The Fermat cubic surface
An infamous cubic surface is the Fermat, defined by the equation \(X_0^3 + X_1^3 + X_2^3 + X_3^3 = 0\). Its automorphism group is \(C_3^{\times 3}\rtimes S_4\), of order \(648\). It is the most symmetric cubic surface.
Lines
The lines on the Fermat admit very easy formulas. Although the automorphism group is much larger, we graph them together with their isotropy as a subgroup of \(S_4\) – this is to mirror the computations found in [B25] and [BR25]. In the following, let \(\zeta\) denote a primitive 6th root of unity. We describe the lines in terms of parametric coordinates \([s:t]\in \mathbb{P}^1\).
| Number | Formula | Isotropy in S_4 |
|---|---|---|
| 1 | \([s\colon -s\colon t \colon \zeta t]\) | \( \langle ((1\ 2) \rangle \) |
| 2 | \([s\colon-s\colon t\colon\zeta^{-1} t]\) | \( \langle ((1\ 2) \rangle \) |
| 3 | \([s\colon\zeta s\colon t\colon- t]\) | \( \langle ((3\ 4) \rangle \) |
| 4 | \([s\colon\zeta^{-1} s\colon t \colon - t]\) | \( \langle ((3\ 4) \rangle \) |
| 5 | \([s\colon t \colon \zeta s\colon -t]\) | \( \langle ((2\ 4) \rangle \) |
| 6 | \([s \colon t \colon \zeta^{-1} s \colon -t]\) | \( \langle ((2\ 4) \rangle \) |
| 7 | \([s \colon t \colon -s \colon \zeta t]\) | \( \langle ((1\ 3) \rangle \) |
| 8 | \([s \colon t \colon -s \colon \zeta^{-1} t]\) | \( \langle ((1\ 3) \rangle \) |
| 9 | \([s \colon t \colon -t \colon \zeta s]\) | \( \langle ((2\ 4) \rangle \) |
| 10 | \([s \colon t \colon -t \colon \zeta^{-1} s]\) | \( \langle ((2\ 4) \rangle \) |
| 11 | \([s \colon t \colon \zeta t \colon -s]\) | \( \langle ((1\ 4) \rangle \) |
| 12 | \([s \colon t \colon \zeta^{-1} t \colon -s]\) | \( \langle ((1\ 4) \rangle \) |
| 13 | \([s\colon \zeta s\colon t\colon \zeta t]\) | \( \langle ((1\ 3)(2\ 4) \rangle \) |
| 14 | \([s\colon \zeta s\colon t\colon \zeta^{-1} t]\) | \( \langle ((1\ 4)(2\ 3) \rangle \) |
| 15 | \([s\colon \zeta^{-1} s\colon t\colon \zeta t]\) | \( \langle ((1\ 4)(2\ 3) \rangle \) |
| 16 | \([s\colon \zeta^{-1} s\colon t\colon \zeta^{-1} t]\) | \( \langle ((1\ 3)(2\ 4) \rangle \) |
| 17 | \([s\colon t\colon \zeta s\colon \zeta t]\) | \( \langle ((1\ 2)(3\ 4) \rangle \) |
| 18 | \([s\colon t\colon \zeta s\colon \zeta^{-1} t]\) | \( \langle ((1\ 4)(2\ 3) \rangle \) |
| 19 | \([s\colon t\colon \zeta^{-1} s\colon \zeta t]\) | \( \langle ((1\ 4)(2\ 3) \rangle \) |
| 20 | \([s\colon t\colon \zeta^{-1} s\colon \zeta^{-1} t]\) | \( \langle ((1\ 2)(3\ 4) \rangle \) |
| 21 | \([s\colon t\colon \zeta t\colon \zeta s]\) | \( \langle ((1\ 2)(3\ 4) \rangle \) |
| 22 | \([s\colon t\colon \zeta^{-1} t\colon \zeta s]\) | \( \langle ((1\ 3)(2\ 4) \rangle \) |
| 23 | \([s\colon t\colon \zeta t\colon \zeta^{-1} s]\) | \( \langle ((1\ 3)(2\ 4) \rangle \) |
| 24 | \([s\colon t\colon \zeta^{-1} t\colon \zeta^{-1} s]\) | \( \langle ((1\ 2)(3\ 4) \rangle \) |
| 25 | \([s\colon -s\colon t\colon -t]\) | \( \langle(1\ 3)(2\ 4), (1\ 2), (3\ 4)\rangle \) |
| 26 | \([s\colon t\colon -s\colon -t]\) | \( \langle(1\ 3)(2\ 4), (1\ 2), (3\ 4)\rangle \) |
| 27 | \([s\colon t\colon -t\colon -s]\) | \( \langle(1\ 3)(2\ 4), (1\ 2), (3\ 4)\rangle \) |
Adjacency
The (complement of the) adjacency matrix of the lines above is available here. We can present the Weyl group of \(E_6\) as a subgroup of \(S_{27}\) in a few ways, either via monodromy computations or just by extracting the automorphism group of the Schläfli graph described by the adjacency matrix above using nauty or any similar tool. The generators for \(W(E_6)\) as a symmetric subgroup are the following:
| Number | Generator | Matrix rep | Order, trace | Atlas notation | Carter notation |
|---|---|---|---|---|---|
| 1 | (13,23)(14,19)(15,18)(16,22)(17,24)(20,21) | | (2,4) | 2C | \[A_1\] |
| 2 | (5,14)(7,15)(9,13)(11,16)(17,27)(21,26) | | (2,4) | 2C | \[A_1\] |
| 3 | (2,6)(4,8)(5,19)(7,18)(9,23)(11,20)(12,25)(16,21)(22,26)(24,27) | | (2,2) | 2B | \[2A_1\] |
| 4 | (5,8)(6,7)(9,12)(10,11)(17,20)(21,24) | | (2,4) | 2C | \[A_1\] |
| 5 | (3,4)(5,10)(6,9)(7,12)(8,11)(13,15)(14,16)(17,24)(18,23)(19,22)(20,21)(26,27) | | (2,0) | 2D | \[3A_1\] |
| 6 | (1,2)(5,9)(6,10)(7,11)(8,12)(13,14)(15,16)(17,21)(18,22)(19,23)(20,24)(26,27) | | (2,-2) | 2A | \[4A_1\] |
Note that every automorphism of \(W(E_6)\) is inner, so we can extract conjugacy classes of the permutations above by picking an arbitrary isomorphism with a matrix group presentation.
In any case we can graph the adjacency matrix and get the (complement of the) Schläfli graph. The colors correspond to the symmetries of the lines under the \(S_4\) action.
Tritangents
All the 45 tritangents on the Fermat (in the labeling above) are as follows:
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
[1, 2, 25],
[1, 6, 12],
[1, 7, 9],
[1, 13, 15],
[1, 18, 23],
[2, 5, 11],
[2, 8, 10],
[2, 14, 16],
[2, 19, 22],
[3, 4, 25],
[3, 5, 9],
[3, 8, 12],
[3, 13, 14],
[3, 19, 23],
[4, 6, 10],
[4, 7, 11],
[4, 15, 16],
[4, 18, 22],
[5, 6, 26],
[5, 15, 24],
[5, 17, 18],
[6, 14, 21],
[6, 19, 20],
[7, 8, 26],
[7, 14, 24],
[7, 17, 19],
[8, 15, 21],
[8, 18, 20],
[9, 10, 27],
[9, 16, 20],
[9, 21, 22],
[10, 13, 17],
[10, 23, 24],
[11, 12, 27],
[11, 13, 20],
[11, 21, 23],
[12, 16, 17],
[12, 22, 24],
[13, 22, 26],
[14, 18, 27],
[15, 19, 27],
[16, 23, 26],
[17, 21, 25],
[20, 24, 25],
[25, 26, 27],
[1, 2, 25],
[1, 6, 12],
[1, 7, 9],
[1, 13, 15],
[1, 18, 23],
[2, 5, 11],
[2, 8, 10],
[2, 14, 16],
[2, 19, 22],
[3, 4, 25],
[3, 5, 9],
[3, 8, 12],
[3, 13, 14],
[3, 19, 23],
[4, 6, 10],
[4, 7, 11],
[4, 15, 16],
[4, 18, 22],
[5, 6, 26],
[5, 15, 24],
[5, 17, 18],
[6, 14, 21],
[6, 19, 20],
[7, 8, 26],
[7, 14, 24],
[7, 17, 19],
[8, 15, 21],
[8, 18, 20],
[9, 10, 27],
[9, 16, 20],
[9, 21, 22],
[10, 13, 17],
[10, 23, 24],
[11, 12, 27],
[11, 13, 20],
[11, 21, 23],
[12, 16, 17],
[12, 22, 24],
[13, 22, 26],
[14, 18, 27],
[15, 19, 27],
[16, 23, 26],
[17, 21, 25],
[20, 24, 25],
[25, 26, 27]
As an \(S_4\)-set, the tritangents have the following symmetries: \(2[S_4/C_2^o] + [S_4/K_4] + [S_4/C_4] + [S_4/C_3] + [S_4/S_4]\).
References:
- [B25] Equivariant enumerative geometry, T. Brazelton, Adv. Math. 2025
- [BR25] Monodromy in the space of symmetric cubic surfaces with a line, T. Brazelton & S. Raman, preprint, 2025.