On the locus of points of high rank

Given a closed subvariety X in a projective space, the rank with respect to X of a point p in this projective space is the least integer r such that p lies in the linear span of some r points of X. Let Wk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W_k$$\end{document} be the closure of the set of points of rank with respect to X equal to k. For small values of k such loci are called secant varieties. This article studies the loci Wk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W_k$$\end{document} for values of k larger than the generic rank. We show they are nested, we bound their dimensions, and we estimate the maximal possible rank with respect to X in special cases, including when X is a homogeneous space or a curve. The theory is illustrated by numerous examples, including Veronese varieties, the Segre product of dimensions (1, 3, 3), and curves. An intermediate result provides a lower bound on the dimension of any GLn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {GL}_n$$\end{document} orbit of a homogeneous form.


Introduction
A general m ×n matrix has rank min{m, n}, and this is the greatest possible rank. The locus of matrices of rank at most r , for r min{m, n}, is well studied: its defining equations are well known, along with its codimension, singularities, and so on.
Also well studied are the loci of tensors of a fixed format and of rank at most r . These, up to closure, are secant varieties of Segre varieties. Despite intense study, defining equations and dimensions of such secant varieties are known only in limited cases, to say nothing of their singularities. For introductory overviews of this, see for example [13,22]. In contrast to the matrix case, however, special tensors may have ranks strictly greater than the rank of a general tensor. The locus of tensors with ranks greater than the generic rank is quite mysterious. In general it is not known what is the dimension of this locus, what are its equations, whether it is irreducible-or even whether it is nonempty.
Similarly, the closure of the locus of symmetric tensors of rank at most r is a secant variety of a Veronese variety. In this case, the dimensions of all such secant varieties are known, although the equations are not known. The same sources [13,22] give introductions to this case as well. But once again, special symmetric tensors may have ranks strictly greater than the rank of a general symmetric tensor. And once again, the locus of such symmetric tensors is almost completely unknown.
Here we study high rank loci for tensors and symmetric tensors, and for more general notions of rank. We consider rank with respect to a nondegenerate, irreducible projective variety X ⊆ P N over an algebraically closed field k of characteristic zero. Let rank = rank X denote rank with respect to X , the function that assigns to each point p ∈ P N the least number r such that p lies in the linear span of some r points of X . See Sect. 2 for more details. For k 1 let W k = rank −1 (k) = { p ∈ P N : rank( p) = k }.
Let g be the generic rank with respect to X . Note that W k = σ k (X ) is the kth secant variety for 1 k g, in particular W 1 = X and W g = P N . We seek to understand the high rank loci, namely, W k for k > g.
We give dimension bounds for the W k and we find containments and noncontainments between the high rank loci and secant varieties. Using these, we can improve previously known upper bounds for rank in the cases where X is a curve (Theorem 3.7) or a projective homogeneous variety (Theorem 3.9). This includes Segre and Veronese varieties, corresponding to tensor rank and symmetric tensor rank. The key result is a nesting statement, that each high rank locus W k for k > g is contained in the next highest one W k−1 , and in fact more strongly the join of W k and X is contained in W k−1 , see Theorem 3.1.
We give a lower bound for the dimension of the locus of symmetric tensors of maximal rank, showing that, even though the maximal value of rank is unknown (!), there is a relatively large supply of symmetric tensors with maximal rank, see Theorem 4.1. Possibly of independent interest, we give a lower bound for the dimension of the GL(V ) orbit of a homogeneous form F ∈ S d (V ), assuming only that F is concise, i.e., cannot be written using fewer variables; other well-known results assume F defines a smooth hypersurface, but we give a bound even if the hypersurface defined by F is singular, reducible, or non-reduced, see Proposition 4.7.
We find the dimension of the locus of 2×2n ×2n tensors of maximal rank, see Proposition 5.4, and we characterize all 2×4×4 tensors with greater than generic rank, see Proposition 5.8.
Finally, in Sect. 6 we let X be a curve contained in a smooth quadric in P 3 . Then the generic rank is g = 2 and the maximal rank is m = 2 or 3. When X is a general curve of bidegree (2, 2) we show that W 3 is a curve of degree 8, disjoint from X , with four points of rank 2 and all the rest of rank 3. Piene has shown that if X is a general curve of bidegree (3,3), then W 3 is empty, i.e., m = 2. We extend this to general curves of bidegree (a, b) with a 4 and b 1.

Background
We work over a closed field k of characteristic zero. For a finite dimensional vector space V , let PV be the projective space of lines through the origin in V , and for q ∈ V , q = 0, let [q] be the corresponding point in PV . A variety X ⊆ PV is a reduced closed subscheme. We deal only with varieties in PV defined over k. Recall that a variety X ⊆ PV is nondegenerate if X is not contained in any proper linear subspace, equivalently if X linearly spans PV .

Ranks and secant varieties
Let X ⊆ PV be a nondegenerate variety. For q ∈ V , q = 0, the rank with respect to X of q, denoted rank X (q) or more simply rank(q), is the least integer r such that q = x 1 +· · ·+ x r for some x i ∈ V with [x i ] ∈ X for 1 i r . Equivalently, rank(q) is the least integer r such that [q] lies in the span of some r distinct, reduced points in X . We extend rank to PV by rank([q]) = rank(q).
For example, tensor rank is rank with respect to a Segre variety, Waring rank is rank with respect to a Veronese variety, and alternating tensor rank is rank with respect to a Grassmannian in its Plücker embedding.
The rank function is subadditive and invariant under multiplication by scalars. In particular, rank( p) − rank(q) rank( p + q) rank( p) + rank(q).
The r th secant variety of X , denoted σ r (X ), is the closure of the union of the planes spanned by r distinct, reduced points in X . Equivalently, σ r (X ) is the closure of the set of points of rank at most r .
There is a unique value g, called the generic rank, such that there is a Zariski open, dense subset of P N of points with rank g. The generic rank is the least value r such that σ r (X ) = P N . (The situation is more complicated over non-closed fields. See for example [6] for the real case.)

Upper bounds for rank
As long as X is nondegenerate, we may choose a basis for V consisting of points x i with [x i ] ∈ X , and then every point in PV can be written as a linear combination of those basis elements. This shows that every point in PV has rank at most dim V . In particular the values of rank are finite and bounded.
Let m be the maximal rank with respect to an irreducible, nondegenerate variety X . Recall the following well-known upper bounds.
Proof For any q / ∈ X , a general plane through q of dimension codim X is spanned by its intersection with X (see argument in [23], or [20,Proposition 18.10]), which is reduced by Bertini's theorem. This plane intersects X in deg X many points; choosing a spanning subset shows rank(q) codim X + 1. This was also observed by Geramita when X is a Veronese variety, corresponding to the case of Waring rank [19, p. 60]. It is false in the positive characteristic case, see [4], and it is false over the real numbers, see [3,6]. (In the positive characteristic case and over the real numbers the bound is codim X + 2.) Proof A general line through q ∈ P N is spanned by two points x, y in the dense open set of points of rank g. So q is a linear combination of x and y, and rank(q) rank(x)+ rank(y) = 2g. If σ g−1 (X ) is a hypersurface, a general line through q contains a point x of rank g − 1 and a point y of rank g. Again rank(q) rank(x) + rank(y) = 2g − 1.
This bound holds over the real numbers and over closed fields in arbitrary characteristic, see [7]. Over the real numbers this bound is sharp, see [6,Theorem 2.10]. We show that, over a closed field k of characteristic zero, it can be improved to m 2g − 1 in some cases, such as when X is a curve or a homogeneous variety. It is an open question whether m 2g − 1 for every variety X over a closed field.

Joins and vertices
We recall some basic notions of joins and vertices of varieties in P N . See [17,Section 4.6] for more details.
, is the closure of the union of all lines spanned by points p, q with p ∈ V 1 , q ∈ V 2 , and p = q. We also use additive notation: Note that if X, Y are irreducible then so is X + Y .
It is well known that Vertex(W ) ⊆ W and Vertex(W ) is a linear space.

General results
Theorem 3.1 Let X ⊆ P N be an irreducible, nondegenerate variety. Let g be the generic rank and m the maximal rank with respect to X . Then for each k, g + 1 k m, W k + X ⊆ W k−1 . In particular W m ⊂ W m−1 ⊂ · · · ⊂ W g+1 ⊂ W g = P N .
Proof Let W be an irreducible component of W m . A general point of W + X has rank m or m −1. If the general point has rank m then W + X ⊆ W m . Since W + X is irreducible, it is contained in one of the irreducible components of W m ; since W ⊆ W + X , it must be W + X = W . But then X ⊆ Vertex(W ), contradicting the nondegeneracy of It is an interesting problem to find an example where the inclusion W k + X ⊆ W k−1 is strict. We give a sharp bound on the dimension of the high rank loci. Theorem 3.6 Let X be an irreducible variety in P N and let g be the generic rank with respect to X . For every k 1, codim W g+k 2k − 1.  Proof First recall that X is nondefective, meaning that for k 1, dim k X = min{ N , 2k − 1}, see e.g. [1, Introduction, Remark 1.6]. Then N > dim(g − 1) X = 2g − 3, and N = dim g X 2g − 1. Hence N ∈ {2g − 1, 2g − 2}.
If N is even, N = 2g − 2, then dim σ g−1 (X ) = 2g − 3 = N − 1. This is the case in which σ g−1 (X ) is a hypersurface. By Theorem 2.1 again, m codim X + 1 = 2g − 2. Second, if X is a projective homogeneous variety in a homogeneous embedding then we can obtain the same improvement. Proof X is the unique closed orbit of G on PV , see for example [18,Claim 23.52]. Since X is G-invariant, so is each rank locus W k . Every G-invariant closed set contains X , in particular X ⊂ W m . The asssertion m = 2g contradicts Corollary 3.4, thus m 2g − 1.
If in addition σ g−1 (X ) is a hypersurface, and m = 2g − 1, then Therefore W g+1 contains an irreducible component equal to σ g−1 (X ). This contradicts the definition of the rank locus W g+1 : general points in σ g−1 (X ) have rank g − 1, but general points in each component of W g+1 have rank g + 1. It follows that whenever σ g−1 (X ) is a hypersurface, we must have m 2g − 2.
The bounds in both Theorems 3.7 and 3.9 are attained when X is a rational normal curve, see Sect. 4.4.

Example 3.10
The maximal rank is strictly less than twice the generic rank in the following cases.
• Waring rank, when X is a Veronese variety.
• Tensor rank, when X is a Segre variety.
• Alternating tensor rank, when X is a Grassmannian in its Plücker embedding.
• Multihomogeneous rank, also called partially symmetric tensor rank, when X is a Segre-Veronese variety.

Veronese varieties
When X = ν d (P n−1 ) ⊂ P N is a Veronese variety, then P N is the projective space of degree d homogeneous forms in n variables and X corresponds to the dth powers. Rank with respect to X is called Waring rank. The Waring rank of a homogeneous form of degree d is the least r such that the form can be written as a sum of r dth powers of linear forms. For example, The main result in this section is a lower bound for the dimension of the maximal rank locus W m with respect to any Veronese variety.
is the Veronese variety and that dim V = n 3. Then every irreducible component of the rank locus W m has dimension at least n+1 2 − 1, then d is even and W is the set of all (d/2)th powers of quadrics. This will be proved in Sect. 4.8. The proof uses a lower bound for the dimension of the orbit of a homogeneous form under linear substitutions of variables, which may be of independent interest, see Sect. 4.6. First, we review some background information on apolarity, conciseness, and generic and maximal Waring rank. We also give a full description of the rank loci with respect to a rational normal curve, that is, a Veronese embedding of P 1 , corresponding to Waring rank of binary forms.

Apolarity
Let S = k[x 1 , . . . , x n ] and let T = k[α 1 , . . . , α n ], called the dual ring of S. We let T act on S by differentiation, with α i acting as partial differentiation by x i . This is called the apolarity action and denoted by the symbol , so that n . If F is homogeneous then F ⊥ is a homogeneous ideal. For more details see for example [21, Section 1.1].

Concise forms
In the terminology of [11]

be written as a homogeneous form in fewer variables, even after a linear change of coordinates
The following are equivalent: F is concise; the projective hypersurface V (F) is not a cone (i.e., has empty vertex); the ideal F ⊥ has no linear elements; the (d − 1)-th derivatives of F span the linear forms. Note that the last two conditions can be checked directly by computation.
Write F for the span of the (d − 1)-th (degree 1) derivatives of F. We have F = ((F ⊥ ) 1 ) ⊥ , that is, F is perpendicular to the space of linear forms in the ideal F ⊥ . Nonzero elements of F (or, elements of a basis of F ) are called essential variables of F. We have F ∈ S d F , see [11, Proposition 1].

Generic and maximal Waring rank
The rank of a quadratic form is equal to its number of essential variables, by diagonalization. Thus, if d = 2, then g = m = n. If n = 2, then g = (d + 2)/2 and m = d by work of Sylvester and others in the 19th century, see for example [21, Section 1.3] and references therein. We will review the n = 2 case in the next section.
For n, d 3 the generic rank is known by the famous Alexander-Hirschowitz Theorem:

Moreover, the last proper secant variety σ g−1 (ν d (P n−1 )) is a hypersurface if and only if
On the other hand, the maximal rank m = m n,d is only known in a few initial cases: m 3,3 = 5, m 3,4 = 7, m 3,5 = 10, m 4,3 = 7. See [10] for details and references. We have m n,d > g n,d when n = 2, when n = 3, and when n = 4 and d is odd [10]. In all other cases, it is an open question whether m n,d > g n,d .

Binary forms
In this case the apolarity method provides a full description of the loci W k , as follows. Let τ (X ) be the tangential variety of X . Some parts of the following statement are well known, see for example [21,Section 1.3], and much (perhaps all) of it is known to experts, but we include the statement here for lack of a clear reference.
In particular, we have the following nested inclusions of irreducible varieties (each one of codimension 1 in the next): Proof Fix S = k[x, y] and V = S 1 , the space of linear forms in S. We identify S d with S d V and the dual ring T = k[α, β] with the symmetric algebra on V * .
For a homogeneous polynomial F ∈ S d V let F ⊥ be the apolar ideal of F. Then F ⊥ = ( , ) is a homogeneous complete intersection with deg = r deg = d+2−r , see for example [21,Theorem 1.44]. Note that both and are homogeneous polynomials in two variables, hence they are products of linear factors. Then (see for example [21,Section 1.3], [15]) F ∈ σ r (X )\σ r −1 (X ); if has all distinct roots, then rank(F) = r ; and if has at least one repeated root, then rank(F) = d + 2 − r .
Note further, that if r < d + 2 − r , the polynomial is unique up to rescaling. In particular, whether it has distinct roots or not does not depend on any choices, so the conditions for rank(F) = r or d + 2 − r are well defined. (If r = d + 2 − r , then the conclusions are the same in both cases.) Still assuming r < d + 2 − r , the uniqueness of determines a well-defined map: The map is surjective and every fiber π −1 The locus of with a double root is an irreducible divisor in P(S r V * ) and (the closure of) its preimage W d+2−r is also irreducible of codimension 1 in σ r (X ).
From this we see that dim σ r (X ) = 2r − 1, so the generic rank g = (d + 1)/2 = (d + 2)/2 . Furthermore, the maximal rank is m = d, and it appears whenever r is 2 and has a double root, so that F is in the span of a double point, i.e., F is in the tangential variety τ (X ) = W m = W d .
In between, where the inclusions are of irreducible varieties, each of codimension 1 in the next. This proves the inclusions displayed in the statement.

Powers of quadratic forms
Fix n, let Q n = x 2 1 + · · · + x 2 n , and consider Q k n , a form of degree d = 2k. Reznick showed every form of degree k in n variables is a derivative of Q k n , see [ thus rank(Q 2 n ) n 2 , so for sufficiently large n, g n, 4 = O(n 3 ) rank(Q 2 n ). For small n, Reznick shows that g n,4 rank(Q 2 n ) g n,4 + 1 for n = 3, 4, 5, 6. There is a similar identity: so rank(Q 3 n ) 4 n 3 + 2 n 2 + n. Hence g n, 6 = O(n 5 ) rank(Q 3 n ) for sufficiently large n.
It would be interesting to determine rank(Q k n ), in particular to determine whether the rank is greater than the generic rank, and whether it is strictly less than the maximal rank.

Orbits of homogeneous forms
When F defines a smooth hypersurface in PV of degree deg F 3, the stabilizer of F in SL(V ) is finite, see for example [24, (2.1)]. In particular the projective orbit In this section we establish a lower bound for the dimension of GL(V )·[F], assuming only that F is concise. Proof If Y = P N , then there is nothing to prove, so assume dim Y < N . First let Y be irreducible. Replacing P N with a subspace if necessary, we may assume that Y is nondegenerate. Consider the vertex-incidence subvariety Z ⊂ Y ×(P N ) * , defined by with its natural projections pr 1 : Z → Y and pr 2 : Z → (P N ) * . By our assumptions pr 2 is dominant, so dim Z N > dim Y . Let W be the image of pr 1 . In particular, by dimension count, for a general point w ∈ W , there is a positive dimensional fiber Since Y is irreducible and nondegenerate, a general point y in Y is contained in some Y ∩ H with H ∈ Z w . Then the line through y and w is contained in Y ∩ H . . Pick a general linear substitution of variables, say x n = a 1 x 1 + · · · + a n−1 x n−1 . Let F (x 1 , . . . , x n−1 ) = F(x 1 , . . . , x n−1 , a 1 x 1 + · · · + a n−1 x n−1 ). Then F essentially depends on n − 1 variables and no fewer.
Proof This follows from Lemma 4.4, since a hypersurface is a cone if and only if its reduced subscheme is a cone. Now we turn our attention to GL(V )-orbits. Lemma 4.6 Let F ∈ P(S d V ) be a polynomial in n = dim V variables, which essentially depends on k variables with 0 < k < n (i.e., F is non-trivial and non-concise).
is onto, and each fiber is isomorphic to GL(V )·[F], proving the dimension claim.
Note that if F 3 = Q k 3 for some quadric in three variables Q 3 , then by the generality of our choices of linear substitutions (or equivalently, of hyperplane sections of the loci (F i = 0)), we also must have F = F n = Q k . Thus it only remains to show the claim of the proposition for n = 3.
Denote by (F 3 ) red ∈ P(S r V 3 ) the homogeneous equation of the reduced algebraic set (F 3 = 0) ⊂ P(V * 3 ) P 2 . Observe that (F 3 ) red essentially depends on three variables, just as F 3 does. In particular, r = deg (F 3 ) red 2, and if r = 2, then (F 3 ) red is a nondegenerate (irreducible) quadric Q 3 , and hence F = Q k and the claim of the proposition is proved. From now on, we assume r 3.
Consider the general line section of the plane curve (F 3 = 0). The degree r = deg (F 3 ) red is the number of distinct points of support of this line section. By Lemma 4.6 again and also dim(GL(V 2 )·[F 2 ]) = 3 since F 2 has r (at least three) distinct roots. Therefore dim GL(V 3 )·[F 3 ] 6 and dim GL(V n )·[F n ] n+1 2 as claimed.

Conciseness of forms of high rank
We will use the following lemma in the next section.

Lemma 4.8 Suppose F ∈ P(S d V ) is a form of maximal rank and d 2. Then F is concise. In particular, a general point in each component of W m is a concise form.
Proof We can generalize the above lemma to show that all forms of greater than generic rank are necessarily concise under certain conditions. For this we use the following simplified bound for the maximal rank.  Proof If n = 2 and F is not concise then rank(F) = 1. If d = 2 then there are no forms with greater than the generic rank, so there is nothing to prove. When n = d = 3 the unique (up to coordinate change) form of greater than generic rank is F = x 2 y + y 2 z, which is concise (see for example [23,Section 8]). Now assume d n + 1. Since 2 (d + n − 1)/n we have hence m n−1,d g n,d . It follows that if F is not concise, then rank(F) m n−1,d g n,d .

Dimensions of maximal rank loci for Veronese varieties
We can now prove Theorem 4.1.
Proof of Theorem 4.1 Pick an irreducible component W ⊆ W m and let F ∈ W be a general form from that component. Then F is concise by Lemma 4.8. The closure GL(V )· F of the orbit of F is contained in W . In particular, by Proposition 4.7,

Tensors of format 2 ×4×4
When X = Seg(P n 1 −1 × · · · ×P n k −1 ) ⊂ P N is a Segre variety, N = n 1 · · · n k − 1, then P N is the projective space of tensors of format n 1 × · · · ×n k and X corresponds to the simple tensors. Rank with respect to X is the usual tensor rank. Tensors of format 2×b×c may be regarded as pencils of matrices, which admit a normal form due to Kronecker. Using this normal form we characterize the loci of 2×4×4 tensors of higher than generic rank. Tensors of format 2×4×4 have generic rank 4 and maximal rank 6; we show that W 6 + X = W 5 and W 5 + X = W 4 , where W 4 = P 31 is the space of all 2×4×4 tensors.

Concise tensors
Fix T ∈ V 1 ⊗ · · · ⊗ V k and for each i let V i ⊆ V i be the image of the induced map V * 1 ⊗ · · · ⊗ V * i ⊗ · · · ⊗ V * k → V i . It is easy to see that rank(T ) dim V i for each i, and also that T ∈ V 1 ⊗ · · · ⊗ V k . In particular, if rank(T ) < max{dim V 1 , . . . , dim V k }, then T is not concise.
Non-conciseness is a closed condition (because it is defined by vanishing of minors of certain matrices; see for example [22,Section 3.4.1]). Hence the locus of non-concise tensors contains the secant variety σ r (X ) for each r < max{dim V 1 , . . . , dim V k }, where X is the Segre variety.
Remark 5.1 Observe that non-square matrices are always non-concise, and more generally if n i > j =i n j for some i, then every tensor of format n 1 × · · · ×n k is non-concise.

Normal form
Let {s, t} be a basis for k 2 and let T ∈ k 2 ⊗k b ⊗k c be a tensor. Identifying k b ⊗k c with the space of b ×c matrices, we can write T = s ⊗ M 1 + t ⊗ M 2 for some b ×c matrices M 1 , M 2 . The tensor T corresponds to the pencil spanned by M 1 and M 2 in P(k b ⊗k c ); changes of basis in the pencil correspond to changes of basis in k 2 . It is convenient to write T as the b ×c matrix s M 1 + t M 2 whose entries are homogeneous linear forms in s and t.
There is a normal form due to Kronecker for tensors T ∈ k 2 ⊗k b ⊗k c , i.e. a representative of the GL(k 2 )×GL(k b )×GL(k c )-orbit of T , or in other words, a convenient choice of basis that makes T particularly "simple". Further, the results of Grigoriev, Ja'Ja' and Teichert calculate the rank of each tensor in normal form, see [9,Section 5]. For simplicity of some calculations, we restrict our considerations to the case of even square matrices. Later we restrict further to the case of 4×4 matrices.
Kronecker's normal form is as follows. Suppose V n = k 2 ⊗k 2n ⊗k 2n and X = Seg(P 1 ×P 2n−1 ×P 2n−1 ) ⊂ PV n is a Segre variety. We encode the tensors in V n as 2n ×2n matrices with entries linear forms in two variables s and t. For a positive integer let L denote the ×( + 1) matrix Let F be an f × f matrix with coefficients in k in Jordan normal form. For λ ∈ k, denote by d λ (F) the number of Jordan blocks of size at least 2 with the eigenvalue λ, and by m(F) the maximum among d λ (F). Given a sequence of matrices M 1 , . . . , M k depending on variables s and t, denote by M 1 ⊕ · · · ⊕ M k the block matrix In the above notation we allow M i to be a zero matrix Z p×q of size p ×q, where p, q 0 are nonnegative integers. Thus, for example, if M 2 is a 0×5 matrix, then M 1 ⊕ M 2 is the matrix M 1 with five columns of zeroes added.

Theorem 5.2 ([9, Proposition 5.1 and Theorem 5.3])
For any tensor T ∈ V n = k 2 ⊗k 2n ⊗k 2n there exists a choice of basis of k 2 , k 2n , and k 2n such that T is represented by a matrix where k, l, f, p, and q are nonnegative integers (possibly zero); each i and η j is a positive integer; Id f is the f × f identity matrix over k; F is an f × f matrix in its Jordan normal form; and Z p×q is the p ×q zero matrix.
Moreover, the rank of T is equal to the sum of the ranks of the blocks in this normal form, where rank(L i ) = i + 1, rank(L η j ) = η j + 1, rank(s Id f +tF) = f + m(F), and rank(Z p×q ) = 0. That is, See also references discussed in [9,Remark 5.4].
It is straightforward to see that one can always further change the coordinates so that one of the eigenvalues of F is 0.
We stress that if T = M 1 ⊕ M 2 , then the rank of T is not necessarily equal to rank(M 1 ) + rank(M 2 ). For example, let M 1 = s t 0 s and M 2 = s+t t 0 s+t . Then, by Theorem 5.2 the rank of T = M 1 ⊕ M 2 is 5, while rank(M 1 ) = rank(M 2 ) = 3.

Generic and maximal rank
For tensors in V n = k 2 ⊗k 2n ⊗k 2n , the generic rank is g = 2n, and general tensors have the normal form (s Id 2n +tF), where F is a diagonal matrix with distinct (generic) eigenvalues. This is because a general pencil contains an invertible matrix, and the blocks L i or L η j have no invertible matrices.
Furthermore, the maximal rank is m = 3n, and any tensor T of maximal rank is of the form (s Id 2n +tF), where F has a unique eigenvalue (which we can assume to be 0) and n Jordan blocks of size 2×2. That is, after reordering of rows and columns, we can write T as s Id n t Id n 0 s Id n . Thus W m = W 3n = G ·[T ], where G = GL(k 2 )×GL(k 2n )×GL(k 2n ) is the automorphism group of X ⊂ PV n , and T = s Id n t Id n 0 s Id n . In particular, W m is irreducible.

Remark 5.3
Note that T = s Id n t Id n 0 s Id n is concise, so Hence W m ⊂ σ g−1 (X ).
We compute the dimension of W m = W 3n . The main technique is to reduce to a system of linear equations via the Lie algebra stabilizer. We illustrate this in some detail in this case, as we will use the same method (with fewer details given) to compute dimensions of other orbits of 2×4×4 tensors in the next section.
Proof Let ρ denote the action of G on V n = k 2 ⊗k 2n ⊗k 2n . The dimension of the orbit G · T is equal to the codimension in G of the stabilizer subgroup of T [26, Section 3.7]. We compute the dimension of the stabilizer subgroup by finding the dimension of its tangent space at the identity e ∈ G. Recall that in the representation dρ of the Lie algebra T e (G) ∼ = End(k 2 )×End(k 2n )×End(k 2n ) on V n , a tangent vector (g 1 , This must vanish identically, which yields the equations B 11 = a I n + A 11 , A 21 = 0, Note that A 11 , A 12 , a, b, d are free, so the stabilizer has dimension 2n 2 + 3. Since dim G = 8n 2 + 4, the affine orbit G · T ⊂ V n has dimension 6n 2 + 1. The projective orbit G ·[T ] ⊂ PV n has dimension one less, since G contains subgroups isomorphic to G m = k * that act on V n as rescaling. So dim W m = dim G ·[T ] = 6n 2 , as claimed. Remark 5.5 This shows that for X as above, some of the intermediate joins W 3n + k X for k ∈ {1, . . . , n − 1} must be highly defective. Indeed, the expected dimension of W 3n + n/2 X is already the dimension of the ambient P 8n 2 −1 , while we know that even W 3n + (n − 1) X does not fill P 8n 2 −1 .

Lemma 5.6
The orbit structure of the action of G on P(V 2 ) is as follows.
• There is no open orbit.
• The only orbits of codimension 1 are the orbits of (classes of) tensors (in their Kronecker normal forms): for pairwise distinct eigenvalues λ i . Two tensors of the form T 4 (λ 1 , λ 2 , λ 3 , λ 4 ) are in the same orbit if and only if the cross-ratios of their eigenvalues λ 1 −λ 2 are equal (after possibly permuting the order of λ i ). Any two tensors of the form T 5 (λ 1 , λ 2 , λ 3 ) are in the same orbit. • There are finitely many orbits of codimension at least 2.
Proof The set of projective classes of nonconcise tensors (i.e. those contained in some P(k 1 ⊗k 4 ⊗k 4 ) or P(k 2 ⊗k 3 ⊗k 4 ) or P(k 2 ⊗k 4 ⊗k 3 )) is G-invariant, of dimension 27 (hence codimension 4), and has only finitely many orbits [9, Section 6]. Thus it is enough to prove the lemma for concise tensors.
To see when tensors of the form T 4 = T 4 (λ 1 , λ 2 , λ 3 , λ 4 ) are in the same orbit, note det T 4 = (s +λ 1 t) · · · (s +λ 4 t) determines four points [−λ i , 1] on P 1 parametrised by s, t. In particular, tensors with different cross-ratios of eigenvalues (up to permutation) cannot be in the same orbit. On the other hand, if there are two sets of eigenvalues with the same cross-ratio, then we can change the coordinates (s, t) on k 2 , and then also rescale columns to get from one tensor to the other. Let G 0 [T 4 ] ⊂ G be the identity component of the stabilizer of [T 4 ] ∈ P(V 2 ). Suppose (g 1 , g 2 , g 3 ) ∈ G 0 [T 4 ] , where g 1 ∈ GL 2 , g 2 ∈ GL 4 and g 3 ∈ GL 4 . The action of g 1 on P 1 must preserve the four points (zeroes of determinant). Thus g 1 = μ 1 Id 2 is a rescaling of the identity. Restricting to the s coordinate, we see that the product g 2 g 3 = μ 2 Id 4 is also a rescaling of the identity, that is g 3 = μ 2 g −1 2 . Hence restricting to the t coordinate, g 2 commutes with a diagonal matrix with pairwise distinct entries. Then it is straightforward to see that g 2 is an invertible diagonal matrix, and any invertible diagonal matrix can occur as g 2 . Thus dim G 0 [T 4 ] = 6 and the dimension of the orbit of [T 4 ] is 30 = 36 − 6, as claimed.
In particular, since a general tensor is of the form T 4 , it lies in an orbit of codimension 1. So there is no open orbit.
It remains to check that there are finitely many other concise orbits and that all these other orbits have codimension at least 2, i.e. dimension at most 29.
For the first part we use the normal form described in Theorem 5.2 and rescaling to fix the eigenvalues. It is straightforward to see that there are 14 concise orbits other than the T 4 and T 5 cases. The second part is an explicit computer calculation of the dimension of the Lie algebra stabilizer for each of the cases above, as in the proof of Proposition 5.4. Representatives for the 14 orbits are listed, along with the dimensions of the orbits and their ranks, in Table 1.
Consider the determinant of s M 1 + t M 2 as a homogeneous polynomial of degree 4 in the variables s, t, whose coefficients are degree 4 homogeneous polynomials a i in 32 variables, the coordinates of V 2 : Every class of a non-concise tensor lies in the support of (Discr), as does every class of a tensor of rank 5 or 6.
Proof The first characterization of the divisor is clear from the definition and properties of the discriminant, while the G-invariance follows from this first characterization. Similarly, T 5 is in the support of (Discr), and so is the closure of its orbit. By Lemma 5.6 the orbit of T 5 is the only orbit of codimension 1 which is contained in the support.
Since there are only finitely many orbits of codimension at least 2, the only irreducible G-invariant divisors are the closures of 30-dimensional orbits. Hence the support of (Discr) is irreducible and equal to G ·[T 5 ].
If T = s M 1 + t M 2 is non-concise, then either the normal form for T involves a block of zeros, so det(s M 1 + t M 2 ) = 0, or else the matrices M 1 and M 2 are linearly dependent, so the determinant has a single root of multiplicity 4. Hence all classes of non-concise tensors lie in the support of (Discr). Table 1 lists the tensors of rank 5 or 6 other than T 5 . The determinant of each tensor listed in the table is zero, or has a multiple root. So the classes of these tensors also lie in (Discr).

High rank loci of 2 × 4 × 4 tensors
In this case the generic rank is g = 4 and the maximal rank is m = 6. Hence the equality W 5 + X = W 4 = PV 2 follows from Corollary 2.6. Let T 6 be the (unique up to a choice of coordinates) tensor of rank 6, and let T 1 be a general tensor of rank 1. Then T 6 + T 1 has rank 5 by Theorem 3.1. A computer calculation shows that determinant of T 6 + T 1 is divisible by s 2 and has two other distinct roots not equal to s. Thus T 6 + T 1 must be of the form T 5 . That is, a general element of the (irreducible) join W 6 + X (where X = P 1 ×P 3 ×P 3 ) is a general element of the irreducible variety W 5 , thus W 6 + X = W 5 as claimed.

Remark 5.9
The proofs above show that for X = P 1 ×P 3 ×P 3 ⊂ P 31 (so that dim X = 7) we have dim W 6 = 24 and dim W 5 = dim(W 6 + X ) = 30. That is, the join W 6 + X is defective (it is expected to fill the ambient space excessively, but it does not).

Curves in quadric surfaces
We study rank with respect to a curve C contained in a smooth quadric surface Q ∼ = P 1 ×P 1 in P 3 . By nondefectivity, the generic rank with respect to C is 2, and by Theorem 2.1 or Theorem 3.7 the maximal rank is at most 3.
If C has bidegree (2, 2) then C is an elliptic normal quartic curve. Bernardi, Gimigliano, and Idà gave a description of W 3 in this case, and more generally studied elliptic normal curves of degree d + 1 in P d , d 3 [5, Theorem 28]. We refine their result in the d = 3 case and show that the maximal rank locus W 3 is a curve of degree 8 disjoint from C.
Piene has shown that if C is a general curve of bidegree (3, 3) then W 3 is empty (the maximal rank is 2), see [25,Theorem 2]. We extend this to general curves of bidegree (a, b), where a 4 and b 1.

Elliptic quartic curve
Let us study the locus W 3 with respect to an elliptic normal curve of degree 4 in P 3 . Proposition 6.1 Let C = Q 1 ∩ Q 2 be a smooth complete intersection of two smooth quadrics in P 3 . The generic rank with respect to C is 2 and the maximal rank is 3. W 3 is a curve of degree 8, disjoint from C and containing the vertices of the four singular quadrics that contain C. Every point of W 3 has rank 3, except those four points, which have rank 2.
Proof Note that C has no trisecant, bitangent, or flex lines, since any such line would have to be contained in every quadric surface that contains C. The quadrics containing C form a pencil with four singular members, which are distinct and irreducible. Let the vertices of those cones be V = {x 1 , . . . , x 4 }. Each vertex x i lies off of C, and each x i has rank 2.
Let x ∈ P 3 \(C ∪ V ) and let π : P 3 P 2 be the projection from x. Then π(C) is an elliptic quartic curve, hence has two singularities, counting with multiplicity. This shows that through each point of P 3 \(C ∪ V ) there are two secant or tangent lines to C. A priori this is counting with multiplicity, but since C has no trisecant or bitangent lines, the two secant or tangent lines are distinct. The point x has rank 3 if and only if no proper secant to C passes through x. Thus the points of rank 3 are exactly those in the intersection of two tangent lines to C, other than the points in V (this is one of the results in [5,Theorem 28]).
Let W • 3 denote the set of points of rank 3. Every point in W 3 = W • 3 lies on at least two tangent lines of C, by semicontinuity of the degree of the projection map from the abstract tangent variety {(x, ) : x ∈ , tangent to C }. But no point of C lies on more than one tangent line. This shows that W 3 is disjoint from C.
Let Q be a smooth quadric containing C and let pr i : Q → P 1 , i = 1, 2, be the two natural projections. Then pr i|C : C → P 1 is a 2:1 morphism with four ramification points. This shows that there are four tangent lines to C in each ruling. The tangent lines to C in the rulings of Q intersect in 16 points which do not lie on C, as each tangent line intersects C only at its point of tangency. Therefore the 16 points of intersection are in W • 3 ∩ Q. On the other hand, if w ∈ W • 3 ∩ Q then the tangent lines passing through w are contained in Q. Hence any such quadric intersects W • 3 in exactly 16 points. This shows that the closure W • 3 is a curve of degree 8. Finally, let Q be a singular quadric containing C. Then it is immediate to realize that the vertex of Q is the only point of Q that lies on more than one tangent line of C. So W 3 contains each vertex x 1 , . . . , x 4 of a singular quadric through C. These are the only points of rank 2 in W 3 .

Remark 6.2
The above proof also recovers the (previously known) fact that general points in P 3 , namely those outside of the tangential variety of C, admit precisely two decompositions as linear combinations of two points in C. This holds more generally for elliptic normal curves of even degree, see [14,Proposition 5.2].

Remark 6.3
The example of the elliptic quartic curve shows that W m can be disjoint from the base variety. Thus the situation as in the proof of Theorem 3.9, when W m ⊃ X , is rather special to the homogeneous spaces.

General curves in a quadric surface
Proposition 6.4 Let a b 1 and let C ⊂ Q be a general curve of type (a, b) in the smooth quadric surface Q. If a 4 then W 3 is empty, that is, the maximal rank m is equal to 2.
Proof First let x ∈ Q, x / ∈ C. Let l be the line in Q through x such that l ·C = a. By generality l ∩ C has points of multiplicity at most 2, so a 3 is enough to imply that l ·C is supported in at least two distinct points. Hence rank(x) = 2.
Next let x / ∈ Q, and suppose rank(x) = 3. Let π : P 3 P 2 be the projection from x. Since x lies on no secant line to C, and not every tangent line to C passes through x, π has degree 1 on C. Then π(C) is a plane curve of degree a + b, and every point of π(C) has multiplicity at most 2, since each line through x intersects Q with multiplicity 2. The projection π(C) has no nodes, only cuspidal singularities. Let H x be the polar hyperplane of Q in x, so y ∈ H x ∩ Q if and only if the tangent plane to Q at y contains x, see for example [20, p. 238], [16, Section 1.1.2]. Let Z x = H x ∩ C.