# Moduli of codimension two linear sections of subadjoint varieties

Yingqi Liu  Yingqi Liu, AMSS, Chinese Academy of Sciences, 55 ZhongGuanCun
East Road, Beijing, 100190, China and University of Chinese Academy of
Sciences, Beijing, China<EMAIL_ADDRESS>
###### Abstract.

Let $\displaystyle G$ be a simple algebraic group of type $\displaystyle
F_{4}$, $\displaystyle E_{6}$, $\displaystyle E_{7}$ or $\displaystyle E_{8}$,
and let $\displaystyle\mathfrak{g}$ be its Lie algebra. The adjoint variety
$\displaystyle X_{ad}\subseteq\mathbb{P}\mathfrak{g}$ is defined as the unique
closed orbit of the adjoint action of $\displaystyle G$ on
$\displaystyle\mathbb{P}\mathfrak{g}$. $\displaystyle X_{ad}$ is a Fano
contact manifold covered by lines in
$\displaystyle{\mathbb{P}}{\mathfrak{g}}$. The subadjoint variety
$\displaystyle S\subseteq{\mathbb{P}}W$ is denoted by the variety of lines on
$\displaystyle X_{ad}$ through a fixed point $\displaystyle x$, where
$\displaystyle W\subseteq T_{x}X$ is taken as the contact hyperplane. It
follows from a result in representation theory of Vinberg that the GIT
quotient space of codimension two linear sections of $\displaystyle S$ is
isomorphic to the weighted projective space
$\displaystyle{\mathbb{P}}(1,3,4,6)$. In this note, we investigate the problem
of finding a geometric interpretation of the above isomorphism. As a main
result, for each $\displaystyle{\mathfrak{g}}$ of the above type, we construct
a natural open embedding of the GIT quotient space of nonsingular codimension
two linear sections of $\displaystyle S$ into
$\displaystyle{\mathbb{P}}(1,3,4,6)$ whose complement is a fixed hypersurface
of degree 12. The key ingredient of our construction is to apply a
correspondence of Bahargava and Ho which relates the above moduli problem to a
moduli problem on curves of genus one.

##  1\. Introduction

Let $\displaystyle G$ be a simply connected simple algebraic group over
$\displaystyle{\mathbb{C}}$, and let $\displaystyle\mathfrak{g}$ be its Lie
algebra. The adjoint variety $\displaystyle
X_{ad}\subseteq\mathbb{P}\mathfrak{g}$ is defined as the unique closed
$\displaystyle G$-orbit of the adjoint action on
$\displaystyle\mathbb{P}\mathfrak{g}$. $\displaystyle X_{ad}$ is a Fano
cantact manifold, and it is covered by lines when $\displaystyle G$ is not of
type $\displaystyle A$ or $\displaystyle C$. In this case, the subadjoint
variety $\displaystyle S\subseteq\mathbb{P}W$ is denoted by the variety of
lines on $\displaystyle X_{ad}$ through a fixed point $\displaystyle x$, where
we identify a line through $\displaystyle x$ with its tangent direction at
$\displaystyle x$, and $\displaystyle W\subseteq T_{x}X$ is taken as the
contact hyperplane. $\displaystyle S$ is a rational homogenous space under the
action of the isotropy subgroup of $\displaystyle G$. In fact let
$\displaystyle H$ be the semi-simple part of the isotropy subgroup of
$\displaystyle G$ at $\displaystyle x$. Then $\displaystyle W$ is an
irreducible subspace under the isotropy action of $\displaystyle H$ on
$\displaystyle T_{x}X$. And $\displaystyle S$ is the unique closed orbit of
the action of $\displaystyle H$ on $\displaystyle\mathbb{P}W$. The contact
structure on $\displaystyle X_{ad}$ gives rise to a $\displaystyle
H$-invariant nondegenerate skew bilinear form $\displaystyle w$ on
$\displaystyle W$, with respect to which $\displaystyle
S\subseteq{\mathbb{P}}W$ is known as a Legendrian variety. The importance of
understanding the geometry of subadjoint varieties (especially as Legendrian
varieties) stems from the celebrated conjecture by Lebrun and Salamon, which
asserts that adjoint varieties are the only examples of Fano contact
manifolds. We refer to [LM07] for more details.

Classical examples of subadjoint varieties are the twisted cubic in
$\displaystyle{\mathbb{P}}^{3}$, and the products
$\displaystyle{\mathbb{P}}^{1}\times{\mathbb{Q}}^{n}$ of a line and a smooth
hyperquadric of dimension $\displaystyle n\geqslant 1$, which correspond to
$\displaystyle{\mathfrak{g}}={\mathfrak{g}}_{2}$ and
$\displaystyle{\mathfrak{g}}={\mathfrak{s}}{\mathfrak{o}}_{n+6}$ respectively.
In this article we focus on the following five subadjoint varieties, where we
use $\displaystyle H_{G},W_{G}$ and $\displaystyle S_{G}$ to express the
objects associated with a fixed group $\displaystyle G$.

$\displaystyle\text{Type of\,\,\,}\mathfrak{g}$ | $\displaystyle H_{G}$ | Rep.$\displaystyle W_{G}$ | $\displaystyle S_{G}\subseteq\mathbb{P}W_{G}$ | $\displaystyle dim(S_{G})$
---|---|---|---|---
$\displaystyle D_{4}$ | $\displaystyle SL_{2}\times SL_{2}\times SL_{2}$ | $\displaystyle\mathbb{C}^{2}\otimes\mathbb{C}^{2}\otimes\mathbb{C}^{2}$ | $\displaystyle\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1},Segre$ | 3
$\displaystyle F_{4}$ | $\displaystyle SP_{6}$ | $\displaystyle\wedge^{3}_{0}\mathbb{C}^{6}$ |  $\displaystyle Lag(3,6)$, Plücker | 6
$\displaystyle E_{6}$ | $\displaystyle SL_{6}$ | $\displaystyle\wedge^{3}\mathbb{C}^{6}$ |  $\displaystyle Gr(3,6)$, Plücker | 9
$\displaystyle E_{7}$ | $\displaystyle Spin_{12}$ | half spin |  $\displaystyle{\mathbb{S}}_{6}$, minimal | 15
$\displaystyle E_{8}$ | $\displaystyle E_{7}$ | minuscule |  $\displaystyle E_{7}/P_{7}$, minimal | 27
Table 1.

The subadjoint varieties listed in Table 1, together with the twisted cubic,
are also known as the varieties in the third row of $\displaystyle geometric$
$\displaystyle Freudenthal$-$\displaystyle Tits$ $\displaystyle magic$
$\displaystyle rectangle$ [Ma13, Section 6]. They can be constructed in a
uniform way as twisted cubic “curves” over Hermitian cubic Jordan algebras
(see 2.3). In this note we focus on the study of their linear sections. It is
known that there are exactly four orbits of the action of $\displaystyle H$ on
$\displaystyle\mathbb{P}W$ (see Theorem 2.12). It follows that all nonsingular
hyperplane sections are isomorphic to each other. Moreover when
$\displaystyle{\mathfrak{g}}$ is not of type $\displaystyle D_{4}$ it is
proved in [CFL23] that these nonsingular hyperplane sections are rigid under
smooth deformation.

The main goal of this note is to describe the GIT quotient space of
codimension two linear sections of the subadjoint variety $\displaystyle
S\subseteq\mathbb{P}W$ listed in Table 1. Under the identification
$\displaystyle W\cong W^{\vee}$ via $\displaystyle w$, for a plane
$\displaystyle[U]\in Gr(2,W)$, we denote
$\displaystyle{\mathbb{P}}U^{\perp}\subseteq{\mathbb{P}}W$ to be the linear
subspace isotropic to $\displaystyle U$, and denote $\displaystyle
S_{U}={\mathbb{P}}U^{\perp}\cap S$ to be the associated linear section. We are
going to study the GIT quotient of the $\displaystyle H$-action on the pair
$\displaystyle(Gr(2,W)),\mathcal{O}_{Gr}(1))$, where
$\displaystyle\mathcal{O}_{Gr}(1)$ is the Plücker line bundle on
$\displaystyle Gr(2,W)$, and we fix the linearization of $\displaystyle H$ on
it provided by the linear action of $\displaystyle H$ on $\displaystyle W$.
Recall that the GIT quotient is given by $\displaystyle
Gr(2,W)^{ss}//H=Proj(R_{G})$, where $\displaystyle R_{G}=\oplus_{k\geqslant
0}H^{0}(Gr(2,W),\mathcal{O}_{Gr}(k))^{H}$ is the ring of $\displaystyle
H$-invariant sections. Note that $\displaystyle R_{G}$ can be naturally
identified with the invariant ring
$\displaystyle\mathbb{C}[\mathbb{C}^{2}\otimes W]^{SL_{2}\times H}$, under
which an invariant section of degree $\displaystyle d$ is identified with an
invariant function of degree $\displaystyle 2d$. A key motivation of our study
is the following description of $\displaystyle R_{G}$ arising from Vinberg’s
theory of $\displaystyle\theta$-groups.

######  Theorem 1.1.

[Vi76] Let $\displaystyle G$ be a simple algebraic group. The invariant ring
$\displaystyle\mathbb{C}[\mathbb{C}^{2}\otimes W]^{SL_{2}\times H}$ is a
polynomial ring. If $\displaystyle G$ is of type $\displaystyle
F_{4},E_{6},E_{7}$ or $\displaystyle E_{8}$, the invariant ring
$\displaystyle\mathbb{C}[\mathbb{C}^{2}\otimes W]^{SL_{2}\times H}$ is a
polynomial ring in four generators, of degree $\displaystyle 2,6,8,12$
respectively. In particular $\displaystyle
Proj(R_{G})\cong\mathbb{P}(1,3,4,6)$ is a weighted projective space of
dimension 3.

######  Remark 1.2.

If $\displaystyle G$ is of type $\displaystyle D_{4}$, then the invariant ring
$\displaystyle\mathbb{C}[\mathbb{C}^{2}\otimes W]^{SL_{2}\times H}$ is a
polynomial ring in four generators, of degree 2,4,4,6 respectively. If we
additionally consider the action of $\displaystyle S_{3}$ on $\displaystyle
W={\mathbb{C}}^{2}\otimes{\mathbb{C}}^{2}\otimes{\mathbb{C}}^{2}$ given by
permutations of factors, then the ring
$\displaystyle\mathbb{C}[\mathbb{C}^{2}\otimes W]^{SL_{2}\times H\times
S_{3}}$ becomes also a polynomial ring in four generators, of degree
$\displaystyle 2,6,8,12$ respectively. We will explain these facts in the
Appendix, see (A.5).

It is remarkable that $\displaystyle Proj(R_{G})$ is independent of
$\displaystyle G$ when $\displaystyle G$ is of type $\displaystyle
F_{4},E_{6},E_{7}$ or $\displaystyle E_{8}$. This motivates us to investigate
the following problem which seeks for a geometric interpretation.

######  Main Problem.

Assume $\displaystyle G=F_{4},E_{6},E_{7}$ or $\displaystyle E_{8}$, then:

1\. Provide a geometric interpretation for the generators of $\displaystyle
R_{G}$, and use it to characterize the semi-stable locus $\displaystyle
Gr(2,W_{G})^{ss}$.

2\. Construct geometric identifications between the GIT quotients
$\displaystyle Gr(2,W_{G})^{ss}//H_{G}$ when $\displaystyle G$ varies.

Following Remark 1.2, our main idea to tackle the problem is to compare it to
the case of $\displaystyle G=D_{4}$. On one direction, there exists an
embedding of representations arsing from the construction of the
representation $\displaystyle(H,W)$ in [LM02] using composition algebras:

######  Proposition 1.3.

(Corollary 2.11) Let $\displaystyle G$ be of type $\displaystyle
F_{4},E_{6},E_{7}$ or $\displaystyle E_{8}$. Then there exists an embedding of
representations
$\displaystyle(Lie(H_{D_{4}}),W_{D_{4}})\subseteq(Lie(H_{G}),W_{G})$. Moreover
let $\displaystyle\overline{H}_{G}\subseteq PGL(W_{G})$ be the image of
$\displaystyle H_{G}$ in $\displaystyle PGL(W_{G})$. Then there is a subgroup
$\displaystyle S_{3}\subseteq\overline{H}_{G}$ which acts on
$\displaystyle\mathbb{P}W_{D_{4}}=\mathbb{P}(\mathbb{C}^{2}\otimes\mathbb{C}^{2}\otimes\mathbb{C}^{2})$
by permutations of factors.

For the converse, to associate an $\displaystyle H_{G}$-equivalent class of
planes in $\displaystyle W_{G}$ to an $\displaystyle H_{D_{4}}\times
S_{3}$-equivalent class of planes in $\displaystyle W_{D_{4}}$, we apply the
following correspondence constructed by Bhargava and Ho in [BH16].

######  Definition 1.4.

The tangent variety $\displaystyle TS$ of $\displaystyle
S\subseteq\mathbb{P}W$ is a quartic hypersurface (Theorem 2.12). A plane
$\displaystyle U\subseteq W$ is called nondegenerate if $\displaystyle TS$
intersects $\displaystyle\mathbb{P}U$ at four distinct points. For a tensor
$\displaystyle v\in\mathbb{C}^{2}\otimes W$, we say it is $\displaystyle
nondegenerate$ if it defines a nondegenerate plane in $\displaystyle W$.

######  Theorem 1.5.

[BH16, Theorem 6.28] There is a one to one correspondence between
nondegenerate $\displaystyle GL_{2}\times H$-orbits on
$\displaystyle\mathbb{C}^{2}\otimes W$ and isomorphic classes of triples
$\displaystyle(C,L,\kappa:C\rightarrow S)$, where $\displaystyle C$ is a
nonsingular projective curve of genus one, $\displaystyle L$ is a line bundle
on $\displaystyle C$ of degree 2, and $\displaystyle\kappa$ is a morphism from
$\displaystyle C$ to $\displaystyle S$ such that
$\displaystyle\kappa^{*}(\mathcal{O}_{S}(1))\cong L^{\otimes 3}$.

The correspondence is built upon some special projective properties of
subadjoint varieties (see Section 2.3). The case where $\displaystyle G$ is of
type $\displaystyle D_{4}$ can be written more concretely as follows. In
particular it provides a geometric interpretation for the generators of the
invariant ring in this case (see Proposition A.2).

######  Proposition 1.6.

[BH16, Section 2.4, Theorem 6.4] In Theorem 1.5, if $\displaystyle G=D_{4}$,
then:

(1) A plane $\displaystyle U$ is nondegenerate if and only if the linear
section $\displaystyle S_{U}$ is nonsingular.

(2) The triple
$\displaystyle(C,L,\kappa:C\rightarrow\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1})$
is the same as the datum $\displaystyle(C,L,L_{i},i=1,2,3)$, where
$\displaystyle(L_{1},L_{2},L_{3})$ is a tuple of three line bundles of degree
2 on $\displaystyle C$, such that $\displaystyle 3L=L_{1}+L_{2}+L_{3}$, and
$\displaystyle L\not=L_{i}$, for any $\displaystyle i=1,2,3$.

(3) There exists a choice of normalization for the generators of
$\displaystyle SL_{2}\times H$-invariants $\displaystyle
a_{2},a_{4}.a_{4}^{\prime},a_{6}$, such that given a nondegenerate tensor in
$\displaystyle{\mathbb{C}}^{2}\otimes W$ corresponding to the datum
$\displaystyle(C,L,L_{i},i=1,2,3)$ as in (2), the Jacobian of $\displaystyle
C$ may be given in normal form as

(1.1) $y^{2}=x^{3}+a_{8}x+a_{12},$

on which the points $\displaystyle P_{1}\doteq
L-L_{1}=(a_{4},a_{6}),P_{2}\doteq
L-L_{2}=(a_{4}^{\prime},a_{6}^{\prime}),P_{3}\doteq
L-L_{3}=(a_{4}^{\prime\prime},a_{6}^{\prime\prime})$ lie, such that
$\displaystyle a_{2}$ is the slope of the line connecting $\displaystyle
P_{1}$ and $\displaystyle P_{2}$, and $\displaystyle
a_{6}^{\prime}=a_{6}+a_{2}(a_{4}^{\prime}-a_{4})$.

The key ingredient of our main result is the following, which relates
nondegenerate planes in $\displaystyle W_{G}$ to nondegenerate planes in
$\displaystyle W_{D_{4}}$.

######  Theorem 1.7.

Let $\displaystyle G=D_{4},F_{4},E_{6},E_{7}$ or $\displaystyle E_{8}$, then:

(1). A plane $\displaystyle U\subseteq W_{G}$ is nondegenerate if and only if
$\displaystyle S_{U}$ is nonsingular. There is a $\displaystyle H$-invariant
section $\displaystyle\sigma_{12}\in H^{0}(Gr(2,W),\mathcal{L}^{\otimes
12})^{H}$, such that $\displaystyle[U]\in Gr(2,W)$ is nondegenerate if and
only if $\displaystyle\sigma_{12}(U)\not=0$.

(2).(Proposition 3.3) Let $\displaystyle U\subseteq W_{G}$ be a nondegenerate
plane, and let $\displaystyle(C,L,\kappa:C\rightarrow S_{G})$ be the
associated datum defined in Theorem 1.5. Then there exists (uniquely) three
line bundles $\displaystyle L_{1},L_{2},L_{3}$ on $\displaystyle C$ of degree
two satsifying:

(2.1) $\displaystyle 3L=L_{1}+L_{2}+L_{3}$, and $\displaystyle L\not=L_{i}$
for $\displaystyle i=1,2,3$;

(2.2) For any two distinct points $\displaystyle p,q$ on $\displaystyle C$,
$\displaystyle p+q\in|L_{i}|$ for some $\displaystyle i$ if and only if their
images $\displaystyle\kappa(p),\kappa(q)$ are isotropic under $\displaystyle
w$, or equivalently $\displaystyle\kappa(p)$ and $\displaystyle\kappa(q)$ can
be connected by at most two lines on $\displaystyle S_{G}$.

Denote the open subset parametrizing nonsingular sections by
$\displaystyle{\mathcal{U}}_{G}=\\{\sigma_{12}\not=0\\}\subseteq Gr(2,W_{G})$.
As a nondegenerate plane $\displaystyle U$ varies in
$\displaystyle{\mathcal{U}}_{G}$, we show that Theorem 1.7(2) induces a family
of three line bundles (Proposition 3.10). This motivates us to define the
moduli functor $\displaystyle{\mathcal{M}}$ (Definition 4.1) which
parametrizes the datum $\displaystyle(C,L,L_{1},L_{2},L_{3})$, where
$\displaystyle C$ is a curve of genus one, $\displaystyle L$ is a degree two
line bundle on $\displaystyle C$, and $\displaystyle\\{L_{1},L_{2},L_{3}\\}$
is a multiset of three degree two line bundles on $\displaystyle C$ satisfying
the condition as in Theorem 1.7(2.1).

It follows from Theorem 1.7(1) that $\displaystyle{\mathcal{U}}_{G}\subseteq
Gr(2,W_{G})^{ss}$. We denote $\displaystyle M_{G}={\mathcal{U}}_{G}//H_{G}$ to
be its GIT quotient. In the case where $\displaystyle G=D_{4}$, we denote
$\displaystyle M=M_{D_{4}}//S_{3}$ to be the GIT quotient by $\displaystyle
S_{3}$. We deduce the following from Proposition 1.6:

######  Proposition 1.8.

(1). For the action of $\displaystyle H_{D_{4}}=SL_{2}\times SL_{2}\times
SL_{2}$ on $\displaystyle(Gr(2,W_{D_{4}}),{\mathcal{O}}_{Gr}(1))$, a plane
$\displaystyle[U]$ is GIT-stable if and only if $\displaystyle[U]$ is
nondegenerate.

(2). There is a natural transformation
$\displaystyle\xi:{\mathcal{M}}\rightarrow h_{M}$, making it a coarse moduli
space of $\displaystyle{\mathcal{M}}$.

(3). The parametrization in Prop 1.6 (3) induces a natural open embedding of
$\displaystyle M$ into $\displaystyle\mathbb{P}(1,3,4,6)$ whose complement is
a hypersurface of degree 12.

Combined with Proposition 1.6 and Proposition 1.8, our main theorem provides
an answer to our Main Problem for nonsingular sections:

######  Main Theorem.

For $\displaystyle G=F_{4},E_{6},E_{7}$ or $\displaystyle E_{8}$, Theorem 1.7
induces a natural datum
$\displaystyle\alpha_{G}\in{\mathcal{M}}({\mathcal{U}}_{G})$, which by
Proposition 1.8(2) gives a morphism
$\displaystyle\xi({\mathcal{U}}_{G})(\alpha_{G}):{\mathcal{U}}_{G}\rightarrow
M$. This morphism is $\displaystyle H_{G}$-invariant, and the induced quotient
morphism $\displaystyle\Phi_{G}:M_{G}\rightarrow M$ is an isomorphism.

######  Remark 1.9.

$\displaystyle\Phi_{G}$ induces a birational map: $\displaystyle
Gr(2,W_{G})^{ss}//H_{G}\dashrightarrow Gr(2,W_{D_{4}})^{ss}//H_{D_{4}}\times
S_{3}$. We conjecture that it can be extended to an isomorphism, which would
provide a complete answer to the Main Problem (2).

To study the moduli of codimension two linear sections of $\displaystyle
S_{G}$ one step further, we ask the following question which has an
affirmative answer in the case where $\displaystyle G=F_{4}$ ([DM22,
Proposition 4]).

######  Question 1.10.

Let $\displaystyle G$ be of type $\displaystyle F_{4},E_{6},E_{7}$ or
$\displaystyle E_{8}$. Let $\displaystyle[U],[U^{\prime}]\in{\mathcal{U}}_{G}$
be two general nondegenerate planes in $\displaystyle W_{G}$, and suppose that
$\displaystyle\phi:S_{U}\rightarrow S_{U^{\prime}}$ is an isomorphism. Dose
there exist $\displaystyle h\in\overline{H}_{G}$ such that
$\displaystyle[U^{\prime}]=h.[U]$ and $\displaystyle\phi=h^{*}$?

The article is organized as follows. In Section 2 we first recall some basic
facts on subadjoint varieties. Then we will prove Proposition 1.3, Theorem
1.7(1) and Proposition 1.8(1). Moreover we shall construct a natural morphism
$\displaystyle\sigma_{G}:M\rightarrow M_{G}$ which will be proved to be a
section of $\displaystyle\Phi_{G}$ defined in the Main Theorem. In Section 3
we will prove Theroem 1.7(2) and show that it induces a natural datum
$\displaystyle\alpha_{G}\in{\mathcal{M}}({\mathcal{U}}_{G})$. In Section 4 we
first verify Proposition 1.8(2), then we finish the proof of our Main Theorem.
In the appendix we will prove Proposition 1.8(3).

######  Notations.

1.All schemes are assumed to be locally noetherian. A variety is defined as an
integral separated scheme of finite type over $\displaystyle\mathbb{C}$. Given
a line bundle $\displaystyle L$ on a variety $\displaystyle X$ and a nonzero
section $\displaystyle\sigma\in H^{0}(X,L)$, we denote the associated
effective Cartier divisor by $\displaystyle[\sigma]$.

2\. Given a scheme $\displaystyle S$, a scheme $\displaystyle X$ over
$\displaystyle S$ is said to be projective over $\displaystyle S$ if
$\displaystyle X$ is a closed subscheme of
$\displaystyle\mathbb{P}_{S}({\mathcal{E}})$ for some finite rank vector
bundle $\displaystyle{\mathcal{E}}$ on $\displaystyle S$. For a projective and
flat family $\displaystyle X$ over $\displaystyle S$, we denote $\displaystyle
Div_{X/S}$ to be the scheme representing the functor of families of relative
effective divisors on $\displaystyle X/S$.

##  2\. Preliminaries

###  2.1. Representation $\displaystyle(H,W)$

To study the projective geometry of subadjoint varieties in details, we review
two perspectives to recover the representation $\displaystyle(H,W)$.

####  2.1.1. Cubic Jordan algebras

The first one is given from cubic Jordan algebras, from which we can express
each subadjoint variety as a cubic twisted “curve” over a cubic Jordan algebra
(2.3).

######  Definition 2.1.

A $\displaystyle Jordan$ $\displaystyle algebra$ $\displaystyle J$ over
$\displaystyle\mathbb{C}$ is a vector space over $\displaystyle\mathbb{C}$
with a bilinear product $\displaystyle\bullet$ satisfying the following
axioms:

$x\bullet y=y\bullet x;\,\,x^{2}\bullet(x\bullet y)=x\bullet(x^{2}\bullet
y),\,\,\,\text{\,for $\displaystyle x,y\in J$}.$

A Jordan algebra is called simple if it has no proper ideals, and it is called
semi-simple if it is the direct sum of simple Jordan algebras. A cubic Jordan
algebra is a Jordan algebra in which every element satisfies a cubic equation.
In the following, we always assume $\displaystyle J$ to be a semi-simple cubic
Jordan algebra.

The cubic equation naturally defines an admissible cubic form $\displaystyle
N$ on $\displaystyle J$(see [Cl03, Definition 2]). We denote its linearization
to be $\displaystyle N:J\times J\times J\rightarrow\mathbb{C}$, with
$\displaystyle N(x)=N(x,x,x)$. The cubic equation also defines a trace form
$\displaystyle Tr(\,,\,)$ on $\displaystyle J$. It is nondegenerate when
$\displaystyle J$ is semi-simple and thus one can define a quadratic adjoint
map $\displaystyle\\#$ on $\displaystyle J$ satisfying $\displaystyle
Tr(x^{\\#},y)=N(x,x,y)$ for all $\displaystyle y\in J$. The following table
lists the Jordan algebras associated to Table 1 (see Theorem 2.4 below), where
we remark that the last four columns exactly correspond to simple cubic Jordan
algebras.

$\displaystyle\mathfrak{g}$ | $\displaystyle D_{4}$ | $\displaystyle F_{4}$ | $\displaystyle E_{6}$ | $\displaystyle E_{7}$ | $\displaystyle E_{8}$
---|---|---|---|---|---
$\displaystyle J$ | $\displaystyle{\mathcal{H}}_{3}(0)$ | $\displaystyle{\mathcal{H}}_{3}(\mathbb{C})$ | $\displaystyle{\mathcal{H}}_{3}(\mathbb{C}\oplus\mathbb{C})$ | $\displaystyle{\mathcal{H}}_{3}(\mathbb{H}_{{\mathbb{C}}})$ | $\displaystyle{\mathcal{H}}_{3}(\mathbb{O}_{{\mathbb{C}}})$
Table 2.

Here for a composition algebra $\displaystyle{\mathbb{A}}$,
$\displaystyle\mathcal{H}_{3}({\mathbb{A}})$ is the $\displaystyle Hermitian$
cubic Jordan algebra over $\displaystyle{\mathbb{A}}$ defined as follows:

$\mathcal{H}_{3}({\mathbb{A}})=\Bigg{\\{}\left(\begin{matrix}\xi_{1}&a&b\\\
\overline{a}&\xi_{2}&c\\\
\overline{b}&\overline{c}&\xi_{3}\end{matrix}\right),\xi_{1},\xi_{2},\xi_{3}\in{\mathbb{C}},a,b,c\in{\mathbb{A}}.\Bigg{\\}}$

Recall that there are exatly four complex composition algebras:
$\displaystyle{\mathbb{A}}={\mathbb{C}},{\mathbb{C}}\oplus{\mathbb{C}},{\mathbb{H}}_{{\mathbb{C}}},{\mathbb{O}}_{{\mathbb{C}}}$,
which are the complexifications of
$\displaystyle{\mathbb{R}},{\mathbb{C}},{\mathbb{H}},{\mathbb{O}}$
respectively.
$\displaystyle{\mathcal{H}}_{3}(0)\subseteq{\mathcal{H}}_{3}({\mathbb{A}})$ is
defined as the subalgebra of diagonal matrices. For the definition of
$\displaystyle N$ and $\displaystyle\\#$ on
$\displaystyle{\mathcal{H}}_{3}({\mathbb{A}})$, see for example [Kr07, Example
05].

Let $\displaystyle J$ be a semi-simple cubic Jordam algebra, we define the
group

$\displaystyle NP(J)=\\{g\in GL(J)|N(gA)=N(A),\forall A\in J\\}.$

To describe the action of $\displaystyle NP(J)$ on $\displaystyle J$, we
recall the following definition.

######  Definition 2.2.

Let $\displaystyle J$ be a semi-simple cubic Jordan algerbra and let
$\displaystyle N$ be the cubic norm on $\displaystyle J$. The rank of an
element $\displaystyle x\in J$ is defined by the following relations:

$\displaystyle\displaystyle\textit{rank}\,\,x=3\,\,$
$\displaystyle\displaystyle\textit{if}\,\,N(x)\not=0;$
$\displaystyle\displaystyle\textit{rank}\,\,x=2\,\,$
$\displaystyle\displaystyle\textit{if}\,\,N(x)=0\,\,\textit{and}\,\,x^{\\#}\not=0;$
$\displaystyle\displaystyle\textit{rank}\,\,x=1\,\,$
$\displaystyle\displaystyle\textit{if}\,\,x^{\\#}=0\,\,\textit{and}\,\,x\not=0;$
$\displaystyle\displaystyle\textit{rank}\,\,x=0\,\,$
$\displaystyle\displaystyle\textit{if}\,\,x=0.$

######  Proposition 2.3.

[Kr07, Proposition 12] Let $\displaystyle{\mathbb{A}}$ be a complex
composition algebra. Let $\displaystyle J={\mathcal{H}}_{3}({\mathbb{A}})$ be
the Hermitian cubic Jordan algebra over $\displaystyle{\mathbb{A}}$. Then the
rank is invariant under the action of $\displaystyle NP(J)$. And
$\displaystyle NP(J)$ acts transitively on the sets of elements of rank 0,1,2
respectively. In the case of rank $\displaystyle 3$, the group acts
transitively on the elements of a given norm $\displaystyle
k\in{\mathbb{C}}^{*}$.

Given a semi-simple cubic Jordan algebra $\displaystyle J$, the set of
$\displaystyle Zorn$ $\displaystyle matrices$ over $\displaystyle J$ is
defined as:

$\mathcal{C}_{J}=\Bigg{\\{}\left(\begin{matrix}a&b\\\
c&d\end{matrix}\right),a,d\in\mathbb{C},\,b,c\in J.\Bigg{\\}}$

The determinant of a Zorn matrix $\displaystyle A=(a,b,c,d)$ is given by:

(2.1) $det(A)=(ad-Tr(b,c))^{2}-4Tr(b^{\\#},c^{\\#})+4aN(c)+4dN(b).$

We then define $\displaystyle D$ to be the symmetric quadrilinear map on
$\displaystyle\mathcal{C}_{J}$ with $\displaystyle D(A,A,A,A)=det(A)$. One can
also define a natural skew bilinear form $\displaystyle w$ on
$\displaystyle\mathcal{C}_{J}$ defined by:

(2.2)
$w(A,A^{\prime})=ad^{\prime}-a^{\prime}d+Tr(b^{\prime},c)-Tr(b,c^{\prime}).$

This form is nondegenerate as the trace form $\displaystyle Tr(\,,)$ on
$\displaystyle J$ is nondegenerate. One then can define the cubic adjoint map
$\displaystyle\flat$ on $\displaystyle\mathcal{C}_{J}$ by the properties
$\displaystyle w(A^{\flat},B)=D(A,A,A,B)$ and
$\displaystyle(A^{\flat})^{\flat}=-det(A)^{2}A$. Let
$\displaystyle\widetilde{H}_{J}$ be the group of linear automorphism of
$\displaystyle\mathcal{C}_{J}$ preserving both det and $\displaystyle w$. Then
the dictionary reads as follows:

######  Theorem 2.4.

[Cl03, Proposition 4.4] and [Kr07, Table 1]. Up to a finite cover or a finite
subgroup, the irreducible representations $\displaystyle(H_{G},W_{G})$ are the
same as the representations
$\displaystyle(\widetilde{H}_{J},\mathcal{C}_{J})$, where $\displaystyle G$
ranges over simple groups which are not of type $\displaystyle A$ or
$\displaystyle C$ and $\displaystyle J$ ranges over semi-simple cubic Jordan
algebras.

For a detailed proof see [Cl03, Section 4]. We give some remarks here for
further discussions.

######  Remark 2.5.

Given a cubic Jordan algebra $\displaystyle J$ associated to a simple group
$\displaystyle G$ as above, denote the connected component of
$\displaystyle\widetilde{H}_{J}$ containing the identity by
$\displaystyle(\widetilde{H}_{J})^{o}$ and denote the image of
$\displaystyle(\widetilde{H}_{J})^{o}$ in $\displaystyle
PGL({\mathcal{C}}_{J})$ by $\displaystyle\overline{H}_{J}$. Then under the
identification $\displaystyle W_{G}\cong{\mathcal{C}}_{J}$, $\displaystyle
H_{G}$ is a finite cover of $\displaystyle(\widetilde{H}_{J})^{o}$ and hence a
finite cover of $\displaystyle\overline{H}_{J}$. The action of $\displaystyle
H_{G}$ on $\displaystyle Gr(2,W_{G})$ is induced by the action of
$\displaystyle\overline{H}_{J}$ on $\displaystyle Gr(2,{\mathcal{C}}_{J})$.

######  Example 2.6.

If $\displaystyle J={\mathcal{H}}_{3}(0)$, then
$\displaystyle{\mathcal{C}}_{J}=\\{A=(a,(b_{1},b_{2},b_{3}),(c_{1},c_{2},c_{3}),d)):a,b_{i},c_{j},d\in{\mathbb{C}}\\}$.
The forms $\displaystyle D$ and $\displaystyle w$ are given by:

$\displaystyle\displaystyle det(A)=$ $\displaystyle\displaystyle
a^{2}d^{2}+b_{1}^{2}c_{1}^{2}+b_{2}^{2}c_{2}^{2}+b_{3}^{2}c_{3}^{2}+4(ac_{1}c_{2}c_{3}+b_{1}b_{2}b_{3}d)$
$\displaystyle\displaystyle-2(ab_{1}c_{1}d+ab_{2}c_{2}d+ab_{3}c_{3}d+b_{1}b_{2}c_{1}c_{2}+b_{2}b_{3}c_{2}c_{3}+b_{1}b_{3}c_{1}c_{3}).$
$\displaystyle\displaystyle w(A,A^{\prime})=$ $\displaystyle\displaystyle
ad^{\prime}-(b_{1}c_{1}^{\prime}+b_{2}c_{2}^{\prime}+b_{3}c_{3}^{\prime})+(b_{1}^{\prime}c_{1}+b_{2}^{\prime}c_{2}+b_{3}^{\prime}c_{3})-a^{\prime}d.$

And we can identify $\displaystyle{\mathcal{C}}_{J}$ with
$\displaystyle{\mathbb{C}}^{2}\otimes{\mathbb{C}}^{2}\otimes{\mathbb{C}}^{2}$
as follows. Take $\displaystyle e_{1},e_{2}$ to be a basis of
$\displaystyle{\mathbb{C}}^{2}$ and write $\displaystyle e_{ijk}=e_{i}\otimes
e_{j}\otimes e_{k}$, then we identify:

$\displaystyle\displaystyle(0,(1,0,0),(0,0,0),0)$
$\displaystyle\displaystyle\leftrightarrow
e_{211},(0,(0,1,0),(0,0,0),0)\leftrightarrow
e_{121},(0,(0,0,1),(0,0,0),0)\leftrightarrow e_{112},$
$\displaystyle\displaystyle(0,(0,0,0),(1,0,0),0)$
$\displaystyle\displaystyle\leftrightarrow
e_{122},(0,(0,0,0),(0,1,0),0)\leftrightarrow
e_{212},(0,(0,0,0),(0,0,1),0)\leftrightarrow e_{221},$
$\displaystyle\displaystyle(1,(0,0,0),(0,0,0),0)$
$\displaystyle\displaystyle\leftrightarrow
e_{111},(0,(0,0,0),(0,0,0),1)\leftrightarrow e_{222}.$

Under the correspondence
$\displaystyle(Lie(H_{G}),W_{G})\cong(Lie(\overline{H}_{J}),{\mathcal{C}}_{J})$,
the subadjoint variety $\displaystyle S=S_{G}$ is realized as a
compactification of the algebra $\displaystyle J$. More precisely,
$\displaystyle S$ is the proper image of the following birational map:

(2.3) $\displaystyle\displaystyle Q_{J}:\mathbb{P}(\mathbb{C}\oplus J)$
$\displaystyle\displaystyle\dashrightarrow\mathbb{P}(\mathbb{C}\oplus J\oplus
J\oplus\mathbb{C})=\mathbb{P}{\mathcal{C}}_{J}$
$\displaystyle\displaystyle[t:\alpha]$
$\displaystyle\displaystyle\rightarrow[t^{3}:t^{2}\alpha:t\alpha^{\\#}:N(\alpha)]$

In particular we can identify $\displaystyle J$ with an open subset of
$\displaystyle S$ as follows:

(2.4) $\displaystyle\displaystyle q_{J}:J$
$\displaystyle\displaystyle\rightarrow S,$ $\displaystyle\displaystyle\alpha$
$\displaystyle\displaystyle\rightarrow[1:\alpha:\alpha^{\\#}:N(\alpha)].$

This identification induces a natural $\displaystyle NP(J)$-action on
$\displaystyle S$. We describe this action geometrically as follows. Denote
$\displaystyle x_{0}=Q_{J}([1:0])$, we naturally identify $\displaystyle
T_{x_{0}}S$ with $\displaystyle J$ through $\displaystyle q_{J}$. Denote the
stabilizer subgroup of the action of $\displaystyle\overline{H}_{J}$ on
$\displaystyle S$ at $\displaystyle x_{0}$ by $\displaystyle
Stab_{x_{0}}(\overline{H}_{J})$. It is a parabolic subgroup which is the
product of a vector group and a Levi subgroup $\displaystyle L_{J}$.
$\displaystyle L_{J}$ is the product of a one parameter subgroup
$\displaystyle\mathbb{G}_{m}$ and a semi-simple subgroup $\displaystyle
M_{J}$, where the action of $\displaystyle\mathbb{G}_{m}$ on
$\displaystyle{\mathcal{C}}_{J}$ is given as follows:

(2.5) $\displaystyle\displaystyle\mathbb{C}^{*}\times{\mathcal{C}}_{J}$
$\displaystyle\displaystyle\rightarrow{\mathcal{C}}_{J}$
$\displaystyle\displaystyle(t,(a,b,c,d))$
$\displaystyle\displaystyle\rightarrow(t^{3}a,tb,t^{-1}c,t^{-3}d).$

And the isotropic action of $\displaystyle M_{J}$ on $\displaystyle
T_{x_{0}}(S)\cong J$ preserves the cubic form on $\displaystyle J$, which
induces an isogeny of $\displaystyle M_{J}$ onto $\displaystyle NP(J)^{o}$,
namely the connected component of $\displaystyle NP(J)$ containing the
identity. In particular from Proposition 2.3 we have the following fact:

######  Corollary 2.7.

Let $\displaystyle G$ be of type $\displaystyle F_{4},E_{6},E_{7}$ or
$\displaystyle E_{8}$. Denote the subadjoint variety associated to
$\displaystyle G$ by $\displaystyle S_{G}\subseteq{\mathbb{P}}W_{G}$, and let
$\displaystyle x$ be a point on $\displaystyle S_{G}$. Then the isotropy
action of the stabilizer subgroup $\displaystyle Stab_{x_{0}}(H_{G})$ on
$\displaystyle{\mathbb{P}}T_{x}S_{G}$ has three orbits.

######  Definition 2.8.

Let $\displaystyle x$ be a point on the subadjoint variety $\displaystyle
S_{G}$. A tangent direction $\displaystyle[w]\in{\mathbb{P}}T_{x}S_{G}$ is
said to be generic if $\displaystyle[w]$ lies in the open orbit of the action
of $\displaystyle Stab_{x}(H_{G})$ on $\displaystyle{\mathbb{P}}T_{x}S_{G}$.

####  2.1.2. Triality

The second one is given from the following construction of exceptional Lie
algebras as well as their special representations in [LM02] using composition
algebras which highlights the $\displaystyle triality$ principle. This
construction enables us to define an $\displaystyle S_{3}$-action on
$\displaystyle{\mathbb{P}}W_{G}$.

######  Definition 2.9.

Let $\displaystyle{\mathbb{A}}$ be a complex composition algebra, denote the
triality group by

$T({\mathbb{A}})=\\{\theta=(\theta_{1},\theta_{2},\theta_{3})\in
SO({\mathbb{A}})^{3},\theta_{3}(xy)=\theta_{1}(x)\theta_{2}(y),\forall
x,y\in{\mathbb{A}}\\}.$

It is known that $\displaystyle T({\mathbb{A}})$ is an algebraic subgroup of
$\displaystyle SO({\mathbb{A}})^{3}$. There are three actions of
$\displaystyle T({\mathbb{A}})$ on $\displaystyle{\mathbb{A}}$ corresponding
to its three projections on $\displaystyle SO({\mathbb{A}})$, and we denote
these representations by
$\displaystyle{\mathbb{A}}_{1},{\mathbb{A}}_{2},{\mathbb{A}}_{3}$. Denote by
$\displaystyle t({\mathbb{A}})$ the Lie algebra of the triality group
$\displaystyle T({\mathbb{A}})$. Let $\displaystyle{\mathbb{A}}$ and
$\displaystyle{\mathbb{A}}^{\prime}$ be two composition algebras, and let

$\mathfrak{g}({\mathbb{A}},{\mathbb{A}}^{\prime})=\mathfrak{t}({\mathbb{A}})\times\mathfrak{t}({\mathbb{A}}^{\prime})\oplus({\mathbb{A}}_{1}\otimes{\mathbb{A}}^{\prime}_{1})\oplus({\mathbb{A}}_{2}\otimes{\mathbb{A}}^{\prime}_{2})\oplus({\mathbb{A}}_{3}\otimes{\mathbb{A}}^{\prime}_{3}).$

Then $\displaystyle\mathfrak{g}({\mathbb{A}},{\mathbb{A}}^{\prime})$ has a
natural structure of Lie algebra, which is semi-simple and whose type is given
by Freudental’s magic square([LM02, Theorem 2.1]). If
$\displaystyle{\mathbb{A}}^{\prime}={\mathbb{H}}_{{\mathbb{C}}}$, then
$\displaystyle\mathfrak{g}({\mathbb{A}},{\mathbb{H}}_{{\mathbb{C}}})=\mathfrak{sp}_{6},\mathfrak{sl}_{6},\mathfrak{so}_{12},$
or $\displaystyle\mathfrak{e}_{7}$, which corresponds to $\displaystyle
Lie(H)$ of the last four rows in Table 1. And we can identify
$\displaystyle\mathfrak{t}({\mathbb{H}}_{{\mathbb{C}}})$ with
$\displaystyle\mathfrak{sl}_{2}\times\mathfrak{sl}_{2}\times\mathfrak{sl}_{2}$
by(cf. [BS01, Lemma 1]):

$\mathfrak{sl}_{2}\times\mathfrak{sl}_{2}\times\mathfrak{sl}_{2}\rightarrow\mathfrak{t}({\mathbb{H}}_{{\mathbb{C}}}):\,\,\,\,(x,y,z)\rightarrow(l_{y}-r_{z},l_{z}-r_{x},l_{y}-r_{x}),$

where we identify
$\displaystyle\mathfrak{sl}_{2}=Im({\mathbb{H}}_{{\mathbb{C}}})$ and
$\displaystyle l_{x},r_{x}$ denote the operators of left and right
multiplication by $\displaystyle x$, respectively. We take the Cartan
decomposition of
$\displaystyle{\mathfrak{s}}{\mathfrak{l}}_{2}=Im({\mathbb{H}}_{{\mathbb{C}}})$
as $\displaystyle Im({\mathbb{H}}_{{\mathbb{C}}})=\langle
h\rangle\oplus\langle x\rangle\oplus\langle y\rangle$, where

$\displaystyle h=\begin{pmatrix}1&0\\\
0&-1\end{pmatrix},x=\begin{pmatrix}0&1\\\
0&0\end{pmatrix},y=\begin{pmatrix}0&0\\\ 1&0\end{pmatrix}.$

We denote by $\displaystyle\alpha\in\langle h\rangle^{\vee}$ the simple root
satisfying $\displaystyle\alpha(h)=2$. And let
$\displaystyle\alpha_{1},\alpha_{2},\alpha_{3}$ be the respective simple roots
of $\displaystyle{\mathfrak{t}}({\mathbb{H}}_{{\mathbb{C}}})$ corresponding to
each factor. Denote by $\displaystyle w_{1},w_{2},w_{3}$ the corresponding
fundamental dominant wights of
$\displaystyle{\mathfrak{t}}({\mathbb{H}}_{{\mathbb{C}}})$. Let $\displaystyle
U_{1},U_{2},U_{3}$ be the natural 2-dimensional representations of our three
copies of $\displaystyle\mathfrak{sl}_{2}$, then

$\displaystyle({\mathbb{H}}_{{\mathbb{C}}})_{1}=U_{2}\otimes
U_{3},({\mathbb{H}}_{{\mathbb{C}}})_{2}=U_{3}\otimes
U_{1},({\mathbb{H}}_{{\mathbb{C}}})_{3}=U_{1}\otimes U_{2}.$

The representation $\displaystyle(H,W)$ can be recovered as follows:

######  Theorem 2.10.

[LM02, Theorem 4.1] Let $\displaystyle
V_{{\mathbb{A}}}={\mathbb{A}}_{1}\otimes U_{1}\oplus{\mathbb{A}}_{2}\otimes
U_{2}\oplus{\mathbb{A}}_{3}\otimes U_{3}\oplus U_{1}\otimes U_{2}\otimes
U_{3}$. Then:

(i) $\displaystyle V_{{\mathbb{A}}}$ has a natural structure of
$\displaystyle\mathfrak{g}({\mathbb{A}},{\mathbb{H}}_{{\mathbb{C}}})$-module,
which is simple and isomorphic to
$\displaystyle(Lie(\widetilde{H}_{\mathcal{H}_{3}({\mathbb{A}})}),{\mathcal{C}}_{\mathcal{H}_{3}({{\mathbb{A}}})})$.

(ii) The action of the subalgebra
$\displaystyle\mathfrak{t}({\mathbb{H}}_{{\mathbb{C}}})\cong\mathfrak{sl}_{2}\times\mathfrak{sl}_{2}\times\mathfrak{sl}_{2}$
on $\displaystyle V_{{\mathbb{A}}}$ is given by:

$(x_{1},x_{2},x_{3}).(a_{i}\otimes u_{i})=a_{i}\otimes
x_{i}.u_{i},\,\,\text{for}\,\,i=1,2,3\,\,\text{and}\,\,(x_{1},x_{2},x_{3}).(u_{1}\otimes
u_{2}\otimes u_{3})=x_{1}.u_{1}\otimes x_{2}.u_{2}\otimes x_{3}.u_{3},$

In particular the sub-representation
$\displaystyle(\mathfrak{t}({\mathbb{H}}_{{\mathbb{C}}}),U_{1}\otimes
U_{2}\otimes U_{3})$ is isomorphic to
$\displaystyle(Lie(H_{D_{4}}),W_{D_{4}})$.

(iii) A highest weight vector of the module
$\displaystyle(\mathfrak{t}({\mathbb{H}}),U_{1}\otimes U_{2}\otimes U_{3})$ is
a highest weight vector of the module
$\displaystyle(\mathfrak{g}({\mathbb{A}},{\mathbb{H}}_{{\mathbb{C}}}),V_{{\mathbb{A}}})$.

Then Proposition 1.3 follows from the following corollary.

######  Corollary 2.11.

Under the identification in Theorem 2.10, there is a subgroup $\displaystyle
S_{3}\subset\overline{H}_{{\mathcal{H}}_{3}({\mathbb{A}})}$ which acts on
$\displaystyle\mathbb{P}(U_{1}\otimes U_{2}\otimes U_{3})$ by permutations of
the factors.

Here we first give the construction of the subgroup and verify some
properties. We will conclude its proof after introducing some projective
properties of subadjoint varieties.

For any composition algebra $\displaystyle{\mathbb{A}}$, there is an
$\displaystyle S_{3}$-action on $\displaystyle\mathfrak{t}({\mathbb{A}})$
generated by the following two automorphisms(cf. [SV00, Proposition 3.6.4,
Lemma 3.5.9]):

$\sigma(t_{1},t_{2},t_{3})=(t_{2},\widehat{t_{3}},\widehat{t_{1}}),\,\,\,\tau(t_{1},t_{2},t_{3})=(t_{3},\widehat{t_{2}},t_{1}),$

where $\displaystyle\widehat{t}(x):=\overline{t(\overline{x})}$ for any
$\displaystyle x\in{\mathbb{A}}$. Now given two composition algebras
$\displaystyle{\mathbb{A}},{\mathbb{A}}^{\prime}$, we can extend the
$\displaystyle S_{3}$-action on
$\displaystyle\mathfrak{t}({\mathbb{A}})\times\mathfrak{t}({\mathbb{A}}^{\prime})$
to an $\displaystyle S_{3}$-action on
$\displaystyle\mathfrak{g}({\mathbb{A}},{\mathbb{A}}^{\prime})$ generated by:

$\displaystyle\displaystyle\sigma:\,\,$
$\displaystyle\displaystyle{\mathbb{A}}_{1}\otimes{\mathbb{A}}^{\prime}_{1}\rightarrow{\mathbb{A}}_{3}\otimes{\mathbb{A}}^{\prime}_{3},\,\,\,{\mathbb{A}}_{3}\otimes{\mathbb{A}}^{\prime}_{3}\rightarrow{\mathbb{A}}_{2}\otimes{\mathbb{A}}^{\prime}_{2},\,\,\,{\mathbb{A}}_{2}\otimes{\mathbb{A}}^{\prime}_{2}\rightarrow{\mathbb{A}}_{1}\otimes{\mathbb{A}}^{\prime}_{1}.$
$\displaystyle\displaystyle a_{1}\otimes
a^{\prime}_{1}\rightarrow\overline{a_{1}}\otimes\overline{a^{\prime}_{1}},\,\,\,a_{3}\otimes
a^{\prime}_{3}\rightarrow\overline{a_{3}}\otimes\overline{a^{\prime}_{3}},\,\,\,a_{2}\otimes
a^{\prime}_{2}\rightarrow a_{2}\otimes a^{\prime}_{2}.$

and

$\displaystyle\displaystyle\tau:\,\,$
$\displaystyle\displaystyle{\mathbb{A}}_{1}\otimes{\mathbb{A}}^{\prime}_{1}\rightarrow{\mathbb{A}}_{3}\otimes{\mathbb{A}}^{\prime}_{3},\,\,\,{\mathbb{A}}_{3}\otimes{\mathbb{A}}^{\prime}_{3}\rightarrow{\mathbb{A}}_{1}\otimes{\mathbb{A}}^{\prime}_{1},\,\,\,{\mathbb{A}}_{2}\otimes{\mathbb{A}}^{\prime}_{2}\rightarrow{\mathbb{A}}_{2}\otimes{\mathbb{A}}^{\prime}_{2}.$
$\displaystyle\displaystyle a_{1}\otimes a^{\prime}_{1}\rightarrow
a_{1}\otimes a^{\prime}_{1},\,\,\,a_{3}\otimes a^{\prime}_{3}\rightarrow
a_{3}\otimes a^{\prime}_{3},\,\,\,a_{2}\otimes
a^{\prime}_{2}\rightarrow-\overline{a_{2}}\otimes\overline{a^{\prime}_{2}}.$

The verification that this defines an action on the Lie algebra is just a
computation. Then one easily checks that under this identification the induced
$\displaystyle S_{3}$-action on
$\displaystyle\mathfrak{sl}_{2}\times\mathfrak{sl}_{2}\times\mathfrak{sl}_{2}$
is generated by:

(2.6) $\sigma(x,y,z)=(y,z,x)\,\,\text{and}\,\,\tau(x,y,z)=(z,y,x).$

Note that for each composition algebra $\displaystyle{\mathbb{A}}$,
$\displaystyle V_{{\mathbb{A}}}$ is the unique irreducible
$\displaystyle\mathfrak{g}({\mathbb{A}},{\mathbb{H}})$-module of dimension
$\displaystyle dim(V_{{\mathbb{A}}})$ up to isomorphisms. Thus the
$\displaystyle S_{3}$-action on
$\displaystyle\mathfrak{g}({\mathbb{A}},{\mathbb{H}})$ induces a homomorphism
$\displaystyle f:S_{3}\rightarrow PGL(V_{{\mathbb{A}}})$ such that for any
$\displaystyle g\in S_{3}$ and any representative $\displaystyle F_{g}$ of
$\displaystyle f(g)$ in $\displaystyle GL(V_{{\mathbb{A}}})$ we have:

(2.7) $F_{g}(X.v)=(g.X).(F_{g}(v)),\,\,\text{for
any}\,\,X\in\mathfrak{g}({\mathbb{A}},{\mathbb{H}})\,\,\text{and}\,\,v\in
V_{{\mathbb{A}}}.$

By Theorem 2.10 the highest weight of the action of $\displaystyle
t({\mathbb{H}}_{{\mathbb{C}}})$ on $\displaystyle V_{{\mathbb{A}}}$ is
$\displaystyle w_{1}+w_{2}+w_{3}$ which corresponds to its action on
$\displaystyle U_{1}\otimes U_{2}\otimes U_{3}$. Thus from (2.6)(2.7) we
deduce that the action of $\displaystyle S_{3}$ on
$\displaystyle{\mathbb{P}}V_{{\mathbb{A}}}$ stabilizes the subspace generated
by a maximal weight vector in $\displaystyle U_{1}\otimes U_{2}\otimes U_{3}$,
and hence it stabilizes $\displaystyle{\mathbb{P}}(U_{1}\otimes U_{2}\otimes
U_{3})$. Furthermore from (2.6)(2.7) one can also easily see that the action
of $\displaystyle S_{3}$ on $\displaystyle{\mathbb{P}}(U_{1}\otimes
U_{2}\otimes U_{3})$ is given by permutation of factors.

###  2.2. Projective geometry of subadjoint varieties

In this subsection, we fix a simple group $\displaystyle G$ of type
$\displaystyle D_{4},F_{4},E_{6},E_{7}$ or $\displaystyle E_{8}$. We consider
the subadjoint variety $\displaystyle S\subseteq\mathbb{P}W$ associated to
$\displaystyle G$. We identify $\displaystyle W$ with
$\displaystyle{\mathcal{C}}_{J}$ where $\displaystyle J$ is the cubic Jordan
algebra associated to $\displaystyle G$. We denote $\displaystyle Sec(S)$ to
be the secant variety of $\displaystyle S$ in $\displaystyle\mathbb{P}W$, and
we denote $\displaystyle TS$ to be the tangential variety of $\displaystyle S$
in $\displaystyle\mathbb{P}W$. We also identify $\displaystyle\mathbb{P}W$
with its dual via the skew form $\displaystyle w$. We summarize some
projective properties of subadjoint varieties as follows.

######  Theorem 2.12.

[Cl03, Section 9] (1). $\displaystyle Sec(S)=\mathbb{P}W$, $\displaystyle
TS=\\{det=0\\}$, $\displaystyle Sing(TS)=\\{\flat=0\\}$.

(2). For any point $\displaystyle x\in S$, the embedded tangent space
$\displaystyle\widehat{T_{x}S}$ is a maximal isotropic subspace in
$\displaystyle W$ with respect to the skew form $\displaystyle w$. For any
point $\displaystyle y\in\mathbb{P}W$, denote by $\displaystyle H_{y}$ the
hyperplane defined by $\displaystyle y$. Then $\displaystyle H_{y}$ is tangent
at $\displaystyle x$ if and only if $\displaystyle
y\in\mathbb{P}\widehat{T_{x}S}$. In other words $\displaystyle TS$ is
identified to the dual variety $\displaystyle S^{*}$.

(3). We have the following special properties:

(3.1) Each point in $\displaystyle\mathbb{P}W\backslash TS$ lies on a unique
secant line.

(3.2) For a point $\displaystyle\alpha=[A]$ in $\displaystyle TS\backslash
Sing(TS)$, the point $\displaystyle x_{\alpha}=[A^{\flat}]$ lies in
$\displaystyle S$, and $\displaystyle\alpha$ lies on a unique tangent line
given by the join of $\displaystyle\alpha$ and $\displaystyle x_{\alpha}$.

(3.3) Each point in $\displaystyle Sing(TS)$ belongs to an infinite number of
tangent lines.

(4). The action of $\displaystyle H$ on $\displaystyle\mathbb{P}W$ has exactly
four orbits, namely $\displaystyle\mathbb{P}W\backslash TS$, $\displaystyle
TS\backslash Sing(TS)$, $\displaystyle Sing(TS)\backslash S$, and
$\displaystyle S$.

Next we review the geometry of lines on subadjoint varieties. For any two
points $\displaystyle p,q$ on $\displaystyle S$, denote by $\displaystyle
l(p,q)$ the minimal number of lines on $\displaystyle S$ connecting
$\displaystyle p$ and $\displaystyle q$.

######  Theorem 2.13.

For any $\displaystyle p,q\in S$, we have $\displaystyle l(p,q)\leqslant 3$.
Moreover:

(i) Two pairs of points $\displaystyle(p,q)$ and
$\displaystyle(p^{\prime},q^{\prime})$ on $\displaystyle S$ are conjugate
under the $\displaystyle H$-action if and only if $\displaystyle
l(p,q)=l(p^{\prime},q^{\prime})$.

(ii) For a pair of points $\displaystyle p=[A],q=[B]$ on $\displaystyle S$,
then $\displaystyle l(p,q)\leqslant 2$ if and only if $\displaystyle
p,q\in\mathbb{P}\widehat{T_{z}S}$ for some $\displaystyle z\in S$, and if and
only if $\displaystyle w(A,B)=0$. For a point $\displaystyle z$ on
$\displaystyle S$, $\displaystyle\mathbb{P}\widehat{T_{z}S}\cap S$ equals to
the union of the lines on $\displaystyle S$ through $\displaystyle z$.

(iii) For any $\displaystyle x\in\mathbb{P}W$, $\displaystyle
x\in\mathbb{P}W\backslash TS$ if and only if $\displaystyle x$ lies on a
secant line $\displaystyle l_{pq}$ of $\displaystyle S$, such that
$\displaystyle l(p,q)=3$; and $\displaystyle x\in TS\backslash Sing(TS)$ if
and only if it lies on a tangent line $\displaystyle l_{p}$ of $\displaystyle
S$ at some point $\displaystyle p\in S$, such that $\displaystyle[T_{p}l_{p}]$
is generic.

As a first corollary we can finish the proof of Corollary 2.11. We first
verify the following:

######  Lemma 2.14.

For the natural embedding
$\displaystyle(Lie(H_{D_{4}}),W_{D_{4}})\subset(Lie(H_{G}),W_{G})$ given in
Theorem 2.10, the determinant form $\displaystyle D_{W_{G}}$ and the skew form
$\displaystyle w_{W_{G}}$ on $\displaystyle W_{G}$ restrict to the associated
forms $\displaystyle D_{W_{4}}$ and $\displaystyle w_{W_{D_{4}}}$ on
$\displaystyle W_{D_{4}}$.

###### Proof.

The fact that $\displaystyle w_{W_{G}}|_{W_{D_{4}}}=w_{W_{D_{4}}}$ follows
from the remark following [LM02, Theorem 4.1]. Denote by $\displaystyle
H^{sc}_{G}$ the simply-connected cover of $\displaystyle H_{G}$. Then we have
a natural embedding $\displaystyle SL_{2}\times SL_{2}\times SL_{2}\subseteq
H^{sc}_{G}$. And by Theorem 2.10(iii) there is an equivariant embedding of
subadjoint varieties
$\displaystyle(S_{D_{4}},{\mathbb{P}}W_{D_{4}})\subseteq(S_{G},{\mathbb{P}}W_{G})$.
Thus from Theorem 2.12(1) and (4), to show $\displaystyle
D_{W_{G}}|_{W_{D_{4}}}=D_{W_{D_{4}}}$, it suffices to show that $\displaystyle
TS_{G}\cap{\mathbb{P}}W_{D_{4}}=TS_{D_{4}}$ or equivalently $\displaystyle
TS_{G}$ does not contain $\displaystyle{\mathbb{P}}W_{D_{4}}$. In fact take
any two points $\displaystyle p=[A],q=[B]$ on $\displaystyle S_{D_{4}}$ with
$\displaystyle l_{S_{D_{4}}}(p,q)=3$. Then from Theorem 2.13(ii) we have
$\displaystyle w_{W_{D_{4}}}(A,B)=w_{W_{G}}(A,B)\not=0$. Thus $\displaystyle
l_{S_{G}}(p,q)=3$ and from Theorem 2.13(iii) we have
$\displaystyle[A+B]\in{\mathbb{P}}W_{D_{4}}\backslash TS_{G}$. ∎

###### Proof of Corollary 2.11.

It suffices to verify that the image of our morphism $\displaystyle
f:S_{3}\rightarrow PGL(V_{{\mathbb{A}}})$ lies in
$\displaystyle\overline{H}_{{\mathcal{H}}_{3}({\mathbb{A}})}$. Recall that for
any $\displaystyle g\in S_{3}$, we have shown that $\displaystyle f(g)$
stabilizes the weight subspace $\displaystyle V_{w_{1}+w_{2}+w_{3}}$ of weight
$\displaystyle w_{1}+w_{2}+w_{3}$. Thus we can choose a representative
$\displaystyle F_{g}\in GL(V_{{\mathbb{A}}})$ of $\displaystyle f(g)$ fixing
$\displaystyle V_{w_{1}+w_{2}+w_{3}}$. Now we verify our claim in three steps.

We first prove that $\displaystyle F_{g}$ fixes the determinant form
$\displaystyle D_{W_{G}}$. Note that the action of $\displaystyle S_{3}$ on
$\displaystyle\mathfrak{g}({\mathbb{A}},{\mathbb{H}})$ naturally induces an
$\displaystyle S_{3}$-action on $\displaystyle H^{sc}_{G}$. And from (2.7) we
have:

(2.8) $F_{g}(h.v)=(g.h).(F_{g}(v)),\,\,\text{for any}\,\,h\in
H^{sc}_{G}\,\,\text{and}\,\,v\in V_{{\mathbb{A}}}.$

In particular this implies that $\displaystyle F_{g}$ stabilizes the
subadjoint variety $\displaystyle S_{G}=H^{sc}_{G}.[V_{w_{1}+w_{2}+w_{3}}]$.
Thus it stabilizes $\displaystyle TS_{G}$ and hence by Theorem 2.12 it acts on
$\displaystyle D$ by a scalar. On the other hand note that $\displaystyle
F_{g}$ acts on $\displaystyle U_{1}\otimes U_{2}\otimes U_{3}$ by a certain
permutation of factors. From Example 2.6 one easily checks that it preserves
the determinant form on $\displaystyle W_{D_{4}}$. Then by the above lemma we
conclude that $\displaystyle F_{g}$ preserves the form $\displaystyle
D_{W_{G}}$.

Next we check that it preserves the skew form $\displaystyle w_{W_{G}}$. As
$\displaystyle w_{W_{G}}$ is a $\displaystyle H_{G}^{sc}$-invariant form,
$\displaystyle F_{g}(w_{W_{G}})$ is also an $\displaystyle
H_{G}^{sc}$-invariant form by (2.8). Then from Theorem 2.13(i) and by the
above lemma it suffices to prove that $\displaystyle F_{g}$ preserves the skew
form on $\displaystyle W_{D_{4}}$, which again can be easily checked from
Example 2.6.

Now if $\displaystyle G$ is not of type $\displaystyle E_{6}$, then
$\displaystyle\widetilde{H}_{J}$ is already connected by [Kr07, Proposition
16.(ii)] and thus $\displaystyle
f(g)\in\overline{H}_{{\mathcal{H}}_{3}({\mathbb{A}})}$. If $\displaystyle G$
is of type $\displaystyle E_{6}$, then the corresponding representation is
$\displaystyle(H^{sc}_{G},W_{G})\cong(SL_{6},\wedge^{3}{\mathbb{C}}^{6})$ and
one can easily check that the natural embedding is given through identifying
$\displaystyle{\mathbb{C}}^{6}=U_{1}\oplus U_{2}\oplus U_{3}$. Then as
$\displaystyle F_{g}$ acts on $\displaystyle U_{1}\otimes U_{2}\otimes U_{3}$
by a certain permutation, one deduces that the action of $\displaystyle F_{g}$
on $\displaystyle\wedge^{3}{\mathbb{C}}^{6}$ comes from an element in
$\displaystyle SL_{6}$ and hence $\displaystyle
f(g)\in\overline{H}_{{\mathcal{H}}_{3}({\mathbb{A}})}$. ∎

We can now prove Theorem 1.7(1).

###### Proof of Theorem 1.7(1).

The $\displaystyle H$-invariant section $\displaystyle\sigma_{12}$ comes from
the determinant form $\displaystyle D$. More precisely we consider the
following $\displaystyle SL_{2}\times H$-invariant function:

(2.9) $W\times W\rightarrow\mathbb{C}:\,\,\,(\alpha,\beta)\rightarrow
disc_{a,b}(det(a\alpha+b\beta)),$

where for a fixed pair $\displaystyle(\alpha,\beta)\in W\times W$,
$\displaystyle disc_{a,b}(det(a\alpha+b\beta))$ is the discriminant of the
quartic binary form $\displaystyle det(a\alpha+b\beta)$ over $\displaystyle
a,b$. Note that $\displaystyle det$ is $\displaystyle H$-invariant and the
discriminant of a binary quartic form is known as an $\displaystyle
SL_{2}$-invariant of degree 12. Thus the above function is an $\displaystyle
SL_{2}\times H$-invariant of degree 24. It defines a $\displaystyle
H$-invariant section $\displaystyle\sigma_{12}\in
H^{0}(Gr(2,W),{\mathcal{O}}_{Gr(2,W)}(12))$ such that for any plane
$\displaystyle[U]\in Gr(2,W)$, $\displaystyle\sigma_{12}([U])\not=0$ if and
only if $\displaystyle[U]$ is nondegenerate.

Given a plane $\displaystyle[U]\in Gr(2,W)$, for any point $\displaystyle x\in
TS\cap{\mathbb{P}}U$, we denote $\displaystyle S_{x}=\\{p\in
S:x\in{\mathbb{P}}\widehat{T_{p}S}\\}$. Then by Theorem 2.12(2):

$\displaystyle\displaystyle Sing(S_{U})=\\{p\in S_{U}:\,\,\text{there exists
some}\,\,x\in{\mathbb{P}}U\,\,\text{such that}\,\,H_{x}\,\,\text{is tangent
at}\,\,p\\}=(\bigcup_{x\in{\mathbb{P}}U\cap TS}S_{x})\cap S_{U}.$

If $\displaystyle[U]$ is nondegenerate, then for any point
$\displaystyle\alpha=[A]\in{\mathbb{P}}U$, we claim that $\displaystyle
w(A^{\flat},-)$ is not identically zero on $\displaystyle U$. Assume
otherwise, then for any point $\displaystyle\beta=[B]\in{\mathbb{P}}U$, we
have $\displaystyle w(A^{\flat},B)=D(A,A,A,B)=0$. But then as $\displaystyle
TS=\\{det=0\\}=\\{[B]:D(B,B,B,B)=0\\}$, we conclude that $\displaystyle TS$
intersects $\displaystyle{\mathbb{P}}U$ at $\displaystyle\alpha=[A]$ of order
at least two, which is a contradiction. Now assume $\displaystyle S_{U}$ is
singular, then there exists a point $\displaystyle\alpha=[A]\in
TS\cap{\mathbb{P}}U$ such that $\displaystyle S_{\alpha}\cap
S_{U}\not=\varnothing$. By the above discussion we have $\displaystyle
A^{b}\not=0$, and hence by Theorem 2.12(3.2) $\displaystyle S_{\alpha}\cap
S_{U}=[A^{\flat}]$. But then from $\displaystyle[A^{\flat}]\in S_{U}$, we
conclude that $\displaystyle w(A^{\flat},-)$ is identically zero on
$\displaystyle U$, which is a contradiction. Conversely assume $\displaystyle
S_{U}$ is nonsingular, then by the above discussion there will be no roots of
$\displaystyle det|_{{\mathbb{P}}U}$ lying in $\displaystyle TS\backslash
Sing(TS)$ and of order at least two. Thus to show $\displaystyle[U]$ is
nondegenerate, it suffices to show that $\displaystyle{\mathbb{P}}U\cap
Sing(TS)=\varnothing$. Assume otherwise, then by Theorem 2.12(3.3) we have
$\displaystyle dim(S_{x})>0$ for any $\displaystyle x\in
Sing(TS)\cap{\mathbb{P}}U$. Then as $\displaystyle S_{x}\cap S_{U}=S_{x}\cap
H_{y}$ for any $\displaystyle y\in{\mathbb{P}}U$ distinct from $\displaystyle
x$, we conclude that it is non-empty and this yields a contradiction. ∎

The main application of Corollary 2.11 is the following:

######  Corollary 2.15.

The embedding
$\displaystyle(Lie(H_{D_{4}}),W_{D_{4}})\subset(Lie(H_{G}),W_{G})$ given in
Theorem 2.10 induces a morphism between GIT quotients:

$\displaystyle\sigma_{G}:M={\mathcal{U}}_{D_{4}}//(H_{D_{4}}\times
S_{3})\rightarrow M_{G}={\mathcal{U}}_{G}//H_{G}$

.

###### Proof.

By Lemma 2.14, the embedding induces an inclusion
$\displaystyle{\mathcal{U}}_{D_{4}}\subseteq{\mathcal{U}}_{G}$. And from
Corollary 2.11 there exists a group homomorphism: $\displaystyle S_{3}\times
H^{sc}_{D_{4}}\rightarrow\overline{H}_{{\mathcal{H}}_{3}({\mathbb{A}})}$ such
that the inclusion is equivariant under the associated group actions. Then
following Remark 2.5 we conclude that it induces a morphism between GIT
quotients. ∎

###  2.3. Bhargava-Ho’s correspondence

In this subsection we first review the Bhargava-Ho’s correspondence in Theorem
1.5. Then we apply it to calculate the GIT stable locus of the action of
$\displaystyle H_{D_{4}}=SL_{2}\times SL_{2}\times SL_{2}$ on $\displaystyle
Gr(2,W_{D_{4}})=Gr(2,{\mathbb{C}}^{2}\otimes{\mathbb{C}}^{2}\otimes{\mathbb{C}}^{2})$.
We also fix a simple group $\displaystyle G$ of type $\displaystyle
D_{4},F_{4},E_{6},E_{7}$ or $\displaystyle E_{8}$, and denote by
$\displaystyle J,W,H$ the associated objects to $\displaystyle G$.

######  Definition 2.16.

Given a nondegenerate plane $\displaystyle U\subseteq W$, there is a natural
diagram of morphisms by Theorem 2.12 (3.1)(3.2):

(2.10)
$\displaystyle{C_{U}}$$\displaystyle{\mathcal{T}}$$\displaystyle{S}$$\displaystyle{\mathbb{P}U}$$\displaystyle{Hilb_{2}(S)}$$\displaystyle\scriptstyle{\pi^{*}\psi_{U}}$$\displaystyle\scriptstyle{\pi_{U}}$$\displaystyle\scriptstyle{\pi}$$\displaystyle\scriptstyle{v}$$\displaystyle\scriptstyle{\psi_{U}}$

where $\displaystyle\pi$ is the universal family of subschemes of length two
on $\displaystyle S$, and $\displaystyle v$ is the evaluation map to
$\displaystyle S$. And the morphism $\displaystyle\psi_{U}$ is defined as
follows.

$\displaystyle\bullet$ For each point $\displaystyle x\in\mathbb{P}U\backslash
TS$, $\displaystyle\psi_{U}(x)\dot{=}x_{p}+x_{q}$ where $\displaystyle
l_{x_{p}x_{q}}$ is the unique secant line through $\displaystyle x$.

$\displaystyle\bullet$ For each point $\displaystyle y\in\mathbb{P}U\cap TS$,
denote by $\displaystyle y_{r}$ the point on $\displaystyle S$ such that
$\displaystyle l_{y_{r},y}$ is the unique tangent line on $\displaystyle S$
through $\displaystyle y$. Then $\displaystyle\psi_{U}(y)$ is defined by the
tangent direction $\displaystyle[T_{y_{r}}(l_{y_{r},y})]$.

Finally $\displaystyle\pi_{U}\dot{=}\psi_{U}^{*}\pi$. Then the datum defined
in Theorem 1.5 is given by the curve $\displaystyle C_{U}$, the morphism
$\displaystyle\kappa_{U}=v\circ\pi^{*}\psi_{U}$, and the line bundle
$\displaystyle L_{U}=\pi_{U}^{*}{\mathcal{O}}_{{\mathbb{P}}U}(1)$.

If $\displaystyle G$ is of type $\displaystyle D_{4}$ then $\displaystyle
H=SL_{2}\times SL_{2}\times SL_{2}$, and $\displaystyle
S={\mathbb{P}}^{1}\times{\mathbb{P}}^{1}\times{\mathbb{P}}^{1}$. The morphism
$\displaystyle\kappa_{U}$ from $\displaystyle C_{U}$ to $\displaystyle S$ is
given by three degree two maps to $\displaystyle{\mathbb{P}}^{1}$, and hence
given by three degree two line bundles on $\displaystyle C_{U}$. Now we can
verify Corollary 1.8(1).

###### Proof of Proposition 1.8(1).

We first show that the action of $\displaystyle H_{D_{4}}$ on
$\displaystyle{\mathcal{U}}_{D_{4}}$ is GIT stable. By Theorem 1.7(1) it is
already semi-stable, thus it suffices to show that the dimension of orbits in
$\displaystyle{\mathcal{U}}_{D_{4}}$ remain invariant. In fact we will show
that $\displaystyle dim(Stab_{H_{D_{4}}}([U]))=0$ for any
$\displaystyle[U]\in{\mathcal{U}}_{D_{4}}$. Take any plane $\displaystyle U$,
denote by $\displaystyle(C_{U},L_{U},L_{1},L_{2},L_{3})$ the associated datum.
Then by Proposition 1.6 we obtain an injective group homomorphism:

$Stab_{H_{D_{4}}}([U])\rightarrow Aut(C_{U},L_{U},L_{1},L_{2},L_{3}).$

As $\displaystyle Aut(C_{U},L_{U})$ is finite, it then suffices to check that
the subgroup whose elements fix $\displaystyle(C_{U},L_{U})$ is finite. From
the construction above this subgroup equals $\displaystyle\\{h\in
H_{D_{4}}=SL_{2}\times SL_{2}\times SL_{2}:h.\kappa_{U}=\kappa_{U}\\}=\\{\pm
1\\}\times\\{\pm 1\\}\times\\{\pm 1\\}$, concluding the proof of our claim.

Conversely we show that any plane $\displaystyle[U]$ lying ouside
$\displaystyle{\mathcal{U}}_{D_{4}}$ is unstable. From the proof of Theorem
1.7(1), there exists a point $\displaystyle x=[A]\in{\mathbb{P}}U\cap
TS_{S_{4}}$ such that $\displaystyle S_{x}\cap S_{U}\not=\varnothing$. Whence
there is a point $\displaystyle z=[\alpha]\in S$, such that $\displaystyle
w(\alpha,-)$ is identically zero on $\displaystyle U$ and $\displaystyle
x\in{\mathbb{P}}\widehat{T_{z}S}$. Now under the birational map (2.3) we can
assume $\displaystyle z=Q_{J}([1:0])$ and thus $\displaystyle
A=(a,b,0,0)\in{\mathcal{C}}_{J}$. Take any point $\displaystyle
y=[B]\in{\mathbb{P}}U$ different from $\displaystyle x$, we can write
$\displaystyle B=(a^{\prime},b^{\prime},c^{\prime},0)\in{\mathcal{C}}_{J}$ by
formula (2.2). Then apply the $\displaystyle{\mathbb{G}}_{m}$-action defined
in (2.5), one checks that $\displaystyle\lim_{t\rightarrow
0}t[U]=\lim_{t\rightarrow 0}t[A\wedge B]$ exists hence $\displaystyle[U]$ is
GIT unstable by the Hilbert-Mumford criterion [MFK94, Theorem 2.1]. ∎

##  3\. Three line bundles of degree two

###  3.1. From $\displaystyle(C,L,\kappa)$ to three line bundles of degree two
on $\displaystyle C$

In this subsection we are going to prove Theorem 1.7.(2). Let $\displaystyle
C$ be a smooth projective curve of genus one. To obtain a multiset of line
bundles from a given morphism from $\displaystyle C$ to $\displaystyle S$, we
first identify a set of degree two line bundles on $\displaystyle C$ by the
effective divisors it defines on the smooth surface $\displaystyle
Hilb_{2}(C)$ as follows. Denote by $\displaystyle\Sigma:C\times C\rightarrow
Hilb_{2}(C)$ the quotient map of the $\displaystyle S_{2}$-action on
$\displaystyle C\times C$. Let $\displaystyle Pic(C)$ be the Picard scheme of
$\displaystyle C$ and denote by $\displaystyle Jac^{2}(C)$ the component
parametrizing line bundles of degree two on $\displaystyle C$. As
$\displaystyle C$ is of genus one, $\displaystyle Jac^{2}(C)$ is also a smooth
projective curve of genus one. Denote by $\displaystyle{\mathcal{L}}^{[2]}$
the universal line bundle of degree two on the family
$\displaystyle\pi_{2}:Jac^{2}(C)\times C\rightarrow Jac^{2}(C)$. Then
$\displaystyle{\mathcal{P}}^{[2]}:={\mathbb{P}}((\pi_{2})_{*}({\mathcal{L}}^{[2]}))$
parametrizes the effective divisors of degree two on $\displaystyle C$, i.e.,
$\displaystyle{\mathcal{P}}^{[2]}\cong Hilb_{2}(C)$. The
$\displaystyle{\mathbb{P}}^{1}$-bundle
$\displaystyle{\mathcal{P}}^{[2]}/Jac^{2}(C)$ induces the following natural
morphism:

$\displaystyle\displaystyle\rho:Jac^{2}(C)$
$\displaystyle\displaystyle\rightarrow Div_{Hilb_{2}(C)},$
$\displaystyle\displaystyle L$ $\displaystyle\displaystyle\rightarrow|L|,$

where for $\displaystyle L\in Jac^{2}(C)$,
$\displaystyle|L|\cong{\mathbb{P}}^{1}$ is its complete linear system, and we
identify an effective divisor in $\displaystyle|L|$ with a point in
$\displaystyle Hilb_{2}(C)$. Recall that for a variety $\displaystyle X$, the
sum of Cartier effective divisors induces a natural addtion map on
$\displaystyle Div_{X}$. Thus for any $\displaystyle m\geqslant 1$ we can
define a morphism $\displaystyle\rho_{m}$ by:

$\displaystyle\displaystyle\rho_{m}:Hilb_{m}(Jac^{2}(C))$
$\displaystyle\displaystyle\rightarrow Div_{Hilb_{2}(C)},$
$\displaystyle\displaystyle\sum_{i=1}^{m}L_{i}$
$\displaystyle\displaystyle\rightarrow\sum_{i=1}^{m}|L_{i}|,$

where we identify $\displaystyle Hilb_{m}(Jac^{2}(C))$ with the $\displaystyle
m$-th symmetric product of $\displaystyle Jac^{2}(C)$ via the Hilbert-Chow
morphism.

######  Lemma 3.1.

For any $\displaystyle m\geqslant 1$, $\displaystyle\rho_{m}$ is a closed
embedding.

###### Proof.

First by definition $\displaystyle\rho_{m}$ is injective. It is proper as
$\displaystyle Hilb_{m}(Jac^{2}(C))$ is proper and $\displaystyle
Div_{Hilb_{2}(C)}$ is separated. It then suffices to show that every point in
the image is nonsingular of dimension $\displaystyle m$. In fact for any
$\displaystyle R\in Hilb_{m}(Jac^{2}(C))$, we claim that $\displaystyle
h^{0}(N_{\rho_{m}(R)/Hilb_{2}(C)})=m$, and $\displaystyle
h^{1}(N_{\rho_{m}(R)/Hilb_{2}(C)})=0$. Given $\displaystyle
R=\sum_{k=1}^{r}a_{k}L_{k}$ with $\displaystyle\sum_{k=1}^{r}a_{k}=m$, and
$\displaystyle L_{i}\not=L_{j}$ for $\displaystyle i\not=j$. Then
$\displaystyle
h^{i}(N_{\rho_{m}(R)/Hilb_{2}(C)})=\sum_{k=1}^{r}h^{i}(N_{\rho_{a_{k}}(a_{k}L_{k})/Hilb_{2}(C)})$.
Thus it suffices to prove the claim for the case where $\displaystyle
R=mL_{1}$ for some $\displaystyle L_{1}\in Jac^{2}(C)$. In this case
$\displaystyle
N_{\rho_{m}(R)/Hilb_{2}(C)}\cong{\mathcal{O}}_{m|L_{1}|}(m|L_{1}|)$. We first
prove by induction of $\displaystyle m\geqslant 1$ that $\displaystyle
h^{1}(m|L_{1}|,{\mathcal{O}}_{m|L_{1}|}(k|L_{1}|))=0$, for any integer
$\displaystyle k$. If $\displaystyle m=1$, then $\displaystyle|L_{1}|$ is a
smooth rational curve, and $\displaystyle C_{1}\doteq\Sigma^{*}(|L_{1}|)$ is a
smooth curve of genus one on $\displaystyle C\times C$, intersecting properly
with the diagonal $\displaystyle\Delta_{C}$ at four points. Then
$\displaystyle|L_{1}|\cdot|L_{1}|=C_{1}\cdot C_{1}/2=(K_{C\times
C}+C_{1})\cdot C_{1}/2=deg(K_{C_{1}})/2=0$. Thus $\displaystyle
h^{1}(|L_{1}|,{\mathcal{O}}_{|L_{1}|}(k|L_{1}|))=h^{1}(|L_{1}|,{\mathcal{O}}_{|L_{1}|})=0$,
for any $\displaystyle k$. Now assume that it is true for some integer
$\displaystyle m\geqslant 1$, then we consider the following exact sequence:

$0\rightarrow{\mathcal{O}}_{|L_{1}|}(-(m-1)|L_{1}|)\rightarrow{\mathcal{O}}_{m|L_{1}|}\rightarrow{\mathcal{O}}_{(m-1)|L_{1}|}\rightarrow
0.$

Tensor it with $\displaystyle{\mathcal{O}}_{m|L_{1}|}(k|L_{1}|)$, then by
assumption and by the case $\displaystyle m=1$ we conclude the induction.
Finally by the Riemann-Roch Theorem, for any $\displaystyle m\geqslant 1$:

$\displaystyle\displaystyle
h^{0}(m|L_{1}|,{\mathcal{O}}_{m|L_{1}|}(m|L_{1}|))$
$\displaystyle\displaystyle=\chi(m|L_{1}|,{\mathcal{O}}_{m|L_{1}|}(m|L_{1}|))$
$\displaystyle\displaystyle=\chi(Hilb^{2}(C\times
C),|mL_{1}|)-\chi(Hilb^{2}(C\times C),{\mathcal{O}}_{Hilb^{2}(C\times C)})$
$\displaystyle\displaystyle=m|L_{1}|.(m|L_{1}|-K_{Hilb_{2}(C\times C)})/2$
$\displaystyle\displaystyle=-|mL_{1}|.K_{Hilb^{2}(C\times
C)}/2=-mC_{1}.(K_{C\times C}-\Delta_{C})/4=m.$

∎

######  Definition 3.2.

Let $\displaystyle C$ be a smooth projective curve of genus one. For an
effective divisor $\displaystyle E$ on $\displaystyle C\times C$, we say it
comes from $\displaystyle m$ line bundles of degree two $\displaystyle R\in
Hilb_{m}(Jac^{2}(C))$ if $\displaystyle E=\Sigma^{*}(\rho_{m}(R))$.

Now we fix $\displaystyle G=D_{4},E_{6},E_{7},E_{8}$ or $\displaystyle F_{4}$.
We denote by $\displaystyle J$ the Jordan algebra associated to G,
$\displaystyle S=S_{G}$ the subadjoint variety, and we identify $\displaystyle
W=W_{G}\cong{\mathcal{C}}_{J}$ as in Theorem 2.4. Given a nondegenerate plane
$\displaystyle U\subseteq W=W_{G}$, we denote by
$\displaystyle(C,L,\kappa:C\rightarrow S)$ the triple given in Theorem 1.5.
That is, in diagram (2.10), we let $\displaystyle C=C_{U},L=L_{U}$, and
$\displaystyle\kappa=\kappa_{U}$. As $\displaystyle S\subseteq\mathbb{P}W$ is
linearly normal, we view the skew bilinear form $\displaystyle
w\in\wedge^{2}W^{\vee}$ as a section of
$\displaystyle{\mathcal{O}}_{S}(1)\boxtimes{\mathcal{O}}_{S}(1)$. We have the
following diagram of morphisms:

(3.1) $\displaystyle{S\times S}$$\displaystyle{C\times
C}$$\displaystyle{C}$$\displaystyle{S}$$\displaystyle{Hilb_{2}(C)}$$\displaystyle{\mathbb{P}^{1}}$$\displaystyle\scriptstyle{p_{j}}$$\displaystyle\scriptstyle{\Sigma}$$\displaystyle\scriptstyle{\kappa\times\kappa}$$\displaystyle\scriptstyle{\pi_{U}}$$\displaystyle\scriptstyle{\kappa}$

where $\displaystyle p_{j}$ is the projection to the $\displaystyle j$-th
factor for $\displaystyle j=1,2$. For a point $\displaystyle P\in C$, we
denote $\displaystyle C^{j}_{p}=p_{j}^{-1}(p)$. And we naturally identify it
with $\displaystyle C$ through the maps $\displaystyle\lambda_{j}:C\rightarrow
C^{j}_{p}:Q\rightarrow(Q,P)\,\,\text{or}\,\,(P,Q)$.

The main result of this section is the following description of
$\displaystyle(\kappa\times\kappa)^{*}(w)$, which implies Theorem 1.7(2)
combined with Theorem 2.13(ii).

######  Proposition 3.3.

$\displaystyle(\kappa\times\kappa)^{*}(w)\not=0$. Moreover
$\displaystyle[(\kappa\times\kappa)^{*}(w)]=3\Delta_{C}+E_{1}+E_{2}+E_{3}$,
where $\displaystyle\Delta_{C}$ is the diagonal, and:

(i) each $\displaystyle E_{i}$ is an irreducible smooth divisor intersecting
with $\displaystyle\Delta_{C}$ properly at four points, and is mapped
isomorphically onto $\displaystyle C$ by each $\displaystyle p_{j}$.

(ii) $\displaystyle E_{k}.E_{l}=0$, for any $\displaystyle 1\leqslant
l,k\leqslant 3$. And $\displaystyle E=E_{1}+E_{2}+E_{3}$ comes from three
degree two line bundles $\displaystyle R_{E}\in Hilb_{3}(Jac^{2}(C))$.

We split the proof into several lemmas. The key lemma is as follows.

######  Lemma 3.4.

For any $\displaystyle P\in C$,
$\displaystyle(\kappa\times\kappa)^{*}(w)_{|_{C^{j}_{P}}}\not=0$, and
$\displaystyle[(\kappa\times\kappa)^{*}(w)]_{|_{C^{j}_{P}}}=3P+\sum_{i=1}^{3}P_{i}$
for some points $\displaystyle P_{1},P_{2},P_{3}$ on $\displaystyle C$.
Moreover if $\displaystyle\pi_{U}$ is ramified at $\displaystyle P$, then
$\displaystyle P_{i}\not=P$, for any $\displaystyle i=1,2,3$.

###### Proof.

First note that
$\displaystyle(\kappa\times\kappa)^{*}({\mathcal{O}}_{S}(1)\boxtimes{\mathcal{O}}_{S}(1))]_{|_{C^{j}_{P}}}\cong
3L$ is of degree 6. Thus it suffices to show that

$\displaystyle(\kappa\times\kappa)^{*}(w)_{|_{C^{j}_{P}}}\not=0\,\,\text{and}\,\,ord_{P}((\kappa\times\kappa)^{*}(w)_{|_{C^{j}_{P}}})\geqslant
3,$

where the equality holds if $\displaystyle\pi_{U}$ is ramified at
$\displaystyle P$. Denote $\displaystyle q=\kappa(P)$. Then by the homogeneity
of $\displaystyle S$ we can assume $\displaystyle q=x_{0}=[A_{0}]$ in (2.3),
where $\displaystyle A_{0}=(1,0,0,0)\in{\mathcal{C}}_{J}$. Denote by
$\displaystyle[w_{x_{0}}]=[w(A_{0},-)]\in\mathbb{P}H^{0}(S,{\mathcal{O}}_{S}(1))$
the class of effective Cartier divisor given by $\displaystyle x_{0}$. Then
one easily sees that
$\displaystyle[(\kappa\times\kappa)^{*}(w)]_{|_{C^{j}_{P}}}=\kappa^{*}([w_{x_{0}}])$.
Now if $\displaystyle P$ is an unramified point of $\displaystyle\pi_{U}$,
denote $\displaystyle\overline{P}$ to be its conjugate. Then by the definition
in (2.10), $\displaystyle\pi_{U}(P)$ lies on the line $\displaystyle
l_{\kappa(P),\kappa(\overline{P})}$. And as $\displaystyle U$ is
nondegenerate, by Theorem 2.13(iii), we conclude that $\displaystyle
w(\kappa(P),\kappa(\overline{P}))\not=0$ hence
$\displaystyle(\kappa\times\kappa)^{*}(w)_{|_{C^{j}_{P}}}\not=0$.

On the other hand, from the birational map (2.3), we can write
$\displaystyle\kappa$ locally near $\displaystyle P$ as follows:

$\displaystyle\displaystyle\kappa_{|{\mathcal{V}}}:{\mathcal{V}}$
$\displaystyle\displaystyle\rightarrow\mathbb{P}(\mathbb{C}\oplus J\oplus
J\oplus\mathbb{C})$ $\displaystyle\displaystyle v$
$\displaystyle\displaystyle\rightarrow[1:f(v):f(v)^{\\#}:N(f(v))],$

where $\displaystyle{\mathcal{V}}$ is a suitable affine open neighbourhood of
$\displaystyle P$ in $\displaystyle C$, and $\displaystyle f$ is the unique
morphism from $\displaystyle{\mathcal{V}}$ to $\displaystyle J$ such that
$\displaystyle\kappa_{|_{{\mathcal{V}}}}$ factors through the natural
embedding of $\displaystyle J$ inside $\displaystyle S$ as in (2.4). In
particular $\displaystyle f(P)=0$. By the definition of $\displaystyle w$ in
(2.2), $\displaystyle\kappa_{|{\mathcal{V}}}^{*}(w_{x_{0}})$ equals the
divisor defined by the regular function $\displaystyle N\circ f$. As
$\displaystyle N$ is a cubic polynomial over $\displaystyle J$, we conclude
that $\displaystyle ord_{P}(\kappa_{|{\mathcal{V}}}^{*}(w_{x_{0}}))\geqslant
3$. Now if $\displaystyle P$ is a ramified point of $\displaystyle\pi_{U}$,
then by the definition of $\displaystyle\kappa$ in (2.10),
$\displaystyle\pi_{U}(P)\in TS\backslash Sing(TS)$ and
$\displaystyle\pi_{U}(P)$ lies on the tangent line $\displaystyle
l_{q,\pi_{U}(P)}$. Moreover the tangent direction
$\displaystyle[Im(d\kappa_{|_{P}})]=[T_{q}l_{q,\pi_{U}(P)}]$ is generic by
Theorem 2.13(3), i.e., $\displaystyle N(f^{{}^{\prime}}(q))\not=0$. This
implies that $\displaystyle\kappa_{|{\mathcal{V}}}^{*}(w_{x_{0}})\not=0$, and
$\displaystyle ord_{P}(\kappa_{|{\mathcal{V}}}^{*}(w_{x_{0}}))=3$ ∎

By the above lemma, $\displaystyle(\kappa\times\kappa)^{*}(w)\not=0$. We write

$\displaystyle
D:=[(\kappa\times\kappa)^{*}(w)]=a\Delta_{C}+\sum_{k=1}^{m}E_{k},$

where each $\displaystyle E_{i}$ is a prime divisor different from
$\displaystyle\Delta$, and $\displaystyle E_{i}$ can be equal to
$\displaystyle E_{j}$ for some $\displaystyle i,j$. And we write
$\displaystyle E=\sum_{k=1}^{m}E_{k}$ as the non-diagonal part of
$\displaystyle D$.

######  Lemma 3.5.

$\displaystyle a=3,D^{2}=72,\Delta_{C}.E=12$, and $\displaystyle E^{2}=0$.
Moreover $\displaystyle\chi(C\times C,{\mathcal{O}}_{E})=0$

###### Proof.

Denote by $\displaystyle p_{j}^{k}$ the restriction of $\displaystyle p_{j}$
to $\displaystyle E_{k}$ for each $\displaystyle j=1,2$ and for any
$\displaystyle k$. Take any $\displaystyle P\in
C\backslash(\bigcup\limits_{k}p_{j}^{k}(E_{i}\cap\Delta_{C}))$, then
$\displaystyle 3\leqslant mult_{P}(D\cap C_{P}^{j})=a\Delta_{C}\cdot
C_{P}^{j}=a$ by Lemma 3.4. On the other hand, take $\displaystyle P$ as a
ramified point of $\displaystyle\pi_{U}$, then also by Lemma 3.4,
$\displaystyle a=a\Delta_{C}\cdot C_{P}^{j}\leqslant
ord_{P}(D_{|_{C_{P}^{j}}})=3$. As $\displaystyle D\in|3L\boxtimes 3L|$,
$\displaystyle D^{2}=2\cdot deg(3L)^{2}=72$.
$\displaystyle\Delta_{C}^{2}=(K_{C\times
C}+\Delta_{C})\cdot\Delta_{C}=deg(K_{\Delta_{C}})=0$ as $\displaystyle C$ is
of genus of one. Now $\displaystyle\Delta_{C}\cdot E=\Delta_{C}\cdot
D=deg((3L\boxtimes 3L)_{|_{\Delta_{C}}})=deg(6L)=12$. Then $\displaystyle
E^{2}=D^{2}-2a\Delta_{C}\cdot E=0$. Finally by the Riemann-Roch Theorem we
have

$\displaystyle\chi(C\times C,{\mathcal{O}}_{E})=\chi(C\times
C,{\mathcal{O}}_{C\times C})-\chi(C\times C,{\mathcal{O}}_{C\times
C}(-E))=-(E+K_{C\times C})\cdot E/2=0$

. ∎

######  Corollary 3.6.

We have $\displaystyle m\leqslant 3$. Moreover each $\displaystyle E_{k}$ is a
smooth genus one curve on $\displaystyle C\times C$, and $\displaystyle
p_{j}^{k}$ is a surjective étale morphism for any $\displaystyle j$. Finally
$\displaystyle E_{k}\cdot E_{l}=0$, for any $\displaystyle 1\leqslant
k,l\leqslant 3$.

###### Proof.

For the first two parts, by Hurwitz’s formula, it suffices to show that each
$\displaystyle E_{k}$ is smooth of genus one and $\displaystyle p_{j}^{k}$ is
surjective. From Lemma 3.4, $\displaystyle D$ does not contain $\displaystyle
C_{p}^{j}$ for any $\displaystyle P$ and $\displaystyle j$. Thus
$\displaystyle p_{j}^{k}$ is surjective for each $\displaystyle k$. Denote by
$\displaystyle d_{j}^{k}=deg(p_{j}^{k})$, then by Lemma 3.5 we have
$\displaystyle\sum_{k=1}^{m}d_{j}^{k}=3$, and hence $\displaystyle m\leqslant
3$. If $\displaystyle d_{j}^{k}=1$ for each $\displaystyle k$, then we
conclude that $\displaystyle m=3$, and each $\displaystyle p_{j}^{k}$ is an
isomorphism. Now assume $\displaystyle m\leqslant 2$. Denote $\displaystyle
q_{E}:\tilde{E}\rightarrow E$ to be the normalization of $\displaystyle E$,
and $\displaystyle
Q_{E}\dot{=}(q_{E})_{*}({\mathcal{O}}_{\tilde{E}})/{\mathcal{O}}_{E}$ the
quotient sheaf with finite support. Then
$\displaystyle\chi(Q_{E})=\chi({\mathcal{O}}_{\tilde{E}})$ as
$\displaystyle\chi(C\times C,{\mathcal{O}}_{E})=0$. If $\displaystyle m=1$
then $\displaystyle\tilde{E}$ is irreducible, so $\displaystyle
1-g(\tilde{E})=\chi({\mathcal{O}}_{\tilde{E}})=\chi(Q_{E})\geqslant 0$. But
$\displaystyle g(\tilde{E})\geqslant g(C)=1$. Thus we conclude that
$\displaystyle g(\tilde{E})=1$ hence $\displaystyle Q_{E}=0$, $\displaystyle
q_{E}$ is an isomorphism, and $\displaystyle E$ is a smooth curve of genus
one. If $\displaystyle m=2$ then we can write
$\displaystyle\tilde{E}=\tilde{E}_{1}\cup\tilde{E}_{2}$, where each
$\displaystyle\tilde{E}_{k}$ is the normalization of $\displaystyle E_{k}$.
And we can assume $\displaystyle d_{1}^{1}=1,d_{1}^{2}=2$. Thus $\displaystyle
f_{1}^{1}$ is an isomorphism and $\displaystyle E_{1}$ is a smooth curve of
genus one. Then
$\displaystyle\chi({\mathcal{O}}_{\tilde{E}})=\chi({\mathcal{O}}_{\tilde{E_{1}}})+\chi({\mathcal{O}}_{\tilde{E_{2}}})=1-g(\tilde{E_{2}})=\chi(Q_{E})\geqslant
0$. By the same discussion as for $\displaystyle m=1$ we conclude that
$\displaystyle E_{2}$ is a smooth curve of genus one. For the last part, for
each $\displaystyle E_{k}$ we have $\displaystyle E_{k}^{2}=(K_{C\times
C}+E_{k})\cdot E_{k}=deg(K_{E_{k}})=0$. Thus
$\displaystyle\sum_{E_{i}\not=E_{j}}E_{i}\cdot
E_{j}=(\sum_{k}E_{k})^{2}=E^{2}=0$, concluding the proof. ∎

######  Lemma 3.7.

Let $\displaystyle F$ be an irreducible smooth curve of genus one on
$\displaystyle C\times C$, different from $\displaystyle\Delta_{C}$. If
$\displaystyle F$ is $\displaystyle S_{2}$-invariant and if it is surjective
étale over $\displaystyle C$ under each projection $\displaystyle p_{j}$, then
it intersects properly with the diagonal $\displaystyle\Delta_{C}$.

###### Proof.

Denote by $\displaystyle\sigma$ the generator of $\displaystyle S_{2}$. Assume
$\displaystyle b=(P,P)\in\Delta_{C}\cap F$, denote
$\displaystyle\mathfrak{m}_{C,P}\,(resp.\,\mathfrak{m}_{F,b})$ to be the
maximal ideal of local ring
$\displaystyle{\mathcal{O}}_{C,P}\,(resp.\,{\mathcal{O}}_{F,b})$, and let
$\displaystyle t_{b}\,(resp.\,f_{b})$ be a parameter of the maximal ideal.
Note that the ideal
$\displaystyle(I_{\Delta_{C}})_{|_{b}}\subseteq{\mathcal{O}}_{C\times C,b}$ is
generated by $\displaystyle t_{b}\otimes 1-1\otimes t_{b}$. As $\displaystyle
F$ is étale over $\displaystyle C$ and $\displaystyle S_{2}$-invariant,
$\displaystyle(I_{\Delta_{C}})_{|_{F,b}}$ is then generated by $\displaystyle
f_{b}-\sigma(f_{b})$. As $\displaystyle\sigma$ fixes $\displaystyle b$,
$\displaystyle\sigma$ acts on $\displaystyle{\mathcal{O}}_{F,b}$ and we can
assume $\displaystyle\sigma(f_{b})=cf_{b}+uf_{b}^{2}$ for some $\displaystyle
c\in\mathbb{C},u\in{\mathcal{O}}_{F,b}$. Then applying
$\displaystyle\sigma^{2}=1$ we get $\displaystyle c=\pm 1$. Thus it suffices
to show that $\displaystyle c=-1$. If otherwise, then we claim that
$\displaystyle\sigma$ fixes $\displaystyle{\mathcal{O}}_{F,b}$, whence
$\displaystyle F$ is fixed and thus $\displaystyle F=\Delta_{C}$, which is a
contradiction. To show the claim, we first prove by induction that
$\displaystyle\sigma$ fixes
$\displaystyle{\mathcal{O}}_{F,b}/\mathfrak{m}_{F,b}^{k}$ for each positive
integer $\displaystyle k$, then the claim follows from the fact that
$\displaystyle{\mathcal{O}}_{F,b}$ is $\displaystyle\sigma$-equivariantly
included in the complete local ring
$\displaystyle\widehat{{\mathcal{O}}}_{F,b}$. Now assume by induction that
$\displaystyle\sigma$ fixes
$\displaystyle{\mathcal{O}}_{F,b}/\mathfrak{m}_{F,b}^{k}$ for some
$\displaystyle k\geqslant 2$, then consider the
$\displaystyle\sigma$-equivariant exact sequence:

$0\rightarrow\mathfrak{m}_{F,b}^{k}/\mathfrak{m}_{F,b}^{k+1}\rightarrow{\mathcal{O}}_{F,b}/\mathfrak{m}_{F,b}^{k+1}\rightarrow{\mathcal{O}}_{F,b}/\mathfrak{m}_{F,b}^{k}\rightarrow
0.$

By assumption for every
$\displaystyle\alpha\in{\mathcal{O}}_{F,b}/\mathfrak{m}_{F,b}^{k+1}$,
$\displaystyle\sigma(\alpha)-\alpha\in\mathfrak{m}_{F,b}^{k}/\mathfrak{m}_{F,b}^{k+1}$.
But $\displaystyle\sigma(\sigma(\alpha)-\alpha)=-(\sigma(\alpha)-\alpha)$ and
$\displaystyle\sigma$ fixes
$\displaystyle\mathfrak{m}_{F,b}^{k}/\mathfrak{m}_{F,b}^{k+1}\cong
Sym^{k}(\mathfrak{m}_{F,b}/\mathfrak{m}_{F,b}^{2})$. Thus we conclude that
$\displaystyle\sigma(\alpha)=\alpha$, the claim is proved. ∎

Recall that $\displaystyle\Sigma$ is a ramified double cover with ramified
divisor $\displaystyle\Delta_{C}$. Then any $\displaystyle S_{2}$-invariant
effective divisor on $\displaystyle C\times C$ whose components do not include
$\displaystyle\Delta_{C}$ must be a pull-back of an effective divisor on
$\displaystyle Hilb_{2}(C)$. Denote by $\displaystyle F$ the effective divisor
on $\displaystyle Hilb_{2}(C)$ such that $\displaystyle\Sigma^{*}(F)=E$.

######  Lemma 3.8.

$\displaystyle F^{2}=0$, and
$\displaystyle\chi(Hilb_{2}(C),{\mathcal{O}}_{F})=3$.

###### Proof.

$\displaystyle F^{2}=\Sigma^{*}(F)\cdot\Sigma^{*}(F)/2=E^{2}/2=0$ by Lemma
3.5. And $\displaystyle F\cdot K_{Hilb_{2}(C)}=E\cdot(K_{C\times
C}-\Delta_{C})/2=-E\cdot\Delta_{C}/2=-6$. Thus by the Riemann-Roch Theorem:

$\chi(Hilb_{2}(C),{\mathcal{O}}_{F})=\chi({\mathcal{O}}_{Hilb_{2}(C)})-\chi({\mathcal{O}}(-F))=-F\cdot(F+K_{Hilb_{2}(C)})/2=3.$

∎

Now we are ready to prove Proposition 3.3.

###### Proof of Proposition 3.3.

First we show that $\displaystyle m=3$. Assume that $\displaystyle m=1$, then
$\displaystyle F$ is a prime divisor. So
$\displaystyle\chi({\mathcal{O}}_{F})=1-h^{1}({\mathcal{O}}_{F})\leqslant 1$,
contradicting Lemma 3.8. If $\displaystyle m=2$ then $\displaystyle
E=E_{1}+E_{2}$ with $\displaystyle E_{1}\not=E_{2}$, as $\displaystyle
d_{j}^{1}\not=d_{j}^{2}$. Then both of them are $\displaystyle
S_{2}$-invariant. Otherwise from $\displaystyle E_{1}\cdot E_{2}=0$ we have
$\displaystyle E_{1}\cdot\Delta_{C}=E_{2}\cdot\Delta_{C}=0$, contradicting
$\displaystyle E\cdot\Delta_{C}=12$. Thus for each $\displaystyle k$ we can
assume $\displaystyle E_{k}=\Sigma^{*}(F_{k})$ for some prime divisor
$\displaystyle F_{k}$ on $\displaystyle Hilb_{2}(C)$. And we have
$\displaystyle F_{1}\cdot F_{2}=E_{1}\cdot E_{2}/2=0$. But then
$\displaystyle\chi({\mathcal{O}}_{F})=\chi({\mathcal{O}}_{F_{1}})+\chi({\mathcal{O}}_{F_{2}})\leqslant
2$, again a contradiction. Thus $\displaystyle m=3$. And from the proof of
Corollary 3.6, we conclude that each $\displaystyle p_{j}^{k}$ is an
isomorphism.

Then we show that each $\displaystyle E_{k}$ intersects properly
$\displaystyle\Delta_{C}$ at four points applying Lemma 3.7. We first show
that each $\displaystyle E_{k}$ is $\displaystyle\sigma$-invariant. Assume
otherwise, then we can assume $\displaystyle\sigma(E_{1})=E_{2}$ with
$\displaystyle E_{1}\not=E_{2}$, and hence $\displaystyle E_{3}$ is
$\displaystyle\sigma$-invariant as $\displaystyle E$ is
$\displaystyle\sigma$-invariant. Then there is a prime divisor $\displaystyle
F^{\prime}$(resp. $\displaystyle F_{3}$) on $\displaystyle Hilb_{2}(C)$ such
that $\displaystyle\Sigma^{*}(F^{\prime})=E_{1}+E_{2}$(resp.
$\displaystyle\Sigma^{*}(F_{3})=E_{3}$), with $\displaystyle
F^{\prime}+F_{3}=F$. Moreover $\displaystyle F^{\prime}\cdot
F_{3}=(E_{1}+E_{2})\cdot E_{3}/2=0$. Thus
$\displaystyle\chi({\mathcal{O}}_{F})=\chi({\mathcal{O}}_{F_{3}})+\chi({\mathcal{O}}_{F^{\prime}})\leqslant
2$, a contradiction. Now we can assume for each $\displaystyle k$ that there
is a prime divisor $\displaystyle F_{k}$ such that
$\displaystyle\Sigma^{*}(F_{k})=E_{k}$, and $\displaystyle
F=F_{1}+F_{2}+F_{3}$. Then by similar calculations as in Lemma 3.8, we
conclude that $\displaystyle
E_{k}\cdot\Delta_{C}/4=\chi({\mathcal{O}}_{F_{k}})\leqslant 1$ for each
$\displaystyle k$. Thus from $\displaystyle\Delta_{C}\cdot E=12$ we get
$\displaystyle\Delta_{C}\cdot E_{k}=4$, and
$\displaystyle\chi({\mathcal{O}}_{F_{k}})=1$ for each $\displaystyle k$. In
particular this implies that $\displaystyle F_{k}\cong\mathbb{P}^{1}$ for each
$\displaystyle k$. Then $\displaystyle\kappa_{k,j}:(p_{j}^{k})^{-1}\circ
S_{|_{E_{k}}}:C\rightarrow F_{k}$ is a ramified double cover of
$\displaystyle\mathbb{P}^{1}$ for each $\displaystyle k$. Denote by
$\displaystyle L_{k}=\kappa_{k,j}^{*}({\mathcal{O}}(1))$. Then one easily
checks that $\displaystyle F_{k}=\rho_{1}(L_{k})$ for each $\displaystyle k$,
and thus $\displaystyle F=\rho_{3}(L_{1}+L_{2}+L_{3})$. The proof is complete.
∎

###  3.2. Relative setting

Throughout this subsection $\displaystyle G$ is supposed to be of type
$\displaystyle F_{4},E_{6},E_{7}$ or $\displaystyle E_{8}$. Let
$\displaystyle{\mathcal{U}}_{G}\subseteq Gr(2,W_{G})$ be the open subset of
nondegenerate planes. Recall from (2.10) that, every nondegenerate plane
$\displaystyle[U]\in{\mathcal{U}}_{G}$ is associated with a datum
$\displaystyle(C_{U},L_{U},\kappa_{U}:C_{U}\rightarrow S_{G})$. As
$\displaystyle U$ varies in $\displaystyle{\mathcal{U}}_{G}$, it then defines
a family of data over $\displaystyle{\mathcal{U}}_{G}$. We formulate it as
follows.

Recall that $\displaystyle\pi:{\mathcal{T}}\rightarrow Hilb_{2}(S)$ is the
universal family of subschemes of length two on $\displaystyle S$. And
$\displaystyle v$ is the evaluation map to $\displaystyle S$. From Theorem
2.12 (3), one can easily see that there is an $\displaystyle
H_{G}$-equivariant morphism $\displaystyle\gamma_{G}:\mathbb{P}W_{G}\backslash
Sing(TS_{G})\rightarrow Hilb_{2}(S_{G})$, such that for every nondegenerate
plane $\displaystyle U$, the morphism $\displaystyle\psi_{U}$ defined in
(2.10) coincides with the composition of $\displaystyle\gamma_{G}$ and the
inclusion of $\displaystyle\mathbb{P}U$ in $\displaystyle\mathbb{P}W_{G}$.
Denote $\displaystyle\eta_{G}:{\mathcal{P}}_{G}\rightarrow{\mathcal{U}}_{G}$
to be the universal bundle of lines in $\displaystyle\mathbb{P}W_{G}$. Let
$\displaystyle{\mathcal{O}}_{{\mathcal{P}}_{G}}(1)$ be the tautological line
bundle of $\displaystyle{\mathcal{P}}_{G}$. And let
$\displaystyle\mu_{G}:{\mathcal{P}}_{G}\rightarrow\mathbb{P}W_{G}$ be the
evaluation map. Then by our definition of nondegenerate planes, $\displaystyle
Im(\mu_{G})$ is included in $\displaystyle\mathbb{P}W_{G}\backslash
Sing(TS_{G})$. Thus one can define a similar diagram of morphisms as (2.10):

(3.2)
$\displaystyle{{\mathcal{C}}_{G}}$$\displaystyle{\mathcal{T}}$$\displaystyle{S}$$\displaystyle{{\mathcal{P}}_{G}}$$\displaystyle{Hilb_{2}(S),}$$\displaystyle\scriptstyle{\pi^{*}\psi_{G}}$$\displaystyle\scriptstyle{\pi_{G}}$$\displaystyle\scriptstyle{\pi}$$\displaystyle\scriptstyle{v}$$\displaystyle\scriptstyle{\psi_{G}}$

where $\displaystyle\psi_{G}=\gamma_{G}\circ\mu_{G}$, and
$\displaystyle\pi_{G}=\psi_{G}^{*}(\pi)$. We denote
$\displaystyle{\mathcal{L}}_{G}=\pi_{G}^{*}({\mathcal{O}}_{{\mathcal{P}}_{G}}(1))$,
$\displaystyle\kappa_{G}=v\circ\pi^{*}\psi_{G}$, and $\displaystyle
K_{G}=\eta_{G}\circ\pi_{G}$. Then one sees that $\displaystyle K_{G}$ is a
smooth proper family of curves of genus one, $\displaystyle{\mathcal{L}}_{G}$
is a family of degree two line bundles on
$\displaystyle{\mathcal{C}}_{G}/{\mathcal{U}}_{G}$. The datum
$\displaystyle(K_{G}:{\mathcal{C}}_{G}\rightarrow{\mathcal{U}}_{G},{\mathcal{L}}_{G},\kappa_{G})$
defines our desired family of nondegenerate morphisms from polarized curves of
genus one to $\displaystyle S_{G}$.

Next we aim to formulate our results in Section 3.1 under this relative
setting. First we fix some notations. For any
$\displaystyle[U]\in{\mathcal{U}}_{G}$, we denote by $\displaystyle E_{U}$ the
non-diagonal part of $\displaystyle(\kappa_{U}\times\kappa_{U})^{*}(w)$,
$\displaystyle F_{U}$ the effective divisor on $\displaystyle Hilb_{2}(C_{U})$
such that $\displaystyle\Sigma_{U}^{*}(F_{U})=E_{U}$,
$\displaystyle\rho_{m,U}=\rho_{m}$ the morphism defined before for
$\displaystyle C=C_{U}$, and $\displaystyle R_{U}\in Hilb_{3}(Jac^{2}(C_{U}))$
such that $\displaystyle\rho_{3,U}(R_{U})=F_{U}$.

As $\displaystyle{\mathcal{C}}_{G}/{\mathcal{U}}_{G}$ is a smooth projective
family of curves of genus one, the functor $\displaystyle
Pic_{({\mathcal{C}}_{G}/{\mathcal{U}}_{G})(\text{\'{e}tale})}$ is represented
by a scheme $\displaystyle Pic_{{\mathcal{C}}_{G}/{\mathcal{U}}_{G}}$[FGI+05,
Theorem 9.4.8(1)]. Moreover for any $\displaystyle k\in{\mathbb{Z}}$, denote
$\displaystyle
Pic_{({\mathcal{C}}_{G}/{\mathcal{U}}_{G})(\text{\'{e}tale})}^{k}$ to be the
étale subsheaf associated to the presheaf whose $\displaystyle T$-point for a
scheme $\displaystyle T$ over $\displaystyle{\mathcal{U}}_{G}$ is represented
by line bundles on
$\displaystyle{\mathcal{C}}_{G}\times_{{\mathcal{U}}_{G}}T/T$ of degree
$\displaystyle k$. Then from the proof of [FGI+05, Theorem 9.4.8(2)],
$\displaystyle
Pic_{({\mathcal{C}}_{G}/{\mathcal{U}}_{G})(\text{\'{e}tale})}^{k}$ is an open
subsheaf represented by an open subscheme $\displaystyle
Pic_{{\mathcal{C}}_{G}/{\mathcal{U}}_{G}}^{k}\subseteq
Pic_{{\mathcal{C}}_{G}/{\mathcal{U}}_{G}}$ which is quasi-projective over
$\displaystyle{\mathcal{U}}_{G}$, and $\displaystyle
Pic_{{\mathcal{C}}_{G}/{\mathcal{U}}_{G}}$ is the disjoint union of
$\displaystyle Pic_{{\mathcal{C}}_{G}/{\mathcal{U}}_{G}}^{k}$ for
$\displaystyle k\in{\mathbb{Z}}$.

######  Lemma 3.9.

For any $\displaystyle k\in{\mathbb{Z}}$, $\displaystyle
Pic_{{\mathcal{C}}_{G}/{\mathcal{U}}_{G}}^{k}$ is a smooth projective family
of curves of genus one over $\displaystyle{\mathcal{U}}_{G}$.

###### Proof.

The fact that $\displaystyle Pic_{{\mathcal{C}}_{G}/{\mathcal{U}}_{G}}$ is
smooth over $\displaystyle{\mathcal{U}}_{G}$ follows from [FGI+05, Proposition
9.5.19] and that $\displaystyle{\mathcal{C}}_{G}/{\mathcal{U}}_{G}$ is a
smooth family of curves. It is projective over
$\displaystyle{\mathcal{U}}_{G}$ by a general fact that for a smooth
projective family of irreducible varieties $\displaystyle X/S$, any closed
subscheme of $\displaystyle Pic_{X/S}$ is projective over $\displaystyle S$ if
it is of finite type(cf. [FGI+05, Exercise 9.5.7]). Finally for any point
$\displaystyle[U]\in{\mathcal{U}}_{G}$, $\displaystyle
Pic_{{\mathcal{C}}_{G}/{\mathcal{U}}_{G}}^{k}|_{[U]}\cong Jac^{k}(C_{U})$ for
each $\displaystyle k\in{\mathbb{Z}}$, which is a smooth projective curve of
genus one. Thus it follows that $\displaystyle
Pic_{{\mathcal{C}}_{G}/{\mathcal{U}}_{G}}^{k}$ is a smooth projective family
of curves of genus one. ∎

Denote by $\displaystyle
S^{m}_{{\mathcal{U}}_{G}}(Pic^{2}_{{\mathcal{C}}_{G}/{\mathcal{U}}_{G}})$ the
family of $\displaystyle m$ line bundles of degree two on
$\displaystyle{\mathcal{C}}_{G}/{\mathcal{U}}_{G}$, which is defined as the
quotient of the permutation action of $\displaystyle S_{m}$ on the m-tuple
relative product of $\displaystyle
Pic^{2}_{{\mathcal{C}}_{G}/{\mathcal{U}}_{G}}/{\mathcal{U}}_{G}$. Then one
easily sees that it is also smooth projective over
$\displaystyle{\mathcal{U}}_{G}$. Over any point
$\displaystyle[U]\in{\mathcal{U}}_{G}$, its fiber is isomorphic to
$\displaystyle Hilb_{m}(Jac^{2}(C_{U}))$ via the Hilbert-Chow morphism. We aim
to prove the following:

######  Proposition 3.10.

There is a section
$\displaystyle{\mathcal{R}}_{G}:{\mathcal{U}}_{G}\rightarrow
S^{3}_{{\mathcal{U}}_{G}}(Pic^{2}_{{\mathcal{C}}_{G}/{\mathcal{U}}_{G}})$,
such that for any $\displaystyle[U]\in{\mathcal{U}}_{G}$,
$\displaystyle{\mathcal{R}}_{G}([U])=R_{U}$.

Consider the action of $\displaystyle S_{2}$ on
$\displaystyle{\mathcal{C}}_{G}\times_{{\mathcal{U}}_{G}}{\mathcal{C}}_{G}$
given by permutations of factors. We define $\displaystyle
S_{G}:{\mathcal{C}}_{G}\times{\mathcal{C}}_{G}\rightarrow
S^{2}_{{\mathcal{U}}_{G}}({\mathcal{C}}_{G})\dot{=}({\mathcal{C}}_{G}\times_{{\mathcal{U}}_{G}}{\mathcal{C}}_{G})//S_{2}$
to be its GIT quotient map. Then we can get a similar diagram as (3.1):

(3.3) $\displaystyle{S\times
S}$$\displaystyle{{\mathcal{C}}_{G}\times{\mathcal{C}}_{G}}$$\displaystyle{{\mathcal{C}}_{G}}$$\displaystyle{S}$$\displaystyle{S^{2}_{{\mathcal{U}}_{G}}({\mathcal{C}}_{G})}$$\displaystyle{{\mathcal{U}}_{G}}$$\displaystyle\scriptstyle{p_{j,G}}$$\displaystyle\scriptstyle{\Sigma_{G}}$$\displaystyle\scriptstyle{\kappa_{G}\times\kappa_{G}}$$\displaystyle\scriptstyle{K_{G}}$$\displaystyle\scriptstyle{\kappa_{G}}$$\displaystyle\scriptstyle{q_{G}}$

By Propostion 3.3,
$\displaystyle(\kappa_{G}\times\kappa_{G})^{*}(\omega)\not=0$. We write
$\displaystyle[(\kappa_{G}\times\kappa_{G})^{*}(\omega)]=a\Delta_{{\mathcal{C}}_{G}}+{\mathcal{E}}_{G}$,
where $\displaystyle\Delta_{{\mathcal{C}}_{G}}$ is the diagonal, and
$\displaystyle{\mathcal{E}}_{G}$ is an effective divisor whose component does
not contain $\displaystyle\Delta_{{\mathcal{C}}_{G}}$. Recall that for a
scheme $\displaystyle X/S$, a relative effective divisor on $\displaystyle
X/S$ is an effective divisor on $\displaystyle X$ which is flat over
$\displaystyle S$.

######  Corollary 3.11.

We have a=3. Moreover $\displaystyle{\mathcal{E}}_{G}$ is a relative effective
divisor on
$\displaystyle({\mathcal{C}}_{G}\times{\mathcal{C}}_{G})/{\mathcal{U}}_{G}$,
and there is a relative effective divisor $\displaystyle{\mathcal{F}}_{G}$ on
$\displaystyle S^{2}_{{\mathcal{U}}_{G}}({\mathcal{C}}_{G})/{\mathcal{U}}_{G}$
such that $\displaystyle{\mathcal{E}}_{G}=\Sigma_{G}^{*}({\mathcal{F}}_{G})$.
For any $\displaystyle[U]\in{\mathcal{U}}_{G}$, we have
$\displaystyle{\mathcal{E}}_{G}|_{[U]}=E_{U}$ and
$\displaystyle{\mathcal{F}}_{G}|_{[U]}=F_{U}$.

###### Proof.

As $\displaystyle\Delta_{{\mathcal{C}}_{G}}$ is not contained in the component
of $\displaystyle{\mathcal{E}}_{G}$, there exists a nondgenerate plane
$\displaystyle[U]\in{\mathcal{C}}_{G}$ such that
$\displaystyle{\mathcal{E}}_{G}|_{[U]}$ does not contain
$\displaystyle\Delta_{C_{U}}$. Then it follows from Proposition 3.3 that
$\displaystyle a=3$. And for any $\displaystyle[U]\in{\mathcal{U}}_{G}$, one
has $\displaystyle{\mathcal{E}}_{G}|_{[U]}=E_{U}$, which is an effective
Cartier divisor on $\displaystyle C_{U}\times C_{U}$. Thus
$\displaystyle{\mathcal{E}}_{G}$ is a relative effective divisor on
$\displaystyle({\mathcal{C}}_{G}\times{\mathcal{C}}_{G})/{\mathcal{U}}_{G}$
[FGI+05, Lemma 9.3.4(iii)]. Moreover as $\displaystyle{\mathcal{E}}_{G}$ is
obviously $\displaystyle S_{2}$-invairant, there is is an effective divisor
$\displaystyle{\mathcal{F}}_{G}$ on $\displaystyle
S^{2}_{{\mathcal{U}}_{G}}({\mathcal{C}}_{G})$ such that
$\displaystyle{\mathcal{E}}_{G}=S_{G}^{*}({\mathcal{F}}_{G})$. Then for each
$\displaystyle[U]\in{\mathcal{U}}_{G}$,
$\displaystyle{\mathcal{F}}_{G}|_{[U]}=F_{U}$. And thus
$\displaystyle{\mathcal{F}}_{G}$ is a relatively effective divisor on
$\displaystyle
S^{2}_{{\mathcal{U}}_{G}}({\mathcal{C}}_{G})/{\mathcal{U}}_{G}$. ∎

Note that there is a natural morphism
$\displaystyle\Theta_{G,m}:S^{m}_{{\mathcal{U}}_{G}}(Pic^{2}_{{\mathcal{C}}_{G}/{\mathcal{U}}_{G}})\rightarrow
Div_{S^{2}_{{\mathcal{U}}_{G}}({\mathcal{C}}_{G})/{\mathcal{U}}_{G}}$ for each
$\displaystyle m\geqslant 1$, such that for any
$\displaystyle[U]\in{\mathcal{U}}_{G}$ we have
$\displaystyle\Theta_{G,m}|_{[U]}=\rho_{m,U}$. Applying Lemma 3.1 we get:

######  Corollary 3.12.

For each $\displaystyle m\geqslant 1$, $\displaystyle\Theta_{G,m}$ is a closed
embedding.

###### Proof.

By Lemma 3.1, $\displaystyle\Theta_{G,m}|_{[U]}$ is a closed embedding for any
$\displaystyle[U]\in{\mathcal{U}}_{G}$. Then it suffices to show that
$\displaystyle\Theta_{G,m}$ is proper. As $\displaystyle
S^{2}_{{\mathcal{U}}_{G}}({\mathcal{C}}_{G})$ is projective and smooth over
$\displaystyle{\mathcal{U}}_{G}$, $\displaystyle
Div_{S^{2}_{{\mathcal{U}}_{G}}({\mathcal{C}}_{G})/{\mathcal{U}}_{G}}$ is a
locally quasi-projective scheme over $\displaystyle{\mathcal{U}}_{G}$[FGI+05,
Theorem 9.3.7]. In particular it is separated over
$\displaystyle{\mathcal{U}}_{G}$. Thus $\displaystyle\Theta_{G,m}$ is proper
as $\displaystyle
S^{m}_{{\mathcal{U}}_{G}}(Pic^{2}_{{\mathcal{C}}_{G}/{\mathcal{U}}_{G}})$ is
proper over $\displaystyle{\mathcal{U}}_{G}$. ∎

###### Proof of Proposition 3.10.

The relative effective divisor $\displaystyle{\mathcal{F}}_{G}$ on
$\displaystyle S^{2}_{{\mathcal{U}}_{G}}({\mathcal{C}}_{G})/{\mathcal{U}}_{G}$
defines a section $\displaystyle\tau_{G}:{\mathcal{U}}_{G}\rightarrow
Div_{S^{2}_{{\mathcal{U}}_{G}}({\mathcal{C}}_{G})/{\mathcal{U}}_{G}}$. From
Corollary 3.11, we have $\displaystyle Im(\tau_{G})\subseteq
Im(\Theta_{G,3})$. Then the proof follows from the fact that
$\displaystyle\Theta_{G,m}$ is a closed embedding and both $\displaystyle
S^{m}_{{\mathcal{U}}_{G}}(Pic^{2}_{{\mathcal{C}}_{G}/{\mathcal{U}}_{G}})$ and
$\displaystyle{\mathcal{U}}_{G}$ are varieties. ∎

##  4\. proof of the main theorem

In this section we are going to prove our Main Theorem. We first give the
precise definition of the moduli functor $\displaystyle{\mathcal{M}}$ as
follows, where we denote by $\displaystyle(Sch/\mathbb{C})_{fppf}$ the site
consisting of the category of locally noetherian schemes over
$\displaystyle\mathbb{C}$ equipped with $\displaystyle fppf$ topology.

######  Definition 4.1.

Denote by $\displaystyle\mathcal{M}$ the sheaf associated to the following
presheaf:

$\displaystyle\displaystyle P\mathcal{M}:(Sch/\mathbb{C})^{op}_{fppf}$
$\displaystyle\displaystyle\rightarrow(Sets)$ $\displaystyle\displaystyle T$
$\displaystyle\displaystyle\rightarrow\\{K:\mathcal{C}\rightarrow
T;{\mathcal{L}},{\mathcal{R}}\\}/\sim.$

Here $\displaystyle K$ is a smooth proper family of curves of genus one.
$\displaystyle{\mathcal{L}}\in Pic({\mathcal{C}}/T)(T)$ is a family of line
bundles of degree two on $\displaystyle{\mathcal{C}}/T$, and
$\displaystyle{\mathcal{R}}:T\rightarrow S^{3}_{T}(Pic^{2}({\mathcal{C}}/T))$
is a section of the third symmetric product of Picard scheme of degree two of
the family C/T. Moreover for any point $\displaystyle t\in T$, denote by
$\displaystyle{\mathcal{R}}_{t}=(L_{1,t},L_{2,t},L_{3,t})$ the corresponding
three line bundles on $\displaystyle{\mathcal{C}}_{t}$. Then we also require
that $\displaystyle{\mathcal{L}}_{t}\cong L_{i,t}$ for any $\displaystyle
i=1,2,3$, and that $\displaystyle L_{1,t}+L_{2,t}+L_{3,t}=3{\mathcal{L}}_{t}$.
Two data $\displaystyle({\mathcal{C}}/T,{\mathcal{L}},{\mathcal{R}})$ and
$\displaystyle({\mathcal{C}}^{\prime}/T,{\mathcal{L}}^{\prime},{\mathcal{R}}^{\prime})$
are said to be equivalent if and only if there is a $\displaystyle
T$-isomorphism
$\displaystyle\Psi:{\mathcal{C}}\rightarrow{\mathcal{C}}^{\prime}$, such that
$\displaystyle\Psi^{*}({\mathcal{L}}^{\prime})\cong{\mathcal{L}}$, and
$\displaystyle\Psi^{*}({\mathcal{R}}^{\prime})={\mathcal{R}}$

We can verify Proposition 1.8(2).

###### Proof of Proposition 1.8(2).

The proof is adapted from the proof of [BH16, Theorem 6.28]. Given a scheme
$\displaystyle T$ and given an equivalent class in $\displaystyle
P{\mathcal{M}}(T)$ represented by
$\displaystyle(K:{\mathcal{C}}/T,{\mathcal{L}},{\mathcal{R}})$. Denote by

${\mathcal{Q}}:Pic^{2}_{{\mathcal{C}}/T}\times_{T}Pic^{2}_{{\mathcal{C}}/T}\times_{T}Pic^{2}_{{\mathcal{C}}/T}\rightarrow
S^{3}_{T}(Pic^{2}_{{\mathcal{C}}/T})$

the quotient map of the $\displaystyle S_{3}$-action. Denote by

${\mathcal{R}}^{*}({\mathcal{Q}}):\widetilde{T}\rightarrow
T,\,\,\text{and}\,\,\,{\mathcal{Q}}^{*}({\mathcal{R}}):\widetilde{T}\rightarrow
Pic^{2}_{{\mathcal{C}}/T}\times_{T}Pic^{2}_{{\mathcal{C}}/T}\times_{T}Pic^{2}_{{\mathcal{C}}/T}$

the pull-backs. Then $\displaystyle\widetilde{T}$ inherits an $\displaystyle
S_{3}$-action, and $\displaystyle{\mathcal{R}}^{*}({\mathcal{Q}})$ is its
quotient map. Moreover $\displaystyle{\mathcal{Q}}^{*}({\mathcal{R}})$
naturally defines three families of line bundles
$\displaystyle{\mathcal{L}}_{1},{\mathcal{L}}_{2},{\mathcal{L}}_{3}$ on the
family
$\displaystyle\widetilde{K}:{\mathcal{C}}_{\widetilde{T}}={\mathcal{C}}\times_{T}\widetilde{T}/\widetilde{T}$
up to an étale cover of $\displaystyle\widetilde{T}$. Then from $\displaystyle
fpqc$ descent of morphisms we can assume that the three line bundles are
defined over $\displaystyle\widetilde{T}$. Denote by
$\displaystyle\widetilde{{\mathcal{L}}}$ the pullback of
$\displaystyle{\mathcal{L}}$ to the family
$\displaystyle{\mathcal{C}}_{\widetilde{T}}/\widetilde{T}$.

Now denote by
$\displaystyle{\mathcal{P}}=\mathbb{P}_{\widetilde{T}}(\widetilde{K}_{*}(\widetilde{{\mathcal{L}}}))$,
let
$\displaystyle\pi_{\widetilde{T}}:{\mathcal{C}}_{\widetilde{T}}\rightarrow{\mathcal{P}}$
be the double cover,
$\displaystyle{\mathcal{P}}_{i}=\mathbb{P}_{\widetilde{T}}(\widetilde{K}_{*}({\mathcal{L}}_{i}))$
for $\displaystyle i=1,2,3$, and
$\displaystyle{\mathcal{O}}_{{\mathcal{P}}}(1)$ the tautological line bundle
on $\displaystyle{\mathcal{P}}$. Choose a Zariski covering $\displaystyle
u:\overline{T}\rightarrow\widetilde{T}$ such that there exists an isomorphism
$\displaystyle\widetilde{K}_{*}({\mathcal{L}}_{i})|_{\overline{T}}\cong{\mathcal{O}}_{\overline{T}}^{\oplus
2}$ for each $\displaystyle i$. It then induces a
$\displaystyle\overline{T}$-isomorphism
$\displaystyle\tau_{i}:{\mathcal{P}}_{i}\times_{T}\overline{T}\cong\overline{T}\times\mathbb{P}^{1}$
for each $\displaystyle i$. Thus it gives a morphism
$\displaystyle\kappa_{\overline{T}}:{\mathcal{C}}_{\overline{T}}={\mathcal{C}}\times_{T}\overline{T}\rightarrow\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1}$,
such that
$\displaystyle{\mathcal{L}}_{i}|_{\overline{T}}\cong(p_{i}\circ\kappa_{\overline{T}})^{*}({\mathcal{O}}_{\mathbb{P}^{1}}(1))$,
where $\displaystyle p_{i}$ is the projection of
$\displaystyle\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1}$ to its
$\displaystyle i$-th factor. By our assumption and the construction (2.10),
there exists a morphism
$\displaystyle\psi_{\overline{T}}:{\mathcal{P}}_{\overline{T}}=\overline{T}\times_{T}{\mathcal{P}}\rightarrow
Hilb_{2}(\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1})$ such that
the ramified double cover $\displaystyle\pi_{\overline{T}}$ is the pull back
of universal subscheme of lenghth two. Denote by $\displaystyle Li$ the
universal scheme of lines connecting two points on
$\displaystyle\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1}$. Let
$\displaystyle
e_{\overline{T}}:Li_{\overline{T}}=\psi_{\overline{T}}^{*}(Li)\rightarrow{\mathcal{P}}_{\overline{T}}$
be its pull back, and let $\displaystyle{\mathcal{O}}_{Li_{\overline{T}}}(1)$
be the tautological line bundle on $\displaystyle Li_{\overline{T}}$. We
describe the above morphisms in the following diagram:

(4.1)
$\displaystyle{{\mathcal{C}}_{\overline{T}}}$$\displaystyle{\mathcal{T}}$$\displaystyle{\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1}}$$\displaystyle{{\mathcal{P}}_{\overline{T}}}$$\displaystyle{Hilb^{2}(\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1})}$$\displaystyle{Li_{\overline{T}}}$$\displaystyle{Li}$$\displaystyle{\mathbb{P}(\mathbb{C}^{2}\otimes\mathbb{C}^{2}\otimes\mathbb{C}^{2})}$$\displaystyle\scriptstyle{\pi^{*}\psi_{\overline{T}}}$$\displaystyle\scriptstyle{\pi_{\overline{T}}}$$\displaystyle\scriptstyle{\pi}$$\displaystyle\scriptstyle{v}$$\displaystyle\scriptstyle{\psi_{\overline{T}}}$$\displaystyle\scriptstyle{e_{\overline{T}}}$$\displaystyle\scriptstyle{e}$

Then by our assumption, $\displaystyle
Li_{\overline{T}}=\mathbb{P}_{{\mathcal{P}}_{\overline{T}}}((\pi_{\overline{T}})_{*}\kappa_{\overline{T}}^{*}({\mathcal{O}}_{\mathbb{P}^{1}}(1)\boxtimes{\mathcal{O}}_{\mathbb{P}^{1}}(1)\boxtimes{\mathcal{O}}_{\mathbb{P}^{1}}(1)))=\mathbb{P}_{{\mathcal{P}}_{\overline{T}}}((\pi_{\overline{T}})_{*}({\mathcal{O}}_{{\mathcal{C}}_{\overline{T}}})\otimes{\mathcal{O}}_{{\mathcal{P}}_{\overline{T}}}(3))$.
As $\displaystyle\pi_{\overline{T}}$ is a ramified double cover,
$\displaystyle(\pi_{\overline{T}})_{*}({\mathcal{O}}_{{\mathcal{C}}_{\overline{T}}})={\mathcal{O}}_{{\mathcal{P}}_{\overline{T}}}\oplus{\mathcal{A}}_{-2}$
for a line bundle $\displaystyle{\mathcal{A}}_{-2}$ of degree
$\displaystyle(-2)$ on the $\displaystyle\mathbb{P}^{1}$-bundle
$\displaystyle{\mathcal{P}}_{\overline{T}}/\overline{T}$. Then the projection
$\displaystyle(\pi_{\overline{T}})_{*}({\mathcal{O}}_{{\mathcal{C}}_{\overline{T}}})\otimes{\mathcal{O}}_{{\mathcal{P}}_{\overline{T}}}(3)\rightarrow{\mathcal{A}}_{-2}\otimes{\mathcal{O}}_{{\mathcal{P}}_{\overline{T}}}(3)$
yields a section $\displaystyle
s_{\overline{T}}:{\mathcal{P}}_{\overline{T}}\rightarrow Li_{\overline{T}}$,
and hence we obtain a map $\displaystyle
l_{\overline{T}}:{\mathcal{P}}_{\overline{T}}\rightarrow\mathbb{P}(\mathbb{C}^{2}\otimes\mathbb{C}^{2}\otimes\mathbb{C}^{2})$.
Restricted to any $\displaystyle v\in\overline{T}$, one sees that the above
construction is the same as [BH16, Construction 6.30]. In particular
$\displaystyle(\l_{\overline{T}})|_{v}$ is a linear embedding for any
$\displaystyle v\in\overline{T}$. Thus it induces a morphism
$\displaystyle\beta_{\overline{T}}:\overline{T}\rightarrow{\mathcal{U}}_{D_{4}}\subseteq
Gr(2,\mathbb{C}^{2}\otimes\mathbb{C}^{2}\otimes\mathbb{C}^{2})$, such that for
any $\displaystyle v\in\overline{T}$, the image of $\displaystyle v$ is
exactly the nondegenerate plane given by the datum
$\displaystyle(\kappa_{\overline{T}}|_{v},{\mathcal{L}}|_{v})$ as in Theorem
1.5. Then we get an induced morphism
$\displaystyle\beta^{\prime}_{\overline{T}}:\overline{T}\rightarrow
M={\mathcal{U}}_{D_{4}}//H_{D_{4}}\times S_{3}$. We now check that this
morphism can be uniquely descended to a morphism from $\displaystyle T$ to
$\displaystyle M$. First if we choose any other Zariski cover
$\displaystyle\overline{T}_{2}\rightarrow\widetilde{T}$, we note that the
pullbacks of $\displaystyle\beta_{\overline{T}}^{\prime}$ and
$\displaystyle\beta_{\overline{T}_{2}}^{\prime}$ to the fiber product
$\displaystyle\overline{T}\times_{\widetilde{T}}\overline{T}_{2}$ differ from
a family of translations in $\displaystyle PGL_{2}\times PGL_{2}\times
PGL_{2}$, i.e., induced by a morphism
$\displaystyle\overline{T}\times_{\widetilde{T}}\overline{T}_{2}\rightarrow
PGL_{2}\times PGL_{2}\times PGL_{2}$. Thus by the universal property of the
GIT quotient by $\displaystyle H_{D_{4}}$ and by the $\displaystyle fpqc$
descent of morphisms, we obtain a unique morphism:
$\displaystyle\beta_{\widetilde{T}}^{\prime}:\widetilde{T}\rightarrow M$. Next
for the finite cover $\displaystyle\widetilde{T}\rightarrow T$, this is the
quotient morphism of the $\displaystyle S_{3}$-action on
$\displaystyle\widetilde{T}$. From our construction one sees that
$\displaystyle\beta_{\widetilde{T}}^{\prime}$ is $\displaystyle
S_{3}$-invariant, whence we obatin a unique morphism
$\displaystyle\beta^{\prime}_{T}:T\rightarrow M$ descending
$\displaystyle\beta^{\prime}_{\widetilde{T}}$. One can then easily check that
the above construction does not depend on the choice of the representing
datum. It induces a natural transformation
$\displaystyle\xi:{\mathcal{M}}\rightarrow h_{M}$ which satisfies the
universal property of coarse moduli spaces by considering the natural family
$\displaystyle({\mathcal{C}}_{D_{4}}/{\mathcal{U}}_{D_{4}},{\mathcal{L}}_{D_{4}},{\mathcal{R}}_{D_{4}})$.
Finally the fact that $\displaystyle\xi(Spec({\mathbb{C}}))$ is bijective
follows from Proposition 1.6(2). ∎

###### Proof of the Main Theorem.

Now given $\displaystyle G=F_{4},E_{6},E_{7}$ or $\displaystyle E_{8}$, we
have constructed in Section 3 a datum associated to
$\displaystyle{\mathcal{U}}_{G}$, namely
$\displaystyle\alpha_{G}=({\mathcal{C}}_{G}/{\mathcal{U}}_{G},{\mathcal{L}}_{G},{\mathcal{R}}_{G})$.
Thus by Proposition 1.8(2) we obtain a morphism
$\displaystyle\xi({\mathcal{U}}_{G})(\alpha_{G}):{\mathcal{U}}_{G}\rightarrow
M$, such that for any nondegenerate plane
$\displaystyle[U]\in{\mathcal{U}}_{G}$, its image is the equivalent class of
nondegenerate planes in
$\displaystyle\mathbb{C}^{2}\otimes\mathbb{C}^{2}\otimes\mathbb{C}^{2}$ given
by the datum $\displaystyle(C_{U},\pi_{U},L_{U})$ as in Theorem 1.7(2). The
morphism is $\displaystyle H_{G}$-invariant as $\displaystyle
H_{G}$-equivalent planes have isomorphic data. We then obtain a morphism
between GIT quotients of nonsingular sections
$\displaystyle\Phi_{G}:M_{G}\rightarrow M$. On the other hand recall from
Section 2.2 that there is a natural embedding of representations:
$\displaystyle(Lie(H_{D_{4}}),W_{D_{4}})\subset(Lie(H_{G}),W_{G})$. For a
plane $\displaystyle[U]\in{\mathcal{U}}_{D_{4}}$, one sees that the associated
datum $\displaystyle(C_{U},L_{U},L_{1},L_{2},L_{3})$ given by Proposition 1.6
is the same as the datum given by Theorem 1.7(2) if we view $\displaystyle[U]$
as a nondegenerate plane in $\displaystyle W_{G}$ as well. In other words we
can conclude that the natural morphism $\displaystyle\sigma_{G}:M\rightarrow
M_{G}$ constructed in Corollary 2.15 is a section of $\displaystyle\Phi_{G}$.
Finally by Theorem 1.1, both $\displaystyle M$ and $\displaystyle M_{G}$ are
varieties of dimension 3. It follows that the section
$\displaystyle\sigma_{G}$ is an isomorphism. ∎

##  Appendix A The open embedding of $\displaystyle
M\subseteq{\mathbb{P}}(1,3,4,6)$

In this appendix we follow [BH16, Section 2.4] to realize $\displaystyle
M=M_{D_{4}}//S_{3}$ as an open subset of $\displaystyle{\mathbb{P}}(1,3,4,6)$
whose complement is a hypersurface of degree 12. The main ingredient is to
consider the following moduli functor as indicated by Proposition 1.6(3):

######  Definition A.1.

Denote by $\displaystyle{\mathcal{M}}^{\prime}$ the sheaf associated to the
following presheaf:

$\displaystyle\displaystyle
P\mathcal{M}^{\prime}:(Sch/\mathbb{C})^{op}_{fppf}$
$\displaystyle\displaystyle\rightarrow(Sets)$ $\displaystyle\displaystyle T$
$\displaystyle\displaystyle\rightarrow\\{J:E/T,\sigma,\sigma_{1},\sigma_{2},\sigma_{3}\\}/\sim.$

Here ($\displaystyle J:E/T,\sigma$) is a smooth proper family of elliptic
curves, namely $\displaystyle J$ is a smooth proper family of irreducible
curves of genus one and $\displaystyle\sigma$ is a section of $\displaystyle
J$. Moreover each $\displaystyle\sigma_{j}$ is a section of $\displaystyle J$
such that

(A.1)
${\mathcal{O}}_{E}(\sum_{j=1}^{3}\sigma_{j}(T))\cong{\mathcal{O}}_{E}(3\sigma(T))\otimes
J^{*}({\mathcal{L}}),$

for some line bundle $\displaystyle{\mathcal{L}}$ on $\displaystyle T$, and
$\displaystyle\sigma(p)\not=\sigma_{j}(p)$ for any geometric point
$\displaystyle p$ of $\displaystyle T$ and for any $\displaystyle j=1,2,3$.

######  Proposition A.2.

$\displaystyle{\mathcal{M}}^{\prime}$ admits a coarse modul space as an open
subset of $\displaystyle{\mathbb{P}}(1,2,2,3)$ whose complement is a
hypersurface of degree 12. More precisely, given a datum
$\displaystyle(E,P,P_{i},i=1,2,3)$ in $\displaystyle
P{\mathcal{M}}^{\prime}(Spec({\mathbb{C}}))$. Write it in normal form as

(A.2) $y^{2}=x^{3}+a_{4}x+a_{6},$

where $\displaystyle P=(0,1)$, and we write $\displaystyle
P_{1}=(a_{2},a_{3}),P_{2}=(a_{2}^{\prime},a_{3}^{\prime}),P_{3}=(a_{2}^{\prime\prime},a_{3}^{\prime\prime})$.
Let $\displaystyle a_{1}$ be the slope of the line connecting $\displaystyle
P_{1}$ and $\displaystyle P_{2}$. Then the correponding point in
$\displaystyle{\mathbb{P}}(1,2,2,3)$ is
$\displaystyle[a_{1}:a_{2}:a_{2}^{\prime}:a_{3}]$. Moreover the following
equality holds:

(A.3) $\displaystyle\displaystyle a_{3}^{\prime}$
$\displaystyle\displaystyle=a_{3}+a_{1}(a_{2}^{\prime}-a_{2}),a_{2}^{\prime\prime}=a_{1}^{2}-a_{2}-a_{2}^{\prime},a_{3}^{\prime\prime}=a_{1}^{3}-3(a_{1}a_{2}-a_{3})-a_{3}-a_{3}^{\prime};$
$\displaystyle\displaystyle a_{4}$
$\displaystyle\displaystyle=a_{1}(a_{3}+a_{3}^{\prime})-(a_{2}^{2}+a_{2}a_{2}^{\prime}+a_{2}^{\prime
2}),a_{6}=a^{2}_{3}-a_{1}a_{2}(a_{3}+a_{3}^{\prime})+a_{2}a_{2}^{\prime}(a_{2}+a_{2}^{\prime}),$

and we have $\displaystyle\Delta(E):=4a_{4}^{3}+27a_{6}^{2}\not=0$.

###### Proof.

We write
$\displaystyle{\mathbb{P}}(1,2,2,3)=Proj({\mathbb{C}}[c_{1},c_{2},c_{2}^{\prime},c_{3}])$,
where $\displaystyle c_{i},c_{j}^{\prime}$ are generators of degree
$\displaystyle i,j$ respectively. Denote $\displaystyle c_{3}^{\prime},c_{4},$
and $\displaystyle c_{6}$ by the same formula as (A.3) in terms of
$\displaystyle c_{1},c_{2},c_{2}^{\prime}$ and $\displaystyle c_{3}$. Let
$\displaystyle
M^{\prime}=D_{+}(4c_{4}^{3}+27c_{6}^{2})\subseteq{\mathbb{P}}(1,2,2,3)$. Then
we aim to construct a natural transformation
$\displaystyle\xi^{\prime}:{\mathcal{M}}^{\prime}\rightarrow h_{M^{\prime}}$
which makes $\displaystyle M^{\prime}$ to be the coarse moduli space of
$\displaystyle{\mathcal{M}}^{\prime}$. To do so, we are going to write a
family of elliptic curves (Zariski locally) in Weierstrass forms. More
precisely we recall the following standard fact:

######  Lemma A.3.

Let $\displaystyle(J:E/T,\sigma)$ be a family of elliptic curves over
$\displaystyle T$. Denote by $\displaystyle D=\sigma(T)$ the effective Cartier
divisor of $\displaystyle E$ and denote by
$\displaystyle{\mathcal{L}}={\mathcal{O}}_{E}(D)$ the associated line bundle
on $\displaystyle E$. Then:

(1) There exists an open covering of $\displaystyle T=\cup_{i}T_{i}$ such that
$\displaystyle J_{*}({\mathcal{L}}^{\otimes{k}})$ and $\displaystyle
J_{*}({\mathcal{L}}^{\otimes{k+1}})/J_{*}({\mathcal{L}}^{\otimes{k}})$ are
free of rank $\displaystyle k$ and $\displaystyle 1$ respectively on each
$\displaystyle T_{i}$ and for any $\displaystyle k\leqslant 6$.

(2) For each $\displaystyle T_{i}$ there exists a choice of a basis of
$\displaystyle J_{*}({\mathcal{L}}^{\otimes{3}})(T_{i})$ namely $\displaystyle
1,x_{i},y_{i}$ such that $\displaystyle x_{i}\in
J_{*}({\mathcal{L}}^{\otimes{2}})(T_{i})\backslash
J_{*}({\mathcal{L}})(T_{i})$, $\displaystyle y_{i}\in
J_{*}({\mathcal{L}}^{\otimes{3}})(T_{i})\backslash
J_{*}({\mathcal{L}}^{\otimes{2}})(T_{i})$ and satsfying:

(A.4) $y_{i}^{2}=x_{i}^{3}+a_{4,i}x_{i}+a_{6,i},$

for some $\displaystyle a_{4,i},a_{6,i}\in{\mathcal{O}}(T_{i})$.

(3) For each $\displaystyle T_{i}$ the sections $\displaystyle
1,x_{i},y_{i}\in{\mathcal{L}}^{\otimes{3}}(J^{-1}(T_{i}))$ induces a closed
embedding of $\displaystyle J^{-1}(T_{i})$ into
$\displaystyle{\mathbb{P}}^{2}_{T_{i}}$ whose image is $\displaystyle
V_{+}(y_{i}^{2}z_{i}-(x_{i}^{3}+a_{4,i}x_{i}z_{i}^{2}+a_{6,i}z_{i}^{3}))$.
Moreover $\displaystyle\sigma|_{T_{i}}=[0:1:0]$ and $\displaystyle E\cap
J^{-1}(T_{i})$ equlas to the hyperplane section $\displaystyle
V_{+}(z_{i})\cap J^{-1}(T_{i})$.

(4) For any $\displaystyle i,i^{\prime}\in I$, we have $\displaystyle
x_{i}|_{T_{i}\cap T_{j}}=n_{i}x_{j}|_{T_{i}\cap T_{j}}$ and $\displaystyle
y_{i}|_{T_{i}\cap T_{j}}=m_{i}y_{j}|_{T_{i}\cap T_{j}}$ for some
$\displaystyle m_{i},n_{i}\in{\mathcal{O}}(T_{i}\cap T_{j})^{*}$ such that
$\displaystyle m_{i}^{2}=n_{i}^{3}$.

Now for a given a datum
$\displaystyle(J:E/T,\sigma,\sigma_{1},\sigma_{2},\sigma_{3})$ representing a
class in $\displaystyle P{\mathcal{M}}^{\prime}(T)$, we can choose
$\displaystyle T=\cup_{i}T_{i}$ and suitable sections $\displaystyle
x_{i},y_{i}$ satisfying the properties in Lemma A.3. From Definition A.1 the
image of $\displaystyle\sigma_{j}|_{T_{i}}$ lies in $\displaystyle
D_{+}(z_{i})$. Thus we can write
$\displaystyle\sigma_{1}|_{T_{i}}=[a_{2,i}:a_{3,i}:1]$,
$\displaystyle\sigma_{2}|_{T_{i}}=[a_{2,i}^{\prime}:a_{3,i}^{\prime}:1]$ and
$\displaystyle\sigma_{3}|_{T_{i}}=[a_{3,i}^{\prime\prime}:a_{3,i}^{\prime\prime}:1]$
for some $\displaystyle
a_{2,i},a_{2,i}^{\prime},a_{3,i},a_{3,i}^{\prime},a_{2,i}^{\prime\prime},a_{3,i}^{\prime\prime}\in{\mathcal{O}}(T_{i})$.
By (A.1) and Lemma A.3(3) we conclude that
$\displaystyle\sigma_{1}(T_{i})+\sigma_{2}(T_{i})+\sigma_{3}(T_{i})=H_{i}\cap
J^{-1}(T_{i})$ for some hyperplane $\displaystyle
H_{i}=V_{+}(\alpha_{i}x_{i}+\beta_{i}y_{i}+\gamma_{i}z_{i})\subseteq{\mathbb{P}}_{T_{i}}^{2}$.
As $\displaystyle\sigma_{j}(p)\not=\sigma(p)=[0:1:0]$ for any $\displaystyle
j$ and for any geometric point $\displaystyle p$ on $\displaystyle T_{i}$, we
conclude that $\displaystyle\beta_{i}$ is invertible in
$\displaystyle{\mathcal{O}}(T_{i})$. And for any point $\displaystyle p\in
T_{i}$, the slope of the lines connecting
$\displaystyle\sigma_{1}(p),\sigma_{2}(p),\sigma_{3}(p)$ in
$\displaystyle{\mathbb{P}}_{k(p)}^{2}$ is given by
$\displaystyle(\beta_{i}^{-1}\alpha_{i})(p)$. Denote by $\displaystyle
a_{1,i}=\beta_{i}^{-1}\alpha_{i}$. Then the equation (A.3) holds for
$\displaystyle
a_{3,i}^{\prime},a_{3,i}^{\prime\prime},a_{2,i}^{\prime\prime},a_{4,i}$ and
$\displaystyle a_{6,i}$ in terms of $\displaystyle
a_{1,i},a_{2,i},a_{2,i}^{\prime},a_{3,i}$. To show this, it suffices to check
on any geometric point $\displaystyle t\in T_{i}$, which follows from [BH16,
Section 2.4]. Here we note that there is a flaw in the expression of
$\displaystyle a_{2}^{\prime\prime}$ in [BH16, Section 2.4]: the coefficient
of the term $\displaystyle a_{1}^{2}$ should be 1 instead of 3, which can be
checked directly. Next we recall the following fact:

######  Lemma A.4.

Let $\displaystyle R$ be a ring, then the closed subscheme of
$\displaystyle{\mathbb{P}}_{R}^{2}$ defined by the Weierstrass equation

$y^{2}z=x^{3}+b_{4}xz^{2}+b_{6}z^{3}$

is an elliptic curve over $\displaystyle R$ if and only if
$\displaystyle\Delta=4b_{4}^{3}+27b_{6}^{2}$ is invertible in $\displaystyle
R$.

Thus we conclude that $\displaystyle\Delta_{i}=4a_{4,i}^{3}+27a_{6,i}^{2}$ is
invertible in $\displaystyle{\mathcal{O}}(T_{i})$ for each $\displaystyle
T_{i}$. Now for each $\displaystyle T_{i}$ we can assign it a morphism
$\displaystyle f_{i}$ to $\displaystyle
M^{\prime}=D_{+}(4c_{4}^{3}+27c_{6}^{2})$ by

$\displaystyle\displaystyle
f_{i}(t)=[a_{1,i}(t):a_{2,i}(t):a_{2,i}^{\prime}(t):a_{3,i}(t)],$

for any point $\displaystyle t\in T_{i}$. Then by Lemma A.3(4) we conclude
that $\displaystyle\\{f_{i}:T_{i}\rightarrow M^{\prime},i\in I\\}$ glues to a
morphism $\displaystyle f:T\rightarrow M^{\prime}$ which does not depend on
the choice of the open covering and the choice of sections $\displaystyle
x_{i},y_{i}$ as well as the choice of the representative of a given class.
Furthermore one can easily check that this induces a natural transformation
$\displaystyle{\mathcal{M}}^{\prime}\rightarrow h_{M^{\prime}}$. On the other
hand, denote by
$\displaystyle\widetilde{M}^{\prime}={\mathbb{C}}^{4}\backslash V(\Delta)$
where $\displaystyle\Delta$ is defined as in (A.3). Then one sees that there
exists a natural datum
$\displaystyle\alpha^{\prime}\in{\mathcal{M}}^{\prime}(\widetilde{M}^{\prime})$
associated to $\displaystyle\widetilde{M}^{\prime}$ given as above. And the
induced morphism from $\displaystyle\widetilde{M}^{\prime}$ to $\displaystyle
M^{\prime}$ is given by the GIT quotient of the correponding
$\displaystyle{\mathbb{C}}^{*}$-action. Then using the datum
$\displaystyle\alpha^{\prime}$ one can easily verify the universal property of
coarse moduli spaces. Finally the fact that
$\displaystyle\xi^{\prime}(Spec({\mathbb{C}}))$ is bijective is clear from our
construction. ∎

Next we relate $\displaystyle M^{\prime}$ to the space $\displaystyle M$ by
constructing a natural datum in $\displaystyle
P{\mathcal{M}}^{\prime}({\mathcal{U}}_{D_{4}})$. Recall from diagram (3.2)
that there is a morphism
$\displaystyle\kappa_{D_{4}}:{\mathcal{C}}_{D_{4}}\rightarrow{\mathbb{P}}^{1}\times{\mathbb{P}}^{1}\times{\mathbb{P}}^{1}$
and there is a line bundle of degree two $\displaystyle{\mathcal{L}}_{D_{4}}$
on $\displaystyle{\mathcal{C}}_{D_{4}}/{\mathcal{U}}_{D_{4}}$. Moreover there
are three line bundles of degree two on
$\displaystyle{\mathcal{C}}_{D_{4}}/{\mathcal{U}}_{D_{4}}$ defined by
$\displaystyle{\mathcal{L}}_{i}=(p_{i}\circ\kappa_{D_{4}})^{*}({\mathcal{O}}_{{\mathbb{P}}^{1}}(1))$
for $\displaystyle i=1,2,3$. By Lemma 3.9, the sheaf $\displaystyle
Pic_{({\mathcal{C}}_{G}/{\mathcal{U}}_{G})(\text{\'{e}tale})}^{2}$ is
represented by a scheme $\displaystyle
Pic^{2}({\mathcal{C}}_{D_{4}}/{\mathcal{U}}_{D_{4}})$ which is a smooth
projective family of curves of genus one over
$\displaystyle{\mathcal{U}}_{D_{4}}$. Thus the line bundles
$\displaystyle{\mathcal{L}}_{D_{4}},{\mathcal{L}}_{1},{\mathcal{L}}_{2},{\mathcal{L}}_{3}$
are given by sections of $\displaystyle
Pic^{2}({\mathcal{C}}_{D_{4}}/{\mathcal{U}}_{D_{4}})/{\mathcal{U}}_{D_{4}}$
which we denote by $\displaystyle\sigma,\sigma_{1},\sigma_{2}$ and
$\displaystyle\sigma_{3}$ respectively. Then
$\displaystyle(Pic^{2}({\mathcal{C}}_{D_{4}}/{\mathcal{U}}_{D_{4}})/{\mathcal{U}}_{D_{4}},\sigma)$
is a family of elliptic curves and from Proposition 1.6(2) we conclude that
the datum
$\displaystyle(Pic^{2}({\mathcal{C}}_{D_{4}}/{\mathcal{U}}_{D_{4}})/{\mathcal{U}}_{D_{4}},\sigma,\sigma_{i},i=1,2,3)$
defines a datum
$\displaystyle\beta_{D_{4}}\in{\mathcal{M}}^{\prime}({\mathcal{U}}_{D_{4}})$.

The open embedding of $\displaystyle M$ into
$\displaystyle{\mathbb{P}}(1,3,4,6)$ is given as follows:

######  Proposition A.5.

Let
$\displaystyle\xi^{\prime}({\mathcal{U}}_{D_{4}})(\beta_{D_{4}}):{\mathcal{U}}_{D_{4}}\rightarrow
M^{\prime}$ be the morphism induced by $\displaystyle\beta_{D_{4}}$ then:

(1). $\displaystyle\xi^{\prime}({\mathcal{U}}_{D_{4}})(\beta_{D_{4}})$ is
$\displaystyle H_{D_{4}}$-invariant and the induced quotient map
$\displaystyle M_{D_{4}}\rightarrow M^{\prime}$ is an isomorphism.

(2). The $\displaystyle S_{3}$-action on $\displaystyle M_{D_{4}}\cong
M^{\prime}$ induces a natural $\displaystyle S_{3}$-action on
$\displaystyle{\mathbb{P}}(1,2,2,3)$. The induced quotient map gives an open
embedding of $\displaystyle M=M_{D_{4}}//S_{3}$ into
$\displaystyle{\mathbb{P}}(1,2,2,3)//S_{3}\cong{\mathbb{P}}(1,3,4,6)$ whose
complement is a hypersurface of degree 12.

###### Proof.

That $\displaystyle\xi^{\prime}({\mathcal{U}}_{D_{4}})(\beta_{D_{4}})$ is
$\displaystyle H_{D_{4}}$-invariant follows from the fact that $\displaystyle
H_{D_{4}}$ naturally acts on the family
$\displaystyle{\mathcal{C}}_{D_{4}}/U_{D_{4}}$ and the line bundles
$\displaystyle{\mathcal{L}},{\mathcal{L}}_{1},{\mathcal{L}}_{2},{\mathcal{L}}_{3}$.
As both $\displaystyle M_{D_{4}}$ and $\displaystyle M^{\prime}$ are normal
varieties, it suffices to show that the quotient map is bijective. By
Proposition 1.6(1) and Corollary 1.9(1), $\displaystyle M_{D_{4}}$
parametrizes isomorphic classes of data consisting of
$\displaystyle(C,L,L_{i},i=1,2,3)$. And the quotient morphism maps an
isomorphic class of $\displaystyle(C,L,L_{i},i=1,2,3)$ to the class of
$\displaystyle(Jac^{2}(C),L,L_{i},i=1,2,3)$. Then one can easily verify the
bijectivity of the quotient map by choosing an isomrophism between a given
curve $\displaystyle C$ of genus one and its Jacobian $\displaystyle
Jac^{2}(C)$.

To see (2), note that for a given class of datum
$\displaystyle(E,P,P_{1},P_{2},P_{3})$ as in Proposition A.2, the induced
$\displaystyle S_{3}$-action on the datum is given by the corresponding
permutation on $\displaystyle P_{1},P_{2}$ and $\displaystyle P_{3}$. Then
following the descripition in Proposition A.2, one sees that the
$\displaystyle S_{3}$-action is induced by an $\displaystyle S_{3}$-action on
$\displaystyle{\mathbb{P}}(1,2,2,3)$ defined by:

$\displaystyle\displaystyle(12).c_{1}$
$\displaystyle\displaystyle=c_{1},(12).c_{2}=c_{2}^{\prime},(12).c_{2}^{\prime}=c_{2},(12).c_{3}=c_{3}+c_{1}(c_{2}^{\prime}-c_{2});$
$\displaystyle\displaystyle(123).c_{1}$
$\displaystyle\displaystyle=c_{1},(123).c_{2}=c_{2}^{\prime},(123).c_{2}^{\prime}=c_{1}^{2}-c_{2}-c_{2}^{\prime},(123).c_{3}=c_{3}+c_{1}(c_{2}^{\prime}-c_{2}).$

Then a direct calculation yields that the invariant subring
$\displaystyle{\mathbb{C}}[c_{1},c_{2},c_{2}^{\prime},c_{3}]^{S_{3}}={\mathbb{C}}[d_{1},d_{3},d_{4},d_{6}]$
is a polynomial ring in four generators $\displaystyle
d_{1},d_{3},d_{4},d_{6}$ of degree 1,3,4,6 respectively. More precisely:

(A.5) $\displaystyle\displaystyle d_{1}$ $\displaystyle\displaystyle=c_{1},$
$\displaystyle\displaystyle d_{3}$
$\displaystyle\displaystyle=c_{1}c_{2}-c_{3},$ $\displaystyle\displaystyle
d_{4}$ $\displaystyle\displaystyle=c_{2}^{2}+c_{2}^{\prime
2}+c_{2}c_{2}^{\prime}-c_{1}^{2}(c_{2}+c_{2}^{\prime}),$
$\displaystyle\displaystyle d_{6}$
$\displaystyle\displaystyle=c_{2}c_{2}^{\prime}(c_{2}+c_{2}^{\prime}-c_{1}^{2}).$

The above construction induces an open embedding of $\displaystyle
M=M_{D_{4}}//S_{3}$ into
$\displaystyle{\mathbb{P}}(1,2,2,3)//S_{3}\cong{\mathbb{P}}(1,3,4,6)$. Note
that the $\displaystyle S_{3}$-action on a given datum
$\displaystyle(C,L,L_{1},L_{2},L_{3})$ fixes the defining equation of the
elliptic curve, whence it fixes $\displaystyle\Delta=4c_{4}^{3}+27c_{6}^{2}$.
This implies that the image of the open embedding of $\displaystyle M$ into
$\displaystyle{\mathbb{P}}(1,3,4,6)$ equals to the principle open subset
$\displaystyle D_{+}(\Delta)$. ∎

## Acknowledgment

I am grateful to Thomas Dedieu and Laurent Manivel for kindly introducing me
this problem, and for their helpful discussions. I am particularly indebted to
Laurent Manivel for valuable suggestions on this article, and for providing me
the important reference [BH16]. I would like to thank Baohua Fu for his
constant support and for his suggestions on this article. Most part of this
article was written during my visit at Toulouse Mathematics Institute in 2023,
supported by a scholarship from the University of Chinese Academy of Sciences.
I would like to thank Toulouse Mathematics Institute for its hospitality.

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