The Variety of Polar Simplices

Kristian Ranestad and Frank-Olaf Schreyer^{1}

Received: April 25, 2012 Revised: January 29, 2013

Communicated by Gavril Farkas

Abstract. A collection of n distinct hyperplanes L_{i} = {l_{i} = 0} ⊂ P^{n−1}, the (n − 1)-dimensional projective space over an algebraically closed field of characteristic not equal to 2, is a polar simplex of a smooth quadric Q^{n−2} = {q = 0}, if each L_{i} is the polar hyperplane of the point p_{i} = ⋂_{j≠i} L_{j}, equivalently, if q = l_{1}^{2} + … + l_{n}^{2} for suitable choices of the linear forms l_{i}. In this paper we study the closure V P S(Q, n) ⊂ Hilb_{n}(P̌^{n−1}) of the variety of sums of powers presenting Q from a global viewpoint: V P S(Q, n) is a smooth Fano variety of index 2 and Picard number 1 when n < 6, and V P S(Q, n) is singular when n ≥ 6.

2010 Mathematics Subject Classification: 14J45, 14M Keywords and Phrases: Fano n-folds, Quadric, polar simplex, syzygies

Contents

1. Introduction  470
2. Apolar subschemes of length n  474
3. A rational parameterization  477
4. The Mukai form  480
5. An open affine subvariety  487
6. Global invariants of V P S(Q, n)  501
References  504

1The first author was partially supported by an UCM-EEA grant under the NILS program, and both authors were supported by Institut Mittag-Leffler.

1. Introduction

Let Q = {q = 0} be a (n − 2)-dimensional smooth quadric defined over the complex numbers, or any algebraically closed field of characteristic not equal to 2. We denote the projective space containing Q by P̌^{n−1} because its dual space P^{n−1} plays the major role in this paper. A collection L_{1} = {l_{1} = 0}, …, L_{n} = {l_{n} = 0} of n hyperplanes is a polar simplex iff each L_{i} is the polar of the point p_{i} = ⋂_{j≠i} L_{j}, equivalently, iff the quadratic equation

q = ∑_{i=1}^{n} l_{i}^{2}

holds for suitable choices of the linear forms l_{i} defining L_{i}. In this paper we study the collection of polar simplices, or equivalently, the variety of sums of powers presenting q from a global viewpoint. We may regard a polar simplex as a point in Hilb_{n}(P^{n−1}). Let V P S(Q, n) ⊂ Hilb_{n}(P^{n−1}) be the closure of the variety of sums of n squares presenting Q. The first main result is:

Theorem 1.1. If 2 ≤ n ≤ 5, then V P S(Q, n) is a smooth rational \binom{n}{2}- dimensional Fano variety of index 2 and Picard number 1. If n ≥ 6, then V P S(Q, n) is a singular rational \binom{n}{2}-dimensional variety.

If n = 2, then V P S(Q, n) = P^{1}, and if n = 3, then V P S(Q, n) is a rational Fano threefold of index 2 and degree 5 (cf.[Muk92]). The quadratic form defines a collineation q : P̌^{n−1} → P^{n−1}, let q^{−1} : P^{n−1} → P̌^{n−1} be the inverse collineation, and Q^{−1} = {q^{−1} = 0} ⊂ P^{n−1} the corresponding quadric. Consider the double Veronese embedding Q^{−1} → P^{\binom{n+1}{2}−2}, and let T Q^{−1} be the image by the Gauss map of tangent spaces Q^{−1} → 𝔾(n − 1, \binom{n+1}{2} − 1). Our second main result is:

Theorem 1.2. V P S(Q, n) has a natural embedding in the Grassmannian variety 𝔾(n−1, \binom{n+1}{2}−1) and contains the image T Q^{−1} of the Gauss map of the quadric Q^{−1} in its Veronese embedding. When n = 4 or n = 5. the restriction of the Plcker line bundle generates the Picard group of V P S(Q, n), and the degree is 310, resp. 395780.

We denote the coordinate ring of P^{n−1} by S = ℂ[x_{1}, …, x_{n}] and the coordinate ring of the dual P̌^{n−1} by T = ℂ[y_{1}, …, y_{n}]. In particular S_{1} = (T_{1})^{∗}, so we may set P^{n−1} = P(T_{1}), the projective space of 1-dimensional subspaces of T_{1} with coordinate functions in S, and P̌^{n−1} = P(S_{1}) with coordinate functions in T. Let q ∈ T = ℂ[y_{1}, …, y_{n}] be a quadratic form defining the smooth (n − 2)- dimensional quadric Q ⊂ P̌^{n−1} = P(S_{1}). Regard [q] as a point in P(T_{2}) and consider the Veronese variety V_{2} ⊂ P(T_{2}) of squares,

V_{2} = {[l^{2}] ∈ P(T_{2})|l ∈ P(T_{1})}.

Then a polar simplex to Q is simply a collection of n points on V_{2} whose linear span contains [q]. Any length n subscheme Γ ⊂ V_{2} whose span in P(T_{2}) contains [q] is called an apolar subschemes of length n to Q. The closure V P S(Q, n) of the polar simplices in Hilb_{n}(P(T_{1})) consists of apolar subschemes of length n. We denote by V AP S(Q, n) the subset of Hilb_{n}(P(T_{1})), with reduced scheme structure, parameterizing all apolar subschemes of length n to Q. Our third main result is:

Theorem 1.3. The algebraic set V AP S(Q, n) is isomorphic to the complete linear section

V AP S(Q, n) = ⟨T Q^{−1}⟩ ∩ 𝔾(n − 1, T_{2}/q) ⊂ P(∧^{n−1}(T_{2}/q))

in the Plücker space. For n ≤ 6 the two subschemes V P S(Q, n) and V AP S(Q, n) coincide. For n ≥ 24, the scheme V AP S(Q, n) has more than one component.

Notice that we do not claim that the linear section ⟨T Q^{−1}⟩ ∩ 𝔾(n − 1, T_{2}/q) is reduced, only that its reduced structure coincides with V AP S(Q, n). The linear span ⟨T Q^{−1}⟩ has dimension \binom{2n−1}{n−1}−\binom{2n−3}{n−2}−1, while the Grassmannian

has dimension (n−1)\binom{n}{2} in \binom{\binom{n+1}{2}−1}{n−1}-dimensional Plücker space. So this linear section is far from a proper linear section when n ≥ 4, i.e. the codimension of V AP S(Q, n) in the Grassmannian is much less than the codimension of its linear span in the Plücker space. We find a covering of V AP S(Q, n) by affine subschemes V_{h}^{aff} (n) that are contractible to a point [Γ_{p}] ∈ V P S(Q, n) (Lemma 5.3). Therefore the apolar subschemes Γ_{p} play a crucial point. Let us explain what they are: The projection of the Veronese variety V_{2} ⊂ P(T_{2}) from [q] ∈ P(T_{2}) is a variety V_{2,q} ⊂ P(T_{2}/q). Since a polar simplex to Q is a collection on n points on V_{2} whose span contains [q], the variety V SP (Q, n) is naturally embedded in and in fact coincides with the variety of (n − 2)-secant spaces of the projected Veronese variety V_{2,q}. The double Veronese embedding of Q^{−1} is a linearly normal subvariety in V_{2,q} that spans P(T_{2}/q). For each point p ∈ Q^{−1} consider the tangent space to Q^{−1} in this embedding. This tangent space intersects V_{2,q} along the subscheme Γ_{p}, and belong to the boundary of variety of (n − 2)-secant spaces of V_{2,q}. The affine subscheme V_{h}^{aff} (n) is contractible to Γ_{p}, but depend only on a hyperplane: It consists of the apolar subschemes that do not intersect a tangent hyperplane h to Q^{−1}. The point p is simply a point on Q^{−1} that does not lie in this hyperplane. Our computations show that the affine scheme V_{h}^{aff} (n) and certain natural subschemes has particularly interesting structure: V_{h}^{aff} (n) is isomorphic to an affine space when n < 6 while V_{h}^{aff} (6) is isomorphic to a 15-dimensional cone over the 10-dimensional spinor variety (Corollary 5.16). Why this spinor variety appears is quite mysterious to us. Recall that Mukai showed that a general canonical curve of genus 7 is a linear section of the spinor variety. Let V_{p}^{loc}(n) ⊂ V AP S(Q, n) be the subscheme of apolar subschemes in V AP S(Q, n) with support at a single point p ∈ Q^{−1}. The subscheme V_{p}^{loc}(n) is naturally contained in V_{p}^{sec}(n), the variety of apolar subschemes in V_{h}^{aff} (n) that contains the point p. We compute these subschemes with Macaulay2 [GS] when n < 6 and find that V_{p}^{loc}(5) is isomorphic to a 3-dimensional cone over the tangent developable of a rational normal sextic curve. This cone is a codimension 3 linear section of the scheme V_{p}^{sec}(5), which is isomorphic to a 6-dimensional cone over the intersection of the Grassmannian 𝔾(2, 5) with a quadric. Mukai showed that a general canonical curve of genus 6 is a linear section of the intersection of 𝔾(2, 5) with a quadric. The appearances in the cases n = 5, 6 of a natural variety whose curve sections are canonical curves is both surprising and unclear to us. The computational results are summarized in Table 1 in Section 5. By the very construction of polar simplices, it is clear that V P S(Q, n) has dimension \binom{n}{2}. On the other hand, the special orthogonal group SO(n, q) that preserves the quadratic form q, acts on the set of polar simplices: If we assume that the symmetric matrix of q with respect to the variables in T is the identity matrix, then regarding SO(n, q) as orthogonal matrices the rows define a polar simplex. Matrix multiplication therefore defines a transitive action of SO(n, q) on the set of polar simplices. By dimension count, this action has a finite stabilizer at a polar simplex. This stabilizer is the subgroup H ⊂ SO(n, q) of rotational symmetries of the hypercube [−1, 1]^{n} ⊂ ℝ^{n} of order 2^{n−1} · n! as suggested by an anonymous referee. We get

Proposition 1.4. V P S(Q, n) is a compactification of the group SO(n, q)/H.

The linear representation of SO(n, q) on T_{2} decomposes

T_{2} = ⟨q⟩ ⊕ T_{2,q},

where the hyperplane P(T_{2,q}) intersect the Veronese variety V_{2} along the Veronese image of Q^{−1}. Therefore we may identify T_{2}/q = T_{2,q} and the projection from [q]: P(T_{2}) −−→ P(T_{2,q}) is an SO(n, q)-equivariant projection. Q^{−1} ⊂ P(T_{2,q}) is a closed orbit, and similarly the image T Q^{−1} of the Gauss map is a closed orbit for the induced representation on the Plücker space of 𝔾(n − 1, T_{2,q}). The linear span of this image is therefore the projectivization of an irreducible representation of SO(n, q). The set of polar simplices form an orbit for the action of SO(n, q), so the linear span of V P S(Q, n) is also the projectivization of an irreducible representation of SO(n, q). Therefore

V P S(Q, n) ⊂ ⟨T Q^{−1}⟩ ∩ 𝔾(n − 1, T_{2,q}).

We show that the intersection ⟨T Q^{−1}⟩∩ 𝔾(n−1, T_{2,q}) parameterizes all apolar subschemes of length n, hence Theorem 1.1. The organization of the paper follows distinct approaches to V P S(Q, n). To start with we introduce the classical notion of apolarity and regard polar simplices as apolar subschemes in P(T_{1}) of length n with respect to q. We use syzygies to characterize these subschemes among elements of the Hilbert scheme. In fact, polar simplices are characterized by their smoothness, the Betti numbers of their resolution, and their apolarity with respect to q. Allowing singular subschemes, we consider all apolar subschemes of length n. We show in Section 2 that these subschemes naturally appear in the closure V P S(Q, n) of the set of polar simplices in the Hilbert scheme. For n > 6 there may be apolar subschemes of length n that do not belong to the closure V P S(Q, n) of the smooth ones. In fact, we show in Section 2 that at least for n ≥ 24, there are nonsmoothable apolar subschemes of length n, i.e. that V P S(Q, n) is not the only component of V AP S(Q, n). The variety V P S(Q, n), in its embedding in 𝔾(n − 1, T_{2,q}), has order one, i.e. through a general point in P(T_{2,q}) there is a unique (n − 2)-dimensional linear space that form the span of an apolar subscheme Γ of length n. This is a generalization of the fact that a general symmetric n × n matrix has n distinct eigenvalues. In Section 3 we use a geometric approach to characterize the generality assumption. The fact that V P S(Q, n) has order one, means that it is the image of a rational map

γ : P(T_{2,q}) −−→ 𝔾(n − 1, T_{2,q}).

In Section 4 we use a trilinear form introduced by Mukai to give equations for the map γ. With respect to the variables in T we may associate a symmetric matrix A to each quadratic form q′ ∈ T_{2,q}. The Mukai form associates to q′ a space of quadratic forms in S_{2} that vanish on all the projectivized eigenspaces of the matrix A. For general q′ these quadratic forms generate the ideal of the unique common polar simplex of q and q′. This is Proposition 4.2. The Mukai form therefore defines the universal family of polar simplices, although it does not extend to the whole boundary. Common apolar subschemes to q and q′, when q′ has rank at most n − 2, form the exceptional locus of the map γ. We do not compute the image of γ in 𝔾(n − 1, T_{2,q}). Instead we compute affine perturbations of [Γ_{p}] in 𝔾(n − 1, T_{2,q}) that correspond to apolar subschemes to Q. These perturbations form the affine subschemes V_{h}^{aff} (n) that cover V AP S(Q, n). In Section 5 we make extensive computations of these affine subschemes. Each once of them is contractible to a point [Γ_{p}] on the subvariety T Q^{−1} ⊂ V P S(Q, n). The question of smoothness of V P S(Q, n) is reduced to a question of smoothness of the affine scheme V_{h}^{aff} (n) at the point [Γ_{p}]. For n ≤ 5 we show that such a point is smooth, while for n ≥ 6, it is singular. The main result of Section 5 is however Theorem 1.3, that V AP S(Q, n) is a linear section of the Grassmannian. In the final Section 6 we return to the geometry of V P S(Q, n) and compute the degree by a combinatorial argument for any n. The Fano-index is computed using the natural P^{n−2}-bundle on V P S(Q, n), obtained by restricting the incidence variety over the Grassmannian, and its birational morphism to P(T_{2,q}). We thank Tony Iarrobino for sharing his insight on Artinian Gorenstein rings with us, Francesco Zucconi for valuable comments on a previous version of this paper. Let us briefly summarize the notation:

• ℂ denotes the field of complex numbers.

• q ∈ T_{2} is a non-degenerate quadratic form, and defines a collineation q : S_{1} → T_{1} and a linear form q : S_{2} → ℂ.

• Q is the quadratic hypersurface {q = 0} ⊂ P(S_{1}).

• q^{−1} ∈ S_{2} is a quadratic form, that defines the collineation q^{−1} : T_{1} → S_{1} inverse to q and a linear form q^{−1} : T_{2} → ℂ.

• Q^{−1} is the quadratic hypersurface {q^{−1} = 0} ⊂ P(T_{1}).

• q^{⊥} ⊂ S_{2} is the kernel of the linear form q : S_{2} → ℂ.

• T_{2,q} is the kernel (q^{−1})^{⊥} of the linear form q^{−1} : T_{2} → ℂ

• π_{q} : P(T_{2}) −−→ P(T_{2,q}) is the projection from [q] ∈ P(T_{2}), and V_{2,q} ⊂ P(T_{2,q}) is the image under this projection of the Veronese variety V_{2} ⊂ P(T_{2}).

2. Apolar subschemes of length n

We follow the approach of [RS00]: The apolarity action is defined as the action of S = ℂ[x_{1}, …, x_{n}] as polynomial differential forms on T = ℂ[y_{1}, …, y_{n}] by setting x_{i} = \frac{∂}{∂y_{i}}. This makes the duality between S_{1} and T_{1} explicit and, in fact, defines a natural duality between T_{i} and S_{i}. The form q ∈ T_{2} define the smooth (n − 2)-dimensional quadric hypersurface

Q = \{ [∑ a_{i}x_{i}] | (∑ a_{i} \frac{∂}{∂y_{i}})^{2}(q) = 0\} ⊂ P(S_{1}).

Apolarity defines a graded Artinian Gorenstein algebra associated to Q:

A^{Q} = ℂ[x_{1}, …, x_{n}]/(q^{⊥})

where

q^{⊥} = {D ∈ S_{2} = ℂ[x_{1}, …, x_{n}]_{2}|D(q) = 0}.

A subscheme Y ⊂ P(T_{1}) is apolar to Q, or equivalently apolar to q, if the space of quadratic forms in its ideal I_{Y,2} ⊂ q^{⊥}. The apolarity lemma (cf.[RS00] 1.3) says that any smooth Γ, [Γ] ∈ Hilb_{n}(P(T_{1})) is a polar simplex with respect to Q ⊂ P(S_{1}) = P̌^{n−1} if and only if I_{Γ,2} ⊂ q^{⊥} ⊂ S_{2}, i.e. Γ is apolar to Q. We drop, for the moment, the smoothness criterium and consider any [Γ] ∈ Hilb_{n}(P(T_{1})), such that Γ is apolar to Q. Notice that since Q is nonsingular, Γ is nondegenerate. But more is known: The following are the graded Betti numbers of A^{Q} and Γ, given in Macaulay2 notation [GS].

Proposition 2.1. a) For a smooth quadric Q ⊂ P̌^{n−1} the syzygies of the apolar Artinian Gorenstein ring A^{Q} are

\begin{matrix}1 & − & … & − & … & − & − \\ − & \frac{n−1}{n+1} \binom{n+2}{2} & … & \frac{k(n−k)}{n+1} \binom{n+2}{k+1} & … & \frac{n−1}{n+1} \binom{n+2}{n} & − \\ − & − & … & − & … & − & 1 \end{matrix}

b) A zero-dimensional nondegenerate scheme Γ ⊂ P^{n−1} of length n has syzygies

\begin{matrix}1 & − & … & − & … & − & − \\ − & \binom{n}{2} & … & k\binom{n}{k+1} & … & (n − 1)\binom{n}{n} & − \end{matrix}

Proof. Eg. [Beh81] and [ERS81] □ Corollary 2.2. The natural morphism

V AP S(Q, n) → 𝔾(\binom{n}{2}, q^{⊥}); Γ ↦ I_{Γ,2} ⊂ q^{⊥}

is injective. Equivalently, there is a natural injective morphism

V AP S(Q, n) → 𝔾(n − 1, T_{2,q}); Γ ↦ I_{Γ,2}^{⊥} ⊂ T_{2,q}

into the variety of (n − 2)-dimensional subspaces of P(T_{2,q}) that intersect the projected Veronese variety V_{2,q} in a scheme of length n. In particular, the Hilbert scheme and Grassmannian compactification in 𝔾(n − 1, T_{2,q}) of the variety of polar simplices coincide.

Proof. Apolarity defines a natural isomorphism q^{⊥} ≅ T_{2,q}^{∗}. Therefore the subspace I_{Γ,2} ⊂ q^{⊥} defines a (n − 1)-dimensional subspace I_{Γ,2}^{⊥} ⊂ T_{2,q}. The intersection P(I_{Γ,2}^{⊥}) ∩ V_{2,q} with the projected Veronese variety is precisely π_{q}(Γ). The variety V P S(Q, n) ⊂ Hilb_{n}(P^{n−1}) is the closure of the set of polar simplices inside the set of apolar subschemes of length n. The former set is irreducible, while the latter set is a closed variety defined by the condition that the generators of the ideal of the subscheme lie in q^{⊥}. By Proposition 2.1, the map Γ ↦ I_{Γ,2} ⊂ q^{⊥} extends to all of V P S(Q, n) as an injective morphism. □

We relate apolarity to polarity with respect to a quadric hypersurface. The classical notion of polarity is the composition of the linear map q^{−1} with apolarity: The polar to a point [l] ∈ P(T_{1}) with respect to Q^{−1} is the hyperplane h_{l} = P(q^{−1}(l)^{⊥}) ⊂ P(T_{1}), where

(q^{−1}(l))^{⊥} = {l′ ∈ T_{1}|l′(q^{−1}(l)) = q^{−1}(l · l′) = 0}.

In particular, the polar hyperplane to l contains l if and only if q^{−1}(l^{2}) = 0, i.e. the point [l] lies on the hypersurface Q^{−1}. Let Γ ⊂ P(T_{1}) be a length n subscheme that contains [l] and is apolar to Q. The subscheme Γ′ ⊂ Γ residual to [l] is defined by the quotient I_{Γ′} = I_{Γ} : (l^{⊥}). Since Γ is non degenerate, Γ′ spans a unique hyperplane. This hyperplane is defined by a unique linear form u′ ∈ S_{1}, and is characterized by the fact that u′ · u(q) = u′q(u) = 0 for all u ∈ l^{⊥}, so it is the hyperplane P(q(l^{⊥})). But

l′ ∈ q(l^{⊥}) ⇔ 0 = q^{−1}(l′)l = q^{−1}(l · l′) = q^{−1}(l)l′ ⇔ l′ ∈ (q^{−1}(l))^{⊥},

so P(q(l^{⊥})) is the polar hyperplane P(q^{−1}(l)^{⊥}) to [l] with respect to Q^{−1}. Thus the subscheme Γ′ residual to [l] in Γ spans the polar hyperplane to [l] with respect to Q^{−1}.

Lemma 2.3. A component of an apolar subscheme has support on Q^{−1} if and only if this component is nonreduced.

Proof. If a component is a reduced point, the residual is contained in the polar hyperplane to this point, so by nondegeneracy the polar hyperplane cannot contain the point. If a component is nonreduced, the residual to the point supporting the component lies in the polar hyperplane to this point, so the point is on Q^{−1}. □

Each component Γ_{0} of an apolar subscheme to q is apolar to a quadratic form q_{0} defined on the span of Γ_{0} and uniquely determined as a summand q. This is the content of the next proposition.

Proposition 2.4. Let Γ = Γ_{1} ∪ Γ_{2} be an apolar subscheme of length n to q that decomposes into two disjoint subschemes Γ_{1} and Γ_{2} of length n_{1} and n_{2}. Let U_{1} ⊂ T_{1} and U_{2} ⊂ T_{1} be subspaces such that Γ_{i} spans P(U_{i}). Then there is a unique decomposition q = q_{1} + q_{2} with q_{i} ∈ (U_{i})^{2}. Furthermore, subschemes Γ_{1} ⊂ P(U_{1}) and Γ_{2} ⊂ P(U_{2}) of length n_{1} and n_{2} are apolar to q_{1} and q_{2} respectively, if and only if Γ_{1} ∪ Γ_{2} is apolar to q.

Proof. Since Γ is nondegenerate, T_{1} = U_{1} ⊕ U_{2}. Let U_{i}^{⊥} ⊂ S_{1} be the space of forms vanishing on U_{i} via apolarity. Then U_{1}^{⊥} are natural coordinates on P(U_{2}) and likewise, U_{2}^{⊥} are natural coordinates on P(U_{1}). Let I_{1} ⊂ (U_{2}^{⊥})^{2} be the quadratic forms generating the ideal of Γ_{1} in P(U_{1}), and likewise I_{2} the quadratic forms generating the ideal of Γ_{2} in P(U_{2}). Then I_{1}⊕I_{2}⊕(U_{1}^{⊥})·(U_{2}^{⊥}) ⊂ S_{2} is the space of quadratic forms in the ideal of Γ. Consider the intersections, q_{2}^{⊥} = q^{⊥} ∩ (U_{1}^{⊥})^{2} and q_{1}^{⊥} = q^{⊥} ∩ (U_{2}^{⊥})^{2}. Since q is non degenerate, q^{⊥} does not contain either of the subspaces (U_{i}^{⊥})^{2}. Therefore q_{2}^{⊥} is a codimension one subspace in (U_{1}^{⊥})^{2} and is apolar to a quadratic form q_{2} ∈ (U_{2})^{2}, unique up to scalar. Similarly, q_{1}^{⊥} is apolar to a unique quadratic from q_{1} ∈ (U_{1})^{2}. The space of quadratic forms q_{1}^{⊥}⊕q_{2}^{⊥}⊕(U_{1}^{⊥})·(U_{2}^{⊥}) is contained in q^{⊥} and is apolar to the subspace ⟨q_{1}, q_{2}⟩ ⊂ T_{2}. Therefore, there are unique nonzero coefficients c_{1} and c_{2} such that q = c_{1}q_{1} + c_{2}q_{2}. Furthermore, each Γ_{i} is apolar to q_{i}, i = 1, 2. It remains only to show the last statement. Assume Γ_{1} and Γ_{2} are apolar to q_{1} and q_{2} respectively. Then Γ_{1}∪Γ_{2} is non degenerate of length n. Let I_{1} ⊂ (U_{2}^{⊥})^{2} be the generators of the ideal of Γ_{1} and I_{2} ⊂ (U_{1}^{⊥})^{2} be the generators of the ideal of Γ_{2}. Then the quadratic forms in

I_{1} ⊕ I_{2} ⊕ (U_{1})^{⊥} · (U_{2})^{⊥}

all lie in the ideal of Γ_{1} ∪ Γ_{2}. The dimension of this space of quadratic forms is

\binom{n_{1}}{2} + \binom{n_{2}}{2} + n_{1} · n_{2} = \binom{n}{2},

so they generate the ideal of Γ_{1} ∪ Γ_{2}. Since all these forms are apolar to q = q_{1} + q_{2}, the subscheme Γ_{1} ∪ Γ_{2} is apolar to q. □

Remark 2.5. By Proposition 2.4, the orbits of SO(n, q) in V ASP (Q, n) are characterized by their components.

We shall return to the set of local apolar subschemes V_{p}^{loc}(n) supported at a point p ∈ Q^{−1} in section 5. Here we show that apolar subschemes of length n to q are all locally Gorenstein.

Lemma 2.6. Let B be a local Artinian ℂ = B/m_{B}-algebra of length n and Φ : SpecB → A^{n−1} ⊂ P^{n−1} the reembedding given by ℂ-basis of m_{B}. The subscheme Im Φ is apolar to a full rank quadric if and only if B is Gorenstein.

Proof. Let ϕ : A = ℂ[x_{1}, .., x_{n−1}] → B be the ring homomorphism corresponding to Φ. Thus ϕ is defined by a linear k-isomorphism ϕ_{1} = A_{≤1} = ⟨1, x_{1}, ..., x_{n−1}⟩ → B. Let π : B → (0 : m_{B}) be the projection onto the socle of B, let ψ : (0: m_{B}) → ℂ be a linear form and consider the bilinear form

A × A ⟶^{ϕ·ϕ} B ⟶^{π} (0 : m_{B}) ⟶^{ψ} ℂ,

where the first map is the composition of ϕ with multiplication. This map extends to the tensor product A⊗A, and the restriction then to the symmetric part

(A_{≤1})^{2} ⊂ A_{≤1} ⊗ A_{≤1}

defines a linear form

β_{ψ} : (A_{≤1})^{2} → ℂ

and an associated quadratic form

q_{ψ} : A_{≤1} → ℂ.

Clearly the kernel of β_{ψ} generate an ideal in A that is apolar to q_{ψ}. On the other hand, B is Gorenstein if and only if the socle is 1-dimensional. So for the lemma, it suffices to prove that q_{ψ} is non degenerate, i.e. has rank n, if and only if the linear form ψ is an isomorphism. But q_{ψ} is degenerate if and only if the kernel of β_{ψ} contains x · A_{≤1} for some nonzero element x ∈ A_{≤1}. Now, β_{ψ}(x · A_{≤1}) = 0 if and only if ϕ(x) · B ∩ (0 : m_{B}) ⊂ ker ψ. Since B is Artinian, ϕ(x) · B ∩ (0 : m_{B}) is a nonzero subspace of (0 : m_{B}), so it suffices to consider elements x, which map to the socle. But then the kernel of β_{ψ} contains x · A_{≤1} precisely when x is in the kernel of ψ and the lemma follows. □

Corollary 2.7. V AP S(Q, n) is reducible for n ≥ 24

Proof. Consider a general graded Artinian Gorenstein algebra B of embedding dimension e and socle in degree 3. The length of B is 2e + 2. By the Macaulay correspondence [Mac16], such algebras are in bijection with homogeneous forms, up to scalars, of degree 3 in e variables, hence depends on \binom{e−1+3}{3} = (e + 2)(e + 1)e/6 − 1 variables. The family of smoothable algebras have dimension at most e(2e + 2) − 1 So for e + 2 > 2 · 6 a general algebra B cannot be smoothable, for trivial reason. In particular, e = 11 hence n = 24 is enough. □

We do not believe the bound n ≥ 24 is sharp.

3. A rational parameterization

In this section we show that through a general point in P(T_{2,q}) there is a unique n-secant (n − 2)-space to the projected Veronese variety V_{2,q}. Furthermore, we give a characterization of the points for which there are more than one, i.e. infinitely many n-secant (n − 2)-spaces to V_{2,q}. If we choose basis a for T_{1} such that the symmetric matrix associated to q is the identity matrix, then the eigenvectors of the symmetric matrix associated to a general quadric q′ are distinct. Thus, the symmetric matrices associated to q and q′ have a unique set of n common 1-dimensional eigenspaces. Hodge & Pedoe [HP52, XIII.8,Theorem II] and Gantmacher [Gan59, Chapter XII, Theorem 3] found canonical forms for any pair of quadratic forms q, q′ as soon as one of them is nonsingular. Here we are concerned with the simplex formed by the set of common eigenspaces and give a geometric formulation and proof.

Proposition 3.1. Let q, q′ ∈ T_{2} be two general quadrics. Then there exists a unique n−simplex {L_{1}, …, L_{n}} polar to both q and q′.

Proof. By the above, it suffices to show the relation between the collection of common eigenspaces of the associated symmetric matrices and the common simplex. So we assume that q, q′ are quadrics of rank n and that

q = ∑_{i=1}^{n} l_{i}^{2}, and q′ = ∑_{i=1}^{n} λ_{i}l_{i}^{2},

where the λ_{i} are pairwise distinct coefficients and L_{i} = {l_{i} = 0}, i = 1, …, n. Let

q_{i} = λ_{i}q − q′, i = 1, …, n.

Then the q_{i} are precisely the quadratic forms of the pencil generated by q and q′ that have rank less than n. The rank of q_{i} is exactly n − 1 since λ_{i} ≠ λ_{j} for i ≠ j, so q_{i} ∈ (U_{i})^{2} for a unique rank n − 1 subspace U_{i} ⊂ T_{1}. The intersection ∩_{i≠j}U_{i} is the 1-dimensional subspace generated by the nonzero linear form l_{j}. These forms are therefore determined uniquely by the pencil generated by q and q′. □

A precise condition for generality in the proposition is given by rank:

Lemma 3.2. A pencil of quadratic forms in n variables have a unique common apolar subscheme of length n if and only if every quadric in the pencil have rank at least n − 1 and some, hence the general quadric has rank n. Furthermore the unique apolar subscheme is curvilinear, i.e. embeddable in a smooth curve.

Proof. Let

⟨q′ + λq⟩_{λ∈A_{ℂ}^{1}}

be a pencil with discriminant Δ ⊂ A_{ℂ}^{1}, a scheme of length n. Consider the incidence

{(D, λ)|D(q′ + λq) = 0} ⊂ P(T_{1}) × A_{ℂ}^{1}

with projections p_{T} and p_{ℂ}. Clearly the fibers of each projection are all linear. Now as in the proof of the proposition, a general length n subscheme of p_{T} (p_{ℂ}^{−1}(Δ) is a common apolar subscheme to the pencil of quadratic forms. Therefore, the common apolar subscheme is unique if and only if p_{T} (p_{ℂ}^{−1}(Δ) is finite, i.e. the corank of any quadric in L is at most 1. In this case both projections restricted to the incidence are isomorphisms onto their images. In particular the apolar subscheme is isomorphic to Δ, so it is curvilinear. □

Remark 3.3. The ideal of the curvilinear image Γ of the map

Spec ℂ[t]/(t^{n}) → P^{n−1}
t ↦ (1 : t : t^{2} : …: t^{n−1}),

is generated by the 2 × 2 minors of

(\begin{matrix}x_{1} & x_{2} & … & x_{n−1} & x_{n} \\ x_{2} & x_{3} & … & x_{n} & 0 \end{matrix}) ,

so the Γ is apolar to the maximal rank quadric

∑_{k=1}^{n} y_{k}y_{n+1−k}.

This remark generalizes to a partial converse of Lemma 3.2.

Lemma 3.4. Any curvilinear nondegenerate zero-dimensional subscheme Γ ⊂ P^{n−1} of length n is apolar to a quadric Q ⊂ P̌^{n−1} of maximal rank.

Proof. Let Γ be a nondegenerated curvilinear subscheme with r components of length n_{1}, …, n_{r} such that n_{1} +…+n_{r} = n. Then Γ is projectively equivalent to Γ′ = Γ_{1} ∪ … ∪ Γ_{r}, where Γ_{i} is the image of

Spec ℂ[t]/(t^{n_{i}}) → P(ℂ^{n_{i}}) ⊂ P(ℂ^{n_{1}} ⊕ … ⊕ ℂ^{n_{r}})
t ↦ (1 : t : t^{2} : …: t^{n_{i}−1}),

where the nonzero coordinates in the image are x_{i,1}, …, x_{i,n_{i}}. The ideal of Γ′ is generated by the 2 × 2 minors of the r matrices

(\begin{matrix}x_{(1,1)} & … & x_{(1,n_{1}−1)} & x_{(1,n_{1})} \\ x_{(1,2)} & … & x_{(1,n_{1})} & 0 \end{matrix}) ⋯ (\begin{matrix}x_{(r,1)} & … & x_{(r,n_{r}−1)} & x_{(r,n_{r})} \\ x_{(r,2)} & … & x_{(r,n_{r})} & 0 \end{matrix}) ,

and the products

x_{(i,k_{i})}x_{(j,k_{j})} for 1 ≤ i < j ≤ r, 1 ≤ k_{i} ≤ n_{i}, 1 ≤ k_{j} ≤ n_{j}.

So Γ′ is apolar to the maximal rank quadric

∑_{i=1}^{r} ∑_{k=1}^{n_{i}} y_{(i,k)}y_{(i,n_{i}+1−k)}.

□

More important to us will be that rank n quadrics have apolar subschemes of length n that are not curvilinear (when n > 3).

Remark 3.5. Consider the rank n quadric

q = 2y_{1}y_{n} + y_{2}^{2} + … + y_{n−1}^{2}.

The subscheme Γ_{p} ⊂ P^{n−1} defined by

(x_{1}^{2}, x_{1}x_{2}, x_{2}^{2} − x_{1}x_{n}, x_{1}x_{3}, …, x_{n−1}^{2} − x_{1}x_{n}),

has degree n and is apolar to q, but it is clearly not curvilinear when n > 3. It contains the tangency locus of the quadric \{q^{−1} = \frac{1}{2}x_{1}x_{n} + \frac{1}{4}x_{2}^{2} +…+ \frac{1}{4}x_{n−1}^{2} = 0} at the point [0 : 0 : ... : 1]. The tangency locus has length n−1 and is defined by

(x_{1}, (x_{2}^{2} − x_{1}x_{n}, x_{2}x_{3}, ..., x_{n−1}^{2} − x_{1}x_{n})).

The subscheme Γ_{p} is itself not contained in the tangent hyperplane {x_{1} = 0}, but it is the unique apolar subscheme to q that contains the first order neighborhood of [0 : 0 : ... : 1] on {q^{−1} = 0}. It will be the focus of our attention in Section 5.

It follows immediately from Proposition 3.1 that there is a rational and dominant map

γ : P(T_{2,q}) −−→ V P S(Q, n) ⊂ 𝔾(n − 1, T_{2,q})

whose general fiber is a n-secant (n−2)-space to the projected Veronese variety V_{2,q}. In the next section we find equations for this map.

4. The Mukai form

Mukai introduced in [Muk92] a trilinear form in his approach to varieties of sums of powers of conics in particular, and to forms of even degree in general (see also [Dol12, Sections 1.4 and 2.1.3]). In this section we show how this form naturally gives equations for the map γ and for the universal family of polar simplices. The main result of this section, Proposition 4.2, gives the equations for the common apolar subscheme of length n of a pencil of quadrics in n variables, whenever this subscheme is unique, cf. Lemma 3.2. Both the quadratic form q ∈ T_{2} and the inverse q^{−1} ∈ S_{2} play a crucial role in the definition of the Mukai form. Recall that the form q defines an invertible linear map q : S_{1} → T_{1}, and q^{−1} defines the inverse map: q^{−1} : T_{1} → S_{1}. In coordinates, if q = (α_{1}y_{1}^{2} + ... + α_{n}y_{n}^{2}), then q^{−1} = (\frac{1}{4α_{1}} x_{1}^{2} + ... + \frac{1}{4α_{n}} x_{n}^{2}). We will arrive at Mukai’s form from

τ ∈ Hom(∧^{2}S_{1} ⊗ T_{2} ⊗ T_{2} ⊗ S_{2}, ℂ)

defined by

z_{1} ∧ z_{2} ⊗ q_{1} ⊗ q_{2} ⊗ α ↦ (z_{1}(q_{1})z_{2}(q_{2}) − z_{2}(q_{1})z_{1}(q_{2}))(α)

where f(g), as above, means f viewed as differential operator applied to g. Interpreting ω = z_{1} ∧ z_{2} ∈ ∧^{2}S_{1} ⊂ Hom(T_{1}, S_{1}), q_{j} ∈ T_{2} ⊂ Hom(S_{1}, T_{1}) and α ∈ S_{2} ⊂ Hom(T_{1}, S_{1}) the expression

(ω ⊗ q_{1} ⊗ q_{2} ⊗ α) ↦ \frac{1}{2}trace (α ∘ q_{2} ∘ ω ∘ q_{1} − α ∘ q_{1} ∘ ω ∘ q_{2})

= \frac{1}{2}trace (ω ∘ q_{1} ∘ α ∘ q_{2} − α ∘ q_{1} ∘ ω ∘ q_{2})

gives an alternative description of τ. In fact f(g) = \frac{1}{2}(trace f ∘ g) holds for f ⊗ g ∈ S_{2} ⊗ T_{2} ⊂ Hom(T_{1}, S_{1}) ⊗ Hom(S_{1}, T_{1}) and trace ((α ∘ q_{2}) ∘ (ω ∘ q_{1})) = trace ((ω ∘ q_{1}) ∘ (α ∘ q_{2})). We now substitute α = q^{−1}. Then \frac{1}{2}trace (ω ∘ q_{1} ∘ q^{−1} ∘ q_{2} − q^{−1} ∘ q_{1} ∘ ω ∘ q_{2}) = 0 for q_{1} = q, and, since the first expression for τ is alternating on T_{2} ⊗ T_{2}, we have \frac{1}{2}trace (ω ∘ q_{1} ∘ q^{−1} ∘ q_{2} − q^{−1} ∘ q_{1} ∘ ω ∘ q_{2}) = 0 for q_{2} = q as well. Thus τ induces a well defined trilinear form

τ_{q} ∈ Hom(∧^{2}S_{1} ⊗ T_{2,q} ⊗ T_{2,q}, ℂ)

on the quotient space T_{2,q} = T_{2}/⟨q⟩. Since T_{2,q}^{∗} = q^{⊥} ⊂ S_{2} = T_{2}^{∗} and

Hom(∧^{2}S_{1} ⊗ T_{2,q} ⊗ T_{2,q}, ℂ) ≅ Hom(T_{2,q}, Hom(∧^{2}S_{1}, q^{⊥}))

we have a second interpretation of τ_{q}. With this interpretation, the image of τ_{q}(q_{1}) ∈ Hom(∧^{2}S_{1}, q^{⊥}) ⊂ Hom(∧^{2}S_{1}, S_{2}) is defined by

ω ↦ [ω ∘ q_{1} ∘ q^{−1} − q^{−1} ∘ q_{1} ∘ ω] ∈ q^{⊥} ⊂ S_{2} ⊂ Hom(T_{1}, S_{1}).

The form τ is alternating on T_{2} ⊗T_{2}, so τ(ω, q′, q′, q^{−1}) = 0 for every ω ∈ ∧^{2}S_{1}. Therefore

τ_{q}(q′)(∧^{2}S_{1}) ⊂ (q′)^{⊥}.

If Q′ is the quadric {q′ = 0} ⊂ P(S_{1}), we may therefore conclude:

Lemma 4.1. Any quadratic form in τ_{q}(q′)(∧^{2}S_{1}) is apolar to both Q and Q′:

τ_{q}(q′)(∧^{2}S_{1}) ⊂ q^{⊥} ∩ (q′)^{⊥}.

□

Notice that the linear space of quadratic forms τ_{q}(q′)(∧^{2}S_{1}) is not all of q^{⊥} ∩ (q′)^{⊥}. It is a special subspace of the intersection. Since τ_{q}(q) = 0, we have τ_{q}(q′) = τ_{q}(q′ + λq) for any λ, so the space τ_{q}(q′)(∧^{2}S_{1}) of quadratic forms depends only on the pencil ⟨q, q′⟩. If the pencil of quadratic forms ⟨q, q′⟩ ⊂ T_{2} contains no forms of corank at least 2, then, by Lemma 3.2, there is a unique common apolar subscheme Γ_{q′} of length n to q and q′. The significance of the form τ_{q} is

Proposition 4.2. Let q′ ∈ T_{2,q}. Then the linear map

τ_{q}(q′) : ∧^{2}S → q^{⊥}

is injective if and only if q and q′ have a unique common apolar subscheme of length n. Furthermore, in this case the image generates the ideal in S of this subscheme.

Proof. Our argument depends on several lemmas, in which we study Im τ_{q}(q′) ⊂ S_{2} by considering the symmetric matrices associated to these quadratic forms with respect to a suitable basis. Thus, we choose coordinates such that q = \frac{1}{2}(y_{1}^{2} + y_{2}^{2} + ... + y_{n}^{2}) and hence q^{−1} = \frac{1}{2}(x_{1}^{2} + x_{2}^{2} + ... + x_{n}^{2}). The symmetric matrices of these quadratic forms with respect to the coordinate basis of T_{1} and S_{1} are both the identity matrix. We denote by A the symmetric matrix of q′, i.e. q′ = \frac{1}{2}(y_{1}, ..., y_{n})A(y_{1}, ..., y_{n})^{t}. For a form ω ∈ ∧^{2}S_{1} there is similarly an associated skew symmetric matrix Λ_{ω}. For a form l ∈ T_{1} we denote by v_{l} the column vector of its coordinates. The quadratic forms in the image τ_{q}(q′) are the forms associated to the symmetric bilinear forms

{ω ∘ q′ ∘ q^{−1} − q^{−1} ∘ q′ ∘ ω|ω ∈ ∧^{2}S_{1}}, so their associated symmetric matrices are

{Λ_{ω}A − AΛ_{ω}|ω ∈ ∧^{2}S_{1}}.

Lemma 4.3. Let [l] ∈ P(T_{1}), then every quadric in τ_{q}(q′)(∧^{2}S) ⊂ q^{⊥} vanishes at the point [l] if and only if there is a quadric q_{λ} = q′ + λq for some λ ∈ ℂ, such that l lies in the kernel of the linear transformation q_{λ} ∘ q^{−1} : T_{1} → T_{1}. Equivalently, in terms of matrices: If v_{l} is the column coordinate vector of l, then v_{l}^{t}(Λ_{ω}A−AΛ_{ω})v_{l} = 0 for every ω ∈ ∧^{2}S_{1} if and only if v_{l} is an eigenvector for the matrix A.

Proof. Note first that the matrix of the linear transformation q_{λ} ∘ q^{−1} , with respect to the coordinate basis of T_{1}, is simply A + λI. Hence, the two parts of the lemma are equivalent. In the matrix notation, if v_{l} is an eigenvector for A with eigenvalue λ, then

v_{l}^{t}(Λ_{ω}A − AΛ_{ω})v_{l} = v_{l}^{t}Λ_{ω}Av_{l} − v_{l}^{t}AΛ_{ω}v_{l}
= v_{l}^{t}Λ_{ω}λv_{l} − λv_{l}^{t}Λ_{ω}v_{l} = 0,

so the if part follows. Conversely, assume that

v_{l}^{t}(Λ_{ω}A − AΛ_{ω})v_{l} = 0

for every skew symmetric n × n matrix Λ_{ω}. Again

v_{l}^{t}(Λ_{ω}A − AΛ_{ω})v_{l} = v_{l}^{t}Λ_{ω}Av_{l} − v_{l}^{t}Λ_{ω}Av_{l},

and since A is symmetric and Λ_{ω} is skewsymmetric, (v_{l}^{t}Λ_{ω}Av_{l})^{t} = −v_{l}^{t}AΛ_{ω}v_{l}, so we deduce that

v_{l}^{t}Λ_{ω}Av_{l} = 0.

But v_{l}^{t}Λ_{ω}u = 0 for every skew symmetric matrix Λ_{ω} only if u is proportional to v_{l}, so we conclude that A(v_{l}) = λv_{l} for some λ. □

Remark 4.4. A point l ∈ T_{1} lies in the kernel of q_{λ} ∘ q^{−1} if and only if q^{−1}(l) lies in the kernel of q_{λ} : S_{1} → T_{1}. Equivalently, {q_{λ} = 0} ⊂ P(S_{1}) is a singular quadric and [q^{−1}(l)] ∈ P(S_{1}) lies in its singular locus.

Corollary 4.5. τ_{q}(q′) is injective only if ⟨q, q′⟩ contains no quadratic form of rank less than n − 1.

Proof. If the quadratic form q_{λ} = q′ + λq has rank less than n − 1, then there are independent forms l, l′ ∈ T_{1} such that ⟨q^{−1}(l), q^{−1}(l′)⟩ is contained in the kernel of q_{λ} : S_{1} → T_{1}. In particular, viewed as differential operators applied to q_{λ},

q^{−1}(l)(q_{λ}) = q^{−1}(l′)(q_{λ}) = 0 ∈ T_{1}.

Let

ω = q^{−1}(l) ∧ q^{−1}(l′) ∈ ∧^{2}S_{1} ⊂ Hom(T_{1}, S_{1}).

Then

ω ⊗ q_{λ} ⊗ q_{2} ⊗ q^{−1} ↦ [q^{−1}(l)(q_{λ}) · q^{−1}(l′)(q_{2}) − q^{−1}(l′)(q_{λ}) · q^{−1}(l)(q_{2})](q^{−1}) = 0

for every q_{2} ∈ T_{2}, so τ_{q}(q′)(ω) = 0 and τ_{q}(q′) is not injective. □

To complete the proof of Proposition 4.2, we assume that q and q′ have a unique common apolar subscheme Γ of length n, i.e. by Lemma 3.2, no quadratic form in ⟨q, q′⟩ has rank less than n − 1. We want to show that τ_{q}(q′) is injective and that the image generates the ideal of Γ. Let Γ = Γ_{1} ∪ ... ∪ Γ_{r} be a decomposition of Γ into its connected components. Then each Γ_{i} is a finite local curvilinear scheme. Let n_{i} be the length of Γ_{i}. By Proposition 2.4, there is a decomposition T_{1} = ⊕_{i}U_{i} such that each Γ_{i} ⊂ P(U_{i}). Furthermore U_{i} has dimension n_{i} and q and q′ have unique decompositions q = q_{1} + ... + q_{r} and q′ = q′_{1} + ... + q′_{r} with q_{i}, q′_{i} ∈ (U_{i})^{2} ⊂ T_{2}. Denote by U′_{i} = ⊕_{j≠i}U_{j}, and let (U′_{i})^{⊥} be the orthogonal subspace of linear forms in S_{1}. Then

∑_{i} U_{i}^{⊥} · (U′_{i})^{⊥} ⊂ S_{2}

generate the ideal of ∪_{i}P(U_{i}) ⊂ P(T_{1}). The linear forms q^{−1}(U_{i}) ⊂ S_{1} are natural coordinates on P(U_{i}). Denote by I_{Γ_{i},2} the quadratic forms in these coordinates in the ideal of Γ_{i}. Then I_{Γ_{i},2} ⊂ (q^{−1}(U_{i}))^{2} ⊂ S_{2} and the space of quadratic forms in the ideal of Γ is

I_{Γ,2} = ∑_{i} I_{Γ_{i},2} + ∑_{i} U_{i}^{⊥} · (U′_{i})^{⊥} ⊂ S_{2}.

We

Claim 4.6.

Im τ_{q}(q′) ⊃ ∑_{i} I_{Γ_{i},2} + ∑_{i} U_{i}^{⊥} · (U′_{i})^{⊥}.

If the claim holds, τ_{q}(q′) is injective, since dim ∧^{2} S_{1} = dimI_{Γ,2} (= \binom{n}{2}), so the equality Im τ_{q}(q′) = I_{Γ,2} holds and the proof of Proposition 4.2 is complete. We use matrices to prove the claim. To interpret the decomposition of q and q′ in terms of matrices, we choose a basis for each U_{i} such that the symmetric matrix associated to each q_{i} is the n_{i} × n_{i} identity matrix. Let A_{i} be the symmetric n_{i} × n_{i} matrix associated to q′_{i}. The union of the bases for the U_{i} form a basis for T_{1} with respect to which the symmetric matrix A of q′ has r diagonal blocks A_{i} and zeros elsewhere. The matrices A_{i} each have a unique eigenvalue λ_{i}, and these eigenvalues are pairwise distinct. Furthermore, each A_{i} has a 1-dimensional eigenspace, so their Jordan form has a unique Jordan block, and we may write A_{i} = λ_{i}I_{n_{i}} + B_{i} with B_{i} a nilpotent symmetric matrix. (See ([DZ04, Theorem 2.3]) for a nice normal form for the matrices B_{i}.) By extending each A_{i} with zeros to n × n matrices, we may write A = ∑ A_{i}. The decomposition T_{1} = ⊕_{i}U_{i} is then defined by U_{i} = ker(λ_{i}I − A)^{n_{i}} ⊂ T_{1}. Denote by U′_{i} = ⊕_{j≠i}U_{j}. Then P(U_{i}) and P(U′_{i}) have complementary dimension in P(T_{1}). We shall use the techniques applied by Gantmacher in the analysis of commuting matrices ([Gan59, Chapter VIII]) to show

Lemma 4.7. Let A be the symmetric matrix of the quadratic form q′ ∈ T_{2,q} as above. Let T_{1} = U_{i} ⊕ U′_{i} be the decomposition associated to the eigenvalue λ_{i}.

Then

U_{i}^{⊥} · (U′_{i})^{⊥} ⊂ Im τ_{q}(q′) ⊂ S_{2}.

Proof. Set d = n_{i} and λ = λ_{i} and choose coordinates such that U_{λ} = ⟨y_{1}, ..., y_{d}⟩ and U′_{λ} = ⟨y_{d+1}, ..., y_{n}⟩. Then (U′_{λ})^{⊥} = ⟨x_{1}, ..., x_{d}⟩ and (U_{λ})^{⊥} = ⟨x_{d+1}, ..., x_{n}⟩. Consider the matrix B of the quadratic form τ_{q}(q′)(x_{i}∧x_{j}) with i ≤ d and j > d. The skew symmetric matrix Λ_{(ij)} of x_{i} ∧ x_{j} has (ij)-th entry 1, consequently (ji)-th entry −1, and 0 elsewhere, and

B = Λ_{(ij)}A − AΛ_{(ij)}.

The nonzero entries in Λ_{(ij)}A are in positions (i, k) with k > d and (j, k) with k ≤ d, while the nonzero entries in AΛ_{(ij)} are in positions (k, i) with k > d and (k, j) with k ≤ d. Therefore the quadratic form τ_{q}(q′)(x_{i} ∧ x_{j}) lies in the space

⟨x_{a}x_{b}⟩_{a≤d<b} = (U′_{λ})^{⊥} · (U_{λ})^{⊥}.

A linear relation between these quadratic forms would correspond to a skew symmetric matrix Λ with nonzero entries only in the rectangular block (ij), i ≤ d, j > d, such that ΛA − AΛ = 0. Write A as a sum A = A_{λ} + A_{μ_{1}} + ... + A_{μ_{s}} where the μ_{i} are the eigenvalues of A distinct from λ. Let Λ be a skew symmetric matrix and let Λ_{λ,μ_{i}} be the rectangular submatrix with rows equal to the nonzero rows of A_{λ} and columns equal to the nonzero columns of A_{μ_{i}}. Then the corresponding submatrix

(ΛA − AΛ)_{λ,μ_{i}} = Λ_{λ,μ_{i}}A_{μ_{i}} − A_{λ}Λ_{λ,μ_{i}}.

So ΛA − AΛ = 0 only if Λ_{λ,μ_{i}}A_{μ_{i}} − A_{λ}Λ_{λ,μ_{i}} = 0 for each μ_{i}. Let μ be one of the μ_{i}, and assume for simplicity U_{μ} = ⟨y_{d+1}, ..., y_{d+e}⟩. Let I_{d} be the diagonal matrix with 1 in the d first entries and 0 elsewhere, and let I_{e} be the diagonal matrix with 1 in the entries d + 1, ..., d + e and 0 elsewhere. Then the special summand A_{λ} of A can be written as a sum A_{λ} = λI_{d} +B_{d} where B_{d} is nilpotent of order d. Likewise, A_{μ} = μI_{e} + B_{e} where B_{e} is nilpotent of order e. So we may write A = λI_{d} + B_{d} + μI_{e} + B_{e} + A′, where A′ = A − A_{λ} − A_{μ}. But then (ΛA − AΛ)_{λ,μ} = 0 only if

Λ_{λ,μ}A_{μ} − A_{λ}Λ_{λ,μ} = 0,

i.e. when

Λ_{λ,μ}(μI_{e} + B_{e}) − (λI_{d} + B_{d})Λ_{λ,μ} = 0.

This is equivalent to

(λ − μ)Λ_{λ,μ} = Λ_{λ,μ}B_{e} − B_{d}Λ_{λ,μ}.

Multiplying both sides by (λ − μ) and substituting on the right hand side (λ − μ)Λ_{λ,μ} with Λ_{λ,μ}B_{e} − B_{d}Λ_{λ,μ} we get

(λ − μ)^{2}Λ_{λ,μ} = (Λ_{λ,μ}B_{e} − B_{d}Λ_{λ,μ})B_{e} − B_{d}(Λ_{λ,μ}B_{e} − B_{d}Λ_{λ,μ})
= (Λ_{λ,μ}(B_{e})^{2} − 2B_{d}Λ_{λ,μ}B_{e} + (B_{d})^{2}Λ_{λ,μ}).

Iterating m = d + e − 1 times we get

(λ − μ)^{m}Λ_{λ,μ} = ∑_{s+t=m}(−1)^{s}\binom{s + t}{t}(B_{d})^{s}Λ_{λ,μ}(B_{e})^{t}.

But on the right hand side either (B_{d})^{s} = 0 or (B_{e})^{t} = 0 when s + t = d + e, so Λ_{λ,μ} = 0. Thus Λ_{λ,μ_{i}} = 0 for all i, and the symmetric matrices AΛ_{ij} − Λ_{ij}A with i ≤ d, j > d are linearly independent. The corresponding quadratic forms therefore are linearly independent in the space ⟨x_{1}, ..., x_{d}⟩×⟨x_{d+1}, ..., x_{n}⟩. Since the dimensions coincides, the quadratic forms span this space, and the lemma follows. □

Next, we consider the case when the symmetric matrix A only has one eigenvalue. Thus we assume that Γ has only one component, the symmetric matrix A of q′ has only one eigenvalue and up to scalars only one nonzero eigenvector. Hence ⟨q, q′⟩ contains exactly one quadratic form of rank n − 1. In particular, by Lemma 3.2, Γ is curvilinear. Without loss of generality we may assume that q′ has rank n − 1 i.e. that the eigenvalue is 0. Then A is nilpotent, and since A is a one-dimensional eigenvector space, A^{n} = 0 and A^{i} ≠ 0 for any i < n.

Lemma 4.8. Let q′ ∈ T_{2,q} be a quadratic form whose associated n × n matrix A is symmetric, nilpotent and has rank n − 1. Then the ideal generated by the quadratic forms τ_{q}(q′) ⊂ q^{⊥} is the ideal of the unique common apolar subscheme Γ of length n of q and q′. Moreover Γ is a local curvilinear subscheme.

Proof. Let Λ be a skew symmetric n × n matrix and think of A and Λ as the matrices of linear endomorphisms of a n-dimensional vector space V. Then we may choose a basis v_{1}, …, v_{n} ∈ V such that Av_{1} = 0 and Av_{i} = v_{i−1} for i = 2, …, n. Let

ρ : Spec(ℂ[t]/t^{n}) → P(V) : t ↦ [v_{1} + tv_{2} + … + t^{n−1}v_{n}]

and set Γ = Im ρ. Then I_{Γ} is generated by \binom{n}{2} quadratic forms. We shall show that the symmetric matrices of these forms coincide with the matrices ΛA − AΛ as Λ varies. We evaluate the quadratic form associated to ΛA − AΛ on the vector v = v_{1} + tv_{2} + … + t^{n−1}v_{n}:

v^{t}(ΛA − AΛ)v = v^{t}ΛAv − v^{t}AΛv.

But

v^{t}ΛAv = (v_{1} + tv_{2} + … + t^{n−1}v_{n})^{t}ΛA(v_{1} + tv_{2} + … + t^{n−1}v_{n})

= (v_{1} + tv_{2} + … + t^{n−1}v_{n})^{t}Λ(tv_{1} + … + t^{n−1}v_{n−1})

= (v_{1} + … + t^{n−2}v_{n−1})^{t}Λ(v_{1} + … + t^{n−2}v_{n−1})t

+ t^{n}v_{n}^{t}Λ(v_{1} + … + t^{n−2}v_{n−1})

= 0

since Λ is skew symmetric and t^{n} = 0. Therefore the quadratic forms with matrices ΛA − AΛ are in the ideal of Γ. They are independent and therefore generate the ideal unless ΛA − AΛ = 0 for some nontrivial Λ. But then Λ and A commute, hence have common eigenvectors. Λ is nontrivial and skew symmetric so it has at least 2 independent eigenvectors, while A has only one, so this is impossible. Clearly, Γ is curvilinear, and any non degenerate local curvilinear subscheme of length n in P(V) is projectively equivalent to it, so the lemma follows. □

To complete the proof of the claim 4.6 and the proof of Proposition 4.2, we consider the common apolar subscheme Γ = Γ_{1} ∪ ... ∪ Γ_{r} to q and q′, and the corresponding decompositions q = ∑ q_{i} and q′ = ∑ q′_{i} as above. By Lemma 4.7,

∑_{i} U_{i}^{⊥} · (U′_{i})^{⊥} ⊂ Im τ_{q}(q′).

Furthermore, applying Lemma 4.8 to each component q_{i} and q′_{i}, the image of τ_{q_{i}}(q′_{i}) in (q^{−1}(U_{i}))^{2} is I_{Γ_{i},2}. But τ_{q_{i}}(q′_{i}) is the restriction of τ_{q}(q′) to ∧^{2}(q^{−1}(U_{i})), so

I_{Γ_{i},2} ⊂ Im τ_{q}(q′) i = 1, ..., r

and the claim and Proposition 4.2 follows. □

By Lemma 4.3, the quadratic forms in Im τ_{q}(q′) vanish in every point on any common apolar subscheme of length n to q and q′. Combined with Proposition 4.2 it may be reasonable to guess that Im τ_{q}(q′) is precisely the quadratic forms in the intersection of the ideals of these common apolar subschemes. We do not have a clear answer and leave this as an open question. We are now ready to analyze our main object V P S(Q, n) in its embedding in 𝔾(n − 1, T_{2,q}), i.e. as the image of the rational map

γ : P(T_{2,q}) −−→ 𝔾(n − 1, T_{2,q}).

We identify the restriction of the Plücker divisor to V P S(Q, n). Let h ⊂ P(T_{1}) be a hyperplane, and denote by H_{h} ⊂ V SQ(Q, n) the set

H_{h} = {[Γ] ∈ V SQ(Q, n)|Γ ∩ h ≠ ∅}.

Lemma 4.9. H_{h} is the restriction to V P S(Q, n) of a Plücker divisor on 𝔾(n − 1, T_{2,q}).

Proof. The hyperplane h ⊂ P(T_{1}) is defined by some l ∈ S_{1}. Let V (l) = {q′ ∈ T_{2}|l(q′) = 0}, then V (l)^{⊥} = l · S_{1} = {l · l′|l′ ∈ S_{1}} ⊂ S_{2}. For any nondegenerate subcheme Γ ⊂ P(T_{1}) of length n, the ideal I_{Γ} ⊂ S contains a reducible quadric l_{1} ·l_{2} only if Γ intersects both hyperplanes {l_{1} = 0} and {l_{2} = 0}. On the other hand the subspace of quadrics I_{Γ,2} ⊂ S_{2} has codimension n, which coincides with the dimension of l · S_{1}. Therefore

I_{Γ,2} ∩ l · S_{1} ≠ {0} ⊂ S_{2} if and only if (I_{Γ,2})^{⊥} ∩ V (l) ≠ {0} ⊂ T_{2}.

Notice that P((I_{Γ,2})^{⊥}) equals the span ⟨Γ⟩ ⊂ P(T_{2}) of Γ in the Veronese embedding. For the lemma we now consider apolar subschemes to q and the projection from P(T_{2}) to P(T_{2,q}). Since q has maximal rank, l(q) ≠ 0, i.e. q ∉ V (l). Thus P(V (l)) is projected isomorphically to its image P(V_{q}(l)) ⊂ P(T_{2,q}). For an apolar subscheme Γ of length n the quadratic form q lies in the linear span of Γ ⊂ P(T_{2}), so this subspace is mapped to the (n − 2)-dimensional linear span of Γ in P(T_{2,q}). We therefore deduce from the above equivalence: If Γ is apolar to q, then the linear span of Γ in P(T_{2,q}) intersects the codimension n linear space P(V_{q}(l)) if and only if Γ intersects the hyperplane h ⊂ P(T_{1}). But the set of (n − 2)-dimensional subspaces in P(T_{2,q}) that intersect a linear space of codimension n form a Plücker divisor, so the lemma follows. □

In the next section we use the special Plücker divisors H_{h} of this lemma to give a local affine description of V P S(Q, n), or better, the variety V AP S(Q, n) of all apolar subschemes of length n.

5. An open affine subvariety

We use a standard basis approach to compute an open affine subvariety of V AP S(Q, n), the variety of all apolar subschemes of length n to Q. Of course this will include our primary object of interest, namely V P S(Q, n). For small n there will be no difference, but for larger n we have already seen that they do not coincide. The distinction between the two will eventually be the main concern in our analysis. The computations in this section extensively use Macaulay2 [GS]. In particular when we show, by direct computation, that V AP S(Q, 6) is irreducible and therefore coincides with V P S(Q, 6) (Corollary 5.16). We choose coordinates such that

Q = {q = 2y_{1}y_{n} + y_{2}^{2} + … + y_{n−1}^{2} = 0},

and consider the apolar subscheme Γ_{p} to q defined by

x_{1}^{2}, x_{1}x_{2}, x_{2}^{2} − x_{1}x_{n}, x_{1}x_{3}, x_{2}x_{3}, … x_{n−1}^{2} − x_{1}x_{n}.

It is of length n and corresponds in the setting of the previous section to the intersection of the projected Veronese variety V_{2,q} with the tangent space T_{p} to v_{2}(Q^{−1}) ⊂ P(T_{2,q}) at the point v_{2}(p) = [y_{n}^{2}] ∈ P(T_{2,q}), where p = [y_{n}] = [0 : …: 1] ∈ P(T_{1}). The tangent space to the Veronese variety V_{2} ⊂ P(T_{2}) at [y_{n}^{2}] is spanned by

⟨y_{1}y_{n}, y_{2}y_{n}, ..., , y_{n−1}y_{n}, y_{n}^{2}⟩.

The quadric Q^{−1} is defined by \frac{1}{2}x_{1}x_{n} + \frac{1}{4}(x_{2}^{2} + ... + x_{n−1}^{2}). Its tangent space in P (T_{1}) at [y_{n}] is defined by x_{1}, so its tangent space in P (T_{2}) at [y_{n}^{2}] is defined by x_{1} inside the tangent space to the Veronese variety. Therefore, the tangent space T_{p} to v_{2}(Q^{−1}) is spanned by

⟨y_{2}y_{n}, ..., y_{n−1}y_{n}, y_{n}^{2}⟩.

The orthogonal space of quadratic forms in S_{2} is spanned by

⟨x_{1}^{2}, x_{1}x_{2}, x_{2}^{2}, x_{1}x_{3}, x_{2}x_{3}, … x_{n−1}^{2}, x_{1}x_{n}⟩

and intersect q^{⊥} precisely in the ideal of Γ_{p} given above.

With reverse lexicographically order on the coordinates x_{1}, ..., x_{n}, the initial ideal of Γ_{p} is generated by the monomials

x_{1}^{2}, x_{1}x_{2}, x_{2}^{2}, x_{1}x_{3}, x_{2}x_{3}, … x_{n−1}^{2}.

In this monomial order, these monomials have the highest order in the ideal of any apolar scheme Γ that does not intersect the hyperplane {x_{n} = 0}. In fact, if the initial ideal of Γ contains x_{i}x_{n}, then x_{n} divides a quadratic form in the ideal of Γ. But if Γ does not intersect {x_{n} = 0}, then Γ would be degenerate. We therefore consider the open subvariety V_{h}^{aff} (n) containing [Γ_{p}] in V AP S(Q, n), parametrizing apolar subschemes Γ of length n with support in D(x_{n}). This is the complement of the divisor H_{h} defined by h = {x_{n} = 0}, the tangent hyperplane to Q^{−1} at [y_{1}] = [1 : 0 : ... : 0] ∈ P(T_{1}). For Γ ∈ V_{h}^{aff} (n) the initial terms of the generators of the ideal I_{Γ} coincide with those of I_{Γ_{p}}. More precisely, the generators of I_{Γ} may be obtained by adding suitable multiples of the monomials x_{i}x_{n}, i ≥ 1 to these initial terms. We may therefore write these generators in the form

x_{1}^{2} − a_{(11,1)}x_{1}x_{n} − a_{(11,2)}x_{2}x_{n} − a_{(11,2)}x_{2}x_{n} − ... − a_{(11,n)}x_{n}^{2},
x_{i}x_{j} − a_{(ij,1)}x_{1}x_{n} − a_{(ij,2)}x_{2}x_{n} − ... − a_{(ij,n)}x_{n}^{2}, 1 ≤ i < j ≤ n − 1,
x_{i}^{2} − x_{1}x_{n} − a_{(ii,2)}x_{2}x_{n} − ... − a_{(ii,n)}x_{n}^{2}, 2 ≤ i ≤ n − 1.

Analyzing these equations of Γ further, we see that the apolarity condition, i.e. that I_{Γ,2} ⊂ q^{⊥}, means that a_{(11,1)} = 0 and that a_{(ij,1)} = 0 when i ≠ j. Therefore they take the form

f_{11} =x_{1}^{2} − a_{(11,2)}x_{2}x_{n} − … − a_{(11,n)}x_{n}^{2},
f_{12} =x_{1}x_{2} − a_{(12,2)}x_{2}x_{n} − … − a_{(12,n)}x_{n}^{2},
f_{22} =(x_{2}^{2} − x_{1}x_{n}) − a_{(22,2)}x_{2}x_{n} − … − a_{(22,n)}x_{n}^{2},
(5.1)  f_{13} =x_{1}x_{3} − a_{(13,2)}x_{2}x_{n} − … − a_{(13,n)}x_{n}^{2},
f_{23} =x_{2}x_{3} − a_{(23,2)}x_{2}x_{n} − … − a_{(23,n)}x_{n}^{2},
f_{33} =(x_{3}^{2} − x_{1}x_{n}) − a_{(33,2)}x_{2}x_{n} − … − a_{(33,n)}x_{n}^{2},
⋮⋮
f_{(n−1)(n−1)} =(x_{n−1}^{2} − x_{1}x_{n}) − a_{((n−1)(n−1),2)}x_{2}x_{n} − … − a_{((n−1)(n−1),n)}x_{n}^{2}.

To insure that these perturbed equations actually define length n subschemes, we ask that the first order relations or syzygies among the generators of I_{Γ_{p}} lift to the entire family. This is in fact precisely the requirement for the perturbation to define a flat family [Art76, Proposition 3.1], and will be pursued below when we find equations for V_{h}^{aff} (n). Here, we introduce weights and a torus action on this family: We give

• x_{n} and a_{(ij,k)}, where 2 ≤ i, j, k ≤ n − 1, weight 1,

• x_{i}, where 2 ≤ i ≤ n − 1, and a_{(ij,n)}, where 2 ≤ i, j ≤ n − 1. weight 2,

• x_{1} and a_{(1i,n)} and a_{(11,i)}, where 2 ≤ i ≤ n − 1, weight 3,

• a_{(11,n)} weight 4.

Notice that with these weights each generator f_{ij} is homogeneous. A ℂ^{∗}-action defined by multiplying each parameter with a constant λ^{w} to the power of its weight, acts on each generator by a scalar multiplication, i.e. on the total family in P(T_{1}) × V_{h}^{aff} (n). This ℂ^{∗}- action induces an action on the family V_{h}^{aff} (n). In particular, if [a] = [a_{(ij,k)}] ∈ V_{h}^{aff} (n) defines a subscheme Γ_{[a]}, then [λ^{w}(a)] = [λ^{w}(a_{ij,k})] ∈ V_{h}^{aff} (n) and defines a subscheme Γ_{[λ^{w}a]}, such that p′ ∈ Γ_{[a]} if and only if λ^{w}(p′) ∈ Γ_{[λ^{w}a]}. Since lim_{λ→0} λ^{w}(a_{ij,k}) = 0, the limit when λ → 0 of the ℂ^{∗}- action is the point in V_{h}^{aff} (n) representing Γ_{p}. Thus we have shown

Lemma 5.1. The affine algebraic set V_{h}^{aff} (n) of apolar subschemes of length n contained in D(x_{n}) coincides with the apolar schemes of length n whose equations are affine perturbations of the equations of Γ_{p}. Furthermore, the family V_{h}^{aff} (n) is contractible to the point [Γ_{p}].

An immediate consequence is the

Corollary 5.2. The apolar subscheme Γ_{p} belongs to V P S(Q, n). In particular, the variety of tangent spaces T Q^{−1} ⊂ 𝔾(n − 1, T_{q,2}) to the Veronese embedding of the quadric Q^{−1} ⊂ P(T_{q,2}) is a subvariety of V P S(Q, n).

Notice that V_{h}^{aff} (n) depends only on h, and not on p. Only the coordinates on V_{h}^{aff} (n) depend on p. On the other hand, the contractible varieties V_{h}^{aff} (n) form a covering of V AP S(Q, n):

Lemma 5.3. If h_{j} = {l_{j} = 0}, j = 1, ..., n^{2} is a collection of tangent hyperplanes to Q^{−1}, so that no subset of n of them have a common point, then the open subvarieties V_{h_{j}}^{aff} (l_{j}) parametrizing apolar subschemes Z of length n with support in D(l_{j}) form a covering of V AP S(Q, n) of isomorphic varieties.

Proof. If an apolar subscheme Γ has k ≤ n components, then the collection of hyperplanes among the {l_{j} = 0} that intersect Γ is at most k(n − 1) < n^{2}, so the V_{h_{j}}^{aff} (l_{j}) form a covering. The last part follows from the homogeneity. □

To find equations for the family V_{h}^{aff} (n), we use the parameters for the generators in (5.1), i.e.

a_{(ij,k)} i, j ∈ {1, …, n − 1}, 2 ≤ k ≤ n,

where we read the first index (ij) as an unordered pair.

It will be useful to write the generators with matrices:

( f_{11} ; f_{12} ; f_{22} ; ⋮ ; f_{(n−1)(n−1)})=(\begin{matrix}1 & 0 & ... & 0 & −a_{(11,2)} & ... & −a_{(11,n)} \\ 0 & 1 & ... & 0 & −a_{(12,2)} & ... & −a_{(12,n)} \\ 0 & 0 & ... & 0 & −a_{(22,2)} & ... & −a_{(22,n)} \\ ⋮ & ⋮ & & ⋮ & ⋮ & & ⋮ \\ 0 & 0 & ... & 1 & ⋮ & ... & −a_{((n−1)(n−1),n)} \end{matrix})·( x_{1}^{2} ; x_{1}x_{2} ; x_{2}^{2} − x_{1}x_{n} ; ⋮ ; x_{n−1}^{2} − x_{1}x_{n} ; x_{2}x_{n} ; ⋮ ; x_{n}^{2}).

We denote by A_{F} the \binom{n}{2}×(\binom{n}{2}+n−1)-dimensional coefficient matrix of these generators. The maximal minors of A_{F} are, of course, precisely the Plücker coordinates for V_{h}^{aff} (n) in 𝔾(\binom{n}{2}, q^{⊥}), or equivalently in 𝔾(n − 1, T_{2,q}). We find the equations of the family by asking that the first order syzygies among the generators of I_{Γ_{p}} lift to the entire family. By [Art76, Proposition 3.1], this is precisely the requirement for the perturbation to define a flat family. We use a standard basis approach (cf. [Sch91]). The syzygies for a subscheme Z in the family are all linear, and the initial terms are inherited from Γ_{p}. Therefore, the difference between syzygies of Z and syzygies of Γ_{p} are only multiples of x_{n}. By the division theorem ([Sch91, Theorem A.3]), every syzygy has the initial term x_{k}(x_{i}x_{j}), where k > j ≥ i, and has the form

x_{k}f_{ij} = ∑_{st} g_{ij}^{st}f_{st}

where f_{ij} is the generator with initial term x_{i}x_{j} and g_{ij}^{st} is a linear form such that g_{ij}^{st}f_{st} has higher order than x_{k}(x_{i}x_{j}). More precisely, we therefore consider products of the generators (f_{ij}) with a first order syzygy for Γ_{p} and add precisely those multiples of x_{n} in the syzygy that eliminates monomials x_{k}x_{l}x_{n} with k ≤ l < n in the product. The relations among the parameters required for the lifting of the syzygies can then be read off as the coefficients of the monomials x_{t}x_{n}^{2}.

Theorem 5.4. The equations defining V_{h}^{aff} (n) all lie in the linear span of the 2 × 2 minors of the coefficient matrix A_{F} of the family of equations f_{ij}. In particular V AP S(Q, n) is a linear section of the Grassmannian 𝔾(n − 1, T_{2,q}).

Remark 5.5. This generalizes the result of Mukai in case n = 3, cf. [Muk92]. For an exposition of different approaches to the n = 3 case, see [Dol12, Section 2.1.3].

Proof. Consider the following first order syzygies of Γ_{p} of rank 2 and 3: R_{i} · S_{i}(m), 1 < i < n, m = 1, 2 :

(x_{1}^{2} x_{1}x_{i} x_{i}^{2} − x_{1}x_{n}) · (\begin{matrix}−x_{i} & x_{n} \\ x_{1} & −x_{i} \\ 0 & x_{1} \end{matrix}) = 0,

where S_{i}(m) is the m-th column vector in the syzygy matrix S_{i}, R_{ij} · S_{ij}(m), 1 < i < j < n, m = 1, …, 4 :

(x_{i}^{2} − x_{1}x_{n} x_{1}x_{i} x_{1}x_{j} x_{i}x_{j} x_{j}^{2} − x_{1}x_{n}) · (\begin{matrix}x_{j} & 0 & 0 & 0 \\ 0 & x_{n} & x_{j} & 0 \\ x_{n} & 0 & −x_{i} & −x_{i} \\ −x_{i} & −x_{j} & 0 & x_{1} \\ 0 & x_{i} & 0 & 0 \end{matrix}) = 0,

and R_{ijk} · S_{ijk}(m), 1 < i < j < k < n, m = 1, 2 :

(x_{i}x_{j} x_{j}x_{k} x_{i}x_{k}) · (\begin{matrix}−x_{k} & −x_{k} \\ x_{i} & 0 \\ 0 & x_{j} \end{matrix}) = 0.

These syzygies are clearly linearly independent, and their cardinality, 2\binom{n}{3}, coincides with the dimension of the space of first order syzygies, according to Proposition 2.1, so they form a basis. We lift these syzygies by adding the multiples of x_{n} in the syzygy matrix, that reduces the product to cubic polynomials with monomials only of the form x_{i}x_{n}^{2}. We denote by S̃_{i}(j) the syzygies obtained from S_{i}(j) this way. Likewise we denote by R̃_{i} the row vector obtained from R_{i} by substituting the entries x_{s}x_{t} by f_{st}. Similarly we get row vectors R̃_{ij}, R̃_{ijk} and column vectors S̃_{ij}(r) and S̃_{ijk}(r). For example, with i = 2 and n = 4 we get

R̃_{2} · S̃_{2}(1) = (f_{11} f_{12} f_{22} f_{13} f_{23} f_{33}) · (\begin{matrix}−x_{2} \\ x_{1} + a_{(12,2)}x_{4} \\ −a_{(11,2)}x_{4} \\ a_{(12,3)}x_{4} \\ −a_{(11,3)}x_{4} \\ 0 \end{matrix})

= (−a_{(12,4)} + a_{(11,2)})x_{1}x_{4}^{2}

+(a_{(11,4)} − a_{(12,2)}^{2} + a_{(22,2)}a_{(11,2)} − a_{(13,2)}a_{(12,3)} + a_{(23,2)}a_{(11,3)})x_{2}x_{4}^{2} +(−a_{(12,3)}a_{(12,2)} + a_{(22,3)}a_{(11,2)} − a_{(13,3)}a_{(12,3)} + a_{(23,3)}a_{(11,3)})x_{3}x_{4}^{2} +(−a_{(12,4)}a_{(12,2)} + a_{(22,4)}a_{(11,2)} − a_{(13,4)}a_{(12,3)} + a_{(23,4)}a_{(11,3)})x_{4}^{3}.

For general i and n we get (with the first pair in the index unordered to simplify presentation of the summation)

\begin{aligned}R̃_{i} · S̃_{i}(1) = & (a_{(11,i)} − a_{(1i,n)})x_{1}x_{n}^{2} + a_{(11,n)}x_{i}x_{n}^{2} \\ & + ∑_{j=2}^{n}(∑_{k=2}^{n−1}(a_{(11,k)}a_{(ik,j)} − a_{(1i,k)}a_{(1k,j)})x_{j}x_{n}^{2}. \end{aligned}

Similarly

R̃_{i} · S̃_{i}(2) = (a_{(1i,i)} − a_{(ii,n)})x_{1}x_{n}^{2} + a_{(1i,n)}x_{i}x_{n}^{2}
+ ∑_{j=2}^{n}(−a_{(11,j)} + ∑_{k=2}^{n−1}(a_{(1i,k)}a_{(ik,j)} − a_{(ii,k)}a_{(1k,j)}))x_{j}x_{n}^{2},
R̃_{ij} · S̃_{ij}(1) = (a_{(ij,i)} − a_{(ii,j)})x_{1}x_{n}^{2} + a_{(ij,n)}x_{i}x_{n}^{2} − a_{(ii,n)}x_{j}x_{n}^{2}
+ ∑_{k=2}^{n}(−a_{(1j,k)} + ∑_{m=2}^{n−1}(a_{(ij,m)}a_{(im,k)} − a_{(ii,m)}a_{(jm,k)}))x_{k}x_{n}^{2},
R̃_{ij} · S̃_{ij}(2) = (a_{(ij,j)} − a_{(jj,i)})x_{1}x_{n}^{2} − a_{(jj,n)}x_{i}x_{n}^{2} + a_{(ij,n)}x_{j}x_{n}^{2}
+ ∑_{k=2}^{n}(−a_{(1i,k)} + ∑_{m=2}^{n−1}(a_{(ij,m)}a_{(jm,k)} − a_{(jj,m)}a_{(im,k)}))x_{k}x_{n}^{2},
R̃_{ij} · S̃_{ij}(3) = (a_{(1j,i)} − a_{(1i,j)})x_{1}x_{n}^{2} + a_{(1j,n)}x_{i}x_{n}^{2} − a_{(1i,n)}x_{j}x_{n}^{2}
+ ∑_{k=2}^{n}(∑_{m=2}^{n−1}(a_{(1j,m)}a_{(im,k)} − a_{(1i,m)}a_{(jm,k)}))x_{k}x_{n}^{2},
R̃_{ij} · S̃_{ij}(4) = (a_{(1j,i)} − a_{(ij,n)})x_{1}x_{n}^{2} − a_{(1j,n)}x_{i}x_{n}^{2}
+ ∑_{k=2}^{n}(∑_{m=2}^{n−1}(a_{(1j,m)}a_{(im,k)} − a_{(ij,m)}a_{(1m,k)}))x_{k}x_{n}^{2},
R̃_{ijk} · S̃_{ijk}(1) = (a_{(ij,k)} − a_{(jk,i)})x_{1}x_{n}^{2} − a_{(jk,n)}x_{i}x_{n}^{2} + a_{(ij,n)}x_{k}x_{n}^{2}
+ ∑_{l=2}^{n}(∑_{m=2}^{n−1}(a_{(ij,m)}a_{(km,l)} − a_{(jk,m)}a_{(im,l)}))x_{l}x_{n}^{2},
R̃_{ijk} · S̃_{ijk}(2) = (a_{(ij,k)} − a_{(ik,j)})x_{1}x_{n}^{2} − a_{(ik,n)}x_{j}x_{n}^{2} + a_{(ij,n)}x_{k}x_{n}^{2}
+ ∑_{l=2}^{n}(∑_{m=2}^{n−1}(a_{(ij,m)}a_{(km,l)} − a_{(ik,m)}a_{(jm,l)}))x_{l}x_{n}^{2}.

The linear relations in the parameters of the family V_{h}^{aff} (n) are precisely the coefficients of x_{1}x_{n}^{2} in these products:

Lemma 5.6. The space of linear forms in the ideal of V_{h}^{aff} (n) is generated by the following forms, where {i, j, k} is any subset of distinct elements in {2, …, n − 1}

a_{(11,i)} − a_{(1i,n)}, a_{(1i,i)} − a_{(ii,n)}, a_{(1j,i)} − a_{(ij,n)}, a_{(ij,j)} − a_{(jj,i)}, a_{(ij,k)} − a_{(jk,i)}.

□

Notice that only the first two occur when n = 3, and only the first four occur when n = 4. Using these linear relations, the quadratic ones all become linear in the 2 × 2 minors of the matrix A_{F} of coefficients a_{(ij,k)}, i.e. linear in the Plücker coordinates. In fact, by a straightforward but tedious derivation from the above presentation, we may write the generators of the ideal V_{h}^{aff} (n) as linear combinations of 2 × 2 minors in the coefficient matrix A_{F} (cf. the documented computer algebra code to perform the computation of ideal generators [RS11]):

Lemma 5.7. Modulo the linear forms the ideal of V_{h}^{aff} (3) is generated by

a_{(11,3)} + a_{(12,3)}a_{(22,2)} − a_{(12,2)}a_{(22,3)}.

□

Lemma 5.8. Modulo the linear forms the ideal of V_{h}^{aff} (4) is generated by

−a_{(12,2)} −a_{(13,3)} +(a_{(23,2)}a_{(22,3)} −a_{(22,2)}a_{(23,3)})+(a_{(33,2)}a_{(23,3)} −a_{(23,2)}a_{(33,3)}),

−a_{(11,2)} + (a_{(12,3)}a_{(23,2)} − a_{(23,3)}a_{(12,2)}) + (a_{(13,3)}a_{(33,2)} − a_{(33,3)}a_{(13,2)}), −a_{(11,3)} + (a_{(12,2)}a_{(22,3)} − a_{(22,2)}a_{(12,3)}) + (a_{(13,2)}a_{(23,3)} − a_{(23,2)}a_{(13,3)}),

(a_{(11,2)}a_{(22,3)} − a_{(12,2)}a_{(12,3)}) + (a_{(11,3)}a_{(23,3)} − a_{(12,3)}a_{(13,3)}), a_{(11,4)} + (a_{(12,4)}a_{(22,2)} − a_{(22,4)}a_{(12,2)}) + (a_{(13,4)}a_{(23,2)} − a_{(23,4)}a_{(13,2)}), a_{(11,4)} + (a_{(12,4)}a_{(23,3)} − a_{(23,4)}a_{(12,3)}) + (a_{(13,4)}a_{(33,3)} − a_{(33,4)}a_{(13,3)}).

□

Lemma 5.9. Modulo the linear forms the ideal of V_{h}^{aff} (n), when n > 4, is generated by the following forms: For i ∈ {2, …, n − 1},

a_{(11,n)} − ∑_{m=2}^{n−1}(a_{(im,n)}a_{(1m,i)} − a_{(1m,n)}a_{(im,i)}),

for any subset {i, j} ⊂ {2, …, n − 1},

a_{(11,i)} − ∑_{m=2}^{n−1}(a_{(jm,n)}a_{(im,j)} − a_{(im,n)}a_{(jm,j)}),

∑_{m=2}^{n−1}(a_{(1m,n)}a_{(im,j)} − a_{(im,n)}a_{(1m,j)})

and

a_{(1i,i)} + a_{(1j,j)} − ∑_{m=2}^{n−1}(a_{(jm,i)}a_{(im,j)} − a_{(im,i)}a_{(jm,j)}),

for any subset {i, j, k} ⊂ {2, …, n − 1},

a_{(1j,k)} − ∑_{m=2}^{n−1}(a_{(jm,i)}a_{(im,k)} − a_{(im,i)}a_{(jm,k)})

and

∑_{m=2}^{n−1}(a_{(jm,n)}a_{(im,k)} − a_{(im,n)}a_{(jm,k)}),

and for any subset {i, j, k, l} ⊂ {2, …, n − 1},

∑_{m=2}^{n−1}(a_{(im,j)}a_{(km,l)} − a_{(km,j)}a_{(im,l)}).

□

Since the open affine sets V_{h}^{aff} (n) cover V AP S(Q, n), we conclude that V AP S(Q, n) is a linear section of the Grassmannian 𝔾(\binom{n}{2}, q^{⊥}). Equivalently, V AP S(Q, n) is projectively equivalent to a linear section of 𝔾(n − 1, T_{2,q}) in its Plücker embedding. This concludes the proof of Theorem 5.4. □

Using the linear relations we may reduce the number of variables, when n > 4, and use as indices the following unordered three element sets:

I = {{11k}|1 < k ≤ n} ∪ {{1jk}|1 < j ≤ k < n} ∪ {{ijk}|1 < i ≤ j ≤ k < n}.

Let R = ℂ[a_{I}|I ∈ I]. We substitute a_{11k} = a_{11,k}, a_{1jk} = a_{1j,k}, a_{ijk} = a_{ij,k} and get:

Lemma 5.10. The ideal of V_{h}^{aff} (n), when n > 4, is generated by the following polynomials in R: For i ∈ {2, …, n − 1},

a_{11n} − ∑_{m=2}^{n−1}(a_{1im}^{2} − a_{11m}a_{iim}),

for any subset {i, j} ⊂ {2, …, n − 1},

a_{11i} − ∑_{m=2}^{n−1}(a_{1jm}a_{ijm} − a_{1im}a_{jjm}),

∑_{m=2}^{n−1}(a_{11m}a_{ijm} − a_{1im}a_{1jm}), a_{1ii} + a_{1jj} − ∑_{m=2}^{n−1}(a_{ijm}^{2} − a_{iim}a_{jjm}),

for any subset {i, j, k} ⊂ {2, …, n − 1},

a_{1jk} − ∑_{m=2}^{n−1}(a_{ijm}a_{ikm} − a_{iim}a_{jkm}), ∑_{m=2}^{n−1}(a_{1jm}a_{ikm} − a_{1im}a_{jkm}),

and for any subset {i, j, k, l} ⊂ {2, …, n − 1},

∑_{m=2}^{n−1}(a_{ijm}a_{klm} − a_{jkm}a_{ilm}).

□

Notice that these generators are all homogeneous in the weights introduced above. The linear parts of the ideal generators define the tangent space of the family V_{h}^{aff} (n) at [Γ_{p}], so another consequence of our computations is the tangent space dimension.

Proposition 5.11. Let L(n) be the space of linear forms spanned by the linear parts of the generators in the ideal of V_{h}^{aff} (n). Then L(3) is spanned by

a_{(11,3)}, a_{(11,2)} − a_{(12,3)} and a_{(12,2)} − a_{(22,3)}.

L(4) is spanned by

a_{(11,4)}, a_{(11,2)}, a_{(12,4)}, a_{(11,3)}, a_{(13,4)},
a_{(12,2)} − a_{(22,4)}, a_{(13,3)} − a_{(33,4)}, a_{(12,2)} + a_{(13,3)},

a_{(12,3)} − a_{(23,4)}, a_{(13,2)} − a_{(12,3)}, a_{(23,3)} − a_{(33,2)} and a_{(22,3)} − a_{(23,2)}. L(n), when n > 4, is spanned by a_{(11,n)} and for any i ∈ {2, …, n − 1},

a_{(11,i)}, a_{(1i,n)}, a_{(1i,i)}, a_{(ii,n)},

for any subset {i, j} ⊂ {2, …, n − 1},

a_{(1j,i)}, a_{(ij,n)}, a_{(ii,j)} − a_{(ij,i)},

and for any subset {i, j, k} ⊂ {2, …, n − 1},

a_{(ij,k)} − a_{(jk,i)}.

In particular

V_{h}^{aff} (3) ≅ A^{3}, V_{p}^{aff} (4) ≅ A^{6}, V_{h}^{aff} (5) ≅ A^{10}.

Proof. The linear parts of the generators can be read off Lemma 5.6 and Lemmas 5.7, 5.8 and 5.9. Notice only that the two term forms

a_{(12,2)} − a_{(22,4)}, a_{(13,3)} − a_{(33,4)}, a_{(12,2)} + a_{(13,3)}

span a three dimensional space, while

a_{(1i,i)} − a_{(ii,n)}, a_{(1j,j)} − a_{(jj,n)}, a_{(1i,i)} + a_{(1j,j)}, for 1 < i < j < n

span the space generated by

a_{(1i,i)}, a_{(ii,n)}, 1 < i < n

when n > 4. For n = 3, the family V_{h}^{aff} (3) has 6 parameters, while there are three independent linear forms in the relations so the tangent space at [Γ_{p}] has dimension 6 − 3 = 3 as expected. In fact V_{h}^{aff} (3) ≅ A^{3} with parameters

a_{(11,2)}, a_{(12,2)}, a_{(22,2)}.

For n = 4 the family V_{h}^{aff} (4) has 18 parameters, while the linear forms in the relations are generated by 12 independent forms, so the tangent space at [Γ_{p}] has dimension 18 − 12 = 6. In fact V_{h}^{aff} (4) ≅ A^{6} with parameters

a_{(12,2)}, a_{(12,3)}, a_{(23,3)}, a_{(22,3)}, a_{(22,2)}, a_{(33,3)}.

For n > 4 we see that all parameters with a 1 or an n in the index are independent forms in the space of linear parts of ideal generators in V_{h}^{aff} (n). Furthermore, the other linear parts, simply expresses that {(ijk)|1 < i ≤ j ≤ k < n} form a natural index set for representatives of the parameters. The cardinality of this index set is simply the cardinality of monomials of degree 3 in n − 2 variables, i.e. \binom{n}{3}. In case n = 5 we again conclude that V_{h}^{aff} (5) ≅ A^{10} with parameters

{a_{ijk}|2 ≤ i ≤ j ≤ k ≤ 4}.
□

Corollary 5.12. The tangent space dimension of V AP S(Q, n) at [Γ_{p}] is \binom{n}{3} when n > 5. When n ≤ 5, V AP S(Q, n) has a finite cover of affine spaces, in particular V AP S(Q, n) is smooth and coincides with V P S(Q, n). □

Remark 5.13. Let Γ be a smooth apolar subscheme to Q consisting of n distinct points. Any subset of n − 2 points in Γ is contained in a pencil of apolar subschemes that form a line in V P S(Q, n) through [Γ]. Thus \binom{n}{2} lines in V P S(Q, n) through [Γ] is contained in the tangent space at [Γ].

We extend this remark and give a conceptual reason for the dimension of the tangent space to V AP S(Q, n) at [Γ_{p}].

Proposition 5.14. Let [Γ_{p}] ∈ V P S(Q, n) ⊂ 𝔾(n − 1, T_{2,q}) be a point on the subvariety T Q^{−1} in its Grassmannian embedding. Then V P S(Q, n) contains the cone over a 3-uple embedding of P^{n−3} with vertex at [Γ_{p}].

Proof. We first identify a cone over a 3-uple embedding of P^{n−3} inside V AP S(Q, n), and then give an explicit description of the apolar subschemes parameterized by this cone in order to show that the cone is contained in V P S(Q, n). Consider the subvariety V_{p}^{vero}(n) ⊂ V_{h}^{aff} (n) parameterizing ideals I_{Γ} with coefficient matrix A_{F} (Γ) = (I A) where the submatrix A = (a_{(ij,k)}) has rank at most 1 and has nonzero entries only in the submatrix A_{0} ⊂ A with entries {a_{(ij,k)}|1 < i ≤ j < n, 1 < k < n}. As above, using the linear relations, we may substitute the parameters a_{(ij,k)} with parameters a_{ijk} whose indices are unordered triples (ijk). In these new parameters the matrix A_{0} takes the form:

A_{0} = (\begin{matrix}a_{222} & a_{223} & ... & a_{22(n−1)} \\ a_{223} & a_{233} & ... & a_{23(n−1)} \\ ... & ... & ... & ... \\ a_{22(n−1)} & a_{23(n−1)} & ... & a_{2(n−1)(n−1)} \\ ... & ... & ... & ... \\ a_{2(n−1)(n−1)} & a_{3(n−1)(n−1)} & ... & a_{(n−1)(n−1)(n−1)} \end{matrix}) .

By Theorem 5.4, the equations of V_{h}^{aff} (n) are linear in the 2 × 2 minors of the coefficient matrix A, so any rank 1 matrix A_{0} defines a point on V_{p}^{vero}(n). The symmetry in the indices explains why the 2 × 2 minors of the matrix define the 3-uple embedding of P^{n−3}. Since the ideal of Γ_{p} correspond to the zero matrix, we conclude that the subvariety V_{p}^{vero}(n) in V AP S(Q, n) is the cone over this 3-uple embedding. To see that V_{p}^{vero}(n) is contained in V P S(Q, n) we show that a general point on V_{p}^{vero}(n) lies in the closure of smooth apolar subscheme to Q. For this, we describe for each general point s ∈ P^{n−3} an apolar subscheme Γ_{s} belonging to V_{p}^{vero}(n). It has two components Γ_{s} = Γ_{s,0}∪p_{s}, the first one Γ_{s,0} of length n−1 and supported at p, while the second component p_{s} is a closed point. We shall show that q has a decomposition q = q_{l} + q(l)^{2} ∈ T_{2} where [q(l)] = p_{s} ∈ P(T_{1}) and q_{l} ∈ (l^{⊥})^{2}. The subscheme Γ_{s,0} is apolar to q_{l} and contains the first order neighborhood of p inside the quadric {q_{l}^{−1} = 0} ⊂ P(l^{⊥}) in the hyperplane polar to p_{s}. Then Γ_{s,0} lies in the closure of smooth apolar subschemes to q_{l}. We conclude by applying Proposition 2.4. Let s = [s_{2} : .. : s_{n−1}] ∈ P^{n−3} and let

||s||^{2} = s_{2}^{2} + … + s_{n−1}^{2}, ⟨s, x⟩ = ∑_{i=2}^{n−1} s_{i}x_{i}, ⟨s, y⟩ = ∑_{i=2}^{n−1} s_{i}y_{i},

then

x_{1}^{2}, …, x_{1}x_{n−1},

x_{i}^{2} − x_{1}x_{n} − s_{i}^{2}⟨s, x⟩x_{n} 1 < i < n, x_{i}x_{j} − s_{i}s_{j}⟨s, x⟩x_{n} 1 < i < j < n defines a subscheme Γ_{s} that belongs to V_{p}^{vero}(n). When ||s||^{2} ≠ 0, then Γ_{s} contains the point

p_{s} = [q(||s||^{2}⟨s, x⟩ + x_{1})] = [||s||^{2}⟨s, y⟩ + y_{n}] ∈ P(T_{1})

Consider the linear subspace L_{s} = {x_{1} = 0} ∩ {⟨s, x⟩ = 0}. The intersection Γ_{s} ∩ L_{s} is the subscheme defined by

x_{2}^{2} = x_{2}x_{3} = … = x_{n−1}^{2} = 0.

This subscheme has length n−2. The union p_{s} ∪(Γ_{s} ∩L_{s}) spans the hyperplane {x_{1} = 0}, so the residual point in Γ_{s} is the pole, with respect to Q^{−1}, of this hyperplane, i.e. the point p. Therefore the subscheme Γ_{s,0} = Γ_{s} \ p_{s} has length n − 1, is supported in p, and contains the first order neighborhood of p in the codimension two linear space L_{s}. The subscheme Γ_{s,0} is apolar to the quadric

q_{s} = (||s||^{2}⟨s, y⟩ + y_{n})^{2} − ||s||^{6}(2y_{1}y_{n} + y_{2}^{2} + ... + y_{n−1}^{2}).

Let l = ||s||^{2}⟨s, x⟩ + x_{1}. Then p_{s} = [q(l)], while

l^{⊥} = ⟨y_{2} − ||s||^{2}s_{i}y_{1}, ..., y_{n−1} − ||s||^{2}s_{i}y_{1}, y_{n}⟩.

Then q_{s} ∈ (l^{⊥})^{2} < 2 and

(q(l))^{2} − q_{s} = ||s||^{6} · q ∈ T_{2}.

According to Proposition 2.4 a subscheme Γ_{0} in P(l^{⊥}) of length n−1 is apolar to q_{s} if and only if Γ = Γ_{0}∪p_{s} is apolar to q. Now, Γ_{s,0} is apolar to q_{s} and contains a first order neighborhood of a point on the smooth quadric {q_{s}^{−1} = 0} in P(l^{⊥}) ⊂ P(T_{1}). By Remark 3.5, the subscheme Γ_{s,0} is a subscheme like Γ_{p}, with respect to q_{s}. Therefore Γ_{s,0} lies in the closure of smooth apolar subschemes to q_{s}. But then Γ_{s} must lie in the closure of smooth apolar subschemes to q. Hence [Γ_{s}] ∈ V P S(q, n).

□

Corollary 5.15. V P S(Q, n) is singular when n ≥ 6.

Proof. The cone with vertex at [Γ_{p}] ∈ T Q^{−1} over the 3-uple embedding of P^{n−3} is contained in the tangent space of V P S(Q, n) at [Γ_{p}], i.e. also in the tangent space of V AP S(Q, n). Since the span of the cone and the tangent space of the latter have the same dimension, they coincide. In particular the tangent space of V P S(Q, n) at [Γ_{p}] has dimension \binom{n}{3}. When n ≥ 6, then \binom{n}{3} > \binom{n}{2} = dimV P S(Q, n) so V P S(Q, n) is singular. □

We pursue the case n = 6 a bit further and show that V AP S(Q, 6) and V P S(Q, 6) coincide. We use the symmetric variables

a_{ijk} = a_{(ij,k)}, 1 ≤ i, j, k ≤ 6

for any permutation of the letters i, j, k. According to Lemma 5.10 we may list the generators explicitly. This list is however not minimal. In fact, a minimal set of generators is given by the following twenty generators in weight 2, four generators in weight 3 and one generator in weight 4. The twenty generators of weight 2 are the generators of weight 2 in Lemma 5.10: For each 1 < k < 6, and each pair {i, j} ⊂ {2, 3, 4, 5} \ {k} the generator

−a_{ij6} + ∑_{m=2}^{5}(a_{ikm}a_{jkm} − a_{ijm}a_{kkm}),

for each pair {i, j} ⊂ {2, 3, 4, 5} the generator

−a_{ii6} − a_{jj6} + ∑_{m=2}^{5}(a_{ijm}a_{ijm} − a_{iim}a_{jjm}),

and additionally the two generators

∑_{m=2}^{5}(a_{23m}a_{45m} − a_{24m}a_{35m}) and ∑_{m=2}^{5}(a_{23m}a_{45m} − a_{25m}a_{34m}).

The last five generators are computed from the list of Lemma 5.10 using Macaulay2 [GS], see the documented code in [RS11]. Of weight 3 we find, for i = 2, 3, 4:

a_{11i} − ∑_{m=2}^{5}(a_{im6}a_{m55} − a_{m56}a_{im5})

and

a_{115} − ∑_{m=2}^{5}(a_{m46}a_{m45} − a_{m56}a_{m44}).

The generator of weight 4 is

a_{116} − ∑_{m=2}^{5} a_{m56}^{2} + ∑_{m=2}^{5} a_{11m}a_{m55}.

The ten parameters with 6 in the index appear linearly in the 20 generators of weight 2, while the five parameters with 11 in the index appear linearly in the five generators of weights 3 and 4. The remaining 10 generators of weight 2 therefore depend only on 20 parameters a_{I}. In fact they depend only on 16 linear forms. It is a remarkable fact that these ten quadratic forms define the 10-dimensional spinor variety. To see this we choose and rename the following 16 forms:

x_{1234} = −a_{353} + a_{252}, x_{15} = −a_{555} + a_{454} + a_{353} + a_{252}, x_{34} = a_{453},

x_{1235} = a_{554} − a_{444} + a_{343} + a_{242}, x_{14} = a_{343} − a_{242}, x_{35} = a_{553} − a_{232}, x_{1245} = −a_{553} − a_{443} + a_{333} − a_{232}, x_{13} = a_{553} − a_{443}, x_{24} = a_{452}, x_{1345} = a_{552} + a_{442} + a_{332} − a_{222}, x_{12} = a_{552} − a_{442}, x_{23} = a_{352}, x_{2345} = a_{454} − a_{353}, x_{45} = a_{554} − a_{242}, x_{25} = −a_{442} + a_{332}, x_{0} = a_{342}.

In these variables the ten quadratic generators takes the form

q_{0} = x_{25}x_{34} − x_{35}x_{24} + x_{45}x_{23} + x_{2345}x_{0},

q_{1} = −x_{45}x_{13} + x_{14}x_{35} − x_{15}x_{34} + x_{1345}x_{0},

q_{2} = x_{45}x_{12} + x_{14}x_{25} + x_{15}x_{24} + x_{1245}x_{0},
q_{3} = −x_{35}x_{12} + x_{13}x_{25} − x_{15}x_{23} + x_{1235}x_{0},
q_{4} = x_{12}x_{34} − x_{13}x_{24} + x_{14}x_{23} + x_{1234}x_{0},
q_{5} = x_{1345}x_{12} + x_{1245}x_{13} + x_{1235}x_{14} + x_{15}x_{1234},
q_{6} = −x_{2345}x_{12} + x_{1245}x_{23} + x_{1235}x_{24} + x_{1234}x_{25},
q_{7} = −x_{2345}x_{13} − x_{1345}x_{23} + x_{1235}x_{34} + x_{1234}x_{35},
q_{8} = −x_{2345}x_{14} − x_{1345}x_{24} − x_{1245}x_{34} + x_{1234}x_{45},
q_{9} = −x_{15}x_{2345} − x_{1345}x_{25} − x_{1245}x_{35} − x_{1235}x_{45}.

The first five express (when x_{0} = 1) the variables x_{ijkl} as quadratic Pfaffians in the x_{st}, while the last five quadrics express the linear syzygies among these Pfaffians. The ten quadratic forms satisfy the following quadratic relation

q_{0}q_{5} + q_{1}q_{6} + q_{2}q_{7} + q_{3}q_{8} + q_{4}q_{9} = 0.

In fact the ten quadratic forms generate the ideal of the 10-dimensional spinor variety embedded in P^{15} by its spinor coordinates [RS00, Section 6],[Muk95].

Corollary 5.16. V_{h}^{aff} (6) is isomorphic to a cone over the ten-dimensional spinor variety embedded in P^{15} by its spinor coordinates. In particular V AP S(Q, 6) is singular, irreducible and coincides with V P S(Q, 6). □

We end this section summarizing some computational results, for small n, of some natural subschemes of V AP S(Q, n). The first is the punctual part V_{p}^{loc}(n) of V AP S(Q, n), i.e. the variety of apolar subschemes in V AP S(Q, n) with support at a single point p. The support p of a local apolar subscheme must lie on Q^{−1} by Lemma 2.3. Therefore we may assume that p = [0 : 0 : ... : 1], and use the equations 5.1. Of course, [Γ_{p}] is then in V_{p}^{loc}(n). Furthermore, V_{p}^{loc}(n) is naturally contained in a second natural subscheme of V AP S(Q, n), namely V_{p}^{sec}(n), the variety of apolar subschemes in V_{h}^{aff} (n) that contains the point p. We will do the explicit computation in the cases where V AP S(Q, n) = V P S(Q, n) is smooth, i.e. when n < 6. An apolar subscheme in V_{h}^{aff} (n) lies in V_{p}^{sec}(n) if and only if the term x_{n}^{2} does not appear in any equation, so V_{p}^{sec}(n) is defined by the equations a_{(ij,n)} = 0 for 1 ≤ i ≤ j < n in V_{h}^{aff} (n). The linear relations then imply that a_{(1i,j)} = 0 for all i and j, and as before that each parameter a_{(ij,k)} with 1 < i, j, k < n may be represented by a parameter

a_{ijk} with 1 < i ≤ j ≤ k < n.

In particular for n = 3 the only parameter left is a_{222}, and V_{p}^{sec}(3) is isomorphic to the affine line. The equations x_{1}^{2} = x_{1}x_{2} = x_{2}^{2} − x_{1}x_{3} − a_{222}x_{2}x_{3} define a scheme supported at p only if a_{222} = 0, so V_{p}^{loc}(n) is a point in the case n = 3. The computation of V_{p}^{sec}(n) follow the same procedure for every n. For a local scheme Γ in V_{p}^{loc}(n) we may set x_{n} = 1 in the equations.

Lemma 5.17. A local scheme Γ supported at p, that belongs to V_{p}^{loc}(n), is Gorenstein. The maximal ideal of its affine coordinate ring is spanned by x_{2}, …, x_{n−1}, x_{1}, and its socle is generated by x_{1}.

Proof. The scheme Γ is Gorenstein by Lemma 2.6. The maximal ideal is certainly generated by x_{1}, x_{2}, …, x_{n−1}, and since Γ is nondegenerate these are linearly independent. Finally, x_{1}x_{i} = 0 for all i by the apolarity condition as soon as p ∈ Γ, so the socle is generated by x_{1}. □

We may now get explicit equations for V_{p}^{loc}(n). If [Γ] ∈ V_{p}^{loc}(n), then by definition m_{p}^{n} = 0. But the maximal ideal is generated by x_{1}, x_{2}, …, x_{n−1}, so this means that any monomial of degree n in the x_{i} must vanish in the coordinate ring of Γ. On the other hand, the equations for Γ define the products

x_{i}x_{j} = ∑_{k=2}^{n−1} a_{ijk}x_{k} and x_{i}^{2} = x_{1} + ∑_{k=2}^{n−1} a_{iik}x_{k}

in this ring. Therefore, by iteration, we get polynomial relations in the parameters a_{ijk}. Imposing the apolarity condition, symmetrizing the parameters and adding the equations for V_{p}^{sec}(n), then after, possibly, saturation we get set theoretic equations for V_{p}^{loc}(n). When n = 4 we have the parameters a_{222}, a_{223}, a_{233}, a_{333} for V_{p}^{sec}(4) and the relation

a_{223}^{2} − a_{222}a_{233} + a_{233}^{2} − a_{223}a_{333}.

Thus V_{p}^{sec}(4) is a quadric hypersurface in A^{4}. For V_{p}^{loc}(4) we first get the subscheme defined by the equations

x_{1}^{2} = x_{1}x_{2} = x_{1}x_{3} = 0

and

x_{2}^{2} = x_{1} + a_{222}x_{2} + a_{223}x_{3}, x_{2}x_{3} = a_{223}x_{2} + a_{233}x_{3}, x_{3}^{2} = x_{1} + a_{233}x_{2} + a_{333}x_{3}.

The coefficient of x_{1} using these relations iteratively to compute x_{2}^{4}, …, x_{3}^{4}, must vanish, so it yields the equations a_{222} +a_{233} = a_{223} +a_{333} = a_{233}^{2} +a_{223}^{2} = 0. The other coefficients give no additional relations, and neither does the equations for V_{p}^{sec}(4), so V_{p}^{loc}(4) is 1-dimensional and consists of a pair of affine intersecting lines. When n = 5, the computation becomes a bit more involved. There are ten parameters a_{ijk}. The equations of V_{p}^{sec}(5) are

a_{234}^{2} − a_{233}a_{244} + a_{334}^{2} − a_{333}a_{344} + a_{344}^{2} − a_{334}a_{444} = 0,
a_{224}a_{234} − a_{223}a_{244} + a_{234}a_{334} − a_{233}a_{344} + a_{244}a_{344} − a_{234}a_{444} = 0,
a_{224}a_{233} − a_{223}a_{234} + a_{234}a_{333} − a_{233}a_{334} + a_{244}a_{334} − a_{234}a_{344} = 0,
a_{234}^{2} + a_{224}^{2} − a_{222}a_{244} + a_{244}^{2} − a_{223}a_{344} − a_{224}a_{444} = 0,
a_{223}a_{224} − a_{222}a_{234} + a_{233}a_{234} + a_{234}a_{244} − a_{223}a_{334} − a_{224}a_{344} = 0,
a_{223}^{2} − a_{222}a_{233} + a_{233}^{2} − a_{223}a_{333} − a_{224}a_{334} + a_{234}^{2} = 0.

They define in A^{10} the affine cone over the intersection of the Grassmannian variety 𝔾(2, 5) with a quadric. For V_{p}^{loc}(5) there are additional equations defining the cone over the tangent developable of a rational normal sextic curve, a

n  V_{p}^{loc}(n)  V_{p}^{sec}(n)  V_{h}^{aff} (n)
dim  degree dim  degree dim degree
3 0  1 1  1 3  1
point  A^{1}  A^{3}
4 1  2 3  2 6  1
two lines  Quadric 3-fold  A^{6}
5 3  10 6  10 10  1
cone over tangent developable cone over 𝔾(2, 5) ∩ Q  A^{10}

of a rational sextic curve

6 15 12
cone over S_{10}

Table 1.

codimension 3 linear section of V_{p}^{sec}(5). The cone over the rational normal curve parameterizes local apolar subschemes that are not curvilinear. For the computations in Macaulay2 [GS], see the documented code in [RS11]. The findings are summarized in Table 1.

6. Global invariants of V P S(Q, n)

We consider V P S(Q, n) as a subscheme of 𝔾(n − 1, T_{2,q}), and the incidence

I_{Q}^{V P S} = {([q′], [Γ])|[q′] ⊂ ⟨Γ⟩} ⊂ P(T_{2,q}) × V P S(Q, n).

The incidence is a projective bundle,

I_{Q}^{V P S} = P(E_{Q}) ⟶^{π} V P S(Q, n),

while the first projection is birational (the rational map γ : P(T_{2,q}) −−→ V P S(Q, n) factors through the inverse of this projection). Denote by L the tautological divisor on P(E_{Q}). It is the pullback of the hyperplane divisor on P(T_{2,q}). When V P S(Q, n) is smooth,

Pic(I_{Q}^{V P S}) ≅ Pic(V P S(Q, n)) ⊕ ℤ[L].

Recall from Lemma 4.9, that the set H_{h} ⊂ V P S(Q, n) of subschemes Γ that intersects a hyperplane h ⊂ P(T_{1}) form a Plücker divisor restricted to V P S(Q, n). Therefore the class of the Plücker divisor coincides with the first Chern class c_{1}(E_{Q}).

Theorem 6.1. i) Pic(V P S(Q, 4) ≅ Pic(V P S(Q, 5)) ≅ ℤ. ii) The ample generator H is very ample, and V P S(Q, 4) and V P S(Q, 5) are Fano-manifolds of index 2. iii) The boundary in V P S(Q, n) consisting of singular apolar subschemes is, when n ≤ 5, an anticanonical divisor.

Proof. i) Let n = 4 or n = 5. Then the Plücker divisor H is very ample by the above. Furthermore, the complement V_{p}^{aff} of the special Plücker divisor defined by a tangent hyperplane to Q^{−1} ⊂ P(T_{1}), the divisor H_{{x_{n}=0}} in the above notation, is isomorphic to affine space by Proposition 5.11. Therefore the Picard group has rank 1 as soon as this special Plücker divisor is irreducible. The tangent hyperplanes to Q^{−1} cover all of P(T_{1}), so the corresponding Plücker divisors cover V P S(Q, n). Furthermore, for any subscheme Γ in V P S(Q, n), there is tangent hyperplane that does not meet Γ, so these special Plücker divisors have no common point on V P S(Q, n). Assume that the special Plücker divisors are reducible, then we may write H = H_{1} + H_{2}, where both H_{1} and H_{2} moves without base points on V P S(Q, n). Since H · l = 1 for every line on V P S(Q, n), only one of the two components can have positive intersection with a line. The other, say H_{2}, must therefore contain every line that it intersects. But this is impossible, since H_{2} must contain all of V P S(Q, n), by the following lemma:

Lemma 6.2. Any two polar simplices Γ and Γ′ are connected by a sequence of lines in V P S(Q, n).

Proof. This is immediate when n = 2. For n > 2, let [l] ∈ Γ and [l′] ∈ Γ′, and let P(U) = h_{l} ∩ h_{l′} ⊂ P(T_{1}) be the intersection of their polar hyperplanes. Then q = l^{2} + l_{1}^{2} + q_{U} = (l′)^{2} + (l′_{1})^{2} + q_{U} for q_{U} ∈ U^{2} and suitable l_{1} and l′_{1}. Let Γ_{U} be a polar simplex for q_{U}. Then Γ is line connected to Γ_{U} ∪ {[l_{1}], [l]} by induction hypothesis. Likewise Γ is line connected to Γ_{U} ∪ {[l′_{1}], [l′]}. Finally Γ_{U} ∪ {[l_{1}], [l]} and Γ_{U} ∪ {[l′_{1}], [l′]} span a line in V P S(Q, n), which completes the induction. □

ii) Since Pic(I_{Q}^{V P S}) ≅ Pic(V P S(Q, n)) ⊕ ℤ[L] we deduce from i) that the birational morphism

σ: I_{Q}^{V P S} → P(T_{2,q})

has an irreducible exceptional divisor. Let E ∈ Pic(I_{Q}^{V P S}) be the class of this exceptional divisor. Then, since the map γ : P(T_{2,q}) −−→ 𝔾(n − 1, T_{2,q}) is defined by polynomials of degree \binom{n}{2}, the size of the minors in the Mukai form, we have

π^{∗}H = \binom{n}{2}L − E and K_{I_{Q}^{V P S}} = −(\binom{n + 1}{2} − 1)L + E.

On the other hand H = c_{1}(E_{Q}) where I_{Q}^{V P S} = P(E_{Q}) is a projective bundle over V P S(Q, n) so

−(\binom{n + 1}{2} − 1)L + E = K_{I_{Q}^{V P S}} = π^{∗}K_{V P S} + π^{∗}(c_{1}(E_{Q})) − (n − 1)L.

Therefore −K_{V P S(Q,n)} = 2H. Finally, since V P S(Q, n) ⊂ 𝔾(n − 1, T_{2,q}) contains lines, H is not divisible. iii) The boundary in V P S(Q, n) consisting of singular apolar subschemes, coincides, by Lemma 2.3, with the set of subschemes Γ ⊂ P(T_{1}) that intersect quadric Q^{−1}. The Plücker divisor H is represented by the divisor of subschemes Γ that intersect a hyperplane in P(T_{1}), so −K = 2H is represented by the boundary. □

Theorem 6.3. Let n > 2 and let V P S(Q, n) ⊂ 𝔾(n − 1, T_{2,q}) be the variety of polar simplices in its Grassmannian embedding, with Plücker divisor H. The V P S(Q, n) has degree

H^{m} = ∑_{λ⊢m} \binom{m}{λ}/(λ^{∗}!) · d_{λ}

where the sum runs over all partitions λ = (λ_{1}, …, λ_{n}) of m = \binom{n}{2} = dimV P S(Q, n) into integers n − 1 ≥ λ_{1} ≥ … ≥ λ_{n} ≥ 0. Here λ^{∗} = (λ_{1}^{∗}, …, λ_{n−1}^{∗}) denotes the sequence λ_{i}^{∗} =| {j | λ_{j} = i} | and λ^{∗}! = Πλ_{i}^{∗}!. Finally

d_{λ} = ∏_{1≤i<j≤n}(D_{i} + D_{j})

is the intersection number of m divisors on the product

P^{n−1−λ_{1}} × … × P^{n−1−λ_{n}}

with D_{i} the pullback of the hyperplane class on the i^{th} component.

Proof. We first show that for \binom{n}{2} general hyperplanes h_{i} ⊂ P(T_{1}), the corresponding Plücker divisors H_{h_{i}} has a proper transverse intersection on the smooth part of V P S(Q, n). Therefore, by properness, the intersection is finite, and, by transversality, it is smooth, so it is a finite set of points. The cardinality is the degree of V P S(Q, n). First, let λ = (λ_{1}, …, λ_{n}) be a partition of m and consider the partition h_{11}, …, h_{1λ_{1}}, …, h_{n1}, …, h_{nλ_{n}} of m general hyperplanes into n sets of size λ_{1}, …, λ_{n}. Let L_{i} = ∩_{j}h_{ij}, it is a linear space of dimension n − 1 − λ_{i}. Consider the product of these linear spaces in the product P(T_{1})^{n}:

L_{1} × ⋯ × L_{n} ⊂ P(T_{1}) × ⋯ × P(T_{1}).

Let Δ ∈ P(T_{1})^{n} be the union of all diagonals and let L^{o} = L_{1} × ⋯ × L_{n} \ Δ ⊂ P(T_{1})^{n}. Then L^{o} parameterizes n-tuples of points Γ = {p_{1}, …, p_{n}} ⊂ P(T_{1}), with p_{i} ∈ L_{i}. Of course, L^{o} has a natural map to the Hilbert scheme of P(T_{1}) that forgets the ordering, so we will identify elements in L^{o} with their image in the Hilbert scheme. Consider the incidence between subschemes Γ ∈ L^{o} and quadratic forms q ∈ T_{2}:

I_{L} = {(Γ, [q])|I_{Γ} ⊂ q^{⊥}} ⊂ L^{o} × P(T_{2}).

This variety is defined by the equations h_{ij}(p_{i}) = 0 and the apolarity, q(I_{Γ}) = 0. Clearly L is a smooth scheme of dimension \binom{n}{2}. The fibers of the projection I_{L} → L are (n − 1)-dimensional projective spaces, so I_{L} is a smooth variety of dimension equal to dimP(T_{2}). The projection I_{L} → P(T_{2}) is clearly onto, so the fibers are finite. Since both spaces are smooth, the general fiber is smooth. Now, Γ ⊂ L^{o} lies in the fiber over [q], precisely when I_{Γ} ⊂ q^{⊥}, i.e. [Γ] ∈ V P S(Q, n) and h_{ij}(p_{i}) = 0, i.e. [Γ] lies in the intersection of all the Plücker hyperplanes H_{h_{ij}}. Since the general fibers are smooth the divisors H_{h_{ij}} intersect transversally in V P S(Q, n), where Q = {q = 0}, and have an isolated intersection point at each point [Γ]. Turning the argument around and considering all partitions, we get that for general hyperplanes h_{1}, …, h_{m} in P(T_{1}) the Plücker hyperplanes H_{h_{i}} has a transversal intersection at a finite number of points in V P S(Q, n) corresponding to smooth apolar subschemes. We proceed to compute the cardinality of the intersection, i.e. the formula given in the theorem. Let [Γ] = [{p_{1}, …, p_{n}}] ∈ V P S(Q, n) be a point in the intersection of the hyperplanes H_{h_{j}}. Then each h_{j} contains some p_{i} ∈ Γ, by the definition of H_{h_{j}}. For each i let λ_{i} be the number of hyperplanes h_{j} that contains p_{i}. The set of positive integers {λ_{1}, ..., λ_{n}} must add up to m: It is at least m by definition, and at most m by the generality assumption discussed above. Therefore the point [Γ] defines a unique partition of the set of hyperplanes {h_{j}}_{j=1}^{m} into subsets {h_{ij}}_{j=1}^{λ_{i}} of cardinality λ_{i}, as above. The factor \binom{m}{λ} in the degree formula counts the number of ordered partitions of m hyperplanes into subsets of cardinality λ_{i}, while λ^{∗}! counts the permutations of the subsets of the same cardinality, i.e. the number of ordered partitions determined by [Γ]. Therefore the remaining factor d_{λ} for each partition should count the number of polar simplices Γ that intersect the n linear subspaces L_{i} = ∩_{j}h_{ij} ⊂ P(T_{1}) of codimension λ_{i}, i = 1, ..., n. Let [Γ] = [{p_{1}, ..., p_{n}}] ∈ V P S(Q, n) and assume that (p_{1}, ..., p_{n}) ∈ L_{1}×⋯×L_{n}. For each pair of linear spaces L_{i}, L_{j} the bilinear form associated to the quadratic form q restricts to a linear form on the product L_{i}×L_{j} that vanishes on (p_{i}, p_{j}). This linear form defines a divisor H_{ij} in the divisor class H_{i} + H_{j}, where H_{i} is the pullback to the product of the hyperplane class on L_{i}. If D_{ij} and D_{i} are the pullbacks of H_{ij}, respectively H_{i}, to the product Π_{i}L_{i}, then Γ ⊂ Π_{i}L_{i} lies in the intersection ∩_{i<j}D_{ij}. Conversely, consider a point (p_{1}, …, p_{n}) ∈ L_{1} × ⋯ × L_{n} that lies in the intersection of the divisors ∩_{i<j}D_{ij}. The projection of this point into P(T_{1}) is a collection of n points Γ = {p_{1}, …, p_{n}}. Let p_{i} = [v_{i}], v_{i} ∈ T_{1}, then the bilinear form q : T_{1} ×T_{1} → ℂ, (q^{−1}(v_{i})(q))(v_{j}) = 0 for every i ≠ j, so the hyperplanes p_{i}^{⊥} ⊂ P(S_{1}) form a polar simplex to Q. Hence [Γ] is a point in V P S(Q, n). Thus d_{λ} counts the number of polar simplices Γ that intersect the n linear subspaces L_{i} and the degree formula follows.

□

The Theorem 1.1 in the introduction follows from Corollary 5.12, Corollary 5.15 and Theorem 6.1. Theorem 1.2 follows from Corollary 2.2, Corollary 5.2, Theorem 6.1 and the degree is computed from Theorem 6.3. Theorem 1.3 follows from Theorem 5.4, Corollary 5.16 and Corollary 2.7.

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Kristian Ranestad Matematisk institutt Universitetet i Oslo PO Box 1053 Blindern NO-0316 Oslo Norway ranestad@math.uio.no Frank-Olaf Schreyer Mathematik und Informatik Universität des Saarlandes Campus E2 4 D-66123 Saarbrücken Germany schreyer@math.uni-sb.de 506

Documenta Mathematica 18 (2013)