Journal of Lie Theory Volume 7 (1997) 165–169 ◯^{C}1997 Heldermann Verlag

Ternary Quartics and 3-dimensional Commutative Algebras

Pavel Katsylo and Dmitry Mikhailov^{∗}

Communicated by E. B. Vinberg

Abstract. We find a connection between 3-dimensional commutative algebras with trivial trace and plane quartics and their bitangents.

1. Introduction

In this paper a structure of a commutative algebra on C^{3} is called a 3-dimensional algebra. Let A be the set of 3-dimensional algebras. Consider A as a linear space. Let A_{0} ⊂ A be the linear subspace of algebras with trivial trace. By definition, η ∈ A_{0} if the contraction of the structure tensor of η is equal to zero.

By P V we denote the projectivization of a vector space V . For v ∈ V, v ≠ 0 we denote by \overline{v} the corresponding point of the projective space P V .

Let η ∈ A_{0} be an algebra with trivial trace. Recall that an element a ∈ C^{3} is called an idempotent if a ≠ 0, a^{2} = a. We say that an element \overline{a} ∈ P C^{3} = P^{2} is a generalized idempotent if a^{2} = λa, where λ ∈ C. Every idempotent defines a generalized idempotent. Every generalized idempotent \overline{a} ∈ P^{2} such that a^{2} ≠ 0 defines uniquely an idempotent a′ ∈ C^{3} such that \overline{a} = \overline{a′}. Define the subscheme X(η) ⊂ P^{2} of the generalized idempotents by the following equation:

a^{2} ∧ a = 0.  (0.1)

Consider the open SL_{3}-invariant subset

A′_{0} = {η ∈ A_{0} | dim X(η) = 0} ⊂ A_{0}.

Lemma 1.1. A′_{0} is nonempty

Consider η ∈ A_{0}. The algebra η defines the quadratic mapping

C^{3} ⟶ C^{3}, a ↦ a^{2}.

ISSN 0949–5932 / $2.50 ◯^{C} Heldermann Verlag This quadratic mapping defines the section η̃ of the vector bundle T_{P^{2}}(1) (see [3], Ch. 1). The scheme of zeros of the section η̃ is X(η). We have

∗Research supported by Max-Planck-Institute fur Mathematik and Grant NMQZ000 of the International Science Foundation.

deg X(η) = c_{2}(T_{P^{2}}(1)) = 7,

for η ∈ A_{0} (see [3], Ch. 1).

Consider the open SL_{3}-invariant subset

A′′_{0} = \{η ∈ A′_{0} | X(η) = \{\overline{a_{1}}, …, \overline{a_{7}}\}, a_{i}^{2} ≠ 0, 1 ≤ i ≤ 7,
every 3 points of X(η) do not lie on a line,
every 6 points of X(η) do not lie on a quadric} ⊂ A′_{0}.

Lemma 1.2. A′′_{0} is nonempty.

Consider the rational SL_{3}-morphism

φ : P A_{0} ⟶ (P^{2})^{(7)}, η ↦ X(η).

We use the standard notation (P^{2})^{(7)} for the 7-th symmetric degree of P^{2}.

Proposition 1.3. φ is a birational isomorphism.

In other words, a 3-dimensional algebra in general position with a trivial trace is uniquely (up to a scalar factor) defined by its generalized idempotents.

Fix η ∈ A′′_{0}. Let a_{1}, …, a_{7} be the idempotents of the algebra η. Let

π = π(η) : Z = Z(η) ⟶ P^{2}

be the blowing up of X(η) in P^{2}. It is well known that Z is a Del Pezzo surface of degree 2. We use some facts on the Del Pezzo surfaces, see [2], Ch. 5, section 4 for details. Let

β = β(η) : Z ⟶ P^{2∗} = P C^{3∗}

be the canonical double covering with a nonsingular quartic Y = Y (η) ⊂ P^{2∗} as the branch locus.

The SL_{3}-module S^{2}A_{0} contains with multiplicity one a submodule isomorphic to S^{4}C^{3}. Therefore, there exists a unique (up to a scalar factor) nontrivial quadratic SL_{3}-mapping

ε : A_{0} ⟶ S^{4}C^{3}.

Lemma 1.4. ε(η) ≠ 0. Consider the quaternary form ε(η) on the space C^{3∗}. This quaternary form defines a quartic Y′ = Y′(η) ⊂ P^{2∗}. Consider the 28 linear forms a_{1}, …, a_{7}, (a_{i}−a_{j})^{2}, 1 ≤ i < j ≤ 7 on the space C^{3∗}. These linear forms define 28 lines A_{1}, …, A_{7}, A_{ij} ∈ P^{2∗}.

Theorem 1.5. 1. Y is isomorphic to Y′.

2. The 28 bitangents to Y′ are A_{1}, …, A_{7}, A_{ij}, 1 ≤ i < j ≤ 7. The authors thank B.Broer for a useful discussion.

2. Proofs

Let e_{1}, e_{2}, e_{3} be the standard basis in C^{3}, and x_{1}, x_{2}, x_{3} the dual basis in C^{3∗}. The group SL_{3} acts canonically in the space S^{a}C^{3} ⊗ S^{b}C^{3∗}, a, b ≥ 0. For a, b ≥ 1 consider the linear SL_{3}-mapping

Δ = ∑ \frac{∂}{∂e_{i}} ⊗ \frac{∂}{∂x_{i}} : S^{a}C^{3} ⊗ S^{b}C^{3∗} ⟶ S^{a−1}C^{3} ⊗ S^{b−1}C^{3∗}.

It is well known that the representation of the group SL_{3} in the space V (a, b) = ker δ is irreducible (see [1], part III, section 13). Assume that V (a, 0) = S^{a}C^{3}, V (0, b) = S^{b}C^{3∗}.

A structure of a commutative algebra on C^{3} is a symmetric bilinear mapping

C^{3} × C^{3} ⟶ C^{3}.

The set of such symmetric bilinear mappings is C^{3} ⊗ S^{2}C^{3∗}. Therefore, the linear space A of 3-dimensional algebras is C^{3} ⊗ S^{2}C^{3∗}. The contraction of structure tensors of algebras is the mapping

Δ : C^{3} ⊗ S^{2}C^{3∗} ⟶ C^{3∗}.

Therefore, the linear space A_{0} of 3-dimensional algebras with trivial trace is V (1, 2).

¿From the Littelwood-Richardson rule we get the following SL_{3}-module decomposition

V (1, 2) ⊗ C^{3} ⊗ C^{3} ≃ C^{3} ⊕ 2V (0, 2) ⊕ 2V (2, 1) ⊕ 2V (1, 3) ⊕ V (3, 2)

(see [1], App. A). Thus there exists a unique (up to a scalar factor) nontrivial trilinear SL_{3}-mapping

μ : V (1, 2) × C^{3} × C^{3} ⟶ C^{3}.

Let us give the explicit form of μ:

μ(e_{i_{1}} ⊗ x_{j_{1}}x_{j_{2}}, e_{i_{2}}, e_{i_{3}}) = Δ^{2}(e_{i_{1}}e_{i_{2}}e_{i_{3}} ⊗ x_{j_{1}}x_{j_{2}}).

The algebraic structure corresponding to η ∈ V (1, 2) is the bilinear symmetric mapping

μ(η, ·, ·) : C^{3} × C^{3} ⟶ C^{3}.

Example 2.1. Consider the algebra

η_{0} = \frac{1}{4}(e_{1} ⊗ x_{3}^{2} + e_{2} ⊗ x_{1}^{2} + e_{3} ⊗ x_{2}^{2}).

The multiplication table of η_{0} is as following:

e_{1} ∗ e_{2} = e_{2} ∗ e_{3} = e_{3} ∗ e_{1} = 0, e_{1}^{2} = e_{2}, e_{2}^{2} = e_{3}, e_{3}^{2} = e_{1}.

It can be easily checked that the subscheme X(η_{0}) ⊂ P^{2} of the generalized idempotents of η_{0} is

X(η_{0}) = \{\overline{θe_{1} + θ^{2}e_{2} + θ^{4}e_{3}} | θ ∈ μ_{7}\},

where μ_{7} = {θ ∈ C | θ^{7} = 1}.

We have η_{0} ∈ A′_{0}. It follows that A′_{0} is nonempty. It can easily be checked that η_{0} ∈ A′′_{′}. Therefore, A′′_{0} is nonempty. This proves of Lemmas 1.1 and 1.2.

It can easily be checked that φ^{−1}(X(η_{0})) = \{\overline{η_{0}}\}. It follows from (0.1) that a fiber of φ of general position is a point in P V (1, 2). Hence φ is a birational isomorphism. This proves Proposition 1.3.

Fix η ∈ A′′_{0} and let a_{1}, …, a_{7} be the idempotents of η.

Consider the cubic mapping

ψ = ψ(η) : C^{3} ⟶ ∧^{2}C^{3} ≃ C^{3∗}, a ↦ a^{2} ∧ a.

Lemma 2.2. Consider a_{i} as a linear form on C^{3∗}. Let Q_{i} = Q_{i}(η) be the cubic corresponding to the cubic form ψ^{∗}(a_{i}). Then the cubic Q_{i} contains each of \overline{a_{1}}, …, \overline{a_{7}} with multiplicity ≥ 2.

Proof. It is obvious that Q_{i} ∋ \overline{a_{1}}, …, \overline{a_{7}}. Let us prove that Q_{i} contains \overline{a_{i}} with multiplicity ≥ 2. We have

ψ^{∗}(a_{i}) : a ↦ a^{2} ∧ a ∧ a_{i} ∈ ∧^{3}C^{3} ≃ C,
ψ^{∗}(a_{i})(a_{i} + tb) = (a_{i} + tb)^{2} ∧ (a_{i} + tb) ∧ a_{i} = a_{i}^{2} ∧ a_{i} ∧ a_{i}+
t((2a_{i} ∗ b) ∧ a_{i} ∧ a_{i} + a_{i}^{2} ∧ b ∧ a_{i}) + o(t) = 0 + t · 0 + o(t)

for any b ∈ C^{3}.

Lemma 2.3. Consider (a_{i} − a_{j})^{2}, i < j as linear forms on C^{3∗}. Let Q_{ij} = Q_{ij}(η) be the cubic corresponding to the cubic form ψ^{∗}((a_{i} − a_{j})^{2}). Then Q_{ij} is the union of the line ⟨\overline{a_{i}}, \overline{a_{j}}⟩ and the quadric containing the points \overline{a_{1}}, …, ˆ \overline{a_{i}}, …, ˆ \overline{a_{j}}, …, \overline{a_{7}}.

Proof. It is obvious that Q_{ij} ∋ \overline{a_{1}}, …, \overline{a_{7}}. We have to prove that Q_{ij} contains the line ⟨\overline{a_{i}}, \overline{a_{j}}⟩. We have

ψ^{∗}((a_{i} − a_{j})^{2}) : a ↦ a^{2} ∧ a ∧ (a_{i} − a_{j})^{2} ∈ ∧^{3}C^{3} ≃ C,

ψ^{∗}((a_{i} − a_{j})^{2})(t_{i}a_{i} + t_{j}a_{j}) = (t_{i}a_{i} + t_{j}a_{j})^{2} ∧ (t_{i}a_{i} + t_{j}a_{j}) ∧ (a_{i} − 2a_{i} ∗ a_{j} + a_{j}) =

t_{i}^{3}a_{i}^{2} ∧ a_{i} ∧ (a_{i} − 2a_{i} ∗ a_{j} + a_{j}) + t_{i}^{2}t_{j}(a_{i}^{2} ∧ a_{j} ∧ (a_{i} − 2a_{i} ∗ a_{j} + a_{j})+ (2a_{i} ∗ a_{j}) ∧ a_{i} ∧ (a_{i} − 2a_{i} ∗ a_{j} + a_{j})) + t_{i}t_{j}^{2}(a_{j}^{2} ∧ a_{i} ∧ (a_{i} − 2a_{i} ∗ a_{j} + a_{j})+ (2a_{j} ∗ a_{i}) ∧ a_{j} ∧ (a_{i} − 2a_{i} ∗ a_{j} + a_{j})) + t_{j}^{3}a_{j}^{2} ∧ a_{j} ∧ (a_{i} − 2a_{i} ∗ a_{j} + a_{j}) = 0.

Consider the rational morphism

Ψ = Ψ(η) : P^{2} ⟶ P^{2∗}, \overline{a} ↦ \overline{ψ(a)}.

It is not defined exactly on X(η). Let

P^{2} ⟵^{π} Z = Z(η) ⟶^{β} P^{2∗}

be the regularization of Ψ, Ψ = β ∘ π^{−1}. It is well known that Z is a Del Pezzo surface of degree 2, π is the blowing up of the seven points \overline{a_{1}}, …, \overline{a_{7}} in P^{2}, β :

Z ⟶ P^{2∗} is a double covering with the nonsingular quartic Y = Y (η) ⊂ P^{2∗} as the branch locus (see [2], Ch. 5).

Let us prove Lemma 1.4 and Theorem 1.5. Consider the nontrivial homogeneous SL_{3}-equivariant mapping of degree 6

\begin{matrix}γ_{i} : V (1, 2) ⟶ V (0, 12) = S^{12}C^{3∗}, \\ γ_{1} : η ↦ ψ^{∗}(ε(η)), γ_{2} : η ↦ (det(\frac{∂ψ^{∗}(e_{i})}{∂x_{j}}))^{2}. \end{matrix}

It can be checked that the SL_{3}-module S^{6}V (1, 2) contains V (0, 12) with multiplicity one. Thus γ_{1} = cγ_{2}, where c ≠ 0. This implies Lemma 1.4 and statement 1. of Theorem 1.5.

The second part of Theorem 1.5 is a corollary of Lemmas 2.2 and 2.3 (see [2], Ch. 5, Section 4)

References

[1]  Fulton W. and Harris J., “Representation Theory,” Springer-Verlag, 1991.
[2]  Hartshorne R., “Algebraic Geometry,” Springer-Verlag, 1977.

[3] Okonek Ch., M. Schneider, and Spindler H., Vector Bundles on Complex Projective Spaces, Birkhauser, 1980.

Independent University of Moscow, Bolshoi Vlasievkii 11, 121002, Moscow, Russia. katsylo@ium.ips.ras.ru

Received December 9, 1994 and in final form January 30, 1997