A FOOTNOTE TO A FOOTNOTE TO A PAPER OF B. SEGRE

LUCA CHIANTINI AND GIORGIO OTTAVIANI

Dedicated to Ciro Ciliberto, for his 70th birthday.

Si canimus silvas, silvae sint consule dignae. (Vergilius)

1. Introduction

The paper is devoted to a detailed study of sextics in three variables having a decomposition as a sum of nine powers of linear forms. This is indeed the unique case of a Veronese image X of the plane which, in the terminology introduced by Ciliberto and the first author in [12], is weakly defective, and non-identifiable: a general sextic of the 9-secant variety of X has two minimal decompositions.

The title originates from a famous paper of 1981, [4], where Arbarello and Cornalba state and prove a result on plane curves with preassigned singularities, which is relevant to extend the studies of B. Segre on special linear series on curves. The result (Theorem 3.2) says that the linear system of sextics with 9 general nodes in ℙ^{2} is the unique non-superabundant system of plane curves with general nodes whose (unique) member is non-reduced. The result is of course relevant also in the theory of interpolation, and in the study of secant varieties to Veronese varieties, with consequences for the Waring decomposition of forms. The matter turns out to be strictly related to the uniqueness of a minimal Waring decomposition of a general form. From this point of view Theorem 3.2 of [4] and Theorem 2.9 of [13] imply that the unique case of general ternary forms of fixed (Waring) rank with a finite number greater than one of minimal decomposition holds for degree 6 and rank 9. Since the unique sextic with 9 general nodes is twice an elliptic curve, it turns out by Proposition 5.2 of [13] that a general ternary sextic of rank 9 has exactly two minimal decompositions. A complete list of cases, in any number of variables, in which the previous phenomenon appears is contained in [15].

Remaining in the case of ternary sextics, already in [4] the authors observe that if the 9 nodes are in special position (e.g. when they are complete intersection of two cubics), then the linear system of nodal curves has also reduced members. Our analysis starts here, and aims to describe the decompositions of specific ternary sextics, with respect to the postulation of the corresponding sets of projective points in ℙ^{2}.

We consider a fixed sextic form F in the polynomial ring R = ℂ[x_{0}, x_{1}, x_{2}] in three variables. In order to effectively produce decompositions of F, the first natural step is to consider the apolar ideal F^{⊥} of F. In general, the apolar ideal F^{⊥} of a form F of degree d is defined via the natural action of the dual ring R^{∨} on R. F^{⊥} is the ideal of elements D ∈ R^{∨} that kill F. A classical theorem by Sylvester says that a set of points is a decompositions of F if and only if the corresponding ideal in R^{∨} is contained in F^{⊥}. Indeed, in order to find properties of the decompositions of a specific form F, it is often sufficient to look at homogeneous pieces of the ideal F^{⊥}, which correspond to the kernels of the (catalecticant) maps (R^{∨})_{k} → R_{d−k} induced by F, and their projective versions C_{F}^{k}. In [22], as well as in [27], it is explained that if the image of C_{F}^{k} has the expected dimension and cuts the corresponding Veronese variety in ℙ(R_{d−k}) in a finite set A, then the set A determines a decomposition of F, and many minimal decompositions can be found in this way.

2000 Mathematics Subject Classification. 14N07.

Consider, in particular, the case of a sextic F of rank 8. In this situation, the image of the catalecticant map of order 3 determines a linear space of codimension 2 in ℙ(R_{3}), which cuts the 3-Veronese of ℙ^{2} in 9 points. The set Z of 9 points is the image of a complete intersection of two cubics in ℙ^{2}, via the Veronese map. Then the 8 points of a minimal decomposition of F are among the points of Z. It is quite easy to find the coefficients of F with respect to the points of Z, and then determine the 8 points that give a minimal decomposition. The procedure shows that there are sextic forms which lie in between forms of rank 8 and forms of rank 9. These are forms of rank 9, with a minimal decomposition Z coming from a complete intersection of two plane cubics. Forms of this sort fill a subvariety W of the secant variety S^{9}(X) (X being the 6-Veronese image of ℙ^{2}) which contains S^{8}(X). We show that a general sextic in W has a unique decomposition with 9 powers of linear forms. From a certain point of view, this is quite surprising. A general sextic F′ of rank 9 has two minimal decompositions of length 9, coming from the existence of an elliptic normal curve in X which spans F′. When F′ sits in W, then there is a pencil of elliptic curves in X which span F′, so one may expect infinitely many minimal decompositions, one for each elliptic curve.

The phenomenon can be investigated in terms of the map s^{9} from the abstract secant variety AS^{9}(X) to S^{9}(X), which is generically 2:1. The component of the ramification locus R is strictly connected with the theory of Terracini loci, introduced in [6]. Roughly speaking, Terracini loci contain finite subsets of X such that the tangent spaces to the points are not independent. A subset A ⊂ X consisting of 9 points belongs to the Terracini locus 𝕋_{9}(X) exactly when the linear span of v_{6}(A) is contained in R. Thus, when a sextic form F has a decomposition A complete intersection of two cubics, which obviously lie in the Terracini locus, then in a neighborhood of F the secant map has degree 2, so it has degree 2 also in F, but the fiber is non-reduced. The subvariety W of S^{9}(X) lies in between S^{9}(X) and S^{8}(X), and it is easy to detect because it is easy to compute the rank and the kernel of the catalecticant map C_{F}^{3} . There are other relevant varieties of S^{9}(X), which can be studied in terms of the geometry of the decomposition. Given a general set A of 9 points in ℙ^{2}, and a general sextic F in the span of v_{6}(A), a second decomposition of F comes out by taking a set of 9 points B linked to A by a cubic and a sextic. In this case, as explained in [2], the sum I_{A} + I_{B} of the homogeneous ideals of A and B determines a subspace of R_{6}, whose orthogonal direction gives coefficients for the (unique) form F in the intersection ⟨v_{6}(A)⟩∩⟨v_{6}(B)⟩. In general, the cubic is uniquely determined by A but the sextic moves and then the set of forms defined by (I_{A}+I_{B})_{6} dominates ⟨v_{6}(A)⟩. On the contrary, when A is complete intersection, then also B is complete intersection. We obtain that starting from a complete intersection A the linked sets B determine only a subvariety of forms in ⟨v_{6}(A)⟩, each having a 2-dimensional set of decompositions. When A varies in the Hilbert stratum of complete intersections of type (3, 3) in ℙ^{2}, we obtain a subvariety W′ ⊂ W of sextics in S^{9}(X) with a 2-dimensional set of decomposition, which can be easily described because for such forms F the kernel of C_{F}^{3} has dimension 3, so that W′ does not contain S^{8}(X), but properly contains S^{7}(X).

Let us notice that the picture is completed by another subvariety R′ of the ramification locus, given by forms F with two different and linked decompositions A, B, both sets of nodes of irreducible sextics. The variety R′ parametrizes forms with a 1-dimensional family of decompositions, and its intersection with W contains W′. The variety R′ is the most mysterious object in the picture, for it cannot be completely described in terms of the catalecticant map C_{F}^{3} and its kernel. While the third section is devoted to the geometry (and, somehow, parametric) description of loci in S^{9}(X), in the last section we provide for most of them a set of equations. Equations for S^{9}(X), W, W′, can be easily described generically by the vanishing of appropriate minors of the catalecticant map C_{F}^{3} . Equations for S^{8}(X) and S^{7}(X) require the determinant of a 27 × 27 matrix A_{f} which represents a flattening, introduced in [24]. It is remarkable that this invariant H_{27} = det A_{f}, joint with the catalecticant matrix C^{3}, allows to describe the equations of all the secant varieties to X = v_{6}(ℙ^{2}). The ramification locus turns out to be described also in terms of det(A_{f}) (see Theorem 4.4). We wonder if a description of equations for R′ can be achieved in terms of minors of A_{f}.

Acknowledgements. We are honored to have the chance of contributing to the volume that celebrates the 70th birthday of Ciro Ciliberto. A master, a guide, and a friend for us.

2. Preliminaries

2.1. Notation. In this section, as in most of the paper, we write ℙ^{n} for the projective space of linear forms in n + 1 variables, with complex coefficients.

In the sequel, by abuse, we will identify a form F with the point in the projective space ℙ(Sym^{d}(ℂ^{n+1})) associated to F, and also with the hypersurface of equation F = 0.

Fix N = −1 + \binom{n+d}{n} and identify ℙ^{N} with ℙ(Sym^{d}(ℂ^{n+1})), the projective space of degree d forms in n + 1 variables.

If we denote with R the polynomial ring R = ℂ[x_{0}, …, x_{n}], then ℙ^{N} is the projective space over the degree d piece R_{d}, while ℙ^{n} is the projective space over R_{1}.

We denote with v_{d} : ℙ^{n} → ℙ^{N} the d-th Veronese map of ℙ^{n} which, in the previous notation, maps L ∈ ℙ^{n} to L^{d}.

A (Waring) expression of length r of F is an equality

F = a_{1}L_{1}^{d} + ⋯ + a_{r}L_{r}^{d}

where the L′_{i}s are linear forms and the a_{i}’s are complex coefficients. Each L_{i} represents a point in ℙ^{n}. The expression is non-redundant if the L_{i}^{d} are linearly independent in Sym^{d}(ℂ^{n+1}), and all the coefficients are non-zero. Non-redundant means that one cannot find a proper subset A′ ⊂ A such that also A′ is a decomposition of F.

We are aware that, since we are working over the algebraically closed field ℂ, we could get rid of the coefficients a_{i}’s in a Waring expression of F. It is convenient for us to maintain the coefficients, because we will compare, below, Waring expressions in which the linear forms L_{i}’s are fixed and the coefficients move.

Given a Waring expression of F, we call (Waring) decomposition of F the finite set A = {L_{1}, …, L_{r}} ⊂ ℙ^{n}. The length of the decomposition A is the cardinality of the finite set A, often denoted with ℓ(A). From the geometric point of view, A is a decomposition of F if and only if F belongs to the linear span ⟨v_{d}(A)⟩ of v_{d}(A), and A is non-redundant if one cannot find a proper subset A′ ⊂ A such that also A′ is a decomposition of F.

The rank of F is the minimal r for which F has a Waring expression of length r. The border rank of F is the minimal r′ such that F is limit of forms of rank r′.

2.2. Catalecticant maps. The projective catalecticant map associated to a form F, as suggested in [22], is defined as follows.

Denote with R^{∨} = ℂ[∂_{0}, …, ∂_{n}] the ring of linear operators. There is a natural contraction map C : R^{∨} × R → R defined by linearity and by

C(∂_{i_{1}} ⋯ ∂_{i_{k}}, F) ↦ \frac{∂^{k}F}{∂_{x_{i_{1}}} ⋯ ∂_{x_{i_{k}}}} .

Notice that C maps (R^{∨})_{k} × R_{d} bilinearly to R_{d−k}.

For fixed F ∈ R_{d} and k ≤ d, define the catalecticant map of order k associated to F as

C_{F}^{k} : (R^{∨})_{k} → R_{d−k} D ↦ C(D, F).

Call k-polar space of F the subspace Im(C_{F}^{k}) of R_{d−k}, and call k-polar projective space of F its projectification P_{F}^{k} = ℙ(Im(C_{F}^{k})).

We will use in the sequel the correspondence between R_{d} and R_{d}^{∨} induced by the monomial basis.

Properties of catalecticant maps in connection with decompositions of a form F are deeply studied in [22]. We recall some basic facts.

Theorem 2.1. If A = {L_{1}, …, L_{r}} ⊂ ℙ^{n} is a decomposition of F, then the projective polar space P_{F}^{k} is contained in the linear span of L_{1}^{d−k}, …, L_{r}^{d−k}. Thus, the projective dimension of P_{F}^{k} is at most equal to rank(F).

Comparing dimensions, one obtains.

Corollary 2.2. In the setting of Theorem 2.1, if moreover P_{F}^{k} has (projective) dimension r − 1, then the points L_{1}^{d−k}, …, L_{r}^{d−k} are linearly independent, and

P_{F}^{k} = ⟨L_{1}^{d−k}, …, L_{r}^{d−k}⟩.

In particular, if dim(P_{F}^{k}) = rank(F), then any minimal decomposition A of F satisfies

v_{d−k}(A) ⊂ P_{F}^{k} ∩ v_{d−k}(ℙ^{n}).

We notice that it is easy to construct the matrix of C_{F}^{k} with respect to the monomial basis, and then check the dimension of the image of C_{F}^{k}.

The Apolarity Theorem (see [22] Section 4) determines a fundamental link between catalecticant maps and decompositions. We will use it in the form below

Theorem 2.3. The direct sum of the kernels of the catalecticant maps C_{F}^{k} is an (artinian, homogeneous) ideal in R^{∨}, called the apolar ideal F^{⊥} of F. If the homogeneous ideal I_{A^{∨}} of a finite set A^{∨}, in the projective space (ℙ^{n})^{∨} associated to R^{∨}, is contained in F^{⊥}, then the corresponding set A ⊂ ℙ^{n} determines a decomposition of F.

Notice that when I_{A^{∨}} ⊂ F^{⊥}, then the k-polar space of F is contained in the span of v_{d−k}(A) for all k. It is well known that the converse fails. Thus in general one cannot hope to find a decompositions of F by looking only to one catalecticant map. Yet, we will see that there are cases in which just one catalecticant map is sufficient.

Remark 2.4. Consider an element D of the kernel of C_{F}^{k}, and an element D′ of the kernel of C_{F}^{k′}. We can consider D, D′ as polynomials in R^{∨}, with degrees k, k′ resp. Assume that D, D′ have no common factors. By the Apolarity Theorem 2.3 the complete intersection of D, D′ determines a decomposition Z of F, of length kk′. Since F^{⊥} is strictly bigger than the ideal generated by D, D′, for F^{⊥} is artinian, one can have decompositions Z′ of F which are different, even disjoint, from Z.

2.3. Hilbert functions. Let Z be a finite, reduced subset of length r in ℙ^{n}. Fix a set of representatives for the coordinates points of Z. The evaluation of polynomials at the chose representatives provides for any degree d a linear map ρ_{d} : R_{d} → ℂ^{r} whose image has dimension which depends only on Z and not on the choice of the representatives. The Hilbert function of Z is the map

h_{Z} : ℤ → ℤ h_{Z}(d) = dim Im(ρ_{d}).

We will often use the difference Dh_{Z}(d) = h_{Z}(d) − h_{Z}(d − 1), and the h-vector of Z, which is the t-uple of non-zero values of Dh_{Z}.

Clearly h_{Z}(i) = Dh_{Z}(i) = 0 when i is negative. Also h_{Z}(0) = Dh_{Z}(0) = 1. Moreover h_{Z}(d) is non decreasing, and h_{Z}(d) = r for all large d. Consequently Dh_{Z} is non-negative and Dh_{Z}(d) = 0 if d is sufficiently large. Other standard properties of h_{Z} and Dh_{Z} are known in the literature, and are collected e.g. in [11]. We list the most relevant of them.

a1) for all d ≫ 0, ∑_{i=0}^{d} Dh_{Z}(i) = h_{Z}(d) = r;

a2) if Dh_{Z}(i) = 0 for some i > 0, then Dh_{Z}(j) = 0 for all j ≥ i;

a3) if Z′ ⊂ Z then for all i, h_{Z′}(i) ≤ h_{Z}(i) and Dh_{Z′}(i) ≤ Dh_{Z}(i).

a4) if Z is contained in a plane curve of degree q, then Dh_{Z}(i) ≤ q for all i. The application of Hilbert functions to tensor analysis is based on the trivial

observation that h_{Z}(1) < min{n + 1, r}, i.e. the evaluation map in degree 1 is not of maximal rank, if and only if Z is linearly dependent and fails to span ℙ^{n}. This implies that when \binom{n+d}{n} ≥ r, then h_{Z}(d) < r if and only if v_{d}(Z) is linearly dependent. It follows that if A, B are two different, non redundant decompositions of the same form F ∈ R_{d}, and we take Z to be the union Z = A ∪ B, then h_{Z}(d) < r, so that

b0) Dh_{Z}(d + 1)> 0.

The next properties of the union Z = A ∪ B, which will be used in the sequel, follow from results in the theory of finite subsets in projective space, as the Cayley-Bacharach theorem, or the Macaulay Maximal Growth principle. For an account, we refer to [11] or [2].

Proposition 2.5. Let A, B be two different, non-redundant decomposition of a form F of degree d in n + 1 variables. Put Z = A ∪ B and assume A ∩ B = ∅.

b1) For all j ≤ d + 1 one has

∑_{i=0}^{j} Dh_{Z}(i) ≤ ∑_{i=d+1−j}^{d+1} Dh_{Z}(i).

b2) The projective dimension of the intersection ⟨v_{d}(A)⟩ ∩ ⟨v_{d}(B)⟩ is equal to (∑_{i>d} Dh_{Z}(i)) − 1. In particular, if Dh_{Z}(d + 1) = 1 and Dh_{Z}(d + 2) = 0, then the intersection contains only F.

b3) Assume n = 2, i.e. F is a ternary form. If for some 0 < i < j one has Dh_{Z}(i) > Dh_{Z}(j) > 0 then Dh_{Z}(j + 1) < Dh_{Z}(j).

2.4. Ramification and the Terracini locus. The study of decompositions of tensors is naturally linked to the geometric theory of secant varieties. We refer to the book [21], or to Section III of the book [8] for an introduction to secant and abstract secant varieties.

In this setting, following [6], one defines the Terracini loci of a variety X as follows.

Definition 2.6. The Terracini locus 𝕋_{r}(X) of a variety X ⊂ ℙ^{N} is the closure, in the symmetric product X^{(r)}, of of the locus of subsets A of cardinality r, contained in the regular part X_{reg}, such that the span of the tangent spaces to X at the points of A has dimension smaller than the expected value r(n + 1) − 1.

The locus 𝕋_{r}(X) is connected to the structure of the natural map (projection) from the abstract secant variety AS^{r}(X) to S^{r}(X),

s^{r} : AS^{r}(X) → S^{r}(X).

Given a linearly independent finite set Y ⊂ X_{reg}, by Terracini’s Lemma and its proof (see [31] or, in a modern setting, Section 5.3 of [23] and Proposition 2.4 of [14]), it follows that the span of the tangent spaces to X at the points of Y correspond to the image of the differential of s^{r}. Since for a general P ∈ ⟨Y ⟩, AS^{r}(X) is smooth of dimension r(n + 1) − 1 at (P, Y) ∈ AS^{r}(X), then Y belongs to 𝕋_{r}(X) if and only if the linear span ⟨Y ⟩ is contained in the ramification locus of the map s^{r}.

3. Plane sextics

In this section we will analyze in details the situation of forms of degree 6 in three variables, i.e. sextic plane curves. We will see how the study of the catalecticant map determines several loci in the secant varieties of the surface X = v_{6}(ℙ^{2}).

Remark 3.1. Sextics in three variables are parameterized by ℙ(Sym^{6}(ℂ^{3})) = ℙ^{27}. By [1], the (Waring) rank of a general ternary sextic is 10. Sextics of (border) rank 9 determine a hypersurface in the space of sextics: the 9-secant variety to the 6-Veronese of ℙ^{2}. Moreover, by [4] and by [13], a general form of rank 9 has exactly two minimal decompositions. Sextics of rank 8 determine an irreducible subvariety of dimension 23 in ℙ^{27}, i.e. the 8-secant variety to v_{6}(ℙ^{2}). By [15], a general sextic of rank 8 has a unique decomposition of length 8.

Following [27], let us look what happens to the catalecticant map of order three applied to a sextic F of rank < 10. We will focus mainly on the cases in which F has rank either 8 or 9.

We start by recalling the following, elementary fact.

Remark 3.2. Let C be a form in the kernel of C_{F}^{k}. Then C corresponds by duality to a form in R (that we continue to denote as C), and its coefficients correspond to the orthogonal to a hyperplane in ℙ(R_{k}^{∨}) that contains P_{F}^{k}. Since the Veronese map and the duality are both given in terms of the monomial basis, then the intersection of P_{F}^{k} with v_{k}(ℙ^{2}) corresponds to a divisor that defines v_{k}, i.e. to a curve of degree k in ℙ^{2}, which is exactly C. Thus P_{F}^{k} ∩ v_{k}(ℙ^{2}) = v_{k}(C).

Proposition 3.3. If F has rank r < 10 then dim(P_{F}^{3}) ≤ r − 1. If dim(P_{F}^{3}) = 8 and the kernel of C_{F}^{3} is generated by an irreducible form C, then F has border rank at most 9, and rank ≥ 9.

Proof. If we take a minimal decomposition A of F, then the ideal of A certainly contains 10 − r independent cubics. Thus from Theorem 2.1 we know that the kernel of C_{F}^{3} has dimension at least 10 − r, so that dim(P_{F}^{3}) ≤ r − 1. Now assume that the dimension of P_{F}^{3} is exactly equal to 8, so that F cannot have rank smaller than 9. P_{F}^{3} is then a hyperplane in ℙ(Sym^{3}(ℂ^{3})) = ℙ^{9} which intersects v_{3}(ℙ^{2}) in v_{3}(C). Since F^{⊥} certainly contains some form C′ independent from C, and by the Apolarity Theorem C ∩ C′ is a decomposition of F, hence F lies in the span of v_{6}(C), which is the Veronese image of an irreducible plane cubic of arithmetic rank 1. Since irreducible curves are never defective, then F has border rank at most 9 with respect to v_{6}(C), hence also with respect to v_{6}(ℙ^{2}). □

Notice that not every general set of 9 points in C determines a decomposition of F. So, it is not sufficient that the span of v_{3}(A) contains P_{F}^{3} for A to be a decomposition of F.

Example 3.4. Let F be a sextic of rank 8. Since 8 points always lie in a pencil of cubic curves, then dim(P_{F}^{3}) ≤ 7. Assume that P_{F}^{3} has dimension exactly 7 and look at the intersection of P_{F}^{3} with the 3-Veronese surface S = v_{3}(ℙ^{2}). Since deg(S) = 9, it is clear that P_{F}^{3} intersects S either in a curve Γ, or in a scheme of length 9. The former case cannot happen. Namely the curve Γ would be the image in v_{3} of a plane curve whose ideal in degree 3 has linear dimension 1, but no such plane curves exist. Thus Z = P_{F}^{3} ∩ S is a subscheme of length 9. It is easy to see (from Section 4 of [27]) that for a general choice of the form F the scheme Z is reduced, hence it consists of 9 distinct points. It follows that any decomposition of length 8 of F is contained in Z, for any set of length 8 sits in two independent cubics, and by the Apolarity Theorem these two cubics are in the kernel of C_{F}^{3} . When Z is reduced, it consists of a minimal decomposition, plus one extra point.

Proposition 3.5. Let F be a sextic ternary form such that dim(P_{F}^{3}) = 7 and Y = P_{F}^{3} ∩ S is reduced. Assume that F has rank 8. Then there exists only one decomposition of F of length 8, i.e. F is identifiable, in the sense of [13].

Proof. Put Z = v_{3}^{−1}(Y). By Example 3.4 every decomposition of length 8 of F is a subset of Z. Since Z is complete intersection of two cubics, then Z imposes independent conditions to curves of degree ≥ 3 + 3 − 2 = 4. In particular, v_{6}(Z) is linearly independent, thus every subset of v_{6}(Z) is linearly independent. Assume that there are two different decompositions A, A′ of length 8. Since A, A′ ⊂ Z by Remark 3.4, then B = A∩A′ has length 7. Since v_{6}(A) and v_{6}(A′) are linearly independent, then the spaces ⟨v_{6}(A)⟩ and ⟨v_{6}(A′)⟩, both containing F, meet exactly in ⟨v_{6}(B)⟩. Then F ∈ ⟨v_{6}(B)⟩, which means that rank(F) ≤ 7, a contradiction. □

In conclusion, we get

Corollary 3.6. Let F be a sextic ternary form such that dim(P_{F}^{3}) = 7 and Y = P_{F}^{3} ∩ S is reduced. Then Z = v_{3}^{−1}(Y) ⊂ ℙ^{2} is a complete intersection of two cubics, and either

• F has rank 8 and a unique decomposition of length 8, contained in Z; or • F has rank 9, and Z is a minimal decomposition of F, i.e. there exists a minimal decomposition of F which is a complete intersection of 2 cubics.

An algorithm that can guarantee that a ternary form F of degree 6 has rank 8, and find a decomposition, is the following (see [27]). Compute dim(P_{F}^{3}). If dim(P_{F}^{3}) > 7, then F cannot have rank 8. Assume that dim(P_{F}^{3}) = 7 (this will be true for general forms of rank 8). Compute two generators D, D′ of the kernel of C_{F}^{3} and compute the coordinates of the points of the complete intersection Z of D and D′ (this can require approximation). Assume that Z has length 9 (this will be true for general forms of rank 8). Then Z = {L_{1}, …, L_{9}}. If we identify each L_{i} with a linear form in R, then there exists a linear combination F = a_{1}L_{1}^{6} + ⋯ + a_{9}L_{9}^{6} and we know that the linear combination is unique, for v_{6}(Z) is linearly independent. Compute the a_{i}’s. By Proposition 3.5, F has rank 8 if and only if one coefficient a_{i} is 0. In this case, by dropping the summand a_{i}L_{i}^{6}, we also obtain the unique decomposition of length 8 of F.

It is known that since a general point in ℙ^{17} has exactly two different decompositions with respect to an elliptic normal curve, then a general sextic of rank 9 has exactly two minimal general decomposition. A geometric way to find the second decomposition, once one of them is known, can be described in terms of liaison. We need first a series of preparatory results on the interaction between Hilbert functions and decompositions.

Lemma 3.7. (Intersection Lemma) Let F ∈ R_{d} be a form with two non-redun-dant decompositions A, B, of length ℓ(A) ≥ ℓ(B). Assume A ∩ B ≠ ∅. Then there exists a form F′ ∈ R_{d} with two non-redundant decompositions A′ ⊂ A, B′ ⊂ B such that A′ ∩ B′ = ∅ and ℓ(A′) ≥ ℓ(B′), and moreover either ℓ(A′) > ℓ(B′) or A′ ≠ A.

Proof. Let A = {L_{1}, …, L_{r}} and B = {L_{1}, …, L_{j}, M_{j+1}, …, M_{r′}} with L′_{i} ∉ A for i = j +1, …, r′ (r = ℓ(A), r′ = ℓ(B)). Thus A∩B = {L_{1}, …, L_{j}} and j > 0. Write

F = a_{1}L_{1}^{d} + ⋯ + a_{j}L_{j}^{d} + a_{j+1}L_{j+1}^{d} + ⋯ + a_{r}L_{r}^{d}
F = b_{1}L_{1}^{d} + ⋯ + b_{j}L_{j}^{d} + b_{j+1}M_{j+1}^{d} + ⋯ + b_{r′}M_{r′}^{d}.

Since A, B are non-redundant, then v_{d}(A), v_{d}(B) are both linearly independent, and the a_{i}’s and b_{i}’s are all non-zero. Consider the form F′ obtained by subtracting from F the first j summands of the second Waring expression above

F′ = (a_{1} − b_{1})L_{1}^{d} + ⋯ + (a_{j} − b_{j})L_{j}^{d} + a_{j+1}L_{j+1}^{d} + ⋯ + a_{r}L_{r}^{d}
F′ = b_{j+1}M_{j+1}^{d} + ⋯ + b_{r′}M_{r′}^{d}.

Then B′ = {M_{j+1}, …, M_{r′}} is a decomposition of F′, of length r′ − j < r′, which is non-redundant since b_{j+1}, …, b_{r′} ≠ 0 and v_{d}(B′) is linearly independent. Let A′ = A \ {L_{i} : i ≤ j and a_{i} = b_{i}}. Then also A′ is a decomposition of F′, which is non-redundant since v_{d}(A′) is linearly independent and a_{j+1}, …, a_{r} ≠ 0. A′ and B′ are clearly disjoint, and ℓ(A′) ≥ r − j ≥ r′ − j = ℓ(B′). Finally, if A′ = A then ℓ(A′) = r > r′ − j. □

The next result tells us where we must look, geometrically, in order to find minimal decompositions of a sextic form. We give a long proof in which we analyze one by one the several cases, just because we want to stress the use of the Hilbert function for tensor analysis. Indeed, some steps could be shortened by using [5] or Section 3 of [2]. See also [26] for an approach to the problem.

Proposition 3.8. Let F be a ternary sextic with two different non-redundant decompositions A, B, with length ℓ(A) ≤ 9 and ℓ(B) ≤ ℓ(A). Assume that no 4 points of A are aligned, and that A does not lie in a conic. Then ℓ(B) = ℓ(A) = 9 and A ∩ B = ∅. Furthermore A∪B lies in a cubic curve C, and it is complete intersection of C and a sextic curve.

Proof. We set r = ℓ(A), r′ = ℓ(B) and we make induction on r, r′. Put Z = A ∪ B and notice that Dh_{Z}(7) > 0, by b0). Since A lies in no conics by assumption, then r ≥ 6. Assume r = 6, so that ℓ(Z) ≤ 12. Since A is not aligned, then Dh_{A}(1) = 2. If Dh_{A}(2) ≤ 1, then A sits in two independent conics, which, by Bezout, is possible only if A has 5 aligned points, a contradiction. Thus Dh_{A}(2) ≥ 2, so that the h-vector of A is either (1, 2, 3), or (1, 2, 2, 1). In the former case, arguing as in cases r = 4, 5, we get Dh_{Z}(3) = 0, a contradiction. In the latter case, we have ∑_{i=0}^{3} Dh_{Z}(i) ≥ 6, thus by b1) also ∑_{i=4}^{7} Dh_{Z}(i) ≥ 6. Since ℓ(Z) ≤ 12, we must have ∑_{i=0}^{3} Dh_{Z}(i) = 6 which, by a3), implies Dh_{Z}(i) = Dh_{A}(i) for i = 0, …, 3. Then Dh_{Z}(3) = 1, so that, by b3), Dh_{Z}(4) = 0, which is not consistent with b0). Assume r = 7, 8. Since A lies in no conics, then Dh_{A}(2) = 3. If Dh_{Z}(3) ≤ 2, then by applying b3) for i = 4, 5 we see that Dh_{Z}(5) = 0, which contradicts b0). Thus Dh_{Z}(3) ≥ 3, so that ∑_{i=0}^{3} Dh_{Z}(i) ≥ 9. Then by b1) also ∑_{i=4}^{7} Dh_{Z}(i) ≥ 9, which is not consistent with ℓ(Z) ≤ 2r. So, we are left with the case r = 9. As above Dh_{Z}(2) = 3, and Dh_{Z}(3) ≤ 2 yields a contradiction. Thus the h-vector of Z starts with (1, 2, 3, q, …) with q = Dh_{Z}(3) ≥ 3. Then ∑_{i=0}^{3} Dh_{Z}(i) = 6 + q, so that by b1) ∑_{i=4}^{7} Dh_{Z}(i) ≥ 6 + q. Since ℓ(Z) ≤ 18, it follows q = 3. This means that ℓ(Z) = 18, i.e. ℓ(B) = 9, and moreover h_{Z}(3) = 9, so Z lies in a cubic curve. By Proposition 2.5, one computes that the unique possibility for the h-vector of Z is (1, 2, 3, 3, 3, 3, 2, 1). By [3] Lemma 5.3 we know that Z has the Cayley-Bacharach property. Since the h-vector of Z is the same than the h-vector of a complete intersection of type (3, 6), by [17] Z is complete intersection of a cubic and a sextic curve. Then A and B are linked by a cubic and a sextic curve. Finally, the case where A ∩ B ≠ ∅ implies, by Lemma 3.7, the existence of a sextic F′ with nonredundant decompositions A′ ⊂ A and B′, such ℓ(B′) ≤ ℓ(A′), and either ℓ(A′) ≤ 8 or ℓ(B′) ≤ 8. The existence of F′ is excluded by the previous argument. □

Given a general sextic F of rank 9, and a decomposition A of F, we can find a second decomposition B of F with a procedure introduced e.g. in [2], Section 4.

Remark 3.9. Let F be a sextic of rank 9, with a decomposition A of length 9. From the kernel of C_{F}^{3} , which has dimension 1 if F is general, we find a cubic C containing A. A resolution of the homogeneous ideal of A is given by

0 → R(−5)^{3} →^{M} R(−4)^{3} ⊕ R(−3) → I_{A} → 0.

M is the Hilbert-Burch matrix of A, whose minors provide generators for I_{A}. The degrees of M are

(\begin{matrix}1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \\ 2 & 2 & 2 \end{matrix}) .

We know that A ∪ B is complete intersection of C with a sextic. The homogeneous ideal of the residue B of A in the complete intersection is obtained by erasing the bottom row of M and adding a column of three quadrics. The maximal minors of the matrix M′ that we obtain generate the homogeneous ideal of a scheme B linked to A in a complete intersection (3, 6). The previous procedure produces a set B which decomposes some form F′ ∈ v_{6}(A). In other words, the intersection ⟨v_{6}(A)⟩ ∩ ⟨v_{6}(B)⟩ is a form F′ in v_{6}(A) which depends on the choice of the three quadrics in M′. One can find F′ by taking the sum I_{A}+I_{B}, which determines in R_{6} a linear subspace H of codimension 1. The coefficients of F′ form a vector orthogonal to H. Thus, in order to find the second decomposition of the given F, one needs to choose the three quadrics Q_{1}, Q_{2}, Q_{3} in M′ so that the orthogonal to H matches with the coefficient of F. In practice, one needs to solve a linear system whose matrix has entries linear in the coefficients of the Q_{i}’s. A convenient way to write this linear system into M2 [19] is to ask that the minors of M′ are apolar to F.

With a procedure similar to the proof of Proposition 3.8, we can prove the following property of sextics of rank 9 with a decomposition A complete intersection of two cubics.

Proposition 3.10. Let F be a sextic ternary form such that dim(P_{F}^{3}) = 7 and Y = P_{F}^{3} ∩ S is reduced. Assume that F has rank 9. Then A = v_{3}^{−1}(Y) is the unique decomposition of F of length 9, i.e. F is identifiable, in the sense of [13].

Proof. From Proposition 3.6 we know that A is a complete intersection of two cubics. In particular, no 4 points of A are aligned and A does not lie in a conic. The h-vector of A is (1, 2, 3, 2, 1). Assume that there exists another decomposition B of length 9. Then by Proposition 3.8 we know that A ∩ B = ∅ and Z = A ∪ B lies in a cubic, so that the h-vector of Z starts with (1, 2, 3, 3, …). Since Z lies in a plane curve of degree 3, by a4) Dh_{Z}(i) ≤ 3 for all i. Next, we claim that Dh_{Z}(4) = 3. Indeed if Dh_{Z}(4) ≤ 2 then by b3) Dh_{Z}(5) ≤ 1 and Dh_{Z}(6) = 0, which contradicts b0). If Dh_{Z}(4) ≥ 4, then ∑_{i=5}^{7} Dh_{Z}(i) < 6, which contradicts b1). The same argument proves that Dh_{Z}(5) = 3, and Dh_{Z}(6) ≤ 2. Then Dh_{Z}(6)+Dh_{Z}(7) = 3. Since by b3) Dh_{Z}(6) > Dh_{Z}(7) ≥ 1, then the h-vector of Z is (1, 2, 3, 3, 3, 3, 2, 1). By [3] Lemma 5.3 we know that Z has the Cayley-Bacharach property CB(6). Since moreover the h-vector of Z is the same than the h-vector of a complete intersection of type (3, 6), by [17] Z is complete intersection of a cubic and a sextic curve. Then A and B are linked by a cubic and a sextic curve. This implies, by [25] that the h-vector of B is (1, 2, 3, 2, 1). Thus B lies in the base locus of a pencil of cubic curves. Since B is a decomposition of F, and dim(P_{F}^{3}) = 7, there is only one pencil of cubic curves that contain B, and the base locus of the pencil is A. Comparing the degrees, we get A = B. □

From a certain point of view, Proposition 3.10 is rather surprising. Namely, as explained in Theorem 3.2 of [4], the 6-Veronese surface v_{6}(ℙ^{2}) is 9-weakly defective, in the sense of [12]: a general hyperplane which is tangent to v_{6}(ℙ^{2}) at 9 points is tangent along an elliptic normal curve C. Thus, by Theorem 2.9 and Proposition 5.2 of [13], F has two minimal decompositions of length 9, both lying in C. Notice that, in general, none of the two decompositions lies in two independent cubic curves. If the form F has a decomposition Z of length 9 which is a complete intersections of two cubics, then there are infinitely many elliptic normal curves in v_{6}(ℙ^{2}) containing v_{6}(Z), so one may expect the existence of infinitely many decompositions of length 9 of F, for any elliptic normal curves v_{6}(Z) could provide a second decomposition Z′ ≠ Z of F. On the contrary, in this case the construction collapses and any elliptic normal curves provides only one decomposition for F, namely A alone.

A motivation for the peculiar behavior of sextics with a decomposition complete intersection of two cubics can be explained in terms of the ramification of the map s^{9} from the abstract secant variety AS^{9}(v_{6}(ℙ^{2})) to S^{9}(v_{6}(ℙ^{2})), and the Terracini locus of v_{6}(ℙ^{2}).

Remark 3.11. For X = v_{6}(ℙ^{2}), by Definition 2.6 a linearly independent set v_{6}(A) of length 9 belongs to 𝕋_{9}(X) if the 9 tangent planes at the points of v_{6}(A) span a subspace of dimension ≤ 25. When A is complete intersection of two cubics Γ_{1}, Γ_{2}, then certainly v_{6}(A) belongs to the Terracini locus. Indeed, there is a 2-dimensional family of sextic curves in ℙ^{2} which are singular at the points of A: the sextics formed by the union of two cubics in the pencil generated by Γ_{1}, Γ_{2}. Any sextic like that corresponds to the intersection of X with a hyperplane which contains the tangent planes to X at the points of v_{6}(A). Thus, the span of the tangent planes at the points of v_{6}(A) has codimension at least 2 in ℙ^{N} = ℙ^{27}. By Remark 3.1, the map s^{9} : AS^{9}(X) → ℙ^{27} is generically 2 : 1, when A is a general set of 9 points in the plane. By Proposition 3.10 we see that general points (F, v_{6}(A)) such that A is complete intersection of two cubics are in the ramification locus of s^{9}. The last observation is consistent with the fact that the differential of s^{9} degenerates at points (F, v_{6}(A)) such that A is complete intersection of two cubics.

Definition 3.12. We denote with R the closure of the set of sextics with a decomposition formed by a set A of nine points which are nodes of an irreducible sextic. By [20], R is irreducible. We denote by W the closure of the set of sextics with a decomposition formed by a set A of nine points, complete intersection of two cubics.

Remark 3.13. The previous discussion proves that, in the new notation, W properly contains S^{8}(X), and both W and R are contained in the Terracini locus 𝕋_{9}(X), which is the closure of the image of the locus of smooth points in AS^{9}(X) in which the differential of the map s^{9} drops rank. Indeed, we will show in the next section that R is the unique component of 𝕋_{9}(X) which intersect the smooth locus of S^{9}(X), while W, whose codimension is bigger than 1, sits into the singular locus of S^{9}(X). We will also prove in the next section that W is not contained in R, which means that the Terracini locus has at least two components.

Remark 3.14. For the 6-Veronese variety X of ℙ^{2}, the Terracini locus 𝕋_{9}(X) corresponds to sets A of 9 points which are singular in a pencil of sextics. General sets in 𝕋_{9}(X) are easy to describe (see Example 5.1 in [6]): just take C to be a reduced elliptic plane sextic whose singularities are nodes. C has a set A of 9 nodes which also sit in a cubic curve D. All sextics in the pencil generated by C and D^{2} are singular at A. It follows that the 9 tangent planes to X at the points of v_{6}(A) span a linear space of dimension 25 in ℙ^{27}, the linear space spanned by X. Notice that a 𝕋_{9}(X) does not contain a general set of 9 points in the plane, and a general A ∈ 𝕋_{9}(X) is not complete intersection of two cubics.

General forms contained in the variety R have only one decomposition with 9 summands. On the other hand, both R and W contain subvarieties whose elements have many decompositions of length 9. Namely, if A lies in the Terracini locus 𝕋_{9}(X), then the liaison procedure introduced in Remark 3.9 determine a second decomposition for some forms in ⟨v_{6}(A)⟩.

Lemma 3.15. Let C be a general (smooth) cubic plane curve and let A be a reduced divisor of degree 9 on C such that 2A ∈ |O_{C}(6)|, but A ∉ |O_{C}(3)|. Then there exists a sextic plane curve G which is singular at the points of A and irreducible.

Proof. Call P_{1}, …, P_{9} the points of A. For each i call ϵ_{i} the double structure on P_{i} contained in C. For each P_{i} choose a scheme μ_{i} of length 2 in ℙ^{2}, supported at P_{i} such that μ_{i} ∉ C. By assumption there exists a sextic G′ such that G′ does not contain C and ϵ_{i} ∈ G′ for all i. Thus, the linear system M of sextics in ℙ^{2} which contain {ϵ_{1}, …, ϵ_{9}} has (projective) dimension 10. Note that M contains the double curve 2C. Since ϵ_{i} ∪ μ_{i} has length 3, then every μ_{i} imposes at most one condition to curves in M. Then there exists a sextic G′′ which contains ϵ_{i} ∪ μ_{i} for all i and G′′ is different from 2C. Since G′′ contains ϵ_{i} ∪ μ_{i}, then G′′ is singular at the points of A. G′′ cannot contain C, because A ∉ |O_{C}(3)|. Thus a general curve G in the pencil generated by 2C and G′′ is irreducible, by Bertini. □

Proposition 3.16. Let A be a set of 9 points in ℙ^{2}, which are nodes of a general reduced elliptic sextic curve G. The generality of G implies that there exists a unique cubic curve C containing A, and moreover C is smooth. Let B be the residue of A in the intersection of C with a general sextic S that passes through A. Then also B is the set of nodes of a reduced sextic G′.

Proof. The first claim is classical. Just to see an argument, observe that for a general set D of 8 points in a general cubic curve C, the divisor 2D on C lies in some divisor D′ of the linear series |O_{C}(6)|. D′ − D consists of 2 points, that determine a linear series L = g_{2}^{1}. Choose a Weierstrass point P_{9} of L. Since L cannot have 8 Weierstrass points, then by monodromy for a general choice of D we can assume that P_{9} ∉ D. Then 2D + 2P_{9} ∈ |O_{C}(6)| and then D ∪ {P_{9}} is contained in the singular locus of a sextic curve, by Lemma 3.15. Since the variety that parametrizes sextic plane curves with 9 singular points is irreducible, we see that on a general cubic C we can find a set of 9 nodes of an irreducible sextic. For the second claim, observe that on C the divisor 2A belongs to |O_{C}(6)|, and also the divisor A + B belongs to |O_{C}(6)|. Thus 2A + 2B belongs to |O_{C}(12)|. Then also 2B ∈ |O_{C}(6)|, and the claim follows from the generality of G, C, S and from Lemma 3.15. □

In other words, the previous proposition says that some forms F of rank 9 with a decomposition A which lies in the Terracini locus T_{9}(X), have a second decomposition B of length 9 which also sits in the Terracini locus T_{9}(X). Thus the subscheme of the abstract 9-secant variety which maps to F is supported at two points, but it has length 4. The closure R′ of the locus of forms F as above provides a proper subvariety of R, whose geometry has non-trivial aspects.

Proposition 3.17. A general sextic in R′ has infinitely many decompositions of length 9. Thus R′ is the locus in R in which the dimension of fibers of the map s^{9} : AS^{9}(X) → S^{9}(X) jumps.

Proof. Fix a general cubic curve C and a general set A ⊂ C of length 9, such that there is an irreducible sextic S singular at the points of A. For a general sextic S′ through A the intersection S′ ∩ C is a set A ∪ B of 18 points, with B disjoint from A. As explained in Proposition 3.8, the linear spans ⟨v_{6}(A)⟩ and ⟨v_{6}(B)⟩ meet in a point F ∈ ℙ^{27} which represents a point in R′. Since any element of R′ ∩ ⟨v_{6}(A)⟩ arises in this way, we get a surjective map form the projective space ℙ over (I_{A})_{6} mod multiples of C, to R′ ∩ ⟨v_{6}(A)⟩. Since A is separated by cubics, then ℙ has dimension 27 − 19 = 8. Since ⟨v_{6}(A)⟩ also has dimension 8, and R′ ∩ ⟨v_{6}(A)⟩ is a proper subvariety, because R ≠ R′, the claim follows. □

The same procedure of liaison produces a special subvariety of W.

Example 3.18. Let A be a set of 9 points, complete intersection of two cubics C_{1}, C_{2} Fix a general cubic curve C_{3} and consider the set B = C_{1} ∩ C_{3}. Then A ∩ B = ∅, and Z = A ∪ B is a complete intersection of a cubic and a sextic curve. Thus the h-vector of Z is (1, 2, 3, 3, 3, 3, 2, 1), By Proposition 2.19 of [2], it follows that the spans ⟨v_{6}(A)⟩ and ⟨v_{6}(B)⟩ meet in exactly one point, corresponding to a sextic form F. Since F has a decomposition given by C_{1} ∩ C_{2}, then F corresponds to a point in the 9-secant variety of X spanned by a set in the Terracini locus. Moreover, from the Apolarity Theorem, C_{1}, C_{2} lie in the apolar ideal of F. For the same reason, looking at the decomposition C_{1} ∩ C_{3} of F, one obtains that also C_{3} lies in F^{⊥}. Thus we have an example of a form F whose apolar ideal contains 3 independent cubics C_{1}, C_{2}, C_{3}. Notice that, by the generality of C_{3}, the intersection C_{1}∩C_{2}∩C_{3} is empty. Thus the image of C_{F}^{3} , which is a ℙ^{6}, will not cut the 3-Veronese surface in ℙ^{9}. In particular, F has not rank 7. By the Apolarity Theorem, a decomposition of F can be found by taking any general pair of cubics in the linear system spanned by C_{1}, C_{2}, C_{3}. Thus F has a 2-dimensional family of decompositions.

Definition 3.19. We denote with W′ the closure of the locus of sextic forms F such that the catalecticant map C_{F}^{3} has a kernel of dimension 3.

A general choice of three cubics C_{1}, C_{2}, C_{3} determines F ∈ W′, by taking the intersection of the spans of v_{6}(C_{1} ∩ C_{2}) and v_{6}(C_{1} ∩ C_{3}) (any general choice of two pairs will be suitable and determine the same F). Thus W′ has the dimension of the Grassmannian of planes in ℙ^{9}, i.e. 21.

We stress that there is no way to determine if F belongs to W′ only by looking at the 9 projective points of a decomposition of F. Once one knows that F has a decomposition A which is complete intersection, and one fixes representatives {L_{1}, …, L_{9}} for the points of A, then the membership of F in W′ depends on the coefficients of a linear combination of powers L_{1}^{6}, …, L_{9}^{6} that determine F in ⟨v_{6}(A)⟩.

Observe that the locus S^{7}(X) lies in the intersection of S^{8}(X) with W′. We will see in the next section that, at least set-theoretically, S^{7}(X) = S^{8}(X) ∩ W′.

Example 3.20. Many forms F in the subvariety W′ defined above can be computed following the procedure introduced in Section 4 of [2], as explained in Remark 3.9. Just to see an example of such a form, consider

F = (x_{0}x_{1}x_{2})^{2}.

One easily sees that F^{⊥} contains three independent cubics, which are x_{0}^{3}, x_{1}^{3}, x_{2}^{3}. We have the explicit decomposition of rank 9 [9]

810(x_{0}x_{1}x_{2})^{2} = ∑_{p,q=0}^{2} e^{2πi(p+q)/3} (x_{0} + e^{2πip/3}x_{1} + e^{2πiq/3}x_{2})^{6} .

In the next section we will compute equations for the 8-th secant variety S^{8}(X). By applying these equations to F, one realizes that F does not belong to S^{8}(X). Thus F has rank (and border rank) 9. This proves that W′ is not contained in S^{8}(X).

The relations among the subvarieties R, R′, W, W′ of S^{9}(X) reflect the rich geometry of secant varieties, as soon as the genus approaches the generic value. The most complicate object to describe remains R′, for which we do not have a set of equations. We hope that the geometric structure of R′ will be clarified, in a future footnote, maybe.

4. Equations for loci in S^{9}.

We introduce a Young flattening (see [24]) to get a more refined study of plane sextics. Let T be the tangent bundle of ℙ^{2}, we consider the rank three bundle E = Sym^{2}T. The space of sections H^{0}(Sym^{2}T) is 27-dimensional and can be identified with the SL(3)-module Γ^{4,2}ℂ^{3}, corresponding to the Young diagram

A presentation of Sym^{2}T can be described as follows. First we recall that T is presented by the following exact sequence

0 → O → ℂ^{3} ⊗ O(1) → T → 0

which dualizes to

0 → T^{∨} → ℂ^{3} ⊗ O(−1) → O → 0

and since T^{∨} = T (−3) we get

0 → T → ℂ^{3} ⊗ O(2) → O(3) → 0

The first and third sequence fit together into the presentation of T

ℂ^{3} ⊗ O(1)  ⟶^{f}  ℂ^{3} ⊗ O(2)
↘  ↗
T

where the horizontal skew-symmetric map f(v) = v ∧ x is given by the matrix

(\begin{matrix}0 & x_{2} & −x_{1} \\ −x_{2} & 0 & x_{0} \\ x_{1} & −x_{0} & 0 \end{matrix})

The second symmetric power Sym^{2}T = E appears in the exact sequence

0 → ℂ^{3} ⊗ O(1) → Sym^{2}ℂ^{3} ⊗ O(2) → Sym^{2}T → 0

and repeating the above argument, thanks to the identification Sym^{2}T^{∨} = Sym^{2}T (−6) we get the presentation

Sym^{2}ℂ^{3} ⊗ O(2)  ⟶^{Sym^{2}f}  Sym^{2}ℂ^{3} ⊗ O(4)
↘  ↗
Sym^{2}T

where the horizontal symmetric map Sym^{2}f(v^{2}) = (v ∧ x)^{2} is given by the matrix

(1) B = (\begin{matrix}0 & 0 & 0 & x_{2}^{2} & −2 x_{1}x_{2} & x_{1}^{2} \\ 0 & −2 x_{2}^{2} & 2 x_{1}x_{2} & 0 & 2 x_{0}x_{2} & −2 x_{0}x_{1} \\ 0 & 2 x_{1}x_{2} & −2 x_{1}^{2} & −2 x_{0}x_{2} & 2 x_{0}x_{1} & 0 \\ x_{2}^{2} & 0 & −2 x_{0}x_{2} & 0 & 0 & x_{0}^{2} \\ −2 x_{1}x_{2} & 2 x_{0}x_{2} & 2 x_{0}x_{1} & 0 & −2 x_{0}^{2} & 0 \\ x_{1}^{2} & −2 x_{0}x_{1} & 0 & x_{0}^{2} & 0 & 0 \end{matrix})

Note that for L = O(6) we have E = E^{∨} ⊗ L hence, as in [24] we have a contraction map

(2)  \begin{matrix}End(Sym^{2}ℂ^{3}) & ⟶^{P_{f}} & End(Sym^{2}ℂ^{3}) \\ ↓ & & ↑ \\ H^{0}(E) & ⟶^{A_{f}} & H^{0}(E^{∨} ⊗ L)^{∨} \end{matrix}

where the horizontal map P_{f} for f = v^{6} ∈ v_{6}(ℙ^{2}) is defined as

P_{v^{6}}(M^{2})(w^{2}) = (M(v) ∧ v ∧ w)^{2}v^{2} ∀M ∈ End(ℂ^{3}), w^{2} ∈ Sym^{2}ℂ^{3}

where M^{2} ∈ End(Sym^{2}ℂ^{3}) is defined by M^{2}(v^{2}) = (M(v))^{2} and then extended by linearity to any N ∈ End(Sym^{2}ℂ^{3}) and to any plane sextic f. Comparing with [28, §2] we see that the invariant det A_{f} of degree 27 of plane sextics has a construction similar to the Aronhold invariant of degree 4 of plane cubics.

The coordinate description of P_{f} is the following. Differentiate the 6 × 6 catalecticant C(f) (given by differentiating on rows and columns by monomials of degree 2) by B in (1) . The output is a 36 × 36 matrix obtained by replacing any entry of B by a 6 × 6 catalecticant block.

The M2 [19] commands to get P_{f} are the following, after the matrix B has been defined as in (1)

  R=QQ[x_0..x_2]
  f=x_0^3+x_1^3+x_2^3+5*x_0*x_1*x_2---any cubic polynomial
  P=diff(transpose basis(2,R),diff(basis(2,R),diff(B,f)))

The map A_{f} is symmetric and could be obtained by a convenient submatrix of P_{f}. An alternative way to get a coordinate description is to employ the M2 package “PieriMaps” by Steven Sam with the M2 commands [19]

loadPackage "PieriMaps"
pieri({6,4,2},{1,1,2,2,3,3},QQ[x_0..x_2]);

although the symmetry is not transparent from the coordinates chosen by the system.

The contraction A_{f} in (2) can be pictorially described (compare with [28]) by

⊗ ∗ ∗ ∗ ∗ ∗ ∗ → ∗ ∗ ; ∗ ∗ ; ∗ ∗ ≃

We have the decomposition

End(Sym^{2}ℂ^{3}) = Γ^{4,2}ℂ^{3} ⊕ Γ^{2,1}V ⊕ ∧^{3}ℂ^{3}

corresponding to

⊗  =  ⊕  ⊕

Note that the last two summands correspond to End(ℂ^{3}).

If f ∈ v_{6}(ℙ^{2}) then rk(A_{f}) = rk(E) = 3, so that if f ∈ S^{8}(v_{6}(ℙ^{2}) then rk(A_{f}) ≤ 3 · 8 = 24 and in particular det A_{f} = 0. This was observed already in [24, Theorem 4.2.9].

Proposition 4.1. (1) The variety W of sextics in S^{9}(v_{6}(ℙ^{2})) such that their Waring decomposition comes from a complete intersection of two cubics is cut by the 9-minors of C^{3}. It has codimension 3 and degree 165 and it coincides with the singular locus of S^{9}(v_{6}(ℙ^{2})).

(2) The secant variety S^{8}(v_{6}(ℙ^{2})) coincides with the reduced structure on W ∩ V (det A_{f}), it has codimension 4 and degree 1485 = 165 · 9.

Proof. First recall that the variety of symmetric 10 × 10 matrices of rank ≤ 8 is irreducible of codimension 3 and degree 165 by Segre formula [30]. Since the variety C_{9}^{3} given by the 9-minors of C^{3} has again codimension 3, being a linear section of the previous one, it has the same degree 165. This variety C_{9}^{3} corresponds to the row with s = 8 of Table 3.1 in [22], indeed with the notations of [22], it is the union of irreducible strata Gor(T) for several Hilbert functions T, and one may check using Conca-Valla formula ([16] or [22, Theorem 4.26]) that only one strata has the maximum dimension. It follows that C_{9}^{3} is irreducible. We know that W is irreducible and contained in the variety given by the 9-minors of C^{3}, since the two varieties have the same dimension, the equality follows, proving i). The claim about the singular locus follows because by direct computation on a random linear subspace the singular locus of S^{9}(v_{6}(ℙ^{2})), which obviously contains W, has codimension 3 and degree 165.

In order to prove ii), recall that if f ∈ S^{8}(v_{6}(ℙ^{2})) then det(A_{f}) = 0. Pick 9 general points p_{i} for i = 0, …, 8 and consider the linear span f = ∑_{i=0}^{8} k_{i}p_{i}^{6} with coefficients k_{i}, in other words we restrict f to a general 9-secant space. The expression det A_{f} is symmetric in k_{i}, has total degree 27 and vanishes when ∏_{i=0}^{8} k_{i} vanishes. An explicit computation with the coordinate expression found by the package “PieriMaps” as described above shows that det(A_{f}) coincides, up to scalar

multiples, with (∏_{i=0}^{8} k_{i})^{3}. We get that the condition that det(A_{f}) vanishes on a general 9-secant space is equivalent to f being of rank 8. This expression is unchanged if f ∈ W, namely if f belongs to a 9-secant space corresponding to a complete intersection of two general cubics. This computation shows that V (det A_{f}) meets W with multiplicity 3 at a general point of the intersection, so that the reduced structure on W ∩ V (det A_{f}) has degree 1485 = 165 · 9. Note the rank of A_{f} drops by 3 on W. The degree of S^{8}(v_{6}(ℙ^{2})) can be computed with Numerical Algebraic Geometry. The M2 package NumericalImplicitization by Cho and Kileel shows indeed that its degree is 1485, which can be certified by the trace test. Since S^{8}(v_{6}(ℙ^{2})) ⊆ (W ∩ V (det A_{f}))_{red}, this proves ii) □

Remark 4.2. ii) of Prop. 4.1 strengthens [24, Theorem 4.2.9] where the 25-minors of A_{f} were considered.

We wish now to find the ramification locus of the 2 : 1 map s^{9} : AS^{9}(v_{6}(ℙ^{2})) → S^{9}(v_{6}(ℙ^{2})). Let p_{1}, …, p_{10} be points in ℙ^{2}. Let C(p_{1}, …, p_{10}) be a multihomogeneous polynomial of degree 3 in the coordinates of the points p_{1}, ..., p_{10} , skewsymmetric with respect to them and that vanishes if and only if p_{1}, ..., p_{10} are on a cubic. A coordinate expression is given by the determinant of the 10 × 10 matrix such that its i-th row is the evaluation on p_{i} of the monomial basis of Sym^{3}ℂ^{3}.

We construct now the Terracini matrix of the points p_{1}, …, p_{9} which defines the tangent space at S^{9}(v_{6}(ℙ^{2}) at any point in ⟨p_{1}^{6}, … p_{9}^{6}⟩. Consider the 3 × 28 Jacobian matrix J of the monomial basis of Sym^{6}ℂ^{3}. Let J(p_{i}) be the evaluation of J at p_{i} and let T (p_{1}, …, p_{9}) be the 27×28 Terracini matrix obtained by stacking

J(p_{1}), …, J(p_{9}). Let R(p_{1}, …, p_{9}; p_{10}) = det ( \frac{T (p_{1}, …, p_{9})}{(p_{10})_{0}^{6}  …  (p_{10})_{2}^{6}} )

be the determinant of the 28 × 28 matrix obtained by stacking T (p_{1}, …, p_{9}) and the monomial basis of Sym^{6}ℂ^{3} evaluated at p_{10}.

R(p_{1}, …, p_{9}; p_{10}) is a multihomogeneous polynomial of degree 15 in the coordinates of the points p_{1}, ..., p_{9}, of degree 6 in the coordinates of p_{10}, skew-symmetric with respect to p_{1} …, p_{9} and that vanishes if and only if p_{10} lies on a sextic singular at p_{1}, ..., p_{9}.

Since C(p_{1}, …, p_{10})^{2} is a (non reduced) sextic singular at p_{1}, …, p_{10} we have the factorization

R(p_{1}, …, p_{9}; p_{10}) = C(p_{1}, …, p_{10})^{2}N(p_{1}, …, p_{9})

where N(p_{1}, …, p_{9}) is a multihomogeneous polynomial of degree 9 in the coordinates of the points p_{1}, ..., p_{9} , symmetric with respect to p_{1} …, p_{9} and that vanishes if and only if there is a reduced sextic singular at p_{1}, ..., p_{9}. Note that the result in [4] that the unique sextic singular at p_{1}, …, p_{9} is the non-reduced curve C(p_{1}, …, p_{9}, p)^{2} in p is equivalent to the fact that T has maximal rank for a general choice of p_{1}, …, p_{9}, which can be checked by a random choice. This proves that N(p_{1}, …, p_{9}) is a nonzero polynomial, which defines the codimension 1 condition that the space of sextics singular at p_{1}, …, p_{9} has dimension ≥ 2. It can be checked that at general p_{1}, …, p_{9} on the hypersurface N(p_{1}, …, p_{9}) = 0 then T has corank 1, while if p_{1}, …, p_{9} are distinct points defined by a general complete intersection then both R(p_{1}, …, p_{9}; p_{10}) and C(p_{1}, …, p_{10}) vanish ∀p_{10} and N(p_{1}, …, p_{9}) ≠ 0.

We note also that the same argument given in the proof of [29, Theor. 45] shows that N(p_{1}, …, p_{9}) is irreducible, since there are no nonzero symmetric cubic invariants of 9 points in ℙ^{2}.

Remark 4.3. For general p_{1}, …, p_{8}, the locus of ninth point p_{9} such that there is a reduced sextic singular at p_{1}, …, p_{8}, p_{9} has two irreducible components, namely the nonic N(p_{1}, …, p_{9}) = 0 and the point p_{9} in the base locus of the linear system of cubics through p_{1}, …, p_{8}. The general sextic arising from the first (resp. second) component is irreducible (resp. reducible) and lies in R (resp. in W), see Definition 3.12. More precisely, R ⊂ S^{9}(v_{6}(ℙ^{2})) is the closure of ∪⟨p_{1}^{6}, …, p_{9}^{6}⟩, where the union is taken for distinct p_{1}, …, p_{9} satisfying N(p_{1}, …, p_{9}) = 0. As noted in Remark 3.13, R and W are two irreducible components of the ramification locus of the 2 : 1 map AS^{9}(v_{6}(ℙ^{2})) → S^{9}(v_{6}(ℙ^{2})). They are distinct since H_{27} vanishes on R and not on W. When N(p_{1}, …, p_{9}) = 0 then Z = {p_{1}, …, p_{9}} is self-linked on the cubic C(p_{1}, …, p_{9}, p) with respect to any irreducible sextic singular at p_{1}, …, p_{9}, which meets the cubic in 2p_{1} + … + 2p_{9}.

Theorem 4.4. Let f = ∑_{i=1}^{9} p_{i}^{6} be a rank 9 sextic, in affine notation. Then

det A_{f} = λN(p_{1}, … p_{9})^{2}

for a nonzero scalar λ ∈ ℂ^{∗}.

Proof. Recall we denoted E = Sym^{2}T and denote Z = {p_{1}, …, p_{9}}. Assume that det A_{f} vanishes, then ker A_{f} is a nonzero subspace of H^{0}(E). Consider the restriction H^{0}(E) ⟶^{j} H^{0}(E_{|Z}), note both sides are 27-dimensional. Assume j is injective. Then by [24, Lemma 5.4.1] we have that H^{0}(I_{Z} ⊗E) = ker A_{f} is nonzero, which is a contradiction since a section vanishing on Z is in the kernel of j. It follows that j is not injective and there is a section s ∈ H^{0}(E) vanishing at Z = {p_{1}, …, p_{9}}. Again by [24, Lemma 5.4.1] the section s belongs to ker A_{f}. Then the contraction

Sym^{2}H^{0}(I_{Z} ⊗ E) → H^{0}(I_{Z} ⊗ E) ⊗ H^{0}(I_{Z} ⊗ E^{∨} ⊗ O(6)) → H^{0}(I_{Z^{2}} ⊗ O(6))

as in [24, Theorem 5.4.3] takes s^{2} to a sextic singular at Z. There are two cases, Z is a complete intersection or N(p_{1}, …, p_{9}) = 0. The first case has codimension 3 by 4.1 and can be excluded in proving a polynomial equality, like in our statement. It follows that det A_{f} = 0 implies generically N(p_{1}, …, p_{9}) = 0, hence a power of N(p_{1}, … p_{9}) is divided by det A_{f}. The multihomogeneous polynomial det A_{f} is symmetric in p_{1}, …, p_{9} and has total degree 27 · 6, hence must have degree 18 in each p_{i}. The result follows since N(p_{1}, …, p_{9}) is irreducible. □

Corollary 4.5. The variety R is the complete intersection of the two hypersurfaces det A_{f} and S^{9}(v_{6}(ℙ^{2})) and has degree 10 · 27 = 270.

Tables about the loci we have studied

In the next table we list some informations about the loci we have studied in ℙ^{27} = ℙ(Sym^{6}ℂ^{3}). The main theme is that (C)_{3} and the invariant H_{27} = det A_{f} are enough to describe all k-secant varieties to v_{6}(ℙ^{2}).

dim  deg  equations
S^{9} 26  10  det C^{3}  9-secant
R 25  270  det C^{3}, H_{27} comp. of ramif. locus of AS^{9} → S^{9}
W 24  165  (C^{3})_{9}  Sing S^{9}, span of compl. inters. 9-ples
S^{8} 23 1485 = (165 · 27)/3 (C^{3})_{9} , H_{27}  8-secant
W′ 21 2640 (C^{3})_{8} sextics with three apolar cubics
S^{7} 20 11880 = (2640 · 27)/6 (C^{3})_{8} , H_{27}  7-secant

The fact that Sing S^{9} = W has been checked computationally, since the singular locus of S^{9} has codimension 3 and degree 165, and contains the variety W given by the 9-minors of C^{3}, hence the equality follows. The following graph links the containments among the several loci.

S^{9}
| ⧹
W  R
| ⧹ |
W′  S^{8}
⧹ |
S^{7}

The right colums is obtained by the left column after cut with the invariant hypersurface H_{27}. The three diagonal links correspond to cut with the invariant H_{27}. The matrix A_{f} drops rank by 1 in the first link, by 3 in the second one, by 6 in the third one. Hence a check on the degrees shows that the intersection with H_{27} does not contain other components in all the three cases.

The following table lists the properties related to Waring decomposition of general member of the loci (brk is the border rank)

rk C^{3} brk rk #{ minimal Waring decompositions}
S^{9}  9  9 9 2
R  9  9 9 1
W  8  9 9 1 (Prop.3.10)
S^{8}  8  8 8 1
W′ 7  9 9 ∞
S^{7}  7  7 7 1

Tables about the dual loci

dim  deg
(S^{7})^{∨} 20  34435125 [10]  irred. sextics with 7 singular pts
(W′)^{∨} 26  83200 [7]  sums of three squares
(S^{8})^{∨} 19  58444767 [10]  irred. sextics with 8 singular pts
W^{∨} 18  \frac{1}{2} \binom{18}{9} = 24310 reducible in 2 cubics, or sums of two squares
R^{∨}  18 57435240 [10, 18]  irred. sextics with 9 singular pts
(S^{9})^{∨} 9  2^{9} = 512  square of a cubic
(S^{7})^{∨}
|
(S^{8})^{∨}  (W′)^{∨}
| ⧹ |
R^{∨}  W^{∨}
⧹  |
(S^{9})^{∨}

Note that the last diagram about dual loci is not dual to the previous one, since by dimensional reasons (W′)^{∨} is not contained in (S^{7})^{∨}. Nevertheless we have the inclusion S^{7} ⊂ W′.

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Luca Chiantini. Dipartimento di Ingegneria dell’Informazione e Scienze Matematiche, Università di Siena, Italy

Email address: luca.chiantini@unisi.it

Giorgio Ottaviani. Dipartimento di Matematica e Informatica ‘Ulisse Dini’, Università di Firenze, Italy

Email address: giorgio.ottaviani@unifi.it