Vanishing Hessian, wild forms and their border VSP

We show that forms with vanishing Hessian and of minimal border rank are wild, i.e. their smoothable rank is strictly larger than their border rank. This discrepancy is caused by the difference between the limit of spans of a family of zero-dimensional schemes and the span of their flat limit. We exhibit an infinite series of wild forms of every degree $d\geq 3$ as well as an infinite series of wild cubics. Inspired by recent work on border apolarity of Buczy\'nska and Buczy\'nski, we study the border varieties of sums of powers $\underline{\mathrm{VSP}}$ of these forms in the corresponding multigraded Hilbert scheme.


Introduction
Notions of ranks abound in the literature, perhaps because of their natural appearance in the realms of algebra and geometry, and in numerous applications of theirs; see [9,10] and references therein for an introduction to the subject.
These ranks vastly generalize matrix rank and yet they are very classical, dating back to the pioneering work of Sylvester. His work featured Waring ranks of binary forms; see [9] for a historical account on the subject. Since then, ever growing research efforts have been devoted to understanding ranks with respect to some special projective varieties X of interest. Last decades have witnessed steady progress on tensor and Waring ranks, i.e. the cases when the projective varieties are the classical Segre and Veronese varieties.
These results have been developed in parallel in their geometric and algebraic aspects. The first are naturally related to secant varieties of X [15, Chapter 1], whereas the second to the theory of apolarity and inverse systems established by Macaulay [9, §1.1].
Interestingly, scheme-theoretic versions of X-ranks have been introduced and studied as well. These latter ones take into account more general zero-dimensional schemes, besides the reduced zero-dimensional ones featured in the X-ranks. This more general framework naturally leads to new notions of X-rank: the smoothable X-rank, and the cactus X-rank; the latter was originally called scheme length [9, Definition 5.1]. We recall their definitions in §2.
One subtle phenomenon is that, for special points, perhaps unexpectedly smoothable ranks may be larger than border ranks. This discrepancy is caused by the difference between the limit of spans of a family of zero-dimensional schemes and the span of their flat limit. The difference between smoothable and border ranks does not appear for general points (forms) of fixed border rank. Therefore, it is a natural and interesting problem to investigate the structure of the instances where these two differ.
As far as we know, Buczyńska and Buczyński [3] were the first authors to bring the difference between these ranks to attention. They introduced the notion of wild forms, i.e. those whose smoothable rank is strictly larger than their border rank. They gave one such a form [3, §4], up to concise minimal border rank direct summands.
On another direction, in recent groundbreaking work, Buczyńska and Buczyński [2] expanded the apolarity theory of X-ranks to border apolarity, which is devised to provide information about border X-ranks. Along the way, they introduced the border varieties of sums of powers VSP, mirroring the classical varieties of sums of powers [13,14].
Inspired by [2, §5.3], we establish a new result on wild forms. To state it, let V be a complex finite-dimensional vector space; given a form F ∈ S d V * , let Hess(F ) denote the determinant of its Hessian matrix. Forms with identically vanishing Hessian have many remarkable geometric and algebraic properties; see [15,Chapter 7] for a detailed and updated exposition about them. Our Theorem 3.5 connects wild forms and vanishing Hessian, following the lines paved by Ottaviani's remark [2, Remark 5.1]: Theorem. Let d ≥ 3 and F ∈ S d V * be a concise form of minimal border rank. Then: For d ≤ 4, the condition Hess(F ) = 0 is equivalent to the Artinian Gorenstein C-algebra T /Ann(F ) having the Strong Lefschetz Property (SLP) [15,Corollary 7.2.21]. Then the result above reads: Theorem. Let 3 ≤ d ≤ 4 and let F ∈ S d V * be a concise form of minimal border rank. Then: T /Ann(F ) does not have SLP =⇒ F is wild.
As an application, we exhibit two infinite series of wild forms. In §5, we give a series of wild forms G d of every degree d ≥ 3, and in §6 a series of wild cubics F n . In particular, this shows the next Theorem. For every d ≥ 3, there exist wild forms of degree d.
Employing Buczyńska-Buczyński's border apolarity theory, we offer a study of border varieties of sums of powers VSP's of these forms in the corresponding multigraded Hilbert schemes. To our knowledge, this is the first attempt to describe such varieties.
For the first series G d , we show that they are projective spaces; see Theorem 5.6. For the second series F n and n ≥ 10, in Theorem 6.9 we prove that they are reducible. This is achieved by relying on the (usual) Hilbert schemes of zero-dimensional schemes on a chain of lines, see §6. We point out that this result on reducibility is also motivated by the fact that establishing this property is usually a delicate and interesting issue even in the context of the classical varieties of sums of powers VSP's.

Structure of the paper.
In §2, we introduce notation and recall the definitions of ranks we need throughout the article. §3 is devoted to the proof of our main result, Theorem 3.5.
In §4, we give the definition of a limiting scheme of a border rank decomposition and its relations with VSP. Theorem 4.3 recalls that the saturation of an ideal in VSP is the ideal of a limiting scheme of a border rank decomposition. Theorem 4.4 shows the correspondence between ideals and border decompositions, in the regime of minimal border rank.
In §5, we introduce the infinite series of degree d forms G d . We show that they are wild in Corollary 5.2. Moreover, we prove that their VSP's are isomorphic to projective spaces; this is achieved in Theorem 5.6.
In §6, we introduce the infinite series of cubics F n . Corollary 6.7 states that they are wild. We show that when n ≥ 10, their VSP's are reducible; see Theorem 6.9.

Preliminaries
Here we introduce notation and definitions we use throughout the paper. We work over the complex numbers. Let V ∼ = C n+1 and V * = x 0 , . . . , x n . Let P n = P(V ) denote the projectivization of V . Let S = S • V * ∼ = C[x 0 , . . . , x n ] be its homogeneous coordinate ring, and T = C[y 0 , . . . , y n ] be its dual ring, i.e. T acts by differentiation on For a homogeneous ideal J ⊂ T , let J d denote its degree d homogeneous component. For J ⊂ T , let J sat denote its saturation. The Hilbert function of J is the numerical function Given a form F ∈ S d V * , Ann(F ) ⊂ T denotes its annihilator or apolar ideal The algebra T /Ann(F ) is a graded Artinian Gorenstein C-algebra; see [9, §2.3] or [15,Theorem 7.2.15]. Let N d = n+d d − 1 and X = ν d (P n ) ⊂ P N d be the d-th Veronese embedding of P n . We only consider ranks with respect to the Veronese variety, although the ensuing definitions may be more generally introduced for any projective variety.
Let R be a zero-dimensional scheme over C; then R = Spec(A) for some finite- For a zero-dimensional scheme R ⊂ P n = P(V ), let R denote its span, i.e.
where I is the saturated ideal defining R in P n .
Definition 2.1 (Border rank). For a point F ∈ P N d , the border rank of F is the minimal integer r such that F ∈ σ r (X), the r-th secant variety of X. The border rank of F is denoted r(F ).
Definition 2.2. A form F ∈ P N d is said to be of minimal border rank when In general, given F ∈ P N d , it might be challenging to produce a border rank decomposition F = lim t→0 To determine border ranks for our infinite series of forms G d and F n , we employ a useful criterion: . Let X be as above. Suppose there exist points z 1 , . . . , z r ∈ X such that dim z 1 , . . . , z r < r − 1. Then the span of the affine cones of Zariski tangent spaces at these points is contained in the r-th secant variety σ r (X), i.e. P T z 1 , . . . , P T zr ⊂ σ r (X). We now recall the scheme-theoretic ranks attached to X.
The smoothable rank of F ∈ P N d is the minimal integer r such that there exists a finite scheme R ⊂ X of length r which is smoothable (in X) and F ∈ R . The smoothable rank of F is denoted sr(F ).
where R(t) is a family of zero-dimensional schemes over the base Spec(C[t ±1 ]) and lim t→0 R(t) denotes its flat limit; see for instance [4,II.3.4].
Definition 2.6 (Cactus rank). The cactus rank of F ∈ P N d is the minimal integer r such that there exists a finite scheme R ⊂ X of length r such that F ∈ R . The cactus rank of F is denoted cr(F ).
The cactus rank was originally called scheme length [9, Definition 5.1].
Definition 2.8. A form F ∈ S d V * is a form with vanishing Hessian if the determinant of its Hessian matrix Hess(F ) = det See [15,Chapter 7] and references therein for a complete introduction to several remarkable algebraic and geometric properties that forms with vanishing Hessian possess.
For any form F ∈ S d V * , Buczyńska and Buczyński [2, §4.1] introduced the border variety of sums of powers VSP(F, r(F )). These varieties live in multigraded Hilbert schemes, which were introduced by Haiman and Sturmfels [7]. We now recall the definition of an irreducible component of the multigraded Hilbert scheme we are concerned with; see [2, §3] for a detailed discussion. Definition 2.9. An ideal J ⊂ T , whose Hilbert polynomial is equal to r ∈ N, is said to have a generic Hilbert function if its Hilbert function satisfies Let Slip r,P n be the irreducible component of the multigraded Hilbert scheme Hilb hr,P n T containing the radical ideals of r distinct points with a generic Hilbert function. Therefore, every ideal in Slip r,P n has a generic Hilbert function being a flat limit of such ideals.
The border variety of sums of powers, or border VSP, of F is VSP(F, r) = J ∈ Slip r,P n | J ⊂ Ann(F ) ⊂ T . One case of interest is when r = r(F ).

Wildness
Remark 3.3. A consequence of wildness of a form F is that all the ideals in VSP(F, r(F )) are not saturated.
Before proving the next result, we introduce another piece of notation. Let W ⊂ U be finite-dimensional vector spaces. Then Proposition 3.4. Let R ⊂ P n be a projective scheme defined by the saturated ideal J such that R = P n . Then, for any d ≥ 1, the affine variety C[(J d ) ⊥ ] has dimension n + 1, i.e. the linear space (J d ) ⊥ is spanned by at least n + 1 algebraically independent forms.
Proof. Choose a linear form z such that J : z = J , i.e. J is saturated with respect to z, or equivalently, there is no associated prime containing z. Such a linear form exists by assumption.
Let S = C[z, x 1 , . . . , x n ] be the homogeneous coordinate ring of the ambient projective space. Up to change of basis, we can present the homogeneous saturated ideal of R as follows: where the i are (possibly zero) linear forms and the h j are forms of degree d j that are quadratic in the variables which is in contradiction with our assumption.
The condition V ∩ W = 0 implies that we have a surjection . . , z d−1 x n (by abuse of notation, the duals of z d−1 x i are denoted in the same way) to an independent set in W ⊥ . Therefore: To show that the forms on the right-hand side are algebraically independent, we dehomogenize them and look at the affine map they induce: Notice that the differential of ϕ at the origin 0 ∈ C n is the identity. Thus there is a local isomorphism between tangent spaces and so the dimension of the image is n. Therefore the dimension of the affine variety C[(J d ) ⊥ ] is n + 1. This is equivalent to the fact that the linear space (J d ) ⊥ is spanned by at least n + 1 algebraically independent degree d forms.
Theorem 3.5. Let d ≥ 3 and F ∈ S d V * be a concise form of minimal border rank. Then: be the linear space spanned by the first derivatives of F . Let I = Ann(F ) d−1 be the homogeneous ideal generated by the degree (d − 1) homogeneous piece of the annihilator of F .
Let R W be the projective scheme in P n defined by I sat , which is a priori possibly empty. As I sat d−1 ⊇ I d−1 , we have: Assume the saturated ideal I sat does not contain any linear form. By definition, this is equivalent to the subscheme R W ⊂ P n spanning the whole P n . By Proposition 3.4, (I sat ) d−1 ⊥ is spanned by at least n + 1 algebraically independent forms. However, W has dimension exactly n + 1, so W = (I sat ) d−1 ⊥ and a basis of W consists of n + 1 algebraically independent forms. Therefore, the derivatives of F must be algebraically independent and so Hess(F ) = 0; see [15, §7.2]. This shows that whenever F is concise and has vanishing Hessian, I sat must contain a linear form. Suppose F is concise with Hess(F ) = 0 and of minimal border rank. Then the Hilbert function of Ann(F ) is as follows: HF(T /Ann(F )) : 1 (n + 1) . . . (n + 1) 1 Now, we show by contradiction the first implication in the statement. Suppose the cactus rank of F satisfies cr(F ) ≤ n + 1.
Let J ⊂ Ann(F ) be any saturated ideal evincing the cactus rank of F , i.e. the zerodimensional scheme defined by J has degree cr(F ). Since its Hilbert function HF(T /J ) is non-decreasing until it stabilizes to the constant polynomial cr(F ) ∈ N [9, Theorem On the other hand, J d−1 ⊂ I and so The inequalities imply J d−1 = I d−1 . Now, I sat ⊂ J sat = J . Hence J contains a linear form, i.e.
On the other hand, since J ⊂ Ann(F ), one has: which is a contradiction. Therefore sr(F ) ≥ cr(F ) > n + 1 = r(F ). Hence F is wild.
Remark 3.6. It is clear that for d = 2 and any n ≥ 1, the condition Hess(F ) = 0 is equivalent to F being not concise. Also, for d = 3 and n ≤ 3, Hess(F ) = 0 is equivalent to F being not concise [15,Theorem 7.1.4]. A complete classification is known up to n ≤ 6; see [15, §7.6] for a detailed discussion.
Proposition 3.7. Keep the notation from Theorem 3.5. Let F be a concise and minimal border rank cubic. If F is not a wild cubic and R W is reduced, then F is a Fermat cubic (up to scaling variables) F = x 3 0 + · · · + x 3 n , and VSP(F, n + 1) is a single point. Proof. Since F is concise, of minimal border rank and it is not wild, the proof of Theorem 3.5 yields that I sat does not contain any linear form. This is equivalent to Since R W is a reduced scheme by assumption, we may find a reduced zero-dimensional subscheme Z ⊂ R W of length n + 1 such that Z = P n . Up to change of basis, we have: Since dim W = n + 1, we have W = x 2 0 , . . . , x 2 n and Thus Ann(F ) 3 = x 3 0 , . . . , x 3 n ⊥ . Then, since every cubic monomial divisible by two distinct variables sits in Ann(F ) 3 , F is a Fermat cubic up to the action of a diagonal matrix. Now, the Hilbert function of I is: HF(T /I) : 1 (n + 1) (n + 1) (n + 1) · · · For k = 0, 1, 2, the dimension of I k is clear from definitions. To see the dimension of I k for k ≥ 3, note that y k 0 , . . . , y k n / ∈ I k and they are the only missing monomials. Let J ∈ VSP(F, n + 1). Then its Hilbert function is HF(T /J ) : 1 (n + 1) (n + 1) (n + 1) · · · , because J ∈ Slip n+1,P n . Since J ⊂ Ann(F ), we have the equality I = J .
Repeating part of the proof above, one shows: Proposition 3.8. Keep the notation from Theorem 3.5. Let F be a concise and minimal border rank cubic. If F is not a wild cubic and I = Ann(F ) 2 is saturated of degree n+1, then VSP(F, n + 1) = {I}. Example 3.9. Let F = x 2 0 x 1 ∈ S 3 C 2 * . In this case, I = Ann(F ) 2 = y 2 1 ⊂ T is the ideal of a 2-jet on a P 1 . Proposition 3.8 gives VSP(F, 2) = {I}. a cuspidal cubic). In both cases, I = Ann(F ) 2 is saturated of degree 3. For F tg , the scheme defined by I is the 2-fat point (of degree 3) in P 2 ; for F cusp , the scheme defined by I is the union of a simple point and a 2-jet.

The limiting scheme
We start with the definition of limiting scheme: Suppose we are given a border rank decomposition for F , i.e. (1) are linear forms. The reduced zero-dimensional scheme whose (closed) points are the L j (t) is denoted R(t). So R(t) is a family over the base Spec(C[t ±1 ]). For each t = 0, the radical ideal defining R(t) is denoted I R(t) . The flat limit Z = lim t→0 R(t) is called the limiting scheme of (1). Note that Z and R(t) (t = 0) have the same Hilbert polynomial; see e.g. [8, Theorem III.9.9].
We have the following corollary from the proof of [3, Proposition 2.6]: Corollary 4.2. Keep the assumptions from Proposition 2.3. Assume F ∈ P T z 1 , . . . , P T zr ⊂ σ r (X).
Then we can find a border rank decomposition for F whose limiting scheme is the smooth scheme supported at the r points {z 1 , . . . , z r }.
We can find curvesẑ 1 (t), . . . ,ẑ r (t) in the affine cone X over X such that z i (0) =ẑ i and dẑ i dt (0) = v z i . Then we have the following border rank decomposition for F : Since {z 1 , . . . , z r } are r distinct points on X, the limiting scheme corresponding to the border rank decomposition above is the smooth scheme supported at the r points {z 1 , . . . , z r }.
The next result is a consequence of Buczyńska-Buczyński's theory: Proof. Any ideal J ∈ VSP(F, r(F )) comes from some border rank decomposition (1); see the proof of [2, Theorem 3.15]. Such a border rank decomposition (1) determines a family of zero-dimensional schemes R(t), each of length r(F ), such that their ideals I R(t) have the generic Hilbert function and J = lim t→0 I R(t) .
By definition of limits, we have Let Z = lim t→0 R(t) be the limiting scheme of the given border rank decomposition (1). Since Z has length r(F ), its ideal has the same degree (or Hilbert polynomial) as J . Since J ⊂ I Z , their saturations coincide. Since I Z is saturated by definition, J sat = I Z .
Theorem 4.4. Suppose F ∈ S d V * is concise and of minimal border rank n + 1. Then every border rank decomposition of F determines an ideal in VSP(F, n + 1).
Proof. Given any border rank decomposition F = lim t→0 H(t), where each H(t) = 1 t s L 1 (t) d + . . . + L n+1 (t) d has rank n + 1, one has that I R(t) ⊂ Ann(H(t)) (t = 0) by the classical Apolarity lemma [9, Lemma 1.15]. Since F is concise, we can find t = 0 such that H(t) is concise. In this case, R(t) consists of n + 1 linearly independent points. Therefore I R(t) has the generic Hilbert function of n + 1 points in P n . Hence J := lim t→0 I R(t) ∈ VSP(F, n + 1).

Wild forms of higher degree and their VSP
Let d ≥ 3 and define the following infinite series of forms of degree d: For d = 3, this coincides up to change of variables with the wild cubic form found in [3, §4]. This infinite series generalizes it.
Note that, for every d ≥ 3, the partial derivatives of G d with respect to the variables v i are algebraically dependent: their relations coincides with the equations of the usual Veronese embedding of degree d − 1 of P 1 in P d−1 . Thus Hess(G d ) = 0 for d ≥ 3. Let Ann(G d ) ⊂ T = C[x 0 , . . . , x d−1 , y 0 , y 1 ] be its annihilator, where x i is dual to v i and y j is dual to u j . Proof. Let d 0 , . . . , d d be d + 1 pairwise distinct linear forms in u 0 , u 1 . They may be viewed as d + 1 distinct points on the degree d rational normal curve ν d (P 1 ) ⊂ P d . They are linearly independent. (This is well-known and can be explicitly checked, for instance, by calculating the corresponding Wronskian matrix at the origin and show it is full rank.) Any other form d d+1 is linearly dependent to those above, because dim S d u 0 , u 1 = d + 1.
Up to change of bases one has denote the affine cone of the Zariski tangent space to ν d (P 1 ) at d i . Therefore Since Proof. By Lemma 5.1, G d has minimal border rank. Moreover, as noticed above, Hess(G d ) = 0. Theorem 3.5 shows that the degree d forms G d are wild. Proof. Let J ∈ VSP(G d , d + 2) and let I = Ann(G d ) ≤d−1 .
Then J ∈ Slip d+2,P d+1 and, by definition, the Hilbert function of J is the generic Hilbert function of d + 2 points in P d+1 . Since J ⊂ Ann(G d ), it follows that J ⊃ I, as they must coincide up to degree d − 1.
Now, consider the Hilbert function of I. The following relations hold in T /I: where c i is a non-zero constant for every i and Thus if J ∈ VSP(G d , d + 2) then J contains I and an ideal Q generated by some element q ∈ (T /I) d+2 . For the converse, let J Q = Ann(G d ) ≤d−1 + Q, for some Q = q such that q ∈ S d+2 y 0 , y 1 . Note that HF(T /J Q , d+2) = d+2. We can apply Gotzmann's Persistence Theorem [6,Theorem 3.8] to conclude that HF(T /J Q , j) = d + 2 for all j ≥ d + 2.
To show that all such J Q are in fact in VSP(G d , d + 2), consider Q = q where q is a form with d + 2 distinct roots. This gives us d + 2 distinct points z 1 , . . . , z d+2 ∈ P( y 0 , y 1 * ). By Lemma 5.1 and Corollary 4.2, we may find a border rank decomposition defining a family of zero-dimensional schemes R(t), whose limiting scheme is the smooth scheme Z supported at z 1 , . . . , z d+2 .
Let I Z be the radical ideal defining the limiting scheme Z. By Theorem 4.4, there is a corresponding ideal J ∈ VSP(G d , d + 2) such that J sat = I Z . Note that I Z = J sat Q . Since J ∈ VSP(G d , d + 2), by the first part of this proof, J = J Q , for some Q = q . Therefore one has J sat Q = I Z . Since J sat Q = J sat Q , we must have Q = Q . In conclusion, we derive J Q = J ∈ VSP(G d , d + 2). Now, consider the morphism This morphism is proper and hence closed. We have shown that the generic point of P((T /I) d+2 ) lies in the image, therefore ψ d is surjective.
Let S d+2 P 1 denote the (d + 2)-fold symmetric product of the projective line. Proposition 5.5 yields the following: Proof. The morphism ψ d in the proof of Proposition 5.5 is surjective. It is also injective because the point [q] uniquely determines the ideal I Q . Since P d+2 is smooth and so normal, the map ψ d is an isomorphism by a variant of Zariski's Main Theorem [11,Corollary 4.6]. The isomorphism follows from the description of the vector space (T /I) d+2 given in the proof of Proposition 5.5.
6. An infinite series of wild cubics and their VSP For every k ≥ 1 and n = 3k + 1, we introduce the following infinite series of cubic forms: Remark 6.1. This infinite series is inspired by the examples appeared in [5], which in turn are a generalization of the Perazzo cubic hypersurface {F 4 = 0} ⊂ P 4 . The latter is exactly the wild cubic found in [3, §4].
Let Ann(F n ) ⊂ T = C[y 0 , . . . , y n ] be the annihilator of F n with y i being dual to x i . Remark 6.2. Note that F 4 coincides with G 3 from §5 up to change of basis. So VSP(F 4 , 5) ∼ = P 5 .
The following combinatorial arrangement of lines is important for us to study the infinite series F n ; Proposition 6.4 provides the motivation for looking at it. Definition 6.3 (Chains of lines). A chain of lines is a collection of distinct lines C 1 , . . . , C m = P 1 such that (up to reindexing) for 1 ≤ i, j ≤ m: Proposition 6.4. The projective scheme whose ideal is (Ann(F n ) 2 ) sat is a chain of lines.
Proof. Let I = Ann(F n ) 2 and let I sat denote its saturation. Let n = 3k + 1 for k ≥ 1. We divide the proof according to the residue (mod 3) of each index 0 ≤ j ≤ n in y j .
• j ≡ 0 (mod 3). The monomial y j y i ∈ I for all i = j + 1. Moreover, y j y 3 j+1 ∈ I, therefore y j ∈ I sat .
• j ≡ 2 (mod 3). In this case, one has y j y i ∈ I for all i = j − 1, j + 2. Note that y j y 2 j−1 , y j y 2 j+2 ∈ I. Therefore y j ∈ I sat .
For i ≡ 1 (mod 3), y i / ∈ I sat . We show that y k i / ∈ I for any k ≥ 2. Assuming on the contrary that y k i ∈ I, one has y k i = s j=0 m j h j , where the h j are the generators of I and m j ∈ T . However, on the right-hand side, every monomial that is divisible by y i is divisible by some other distinct y j as well.
We show that y i y i+3 / ∈ I sat . On the contrary, suppose y i y i+3 ∈ I sat . Thus, there exists k > 1 such that (y i y i+3 ) k ∈ I. Using the same argument as above, we see that every monomial in I, that is divisible by y i y i+3 , must be divisible by some other distinct y j as well.
In conclusion, the saturated ideal I sat is generated by: The projective scheme defined by I sat is a chain of k lines C k , as in Definition 6.3. Its irreducible components are lines L i,j , such that |i − j| = 3 and i ≡ 1 (mod 3), whose ideal is defined by J i,j = y k | k = i, j . This concludes the proof.
Notation. In the following, let C k denote the chain of lines appearing in the proof of Proposition 6.4, corresponding to the cubic F n when n = 3k + 1. Moreover, C k comes equipped with an ordering on its components C h k induced by the lexicographic order on the variables y 1 > y 4 > · · · > y 3k+1 . Therefore, the h-th component of C k refers to the line where the homogeneous coordinates are y 3h−2 and y 3h+1 . Proposition 6.5. Let n ≥ 4. The form F n has minimal border rank n + 1.
Proof. By conciseness, r(F n ) ≥ n + 1. We show the opposite inequality as follows. Let n = 3k + 1 and consider the powers of linear forms i,j depends only on corresponding two variables. Note that there are n + 1 of such forms. Moreover, we require 3 i,j = x 3 3i−2 , x 3 3i+1 for 1 ≤ i ≤ k − 1, where the latter cubes correspond to the intersections between the i-th component of C k and the other lines in C k . This may be regarded as a configuration of points on the lines C i k of the chain C k ; an instance of this is depicted in Figure 1.
Write P T 3 for the affine cone of the Zariski tangent space to ν 3 (P n ) at the point 3 . As in the proof of Proposition 5.1, for d = 3, we have More generally, for each 1 ≤ i ≤ k, one has: The forms 3 i,j are linearly dependent. Indeed, consider the forms on the first component C 1 k of the chain: 3 1,1 , 3 1,2 , 3 1,3 and 3 1,4 are linearly independent and span every cubic form defined on C 1 k . Thus the cube x 3 4 may be written as On the line C 2 k , we have the linear relation: So we may rewrite the cube x 3 7 as a linear combination of the 3 i,j with 1 ≤ i ≤ 2. Proceed similarly up to C k−1 k , where we express x 3 3k−2 as a linear combination of 3 Finally, on the line C k k use the linear relation among the five cubic forms x 3 3k−2 , 3 k,1 , 3 k,2 , 3 k,3 , and 3 k,4 . This process gives a linear relation among the 3 i,j . Now, to conclude, use Proposition 2.3 which yields r(F n ) ≤ n + 1.
Corollary 6.7. The cubics F n are wild.
Proof. Their Hessian is vanishing, as the partial derivatives are algebraically dependent. By Proposition 6.5, F n has minimal border rank. Theorem 3.5 shows that the cubics F n are wild. Lemma 6.8. Let n = 3k + 1 and let C k be the chain of lines from Proposition 6.4. Let J ∈ VSP(F n , n + 1). Then: (i) there exist forms q 1 , . . . , q s in the variables y 3h+1 's such that J = Ann(F ) 2 + q 1 , . . . , q s ; (ii) the ideal J sat defines a projective scheme that is a zero-dimensional scheme of length n + 1 supported on C k ; (iii) each form q j has degree at most n + 1; (iv) there exists a proper map where the latter is the Hilbert scheme of zero-dimensional schemes of length n+1 supported on the reducible curve C k ⊂ P n .
Proof. (i). From the description of I = Ann(F ) 2 in the proof of Proposition 6.4, it is straightforward to see that each degree d ≥ 4 graded piece of the quotient ring T /I has a basis of monomials {m 1 , . . . , m h }, where each m j is a monomial in two variables y 3h−2 and y 3h+1 ; each such a pair of variables corresponds to a unique line in the chain C k . Since a necessary condition for membership of an ideal J in VSP(F n , n + 1) is possessing a generic Hilbert function, we add forms q j (in the variables y 3h+1 's) to J until the ideal reaches a generic Hilbert function. (ii). Let I = Ann(F ) 2 . Note that the Hilbert polynomial of J sat is n + 1. Moreover, J sat ⊃ I sat . By Proposition 6.4, I sat is the ideal defining the projective scheme C k . Therefore J sat defines a zero-dimensional scheme of length n + 1 supported on C k . (iii). The Castelnuovo-Mumford regularity of a zero-dimensional scheme of length n + 1 is at most n + 1 [9, Theorem 1.69], and the degrees of the generators q j are bounded above by the regularity. (iv). By (ii), the projective scheme defined by J sat is a zero-dimensional scheme of length n + 1 supported on C k . Define the map: ψ k : VSP(F n , n + 1) → Hilb n+1 (C k ), Then this is a well-defined morphism. It is proper because it is projective. Theorem 6.9. Let n = 3k + 1 ≥ 10. The variety VSP(F n , n + 1) is reducible.
Proof. Since ψ k is proper by Lemma 6.8(iv), if VSP(F n , n + 1) were irreducible, then the image ψ k (VSP(F n , n + 1)) would be closed and irreducible. We show next that ψ k (VSP(F n , n + 1)) has at least two irreducible components.
As in proof of Proposition 6.5, F n is in the span of the affine cones of Zariski tangent spaces of ν 3 (P n ) at the following points: · · · · · · · · · 3 a,1 , 3 a,2 , 3 a,3 , 3 a,4 , · · · · · · · · · 3 b,1 , 3 b,2 , 3 b,3 , 3 b,4 , · · · · · · · · · 3 k,2 , 3 k,3 , 3 k,4 . Here we generalize the original configuration described in Proposition 6.5, where the forms 3 1,1 and 3 k,1 are replaced by 3 a,1 and 3 b,1 , with 1 ≤ a < b ≤ n − 3 and a ≡ b ≡ 1 (mod 3). Note that, if n ≥ 10, there are at least two such pairs (a, b). To see that the forms 3 i,j above are linearly dependent, perform the same procedure presented in the proof of Proposition 6.5, starting from the a-th component and ending at the b-th component of C k . This produces a linear relation among the 3 i,j with a ≤ i ≤ b. Thus, by Corollary 4.2 and Theorem 4.4, we may find a border rank decomposition given by a family of schemes R(t) over the base Spec(C[t ± ]), whose limiting scheme is supported on the points 3 i,j above, with a corresponding ideal J (a,b) ∈ VSP(F n , n + 1). Thus ψ k (J (a,b) ) ∈ Hilb n+1 (C k ).
As before, let C h k denote the h-th component of C k . Let Hilb n+1 (C k ) (a,b) be the irreducible component of Hilb n+1 (C k ) defined by: Hilb n+1 (C k ) (a,b) = {Z ⊂ C k | Z smooth, Z ∩ C a k = Z ∩ C b k = 4, Z ∩ C j k = 3, j = a, b}.
Consider the components Hilb n+1 (C k ) (a,b) . Notice that dim Hilb n+1 (C k ) (a,b) = 3k + 2 = n + 1. To see this, let Z ∈ Hilb n+1 (C k ) (a,b) be a general point and so Z is a smooth zero-dimensional scheme. The normal bundle N Z/C k has n + 1 global sections (an affine coordinate at each smooth point of Z). Therefore n + 1 = h 0 (N Z/C k ) = dim T Z Hilb n+1 (C k ) = = dim T Z Hilb n+1 (C k ) (a,b) = dim Hilb n+1 (C k ) (a,b) .
Since Hilb n+1 (C k ) (1,2) and Hilb n+1 (C k ) (1,3) have the same maximal dimension, they are two distinct irreducible components of the Hilbert scheme Hilb n+1 (C k ). As the image ψ k (VSP(F n , n + 1)) is closed, it contains both of them; therefore the image is reducible. In conclusion, VSP(F n , n + 1) must be reducible. i.e. the generic fiber of ρ is a single point. We do not know whether VSP (F 7 , 8) is irreducible or not. The locus where the morphism is injective is then birational to P 8 . If it is irreducible and 8-dimensional, it cannot be isomorphic to P 8 : the isomorphic fibers ρ −1 (x 4 1 ) and ρ −1 (x 4 7 ) both contain a linear space P 4 . Thus these two linear spaces do not intersect. Question 6.11. Is VSP(F 7 , 8) irreducible? Are the irreducible components of the border varieties VSP(F n , n + 1) rational? More generally, it would be interesting to analyze rationality and unirationality of (the irreducible components of) border varieties of sums of powers VSP's alike in the context of VSP's; see for instance [12] for several results in this direction.