La famille de Weyl

A theta characteristic on $V$ is a quadratic form with polar form $B$. They are the 64 forms $q_a=q_0+B(a,\cdot)$, $a\in V$; $q_a$ is odd when $q_0(a)=1$, and there are 28 odd and 36 even ones.

The nine sets, the odd theta characteristics (28), the even ones (36), the nonzero vectors (63), the split Cayley hexagons whose lines are isotropic lines (120), the Lagrangian subspaces (135), the Aronhold heptads (288), the isotropic planes (315), the non-isotropic planes (336) and the Desarguesian symplectic spreads (960), are transitive; each stabilizer fixes no other point of its set and is the only class of subgroups of its order. So each is a rigid object. Moreover:

(a) of the 3276 triples of odd forms, 1260 are syzygetic and 2016 azygetic; (b) the syzygetic tetrads $\{a,b,c,a+b+c\}$, 315 of them, correspond bijectively to the isotropic planes by $\{a,b,c,d\}\mapsto\{a+b,a+c,a+d\}$; (c) the Aronhold heptads are the 7-sets of odd forms with every triple azygetic, the sum of a heptad’s seven forms is an even form, and each even form is the sum of exactly 8 heptads.

(a) The stabilizer of an odd form $t^*$, a group $W(E_6)$, has orbits $1+27$ on the odd forms, 36 on the even, $27+36$ on the nonzero vectors, the 27 being the singular points of $q_{t^*}$, and $45+270$ on the isotropic planes, the 45 being the totally singular lines of $q_{t^*}$, which with the 27 points form the generalized quadrangle $GQ(2,4)$.

(b) The stabilizer of the even form $q_0$, a group $S_8$, has orbits 28 on the odd forms, $1+35$ on the even, $28+35$ on the nonzero vectors, $30+105$ on the Lagrangians, the 30 being the planes totally singular for $q_0$, and $8+280$ on the heptads.

(a) For $n=6,7,8$ there are 27, 56, 240 exceptional classes and 72, 126, 240 roots, and $|W(E_n)|=51840$, 2903040, 696729600. The Geiser involution ($n=7$) and the Bertini involution $E\mapsto-2K-E$ ($n=8$) are central in $W$, acting as $-1$ on $K^\perp$; the centre of $W(E_6)$ is trivial.

(e) Under $W(E_7)\subset W(E_8)$ the exceptional classes split as $240=1+1+56+56+126$ and the Bertini pairs as $120=1+56+63$. Under $W(E_6)\subset W(E_7)$ the exceptional classes split as $56=1+1+27+27$, the bitangents as $28=1+27$, the roots as $126=72+27+27$ and the pairs of roots as $63=36+27$.

Let $Q_6=K^\perp\subset I^{1,6}$, the lattice $E_6$. (a) In characteristic 2, $Q_6/2Q_6$ with $q=x\cdot x/2\bmod2$ is a nondegenerate elliptic quadratic space of dimension 6 on which $W(E_6)$ acts faithfully, so $W(E_6)\cong O^-(6,2)$. The map $E\mapsto\langle3E+K\rangle$ carries the 27 lines onto the 27 singular points and the 45 tritangent trios onto the 45 totally singular lines, and the 36 pairs of roots go onto the 36 nonsingular points.

(b) In characteristic 3, $V_3=Q_6/3Q_6^*$ is 5-dimensional, $x\cdot y\bmod3$ is nondegenerate on it, and $W(E_6)$ acts faithfully on its 121 points, so $W(E_6)\cong\SO(5,3)$. The points form three orbits: 36 with $x\cdot x\equiv1$, the pairs of roots; 45 with $x\cdot x\equiv-1$, onto which $\{E_1,E_2,E_3\}\mapsto\langle E_1-E_2\rangle$ carries the tritangent trios; and 40 singular, onto which the subsystems $3A_2$ map bijectively.

(a) The six points of contact of three bitangents with labels $a,b,c$ lie on a conic if and only if $q_0(a+b+c)=1$, which holds for 1260 of the 3276 triples; four bitangents whose labels sum to 0 have their eight points of contact on a conic, and these are the 315 syzygetic tetrads.

(b) Sending a pair of bitangents to the sum of their labels identifies $J(X)[2]$ with $V$, $G$-equivariantly, and the Weil pairing with $B$. The theta characteristic of the bitangent $(p,L)$ is $q_{(p;L)}$, so the 64 theta characteristics of $X$ are the forms $q_{(p;L)}$ with $p$ and $L$ arbitrary, odd exactly when $p\neq0$, $L\neq0$ and $p\notin L$.

(c) $q_0$ is the only $G$-invariant theta characteristic, and it is even. $V_{\mathrm{pt}}=\{(p;0)\}$ and $V_{\mathrm{ln}}=\{(0;L)\}$ are the only $G$-invariant Lagrangian subspaces of $J(X)[2]$: the natural module of $\GL(3,2)$ and its dual.

(a) The four bitangents through a centre have labels summing to 0 and form a syzygetic tetrad, paired as above. (b) Every Coxeter edge $\{u,v\}$ spans an isotropic line whose third point $u+v$ is a flag class; for each $u$, nine odd $v$ have this property, and the three Coxeter neighbours among them are those for which the cross classes $(x_u;y_v)$ and $(x_v;y_u)$ are odd. (c) Exactly one split Cayley hexagon is $G$-invariant. Its lines are the 21 lines $\{(p;0),(0;L),(p;L)\}$, $p\in L$, and the 42 lines $\{u,v,u+v\}$ over the Coxeter edges; its 35 points with $q_0=0$ form a geometric hyperplane, and its collinearity graph on the other 28 points, the bitangents, is the Coxeter graph.

(a) All 28 bitangents are defined over $\Q(\zeta_7)$, and $\sigma_a$ acts on them as $m_a$ acts on their labels in $\Proj^1(\F_7)$. The action is faithful, so the bitangents generate $\Q(\zeta_7)$, and the image is the cyclic group $\{m_a\}$ of order 6 fixing 0 and $\infty$; $\mathrm{Gal}(\Q(\zeta_7)/\Q(\sqrt{-7}))$ acts by elements of $G$, and complex conjugation by an outer automorphism.

(b) The Galois orbits on the bitangents have sizes 1,3,6,6,6,6. The line $x+y+z=0$ is the only rational bitangent; its three Coxeter neighbours $\{1,6\}$, $\{2,5\}$, $\{3,4\}$ form an orbit, each defined over $\Q(\zeta_7+\zeta_7^{-1})$, and these four are the real bitangents.

(c) On $J(X)[2]$ the Galois group fixes exactly one nonzero class, an antiflag class, and exactly two theta characteristics, $q_0$ and $x+y+z=0$. Complex conjugation fixes exactly 7 nonzero classes, so by Zeuthen’s classification the real locus of $X$ is a single oval.