The dual of the invariant quintic

E. Freitag Mathematisches Institut Universität Heidelberg Im Neuenheimer Feld 288 69120 Heidelberg E-mail: freitag@mathi.uni-heidelberg.de home page: http://www.rzuser.uni-heidelberg.de/∼t91/

B. Hunt Max-Planck-Institut für Mathematik in den Naturwissenschaften 04103 Leipzig E-mail: Bruce.Hunt@mis.mpg.de

Februar, 1998

Abstract

A very remarkable object is the unique quintic (hypersurface of degree 5) I 5 ⊂ P^{5}ℂ, which is invariant under the natural action of the Weyl group W (E_{6}) of the lattice E_{6}. This quintic has been investigated in [H], chapter 6, by the second author and also a certain amount of information about the dual variety of I 5 has been obtained. An explicit determination of this dual and even of its degree seemed to be an unsolvable problem along these lines.

In the paper [FH] a certain hypersurface I 32 ⊂ P^{5}ℂ of degree 32, which is also invariant under W (E_{6}), has been discovered in connection with the investigation of certain modular varieties. Computer calculations by means of MACAULEY led the second author to the conjecture that I 32 is the dual of I 5. Using the MAPLE system the first author was able to verify this conjecture up to a linear transformation which intertwines the representation of W (E_{6}) with its contragredient.

Introduction: The dual variety

For a good comprehensive introduction into the concept of the dual variety we refer to [K].

Let V be a finite dimensional complex vector space and

X ⊂ P (V)

a closed subvariety (reduced and irreducible). The dual variety

X^{⊥} ⊂ P (V^{⊥}), V^{⊥} = Hom_{ℂ}(V, ℂ)

is the closure of the set of all hyperplanes of P (V) which are tangent to X at a smooth point. The biduality theorem states

(X^{⊥})^{⊥} = X.

In general the dimensions of X and X^{⊥} will be different. But if X ⊂ P (V) is a hypersurface, except for some exceptional cases (ruled hypersurfaces), the dual variety X^{⊥} ⊂ P (V^{⊥}) is also a hypersurface and hence of the same dimension as X. The biduality theorem then implies that the Gauss map

G : X − − > X^{⊥}

is birational. (The Gauss map attaches to every smooth point of X its tangent hyperplane.)

We assume now V = ℂ^{n} and identify V^{⊥} with ℂ^{n} in the ususal manner. The hypersurfaces X, X^{⊥} are given as zero sets of irreducible polynomials P, P^{⊥} ∈ ℂ[X_{0}, …, X_{n}] say of degree d and d^{⊥}. These polynomials of course are determined only up to constant factors. We call P and P^{⊥} a pair of dual polynomials. The duality relation is simply

P^{⊥}(Grad P) ≡ 0 mod P,

where

Grad P = (\frac{∂P}{∂X_{0}} , …, \frac{∂P}{∂X_{n}} )

is the gradient of P. It is quite obvious that it is difficult to determine P^{⊥} explicitly from this equation because this involves complicated elimination. Even by means of expert systems like MACAULEY one has a chance only in special cases of low dimension and degree. The chance is better of one knows P^{⊥} conjecturally from elsewhere. But even then it may be difficult to decide whether P^{⊥}(Grad P) is divisible by P.

Here is a case where we succeeded to find a rather big dual polynomial due to some lucky circumstances.

The invariant polynomials

Let 𝔱 = ℝ^{6} be a Cartan algebra of a compact form of the Liealgabra E_{6} over ℝ. We denote by Y_{1}, …, Y_{6} the standard basis of the dual space of 𝔱, i.e. Y_{i}(x_{1}, …, x_{6}) = x_{i}. All that we have to know about E_{6} is that the roots are

±(Y_{i} ± Y_{j}), 2 ≤ i < j ≤ 6,
±\frac{1}{2}(Y_{1} ± Y_{2} ± Y_{3} ± Y_{4} ± Y_{5} ± Y_{6}), even number of “−” signs inside the paranthesis

A system of simple roots is

\begin{matrix}W_{1} = & −\frac{1}{2}(−Y_{1} + Y_{2} + Y_{3} + Y_{4} + Y_{5} + Y_{6}) \\ W_{2} = & Y_{2} − Y_{3} \\ W_{3} = & −\frac{1}{2}(Y_{1} + Y_{2} + Y_{3} − Y_{4} − Y_{5} − Y_{6}) \\ W_{4} = & Y_{3} − Y_{4} \\ W_{5} = & Y_{4} − Y_{5} \\ W_{6} = & Y_{5} − Y_{6} \end{matrix}

The root lattice is the lattice Λ ⊂ 𝔱^{∗} generated by the roots. There exists a unique symmetric bilinear form such that all root vectors have norm 2. The Gram matrix of Λ with respect to the simple roots is

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

The Weyl group W (E_{6}) is a subgroup of index two in the full group of automorphisms of (Λ, ⟨ , ⟩). It is characterized by the fact that it does not contain a ↦ −a. The intersection of W (E_{6}) with the special linear group is a simple subgroup of index 2. Its order is 25, 920. The dual lattice Λ^{∗} of Λ is also considered a sublattice of 𝔱^{∗} using ⟨ , ⟩. This lattice has 27 pairs of minimal vectors ±a. Their norm is 4/3.

1 Proposition. The following 27 “fundamental weights” are permuted under the Weyl group of E_{6}. Dividing them by 6 one obtains a system of representatives of the pairs of minimal vectors of Λ^{∗}.

U_{0} = Y_{1} + 3 · Y_{2} + 3 · Y_{3} + 3 · Y_{4} + 3 · Y_{5} + 3 · Y_{6}
U_{1} = Y_{1} − 3 · Y_{2} − 3 · Y_{3} + 3 · Y_{4} + 3 · Y_{5} + 3 · Y_{6}
U_{2} = Y_{1} − 3 · Y_{2} + 3 · Y_{3} − 3 · Y_{4} + 3 · Y_{5} + 3 · Y_{6}
U_{3} = Y_{1} − 3 · Y_{2} + 3 · Y_{3} + 3 · Y_{4} − 3 · Y_{5} + 3 · Y_{6}
U_{4} = Y_{1} − 3 · Y_{2} + 3 · Y_{3} + 3 · Y_{4} + 3 · Y_{5} − 3 · Y_{6}
U_{5} = Y_{1} + 3 · Y_{2} − 3 · Y_{3} − 3 · Y_{4} + 3 · Y_{5} + 3 · Y_{6}

U_{6} = Y_{1} + 3 · Y_{2} − 3 · Y_{3} + 3 · Y_{4} − 3 · Y_{5} + 3 · Y_{6}
U_{7} = Y_{1} + 3 · Y_{2} − 3 · Y_{3} + 3 · Y_{4} + 3 · Y_{5} − 3 · Y_{6}
U_{8} = Y_{1} + 3 · Y_{2} + 3 · Y_{3} − 3 · Y_{4} − 3 · Y_{5} + 3 · Y_{6}
U_{9} = Y_{1} + 3 · Y_{2} + 3 · Y_{3} − 3 · Y_{4} + 3 · Y_{5} − 3 · Y_{6}
U_{10} = Y_{1} + 3 · Y_{2} + 3 · Y_{3} + 3 · Y_{4} − 3 · Y_{5} − 3 · Y_{6}
U_{11} = Y_{1} + 3 · Y_{2} − 3 · Y_{3} − 3 · Y_{4} − 3 · Y_{5} − 3 · Y_{6}
U_{12} = Y_{1} − 3 · Y_{2} + 3 · Y_{3} − 3 · Y_{4} − 3 · Y_{5} − 3 · Y_{6}
U_{13} = Y_{1} − 3 · Y_{2} − 3 · Y_{3} + 3 · Y_{4} − 3 · Y_{5} − 3 · Y_{6}
U_{14} = Y_{1} − 3 · Y_{2} − 3 · Y_{3} − 3 · Y_{4} + 3 · Y_{5} − 3 · Y_{6}
U_{15} = Y_{1} − 3 · Y_{2} − 3 · Y_{3} − 3 · Y_{4} − 3 · Y_{5} + 3 · Y_{6}
U_{16} = 4 · Y_{1}
U_{17} = −2 · Y_{1} + 6 · Y_{2}
U_{18} = −2 · Y_{1} − 6 · Y_{2}
U_{19} = −2 · Y_{1} + 6 · Y_{3}
U_{20} = −2 · Y_{1} − 6 · Y_{3}
U_{21} = −2 · Y_{1} + 6 · Y_{4}
U_{22} = −2 · Y_{1} − 6 · Y_{4}
U_{23} = −2 · Y_{1} + 6 · Y_{5}
U_{24} = −2 · Y_{1} − 6 · Y_{5}
U_{25} = −2 · Y_{1} + 6 · Y_{6}
U_{26} = −2 · Y_{1} − 6 · Y_{6}

The polynomials

J_{k} := ∑_{i=0}^{26} U_{i}^{k}

are invariant under the Weyl group W (E_{6}). It is known that the ring of all invariant polynomials is the polynomial ring

ℂ[Y_{1}, … Y_{6}]W(E^{6}) = ℂ[J_{2}, J_{5}, J_{6}, J_{8}, J_{9}, J_{12}].

The polynomial J_{2} is nothing but the Killing form up to a constant factor. The next one is the polynomial of our interest, J_{5}, and the hypersurface it defines,

I 5 := {J_{5} = 0}.

A quick way to write it explicitly is:

2 Remark. Let E_{i} denote the i^{th} elementary symmetric polynomial in the five variables Y_{2}, …, Y_{6}. Then

J_{5} = 720(Y_{1}^{5} − 6Y_{1}^{3}E_{1} − 27Y_{1}(E_{1}^{2} − 4E_{2}) + 648Y_{2}Y_{3}Y_{4}Y_{5}Y_{6})

Our goal is to find the dual polynomial (which is determined up to a constant factor). By the way the degree of the dual polynomial would be 5 · 4^{4} = 1280 if the quintic where nonsingular [K]. But singularities have the tendency to lower the degree. In our case it turns out to be 32.

3 Theorem. The dual polynomial of J_{5} has degree 32. It is obtained from the following polynomial J by the replacement Y_{1} ↦ 3Y_{1}

J= 2^{32} ·3^{16} ·5·7^{2} ·J_{8}^{4} −2^{34} ·3^{14} ·5·7^{2} ·J_{5}^{2}J_{6}J_{8}^{2} +2^{38} ·3^{13} ·5·7·J_{5}^{3}J_{8}J_{9} −2^{39} ·3^{12} ·7^{2} ·J_{5}^{4}J_{12} +2^{34} · 3^{10} · 7^{2} · 61 · J_{5}^{4}J_{6}^{2} −2^{35} · 3^{14} · 5 · 7^{2} · J_{2}J_{6}J_{8}^{3} −2^{35} · 3^{14} · 5^{2} · 7 · J_{2}J_{5}J_{8}^{2}J_{9} +2^{38} · 3^{12} · 5 · 7^{2} · J_{2}J_{5}^{2}J_{8}J_{12} +2^{36} · 3^{10} · 5 · 7^{3} · J_{2}J_{5}^{2}J_{6}^{2}J_{8} −2^{36} · 3^{11} · 5 · 7 · 13 · J_{2}J_{5}^{3}J_{6}J_{9} +2^{35} · 3^{10} · 7^{2} · 13 · J_{2}J_{5}^{6} −2^{34} · 3^{12} · 5^{2} · 7^{2} · J_{2}^{2}J_{8}^{2}J_{12} +2^{33} · 3^{10} · 5 · 7^{2} · 113 · J_{2}^{2}J_{6}^{2}J_{8}^{2} +2^{35} · 3^{11} · 5^{2} · 7 · 17 · J_{2}^{2}J_{5}J_{6}J_{8}J_{9} −2^{35} · 3^{11} · 5 · 7^{3} · J_{2}^{2}J_{5}^{2}J_{6}J_{12} −2^{34} · 3^{9} · 5^{2} · 7^{2} · J_{2}^{2}J_{5}^{2}J_{6}^{3} −2^{35} · 3^{10} · 7^{2} · 29 · J_{2}^{2}J_{5}^{4}J_{8} +2^{35} · 3^{10} · 5^{4} · J_{2}^{3}J_{8}J_{9}^{2} +2^{36} · 3^{10} · 5^{2} · 7^{2} · J_{2}^{3}J_{6}J_{8}J_{12} −2^{35} · 3^{8} · 5 · 7^{2} · 41 · J_{2}^{3}J_{6}^{3}J_{8} −2^{36} · 3^{9} · 5^{3} · 7 · J_{2}^{3}J_{5}J_{9}J_{12} −2^{35} · 3^{7} · 5^{2} · 7 · 193 · J_{2}^{3}J_{5}J_{6}^{2}J_{9} −2^{31} · 3^{12} · 5 · 7^{2} · J_{2}^{3}J_{5}^{2}J_{8}^{2} +2^{33} · 3^{8} · 7^{2} · 367 · J_{2}^{3}J_{5}^{4}J_{6} +2^{34} · 3^{8} · 5^{3} · 7^{2} · J_{2}^{4}J_{12}^{2} +2^{26} · 3^{10} · 5^{3} · 7^{4} · J_{2}^{4}J_{8}^{3} −2^{33} · 3^{8} · 5^{4} · 13 · J_{2}^{4}J_{6}J_{9}^{2} −2^{34} · 3^{6} · 5^{2} · 7^{2} · 41 · J_{2}^{4}J_{6}^{2}J_{12} +2^{32} · 3^{4} · 5 · 7^{2} · 41^{2} · J_{2}^{4}J_{6}^{4} −2^{27} · 3^{14} · 5 · 7^{2} · J_{2}^{4}J_{5}^{2}J_{6}J_{8} +2^{30} · 3^{7} · 5^{2} · 7 · 11 · 31 · J_{2}^{4}J_{5}^{3}J_{9} −2^{27} · 3^{8} · 5^{3} · 7^{2} · 139 · J_{2}^{5}J_{6}J_{8}^{2} −2^{28} · 3^{7} · 5^{4} · 7 · 59 · J_{2}^{5}J_{5}J_{8}J_{9} +2^{30} · 3^{6} · 5^{3} · 7^{2} · J_{2}^{5}J_{5}^{2}J_{12} +2^{28} ·3^{4} ·5·7^{2} ·10903·J_{2}^{5}J_{5}^{2}J_{6}^{2} −2^{27} ·3^{6} ·5^{4} ·7^{4} ·J_{2}^{6}J_{8}J_{12} +2^{26} ·3^{4} ·5^{3} ·7^{2} ·11^{2} ·41·J_{2}^{6}J_{6}^{2}J_{8} +2^{27} · 3^{5} · 5^{4} · 7 · 457 · J_{2}^{6}J_{5}J_{6}J_{9} −2^{24} · 3^{4} · 7^{2} · 179 · 593 · J_{2}^{6}J_{5}^{4} +2^{27} · 3^{8} · 5^{5} · J_{2}^{7}J_{9}^{2} +2^{28} ·3^{4} ·5^{4} ·7^{2} ·41·J_{2}^{7}J_{6}J_{12} −2^{27} ·3^{2} ·5^{3} ·7^{2} ·41^{2} ·J_{2}^{7}J_{6}^{3} +2^{23} ·3^{4} ·5·7^{2} ·64937·J_{2}^{7}J_{5}^{2}J_{8} +2^{17} · 3^{4} · 5^{2} · 7^{2} · 659947 · J_{2}^{8}J_{8}^{2} −2^{18} · 3^{2} · 5 · 7^{2} · 239 · 23747 · J_{2}^{8}J_{5}^{2}J_{6} −2^{19} · 3^{2} · 5^{2} · 7^{2} · 561947 · J_{2}^{9}J_{6}J_{8} −2^{19} · 3^{5} · 5^{3} · 7 · 17 · 179 · J_{2}^{9}J_{5}J_{9} −2^{18} · 3^{4} · 5^{3} · 7^{2} · 11 · 67 · J_{2}^{10}J_{12} +2^{17}·5^{2}·7^{2}·19·23·41·109·J_{2}^{10}J_{6}^{2} +2^{16}·3^{2}·5^{3}·7^{2}·4373·J_{2}^{11}J_{5}^{2} +2^{10}·3^{2}·5^{4}·7^{4}·11·67·J_{2}^{12}J_{8} −2^{11} · 5^{4} · 7^{2} · 11 · 41 · 67 · J_{2}^{13}J_{6} +5^{3} · 7^{2} · 11^{2} · 67^{2} · J_{2}^{16}.

We denote by J̃ the polynomial obtained from J replacing Y_{1} ↦ 3Y_{1}. Similarly the polynomial obtained from J_{k} by this transformation is denoted by J̃_{k}. The meaning of this transformation is that it intertwines the dual representation of W (E_{6}) with the original one. A simple observation states that P (Grad Q) is invariant under W (E_{6}) if Q is invariant under W (E_{6}) and P is invariant under the dual representation (i.e. invariant under the transposed matrices). We obtain that J̃_{k}(Grad J_{5}) can be written as a polynomial in the invariants J_{2}, J_{5}, J_{6}, J_{8}, J_{9}, J_{12}. As soon as one has found these polynomials explicitly for k = 2, 5, 6, 8, 9, 12 one has a representation of J̃(Grad J_{5}) as a polynomial in the invariants. The duality relation means that this polynomial is divisible by J_{5}. This means nothing other than that it vanishes after replacing J_{5} by 0. The explicit expressions for J̃_{k}(Grad J_{5}) in the six cases as polynomials in the invariants have been found by means of a computer calculation. The expressions can be found in http://www.rzuser.uni-heidelberg.de/∼t91/programme/gradient.* The most involved calculations concerns the case k = 12. The degree of J̃_{12}(Grad J_{5}) is 48 and there are 248 monomials of this degree. We choose 350 special values at random and had to solve 350 equations with 248 variables. The problem is that the solution consists of huge rational numbers with denominators. Therefore we solved the equations modulo several primes (actually 15 such beginning at 100003). To reconstruct the rational numbers we made an a priori assumption about the their size. This a priori assumption is justified later because it is easy to verify the result by direct calculation. The vanishing of J̃(Grad J_{5}) after making the substitution J_{5} = 0 again can be verified by inserting enough special values.

In [Hu] it has been proved that J_{5} defines a rational variety. We obtain

4 Corollary. The hypersurface I 32 defined by J is rational.

Literature

[FH] Freitag, E. Hermann, C.F.: Some modular varieties of low dimension, preprint, preliminary version available on the net at http://www.

rzuser.uni-heidelberg.de/∼t91/skripten/preprints/orth.dvi.

[H] Hunt, B.: The geometry of som special arithmetic quotients, Lecture Notes in Math. 1637 Springer Verlag Berlin-Heidelberg (1996)

[K] Kleiman, S:L.: Concerning the dual variety, in 18th Scandinavian Congress of Mathematics, 1980, E. Balslev (ed.). Progress in Math. 11 Birkhäuser, 386-396 (1981)