1.2pt

# Radiant toric varieties and unipotent group actions

Ivan Arzhantsev  National Research University Higher School of Economics,
Faculty of Computer Science, Pokrovsky Boulevard 11, Moscow, 109028 Russia
<EMAIL_ADDRESS>,  Alexander Perepechko  Kharkevich Institute for Information
Transmission Problems, 19 Bolshoy Karetny per., Moscow, 127994 Russia
National Research University Higher School of Economics, Faculty of Computer
Science, Pokrovsky Boulevard 11, Moscow, 109028 Russia<EMAIL_ADDRESS>and  Kirill
Shakhmatov  National Research University Higher School of Economics, Faculty
of Computer Science, Pokrovsky Boulevard 11, Moscow, 109028 Russia
<EMAIL_ADDRESS>
###### Abstract.

We consider complete toric varieties $X$ such that a maximal unipotent
subgroup $U$ of the automorphism group $\mathrm{Aut}(X)$ acts on $X$ with an
open orbit. It turns out that such varieties can be characterized by several
remarkable properties. We study the set of Demazure roots of the corresponding
complete fan, describe the structure of a maximal unipotent subgroup $U$ in
$\mathrm{Aut}(X)$, and find all regular subgroups in $U$ that act on $X$ with
an open orbit.

###### Key words and phrases:

Toric variety, polyhedral fan, automorphism group, maximal unipotent subgroup,
Demazure root, additive action, locally nilpotent derivation

###### 2010 Mathematics Subject Classification:

Primary 14J50, 14M25; Secondary 14M17, 20G07, 32M12

Supported by the grant RSF-DST 22-41-02019

## Introduction

Let $X$ be a complete toric variety with an acting torus $T$. All varieties
and actions are defined over an algebraically closed field $\mathbb{K}$ of
characteristic zero. It is well known that the automorphism group
$\mathrm{Aut}(X)$ is a linear algebraic group and $T$ is a maximal torus in
$\mathrm{Aut}(X)$. An algebraic subgroup $H$ in $\mathrm{Aut}(X)$ is called
_regular_ if $H$ is normalized by the torus $T$.

We say that $X$ is _radiant_ if a maximal unipotent subgroup $U$ of the
automorphism $\mathrm{Aut}(X)$ acts on $X$ with an open orbit. The aim of this
paper is to show that radiant toric varieties have important distinguishing
features that make this class of varieties attractive for study.

Let $\mathbb{G}_{a}$ and $\mathbb{G}_{m}$ stand for the additive and the
multiplicative group of the ground field $\mathbb{K}$, respectively. The
groups $\mathbb{G}_{a}$ and $\mathbb{G}_{m}$ are commutative one-dimensional
linear algebraic groups, and $\mathbb{G}_{a}$ is unipotent. Moreover, any
commutative unipotent linear algebraic group is isomorphic to the vector group
$\mathbb{G}_{a}^{n}=\mathbb{G}_{a}\times\ldots\times\mathbb{G}_{a}$ ($n$
times). An _additive action_ on a complete algebraic variety $X$ is a faithful
regular action $\mathbb{G}_{a}^{n}\times X\to X$ with an open orbit. Since the
group $\mathbb{G}_{a}^{n}$ is isomorphic to the affine space $\mathbb{A}^{n}$
as an algebraic variety, we obtain an open embedding of $\mathbb{A}^{n}$ into
$X$ such that the action of $\mathbb{G}_{a}^{n}$ on $\mathbb{A}^{n}$ by
translations extends to an action on $X$. This shows that a variety with an
additive action is an equivariant completion of the affine space, and vice
versa.

If $X$ is toric we say that an additive action $\mathbb{G}_{a}^{n}\times X\to
X$ is _normalized_ if the image of the group $\mathbb{G}_{a}^{n}$ in
$\mathrm{Aut}(X)$ is a regular subgroup. Let us say that a regular
$\mathbb{G}_{a}$-subgroup in $\mathrm{Aut}(X)$ is a _root_ subgroup. It is
easy to see that a normalized additive action is an additive action given by
$n$ pairwise commuting root subgroups in $\mathrm{Aut}(X)$.

It is well known that any complete toric variety $X$ corresponds to a complete
fan $\Sigma=\Sigma_{X}$ in the lattice $N$ of one-parameter subgroups of the
acting torus $T$. Let $\rho_{1},\ldots,\rho_{m}$ be the rays of the fan
$\Sigma$. We say that the fan $\Sigma$ is _bilateral_ if there is a basis
$p_{1},\ldots,p_{n}$ of the lattice $N$ such that, up to renumbering, the rays
$\rho_{1},\ldots,\rho_{n}$ are generated by the vectors $p_{1},\ldots,p_{n}$,
respectively, and the remaining rays $\rho_{n+1},\ldots,\rho_{m}$ lie in the
negative orthant with respect to this basis.

The following result is proved in [3, Theorem 4.1 and Corollary 1].

######  Theorem A.

Let $X$ be a complete toric variety. The following conditions are equivalent:

  1. (i)

the variety $X$ is radiant;

  2. (ii)

the variety $X$ admits an additive action;

  3. (iii)

the variety $X$ admits a normalized additive action;

  4. (iv)

the fan $\Sigma_{X}$ is bilateral.

In particular, radiant toric varieties are precisely toric equivariant
completions of affine spaces, cf. [4].

One more characterization of radiant projective toric varieties was proposed
by Fu and Hwang [12]. Let $X\subseteq\mathbb{P}(V)$ be a closed subvariety.
For a nonsingular point $x\in X$, a $\mathbb{G}_{m}$-action on $X$ coming from
a multiplicative subgroup of $\mathrm{GL}(V)$ is said to be of _Euler type_ at
$x$ if $x$ is an isolated fixed point of the restricted
$\mathbb{G}_{m}$-action on $X$ and the induced $\mathbb{G}_{m}$-action on the
tangent space $T_{x}X$ is by scalar multiplication. The subvariety
$X\subseteq\mathbb{P}(V)$ is _Euler-symmetric_ if for a general point $x\in
X$, there exists a $\mathbb{G}_{m}$-action on $X$ of Euler type at $x$. It is
proved in [12, Theorem 3.7(i)] that every Euler-symmetric variety admits an
additive action. If $X$ is a projective toric variety, it is shown in [16,
Theorem 3] that $X$ admits an additive action if and only if $X$ is Euler-
symmetric with respect to some embedding into a projective space. Moreover, in
this case $X$ is Euler-symmetric with respect to any linearly non-degenerate
linearly normal embedding into a projective space.

Lattice polytopes that correspond to projective radiant toric varieties are
described in [3, Theorem 5.2]. Examples of non-projective radiant toric
varieties are given in [17]; see also Example 4.7 below.

It is shown in [3, Theorem 3.6] that a normalized additive action on a radiant
toric variety is unique up to isomorphism. Moreover, it follows from [11,
Corollary 4] that any additive action on a radiant toric variety is isomorphic
to a normalized additive action if and only if the maximal unipotent subgroup
$U$ in $\mathrm{Aut}(X)$ is commutative. In this case $U$ is the only
candidate up to conjugation for a subgroup in $\mathrm{Aut}(X)$ defining an
additive action on $X$.

Let us describe the content of the paper. In Section 1 we discuss basic facts
on toric varieties and Cox rings and recall a description of the automorphism
group of a complete toric variety due to Demazure. Section 2 is devoted to the
study of Demazure roots of a bilateral fan. Unlike classical root systems, the
set of Demazure roots of a complete polyhedral fan may seem rather random and
unsymmetric. However, a more detailed analysis shows that in the case of a
bilateral fan, the set of Demazure roots has many nice properties. The study
of such properties was started by Dzhunusov [10, 11]. We develop these results
and find new features of Demazure roots. With any bilateral fan $\Sigma$ we
associate its ray matrix $A$. We show that Demazure roots of $\Sigma$ admit an
explicit description in terms of columns of the matrix $A$. This allows both
to study properties of Demazure roots and to construct bilateral fans with
prescribed Demazure roots.

In Section 3, using results on Demazure roots, we give a description of a
maximal unipotent subgroup of the automorphism group $\mathrm{Aut}(X)$ of a
radiant toric variety $X$ as a semidirect product of some explicitly presented
unipotent groups. The next step is to find all regular unipotent subgroups in
$\mathrm{Aut}(X)$ which act on $X$ with an open orbit. It is done in Section
4. We define a _principal unipotent subgroup_ of $\mathrm{Aut}(X)$ as a
regular commutative unipotent subgroup that acts on $X$ with an open orbit. It
is proved that a regular unipotent subgroup $U$ in $\mathrm{Aut}(X)$ acts on
$X$ with an open orbit if and only if $U$ contains a principal unipotent
subgroup. We come to considering two types of radiant toric varieties: the
ones that admit a faithful action of a non-commutative unipotent group and the
ones that do not. Specific properties of radiant toric varieties of each type
are studied.

In Section 5 we compute the center of a regular unipotent subgroup $U$ acting
on $X$ with an open orbit. With any such $U$ we associate a directed graph
$\Gamma$ and describe the central series and the derived subgroups of $U$ in
terms of $\Gamma$. There results allow to find the nilpotency class and the
derived length of a regular unipotent subgroup.

It is shown in Section 6 that the open orbit of a non-commutative regular
unipotent subgroup has smaller dimension than the subgroup itself. This
imposes certain restrictions on equivariant toric completions of unipotent
groups. Finally, in Section 7 we describe smooth radiant toric surfaces. It
turns out that the Picard number of a smooth non-radiant toric surface is at
least 4. We expect further classification results on smooth radiant toric
varieties of small dimensions.

##  1\. Preliminaries

In this section we recall basic facts on toric varieties. Hereinafter we fix
an algebraically closed field $\mathbb{K}$ of characteristic zero. By word
‘variety’ we mean a separated algebraic variety over $\mathbb{K}$.

###  1.1. Toric varieties and Cox rings

Let $T=(\mathbb{K}^{\times})^{n}$ be an algebraic torus of rank $n$. A normal
irreducible variety $X$ is called _toric_ , if there is a faithful action of
$T$ on $X$ with an open orbit. There is a well-known combinatorial description
of toric varieties. We briefly recall it in this subsection and refer to
standard textbooks [8, 13] on toric varieties for details.

Let $N\cong\mathbb{Z}^{n}$ be the lattice of one-parameter subgroups of $T$
and $M=\mathrm{Hom}_{\mathbb{Z}}(N,\mathbb{Z})\cong\mathbb{Z}^{n}$ be the
character lattice of $T$. Denote by $N_{\mathbb{Q}}$ and $M_{\mathbb{Q}}$ the
associated $\mathbb{Q}$-vector spaces $N\otimes_{\mathbb{Z}}\mathbb{Q}$ and
$M\otimes_{\mathbb{Z}}\mathbb{Q}$. Let
$\langle\cdot,\cdot\rangle:M_{\mathbb{Q}}\times N_{\mathbb{Q}}\to\mathbb{Q}$
be the pairing of dual vector spaces $N_{\mathbb{Q}}$ and $M_{\mathbb{Q}}$.

A _cone_ in $N$ is a convex polyhedral cone in $N_{\mathbb{Q}}$. A cone is
called _strongly convex_ if it contains no nonzero linear subspace. Recall
that the dual cone $\sigma^{\vee}$ to a cone $\sigma$ in $N$ is defined by

$\sigma^{\vee}=\\{u\in M_{\mathbb{Q}}\ |\ \langle u,v\rangle\geq 0\ \forall
v\in\sigma\\}.$

Given a strongly convex cone $\sigma$ in $N$ one can consider the finitely
generated $\mathbb{K}$-algebra $\mathbb{K}[\sigma^{\vee}\cap M]$ graded by the
semigroup of lattice points of the cone $\sigma^{\vee}$:

$\mathbb{K}[\sigma^{\vee}\cap M]=\bigoplus_{u\in\sigma^{\vee}\cap
M}\mathbb{K}\chi^{u},$

where $\chi^{u}\cdot\chi^{u^{\prime}}=\chi^{u+u^{\prime}}$. Denote by
$X(\sigma)$ the corresponding affine variety
$\mathrm{Spec}(\mathbb{K}[\sigma^{\vee}\cap M])$. It is well-known that given
an affine variety $X$ there is a bijection between faithful $T$-actions on $X$
and effective $M$-gradings on $\mathbb{K}[X]$. The aforementioned
$\sigma^{\vee}\cap M$-grading on $\mathbb{K}[\sigma^{\vee}\cap M]$ corresponds
to a faithful $T$-action on $X(\sigma)$ with an open orbit. Therefore,
$X(\sigma)$ is a toric variety. Moreover, every affine toric variety $X$
arises this way.

A fan in $N$ is a finite collection $\Sigma$ of strongly convex cones in $N$
such that for all $\sigma_{1},\sigma_{2}\in\Sigma$ every face of $\sigma_{1}$
is an element of $\Sigma$ and the intersection $\sigma_{1}\cap\sigma_{2}$ is a
face of both $\sigma_{1}$ and $\sigma_{2}$. There is a one-to-one
correspondence between toric varieties with an acting torus $T$ and fans in
$N$. Namely, given a fan $\Sigma$ in $N$ the corresponding toric variety
$X(\Sigma)$ is a union of affine charts $X(\sigma),\sigma\in\Sigma$, where any
two charts $X(\sigma_{1})$ and $X(\sigma_{2})$ are glued along their open
subset $X(\sigma_{1}\cap\sigma_{2})$. Every toric variety arises this way.

Denote by $|\Sigma|$ the _support_ of a fan $\Sigma$ in $N$:

$|\Sigma|=\bigcup_{\sigma\in\Sigma}\sigma.$

A fan $\Sigma$ in $N$ is called complete if $|\Sigma|=N_{\mathbb{Q}}$. The
corresponding toric variety $X(\Sigma)$ is complete if and only if the fan
$\Sigma$ is complete.

Let us recall basic ingredients of the Cox construction; see [2, Chapter 1]
for more details. Let $X$ be a normal variety with free finitely generated
divisor class group $\mathrm{Cl}(X)$ and only constant invertible regular
functions. Let $\mathrm{WDiv}(X)$ be the group of Weil divisors on $X$ and
$K\subseteq\mathrm{WDiv}(X)$ be a subgroup which maps onto $\mathrm{Cl}(X)$
isomorphically. The _Cox ring_ of the variety $X$ is defined as a $K$-graded
algebra

$R(X)=\bigoplus_{D\in K}H^{0}(X,D),$

where $H^{0}(X,D)=\\{f\in\mathbb{K}(X)^{\times}\ |\ \mathrm{div}(f)+D\geq
0\\}\cup\\{0\\}$, the multiplication on homogeneous components coincides with
multiplication in $\mathbb{K}(X)$ and extends to $R(X)$ by linearity. Up to
isomorphism the graded ring $R(X)$ does not depend on the choice of a subgroup
$K$. This construction may also be generalized to the case when
$\mathrm{Cl}(X)$ is a finitely generated group with torsion.

Consider a fan $\Sigma$ in $N$ and the corresponding toric variety
$X=X(\Sigma)$. One-dimensional cones in $\Sigma$ are called rays. We denote by
$\Sigma(1)=\\{\rho_{1},\dots,\rho_{m}\\}$ the set of rays in $\Sigma$. Let us
denote by $p_{l}$ the generator of the semigroup $\rho_{l}\cap N$. Assume that
$X$ has only constant invertible regular functions. This is equivalent to the
condition $\langle p_{1},\dots,p_{m}\rangle_{\mathbb{Q}}=N_{\mathbb{Q}}$.

There is a one-to-one correspondence between cones $\sigma\in\Sigma$ and
$T$-orbits $O(\sigma)$ in $X$ such that $\dim
O(\sigma)=n-\dim\langle\sigma\rangle$. In particular, each ray
$\rho_{l}\in\Sigma(1)$ corresponds to a prime $T$-invariant Weil divisor
$D_{l}=\overline{O(\rho_{l})}$ on $X$. The divisor class group
$\mathrm{Cl}(X)$ of $X$ is generated by classes $[D_{1}],\dots,[D_{m}]$.
Moreover, there is a short exact sequence

${0}$${M}$${\mathbb{Z}^{m}}$${\mathrm{Cl}(X)}$${0,}$ (1)

where the map $M\to\mathbb{Z}^{m}$ is given by $u\mapsto(\langle
u,p_{1}\rangle,\dots,\langle u,p_{m}\rangle)$ and the map
$\mathbb{Z}^{m}\to\mathrm{Cl}(X)$ is given by $(c_{1},\dots,c_{m})\mapsto
c_{1}[D_{1}]+\dots+c_{m}[D_{m}]$. In [7] it is proved that the Cox ring $R(X)$
is a $\mathrm{Cl}(X)$-graded polynomial algebra
$\mathbb{K}[x_{1},\dots,x_{m}]$ with $\deg(x_{l})=[D_{l}]$ for each
$l=\overline{1,m}$.

One can easily characterize radiant toric varieties in terms of Cox rings.
Namely, a complete toric variety $X$ is radiant if and only if the group
$\mathrm{Cl}(X)$ is free and there is a positive integer $n<m$ such that we
can reorder the variables in $\mathbb{K}[x_{1},\dots,x_{m}]$ in such a way
that $\deg(x_{n+1}),\ldots,\deg(x_{m})$ form a basis in $\mathrm{Cl}(X)$ and
for $j=\overline{1,n}$ we have $\deg(x_{j})=\sum_{k=n+1}^{m}a_{kj}\deg(x_{k})$
for some non-negative integers $a_{kj}$; cf. [11, Corollary 3].

The $\mathrm{Cl}(X)$-grading on $R(X)$ gives rise to an action of quasitorus
$G=\mathrm{Hom}_{\mathbb{Z}}(\mathrm{Cl}(X),\mathbb{K}^{\times})$ on
$\mathbb{K}^{m}=\mathrm{Spec}(R(X))$. Note that if we apply
$\mathrm{Hom}_{\mathbb{Z}}(-,\mathbb{K}^{\times})$-functor to short exact
sequence (1), we obtain an inclusion
$G\hookrightarrow(\mathbb{K}^{\times})^{m}$. Denote by $Z$ the closed subset
of $\mathbb{K}^{m}$ defined by equations

$\prod_{\rho_{l}\not\in\sigma(1)}x_{l}=0\quad\forall\sigma\in\Sigma.$

The subset $Z$ is invariant under the action of the group $G$, thus there is
an action of $G$ on the variety $\widehat{X}=\mathbb{K}^{m}\setminus Z$. The
variety $X$ is isomorphic to the good categorical quotient $\widehat{X}/\\!/G$
and the isomorphism $X\cong\widehat{X}/\\!/G$ is compatible with torus:
$X\supset T\cong(\mathbb{K}^{\times})^{m}/\\!/G=(\mathbb{K}^{\times})^{m}/G$.

###  1.2. Demazure roots and root subgroups

Hereinafter we fix a complete fan $\Sigma$ in $N$ with rays
$\rho_{1},\dots,\rho_{m}$ and their primitive generators $p_{1},\dots,p_{m}$
and let $X=X(\Sigma)$, $R=R(X)$. Denote by $\mathfrak{R}_{l}\subseteq M,\
l=\overline{1,m}$ the set of vectors $e\in M$ such that

$\langle e,p_{l}\rangle=-1\quad\text{and}\quad\langle e,p_{s}\rangle\geq 0\
\text{for all}\ s\neq l.$

The elements of the set $\mathfrak{R}=\bigsqcup_{l=1}^{m}\mathfrak{R}_{l}$ are
called Demazure roots of $\Sigma$; cf. [9, Définition 4]. The roots in
$\mathfrak{S}=\mathfrak{R}\cap-\mathfrak{R}$ are called _semisimple_ , and the
ones in $\mathfrak{U}=\mathfrak{R}\setminus\mathfrak{S}$ are called
_unipotent_.

Let $e\in\mathfrak{R}_{l}$. Denote by $\partial_{e}$ the locally nilpotent
derivation on $R$:

$\partial_{e}=\left(\prod_{s\neq l}x_{s}^{\langle
e,p_{s}\rangle}\right)\frac{\partial}{\partial
x_{l}}=x^{\theta(e)}\frac{\partial}{\partial x_{l}},$

where the map $\theta:\mathfrak{R}_{l}\to\mathbb{Z}^{m}$ is defined by

$\theta(e)=\big{(}\langle e,p_{1}\rangle,\dots,\langle
e,p_{l-1}\rangle,0,\langle e,p_{l+1}\rangle,\dots,\langle
e,p_{m}\rangle\big{)}.$

The action of operators $\exp(\alpha\partial_{e}),\alpha\in\mathbb{K}$ on $R$
gives rise to an action of the group $U_{e}\cong\mathbb{G}_{a}$ on
$\widehat{X}$ that commutes with the action of $G$. This induces a
$U_{e}$-action on $\widehat{X}/\\!/G=X$ so $U_{e}\subseteq\mathrm{Aut}(X)$.
The subgroups $U_{e},\ e\in\mathfrak{R}$ are called root subgroups of
$\mathrm{Aut}(X)$.

Let $e\in\mathfrak{R}$. It is easy to see that as a subgroup of
$\mathrm{Aut}(\widehat{X})$ the group $U_{e}$ is normalized by
$(\mathbb{K}^{\times})^{m}$, so $U_{e}\subseteq\mathrm{Aut}(X)$ is normalized
by the acting torus $T$. Conversely, it is known that every
$\mathbb{G}_{a}$-subgroup of $\mathrm{Aut}(X)$ which is normalized by $T$ has
the form $U_{e}$ for some $e\in\mathfrak{R}$; see [9, Théorème 3] and [15,
Proposition 3.14].

In the case of a radiant toric variety $X$ there is a normalized additive
action on $X$ given by the tuple of pairwise commuting homogeneous locally
nilpotent derivations $\partial_{1},\ldots,\partial_{n}$ on the ring
$R(X)=\mathbb{K}[x_{1},\dots,x_{m}]$, where

$\partial_{j}=x_{n+1}^{a_{n+1j}}\ldots x_{m}^{a_{mj}}\frac{\partial}{\partial
x_{j}};$

see [3, Lemma 3.5].

###  1.3. The automorphism group of a complete toric variety

In addition to quotient realization, the Cox ring $R$ contains information
about the automorphism group of $X$. Let $\mathrm{Aut}_{g}(R)$ be the group of
graded automorphisms of the algebra $R$. If $\phi\in\mathrm{Aut}_{g}(R)$, then
$\phi$ commutes with the action of the group $G$ and thus $\phi$ induces an
automorphism of $X$. In fact, there is an isomorphism
$\mathrm{Aut}^{0}(X)\cong\mathrm{Aut}_{g}(R)/G$. Moreover, $\mathrm{Aut}(X)$
is a linear algebraic group generated by the acting torus $T$, the root
subgroups $U_{e},e\in\mathfrak{R}$ and a finite group; see [7, Section 4].

We recall basic facts on the structure of the group $\mathrm{Aut}(X)$ proved
by Demazure [9]; see also [15, Section 3.4]. The torus $T$ is a maximal torus
in $\mathrm{Aut}(X)$ and $\mathfrak{R}$ is a root system of $\mathrm{Aut}(X)$
with respect to $T$. The unipotent radical $G_{u}$ of $\mathrm{Aut}^{0}(X)$ is
generated by the subgroups $U_{e},e\in\mathfrak{U}$, while there exists a
connected reductive subgroup $G_{\text{red}}$ with $T$ as a maximal torus such
that $\mathfrak{S}$ is the root system for $G_{\text{red}}$ and
$\mathrm{Aut}^{0}(X)=G_{u}\rtimes G_{\text{red}}$. It follows that if we pick
a system of positive roots $\mathfrak{S}^{+}$ with
$\mathfrak{S}=\mathfrak{S}^{+}\sqcup\mathfrak{S}^{-}$, then the subgroup
generated by $G_{u}$ and all subgroups $U_{e},e\in\mathfrak{S}^{+}$ is a
maximal unipotent subgroup in $\mathrm{Aut}(X)$.

By [5], for any connected (not necessary linear) algebraic group $F$ of
dimension $n$ there exists a smooth projective variety $X$ of dimension $2n$
such that the connected component of the group $\mathrm{Aut}(X)$ is isomorphic
to $F$. At the same time, the class of connected linear algebraic groups that
can be realized as the connected component of the group $\mathrm{Aut}(X)$ of a
complete toric variety $X$ is quite constrained. In particular, it is proved
in [9, Proposition 3.3] that the semisimple part of such a group is a group of
type $A$. One of the aims of this work is to study further specific properties
of the group $\mathrm{Aut}(X)$ in the case of a radiant toric variety $X$.

###  1.4. Regular subgroups

We say that an algebraic subgroup $H\subseteq\mathrm{Aut}(X)$ is _regular_ if
it is normalized by the acting torus $T$. In particular, root subgroups in
$\mathrm{Aut}(X)$ are precisely regular $\mathbb{G}_{a}$-subgroups.

Note that a regular unipotent subgroup $U$ is defined by the root subgroups
which are contained in $U$. Denote by $\mathfrak{R}(U)$ the set of all roots
$e\in\mathfrak{R}$ such that $U_{e}\subseteq U$. We call elements of the set
$\mathfrak{R}(U)$ the _roots_ of the group $U$. Conversely, given a subset
$\mathfrak{M}\subseteq\mathfrak{R}$ such that the subgroups
$U_{e},e\in\mathfrak{M}$ generate a unipotent subgroup, we denote this
subgroup by $U(\mathfrak{M})$. Clearly, if $U$ is a regular unipotent
subgroup, then $U(\mathfrak{R}(U))=U$. If
$\mathfrak{M}=\mathfrak{R}(U(\mathfrak{M}))$, then we call such a set of roots
_saturated_ , see Definition 4.1.

On the other hand, one might have
$\mathfrak{R}(U(\mathfrak{M}))\supsetneq\mathfrak{M}$ for a subset
$\mathfrak{M}\subseteq\mathfrak{R}$. This can happen due to the fact that if
$e,e^{\prime}\in\mathfrak{R}$ and $e+e^{\prime}\in\mathfrak{R}$, then $\langle
U_{e},U_{e^{\prime}}\rangle\supset U_{e+e^{\prime}}$.

######  Example 1.1.

Let $\Sigma$ be a fan of projective plane $\mathbb{P}^{2}$ so that $n=2$,
$m=3$ and ${p_{1}=(1,0)}$, ${p_{2}=(0,1)}$, $p_{3}=(-1,-1)$. Take the dual
basis in $M$ and consider two roots $\mathfrak{M}=\\{(-1,1),(0,-1)\\}$. As
$(3\times 3)$-matrices acting on homogeneous coordinates
$[z_{0}:z_{1}:z_{2}]$, the elements of the subgroups $U_{(-1,1)}$ and
$U_{(0,-1)}$ are

$u_{(-1,1)}(\alpha)=\begin{pmatrix}1&\alpha&0\\\ 0&1&0\\\
0&0&1\end{pmatrix}\text{ and }\ u_{(0,-1)}(\beta)=\begin{pmatrix}1&0&0\\\
0&1&\beta\\\ 0&0&1\end{pmatrix},$

respectively. Clearly, these root subgroups generate a unipotent subgroup
$U(\mathfrak{M})$. However, one can check that

$u_{(-1,1)}(\alpha)u_{(0,-1)}(1)u_{(-1,1)}(-\alpha)u_{(0,-1)}(-1)=\begin{pmatrix}1&0&\alpha\\\
0&1&0\\\ 0&0&1\end{pmatrix},$

which is an element of the subgroup $U_{(-1,0)}$. It shows that

$\mathfrak{R}(U(\mathfrak{M}))=\mathfrak{M}\cup\\{(-1,0)\\}\supsetneq\mathfrak{M}.$

Let $U$ be a regular unipotent subgroup in $\mathrm{Aut}(X)$ with
$\mathfrak{R}(U)=\\{e_{1},\ldots,e_{k}\\}$, and $U_{e_{1}},\ldots,U_{e_{k}}$
be all root subgroups in $U$. Then the multiplication map

$U_{e_{1}}\times\ldots\times U_{e_{k}}\to U,\quad(u_{1},\ldots,u_{k})\mapsto
u_{1}\ldots u_{k}$

is an isomorphism of affine varieties.

##  2\. Demazure roots of bilateral fans

A complete fan $\Sigma$ in a lattice $N$ of rank $n$ is called _bilateral_ if
we can order the rays $\rho_{1},\ldots,\rho_{m}$ of $\Sigma$ so that the
primitive vectors $p_{1},\dots,p_{n}$ form a basis of $N$ and for any
$k=\overline{n+1,m}$ we have

$p_{k}=-\sum_{j=1}^{n}a_{kj}p_{j},\text{ where }a_{kj}\in\mathbb{Z}_{\geq 0}.$

In this section we continue the study of Demazure roots of a bilateral fan
started in [3, 10, 11].

Let $q_{1},\dots,q_{n}$ be the basis of the lattice
$M=\mathrm{Hom}(N,\mathbb{Z})$ dual to $p_{1},\dots,p_{n}$. One can easily see
that every root $e\in\mathfrak{R}_{i}$, $i=\overline{1,n}$ has the form

$e=-q_{i}+\sum_{j\neq i}b_{j}q_{j},\text{ where }b_{j}=\langle
e,p_{j}\rangle\in\mathbb{Z}_{\geq 0}.$

Moreover, we have $-q_{i}\in\mathfrak{R}_{i}$ for every $i=\overline{1,n}$.

A root $e\in\mathfrak{R}$ is called _basic_ if $e=-q_{i}$ for some
$i=\overline{1,n}$. We also call a root of the form $-q_{i}+q_{j}$
_elementary_ , where $i,j=\overline{1,n}$ and $i\neq j$ . All other roots in
$\bigsqcup_{i=1}^{n}\mathfrak{R}_{i}$ are called _special_. Finally, roots in
$\bigsqcup_{k=n+1}^{m}\mathfrak{R}_{k}$ are called _detached_.

Note that if $e\in\mathfrak{R}_{i}\cap\mathfrak{S}$ for some
$i=\overline{1,n}$, then $e$ is either basic or elementary. In [10,
Proposition 2] it is proved that
$\mathfrak{R}_{k}\subseteq\\{q_{1},\dots,q_{n}\\}$ for all
$k=\overline{n+1,m}$. So, the set $\mathfrak{R}$ is composed of

  * •

basic roots in $\bigsqcup_{i=1}^{n}\mathfrak{R}_{i}$;

  * •

elementary roots in $\bigsqcup_{i=1}^{n}\mathfrak{R}_{i}$;

  * •

special roots in $\bigsqcup_{i=1}^{n}\mathfrak{R}_{i}$, they are unipotent;

  * •

detached roots in $\bigsqcup_{k=n+1}^{m}\mathfrak{R}_{k}$, they are
semisimple.

######  Lemma 2.1.

Consider roots $e\in\mathfrak{R}_{i}$ and $e^{\prime}\in\mathfrak{R}_{j}$ for
some $i,j=\overline{1,n}$, $i\neq j$ such that $\langle
e^{\prime},p_{i}\rangle=0$. Denote $d=\langle e,p_{j}\rangle$.

  1. (1)

If $d>0$, then $e+e^{\prime}\in\mathfrak{R}_{i}$ and
$[\partial_{e},\partial_{e^{\prime}}]=-d\partial_{e+e^{\prime}}$.

  2. (2)

If $d=0$, then $[\partial_{e},\partial_{e^{\prime}}]=0$.

###### Proof.

Let us prove that $e+e^{\prime}\in\mathfrak{R}_{i}$ when $d>0$. Clearly, the
conditions $\langle e+e^{\prime},p_{i}\rangle=-1$ and $\langle
e+e^{\prime},p_{l}\rangle\geq 0$ are satisfied for $l\neq i,j$. We have
$\langle e+e^{\prime},p_{j}\rangle=\langle e,p_{j}\rangle+\langle
e^{\prime},p_{j}\rangle=d-1\geq 0$ so $e+e^{\prime}\in\mathfrak{R}_{i}$.

Write $\partial_{e}=fx_{j}^{d}\frac{\partial}{\partial x_{i}}$ and
$\partial_{e^{\prime}}=h\frac{\partial}{\partial x_{j}}$ for some monomials
$f$ and $h$ that do not depend on $x_{i}$ and $x_{j}$. Then
$[\partial_{e},\partial_{e^{\prime}}]=fh[x_{j}^{d}\frac{\partial}{\partial
x_{i}},\frac{\partial}{\partial x_{j}}]$. If $d>0$, then
$[\partial_{e},\partial_{e^{\prime}}]=-dfhx_{j}^{d-1}\frac{\partial}{\partial
x_{i}}=-d\partial_{e+e^{\prime}}$. Otherwise $d=0$ and
$[\partial_{e},\partial_{e^{\prime}}]=0$. ∎

Now let us consider the following partial preorder on the set
$C=\\{1,\dots,n\\}$. We say that $i\succcurlyeq j$ if $a_{ki}\geq a_{kj}$ for
all $k=\overline{n+1,m}$. Further, we let $i\asymp j$ if $a_{ki}=a_{kj}$ for
all $k=\overline{n+1,m}$. The relation $\asymp$ is an equivalence relation on
$C$ and the preorder $\succcurlyeq$ induces a partial order on the set of
equivalence classes.

One can regard the defined preorder as a preorder on the set of columns of a
matrix. Namely, we write the coordinates of the vectors
$-p_{n+1},\dots,-p_{m}$ as rows of an $(m-n)\times n$ matrix:

$A=\begin{pmatrix}a_{n+1\,1}&\ldots&a_{n+1\,n}\\\ \vdots&&\vdots\\\
a_{m1}&\ldots&a_{mn}\end{pmatrix}.$

We say that $A$ is the _ray matrix_ of the bilateral fan $\Sigma$. Let
$v_{1},\dots,v_{n}$ be the columns of the matrix $A$. Then $i\succcurlyeq j$
if and only if $v_{i}\geq v_{j}$ (coordinate-wise), and $i\asymp j$ if and
only if $v_{i}=v_{j}$.

The rows of $A$ are nonzero pairwise distinct primitive integer vectors with
non-negative entries. Since the fan $\Sigma$ is complete, the matrix $A$
contains no zero column: indeed, if there is $i\in C$ such that $a_{ki}=0$ for
all $k>n$, then every cone from $\Sigma$ is contained in the half-space
$\\{v\in N_{\mathbb{Q}}\ |\ \langle q_{i},v\rangle\geq 0\\}$, a contradiction.
Conversely, any matrix $A$ satisfying these conditions is the ray matrix of
some bilateral fan $\Sigma$.

The ray matrix $A$ does not define the bilateral fan $\Sigma$. But it
determines the set of rays $\Sigma(1)$ and thus the set of Demazure roots
$\mathfrak{R}$. In fact, it is convenient to describe the set of Demazure
roots in terms of the matrix $A$. Let us collect the corresponding results in
the following proposition.

######  Proposition 2.2.

Let $\Sigma$ be a bilateral fan and $A$ be its ray matrix. Then

  1. (1)

a vector $e=-q_{i}+\sum_{j\neq i}b_{j}q_{j}$, $b_{j}\in\mathbb{Z}_{\geq 0}$ is
a root if and only if $-Ae\geq 0$;

  2. (2)

a vector $e\in M$ is a detached root in $\mathfrak{R}_{k}$ if and only if
$e=q_{i}$, where the column $v_{i}$ is a vector of length $(m-n)$ with a $1$
in the $(k-n)^{\text{th}}$ coordinate and $0$’s elsewhere.

###### Proof.

  1. (1)

Since $-Ae=v_{i}-\sum_{j\neq i}b_{j}v_{j}$, the condition $-Ae\geq 0$ is just
the system of conditions $\langle e,p_{k}\rangle\geq 0$, $k>n$ written in
vector form.

  2. (2)

Let $e=\sum_{j=1}^{n}w_{j}q_{j}$. Then $e\in\mathfrak{R}_{k}$ for some $k>n$
if and only if all $w_{j}=\langle e,p_{j}\rangle$ are non-negative and the
vector $Ae=\sum_{j=1}^{n}w_{j}v_{j}$ has one coordinate 1 and all other
coordinates non-positive. Since all vectors $v_{j}$ are nonzero and have non-
negative coordinates, this is the case if and only if all coordinates $w_{j}$
but one are $0$’s, one coordinate $w_{i}$ equals $1$, and the vector $v_{i}$
also has one coordinate equal to $1$ and others equal to $0$. Then $e=q_{i}$
and $\langle e,p_{k}\rangle=-1$ means that $v_{i}$ has $1$ in the
$(k-n)^{\text{th}}$ coordinate.

∎

######  Example 2.3.

Let $X=\mathbb{F}_{1}\times\mathbb{P}^{1}$, where $\mathbb{F}_{1}$ is the
Hirzebruch surface, so the ray matrix of $X$ equals

$A=\begin{pmatrix}1&1&0\\\ 1&0&0\\\ 0&0&1\end{pmatrix}.$

We see that

$\mathfrak{R}_{1}=\\{-q_{1},\ -q_{1}+q_{2}\\},\
\mathfrak{R}_{2}=\\{-q_{2}\\},\ \mathfrak{R}_{3}=\\{-q_{3}\\},$
$\mathfrak{R}_{4}=\\{q_{2}\\},\ \mathfrak{R}_{5}=\emptyset,\
\mathfrak{R}_{6}=\\{q_{3}\\}.$

The next observation follows from Lemma 2.1; it is also proved in [11, Lemma
2].

######  Lemma 2.4.

For any $i=\overline{1,n}$ if $\mathfrak{R}_{i}\neq\\{-q_{i}\\}$, then
$\mathfrak{R}_{i}$ contains an elementary root. Moreover, if $\langle
e,p_{j}\rangle>0$ for some $e\in\mathfrak{R}_{i}$ and $j=\overline{1,n}$, then
$-q_{i}+q_{j}\in\mathfrak{R}_{i}$.

We proceed with the following result.

######  Proposition 2.5.

Take $i,j\in C$ with $i\neq j$. Then

  1. (1)

$i\succcurlyeq j$ if and only if $\mathfrak{R}_{i}$ contains the elementary
root $-q_{i}+q_{j}$;

  2. (2)

if $i\asymp j$, then the translation map $\xi:M\to M$ given by
$\xi(e)=-q_{i}+q_{j}+e$ induces a bijection
$\mathfrak{R}_{j}\setminus\\{-q_{j}+q_{i}\\}\to\mathfrak{R}_{i}\setminus\\{-q_{i}+q_{j}\\}$.

###### Proof.

  1. (1)

By 2.2.(1) the vector $-q_{i}+q_{j}$ is a root if and only if
$-A(-q_{i}+q_{j})=v_{i}-v_{j}\geq 0$ and by definition $i\succcurlyeq j$ if
and only if $v_{i}\geq v_{j}$.

  2. (2)

Since the inverse of $\xi$ is the map $e\mapsto-q_{j}+q_{i}+e$, due to the
symmetry of the assertion it suffices to prove that if $e\in\mathfrak{R}_{j}$
and $\langle e,p_{i}\rangle\neq 0$, then $e=-q_{j}+q_{i}$.
Let $e=-q_{j}+\sum_{l\neq j}b_{l}q_{l}\in\mathfrak{R}_{j}$, where
$b_{i}=\langle e,p_{i}\rangle>0$. We have

$-Ae=v_{j}-\sum_{l\neq j}b_{l}v_{l}=-(b_{i}-1)v_{i}-\sum_{l\neq
i,j}b_{l}v_{l}$

since $v_{i}=v_{j}$ by definition of $\asymp$. We obtain that a non-negative
vector $-Ae$ equals the sum of non-positive vectors. It follows that
$(b_{i}-1)v_{i}=0$ and $b_{l}v_{l}=0$ for all $l\neq i,j$. But all $v_{s}$ are
nonzero, so $b_{i}=1$ and $b_{l}=0$ for all $l\neq i,j$. We obtain
$e=-q_{j}+q_{i}$.

∎

The following lemma contains one more property of Demazure roots that will be
needed in the next section.

######  Lemma 2.6.

For $i\in C$ we have $q_{i}\in\mathfrak{R}$ if and only if all roots in
$\mathfrak{R}_{i}$ are semisimple. In particular, the set $\mathfrak{R}_{i}$
contains no special root.

###### Proof.

If the root $-q_{i}\in\mathfrak{R}_{i}$ is semisimple, then
$q_{i}\in\mathfrak{R}$.

Conversely, assume that $q_{i}\in\mathfrak{R}$. Then $q_{i}$ is detached and
the vector $v_{i}$ is as in Proposition 2.2.(2). Let $e=-q_{i}+\sum_{s\neq
i}b_{s}q_{s}$ be a root in $\mathfrak{R}_{i}$. Then

$-Ae=v_{i}-\sum_{s\neq i}b_{s}v_{s}\geq 0.$

Since all vectors $v_{s}$ are non-zero, $\sum_{s\neq i}b_{s}$ equals either
$0$ or $1$. In the first case all $b_{s}$ equal $0$ and the root $e=-q_{i}$ is
semisimple. Otherwise we have $v_{i}=v_{j}$ and $e=-q_{i}+q_{j}$ for some
$1\leq j\neq i\leq n$, so $i\asymp j$. In this case the vector $-e$ is a root
as well, so the root $e$ is semisimple. ∎

Let $C_{1},\dots,C_{r}$ be the equivalence classes of the relation $\asymp$.
By Proposition 2.5 an elementary root $-q_{i}+q_{j}$ is semisimple if and only
if $i$ and $j$ are in the same equivalence class. By renumbering the classes
we may assume that $C_{l}\succ C_{s}$ implies $l<s$. Furthermore, by
renumbering the vectors $p_{1},\dots,p_{n}$ we may assume that
$C_{1},\dots,C_{r}$ is an (element-wise) increasing sequence of segments of
$C$. In other words, we have the following:

$\begin{array}[]{c}\text{the sequence }C_{1},\dots,C_{r}\text{ is
}\succcurlyeq\text{-non-increasing \ and}\\\ \text{there exist numbers
}0=c_{0}<c_{1}<\ldots<c_{r}=n\text{ such that for all }s=\overline{1,r}\\\
\text{we have }C_{s}=\\{c_{s-1}+1,c_{s-1}+2,\dots,c_{s}\\}.\end{array}$ (2)

######  Lemma 2.7.

Assume that the condition (2) holds. If $e\in\mathfrak{R}_{i}$ is a unipotent
root, then $e=-q_{i}+\sum_{j>i}\langle e,p_{j}\rangle q_{j}$.

###### Proof.

The case of basic $e$ is trivial. Assume that $\langle e,p_{j}\rangle>0$ for
some $j\in C$. By Lemma 2.4, we have $-q_{i}+q_{j}\in\mathfrak{R}_{i}$. So
$i\succcurlyeq j$ by Proposition 2.5.(1).

If $i\asymp j$, then by Proposition 2.5.(2) and the fact that $e$ is unipotent
we have $e+q_{i}-q_{j}\in\mathfrak{R}_{j}$. It implies $\langle
e,p_{j}\rangle=0$, a contradiction. We conclude that $i\succ j$ and thus
$j>i$. ∎

##  3\. The structure of a maximal unipotent subgroup

Let us fix a radiant toric variety $X=X(\Sigma)$ with a bilateral fan $\Sigma$
and keep the notation of the previous section. Let us also assume the
condition (2). We are going to describe the structure of a maximal unipotent
subgroup of the automorphism group $\mathrm{Aut}(X)$.

Given $v\in N$ such that $\langle e,v\rangle\neq 0\ \forall e\in\mathfrak{S}$,
we denote

$\mathfrak{S}^{+}_{v}=\\{e\in\mathfrak{S}\ |\ \langle e,v\rangle>0\\}\text{
and }\mathfrak{R}^{+}_{v}=\mathfrak{S}^{+}_{v}\sqcup\mathfrak{U}.$

Recall that $U(\mathfrak{R}^{+}_{v})$ is a maximal (regular) unipotent
subgroup in $\mathrm{Aut}(X)$ and
$\mathfrak{R}(U(\mathfrak{R}^{+}_{v}))=\mathfrak{R}^{+}_{v}$. Any maximal
regular unipotent subgroup is obtained this way for a suitable $v$. Let us
define $U_{\mathrm{max}}=U(\mathfrak{R}^{+}_{v})$ with $v$ such that

$\langle q_{1},v\rangle<\dots<\langle q_{n},v\rangle<0.$ (3)

######  Lemma 3.1.

Assume that the condition (2) is satisfied and $v\in N$ is chosen according to
(3). Then a root $e\in\mathfrak{R}$ lies in $\mathfrak{R}^{+}_{v}$ if and only
if it has the form $e=-q_{i}+\sum_{j>i}\langle e,p_{j}\rangle q_{j}$ for some
$i\in C$.

###### Proof.

All unipotent roots lie in $\mathfrak{R}^{+}_{v}$ and have the desired form by
Lemma 2.7. A semisimple root is basic, elementary or detached. In the first
case it is in $\mathfrak{R}^{+}_{v}$ and has the desired form. In the second
case $-q_{i}+q_{j}\in\mathfrak{R}^{+}_{v}$ if and only if $j>i$ due to (3). In
the third case it is not in $\mathfrak{R}^{+}_{v}$ and is not of the
corresponding form. ∎

Hereinafter we fix a vector $v$ satisfying the condition (3). Denote
$\mathfrak{R}^{+}=\mathfrak{R}^{+}_{v}$ and
$\mathfrak{R}_{i}^{+}=\mathfrak{R}_{i}\cap\mathfrak{R}^{+}$ for all $i\in C$.
In particular, we have
$\mathfrak{R}^{+}=\mathfrak{R}(U_{\mathrm{max}})=\bigsqcup_{i=1}^{n}\mathfrak{R}_{i}^{+}$.

######  Remark 3.2.

By Lemma 2.4, if for $i\in C$ there is a root $e\in\mathfrak{R}_{i}$ with
$\langle e,p_{j}\rangle>0$ for some $j\in C$, then $i\succcurlyeq j$. We
conclude that if $i$ lies in a $\succcurlyeq$-minimal equivalence class
$C_{s}$ and $i$ is a maximal number in $C_{s}$, then
$\mathfrak{R}_{i}^{+}=\\{-q_{i}\\}$. In particular,
$\mathfrak{R}_{n}^{+}=\\{-q_{n}\\}$.

Given a subset $C^{\prime}\subseteq C$, we denote

$U(C^{\prime})=U\left(\bigsqcup_{i\in C^{\prime}}\mathfrak{R}_{i}^{+}\right).$

For example, $U(i)=U(\mathfrak{R}_{i}^{+})=\langle U_{e}\ |\
e\in\mathfrak{R}_{i}^{+}\rangle$. Note that if
$C^{\prime}=\\{i_{1},\dots,i_{k}\\}$, then $U(C^{\prime})$ acts trivially on
coordinates $x_{j}$ in the Cox ring $R(X)$ for all $j\neq i_{1},\dots,i_{k}$;
see Subsection 1.2. In particular, if $I$ and $J$ are disjoint subsets of $C$,
then $U(I)\cap U(J)=\\{\mathrm{id}\\}$. For
$e\in\mathfrak{R}^{+},\alpha\in\mathbb{K}$ denote
$u_{e}(\alpha)=\exp(\alpha\partial_{e})\in U_{e}$.

######  Proposition 3.3.

  1. (1)

The subgroup $U(i)$ is commutative of dimension $|\mathfrak{R}_{i}^{+}|$.

  2. (2)

The subgroup $U(i)$ is normalized by the subgroup $U(j)$ for $j>i$. Moreover,
if $e\in\mathfrak{R}_{i}^{+}$, $e^{\prime}\in\mathfrak{R}_{j}^{+}$, and
$d=\langle e,p_{j}\rangle$, then

$u_{e^{\prime}}(\alpha^{\prime})^{-1}u_{e}(\alpha)u_{e^{\prime}}(\alpha^{\prime})=\prod_{k=0}^{d}u_{e+ke^{\prime}}\left(\binom{d}{k}\alpha(\alpha^{\prime})^{k}\right)\in
U(i)$

for all $\alpha,\alpha^{\prime}\in\mathbb{K}$.

###### Proof.

The first assertion follows from the observation that the sum of two roots in
$\mathfrak{R}_{i}$ is not a root. For the second assertion write

$u_{e^{\prime}}(-\alpha^{\prime})u_{e}(\alpha)u_{e^{\prime}}(\alpha^{\prime})\cdot
x_{i}=x_{i}+\alpha\left(x_{j}+\alpha^{\prime}x^{\theta(e^{\prime})}\right)^{d}\frac{x^{\theta(e)}}{x_{j}^{d}}=$
$=x_{i}+\sum_{k=0}^{d}\binom{d}{k}\alpha(\alpha^{\prime})^{k}\frac{x^{\theta(e)+k\theta(e^{\prime})}}{x_{j}^{k}}=x_{i}+\sum_{k=0}^{d}\binom{d}{k}\alpha(\alpha^{\prime})^{k}x^{\theta(e+ke^{\prime})}=$
$=\prod_{k=0}^{d}u_{e+ke^{\prime}}\left(\binom{d}{k}\alpha(\alpha^{\prime})^{k}\right)\cdot
x_{i}.$

∎

######  Remark 3.4.

It is clear from the description of root subgroups given in Subsection 1.2
that generic orbits of the action of the subgroup $U(i)$ on $X$ are one-
dimensional.

Let $U\subseteq\mathrm{Aut}(X)$ be a regular unipotent subgroup such that
$\mathfrak{M}=\mathfrak{R}(U)\subseteq\mathfrak{R}^{+}$. For each
$i=1,\dots,n$ denote $\mathfrak{M}_{i}=\mathfrak{M}\cap\mathfrak{R}_{i}^{+}$.

######  Corollary 3.5.

We have

$U=\Big{(}\big{(}\cdots(U(\mathfrak{M}_{n})\ltimes
U(\mathfrak{M}_{n-1}))\dots\big{)}\ltimes U(\mathfrak{M}_{2})\Big{)}\ltimes
U(\mathfrak{M}_{1}).$

In particular,
$U_{\mathrm{max}}\cong\Big{(}\big{(}\cdots(\mathbb{G}_{a}\ltimes\mathbb{G}_{a}^{|\mathfrak{R}_{n-1}^{+}|})\dots\big{)}\ltimes\mathbb{G}_{a}^{|\mathfrak{R}_{2}^{+}|}\Big{)}\ltimes\mathbb{G}_{a}^{|\mathfrak{R}_{1}^{+}|}$.

###### Proof.

Applying the second claim of Proposition 3.3 to all pairs $(1,j)$, $j>1$ we
obtain $U=U(\mathfrak{M}_{2}\cup\ldots\cup\mathfrak{M}_{n})\ltimes
U(\mathfrak{M}_{1})$. Similarly,
$U(\mathfrak{M}_{2}\cup\ldots\cup\mathfrak{M}_{n})=U(\mathfrak{M}_{3},\ldots,\mathfrak{M}_{n})\ltimes
U(\mathfrak{M}_{2})$, and so on. At the end we obtain

$U=\Big{(}\big{(}\cdots(U(\mathfrak{M}_{n})\ltimes
U(\mathfrak{M}_{n-1}))\dots\big{)}\ltimes U(\mathfrak{M}_{2})\Big{)}\ltimes
U(\mathfrak{M}_{1}),$

which proves the assertion. If $U=U_{\mathrm{max}}$, then
$\mathfrak{M}=\mathfrak{R}^{+}$ and
$U(\mathfrak{M}_{i})=U(i)=\mathbb{G}_{a}^{|\mathfrak{R}_{i}^{+}|}$ by the
first assertion of Proposition 3.3. Moreover, $|\mathfrak{R}_{n}^{+}|=1$ by
Remark 3.2. ∎

Now let us fix positive integers $k>l$ and consider the subgroup $U_{k,l}$ of
the group $U_{k}$ of all unitriangular $(k\times k)$-matrices such that

$(\beta_{ij})_{i,j=1}^{k}\in U_{k,l}\iff\beta_{ij}=0\text{ for all }j>i>l.$

In other words, the group $U_{k,l}$ consists of matrices of the form

$\left(\begin{array}[]{c|c}\begin{matrix}1&&&\\\ &\ddots&\text{\Huge*}&\\\
&&\ddots&\\\ \text{\huge 0}&&&1\end{matrix}&\text{\Huge*}\\\
\hline\cr\text{\huge 0}&\begin{matrix}1&&&\\\ &\ddots&&\text{\huge 0}\\\
&&\ddots&\\\ \text{\huge 0}&&&1\end{matrix}\end{array}\right)$

In particular, the subgroup $U_{k,k-1}$ coincides with the group $U_{k}$. Note
also that the group $U_{k,l}$ is commutative if and only if $l=1$.

######  Proposition 3.6.

Let $C_{s}=\\{c_{s-1}+1,c_{s-1}+2,\dots,c_{s}\\}$ be an equivalence class in
$C$. Denote $c=c_{s-1}+1$. Then

$U(C_{s})\cong U_{k,l}$

with $k=|\mathfrak{R}_{c}^{+}|+1$ and $l=|C_{s}|$.

###### Proof.

Denote by $E$ the identity $(k\times k)$-matrix. Let $E_{i,j}$ be a $(k\times
k)$-matrix with a unique nonzero entry on $i^{\text{th}}$ row and
$j^{\text{th}}$ column equal to $1$. For each $i\leq l$ the matrices
$E+\sum_{j>i}\beta_{ij}E_{i,j}$ form a subgroup $H_{i}$ in $U_{k,l}$
isomorphic to $\mathbb{G}_{a}^{k-i}$:

$\left(E+\sum_{j>i}\beta_{ij}E_{i,j}\right)\cdot\left(E+\sum_{j>i}\beta^{\prime}_{ij}E_{i,j}\right)=E+\sum_{j>i}(\beta_{ij}+\beta^{\prime}_{ij})E_{i,j}.$

Further, for $l\geq i^{\prime}>i$ we have

$\left(E+\sum_{j>i^{\prime}}\beta_{i^{\prime}j}E_{i^{\prime},j}\right)\cdot\left(E+\sum_{j>i}\beta_{ij}E_{i,j}\right)=E+\sum_{j>i^{\prime}}\beta_{i^{\prime}j}E_{i^{\prime},j}+\sum_{j>i}\beta_{ij}E_{i,j},$

so every element of $U_{k,l}$ is uniquely decomposed as the product
$h_{l}h_{l-1}\cdots h_{1}$, where $h_{i}\in H_{i}$ for all $i$. Finally, for
$1\leq i<j\leq k$ and $i<i^{\prime}<j^{\prime}\leq k$ write

$(E+\beta^{\prime}E_{i^{\prime},j^{\prime}})^{-1}(E+\beta
E_{i,j})(E+\beta^{\prime}E_{i^{\prime},j^{\prime}})=(E-\beta^{\prime}E_{i^{\prime},j^{\prime}})(E+\beta
E_{i,j})(E+\beta^{\prime}E_{i^{\prime},j^{\prime}})=$ $=E+\beta
E_{i,j}+\delta_{ji^{\prime}}\beta\beta^{\prime}E_{i,j^{\prime}}.$

In particular, $U_{k,l}=\Big{(}\big{(}\cdots(H_{l}\ltimes
H_{l-1})\dots\big{)}\ltimes H_{2}\Big{)}\ltimes H_{1}$.

By Corollary 3.5 the group $U(C_{s})$ is a semidirect product of the subgroups
$U(c)$, $U(c+1)$, $U(c+2),\dots$ with the action on normal subgroups as in
Proposition 3.3. Note that if $i\in C_{s}$, then
$|\mathfrak{R}_{i}^{+}|=|\mathfrak{R}_{c}^{+}|-(i-c)=k-(i-c+1)$ by Proposition
2.5.(2) and Lemma 3.1.

Denote by $\mathfrak{T}$ the set of roots in $\mathfrak{R}^{+}$, which are not
elementary semisimple. Let us enumerate roots
$e\in\mathfrak{R}_{c}^{+}\cap\mathfrak{T}$ by numbers $j_{e}$, $l+1\leq
j_{e}\leq k$. By Proposition 2.5.(2), for every $i\in C_{s}$ the map $e\mapsto
e-q_{c}+q_{i}$ establishes a bijection between roots in
$\mathfrak{R}_{i}^{+}\cap\mathfrak{T}$ and roots in
$\mathfrak{R}_{c}^{+}\cap\mathfrak{T}$. We let $j_{e}$ be equal to
$j_{e^{\prime}}$, where the root $e^{\prime}$ corresponds to $e$ under this
bijection.

We conclude that a map $\phi:U(C_{s})\to U_{k,l}$ defined on generators as

$\phi(u_{-q_{i}+q_{j}}(\alpha))=E+\alpha E_{i-c+1,j-c+1}\text{ with }i,j\in
C_{s};$ $\phi(u_{e}(\alpha))=E+\alpha E_{i-c+1,j_{e}}\text{ for other roots
}e\in\mathfrak{R}_{i}^{+}$

is the required isomorphism. ∎

######  Theorem 3.7.

Let $U_{\mathrm{max}}$ be a maximal unipotent subgroup of the automorphism
group $\mathrm{Aut}(X)$ of a radiant toric variety $X$. Assume that the
condition (2) holds for the equivalence classes $C_{1},\dots,C_{r}$ of the
relation $\asymp$. Then

$U_{\mathrm{max}}\cong\Big{(}\big{(}\cdots\big{(}U_{k_{r},l_{r}}\ltimes
U_{k_{r-1},l_{r-1}}\big{)}\dots\big{)}\ltimes U_{k_{2},l_{2}}\Big{)}\ltimes
U_{k_{1},l_{1}},$

where $k_{s}=|\mathfrak{R}_{c_{s-1}+1}^{+}|+1$ and $l_{s}=|C_{s}|$,
$s=\overline{1,r}$.

###### Proof.

By the same argument as in the proof of Corollary 3.5, grouping together
subsets $\mathfrak{R}_{i}^{+}$, $i\in C_{s}$ for each $s=\overline{1,r}$, we
obtain a decomposition

$U_{\mathrm{max}}=\Big{(}\big{(}\cdots\big{(}U(C_{r})\ltimes
U(C_{r-1})\big{)}\dots\big{)}\ltimes U(C_{2})\Big{)}\ltimes U(C_{1}).$

By Proposition 3.6 we have $U(C_{s})\cong U_{k_{s},l_{s}}$. This proves the
theorem. ∎

######  Example 3.8.

Let $X$ be the projective space $\mathbb{P}^{n}$ so that the number of rays of
the fan $\Sigma$ equals $n+1$ and $p_{n+1}=-(p_{1}+\dots+p_{n})$. We have
$1\asymp 2\asymp\dots\asymp n$, so the whole set $C$ is an equivalence class.
For each $i\in C$ the set of roots $\mathfrak{R}_{i}^{+}$ consists of vectors

$-q_{i},-q_{i}+q_{i+1},-q_{i}+q_{i+2}\dots,-q_{i}+q_{n}.$

In particular, we have $|\mathfrak{R}_{1}^{+}|=n$, so $U_{\mathrm{max}}\cong
U_{n+1,n}$ is the group $U_{n+1}$. Of course, this also follows from the well-
known fact $\mathrm{Aut}(\mathbb{P}^{n})=\mathrm{PGL}(n+1)$.

######  Example 3.9.

Let $X$ be as in Example 2.3. Then

$U_{\mathrm{max}}=(U(3)\ltimes U(2))\ltimes
U(1)\cong(\mathbb{G}_{a}\ltimes\mathbb{G}_{a})\ltimes\mathbb{G}_{a}^{2}.$

In fact, for this particular case $U(1,2)\cong U_{3}$ and the subgroup $U(3)$
commutes with the subgroup $U(1,2)$, so we have $U_{\mathrm{max}}=U(3)\times
U(1,2)\cong\mathbb{G}_{a}\times U_{3}$.

Finishing this section, let us describe a maximal unipotent subgroup
$U_{\mathrm{ss}}$ of the semisimple part of the group $\mathrm{Aut}(X)$ as a
subgroup of $U_{\mathrm{max}}$. The roots of $U_{\mathrm{ss}}$ are the
semisimple roots in $\mathfrak{R}^{+}$. For any subset $S\subseteq C$ denote
by $\mathfrak{R}(S)$ the set of roots $\bigsqcup_{i\in S}\mathfrak{R}_{i}$ and
let $\mathfrak{R}^{+}(S)=\mathfrak{R}(S)\cap\mathfrak{R}^{+}$. For each
$s=\overline{1,r}$ let $U^{s}\subseteq U(C_{s})$ be the subgroup such that
$\mathfrak{R}(U^{s})=\mathfrak{R}^{+}(C_{s})\cap\mathfrak{S}$. By Theorem 3.7
there is a decomposition

$U_{\mathrm{ss}}=\Big{(}\big{(}\cdots\big{(}U^{r}\ltimes
U^{r-1}\big{)}\dots\big{)}\ltimes U^{2}\Big{)}\ltimes U^{1}.$

Denote by $U_{k,l}^{(1)}$ (resp. $U_{k,l}^{(2)}$) the subgroup of $U_{k,l}$
consisting of all matrices that differ from the unit matrix at most in the
upper left $(l\times l)$-square (resp. at most in the upper right
$(l\times(k-l))$-rectangle). Then $U_{k,l}=U_{k,l}^{(1)}\ltimes
U_{k,l}^{(2)}$, the subgroup $U_{k,l}^{(1)}$ is isomorphic to $U_{l}$, and the
subgroup $U_{k,l}^{(2)}$ is commutative.

######  Proposition 3.10.

In the above notation, the following holds.

  1. (1)

$U_{\mathrm{ss}}=U^{1}\times\ldots\times U^{r}$.

  2. (2)

For each $s=\overline{1,r}$ we have $U^{s}=U(C_{s})\cong U_{k_{s}}$ if all
roots in $\mathfrak{R}(C_{s})$ are semisimple, and $U^{s}\cong
U_{k_{s},l_{s}}^{(1)}\cong U_{l_{s}}$ otherwise.

###### Proof.

  1. (1)

Since we already have a semidirect product decomposition of $U_{\mathrm{ss}}$,
it suffices to prove that each two subgroups $U^{s}$ and $U^{s^{\prime}}$,
$s\neq s^{\prime}$ commute with each other. It follows from the facts that a
positive semisimple root is either basic or elementary and the sum of two such
roots from different components is not a root.

  2. (2)

If all roots in $\mathfrak{R}(C_{s})$ are semisimple, then
$U^{s}=U(C_{s})\cong U_{k_{s},l_{s}}\cong U_{k_{s}}$, since $l_{s}=k_{s}-1$ by
definition. Assume that there are unipotent roots in $\mathfrak{R}(C_{s})$. By
Lemma 2.6 for each $i\in C_{s}$ we have $-q_{i}\in\mathfrak{U}$ and due to the
construction of the isomorphism $U(C_{s})\cong U_{k_{s},l_{s}}$ in the proof
of Proposition 3.6 we see that $U^{s}\cong U_{k_{s},l_{s}}^{(1)}$.

∎

We see that each subgroup $U^{s}$ is the group of all unitriangular matrices:
it is isomorphic either to $U_{k_{s}}$ or to $U_{l_{s}}$. This is a
manifestation of a famous theorem due to Demazure: the semisimple part of the
automorphism group of a complete toric variety is a group of type $A$; see [9,
Proposition 3.3]. The number of simple components in the semisimple part of
the automorphism group $\mathrm{Aut}(X)$ equals the number of such classes
$C_{s}$ that either $C_{s}$ contains at least two elements or the
corresponding basic root $-q_{i}$ is semisimple.

The center of $U_{\mathrm{max}}$ is described in Remark 5.2 below.

##  4\. Regular unipotent subgroups

In this section we study regular unipotent subgroups of the automorphism group
$\mathrm{Aut}(X)$ of a radiant toric variety $X$ that act on $X$ with an open
orbit. We keep the notation of the previous sections.

######  Definition 4.1.

We say that a subset of roots $\mathcal{A}\subseteq\mathfrak{R}$ is
_saturated_ with respect to a subset $\mathcal{B}\subseteq\mathfrak{R}$ if for
any $a\in\mathcal{A}$ and $b\in\mathcal{B}$ the sum $a+b$ is either contained
in $\mathcal{A}$ or is not a root.

Recall that $\mathfrak{R}(U_{\mathrm{max}})=\mathfrak{R}^{+}$.

######  Lemma 4.2.

A subset $\mathfrak{M}\subseteq\mathfrak{R}^{+}$ equals $\mathfrak{R}(U)$ for
some regular subgroup $U\subseteq U_{\mathrm{max}}$ if and only if for each
$i\in C$ the subset $\mathfrak{M}_{i}$ is saturated with respect to
$\bigsqcup_{j>i}\mathfrak{M}_{j}$.

###### Proof.

Observe that a subset $\mathfrak{M}\subseteq\mathfrak{R}^{+}$ equals
$\mathfrak{R}(U)$ for some regular subgroup $U\subseteq U_{\mathrm{max}}$ if
and only if the linear span of derivations $\partial_{e}$, $e\in\mathfrak{M}$
is a Lie subalgebra in the tangent algebra of the group $U_{\mathrm{max}}$. By
Lemmas 2.1 and 3.1, this is the case if and only if for all
$e\in\mathfrak{M}_{i}$ and $e^{\prime}\in\mathfrak{M}_{j}$, $j>i$ with
$\langle e,p_{j}\rangle>0$ we have $e+e^{\prime}\in\mathfrak{M}_{i}$. ∎

######  Definition 4.3.

A _principal unipotent subgroup_ of the automorphism group $\mathrm{Aut}(X)$
of a radiant toric variety $X$ is a regular commutative unipotent subgroup
that acts on $X$ with an open orbit.

The action of a subgroup of $\mathrm{Aut}(X)$ on $X$ is faithful, so a
principal unipotent subgroup in $\mathrm{Aut}(X)$ is precisely a subgroup that
defines a normalized additive action on $X$. Since every two such actions are
equivalent (see [3, Theorem 3.6]), every two principal unipotent subgroups in
$\mathrm{Aut}(X)$ are conjugate. As we have seen in Subsection 1.2,
$U(\\{-q_{1},\ldots,-q_{n}\\})$ is an example of a principal unipotent
subgroup.

######  Theorem 4.4.

Let $X$ be a radiant toric variety. A regular unipotent subgroup $U$ in
$U_{\mathrm{max}}$ with $\mathfrak{M}=\mathfrak{R}(U)$ acts on $X$ with an
open orbit if and only if $U$ contains a principal unipotent subgroup. More
precisely, the subgroup $U$ acts on $X$ with an open orbit if and only if
$-q_{1},\ldots,-q_{n}\in\mathfrak{M}$.

###### Proof.

If a subgroup $U(\mathfrak{M})$ acts on $X$ with an open orbit, then the set
$\mathfrak{M}_{i}$ is nonempty for each $i$. Indeed, if such intersection is
empty, then any orbit of the group $U(\mathfrak{M})$ on the spectrum of the
Cox ring $R(X)$ is contained in the subvariety, where the coordinates
$x_{i},x_{n+1},\ldots,x_{m}$ are constant. So, the dimension of any orbit is
less than $n$, and it can not project to an open orbit on $X$.

Let us prove by induction that $-q_{1},\ldots,-q_{n}\in\mathfrak{M}$. By
Remark 3.2, we have $-q_{n}\in\mathfrak{M}$. Assuming for some $i<n$ that
$-q_{i+1},\ldots,-q_{n}\in\mathfrak{M}$, we take an arbitrary element of
$\mathfrak{M}_{i}$, say $-q_{i}+\sum_{j>i}b_{j}q_{j}$, and apply consecutively
Lemma 2.1 for each $e^{\prime}=-q_{i+1},\ldots,-q_{n}$. Then
$-q_{i}\in\mathfrak{M}$, and the direct implication follows.

In Subsection 1.2 we have already seen that locally nilpotent derivations
$\partial_{-q_{1}},\dots,\partial_{-q_{n}}$ give rise to an additive action on
$X$, so the inverse implication is straightforward. ∎

This fact and the results of [11] motivate the following definition.

######  Definition 4.5.

We say that a radiant toric variety $X$ is of _Type I_ , if a maximal
unipotent subgroup in $\mathrm{Aut}(X)$ is commutative. All other radiant
toric varieties are assigned to _Type II_.

In Type I, the maximal unipotent subgroup in $\mathrm{Aut}(X)$ is a principal
unipotent subgroup. By [11, Corollary 4], radiant toric varieties of Type I
are precisely complete toric varieties that admit a unique additive action. On
the other hand, Type II consists exactly of complete toric varieties that
admit a faithful action of a non-commutative unipotent group with an open
orbit.

Clearly, the direct product $X_{1}\times\ldots\times X_{s}$ of radiant toric
varieties is again a radiant toric variety, and it is of Type I if and only if
all factors $X_{i}$ are of Type I. Let us give one more result in this
direction.

######  Proposition 4.6.

Every radiant toric variety of Type I admits a unique decomposition
${X\cong(\mathbb{P}^{1})^{b}\times Y}$, where $b$ is a non-negative integer
and $Y$ is a radiant toric variety of Type I such that the connected component
$\mathrm{Aut}(Y)^{0}$ is solvable.

###### Proof.

Since $X$ is of Type I, besides basic roots $-q_{1},\ldots,-q_{n}$ the set
$\mathfrak{R}$ may contain only some detached roots $q_{i}$.

Assume that $q_{1}\in\mathfrak{R}_{n+1}$. By Proposition 2.2.(2), in the ray
matrix $A$ we have $a_{n+1\,1}=1$ and $a_{k\,1}=0$ for all
$k=\overline{n+2,m}$. If $a_{n+1\,s}>0$ for some $s=\overline{2,n}$, then by
Proposition 2.2.(1) we obtain $-q_{s}+q_{1}\in\mathfrak{R}_{s}$, a
contradiction. So, we have $a_{n+1\,s}=0$ for all $s=\overline{2,n}$.

It means that $\Sigma=\Sigma_{1}\oplus\Sigma_{2}$, where $\Sigma_{1}$ is the
fan generated by $p_{1}$ and $-p_{1}$ and $\Sigma_{2}$ is a supplementary
subfan. It implies that $X\cong\mathbb{P}^{1}\times X(\Sigma_{2})$. Repeating
this procedure, we come to the decomposition $X\cong(\mathbb{P}^{1})^{b}\times
Y$, where the fan of the toric variety $Y$ has no detached roots. We conclude
that $\mathrm{Aut}(Y)^{0}\cong T\ltimes U$, where $T$ is the acting torus of
$Y$ and $U$ is a principal unipotent subgroup of $\mathrm{Aut}(Y)$. In
particular, the group $\mathrm{Aut}(Y)^{0}$ is solvable.

In any decomposition of this form, $b$ is the number of pairs of semisimple
roots of $\Sigma$, and the fan of $Y$ is the subfan of $\Sigma$ obtained as an
intersection with the subspace generated by the vectors $p_{i}$ such that
$\mathfrak{R}_{i}$ contains no semisimple root. This shows that the
decomposition is unique. ∎

Let us give an example of a smooth non-projective radiant toric variety
$X(\Sigma)$ of dimension $n$ for each $n\geq 3$; see [17] for details.

######  Example 4.7.

The set of primitive vectors on the rays of $\Sigma$ consists of a basis
$p_{1},\dots,p_{n}$ of the lattice $N$, the vector $w=-p_{1}-\ldots-p_{n}$,
and the vectors $u_{i}=p_{i}+w$, $i\in C=\\{1,\dots,n\\}$. The maximal cones
of $\Sigma$ are constructed as follows. For each integer $k$ denote by $p_{k}$
and $u_{k}$ the vectors $p_{i}$ and $u_{i}$ correspondingly, where $k$ and $i$
are equal modulo $n$ and $i\in C$. Now for each $i\in C$ write a sequence of
vectors

$p_{i},p_{i+1},\dots,p_{i+n-2},u_{i},u_{i+1},\dots,u_{i+n-2},w.$

For each (continuous) segment $S$ of $n$ elements of this sequence, the cone
generated by elements of $S$ lies in $\Sigma$. Lastly, $\Sigma$ also contains
the cone generated by $p_{1},\ldots,p_{n}$.

The toric variety $X(\Sigma)$ is of Type I. Indeed, the ray matrix $A$ equals

$\begin{pmatrix}0&1&\dots&1\\\ 1&0&\dots&1\\\ \vdots&&\ddots&\vdots\\\
1&1&\dots&0\\\ 1&1&\dots&1\\\ \end{pmatrix},$

so each two columns of $A$ are incomparable. In the case $n=3$ the orbit
structure of action of $U_{\mathrm{max}}$ on $X$ is described in [17]. This
description shows that the number of orbits of a maximal unipotent subgroup
$U_{\mathrm{max}}$ on a radiant toric variety can be infinite.

######  Problem 4.8.

Characterize bilateral fans $\Sigma$ such that the action
$U_{\mathrm{max}}\curvearrowright X(\Sigma)$ has a finite number of orbits.

Since a closed subvariety of a projective variety is projective, the direct
product of complete varieties is projective if and only if each factor is
projective. This shows that the product $X\times\mathbb{P}^{2}$, where $X$ is
a variety from Example 4.7, is an example of a smooth non-projective radiant
toric variety of Type II.

Now we come to an algorithm that allows to construct all regular unipotent
subgroups in $\mathrm{Aut}(X)$ that act on $X$ with an open orbit.

######  Algorithm 4.9.

We sequentially construct a subset $\mathfrak{M}\subseteq\mathfrak{R}^{+}$,
which is the set of roots of a regular unipotent subgroup of
$\mathrm{Aut}(X)$. For this we produce subsets
$\mathfrak{M}_{i}\subseteq\mathfrak{R}_{i}^{+}$ step by step starting from
$i=n$ down to $i=1$. By Theorem 4.4 we start with
$\mathfrak{M}_{n}=\mathfrak{R}_{n}^{+}=\\{-q_{n}\\}$.

Assume that we constructed subsets
$\mathfrak{M}_{i+1},\mathfrak{M}_{i+2},\dots,\mathfrak{M}_{n}$ for some $i\in
C$ such that for each $j>i$ the set $\mathfrak{M}_{j}$ is saturated with
respect to $\bigsqcup_{k>j}\mathfrak{M}_{k}$. Let us construct the set
$\mathfrak{M}_{i}$. Once again by Theorem 4.4 we have to start with
$\mathfrak{M}_{i}=\\{-q_{i}\\}$. Choose some subset of roots
$E\subseteq\mathfrak{R}_{i}^{+}$ that we wish to add to $\mathfrak{M}_{i}$.
Since we want to preserve the condition of saturation, we also need to add to
$\mathfrak{M}_{i}$ all sums of the form $e+\sum_{s}e_{s}$, where $e\in E$,
$e_{s}\in\bigsqcup_{j>i}\mathfrak{M}_{j}$, and
$e+\sum_{s}e_{s}\in\mathfrak{R}_{i}^{+}$. After that we move to
$\mathfrak{M}_{i-1}$. In short, we iterate over the collection of subsets of
$\mathfrak{R}_{i}^{+}$ which are saturated with respect to
$\bigsqcup_{k>i}\mathfrak{M}_{k}$ and recursively repeat the procedure for
$\mathfrak{M}_{i-1}$.

Assume that we constructed $\mathfrak{M}_{1},\dots,\mathfrak{M}_{n}$. Let
$\mathfrak{M}=\mathfrak{M}_{1}\sqcup\ldots\sqcup\mathfrak{M}_{n}$. By Theorem
4.4 there exists a regular unipotent subgroup $U\subseteq\mathrm{Aut}(X)$ such
that $\mathfrak{R}(U)=\mathfrak{M}$ and $U$ acts on $X$ with an open orbit.
All regular unipotent subgroups that act on $X$ with an open orbit are
obtained this way for a suitable choice of intermediate subsets
$\mathfrak{M}_{i}$.

######  Example 4.10.

Let $n=3$, $m=4$, and $p_{4}=(-3,-2,-1)$. In this case $X$ is a weighted
projective space $\mathbb{P}(1,2,3)$. We have $1\succ 2\succ 3$ and

$\mathfrak{R}_{3}^{+}=\\{-q_{3}\\},$ $\mathfrak{R}_{2}^{+}=\\{-q_{2},\
-q_{2}+q_{3},\ -q_{2}+2q_{3}\\},$ $\mathfrak{R}_{1}^{+}=\\{-q_{1},\
-q_{1}+q_{2},\ -q_{1}+q_{2}+q_{3},\ -q_{1}+q_{3},\ -q_{1}+2q_{3},\
-q_{1}+3q_{3}\\}.$

By Corollary 3.5, we obtain
$U_{\mathrm{max}}\cong\big{(}\mathbb{G}_{a}\ltimes\mathbb{G}_{a}^{3}\big{)}\ltimes\mathbb{G}_{a}^{6}$.

Let us apply Algorithm 4.9. We start with $\mathfrak{M}_{3}=\\{-q_{3}\\}$. The
following cases are possible for $\mathfrak{M}_{2}$.

  1. (1)

If $\mathfrak{M}_{2}=\\{-q_{2}\\}$, then a subset
$\mathfrak{M}_{1}\subseteq\mathfrak{R}_{1}^{+}$ satisfies the constraint

$-q_{1}+b_{2}q_{2}+b_{3}q_{3}\in\mathfrak{M}_{1}\implies$
$-q_{1}+b^{\prime}_{2}q_{2}+b^{\prime}_{3}q_{3}\in\mathfrak{M}_{1}\ \text{for
all}\ b^{\prime}_{2}\leq b_{2},\ b^{\prime}_{3}\leq b_{3}.$

This case gives 11 subgroups: one of dimension 3, two of dimension 4, two of
dimension 5, three of dimension 6, two of dimension 7, and one of dimension 8.

  2. (2)

If $\mathfrak{M}_{2}=\\{-q_{2},-q_{2}+q_{3}\\}$, then
$\mathfrak{M}_{1}\subseteq\mathfrak{R}_{1}^{+}$ satisfies the constraint from
the first case and two additional ones:

$-q_{1}+q_{2}\in\mathfrak{M}_{1}\implies-q_{1}+q_{3}\in\mathfrak{M}_{1},$
$-q_{1}+q_{2}+q_{3}\in\mathfrak{M}_{1}\implies-
q_{1}+2q_{3}\in\mathfrak{M}_{1}.$

This case gives 9 subgroups: two for each of dimensions 6, 7, 8 and one for
each of dimensions 4, 5, and 9.

  3. (3)

If $\mathfrak{M}_{2}=\mathfrak{R}_{2}^{+}$, then there are two additional
constraints on $\mathfrak{M}_{1}\subseteq\mathfrak{R}_{1}^{+}$:

$-q_{1}+q_{2}\in\mathfrak{M}_{1}\implies-q_{1}+2q_{3}\in\mathfrak{M}_{1},$
$-q_{1}+q_{2}+q_{3}\in\mathfrak{M}_{1}\implies-
q_{1}+3q_{3}\in\mathfrak{M}_{1}.$

This case gives 7 subgroups: two of dimension 8 and one for each of dimensions
5, 6, 7, 9, and 10.

We conclude that there are 27 regular unipotent subgroups in
$U_{\mathrm{max}}$ that act on $\mathbb{P}(1,2,3)$ with an open orbit.

##  5\. Centers and central series

We proceed with a computation of the center of a regular unipotent subgroup.
Let $U$ be a regular unipotent subgroup of $\mathrm{Aut}(X)$ that acts on $X$
with an open orbit. Denote

$C(U)=\\{i\in C\mid\langle e,p_{i}\rangle\leq 0\ \forall
e\in\mathfrak{R}(U)\\}.$

######  Proposition 5.1.

The center $Z(U)$ of $U$ equals $\prod_{i\in C(U)}U_{-q_{i}}$. In particular,
$Z(U)$ is contained in a principal unipotent subgroup.

###### Proof.

Consider the center $\mathfrak{z}(\mathfrak{u})$ of the tangent algebra
$\mathfrak{u}=\mathrm{Lie}(U)$. Let
$\delta=\sum_{e\in\mathfrak{R}(U)}\alpha_{e}\partial_{e}\in\mathfrak{z}(\mathfrak{u})$.
Then by Lemma 2.1 for all $i\in C$ we have

$0=[\delta,\partial_{-q_{i}}]=\sum_{e:\langle
e,p_{i}\rangle>0}-\alpha_{e}\langle e,p_{i}\rangle\partial_{e-q_{i}}.$

So, $\alpha_{e}=0$ for any non-basic $e$, hence
$\delta=\sum_{i=1}^{n}\alpha_{-q_{i}}\partial_{-q_{i}}$. Furthermore, if
$-q_{j}+q_{k}\in\mathfrak{R}(U)$ for some $j\neq k$, then by Lemma 2.1

$0=[\delta,\partial_{-q_{j}+q_{k}}]=[\alpha_{-q_{k}}\partial_{-q_{k}},\partial_{-q_{j}+q_{k}}]=\alpha_{-q_{k}}\langle-
q_{j}+q_{k},p_{k}\rangle\partial_{-q_{j}}=\alpha_{-q_{k}}\partial_{-q_{j}}.$

By Lemma 2.4, we have $\delta=\sum_{i\in C(U)}\alpha_{i}\partial_{-q_{i}}$.
Conversely, every $\delta\in\mathfrak{u}$ of such form lies in
$\mathfrak{z}(\mathfrak{u})$. We conclude with the fact that
$\mathrm{Lie}(Z(U))=\mathfrak{z}(\mathfrak{u})$. ∎

######  Remark 5.2.

If $U=U_{\mathrm{max}}$, then $C(U)$ is the set of minimal numbers of classes
$C_{s}$ that are maximal with respect to the partial preorder $\succcurlyeq$.

In the rest of this section we describe the lower and upper central series of
a regular unipotent subgroup $U$ in terms of a directed graph on the set of
roots of $U$.

######  Definition 5.3.

Given a regular unipotent subgroup $U\subseteq U_{\mathrm{max}}$ with
$\mathfrak{M}=\mathfrak{R}(U)$, we define a directed graph
$\Gamma(\mathfrak{M})$ with $\mathfrak{M}$ as the set of vertices as follows.
There is an arrow from a root $a\in\mathfrak{M}$ to a root $b\in\mathfrak{M}$
if and only if there exists $e\in\mathfrak{M}$ such that $b=a+e$.

Let $k$ be a non-negative integer. We denote by $\mathfrak{M}^{\uparrow k}$
the subset of roots $b$ in $\mathfrak{M}$ such that there is a path of length
$k$ in $\Gamma(\mathfrak{M})$ ending in $b$. Further, let us denote by
$\mathfrak{M}^{\downarrow k}$ the subset of roots $a$ in $\mathfrak{M}$ such
that any path in $\Gamma(\mathfrak{M})$ starting in $a$ is of length less than
$k$.

######  Remark 5.4.

If there is an arrow from $a\in\mathfrak{M}_{i}$ to
$b=a+e\in\mathfrak{M}_{j}$, then $i\geq j$ and
$[\partial_{a},\partial_{e}]=d\partial_{b}$ for some $d\neq 0$ by Lemma 2.1.

We claim that the graph $\Gamma(\mathfrak{M})$ is acyclic. Assume to the
contrary that there is a cycle $S=(b_{0}=a,b_{1},\dots,b_{l-1},b_{l}=a)$. Then
each vertex of this cycle lies in the same $\mathfrak{M}_{i}$. For each
$k=\overline{1,l}$ denote $e_{k}=b_{k}-b_{k-1}$. We have
$a=a+\sum_{k=1}^{l}e_{k}$ or, equivalently, $\sum_{k=1}^{l}e_{k}=0$. This is a
contradiction, since due to Lemma 3.1 the sum of positive roots cannot be
zero.

######  Proposition 5.5.

Let $U$ be a regular unipotent subgroup and $l$ be the length of the longest
path in $\Gamma(\mathfrak{M})$, where $\mathfrak{M}=\mathfrak{R}(U)$. Then the
lower (resp. upper) central series of $U$ consists of subgroups
$U(\mathfrak{M}^{\uparrow k})$ (resp. $U(\mathfrak{M}^{\downarrow k})$) for
$k=\overline{0,l+1}$ in this order.

###### Proof.

We proceed by induction on $k$. If $k=0$, then $\mathfrak{M}^{\uparrow
k}=\mathfrak{M}$ and $\mathfrak{M}^{\downarrow k}=\emptyset$, so
$U(\mathfrak{M}^{\uparrow k})=U$ and $U(\mathfrak{M}^{\downarrow
k})=\\{\mathrm{id}\\}$ are the first subgroups of the corresponding central
series. Assume that $U(\mathfrak{M}^{\uparrow k})$ and
$U(\mathfrak{M}^{\downarrow k})$ are the $k^{\text{th}}$ subgroups of the
lower and upper central series, respectively. We use the fact that the tangent
algebra of the commutator of two closed connected subgroups is the commutator
of their tangent algebras; see [18, Theorem 24.5.11]. So the tangent algebra
$\mathrm{Lie}([U,U(\mathfrak{M}^{\uparrow k})])$ is spanned by all elements
$[\partial_{e},\partial_{a}]$, where $e\in\mathfrak{M}$ and
$a\in\mathfrak{M}^{\uparrow k}$. By Remark 5.4, they span the subspace
$\langle\partial_{b}\mid b\in\mathfrak{M}^{\uparrow k+1}\rangle$, hence
$[U,U(\mathfrak{M}^{\uparrow k})]=U(\mathfrak{M}^{\uparrow k+1})$.

Similarly, consider $h\in U$ such that $ghg^{-1}h^{-1}\in
U(\mathfrak{M}^{\downarrow k})$ for all $g\in U$. Then $h=\exp(\partial)$ for
some $\partial\in\mathrm{Lie}(U)$ and
$[\partial,\partial_{e}]\in\mathrm{Lie}(U(\mathfrak{M}^{\downarrow k}))$ for
all $e\in\mathfrak{M}$. So, $\partial$ is a linear combination of elements
$\\{\partial_{u}\mid u\in\mathfrak{M}^{\downarrow k+1}\\}$.

Finally, $\mathfrak{M}^{\uparrow l}\supsetneq\mathfrak{M}^{\uparrow
l+1}=\emptyset$ and $\mathfrak{M}^{\downarrow
l}\subsetneq\mathfrak{M}^{\downarrow l+1}=\mathfrak{M}$. ∎

######  Corollary 5.6.

The nilpotency class of $U_{\mathrm{max}}$ equals $l+1$, where $l$ is the
length of the longest path in $\Gamma(\mathfrak{R}^{+})$.

######  Remark 5.7.

The upper and lower central series of a regular unipotent group $U$ may not
coincide. Indeed, let $\mathfrak{M}_{2}=\\{-q_{2}\\}$ and
$\mathfrak{M}_{1}=\\{-q_{1},-q_{1}+q_{2},-q_{1}+2q_{2}\\}$. Then
$\mathfrak{M}^{\uparrow 1}=\\{-q_{1},-q_{1}+q_{2}\\}$, but
$\mathfrak{M}^{\downarrow 2}=\\{-q_{1},-q_{1}+q_{2},-q_{2}\\}$.

######  Corollary 5.8.

The derived length of $U_{\mathrm{max}}$ equals the minimal number $k$ such
that

$\underbrace{\Big{(}\cdots\big{(}(\mathfrak{R}^{+})^{\uparrow
1}\big{)}^{\uparrow 1}\cdots\Big{)}^{\uparrow 1}}_{k\text{ times}}=\emptyset$

###### Proof.

The claim is a direct implication of the equality
$\mathfrak{R}([U,U])=\mathfrak{R}(U)^{\uparrow 1}$ for any regular subgroup
$U\subseteq U_{\mathrm{max}}$. ∎

In particular, $U_{\mathrm{max}}$ is commutative if and only if the graph
$\Gamma(\mathfrak{R}^{+})$ has no arrow, and $U_{\mathrm{max}}$ is metabelian
if and only if the graph $\Gamma((\mathfrak{R}^{+})^{\uparrow 1})$ has no
arrow.

######  Example 5.9.

For the maximal unipotent subgroup $U_{\mathrm{max}}$ treated in Example 4.10
the longest path has length $4$ (see Figure 1), so the nilpotency class is
$5$. The derived series consists of $U_{\mathrm{max}}$,
$U(\mathfrak{R}^{\prime})$ with $\mathfrak{R}^{\prime}$ in Figure 2, and
$U(\\{-q_{1},-q_{1}+q_{3}\\})$, so the derived length equals $3$.

$q_{1}$$q_{2}$$q_{3}$$\mathfrak{R}_{3}$$\mathfrak{R}_{1}$$\mathfrak{R}_{2}$$-q_{3}$$-q_{2}$$-q_{1}$$-q_{2}+q_{3}$$-q_{2}+2q_{3}$white$-q_{1}+q_{3}$$-q_{1}+2q_{3}$$-q_{1}+3q_{3}$$-q_{1}+q_{2}$$-q_{1}+q_{2}+q_{3}$
Figure 1. The graph $\Gamma(\mathfrak{R})$ from Example 4.10. Here each third
of the plane represents the elements of $\mathfrak{R}_{i}$, $i=1,2,3$ in
suitable coordinates. The inner arrows are depicted dashed and outer ones
dotted.
$q_{1}$$q_{2}$$q_{3}$$\mathfrak{R}_{3}$$\mathfrak{R}_{1}$$\mathfrak{R}_{2}$$-q_{2}$$-q_{1}$$-q_{2}+q_{3}$white$-q_{1}+q_{3}$$-q_{1}+2q_{3}$$-q_{1}+3q_{3}$$-q_{1}+q_{2}$
Figure 2. The graph $\Gamma(\mathfrak{R}^{\prime})$ with
$\mathfrak{R}^{\prime}=\mathfrak{R}([U_{\mathrm{max}},U_{\mathrm{max}}])$ in
Example 4.10.

We end this section with the observation that the lower central series is
defined by a subgraph of $\Gamma(\mathfrak{M})$, which has simpler structure.

######  Definition 5.10.

We call an arrow in $\Gamma(\mathfrak{M})$ from $a$ to $b$ _inner_ , if
$a,b\in\mathfrak{M}_{i}$ for some $i$, and _outer_ otherwise.

######  Lemma 5.11.

Assume that there is an inner arrow $a\to b$ and an outer one $b\to c$ in
$\Gamma(\mathfrak{M})$. Then there exist an outer arrow $a\to b^{\prime}$ and
an inner one $b^{\prime}\to c$ for some vertex
$b^{\prime}\in\Gamma(\mathfrak{M})$.

###### Proof.

Let $a,b\in\mathfrak{M}_{j}$, $c\in\mathfrak{M}_{k}$, and the root $d=b-a$
belong to $\mathfrak{M}_{i}$. Then $i>j>k$ and the root $e=c-b$ belongs to
$\mathfrak{M}_{k}$.

Let us take $b^{\prime}=a+e$ and prove that $b^{\prime}\in\mathfrak{M}_{k}$.
The root $d$ has $-1$ at $i^{\text{th}}$ coordinate and $0$’s at
$j^{\text{th}}$ and $k^{\text{th}}$ coordinates. The roots $a,b$ have $-1$ at
$j^{\text{th}}$ coordinate and $0$ at $k^{\text{th}}$ coordinate. The roots
$c,e$ have $-1$ at $k^{\text{th}}$ coordinate and $e$ has positive
$j^{\text{th}}$ coordinate. These observations imply that $b^{\prime}$ has
$-1$ at $k^{\text{th}}$ coordinate and non-negative elsewhere.

So, by Proposition 2.2, since $-Ab^{\prime}=(-Aa)+(-Ae)$ has non-negative
coordinates, $b^{\prime}$ is indeed a root in $\mathfrak{M}_{k}$. The claim
follows. ∎

######  Proposition 5.12.

For any path ending in a vertex $v\in\Gamma(\mathfrak{M})$ there exists a path
of the same length, which ends in $v$ and consists of inner arrows.

###### Proof.

Consider a path $a_{1}\to\cdots\to a_{k}\to v$ of length $k$, which contains
an outer arrow. By Lemma 5.11, we can swap inner and outer arrows until the
first arrow $a_{1}\to a_{2}^{\prime}$ is outer.

Let $a_{2}^{\prime}=a_{1}+a_{1}^{\prime}$. Then the arrow $a_{1}^{\prime}\to
a_{2}^{\prime}$ is inner since $a_{1}^{\prime}$ and $a_{2}^{\prime}$ lie in
the same $\mathfrak{R}_{i}$. We obtain a path from $a_{1}^{\prime}$ to $v$ of
length $k$ with a smaller number of outer arrows. Proceeding in this manner,
we construct a desired path. ∎

######  Corollary 5.13.

The subgroups $U(\mathfrak{M}^{\uparrow i})$ do not change if we substitute
the initial graph $\Gamma(\mathfrak{M})$ by the subgraph of inner arrows.

######  Remark 5.14.

Proposition 5.12 shows that there is a (commutative) subgroup
$U(\mathfrak{M}_{i})$ in $U(\mathfrak{M})$ such that the intersections of the
subgroups in the lower central series with $U(\mathfrak{M}_{i})$ are pairwise
distinct. In particular, the nilpotency class of $U(\mathfrak{M})$ does not
exceed $\max_{i}|\mathfrak{M}_{i}|$.

##  6\. Toric equivariant completions of unipotent groups

In this section we consider equivariant completions of unipotent groups by
toric varieties. The following proposition shows that a complete toric variety
$X$ can not be realized as an equivariant completion of a non-commutative
regular unipotent subgroup.

######  Proposition 6.1.

Consider a complete toric variety $X$ with an acting torus $T$. Let a regular
unipotent subgroup $U$ of $\mathrm{Aut}(X)$ act on $X$ with an open orbit $O$
so that the action $U\curvearrowright O$ has trivial stabilizers. Then the
group $U$ is commutative.

###### Proof.

We may assume that $U$ is contained in $U_{\mathrm{max}}$. If $U$ is not
commutative, then by Theorem 4.4 $U$ contains a principal unipotent subgroup
as a proper subgroup. By Lemma 2.4 there is an elementary root $-q_{i}+q_{j}$
in $\mathfrak{R}(U)$. Then the one-parameter subgroups $U_{-q_{i}}$ and
$U_{-q_{i}+q_{j}}$ in $U$ acting on the spectrum of the Cox ring $R(X)$ change
only the coordinate $x_{i}$. This shows that generic orbits of $U_{-q_{i}}$
and $U_{-q_{i}+q_{j}}$ on $X$ coincide, and the action $U\curvearrowright O$
has non-trivial stabilizers, a contradiction. ∎

The next example demonstrates that the regularity condition is crucial.

######  Example 6.2.

For $n\geq 3$ we consider the subgroup

$U=\left\\{\left.\left(\begin{array}[]{cccccc}1&0&0&0&\cdots&0\\\
a_{1}&1&0&0&\cdots&0\\\ a_{2}&a_{n}&1&0&\cdots&0\\\ a_{3}&0&0&1&\cdots&0\\\
\vdots&&&&\ddots\\\ a_{n}&0&0&0&&1\\\
\end{array}\right)\right|a_{1},\ldots,a_{n}\in\mathbb{K}\right\\}\subset\mathrm{GL}(n+1,\mathbb{K}).$

with the action on $\mathbb{P}^{n}$ given in homogeneous coordinates by

$(a_{1},\ldots,a_{n})\cdot[z_{0}:\ldots:z_{n}]=[z_{0}:z_{1}+a_{1}z_{0}:z_{2}+a_{2}z_{0}+a_{n}z_{1}:z_{3}+a_{3}z_{0}:\ldots:z_{n}+a_{n}z_{0}].$

This subgroup is neither commutative nor regular. It acts on $\mathbb{P}^{n}$
with an open orbit and generic stabilizers are trivial.

The theory of equivariant completions of commutative unipotent groups is
nothing but the theory of additive actions. In the toric case this theory is
rather well-developed, see [3, 4, 10, 11, 14]. It will be interesting to study
(toric) equivariant completions of non-commutative unipotent groups.
Observations given above may serve as first steps towards this goal.

It is natural here to draw an analogy with the case of flag varieties. Take a
simple linear algebraic group $G$ and a parabolic subgroup $P$ in $G$. The
homogeneous space $G/P$ is a projective variety called a flag variety of the
group $G$. This variety contains an open Bruhat cell $O$, which is an orbit of
the unipotent radical $U$ of the opposite parabolic subgroup $P^{-}$, and the
action of $U$ on $O$ has trivial stabilizers. So $G/P$ is an equivariant
completion of $U$. Moreover, $U$ may be regarded as a regular unipotent
subgroup of $\mathrm{Aut}(G/P)$ since $U$ is normalized by a maximal torus in
$G$.

At the same time, if $G$ is the connected component of the group
$\mathrm{Aut}(G/P)$ (this is almost always the case), then $G/P$ admits an
additive action if and only if the group $U$ is commutative, and the list of
such cases is very short (see [1]). In particular, it may happen only if the
parabolic subgroup $P$ is maximal.

It is proved in [6, Theorem 1.1] that $G/P$ can be realized as an equivariant
completion of the unipotent radical $U$ in a unique way up to isomorphism,
provided $G/P$ is not isomorphic to a projective space.

##  7\. The case of toric surfaces

In this section we illustrate the results obtained above in the case of
surfaces. We keep the notation of the previous sections. Let $X$ be a radiant
toric surface; here $n=2$ and so $C=\\{1,2\\}$.

If 1 and 2 are $\succcurlyeq$-incomparable, then $\mathfrak{R}$ consists of
two basic roots and $U_{\mathrm{max}}=\mathbb{G}_{a}^{2}$ is the only
unipotent group that acts faithfully with an open orbit on $X$. Equivalently,
$X$ is a radiant toric variety of type I.

Assume now that $1\succcurlyeq 2$. Then

$\mathfrak{R}_{2}^{+}=\\{-q_{2}\\},\ \mathfrak{R}_{1}^{+}=\\{-q_{1},\
-q_{1}+q_{2},\ \dots,\ -q_{1}+dq_{2}\\}$

for some $d\geq 1$. By definition of a Demazure root, the number $d$ is
determined by the following conditions on elements of the ray matrix $A$:

$a_{k1}\geq da_{k2}\quad\text{for}\ \text{all}\ 3\leq k\leq
m\quad\text{and}\quad a_{k1}<(d+1)a_{k2}\quad\text{for}\ \text{some}\ 3\leq
k\leq m.$

In this case the group $U_{\mathrm{max}}$ is metabelian of nilpotency class
$d+1$ and

$U_{\mathrm{max}}\cong\mathbb{G}_{a}\ltimes\mathbb{G}_{a}^{d+1}.$

Applying Algorithm 4.9 to this setting, we see that there are $d+1$ non-
isomorphic regular unipotent subgroups that act faithfully with an open orbit
on $X$:

$U_{-q_{2}}\ltimes\big{(}U_{-q_{1}}\times U_{-q_{1}+q_{2}}\times\cdots\times
U_{-q_{1}+lq_{2}}\big{)}\cong\mathbb{G}_{a}\ltimes\mathbb{G}_{a}^{l+1},\quad
0\leq l\leq d.$

By Proposition 3.3.(2) the action on normal subgroup is defined by

$u_{-q_{2}}(-\alpha^{\prime})u_{-q_{1}+kq_{2}}(\alpha)u_{-q_{2}}(\alpha^{\prime})=$
$=\prod_{i=0}^{k}u_{-q_{1}+(k-i)q_{2}}\left(\binom{k}{i}\alpha(\alpha^{\prime})^{i}\right)=\prod_{j=0}^{k}u_{-q_{1}+jq_{2}}\left(\binom{k}{j}\alpha(\alpha^{\prime})^{k-j}\right).$

Now let us take a closer look at smooth toric surfaces. Such a surface is
given by primitive vectors $p_{0},p_{1},\ldots,p_{m},p_{m+1}$ with
$p_{0}=p_{m}$ and $p_{1}=p_{m+1}$ such that any pair $p_{s},p_{s+1}$ forms a
basis of the lattice $N=\mathbb{Z}^{2}$. In this case
$p_{s-1}+p_{s+1}=c_{s}p_{s}$ for some integer numbers $c_{s}$,
${s=\overline{1,m}}$; see [15, Section 1.7] or [13, Section 2.5]. Clearly, the
sequence $(c_{1},\ldots,c_{m})$ determines a smooth complete toric surface
uniquely.

######  Proposition 7.1.

Let $X$ be a smooth complete toric surface given by a sequence
$(c_{1},\ldots,c_{m})$. Then $X$ is radiant if and only if the sequence
$(c_{1},\ldots,c_{m},c_{1})$ contains at least two standing nearby non-
positive numbers.

###### Proof.

A complete toric surface $X$ is radiant if and only if for some $s$ all the
vectors $p_{i}$, $i\neq s,s+1$ are contained in the negative orthant with
respect to the basis $p_{s},p_{s+1}$. We have

$p_{s-1}=c_{s}p_{s}-p_{s+1},\quad p_{s+2}=c_{s+1}p_{s+1}-p_{s},$

and all other vectors $p_{i}$ lie between $p_{s-1}$ and $p_{s+2}$. This
implies the claim. ∎

For $m=3$ we have only the projective plane $\mathbb{P}^{2}$ represented by
the sequence $(-1,-1,-1)$. The case $m=4$ corresponds to Hirzebruch surfaces
$\mathbb{F}_{q}$ represented by sequences $(0,q,0,-q)$ for some non-negative
integer $q$. Since any smooth complete toric surface can be obtained either
from $\mathbb{P}^{2}$ or from $\mathbb{F}_{q}$ by a sequence of blow-ups at
$T$-fixed points, any sequence $(c_{1},\ldots,c_{m})$ can be obtained either
from $(-1,-1,-1)$ or from $(0,q,0,-q)$ by a series of operations, which add
$1$ to some elements $c_{s}$ and $c_{s+1}$ (we assume that $c_{m+1}=c_{1}$)
and insert $1$ between them; see [15, Section 1.7] or [13, Section 2.5] for
details.

Proposition 7.1 and the description of sequences $(c_{1},\ldots,c_{m})$ given
above imply that for $m\leq 5$ all smooth complete toric surfaces are radiant.
For $m=6$ we have the blow-up of $\mathbb{P}^{2}$ at three $T$-fixed points
corresponding to the sequence $(1,1,1,1,1,1)$, which is not radiant.

It is an interesting problem to study smooth radiant toric varieties in higher
dimensions.

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