# Minimal rational curves
on complete symmetric varieties

Michel Brion111Université Grenoble Alpes, Institut Fourier, CS 40700, 38058
Grenoble Cedex 9, France<EMAIL_ADDRESS>Shin-young
Kim222Center for Geometry and Physics, Institute for Basic Science, Pohang
37673, Korea<EMAIL_ADDRESS>and Nicolas Perrin333École Polytechnique,
Centre de Mathématiques Laurent Schwartz, 91128 Palaiseau Cedex, France,
<EMAIL_ADDRESS>
###### Abstract

We describe the families of minimal rational curves on any complete symmetric
variety, and the corresponding varieties of minimal rational tangents (VMRT).
In particular, we prove that these varieties are homogeneous and that for non-
exceptional irreducible wonderful varieties, there is a unique family of
minimal rational curves, and hence a unique VMRT. We relate these results to
the restricted root system of the associated symmetric space. In particular we
answer by the negative a question of Hwang: for certain Fano wonderful
symmetric varieties, the VMRT has two connected components.

##  1 Introduction

Let $X$ be a projective uniruled variety over the field of complex numbers. An
irreducible family ${\mathcal{K}}$ of rational curves on $X$ is called a
_covering family_ if there is a member of ${\mathcal{K}}$ passing through a
general point $x\in X$. If in addition the subfamily ${\mathcal{K}}_{x}$ of
curves in ${\mathcal{K}}$ passing through $x$ is proper, then ${\mathcal{K}}$
is called a _minimal family of rational curves_.

These curves play a prominent role in the study of the variety $X$. There is a
rational map $\tau_{x}:{\mathcal{K}}_{x}\dasharrow{\mathbb{P}}(T_{x}X)$
sending a curve to its tangent direction at $x$ and its image
${\mathcal{C}}_{x}\subset{\mathbb{P}}(T_{x}X)$ is an important invariant of
$X$ called _the variety of minimal rational tangents_ or VMRT of $X$, see
[Hw01], [HM04] and references therein.

The VMRT of projective rational homogeneous spaces $G/P$ for $G$ reductive and
$P$ a parabolic subgroup are well understood. For example, if $G/P$ has Picard
rank $1$, then there is a unique VMRT which characterizes $G/P$ and was used
to prove its rigidity in [HM05], with the unique exception of $B_{3}/P_{2}$
which admits an explicit degeneration constructed in [PP10]. If the Picard
number of $G/P$ is greater than $1$, there are several minimal families of
rational curves.

In [BF15], the authors consider another case where $X$ has a large Picard
group, namely the wonderful compactifications of adjoint semisimple groups.
Suprisingly, they prove that there is a unique minimal family of rational
curves for any such wonderful compactification and also that the corresponding
VMRT is a rational homogeneous variety. These results were used in [FL20] to
prove the rigidity of wonderful compactifications of groups, under the
condition that the special fiber is Fano.

In this paper we generalize the results of [BF15] and describe the minimal
families of rational curves on any complete symmetric variety. Rigidity of
symmetric varieties of Picard number $1$ has already attracted some attention
(see [KP19] and [CFL22]), we hope that our results will open new directions
for higher Picard numbers.

To state our main results, we recall basic definitions and properties of
complete symmetric varieties. Let $G$ be a connected reductive group and let
$\sigma$ be a group involution of $G$. A symmetric subgroup is a closed
subgroup $H\subset G$ such that $G^{\sigma,0}\subset H\subset G^{\sigma}$. The
homogeneous space $G/H$ is a _symmetric space_. We denote by $\mathfrak{g}$
and $\mathfrak{h}$ the Lie algebras of $G$ and $H$. Note the decomposition
$\mathfrak{g}=\mathfrak{h}\oplus\mathfrak{p}$ as $H$-representations, where
$\mathfrak{h}=\mathfrak{g}^{\sigma}$ and
$\mathfrak{p}=\mathfrak{g}^{-\sigma}$.

Consider the normalizer $N=\operatorname{N}_{G}(H)$; the quotient $G/N$ is the
_adjoint homogeneous space_ of the symmetric space $G/H$. This last space
admits by [deCP83] a unique wonderful compactification $X_{\rm ad}$. This is a
smooth projective $G$-variety having an open dense orbit $G\cdot x_{\rm
ad}=X_{\rm ad}^{0}\simeq G/N$, such that the boundary $\partial X_{\rm
ad}=X_{\rm ad}\setminus X^{0}_{\rm ad}$ is a simple normal crossing divisor:
$\partial X_{\rm ad}=X_{\rm ad}^{1}\cup\cdots\cup X_{\rm ad}^{r}$ where
$X_{\rm ad}^{i}$ is a prime $G$-stable divisor for all $i\in[1,r]$.
Furthermore for any $y,z\in X_{\rm ad}$ we have $G\cdot y=G\cdot z$ if and
only if $\\{i\ |y\in X_{\rm ad}^{i}\\}=\\{i\ |z\in X_{\rm ad}^{i}\\}$. The
integer $r$ is _the rank_ of $G/N$. A _complete symmetric variety_ is a smooth
proper $G$-variety $X$ having a dense orbit $G\cdot x=X^{0}\simeq G/H$ such
that the natural map $G/H\to G/N\subset X_{\rm ad}$ extends to a
$G$-equivariant morphism $\pi:X\to X_{\rm ad}$. The boundary $\partial
X=X\setminus X^{0}$ is also a simple normal crossing divisor with $G$-stable
prime components.

Let $X$ be a complete symmetric variety with base point $x$ and map $\pi:X\to
X_{\rm ad}$, and let ${\mathcal{K}}$ be a minimal family of rational curves on
$X$. We will prove the following results.

######  Theorem 1.1 (Theorem 5.1).

${\mathcal{K}}_{x}$ is smooth and
$\tau_{x}:{\mathcal{K}}_{x}\to{\mathcal{C}}_{x}$ is an isomorphism.

In particular, understanding the VMRT as an abstract variety is equivalent to
understanding ${\mathcal{K}}_{x}$. If the map $\pi$ contracts curves of the
family ${\mathcal{K}}$, then the description of the VMRT follows easily from
the case of toric varieties treated in [CFH14], see Lemma 3.4.

######  Theorem 1.2.

If $\pi$ contracts a curve of ${\mathcal{K}}$, then ${\mathcal{C}}_{x}$ is a
linear subspace of ${\mathbb{P}}(\mathfrak{p})$.

We are therefore left to consider curves not contracted by $\pi$.

######  Theorem 1.3 (Proposition 3.6).

Assume that $\pi$ contracts no curve in ${\mathcal{K}}$ and let
$C\in{\mathcal{K}}$.

  1. 1.

There exists a unique minimal family of rational curves ${\mathcal{L}}$ in
$X_{\rm ad}$ such that $\pi$ maps curves of ${\mathcal{K}}$ to curves of
${\mathcal{L}}$.

  2. 2.

We have $1\leq\partial X\cdot C\leq\partial X_{\rm ad}\cdot\pi(C)\leq 2$.

  3. 3.

If $\partial X\cdot C=\partial X_{\rm ad}\cdot\pi(C)$ for some
$C\in{\mathcal{K}}$, then any component of ${\mathcal{K}}_{x}$ is isomorphic
to a component of ${\mathcal{L}}_{x_{\rm ad}}$.

One of the key ingredients for proving the above results is what we call
highest weight curves (see Subsection 2.6). Given a Borel subgroup $B_{H}$ of
$H$, the Borel Fixed Point Theorem implies that any irreducible component of
${\mathcal{K}}_{x}$ contains a $B_{H}$-fixed point $C$. Moreover, if $C$ is
not contracted by $\pi$, then $C$ is mapped to a $B_{H}$-fixed point $C_{\rm
ad}$ in ${\mathcal{L}}_{x_{\rm ad}}$ that determines the associated component
of ${\mathcal{L}}_{x_{\rm ad}}$. Furthermore, the tangent space at $x_{\rm
ad}$ of the highest weight curve $C_{\rm ad}$ in ${\mathcal{L}}_{x_{\rm ad}}$
is a highest weight line in $T_{x_{\rm ad}}(X_{\rm ad})$.

In view of the above results, we focus on wonderful compactifications of
adjoint symmetric spaces. Decomposing $G$ into a product of irreducible
$\sigma$-stable factors, we obtain a decomposition of $G/N$ into a product of
_irreducible symmetric spaces_. There are three possible types for these
irreducible factors (since $N$ contains the center of $G$, we may assume that
$G$ is of adjoint type for this list, see Subsection 2.5 for more details):

  1. 1.

_Group type_ : $(H\times H)/{\rm diag}(H)$, where $H$ is simple adjoint.

  2. 2.

_Hermitian type_ : $G/\operatorname{N}_{G}(L)$, where $G$ is simple adjoint
and $L\subset G$ is a Levi subgroup.

  3. 3.

_Simple type_ : $G/H$, where $G$ is simple adjoint and $H^{0}$ is simple.

Given a highest weight curve $C$ on $X$, we prove that there is a unique
irreducible factor $X_{C}$ of $X_{{\rm ad}}$ such that the composition of
$\pi:X\to X_{{\rm ad}}$ with the projection $X_{{\rm ad}}\to X_{C}$ sends $C$
isomorphically to its image. We may thus replace $X_{\rm ad}$ by $X_{C}$ and
assume that $X_{\rm ad}$ is irreducible. In particular $G/N$ is as in one of
the above three cases. To understand the geometry of the irreducible factors,
we use the _restricted root system_.

There exists a maximal torus $T_{{\rm s}}$, called of _split type_ , such that
$T_{{\rm s}}$ is $\sigma$-stable and $S=\\{t\in T_{{\rm s}}\ |\
\sigma(t)=t^{-1}\\}^{0}$ has maximal dimension. The root system $R$ of
$(G,T_{{\rm s}})$ is stable under the action of $\sigma$ and there is a basis
$\Delta$ of $R$ such that, for $\alpha\in\Delta$, either
$\sigma(\alpha)=\alpha$ or $\sigma(\alpha)<0$. Set
$\Delta_{1}=\\{\alpha\in\Delta\ |\ \sigma(\alpha)<0\\}$ and
${\overline{\alpha}}=\alpha-\sigma(\alpha)$. The set
$\overline{R}=\\{{\overline{\alpha}}\ |\ \alpha\in R\\}$ is a (possibly non-
reduced) root system with basis $\overline{\Delta}=\\{{\overline{\alpha}}\ |\
\alpha\in\Delta_{1}\\}$ called the _restricted root system_ of the symmetric
space. The rank of $\overline{R}$ is the rank $r$ of $G/H$. In Subsection 4.2,
we relate curves and divisors in $X_{\rm ad}$ to the restricted root system
(the results are probably well known to the experts but we could not find a
convenient reference). Let $\overline{R}^{\vee}$ be the dual root system of
$\overline{R}$ with coroot lattice ${\mathbb{Z}}\overline{\Delta}^{\vee}$ and
denote by $A_{1}(X)$ the Chow group of curves modulo rational equivalence. We
prove the following result.

######  Proposition 1.4 (Proposition 4.19).

There is a surjective ${\mathbb{Z}}$-linear map $\psi:A_{1}(X_{\rm
ad})\to{\mathbb{Z}}\overline{\Delta}^{\vee}$ such that:

  1. 1.

The image of the monoid of effective curves is the monoid spanned by coroots.

  2. 2.

The image of the monoid of curves having non-negative intersection with any
component of $\partial X_{\rm ad}$ is the intersection of
${\mathbb{Z}}\overline{\Delta}^{\vee}$ with the monoid of dominant
cocharacters.

If the map $\psi$ is not injective, then $X_{\rm ad}$ is called _exceptional_.
Since the class of a curve $C$ in a covering family ${\mathcal{L}}$ of
rational curves is effective and has non-negative intersection with any
component of $\partial X_{\rm ad}$, it has to be contained in the intersection
of the monoids spanned by coroots and by the dominant cocharacters. There is a
unique minimal such coroot ${\overline{\Theta}^{\vee}}$, the coroot of the
highest root ${\overline{\Theta}}\in\overline{R}$. This gives a very natural
candidate for classes of minimal families. Indeed we prove the following
result.

######  Theorem 1.5 (Corollary 4.20).

Assume that $X_{\rm ad}$ is irreducible.

  1. 1.

If $X_{\rm ad}$ is not exceptional, there is a unique minimal family
${\mathcal{L}}$ and the class of any $C_{\rm ad}\in{\mathcal{L}}$ satisfies
$\psi([C_{\rm ad}])={\overline{\Theta}}^{\vee}$.

  2. 2.

If $X_{\rm ad}$ is exceptional, then there are exactly two minimal families
${\mathcal{L}}^{+}$ and ${\mathcal{L}}^{-}$ and the class of any $C_{\rm
ad}^{\pm}\in{\mathcal{L}}^{\pm}$ satifies $\psi([C_{\rm
ad}^{\pm}])={\overline{\Theta}}^{\vee}$.

Note that if $C_{\rm ad}$ is a highest weight curve in ${\mathcal{L}}_{x_{\rm
ad}}$, then its $H$-orbit $H\cdot C_{\rm ad}$ is contained in
${\mathcal{L}}_{x_{\rm ad}}$. We describe the family ${\mathcal{L}}_{x_{\rm
ad}}$ by comparing the dimension of this orbit with the dimension of the
family ${\mathcal{L}}_{x_{\rm ad}}$ of curves whose class is described by the
previous result. To compute the dimension of the $H$-orbits, we prove that the
tangent line $T_{x_{\rm ad}}C_{\rm ad}$ lies in very specific nilpotent orbits
in $\mathfrak{g}$. Let ${\mathcal{O}}_{\min}$ be the minimal non-zero
nilpotent orbit in $\mathfrak{g}$ and ${\mathcal{O}}_{{{\rm sum}},\sigma}$ be
the nilpotent orbit of $e_{\Theta}-\sigma(e_{\Theta})$, where $\Theta$ is the
highest root of $G$ and $e_{\Theta}\in\mathfrak{g}_{\Theta}\setminus\\{0\\}$.
Let $m\in T_{x_{\rm ad}}C_{\rm ad}\setminus\\{0\\}$, we prove the following in
Corollary 4.25.

######  Proposition 1.6.

We have $m\in{\mathcal{O}}_{\min}$ if $\sigma(\Theta)=-\Theta$ and
$m\in{\mathcal{O}}_{{{\rm sum}},\sigma}$ otherwise.

Using results of Kostant and Rallis [KR71] we prove that the orbit $H\cdot m$
is Lagrangian in the nilpotent orbit $G\cdot m$ (equipped with the Kirillov-
Kostant-Souriau invariant symplectic structure) and we are able to compute the
dimension of these orbits. We obtain:

######  Theorem 1.7 (Theorem 4.43).

If the restricted root system $\overline{R}$ is not of type ${\rm A}_{r}$,
then $\partial X_{\rm ad}\cdot C_{\rm ad}=1$ and $\dim H\cdot C_{\rm
ad}=\dim{\mathcal{L}}_{x_{\rm ad}}$. Otherwise, we have $\partial X_{\rm
ad}\cdot C_{\rm ad}=2$ and $\dim H\cdot C_{\rm ad}=\dim{\mathcal{L}}_{x_{\rm
ad}}-1$.

  1. 1.

If $\overline{R}$ is not of type ${\rm A}_{r}$, then ${\mathcal{L}}_{x_{\rm
ad}}=H\cdot C_{\rm ad}$. Furthermore, ${\mathcal{L}}_{x_{\rm ad}}$ has two
components if $X$ is Hermitian non-exceptional and is irreducible otherwise.

  2. 2.

If $\overline{R}$ is of type ${\rm A}_{1}$, then ${\mathcal{L}}_{x_{\rm
ad}}\simeq{\mathbb{P}}(\mathfrak{p})$.

  3. 3.

If $\overline{R}$ is of type ${\rm A}_{r}$ with $r\geq 2$, then there exists a
$G$-equivariant birational morphism $X_{\rm ad}\to{\mathbb{P}}(V)$, for some
irreducible $G$-representation $V$, and ${\mathcal{L}}_{x_{\rm ad}}$ is
isomorphic to the closed $G$-orbit in ${\mathbb{P}}(V)$. The orbit $H\cdot
C_{\rm ad}$ is a prime divisor in ${\mathcal{L}}_{x_{\rm ad}}$.

  4. 4.

The orbits $H\cdot C_{\rm ad}$ and the variety ${\mathcal{L}}_{x_{\rm ad}}$
are described in Table 1.

By results of Ruzzi [Ru12], the wonderful compactifications of irreducible
symmetric spaces are weak Fano varieties and most of them are Fano, with
exceptions classified in loc. cit., Table 2 (see also §6.3). As a consequence,
the wonderful compactifications of all Hermitian non-exceptional symmetric
spaces are Fano, except in type CI. We thus obtain the following result.

######  Corollary 1.8.

Let $X_{\rm ad}$ be the wonderful compactification of a Hermitian non-
exceptional symmetric space not of type ${\rm CI}$ and whose restricted root
system is not of type ${\rm A}_{r}$. Then $X_{\rm ad}$ is Fano and its VMRT
has two irreducible components.

This answers by the negative a question of Hwang, see [Hw01, Section 5,
Question 2]: there exists a Fano variety with non-irreducible VMRT of positive
dimension. There are four types in the classification leading to
counterexamples: AIII, BDI, DIII and EVII. Note that if the VMRT has positive
dimension, these Fano varieties have Picard number at least $2$ so that the
question for prime Fano varieties remains open.

In Table 1, we also give the embedding of ${\mathcal{L}}_{\rm ad}\simeq{\rm
VMRT}(X_{\rm ad})$ in ${\mathbb{P}}(\mathfrak{p})$. All VMRT are disjoint
unions of projective rational homogeneous varieties, which are in turn
products of homogeneous varieties of Picard rank one. In most cases, the
embedding is the minimal embedding. For two cases (types AI and CI), the
embedding is twice the minimal embedding. There is also a mixed case in type
G.

From these results and Theorem 1.3, we obtain a full description of the VMRT
of any complete symmetric variety $X$. We refer to Theorem 5.2 for more
details.

###### Contents

  1. 1 Introduction
  2. 2 Rational curves and symmetric spaces
    1. 2.1 Families of rational curves
    2. 2.2 Almost homogeneous varieties
    3. 2.3 Symmetric spaces
    4. 2.4 The normalizer of a symmetric subgroup
    5. 2.5 The adjoint symmetric space
    6. 2.6 Highest weight curves
  3. 3 Complete symmetric varieties
    1. 3.1 Wonderful and complete symmetric varieties
    2. 3.2 Relation between minimal families
    3. 3.3 Minimal families of group and Hermitian types
    4. 3.4 Some cases of simple type
  4. 4 Minimal families on wonderful symmetric varieties
    1. 4.1 Restricted root system
    2. 4.2 Divisors and restricted root system
    3. 4.3 Curves classes of the minimal families
    4. 4.4 Contact structure
    5. 4.5 Wonderful symmetric varieties of type ${\rm A}_{r}$
    6. 4.6 Minimal families on wonderful compactifications
  5. 5 Minimal families on complete symmetric varieties
  6. 6 Appendix
    1. 6.1 Restricted root systems
    2. 6.2 Marked Kac diagrams
    3. 6.3 Some examples
    4. 6.4 Classification table

Acknowledgements. The authors thank the Institut Fourier for hosting several
meetings that led to this work. They also thank the Tsinghua Sanya
International Mathematics Forum (TSIMF) and Prof. Baohua Fu for his
invitation. The stay of the authors in TSIMF was very influential for this
work. They finally thank the Centre Mathématique Laurent Schwartz (CMLS) at
École Polytechnique for hosting the final meeting which led to this work. S.K.
was partially supported by NRF Korea (2018R1A6A3A03012791) and the Institute
for Basic Science (IBS-R003-D1). N.P. was partially supported by ANR Catore
(ANR-18-CE40-0024) and ANR FanoHK (ANR-20-CE40-0023).

##  2 Rational curves and symmetric spaces

In this section, we recall basic results on rational curves on uniruled
varieties and then specialise to the case of almost homogeneous varieties. We
also introduce symmetric homogeneous spaces and their adjoint symmetric space,
and we obtain the existence of highest weight curves and their basic
properties.

###  2.1 Families of rational curves

In this subsection, we recall some notions and results on rational curves,
after [Ko96, II.2.2, II.2.3] and [BK21, §2.1, §2.2].

Let $X$ be a smooth projective variety. Consider the scheme of morphisms
$\operatorname{Hom}({\mathbb{P}}^{1},X)$ and the open subscheme
$\operatorname{Hom}_{{\rm bir}}({\mathbb{P}}^{1},X)$ consisting of morphisms
which are birational onto their image. The (normalized) _space of rational
curves_ $\operatorname{RatCurves}(X)$ is the quotient of the normalization
$\operatorname{Hom}^{{\rm n}}_{{\rm bir}}({\mathbb{P}}^{1},X)$ by the free
action of ${\rm Aut}({\mathbb{P}}^{1})={\rm PGL}_{2}$ via reparametrization.
We have a universal family

$\rho:\operatorname{Univ}(X)\longrightarrow\operatorname{RatCurves}(X)$

which is a ${\mathbb{P}}^{1}$-bundle, and an evaluation map

$\mu:\operatorname{Univ}(X)\longrightarrow X$

such that the morphism
$\rho\times\mu:\operatorname{Univ}(X)\to\operatorname{RatCurves}(X)\times X$
is finite.

Let $f\in\operatorname{Hom}_{{\rm bir}}({\mathbb{P}}^{1},X)$ with image
$C\subset X$. We say that $C$ is _free_ if the pull-back $f^{*}(T_{X})$ is
globally generated, where $T_{X}$ denotes the tangent bundle. Every free
morphism yields a smooth point of $\operatorname{Hom}_{{\rm
bir}}({\mathbb{P}}^{1},X)$, and hence of $\operatorname{RatCurves}(X)$. Also,
we say that $C$ is _embedded_ if $f$ is an immersion. The free (resp. embedded
free) curves form smooth open subschemes $\operatorname{RatCurves}_{{\rm
emfr}}(X)\subset\operatorname{RatCurves}_{{\rm fr}}(X)$ of the space of
rational curves.

A _family of rational curves_ on $X$ is a component ${\mathcal{K}}$ of
$\operatorname{RatCurves}(X)$. We then have a universal family
$\rho:{\mathcal{U}}=\rho^{-1}({\mathcal{K}})\to{\mathcal{K}}$ which is again a
${\mathbb{P}}^{1}$-bundle, and an evaluation map $\mu:{\mathcal{U}}\to X$. For
any $x\in X$, let ${\mathcal{U}}_{x}=\mu^{-1}(x)$ and
${\mathcal{K}}_{x}=\rho({\mathcal{U}}_{x})$; then ${\mathcal{K}}_{x}$ is the
subfamily of curves through $x$. The restriction
$\rho_{x}:{\mathcal{U}}_{x}\to{\mathcal{K}}_{x}$ is finite, and is an
isomorphism above the smooth open subset of embedded free curves (see [BK21,
Lem. 2.1]).

The family ${\mathcal{K}}$ is _covering_ if ${\mathcal{K}}_{x}$ is non-empty
for $x$ general. If in addition ${\mathcal{K}}_{x}$ is projective for $x$
general, we say that ${\mathcal{K}}$ is _minimal_.

By sending every embedded free curve in ${\mathcal{K}}_{x}$ to its tangent
direction at $x$, we obtain a morphism $\tau_{x}:{\mathcal{K}}_{{\rm
emfr},x}\to{\mathbb{P}}(T_{x}X)$, where ${\mathbb{P}}(T_{x}X)$ denotes the
projectivization of the tangent space. We will view $\tau$ as a rational map
${\mathcal{K}}_{x}\dasharrow{\mathbb{P}}(T_{x}X)$, defined at every curve
which is smooth at $x$. The closure of the image of $\tau$ is denoted by
${\mathcal{C}}_{x}$ and called the _variety of tangents_ of ${\mathcal{K}}$ at
$x$.

Let ${\mathcal{K}}$ be a minimal family of rational curves on $X$. By [Ke02,
Thm. 3.3], for a general point $x$, there are only finitely many curves in
${\mathcal{K}}_{x}$ which are singular at $x$. Thus, $\tau_{x}$ is defined
along every positive-dimensional irreducible component of ${\mathcal{K}}_{x}$.
In view of [Ke02, Thm. 3.4], $\tau_{x}$ extends to a finite morphism

$\tau_{x}^{{\rm n}}:{\mathcal{K}}_{x}^{{\rm
n}}\longrightarrow{\mathbb{P}}(T_{x}X),$

where ${\mathcal{K}}_{x}^{{\rm n}}$ denotes the normalization. Moreover,
$\tau_{x}^{{\rm n}}$ is birational onto its image by [HM04, Thm. 1]. The image
${\mathcal{C}}_{x}$ is called the _variety of minimal tangents_ of
${\mathcal{K}}$ at $x$ (VMRT).

Next, we consider covariance properties of families under a morphism of smooth
projective varieties $\pi:X\to Y$. Let ${\mathcal{K}}$ be a family of rational
curves on $X$. Assume that some $C\in{\mathcal{K}}$ is represented by a free
morphism $f:{\mathbb{P}}^{1}\to X$ which is birational onto its image, and
such that the composition $\pi\circ f:{\mathbb{P}}^{1}\to Y$ is free and
birational onto its image as well. Let $D$ be the corresponding rational curve
in $Y$, and ${\mathcal{L}}$ the family on $Y$ containing the free rational
curve $D$. Finally, let $x=f(0)\in C$ and $y=\pi(x)\in D$.

######  Lemma 2.1.

With the above notation and assumptions, the morphism $\pi:X\to Y$ induces
rational maps

$\pi_{*}:{\mathcal{K}}\dasharrow{\mathcal{L}},\quad\pi_{*,x}:{\mathcal{K}}_{x}\dasharrow{\mathcal{L}}_{y}$

which are defined at $C$ and send $C$ to $D$. If the differential
$d\pi_{x}:T_{x}X\to T_{y}Y$ is injective, then so is the differential of
$\pi_{*,x}$ at $C$.

###### Proof.

Composing by $\pi$ yields a morphism
$\operatorname{Hom}({\mathbb{P}}^{1},X)\to\operatorname{Hom}({\mathbb{P}}^{1},Y)$
which is ${\rm Aut}({\mathbb{P}}^{1})$-equivariant, and hence an equivariant
rational map $\operatorname{Hom}_{{\rm
fr}}({\mathbb{P}}^{1},X)\dasharrow\operatorname{Hom}_{{\rm
fr}}({\mathbb{P}}^{1},Y)$ which is defined at $f$. This readily yields the
rational map $\pi_{*}$. The rational map $\pi_{*,x}$ is obtained from the
analogous morphism $\operatorname{Hom}({\mathbb{P}}^{1},X;0\mapsto
x)\to\operatorname{Hom}({\mathbb{P}}^{1},Y;0\mapsto y)$ with the notation of
[Ko96, II.1]. By loc. cit., II.2.3, the differential of the above morphism at
$f$ is identified with the natural map
$H^{0}({\mathbb{P}}^{1},(f^{*}T_{X})(-1))\to
H^{0}({\mathbb{P}}^{1},(f^{*}\pi^{*}T_{Y})(-1))$. This implies the final
assertion as $d\pi$ is injective on an open dense subset of $X$. ∎

In the opposite direction, assume that $\pi$ _contracts_ a curve
$C\in{\mathcal{K}}$, i.e., the composition
$\rho^{-1}(C)\stackrel{{\scriptstyle\mu}}{{\longrightarrow}}X\stackrel{{\scriptstyle\pi}}{{\longrightarrow}}Y$
is constant; then $\pi$ contracts all the curves in ${\mathcal{K}}$. With this
terminology, we may recall a useful observation ([BK21, Lem. 2.3]):

######  Lemma 2.2.

Consider two smooth projective varieties $Y$, $Z$, and let $X:=Y\times Z$ with
projections $p:X\to Y$, $q:X\to Z$.

  1. 1.

The pull-back map $p^{*}:\operatorname{Hom}({\mathbb{P}}^{1},Y)\times
Z\to\operatorname{Hom}({\mathbb{P}}^{1},X)$, $(f,z)\to(t\mapsto(f(t),z))$
induces a closed immersion $\operatorname{RatCurves}(Y)\times
Z\to\operatorname{RatCurves}(X)$ with image a union of components.

  2. 2.

The map $p^{*}$ sends covering (resp. minimal) families to covering (resp.
minimal) families.

  3. 3.

A family of rational curves ${\mathcal{K}}$ on $X$ is the pull-back of a
family on $Y$ if and only if $q$ contracts some curve in ${\mathcal{K}}$.

  4. 4.

Every family of minimal rational curves on $X$ is the pull-back of a unique
family of minimal rational curves on $Y$ or $Z$.

###  2.2 Almost homogeneous varieties

We now assume that $X$ is _almost homogeneous_ , i.e., it is equipped with an
action of a connected linear algebraic group $G$, and contains an open
$G$-orbit $X^{0}$. We recall and slightly generalize results from [BF15, §2]
and [BK21, §2.3].

Choose a base point $x\in X^{0}$, and denote by $H=G_{x}$ its isotropy group.
Then the orbit $X^{0}=G\cdot x$ is identified with the homogeneous space
$G/H$, and the pair $(X,x)$, with an equivariant embedding of this homogeneous
space. Denoting by $\mathfrak{g}$ (resp. $\mathfrak{h}$) the Lie algebra of
$G$ (resp. $H$), the tangent space $T_{x}X$ is identified with the quotient
$\mathfrak{g}/\mathfrak{h}$ as a representation of $H$ (the _isotropy
representation_).

Since $G$ is a rational variety, $X$ is unirational; as a consequence,
covering families exist. Also, $G$ acts on $\operatorname{RatCurves}(X)$ and
on $\operatorname{Univ}(X)$ so that $\rho$ and $\mu$ are equivariant. Since
$G$ is connected, it stabilizes every family ${\mathcal{K}}$, as well as the
open subset ${\mathcal{K}}^{0}$ consisting of curves which meet $X^{0}$. Every
such curve is free (see e.g. [BF15, Lem. 2.1(i)]); thus, ${\mathcal{K}}^{0}$
is smooth. The subgroup $H\subset G$ acts compatibly on ${\mathcal{U}}_{x}$,
${\mathcal{K}}_{x}$, ${\mathbb{P}}(T_{x}X)$ and ${\mathcal{C}}_{x}$.

We now obtain a variant of [BK21, Lem. 2.4]:

######  Lemma 2.3.

A family of rational curves ${\mathcal{K}}$ on $X$ is covering if and only if
${\mathcal{U}}_{x}$ is non-empty; equivalently, ${\mathcal{K}}_{x}$ is non-
empty. Under these assumptions, ${\mathcal{U}}_{x}$ is smooth and its
components are permuted transitively by $H$.

###### Proof.

The morphism $\mu$ restricts to a $G$-equivariant morphism

$\mu^{0}:{\mathcal{U}}^{0}=\mu^{-1}(X^{0})\longrightarrow X^{0}=G/H$

with fiber at $x$ being ${\mathcal{U}}_{x}$. This yields an isomorphism
${\mathcal{U}}^{0}\simeq G\times^{H}{\mathcal{U}}_{x}$, where the right-hand
side denotes the quotient of $G\times{\mathcal{U}}_{x}$ by the $H$-action via
$h\cdot(g,z)=(gh^{-1},h\cdot z)$. Since ${\mathcal{K}}^{0}$ is smooth, so are
${\mathcal{U}}^{0}$ and hence ${\mathcal{U}}_{x}$. Also, ${\mathcal{U}}^{0}$
is irreducible; thus, $H$ acts transitively on the components of
${\mathcal{U}}_{x}$. ∎

Next, let $\pi:X\to Y$ be a surjective morphism, where $Y$ is a smooth
projective variety. Assume that $Y$ is equipped with a $G$-action such that
$\pi$ is equivariant. Let $y=\pi(x)$ and $Y^{0}=G\cdot y$; then
$Y^{0}=\pi(X^{0})$ is open in $Y$. We now have the following variants of
[BK21, Lem. 2.6, Rem. 2.7]:

######  Lemma 2.4.

Keep the above notation and assumptions, and consider a covering family of
rational curves ${\mathcal{K}}$ on $X$. Assume that there exists
$C\in{\mathcal{K}}^{0}$ such that $\pi|_{C}$ is birational onto its image $D$.
Then:

  1. 1.

$D\in{\mathcal{L}}$ for a unique covering family ${\mathcal{L}}$ of rational
curves on $Y$.

  2. 2.

$\pi$ induces a $G$-equivariant rational map
$\pi_{*}:{\mathcal{K}}\dasharrow{\mathcal{L}}$, which is defined at $C$ and
satisfies $\pi_{*}(C)=D$, and an $H$-equivariant rational map

$\pi_{*,x}:{\mathcal{K}}_{x}\dasharrow{\mathcal{L}}_{y},\quad C\longmapsto D.$

  3. 3.

We have a commutative diagram of $H$-equivariant rational maps

$\textstyle{{\mathcal{K}}_{x}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{*,x}}$$\scriptstyle{\tau_{x}}$$\textstyle{{\mathcal{L}}_{y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau_{y}}$$\textstyle{{\mathbb{P}}(T_{x}X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d\pi_{x}}$$\textstyle{{\mathbb{P}}(T_{y}Y).}$

###### Proof.

(1) Replacing $C$ with a translate $g\cdot C$ for some $g\in G$, we may assume
that $x\in C$. Then the assertion follows from Lemma 2.3.

(2) This is a consequence of Lemma 2.1, except for the equivariance assertions
which are easily checked.

(3) This follows readily from the definitions. ∎

######  Remark 2.5.

If $\pi$ is birational (equivalently, it induces an isomorphism $X^{0}\to
Y^{0}$), then the assumptions of Lemma 2.4 hold and moreover $\pi_{*,x}$ is an
immersion. Indeed, $\pi_{*,x}$ is clearly an injective morphism. Moreover, the
differential of $\pi_{*,x}$ at every $C\in{\mathcal{K}}_{x}$ is injective by
Lemma 2.1.

Still considering a covering family of rational curves ${\mathcal{K}}$ on $X$,
we now assume that $\pi$ contracts some curve in ${\mathcal{K}}$, and hence
all curves in ${\mathcal{K}}$. Let

$X\stackrel{{\scriptstyle\pi^{\prime}}}{{\longrightarrow}}Y^{\prime}\stackrel{{\scriptstyle\eta}}{{\longrightarrow}}Y$

be the Stein factorization of $\pi$, where $Y^{\prime}$ is a normal projective
variety (possibly singular), $\pi^{\prime}$ is a contraction (that is,
$\pi^{\prime}_{*}({\mathcal{O}}_{X})={\mathcal{O}}_{Y^{\prime}}$), and $\eta$
is finite surjective. Then there is a unique action of $G$ on $Y^{\prime}$
such that $\pi^{\prime}$ and $\eta$ are equivariant. Let
$y^{\prime}=\pi^{\prime}(x)$ and $I=G_{y^{\prime}}$; then $H\subset I\subset
G$ and the orbit $G\cdot y^{\prime}\simeq G/I$ is open in $Y^{\prime}$. Also,
let $F=\pi^{\prime-1}(y^{\prime})$; then $F$ is the connected component of $x$
in the fiber $\pi^{-1}(y)$, and hence is a smooth projective variety (by
generic smoothness). Moreover, $F$ is stable by $I$ and contains $I\cdot x$ as
its open orbit. Clearly, every curve in ${\mathcal{K}}_{x}$ is contained in
$F$.

######  Lemma 2.6.

Keep the above notation and assumptions, and assume that $I$ normalizes $H$.
Then ${\mathcal{K}}_{x}$ is irreducible and there exists a unique covering
family of rational curves ${\mathcal{L}}$ on $F$ such that
${\mathcal{K}}_{x}={\mathcal{L}}_{x}$. Moreover,
${\mathcal{K}}^{0}=G\cdot{\mathcal{L}}^{0}$.

###### Proof.

Note that $H$ acts trivially on $I/H$, since $H\triangleleft I$. Thus, $H$
acts trivially on $F$, and hence on ${\mathcal{U}}_{x}$. As
${\mathcal{U}}^{0}\simeq G\times^{H}{\mathcal{U}}_{x}$, we obtain
${\mathcal{U}}^{0}\simeq G/H\times{\mathcal{U}}_{x}$. Since
${\mathcal{U}}^{0}$ is irreducible, so are ${\mathcal{U}}_{x}$ and
${\mathcal{K}}_{x}$.

The inclusion $\iota:F\to X$ induces compatible immersions

$\operatorname{RatCurves}_{{\rm
fr}}(F)\longrightarrow\operatorname{RatCurves}_{{\rm
fr}}(X),\quad\operatorname{Univ}_{{\rm
fr}}(F)\longrightarrow\operatorname{Univ}_{{\rm fr}}(X),$

since they are injective and their differentials are injective as well. It
follows that ${\mathcal{K}}_{x}$ is an irreducible component of
$\mu_{F}^{-1}(x)$, where $\mu_{F}:\operatorname{Univ}(F)\to F$ denotes the
evaluation map. So ${\mathcal{K}}_{x}$ is an irreducible component of
${\mathcal{L}}_{x}$ for a unique family of rational curves ${\mathcal{L}}$ on
$F$. Since $(F,x)$ is an equivariant embedding of the homogeneous space
$I^{0}/I^{0}\cap H$ and $I^{0}\cap H\triangleleft I^{0}$, we see that
${\mathcal{L}}_{x}$ is irreducible. Thus,
${\mathcal{K}}_{x}={\mathcal{L}}_{x}$ and
${\mathcal{L}}^{0}=I^{0}\cdot{\mathcal{L}}_{x}=I^{0}\cdot{\mathcal{K}}_{x}$,
so that
${\mathcal{K}}^{0}=G\cdot{\mathcal{K}}_{x}=G\cdot{\mathcal{L}}_{x}=G\cdot{\mathcal{L}}^{0}$.
∎

######  Remark 2.7.

The above assignement ${\mathcal{K}}\mapsto{\mathcal{L}}$ yields a bijection
between covering families of rational curves on $X$ which are contracted by
$\pi$, and covering families of rational curves on $F$. This restricts to a
bijection between minimal families.

###  2.3 Symmetric spaces

In this subsection, we recall some basic facts on symmetric spaces, after
[Ti11, §26] and its references. We begin with some notation and conventions
which will be used throughout the sequel.

Let $G$ be a connected reductive algebraic group. Let $T\subset G$ be a
maximal torus, and $B\subset G$ a Borel subgroup containing $T$. We denote the
character group of $T$ by $\mathfrak{X}=\mathfrak{X}(T)$, and the root system
of $(G,T)$ by $R=R(G,T)\subset\mathfrak{X}$. The roots of $(B,T)$ form the set
of positive roots $R^{+}$, with basis $\Delta$ (the set of simple roots). The
Weyl group of $(G,T)$ is denoted by $W$.

Recall the decomposition of Lie algebras
$\mathfrak{g}=\mathfrak{t}\oplus\bigoplus_{\alpha\in R}\mathfrak{g}_{\alpha}$.
For any $\alpha\in R$, we denote by $U_{\alpha}$ the closed subgroup of $G$
with Lie algebra $\mathfrak{g}_{\alpha}$, and by $G_{\alpha}$ the subgroup of
$G$ generated by $U_{\alpha}$ and $U_{-\alpha}$. Then $G_{\alpha}$ is a closed
subgroup of $G$, isomorphic to $\operatorname{SL}_{2}$ or
$\operatorname{PSL}_{2}$.

Next, let $\sigma$ be a group involution of $G$. Denote by $G^{\sigma}$ the
fixed point subgroup, and by $G^{\sigma,0}$ its neutral component. Let $H$ be
a subgroup of $G$ such that $G^{\sigma,0}\subset H\subset G^{\sigma}$; then we
say that $H$ is a _symmetric subgroup_ of $G$, and the homogeneous space $G/H$
is a _symmetric space_.

The group $H$ is reductive; equivalently, the variety $G/H$ is affine. Also,
$\sigma$ induces an involution of $\mathfrak{g}$, still denoted by $\sigma$
for simplicity. The Lie algebra of $H$ satisfies
$\mathfrak{h}=\mathfrak{g}^{\sigma}$ and
$\mathfrak{g}=\mathfrak{h}\oplus\mathfrak{p}$, where

$\mathfrak{p}=\mathfrak{g}^{-\sigma}=\\{x\in\mathfrak{g}~{}|~{}\sigma(x)=-x\\}$

is a $G^{\sigma}$-stable complement of $\mathfrak{h}$ in $\mathfrak{g}$.

Recall that $\sigma$ stabilizes a maximal torus $T$ of $G$. Thus, $\sigma$
acts on the character group $\mathfrak{X}$ and stabilizes the root system $R$;
it also acts on the Weyl group $W$ by conjugation. We may choose a scalar
product $(-,-)$ on the real vector space
$\mathfrak{X}_{{\mathbb{R}}}=\mathfrak{X}\otimes_{\mathbb{Z}}{\mathbb{R}}$
which is invariant under $W$ and $\sigma$.

######  Definition 2.8.

For $\alpha\in R$, one of the following cases occurs:

  1. 1.

$\sigma(\alpha)=\alpha$ and $\sigma$ fixes pointwise $\mathfrak{g}_{\alpha}$.
Then $\alpha$ is called a _compact imaginary_ root.

  2. 2.

$\sigma(\alpha)=\alpha$ and $\sigma$ acts on $\mathfrak{g}_{\alpha}$ by $-1$.
Then $\alpha$ is _non-compact imaginary_.

  3. 3.

$\sigma(\alpha)=-\alpha$. Then $\alpha$ is _real_.

  4. 4.

$\sigma(\alpha)\neq\pm\alpha$. Then $\alpha$ is _complex_.

Recall that any two maximal tori of $G$ are conjugate and hence
$\mathfrak{X}$, $R$ and $W$ are independent of the choice of $T$. But the
action of $\sigma$ on these objects depends on the choice of the
$\sigma$-stable torus $T$, up to conjugacy by $H$. We now consider two special
conjugacy classes of $\sigma$-stable maximal tori, that we call of fixed
(resp. split) type. These are constructed as follows.

#### Maximal tori of fixed type.

Choose a maximal torus $T_{H}$ of $H$; then its centralizer
$T=\operatorname{C}_{G}(T_{H})$ is a $\sigma$-stable maximal torus of $G$ and
we have $T_{H}=T^{\sigma,0}$. Moreover, $T$ is contained in a $\sigma$-stable
Borel subgroup $B$ of $G$; then $B_{H}=B^{\sigma,0}$ is a Borel subgroup of
$H$ (see [Ti11, Lem. 26.7]). Thus, $B_{H}=U_{H}T_{H}$, where $U_{H}=U\cap H$
is a maximal unipotent subgroup of $H$.

The action of $\sigma$ on the root system $R$ stabilizes $R^{+}$. In
particular, there are no real roots. The subset of simple roots $\Delta$ is
$\sigma$-stable as well.

Clearly, the maximal tori obtained in this way are exactly those containing a
maximal $\sigma$-fixed torus (i.e., a subtorus $S\subset G$ such that
$\sigma(s)=s$ for all $s\in S$); they are all conjugate under
$H^{0}=G^{\sigma,0}$. We call every such maximal torus _of fixed type_ and
denote it by $T_{{\rm f}}$.

#### Maximal tori of split type.

In the opposite direction, a subtorus $S\subset G$ is called _$\sigma$ -split_
if $\sigma(s)=s^{-1}$ for all $s\in S$. Choose such a torus $S$ maximal for
this property; then $L=\operatorname{C}_{G}(S)$ satisfies $[L,L]\subset
H^{0}$. As a consequence, every maximal torus of $G$ containing $S$ is
$\sigma$-stable. We call every such maximal torus _of split type_ and denote
it by $T_{{\rm s}}$. Also, $L$ is a Levi subgroup of a minimal $\sigma$-split
parabolic subgroup $P$, that is, $P$ is a parabolic subgroup of $G$ which is
opposite to $\sigma(P)$, and minimal for this property. Denote by
$\operatorname{R}_{u}(P)$ the unipotent radical of $P$; then by the Iwasawa
decomposition, the morphism

$\iota:\operatorname{R}_{u}(P)\times S/S\cap H\longrightarrow
G/H,\quad(g,z)\longmapsto g\cdot z$

is an open immersion with image $P/P\cap H$, the $P$-orbit of the base point.
Moreover, $\iota$ is $P$-equivariant, where $P=\operatorname{R}_{u}(P)\rtimes
L$ acts on $\operatorname{R}_{u}(P)\times S/S\cap H$ via
$(u,l)\cdot(g,z)=(ulgl^{-1},l\cdot z)$, and on $G/H$ via left multiplication.
Finally, recall that the maximal split tori are all conjugate under $H^{0}$,
as well as the minimal split parabolic subgroups and the maximal tori of split
type.

The maximal tori of fixed type will be used in the rest of this section and in
Section 3. Those of split type, and the corresponding restricted root system,
feature prominently in the subsequent sections.

###  2.4 The normalizer of a symmetric subgroup

We keep the notation of §2.3 and denote by $Z=Z(G)$ the center of $G$, with
Lie algebra $\mathfrak{z}$. Since $Z$ is $\sigma$-stable, we have
$\mathfrak{z}=(\mathfrak{z}\cap\mathfrak{h})\oplus(\mathfrak{z}\cap\mathfrak{p})$.
Also, we denote by $N=\operatorname{N}_{G}(G^{\sigma})$ the normalizer of
$G^{\sigma}$ in $G$ and by $G_{{\rm ad}}=G/Z$ the adjoint group.

######  Lemma 2.9.

  1. 1.

$N=\\{g\in G~{}|~{}\sigma(g)g^{-1}\in Z\\}$.

  2. 2.

$N=\operatorname{N}_{G}(G^{\sigma,0})=\operatorname{N}_{G}(\mathfrak{h})$.

  3. 3.

$N^{0}=Z^{0}H^{0}$.

  4. 4.

$Z^{0}T_{H}$ is a maximal torus of $N$ for any maximal torus $T_{H}$ of $H$.

###### Proof.

(1) This is obtained in [Vu90, Lem. 1] (see also [deCP83, I.7]); we recall the
argument for completeness.

Let $g\in G$ such that $\sigma(g)g^{-1}\in Z$. For any $h\in G^{\sigma}$, we
have $\sigma(ghg^{-1})=\sigma(g)h\sigma(g)^{-1}=ghg^{-1}$, that is,
$ghg^{-1}\in G^{\sigma}$. So $g\in N$. For the converse, observe that $N$ is
reductive and normalized by $\sigma$. The corresponding semi-direct product
$N\rtimes\langle\sigma\rangle$ is a reductive algebraic group, which acts
linearly on $\mathfrak{g}$ and stabilizes $\mathfrak{h}$. Thus, $\mathfrak{h}$
has an $N\rtimes\langle\sigma\rangle$-stable complement, which must be
$\mathfrak{p}$. In particular, $\mathfrak{p}$ is $N$-stable; thus,
$\operatorname{Ad}(N)$ commutes with $\sigma$. So
$\operatorname{Ad}(\sigma(g)g^{-1})=\sigma\operatorname{Ad}(g)\sigma^{-1}\operatorname{Ad}(g)^{-1}={\rm
id}$ for any $g\in N$, that is, $\sigma(g)g^{-1}\in Z$.

(2) Clearly, we have
$N=\operatorname{N}_{G}(G^{\sigma})\subset\operatorname{N}_{G}(G^{\sigma,0})=\operatorname{N}_{G}(\mathfrak{h})$.
Moreover, $\operatorname{N}_{G}(\mathfrak{h})$ is reductive and normalized by
$\sigma$. Arguing as in the proof of (1), it follows that $\sigma(g)g^{-1}\in
Z$ for any $g\in\operatorname{N}_{G}(\mathfrak{h})$, and hence $g\in N$.

(3) Denote by $\mathfrak{n}$ the Lie algebra of $N$. Then (1) yields that
$\mathfrak{n}=\\{x\in\mathfrak{g}~{}|~{}\sigma(x)-x\in\mathfrak{z}\\}$. Using
the $\sigma$-stable decomposition
$\mathfrak{g}=\mathfrak{z}\oplus[\mathfrak{g},\mathfrak{g}]$, it follows that
$\mathfrak{n}=\mathfrak{z}\oplus[\mathfrak{g},\mathfrak{g}]^{\sigma}=\mathfrak{z}+\mathfrak{h}$.
This yields the assertion.

(4) This follows readily from (3). ∎

######  Lemma 2.10.

Let $S$ be a maximal $\sigma$-split torus of $G$.

  1. 1.

$N=G^{\sigma,0}(N\cap S)$.

  2. 2.

$H=G^{\sigma,0}(H\cap S)$ and $H\cap S$ is an elementary abelian $2$-group.

  3. 3.

$N=\operatorname{N}_{G}(H)$ and $N/H\simeq N\cap S/H\cap S$. In particular,
$N/H$ is diagonalizable.

###### Proof.

(1) Let $P$ be a minimal $\sigma$-split parabolic subgroup of $G$ containing
$S$. Then $P_{{\rm ad}}=P/Z$ is a minimal $\sigma$-split parabolic subgroup of
$G_{{\rm ad}}$, containing $S_{{\rm ad}}=S/S\cap Z$ which is a maximal
$\sigma$-split torus of $G_{{\rm ad}}$. By the Iwasawa decomposition, the
morphism

$\operatorname{R}_{u}(P_{{\rm ad}})\times S_{{\rm ad}}/S_{{\rm
ad}}^{\sigma}\longrightarrow G_{{\rm ad}}/G_{{\rm
ad}}^{\sigma},\quad(g,z)\longmapsto g\cdot z$

is an open immersion. As a consequence, the multiplication map
$\operatorname{R}_{u}(P)\times SN\to G$ is an open immersion as well. Its
image is $\operatorname{R}_{u}(P)SN=PN$, the open orbit of $P\times N$ in $G$.
Likewise, $PG^{\sigma,0}$ is the open orbit of $P\times G^{\sigma,0}$ in $G$.
Let $g\in N$; then the orbit $PgG^{\sigma,0}=PG^{\sigma,0}g$ is open in $G$.
So $g\in PG^{\sigma,0}$, and hence $PN=PG^{\sigma,0}$. It follows that
$SN=SG^{\sigma,0}$; this yields the assertion.

(2) The first assertion follows readily from (1). For the second assertion,
just note that every $g\in S\cap H$ satisfies $g^{-1}=\sigma(g)=g$.

(3) Clearly, we have
$\operatorname{N}_{G}(H)\subset\operatorname{N}_{G}(H^{0})=\operatorname{N}_{G}(G^{\sigma,0})$.
Moreover, $\operatorname{N}_{G}(G^{\sigma,0})=N$ in view of Lemma 2.9.2. Also,
by combining (1) and (2) above, we see that $N$ normalizes $H$, since $N\cap
S$ normalizes $G^{\sigma,0}$ and centralizes $H\cap S$. Thus,
$\operatorname{N}_{G}(H)=N$. By (1) again, we have $N=H(N\cap S)$, and hence
$N/H\simeq(N\cap S)/(H\cap S)$ is diagonalizable. ∎

###  2.5 The adjoint symmetric space

Recall that $G_{{\rm ad}}=G/Z$ is the adjoint group, and let

$q:G\longrightarrow G_{{\rm ad}}$

be the quotient homomorphism. Then $\sigma$ induces an involution of $G_{{\rm
ad}}$, still denoted by $\sigma$ for simplicity. The homogeneous space
$G_{{\rm ad}}/G_{{\rm ad}}^{\sigma}$ is called an _adjoint symmetric space_.
By Lemma 2.9, we have $N=q^{-1}(G_{{\rm ad}}^{\sigma})$; this yields an
isomorphism

$G/N\simeq G_{{\rm ad}}/G_{{\rm ad}}^{\sigma}.$

Moreover, $G_{{\rm ad}}^{\sigma}$ is its own normalizer in $G_{{\rm ad}}$.

By decomposing $G_{\rm ad}$ into a product of indecomposable $\sigma$-stable
factors, we obtain a decomposition of $G/N$ into a product of _irreducible
symmetric spaces_. These fall into three types:

  1. 1.

(group) $(H\times H)/{\rm diag}(H)$, where $H$ is simple adjoint.

  2. 2.

(Hermitian) $G/\operatorname{N}_{G}(L)$, where $G$ is simple adjoint and
$L\subset G$ is a Levi subgroup.

  3. 3.

(simple) $G/H$, where $G$ is simple adjoint and $H^{0}$ is simple.

In type (1), we have $\sigma(x,y)=(y,x)$ for all $x,y\in H$. Thus, $G/N$ is
just the group $H$ on which $H\times H$ by left and right multiplication. The
isotropy representation $\mathfrak{p}$ is the adjoint representation of $H$ in
$\mathfrak{h}$. This is an irreducible representation with highest weight the
highest root $\Theta$.

In type (2), we have $L=P\cap Q$, where $P$ and $Q$ are opposite maximal
parabolic subgroups of $G$. Moreover, $\sigma$ is the conjugation ${\rm
Int}(c)$, where $c\in Z(L)$ and $c^{2}\in Z(G)$. Denote by $\alpha$ the unique
simple root which is not a root of $L$; then $\alpha$ has coefficient $1$ in
the expansion of the highest root $\Theta$ as a linear combination of simple
roots. We have $\mathfrak{p}=\mathfrak{u}_{P}\oplus\mathfrak{u}_{Q}$, where
$\mathfrak{u}_{P}$ (resp. $\mathfrak{u}_{Q}$) denotes the Lie algebra of
$\operatorname{R}_{u}(P)$ (resp. $\operatorname{R}_{u}(Q)$). Moreover, the
representations $\mathfrak{u}_{P}$, $\mathfrak{u}_{Q}$ of $L$ are irreducible
and dual to each other (see e.g. [RRS92, §5.5] for these results). Their
highest weights relative to $L$ are $\Theta$, $-\alpha$; they are linearly
independent unless $G=\operatorname{PSL}_{2}$.

We say that $G/\operatorname{N}_{G}(L)$ is (Hermitian) _exceptional_ if $P$
and $Q$ are not conjugate in $G$. Then $\operatorname{N}_{G}(L)=L$, and hence
$G/\operatorname{N}_{G}(L)$ may be identified with the open $G$-orbit in
$G/P\times G/Q$ on which $G$ acts diagonally. In the non-exceptional case,
where $P$ and $Q$ are conjugate in $G$, the group $\operatorname{N}_{G}(L)/L$
has order $2$ and exchanges $P$ and $Q$. Moreover, $G/\operatorname{N}_{G}(L)$
may be identified with the open $G$-orbit in the symmetric square
$(G/P)^{(2)}$ (the quotient of $G/P\times G/P$ by the involution
$(y,z)\mapsto(z,y)$).

In type (3), $\mathfrak{p}$ is irreducible as a representation of $H^{0}$,
with a non-zero highest weight.

###  2.6 Highest weight curves

Throughout this subsection, we choose a maximal torus $T_{H}\subset H$ and a
Borel subgroup $B_{H}\subset H$ containing $T_{H}$. Recall that
$T=\operatorname{C}_{G}(T_{H})$ is a maximal torus of fixed type of $G$. We
first obtain a generalization of [BF15, Lem. 2.2]:

######  Lemma 2.11.

Let $C$ be an irreducible $B_{H}$-stable curve in $G/H$ through the base point
$x$.

  1. 1.

Either $C$ is contained in $Z^{0}\cdot x$, or $B_{H}$ acts non-trivially on
$C$.

  2. 2.

In the latter case, $C$ is smooth and $B_{H}$-equivariantly isomorphic to its
tangent line at $x$, which is the $T_{H}$-weight space
$\mathfrak{p}_{\lambda}$ for a unique non-zero highest weight $\lambda$ of
$\mathfrak{p}$. Moreover, $\lambda$ determines $C$ uniquely, and the
stabilizer of $C$ in $H$ equals the stabilizer of the weight space
$\mathfrak{p}_{\lambda}$.

###### Proof.

(1) Assume that $C$ is fixed pointwise by $B_{H}$. Then the orbit $H^{0}\cdot
y$ is complete for any $y\in C$. Since $G/H$ is affine, this orbit must be a
point, i.e., $y$ is fixed pointwise by $H^{0}$. Let $g\in G$ such that
$y=g\cdot x$, then $g^{-1}H^{0}g\cdot x=x$, i.e., $g^{-1}H^{0}g\subset H$. So
$g\in\operatorname{N}_{G}(H^{0})=N$ (Lemma 2.9). Thus, $C\subset N\cdot x$. As
$C$ is connected and contains $x$, it follows that $C\subset N^{0}\cdot x$.
But $N^{0}\cdot x=Z^{0}\cdot x$ by Lemma 2.9 again; this yields the assertion.

(2) This is obtained by arguing as in the proof of [BF15, Lem. 2.2(i)]. We
provide details for the reader’s convenience.

Since $C$ is not fixed pointwise by $B_{H}$, it contains an open orbit
$B_{H}\cdot y$, where $y\neq x$. Thus, the isotropy group $B_{H,y}$ has
codimension $1$ in $B_{H}$. We thus have $B_{H,y}^{0}=U_{H,y}\rtimes S$ for
some subtorus $S$ of $B_{H}$. Replacing $y$ with a $B_{H}$-translate, we may
assume that $S\subset T_{H}$.

If $S=T_{H}$, then $B_{H}\cdot y=U_{H}\cdot y$ is closed in $G/H$, since the
latter is an affine variety. But $x\in\overline{B_{H}\cdot
y}\setminus\\{y\\}$, a contradiction. For dimension reasons, it follows that
$S$ is a subtorus of codimension $1$ of $T_{H}$, and $U_{H}\subset B_{H,y}$.
As a consequence, $C$ is fixed pointwise by $U_{H}$, since the latter is a
normal subgroup of $B_{H}$. Thus, $T_{H}\cdot y$ is open in $C$.

In particular, $x\in\overline{H\cdot y}$. So $C$ is contained in the fiber at
$x$ of the geometric invariant theory quotient $G/H\to
H\backslash\\!\\!\backslash G/H$ of the smooth affine $H$-variety $G/H$. By a
corollary of Luna’s slice theorem (see [Lu73, II.1, III.1]), this fiber is
$H$-equivariantly isomorphic to the nilcone ${\mathcal{N}}$ of $\mathfrak{p}$
(the fiber at $0$ of the quotient $\mathfrak{p}\to\mathfrak{p}/\\!\\!/H$; it
consists of the points $z\in\mathfrak{p}$ such that $0\in\overline{H\cdot
z}$). Thus, $C$ is $B_{H}$-equivariantly isomorphic to a $B_{H}$-stable curve
$D$ in ${\mathcal{N}}$. Moreover, $C$ and $D$ have the same stabilizer in $H$.

As $U_{H}$ fixes $D$ pointwise, we have
$D\subset{\mathcal{N}}\cap\mathfrak{p}^{U_{H}}$. Also,
$\mathfrak{p}=(\mathfrak{p}\cap\mathfrak{z})\oplus(\mathfrak{p}\cap[\mathfrak{g},\mathfrak{g}])$
and the projection $\mathfrak{p}\to\mathfrak{p}\cap\mathfrak{z}$ is
$H$-invariant, hence sends ${\mathcal{N}}$ to $0$. it follows that
$D\subset(\mathfrak{p}\cap[\mathfrak{g},\mathfrak{g}])^{U_{H}}$.

So we may assume that $G/H$ is an adjoint symmetric space. Using the product
decomposition of these spaces, we see that $D$ is a highest weight line from a
unique indecomposable factor of $G/H$, and is uniquely determined by its
weight. ∎

We say that a curve $C$ as in Lemma 2.11.2 is a _highest weight curve_. The
corresponding highest weight $\lambda$ satisfies $\lambda=\alpha|_{T_{H}}$ for
some root $\alpha$, since the non-zero weights of $T_{H}$ in $\mathfrak{p}$
are restrictions of non-zero weights of $T$ in $\mathfrak{g}$. Let $S={\rm
Ker}(\lambda)^{0}=({\rm Ker}(\alpha)\cap T_{H})^{0}$; then $S$ is a subtorus
of codimension $1$ of $T_{H}$, and fixes $C$ pointwise. Thus, the centralizer
$\operatorname{C}_{G}(S)$ is a $\sigma$-stable connected reductive subgroup of
$G$ containing $T$. Moreover, $\operatorname{C}_{H}(S)$ is a symmetric
subgroup of $\operatorname{C}_{G}(S)$ containing $T_{H}$, and $C$ is a highest
weight curve of the symmetric space
$\operatorname{C}_{G}(S)/\operatorname{C}_{H}(S)$.

Recall the following easy result (see [Sp83, §2] and [Br99, Lem. 2.5]):

######  Lemma 2.12.

With the above notation and assumptions, the adjoint symmetric space of
$\operatorname{C}_{G}(S)/\operatorname{C}_{H}(S)$ is one of the following:

  1. (${\rm A}_{1}$)

$\operatorname{PSL}_{2}/N$, where $N$ denotes the normalizer of the diagonal
torus in $\operatorname{PSL}_{2}$.

  2. (${\rm A}_{1}\times{\rm A}_{1}$)

$(\operatorname{PSL}_{2}\times\operatorname{PSL}_{2})/{\rm
diag}(\operatorname{PSL}_{2})$. Then $\sigma(\alpha)$ is strongly orthogonal
to $\alpha$.

  3. (${\rm A}_{2}$)

$\operatorname{PSL}_{3}/\operatorname{SO}_{3}$, where $\operatorname{SO}_{3}$
denotes the special orthogonal group. Then $\alpha+\sigma(\alpha)\in R$.

Moreover, $\alpha$ is non-compact imaginary in types $({\rm A}_{1})$ and
$({\rm A}_{2})$. If the involution $\sigma$ is inner (equivalently, $T_{H}$ is
a maximal torus of $G$), then only type $({\rm A}_{1})$ occurs.

Still considering a highest curve $C$ of weight $\lambda$, we now obtain a
description of $C$ and its tangent line $T_{x}C\subset T_{x}G/H$ (using the
identifications $T_{x}G/H=\mathfrak{g}/\mathfrak{h}=\mathfrak{p}$) in the
above three types.

######  Proposition 2.13.

In type $({\rm A}_{1})$, there is a unique root $\alpha$ such that
$\lambda=\alpha|_{T_{H}}$. Moreover, $C=U_{\alpha}\cdot x$ and
$T_{x}C=\mathfrak{g}_{\alpha}$.

In type $({\rm A}_{1}\times{\rm A}_{1})$, there are exactly two roots
$\alpha$, $\beta$ such that $\lambda=\alpha|_{T_{H}}=\beta|_{T_{H}}$.
Moreover, $\alpha$ and $\beta=\sigma(\alpha)$ are the simple roots of
$(\operatorname{C}_{G}(S),T)$. We have $C=U_{\alpha}\cdot x=U_{\beta}\cdot x$
and $T_{x}C={\mathbb{C}}(e_{\alpha}-\sigma(e_{\alpha}))$, where
$e_{\alpha}\in\mathfrak{g}_{\alpha}\setminus\\{0\\}$.

In type $({\rm A}_{2})$, there is a unique root $\alpha$ such that
$\lambda=\alpha|_{T_{H}}$. Moreover, $\alpha=\alpha_{1}+\alpha_{2}$, where
$\alpha_{1}$ and $\alpha_{2}=\sigma(\alpha_{1})$ are the simple roots of
$(\operatorname{C}_{G}(S),T)$. Also, $C=U_{\alpha}\cdot x$ and
$T_{x}C=\mathfrak{g}_{\alpha}$.

###### Proof.

In type $({\rm A}_{1})$, there are two highest weight curves in
$\operatorname{C}_{G}(S)/\operatorname{C}_{H}(S)$, namely, $U_{\alpha}\cdot x$
and $U_{-\alpha}\cdot x$.

In type $({\rm A}_{1}\times{\rm A}_{1})$, recall that the adjoint symmetric
space of $\operatorname{C}_{G}(S)/\operatorname{C}_{H}(S)$ is the group
$(\operatorname{PSL}_{2}\times\operatorname{PSL}_{2})/{\rm
diag}(\operatorname{PSL}_{2})=\operatorname{PSL}_{2}$. So the roots of
$(\operatorname{C}_{G}(S),T)$ are $\pm\alpha$, $\pm\sigma(\alpha)$. Moreover,
$U_{\alpha}\cdot x$ is the unique highest weight curve; it is identified with
the standard unipotent subgroup $U\subset\operatorname{PSL}_{2}$, and likewise
for $U_{\sigma(\alpha)}\cdot x$.

In type $({\rm A}_{2})$, the adjoint symmetric space of
$\operatorname{C}_{G}(S)/\operatorname{C}_{H}(S)$ is
$\operatorname{PSL}_{3}/\operatorname{SO}_{3}$; one checks that the highest
weight of its isotropy representation is $(\alpha_{1}+\alpha_{2})|_{T_{H}}$,
where $\alpha_{1}$, $\alpha_{2}$ are the simple roots of
$\operatorname{PSL}_{3}$. It follows that $\alpha=\alpha_{1}+\alpha_{2}$, and
$\sigma(\alpha_{1})=\alpha_{2}$. Finally, $U_{\alpha}\cdot x$ is an
irreducible curve in $G/H$, stable by $T_{H}$ and fixed by $U_{H}$ (since the
latter commutes with $U_{\alpha}$ and fixes $x$). Thus, $U_{\alpha}\cdot x$ is
the highest weight curve in $\operatorname{C}_{G}(S)/\operatorname{C}_{H}(S)$.

This yields the assertions on roots and highest weight curves. Those on their
tangent lines are readily verified. ∎

######  Corollary 2.14.

Let $C$ be a highest weight curve of weight $\lambda$. Then there exists
$\alpha\in R$ such that $\lambda=\alpha|_{T_{H}}$ and $C=U_{\alpha}\cdot x$.

######  Corollary 2.15.

We have the following alternative for a simple group $G$:

  1. 1.

$T_{x}C=\mathfrak{g}_{\alpha}$ for a long root $\alpha$, or

  2. 2.

$T_{x}C=\mathfrak{g}_{\alpha}$ for a short root $\alpha$, or

  3. 3.

$T_{x}C$ is spanned by $e_{\alpha}-\sigma(e_{\alpha})$, where $\alpha\in R$ is
strongly orthogonal to $\sigma(\alpha)$. Moreover, $G$ is simply-laced.

###### Proof.

In view of Proposition 2.13, we only have to show that $G$ is simply laced in
case (3). Then $\sigma(\alpha)\neq\alpha$, and hence $\sigma$ acts non-
trivially on $R$. As $\sigma$ stabilizes the set $\Delta$ of simple roots, it
induces a non-trivial automorphism of the Dynkin diagram. But this only occurs
for $G$ simply-laced. ∎

######  Remark 2.16.

The three cases in Corollary 2.15 do occur (see Table 1 for the notation on
types):

  1. 1.

In type AI with $G=\operatorname{SL}_{n}$ and $H=\operatorname{SO}_{n}$, then
$T_{x}C=\mathfrak{g}_{\Theta}$ with $\Theta$ the highest root.

  2. 2.

In type BII with $G=\operatorname{SO}_{2n+1}$ and $H=S({\rm O}_{1}\times{\rm
O}_{2n})$, then $T_{x}C=\mathfrak{g}_{\theta}$ with $\theta$ the highest short
root.

  3. 3.

In type AII with $G=\operatorname{SL}_{2n}$ and $H={\rm Sp}_{2n}$, then
$T_{x}C$ is spanned by $e_{\Theta-\alpha_{1}}-e_{\Theta-\alpha_{2n-1}}$ where
$\Theta$ is the highest root and $\alpha_{1}$ and $\alpha_{2n-1}$ are simple
roots labeled as in [Bo68].

  4. 4.

Clearly, case (2) does not occur for simply-laced groups.

  5. 5.

In type $G_{2}$, none of cases (2) and (3) do occur. Indeed
$\sigma(\alpha)=\alpha$ for any root $\alpha\in R$. Therefore, every root is
imaginary. An easy computation shows that the highest non-compact root must be
long.

Next, we assume that $G/H$ is irreducible; in particular, $G$ is simple or
$G=H\times H$ with $H$ simple. We show that $T_{x}C$ is contained in a
nilpotent orbit of a very special type, defined as follows:

######  Definition 2.17.

Let $G/H$ be an irreducible symmetric space.

  1. 1.

If $G=H\times H$, then set
${\mathcal{O}}_{\min}=G\cdot(e,-e)\subset\mathfrak{g}=\mathfrak{h}\oplus\mathfrak{h}$
where $e\in\mathfrak{h}$ is a highest weight vector for $H$.

  2. 2.

If $G$ is simple, define a _nilpotent orbit_ ${\mathcal{O}}_{\min}$ and a
_type of nilpotent orbits_ ${\mathcal{O}}_{{\rm sum}}$ in $\mathfrak{g}$ as
follows.

    1. (a)

${\mathcal{O}}_{\min}=G\cdot e$ where $e$ is a highest weight vector in
$\mathfrak{g}$.

    2. (b)

A nilpotent orbit ${\mathcal{O}}$ is of type ${\mathcal{O}}_{{{\rm sum}}}$ if
${\mathcal{O}}=G\cdot(e_{1}+e_{2})$, where $e_{i}\in\mathfrak{g}_{\alpha_{i}}$
is a root vector with $\alpha_{1}$ and $\alpha_{2}$ two strongly orthogonal
long roots.

######  Remark 2.18.

There is a unique nilpotent orbit of type ${\mathcal{O}}_{{\rm sum}}$ except
for $G$ of type ${\rm B}_{n}$ or ${\rm D}_{n}$, in which case there are two
possible nilpotent orbits.

######  Proposition 2.19.

With the above notation, $T_{x}C\setminus\\{0\\}$ is contained in
${\mathcal{O}}_{\min}$ or in a nilpotent orbit of type ${\mathcal{O}}_{{\rm
sum}}$.

This follows by combining Corollary 2.15, Remark 2.16.5 and the next result.

######  Lemma 2.20.

Assume that $G$ is not simply-laced and not of type $G_{2}$. Let
$e_{\alpha}\in\mathfrak{g}_{\alpha}\setminus\\{0\\}$ with $\alpha$ a short
root. Then $e_{\alpha}$ belongs to a nilpotent orbit of type
${\mathcal{O}}_{{\rm sum}}$.

###### Proof.

Since all short roots are in the same orbit under the action of the Weyl
group, we may assume that $\alpha$ is the highest short root $\theta$.

There exists a simple root $\beta$ such that $\theta+\beta$ is a root; then
$\theta+\beta$ must be a long root. We claim that
$\langle\beta^{\vee},\theta+\beta\rangle=2$. Indeed, since $\theta$ is a
dominant weight, we have $\langle\beta^{\vee},\theta\rangle\geq 0$ and we get
$\langle\beta^{\vee},\theta+\beta\rangle\geq\langle\beta^{\vee},\beta\rangle=2$.
Since $G$ is not of type $G_{2}$, we must have equality:
$\langle\beta^{\vee},\theta+\beta\rangle=2$.

We get $s_{\beta}(\theta+\beta)=\theta-\beta$; thus, $\theta-\beta$ is a long
root. Denote by $(-,-)$ a scalar product on $\mathfrak{X}_{{\mathbb{R}}}$
which is invariant under $W$ and $\sigma$, and such that long roots have
length $2$. We have
$(\theta-\beta,\theta-\beta)=2=(\theta+\beta,\theta+\beta)$. This gives
$(\theta,\beta)=0$ and $(\theta,\theta)=1=(\beta,\beta)$, so both $\theta$ and
$\beta$ are short roots. We also get
$(\theta-\beta,\theta+\beta)=(\theta,\theta)-(\beta,\beta)=0$, so
$\theta-\beta$ and $\theta+\beta$ are long orthogonal roots. We check that
they are strongly orthogonal: their sum is $\theta-\beta+\theta+\beta=2\theta$
which is not a root, and their difference $\theta+\beta-(\theta-\beta)=2\beta$
is not a root either.

We are left to prove that $e_{\theta}$ and $e_{\theta-\beta}+e_{\theta+\beta}$
are in the same $G$-orbit in $\mathfrak{g}$. The group $G_{\beta}$ (generated
by $U_{\pm\beta}$) acts on $\mathfrak{g}$ and stabilizes the subspace
$V=\mathfrak{g}_{\theta-\beta}\oplus\mathfrak{g}_{\theta}\oplus\mathfrak{g}_{\theta+\beta}$
on which it acts via the adjoint representation. Moreover, $G_{\beta}$ acts
with two orbits in the projective space ${\mathbb{P}}(V)$: the minimal orbit
and its complement. The point $[e_{\theta}]$ is in this last orbit, which also
contains $[e_{\theta-\beta}+e_{\theta+\beta}]$. This yields the assertion,
since nilpotent orbits are stable under non-trivial homotheties. ∎

######  Remark 2.21.

In Proposition 4.23 and Corollary 4.25, we will give a more precise statement
describing the nilpotent orbit containing $T_{x}C$ for $C\in{\mathcal{K}}_{x}$
with ${\mathcal{K}}$ a minimal family.

##  3 Complete symmetric varieties

In this section, we recall the definitions of wonderful symmetric varieties
and complete symmetric varieties and we describe their relations, especially
how to compare their respective minimal families. We then recall the result of
[BF15] on minimal families on wonderful compactifications of groups. We end
the section by a description of the minimal families on complete symmetric
varieties in the group type, in the Hermitian type, and in some cases of
simple type.

###  3.1 Wonderful and complete symmetric varieties

We use the notation of Subsections 2.3 and 2.4. In particular, $G$ denotes a
connected reductive group, $H$ a symmetric subgroup relative to an involution
$\sigma$, and $N$ the normalizer of $H$ in $G$. We denote by $x$ (resp.
$x_{\rm ad}$) the base point of the homogeneous space $G/H$ (resp. $G/N$). The
natural morphism

$\pi:G/H\longrightarrow G/N,\quad x\longmapsto x_{\rm ad}$

is a principal bundle under $N/H$. Moreover, $N/H$ is diagonalizable by Lemma
2.10. We have the “Stein factorization” of $\pi$ as

$G/H\stackrel{{\scriptstyle\pi^{\prime}}}{{\longrightarrow}}G/N^{0}H\stackrel{{\scriptstyle\eta}}{{\longrightarrow}}G/N,$

where $\pi^{\prime}$ is a principal bundle under the torus $N^{0}H/H\simeq
Z^{0}/H\cap Z^{0}$ (Lemma 2.9), and $\eta$ is a principal bundle under
$N/N^{0}H$, a finite abelian group (Lemma 2.10).

By [deCP83], the adjoint symmetric space $G/N=G_{{\rm ad}}/G_{{\rm
ad}}^{\sigma}$ admits a wonderful equivariant embedding that we denote by
$X_{\rm ad}$, with base point $x_{\rm ad}$. We say that $X_{{\rm ad}}$ is a
_wonderful symmetric variety_.

We now recall from [LP90, §3.3] how to obtain $X_{{\rm ad}}$ from the
wonderful $G_{{\rm ad}}\times G_{{\rm ad}}$-equivariant embedding
$\overline{G_{\rm ad}}$ of $G_{\rm ad}=(G_{{\rm ad}}\times G_{{\rm ad}})/{\rm
diag}(G_{{\rm ad}})$. We begin with a general construction: the morphism

$G\longrightarrow G,\quad g\longmapsto\sigma(g)g^{-1}$

factors through a closed immersion $\iota:G/G^{\sigma}\to G$ which sends the
base point $x$ to the neutral element $e$. The image of $\iota$ is a connected
component of the fixed locus $G^{-\sigma}$, where $-\sigma$ denotes the
involution $g\mapsto\sigma(g^{-1})$ of $G$ (viewed as a variety). Note that
$\iota$ is equivariant for the natural action of $G$ on $G/G^{\sigma}$, and
the $G$-action on itself via twisted conjugation, defined by $g_{1}\cdot
g_{2}:=\sigma(g_{1})g_{2}g_{1}^{-1}$. Also, the differential of $\iota$ at $x$
is identified with the inclusion $\mathfrak{p}\hookrightarrow\mathfrak{g}$.

This construction applies to the involution $\sigma$ of $G_{{\rm ad}}$;
moreover, $-\sigma$ extends uniquely to an involution of $\overline{G_{\rm
ad}}$ that we still denote by $-\sigma$, and $\iota$ extends uniquely to a
closed immersion

$\overline{\iota}:X_{{\rm ad}}\hookrightarrow\overline{G_{\rm ad}}$

which identifies $X_{{\rm ad}}$ with a component of $(\overline{G_{\rm
ad}})^{-\sigma}$.

######  Definition 3.1.

A _complete symmetric variety_ is a smooth projective equivariant embedding
$(X,x)$ of $G/H$ such that the morphism $\pi:G/H\to G/N$ extends to a morphism
$X\to X_{\rm ad}$; equivalently, $X$ is toroidal in the sense of [Ti11, §29]).

We still denote by $\pi:X\to X_{\rm ad}$ this extension, which is of course
unique and hence $G$-equivariant. The boundary $\partial X=X\setminus X^{0}$
is a divisor with simple normal crossings. We will use the following relation
between the canonical divisors of $X$ and $X_{{\rm ad}}$:

######  Lemma 3.2.

With the above notation, we have the equality of divisor classes

$K_{X}+\partial X=\pi^{*}(K_{X_{{\rm ad}}}+\partial X_{{\rm ad}}).$

###### Proof.

Recall from [deCP83] that $X_{{\rm ad}}$ is isomorphic to the $G$-orbit
closure of $\mathfrak{h}$ in the Grassmannian of subspaces of $\mathfrak{g}$.
Moreover, $K_{X_{{\rm ad}}}+\partial X_{{\rm ad}}$ is the hyperplane class $H$
in the corresponding Plücker embedding (see [Ti11, Prop. 30.8]). Also,
$K_{X}+\partial X=\pi^{*}(H)$ by loc. cit. ∎

We will also use the following description of the general fibers of $\pi$; by
equivariance, it suffices to describe the fiber at $x$. In view of the Stein
factorization, $\pi$ is the composition of a contraction $\pi^{\prime}:X\to
X^{\prime}$ as discussed after Remark 2.5, and a finite surjective equivariant
morphism $\eta:X^{\prime}\to X_{\rm ad}$. The pair
$(X^{\prime},x^{\prime}:=\pi^{\prime}(x))$ is a normal projective equivariant
embedding (possibly singular) of $G/N^{0}H=G/Z^{0}H$ (a symmetric space under
$G/Z^{0}$). Moreover, the fiber of $\pi$ at $x$ is isomorphic to the
associated bundle $N\times^{N^{0}H}F$, where $F$ denotes the fiber of
$\pi^{\prime}$ at $x$. The group $N^{0}H$ acts on $F$ via its quotient torus
$N^{0}H/H\simeq N^{0}/H\cap N^{0}\simeq Z^{0}/H\cap Z^{0}$ (where the second
isomorphism follows from Lemma 2.9), and $F$ is a smooth projective toric
variety under that torus.

###  3.2 Relation between minimal families

In this subsection, we consider a complete symmetric variety $X$ with base
point $x$. We will reduce somehow the description of minimal families of
rational curves on $X$ to the cases where $X$ is a smooth projective toric
variety or a wonderful symmetric variety. A key notion is that of a _highest
weight curve_ , i.e., an irreducible curve $C\subset X$ through $x$ which is
stable under the Borel subgroup $B_{H}$; equivalently, $C\cap X^{0}$ is a
highest weight curve in the sense of §2.6.

By Corollary 2.14, we have $C=\overline{U_{\alpha}\cdot x}$ for some root
$\alpha$. In view of [BF15, Lem. 2.1 (i), Lem. 2.4], this yields:

######  Lemma 3.3.

Let $C$ be a highest weight curve. Then $C$ is an embedded free rational
curve.

We now obtain an alternative for minimal families:

######  Lemma 3.4.

Let ${\mathcal{K}}$ be a family of minimal rational curves on $X$.

  1. 1.

Either ${\mathcal{K}}$ is contracted by $\pi$, or ${\mathcal{K}}_{x}$ contains
a highest weight curve.

  2. 2.

In the former case, ${\mathcal{K}}_{x}$ is a minimal family of rational curves
on the toric variety $F$. Moreover, the tangent map $\tau_{x}$ is an
isomorphism of ${\mathcal{K}}_{x}$ with a linear subspace of
${\mathbb{P}}(\mathfrak{p}\cap\mathfrak{z})$.

###### Proof.

(1) This follows from Lemma 2.11 together with Borel’s fixed point theorem,
which yields the existence of a $B_{H}$-fixed point in ${\mathcal{K}}_{x}$.

(2) The first assertion is a consequence of Lemma 2.6. The second assertion
follows from [CFH14, Cor. 2.5]. ∎

######  Proposition 3.5.

Let $X=X_{{\rm ad}}$ be a wonderful symmetric variety, and ${\mathcal{K}}$ a
family of rational curves on $X$. Then ${\mathcal{K}}$ is minimal if and only
if ${\mathcal{K}}_{x}$ contains a highest weight curve.

If $X$ is irreducible, then it has a unique minimal family unless $X$ is
Hermitian exceptional; in that case, there are two minimal families.

###### Proof.

If ${\mathcal{K}}$ is minimal, then ${\mathcal{K}}_{x}$ contains a highest
weight curve by Lemma 3.4.

For the converse, using Lemma 2.2 and the structure of adjoint symmetric
spaces (§2.5), we may assume that $X$ is irreducible.

For $X$ not Hermitian, there is a unique highest weight curve $C$. Let
${\mathcal{L}}$ be a minimal family of rational curves on $X$. Then
$C\in{\mathcal{L}}$ by the above step, and hence
${\mathcal{K}}={\mathcal{L}}$.

For $X$ Hermitian, there are two highest weight curves, with highest weights
$\Theta$ and $-\alpha$. Consider a Chevalley involution of $(G,T)$, i.e., an
involution $\tau$ of $G$ such that $\tau(t)=t^{-1}$ for all $t\in T$. Then
$\tau$ commutes with $\sigma={\rm Int}(c)$, since $\tau(c)=c^{-1}=cz$ for some
$z\in Z$. Thus, $\tau$ induces an involution $\iota$ of $X$ fixing $x$. Also,
$\tau$ sends every root to its opposite; in particular,
$\tau(\Theta)=-\Theta$. Choose a representative $g\in\operatorname{N}_{L}(T)$
of the longest element of the Weyl group of $(L,T)$. Then $g\circ\iota$ is an
automorphism of $X$ which fixes $x$ and exchanges the two highest weight
curves. So each of these curves is contained in a minimal family, and hence
${\mathcal{K}}$ is minimal.

This completes the proof of the first assertion. For the second assertion, it
only remains to show that there are two minimal families if $X$ is Hermitian
exceptional. But then the open immersion $G/N\to G/P\times G/Q$ extends to a
morphism $X\to G/P\times G/Q$, and the resulting projection $X\to G/P$
contracts the highest weight curve with weight $\Theta$ but not the other one.
So these two curves cannot be in the same family. (In the non-exceptional
case, they are exchanged by any element of $\operatorname{N}_{G}(L)\setminus
L$). ∎

######  Proposition 3.6.

Let ${\mathcal{K}}$ be a minimal family of rational curves on $X$ containing a
highest weight curve $C$.

  1. 1.

${\mathcal{K}}_{x}$ consists of embedded free curves; it is smooth and
equidimensional, of dimension $-K_{X}\cdot C-2$.

  2. 2.

There is a unique minimal family of rational curves ${\mathcal{L}}$ on
$X_{{\rm ad}}$ and a commutative diagram of $H$-equivariant rational maps

$\textstyle{{\mathcal{K}}_{x}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{*,x}}$$\scriptstyle{\tau_{x}}$$\textstyle{{\mathcal{L}}_{x_{{\rm
ad}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau_{x_{{\rm
ad}}}}$$\textstyle{{\mathbb{P}}(T_{x}X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d\pi_{x}}$$\textstyle{{\mathbb{P}}(T_{x_{{\rm
ad}}}X_{{\rm ad}})}$

where $\tau_{x}$ and $\tau_{x_{{\rm ad}}}$ are finite and birational onto
their image, and $\pi_{*,x}$ is a finite morphism. If $\pi$ is birational,
then $\pi_{*,x}$ is finite and birational onto its image as well.

  3. 3.

We have $1\leq\partial X\cdot C\leq\partial X_{{\rm ad}}\cdot\pi(C)$.
Moreover, $\partial X\cdot C=\partial X_{{\rm ad}}\cdot\pi(C)$ if and only if
the image of $\pi_{*,x}$ is a union of components of ${\mathcal{L}}_{x_{{\rm
ad}}}$.

  4. 4.

If each connected component of ${\mathcal{L}}_{x_{{\rm ad}}}$ is a unique
$N^{0}$-orbit, then $\pi_{*,x}$ sends each component of ${\mathcal{K}}_{x}$
isomorphically to a component of ${\mathcal{L}}_{x_{{\rm ad}}}$.

###### Proof.

(1) The open subset ${\mathcal{K}}_{{\rm emfr},x}$ is $B_{H}$-stable, and
contains every $B_{H}$-fixed point by Lemma 2.11.2 and Lemma 3.3. Using
Borel’s fixed point theorem, it follows that ${\mathcal{K}}_{{\rm emfr},x}$ is
the whole ${\mathcal{K}}_{x}$. Thus, ${\mathcal{K}}_{x}$ is smooth; it is
equidimensional by Lemma 2.3. The assertion on its dimension follows from
[Ko96, II.3.2].

(2) By Lemma 2.11.2 again, $\pi|_{C}$ is birational to its image $D$. In view
of Lemma 2.1, this yields a commutative diagram of rational maps

$\textstyle{{\mathcal{K}}_{x}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{*,x}}$$\scriptstyle{\tau_{x}}$$\textstyle{{\mathcal{L}}_{x_{{\rm
ad}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau_{x_{{\rm
ad}}}}$$\textstyle{{\mathbb{P}}(T_{x}X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d\pi_{x}}$$\textstyle{{\mathbb{P}}(T_{x_{{\rm
ad}}}X_{{\rm ad}})}$

for a unique covering family of rational maps ${\mathcal{L}}$ on $X_{{\rm
ad}}$. Since ${\mathcal{K}}_{x}$ is smooth, $\tau_{x}$ is finite and
birational onto its image. Also, $\pi_{*,x}$ is a morphism since it is
$B_{H}$-equivariant and defined at every $B_{H}$-fixed point. Moreover,
${\mathcal{L}}$ contains the highest weight curve $D$, and hence is minimal by
Proposition 3.5. Thus, $\tau_{x_{{\rm ad}}}$ is also a morphism, and is finite
and birational onto its image as well.

The rational map
$d\pi_{x}:{\mathbb{P}}(T_{x}X)\dasharrow{\mathbb{P}}(T_{x_{{\rm ad}}}X_{{\rm
ad}})$ is a linear projection, and hence yields an affine morphism on its
domain of definition. As a consequence, the fibers of $\pi_{*,x}$ are affine;
thus, $\pi_{*,x}$ is a finite morphism.

The final assertion follows from Remark 2.5.

(3) Since $X^{0}$ is affine, $C$ intersects $\partial X$ and hence $\partial
X\cdot C\geq 1$.

By (1), we have $\dim({\mathcal{K}}_{x})=-K_{X}\cdot C-2$ and
$\dim({\mathcal{L}}_{x_{{\rm ad}}})=-K_{X_{{\rm ad}}}\cdot\pi(C)-2$. Since
$\pi_{*,x}:{\mathcal{K}}_{x}\to{\mathcal{L}}_{x_{{\rm ad}}}$ is finite, it
follows that

$K_{X}\cdot C\geq K_{X_{{\rm ad}}}\cdot\pi(C),$

with equality if and only if the image of $\pi_{*,x}$ is a union of components
of ${\mathcal{L}}_{x_{{\rm ad}}}$. Moreover,

$(K_{X}+\partial X)\cdot C=(K_{X_{{\rm ad}}}+\partial X_{{\rm
ad}})\cdot\pi(C)$

by Lemma 3.2 and the projection formula. This yields the remaining statements.

(4) By assumption, each component of ${\mathcal{L}}_{x_{{\rm ad}}}$ is
homogeneous under $N^{0}$, and hence under $H^{0}$ (Lemma 2.9). The
corresponding isotropy group is a parabolic subgroup of $H^{0}$; thus, it is
connected. As $\pi_{*,x}$ is finite and $H^{0}$-equivariant, this yields the
assertion. ∎

######  Example 3.7.

Let $G=\operatorname{SO}_{n}$, where $n\geq 3$, and let $\sigma$ be the
conjugation by $c={\rm diag}(1,\ldots,1,-1)\in{\rm O}_{n}$. Let
$H=G^{\sigma,0}=\operatorname{SO}_{n-1}$. Then $N=G^{\sigma}={\rm O}_{n-1}$
embedded in $\operatorname{SO}_{n}$ via $g\mapsto(g,\det(g))$, and
$\mathfrak{p}={\mathbb{C}}^{n-1}$ on which ${\rm O}_{n-1}$ acts via its
standard representation. Also, $G/H$ has a unique smooth projective
equivariant embedding: the quadric
${\mathbb{Q}}_{n-1}\subset{\mathbb{P}}^{n}={\mathbb{P}}({\mathbb{C}}^{n}\oplus{\mathbb{C}})$,
where $\operatorname{SO}_{n}$ acts on
${\mathbb{P}}({\mathbb{C}}^{n}\oplus{\mathbb{C}})$ via its standard
representation on ${\mathbb{C}}^{n}$. Moreover, $X_{{\rm
ad}}={\mathbb{P}}^{n-1}={\mathbb{P}}({\mathbb{C}}^{n})$ and $\pi:X\to X_{{\rm
ad}}$ is a ramified double cover induced by the linear projection
${\mathbb{P}}({\mathbb{C}}^{n}\oplus{\mathbb{C}})\dasharrow{\mathbb{P}}({\mathbb{C}}^{n})$.

If $n\geq 4$ then $X$ has a unique minimal family of rational curves
${\mathcal{K}}$; it consists of the lines in ${\mathbb{Q}}_{n-1}$. Moreover,
$\pi_{*}$ sends ${\mathcal{K}}$ to the family ${\mathcal{L}}$ of lines in
${\mathbb{P}}^{n-1}$, and
$\pi_{*;x}:{\mathcal{K}}_{x}\to{\mathcal{L}}_{x_{{\rm ad}}}$ is identified
with the inclusion
${\mathbb{Q}}_{n-3}\subset{\mathbb{P}}^{n-2}={\mathbb{P}}(\mathfrak{p})$,
compatibly with the action of ${\rm O}_{n-1}=N$.

If $n=3$ then $X={\mathbb{Q}}^{2}\simeq{\mathbb{P}}^{1}\times{\mathbb{P}}^{1}$
has two minimal families of rational curves, the fibers of the two projections
to ${\mathbb{P}}^{1}$. For both families, $\pi_{*,x}$ identifies
${\mathcal{K}}_{x}$ with a point in ${\mathbb{P}}^{1}$.

###  3.3 Minimal families of group and Hermitian types

We still consider a complete symmetric variety $X$ with base point $x$, and a
minimal family of rational curves ${\mathcal{K}}$ on $X$; we assume that
${\mathcal{K}}$ contains a highest weight curve $C$. As explained above, there
is a unique indecomposable factor $X_{C}$ of $X_{{\rm ad}}$ such that the
composition of $\pi:X\to X_{{\rm ad}}$ with the projection $X_{{\rm ad}}\to
X_{C}$ sends $C$ isomorphically to its image. In this subsection, we will
handle in details the cases where $X_{C}$ is of group or Hermitian types.

We first handle the group type, where $X_{C}$ is the wonderful completion of
an adjoint simple group $H_{C}$. The Lie algebra of $H_{C}$ is denoted by
$\mathfrak{h}_{C}$, and we still denote by $C$ the highest weight curve in
$X_{C}$. By the main result of [BF15], $X_{C}$ has a unique minimal family of
rational curves ${\mathcal{L}}$. Moreover, the tangent map
$\tau_{x_{C}}:{\mathcal{L}}_{x_{C}}\to{\mathbb{P}}(\mathfrak{h}_{C})$ is an
$H_{C}$-equivariant isomorphism to its image ${\mathcal{C}}_{x_{C}}$. If
$H_{C}$ is of type ${\rm A}_{r}$ with $r\geq 2$, i.e., $H_{C}\simeq{\rm
PGL}(V)$ where $V$ is a vector space of dimension $r+1\geq 3$, then
$\mathfrak{h}_{C}\simeq{\rm End}(V)/{\mathbb{C}}{\rm id}$ and
${\mathcal{C}}_{x_{C}}$ is isomorphic to
${\mathbb{P}}(V)\times{\mathbb{P}}(V^{\vee})$ embedded in
${\mathbb{P}}(\mathfrak{h}_{C})$ via the Segre embedding

${\mathbb{P}}(V)\times{\mathbb{P}}(V^{\vee})\hookrightarrow{\mathbb{P}}(V\otimes
V^{\vee})={\mathbb{P}}({\rm End}(V))$

followed by the linear projection ${\mathbb{P}}({\rm
End}(V))\dasharrow{\mathbb{P}}({\rm End}(V)/{\mathbb{C}}{\rm id})$. In all
other types, we have
${\mathcal{C}}_{x_{C}}={\mathbb{P}}({\mathcal{O}}_{C,\min})$, the
projectivization of the minimal nilpotent orbit in
${\mathbb{P}}(\mathfrak{h}_{C})$.

######  Proposition 3.8.

If $X_{C}$ is of group type, then $\partial X\cdot C$ equals $1$ or $2$. In
the former case, every component of ${\mathcal{K}}_{x}$ is isomorphic to
${\mathbb{P}}({\mathcal{O}}_{C,\min})$. In the latter case, we have
$H_{C}\simeq{\rm PGL}(V)$, where $\dim(V)\geq 3$, and every component of
${\mathcal{K}}_{x}$ is isomorphic to
${\mathbb{P}}(V)\times{\mathbb{P}}(V^{\vee})$.

###### Proof.

Recall from Proposition 3.6 that $1\leq\partial X\cdot C\leq\partial X_{{\rm
ad}}\cdot\pi(C)$. Moreover, the line bundle on $X_{{\rm ad}}$ associated with
the divisor $\partial X_{{\rm ad}}$ equals ${\mathcal{L}}_{X_{{\rm
ad}}}(\alpha_{1}+\cdots+\alpha_{r})$ with the notation of [BF15, §3]. By
combining [BF15, Lem. 3.3, Lem. 3.4], it follows that $\partial X_{{\rm
ad}}\cdot\pi(C)=2$ if $H_{C}$ is of type ${\rm A}_{r}$ where $r\geq 2$;
otherwise, $\partial X_{{\rm ad}}\cdot\pi(C)=1$.

In the latter case, we must have $\partial X\cdot C=\partial X_{{\rm
ad}}\cdot\pi(C)$. So every component of ${\mathcal{K}}_{x}$ is isomorphic to
the orbit $H_{C}\cdot C$, by Proposition 3.6 again. Moreover, $H_{C}\cdot
C={\mathbb{P}}({\mathcal{O}}_{C,\min})$.

In the former case,
${\mathcal{L}}_{x_{C}}={\mathbb{P}}(V)\times{\mathbb{P}}(V^{\vee})$ consists
of two orbits of $H_{C}={\rm PGL}(V)$: a closed orbit of codimension $1$ (the
incidence variety, isomorphic to ${\mathbb{P}}({\mathcal{O}}_{C,\min})$), and
an open orbit isomorphic to $\operatorname{SL}_{r+1}/{\rm GL}_{r}$, and hence
simply connected.

If $\partial X\cdot C=2$, then by Proposition 3.6 again, we get a finite
surjective $H$-equivariant morphism
$\pi_{*,x}:{\mathcal{K}}_{x}\to{\mathbb{P}}(V)\times{\mathbb{P}}(V^{\vee})$.
Since the open orbit in the right-hand side is simply connected, it follows
that $\pi_{*,x}$ is birational on each component, and hence an isomorphism in
view of Zariski’s main theorem.

On the other hand, if $\partial X\cdot C=1$, then the image of $\pi_{*,x}$ is
the closed orbit and we conclude as above. ∎

######  Example 3.9.

Assume that $\pi:X\to X_{{\rm ad}}$ is birational and $X_{{\rm ad}}$ is the
wonderful completion of ${\rm PGL}(V)$, where $\dim(V)\geq 3$. Then the
highest weight curve $C_{{\rm ad}}\in X_{{\rm ad}}$ intersects a unique ${\rm
PGL}(V)\times{\rm PGL}(V)$-orbit ${\mathcal{O}}_{1,r}$ of codimension $2$ in
$X_{{\rm ad}}$ (see [BF15, Lem.3.4]).

If $\pi$ is an isomorphism over ${\mathcal{O}}_{1,r}$, then the minimal family
${\mathcal{K}}$ on $X$ satisfies $\partial X\cdot C=2$ and
${\mathcal{K}}_{x}={\mathbb{P}}(V)\times{\mathbb{P}}(V^{\vee})$. Indeed, $\pi$
is an isomorphism over an open neighborhood of ${\mathcal{O}}_{1,r}$ in
$X_{{\rm ad}}$, stable by ${\rm PGL}(V)\times{\rm PGL}(V)$, and every curve in
${\mathcal{L}}_{x_{{\rm ad}}}$ intersects such a neighborhood.

On the other hand, if $\pi$ is not an isomorphism over ${\mathcal{O}}_{1,r}$,
then $\partial X\cdot C=1$ and ${\mathcal{K}}_{x}$ is the incidence variety
${\mathbb{P}}({\mathcal{O}}_{\min})$; moreover, we have $\partial X_{{\rm
ad}}\cdot C_{{\rm ad}}=2$ and ${\mathcal{L}}_{x_{{\rm
ad}}}={\mathbb{P}}(V)\times{\mathbb{P}}(V^{\vee})$. Indeed, $\pi$ factors
through the blow-up $\varphi:X^{\prime}\to X_{{\rm ad}}$ of
$\overline{{\mathcal{O}}_{1,r}}$ in $X_{{\rm ad}}$. Using Proposition 3.6, we
may thus assume that $X=X^{\prime}$. Then $K_{X}=\pi^{*}(K_{X_{{\rm ad}}})+E$,
where $E$ denotes the exceptional divisor. Thus,

$K_{X}\cdot C=K_{X_{{\rm ad}}}\cdot C_{{\rm ad}}+E\cdot C>K_{X_{{\rm
ad}}}\cdot C_{{\rm ad}},$

since $C$ intersects $E$. It follows that
$\dim({\mathcal{K}}_{x})<\dim({\mathcal{L}}_{x_{{\rm ad}}})$, and we conclude
by Proposition 3.6 again.

Next, we handle the Hermitian type, where $X_{C}$ is the wonderful completion
of the symmetric space $G_{C}/\operatorname{N}_{G_{C}}(L_{C})$ for a simple
factor $G_{C}$ of $G_{{\rm ad}}$ with a Levi subgroup $L_{C}\subset G_{C}$. We
may then assume that $G=G_{C}$.

######  Proposition 3.10.

If $X_{C}$ is of Hermitian type but not of type ${\rm PGL}_{2}/N$, then
$\partial X\cdot C=1$ and every component of ${\mathcal{K}}_{x}$ is isomorphic
to the orbit $L_{C}\cdot C$.

###### Proof.

By Proposition 3.6, we may assume that $X=X_{C}$. We now view $X$ as a
subvariety of $\overline{G}$ (as recalled in §3.1), and use the description of
minimal rational curves in $\overline{G}$, as in the proof of Proposition 3.8.

With the notation of §2.6, the highest weight curves are
$C_{\Theta}:=\overline{U_{\Theta}\cdot x}$ and
$C_{-\alpha}:=\overline{U_{-\alpha}\cdot x}$ (indeed, these curves are
irreducible, stable by $B_{L}$ and distinct). As observed in the proof of
Proposition 3.5, these curves are exchanged by an automorphism of $X$ fixing
$x$; thus, we may assume that $C=C_{\Theta}$.

By [RRS92, §5.5], $\sigma$ is the inner involution ${\rm Int}(c)$, where $c\in
T$ satisfies $\alpha(c)=-1$ and $\beta(c)=1$ for all simple roots
$\beta\neq\alpha$. In particular, the roots $\Theta$ and $-\alpha$ are non-
compact imaginary. Thus, the closed immersion

$\iota:G/\operatorname{N}_{G}(L)\longrightarrow G,\quad
g\operatorname{N}_{G}(L)\longmapsto\sigma(g)g^{-1}$

induces isomorphisms

$U_{\Theta}\cdot
x\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}U_{\Theta},\quad
U_{-\alpha}\cdot
x\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}U_{-\alpha}.$

So $\overline{\iota}:X\to\overline{G}$ sends $C_{\Theta}$, $C_{-\alpha}$
isomorphically to the corresponding root curves considered in [BF15, §3].
Since $\Theta$ and $-\alpha$ are long roots, these root curves are minimal;
hence $\overline{\iota}$ sends ${\mathcal{K}}_{x}$ to the unique family
${\mathcal{L}}_{\overline{G},e}$ of minimal rational curves through $e$ in
$\overline{G}$. Moreover, $\overline{\iota}({\mathcal{K}}_{x})$ is contained
in the fixed locus ${\mathcal{L}}_{\overline{G},e}^{-\sigma}$.

If $G$ is not of type $A_{r}$, where $r\geq 2$, then the tangent map
$\tau_{e}$ identifies ${\mathcal{L}}_{\overline{G},e}$ with
${\mathbb{P}}({\mathcal{O}}_{\min})$. Since $d\iota_{x}$ identifies $T_{x}X$
with $\mathfrak{p}$, we see that

$\overline{\iota}({\mathcal{K}}_{x})\subset{\mathbb{P}}({\mathcal{O}}_{\min}\cap\mathfrak{p})\subset{\mathbb{P}}({\mathcal{O}}_{\min})^{\sigma}.$

By [Ri82, Thm. A], the right-hand side is a finite union of closed orbits of
$G^{\sigma,0}=L$. We conclude that the component of $C$ in ${\mathcal{K}}_{x}$
is $L\cdot C$.

Otherwise, $G={\rm PGL}(V)$ where $\dim(V)=r+1$, and $\tau_{e}$ yields an
isomorphism

${\mathcal{L}}_{\overline{G},e}\simeq{\mathbb{P}}(V)\times{\mathbb{P}}(V^{\vee})\subset{\mathbb{P}}({\rm
End}(V)/{\mathbb{C}}{\rm id})={\mathbb{P}}(\mathfrak{g}),$

equivariantly for the action of $-\sigma$. Consider the $\sigma$-eigenspace
decomposition $V=V_{1}\oplus V_{-1}$. Then $L$ is the image of ${\rm
GL}(V_{1})\times{\rm GL}(V_{-1})$ in ${\rm PGL}(V)$; also,
${\mathbb{P}}(V)^{\sigma}={\mathbb{P}}(V_{1})\sqcup{\mathbb{P}}(V_{-1})$ and
likewise for ${\mathbb{P}}(V^{\vee})^{\sigma}$. Moreover, the image of
$\iota({\mathcal{C}})$ under $\tau_{e}$ is contained in

${\mathbb{P}}(\mathfrak{p})={\mathbb{P}}(\operatorname{Hom}(V_{1},V_{-1})\oplus\operatorname{Hom}(V_{-1},V_{1})).$

It follows that $\iota({\mathcal{K}}_{x})$ is contained in
$({\mathbb{P}}(V_{1}^{\vee})\times{\mathbb{P}}(V_{-1}))\sqcup({\mathbb{P}}(V_{-1}^{\vee})\times{\mathbb{P}}(V_{1}))$.
As a consequence, the component of $C$ in ${\mathcal{K}}_{x}$ is $L\cdot C$ in
this case, too.

We now show that $\partial X\cdot C=1$. Consider first the exceptional case,
where $H=L=P\cap Q$. Then we have a $G$-equivariant birational morphism
$\varphi:X\to G/P\times G/Q=:Y$ which sends $x$ to the base point $y=(P,Q)$.
Since $P$ is a maximal parabolic subgroup of $G$ associated with a long root,
$G/P$ has a unique family of minimal rational curves ${\mathcal{L}}$.
Moreover, denoting by $P$ the base point of the homogeneous space $G/P$ and by
$D$ the Schubert line in that space (i.e., the unique irreducible $B$-stable
curve), we have that ${\mathcal{L}}_{P}=L\cdot D$ (see e.g. [BK21, Prop.
3.3]). The projection $p:X\to G/P$ sends $C$ isomorphically to $D$, and yields
an isomorphism $p_{*}:L\cdot C\to L\cdot D$ which identifies $L\cdot C$ with
the variety of lines in $G/P$ through its base point. Since $\dim(L\cdot
C)=\dim({\mathcal{K}}_{x})=-K_{X}\cdot C-2$ and $\dim(L\cdot D)=-K_{G/P}\cdot
D-2$, we obtain

$K_{X}\cdot C=K_{G/P}\cdot D=K_{Y}\cdot\varphi(C)=\varphi^{*}(K_{Y})\cdot C$

by using the projection formula. On the other hand, we have
$K_{X}=\varphi^{*}(K_{Y})+\sum_{i}a_{i}E_{i}$, where the $E_{i}$ are the
exceptional divisors of $\varphi$ and the $a_{i}$ are positive integers. Since
$C$ is not contained in any $E_{i}$, it follows that $E_{i}\cdot C=0$ for all
$i$. Also, the boundary of $Y$ is an irreducible divisor $E$, and $\partial
X=E^{\prime}+\sum_{i}E_{i}$, where $E^{\prime}$ denotes the strict transform
of $E$. This yields $\partial X\cdot C=E^{\prime}\cdot C=E\cdot D$ by the
projection formula again. Since $D\subset G/P\times\\{Q\\}$, where we still
denote by $Q$ the base point of $G/Q$, and $E\cap(G/P\times\\{Q\\})$ is
identified with the Schubert divisor in $G/P$, we obtain $E\cdot D=1$. This
yields the assertion in that case.

Next, we consider the non-exceptional case, where $H=\operatorname{N}_{G}(L)$
contains $L$ as a subgroup of index $2$. Then there exists a smooth toroidal
equivariant embedding $X^{\prime}$ of $G/L$ such that the natural map $G/L\to
G/H$ extends to a morphism $\psi:X^{\prime}\to X$. By Lemma 3.2, we have
$K_{X^{\prime}}+\partial X^{\prime}=\psi^{*}(K_{X}+\partial X)$. Moreover, $C$
lifts uniquely to a highest weight curve $C^{\prime}\subset X^{\prime}$, and
the corresponding minimal families have isomorphic components by Proposition
3.6. Taking dimensions, we obtain $K_{X^{\prime}}\cdot C^{\prime}=K_{X}\cdot
C=\psi^{*}(K_{X})\cdot C^{\prime}$. As a consequence, we have $\partial
X^{\prime}\cdot C^{\prime}=\psi^{*}(\partial X)\cdot C^{\prime}=\partial
X\cdot C=1$. ∎

Proposition 3.10 leaves out the case of type ${\rm PGL}_{2}/N$, which is
easily treated:

######  Lemma 3.11.

If $X_{C}$ is of type ${\rm PGL}_{2}/N$, then $\partial X\cdot C$ equals $1$
or $2$. In the former case, ${\mathcal{K}}_{x}$ is finite. In the latter case,
every component of ${\mathcal{K}}_{x}$ is a projective line.

###### Proof.

Note that ${\rm PGL}_{2}/N$ has a unique projective equivariant embedding,
namely, ${\mathbb{P}}^{2}$ on which ${\rm PGL}_{2}$ acts via the
projectivization of its adjoint representation. Thus, $\partial X_{C}$ is a
conic, with $C$ as a tangent line so that $\partial X_{C}\cdot C=2$. Also, the
minimal rational curves on $X_{C}$ are just lines, and those through a given
point form a ${\mathbb{P}}^{1}$. This yields the statement by using
Proposition 3.6 as in the proof of the above proposition. ∎

######  Corollary 3.12.

Let $G$ be a simple adjoint group, and $X$ the wonderful embedding of a
Hermitian symmetric space $G/\operatorname{N}_{G}(L)$. Denote by $C_{\Theta}$
and $C_{-\alpha}$ the highest weight curves in $X$, indexed by their weight.

  1. 1.

If $X$ is exceptional, then it has two minimal families of rational curves
${\mathcal{K}}^{+}$, ${\mathcal{K}}^{-}$. Moreover,
${\mathcal{K}}_{x}^{+}=L\cdot C_{\Theta}$ and ${\mathcal{K}}_{x}^{-}=L\cdot
C_{-\alpha}$.

  2. 2.

If $X$ is non-exceptional, then it has a unique minimal family
${\mathcal{K}}$. Moreover, ${\mathcal{K}}_{x}=\operatorname{N}_{G}(L)\cdot
C_{\Theta}=L\cdot C_{\Theta}\sqcup L\cdot C_{-\alpha}$ unless
$G/\operatorname{N}_{G}(L)={\rm PGL}_{2}/N$.

###  3.4 Some cases of simple type

In this subsection we use previous techniques and the results in the group
case to briefly describe the unique minimal family in some of the cases of
simple type. We refer to Table 1 for the different cases of the
classification. A different approach, working in all cases is developed in
Section 4.

Consider a highest weight curve $C$ of simple type, and denote by $\lambda$
its weight relative to $B_{H}$. Then $\lambda$ is the highest weight of the
representation of $H^{0}$ in $\mathfrak{p}$, and hence is the restriction to
$T_{H}$ of some $\alpha\in R^{+}$ (not necessarily unique). Moreover, $S:={\rm
ker}(\lambda)^{0}$ is a subtorus of codimension $1$ in $T_{H}$, and $C$ is an
irreducible curve in $(G/H)^{S}$ through $x$, stable by the Borel subgroup
$\operatorname{C}_{B_{H}}(S)$ of $\operatorname{C}_{H}(S)$. So $C$ is a
highest weight curve of the symmetric space
$\operatorname{C}_{G}(S)/\operatorname{C}_{H}(S)=(\operatorname{C}_{G}(S)/S)/(\operatorname{C}_{H}(S)/S)$,
where $\operatorname{C}_{H}(S)/S$ has rank $1$. We now apply Lemma 2.12: the
adjoint symmetric space of $\operatorname{C}_{G}(S)/S$ is of type
$(\textrm{A}_{1})$, $(\textrm{A}_{1}\times\textrm{A}_{1})$ or
$(\textrm{A}_{2})$. If $\sigma$ is inner, then only type $(\textrm{A}_{1})$
may occur.

######  Lemma 3.13.

Assume that $\lambda=\alpha|_{T_{H}}$ for a unique root $\alpha$ (i.e., type
$({\rm A}_{1}\times{\rm A}_{1})$ is excluded), and $\alpha$ is long. Then the
component of ${\mathcal{K}}_{x}$ containing $C$ admits a finite equivariant
morphism to ${\mathcal{K}}_{\overline{G_{\rm ad}},{\rm id}}^{-\sigma}$.

###### Proof.

By assumption, $C=\overline{U_{\alpha}\cdot x}$, where $\alpha$ is non-compact
imaginary. Thus, the image of $C$ under the morphism
$\psi:X\to\overline{G_{\rm ad}}$, obtained by composing $\phi:X\to X_{\rm ad}$
with $\iota:X_{\rm ad}\to\overline{G_{\rm ad}}$, is just the closure of
$U_{\alpha}$; since $\alpha$ is long, this is a minimal rational curve on
$\overline{G_{\rm ad}}$. This yields the assertion by arguing as in the proof
of Proposition 3.6. ∎

The assumptions of the lemma hold if and only if $T_{x}C\setminus\\{0\\}$ is
contained in ${\mathcal{O}}_{\min}$ (as follows by combining Proposition 2.13,
Corollary 2.15 and Lemma 2.20).

######  Proposition 3.14.

Assume that $T_{x}C\setminus\\{0\\}\subset{\mathcal{O}}_{\min}$.

  1. 1.

If $\partial X\cdot C=1$, then the component of ${\mathcal{K}}_{x}$ containing
$C$ is $H^{0}\cdot C$.

  2. 2.

If $\partial X\cdot C=2$, then $X$ is of type AI with $G={\rm PGL}_{r+1}$ and
${\mathcal{K}}_{x}\simeq{\mathbb{P}}^{r}$.

###### Proof.

If $G$ is not of type ${\rm A}_{r}$ or if $X$ is Hermitian (but not of type
${\rm PGL}_{2}/N$), we may argue as in the proof of Proposition 3.10 proving
that the component of ${\mathcal{K}}_{x}$ containing $C$ is $H^{0}\cdot C$ and
that $\partial X\cdot C=1$.

If $G$ is of type ${\rm A}_{r}$ and $X$ is not Hermitian, then $X$ is of type
AI, with $G={\rm PGL}(V)$ such that $\dim V=r+1$, $r\geq 2$ and
$\sigma(g)=(g^{t})^{-1}$. In this case, the minimal family
${\mathcal{K}}_{\overline{G_{\rm ad}},{\rm id}}$ identifies with
${\mathbb{P}}(V)\times{\mathbb{P}}(V^{\vee})$ and the involution $-\sigma$
acts via $(-\sigma)([v],[H])=([H^{\perp}],[v^{\perp}])$ where the
orthogonality is taken with respect to the standard scalar product. We thus
have ${\mathcal{K}}_{\overline{G_{\rm ad}},{\rm
id}}^{-\sigma}\simeq{\mathbb{P}}(V)$. If $\partial X\cdot C=1$, then
$\dim{\mathcal{K}}_{x}=\dim{\mathbb{P}}(V)-1=\dim H\cdot C$ and the result
follows as above. If $\partial X\cdot C=2$, then
$\dim{\mathcal{K}}_{x}=\dim{\mathbb{P}}(V)$ proving the result. ∎

######  Remark 3.15.

In Table 1, we list the nilpotent orbits containing $T_{x}C\setminus\\{0\\}$
(see the column“$\sigma(\Theta)=-\Theta$”, the condition
$T_{x}C\setminus\\{0\\}\subset{\mathcal{O}}_{\min}$ being equivalent to
$\sigma(\Theta)=-\Theta$ by Corollary 4.25). In particular, the above
proposition settles all cases except the following symmetric spaces: AII, BII,
CII, DII, EIV and FII. We will deal with all cases in the next section.

##  4 Minimal families on wonderful symmetric varieties

In this section we deal with wonderful embeddings of adjoint irreducible
symmetric spaces. We may therefore assume that $G$ is semisimple and replacing
$G$ by its universal cover, we may further assume that $G$ is simply
connected. Note that in this case $G^{\sigma}$ is connected, so that the group
$H$ satisfies $G^{\sigma}\subset H\subset\operatorname{N}_{G}(G^{\sigma})$ and
$\operatorname{N}_{G}(G^{\sigma})/G^{\sigma}$ is finite (Lemma 2.9). Set
$N=\operatorname{N}_{G}(G^{\sigma})$; then $N=\operatorname{N}_{G}(H)$ by
Lemma 2.10.

Since $G/H$ is irreducible, $\sigma$ acts transitively on the simple factors
of $G$. Therefore, $G/N$ is of group, Hermitian, or simple type. We will
consider its wonderful embedding $X=X_{\rm ad}$.

We start with reminders on restricted root systems (Subsection 4.1) and their
connection to curves and divisors on $X$ (Subsection 4.2). Many results on
these topics are well known but we could not find a good reference, so we
included proofs for the convenience of the reader. From this we obtain an
explicit description of the classes of curves in minimal families on $X$
(Subsection 4.3). We then compute the dimension of these minimal families
${\mathcal{K}}$ using the contact structures on projectivised nilpotent orbits
(Subsection 4.4). It turns out that in all cases except for $X$ of restricted
type $\textrm{A}_{r}$, the family ${\mathcal{K}}_{x}$ has the same dimension
as the orbit $N\cdot C$ where $C\in{\mathcal{K}}_{x}$ is a highest weight
curve, which in turn implies that ${\mathcal{K}}_{x}=N\cdot C$. We deal with
$X$ of restricted type $\textrm{A}_{r}$ separately (Subsection 4.5). We
conclude with a full description of ${\mathcal{K}}_{x}$ (Subsection 4.6).

###  4.1 Restricted root system

Let us first recall a few facts on the restricted root system; we refer to
[Vu90] and [Ti11] for details. Let $T_{{\rm s}}$ be a maximal torus of split
type and $S\subset T_{{\rm s}}$ be its maximal split subtorus: $S=\\{t\in
T_{{\rm s}}\ |\ \sigma(t)=t^{-1}\\}^{0}$. Let $R$ be the root system
associated to the pair $(G,T_{{\rm s}})$. Then $\sigma$ acts on $R$. Set
$\overline{S}=S/S^{\sigma}$,
$\overline{\mathfrak{X}}=\mathfrak{X}(\overline{S})$ and
$\overline{\chi}=\chi-\sigma(\chi)$ for $\chi\in\mathfrak{X}(T_{{\rm s}})$. We
have an identification $\overline{\mathfrak{X}}=\\{\overline{\chi}\ |\
\chi\in\mathfrak{X}(T_{{\rm s}})\\}$. Define the subset
$\overline{R}\subset\overline{\mathfrak{X}}$ via

$\overline{R}=\\{{\overline{\alpha}}\ |\ \alpha\in R\\}.$

Then $\overline{R}$ is an irreducible root system called the _restricted root
system_. It may be non-reduced (see Remark 4.1 below).

Recall that $L=\operatorname{C}_{G}(S)$ is the Levi subgroup containing
$T_{{\rm s}}$ of a parabolic subgroup $P\subset G$ and that $\sigma(P)$ is the
opposite parabolic subgroup to $P$ with common Levi subgroup $L$. Let $B_{{\rm
s}}\subset P$ be a Borel subgroup and let $\Delta\subset R^{+}\subset R$ be
the sets of simple roots and positive roots defined by $B_{{\rm s}}$. Then for
$\alpha\in R^{+}$, we have $\sigma(\alpha)=\alpha$ if and only if $\alpha$ is
a root of $L$; moreover, if $\sigma(\alpha)\neq\alpha$, then
$\sigma(\alpha)<0$. Set $\Delta_{1}=\\{\alpha\in\Delta\ |\
\sigma(\alpha)<0\\}$ and $\Delta_{0}=\Delta\setminus\Delta_{1}$. Then
$\sigma(\alpha)=\alpha$ for any $\alpha\in\Delta_{0}$. Define
$\overline{\Delta}\subset\overline{\mathfrak{X}}$ via

$\overline{\Delta}=\\{{\overline{\alpha}}\ |\ \alpha\in\Delta_{1}\\}.$

Then $\overline{\Delta}$ is a basis of
$\overline{R}\subset\overline{\mathfrak{X}}$. In particular
$|\overline{\Delta}|={\rm rk}(\overline{\mathfrak{X}})=\dim T_{{\rm s}}=r$ is
the rank of $X$. Furthermore, there exists a length-preserving involution
${\overline{\sigma}}$ on $\Delta$, preserving $\Delta_{1}$ and acting
trivially on $\Delta_{0}$, such that for any $\alpha\in\Delta_{1}$, we have

$\sigma(\alpha)+{\overline{\sigma}}(\alpha)=-\sum_{\beta\in\Delta_{0}}c_{\beta}\beta$

with $c_{\beta}\in{\mathbb{Z}}_{\geq 0}$. In particular, if
${\overline{\sigma}}(\alpha)\neq\alpha$, then
$\langle\alpha^{\vee},\sigma(\alpha)\rangle\geq 0$. Note that for
$\alpha,\beta\in\Delta_{1}$, we have
${\overline{\alpha}}={\overline{\beta}}\Leftrightarrow(\beta=\alpha\textrm{ or
}\beta={\overline{\sigma}}(\alpha))$.

A root $\alpha\in\Delta_{1}$ such that $\sigma(\alpha)\neq-\alpha$ and
$\langle\alpha^{\vee},\sigma(\alpha)\rangle\neq 0$ is called _exceptional_. If
$\alpha$ is exceptional, then $\beta={\overline{\sigma}}(\alpha)\neq\alpha$ is
also exceptional and one of the following two conditions is satisfied: either
$\sigma(\alpha)\neq-\beta$, or $\sigma(\alpha)=-\beta$ and
$\langle\alpha^{\vee},\beta\rangle\neq 0$ (see [deCS99, Lemma 4.3]). If there
exists an exceptional root, then $\overline{R}$ and $X$ are called
_exceptional_. This definition is equivalent to the one given in Subsection
2.5, see for example [deCS99, Lemma 4.7]. Note that by [deCS99, Lemma 4.7],
there are at most two exceptional roots (thus, of the form $\alpha$ and
${\overline{\sigma}}(\alpha)$).

######  Remark 4.1.

If $\overline{R}$ is exceptional, it is non-reduced. In fact, for $\alpha$
exceptional, we have $\langle\alpha^{\vee},\sigma(\alpha)\rangle\neq 0$. As
$\alpha$ and ${\overline{\sigma}}(\alpha)$ are different but of the same
length, we have $\langle\alpha^{\vee},\sigma(\alpha)\rangle=1$. Thus,
$\gamma=\alpha-\sigma(\alpha)\in R$ and
${\overline{\gamma}}=2{\overline{\alpha}}$. In particular,
${\overline{\alpha}},2{\overline{\alpha}}\in\overline{R}$ and $\overline{R}$
is non-reduced.

######  Example 4.2.

There are non-reduced restricted root systems which are non-exceptional
(actually only two families: types CII and FII, see Appendix). For example, if
$G={\rm Sp}_{6}$, there exists an involution $\sigma$ such that
$G^{\sigma}={\rm Sp}_{2}\times{\rm Sp}_{4}$ and with the labeling of the
simple roots as in Bourbaki [Bo68], we have
$\Delta_{0}=\\{\alpha_{1},\alpha_{3}\\}$ and $\Delta_{1}=\\{\alpha_{2}\\}$.
Set $\alpha=\alpha_{2}$, then ${\overline{\sigma}}(\alpha)=\alpha$,
$\sigma(\alpha)=-(\alpha_{1}+\alpha_{2}+\alpha_{3})$ and
${\overline{\alpha}}=\alpha_{1}+2\alpha_{2}+\alpha_{3}=\gamma\in R$. We have
${\overline{\gamma}}=2{\overline{\alpha}}$; thus,
$\overline{R}=\\{-2{\overline{\alpha}},-{\overline{\alpha}},{\overline{\alpha}},2{\overline{\alpha}}\\}$
but $G/G^{\sigma}$ is not exceptional.

Let $\alpha\in R^{+}$ such that $\sigma(\alpha)<0$. The roots $\alpha$ and
$\sigma(\alpha)$ have the same length. As explained in [Vu90, Lemme 2.3],
three cases occur and the coroot ${\overline{\alpha}}^{\vee}$ is defined
accordingly:

  1. 1.

If $\sigma(\alpha)=-\alpha$, then
${\overline{\alpha}}^{\vee}=\frac{1}{2}\alpha^{\vee}$.

  2. 2.

If $\langle\alpha^{\vee},\sigma(\alpha)\rangle=0$, then
${\overline{\alpha}}^{\vee}=\frac{1}{2}(\alpha^{\vee}-\sigma(\alpha)^{\vee})$.

  3. 3.

If $\langle\alpha^{\vee},\sigma(\alpha)\rangle=1$, then
${\overline{\alpha}}^{\vee}=\alpha^{\vee}-\sigma(\alpha)^{\vee}$.

Case (3) above actually occurs if and only if $\overline{R}$ is non-reduced,
see Proposition 4.3.5 below. In the next proposition, we summarise the results
on restricted root systems needed for the study of curves and divisors on $X$.
These results might be well known to the experts but we could not find a good
reference, so we included a proof and further results on restricted root
systems in Subsection 6.1 in the Appendix.

######  Proposition 4.3.

Let $\Theta$ be the highest root of $R$ and $w_{0}\in W$ be the longest
element.

  1. 1.

${\overline{\Theta}}$ is the highest root of $\overline{R}$, the actions of
$\sigma$ and $w_{0}$ on roots commute, and we have
$w_{0}({\overline{\Theta}})=-{\overline{\Theta}}$.

  2. 2.

If $\sigma(\Theta)\neq-\Theta$, then $\Theta$ and $\sigma(\Theta)$ are
strongly orthogonal long roots.

  3. 3.

If $\alpha\in\Delta_{1}$ is exceptional, its coefficient in the expansion of
$\Theta$ in simple roots is $1$.

  4. 4.

If $\overline{R}$ is not of type ${\rm A}_{1}$, then there exists
${\overline{\alpha}}\in\overline{\Delta}$ with
$\langle{\overline{\Theta}}^{\vee},{\overline{\alpha}}\rangle=1$ and
$2{\overline{\Theta}}^{\vee}$ is an indivisible cocharacter of $\overline{S}$.

  5. 5.

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

###### Proof.

(1) This is Lemma 6.1, Lemma 6.2 and Corollary 6.3. (2) This is Proposition
6.12.4. (3) This is the last statement in Corollary 6.9. (4) This is
Proposition 6.12.2-3. (5) This is Proposition 6.4. ∎

We end this subsection with a piece of notation. For
${\overline{\alpha}}\in\overline{\Delta}$, we denote by
${\widehat{\alpha}}^{\vee}$ the simple root of $\overline{R}^{\vee}$ colinear
to ${\overline{\alpha}}^{\vee}$. Note that if
$2{\overline{\alpha}}\not\in\overline{R}$, then
${\widehat{\alpha}}^{\vee}={\overline{\alpha}}^{\vee}$ but if
$2{\overline{\alpha}}\in\overline{R}$, we have
${\widehat{\alpha}}^{\vee}=\frac{1}{2}{\overline{\alpha}}^{\vee}=(2{\overline{\alpha}})^{\vee}$.
In particular, for $\alpha\in\Delta_{1}$, Proposition 4.3.5 above implies that

${\widehat{\alpha}}^{\vee}=\left\\{\begin{array}[]{ll}{\overline{\alpha}}^{\vee}&\textrm{
if $\langle\alpha^{\vee},\sigma(\alpha)\rangle\neq 1$},\\\
\frac{1}{2}{\overline{\alpha}}^{\vee}&\textrm{ if
$\langle\alpha^{\vee},\sigma(\alpha)\rangle=1$.}\\\ \end{array}\right.$

We will also need the following result proved in Lemma 6.11.

######  Lemma 4.4.

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

###  4.2 Divisors and restricted root system

We relate the Picard group of $X$ (viewed as the group of divisors up to
linear equivalence), to the restricted root system $\overline{R}$. We will
need some definitions from the theory of spherical varieties, we refer to
[Pe14] for further details. The variety $X$ is spherical: it is a normal
$G$-variety such that $B_{{\rm s}}$ has a dense orbit. This implies that
$B_{{\rm s}}$ acts on $X$ with finitely many orbits. In particular there are
finitely many prime $B_{{\rm s}}$-stable divisors in $X$. The boundary
$\partial X=X_{1}\cup\cdots\cup X_{r}$ with $r$ the rank of $X$ is the union
of the prime $G$-stable divisors. The prime $B_{{\rm s}}$-stable divisors
which are not $G$-stable are called colors. We denote by ${\mathcal{D}}_{X}$
the set of colors and by ${\mathcal{V}}_{X}=\\{X_{1},\ldots,X_{r}\\}$ the set
of prime $G$-stable divisors.

We start with a description of prime $G$-stable divisors. Let $i:Y\to X$ be
the inclusion of the closed $G$-orbit in $X$. Recall that $Y=G/P$ and that for
any character $\lambda$ of $P$, we have a homogeneous line bundle
${\mathcal{L}}_{Y}(\lambda)=G\times^{P}{\mathbb{C}}_{-\lambda}$ on $G/P$,
where ${\mathbb{C}}_{-\lambda}$ is the $1$-dimensional $P$-representation of
weight $-\lambda$. Let $B^{-}_{{\rm s}}$ be the Borel subgroup containing
$T_{{\rm s}}$ opposite to $B_{{\rm s}}$ and let $z\in Y$ be the unique
$B^{-}_{{\rm s}}$-stable point in $Y$. By [deCP83, Proposition 8.1 and
Corollary 8.2], we have the following result.

######  Proposition 4.5.

  1. 1.

The map $i^{*}:{\rm Pic}(X)\to{\rm Pic}(Y)$ is injective.

  2. 2.

For any $i\in[1,r]$, the torus $T_{{\rm s}}$ acts on $T_{z}X/T_{z}X_{i}$ with
weight ${\overline{\alpha}}_{i}\in\overline{\Delta}$.

  3. 3.

We have
${\mathcal{O}}_{X}(X_{i})|_{Y}={\mathcal{L}}_{Y}({\overline{\alpha}}_{i})$.

  4. 4.

The map
${\mathcal{V}}_{X}\to\overline{\Delta},X_{i}\mapsto{\overline{\alpha}}_{i}$ is
bijective.

######  Remark 4.6.

We set $X_{{\overline{\alpha}}_{i}}:=X_{i}$ for $i\in[1,r]$ so that
$X_{\overline{\beta}}$ is well defined for
${\overline{\beta}}\in\overline{\Delta}$.

Next we want to relate colors and restricted roots. This is more difficult,
since there may be more colors than restricted roots as Propositions 4.7 and
4.8 below show. The following proposition holds for any projective spherical
variety with a unique closed orbit (see [Pe14, Theorem 3.2.4])

######  Proposition 4.7.

We have ${\rm Pic}(X)=\bigoplus_{D\in{\mathcal{D}}_{X}}{\mathbb{Z}}[D]$.

The rank of the Picard group is determined as follows (see [deCP83, Theorem
7.6]).

######  Proposition 4.8.

We have ${\rm Pic}(X)={\mathbb{Z}}^{r+s}$ where $r$ is the rank of $X$ and $s$
is the number of restricted simple roots
${\overline{\gamma}}\in\overline{\Delta}$ such that there exists a pair of
exceptional simple roots $\alpha,\beta={\overline{\sigma}}(\alpha)$ with
${\overline{\alpha}}={\overline{\gamma}}={\overline{\beta}}$.

######  Remark 4.9.

In particular we have the following formula:

${\rm rk}({\rm Pic}(X))=\left\\{\begin{tabular}[]{ll}$r$&if $X$ is not
exceptional,\\\ $r+1$&if $X$ is exceptional.\\\ \end{tabular}\right.$

The proof of Proposition 4.8 in [deCP83, Theorem 7.6]) suggests a
correspondence between colors and restricted roots. We give a description of
this using results of Luna. For $\alpha\in\Delta$, recall that $G_{\alpha}$
denotes the subgroup of $G$ generated by $U_{\alpha}$ and $U_{-\alpha}$, and
set ${\mathcal{D}}_{X}(\alpha)=\\{D\in{\mathcal{D}}_{X}\ |\ G_{\alpha}\cdot
D\neq D\\}$. Note that if $\sigma(\alpha)=\alpha$, then
${\mathcal{D}}_{X}(\alpha)=\emptyset$ (see [Lu01]).

######  Proposition 4.10.

For any $\alpha\in\Delta_{1}$, the set ${\mathcal{D}}_{X}(\alpha)$ consists of
a unique element. Moreover, for $\alpha,\beta\in\Delta_{1}$, we have
${\mathcal{D}}_{X}(\alpha)={\mathcal{D}}_{X}(\beta)$ only if
$\langle\alpha^{\vee},\beta\rangle=0$ and $\sigma(\alpha)=-\beta$.

###### Proof.

This is a consequence of a result of Luna which holds true for any wonderful
variety (see [Lu01, Section 1.4]). Luna proves that three cases, called $(a)$,
$(a^{\prime})$ and $(b)$, occur. In cases $(a^{\prime})$ and $(b)$, the set
${\mathcal{D}}_{X}(\alpha)$ consists of a unique element, while in case $(a)$
the set ${\mathcal{D}}_{X}(\alpha)$ consists of two elements. We prove that
case $(a)$ does not occur: in this case, we have
$\alpha={\overline{\gamma}}=\gamma-\sigma(\gamma)$ for some $\gamma\in\Delta$.
Thus, $\sigma(\alpha)=-\alpha$ and
${\overline{\alpha}}=2\alpha=2{\overline{\gamma}}\in\overline{\Delta}$. In
particular, ${\overline{\gamma}},2{\overline{\gamma}}\in\overline{\Delta}$,
which contradicts the fact that $\overline{\Delta}$ is a basis of
$\overline{R}$.

For $\alpha,\beta\in\Delta_{1}$, there are, according to [Lu01, Proposition
3.2], the following possibilities to have
${\mathcal{D}}_{X}(\alpha)\cap{\mathcal{D}}_{X}(\beta)\neq\emptyset$:

  * •

Both $\alpha$ and $\beta$ are in $\overline{\Delta}$, in which case we may
have $|{\mathcal{D}}_{X}(\alpha)\cup{\mathcal{D}}_{X}(\beta)|=3$.

  * •

$\langle\alpha^{\vee},\beta\rangle=0$ and $\alpha+\beta\in\overline{\Delta}$
or $\frac{1}{2}(\alpha+\beta)\in\overline{\Delta}$.

The first case does not occur by the above argument. If
$\langle\alpha^{\vee},\beta\rangle=0$ and
$\frac{1}{2}(\alpha+\beta)\in\overline{\Delta}$ or
$\alpha+\beta\in\overline{\Delta}$, then there exists $\gamma\in\Delta_{1}$
such that
${\overline{\gamma}}=\gamma-\sigma(\gamma)=\frac{1}{2}(\alpha+\beta)$ or
${\overline{\gamma}}=\gamma-\sigma(\gamma)=\alpha+\beta$. Write

$\sigma(\gamma)+{\overline{\sigma}}(\gamma)=-\sum_{\delta\in\Delta_{0}}c_{\delta}\delta.$

Then, we have

$\frac{1}{2}(\alpha+\beta)=\gamma+{\overline{\sigma}}(\gamma)+\sum_{\delta\in\Delta_{0}}c_{\delta}\delta\
\ \ \textrm{ or }\ \ \
\alpha+\beta=\gamma+{\overline{\sigma}}(\gamma)+\sum_{\delta\in\Delta_{0}}c_{\delta}\delta.$

In the first case, this implies $\alpha=\beta$ and
$\gamma+{\overline{\sigma}}(\gamma)\leq\alpha$, which is impossible. In the
second case, we get that $\gamma$ equals $\alpha$ or $\beta$. Assume for
example that $\gamma=\alpha$, then we have
${\overline{\sigma}}(\alpha)={\overline{\sigma}}(\gamma)=\beta$ and
$c_{\delta}=0$ for all $\delta\in\Delta_{0}$. We thus have
$\sigma(\alpha)=\sigma(\gamma)=-{\overline{\sigma}}(\gamma)=-\beta$. Note that
${\overline{\alpha}}={\overline{\beta}}$. ∎

######  Remark 4.11.

If $X$ is not the wonderful compactification of an adjoint symmetric space,
then there may be some simple roots $\alpha\in\Delta_{1}$ with
$|{\mathcal{D}}_{X}(\alpha)|=2$. A typical example is the case
$G=\operatorname{SL}_{2}$ and $H=T$ a maximal torus. Then $N=N_{G}(T)$ is the
normalizer of the torus. There is a unique projective compactification of
$G/H$ given by ${\mathbb{P}}^{1}\times{\mathbb{P}}^{1}$ with two $B$-stable
divisors $D^{+}$ and $D^{-}$ and both are such that
$\operatorname{SL}_{2}\cdot D^{\pm}=X$. We thus have
${\mathcal{D}}_{X}(\alpha)=\\{D^{+},D^{-}\\}$, where $\alpha$ is the unique
simple root of $G$. The wonderful compactification $X_{\rm ad}$ of $G/N$ is
the quotient of $X$ by the involution exchanging the two factors and is
isomorphic to ${\mathbb{P}}^{2}$ with a unique $B$-stable divisor $D$, so that
${\mathcal{D}}_{X_{\rm ad}}(\alpha)=\\{D\\}$.

The restricted root system is
$\overline{R}=\\{-{\overline{\alpha}},{\overline{\alpha}}\\}$ in both cases.
But for a maximal split torus $T_{{\rm s}}$, we have
${\overline{\alpha}}\in\mathfrak{X}(T_{{\rm s}}/H\cap T_{{\rm s}})$ while
${\overline{\alpha}}\not\in\mathfrak{X}(T_{{\rm s}}/N\cap T_{{\rm s}})$ but
$2{\overline{\alpha}}\in\mathfrak{X}(T_{{\rm s}}/N\cap T_{{\rm s}})$.

By Proposition 4.10, we may define a map
$\zeta:\Delta_{1}\to{\mathcal{D}}_{X}$ by $\zeta(\alpha)=D$ with
$D\in{\mathcal{D}}_{X}(\alpha)$.

######  Lemma 4.12.

The map $\zeta:\Delta_{1}\to{\mathcal{D}}_{X}$ is surjective.

###### Proof.

For $D\in{\mathcal{D}}_{X}$, there exists $\alpha\in\Delta_{1}$ such that
$D\in{\mathcal{D}}_{X}(\alpha)$. Indeed, since $G$ is generated by the
$G_{\alpha}$ for $\alpha\in\Delta$ and since $D$ is not $G$-stable, there
exists at least one $\alpha\in\Delta$ with $G_{\alpha}\cdot D\neq D$.
Furthermore, by [Lu01], we have $\alpha\not\in\Delta_{0}$; thus,
$\alpha\in\Delta_{1}$. ∎

######  Corollary 4.13.

The map
$\tau:{\mathcal{D}}_{X}\to\overline{\Delta},D\mapsto{\overline{\alpha}}$, with
$\alpha\in\Delta_{1}$ such that $D\in{\mathcal{D}}_{X}(\alpha)$, is well
defined and surjective.

###### Proof.

The restricted root ${\overline{\alpha}}$ does not depend on the choice of
$\alpha$ with $D\in{\mathcal{D}}_{X}(\alpha)$: if
$D\in{\mathcal{D}}_{X}(\alpha)\cap{\mathcal{D}}_{X}(\beta)$, then
$\beta=-\sigma(\alpha)$ and ${\overline{\beta}}={\overline{\alpha}}$ by
Proposition 4.10. For the surjectivity, note that the composition
$\tau\circ\zeta$ is surjective since
$\tau\circ\zeta(\alpha)={\overline{\alpha}}$. ∎

######  Proposition 4.14.

The map $\tau$ is injective except if $X$ is exceptional, in which case the
only non-trivial fiber is
$\tau^{-1}({\overline{\alpha}})=\\{\zeta(\alpha),\zeta(\beta)\\}$, where
$\alpha,\beta$ are the exceptional roots.

###### Proof.

Assume first that $\alpha$ is exceptional and set
$\beta={\overline{\sigma}}(\alpha)$. Then
$\tau(\zeta(\alpha))={\overline{\alpha}}={\overline{\beta}}=\tau(\zeta(\beta))$.
If $\zeta(\alpha)=\zeta(\beta)$, then
${\mathcal{D}}_{X}(\alpha)\cap{\mathcal{D}}_{X}(\beta)\neq\emptyset$. We thus
have $\beta=-\sigma(\alpha)$ and
$\langle\alpha^{\vee},\sigma(\alpha)\rangle=-\langle\alpha^{\vee},\beta\rangle=0$,
a contradiction with the fact that $\alpha$ is exceptional. Therefore,
$\zeta(\alpha)\neq\zeta(\beta)$ and $\tau$ is not injective.

On the other hand, the map $\tau$ is surjective and $|{\mathcal{D}}_{X}|={\rm
rk}({\rm Pic}(X))=r+s=|\overline{\Delta}|+s$ with $s$ the number of pairs of
exceptional roots $(\alpha,{\overline{\sigma}}(\alpha))$. The result follows
from this. ∎

######  Corollary 4.15.

Let $\alpha,\beta\in\Delta_{1}$ with $\alpha\neq\beta$. Then
$\zeta(\alpha)=\zeta(\beta)$ if and only if $\beta=-\sigma(\alpha)$ and
$\langle\alpha^{\vee},\beta\rangle=0$.

###### Proof.

Assume that $\zeta(\alpha)=\zeta(\beta)$, then
${\mathcal{D}}_{X}(\alpha)\cap{\mathcal{D}}_{X}(\beta)\neq\emptyset$ and the
result follows from Proposition 4.10. Conversely, if $\beta=-\sigma(\alpha)$
and $\langle\alpha^{\vee},\beta\rangle=0$, then
${\overline{\alpha}}={\overline{\beta}}$. If $\zeta(\alpha)\neq\zeta(\beta)$,
then $(\alpha,\beta)$ is a pair of exceptional roots. In particular, we would
have
$0\neq\langle\alpha^{\vee},\sigma(\alpha)\rangle=-\langle\alpha^{\vee},\beta\rangle=0$,
a contradiction. Thus $\zeta(\alpha)=\zeta(\beta)$. ∎

######  Remark 4.16.

We get a characterisation of simple roots having the same restricted root: For
$\alpha,\beta\in\Delta_{1}$ with $\alpha\neq\beta$, we have
${\overline{\alpha}}={\overline{\beta}}$ if and only if either
($(\alpha,\beta)$ is a pair of exceptional roots with
${\overline{\sigma}}(\alpha)=\beta$), or ($\beta=-\sigma(\alpha)$ and
$\langle\alpha^{\vee},\beta\rangle=0$).

We now compute the restrictions $i^{*}{\mathcal{O}}_{X}(D)$ for
$D\in{\mathcal{D}}_{X}$. The next proposition is a direct application of
results in [Lu97]. For $\alpha\in\Delta$, let $\varpi_{\alpha}$ be the
fundamental weight associated to $\alpha$.

######  Proposition 4.17.

Let $\alpha\in\Delta_{1}$ and let $\lambda_{\alpha}\in\mathfrak{X}(T_{{\rm
s}})$ be such that
$i^{*}{\mathcal{O}}_{X}(\zeta(\alpha))={\mathcal{L}}_{Y}(\lambda_{\alpha})$.
Then we have

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

###### Proof.

For $\beta\in\Delta_{1}$, let $Y_{\beta}\subset Y$ be the Schubert curve dual
to ${\mathcal{O}}_{Y}(\varpi_{\beta})$. Then by [Lu97, Lemma 3.1.1 and Lemma
3.1.2], we have

$Y_{\beta}\cdot
i^{*}\zeta(\alpha)=\left\\{\begin{array}[]{ll}2\delta_{\alpha,\beta}&\textrm{
if $\sigma(\alpha)=-\alpha$,}\\\ \delta_{\zeta(\alpha),\zeta(\beta)}&\textrm{
otherwise.}\\\ \end{array}\right.$

The result follows from this and from the facts that if
$|\zeta^{-1}(\zeta(\alpha))|>1$, then
$\zeta^{-1}(\zeta(\alpha))=\\{\alpha,{\overline{\sigma}}(\alpha)\\}$ and that,
by Corollary 4.15, this occurs if and only if
$\sigma(\alpha)=-{\overline{\sigma}}(\alpha)$ and
$\langle\alpha^{\vee},\sigma(\alpha)\rangle=0$. ∎

The above proof implicitly uses the fact that there exists a family of
irreducible $B$-stable curves $(C_{D})_{D\in{\mathcal{D}}_{X}}$ such that the
classes $[C_{D}]\in A_{1}(X)$ form the dual basis to the basis
$([D])_{D\in{\mathcal{D}}_{X}}$ of ${\rm Pic}(X)$ (see [Lu97, Lemma 3.1.2]).
Recall the definition of ${\widehat{\alpha}}$ for
${\overline{\alpha}}\in\overline{\Delta}$ and the notation
$X_{\overline{\beta}}$ for ${\overline{\beta}}\in\overline{\Delta}$ from
Remark 4.6.

######  Corollary 4.18.

We have $X_{\overline{\beta}}\cdot
C_{\zeta(\alpha)}=\langle{\widehat{\alpha}}^{\vee},{\overline{\beta}}\rangle$
for all $\alpha,\beta\in\Delta_{1}$.

###### Proof.

Recall that
$i^{*}{\mathcal{O}}_{X}(X_{\overline{\beta}})={\mathcal{L}}_{Y}({\overline{\beta}})$.
Note that we have
${\overline{\beta}}=\sum_{\gamma\in\Delta}\langle\gamma^{\vee},{\overline{\beta}}\rangle\varpi_{\gamma}$.
We get

$X_{\overline{\beta}}\cdot
C_{\zeta(\alpha)}=\left\\{\begin{array}[]{ll}\frac{1}{2}\langle\alpha^{\vee},{\overline{\beta}}\rangle&\textrm{
if $\sigma(\alpha)=-\alpha$,}\\\
\langle\alpha^{\vee},{\overline{\beta}}\rangle&\textrm{ otherwise.}\\\
\end{array}\right.$

Note that
$\langle\alpha^{\vee},{\overline{\beta}}\rangle=\langle-\sigma(\alpha)^{\vee},{\overline{\beta}}\rangle$
so that if $\sigma(\alpha)=-{\overline{\sigma}}(\alpha)$ and
$\langle\alpha^{\vee},\sigma(\alpha)\rangle=0$, we have
$\langle\alpha^{\vee},{\overline{\beta}}\rangle\varpi_{\alpha}+\langle-\sigma(\alpha)^{\vee},{\overline{\beta}}\rangle\varpi_{-\sigma(\alpha)}=\langle\alpha^{\vee},{\overline{\beta}}\rangle\lambda_{\alpha}$.

We now compare the above values to
$\langle{\widehat{\alpha}}^{\vee},{\overline{\beta}}\rangle$. If
$\sigma(\alpha)=-\alpha$, then
${\widehat{\alpha}}^{\vee}={\overline{\alpha}}^{\vee}=\frac{1}{2}\alpha^{\vee}$
proving the first case. If $\sigma(\alpha)\neq-\alpha$, we have two
possibilities: either $\langle\alpha^{\vee},\sigma(\alpha)\rangle=0$ or
$\langle\alpha^{\vee},\sigma(\alpha)\rangle=1$. In the former case, we have
${\widehat{\alpha}}^{\vee}={\overline{\alpha}}^{\vee}=\frac{1}{2}(\alpha^{\vee}-\sigma(\alpha)^{\vee})$;
thus,
$\langle{\widehat{\alpha}}^{\vee},{\overline{\beta}}\rangle=\frac{1}{2}(\langle\alpha^{\vee},{\overline{\beta}}\rangle+\langle-\sigma(\alpha)^{\vee},{\overline{\beta}}\rangle)=\langle\alpha^{\vee},{\overline{\beta}}\rangle$.
Finally, if $\langle\alpha,\sigma(\alpha)\rangle=1$, then
${\widehat{\alpha}}^{\vee}=\frac{1}{2}{\overline{\alpha}}^{\vee}=\frac{1}{2}(\alpha^{\vee}-\sigma(\alpha)^{\vee})$
and the result follows as before. ∎

Recall that ${\rm Pic}(X)=\oplus_{D\in{\mathcal{D}}_{X}}{\mathbb{Z}}[D]$.
Furthermore, by [Pe14, Theorem 3.2.9] the monoid ${\rm Nef}(X)$ of nef
divisors is given by ${\rm
Nef}(X)=\oplus_{D\in{\mathcal{D}}_{X}}{\mathbb{Z}}_{\geq 0}[D]$. It coincides
with the monoid of globally generated divisors. On the curve side, we have
$A_{1}(X)=\oplus_{D\in{\mathcal{D}}_{X}}{\mathbb{Z}}[C_{D}]$ and the monoid of
effective classes of curves ${\rm NE}(X)$ is given by ${\rm
NE}(X)=\oplus_{D\in{\mathcal{D}}_{X}}{\mathbb{Z}}_{\geq 0}[C_{D}]$ (see [Pe18]
for more on curves on spherical varieties). Furthermore, we have a
${\mathbb{Z}}$-linear map

$\psi:A_{1}(X)\to{\mathbb{Z}}\overline{\Delta}^{\vee}$

defined by $\psi(C_{D})={\widehat{\alpha}}^{\vee}$, for
$D\in{\mathcal{D}}_{X}(\alpha)$. By Corollary 4.18, we have
$X_{\overline{\beta}}\cdot C=\langle\psi(C),{\overline{\beta}}\rangle$ for all
$[C]\in{\rm NE}(X)$.

######  Proposition 4.19.

The map $\psi:A_{1}(X)\to{\mathbb{Z}}\overline{\Delta}^{\vee}$ is surjective.

  1. 1.

The image of the monoid of effective curves is the monoid spanned by coroots.

  2. 2.

The image of the monoid of curves having non-negative intersection with any
component of $\partial X_{\rm ad}$ is the intersection of
${\mathbb{Z}}\overline{\Delta}^{\vee}$ with the monoid of dominant
cocharacters.

###### Proof.

The surjectivity follows from the surjectivity of $\tau$. The monoid of
effective curves is spanned by the set $\\{C_{D}\ |\ D\in{\mathcal{D}}_{X}\\}$
whose image by $\psi$ is $\overline{\Delta}^{\vee}$; this proves (1). Part (2)
follows from Corollary 4.18 ∎

Recall that a curve class $\gamma\in A_{1}(X)$ is covering if there exists a
curve $C$ of class $\gamma$ passing through a general point $x\in X$. Note
that this implies that $\gamma\cdot X_{\overline{\beta}}\geq 0$ for all
${\overline{\beta}}\in\overline{\Delta}$. We call a class $\gamma\in{\rm
NE}(X)$ _virtually covering_ if $X_{\overline{\beta}}\cdot\gamma\geq 0$ for
all ${\overline{\beta}}\in\overline{\Delta}$.

######  Corollary 4.20.

  1. 1.

If $X$ is non-exceptional, then there is a unique virtually covering curve
class $\gamma_{0}\in{\rm NE}(X)$ which is minimal in this monoid. Moreover, we
have $\psi(\gamma)={\overline{\Theta}}^{\vee}$.

  2. 2.

If $X$ is exceptional, then there are exactly two minimal virtually covering
curve classes $\gamma^{+}_{0},\gamma^{-}_{0}\in{\rm NE}(X)$ and we have
$\psi(\gamma^{+}_{0})={\overline{\Theta}}^{\vee}=\psi(\gamma^{-}_{0})$.

###### Proof.

The image by $\psi$ of an effective and virtually covering curve class is in
the intersection of the monoid generated by coroots in $\overline{R}$ and the
dominant chamber. There is a unique minimal such element: the coroot of the
highest root of $\overline{R}$. Since ${\overline{\Theta}}$ is the highest
root of $\overline{R}$ by Proposition 4.3.1, the element
${\overline{\Theta}}^{\vee}$ is the smallest possible image by $\psi$ of an
effective and virtually covering curve class.

If $X$ is non-exceptional, then $\psi$ is injective and this proves (1). To
prove (2), we are left to prove that if $X$ is exceptional, there are exactly
two classes $\gamma^{+}$ and $\gamma^{-}$ in ${\rm NE}(X)$ such that
$\psi(\gamma^{+})={\overline{\Theta}}^{\vee}=\psi(\gamma^{-})$. But the kernel
of $\psi$ is
${\mathbb{Z}}([C_{D_{\alpha}}]-[C_{D_{{\overline{\sigma}}(\alpha)}}])$ with
$\alpha$ an exceptional root. Since the coefficient of
${\widehat{\alpha}}^{\vee}$ in ${\overline{\Theta}}^{\vee}$ is $1$ by Lemma
4.4, there are exactly two classes in ${\rm NE}(X)$ that are mapped to
${\overline{\Theta}}^{\vee}$ via $\psi$, namely, $\gamma^{+}$ with coefficient
$1$ in $C_{\alpha}$ and $0$ on $C_{{\overline{\sigma}}(\alpha)}$, and
$\gamma^{-}$ with coefficient $0$ in $C_{\alpha}$ and $1$ on
$C_{{\overline{\sigma}}(\alpha)}$. ∎

###  4.3 Curves classes of the minimal families

In this subsection, we prove that the curve classes $\gamma_{0}$,
$\gamma_{0}^{+}$ and $\gamma_{0}^{-}$ are covering and are therefore the
classes of minimal rational curves on $X$.

We will need a few more results on $X$. Recall that $x\in X^{0}$ is our base
point and that $r$ is the rank of $X$. The $G$-orbits in $X$ are indexed by
the subsets $I\subset[1,r]$ via ${\mathcal{O}}_{I}=\\{x^{\prime}\in X\ |\
x^{\prime}\in X_{i}\Leftrightarrow i\in I\\}$.

The local structure theorem associated to the closed orbit $Y$ gives the
following: there exists an affine $P$-stable open subset $X_{Y,B}\subset X$
containing $x$ with $X_{Y,B}\cap Y\neq\emptyset$ and a $P$-equivariant
isomorphism $X_{Y,B}\simeq\operatorname{R}_{u}(P)\times{\mathbb{A}}^{r}$. The
closure of $T_{{\rm s}}\cdot x=S\cdot x$ in $X_{Y,B}$ is $T_{{\rm
s}}$-equivariantly isomorphic to ${\mathbb{A}}^{r}$, where the torus $T_{{\rm
s}}$ acts linearly with weights
$\overline{\Delta}=({\overline{\alpha}}_{i})_{i\in[1,r]}$. The prime
$G$-divisor $X_{{\overline{\alpha}}_{i}}$ is defined in $X_{Y,B}$ by the
vanishing of the coordinate with weight ${\overline{\alpha}}_{i}$ in
${\mathbb{A}}^{r}$.

Recall that $i:Y\to X$ denotes the inclusion of the closed $G$-orbit and that
the map $i^{*}:{\rm Pic}(X)\to{\rm Pic}(Y)$ is injective. Let

$\mathfrak{X}_{X}(T_{{\rm s}})=\\{\lambda\in\mathfrak{X}(T_{{\rm s}})\ |\
{\mathcal{L}}_{Y}(\lambda)\in{\rm Im}\,i^{*}{\rm Pic}(X)\subset{\rm
Pic}(Y)\\}.$

For $\lambda\in\mathfrak{X}_{X}(T_{{\rm s}})$, we write
${\mathcal{L}}_{X}(\lambda)$ for the line bundle such that
$i^{*}{\mathcal{L}}_{X}(\lambda)={\mathcal{L}}_{Y}(\lambda)$ (see [deCP83, End
of 8.1] for these results).

Given a cocharacter
$\eta^{\vee}:{\mathbb{G}}_{m}\to\overline{S}=S/S^{\sigma}$, we say that
$\eta^{\vee}$ is dominant if
$\langle\eta^{\vee},{\overline{\alpha}}\rangle\geq 0$ for all
$\alpha\in\Delta$. A cocharacter $\eta^{\vee}$ defines a map
${\mathbb{C}}^{\times}\to X,t\mapsto\eta^{\vee}(t)\cdot x$. This map extends
to a morphism $\eta^{\vee}:{\mathbb{P}}^{1}\to X$.

The following lemma generalizes [BF15, Lemma 3.1] to the case of wonderful
compactifications of adjoint irreducible symmetric spaces.

######  Lemma 4.21.

Let $\eta^{\vee}:{\mathbb{G}}_{m}\to S$ be a dominant cocharacter,
$\eta^{\vee}:{\mathbb{P}}^{1}\to X$ the corresponding morphism, and
$C_{\eta^{\vee}}$ its image.

  1. 1.

We have $\eta^{\vee}(0)\in{\mathcal{O}}_{I}$, where $I:=\\{i\in[1,r]\ |\
\langle\eta^{\vee},{\overline{\alpha}}_{i}\rangle\neq 0\\}$.

  2. 2.

We have $\eta^{\vee}(\infty)\in{\mathcal{O}}_{J}$, where $J:=\\{j\in[1,r]\ |\
\langle\eta^{\vee},w_{0}({\overline{\alpha}}_{j})\rangle\neq 0\\}$.

  3. 3.

The morphism $\eta^{\vee}:{\mathbb{P}}^{1}\to C_{\eta^{\vee}}$ is an
isomorphism if and only if there exists $i\in[1,r]$ such that
$\langle\eta^{\vee},{\overline{\alpha}}_{i}\rangle=1$.

  4. 4.

For $\lambda\in\mathfrak{X}_{X}(T_{{\rm s}})$, we have ${\rm
deg}(\eta^{\vee})^{*}{\mathcal{L}}_{X}(\lambda)=\langle\eta^{\vee},\lambda-
w_{0}\lambda\rangle$.

###### Proof.

(1) Since $\eta^{\vee}$ is dominant, its extends to a morphism
${\mathbb{A}}^{1}\to X_{Y,B}\cap\overline{T}$ defined by
$t\mapsto(t^{\langle\eta^{\vee},{\overline{\alpha}}_{i}\rangle})_{i\in[1,r]}$,
where $X_{Y,B}\cap\overline{T}$ is identified with ${\mathbb{A}}^{r}$ as
above. In particular, $\eta^{\vee}(0)\in X_{Y,B}$ and vanishes on the
coodinates with indices in $\\{i\in[1,r]\ |\
\langle\eta^{\vee},{\overline{\alpha}}_{i}\rangle\neq 0\\}$. Moreover, the
morphism $\eta^{\vee}:{\mathbb{P}}^{1}\to C_{\eta^{\vee}}$ is a local
isomorphism at $\eta^{\vee}(0)$ if and only if there exists $i$ such that
$\langle\eta^{\vee},{\overline{\alpha}}_{i}\rangle=1$.

(2) Consider the open affine subset $w_{0}\cdot X_{Y,B}$ of $X$. It is
isomorphic to $\operatorname{R}_{u}(P)^{w_{0}}\times{\mathbb{A}}^{r}$ with a
linear action of $T_{{\rm s}}$ on ${\mathbb{A}}^{r}$ with weights
$w_{0}(\overline{\Delta})$. All these weights are non-negative linear
combinations of negative roots. In particular, the one-parameter subgroup
$-\eta^{\vee}$ acts with non-negative weights on ${\mathbb{A}}^{r}$, and hence
extends to a morphism ${\mathbb{A}}^{1}\to w_{0}\cdot X_{Y,B}$,
$t\mapsto(t^{\langle\eta^{\vee},-w_{0}({\overline{\alpha}}_{i})\rangle})_{i\in[1,r]}$.
It follows that $\eta^{\vee}(\infty)=(-\eta^{\vee})(0)\in w_{0}\cdot X_{Y,B}$
and as above, $\eta^{\vee}(\infty)\in{\mathcal{O}}_{J}$. Moreover, the
morphism $\eta^{\vee}:{\mathbb{P}}^{1}\to C_{\eta^{\vee}}$ is a local
isomorphism at $\eta^{\vee}(\infty)$ if and only if there exists $i$ such that
$\langle\eta^{\vee},-w_{0}({\overline{\alpha}}_{i})\rangle=1$. Note that
$-w_{0}({\overline{\alpha}}_{i})=-w_{0}(\alpha_{i})+w_{0}(\sigma(\alpha_{i}))=-\overline{w_{0}(\alpha_{i})}$
and since $-w_{0}$ permutes $\Delta_{1}$, by Proposition 4.3.1, the previous
condition is true if and only if there exists $i$ such that
$\langle\eta^{\vee},{\overline{\alpha}}_{i}\rangle=1$.

3\. Follows from the above conditions at $\eta^{\vee}(0)$ and
$\eta^{\vee}(\infty)$.

4\. The pull-back of ${\mathcal{L}}_{X}(\lambda)$ to $X_{Y,B}$ (resp.
$w_{0}\cdot X_{Y,B}$) has a trivializing section of weight $\lambda$ (resp.
$w_{0}(\lambda)$). As a consequence, the line bundle
$(\eta^{\vee})^{*}{\mathcal{L}}_{X}(\lambda)$ is a
${\mathbb{G}}_{m}$-linearized line bundle on ${\mathbb{P}}^{1}$ with weights
$\langle\eta^{\vee},\lambda\rangle$ at $0$ and
$\langle\eta^{\vee},w_{0}\lambda\rangle$ at $\infty$. Since the degree of such
a line bundle is the difference of its weights, this yields our assertion. ∎

We now apply the above result to $\eta^{\vee}={\overline{\Theta}}^{\vee}$.

######  Corollary 4.22.

Consider the morphism ${\overline{\Theta}}^{\vee}:{\mathbb{P}}^{1}\to X$.

  1. 1.

If $\overline{R}$ is not of type ${\rm A}_{1}$, then
${\overline{\Theta}}^{\vee}$ is an isomorphism onto its image.

  2. 2.

If $\overline{R}$ is of type ${\rm A}_{1}$, then ${\overline{\Theta}}^{\vee}$
has degree $2$ over its image.

  3. 3.

The push-forward class is given as follows:

${\overline{\Theta}}^{\vee}_{*}[{\mathbb{P}}^{1}]=\left\\{\begin{array}[]{ll}2\gamma_{0}&\textrm{
if $X$ is non-exceptional,}\\\ \ \\!\\!\gamma_{0}^{+}+\gamma_{0}^{-}&\textrm{
if $X$ is exceptional.}\\\ \end{array}\right.$

###### Proof.

(1) If $\overline{R}$ is not of type $\textrm{A}_{1}$, then Proposition 4.3.4
implies that there exists a simple root
${\overline{\alpha}}\in\overline{\Delta}$ such that
$\langle{\overline{\Theta}}^{\vee},{\overline{\alpha}}\rangle=1$. Therefore,
${\overline{\Theta}}^{\vee}:{\mathbb{P}}^{1}\to X$ is an isomorphism onto its
image.

(2) If $\overline{R}$ is of type $\textrm{A}_{1}$, then
${\overline{\Theta}}^{\vee}$ induces a map ${\mathbb{G}}_{m}\to
X_{Y,B}\cap\overline{T}={\mathbb{A}}^{1},t\mapsto t^{2}$ which is of degree
$2$ onto its image.

(3) For $\lambda\in\mathfrak{X}_{X}(T_{{\rm s}})$, we have
$[{\mathcal{L}}_{X}(\lambda)]\cdot{\overline{\Theta}}^{\vee}_{*}[{\mathbb{P}}^{1}]={\rm
deg}({\overline{\Theta}}^{\vee})^{*}{\mathcal{L}}_{X}(\lambda)=\langle{\overline{\Theta}}^{\vee},\lambda-
w_{0}(\lambda)\rangle=\langle{\overline{\Theta}}^{\vee},\lambda\rangle-\langle
w_{0}({\overline{\Theta}})^{\vee},\lambda\rangle=2\langle{\overline{\Theta}}^{\vee},\lambda\rangle$
by Proposition 4.3.1. This proves the result for $X$ non-exceptional, since
$X_{\overline{\beta}}\cdot
C_{0}=\langle{\overline{\Theta}}^{\vee},{\overline{\beta}}\rangle$,
${\mathcal{O}}_{X}(X_{\overline{\beta}})={\mathcal{L}}_{X}({\overline{\beta}})$
and $({\overline{\beta}})_{{\overline{\beta}}\in\overline{\Delta}}$ generates
$\mathfrak{X}_{X}(T_{{\rm s}})\otimes_{\mathbb{Z}}{\mathbb{Q}}$.

If $X$ is exceptional, then by the same argument, we have that the class of
the image and the class $\gamma_{0}^{+}+\gamma_{0}^{-}$ agree on all boundary
divisors. We therefore only need to check that they agree on $D_{\alpha}$ for
$\alpha$ an exceptional simple root. Assume that $\gamma_{0}^{+}$ is dual to