Symmetric tensor rank with a tangent vector: a generic uniqueness theorem

Let $X_{m,d}\subset \mathbb {P}^N$, $N:= \binom{m+d}{m}-1$, be the order $d$ Veronese embedding of $\mathbb {P}^m$. Let $\tau (X_{m,d})\subset \mathbb {P}^N$, be the tangent developable of $X_{m,d}$. For each integer $t \ge 2$ let $\tau (X_{m,d},t)\subseteq \mathbb {P}^N$, be the joint of $\tau (X_{m,d})$ and $t-2$ copies of $X_{m,d}$. Here we prove that if $m\ge 2$, $d\ge 7$ and $t \le 1 + \lfloor \binom{m+d-2}{m}/(m+1)\rfloor$, then for a general $P\in \tau (X_{m,d},t)$ there are uniquely determined $P_1,...,P_{t-2}\in X_{m,d}$ and a unique tangent vector $\nu$ of $X_{m,d}$ such that $P$ is in the linear span of $\nu \cup \{P_1,...,P_{t-2}\}$, i.e. a degree $d$ linear form $f$ associated to $P$ may be written as $$f = L_{t-1}^{d-1}L_t + \sum_{i=1}^{t-2} L_i^d$$ with $L_i$, $1 \le i \le t$, uniquely determined (up to a constant) linear forms on $\mathbb {P}^m$.

with L i linear forms on P m (v i vectors over a vector field of dimension m + 1 respectively), 1 ≤ i ≤ t, that are uniquely determined (up to a constant).

Introduction
In this paper we want to address the question of the uniqueness of a particular decomposition for certain given homogeneous polynomials. An analogous question can be rephrased in terms of uniqueness of a particular tensor decomposition of certain given symmetric tensors. In fact, given a homogeneous polynomial f of degree d in m + 1 variables defined over an algebraically closed field K, there is an obvious way to associate a symmetric tensor T ∈ S d (V K ), with dim(V K ) = m + 1, to the form f . We will always work over an algebraically closed field K such that char(K) = 0. Fix integers m ≥ 2 and d ≥ 3. Let j m,d : P m ֒→ P N , N := m+d m − 1, be the order d Veronese embedding of P m and set X m,d := j m,d (P m ) (we often write X instead of X m,d ). Let K[x 0 , . . . , x m ] d be the polynomial ring of homogeneous degree d polynomials in m + 1 variables over K and let V * K be the dual space of V K . Since obviously P m ≃ P(K[x 0 , . . . , x m ] 1 ) ≃ P(V * K ), an element of the Veronese variety X m,d can be interpreted either as the projective class of a d-th power of a linear form L ∈ K[x 0 , . . . , x m ] 1 or as the projective class of a symmetric tensor For each integer t such that 1 ≤ t ≤ N let σ t (X) denote the closure in P N of the union of all (t − 1)-dimensional linear subspaces spanned by t points of X (the t-secant variety of X). From this definition one can understand that the generic element of σ t (X m,d ) can be interpreted either as For a given form f (or a symmetric tensor T ), the minimum integer t for which there exists such a decomposition is called the symmetric rank of f (or of T ). Finding those v i 's, i = 1, . . . , t such that T = v ⊗d 1 + · · · + v ⊗d t , with t the symmetric rank of T , is known as the Tensor Decomposition problem and it is a generalization of the Singular Value Decomposition problem for symmetric matrices (i.e. if T ∈ S 2 (V * K )). The existence and the possible uniqueness of the decompositions of a form f as L d 1 + · · · + L d t with t minimal is studied in certain cases in [6], [8], [10], [11]. Let τ (X) ⊆ P N be the tangent developable of X, i.e. the closure in P N of the union of all embedded tangent spaces T P X, P ∈ X. Obviously τ (X) ⊆ σ 2 (X) and τ (X) is integral. Since d ≥ 3, the variety τ (X) is a divisor of σ 2 (X) ( [5], Proposition 3.2). An element in τ (X m,d ) can be described both as [f ] ∈ P(K[x 0 , . . . , x m ] d ) for which there exists two linear forms L 1 , [4] The secant variety σ t (X), t ≥ 2, is the join of t copies of X. For each integer t ≥ 3 let τ (X, t) ⊆ P N be the join of τ (X) and t − 2 copies of X. We recall that min{N, t(m+1)−2} is the expected dimension of τ (X, t), while min{N, t(m+1)−1} is the expected dimension of σ t (X). In the range of triples (m, d, t) we will meet in this paper both τ (X, t) and σ t (X) have the expected dimensions and hence τ (X, t) is a divisor of σ t (X). An element in τ (X m,d , t) can be described both as [3], it is natural to ask the following question.
For non weakly (t − 1)-degenerate subvarieties of P N the corresponding question is true by [8], Proposition 1.5. Here we answer it for a large set of triples of integers (m, d, t) and prove the following result.
In terms of homogeneous polynomials Theorem 1 may be rephrased in the following way.
and let f be a homogeneous degree d form in K[x 0 , . . . , x m ] associated to P . Then f may be written in a unique way In the statement of Theorem 2 the form f is uniquely determined only up to a non-zero scalar, and (as usual in this topic) " uniqueness " may allow not only a permutation of the forms L 1 , . . . , L t−2 , but also a scalar multiplication of each L i .
In terms of symmetric tensors Theorem 1 may be rephrased in the following way.
be a symmetric tensor associated to P . Then T may be written in a unique way As above, in the statement of Theorem 3 the tensor T and the vectors v i 's are uniquely determined only up to non-zero scalars.
To prove Theorem 1, and hence Theorems 2 and 3, we adapt the notion and the results on weakly defective varieties described in [6]. It is easy to adapt [6] to joins of different varieties instead of secant varieties of a fixed variety if a general tangent hyperplane is tangent only at one point ( [7]). However, a general tangent space of τ (X) is tangent to τ (X) along a line, not just at the point of tangency. Hence a general hyperplane tangent to τ (X, t), t ≥ 3, is tangent to τ (X, t) at least along a line. We prove the following result. /(m+1)⌋. Assume t ≤ β +1. Let P be a general point of τ (X, t). Let P 1 , . . . , P t−2 ∈ X and Q ∈ τ (X) be the points such that P ∈ {P 1 , . . . , P t−2 , Q} . Let ν be the tangent vector of X such that Q is a point of ν \ ν red . Let H ⊂ P N be a general hyperplane containing the tangent space T P τ (X, t) of τ (X, t). Then H is tangent to X only at the points P 1 , . . . , P t−2 , ν red , the scheme H ∩ X has an ordinary node at each P i , and H is tangent to τ (X) \ X only along the line ν .

Preliminaries
Notation 1. Let Y be an integral quasi-projective variety and Q ∈ Y reg . Let {kQ, Y } denote the (k − 1)-th infinitesimal neighborhood of Q in Y , i.e. the closed subscheme of Y with (I Q ) k as its ideal sheaf. If Y = P m , then we write kQ instead of {kQ, P m }. The scheme {kQ, Y } will be called a k-point of Y . We also say that a 2-point is a double point, that a 3-point is a triple point and a 4-point is a quadruple point.
We give here the definition of a (2, 3)-point as it is in [5], p. 977.
. , x m ] be the reduced ideal of a simple point Q ∈ P m , and let l ⊂ K[x 0 , . . . , x m ] be the ideal of a reduced line L ⊂ P m through Q. We say that Z(Q, L) is a (2, 3)-point if it is the zero-dimensional scheme whose representative ideal is (q 3 + l 2 ).
We recall the notion of weak non-defectivity for an integral and non-degenerate projective variety Y ⊂ P r (see [6]). For any closed subscheme Z ⊂ P r set: The contact locus H Z of H is the union of all irreducible components of H c containing at least one point of Z red . We use the notation H Z only in the case Z red ⊂ Y reg .
Fix an integer k ≥ 0 and assume that σ k+1 (Y ) doesn't fill up the ambient space P r . Fix a general (k + 1)-uple of points in Y i.e. (P 0 , . . . , P k ) ∈ Y k+1 and set The following definition of weakly k-defective variety coincides with the one given in [6].

Definition 2.
A variety Y ⊂ P r is said to be weakly k-defective if dim(H Z ) > 0 for Z as in (2).
In [6], Theorem 1.4, it is proved that if Y ⊂ P r is not weakly k-defective, then H Z = Z red and that Sing(Y ∩ H) = (Sing(Y ) ∩ H) ∪ Z red for a general Z = ∪ k i=0 {2P i , Y } and a general H ∈ H(−Z). Notice that Y is weakly 0-defective if and only if its dual variety Y * ⊂ P r * is not a hypersurface.
In [7] the same authors considered also the case in which Y is not irreducible and hence its joins have as irreducible components the joins of different varieties. Lemma 1. Fix an integer y ≥ 2, an integral projective variety Y , L ∈ Pic(Y ) and Proof. Let u : Y ′ → Y denote the blowing-up of Y at P and E := u −1 (P ) the exceptional divisor. Since dim(Y ) = x, we have E ∼ = P x−1 . Set R := u * (L). For each integer t ≥ 0 we have u * (R(−tE)) ∼ = I tP ⊗ L. Thus the push-forward u * induces an isomorphism between the linear system |R(−tE)| on Y ′ and the linear system Consider on Y ′ the exact sequence: Our hypothesis implies that h 0 (Y, I yP ⊗L) = h 0 (Y, L)− x+y−1 x . Thus our assump- (3) gives the surjectivity of the restriction map ρ : H 0 (Y ′ , M ) → H 0 (E, M | E ). Since y ≥ 0, the line bundle M |E is spanned. Thus the surjectivity of ρ implies that M is spanned at each point of E. Hence M is spanned in a neighborhood of E. Bertini's theorem implies that a general F ′ ∈ |M | is smooth in a neighborhood of E. Since F is general and |M | ∼ = |I yP ⊗ L|, P is an isolated singular point of F .

τ (X, t) is not weak defective
In this section we fix integers m ≥ 2, d ≥ 3 and set N = m+d m − 1 and X := X m,d . The variety τ (X) is 0-weakly defective, because a general tangent space of τ (X) is tangent to τ (X) along a line. Terracini's lemma for joins implies that a general tangent space of τ (X, t) is tangent to τ (X, t) at least along a line (see Remark 3). Thus τ (X, t) is weakly 0-defective. To handle this problem and prove Theorem 1 we introduce another definition, which is tailor-made to this particular case. As in [5] we want to work with zero-dimensional schemes on X, not on τ (X) or τ (X, t). We consider X = j m,d (P m ) and the 0-dimensional scheme Z ⊂ X which is the image (via j m,d ) of the general disjoint union of t − 2 double points and one (2, 3)-point of P m , in the case of [5] (see Definition 1). We will often work by identifying X with P m , so e.g. notice that H(−∅) is just |O P m (d)|.
Remark 2. Fix P ∈ X and Q ∈ T P X \ {P }. Any two such pairs (P, Q) are projectively equivalent for the natural action of Aut(P m ). We have Q ∈ τ (X) reg and T Q τ (X) ⊃ T P X. Set D := {P, Q} . It is well-known that D \ {P } is the set of all O ∈ τ (X) reg such that T Q τ (X) = T O τ (X) (e.g. use that the set of all g ∈ Aut(P m ) fixing P and the line containing P associated to the tangent vector induced by Q acts transitively on T P X \ D). We are now ready for the following lemma. To prove part (ii) of the lemma we need to prove that dim(H Z ) = 0 for a general H ∈ H(−Z). Since W Z 1 and h 1 (P m , I Z1 (d)) = 0, we have H(−W ) = ∅. Since W red = Z red and Z ⊂ W , to prove parts (ii) and (iii) of the lemma it is sufficient to prove dim((H W ) c ) = 0 for a general H W ∈ H(−W ), where W is as above and (H W ) c is as in Notation 2. Assume that this is not true, therefore: (1) either the contact locus (H W ) c contains a positive-dimensional component J i containing some of the P i 's, for 1 ≤ i ≤ t − 2, (2) or the contact locus (H W ) c contains a positive-dimensional irreducible component T containing Q.
Here we assume the existence of a positive dimensional component J i ⊂ (H W ) c containing one of the P i 's, say for example J t−2 ∋ P t−2 . Thus a general element of |I W (d)| is singular along a positive-dimensional irreducible algebraic set containing P t−2 . Let w : M → P m denote the blowing-up of P m at the points O, P 1 , . . . , P t−3 . Set E 0 := w −1 (O) and E i : , the point A and the integer y = 2 we get a contradiction.
(b) Here we prove the non-existence of a positive-dimensional T ⊂ (H W ) c containing O. Let w 1 : M 1 → P m denote the blowing-up of P m at the points Since h 1 (P m , I Z2 (d)) = 0 and |I Z2 (d)| ⊂ |I Z (d)|, by Lemma 1 (with y = 3) we get a contradiction.
In [3], Lemmas 5 and 6, we proved the following two lemmas:  We will use the following set-up.