , z u ] such that if the coordinates of Q are as above, then P and Q verify the systems (S h ). If Q verifies (S h ), then the coordinates of P are such that:

     0 • • • • • • 0 -a 2 a 1 • • • 0 . . . . . . -a u 0 • • • a 1         z 1 . . . z u    = 0,
that is to say a h z 1 = z h for h = 2, . . . , u.

The solution of such a system is the point P we are looking for, i.e. P : [a 1 , . . . , a u ].

The ideal of projections of Veronese surfaces from points

) if t/2 ≤ k ≤ t, or (E 2 ) if 0 ≤ k < t/2, describe set theoretically X Z,d .
Proof. Obviously X Z,d is contained in the support of the variety defined by the equations in statement of the theorem.

In order to prove the other inclusion we need to prove that if a point Q verifies all the equations required in the statement, then Q ∈ X Z,d .

If Q : [x h;i,j , x h,l ] annihilates the equations (XM), then, by Proposition 4.3, there exists a point P : [z 1 , . . . , z u ] such that P and Q verify the systems (S h ). Solving those systems in the variables xh;i,j , x h,l allows to write the point Q depending on the z 1 , . . . , z u . We do not write the computations for sake of simplicity, but what it turns out is that there exist ci,j , c l ∈ K, with i = 1, 2, 3, j = 1, . . . , t -k + 1 and l = 1, . . . , 2k -t (only if t/2 ≤ k ≤ t) such that the coordinates xh;i,j , x h,l of Q are xh;i,j = ci,j z h and x h,l = c l z h :

Q : [x h;i,j , x h,l ] = [c i,j z h , c l z h ].
Since such a Q, by hypothesis, verifies the equations (Cat), then there exists an unique point R :

[w 1 , w 2 , w 3 ] ∈ P 2 such that z 1 = w d-t-1 1 , z 2 = w d-t-2 1 w 2 , . . . , w d-t-1 3 , therefore Q : [c i,j w α , c l w α ] with |α| = d -t -1.
Assume that R / ∈ Z, that corresponds to assuming that Q lies in the open set given by the image of ϕ Jd minus the exceptional divisors of Bl Z (P 2 ). Now, if t/2 ≤ k ≤ t, the point Q verifies also the equations (E 1 ), while if 0 ≤ k < t/2 the point Q verifies the equations (E 2 ). Therefore if t/2 ≤ k ≤ t, then ci,j = b Gi,j and c l = bG l for i = 1, 2, 3, j = 1, . . . , t -k + 1 and l = 1, . . . , 2k -t; if 0 ≤ k < t/2, then ci,j = b Gi,j for i = 1, 2, 3 and j = 1, . . . , t -k + 1, for some b ∈ K. This proves that Q ∈ X Z,d . Now we want to construct a weak generic hypermatrix of indeterminates A in the variables xh;i,j , x h,l in such a way that the vanishing of its 2-minors coincide with the equations (XM) and (Cat). Then I 2 (A) will be a prime ideal because of Proposition 3.10. so it will only remain to show that the generators of I 2 (A), together with the equations either (E 1 ) or (E 2 ), are generators for the defining ideal of X Z,d .

Let C = (c i1,i2 ) ∈ M 3,d-t-3 (K) be the Catalecticant matrix defined in (18). Let the xh;i,j and the x h,l be defined as in (11). For all i 1 = 1, 2, 3, i 2 = 1, . . . , d -t -3 and i 3 = 1, . . . , r where r

= 2t -k + 3 if t/2 ≤ k ≤ t and r = 3(t -k + 1) if 0 ≤ k < t, construct the hypermatrix A = (a i1,i2,i3 ) (19)
in the following way:

a i1,i2,i3 = xh,i,j if c i1,i2 = z h for h = 1, . . . , d-t+1 2
, and i 3 = 1, . . . , 3(t -k + 1) is the position of the index (i, j) after having ordered the Gi,j with the lexicographic order,

a i1,i2,i3 = x h,i3-3(t-k+1) if c i1,i2 = z h for h = 1, . . . , d-t+1 2 and i 3 -3(t -k + 1) = 1, . . . , 2k -t if t/2 ≤ k ≤ t.
Proposition 4.5. The hypermatrix A defined in ( 19) is a weak generic hypermatrix of indeterminates.

Proof. We need to verify that all the properties of weak generic hypermatrices hold for such an A.

1. The fact that A = (x h;i,j , x h,l ) is a hypermatrix of indeterminates is obvious.

2. The variable x1,1,1 appears only in position a 1,1,1 .

3. The ideals of 2-minors of the sections obtained fixing the third index of A are prime ideals because those sections are Catalecticant matrices and their 2-minors are the equations of a Veronese embedding of P 2 . The sections obtained fixing either the index i 1 or the index i 2 are generic matrices of indeterminates, hence their 2-minors generate prime ideals.

Corollary 4.6. Let A be defined as in (19). The ideal I 2 (A) is a prime ideal.

Proof. This corollary is a consequence of Proposition 4.5 and of Proposition 3.10 . Now, we need to prove that the vanishing of the 2-minors of the hypermatrix A defined in (19) coincide with the equations (XM) and (Cat).

Theorem 4.7. Let X Z,d be as in Theorem 4.4, then the ideal I(X Z,d ) ⊂ K[x h;i,j , x h,l ], with h = 1, . . . , d-t+1 2 , i = 1, 2, 3, j = 1 . . . , t -k + 1 and l = 1, . . . , 2k -t is generated by all the 2-minors of the hypermatrix A defined in ( 19) and the linear formss appearing either in (E 1 ) if t/2 ≤ k ≤ t or in (E 2 ) if 0 ≤ k < t/2. Proof. In Corollary 4.6 we have shown that I 2 (A) is a prime ideal; in Theorem 4.4 we have proved that the equations (XM), (Cat) and either the equations (E 1 ) if t/2 ≤ k ≤ t or the equations (E 2 ) if 0 ≤ k < t/2 define X Z,d set-theoretically. Then we need to prove that the vanishing of the 2-minors of A coincide with the equations (XM) and (Cat) and that either (I 2 (A), (E 1 )) for t/2 ≤ k ≤ t, or (I 2 (A), (E 2 )) is actually equal to I(X Z,d ) for 0 ≤ k ≤ t/2. Denote with I the ideal defined by I 2 (A) and the polynomials appearing either in (E 1 ) in one case or in (E 2 ) in the other case. Denote also V the variety defined by I.

The inclusion V ⊆ X Z,d is obvious because, by construction of A, the ideal I 2 (A) contains the equations (XM) and (Cat), therefore I contains the ideal defined by (XM ), (Cat) and either (E 1 ) or (E 2 ).

For the other inclusion it is sufficient to verify that each 2-minor of A appears either in (XM) or in (Cat). This is equivalent to prove that if Q ∈ X Z,d then Q ∈ V, i.e. if Q ∈ X Z,d then Q annihilates all the polynomials appearing in I.

An element of I 2 (A) with A = (a i1,i2,i3 ) is, by definition of a 2-minor of a hypermatrix, one of the following:

1. a i1,i2,i3 a j1,j2,j3 -a j1,i2,i3 a i1,j2,j3 , 2. a i1,i2,i3 a j1,j2,j3 -a i1,j2,i3 a j1,i2,j3 , 3. a i1,i2,i3 a j1,j2,j3 -a i1,i2,j3 a j1,j2,i3 .

We write for brevity z i1,i2 instead of z h if (i 1 , i 2 ) is the position occupied by z h in the catalecticant matrix C defined in (18). We also rename the Gi,j 's and the G l 's with G l := Gi,j if l = 1, . . . , 3(t -k + 1) is the position of (i, j) ordered with the lexicographic order, and G l := G l-3(t-k+1) if l -3(t -k + 1) = 1, . . . , 2k -t.

With this notation we evaluate those polynomials on Q ∈ X Z,d .

1. a i1,i2,i3 a j1,j2,j3 -a j1,i2,i3 a i1,j2,j3 = G i3 G j3 (z i1,i2 z j1,j2 -z j1,i2 z i1,j2 ) that vanishes on X Z,d because, by definition, z 1 = w d-t-1 1 , z 2 = w d-t-2 1 w 2 , . . ., z u = w d-t-1

3

, hence the z i,j 's vanish on the equations of the Veronese surface Y 2,d-t-1 . The polynomial inside the parenthesis above is a minor of the catalecticant matrix defining such a surface, so the minor of A that we are studying vanishes on X Z,d .

2. The above holds also for the case a i1,i2,i3 a j1,j2,j3 -a i1,j2,i3 a j1,i2,j3 .

3. a i1,i2,i3 a j1,j2,j3 -a i1,i2,j3 a j1,j2,i3 = z i1,i1 G i3 z j1,j2 G j3 -z i1,i2 G j3 z j1,j2 G i3 = 0, evidently. This proves that the vanishing of the 2-minors of A coincides with the equations (XM) and (Cat).

For the remaining part of the proof, we work as in [START_REF] Hà | Box-shaped matrices and the defining ideal of certain blowup surface[END_REF]), proof of Theorem 2.6. Consider, with the previous notation, the sequence of surjective ring homomorphisms:

K[x i,j ] φ → K[w α t j ] ψ → K[w α G j ] x i,j → w α t j → w α G j
where the exponent α appearing in φ(x i,j ) is the triple-index that is in position i after having ordered the w's with the lexicographic order. The ideal I 2 (A) is prime, so I 2 (A) ⊆ ker(φ). Let J ⊂ K[w α t j ] be the ideal generated by the images via φ of the equations appearing either in (E 1 ) or in (E 2 ). The generators of J are zero when t j = G j , then K[w α t j ]/J ≃ K[w α G j ]. Hence J = ker(ψ). Since it is almost obvious that a set of generators for ker(ψ •φ) can be chosen as the generators of ker(φ) together with the preimages via φ of the generators of ker(ψ), then I = ker(ψ • φ). This is equivalent to the fact that I(X Z,d ) = I.

Projection of Veronese varieties

Here we want to generalize the results of the previous section to projections of Veronese varieties from a particular kind of irreducible and smooth varieties V ⊂ P n of codimension 2.

Since we want to generalize the case of s general points in P 2 , we choose V of degree s = t+1 2 + k ≤ d 2 for some non negative integers t, k, d such that 0 < t < d -1 and 0 ≤ k ≤ t.

Moreover we want to define the ideal I(V ) ⊂ K[x 0 , . . . , x n ] of V as we defined J ⊂ K[x 0 , x 1 , x 2 ] in Section 4.1 (with the obvious difference that the elements of I(V ) belong to K[x 0 , . . . , x n ] instead to K[x 0 , x 1 , x 2 ]). To be precise: let L i,j ∈ K[x 0 , . . . , x n ] 1 be generic linear forms, and let Q h,l ∈ K[x 0 , . . . , x n ] 2 be generic quadratic forms for i, h = 1, . . . , k, j = 1, . . . , 2k -t and l = 1, . . . , t -k + 1 if t/2 ≤ k ≤ t; and for i = 1, . . . , t -2k,

  e. the zero ideal (in fact the 2-minors of A do not exist). If t i=1 d i = 2 the two possible cases for the Segre-Veronese varieties are either S

  If we rename the variables in V J1 and in V J2 as: (y 1,...,1 , . . . , y n h 1 ,...,n hs ) = (y 1 , . . . , y N1 ), with N 1 = n h1 • • • n hs , and (z 1,...,1 , . . . , z n k 1

• • • a kt-s,lt-s for l m = 1, . . . , n m and m = 1, . . . , t -s.

  minor of the tensor T .

	Example: The 2-minors of a hypermatrix A = (a i1,...,it ) 1≤ij ≤nj , j=1,...,t are all of the form:
	a i1,...,im,...,it a l1,...,lm,...,lt -a i1,...,lm,...,it a l1,...,im,...,lt
	for 1 ≤ i

j , l j ≤ n j , j = 1, . . . , t and 1 ≤ m ≤ t. Definition 2.5. Let A be a hypermatrix whose entries are in K[u 1 , . . . , u r ]. The ideal I d (A) is the ideal generated by all d-minors of A.

  3.1 Definitions and RemarksBefore defining a Segre-Veronese variety we recall that a Veronese variety Y n,d is the d-uple embedding of P n into P ( n+d d )-1 , via the linear system associated to the sheaf O(d), with d > 0.Definition 3.1. A hypermatrix A = (a i1,...,i d ) 1≤ij ≤n, j=1,...,d is said to be "supersymmetric" if a i1,...,i d = a i σ(1) ,...,i σ(d) for all σ ∈ S d where S d is the permutation group of {1, . . . , d}.

j, l = 1, . . . , t -k + 1 and h = 1, . . . , k if 0 ≤ k < t/2. Define the matrix L either as in (7) or as in (8). The forms F j and G l are the maximal minors of L as previously. For each index j there exist n + 1 forms Gi,j = w i F j with i = 0, . . . , n, because now w = (w 0 , . . . , w n ). Then the degree d part of I(V ) is defined as J d in (9) if t/2 ≤ k ≤ t and as J d in (10) if 0 ≤ k < t/2. This will be the scheme:

Remark: Let W ⊂ P n be a variety of codimension 2 in P n . Let Y W be the blow up of P n along W . Let E be the exceptional divisor of the blow up and H the strict transform of a generic hyperplane. In [Cop] (Theorem 1) it is proved that if W is smooth, irreducible and scheme-theoretically generated in degree at most λ ∈ Z + , then |dH -E| is very ample on the blow up Y W for all d ≥ λ + 1.

A consequence of those remarks is the following: Proposition 5.1. Let V ⊂ P n be defined as in ( 20), and let d > t + 1. If E is the exceptional divisor of the blow up Y V of P n along V and H is the strict transform of a generic hyperplane of P n , then |dH -E| is very ample.

) be the image of the morphism associated to |dH -E|. The arguments and the proofs used to study the ideal I(X Z,d ) in the previous section can all be generalized to

2 . Now let S ′ be the coordinate ring on P(H 0 (O Y V (dH -E))), constructed as K[x i,j , x h,l ] in the previous section: S ′ = K[x i,j , x h,l ] with i = 0, . . . , n; j = 1, . . . , t -k + 1; h = 1, . . . , n+d-t-1 2 and l = 1, . . . , 2k -t only if t/2 ≤ k ≤ t (in the other case the variables x h,l do not exist).

Let (E ′ ) and (E ′′ ) be the equations in S ′ corresponding to (E 1 ) and (E 2 ), respectively. Let C ′ be the catalecticant matrix used to define the Veronese variety Y n,d-t-1 .

The hypermatrix A ′ that we are going to use in this case is the obvious generalization of the hypermatrix A defined in (19); clearly one has to substitute C with C ′ . Now the proof of the fact that I 2 (A ′ ) ⊂ S ′ is a prime ideal is analogous to that one of Corollary 4.6, and pass through the fact that A ′ is a weak generic hypermatrix, hence we get the following: Theorem 5.2. Let (V, I(V )) ⊂ (P n , K[x 0 , . . . , x n ]) be defined as in (20), let Y V be the blow up of P n along V and let X V,d be the image of Y V via |dH -E|, where d

2 , H is a generic hyperplane section of P n and E is the exceptional divisor of the blow up. The ideal I(X V,d ) ⊂ S ′ is generated by all the 2-minors of the hypermatrix A ′ and the polynomials appearing either in (E ′ ) if t/2 ≤ k ≤ t or in (E ′′ ) if 0 ≤ k < t/2, where S ′ , A ′ , (E ′ ) and (E ′′ ) are defined as above.