ON THE DIMENSION OF CONTACT LOCI AND THE IDENTIFIABILITY OF TENSORS

EDOARDO BALLICO, ALESSANDRA BERNARDI AND LUCA CHIANTINI

Abstract. Let X ⊂ ℙ^{r} be an integral and non-degenerate variety. Set n := dim(X). We prove that if the (k + n − 1)-secant variety of X has (the expected) dimension (k + n − 1)(n + 1) − 1 < r and X is not uniruled by lines, then X is not k-weakly defective and hence the k-secant variety satisfies identifiability, i.e. a general element of it is in the linear span of a unique S ⊂ X with ♯(S) = k. We apply this result to many Segre-Veronese varieties and to the identifiability of Gaussian mixtures G_{1,d}. If X is the Segre embedding of a multiprojective space we prove identifiability for the k-secant variety (assuming that the (k +n−1)-secant variety has dimension (k + n − 1)(n + 1) − 1 < r, this is a known result in many cases), beating several bounds on the identifiability of tensors.

1. Introduction

The computation of the decompositions of tensors and the related evaluation of their ranks recently received great interest from people working in algebraic geometry. Indeed, a geometric analysis on projective spaces of tensors, Segre embeddings and their secant varieties produced currently several new results in the field.

From the geometric point of view, the study of decompositions and ranks can be turned to a general problem. For any (irreducible) algebraic variety X which spans a projective space ℙ^{r}, one can define the X-rank r_{X}(q) of a point q ∈ ℙ^{r} as the minimal cardinality of a finite set S ⊂ X such that q ∈ ⟨S⟩, where ⟨ ⟩ denotes the linear span. Any set S for which the minimum is attained is a minimal X-decomposition of q.

In tensor analysis, one interprets ℙ^{r} as a space of tensors of given type (n_{1} + 1) × ⋯ × (n_{s} + 1) (mod scalars), and X as the locus of tensors of rank 1. In geometric terms, X is the multiprojective product ℙ^{n_{1}} ×⋯×ℙ^{n_{s}}, naturally embedded in ℙ^{r} via the Segre map. Then, the notions of X-rank and minimal decompositions match with the usual multilinear algebra notion. The set of tensors of fixed rank k corresponds to an open subset of the k-th secant variety σ_{k}(X) of X. Properties of the rank decompositions can be derived from geometric properties of X, which determine some features of σ_{k}(X). Examples of such analysis are contained in several papers, e.g. see [8, 10, 25, 26, 36, 41], and the book [38].

In this paper we study the problem of the identifiability of generic points q ∈ ℙ^{r} of given X-rank k. A point q is X-identifiable when the set S(q, X) of minimal X-decompositions of q is a singleton; in any case the cardinality of S(q, X) is called the secant degree of q or the k-th secant degree of q. In tensor analysis the identifiability of a tensor is fundamental for algorithms of data compression and for the analysis of mixture models (see [33, 34, 35, 43]). We will say that X is k-identifiable if a generic point of σ_{k}(X) is X-identifiable.

We first work over an algebraically closed field of characteristic 0. Our investigation is based on the notion of weak defectivity introduced by the last author and C. Ciliberto in [25]. A projective variety X is weakly k-defective if hyperplanes of ℙ^{r}, which are tangent to X at k general points, are indeed tangent along a positive dimensional subvariety. It turns out (see [26, Theorem 2.5]) that when r > k(n + 1) − 1 then X is k-identifiable, unless it is weakly k-defective (unfortunately the reverse is not true). Therefore the knowledge on the non-k weak defectivity provides an effective way to determine geometrically the identifiability of generic points. This approach has been applied to tensors and linear systems of tensors in a series of papers ([1, 16, 18, 19, 29, 30, 31]), where identifiability problems are partially or totally solved.

A challenging task is therefore to get criterions that can establish identifiability (or not) in the cases when weak defectivity arises. To this purpose there is a related notion that is very helpful, meaning the tangential weak defectivity. Our new analysis starts with the following observation (see [27]). If X is a weakly k-defective variety of dimension n, then a general choice of a set A of k points on X determines a positive dimensional tangential k-contact locus Γ_{k}(A): the tangency locus of the span of the tangent spaces to X at the points of A. If for a general choice of A we have that dim Γ_{k}(A) > 0 than X is tangential k-weakly defective. It turns out that the k-th secant degree of a general element of X is related with the k-th secant degree of the contact locus Γ_{k}(A) for a general choice of A (cf. Lemma 4.2, Corollary 4.3, [26, Theorem 2.4] and part (iii) of [27, Proposition 3.9]). We will exploit in more details the relation in Section 4.

Moreover, when r ≥ (k + 1)(n + 1) − 1, then the weak k-defectivity of X implies the weak (k + 1)-defectivity. Moreover, we prove that, under some geometric assumption on X, if we take a generic subset A as above and let A′ be obtained by adding to A a generic point of X, then the dimension of the tangential contact locus Γ_{k+1}(A′) is strictly bigger than the dimension of Γ_{k}(A). As the dimension of contact loci is bounded by n − 1, by repeating the previous construction we obtain the following result.

Theorem. (see Theorem 5.1). Let X ⊂ ℙ^{r} be an integral and non-degenerate n-dimensional variety and let k be a positive integer. Assume that X is not uniruled by lines and that dim σ_{k+n−1}(X) = (k + n − 1)(n + 1) − 1 < r. Then X is not weakly j-defective for every j ≤ k.

It follows from the previous analysis that, with the assumptions of Theorem 5.1, X is jidentifiable for every j ≤ k.

The result can be directly applied e.g. to Segre-Veronese embeddings of products of projective spaces, provided that the multidegree (d_{1}, …, d_{s}) satisfies d_{i} > 1 for all i. When some degree d_{i} is 1, as in the case of Segre embeddings (where d_{i} = 1 for all i), then Theorem 5.1 does not apply directly. However, for the Segre product we can modify the argument of Theorem 5.1, and prove that the same conclusion holds when for each i ∈ {1, …, s} there is j ∈ {1, …, s} \ {i} such that n_{j} = n_{i} (see Theorem 5.3).

Combining Theorem 5.3 with the main result in [23], for the Segre embedding X of s ≥ 5 copies of ℙ^{1}, we find that X is not weakly k-defective, hence it is k-identifiable, whenever (k+s−1)(s+1) < 2^{s} − 1 (see Corollary 5.4).

Corollary 5.4 improves our previous knowledge on the weak defectivity of Segre embeddings of many copies of ℙ^{1}, hence on the identifiability of generic binary tensors. Indeed it extends the range where we know that k-identifiability holds up to k ≤ \frac{s^{2}−1}{s+1} − s, which beats all previously known bounds (see Section 4 of [20]) as s grows. Since identifiability cannot hold, for trivial dimensional reasons, when k > s^{2}/(s + 1) − 1, then we are quite close to a complete analysis of the generic identifiability of binary tensors. We also get almost optimal results for the Segre embedding of (ℙ^{m})^{s}, m > 1, see Corollary 5.5) (we use the very strong [9, Theorem 3.1]).

See Example 5.8 for the case of Gaussian mixtures G_{1,d}.

When the assumptions of Theorem 5.1 does not hold, one cannot conclude the non weak kdefectivity of X. Yet, even if X is weakly k-defective, it can still be k-identifiable. We prove the following result.

Theorem. (see Theorem 5.9) Let X be the Segre embedding of a product ℙ^{n_{1}} × ⋯ × ℙ^{n_{q}} in ℙ^{r}, with n = ∑ n_{i} = dim(X). Let k be a positive integer. Assume that dim σ_{k+n−1}(X) = (k + n − 1)(n + 1) − 1 < r (in particular X is not (k + n − 1)-defective). Then X is j-identifiable for every j ≤ k.

It would be interesting to overcame the assumption “X not uniruled by lines ” of Theorem 5.3 in other cases, apart the Segre varieties covered by Theorem 5.9, because it would expand the known cases of identifiability (e.g. for Grassmannians see [6], [17], [21], [22]; for the tangential variety of the Veronese varieties see [7]). We just recall that conjecturally only four Grassmannians have some defective secant variety.

We stress that, in particular, Remark 5.10 suggests that if one is interested in the k-identifiability of some specific Segre product then the computational procedure induced by Theorem 5.9 can be faster than previously known direct procedures.

In the last section of the paper we work over an algebraically closed base field 𝕂 of arbitrary characteristic. To include varieties X defined in positive characteristic, we will assume that X is not very strange (see Definition 6.1). Under this mild assumption, we prove the following.

Theorem 1.1. Let M be an integral projective variety and L, R line bundles on M with L very ample. Set n := dim M. Fix a linear subspace V ⊆ H^{0}(L) such that the associated map j : M → ℙ^{r} is an embedding with image not very strange. Fix an integer k ≤ min{h^{0}(R), dim V − n − 2}. Let W ⊆ H^{0}(L ⊗ R) be the image of the multiplication map V ⊗ H^{0}(R) → H^{0}(L ⊗ R) and let X ⊂ ℙ^{r}, r = dim W − 1, be the embedding induced by W. Then σ_{k}(X) has dimension kn + k − 1 and ♯(S(q, X)) = 1 for a general q ∈ σ_{k}(X).

We will apply Theorem 1.1 to the Segre-Veronese embeddings of a multiprojective space in arbitrary characteristic (see Example 6.3), where we do not have an analogue of [26, Theorem 2.5], hence the analysis of the previous section does not apply. Notice that Theorem 1.1 does not apply to Segre embeddings, because it requires L very ample (or at least birationally very ample) and h^{0}(X, R) > 1.

The authors wish to thank C. Bocci, for his help to understand the case of almost unbalanced tensors (see Example 4.10 below). The authors also thank the anonimous referee for pointing out an improvement in the formula after Corollary 5.4.

2. General notation

In this section we work with projective spaces ℙ^{r} defined over an algebraically closed field 𝕂. Let X ⊂ ℙ^{r} be an integral and non-degenerate n-dimensional variety. We will denote with X_{reg}

the set of regular points in X. For any x ∈ X_{reg} we will denote with T_{x}X the tangent space to X at x.

For any integer k ≥ 2 we will denote with σ_{k}(X) the k-th secant variety of X which is an irreducible projective variety. If k(n + 1) − 1 ≤ r, then k(n + 1) − 1 is the expected dimension of σ_{k}(X) and X is k-defective if dim σ_{k}(X) < k(n + 1) − 1.

We will often use the fact that if X is not k-defective, then, for all k′ ≤ k, X is not k′-defective.

Definition 2.1. For q ∈ ℙ^{r} general, the X-rank r_{X}(q) of a point q ∈ ℙ^{r} is the minimal cardinality of a finite set A ⊂ X such that q ∈ ⟨A⟩.

We define S_{X}(q) as the collection of subsets A ⊂ X of cardinality r_{X}(q) such that q ∈ ⟨A⟩. When S_{X}(q) is finite, we will write α_{X}(q) for the cardinality of S_{X}(q).

Remark 2.2. Since X is irreducible, then the secant varieties σ_{k}(X) are irreducible. Moreover a general point of σ_{k}(X) has X-rank k.

By definition, the set S_{X}(q) is finite for q ∈ σ_{k}(X) general if and only if dim σ_{k}(X) = k(n + 1) − 1 ≤ r. In this case, we will write simply α_{k} for the cardinality of S_{X}(q), where q ∈ σ_{k}(X) is generic. Such an α_{k} is often called the k-th secant degree of X.

The following result holds (with the same elementary proof) in arbitrary characteristic.

Proposition 2.3. Assume dim σ_{k+1}(X) = (k+1)(n+1)−1 and α_{k} > 1. Fix a general q ∈ σ_{k+1}(X), any A ∈ S(q) and any a ∈ A. There are at least α_{k} elements B_{1} …, B_{α_{k}} ∈ S(q) (one of them being A) such that a ∈ B_{i} for all i = 1, …, α_{k}.

Proof. Since dim σ_{k+1}(X) = (k + 1)(n + 1) − 1, we have dim σ_{k}(X) = k(n + 1) − 1 and hence α_{k} is a well-defined finite positive integer. Fix a general q ∈ σ_{k+1}(X) and take any A ∈ S(q) and any a ∈ A. Set E := A \ {a}. Since q is general in σ_{k+1}(X) and dim σ_{k+1}(X) = (k + 1)(n + 1) − 1, a dimensional count gives that for each D ∈ S(q) the set D is a general element of the (k + 1)- symmetric product of X. Thus E is a general subset of X with cardinality k. Since ♯(A) = r_{X}(q), we have q ∉ ⟨A′⟩ for any A′ ⊊ A; in particular q ∉ E. Thus ⟨{a, q}⟩ is a line, which meets ⟨E⟩ in a unique point q′ (because ⟨E⟩ + ⟨a, q⟩ = ⟨A⟩) and q′ ∉ ⟨E′⟩ for any E′ ⊊ E. For a general q ∈ ⟨A⟩, q′ is a general element of ⟨E⟩. Since E is general, q′ is general in σ_{k}(X). Thus r_{X}(q′) = k, E ∈ S(q′) and ♯(S(q′)) = α_{k} > 1. Fix any F ∈ S(q′). We have q ∈ ⟨{F, a}⟩. Since r_{X}(q) = k + 1, we have a ∉ F and F ∪ {a} ∈ S(q). Now we have exactly α_{k} such F ’s, say F_{1}, …, F_{α_{k}}, the B_{i} of the statement are B_{i} = {F_{i}, a} for i = 1, …, α_{k} concluding the proof. □

3. Weak defectivity and identifiability

In this section, as well as in the next two sections, we assume that the algebraically closed base field 𝕂 has characteristic 0.

For a subset A of cardinality k contained in the regular locus X_{reg} we set:

M_{A} := ⟨∪_{x∈A}T_{x}X⟩.

By Terracini’s lemma ([8, Corollary 1.11]), for a general choice of A the space M_{A} is the tangent space to σ_{k}(X) at a general point u ∈ ⟨A⟩. Thus dim M_{A} = min{k(n + 1) − 1, dim⟨X⟩} for A general, unless X is k defective.

Definition 3.1. The tangential k-contact locus Γ_{k}(A) is the closure in X of the union of all the irreducible components which contain at least one point of A, of the locus of points of X_{reg} where M_{A} is tangent to X. We will write t_{k} := dim Γ_{k}(A) for a general choice of A. We say that X is weakly k-defective if t_{k} > 0. We will write just Γ_{k} instead of Γ_{k}(A), when A is general.

With the terminology just introduced we have the following remark:

Remark 3.2. [25, Remark 3.1(iii) at p. 159] If σ_{k+1}(X) ⊊ ℙ^{r} has the expected dimension (k +1) dim X +(k +1)−1 < r and X is weakly k-defective, then X is also weakly (k +1)-defective.

We will need the following result of [27].

Proposition 3.3. For a general choice of A ⊂ X_{reg}, the algebraic set Γ_{k} := Γ_{k}(A) is equidimensional and either it is irreducible or it has exactly k irreducible components, each of them containing a different point of A.

Proof. It is Proposition 3.9 of [27]. (Notice that our σ_{k}(X) is called the (k − 1)-th secant variety in [27], so that all the k’s there should be here replaced by k + 1, in order to apply the results of [27] to our terminology). □

Definition 3.4. Following [26] and [27], we say that Γ_{k} is of the type I (resp. type II ) if Γ_{k} is irreducible (resp. it is not irreducible) for a general choice of A.

The fundamental technical results of the present paper is the following lemma. It shows how one can control the growth of the dimension of contact loci. Mainly, one must take care of the case of contact loci of type II, which are not irreducible.

Lemma 3.5. Let X ⊂ ℙ^{r} be an irreducible projective variety such that dim σ_{k+1}(X) = (k + 1)(n + 1) − 1 < r and assume that X is weakly k-defective. Then X is weakly (k + 1)-defective. Let B (resp. A) be a general subset of k (resp. k + 1) points of X. Call Γ_{k} (resp. Γ_{k+1}) the tangential contact locus of X at the points of B (resp. A). Then either:

• dim Γ_{k} < dim Γ_{k+1}; or

• both Γ_{k} and Γ_{k+1} have type II, each of the irreducible components of Γ_{k} and Γ_{k+1} is a linear space and these components are linearly independent, i.e.

dim⟨Γ_{k+1}⟩ = dim⟨Γ_{k}⟩ + dim Γ_{k} + 1 = (k + 1)(dim Γ_{k} + 1) − 1.

Proof. Since dim σ_{k+1}(X) = (k + 1)(n + 1) − 1, we have dim σ_{k}(X) = k(n + 1) − 1. By Remark 3.2, X is weakly (k + 1)-defective. By [25, first two lines after Definition 1.2], X is not i-defective for i = 1, …, k + 1.

By [27, Proposition 3.9] we know that both Γ_{k} and Γ_{k+1} are equidimensional, that A is in the smooth locus of Γ_{k+1}, that B is in the smooth locus of Γ_{k} and that either Γ_{k} (resp. Γ_{k+1}) is irreducible or it has exactly k (resp. k + 1) irreducible components, each of them containing exactly one point of B (resp. A).

Since B is general, we may assume that B ⊂ A. Thus Γ_{k} ⊆ Γ_{k+1}. Since Γ_{k} is a proper subvariety of X and Γ_{k+1} contains B plus a general point of X, then Γ_{k} ≠ Γ_{k+1}. Thus if either Γ_{k} or Γ_{k+1} is irreducible, then t_{k} < t_{k+1} (where t_{k} is defined in Definition 3.1).

Now assume that both Γ_{k} and Γ_{k+1} are reducible, hence of type II, and that t_{k} = t_{k+1} =: t. Let B = {P_{1}, …, P_{k}} and A = {P_{k+1}} ∪ B. Call Γ^{i} the component of Γ_{k+1} passing through P_{i}, so that also Γ_{k} = Γ^{1} ∪ ⋯ ∪ Γ^{k}. Call Π^{i} the linear span of Γ^{i}. Of course, by the generality of A, all the Π^{i}’s have the same dimension. Assume that dim Π^{i} > dim Γ^{i} and let us show that we get a contradiction.

For i = 2, …, k + 1 call L_{i} the span of Π^{1} ∪ ⋯ ∪ Π^{i}, which is also the span of Γ^{1} ∪ ⋯ ∪ Γ^{i}. By [26, Proposition 2.5] we know that, for all i, dim(L_{i}) ≥ it + i − 1, since X is not i-defective. Moreover by [27, Proposition 3.9] we know that dim(L_{k}) = kt+k −1 and dim(L_{k+1}) = (k +1)t+k. If dim(L_{k−1}) > (k − 1)t + k − 2, then by Grassmann formula, the intersection of L_{k−1} and Π_{k} has dimension greater than (k − 1)t + k − 2 + dim(Π_{k}) − kt − k + 1 = dim(Π_{k}) − t − 1, so that, by Grassmann formula again,

dim(L_{k+1}) < dim(L_{k}) + dim(Π_{k+1}) − dim(Π_{i}) + t + 1 = (k + 1)t + k,

a contradiction. Thus dim(L_{k−1}) = (k − 1)t + k − 2. Arguing by induction, we obtain that dim(L_{i}) = it + i − 1 for i = 2, …, k + 1. In particular dim(Π^{1} ∩ Π^{2}) = 2 dim(Π^{1}) − 2t − 1 ≥ 0. By the generality of A again, dim(Π^{i} ∩ Π^{j}) = dim(Π^{1} ∩ Π^{2}) for all i, j. Thus Π^{3} intersects L_{2} in dimension at least dim Π^{1} ∩ Π^{2}). It follows that L_{3} has dimension

dim(L_{3}) ≤ dim(L_{2}) + dim(Π^{3}) − 2 dim(Π^{1}) + 2t + 1 = (2t + 1) − dim(Π^{i}) + 2t + 1 < 3t + 2,

which yields the claimed contradiction. □

4. Remarks on the secant degree of contact loci

We collect in this sections some remarks on the general tangential contact loci Γ_{k} of X, which clarify the relations between the k-th secant degree of X and the k-th secant degree of a general Γ_{k}. To do that, we need to distinguish the case where Γ_{k} is of type I (i.e. irreducible) from the case where Γ_{k} is of type II.

Notation 4.1. For the rest of the section, let X ⊂ ℙ^{r} be an integral and non-degenerate ndimensional variety and let k be a positive integer. Assume that X is weakly k-defective with

dim σ_{k+n−1}(X) = (k + n − 1)(n + 1) − 1 < r.

Call Γ_{k} the tangential k-contact locus of X at a general set A = {P_{1}, …, P_{k}} of k points of X, and call Π_{k} the linear span of Γ_{k}.

The following technical result is the analogue of [26, Theorem 2.4] for the tangential contact locus Γ_{k}. It is implicitly contained in part (iii) of [27, Proposition 3.9], but we prefer to make it explicit here.

Lemma 4.2. For a general Q ∈ ⟨A⟩ ⊂ Π_{k}, all subsets B ⊂ X of cardinality k such that Q ∈ ⟨B⟩ are contained in the span of Γ_{k}.

Proof. Notice that, by the generality of A, Q is also a general point in the secant variety σ_{k}(X). Since Q is general, the tangent space to σ_{k}(X) at Q is also the span of the tangent spaces to X at the points of B, by the Terracini’s Lemma. The claim now follows from [27, proposition 3.9 (iii)]. □

Corollary 4.3. The k-secant degree of a projective variety X is equal to the k-secant degree of the intersection of X with the span of its tangential k-contact locus.

When the general contact locus of X is of type I, we can give a more explicit statement.

Proposition 4.4. Assume that Γ_{k} is irreducible (i.e. type I). Assume the existence of a component W of Π_{k} ∩ X, different from Γ_{k}, such that Π_{k} is tangent to X along W. Then for all a, b with a > 0 and a + b = k, the join of the a-th secant variety of W with the b-th secant variety of Γ_{k} cannot cover Π_{k}.

Proof. Assume on the contrary that the join covers Π_{k}. Since σ_{k}(X) has the expected dimension and it is equal to the union of the spaces Π_{k}’s, when the k-th points P_{1}, …, P_{k} vary on X, then for dimensional reasons there cannot be a subvariety Y ⊊ X which contains all the respective components W ’s. Thus, for P_{1}, …, P_{k} general, a general point Q of W is a general point of X. Hence the contact variety of Q, P_{2}, …, P_{k} has the same properties of Γ_{k}. In particular Π_{k} is tangent to X along an irreducible variety of the same dimension of Γ_{k}, which contains P_{2}, …, P_{k} and Q. Since Q ∉ Γ_{k}, this is excluded by the generality of P_{1}, …, P_{k}. □

Corollary 4.5. Assume that Γ_{k} is irreducible (i.e. type I). Then the secant degree of X is equal to the secant degree of the general contact locus Γ_{k}.

The situation becomes completely different when the general Γ_{k} is of type II.

Remark 4.6. Assume that dim σ_{k+n}(X) = (k + n)(n + 1) − 1 < r, so that X is also weakly (k + 1)-defective. Assume that a general tangential contact locus Γ_{k+1} is formed by k + 1 linearly independent linear spaces. Then also the general tangential contact locus Γ_{k} is formed by k linearly independent linear spaces.

Indeed, with no loss of generality, as in the proof of Lemma 3.5, we may assume Γ_{k} ⊂ Γ_{k+1}. Thus Γ_{k} is of type II, and all its components are contained in one component of Γ_{k+1}. But clearly if one takes irreducible varieties Γ^{i}, i = 1, …, k, of the same dimension t_{k}, such that each Γ^{i} spans a linear space Π^{i} of dimension > t_{k}, and the linear spaces Π^{i}’s are linearly independent, then the span of the union of the Γ^{i}’s cannot have dimension kd + k − 1.

We point out the following example.

Example 4.7. ([18, Proposition 1.7]) Let X be the Grassmanniann G := 𝔾(ℙ^{2}, ℙ^{7}) and let k = 3. Then X is weakly 3-defective and, for a general choice of P_{1}, P_{2}, P_{3} ∈ X, Γ_{3} is the union of three disjoint linear spaces L_{1}, L_{2}, L_{3} of dimension 3, such that L_{i} contains P_{i}. The span Π_{3} of Γ_{3} is also tangent to X along a linear space W = ℙ^{5}, which misses the P_{i}’s and intersects each L_{i} along a line.

One computes soon that the join of L_{i}, L_{j}, W cannot fill Π_{3}, for obvious dimensional reasons. It follows that the secant degree of X equals the secant degree of Γ_{3}, which is 1. Hence the generic 3-identifiability of G holds.

We can derive, as in the previous example a general statement for the case where the tangential contact variety Γ_{k} is of type II.

Proposition 4.8. Assume that the general k-tangential contact locus Γ_{k} is of type II, i.e. it is the union of components Γ^{1}, …, Γ^{k}. Assume that Π_{k} is tangent to X along Γ_{k} ∪ Γ, where Γ ⊄ Γ_{k}. If ⟨Γ^{1} ∪⋯∪Γ̂^{I} ∪⋯∪Γ^{k} ∪Γ⟩ ≠ Π_{k}, where with Γ̂^{I} we indicate any non empty subset of {Γ^{1}, …, Γ^{k}}, then the k-th secant degree of X is equal to the k-th secant degree of Γ_{k}.

Corollary 4.9. In the assumptions of the previous proposition, assume the Γ^{i}’s are linear subspaces of ℙ^{r}, and linearly independent. Then X is j-identifiable for any j ≤ k.

Proof. Clearly if X is k-identifiable then it is also j-identifiable for every j ≤ k, hence we need simply to show that X is k-identifiable. This last follows since the union of linearly independent linear spaces is identifiable. □

On the other hand, there are cases (where X is an almost unbalanced products of projective spaces) in which Γ_{k} has type II and the previous proposition does not apply, moreover the k-th secant degree of Γ_{k} is different from the k-th secant degree of X.

A first example is the following.

Example 4.10. Consider the Segre variety X of ℙ^{2}×ℙ^{2}×ℙ^{5} and let k = 5. One computes that the general tangential contact locus Γ_{5} consist of 5 linearly independent linear subspaces Γ^{1} ∪ ⋯ ∪ Γ^{5}, where each Γ_{i} is a ℙ^{4} which passes through one of the points P_{i}.

The span Π_{5} of Γ_{5} is a ℙ^{24} which is tangent to X along an extra Γ = ℙ^{4}. Γ is linearly independent from any subset of 4 among the Γ^{i}’s. Thus the 5-th secant degree of Γ_{5} is 1, but the secant degree of X equals the secant degree of Γ_{5} ∪ Γ, hence it is 6.

A similar situation holds for the almost unbalanced cases defined in Section 8 of [20].

5. Applications to concrete examples

We show below how Lemma 3.5 applies, thanks to the observations contained in Section 4, to conclude the identifiability of some relevant cases of Segre products, Veronese varieties, Segre-Veronese varieties and moment varieties.

Non-trivial Segre products X ⊂ ℙ^{r} , X image of ℙ^{a_{1}} × ⋯ × ℙ^{a_{s}}, contain, for each j = 1, …, s, linear spaces L_{1}, …, L_{k}, of dimension a_{j} passing through k general points, which are linearly independent, provided that k(a_{j} + 1) is smaller than (r + 1). We do not know examples where these spaces are contact loci of weakly defective Segre products. In any case, the previous lemma, together with the analysis of weakly defective varieties contained in [26] and [29], yield the following.

Theorem 5.1. Let X ⊂ ℙ^{r} be an integral and non-degenerate n-dimensional variety and let k be a positive integer. Assume that X is not uniruled by lines and that dim σ_{k+n−1}(X) = (k + n − 1)(n + 1) − 1 < r (in particular X is not (k + n − 1)-defective). Then X is not weakly j-defective (hence it is j-identifiable) for every j ≤ k.

Proof. If X is not weakly j-defective, then the claim follows by [26]. Assume that X is weakly j-defective. Hence X is also weakly (j + i)-defective for i = 1, …, n − 1. Since X is not uniruled, then Γ_{j+i} cannot be a union of linear spaces of positive dimension, so by Lemma 3.5 we have:

1 ≤ dim Γ_{j} < ⋯ < dim(Γ_{j+n−1}).

It follows that dim(Γ_{j+n−1}) ≥ n, a contradiction. □

Theorem 5.1 is applicable to any embedding X ⊂ ℙ^{r} with O_{X}(1) ≅ L^{⊗2} ⊗ M with L an ample line bundle and M a nef line bundle. In particular it is applicable to any Veronese embedding of ℙ^{n}, since in this case dim σ_{i}(X) is known for all i by a famous theorem of Alexander and Hirschowitz ([10]). However, the result that we get from Theorem 5.1 for symmetric tensors is weaker than the results of [36, 31], where all the non-identifiable Veronese varieties are classified.

Let us turn to the study of Segre varieties. Fix integer s ≥ 2, n_{i} > 0, 1 ≤ i ≤ s. Set n := n_{1} + ⋯ + n_{s} and r := −1 + ∏_{i=1}^{s}(n_{i} + 1) and let X ⊂ ℙ^{r} be the (n-dimensional) Segre embedding of the multiprojective space ℙ^{n_{1}} × ⋯ × ℙ^{n_{s}}. Since X is uniruled by lines, we cannot apply Theorem 5.1 directly. On the other hand, the structure of linear spaces contained in Segre embeddings of multiprojective spaces is well known.

Remark 5.2. Let π_{i} : X → ℙ^{n_{i}} denote the projection of the Segre variety X into its i-th factor. For any i = 1, …, s let O_{X}(ϵ_{i}) be the line bundle π_{i}^{∗}(O_{ℙ^{n_{i}}} (1)). Each O_{X}(ϵ_{i}) is a spanned line bundle and O_{X}(1) = ⊗_{i=1}^{s}O_{X}(1). Let L ⊂ X be any linear space of positive dimension. Since O_{L}(1) ≅ ⊕_{i=1}^{s}O_{X}(ϵ_{i})_{|L}, there is j ∈ {1, …, s} such that O_{X}(ϵ_{i})_{|L} ≅ O_{L} for all i ≠ j, i.e. L is contained in a fiber of π_{j}. Fix an integer k ≥ 1 such that dim σ_{k+1}(X) = (k + 1)(n + 1) − 1 < r and assume that X is weakly (k + 1)-defective, with general tangential (k + 1)-contact locus Γ_{k+1} equal to a union of linear spaces, say L_{1}, …, L_{k}, L_{k+1}, of dimension ϵ_{k} > 0. Thus for each i = 1, …, k + 1 there is a unique integer j(i) ∈ {1, …, s} such that π_{j(i)}(L_{i}) is a point. We get a function ϕ_{k+1} : {1, …, k + 1} → {1, …, s} defined by the formula ϕ_{k+1}(i) := j(i). The function ϕ_{k+1} cannot depend on the choice of k + 1 general points. Then, moving independently the points, by monodromy we get that this function ϕ_{k+1} must be invariant for the action of the full symmetric group on {1, …, k + 1}. Thus j(i) = j(1) for all i.

Using the previous remark, we can prove the following.

Theorem 5.3. Let X be the Segre embedding of ℙ^{n_{1}}×⋯×ℙ^{n_{k}} in ℙ^{r} as above. Assume that for each i ∈ {1, …, s} there is j ∈ {1, …, s} \ {i} such that n_{j} = n_{i}. Assume also that dim σ_{k+n−1}(X) = (k + n − 1)(n + 1) − 1 < r. Then X is not weakly j-defective (hence it is j-identifiable) for every j ≤ k.

Proof. Assume that X is weakly j-defective. Thus it is also weakly (j + i)-defective, for i = 0, …, n − 1.

Each general contact locus Γ_{j+i} has dimension t_{j+i} ≤ n − 1 and so there cannot be a strictly increasing sequence of integer t_{j}, …, t_{j+n−1}. By Lemma 3.5 there is an integer h ∈ {j, …, j+n−1} such that Γ_{h} is a union L_{1} ∪ ⋯ ∪ L_{h} of h independent linear spaces of dimension t_{h}. Thus, by Remark 5.2, there is an integer m ∈ {1, …, s} such that, for all i = 1, …, h, π_{m}(L_{i}) is a linear space of dimension dim L_{i}, while π_{x}(L_{i}) is a point for all x ∈ {1, …, s} \ {m}.

Notice that the general contact locus Γ_{h} is invariant for linear automorphisms of ℙ^{r} preserving X, in the sense that for any automorphism ϕ : ℙ^{r} → ℙ^{r} with ϕ(X) = X the set ϕ(Γ_{h}) may be taken as a general contact locus. Thus π_{x}(h(L_{i})) is a point for each x ∈ {1, …, s} \ {m}, while π_{m}(h(L_{i}) is a linear space of dimension dim Π_{i}. By assumption there is z ∈ {1, …, s} \ {j} with n_{z} = n_{j}. Hence there is an automorphism ψ of X exchanging the m-th factor and the zth factor. This automorphism extends to a linear automorphism ϕ : ℙ^{r} → ℙ^{r} with ϕ(X) = X, because the embedding X ↪ ℙ^{r} is induced by the complete linear system |O_{X}(1, …, 1)| and ψ^{∗}(O_{X}(1, …, 1)) ≅ O_{X}(1, …, 1). We would have that π_{z}(L_{i}) is a linear space of dimension dim L_{i}, which yields a contradiction. □

The condition that for each i ∈ {1, …, s} there is j ∈ {1, …, s} \ {i} such that n_{j} = n_{i} restricts the range of application of Theorem 5.3. However, the theorem applies to the products of many copies of a given ℙ^{q}, i.e. to the case of cubic tensors (provided that one knows the non-defectivity of the corresponding Segre embedding).

In particular, in the case n_{i} = 1 for all i, as an immediate corollary of [23] and Theorem 5.3, we obtain the following result.

Corollary 5.4. Fix an integer s ≥ 5 and an integer k > 0 such that (k + s − 1)(s + 1) ≤ 2^{s} − 1. Let X ⊂ ℙ^{2^{s}−1} be the Segre embedding of (ℙ^{1})^{s}. Then X is not weakly k-defective, hence it is k-identifiable.

Thus, we obtain that binary tensors, i.e. tensors of type 2 × ⋯ × 2 (s times), are k-identifiable for

k ≤ \frac{2^{s} − 1}{s + 1} − s + 1.

This bound is very near to k_{0} = ⌈2^{s}/(s + 1)⌉ is the maximal integer for which k-identifiability can hold.

As an immediate corollary of [9, Theorem 3.1] we get the following result.

Corollary 5.5. Take X = (ℙ^{m})^{s} ⊂ ℙ^{r}, m > 1, r + 1 = (m + 1)^{s}. Set a := ⌊(m + 1)^{s}/(ms + 1)⌋ and let ϵ be the only integer such that 0 ≤ ϵ ≤ m and a ≡ ϵ (mod m + 1). If either ϵ > 0 or (k + ms)(sm + 1) ≤ (m + 1)^{s} − 1, set a′ := a − ϵ. If ϵ = 0 and (ms + 1)a = (m + 1)^{s} set a′ := a − 1. For any positive integer k + ms − 1 ≤ a′ X is not weakly k-defective and hence it is k-identifiable.

Take X as in Corollary 5.5. Note that if k ≥ a + 2, then σ_{a+2}(X) has expected dimension > r and hence identifiability fails for σ_{a+2}(X). In the special case s = 3, in Corollary 3.10 it is sufficient to assume 3m(k + 3m − 1) < (m + 1)^{3}, because the secant varieties of (ℙ^{m})^{3} are not defective, by [39].

Complete results on the defectivity of Segre products with more than 3 factors, except for the case of many copies of ℙ^{1} and for (ℙ^{m})^{3} ([39]) and an almost complete for [9], are not available. As far as we know, the best procedure to find theoretically the non defectivity of Segre products is the inductive method described in [5].

For Segre-Veronese varieties, which correspond to rank 1 partially symmetric tensors, we can join Theorem 5.1 and Theorem 5.3 to get:

Theorem 5.6. Fix integer s ≥ 2, n_{i} > 0, d_{i} > 0, 1 ≤ i ≤ s. Set n := n_{1} + ⋯ + n_{s} and r = −1+∏ \binom{d_{i}+n_{i}}{d_{i}}. Let X ⊂ ℙ^{r} be the (n-dimensional) Segre-Veronese embedding of the multiprojective space ℙ^{n_{1}} ×⋯×ℙ^{n_{s}}, via the linear system L_{1}^{d_{1}} ⊗⋯⊗L_{s}^{d_{s}}, where each L_{i} is the pull-back of O_{ℙ^{n_{i}}} (1) in the projection onto the i-th factor.

Assume that dim σ_{k+n−1}(X) = (k + n − 1)(n + 1) − 1 < r. Assume also that either:

• d_{i} > 1 for all i; or

• for all i with d_{i} = 1 there exists j ≠ i such that d_{j} = 1 and n_{i} = n_{j}.

Then X is not weakly j-defective (hence it is j-identifiable) for every j ≦ k.

Proof. Notice that X is not uniruled by lines, unless d_{i} = 1 for some i. When some d_{i} is 1, one can still apply the trick introduced in the proof of Theorem 5.3. □

Example 5.7. C. Araujo, A. Massarenti and R. Rischter proved that if h ≤ a^{⌊log_{2}(d−1)⌋}, then σ_{h+1}(X) has the expected dimension (see [13, Theorem 1.1] for a more precise result).

For the case s = 2 there is a very strong conjecture and several results supporting it ([1, 2, 4]). When s = 2 H. Abo and M. C. Brambilla give a list of 9 classes of secant varieties, which are known to be defective and they conjecture that they are the only defective ones for s = 2 ([4, Conjecture 5.5]. See [8, Table 2] for the list of known (in 2010) non-defective cases when s = 2.

H. Abo and M. C. Brambilla also gave many examples of defective secant varieties when s = 3, 4 ([3]). See [4, §5] for a general conjecture for the non-defectivity of Segre-Veronese varieties. If d_{i} ≥ 2 for all i, then X is not uniruled by lines, so we may apply Theorem 5.1 to obtain the non weak defectivity.

Example 5.8. C. Améndola, J.-C. Faugère, K. Ranestad and B. Sturmfels ([11, 12]) studied the Gaussian moment variety G_{n,d} ⊂ ℙ^{N}, N = \binom{n+d}{n} − 1, whose points are the vectors of all moments of degree ≤ d of an n-dimensional Gaussian distribution. We have dim G_{n,d} = n(n + 3)/2. They proved that all the secant varieties of G_{1,d} are non-defective, but σ_{2}(G_{n,d}) is defective for all n ≥ 3 ([12, Theorem 13 and Remark 15]). To apply Theorem 5.1 to G_{1,d} we need prove that G_{1,d} is not uniruled by lines. If so, Theorem 5.1 may be applied to σ_{k}(G_{1,d}) for all k with 2(k+1) < d. We may assume d ≥ 4. The variety G_{1,d} ⊂ ℙ^{d} is described in [11] and [12]; it is an arithmetically Cohen-Macaulay surface of degree \binom{d}{2} ([11, Proposition 1 and the last part of its proof and Corollary 2]) and its singular locus is a line L ⊂ G_{1,d} ([12, Lemma 4]). Let ℓ_{L} : ℙ^{d} \ L → ℙ^{d−2} the note the linear projection of from L and let π : G_{1,d} \ L → ℙ^{d−2} be the restriction of ℓ_{L}. Assume that G_{1,d} is ruled by line and call Δ the one-dimensional family of lines giving the ruling. Since the Grassmannian are complete, Δ coves G_{1,d} and hence the closure of the image of π would be an integral and non-degenerate curve Y ⊂ ℙ^{d−2} such that a general L ∈ Δ is mapped to a general point of Y. The surface G_{1,d} is covered by a family of degree d rational normal curves of ℙ^{d}, each of them tangent to L at a different point. Hence Y must be the rational normal curve of degree d − 2 and X would be contained in a degree d − 2 3-dimensional cone T ⊂ ℙ^{d} with vertex L. Since deg(G_{1,d}) = \binom{d}{2} > 3(d − 2), Bezout theorem gives that every cubic hypersurface containing G_{1,d} contains T, contradicting the fact that G_{1,d} is cut out by cubics ([11, Proposition 1]).

For Segre products, the most general result that we can obtain with our techniques is resumed in the following theorem. It can be applied to a wide class of Segre products, not covered by Theorem 5.3.

Theorem 5.9. Let X be the Segre embedding of a product ℙ^{n_{1}} × ⋯ × ℙ^{n_{q}} in ℙ^{r}, with n = ∑ n_{i} = dim(X). Let k be a positive integer. Assume that dim σ_{k+n−1}(X) = (k + n − 1)(n + 1) − 1 < r (in particular X is not (k + n − 1)-defective). Then X is j-identifiable for every j ≤ k.

Proof. Assume that X is weakly k-defective. Then, as in the proof of Theorem 5.1, there exists some j, k ≤ j ≤ k + n − 2, such that the general tangential contact loci Γ_{j} and Γ_{j+1} have the same dimension. Thus, by Lemma 3.5 and its proof, the contact loci Γ_{j} and Γ_{j+1} are formed by a union of linearly independent linear spaces L_{i}, say of dimension s. By Remark 5.2, the L_{i}’s are contained in some component of the product, which, as in the proof of Theorem 5.6, is the same for all i’s. Say for simplicity that the L_{i}’s sits in the first component. Call Y the Segre embedding of ℙ^{n_{2}} × ⋯ × ℙ^{n_{q}} and call Q_{i} ∈ Y the point to which L_{i} maps in the projection X → Y. The Q_{i}’s are general points of Y. Since X is not j + 1 defective, the span M of the L_{i}’s has dimension (j + 1)(s + 1) − 1. If M meets X only in the union of the L_{i}’s, which are linearly independent subspaces, then by [19, Example 2.2] and Lemma 4.2 X is (j + 1)-identifiable. Otherwise there is a point P ∈ (M ∩ X) \ ⋃ L_{i}. The point P cannot belong to the same fiber over Y of some L_{i}, for otherwise M contains the span of L_{i} and P, and its dimension cannot be (j +1)(s+1)−1. Thus the projection of M ∩ X to Y contains a point Q different from the Q_{i}’s. It follows that for a general choice of j points Q_{1}, …, Q_{j} in Y the span of the Q_{i}’s meets Y in a point Q different from the Q_{i}’s. By the trisecant lemma ([14, p. 109], [24, Example 1.8], [25, Proposition 2.6]), this means that the codimension of Y in its span is at most j − 1. Since the codimension of Y is Π(n_{i} + 1) − ∑ n_{i} − 1 (where the product and the sum range from 2 to q), and moreover (j + 1)(n + 1) < Π_{i=1}^{q}(a_{1} + 1), after a short computation we get that:

a_{1} > Π_{i=2}^{q}(a_{i} + 1) − ∑_{i=2}^{q} a_{i},

i.e. Y is unbalanced in the sense of [20, Section 8]. Since we also have j ≥ a_{1}, it follows by [5, Theorem 4.4] that either σ_{j+1}(X) covers ℙ^{r}, or X is (j + 1)-defective. In both cases we have a contradiction. □

We should remark that the inductive method of [5] is extended to the study of weak defectivity in [20]. The results contained in [20] are, in several cases, improved by the results that one can get via Theorem 5.9.

From the computational point of view, let us point out the following:

Remark 5.10. For specific examples of Segre products X, the stacked hessian method introduced in [30] can provide a positive answer to the k-identifiability of X. The stacked hessian method requires the computations of the derivatives of the span of k tangent spaces at general points, when the points move.

In order to prove the non (k + n)-defectivity of X, by the Terracini’s lemma, one has just to compute the dimension of the span of k + n general tangent spaces (see e.g. [32, Section 6]). When n is not too big with respect to k, this second method, together with Theorem 5.9, can provide the k-identifiability of X with a faster procedure.

6. Positive characteristic

In this section we work over an algebraically closed field 𝕂 of arbitrary characteristic.

Definition 6.1. Let X ⊂ ℙ^{r} be an integral and non-degenerate variety. Set n := dim(X). X is said to be very strange if for a general codimension n linear subspace M ⊂ ℙ^{r} the set X ∩ M is not in linearly general position in M, i.e. there is a hyperplane of M containing at least r − n + 2 points of M ∩ X ([42]).

We recall that X ⊂ ℙ^{r} is not very strange if a general 0-dimensional section of X is in uniform position ([14, Ch. 2],[42, Introduction]). Hence X ⊂ ℙ^{r} is not very strange if either char(𝕂) = 0 or X is non-singular in codimension 1 and r ≥ n + 3 ([42, Corollaries 1.6 and 2.2 and Theorem 0.1]). Thus none of the examples of the last section is very strange. X is not very strange if a general curve section of X is not strange ([42, Lemma 1.1]) and in particular X is not very strange if char(𝕂) > deg(X).

Lemma 6.2. Let Y ⊂ ℙ^{r} be an integral and non-degenerate variety, which is not very strange. Set n := dim(Y). Fix a general S ⊂ Y with ♯(S) ≤ r − n − 1. Then S is the set-theoretic base locus of the linear system on Y induced by H^{0}(ℙ^{r}, I_{S}(1)).

Proof. Let N ⊂ ℙ^{r} be a general linear space of dimension n − r − 1. By Bertini’s theorem the scheme N ∩ Y is a finite set of deg(Y) points. Since Y is not very strange, the set N ∩ Y is in linearly general position in N. Hence for any E ⊂ N ∩ Y with ♯(E) = ♯(S) ≤ n − r − 1, the restriction of U := H^{0}(ℙ^{r}, I_{E}(1)) to N ∩ Y has E as its set-theoretic base locus. Since N is a linear space the restriction of U to Y has base locus contained in N ∩ Y. Thus E is the base locus of the restriction of U to T. Since Y is integral and non-degenerate, a general subset A ⊂ Y with cardinality at most r − n spans a general subspace of ℙ^{r} with dimension ♯(A) − 1. Hence E is a general subset of Y with cardinality ♯(S). □

Proof of Theorem 1.1: Fix a general q ∈ σ_{k}(X). If dim σ_{k}(X) < kn + k − 1, then a dimensional count gives that S(q, X) is infinite.

Hence to prove Theorem 1.1 it is sufficient to find a contradiction to the existence of A, B ∈ S(q, X) with A ≠ B. Set E := B \ A ∪ B and Z := A ∪ E. We have h^{1}(ℙ^{r}, I_{Z}(1)) > 0, because q ∈ ⟨A⟩∩⟨B⟩ and q ∉ ⟨A∩B⟩ ([15, Lemma 1]), i.e. Z does not impose ♯(Z) independent conditions to W. We may also assume that A and B are general in their irreducible component of the constructible set S(q, X). Since q is general, we may assume that A (resp. B) is a general subset of X with cardinality s, but of course A∪B is far from being general (in general). We identify X with M, so that L and R are line bundles on X and O_{X}(1) ≅ L⊗R. Since A is general and ♯(A) ≤ dim V, the set of all f ∈ V vanishing at all points of A is a linear subspace of V with dimension dim(V)−s. Hence h^{1}(ℙ^{r}, I_{A}(1)) = 0. Since A is general, by Lemma 6.2 there is a hypersurface T ∈ |V | with A ⊂ T and E ∩ T = ∅. Let f ∈ V \ {0} be an equation of T. The multiplication by f induces an isomorphism between H^{0}(X, R) and the linear subspace W (−T) := H^{0}(X, I_{T} ⊗ L ⊗ R) ∩ W of all g ∈ W vanishing on T. Moreover, for any finite subset S ⊂ X \ T this isomorphism sends H^{0}(X, I_{S} ⊗ R) onto W (−T − S) = {h ∈ W : h_{|T ∪S} ≡ 0}. Since ♯(E) ≤ ♯(B) ≤ h^{0}(R) and B is general in X, we have h^{0}(X, I_{E} ⊗ R) = h^{0}(X, R) − ♯(E), i.e. E impose s independent conditions to H^{0}(X, R). Since T ∩ E = ∅, we get dim(W (−T − E)) = dim(W (−T)) − ♯(E). Hence H^{0}(ℙ^{r}, I_{Z}(1)) = dim W − ♯(Z), i.e. h^{1}(ℙ^{r}, I_{Z}(1)) = 0, a contradiction. □

We give an example of application of Theorem 1.1.

Example 6.3. Take X = ℙ^{n_{1}} × ⋯ × ℙ^{n_{s}} and positive integers d_{i} > 0 and let X ⊂ ℙ^{r}, r = −1+∏_{i=1}^{s} \binom{n_{i}+d_{i}}{n_{i}}. Write d_{i} = b_{i}+c_{i} with b_{i} > 0, c_{i} ≥ 0 and c_{j} > 0 for some j. Set n := n_{1}+⋯+n_{s}. If we fix any integer k > 0 such that

k ≤ min\{−2 − n + ∏_{i=1}^{s} \binom{n_{i} + b_{i}}{n_{i}}, ∏_{i=1}^{s} \binom{n_{i} + c_{i}}{n_{i}}\},

then Theorem 1.1 applies and proves the identifiability of a general q ∈ ℙ^{r} of X-rank k.

References

[1] H. Abo, On non-defectivity of certain Segre-Veronese varieties. J. Symb. Comput. 45 (2010), no. 12, 1254–1269.

[2] H. Abo, and M. C. Brambilla, Secant varieties of Segre-Veronese varieties ℙ^{m} × ℙ^{n} embedded by O(1, 2). Experim. Math. 18 (2009), no. 3, 369–384.

[3] H. Abo, and M. C. Brambilla, New examples of defective secant varieties of Segre-Veronese varieties. Collect. Math. 63 (2012), no. 3, 287–297.

[4] H. Abo, and M. C. Brambilla, On the dimensions of secant varieties of Segre-Veronese varieties. Ann. Mat. Pura Appl. 192 (2013), no. 1, 61–92.

[5] H. Abo, G. Ottaviani, and C. Peterson, Induction for secant varieties of Segre varieties. Trans. Amer. Math. Soc. 361 (2009), no. 2, 767–792.

[6] H. Abo, G. Ottaviani, and C. Peterson, Non-defectivity of Grassmannians of planes. J. Alg. Geom. 21 (2012), 1–20.

[7] H. Abo, and N. Vannieuwenhoven, Most secant varieties of tangential varieties to Veronese varieties are nondefective. Trans. Amer. Math. Soc. 370 (2018), no. 1, 393–420.

[8] B. Ådlandsvik, Joins and higher secant varieties. Math. Scand. 62 (1987), 213-222.

[9] T. Aladpoosh, and H. Haghighi, On the dimension of higher secant varieties of Segre varieties ℙ^{n} × ⋯ × ℙ^{n}. J. Pure Appl. Algebra 215 (2011), no. 5, 1040–1052.

[10] J. Alexander, and A. Hirschowitz, Polynomial interpolation in several variables. J. Alg. Geom. 4 (1995), no. 2, 201–222.

[11] C. Améndola, J.-C. Faugère, and B. Sturmfels, Moment varieties of Gaussian mixtures. J. Alg. Stat. 7 (2016), no. 1, 14–28.

[12] C. Améndola, K. Ranestad, and B. Sturmfels, Algebraic Identifiability of Gaussian Mixtures. IMRN to appear. Preprint arXiv:1612.01129 (2016).

[13] C. Araujo, A. Massarenti, and R. Rischter, On nonsecant defectivity of Segre-Veronese varieties. Preprint arXiv:1611.01674 (2016).

[14] E. Arbarello, M. Cornalba, P. Griffiths, and J. Harris, Geometry of algebraic curves, Vol. I. Springer, Berlin - Heidelberg - New York, 1985.

[15] E. Ballico and A. Bernardi, Decomposition of homogeneous polynomials with low rank, Math. Z. 271 (2012), 1141–1149.

[16] E. Ballico, A. Bernardi, M.V. Catalisano, and L. Chiantini, Grassman secants, identifiability, and linear systems of tensors. Linear Algebra Appl. 438 (2013), 121–135.

[17] K. Baur, J. Draisma, and W. de Graaf, Secant dimensions of minimal orbits: computations and conjectures. Experim. Math. 16 (2007), no. 2, 239–250.

[18] A. Bernardi, and D. Vanzo, A new class of non-identifiable skew symmetric tensors. Preprint arXiv:1606.04158 (2016).

[19] C. Bocci, and L. Chiantini, On the identifiability of binary Segre products. J. Algebraic Geometry 22 (2013), 1–11.

[20] C. Bocci, L. Chiantini, and G. Ottaviani, Refined methods for the identifiability of tensors. Ann. Mat. Pura Appl. 193 (2014), no. 6, 1691–1702.

[21] A. Boralevi, A note on secants of Grassmannians. Rend. Istit. Mat. Univ. Trieste 45 (2013), 67–72.

[22] M. V. Catalisano, A. V. Geramita, and A. Gimigliano, Secant Varieties of Grassmann Varieties. Proc. Amer. Math. Soc. 183 (2004), 633–642.

[23] M. V. Catalisano, A. V. Geramita, and A. Gimigliano, Secant varieties of ℙ^{1} × ⋯ × ℙ^{1} (n-times) are NOT defective for n ≥ 5. J. Alg. Geom. 20 (2011), 295–327.

[24] L. Chiantini, and C. Ciliberto, A few remarks on the lifting problem. Astérisque 218 (1993), 95–109.

[25] L. Chiantini, and C. Ciliberto, Weakly defective varieties. Trans. Amer. Math. Soc. 454 (2002), no. 1, 151–178.

[26] L. Chiantini, and C. Ciliberto, On the concept of k-secant order of a variety. J. London Math. Soc. 73 (2006), no. 2, 436–454.

[27] L. Chiantini, and C. Ciliberto, On the dimension of secant varieties. J. Europ. Math. Soc. 73 (2006), no. 2, 436–454.

[28] L. Chiantini, M. Mella, and G. Ottaviani, One example of general unidentifiable tensors. J. Algebr. Stat. 5 (2014), no. 1, 64–71.

[29] L. Chiantini, and G. Ottaviani, On generic identifiability of 3-tensors of small rank. SIAM J. Matrix Anal. Applic. 33 (2012), 1018–1037.

[30] L. Chiantini, G. Ottaviani, and N.Vanniuwenhoven, An algorithm for generic and low-rank specific identifiability of complex tensors. SIAM J. Matrix Anal. Applic. 35 (2014), 1265–1287.

[31] L. Chiantini, G. Ottaviani, and N.Vanniuwenhoven, On identifiability of symmetric tensors of subgeneric rank. Trans. Amer. Math. Soc. 369 (2017), no. 2, 4021–4042.

[32] L. Chiantini, G. Ottaviani, and N.Vanniuwenhoven, Effective criteria for specific identifiability of tensors and forms. SIAM J. Matrix Anal. Applic. 38 (2017), 656–681.

[33] I. Domanov, and L. De Lathauwer, On the uniqueness of the canonical polyadic decomposition of third-order tensors–part I: Basic results and unique- ness of one factor matrix. SIAM J. Matrix Anal. Appl. 34 (2013), 855–875.

[34] I. Domanov, and L. De Lathauwer, On the uniqueness of the canonical polyadic decomposition of third-order tensors–part II: Uniqueness of the overall decomposition. SIAM J. Matrix Anal. Appl. 34 (2013), 876–903.

[35] I. Domanov, and L. De Lathauwer, Generic uniqueness conditions for the canonical polyadic decomposition and INDSCAL. SIAM J. Matrix Anal. Applic. 36 (2015), no. 4, 1567-1589.

[36] F. Galuppi, and M. Mella, Identifiability of homogeneous polynomials and Cremona Transformations. Preprint arXiv:1606.06895 (2016).

[37] J. B. Kruskal, Three-way arrays: rank and uniqueness of trilinear decompositions, with application to arithmetic complexity and statistics. Linear Algebra Appl. 18 (1977), no. 2, 95–138.

[38] J.M. Landsberg, Tensors: Geometry and Applications Graduate Studies in Mathematics, Amer. Math. Soc. Providence, 128 (2012).

[39] T. Lickteig, Typical tensorial rank. Linear Algebra Appl. 69 (1985), 95–120.

[40] A. Massarenti, and R. Rischter, Non-secant defectivity via osculating projections. Ann. Scuola Norm. Sup. to appear. Preprint arXiv:1610.09332v1 (2016).

[41] M. Mella, Singularities of linear systems and the Waring problem. Trans. Amer. Math. Soc. 358 (2006), no. 12, 5523–5538.

[42] J. Rathmann, The uniform position principle for curves in characteristic p. Math. Ann. 276 (1987), no. 4, 565–579.

[43] N. D. Sidiropoulos, and R. Bro, On the uniqueness of multilinear decomposition of N-way arrays. J. Chemometrics 14 (2000), 229–239.

(Edoardo Ballico) Dipartimento di Matematica, Univ. Trento, Italy

   E-mail address: edoardo.ballico@unitn.it

(Alessandra Bernardi) Dipartimento di Matematica, Univ. Trento, Italy

   E-mail address: alessandra.bernardi@unitn.it

(Luca Chiantini) Dipartimento di Ingegneria dell’Informazione e Scienze Matematiche, Univ. Siena, Italy

   E-mail address: luca.chiantini@unisi.it