On the Equations Defining Some Hilbert Schemes

Abstract

We work out details of the extrinsic geometry for two Hilbert schemes of some contemporary interest: the Hilbert scheme $\text {Hilb}^{2} \mathbb {P}^{2}$ of two points on $\mathbb {P}^{2}$ and the dense open set parametrizing non-planar clusters in the punctual Hilbert scheme $\text {Hilb}^{4}_{0}(\mathbb {A}^{3})$ of clusters of length four on $\mathbb {A}^{3}$ with support at the origin. We find explicit equations in projective, respectively affine, embeddings for these spaces. In particular, we answer a question of Bernd Sturmfels who asked for a description of the latter space that is amenable to further computations. While the explicit equations we find are controlled in a precise way by the representation theory of SL₃, our arguments also rely on computer algebra.

1 Introduction

The Hilbert scheme $\text {Hilb}^{2} \mathbb {P}^{2}$ of two points on the projective plane has a projective embedding

$$ \text{Hilb}^{2} \mathbb{P}^{2} \hookrightarrow \mathbb{P}^{14} $$

(1)

defined as a composition of a natural map into Gr(2,6) followed by the Plücker embedding (for details, see Section 2.1). On the other hand, the Hilbert scheme $\text {Hilb}^{4}(\mathbb {A}^{3})$ of 4 points on affine 3-space has a distinguished affine open subset $\text {Hilb}^{4}(\mathbb {A}^{3})_{\text {np}}$ of non-planar clusters, which has an affine embedding

$$ \text{Hilb}^{4}(\mathbb{A}^{3})_{\text{np}} \hookrightarrow \mathbb{A}^{15},$$

whose image is the cone over the same Grassmannian (see Section 3.1). This latter Hilbert scheme has a natural closed subset, the space $\text {Hilb}^{4}_{0}(\mathbb {A}^{3})_{\text {np}}$ of non-planar, punctual clusters, where punctual means that the scheme-theoretic support of the subscheme parametrized is at a single point (chosen to be the origin; see Section 3.2 for details). We thus obtain an affine embedding

$$ \text{Hilb}^{4}_{0}(\mathbb{A}^{3})_{\text{np}} \hookrightarrow \mathbb{A}^{15}. $$

(2)

The connection between these Hilbert schemes goes back to Tikhomirov’s [18, Theorem 3]: the affine cone over $\text {Hilb}^{2} \mathbb {P}^{2}\subset \mathbb {P}^{14}$ is the singular locus of $\text {Hilb}^{4}_{0}(\mathbb {A}^{3})_{\text {np}}$.

Our aim in this paper is twofold: we give explicit equations to describe the images of the embeddings (1) and (2), and we recover this relationship between the spaces. We begin in Section 2.2 by identifying various spaces of interest as PGL₃-orbit closures in Gr(2,6). Their defining polynomials arise from the representation theory of SL₃ (see Section 2.3). We use computer algebra to derive specific polynomials, as well as to check various properties of the resulting systems of equations, in particular that they define reduced ideals. We also provide a more synthetic way to derive the equations in Section 2.4.

Our main result is Theorem 6 in Section 3.2. Section 3.3 presents an explicit calculation to illuminate, and yet again reprove, some of the main results. The explicit polynomials defining the embeddings (1) and (2) are listed in the Appendix.

2 The Hilbert Scheme of Two Points on the Projective Plane

2.1 Basics on the Hilbert Scheme of Two Points on the Projective Plane

Let U be a 3-dimensional vector space. In this section, we recall some well-known facts about the Hilbert scheme $\text {Hilb}^{2} \mathbb {P}(U)$ of two points on the projective plane $\mathbb {P}(U)$. We refer to [1, Sections 2 and 3] for more details and further references.

Let $h\colon \text {Hilb}^{2} \mathbb {P}(U) \to \text {Sym}^{2} \mathbb {P}(U)$ be the Hilbert–Chow morphism, $H=h^{\ast }(\mathcal {O}(1))$ the pullback of the natural generator of $\text {Pic}(\text {Sym}^{2} \mathbb {P}(U))$. Let ${{\varDelta }}\subset \text {Sym}^{2} \mathbb {P}(U)$ be the diagonal and B = h^− 1(Δ) the exceptional locus of h, a threefold ruled over ${{\varDelta }}\cong \mathbb {P}(U)$. It is known that

$$ \text{Pic}(\text{Hilb}^{2} \mathbb{P}(U))\cong \mathbb{Z}[H]\oplus \mathbb{Z}\left[\frac{B}{2}\right]. $$

Let ${\mathscr{L}}$ denote the line bundle $\mathcal {O}(2)$ on $\mathbb {P}(U)$. There is an associated rank two bundle ${\mathscr{L}}^{[2]}$ on $\text {Hilb}^{2} \mathbb {P}(U)$ whose fiber over a length 2 subscheme $\zeta \subset \mathbb {P}(U)$ is $H^{0}(\zeta ,{\mathscr{L}})$. Its space of sections is

$$ H^{0}\left( \text{Hilb}^{2} \mathbb{P}(U),\mathcal{L}^{[2]}\right)\cong H^{0}(\mathbb{P}(U),\mathcal{L})=S^{2}U^{\ast} $$

and it is globally generated. Its determinant

$$ D=\det(\mathcal{L}^{[2]}) = 2[H]-\left[\frac{B}{2}\right]\in\text{Pic}(\text{Hilb}^{2} \mathbb{P}(U)) $$

is very ample and yields a PGL(U)-equivariant chain of embeddings

$$ \varphi_{D}\colon~\text{Hilb}^{2} \mathbb{P}(U) \hookrightarrow \text{Gr}(4,S^{2}U^{\ast}) = \text{Gr}(2,S^{2}U) \hookrightarrow \mathbb{P}(\wedge^{2}S^{2} U)\cong\mathbb{P}^{14}. $$

The image of φ_D is of degree

$$ D^{4}= 16 H^{4} - 16 H^{3}B + 6 H^{2}B^{2} - H B^{3} + \frac{B^{4}}{16}=21 $$

which arises from the known intersection numbers [15] on the Hilbert scheme, namely

$$ (H^{4},H^{3}B,H^{2}B^{2},HB^{3},B^{4}) = (3,0,-8,-24,-48). $$

The embedding φ_D restricts to a PGL(U)-equivariant chain of embeddings

$$ \varphi_{D}|_{B}\colon~B\hookrightarrow \text{Gr}(2,S^{2} U)\hookrightarrow \mathbb{P}(\wedge^{2}S^{2} U) \cong \mathbb{P}^{14} $$

of degree

$$ (D|_{B})^{3}= D^{3}B = 8 H^{3} B - 6 H^{2}B^{2} + \frac{3 HB^{3}}{2}- \frac{B^{4}}{8}=18. $$

2.2 Pencils of Conics and Orbit Structure

Consider the space Gr(2, S²U) parametrizing pencils of conics on U^∗. The decomposition of this space into PGL(U)-orbits is classical, see e.g. [14]. There are eight types with normal forms as in the following table, yielding a decomposition of the Grassmannian Gr(2, S²U) into PGL(U)-orbits $\mathcal {O}_{i}$ (and sometimes $\mathcal {O}^{\prime }_{i}$) of dimension i.

$$\begin{array}{ll} \mathcal{O}_{3} \quad\qquad & \langle x^{2},xy\rangle \\ \mathcal{O}_{4} & \langle x^{2},y^{2}\rangle \\ \mathcal{O}^{\prime}_{4} & \langle xy,xz\rangle \\ \mathcal{O}_{5} & \langle x^{2}, y^{2}+xz\rangle \\ \mathcal{O}_{6} & \langle x^{2}, yz\rangle \\ \mathcal{O}^{\prime}_{6} & \langle x^{2}+yz, xz\rangle \\ \mathcal{O}_{7} & \langle x^{2}+y^{2}, xz\rangle \\ \mathcal{O}_{8} & \langle x^{2}+y^{2}, x^{2}+z^{2}\rangle \end{array} $$

Let Y_i, $Y_{i}^{\prime }$ denote the closure of $\mathcal {O}_{i}$, $\mathcal {O}_{i}^{\prime }$ in Gr(2, S²U). Clearly, Y₈ = Gr(2, S²U). The Hasse diagram, where $\mathcal {O}_{i}$ is connected to $\mathcal {O}_{j}$ if the latter is open in $Y_{i}\setminus \mathcal {O}_{i}$, is the following:

Proposition 1

• 1. Y₆ ⊂Gr(2, S²U) is birational to a $\mathbb {P}^{4}$-bundle over $\mathbb {P}^{2}$.

• 2. Y₅ ⊂Gr(2, S²U) is birational to a quadric bundle over $\mathbb {P}^{2}$ and is a non-normal projective variety with singular locus equal to Y₄.

• 3. $Y^{\prime }_{4}\simeq \mathbb {P}(U)\times \mathbb {P}(U^{\ast })$ which is embedded in $\mathbb {P}^{14}$ via the linear series $\mathcal {O}(2,1)$.

• 4. The chain of embeddings

  $$ Y_{3}\subset Y_{4} \subset \text{Gr}(2,S^{2} U)\subset \mathbb{P}(\wedge^{2}S^{2} U) \cong \mathbb{P}^{14} $$

  is isomorphic to the chain

  $$ B\subset \text{Hilb}^{2} \mathbb{P}(U) \subset \text{Gr}(2,S^{2} U)\subset \mathbb{P}(\wedge^{2}S^{2} U) \cong \mathbb{P}^{14} $$

  discussed in the previous section.

Proof

First, note that the variety Y₆ parametrizes pencils of conics containing a double line. We thus have a birational model $\widetilde {Y}_{6}\to Y_{6}$ parametrizing pairs (P, ℓ) of a pencil P of conics on U^∗ and a line $\ell \subset \mathbb {P}(U^{\ast })$ such that ℓ² belongs to P. For the bundle $E=S^{2}U/\mathcal {O}(-2)$ over $\mathbb {P}(U)$, we see that $\widetilde {Y}_{6}=\mathbb {P}(E)$, a $\mathbb {P}^{4}$-bundle over $\mathbb {P}^{2}$.

Next, Y₅ parametrizes pencils generated by a double line and a conic tangent to this line. Consider the birational model $\widetilde {Y}_{5}\to Y_{5}$ defined as the preimage of Y₅ in $\widetilde {Y}_{6}$. Let $Q=U/\mathcal {O}(-1)$ which is a bundle over $\mathbb {P}(U)$. Note that there is a natural morphism $E\rightarrow S^{2}Q$ and hence a natural rank three quadratic form $S^{2}E\rightarrow S^{2}(S^{2}Q)\rightarrow \det (Q)^{2}$. Then, $\widetilde {Y}_{5}\subset \widetilde {Y}_{6}=\mathbb {P}(E)$ is the corresponding quadric bundle over $\mathbb {P}(U)$. The morphism $\tilde {Y}_{5}\to Y_{5}$ is birational and is a double cover over Y₄. Thus, Y₅ is not normal along Y₄ and, since the latter is the complement of the open orbit in Y₅, it has to be the singular locus.

The variety $Y^{\prime }_{4}$ parametrizes pencils of reducible conics containing a fixed line ℓ. Such a pencil is of the form ℓL where L is a hyperplane of U. Hence, we have an isomorphism $Y^{\prime }_{4}\cong \mathbb {P}(U)\times \mathbb {P}(U^{\ast })$. The restriction of the tautological bundle of Gr(2, S²U) is the product of the rank one tautological bundle on $\mathbb {P}(U)$ with the rank two tautological bundle on the dual $\mathbb {P}(U^{\ast })$. Its determinant is $\mathcal {O}(2,1)$ showing that the degree of $Y^{\prime }_{4}$ is $(2h+h^{\prime })^{4}=24$.

In this description, we see that $Y_{3}\subset Y^{\prime }_{4}=\mathbb {P}(U)\times \mathbb {P}(U^{\ast })$ is the incidence quadric, the full flag variety of U. Its degree is $(h+h^{\prime })(2h+h^{\prime })^{3}=18$.

To conclude, consider the variety Y₄ parametrizing pencils generated by two double lines and their degeneracies. Its dense open subset $\mathcal {O}_{4}$ is isomorphic to $\text {Sym}^{2}\mathbb {P}(U)-{{\varDelta }}$ where Δ denotes the diagonal. For dimension reasons, its closure Y₄ must be $\text {Hilb}^{2} \mathbb {P}(U)\subset \text {Gr}(2,S^{2} U)$. Finally, note that the diagonal in $\text {Hilb}^{2} \mathbb {P}(U)=Y_{4}$ is $\mathbb {P} (T_{\mathbb {P}(U)})\cong F\cong Y_{3}$ as claimed which completes the proof. □

2.3 Explicit Equations

Consider the group $\text {SL}(U)\cong \text {SL}_{3}(\mathbb {C})$. Denote by S_{a, b} the representation of SL(U) with highest weight (a − b)ω₁ + bω₂, where ω₁ and ω₂ are the fundamental weights and a ≥ b ≥ 0. Clearly, $S_{0,0}= \mathbb {C}$ is the trivial module, S_1,0 = U is the three-dimensional “vector” representation, and S_1,1 = U^∗ is its dual. Consider the SL(U)-module W = ∧²S²U^∗ = S_{3, 2} of dimension 15. Its second symmetric square splits into irreducibles as

$$ S^{2} W \cong S_{6,4}\oplus S_{4,3}\oplus S_{4,0}\oplus S_{3,1}\oplus S_{2, 2} $$

(3)

of dimension 120 = 60 + 24 + 15 + 15 + 6. This decomposition can be easily checked computationally, e.g., via the SchurRings [16] package in Macaulay2 [9].

Consider the 14-dimensional projective space $\mathbb {P} W^{\ast } = \mathbb {P} (\wedge ^{2} S^{2} U)\cong \mathbb {P}^{14}$ with coordinate ring S^∗(W). Regard elements of S²W as quadrics on $\mathbb {P} W^{\ast }$. Using the decomposition (3) above, define ideals of the ring S^∗(W) generated by spaces of quadrics as follows:

$$ \begin{array}{ll} I_{8} & = \langle S_{3,1}\rangle \lhd S^{\ast}(W),\\ I_{5} & = \langle S_{3,1}, S_{2, 2}\rangle \lhd S^{\ast}(W),\\ I_{4} & = \langle S_{3,1}, S_{2, 2}, S_{4,3}\rangle \lhd S^{\ast}(W),\\ I_{3} & = \langle S_{3,1}, S_{2, 2}, S_{4,3}, S_{4,0}\rangle \lhd S^{\ast}(W). \end{array} $$

(4)

These ideals are generated by 15, 21,45 and 60 quadrics, respectively. The first ideal I₈ consists of quadrics parametrized by S_3,1 = ∧²S²U itself and it is well known that its vanishing locus is the Grassmannian:

$$ \mathbb{V}(I_{8})=\text{Gr}(2,S^{2} U)\subset \mathbb{P}(\wedge^{2}S^{2} U). $$

Proposition 2

For i = 3, 4, 5 we have, using the notations of the previous section,

$$ \mathbb{V}(I_{i})\cong Y_{i}\subset\text{Gr}(2,S^{2} U)\subset \mathbb{P}(\wedge^{2}S^{2} U) \cong \mathbb{P}^{14}. $$

In particular, the ideals of the orbit closures Y₃, Y₄, Y₅ ⊂Gr(2, S²U) discussed before are generated by quadrics.

Proof

Our proof uses computational methods, with some details omitted.

Fix a basis {v_i} of U^∗. The key to finding explicit equations is to derive an explicit form of the decomposition (3) yielding basis elements for each of the modules on the right hand side in terms of the obvious basis of the left hand side $S^{2} W \cong S^{2}\left (\wedge ^{2} S^{2} U^{\ast }\right )$ consisting of symmetric pairs of elements of the form [(v_i ⊗ v_j) ∧ (v_k ⊗ v_l)]. This can be done using SLA [6] in GAP [17]. We obtain explicit generators of these four ideals, which are listed in the Appendix. As expected, the 15-dimensional space S_3,1 has a basis whose elements can readily be identified with the quadratic Plücker relations defining $\mathbb {V}(I_{8})=\text {Gr}(2,6)\subset \mathbb {P}^{14}$.

Given the explicit polynomial generators, the dimension and degree of the remaining ideals can be computed using Macaulay2 [9], namely:

We next claim that each of these ideals defines an irreducible and reduced subscheme in $\mathbb {P}^{14}$.

We first verify that each top-dimensional component in each scheme is irreducible of multiplicity 1. To that end, certifiable witness point sets can be computed in numerical algebraic geometry using Bertini [2] and alphaCertified [12] via well-constrained subsystems [8]. Each witness point set consists of degree-many points along a complimentary dimensional linear space, which are nonsingular with respect to the resulting system. Certifiable monodromy loops following [10] yield that all of the witness points for each system are smoothly connected, showing that each of the top-dimensional components is irreducible of multiplicity 1 with respect to the corresponding ideal.

In each case, showing that the scheme is irreducible and reduced is now equivalent to showing that the top-dimensional irreducible component has the same Hilbert series as the entire scheme; this precludes the existence of embedded components. This was verified using [5] with certified numerical Hilbert function computations [11]. In fact, this also showed that each scheme is arithmetically Cohen–Macaulay via [5].

The ideals I₅ ⊂ I₄ ⊂ I₃ thus define SL(3)-invariant irreducible reduced subschemes of $\mathbb {V}(I_{8})=\text {Gr}(2,6)\subset \mathbb {P}^{14}$ of dimensions 5,4,3 respectively. As there is only one five-dimensional orbit closure in Gr(2,6) by our results in the previous section, we must have $\mathbb {V}(I_{5})=Y_{5}\subset \mathbb {P}^{14}$, and then necessarily $\mathbb {V}(I_{4})=Y_{4}\supset \mathbb {V}(I_{3})=Y_{3}$ as claimed. □

Remark 1

The 45 quadrics defining $Y_{4}\cong \text {Hilb}^{2} \mathbb {P}^{2} \subset \text {Gr}(2,6)$ were also determined, using a different method, in [3, Sect. 2]. That paper also studied the corresponding tropicalization.

Remark 2

In the language of the previous section, the quadrics in S_3,1 and S_4,0 define a four-dimensional variety of degree 24 inside of Gr(2, S²U) which is $Y_{4}^{\prime }\cong \mathbb {P}^{2}\times \mathbb {P}^{2}$ embedded by $\mathcal {O}(2,1)$ as above.

2.4 A Different Way to Derive Equations for Orbit Closures

We explain here an alternative, synthetic way to re-derive the quadratic equations of the orbit closures obtained above by a computer-based calculation. In this section, we will use the language of GL(U)-modules to respect the full symmetry of the problem. By a slight abuse of notation, we will use ψ(−) as shorthand for the quadratic form associated to a symmetric bilinear form ψ(−,−).

The Equations of Y ₅

The equations of $Y_{5}\subset \mathbb {P}^{14}$ are the Plücker equations together with another irreducible module of quadratic equations that is GL(U)-isomorphic to $S^{2} U^{\ast }\otimes \det (U^{\ast })^{2}$. Those equations have a simple description in terms of the discriminant of ternary quadratic forms (or symmetric tensors), which is a GL(U)-equivariant map

$$ \delta:\quad S^{3}(S^{2} U)\rightarrow \det (U)^{2}. $$

Polarizing yields a morphism

$$ {{\varDelta}} :~ S^{2}(S^{2} U)\rightarrow S^{2}U^{\ast}\otimes\det (U)^{2}. $$

Using the same trick for U^∗ and twisting by $\det (U)$ appropriately yields another GL(U)-equivariant map

$$ {{\varDelta}}^{\ast} : S^{2}(S^{2} U^{\ast}\otimes\det (U)^{2})\rightarrow S^{2}U\otimes\det (U)^{2}. $$

In order to describe the equations of Y₅ ⊂Gr(2, S²U), since it is the unique irreducible GL(U)-component of S²(∧²S²U^∗) isomorphic to $S^{2} U^{\ast }\otimes \det (U^{\ast })^{2}$, it suffices to exhibit a nonzero morphism

$$ {{\varPsi}}\colon S^{2}(\wedge^{2} S^{2} U)\to S^{2} U\otimes \det (U)^{2} $$

from the dual of the former to the dual of the latter. For q₁, q₂, q₃, q₄ belonging to S²U, we claim that the following formula defines such a morphism:

$$ {{\varPsi}}(q_{1}\wedge q_{2}, q_{3}\wedge q_{4}) = {{\varDelta}}^{\ast}({{\varDelta}}(q_{1},q_{3}), {{\varDelta}}(q_{2},q_{4})) - {{\varDelta}}^{\ast}({{\varDelta}}(q_{1},q_{4}), {{\varDelta}}(q_{2},q_{3})). $$

Since this is nonzero (see below) and has all the required properties, it expresses in a compact form all the equations of Y₅ apart from the Plücker relations. In the following statement, we use the same notations Ψ and Δ for the quadratic forms associated to the symmetric bilinear forms defined by these symbols.

Proposition 3

The subvariety Y₅ ⊂Gr(2, S²U) of the Grassmannian Gr(2, S²U) is defined by

$$ {{\varPsi}}(q_{1}\wedge q_{2}) = {{\varDelta}}^{\ast}({{\varDelta}}(q_{1}), {{\varDelta}}(q_{2}))-{{\varDelta}}^{\ast}({{\varDelta}}(q_{1},q_{2}))=0 $$

for q₁, q₂ ∈ S²U.

As a sanity check, let us evaluate Ψ on the representatives of the PGL₃-orbits of G(2, S²U). For this we normalize Δ by letting Δ(u², v²) = (u ∧ v)².

• For $\mathcal {O}_{5}$, we let q₁ = x² and q₂ = y² + xz. Hence, Δ(q₁) = 0 and Δ(q₁, q₂) = (x ∧ y)² has rank one, so that ${{\varPsi }}(q_{1}\wedge q_{2}) =-{{\varDelta }}^{\ast }({{\varDelta }}(q_{1},q_{2}))=0$.

• For $\mathcal {O}_{6}$, we let q₁ = x² and q₂ = yz. Hence, Δ(q₁) = 0 and writing 4q₂ = (y + z)² − (y − z)², we get Δ(q₁, q₂) = (x ∧ y)(x ∧ z). Since this has rank two, ${{\varPsi }}(q_{1}\wedge q_{2}) =-{{\varDelta }}^{\ast }({{\varDelta }}(q_{1},q_{2}))\ne 0$.

• For $\mathcal {O}^{\prime }_{4}$, we let q₁ = xy, q₂ = xz, u = x ∧ y, and v = x ∧ z. Then, 2Δ(q₁) = −u², 2Δ(q₂) = −v², and 2Δ(q₁, q₂) = −uv. Therefore, ${{\varDelta }}^{\ast }({{\varDelta }}(q_{1}), {{\varDelta }}(q_{2}))={{\varDelta }}^{\ast }(u^{2}, v^{2})/4=(u\wedge v)^{2}/4$ together with ${{\varDelta }}^{\ast }({{\varDelta }}(q_{1},q_{2}))={{\varDelta }}^{\ast }(uv)/4=-(u\wedge v)^{2}/8$ yields Ψ(q₁ ∧ q₂) = 3(v ∧ w)²/8 ≠ 0.

The Other Quadratic Equations

Consider the remaining modules of equations defining $Y_{4}, Y_{3}\subset \mathbb {P}^{14}$. These irreducible modules are generated by highest weight vectors. We can write down explicit quadratic polynomials on the Grassmannian in terms of those highest weight vectors as follows.

The 15-dimensional module is the irreducible GL(U)-module S⁴(∧²U^∗), whose highest weights vectors are the tensors of the form (e ∧ f)⁴ for e, f ∈ U^∗. We want to associate to such a vector a quadratic polynomial P on the cone of tensors of the form q₁ ∧ q₂ for q₁, q₂ ∈ S²U. In other words, we need to find a polynomial in e, f, q₁, q₂, which has degree four in e and f, degree two in q₁ and q₂, invariant under the action of GL(U), and compatible with the skew-symmetry conditions. By the Fundamental Theorems of Invariant Theory, such an invariant polynomial has to be expressed in terms of contractions of q₁ and q₂ by e and f. A straightforward computation shows that up to scalar, there is only one possibility, namely

$$ \begin{array}{@{}rcl@{}} P(q_{1}\wedge q_{2}) &=& \big(q_{1}(e)q_{2}(f)-q_{2}(e)q_{1}(f)\big)^{2} \\ &&+4\big(q_{1}(e,f)q_{2}(f)-q_{2}(e,f)q_{1}(f)\big) \big(q_{1}(e,f)q_{2}(e)-q_{2}(e,f)q_{1}(e)\big). \end{array} $$

One can easily test this polynomial on representatives of the orbits in Gr(2, S²U) and check that it vanishes identically only on the orbit $\mathcal {O}_{3}=Y_{3}$.

The second, 24-dimensional module is the GL(U)-submodule of S³U^∗⊗∧²U^∗⊗∧³U^∗ whose highest weights vectors are the tensors of the form e³(e ∧ f)(e ∧ f ∧ g) for e, f, g ∈ U^∗. Again, we need to associate to such a vector a quadratic polynomial Q on the cone of tensors of the form q₁ ∧ q₂ for q₁, q₂ ∈ S²U. Thus, we need to find a polynomial in e, f, g, q₁, and q₂ which has degree five in e, degree two in f, degree one in g, degree two in q₁ and q₂, invariant under the action of GL(U), and compatible with the skew-symmetry conditions. Another straightforward computation shows that up to scalar, there is only one possibility, namely

$$ \begin{array}{@{}rcl@{}} Q(q_{1}\wedge q_{2}) &=& \Big(q_{1}(e,g)q_{2}(e)-q_{2}(e,g)q_{1}(e)\Big)\Big(q_{1}(e)q_{2}(f)-q_{2}(e)q_{1}(f)\Big)\\ &&+ \Big(q_{1}(e,f)q_{2}(e)-q_{2}(e,f)q_{1}(e)\Big)\\ &&\times\Big(q_{1}(f,g)q_{2}(e)-q_{2}(f,g)q_{1}(e)+q_{1}(e,g)q_{2}(e,f)-q_{2}(e,g)q_{1}(e,f)\Big). \end{array} $$

Again, one can easily test this polynomial on representatives of the orbits in Gr(2, S²U) and check that it vanishes identically only on $\mathcal {O}_{3}$ and $\mathcal {O}_{4}$. We deduce

Proposition 4

The subvarieties Y₃ ⊂ Y₄ ⊂Gr(2, S²U) of the Grassmannian Gr(2, S²U) are defined by the sets of equations

$$ {{\varPsi}}(q_{1}\wedge q_{2}) = Q(q_{1}\wedge q_{2})=0 $$

and

$$ {{\varPsi}}(q_{1}\wedge q_{2}) = P(q_{1}\wedge q_{2})=Q(q_{1}\wedge q_{2})=0 $$

for q₁, q₂ ∈ S²U, respectively.

3 The Punctual Hilbert Scheme of Four Points on $\mathbb {A}^{3}$

3.1 Non-planar Clusters of Length Four on Affine Three-space

Recall the fixed three-dimensional vector space U from the previous section. In this section, we will think of its dual T = U^∗ as a copy of affine 3-space $\mathbb {A}^{3}$ with ring of functions $\mathbb {C}[T]=\text {Sym}^{\bullet } U$.

Let $\text {Hilb}^{m}(T)\cong \text {Hilb}^{m}(\mathbb {A}^{3})$ denote the Hilbert scheme of m points on affine three-space. The additive structure on T yields a center-of-mass morphism c: Hilb^m(T) → T. As is true in all dimensions, Hilb^m(T) is a nonsingular variety for m ≤ 3. It is also known [13] that Hilb⁴(T) is an irreducible and reduced variety of dimension 12, singular along the locus of length-four subschemes of T given by the squares $[\mathbf {m}_{p}^{2}]\in \text {Hilb}^{4}(T)$ of maximal ideals of points p ∈ T. It has a dense affine open subset Hilb⁴(T)_np containing all its singularities defined by the condition that the clusters parametrized by its points are non-planar, i.e., not scheme-theoretically contained in a plane.

Although the following was already proved in [7], we present a variant of the proof suited to the present narrative.

Theorem 5

There is an SL(U)-equivariant isomorphism

$$ \text{Hilb}^{4}(T)_{\text{np}} \cong T\times \mathcal{C}\text{Gr}(2,S^{2}U), $$

where $\mathcal {C}\text {Gr}(2,S^{2} U)\subset \wedge ^{2}S^{2} U$ is the affine cone over the Grassmannian $\text {Gr}(2,S^{2}U)\subset \mathbb {P}(\wedge ^{2}S^{2} U) \cong \mathbb {P}^{14}$.

Proof

If $I_{\xi }\lhd \mathbb {C}[T]=\text {Spec} \text {Sym}^{\bullet } U$ is the ideal corresponding to a point ξ ∈Hilb⁴(T)_np, then the map $\mathbb {C}\oplus U\rightarrow \mathbb {C}[T]/I_{\xi }$ is an isomorphism of vector spaces. The resulting algebra structure on the vector space $\mathbb {C}\oplus U$ is encoded by two symmetric bilinear maps:

$$ a :~ U\otimes U\rightarrow \mathbb{C} \qquad \text{and} \qquad m :~ U\otimes U\rightarrow U. $$

Requiring the product on $\mathbb {C}\oplus U$ to be associative leads to the following equations for any x, y, z ∈ U:

$$ \begin{array}{@{}rcl@{}} a(x,m(y,z)) &=& a(y, m(x,z)), \end{array} $$

(5)

$$ \begin{array}{@{}rcl@{}} m(x,m(y,z)) +a(y,z)x&=& m(y, m(x,z))+a(x,z)y. \end{array} $$

(6)

Using the highest weight notation for SL₃-modules introduced in the previous section, first observe that m is a tensor in S²U^∗⊗ U≅U^∗⊕ S_{3, 2} and the projection to T = U^∗ is the center of mass map c. Using a translation by the evident action of U^∗ on the whole setup, it suffices to restrict to the case where the center of mass of the ideal I_ξ is at the origin in T so that m ∈ S_{3, 2}.

Equation (6) shows that a(y, z)x − a(x, z)y is determined by m, x, y, and z which implies that a is uniquely determined by m and depends quadratically on it. As confirmed by [16], there exists a unique SL(U)-equivariant map up to scale, namely

$$ {{\varTheta}} :~ S^{2}(S_{3, 2}) \rightarrow S^{2} U^{\ast}. $$

Fixing the correct normalization, (6) implies that a = Θ(m). Now, for any x, y, z ∈ U, we can rewrite (5)–(6) in the following form:

$$ \begin{array}{@{}rcl@{}} {{\varTheta}}(m)(x,m(y,z)) &=& {{\varTheta}}(m)(y, m(x,z)), \end{array} $$

(7)

$$ \begin{array}{@{}rcl@{}} m(x,m(y,z)) +{{\varTheta}}(m)(y,z)x &=& m(y, m(x,z))+{{\varTheta}}(m)(x,z)y. \end{array} $$

(8)

These equations are families of cubic and quadratic equations on m ∈ S_{3, 2}. We claim that

• (a) the quadratic equation (8) on m are equivalent to the Plücker equations on ∧²S²U;

• (b) the cubic equation (7) are implied by the quadratic ones.

In order to prove (a), recall the SL₃-decomposition

$$ S^{2} S_{3, 2} \cong S_{6,4}\oplus S_{4,3}\oplus S_{4,0}\oplus S_{3,1}\oplus S_{2, 2} $$

(9)

already used above in (3). Equation (8) asks for the vanishing of a cubic tensor in x, y, and z, skew-symmetric in x and y, and takes values in U, i.e., an element of the SL₃-decomposition

$$ \text{Hom}(\wedge^{2} U\otimes U,U)\cong 2S_{1,1}\oplus S_{3,1}\oplus S_{2, 2}. $$

(10)

Comparing (9) with (10), the common terms are the last two irreducible components; they are the only ones imposing non-trivial conditions. The component S_{2, 2} has already been taken into account by letting a = Θ(m). The remaining conditions are the quadratic equations parametrized by S_3,1. This means that we get the quadratic equations spanning the ideal $I_{8} \lhd S^{\ast }(\wedge ^{2}S^{2}U^{\ast })$ from (4), parametrized by ∧²S²U, which are precisely the Plücker equations.

We conclude that Hilb⁴(T)_np is contained in $T\times \mathcal {C}\text {Gr}(2,S^{2}U)$ which implies, by a dimension count, that these two reduced schemes must be equal. Since the ideal of $\mathcal {C}\text {Gr}(2,S^{2}U)$ is radical, we also deduce claim (b): the cubic relations (7) do not impose any further conditions on m. □

3.2 The Space of Non-planar Punctual Clusters

Let $\text {Hilb}^{m}_{0}(T)\subset \text {Hilb}^{m}(T)$ denote the punctual Hilbert scheme of m points, the subscheme of the Hilbert scheme Hilb^m(T) given by the condition that the support of the zero-dimensional subscheme being parametrized is at the origin 0 ∈ T. It carries a natural projective scheme structure as the scheme-theoretic fiber of the Hilbert–Chow morphism Hilb^m(T) → S^m(T) over m ⋅ 0 ∈ S^m(T), but here we consider it in its reduced scheme structure.

Clearly, $\text {Hilb}^{2}_{0}(T)\cong \mathbb {P}(T)$, and $\text {Hilb}^{3}_{0}(T)$ and $\text {Hilb}^{4}_{0}(T)$ are known to be irreducible but singular projective varieties of dimensions 4 and 6 respectively. Descriptions of these spaces as well as natural desingularizations are given in [18] from a sheaf-theoretic perspective. We will describe the affine open set $\text {Hilb}^{m}_{0}(T)_{\text {np}}\subset \text {Hilb}^{4}_{0}(T)$ obtained by intersecting $\text {Hilb}^{4}_{0}(T)$ with the set of non-planar clusters. Note that $\text {Hilb}^{m}_{0}(\mathbb {A}^{3})_{\text {np}}$ is dense in $\text {Hilb}^{4}_{0}(\mathbb {A}^{3})$ and forms an affine neighbourhood of its most interesting point $[\mathbf {m}_{0}^{2}]\in \text {Hilb}^{4}_{0}(\mathbb {A}^{3})$, the length-four subscheme of $\mathbb {A}^{3}$ given by the square of the maximal ideal of the origin $0\in \mathbb {A}^{3}$. The following is our main result.

Theorem 6

The reduced space $\text {Hilb}^{4}_{0}(T)_{\text {np}}$ of non-planar, punctual clusters of length 4 on T is SL(U)-equivariantly isomorphic to the cone $\mathcal {C} Y_{5}\subset \wedge ^{2}S^{2} U\cong \mathbb {A}^{15}$ over the projective variety $Y_{5}\subset \mathbb {P}(\wedge ^{2}S^{2} U)$ described in Propositions 1–2. In particular, $\text {Hilb}^{4}_{0}(T)_{\text {np}}$ is a non-normal subvariety of $\mathbb {A}^{15}$ cut out by 21 explicitly computable quadrics and has a codimension one singular locus isomorphic to the affine cone $\mathcal {C} \text {Hilb}^{2}(\mathbb {P}^{2})\subset \wedge ^{2}S^{2} U$ with the apex of the cone corresponding to the distinguished ideal $[\mathbf {m}_{0}^{2}]\in \text {Hilb}^{4}_{0}(T)$.

Proof

The classification of PGL(U)-orbits in Gr(2, U) explained in Section 2.2 above shows that $\mathcal {C} Y_{5}$ is the only six-dimensional SL(U)-stable subvariety of $\mathcal {C}\text {Gr}(2,U)$. □

Remark 3

The description of the singular locus of (a neighbourhood of $[\mathbf {m}_{0}^{2}]$ in) Hilb⁴(T)₀ is not a new result as it was also obtained in [18] using sheaf-theoretic methods. The main advantage of our approach is that we can describe all these spaces by explicit affine equations which may be useful in applications [4].

Remark 4

It is also possible to prove this result using the language used in [7] followed by some computer calculations. In coordinates, we can represent a non-planar cluster of length four using three four-by-four matrices φ₁, φ₂, and φ₃ which describe the action of the coordinate functions of T on the 4-dimensional vector space $\mathbb {C}\oplus U$. The conditions for a triple (φ_i) to describe a cluster become explicit equations which reduce to the Plücker relations. To describe a punctual cluster based at 0, the matrices φ_i should additionally be nilpotent. It can be checked computationally that the reduced ideal of conditions arising from tr ∧^kφ_i = 0 for 1 ≤ k ≤ 4 are generated by the linear relations trφ_i = 0 and the quadratic conditions tr ∧²φ_i = tr ∧²(φ_i + φ_j) = 0, the latter being the 6 extra quadrics of I₅ defining Y₅ inside Gr(2, S²U) in Proposition 2 above.

Remark 5

The orbit decomposition of Gr(2, S²U) from Section 2.2 yields a decomposition of $\mathcal {C}\text {Gr}(2,S^{2}U)$. It is easy to check that the various orbits correspond to different length four subschemes in T as follows.

• 1. $\mathcal {C}O_{8}$ parametrizes four general points;

• 2. $\mathcal {C}O_{7}$ parametrizes two reduced general points p, q ∈ T and a degree two scheme at − (p + q)/2;

• 3. $\mathcal {C}O_{6}$ parametrizes two degree two schemes supported on opposite general points;

• 4. $\mathcal {C}O^{\prime }_{6}$ parametrizes one reduced point at p ∈ T and a degree three scheme supported on − p/3;

• 5. $\mathcal {C}O_{5}$ parametrizes an open subset of $\text {Hilb}^{4}_{0}(T)$;

• 6. $\mathcal {C}O_{4}$ parametrizes clusters with normal form xy = z, all other products being equal to zero;

• 7. $\mathcal {C}O^{\prime }_{4}$ parametrizes one reduced point p and a fat point in a plane at − p/3;

• 8. $\mathcal {C}O_{3}$ parametrizes clusters with normal form x² = z, all other products being equal to zero.

3.3 A Concrete Computation

Starting from a non-planar scheme with center of mass 0 corresponding to an ideal $I\lhd \mathbb {C}[T]$, recall that we get an algebra structure on the vector space $\mathbb {C}\oplus U$ which is partly encoded by a symmetric bilinear multiplication map

$$ m\in \text{Hom}(S^{2} U,U)\cong S^{2}U^{\ast}\otimes \wedge^{2}U^{\ast}\otimes \det(U). $$

Apply to m the morphism

$$ \begin{array}{@{}rcl@{}} {{\varGamma}} : ~S^{2}U^{\ast}\otimes \wedge^{2}U^{\ast}& \rightarrow & \wedge^{2}(S^{2}U^{\ast})\\ a^{2}\otimes b\wedge c & \mapsto & ab\wedge ac. \end{array} $$

Then, the discussion above shows that ${{\varGamma }} (m)\in \wedge ^{2}(S^{2}U^{\ast })\otimes \det (U)$ must be a decomposable tensor.

To see this in a concrete example, fix a basis {x, y, z} of U and consider the reduced scheme

$$ \zeta= \{(-1,0,0), (0,-1,0), (0,0,-1), (1,1,1)\}\subset T. $$

In the ring $\mathbb {C}[x,y,z]/I_{\zeta }$, we have

$$ x^{2} = \frac{1}{2}(1-x+y+z), \quad y^{2} = \frac{1}{2}(1+x-y+z), \quad z^{2} = \frac{1}{2}(1-x+y-z), $$

$$ xy = yz =zx = \frac{1}{4}(1+x+y+z). $$

The tensor m is obtained by keeping the degree one part of the right hand side in these equations. In terms of the dual basis {e, f, g} of U^∗, note that the basis of S²U^∗ dual to the basis {x², y², z², yz, xz, xy} of S²U is {e², f², g², 2fg, 2eg, 2ef}. So, as a tensor,

$$ \begin{array}{@{}rcl@{}} 4m & = & 2e^{2}\otimes (-x+y+z)+2f^{2}\otimes (x-y+z)+2g^{2}\otimes (x+y-z)\\ &&+2(fg+ge+ef)\otimes (x+y+z) \\ & = & (e^{2}+f^{2}+g^{2}+(e+f+g)^{2})\otimes (x+y+z)-4(e^{2}\otimes x+f^{2}\otimes y+g^{2}\otimes z). \end{array} $$

Note that up, to a common factor, x identifies with f ∧ g, y with g ∧ e and z with e ∧ f. Substituting these expressions and applying Γ, we get, after letting h = e + f + g,

$$ \begin{array}{@{}rcl@{}} 4{{\varGamma}} (m) &= & ef\wedge eg+eg\wedge e^{2}+e^{2}\wedge ef+ f^{2}\wedge fg+fg\wedge fe+fe\wedge f^{2}\\ &&+gf\wedge g^{2}+g^{2}\wedge ge+ge\wedge gf \\ &&+hf\wedge hg+hg\wedge he+he\wedge hf-4(ef\wedge eg+fg\wedge fe+ge\wedge gf) \\ & =& hf\wedge hg+hg\wedge he+he\wedge hf-(ef\wedge eg+fg\wedge fe+ge\wedge gf) \\ &&+he\wedge (ef-eg)+hf\wedge (fg-ef)+hg\wedge (eg-fg), \end{array} $$

and the final result of our computation is

$$ 4{{\varGamma}} (m)=(he-hf +eg-fg )\wedge (hf-hg+ef-eg). $$

This is, as expected, a decomposable tensor. Note that since the GL₃-orbit of our scheme is open in the subvariety of Hilb⁴(T) parametrizing schemes with center of mass at the origin, this yields another proof of Theorem 5.

Moreover, observe that we can rewrite he − hf + eg − fg = e² − f² + 2eg − 2fg = (e + g)² − (f + g)², and similarly hf − hg + ef − eg = (e + f)² − (f + g)², so that

$$ 4{{\varGamma}} (m)=(e+f)^{2}\wedge (f+g)^{2}+(f+g)^{2}\wedge (g+e)^{2}+(g+e)^{2}\wedge (e+f)^{2}. $$

This leads to a down-to-earth interpretation of the map

$$ \pi\colon \text{Hilb}^{4}(T)_{\text{np}} \mapsto \mathcal{C}\text{Gr}(2,S^{2}U) $$

from Theorem 5 on the open set of reduced subschemes.

Proposition 7

Consider a finite subscheme ζ ⊂ T consisting of four non-coplanar reduced points p_i ∈ T for $i=1,\dots ,4$ with center of mass p₀. The squares of the six vectors p_ij = p_i + p_j − 2p₀ ∈ T are three tensors $p_{12}^{2}=p_{34}^{2}$, $p_{13}^{2}=p_{24}^{2}$ and $p_{14}^{2}=p_{23}^{2}$ in S²T = S²U^∗ and

$$ \pi(\zeta)= \left( p_{12}^{2}\wedge p_{13}^{2}+p_{13}^{2}\wedge p_{14}^{2}+p_{14}^{2}\wedge p_{12}^{2}\right)\otimes \omega^{-1}, $$

where the twist $\omega \in \det (U^{\ast })$ is given by

$$ \omega = p_{1}\wedge p_{2}\wedge p_{3}- p_{2}\wedge p_{3}\wedge p_{4}+p_{3}\wedge p_{4}\wedge p_{1}-p_{4}\wedge p_{1}\wedge p_{2}. $$

Note that ω is invariant under a common translation of the four points and is non-zero exactly when ζ ⊂ T is non-planar. Moreover, permuting the four points multiplies ω by the sign of the permutation but π(ζ) itself remains invariant. Thus, it only depends on ζ and not on the order of the four points.

Proof of Proposition 7

The expression for π(ζ) depends equivariantly on ζ and yields the correct expression when

$$ \zeta = (-e,-f,-g,e+f+g). $$

Since the orbit of ζ in Hilb⁴(T)_np is dense, this expression must be correct everywhere. □

Appendix: Explicit Polynomials

The following set of Macaulay2 commands generates the ideals I₈, I₅, I₄ and I₃ in (4). As proved in the main body of the paper, the ideal I₄ defines the projective image of the embedding (1), whereas I₅ defines the affine image of the embedding (2).

S=QQ[a,b,c,d,e,f,g,h,i,j,k,l,m,n,o];

I8=ideal(a⋆j-b⋆g+c⋆f, a⋆k-b⋆h+d⋆f, a⋆l-b⋆i+e⋆f, a⋆m-c⋆h+d⋆g,a⋆n-c⋆i+e⋆g,

a⋆o-d⋆i+e⋆h, b⋆m-c⋆k+d⋆j, b⋆n-c⋆l+e⋆j, b⋆o-d⋆l+e⋆k, c⋆o-d⋆n +e⋆m,f⋆m-g⋆k+h⋆j,

f⋆n-g⋆l+i⋆j, f⋆o-h⋆l+i⋆k, g⋆o-h⋆n+i⋆m, j⋆o-k⋆n+l⋆m);

I5=I8+ideal(2⋆d⋆o-e⋆n-2⋆f⋆o-2⋆h⋆l+i⋆i-2⋆j⋆l + 3⋆k⋆k, 2⋆a⋆i-2⋆a⋆k-2⋆b⋆h + 2⋆b⋆j-c⋆e+d⋆d

+ 3⋆f⋆f, c⋆n-2⋆d⋆m-2⋆f⋆m-2⋆g⋆i-2⋆g⋆k + 3⋆h⋆h+j⋆j, a⋆n-2⋆b⋆m-c⋆k+d⋆h+d⋆j-e⋆g+f⋆h-f⋆j,

2⋆a⋆o-b⋆n-c⋆l+d⋆i+d⋆k-e⋆h+f⋆i-f⋆k, c⋆o-e⋆m-f⋆n-2⋆g⋆l+h⋆i+h⋆k-i⋆j+j⋆k);

I4=I5+ideal(a⋆d+a⋆f-b⋆c, a⋆e-b⋆d+b⋆f, g⋆n-h⋆m-j⋆m, c⋆m-g⋆h +g⋆j, e⋆o-i⋆l+k⋆l,

i⋆o+k⋆o-l⋆n, 3⋆a⋆i+a⋆k-b⋆h-3⋆b⋆j-2⋆d⋆f, 2⋆a⋆k+b⋆h-3⋆b⋆j-3⋆c⋆e + 3⋆d⋆d-d⋆f-6⋆f⋆f,

2⋆a⋆l+b⋆i-3⋆b⋆k-e⋆f, 2⋆a⋆m+c⋆h-d⋆g-3⋆f⋆g, a⋆n + 2⋆b⋆m-c⋆i+c⋆k + 3⋆d⋆h-d⋆j-2⋆e⋆g-6⋆f⋆h,

2⋆b⋆o+d⋆l-e⋆k-3⋆f⋆l, 2⋆a⋆o + 3⋆b⋆n+d⋆i + 3⋆d⋆k-e⋆h-3⋆e⋆j-6⋆f⋆i -6⋆f⋆k,

2⋆b⋆m-3⋆c⋆i+c⋆k + 6⋆d⋆h-d⋆j-3⋆e⋆g-6⋆f⋆h + 3⋆f⋆j, b⋆n-c⋆l + 3⋆d⋆k -2⋆e⋆j-3⋆f⋆i,

3⋆c⋆n-9⋆d⋆m + 5⋆f⋆m + 12⋆g⋆i-2⋆g⋆k-6⋆h⋆h-7⋆h⋆j + 3⋆j⋆j, c⋆o + 2⋆d⋆n+e⋆m-3⋆h⋆i + 3⋆j⋆k,

3⋆d⋆m-f⋆m-3⋆g⋆i+g⋆k + 2⋆h⋆j, 2⋆d⋆o+e⋆n-h⋆l-i⋆i-i⋆k+j⋆l + 2⋆k⋆k,

3⋆d⋆n + 3⋆e⋆m-f⋆n-2⋆g⋆l-6⋆h⋆i + 6⋆h⋆k-i⋆j + 3⋆j⋆k, 3⋆e⋆n-2⋆f⋆o-h⋆l-3⋆i⋆i+i⋆k-3⋆j⋆l + 6⋆k⋆k,

2⋆g⋆o+h⋆n-i⋆m-3⋆k⋆m, 3⋆h⋆o+j⋆o-k⋆n-2⋆l⋆m, 3⋆a⋆h-a⋆j-2⋆b⋆g-c⋆f);

I3=I4+ideal(4⋆a⋆g-c⋆c, 4⋆b⋆l-e⋆e, 4⋆m⋆o-n⋆n, 2⋆a⋆h+a⋆j+b⋆g-c⋆d, a⋆l+b⋆i + 2⋆b⋆k-d⋆e,

a⋆m+c⋆j-d⋆g + 2⋆f⋆g, b⋆o+d⋆l-e⋆i + 2⋆f⋆l, g⋆o-i⋆m-j⋆n + 2⋆k⋆m, 2⋆h⋆o-i⋆n-j⋆o+l⋆m,

2⋆a⋆i + 4⋆a⋆k + 4⋆b⋆h + 2⋆b⋆j-c⋆e-2⋆d⋆d, c⋆n-2⋆d⋆m + 4⋆f⋆m-2⋆g⋆i + 4⋆g ⋆k-2⋆j⋆j,

2⋆d⋆o-e⋆n + 4⋆f⋆o + 4⋆h⋆l-2⋆i⋆i-2⋆j⋆l, a⋆n+b⋆m + 2⋆c⋆k-2⋆d⋆h+d⋆j -e⋆g + 4⋆f⋆h + 2⋆f⋆j,

a⋆o+b⋆n+c⋆l-d⋆i + 2⋆d⋆k-2⋆e⋆h + 2⋆f⋆i + 4⋆f⋆k, c⋆o-e⋆m + 2⋆f⋆n+g⋆l-2⋆h⋆i + 4⋆h⋆k-i⋆j-2⋆j⋆k);