$D_{\alpha}$ while $\gamma_{0}^{-}$ is dual to
$D_{{\overline{\sigma}}(\alpha)}$. We get
$2\langle{\overline{\Theta}}^{\vee},\lambda_{\alpha}\rangle=\langle\Theta^{\vee},\lambda_{\alpha}\rangle=1=D_{\alpha}\cdot(\gamma_{0}^{+}+\gamma_{0}^{-})$
by Proposition 4.3.3. Similarly, we have
$2\langle{\overline{\Theta}}^{\vee},\lambda_{{\overline{\sigma}}(\alpha)}\rangle=\langle\Theta^{\vee},\lambda_{{\overline{\sigma}}(\alpha)}\rangle=1=D_{{\overline{\sigma}}(\alpha)}\cdot(\gamma_{0}^{+}+\gamma_{0}^{-})$.
∎

Recall the definitions of the nilpotent orbits ${\mathcal{O}}_{\min}$ and of
type ${\mathcal{O}}_{{\rm sum}}$ from Definition 2.17. For $G$ simple with
maximal torus $T_{s}$ of split type such that $\sigma(\Theta)\neq-\Theta$,
define the nilpotent orbit ${\mathcal{O}}_{{{\rm sum}},\sigma}$ by
${\mathcal{O}}_{{{\rm sum}},\sigma}=G\cdot(e_{\Theta}-\sigma(e_{\Theta}))$
with $e_{\Theta}\in\mathfrak{g}_{\Theta}\setminus\\{0\\}$. Note that by
Proposition 4.3.3, the nilpotent orbit ${\mathcal{O}}_{{{\rm sum}},\sigma}$ is
indeed of type ${\mathcal{O}}_{{\rm sum}}$.

######  Proposition 4.23.

There exists a smooth rational curve $C$ in $X$ such that $x\in C$ and
$[C]=\gamma$, $\gamma_{0}^{+}$ or $\gamma_{0}^{-}$.

Furthermore, we have $T_{x}C\setminus\\{0\\}\subset{\mathcal{O}}_{\min}$ if
$\sigma(\Theta)=-\Theta$, and
$T_{x}C\setminus\\{0\\}\subset{\mathcal{O}}_{{{\rm sum}},\sigma}$ otherwise.

###### Proof.

If $\sigma(\Theta)=-\Theta$, pick
$e=e_{\Theta}\in\mathfrak{g}_{\Theta}\setminus\\{0\\}$. If
$\sigma(\Theta)\neq-\Theta$, pick $e=e_{\Theta}-\sigma(e_{\Theta})$ with
$e_{\Theta}$ as above. Note that $e\in{\mathcal{O}}_{\min}$ for
$\sigma(\Theta)=-\Theta$ and that, by Proposition 4.3.2, $e$ is in a nilpotent
orbit of type ${\mathcal{O}}_{{\rm sum}}$ for $\sigma(\Theta)\neq-\Theta$. Set
$f=\sigma(e)$. We may choose $e$ so that
$h=[e,f]=2{\overline{\Theta}}^{\vee}$. Then $(e,h,f)$ is a
$\mathfrak{sl}_{2}$-triple. The cocharacter $h$ induces a morphism
$h:{\mathbb{P}}^{1}\to X$ which factors through
${\overline{\Theta}}^{\vee}:{\mathbb{P}}^{1}\to X$

$\textstyle{{\mathbb{P}}^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h}$$\scriptstyle{2:1}$$\textstyle{{\mathbb{P}}^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\overline{\Theta}}^{\vee}}$$\textstyle{X}$

so that the vertical map is a double cover. Note in particular that both maps
$h$ and ${\overline{\Theta}}^{\vee}$ have the same image $C^{\prime}$ in $X$.

Denote by $G(h)$ the closed subgroup of $G$ with Lie algebra $\langle
e,h,f\rangle$. Then $G(h)$ is isomorphic to $\operatorname{SL}_{2}$ or ${\rm
PGL}_{2}$, and $\sigma$ acts non-trivially on $G(h)$. In particular, we have
an isogeny $\operatorname{SL}_{2}\to G(h)$ and $\sigma$ lifts to a unique
involution on $\operatorname{SL}_{2}$. Let $T^{\prime}$ be a maximal torus of
$\operatorname{SL}_{2}$ fixed pointwise by $\sigma$ and let $X^{\prime}$ be
the closure of $G(h)\cdot x$ in $X$. Then $X^{\prime}$ is either isomorphic to
${\mathbb{P}}^{1}\times{\mathbb{P}}^{1}$ (the unique proper embedding of
$\operatorname{SL}_{2}/T^{\prime}$) or to ${\mathbb{P}}^{2}$ (the unique
proper embedding of
$\operatorname{SL}_{2}/N_{\operatorname{SL}_{2}}(T^{\prime})$). Note that in
any case $x\in X^{\prime}$; thus, $X^{\prime}\cap X^{0}$ is non-empty. In the
first case, the curve $C^{\prime}$ is linearly equivalent to the diagonal
curve $D$. In the second case, the curve $C^{\prime}$ is a line in
${\mathbb{P}}^{2}$. In both cases, there exists a line $L$ through $x$ in
$X^{\prime}$ so that $T_{x}L$ is equal to $\langle e\rangle$, as a subset of
$T_{x}X$ identified to $\mathfrak{p}$.

If $\overline{R}$ is of type $\textrm{A}_{1}$, then $X$ is non-exceptional and
${\overline{\Theta}}^{\vee}$ has degree $2$ onto its image $C^{\prime}$. In
particular
$[C^{\prime}]=\frac{1}{2}{\overline{\Theta}}^{\vee}_{*}[{\mathbb{P}}^{1}]=\gamma$
and by minimality, $\gamma_{0}$ has to be the class of a line in $X^{\prime}$.
Note that this implies that we are in the case $X^{\prime}={\mathbb{P}}^{2}$.

Assume that $\overline{R}$ is not of type $\textrm{A}_{1}$. Then
${\overline{\Theta}}^{\vee}:{\mathbb{P}}^{1}\to X$ is an isomorphism onto its
image $C^{\prime}$. Note also that $h\in\mathfrak{X}^{\vee}$ is indivisible as
a cocharacter of $T_{{\rm s}}$ by Proposition 4.3.4; thus, $h(-1)$ is non-
central in $G$. Since $h(-1)$ is central in $G(h)$, this group is isomorphic
to $\operatorname{SL}_{2}$ and since $h(-1)$ is non-central in $G$, we have
$G(h)^{\sigma}=T^{\prime}$ and
$X^{\prime}={\mathbb{P}}^{1}\times{\mathbb{P}}^{1}$. Therefore,
$[C^{\prime}]=[D]$. Define $[C^{+}]$ and $[C^{-}]$ to be the classes of the
two rulings in ${\mathbb{P}}^{1}\times{\mathbb{P}}^{1}\subset X$. Then
$[C^{+}]+[C^{-}]=[D]=[C^{\prime}]$. Since any ruling meets $X^{\prime}\cap
X^{0}=X^{\prime}\setminus D$, the classes $[C^{+}]$ and $[C^{-}]$ are
virtually covering. This implies that $[C^{+}]=[C^{-}]=\gamma_{0}$ if $X$ is
non-exceptional and (up to exchanging the two rulings)
$[C^{+}]=\gamma_{0}^{+}$ and $[C^{-}]=\gamma_{0}^{-}$ if $X$ is exceptional. ∎

We may now complete Proposition 3.5 by computing the classes of curves in
minimal families.

######  Corollary 4.24.

  1. 1.

If $X$ is non-exceptional, there exists a unique family ${\mathcal{K}}$ of
minimal rational curves and for $C\in{\mathcal{K}}$, we have $[C]=\gamma_{0}$.

  2. 2.

If $X$ is exceptional, there exists two families ${\mathcal{K}}^{+}$ and
${\mathcal{K}}^{-}$ of minimal rational curves and for
$C\in{\mathcal{K}}^{\pm}$, we have $[C]=\gamma_{0}^{\pm}$.

###### Proof.

Follows from Proposition 3.5, Corollary 4.20 and Proposition 4.23. ∎

######  Corollary 4.25.

Let ${\mathcal{K}}$ be a minimal family, and $C\in{\mathcal{K}}_{x}$. If
$\sigma(\Theta)=-\Theta$, then we have
$T_{x}C\setminus\\{0\\}\subset{\mathcal{O}}_{\min}$. Otherwise,
$T_{x}C\setminus\\{0\\}\subset{\mathcal{O}}_{{{\rm sum}},\sigma}$.

We now compute the dimensions of the minimal families ${\mathcal{K}}$ and
${\mathcal{K}}^{\pm}$ and of the nilpotent orbits ${\mathcal{O}}_{\min}$ and
${\mathcal{O}}_{{{\rm sum}},\sigma}$. Let $\rho$ be the half-sum of positive
roots in $G$ and $\rho_{P}$ be the half-sum of positive roots in $P$. Set
$\kappa=2\rho-2\rho_{P}$. We have $\kappa=\sum_{\alpha\in
R^{+},\sigma(\alpha)<0}\alpha$. Let
$\Sigma=\sum_{{\overline{\alpha}}\in\overline{\Delta}}{\overline{\alpha}}$ be
the sum of all restricted simple roots.

######  Theorem 4.26.

Let ${\mathcal{K}}$ be a minimal family, let $C\in{\mathcal{K}}_{x}$ and $m\in
T_{x}C\setminus\\{0\\}$. We have

$\dim{\mathcal{K}}_{x}=\langle{\overline{\Theta}}^{\vee},\kappa+\Sigma\rangle-2\textrm{
and }\dim G\cdot m=2\langle{\overline{\Theta}}^{\vee},\kappa\rangle.$

In particular, $\dim{\mathcal{K}}_{x}=\frac{1}{2}\dim G\cdot
m-1+(\langle{\overline{\Theta}}^{\vee},\Sigma\rangle-1)$.

######  Remark 4.27.

Note that the value of $\langle{\overline{\Theta}}^{\vee},\Sigma\rangle$
depends on the type of $\overline{R}$ as follows:

$\partial X\cdot
C=\langle{\overline{\Theta}}^{\vee},\Sigma\rangle=\left\\{\begin{array}[]{cl}2&\textrm{if
$\overline{R}$ is of type ${\rm A}_{r}$ with $r\geq 1$,}\\\
1&\textrm{otherwise.}\\\ \end{array}\right.$

###### Proof.

Recall from Proposition 3.6 that $\dim{\mathcal{K}}_{x}=-K_{X}\cdot C-2$. By
adjunction formula, we have $(-K_{X})|_{Y}=-K_{Y}+\partial X|_{Y}$. Since
${\mathcal{L}}_{Y}(-K_{Y})={\mathcal{L}}_{Y}(\kappa)$ and
$i^{*}{\mathcal{L}}_{X}(\partial X)={\mathcal{L}}_{Y}(\Sigma)$, we get
$i^{*}{\mathcal{L}}_{X}(-K_{X})={\mathcal{L}}_{Y}(\kappa+\Sigma)$ and
$\dim{\mathcal{K}}_{x}=\langle{\overline{\Theta}}^{\vee},\kappa+\Sigma\rangle-2$.

We now compute the dimension of $G\cdot m$. Assume first that
$\sigma(\Theta)=-\Theta$, then $G\cdot m={\mathcal{O}}_{\min}$. It is a
classical result that
$\dim{\mathcal{O}}_{\min}=2\langle\Theta^{\vee},\rho\rangle$ (see for example
[CP11] or [BF15, Lemma 4.1]). On the other hand, we have
${\overline{\Theta}}^{\vee}=\frac{1}{2}\Theta^{\vee}$. Thus,
$\langle{\overline{\Theta}}^{\vee},\kappa\rangle=\langle\Theta^{\vee},\rho\rangle-\langle\Theta^{\vee},\rho_{P}\rangle$.
Since $\sigma(\rho_{P})=\rho_{P}$, we get
$\langle\Theta^{\vee},\rho_{P}\rangle=\langle\sigma(\Theta),\sigma(\rho_{P})\rangle=-\langle\Theta^{\vee},\rho_{P}\rangle$
thus $\langle\Theta^{\vee},\rho_{P}\rangle=0$ and $\dim G\cdot
m=2\langle\Theta^{\vee},\rho\rangle=2\langle{\overline{\Theta}}^{\vee},\kappa\rangle$.

Assume now that $\sigma(\Theta)\neq-\Theta$, then we have $G\cdot
m={\mathcal{O}}_{{{\rm sum}},\sigma}$. To compute the dimension of the orbit
${\mathcal{O}}_{{{\rm sum}},\sigma}$, we recall some facts on nilpotent
elements obtained as sums of weight vectors (see [Pa99] and [FR08]). We may
assume $m=e\in\mathfrak{g}$ is a nilpotent element which can be written as a
sum $e=e_{\Theta}+e_{-\sigma(\Theta)}$ with
$e_{\Theta}\in\mathfrak{g}_{\Theta}\setminus\\{0\\}$ and
$e_{-\sigma(\Theta)}\in\mathfrak{g}_{-\sigma(\Theta)}\setminus\\{0\\}$. Recall
that $\Theta$ and $-\sigma(\Theta)$ are strongly orthogonal long roots. Let
$f,h\in\mathfrak{g}$ such that $(e,h,f)$ is a $\mathfrak{sl}_{2}$-triple. We
may assume that
$h=\Theta^{\vee}-\sigma(\Theta)^{\vee}=2{\overline{\Theta}}^{\vee}$ and
$f=\sigma(e)$. Then $h$ induces a grading on
$\mathfrak{g}=\mathfrak{g}(-2)\oplus\mathfrak{g}(-1)\oplus\mathfrak{g}(0)\oplus\mathfrak{g}(1)\oplus\mathfrak{g}(2)$,
where $f\in\mathfrak{g}(-2)$ and $e\in\mathfrak{g}(2)$. Furthermore, if $Q$ is
the subgroup of $G$ with Lie algebra
$\mathfrak{g}(0)\oplus\mathfrak{g}(1)\oplus\mathfrak{g}(2)$, then $Q$ is a
parabolic subgroup and if $L$ is the subgroup of $G$ with Lie algebra
$\mathfrak{g}(0)$, then $L$ is a Levi subgroup of $Q$ and the orbit $L\cdot e$
is dense in $\mathfrak{g}(2)$. In addition, we have $G\cdot
e=G\times^{Q}(L\cdot e)$ so that $\dim G\cdot
e=\dim(G/Q)+\dim\mathfrak{g}(2)=\dim\mathfrak{g}(1)+2\dim\mathfrak{g}(2)$.

If $\sigma(\alpha)=\alpha$, then
$\langle\Theta^{\vee}-\sigma(\Theta^{\vee}),\alpha\rangle=\langle\Theta^{\vee},\alpha\rangle-\langle\Theta^{\vee},\sigma(\alpha)\rangle=0$
and $\langle\Theta^{\vee}-\sigma(\Theta^{\vee}),\rho_{P}\rangle=0$. If
$\alpha\in R^{+}$ is such that $\mathfrak{g}_{\alpha}\subset\mathfrak{g}(i)$
for $i\geq 0$, then
$\langle\Theta^{\vee}-\sigma(\Theta^{\vee}),\alpha\rangle=i$, thus

$\langle\Theta^{\vee}-\sigma(\Theta^{\vee}),\kappa\rangle=\langle\Theta^{\vee}-\sigma(\Theta^{\vee}),2\rho\rangle=\sum_{i=0}^{2}\sum_{\alpha\in
R^{+},\
\mathfrak{g}_{\alpha}\subset\mathfrak{g}(i)}i=\dim\mathfrak{g}(1)+2\dim\mathfrak{g}(2).$

This proves the result, since
$\langle\Theta^{\vee}-\sigma(\Theta^{\vee}),\kappa\rangle=2\langle{\overline{\Theta}}^{\vee},\kappa\rangle$.
∎

###  4.4 Contact structure

In this subsection, we compute the dimension of $H\cdot m$ for $m\in
T_{x}C\setminus\\{0\\}$, $C\in{\mathcal{K}}_{x}$ and ${\mathcal{K}}$ a minimal
family. We first gather some facts on orbits associated with symmetric spaces,
and in particular prove that orbits of symmetric subgroups of $G$ are
Lagrangian subvarieties in nilpotent $G$-orbits. Recall the following general
definitions.

######  Definition 4.28.

Let $\widehat{M}$ be a smooth complex variety of dimension $2n+2$ and let $M$
be a smooth complex variety of dimension $2n+1$.

  1. 1.

A symplectic structure on $\widehat{M}$ is a closed symplectic form
$\omega:T\widehat{M}\times T\widehat{M}\to\widehat{M}\times{\mathbb{C}}$.

  2. 2.

A contact structure on $M$ is an everywhere non-vanishing map
$\eta:TM\to{\mathcal{L}}$, where ${\mathcal{L}}$ is a line bundle, such that
the bilinear form $\theta_{\eta}:D\times D\rightarrow TM/D$ defined by
$(u,v)\mapsto[u,v]\ (\text{mod}\ D)$ on $D:={\rm Ker}\,\eta$ is non-degenerate
for all $m\in M$.

If $\eta:TM\to{\mathcal{L}}$ is a contact structure on $M$, then there is a
natural symplectic structure $\omega$ defined by $\omega=d(p^{*}\eta)$ on
$\widehat{M}={\mathcal{L}}^{\times}$, where ${\mathcal{L}}^{\times}$ is the
${\mathbb{C}}^{\times}$-bundle over $M$ with structure map $p:\widehat{M}\to
M$, associated to ${\mathcal{L}}$.

######  Definition 4.29.

A symplectic structure $\omega$ on $\widehat{M}$ is induced by a contact
structure $\eta:TM\to{\mathcal{L}}$ on $M$ if
$\widehat{M}={\mathcal{L}}^{\times}$ and $\omega=d(p^{*}\eta)$.

The most famous examples of the above structures are given by coadjoint orbits
in the dual $\mathfrak{g}^{\vee}$ of the Lie algebra $\mathfrak{g}$ of a
connected reductive group $G$. For later purposes, we present a (non-
canonical) version of _Kostant-Kirillov form_ which takes place in
$\mathfrak{g}$ the Lie algebra and not $\mathfrak{g}^{\vee}$. If
$\mathfrak{g}$ is semisimple, the Killing form identifies $\mathfrak{g}$ with
$\mathfrak{g}^{\vee}$ and the construction is canonical.

######  Example 4.30.

Choose an invariant non-degenerate bilinear form $B$ on $\mathfrak{g}$. Let
$m$ be a non-zero element in $\mathfrak{g}$ and let $\widehat{M}_{m}=G\cdot m$
and $M_{m}=G\cdot[m]\subset{\mathbb{P}}(\mathfrak{g})$ be the orbits of
$m\in\mathfrak{g}$ and of $[m]\in{\mathbb{P}}(\mathfrak{g})$ under the adjoint
action. Let $G_{m}$ be the isotropy subgroup of $G$ at $m$, with Lie algebra
$\mathfrak{g}_{m}$. Define the anti-symmetric bilinear form $B_{m}$ on
$\mathfrak{g}$ by $B_{m}(y,z)=B(m,[y,z])$. We have ${\rm
Ker}\,B_{m}=\mathfrak{g}_{m}$; thus, $B_{m}$ descends to a symplectic form
$\omega_{m}\colon\mathfrak{g}/\mathfrak{g}_{m}\times\mathfrak{g}/\mathfrak{g}_{m}\rightarrow\mathbb{C}$
at $m\in\mathfrak{g}$. By the Jacobi identity, the form $\omega_{m}$ is
closed.

If $m$ is such that the orbit $\widehat{M}_{m}=G.m$ is the cone in
$\mathfrak{g}$ over $M_{m}=G\cdot[m]\subset{\mathbb{P}}(\mathfrak{g})$ (i.e.,
the affine cone minus the origin), then the arguments in [Be98, Proposition
2.2] adapt verbatim and yield a contact structure $\eta$ on $M_{m}$ which
induces the symplectic form $\omega_{m}$. In particular, if $m$ is a nilpotent
element in $\mathfrak{g}$, then the existence of an $\mathfrak{sl}_{2}$-triple
containing $m$ ensures that $\widehat{M}_{m}$ is the cone over $M_{m}$ (see
[Be98, Paragraph (2.4)]).

Given a symplectic structure on a variety $\widehat{M}$ or a contact structure
on a variety $M$, it is natural to ask for Lagrangian or Legendrian
subvarieties; we recall their definitions. A Lagrangian subspace in a
symplectic vector space $V$ of dimension $2m$ is an isotropic subspace of
maximal dimension, _i.e._ , of dimension $m$.

######  Definition 4.31.

Let $\widehat{M}$ have a symplectic structure $\omega$. A smooth subvariety
$\widehat{L}\subset\widehat{M}$ is called Lagrangian if, for all
$m\in\widehat{L}$, the subspace $T_{m}\widehat{L}\subset T_{m}\widehat{M}$ is
Lagrangian for the symplectic form $\omega_{m}$ on $T_{m}\widehat{M}$.

######  Definition 4.32.

Let $M$ have a contact structure $\eta$ and let $p:\widehat{M}\to M$ be the
${\mathbb{C}}^{\times}$-bundle ${\mathcal{L}}^{\times}$ associated to the line
bundle ${\mathcal{L}}$ with symplectic form $\omega=d(p^{*}\eta)$. A smooth
subvariety $L\subset M$ is called Legendrian if $\widehat{L}=p^{-1}(L)$ is
Lagrangian in $\widehat{M}$.

######  Example 4.33.

Let $G$ be simple and $\mathfrak{g}$ its Lie algebra. Let $m\in\mathfrak{g}$
be a highest weight vector. Set ${\mathcal{O}}_{\min}=G\cdot m$ and
${\mathbb{P}}({\mathcal{O}}_{\min})=G\cdot[m]\subset{\mathbb{P}}(\mathfrak{g})$.
The latter is called the adjoint variety of $G$. It is the unique closed orbit
of $G$ in ${\mathbb{P}}(\mathfrak{g})$ under the adjoint action. Let
${\mathbb{L}}_{G}$ be the set of lines passing through a given point of
${\mathbb{P}}({\mathcal{O}}_{\min})$. Then ${\mathbb{L}}_{G}$ is a smooth
Legendrian variety in its linear span and is homogeneous under the isotropy
subgroup $G_{m}$ (see [LM07]). This Legendrian variety ${\mathbb{L}}_{G}$ is
called the subadjoint variety. Note that in type $\operatorname{C}_{n}$ we
have ${\mathbb{L}}_{G}=\emptyset$: the subadjoint variety is empty, since
${\mathbb{P}}({\mathcal{O}}_{\min})$ is the second Veronese embedding of
${\mathbb{P}}^{2n-1}$ and hence contains no line. We will see in §6.3 that
${\mathbb{L}}_{G}$ can be recovered as the VMRT of a specific wonderful
adjoint symmetric variety for $G$.

Let $H\subset G$ be a symmetric subgroup with group involution $\sigma$. The
following result is well known, we include a proof for the convenience of the
reader.

######  Proposition 4.34 ([KR71, Proposition 5]).

Let $m\in\mathfrak{p}$. Set $\widehat{L}_{m}:=H\cdot m$ and
$L_{m}:=H\cdot[m]$. Then the variety $\widehat{L}_{m}$ is Lagrangian in
$\widehat{M}_{m}$; in particular, $\dim G\cdot m=2\dim H\cdot m$.

If $\widehat{L}_{m}$ is the cone over $L_{m}$ (or equivalently is stable by
non-trivial homotheties), then the variety $L_{m}$ is Legendrian in $M_{m}$,
in particular $\dim G\cdot[m]=2\dim H\cdot[m]+1$.

###### Proof.

Since $\sigma(m)=-m$, the action of $\sigma$ on $\mathfrak{g}$ restricts to an
action on $\mathfrak{g}_{m}$. Since $\sigma$ is semisimple, we thus have
$\mathfrak{g}_{m}=\mathfrak{h}_{m}\oplus\mathfrak{p}_{m}$ and
$\mathfrak{g}/\mathfrak{g}_{m}=\mathfrak{h}/\mathfrak{h}_{m}\oplus\mathfrak{p}/\mathfrak{p}_{m}$
with $\mathfrak{h}_{m}=\mathfrak{h}\cap\mathfrak{g}_{m}$ and
$\mathfrak{p}_{m}=\mathfrak{p}\cap\mathfrak{g}_{m}$. Let
$u,v\in\mathfrak{h}/\mathfrak{h}_{m}$ (resp.
$u,v\in\mathfrak{p}/\mathfrak{p}_{m}$) and let $y$ and $z$ in $\mathfrak{h}$
(resp. in $\mathfrak{p}$) be representatives of $u$ and $v$. In Example 4.30,
the form $B$ can be chosen to be $\sigma$-invariant so that we get

$\begin{array}[]{rcl}\omega_{m}(u,v)&=&\omega_{m}(d\sigma(u),d\sigma(v))\\\
&=&B(m,[d\sigma(y),d\sigma(z)])\\\
&=&-B(d\sigma(m),[d\sigma(y),d\sigma(z)])\\\ &=&-B(d\sigma(m),d\sigma[y,z])\\\
&=&-B(m,[y,z])\\\ &=&-\omega_{m}(u,v).\end{array}$

Hence $\omega_{m}(u,v)=0$ and both $\mathfrak{h}/\mathfrak{h}_{m}$ and
$\mathfrak{p}/\mathfrak{p}_{m}$ are isotropic and therefore Lagrangian in
$\mathfrak{g}/\mathfrak{g}_{m}$. This proves the first part. If $H\cdot m$ is
stable under nontrivial homotheties, then the same holds true for $G\cdot m$.
The result follows from this and the first part. ∎

######  Corollary 4.35.

Let ${\mathcal{K}}$ be a minimal family, let $C\in{\mathcal{K}}_{x}$ and let
$m\in T_{x}C\setminus\\{0\\}$. Then $\widehat{L}_{m}$ is the cone over
$L_{m}$, in particular $\dim H\cdot[m]=\frac{1}{2}\dim G\cdot m-1$.

###### Proof.

Since the $H$-weight of $m$ is non-trivial, the orbit $H\cdot m$ is the cone
over $H\cdot[m]$. The result follows. ∎

Assume that $\overline{R}$ is not of type ${\rm A}_{r}$ and let
$C\in{\mathcal{K}}_{x}$ with ${\mathcal{K}}$ a minimal family. We obtain the
following description of ${\mathcal{K}}_{x}$.

######  Theorem 4.36.

We have ${\mathcal{K}}_{x}=H\cdot C$. Furthermore, if $X$ is Hermitian non-
exceptional, then ${\mathcal{K}}_{x}$ has two components. Otherwise,
${\mathcal{K}}_{x}$ is irreducible.

###### Proof.

Since $\overline{R}$ is not of type ${\rm A}_{r}$, we have $\partial X\cdot
C=1$ (see Remark 4.27). By Theorem 4.26 and Corollary 4.35, we have $\dim
H\cdot C=\dim{\mathcal{K}}_{x}$. If $C$ is not Hermitian or Hermitian
exceptional, then there exists a unique highest weight curve and
${\mathcal{K}}_{x}$ is irreducible, proving the result. If $X$ is Hermitian
non-exceptional, then ${\mathcal{K}}_{x}$ contains two highest weight curves
which are exchanged by $H$; the result follows. ∎

In Proposition 4.34, the condition that $\widehat{L}_{m}$ is the cone over
$L_{m}$ is non-empty.

######  Example 4.37.

There are non-nilpotent elements $m\in\mathfrak{p}$ for which $H\cdot m$ is
not invariant under non-trivial homotheties. For example, let
$G=\operatorname{SL}_{n}$ and $\sigma(g)=\ ^{t}\\!g^{-1}$. Then
$\mathfrak{g}=\mathfrak{sl}_{n}$ is the Lie algebra of traceless matrices,
$\mathfrak{h}$ is the Lie subalgebra of antisymmetric matrices, and
$\mathfrak{p}$ is the subspace of traceless symmetric matrices. Define
$m\in\mathfrak{p}$ as block-matrix as follows:

$m=\left(\begin{array}[]{cc}A&0\\\ 0&0\\\ \end{array}\right)\textrm{ with
}A=\left(\begin{array}[]{cc}0&1\\\ 1&0\\\
\end{array}\right)\in\mathfrak{sl}_{2}.$

For $\lambda\in{\mathbb{C}}^{\times}$, if $\lambda m\in H\cdot m$, then
$\lambda=\pm 1$. Thus, $H\cdot m$ is not the cone over $H\cdot[m]$.

We conclude this subsection by the following related result, which follows
from [Ri82, Theorem A]; we provide a direct proof for the reader’s
convenience.

######  Lemma 4.38.

The orbit $H\cdot m$ (resp. $H\cdot[m]$) is open in $(G\cdot m)^{-\sigma}$
(resp. $(G\cdot[m])^{\sigma}$). In particular, $H\cdot m$ (resp. $H\cdot[m]$)
is a union of connected components of $(G\cdot m)^{-\sigma}$ (resp.
$(G\cdot[m])^{\sigma}$). Moreover, $\dim(H\cdot m)=\dim(G\cdot m)^{-\sigma}$
and $\dim(H\cdot[m])=\dim(G\cdot[m])^{\sigma}$.

###### Proof.

Note that $m$ is fixed for $-\sigma$; therefore, $\sigma([m])=[m]$. We thus
have inclusions $H\cdot m\subset(G\cdot m)^{-\sigma}$ and
$H\cdot[m]\subset(G\cdot[m])^{\sigma}$. To prove the openness, we only need to
check that the tangent spaces agree. We deal with $H\cdot[m]$, the other case
works in a similar way. Since $\sigma$ is semisimple, we have
$T_{m}(G\cdot[m])^{\sigma}=(\mathfrak{g}\cdot[m])^{\sigma}=\mathfrak{g}^{\sigma}\cdot[m]=\mathfrak{h}\cdot[m]=T_{m}(H\cdot[m])$.

Since every $H$-orbit in $(G\cdot m)^{-\sigma}$ is open, there are only
finitely many such orbits and these orbits are also closed, proving the last
statement. ∎

###  4.5 Wonderful symmetric varieties of type ${\rm A}_{r}$

As the above discussion shows, the case of symmetric spaces whose restricted
root system is of type $A_{r}$ with $r\geq 1$ will present a different
feature: the family ${\mathcal{K}}_{x}$ has dimension one more than the orbit
$H\cdot C$ for $C\in{\mathcal{K}}_{x}$. In this section we prove that
${\mathcal{K}}_{x}$ is a rational projective homogeneous space.

Assume that the restricted root system $\overline{R}$ is of type ${\rm A}_{r}$
and let $({\overline{\alpha}}_{i})_{i\in[1,r]}$ be the simple roots of
$\overline{R}$ (labeled as in Bourbaki [Bo68]). For $i\in[1,r]$, let
$\alpha_{i}\in R$ be a simple root such that
$\alpha_{i}-\sigma(\alpha_{i})={\overline{\alpha}}_{i}$. For $\beta\in\Delta$,
let $\varpi_{\beta}$ be the associated fundamental weight of $R$. For each
$i\in[1,r]$, recall the definition of the dominant weight
$\lambda_{i}:=\lambda_{\alpha_{i}}$ from Proposition 4.17:

$\lambda_{i}=\left\\{\begin{array}[]{ll}2\varpi_{\alpha_{i}}&\textrm{ if
$\sigma(\alpha_{i})=-\alpha_{i}$,}\\\
\varpi_{\alpha_{i}}+\varpi_{{\overline{\sigma}}(\alpha_{i})}&\textrm{ if
$\sigma(\alpha_{i})=-{\overline{\sigma}}(\alpha_{i})$ and
$\langle\alpha_{i}^{\vee},\sigma(\alpha_{i})\rangle=0$,}\\\
\varpi_{\alpha_{i}}&\textrm{ otherwise.}\\\ \end{array}\right.$

We now list the different symmetric spaces (up to finite cover) whose
restricted root system is of type ${\rm A}_{r}$, the corresponding dominant
weights $\lambda_{1}$, the irreducible $G$-representations $V_{\lambda_{1}}$
(which will feature prominently in the rest of this section), and the
corresponding $H$-representations $\mathfrak{p}$.

$G/H$ | Rank | $\lambda_{1}$ | $V_{\lambda_{1}}$ | $\mathfrak{p}$
---|---|---|---|---
$\operatorname{SL}_{r+1}\times\operatorname{SL}_{r+1}/\operatorname{SL}_{r+1}$ | $r$ | $(\varpi_{1},0)+(0,\varpi_{r})$ | ${\rm End}({\mathbb{C}}^{r+1})$ | $\mathfrak{sl}_{r+1}$
$\operatorname{SL}_{r+1}/{\rm SO}_{r+1}$ | $r$ | $2\varpi_{1}$ | $S^{2}({\mathbb{C}}^{r+1})$ | $S^{2}({\mathbb{C}}^{r+1})_{0}$
$\operatorname{SL}_{2r+2}/{\rm Sp}_{2r+2}$ | $r$ | $\varpi_{2}$ | $\Lambda^{2}({\mathbb{C}}^{2r+2})$ | $\Lambda^{2}({\mathbb{C}}^{2r+2})_{0}$
${\rm SO}_{n}/{\rm S}({\rm O}_{1}\times{\rm O}_{n-1})$ | $1$ | $\varpi_{1}$ | ${\mathbb{C}}^{n}$ | ${\mathbb{C}}^{n-1}$
$E_{6}/F_{4}$ | $2$ | $\varpi_{1}$ | ${\mathbb{C}}^{27}$ | ${\mathbb{C}}^{26}$

Here $S^{2}({\mathbb{C}}^{r+1})_{0}$ denotes the
$\operatorname{SO}_{r+1}$-stable complement of ${\mathbb{C}}q$ in
$S^{2}({\mathbb{C}}^{r+1})$ with $q$ being the standard quadratic form, and
$\Lambda^{2}({\mathbb{C}}^{2r+2})_{0}$ denotes the ${\rm Sp}_{2r+2}$-stable
complement of ${\mathbb{C}}\omega$ in $\Lambda^{2}({\mathbb{C}}^{2r+2})$ with
$\omega$ being the standard symplectic form. As a consequence of this
classification, we see that $V_{\lambda_{1}}=\mathfrak{p}\oplus{\mathbb{C}}$
as $H$-representations.

Note that since $\overline{R}$ is reduced, none of the symmetric spaces we
consider is exceptional. Let $\overline{R}^{\vee}$ be the dual root system,
let $(\overline{\alpha}_{i}^{\vee})_{i\in[1,r]}$ be the simple coroots and let
$C^{\vee}$ be the dominant chamber of $\overline{R}^{\vee}$. Then the weights
$(\lambda_{i})_{i\in[1,r]}$ are the fundamental weights of $\overline{R}$. The
cone $C^{\vee}$ is thus generated by the fundamental coweights
$(\lambda_{i}^{\vee})_{i\in[1,r]}$.

Next, using the above list, we obtain a geometric construction of the
wonderful compactification:

######  Proposition 4.39.

Let $G/H$ be an adjoint irreducible symmetric space with restricted root
system of type ${\rm A}_{r}$ and let $\lambda_{1}$ be as above.

  1. 1.

The group $G$ acts on ${\mathbb{P}}(V_{\lambda_{1}})$ with $r+1$ orbits whose
closures $(Z_{i})_{i\in[1,r+1]}$ satisfy the following inclusions:
$Z_{1}\subsetneq Z_{2}\subsetneq\cdots\subsetneq
Z_{r+1}={\mathbb{P}}(V_{\lambda_{1}})$. The open orbit is isomorphic to $G/H$.

  2. 2.

The join $J(Z_{1},Z_{i})$ (i.e., the union of lines joining $Z_{1}$ and
$Z_{i}$) equals $Z_{i+1}$ for all $i$.

  3. 3.

The wonderful compactification $X$ is equipped with a birational
$G$-equivariant morphism $f:X\to{\mathbb{P}}(V_{\lambda_{1}})$. If $r=1$, then
$f$ is an isomorphism. If $r\geq 2$, then $f$ is the composition of the blow-
ups of the strict transforms of the orbit closures $Z_{1}$, $\ldots$, $Z_{r}$
in this order. Moreover, these strict transforms are smooth.

###### Proof.

(1) and (2) In all cases except the last one, the $G$-orbits are given by the
rank of matrices (plain matrices, symmetric matrices or skew-symmetric
matrices) and the assertions follows from this. The case of $E_{6}/F_{4}$ is a
classical result (see e.g. [LM01, Proposition 4.1]).

(3) This is again a classical result in the first three cases, see [Va84,
Theorem 1] for
$\operatorname{SL}_{r+1}\times\operatorname{SL}_{r+1}/\operatorname{SL}_{r+1}$
(then $X$ is the moduli space of complete collineations) and [Th99, Theorems
10.1, 11.1] for $\operatorname{SL}_{r+1}/\operatorname{SO}_{r+1}$ and
$\operatorname{SL}_{2r+2}/{\rm Sp}_{2r+2}$ (complete quadrics and complete
skew forms). The next case of $\operatorname{SO}_{n}/{\rm S}({\rm
O}_{1}\times{\rm O}_{n-1})$ is easy, as we then have
$X={\mathbb{P}}^{n-1}={\mathbb{P}}(V_{\lambda_{1}})$.

For $E_{6}/F_{4}$, we have $Z_{1}\subset Z_{2}\subset
Z_{3}={\mathbb{P}}(V_{\lambda_{1}})$, where $Z_{1}$ is smooth and $Z_{2}$ is a
prime divisor. Denote by $\varphi:X^{\prime}\to{\mathbb{P}}(V_{\lambda_{1}})$
the blow-up along $Z_{1}$. Then $X^{\prime}$ is a smooth projective
equivariant compactification of $G/H$, and its boundary is the union of two
prime divisors: the exceptional divisor $X^{\prime}_{1}$ and the strict
transform $X^{\prime}_{2}$ of $Z_{2}$. Moreover, $X^{\prime}_{2}\setminus
X^{\prime}_{1}=Z_{2}\setminus Z_{1}$ is a unique $G$-orbit. It suffices to
show that $X^{\prime}_{1}\setminus X^{\prime}_{2}$ and $X^{\prime}_{1}\cap
X^{\prime}_{2}$ are $G$-orbits as well: then $X^{\prime}$ is a smooth
projective embedding of $G/H$ with a unique closed orbit of codimension $2$,
and hence is isomorphic to $X$ by the classification of embeddings of $G/H$.
We identify $Z_{1}$ to $G/P_{1}$, where $P_{1}$ is the maximal parabolic
subgroup of $G$ associated with the fundamental weight
$\lambda_{1}=\varpi_{1}$. Denote by $M$ the normal space to $Z_{1}$ in
${\mathbb{P}}(V_{\lambda_{1}})$ at the base point of $G/P_{1}$. Then $M$ is a
representation of $P_{1}$, and the $G$-variety $X^{\prime}_{1}$ is isomorphic
to the homogeneous projective bundle $G\times^{P_{1}}{\mathbb{P}}(M)$. Thus,
the $G$-orbits in ${\mathbb{P}}(V_{\lambda_{1}})$ correspond bijectively to
the $P$-orbits in ${\mathbb{P}}(M)$. So it suffices in turn to show that
$P_{1}$ acts on ${\mathbb{P}}(M)$ with two orbits. But the Levi subgroup
$L_{1}$ of $P_{1}$ is isomorphic to
$\operatorname{SO}_{10}\times{\mathbb{C}}^{*}$ up to finite cover, and
$M={\mathbb{C}}^{10}$ where $\operatorname{SO}_{10}$ acts via its standard
representation and ${\mathbb{C}}^{*}$ acts by scalar multiplication.
Therefore, $L_{1}$ acts on ${\mathbb{P}}(M)$ with two orbits: a quadric and
its complement. As $P_{1}$ does not act transitively on ${\mathbb{P}}(M)$, it
acts with two orbits as well. ∎

######  Remark 4.40.

The above statements (1) and (2) can be proved in a uniform way using Jordan
algebras: the representation $V_{\lambda_{1}}$ has the structure of a Jordan
algebra with structure group $G$ and the stabiliser of the unit element is
$H$. The above orbit structure is then explained by the notion of rank for
elements in a Jordan algebra. We refer to [Sp98] and [BP22] for more on Jordan
algebras. We were however not able to fully relate symmetric spaces with
restricted root systems of type ${\rm A}_{r}$ to Jordan algebras without using
a case by case check, so we refrained from using them.

######  Theorem 4.41.

Let $X$ be the wonderful compactification of an adjoint irreducible symmetric
space with restricted root system of type ${\rm A}_{r}$.

  1. 1.

There is a unique minimal family ${\mathcal{K}}$.

  2. 2.

The tangent map ${\mathcal{K}}_{x}\to{\mathcal{C}}_{x}$ is an isomorphism.

  3. 3.

If $r=1$, then ${\mathcal{C}}_{x}\simeq{\mathbb{P}}(\mathfrak{p})$.

  4. 4.

If $r\geq 2$, then ${\mathcal{K}}_{x}$ is isomorphic to the closed $G$-orbit
in ${\mathbb{P}}(V_{\lambda_{1}})$.

###### Proof.

If $r=1$, then $X={\mathbb{P}}(V_{\lambda_{1}})$ with
$V_{\lambda_{1}}=\mathfrak{p}\oplus{\mathbb{C}}$ and $x=[{\mathbb{C}}]$. Thus
there is a unique minimal covering family ${\mathcal{K}}$ and it consists of
lines in $X$. The result follows in this case.

If $r\geq 2$, because of the description of $X$ as a successive blow-up in
Proposition 4.39 and arguing as in [BF15, Proposition 5.1] for the group case
of type ${\rm A}_{r}$, the result follows from Proposition 9.7 in [FH12]: the
family ${\mathcal{K}}_{x}$ is the set of lines in
${\mathbb{P}}(V_{\lambda_{1}})$ passing through a general point and meeting
the closed orbit. The tangent map ${\mathcal{K}}_{x}\to{\mathcal{C}}_{x}$ is
an isomorphism and the VMRT is therefore isomorphic to the closed orbit in
$X$. ∎

###  4.6 Minimal families on wonderful compactifications

We summarise our results. Let $X$ be the wonderful compactification of an
adjoint irreducible symmetric space $G/H$ with base point $x$ and let
${\mathcal{K}}$ be a minimal family.

######  Theorem 4.42.

The tangent map ${\mathcal{K}}_{x}\to{\mathcal{C}}_{x}$ is an isomorphism.

######  Theorem 4.43.

  1. 1.

Any irreducible component of ${\mathcal{K}}_{x}$ contains a unique highest
weight curve $C$. Moreover, ${\mathcal{K}}_{x}$ is equidimensional, of
dimension $\langle{\overline{\Theta}}^{\vee},\kappa\rangle+\partial X\cdot
C-1=\dim H\cdot C+\partial X\cdot C-1$.

  2. 2.

We have $\partial X\cdot C\in\\{1,2\\}$. Moreover, $\partial X\cdot C=2$ if
and only if the restricted root system is of type ${\rm A}_{r}$.

  3. 3.

Assume that $\partial X\cdot C=1$. Then ${\mathcal{K}}_{x}=H\cdot C$.
Furthermore, if $X$ is Hermitian non-exceptional, then ${\mathcal{K}}_{x}$ has
two components. Otherwise, ${\mathcal{K}}_{x}$ is irreducible.

  4. 4.

Assume that $\partial X\cdot C=2$, so that the restricted root system of $X$
is of type ${\rm A}_{r}$.

    1. (a)

If $r=1$, then ${\mathcal{K}}_{x}\simeq{\mathbb{P}}(\mathfrak{p})$.

    2. (b)

If $r\geq 2$, then there exists a $G$-equivariant birational morphism
$X\to{\mathbb{P}}(V)$ for some irreducible $G$-representation $V$ and
${\mathcal{K}}_{x}$ is isomorphic to the closed $G$-orbit in
${\mathbb{P}}(V)$. The orbit $H\cdot C$ is a prime divisor in
${\mathcal{K}}_{x}$.

  5. 5.

The orbits $H\cdot C$ are described in Table 1.

###### Proof.

We start with Theorem 4.43.

(1) Follows from Theorem 4.26.

(2) Follows from Remark 4.27.

(3) Follows from Theorem 4.36.

(4) Follows from (2) and Theorem 4.41.

Next, we prove Theorem 4.42. This result follows from (4) for $\overline{R}$
of type ${\rm A}_{r}$. Assume that $\overline{R}$ is not of type ${\rm
A}_{r}$. By Proposition 3.6, the tangent map
$\tau_{x}:{\mathcal{K}}_{x}\to{\mathcal{C}}_{x}$ is $H$-equivariant, finite
and birational. Furthermore by (3), the variety ${\mathcal{K}}_{x}$ is
$H$-homogeneous. Thus, ${\mathcal{C}}_{x}$ is $H$-homogeneous as well, and
$\tau_{x}$ is bijective. ∎

##  5 Minimal families on complete symmetric varieties

We are now in position to prove our main results. Let $X$ be a complete
symmetric variety and let ${\mathcal{K}}$ be a family of minimal rational
curves on $X$. Let $\pi:X\to X_{\rm ad}$ be the map from $X$ to the wonderful
compactification of the associated adjoint symmetric space.

######  Theorem 5.1.

The tangent map ${\mathcal{K}}_{x}\to{\mathcal{C}}_{x}$ is an isomorphism and
${\mathcal{K}}_{x}$ is smoth.

######  Theorem 5.2.

Let $C\in{\mathcal{K}}_{x}$.

  1. 1.

If $C$ is contracted by $\pi$, then ${\mathcal{K}}_{x}$ is isomorphic to a
linear subspace of ${\mathbb{P}}(\mathfrak{p}\cap\mathfrak{z})$.

Assume that $\pi$ does not contract $C$.

  2. 2.

The map $\pi$ induces an isomorphism between $C$ and its image $D:=\pi(C)$ and
there exists a unique indecomposable factor $X_{C}$ of $X_{\rm ad}$ such that
the composition map $\pi_{C}:X\to X_{\rm ad}\to X_{C}$ does not contract $C$.

  3. 3.

We have $\partial X_{C}\cdot C\in\\{1,2\\}$.

  4. 4.

If $\partial X\cdot C=1$, then ${\mathcal{K}}_{x}=H\cdot C$. Moroever, the
components of ${\mathcal{K}}_{x}$ are isomorphic to the components of $H\cdot
D$.

  5. 5.

If $\partial X\cdot C=2$, then the restricted root system of $X_{C}$ is of
type ${\rm A}_{r}$.

    1. (a)

If $r=1$, then ${\mathcal{K}}_{x}\simeq{\mathbb{P}}(\mathfrak{p}_{C})$.

    2. (b)

If $r\geq 2$, then there is a $G$-equivariant birational morphism
$X_{C}\to{\mathbb{P}}(V)$ for some irreducible $G$-representation $V$ and
${\mathcal{K}}_{x}$ is isomorphic to the closed $G$-orbit in
${\mathbb{P}}(V)$. The orbit $H\cdot C$ is a prime divisor in
${\mathcal{K}}_{x}$.

  6. 6.

The orbits $H\cdot D$ are described in Table 1.

###### Proof.

We start with the proof of Theorem 5.2.

(1) Follows from Lemma 3.4.

(2) Follows from Proposition 3.6.

(3) Follows from Proposition 3.6 and Theorem 4.43.

(4) If $\partial X\cdot C=1$ and $X_{C}$ is not of type ${\rm A}_{r}$, then
$\partial X\cdot C=\partial X_{C}\cdot D$. Moreover, there is a unique minimal
family of rational curves ${\mathcal{L}}$ containing $D=\pi(C)$ and a finite
$H$-equivariant map $\pi_{*,x}:{\mathcal{K}}_{x}\to{\mathcal{L}}_{\pi_{C}(x)}$
(Proposition 3.6 again). By Theorem 4.43, we have
${\mathcal{L}}_{\pi_{C}(x)}=H\cdot D$; in particular, every component of
${\mathcal{L}}_{\pi_{C}(x)}$ is homogeneous under $H^{0}$. Using Proposition
3.6 once more, it follows that $\pi_{*,x}$ induces an isomorphism on
components.

If $X_{C}$ is of type ${\rm A}_{r}$, then $\partial X_{C}\cdot D=2$. Thus, the
image of ${\mathcal{K}}_{x}$ in ${\mathcal{L}}_{\pi_{C}(x)}$ has codimension
$1$ and must be equal to $H\cdot D$. The result follows from this by a similar
argument as in the previous case.

(5) If $\partial X\cdot C=2$, then the restricted root system of $X_{C}$ is of
type ${\rm A}_{r}$ and $\partial X\cdot C=\partial X_{C}\cdot D$. Again, there
is a unique minimal family of rational curves ${\mathcal{L}}$ containing $D$,
and a finite $H$-equivariant map
$\pi_{*,x}:{\mathcal{K}}_{x}\to{\mathcal{L}}_{\pi_{C}(x)}$. Moreover,
${\mathcal{L}}_{\pi_{C}(x)}$ is irreducible and has the same dimension as
${\mathcal{K}}_{x}$; thus, $\pi_{*,x}$ is surjective. As $X_{C}$ is not
Hermitian, ${\mathcal{K}}_{x}$ is irreducible as well. So $\pi_{*,x}$ is an
isomorphism.

Proof of Theorem 5.1. By Proposition 3.6, ${\mathcal{K}}_{x}$ is smooth and
the tangent map ${\mathcal{K}}_{x}\to{\mathcal{C}}_{x}$ is finite, birational
and $H$-equivariant, therefore an isomorphism if ${\mathcal{K}}_{x}=H\cdot C$.
If ${\mathcal{K}}_{x}$ is not $H$-homogeneous, then $X_{C}$ is of type ${\rm
A}_{r}$ and the result follows from (5). ∎

##  6 Appendix

The goal of this appendix is twofold: we first prove basic results on
restricted root systems used to describe curves and divisors on wonderful
compactifications. We also obtain characterisations of exceptional wonderful
varieties useful to establish Table 1. We then give an easy way to describe,
using the Kac diagram of the symmetric space, the components of the $H$-orbit
$H\cdot C$ in ${\mathcal{K}}_{x}$, where $C$ is a highest weight curve.
Finally, in Table 1, we give a list, based on the classification of symmetric
spaces, of minimal families and VMRT of wonderful symmetric varieties.

###  6.1 Restricted root systems

In this subsection we prove useful results on restricted root systems that
might be well known to experts, but for which we could not find a good
reference.

######  Lemma 6.1.

The restricted root ${\overline{\Theta}}$ is the highest root of
$\overline{R}$.

###### Proof.

For $\alpha\in R$, write $\alpha=\sum_{\beta\in\Delta}c_{\beta}\beta$ with all
the $c_{\beta}$ of the same sign. We have
${\overline{\alpha}}=\alpha-\sigma(\alpha)=\sum_{\beta\in\Delta_{1}}c_{\beta}(\beta-\sigma(\beta))=\sum_{\beta\in\Delta_{1}}c_{\beta}{\overline{\beta}}$
and the result follows from this. ∎

######  Lemma 6.2.

Let $w_{0}\in W$ be the longest element, then the actions of $\sigma$ and
$w_{0}$ on roots commute. In particular, $\sigma(w_{0})=w_{0}$.

###### Proof.

Note that $-w_{0}$ is an involution and preserves $R^{+}$ and thus $\Delta$.
Furthermore, if $-w_{0}\neq{\rm Id}$, then ${\overline{\sigma}}$ is either
equal to ${\rm Id}$ or to $-w_{0}$ since there is at most one non-trivial such
involution. Therefore, in any case, ${\overline{\sigma}}$ and $-w_{0}$
commute; thus, ${\overline{\sigma}}$ and $w_{0}$ commute. In particular
$w_{0}(\Delta_{0})=-\Delta_{0}$ and $w_{0}(L)=L$.

Recall that $\sigma=-w_{L}{\overline{\sigma}}$ where $w_{L}$ is the longest
element in the Weyl group $W_{L}$ of the pair $(L,T_{{\rm s}})$, for this see
[Ti11, Page 149]. Therefore
$w_{0}\sigma=-w_{0}w_{L}{\overline{\sigma}}=-w_{w_{0}(L)}w_{0}{\overline{\sigma}}=-w_{L}{\overline{\sigma}}w_{0}=\sigma
w_{0}$. The result follows. ∎

######  Corollary 6.3.

We have $w_{0}({\overline{\Theta}})=-{\overline{\Theta}}$.

###### Proof.

We have
$w_{0}({\overline{\Theta}})=w_{0}(\Theta)-w_{0}(\sigma(\Theta))=w_{0}(\Theta)-\sigma(w_{0}(\Theta))=-\Theta+\sigma(\Theta)=-{\overline{\Theta}}$.
∎

We now prove a characterisation of non-reduced restricted root systems.

######  Proposition 6.4.

Let $\alpha\in\Delta_{1}$. We have the equivalence:
${\overline{\alpha}},2{\overline{\alpha}}\in\overline{R}$ $\Leftrightarrow$
$\langle\alpha^{\vee},\sigma(\alpha)\rangle=1$.

###### Proof.

(1) If $\langle\alpha^{\vee},\sigma(\alpha)\rangle=1$, then
$\beta=\alpha-\sigma(\alpha)={\overline{\alpha}}$ is a root of $R$ and
$\sigma(\beta)=-\beta$. Thus,
${\overline{\beta}}=2\beta=2{\overline{\alpha}}\in\overline{R}$.

Conversely, assume that
${\overline{\alpha}},2{\overline{\alpha}}\in\overline{R}$. As above, there are
three possibilities: $\sigma(\alpha)=-\alpha$,
$\langle\alpha^{\vee},\sigma(\alpha)\rangle=0$ or
$\langle\alpha^{\vee},\sigma(\alpha)\rangle=1$. We rule out the first two
cases.

If $\sigma(\alpha)=-\alpha$, then ${\overline{\alpha}}=2\alpha$ and
$2{\overline{\alpha}}=4\alpha\in\overline{R}$. In particular, we should have a
root $\gamma\in R$ with
$\gamma-\sigma(\gamma)={\overline{\gamma}}=2{\overline{\alpha}}=4\alpha$. This
implies that $\gamma=\lambda\alpha+\sum_{\beta\in\Delta_{0}}c_{\beta}\beta$
with $\lambda>0$ and $c_{\beta}\geq 0$. But
$\sigma(\gamma)=-\lambda\alpha+\sum_{\beta\in\Delta_{0}}c_{\beta}\beta$ and
since $\gamma$ is a root, we have $c_{\beta}=0$ for all $\beta$ and
$\gamma=2\alpha\not\in R$, a contradiction.

Assume that $\langle\alpha^{\vee},\sigma(\alpha)\rangle=0$ and write
$\sigma(\alpha)=-{\overline{\sigma}}(\alpha)-\sum_{\beta\in\Delta_{0}}c_{\beta}\beta$.
Then we have

$0=\langle\alpha^{\vee},\sigma(\alpha)\rangle=-\langle\alpha^{\vee},{\overline{\sigma}}(\alpha)\rangle-\sum_{\beta\in\Delta_{0}}c_{\beta}\langle\alpha^{\vee},\beta\rangle.$

If ${\overline{\sigma}}(\alpha)\neq\alpha$, then all terms of the RHS of the
above equation are non-negative, thus vanish. In particular,
$\langle\alpha^{\vee},\beta\rangle=0$ for all $\beta\in\Delta_{0}$ with
$c_{\beta}\neq 0$. We get
$\langle{\overline{\sigma}}(\alpha)^{\vee},\beta\rangle=0$ and
$\langle\sigma(\alpha)^{\vee},\beta\rangle=0$ for all $\beta$ with
$c_{\beta}\neq 0$. Denoting by $(-,-)$ a $\sigma$-invariant scalar product, we
have
$(\sigma(\alpha)+{\overline{\sigma}}(\alpha),\sigma(\alpha)+{\overline{\sigma}}(\alpha))=(\sigma(\alpha)+{\overline{\sigma}}(\alpha),-\sum_{\beta}c_{\beta}\beta)=0$;
thus, $\sigma(\alpha)=-{\overline{\sigma}}(\alpha)$. Let $\gamma\in R$ such
that ${\overline{\gamma}}=2{\overline{\alpha}}$. Then
$\gamma-\sigma(\gamma)=2(\alpha+{\overline{\sigma}}(\alpha))$. Therefore,
$\gamma=\lambda\alpha+\mu{\overline{\sigma}}(\alpha)+\sum_{\beta\in\Delta_{0}}d_{\beta}\beta$
with $\lambda,\mu,d_{\beta}\geq 0$ and $\lambda+\mu>0$. Then
$\sigma(\gamma)=-\lambda{\overline{\sigma}}(\alpha)-\mu\alpha+\sum_{\beta\in\Delta_{0}}d_{\beta}\beta$
is a root; thus, $d_{\beta}=0$ for all $\beta\in\Delta_{0}$ and
$\gamma=\lambda\alpha+\mu{\overline{\sigma}}(\alpha)$ with $\lambda+\mu=2$.
However there is no such root $\gamma$, since
$\langle\alpha^{\vee},{\overline{\sigma}}(\alpha)\rangle=-\langle\alpha^{\vee},\sigma(\alpha)\rangle=0$,
thus, $\alpha$ and ${\overline{\sigma}}(\alpha)$ are not adjacent in the
Dynkin diagram and the support of $\gamma$ is therefore disconnected.

Finally, assume that ${\overline{\sigma}}(\alpha)=\alpha$. Then
$\sigma(\alpha)=-\alpha-\sum_{\beta\in\Delta_{0}}c_{\beta}\beta\neq-\alpha$.
Again, let $\gamma\in R$ with ${\overline{\gamma}}=2{\overline{\alpha}}$. Then
$\gamma-\sigma(\gamma)=2(\alpha-\sigma(\alpha))$. We have
$({\overline{\gamma}},{\overline{\gamma}})=4({\overline{\alpha}},{\overline{\alpha}})=8(\alpha,\alpha)$.
On the other hand, we have

$({\overline{\gamma}},{\overline{\gamma}})=\left\\{\begin{array}[]{ll}4(\gamma,\gamma)&\textrm{
if $\sigma(\gamma)=-\gamma$},\\\ 2(\gamma,\gamma)&\textrm{ if
$\langle\gamma^{\vee},\sigma(\gamma)\rangle=0$},\\\ (\gamma,\gamma)&\textrm{
if $\langle\gamma^{\vee},\sigma(\gamma)\rangle=1$}.\\\ \end{array}\right.$

We get

$(\gamma,\gamma)=\left\\{\begin{array}[]{ll}2(\alpha,\alpha)&\textrm{ if
$\sigma(\gamma)=-\gamma$},\\\ 4(\alpha,\alpha)&\textrm{ if
$\langle\gamma^{\vee},\sigma(\gamma)\rangle=0$},\\\ 8(\alpha,\alpha)&\textrm{
if $\langle\gamma^{\vee},\sigma(\gamma)\rangle=1$}.\\\ \end{array}\right.$

Since $R$ is a reduced root system, only the first case can occur; thus,
$\sigma(\gamma)=-\gamma$, the root $\gamma$ is long and $\alpha$ is short. We
thus have $2\gamma={\overline{\gamma}}=2{\overline{\alpha}}$; thus,
$\gamma={\overline{\alpha}}$. By Lemma 6.7 below, the root $\gamma$ is
dominant on its support and $\alpha$ is non-orthogonal to $\gamma$.
Furthermore, $\gamma$ is long and $\alpha$ is short, so the support of
$\gamma$ generates a non-simply laced root system. Therefore, $\gamma$ is the
highest root of a root system of type $B_{n}$, $C_{n}$, $F_{4}$ or $G_{2}$.
Since $\alpha$ is short and is the unique simple root which is non-orthogonal
to $\gamma$, the above root system must be of type $C_{n}$ with
$\alpha=\alpha_{1}$ and
$\sigma(\alpha_{1})=-(\alpha_{1}+2\alpha_{2}+\cdots+2\alpha_{n-1}+\alpha_{n})$.
Note that $\sigma(\alpha_{i})=\alpha_{i}$ for $i>1$. However, if
$\delta=\alpha_{1}+\cdots+\alpha_{n}$, then
$\sigma(\delta)=-(\alpha_{1}+\cdots+\alpha_{n-1})$ and
$\delta+\sigma(\delta)=\alpha_{n}$. Let $x_{\delta}\in\mathfrak{g}_{\delta}$
be non-zero and set $x=[x_{\delta},x_{\sigma(\delta)}]$. Then
$x\in\mathfrak{g}_{x_{\alpha_{n}}}$ is non-zero and $\sigma(x)=-x$, which is
impossible since $T_{{\rm s}}$ is of split type. ∎

######  Corollary 6.5.

We have the equivalences:

$\textrm{$\overline{R}$ is non-reduced
}\Leftrightarrow\exists\alpha\in\Delta_{1},\langle\alpha^{\vee},\sigma(\alpha)\rangle=1\Leftrightarrow\exists\alpha\in
R,\langle\alpha^{\vee},\sigma(\alpha)\rangle=1.$

###### Proof.

If $\overline{R}$ is non-reduced then it is of type $\textrm{BC}_{r}$ and
there exists $\alpha\in\Delta_{1}$ with $\alpha,2\alpha\in\overline{R}$. The
second implication from left to right is clear. If $\alpha\in R$ is such that
$\langle\alpha^{\vee},\sigma(\alpha)\rangle=1$, then
${\overline{\alpha}}=\alpha-\sigma(\alpha)=\gamma$ is a root and
${\overline{\gamma}}=2\gamma=2{\overline{\alpha}}\in\overline{R}$; thus,
$\overline{R}$ is non-reduced. ∎

######  Corollary 6.6.

Let $\alpha\in\Delta_{1}$ such that
${\overline{\alpha}},2{\overline{\alpha}}\in\overline{R}$.

  1. 1.

The root ${\overline{\alpha}}$ is the unique root of $\overline{\Delta}$ with
${\overline{\alpha}},2{\overline{\alpha}}\in\overline{R}$.

  2. 2.

The variety $X$ is exceptional if and only if
${\overline{\sigma}}(\alpha)\neq\alpha$.

###### Proof.

(1) Assume that $\alpha\in\Delta_{1}$ is such that
${\overline{\alpha}},2{\overline{\alpha}}\in\overline{R}$. Then
${\overline{\alpha}}\in\overline{\Delta}$ is the unique simple root whose
double is a root in the root system $\textrm{BC}_{r}$ and is therefore unique.

(2) If $X$ is exceptional, then ${\overline{\sigma}}(\alpha)\neq\alpha$ by
definition. If ${\overline{\sigma}}(\alpha)\neq\alpha$, then since
$\langle\alpha^{\vee},\sigma(\alpha)\rangle=1$ by Proposition 6.4, the result
follows from [deCS99, Lemma 4.3].

If $X$ is exceptional then for $\alpha\in\Delta_{1}$ an exceptional root, we
have ${\overline{\sigma}}(\alpha)\neq\alpha$ by definition and
${\overline{\alpha}},2{\overline{\alpha}}\in\overline{R}$ by Remark 4.1. If
$\alpha\in\Delta_{1}$ is such that
${\overline{\alpha}},2{\overline{\alpha}}\in\overline{R}$, then
$\langle\alpha^{\vee},\sigma(\alpha)\rangle=1$. If furthermore
${\overline{\sigma}}(\alpha)\neq\alpha$ then $\alpha$ is exceptional by
[deCS99, Lemma 4.3] again. ∎

For $\alpha\in\Delta_{1}$, write
${\overline{\alpha}}=\alpha-\sigma(\alpha)=\alpha+{\overline{\sigma}}(\alpha)+\sum_{\beta\in\Delta_{0}}c_{\beta}\beta$
and define the support of ${\overline{\alpha}}$ by
$\operatorname{Supp}({\overline{\alpha}})=\\{\alpha,{\overline{\sigma}}(\alpha),\beta\
|\ \beta\in\Delta_{0}\textrm{ with }c_{\beta}>0\\}$.

######  Lemma 6.7.

Let $\alpha\in\Delta_{1}$, then ${\overline{\alpha}}$ is dominant on
$\operatorname{Supp}({\overline{\alpha}})$. More precisely, we have

$\langle\alpha^{\vee},{\overline{\alpha}}\rangle>0,\langle{\overline{\sigma}}(\alpha)^{\vee},{\overline{\alpha}}\rangle>0\textrm{
and }\langle\beta^{\vee},{\overline{\alpha}}\rangle=0\textrm{ for
$\beta\in\operatorname{Supp}({\overline{\alpha}})$}\cap\Delta_{0}.$

###### Proof.

For $\beta\in\Delta_{0}$, we have
$\langle\beta^{\vee},{\overline{\alpha}}\rangle=\langle\sigma(\beta)^{\vee},\sigma({\overline{\alpha}})\rangle=\langle\beta^{\vee},-{\overline{\alpha}}\rangle$;
thus, $\langle\beta^{\vee},{\overline{\alpha}}\rangle=0$. We have
$\langle\alpha^{\vee},{\overline{\alpha}}\rangle=\langle\alpha^{\vee},\alpha-\sigma(\alpha)\rangle=2-\langle\alpha^{\vee},\sigma(\alpha)\rangle$
and since $\alpha$ and $\sigma(\alpha)$ have the same length, we have
$\langle\alpha^{\vee},\sigma(\alpha)\rangle\leq 1$ proving the result. ∎

Recall that $\Theta$ denotes the highest root of $R$, and $\theta$ the highest
short root (if $R$ is simply laced, then $\Theta=\theta$ and all roots are
long and short).

######  Lemma 6.8.

We have the following equivalences

  1. 1.

$\sigma(\Theta)=-\Theta$ $\Leftrightarrow$ there exists a long root $\alpha$
with $\sigma(\alpha)=-\alpha$.

  2. 2.

$\sigma(\theta)=-\theta$ $\Leftrightarrow$ there exists a short root $\alpha$
with $\sigma(\alpha)=-\alpha$.

###### Proof.

The implications from left to right in (1) and (2) are clear. We prove the
converse implications. The proofs in both cases are similar.

We first prove that the converse implications in (1) and (2) are implied by
the following claim: if $\sigma(\alpha)=-\alpha$ and there exists
$\beta\in\Delta$ with $s_{\beta}(\alpha)>\alpha$, then there exists $w\in W$
with $\sigma(w)=w$ such that $w(\alpha)>\alpha$ (recall that $\sigma$ acts on
$W$ by conjugaison).

Assume that the above claim is true. We prove the converse implications in (1)
and (2) by induction on roots for the natural order on roots: if $\alpha$ is a
positive root such that $\sigma(\alpha)=-\alpha$ and $\alpha$ is not maximal,
we produce a root $\alpha^{\prime}>\alpha$ with the same length as $\alpha$
and such that $\sigma(\alpha^{\prime})=-\alpha^{\prime}$. Indeed, if $\alpha$
is not maximal, then there exists $\beta\in\Delta$ with
$s_{\beta}(\alpha)>\alpha$. By the above claim, there exists $w\in W$ with
$\sigma(w)=w$ and $w(\alpha)>\alpha$. We thus have
$\sigma(w(\alpha))=\sigma(w)(\sigma(\alpha))=w(-\alpha)=-w(\alpha)$ with
$w(\alpha)>\alpha$. Therefore, if the claim is true, we get by induction that
$\sigma(\alpha^{\prime})=-\alpha^{\prime}$ for $\alpha^{\prime}$ the highest
short root with the same length as $\alpha$, proving the implication from
right to left of (1) and (2).

We now prove our claim, so let $\alpha\in R$ such that
$\sigma(\alpha)=-\alpha$ and $\beta\in\Delta$ with $s_{\beta}(\alpha)>\alpha$.
In particular $\langle\beta^{\vee},\alpha\rangle<0$. We have four possible
cases: $\sigma(\beta)=\beta$, $\sigma(\beta)=-\beta$,
$\langle\beta^{\vee},\sigma(\beta)\rangle=0$ or
$\langle\beta^{\vee},\sigma(\beta)\rangle=1$.

If $\sigma(\beta)=\beta$, then
$\langle\beta^{\vee},\alpha\rangle=\langle\sigma(\beta)^{\vee},\sigma(\alpha)\rangle=-\langle\beta^{\vee},\alpha\rangle$;
thus, $\langle\beta^{\vee},\alpha\rangle=0$ a contradiction, so this case does
not occur.

If $\sigma(\beta)=-\beta$, then $w=s_{\beta}$ works since $\sigma(w)=w$.

If $\langle\beta^{\vee},\sigma(\beta)\rangle=0$, then set
$w=s_{\beta}s_{\sigma(\beta)}$. Since $s_{\beta}$ and $s_{\sigma(\beta)}$
commute, we have $\sigma(w)=w$. Furthermore, we have
$w(\alpha)=\alpha-\langle\beta^{\vee},\alpha\rangle\beta-\langle\sigma(\beta)^{\vee},\alpha\rangle\sigma(\beta)=\alpha-\langle\beta^{\vee},\alpha\rangle\beta+\langle\beta^{\vee},\alpha\rangle\sigma(\beta)$.
But since $\sigma(\beta)\neq\beta$, we have $\sigma(\beta)<0$ and
$w(\alpha)>\alpha$.

Finally, if $\langle\beta^{\vee},\sigma(\beta)\rangle=1$, define
$\delta=s_{\sigma(\beta)}(\beta)=\beta-\sigma(\beta)={\overline{\beta}}$; then
$\delta$ is a root. Let $w=s_{\delta}$. We have $\sigma(\delta)=-\delta$;
thus, $\sigma(w)=w$. Furthermore, we have
$w(\alpha)=\alpha-\langle\delta^{\vee},\alpha\rangle\delta=\alpha-\langle
s_{\sigma(\beta)}(\beta)^{\vee},\alpha\rangle(\beta-\sigma(\beta))$ and
$\langle
s_{\sigma(\beta)}(\beta)^{\vee},\alpha\rangle=\langle\beta^{\vee},s_{\sigma(\beta)}(\alpha)\rangle=\langle\beta^{\vee},\alpha-\langle\sigma(\beta)^{\vee},\alpha\rangle\sigma(\beta)\rangle=\langle\beta^{\vee},\alpha\rangle-\langle\sigma(\beta)^{\vee},\alpha\rangle=2\langle\beta^{\vee},\alpha\rangle$.
Thus,
$w(\alpha)=\alpha-2\langle\beta^{\vee},\alpha\rangle(\beta-\sigma(\beta))$ and
since $\sigma(\beta)<0$, we get $w(\alpha)>\alpha$. ∎

######  Corollary 6.9.

Assume that $\overline{R}$ is non-reduced. We have the equivalences:

$X\textrm{ is exceptional }\Leftrightarrow\ \sigma(\Theta)=-\Theta\
\Leftrightarrow\textrm{ $R$ is simply laced.}$

Furthermore, if $X$ is exceptional and $\alpha$ is an exceptional root, then
its coefficient in the expansion of $\Theta$ as a linear combination of simple
roots is equal to $1$.

###### Proof.

Assume that $\overline{R}$ is non-reduced and let $\alpha\in\Delta_{1}$ such
that ${\overline{\alpha}},2{\overline{\alpha}}\in\overline{R}$. Then
$\langle\alpha^{\vee},\sigma(\alpha)\rangle=1$ and
$\gamma=\alpha-\sigma(\alpha)={\overline{\alpha}}$ is a root such that
$\sigma(\gamma)=-\gamma$. Therefore we either have $\sigma(\Theta)=-\Theta$ or
$\sigma(\theta)=-\theta$.

If $X$ is exceptional, then ${\overline{\sigma}}(\alpha)\neq\alpha$ by
Corollary 6.6 thereforen ${\overline{\sigma}}$ is a non-trivial involution of
the Dynkin diagram and this implies that $R$ is simply laced. In particular
$\sigma(\Theta)=-\Theta$ (since $\Theta=\theta$). If $\alpha\in\Delta_{1}$ is
exceptional, then ${\overline{\sigma}}(\alpha)\neq\alpha$ and the coefficients
of such roots in $\Theta$ are always equal to $1$.

On the other hand if $X$ is non-exceptional, then
${\overline{\sigma}}(\alpha)=\alpha$ and $\gamma=\alpha-\sigma(\alpha)$ is
dominant on its support and bigger than $2\alpha$. If $\gamma$ is long then it
is the highest root of ${\rm Supp}(\gamma)$, but this is impossible by the
discussion on pages 150–151 in [Ti11]. Therefore, $\gamma$ is short and $R$ is
not simply laced. Assume that $\sigma(\Theta)=-\Theta$ and let $(-,-)$ be a
$(W,\sigma)$-invariant scalar product on $\mathfrak{X}_{{\mathbb{R}}}$ such
that long roots have length $2$. We have
$({\overline{\Theta}},{\overline{\Theta}})=4(\Theta,\Theta)=8$ and
${\overline{\gamma}}=2\gamma=2{\overline{\alpha}}$ is such that
$({\overline{\gamma}},{\overline{\gamma}})=4(\gamma,\gamma)=4$. A
contradiction since in $\overline{R}$ all roots which are the double of
another root have the same length. Therefore, $\sigma(\Theta)\neq-\Theta$. ∎

######  Corollary 6.10.

$G/H$ is exceptional if and only if $G$ is simply laced and $\overline{R}$ is
non-reduced.

Recall the definition of ${\widehat{\alpha}}$ from the end of Subsection 4.1.

######  Lemma 6.11.

Assume that $\alpha\in\Delta_{1}$ is an exceptional root, then the coefficient
of ${\widehat{\alpha}}^{\vee}$ in the expansion of
${\overline{\Theta}}^{\vee}$ is terms of simple coroots of $\overline{R}$ is
equal to $1$.

###### Proof.

Note that $R$ is simply laced, so $\alpha$ and $\Theta$ have the same length.
Since $\alpha$ is exceptional, we have
${\overline{\alpha}},2{\overline{\alpha}}\in\overline{R}$ and
${\widehat{\alpha}}^{\vee}=\frac{1}{2}{\overline{\alpha}}^{\vee}=\frac{1}{2}(\alpha^{\vee}-\sigma(\alpha)^{\vee})$.
On the other hand, since $X$ is exceptional, we have $\sigma(\Theta)=-\Theta$
so that ${\overline{\Theta}}^{\vee}=\frac{1}{2}\Theta^{\vee}$. Since the
coefficient of $\alpha$ in the expansion of $\Theta$ in terms of simple roots
is equal to $1$, then same is true for the coefficient of
${\widehat{\alpha}}^{\vee}$ in the expansion of ${\overline{\Theta}}^{\vee}$
in terms of simple coroots. ∎

Note that if $R$ is not of type ${\rm A}_{1}$, there always exists a simple
root $\alpha_{{\rm adj}}\in\Delta$ such that
$\langle\Theta^{\vee},\alpha_{{\rm adj}}\rangle=1$. Such a simple root
$\alpha_{{\rm adj}}$ is unique if $R$ is not of type ${\rm A}_{r}$. In type
${\rm A}_{r}$ with $r\geq 2$, there are two such simple roots: $\alpha_{1}$
and $\alpha_{r}$ with simple roots labeled as in [Bo68].

######  Proposition 6.12.

Assume that $R$ is not of type ${\rm A}_{1}$ and let $\alpha_{{\rm
adj}}\in\Delta$ be any simple root such that
$\langle\Theta^{\vee},\alpha_{{\rm adj}}\rangle=1$.

  1. 1.

We have the equivalences: $\sigma(\Theta)\neq-\Theta$ $\Leftrightarrow$
$\sigma(\alpha_{{\rm adj}})=\alpha_{{\rm adj}}$
$\Leftrightarrow\langle\Theta^{\vee},\sigma(\Theta)\rangle=0$.

  2. 2.

If $\overline{R}$ is not of type ${\rm A}_{1}$, there exists a simple root
${\overline{\alpha}}\in\overline{\Delta}$ such that
$\langle{\overline{\Theta}}^{\vee},{\overline{\alpha}}\rangle=1$.

  3. 3.

If $\overline{R}$ is not of type ${\rm A}_{1}$, then
$2{\overline{\Theta}}^{\vee}$ is indivisible as a cocharacter of $T_{{\rm
s}}$.

  4. 4.

If $\sigma(\Theta)\neq-\Theta$, then $-\sigma(\Theta)$ is the highest root of
a connected component of the subsystem $R^{\perp}$ of $R$ generated by simple
roots orthogonal to $\Theta$.

###### Proof.

(1) Note that since $\Theta$ is dominant and since $\sigma(\Theta)<0$
(otherwise $P=G$ and $\sigma$ is trivial), we have
$\langle\Theta^{\vee},\sigma(\Theta)\rangle\leq 0$ and therefore either
$\sigma(\Theta)=-\Theta$ or $\langle\Theta^{\vee},\sigma(\Theta)\rangle=0$. We
therefore only need to prove the first equivalence. If
$\sigma(\Theta)=-\Theta$, then $\langle\Theta^{\vee},\sigma(\alpha_{{\rm
adj}})\rangle=\langle\sigma(\Theta^{\vee}),\alpha_{{\rm
adj}}\rangle=-\langle\Theta^{\vee},\alpha_{{\rm adj}}\rangle=-1$ therefore
$\sigma(\alpha_{{\rm adj}})<0$. Conversely, if
$\langle\Theta^{\vee},\sigma(\Theta)\rangle=0$, then $\alpha_{{\rm adj}}$ does
not occur in the support of $\sigma(\Theta)$. Since $\sigma(\Theta)$ is a
negative root we have $\langle\alpha_{{\rm
adj}}^{\vee},\sigma(\Theta)\rangle\geq 0$ and thus
$\langle\sigma(\Theta)^{\vee},\alpha_{{\rm adj}}\rangle\geq 0$. We get
$\langle\Theta^{\vee},\sigma(\alpha_{{\rm
adj}})\rangle=\langle\sigma(\Theta)^{\vee},\alpha_{{\rm adj}}\rangle\geq 0$;
thus, $\sigma(\alpha_{{\rm adj}})>0$ and $\sigma(\alpha_{{\rm
adj}})=\alpha_{{\rm adj}}$.

(2) If $\overline{R}$ is reduced then the result follows, since
${\overline{\Theta}}$ is the highest root of $\overline{R}$: take
${\overline{\alpha}}={\overline{\alpha}}_{{\rm adj}}\in\overline{\Delta}$ a
simple root such that
$\langle{\overline{\Theta}}^{\vee},{\overline{\alpha}}_{{\rm adj}}\rangle=1$.

If $\overline{R}$ is non-reduced, then ${\overline{\Theta}}$ is the highest
root; therefore, there exists a root ${\overline{\beta}}$ such that
${\overline{\Theta}}=2{\overline{\beta}}$. We have
$\langle{\overline{\Theta}}^{\vee},{\overline{\beta}}\rangle=\frac{1}{2}\langle{\overline{\Theta}}^{\vee},{\overline{\Theta}}\rangle=1$.
Since ${\overline{\Theta}}^{\vee}$ is dominant, this implies the result.

(3) If $\sigma(\Theta)=-\Theta$, then
$2{\overline{\Theta}}^{\vee}=\Theta^{\vee}$ and we have $\langle
2{\overline{\Theta}}^{\vee},\alpha_{{\rm adj}}\rangle=1$. If
$\langle\Theta^{\vee},\sigma(\Theta)\rangle=0$, then $\langle
2{\overline{\Theta}}^{\vee},\alpha_{{\rm
adj}}\rangle=\langle\Theta^{\vee}-\sigma(\Theta)^{\vee},\alpha_{{\rm
adj}}\rangle=\langle\Theta^{\vee},\alpha_{{\rm adj}}\rangle=1$ (we use (4)
below for the last equality).

(4) If $\sigma(\Theta)\neq-\Theta$, then
$\langle\Theta^{\vee},\sigma(\Theta)\rangle=0$ and $-\sigma(\Theta)\in
R^{\perp}$ (the subsystem generated by simple roots orthogonal to $\Theta$).
Let $\alpha\in R^{\perp}$. If $\sigma(\alpha)=\alpha$, then
$\langle-\sigma(\Theta)^{\vee},\alpha\rangle=-\langle\Theta^{\vee},\alpha\rangle=0$.
If $\alpha\in\Delta_{1}$, then $\sigma(\alpha)<0$ and
$\langle-\sigma(\Theta)^{\vee},\alpha\rangle=-\langle\Theta^{\vee},\sigma(\alpha)\rangle\geq
0$; thus, $-\sigma(\Theta)$ is dominant in $R^{\perp}$ and the result follows,
since $-\sigma(\Theta)$ and $\Theta$ are long roots. ∎

###  6.2 Marked Kac diagrams

Our description of the components of $H\cdot C$ is based on the fact that
$H\cdot C\simeq H\cdot[m]\subset{\mathbb{P}}(\mathfrak{p})$ for $m\in
T_{x}C\setminus\\{0\\}$ (Lemma 2.11) together with the following result.

######  Lemma 6.13 (Lemma 26.8 of [Ti11]).

The simple roots of $H^{0}$ and the lowest weights of $\mathfrak{p}$ with
respect to the $H^{0}$-representation form a affine simple root system.

Furthermore, the lowest weights of $\mathfrak{p}$ together with the Dynkin
diagram of $H^{0}$ can be encoded in the so-called Kac diagram of $G/H$. We
refer to [Ti11, Sections 26.3 and 26.5] for more on these diagrams.

######  Proposition 6.14.

Let $X$ be the wonderful compactification of an adjoint irreducible symmetric
space. The irreducible components of the orbits $H\cdot C$, where $C$ runs
over the highest weight curves on $X$, are exactly the homogeneous spaces
$H^{0}/Q_{\delta}$, where $\delta$ is a white node in the Kac diagram, and
$Q_{\delta}$ denotes the parabolic subgroup of $H^{0}$ associated to the set
of simple roots of $H$ not adjacent to $\delta$.

###### Proof.

The result follows from the fact that $m$ is a highest weight vector of
$\mathfrak{p}$, because this highest weight is conjugate in $H^{0}$ to a
lowest weight of $\mathfrak{p}$ corresponding to a white node $\delta$. ∎

######  Remark 6.15.

We make the following observations.

  1. 1.

There are two white nodes in the Kac diagram if and only if $X$ is Hermitian.

  2. 2.

If $X$ is Hermitian, the two corresponding parabolic subgroups are conjugated
by an automorphism of $G$. This automorphism is an outer automorphism if and
only if $X$ is exceptional.

We call a Kac diagram with a marked white node a _Marked Kac Diagram_.

######  Example 6.16.

We illustrate the above proposition by a few examples. We picture the Kac
diagram on the left and on the right we picture the Dynkin diagram of $H^{0}$
with the simple roots that are not roots of $H^{0}_{C}$ crossed.

  1. 1.

$G/H=\operatorname{SL}_{8}\times\operatorname{SL}_{8}/(Z(G)\cdot\operatorname{SL}_{8})$
and $H^{0}/H^{0}_{C}\simeq\operatorname{Flag}(1,6)$ as $H^{0}$-varieties.

$\times$$\times$

  2. 2.

$G/H=F_{4}/B_{4}$ and $H^{0}/H^{0}_{C}\simeq{\rm OG}(4,9)$ as
$H^{0}$-varieties.

$<$

$<$

$\times$

  3. 3.

$G/H=\operatorname{SL}_{8}/S({\rm GL}_{3}\times{\rm GL}_{5})$ and
$H^{0}/H^{0}_{C}\simeq{\mathbb{P}}^{2}\times{\mathbb{P}}^{4}$ as varieties..

\begin{picture}(2.0,2.2)\par\put(3.0,2.0){\circle{0.3}}
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\par\put(11.0,0.5){\circle*{0.3}} \put(13.0,0.5){\circle*{0.3}}
\put(9.7,0.32){$\times$} \put(15.8,0.32){$\times$}
\put(14.0,0.5){\circle*{0.3}} \put(15.0,0.5){\circle*{0.3}}
\par\put(10.0,1.5){\line(1,0){1.0}} \put(13.0,1.5){\line(1,0){1.0}}
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\par\put(10.0,1.5){\circle*{0.3}} \put(14.0,1.5){\circle*{0.3}}
\put(10.8,1.32){$\times$} \put(12.7,1.32){$\times$}
\put(15.0,1.5){\circle*{0.3}} \put(16.0,1.5){\circle*{0.3}}
\par\put(17.0,0.3){$\mathbb{P}^{2}\times{\mathbb{P}^{4}}^{\vee}$}
\put(16.8,1.3){${\mathbb{P}^{2}}^{\vee}\times\mathbb{P}^{4}$} \end{picture}

$\times$$\times$

$\times$$\times$

$\mathbb{P}^{2}\times{\mathbb{P}^{4}}^{\vee}$${\mathbb{P}^{2}}^{\vee}\times\mathbb{P}^{4}$

  4. 4.

$G/H={\rm Sp}_{12}/{\rm GL}_{6}$ and $H^{0}/H^{0}_{C}\simeq{\mathbb{P}}^{5}$
as varieties.

$>$

$<$

$\times$$\times$

${\mathbb{P}^{5}}^{*}$$\mathbb{P}^{5}$

###  6.3 Some examples

We describe some families of examples. Recall that $G_{\rm ad}=G/Z(G)$.

#### Hermitian types.

Assume that $G/H$ is of Hermitian type. The involution $\sigma$ is given on
$G_{\rm ad}$ by conjugation with respect to $\varpi_{\alpha}^{\vee}(-1)$,
where $\alpha$ is a simple cominuscule root (appearing with coefficient $1$ in
$\Theta$). In this case, $\sigma(\Theta)=-\Theta$ and any irreducible
component of $H\cdot C$ is a smooth irreducible Schubert variety in
${\mathbb{P}}({\mathcal{O}}_{\min})$, of dimension
$\frac{1}{2}(\dim{\mathcal{O}}_{\min}-1)$. The exceptional cases correspond to
the simple cominuscule roots $\alpha$ which are sent to different simple
cominuscule roots by a Dynkin diagram involution.

#### Subadjoint case.

Let $\aleph=\\{\alpha\in\Delta\ |\ \langle\Theta^{\vee},\alpha\rangle\neq
0\\}$. Then $|\aleph|=1$ except in type $\textrm{A}_{r}$ with $r\geq 2$, where
$|\aleph|=2$. Let
$\varpi^{\vee}(-1)=\prod_{\alpha\in\aleph}\varpi_{\alpha}^{\vee}(-1)$. Define
the involution $\sigma$ on $G_{\rm ad}$ by conjugation by $\varpi^{\vee}(-1)$.
Note that $G/G^{\sigma}$ is not Hermitian, except in type ${\rm A}_{r}$. We
have
$\mathfrak{h}=\mathfrak{g}^{\sigma}=\mathfrak{g}_{\Theta}\oplus\mathfrak{k}$
where $\mathfrak{g}_{\Theta}=\langle e,h,f\rangle$ with
$e\in\mathfrak{g}_{\Theta}\setminus\\{0\\}$,
$f\in\mathfrak{g}_{-\Theta}\setminus\\{0\\}$ and $h=[e,f]$ (in particular,
$\mathfrak{g}_{\Theta}\simeq\mathfrak{sl}_{2}$), and $\mathfrak{k}$ is a
reductive Lie subalgebra of $\mathfrak{g}$. Let $G_{\Theta}$ and $K$ be the
closed connected subgroups of $G$ with Lie algebras $\mathfrak{g}_{\Theta}$
and $\mathfrak{k}$. Then $G_{\Theta}$ is isomorphic to $\operatorname{SL}_{2}$
or $\operatorname{PSL}_{2}$; moreover, $G^{\sigma,0}=G_{\Theta}K$ and
$G_{\Theta}\cap K$ is finite. Furthermore, we have
$\mathfrak{p}={\mathbb{C}}^{2}\otimes V_{K}$, where ${\mathbb{C}}^{2}$ is the
standard representation of $\mathfrak{sl}_{2}$, and $V_{K}$ is a
$K$-representation which is irreducible in all cases except in type ${\rm
A}_{r}$. In type ${\rm A}_{r}$, we have $V_{K}=V^{+}_{K}\oplus V^{-}_{K}$
which are dual irreducible representations. Let
$u^{+},u^{-}\in{\mathbb{C}}^{2}$ be a highest and a lowest weight vector for
$G_{\Theta}$ and let $v\in V_{K}$ be a highest weight for $K$. We identify
$V_{K}$ to the subspace $\langle u^{-}\rangle\otimes V_{K}$ of $\mathfrak{p}$.

We have ${\mathcal{O}}_{\min}=G\cdot e$. We will use the following isomorphism
of $T$-representations: $T_{e}{\mathcal{O}}_{\min}=\langle f,h\rangle\oplus
V_{K}\subset\mathfrak{g}$. Note that the symplectic form $\omega_{e}$ on
$T_{e}{\mathcal{O}}_{\min}$ restricts to orthogonal symplectic forms on
$\langle f,h\rangle$ and $V_{K}$. Recall the definition of the subadjoint
variety ${\mathbb{L}}_{G}$ as the set of lines in
${\mathbb{P}}({\mathcal{O}}_{\min})$ passing through $[e]$. In particular,
${\mathbb{L}}_{G}\subset{\mathbb{P}}(T_{e}{\mathcal{O}}_{\min})$. We have
${\mathbb{L}}_{G}={\mathbb{P}}({\mathcal{O}}_{\min})\cap{\mathbb{P}}(V)$. In
particular ${\mathbb{L}}_{G}=\emptyset$ in type $\operatorname{C}_{r}$, since
${\mathbb{P}}({\mathcal{O}}_{\min})\cap{\mathbb{P}}(\mathfrak{p})=\emptyset$
in this case. In the other cases, ${\mathbb{L}}_{G}=K\cdot[v]$ is the closed
$K$-orbit in ${\mathbb{P}}(V_{K})$, and spans this projective space. Note that
in type ${\rm A}_{r}$, the variety ${\mathbb{L}}_{G}$ has two connected
components given by the closed $K$-orbits in ${\mathbb{P}}(V^{+}_{K})$ and
${\mathbb{P}}(V^{-}_{K})$. Let $\mathfrak{l}_{G}=T_{[v]}{\mathbb{L}}_{G}$,
then $\mathfrak{l}_{G}$ is a Lagrangian subspace in $V_{K}$. We will recover
this fact using the VMRT of $X_{\rm ad}$, the wonderful compactification of
the adjoint symmetric space $G/N_{G}(G^{\sigma})$.

Assume that $G$ is not of type $\operatorname{C}_{r}$. Set $M={\rm
VMRT}(X_{\rm ad})$ and let $\widehat{M}$ be the cone over $M$ in
$\mathfrak{p}\subset\mathfrak{g}$. We have
$M={\mathbb{P}}({\mathcal{O}}_{\min})\cap{\mathbb{P}}(\mathfrak{p})$: the
highest weight $m^{+}=u^{+}\otimes v$ of $\mathfrak{p}$ lies in
${\mathcal{O}}_{\min}$ and $M=(G_{\Theta}\times
K)\cdot[m]={\mathbb{P}}^{1}\times{\mathbb{L}}_{G}$. We get that
${\mathbb{L}}_{G}=M\cap{\mathbb{P}}(V_{K})$.

Let $m=u^{-}\otimes v\in V$. Note that $[m]\in{\mathbb{L}}_{G}\subset M$.
There exists a Weyl group element $s\in W$ such that $[m]=s\cdot[e]$ and
$s(V_{K})=V_{K}$. Thus, $T_{m}{\mathcal{O}}_{\min}=\langle
s(f),s(h)\rangle\oplus V_{K}$. Furthermore, $\omega_{s(e)}$ restricts to
orthogonal symplectic forms on $\langle s(f),s(h)\rangle$ and $V_{K}$. We have
$T_{m}\widehat{M}=T_{m}{\mathcal{O}}_{\min}\cap\mathfrak{p}$ and this space is
Lagrangian for $\omega_{s(e)}$. On the other hand, we have
$T_{m}\widehat{M}=\langle s(f)\rangle\oplus\mathfrak{l}_{G}$. This implies
that $\mathfrak{l}_{G}$ is a Lagrangian subspace of $V_{K}$.

#### Non-Fano cases.

The wonderful compactifications $X_{\rm ad}$ of adjoint irreducible symmetric
spaces are not always Fano. The Fano and non-Fano cases have been classified
in [Ru12, Theorem 2.1, Table 2]. We summarise the results here: The types for
which $X_{\rm ad}$ is not Fano are CI, DI, EI, EV, EVIII, FI and G. An easy
way to find them is to use both the restricted root sytem and the Satake
diagram (see [Ti11, Table 26.3]): the non-Fano cases are those for which the
restricted root system is not of type A nor of type B and the Satake diagram
has only white nodes and no arrow.

###  6.4 Classification table

We list all symmetric spaces $G/H$ (up to finite coverings) with $X_{\rm ad}$
irreducible, their varieties of minimal rational tangents ${\mathcal{C}}_{x}$
and the restriction of ${\mathcal{O}}_{{\mathbb{P}}(\mathfrak{p})}(1)$ to the
VMRT giving the embedding
${\mathcal{C}}_{x}\subset{\mathbb{P}}(\mathfrak{p})$. For
${\mathcal{C}}_{1}\bigsqcup{\mathcal{C}}_{2}$, the notation ${\mathcal{O}}(1)$
corresponds to the embedding in
${\mathbb{P}}(H^{0}({\mathcal{C}}_{1},{\mathcal{O}}_{{\mathcal{C}}_{1}}(1))\oplus
H^{0}({\mathcal{C}}_{2},{\mathcal{O}}_{{\mathcal{C}}_{2}}(1)))$. The
penultimate column describes the orbit $G\cdot m$ for $m\in
T_{x}C\setminus\\{0\\}$ with $C\in{\mathcal{K}}_{x}$ and ${\mathcal{K}}$ a
minimal family.

Some notations. H.n.e = Hermitian non-exceptional. H.e = Hermitian
exceptional. $Q_{n}$ = smooth quadric of dimension $n$. ${\rm Gr}(a,b)$ =
Grassmannian of vector subspaces of dimension $a$ in ${\mathbb{C}}^{b}$. ${\rm
OG}(a,b)$ = closed subset of ${\rm Gr}(a,b)$ of isotropic subspaces for a non-
degenerate quadratic form on ${\mathbb{C}}^{b}$ (with $a<2b$). ${\rm
OG}(b,2b)$ = a connected component of the Grassmannian of maximal isotropic
subspaces in ${\mathbb{C}}^{2b}$ for a non-degenerate quadratic form. ${\rm
IG}(a,2b)$ = closed subset of ${\rm Gr}(a,b)$ of isotropic subspaces for a
non-degenerate symplectic form on ${\mathbb{C}}^{2b}$. ${\rm LG}(b,2b)$ =
Grassmannian of maximal isotropic subspaces in ${\mathbb{C}}^{2a}$ for a non-
degenerate symplectic form. $\operatorname{Flag}(1,r)$ = nested subspaces of
dimension $1$ and $r$ in ${\mathbb{C}}^{r+1}$.

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$\begin{array}[]{|c|c|c|c|c|cc|c|c|c|c|}\textrm{Type}&G/H&{\rm
Condition}&\overline{R}&H\cdot C&{\rm
VMRT}&{\mathcal{O}}_{{\mathbb{P}}(\mathfrak{p})}(1)&\sigma(\Theta)=-\Theta&{\rm
Herm/Exc}&{\rm Fano}\\\ \hline\cr\textrm{Group}&H\times
H/H&\textrm{Type}(H)\neq\textrm{A}_{r}&\textrm{Type of
$H$}&{\mathbb{P}}({\mathcal{O}}_{\min,H})&H\cdot C&{\mathcal{O}}(1)&{\rm
yes}&&{\rm yes}\\\
\textrm{Group}&\operatorname{SL}_{r+1}\times\operatorname{SL}_{r+1}/\operatorname{SL}_{r+1}&r\geq
2&\textrm{A}_{r}&\operatorname{Flag}(1,r)&{\mathbb{P}}^{r}\times{\mathbb{P}}^{r}&{\mathcal{O}}(1,1)&{\rm
yes}&&{\rm yes}\\\
\textrm{Group}&\operatorname{SL}_{2}\times\operatorname{SL}_{2}/\operatorname{SL}_{2}&&\textrm{A}_{1}&{\mathbb{P}}^{1}&{\mathbb{P}}^{2}&{\mathcal{O}}(1)&{\rm
yes}&&{\rm yes}\\\ \textrm{A
I}&\operatorname{SL}_{r+1}/\operatorname{SO}_{r+1}&r\geq
2&\textrm{A}_{r}&Q_{r-1}&{\mathbb{P}}^{r}&{\mathcal{O}}(2)&{\rm yes}&&{\rm
yes}\\\ \textrm{A
I}&\operatorname{SL}_{2}/\operatorname{SO}_{2}&&\textrm{A}_{1}&\\{{\rm
pt}\\}\bigsqcup\\{{\rm pt}\\}&{\mathbb{P}}^{1}&{\mathcal{O}}(1)&{\rm
yes}&\textrm{H.n.e}&{\rm yes}\\\ \textrm{A II}&\operatorname{SL}_{2r+2}/{\rm
Sp}_{2r+2}&r\geq 2&\textrm{A}_{r}&{\rm IG}(2,2r+2)&{\rm
Gr}(2,2r+2)&{\mathcal{O}}(1)&{\rm no}&&{\rm yes}\\\ \textrm{A
III}&\operatorname{SL}_{n}/S({\rm GL}_{r}\times{\rm GL}_{n-r})&1\leq
r<n/2&\textrm{BC}_{r}&{\mathbb{P}}^{r-1}\times{\mathbb{P}}^{n-r-1}&H\cdot
C&{\mathcal{O}}(1,1)&{\rm yes}&\textrm{H.e}&{\rm yes}\\\ \textrm{A
III}&\operatorname{SL}_{2r}/S({\rm GL}_{r}\times{\rm
GL}_{r})&&\textrm{C}_{r}&({\mathbb{P}}^{r-1})^{2}\bigsqcup({\mathbb{P}}^{r-1})^{2}&H\cdot
C&{\mathcal{O}}(1,1)&{\rm yes}&\textrm{H.n.e}&{\rm yes}\\\ \textrm{BD
I}&\operatorname{SO}_{n}/S({\rm O}_{r}\times{\rm O}_{n-r})&3\leq
r\leq\frac{n-1}{2}&\textrm{B}_{r}&Q_{r-2}\times Q_{n-r-2}&H\cdot
C&{\mathcal{O}}(1,1)&{\rm yes}&&{\rm yes}\\\ \textrm{BD
I}&\operatorname{SO}_{n}/S({\rm O}_{2}\times{\rm
O}_{n-2})&&\textrm{B}_{2}&Q_{n-4}\bigsqcup Q_{n-4}&H\cdot
C&{\mathcal{O}}(1)&{\rm yes}&\textrm{H.n.e}&{\rm yes}\\\ \textrm{BD
II}&\operatorname{SO}_{n}/S({\rm O}_{1}\times{\rm
O}_{n-1})&&\textrm{A}_{1}&Q_{n-3}&{\mathbb{P}}^{n-2}&{\mathcal{O}}(1)&{\rm
no}&&{\rm yes}\\\ \textrm{C I}&{\rm Sp}_{2r}/{\rm GL}_{r}&r\geq
3&\textrm{C}_{r}&{\mathbb{P}}^{r-1}\bigsqcup{\mathbb{P}}^{r-1}&H\cdot
C&{\mathcal{O}}(2)&{\rm yes}&\textrm{H.n.e}&{\rm no}\\\ \textrm{C II}&{\rm
Sp}_{2n}/{\rm Sp}_{2r}\times{\rm Sp}_{2n-2r}&1\leq
r\leq\frac{(n-1)}{2}&\textrm{BC}_{r}&{\mathbb{P}}^{2r-1}\times{\mathbb{P}}^{2n-2r-1}&H\cdot
C&{\mathcal{O}}(1,1)&{\rm no}&&{\rm yes}\\\ \textrm{C II}&{\rm Sp}_{4r}/{\rm
Sp}_{2r}\times{\rm Sp}_{2r}&r\geq
2&\textrm{C}_{r}&{\mathbb{P}}^{2r-1}\times{\mathbb{P}}^{2r-1}&H\cdot
C&{\mathcal{O}}(1,1)&{\rm no}&&{\rm yes}\\\ \textrm{D
I}&\operatorname{SO}_{2r}/S({\rm O}_{r}\times{\rm O}_{r})&r\geq
4&\textrm{D}_{r}&Q_{r-2}\times Q_{r-2}&H\cdot C&{\mathcal{O}}(1,1)&{\rm
yes}&&{\rm no}\\\ \textrm{D III}&\operatorname{SO}_{4r}/{\rm
GL}_{2r}&&\textrm{C}_{r}&{\rm Gr}(2,2r)\bigsqcup{\rm Gr}(2,2r)&H\cdot
C&{\mathcal{O}}(1)&{\rm yes}&\textrm{H.n.e}&{\rm yes}\\\ \textrm{D
III}&\operatorname{SO}_{4r+2}/{\rm GL}_{2r+1}&&\textrm{BC}_{r}&{\rm
Gr}(2,2r+1)&H\cdot C&{\mathcal{O}}(1)&{\rm yes}&\textrm{H.e.}&{\rm yes}\\\
\textrm{E I}&E_{6}/C_{4}&&\textrm{E}_{6}&{\rm LG}(4,8)&H\cdot
C&{\mathcal{O}}(1)&{\rm yes}&&{\rm no}\\\ \textrm{E II}&E_{6}/A_{5}\times
A_{1}&&\textrm{F}_{4}&{\rm Gr}(3,6)\times{\mathbb{P}}^{1}&H\cdot
C&{\mathcal{O}}(1,1)&{\rm yes}&&{\rm yes}\\\ \textrm{E
III}&E_{6}/D_{5}\times{\mathbb{C}}^{*}&&\textrm{BC}_{2}&{\rm OG}(5,10)&H\cdot
C&{\mathcal{O}}(1)&{\rm yes}&\textrm{H.e.}&{\rm yes}\\\ \textrm{E
IV}&E_{6}/F_{4}&&\textrm{A}_{2}&F_{4}/P_{4}&E_{6}/P_{6}&{\mathcal{O}}(1)&{\rm
no}&&{\rm yes}\\\ \textrm{E V}&E_{7}/A_{7}&&\textrm{E}_{7}&{\rm
Gr}(4,8)&H\cdot C&{\mathcal{O}}(1)&{\rm yes}&&{\rm no}\\\ \textrm{E
VI}&E_{7}/D_{6}\times A_{1}&&\textrm{F}_{4}&{\rm
OG}(6,12)\times{\mathbb{P}}^{1}&H\cdot C&{\mathcal{O}}(1,1)&{\rm yes}&&{\rm
yes}\\\ \textrm{E
VII}&E_{7}/E_{6}\times{\mathbb{C}}^{*}&&\textrm{C}_{3}&E_{6}/P_{1}\bigsqcup
E_{6}/P_{6}&H\cdot C&{\mathcal{O}}(1)&{\rm yes}&\textrm{H.n.e.}&{\rm yes}\\\
\textrm{E VIII}&E_{8}/D_{8}&&\textrm{E}_{8}&{\rm OG}(8,16)&H\cdot
C&{\mathcal{O}}(1)&{\rm yes}&&{\rm no}\\\ \textrm{E IX}&E_{8}/E_{7}\times
A_{1}&&\textrm{F}_{4}&E_{7}/P_{7}\times{\mathbb{P}}^{1}&H\cdot
C&{\mathcal{O}}(1,1)&{\rm yes}&&{\rm yes}\\\ \textrm{F I}&F_{4}/C_{3}\times
A_{1}&&\textrm{F}_{4}&{\rm LG}(3,6)\times{\mathbb{P}}^{1}&H\cdot
C&{\mathcal{O}}(1,1)&{\rm yes}&&{\rm no}\\\ \textrm{F
II}&F_{4}/B_{4}&&\textrm{BC}_{1}&{\rm OG}(4,9)&H\cdot C&{\mathcal{O}}(1)&{\rm
no}&&{\rm yes}\\\ \textrm{G}&G_{2}/A_{1}\times
A_{1}&&\textrm{G}_{2}&{\mathbb{P}}^{1}\times{\mathbb{P}}^{1}&H\cdot
C&{\mathcal{O}}(1,3)&{\rm yes}&&{\rm no}\\\ \hline\cr\end{array}$ Table 1:
Wonderful compactifications and their VMRT.