Lines in affine simlicial toric varieties

Shulim Kaliman

Department of Mathematics

University of Miami

Coral Gables, FL 33124, U.S.A.

ABSTRACT. We prove that up to automorphisms ${\mathbb{A}}_{\rm\bf k}^{1}$
admits a unique embedding into the regular part of ${\mathbb{A}}_{\rm\bf
k}^{n}/G$ where ${\rm\bf k}$ is an algebraically closed field of
characteristic zero, $n\geq 4$, $G$ is a finite subgroup of $\mathop{\rm
SL}_{n}({\rm\bf k})$ acting naturally on ${\mathbb{A}}_{\rm\bf k}^{n}$ and
${\mathbb{A}}_{\rm\bf k}^{n}/G$ is smooth in codimension 2.

##  1 Introduction

Let $\varphi:C\to C^{\prime}$ be an isomorphism of two smooth polynomial
curves contained in the regular part $Y_{\rm reg}$ of an affine algebraic
variety $Y$ over an algebraically closed field ${\rm\bf k}$ of characteristic
zero. It may happen that $\varphi$ extends to an automorphism of $Y$ and our
aim is to describe some affine algebraic varieties for which this extension
takes place. This problem was studied in several papers [AMo], [Su],
[Cr],[Je], [St],[FS], [Ka20] and [AZ]. It turns out that the answer is
positive for some classes of flexible varieties of dimension $n\geq 4$ where
$Y$ is flexible if the subgroup $\mathop{\rm SAut}(Y)$ of the automorphism
group $\operatorname{{\rm Aut}}(Y)$ of $Y$ generated by all one-parameter
unipotent subgroups acts transitively on $Y_{\rm reg}$. Say, this is so if
$Y={\mathbb{A}}^{n}$ with $n\geq 4$ [Cr],[Je]. For $n=3$ the answer is unknown
but for $n=2$ the famous Abhyankar-Moh-Suzuki theorem [AMo],[Su] states that
an isomorphism of two smooth plane polynomial curves always extends to an
automorphism of the plane ${\mathbb{A}}^{2}$. Perhaps, ${\mathbb{A}}^{2}$ is
the only example of a two-dimensional flexible variety with this property. If
$Y$ is an affine simplicial toric variety ${\mathbb{A}}^{2}/G$ where $G$ is a
finite subgroup of $\mathop{\rm SL}_{2}({\rm\bf k})$ acting naturally on
${\mathbb{A}}^{2}$, then Arzhantsev and Zaidenberg [AZ] showed that the answer
is negative. They actually classified up to automorphisms all smooth
polynomial curves in $Y_{\rm reg}$ (there are only finite number of
isomorphism classes of such curves). In this paper we study the case when $Y$
is an affine simplicial toric variety of dimension $n\geq 4$ (i.e.,
$Y={\mathbb{A}}^{n}/G$ where $G$ is a finite subgroup of $\mathop{\rm
SL}_{n}({\rm\bf k})$ acting naturally on ${\mathbb{A}}^{n}$). We show that the
answer to this extension problem is positive under the assumption of
smoothness in codimension 2. Furthermore, recall that given a subvariety $Z$
of $Y$ with defining ideal $I$ in the algebra ${\rm\bf k}[Y]$ of regular
functions on $Y$ its $k$-infinitesimal neighborhood is the scheme with the
defining ideal $I^{k}$. In particular, if $W$ is another subvariety of $Y$
with defining ideal $J$, then an isomorphism ${\mathcal{Z}}\to{\mathcal{W}}$
of $k$-infinitesimal neighborhoods of $Z$ and $W$ is determined by an
isomorphism of algebras $\frac{{\rm\bf k}[Y]}{I^{k}}\to\frac{{\rm\bf
k}[Y]}{J^{k}}$. There are natural obstacles for extending such isomorphisms to
automorphisms of $Y$. Say, let $Z=W$ be a strict complete intersection given
in $Y$ by $u_{1}=\ldots=u_{k}=0$. Then an automorphism $\psi:\frac{{\rm\bf
k}[Y]}{I^{k}}\to\frac{{\rm\bf k}[Y]}{I^{k}}$ over $\frac{{\rm\bf k}[Y]}{I}$ is
given by polynomials $f_{1},\ldots,f_{k}$ in $u_{1},\ldots,u_{k}$ over
$\frac{{\rm\bf k}[Y]}{I}$ of degree at most $k-1$. If $Y$ does not admit
nonconstant invertible functions and $\psi$ is extendable to an automorphism
of $Y$, then one can see that the Jacobian $\det\left[\frac{\partial
f_{i}}{\partial u_{j}}\right]_{i,j=1}^{k}$ must be equal to a nonzero constant
modulo $I^{k-1}$ in which case we say that $\psi$ has a nonzero constant
Jacobian. There is also a notion of a nonzero constant Jacobian of an
isomorphism ${\mathcal{Z}}\to{\mathcal{W}}$ in the case when both $Z$ and $W$
are smooth polynomial curve in a normal toric variety $Y$ contained in $Y_{\rm
reg}$ (see Definition 5.4). The question when such isomorphisms with nonzero
constant Jacobians are extendable to automorphisms of $Y$ was considered in
[KaUd] and [Ud]. In combinations with the results of [KaUd] and [Ud] we get
the following (Corollary 5.5).

######  Theorem 1.1.

Let $Y$ be an affine simplicial toric variety smooth in codimension 2 such
that $\dim Y\geq 4$ and $\varphi:{\mathcal{C}}_{1}\to{\mathcal{C}}_{2}$ be an
isomorphism of $k$-infinitesimal neighborhoods of two smooth polynomial curves
contained in $Y_{\rm reg}$ such that the Jacobian of $\varphi$ is a nonzero
constant. Then $\varphi$ extends to an automorphism of $Y$.

Since the paper is heavily based on the technique developed in [Ka20] which
was later clarified in [KaUd] we put some necessary facts from [Ka20] and
[KaUd] in Appendix. In particular, one can find there formal definitions of
locally nilpotent vector fields and flexible varieties. In Section 2 we
introduce notations for toric varieties which are used freely throughout the
paper and prove some simple facts about normal affine toric varieties. In
Section 3 we give an analogue of Holme’s theorem [Hol] about closed embeddings
of affine varieties into ${\mathbb{A}}^{n}$ for embeddings into normal toric
varieties without torus factors. In particular, we prove that if an affine
curve can be embedded as a closed subvariety into some ${\mathbb{A}}^{n}$
(where $n$ is always at least 4), then it can be also embedded as a closed
subvariety into a normal affine toric variety of dimension $n$ with no torus
factors so that its image is contained in the regular part of the toric
variety. In Section 4 we study locally nilpotent vector fields on normal
affine toric varieties with no torus factors. The properties of locally
nilpotent vector fields are crucial for us since compositions of elements of
the flows of such vector fields produce automorphisms that extend isomorphisms
of smooth polynomial curves. In Section 5 we prove our main result.

##  2 Notations and Preliminaries

We suppose that readers are familiar with toric varieties and all information
about toric varieties which is used below can be found in the book of Cox,
Little and Schrenck [CLS]). We fix the following notations for the rest of the
paper.

\- $N\simeq\mathbb{Z}^{n}$ \- the standard lattice in $\mathbb{R}^{n}$;

\- $M=\operatorname{{\rm Hom}}_{\mathbb{Z}}(N,\mathbb{Z})$ \- the lattice dual
to $N$;

\- $<m,u>$ \- pairing of $m\in M$ or
$M_{\mathbb{R}}=M\otimes_{\mathbb{Z}}\mathbb{R}$ with $u\in N$ or
$N_{\mathbb{R}}=N\otimes_{\mathbb{Z}}\mathbb{R}$;

\- $\sigma$ \- a rational convex polyhedral cone in $N_{\mathbb{R}}$;

\- $\sigma^{\vee}$ \- the dual cone of $\sigma$ in $M_{\mathbb{R}}$;

\- $\gamma^{\bot}$ \- the set $\\{m\in M_{\mathbb{R}}|<m,\gamma>=0\\}$ where
$\gamma$ is any face of $\sigma$;

\- $X_{\sigma}$ \- the toric variety of $\sigma$, i.e., $X_{\sigma}$ is the
spectrum of the group algebra of the semigroup $\sigma^{\vee}\cap M$;

\- $\mathbb{T}=\operatorname{{\rm Hom}}(M,\mathbb{G}_{m})$ \- the torus acting
on $X_{\sigma}$;

– $\\{\varrho_{1},\ldots,\varrho_{r}\\}$ \- the set of extremal rays of
$\sigma$ (by abusing notations we also denote by $\varrho_{i}$ the ray
generator, i.e., the primitive lattice vector on the corresponding ray);

\- $\sigma(k)$ \- the set of $k$-dimensional faces of $\sigma$ (e.g.,
$\varrho_{i}\in\sigma(1)$);

\- $O_{i}$ \- the $\mathbb{T}$-orbit in $X_{\sigma}$ corresponding to
$\varrho_{i}$ by the Orbit-Cone correspondence [CLS, Theorem 3.2.6];

\- $D_{i}$ \- the irreducible $\mathbb{T}$-invariant Weil divisor in
$X_{\sigma}$ containing $O_{i}$ as an open subset (i.e., $D_{i}$ is the
spectrum of the semigroup algebra of
$\tau_{i}=\varrho^{\bot}\cap\sigma^{\vee}\cap M$);

\- $H_{i}$ \- the $\mathbb{G}_{m}$-subgroup of $\mathbb{T}$ corresponding to
$\varrho_{i}$, i.e., $H_{i}$ is a unique $\mathbb{G}_{m}$-subgroup of
$\mathbb{T}$ that acts trivially on $D_{i}$ and for $t\in H_{i}$ one has
$t.\chi^{m}=t^{<m,\varrho_{i}>}\chi^{m}$.

We would like to remind that since $\sigma^{\vee}\cap M$ and
$\tau_{i}=\varrho^{\bot}\cap\sigma^{\vee}\cap M$ are saturated affine
semigroups the varieties $X_{\sigma}$ and $D_{i}$ are always normal (e.g., see
[CLS, Theorem 1.3.5]). Furthermore, we consider only the case when
$X_{\sigma}$ has no torus factors (or, equivalently, every invertible function
on $X_{\sigma}$ is constant).

Let $X_{\Sigma}$ be the toric variety of a fan $\Sigma$ and $r$ be the
cardinality of one-dimensional cones in $\Sigma$ (in particular, if
$\Sigma=\sigma$ then $r$ is the number of the ray generators $\varrho$ of
$\sigma$). If torus factors are absent, then by [CLS, Theorem 5.1.10] there
exists a subgroup $G$ of $\mathbb{G}_{m}^{r}$ (which is a product of a torus
and a finite group) and a closed subvariety $Z(\Sigma)$ of ${\mathbb{A}}^{r}$
such that $Z(\Sigma)$ is invariant under the natural action of $G$ on
${\mathbb{A}}^{r}$ and $X_{\Sigma}$ is isomorphic to
$({\mathbb{A}}^{r}\setminus Z(\Sigma))//G$ (while $\mathbb{G}_{m}^{r}/G$ is
isomorphic to the torus $\mathbb{T}$ acting on $X_{\Sigma}$). We are dealing
with the situation when $\Sigma=\sigma$ and in this case $Z(\sigma)$ is empty
by construction (see the definition of $Z(\sigma)$ on [CLS, page 206]). Thus,
we have the quotient morphism

$\pi:{\mathbb{A}}^{r}\to{\mathbb{A}}^{r}//G\simeq X_{\sigma}$ (1)

which is $\mathbb{G}_{m}^{r}$-equivariant. In connection with this formula we
fix the following notations.

\- $x_{1},\ldots,x_{r}$ \- a fixed coordinate system on ${\mathbb{A}}^{r}$;

\- ${\tilde{T}}=\mathbb{G}_{m}^{r}$\- the standard torus (with respect to the
coordinate system) acting on ${\mathbb{A}}^{r}$;

\- ${\tilde{D}}_{i}$ \- the hyperplane in ${\mathbb{A}}^{r}$ given by
$x_{i}=0$;

\- ${\tilde{H}}_{i}$ \- the $\mathbb{G}_{m}$-subgroup of ${\tilde{T}}$ acting
trivially on ${\tilde{D}}_{i}$, i.e., this action is the flow of the
semisimple vector field $x_{i}\frac{\partial}{\partial x_{i}}$;

\- $U$ \- the subset of $X_{\sigma}$ consisting of all point $u\in X_{\sigma}$
for which $\pi^{-1}(u)$ is a $G$-orbit (in particular, this orbit is closed in
${\mathbb{A}}^{r}$);

\- $U_{0}$ \- the subset of $X_{\sigma}$ consisting of all point $u\in U$ for
which $\pi^{-1}(u)$ is naturally isomorphic to $G$.

Let us list some properties of the morphism $\pi:{\mathbb{A}}^{r}\to
X_{\sigma}$ and the objects introduced before.

######  Lemma 2.1.

(i) The morphism $\pi$ is an almost geometric quotient, i.e., $U$ is an open
dense subset of $X_{\sigma}$ and, consequently, general orbits of $G$ in
${\mathbb{A}}^{r}$ are closed and isomorphic to $G$. Furthermore, for every
$u$ in $U$ the fiber $\pi^{-1}(u)$ is isomorphic as a homogeneous $G$-space to
$G/F$ where $F$ is a finite subgroup of $G$.

(ii) The group $G\cap{\tilde{H}}_{i}$ is trivial.

(iii) Let $\theta\in\sigma(k)$ be regular, i.e., the set
$\\{\varrho_{i_{1}},\ldots,\varrho_{i_{k}}\\}$ of the generators of the
extremal rays of $\theta$ can be extended to a basis of $N$. Then $G$ meets
the group ${\tilde{F}}$ generated by
${\tilde{H}}_{i1},\ldots,{\tilde{H}}_{i_{k}}$ at identity only.

(iv) The homomorphism $\pi^{*}:{\rm\bf k}[X_{\sigma}]\to{\rm\bf
k}[x_{1},\ldots,x_{r}]$ induced by $\pi$ is determined by the formula

$\pi^{*}(\chi^{m})=\prod_{l=1}^{r}x_{l}^{<m,\varrho_{l}>}.$ (2)

(v) The image of ${\tilde{H}}_{i}$ in $\mathbb{T}=\frac{{\tilde{T}}}{G}$
coincides with $H_{i}$ and $\pi({\tilde{D}}_{i})=D_{i}$.

###### Proof.

For the fist statement of (i) see [CLS, Theorem 5.1.10]. This implies that
$\dim G=\dim A^{r}-\dim X_{\sigma}=r-n$. For every $u\in U$ one has
$\dim\pi^{-1}(u)=r-n$ since it cannot be less by Chevalley’s theorem [Ha,
Chapter II, Exercise 3.22] and it cannot be larger since $\pi^{-1}(u)$ is a
$G$-orbit. Being a homogeneous space $\pi^{-1}(u)$ is of the form $G/F$ where
$F$ is a subgroup of $G$ and the condition on the dimension implies that $F$
is finite. This concludes (i).

By [CLS, Lemma 5.1.1] we have

$G=\\{\bar{t}=(t_{1},\ldots,t_{r})\in\mathbb{G}_{m}^{r}|\,\prod_{i=1}^{r}t_{i}^{<\varrho,m>}=1\text{
for all }m\in M\\}.$ (3)

Suppose that $G\cap{\tilde{H}}_{i}$ contains a finite subgroup of $d$-roots of
unity. Assume that $d>1$ and that $\varepsilon$ is a primitive $d$-root of
unity. Then Formula (3) implies that $\varepsilon^{<m,\varrho_{i}>}=0$ for
every $m\in M$, i.e., $<m,\varrho_{i}>$ is divisible by $d$. Hence,
$\frac{\varrho_{i}}{d}\in N$ contrary to the fact that $\varrho_{i}$ is a
primitive vector in the lattice $N$. This yields (ii).

In (iii) let $G\cap{\tilde{F}}$ contains a subgroup isomorphic to the group of
$d$-roots of unity. Then the similar argument implies that for a collection
$\\{l_{1},\ldots,l_{s}\\}$ of integers with greatest common divisor 1 the sum
$\sum_{s=1}^{k}l_{s}\varrho_{i_{s}}$ is divisible by $d$. However, if $d>1$,
then this is contrary to the fact that the set
$\\{\varrho_{i_{1}},\ldots,\varrho_{i_{k}}\\}$ is extendable to a basis of
$N$. Thus, we have (iii).

For (iv) see [CLS, page 209]. Statement (v) follows from the explicit
construction of $\pi$ in [CLS, Proposition 5.1.9] which implies, in particular
(iv). Vice versa, (v) can be also illustrated by Formula (2). Indeed, this
formula implies that for every
$\chi^{m},\,m\in\tau_{i}=\varrho^{\bot}\cap\sigma^{\vee}\cap M$ the function
$\pi^{*}(\chi^{m})$ is independent of $x_{i}$, i.e., it is fixed under the
${\tilde{H}}_{i}$-action. In particular, $\chi^{m}$ is fixed under the action
of the image $H_{i}^{\prime}$ of ${\tilde{H}}_{i}$ in
$\mathbb{T}=\frac{{\tilde{T}}}{G}$ (which is isomorphic to ${\tilde{H}}_{i}$
by (ii)). Hence, the $H_{i}^{\prime}$-action on $D_{i}$ is trivial. Since
$H_{i}$ is a unique $\mathbb{G}_{m}$-subgroup of $\mathbb{T}$ with this
property we see that $H_{i}^{\prime}=H_{i}$. In particular, the divisor
$\pi^{-1}(D_{i})$ must be fixed under the ${\tilde{H}}_{i}$-action which
implies that $\pi^{-1}(D_{i})={\tilde{D}}_{i}$ and we are done. ∎

Recall that given an affine algebraic group $H$ acting on an affine variety
$Z$, an affine variety $Y$ with a trivial $H$-action and an $H$-equivariant
morphism $\varrho:Z\to Y$ we call $Z$ together with the morphism $\varrho:Z\to
Y$ a principal $H$-bundle if for every $y\in Y$ there exists an étale morphism
$\varphi_{y}:W_{y}\to Y$ such that $y\in{\rm Im}\,\varphi_{y}$ and
$W_{y}\times_{Y}Z$ becomes a trivial principal $H$-bundle under the natural
$H$-action. If each $\varphi_{y}$ is injective, then we call $Z$ a locally
trivial principal $H$-bundle.

######  Proposition 2.2.

(1) The morphism $\pi|_{\pi^{-1}(U_{0})}:\pi^{-1}(U_{0})\to U_{0}$ is a
principal $G$-bundle (in particular, $\pi$ is smooth over $U_{0}$).
Furthermore, if $G$ is connected, then this principal $G$-bundle is locally
trivial.

(2) Let $E$ be the subset of ${\mathbb{A}}^{r}$ consisting of all
$\bar{x}=(x_{1},\ldots,x_{r})\in{\mathbb{A}}^{r}$ such that at most one
coordinate $x_{i}$ is equal to zero. Then $\pi(E)\subset U_{0}$.

(3) $U_{0}$ is the regular part of $X_{\sigma}$.

###### Proof.

If $u\in U_{0}$ and $w\in\pi^{-1}(u_{0})$, then by the Luna étale slice
theorem [Lu] (see also [PV, Theorem 6.4]) there exists a smooth subvariety $V$
of $X_{\sigma}$ transversal to $\pi^{-1}(u)$ at $w$ such that $\pi|_{V}:V\to
U_{0}$ is étale. This implies that $U_{0}$ is contained in the regular part of
$X_{\sigma}$ and that $\pi$ is smooth over $U_{0}$. Furthermore, the natural
$G$-action makes $V\times_{U_{0}}{\mathbb{A}}^{r}$ a trivial $G$-bundle.
Hence, $\pi^{-1}(U_{0})\to U_{0}$ is a principal $G$-bundle. Recall also that
if $G$ is connected, then $G$ is a special group in the sense of Serre (see,
[Gro58, Def. 2 and page 16]) and for any special group $K$ every $K$-principal
bundle is locally trivial [Gro58, Theorem 3]. Thus, we have (1).

For (2) and (3) we need to recall that by Orbit-Cone correspondence [CLS,
Theorem 3.2.6] every $\theta\in\sigma(k)$ corresponds to a $\mathbb{T}$-orbit
$O(\theta)\subset X_{\sigma}$ of dimension $n-k$ where $O(\theta)$ is the
orbit of a so-called distinguished point. The description of this point [CLS,
page 116] implies that $O(\theta)$ consists of all points $u\in X_{\sigma}$
such that $\chi^{m}(u)\neq 0$ if and only if $m\in\theta^{\bot}\cap M$. In
particular, the ring ${\rm\bf k}[R]$ of regular functions on the closure $R$
of ${O(\theta)}$ in $X_{\sigma}$ can be viewed as the semigroup algebra of
$\theta^{\bot}\cap M$. Let $\varrho_{i_{1}},\ldots,\varrho_{i_{k}}$ be the
extremal rays generating $\theta$ (i.e.,
$\theta^{\bot}=\varrho_{i_{1}}^{\bot}\cap\ldots\cap\varrho_{i_{k}}^{\bot}$)
and $F$ be the subgroup of $\mathbb{T}$ generated by
$H_{i_{1}},\ldots,H_{i_{k}}$. Note that the natural inclusion ${\rm\bf
k}[R]\hookrightarrow{\rm\bf k}[X_{\sigma}]$ makes ${\rm\bf k}[R]$ the subring
of $F$-invariants and $R$ is given in $X_{\sigma}$ by the ideal generated by
$\\{\chi^{m}|m\in(\sigma^{\bot}\setminus\theta^{\bot})\cap M\\}$. Hence, $R$
is the fixed point set of the $F$-action since for every $v\in
X_{\sigma}\setminus R$ one can find
$m\in(\sigma^{\bot}\setminus\theta^{\bot})\cap M$ with $\chi^{m}(v)\neq 0$.
The difference between the points of $O(\theta)$ and $R\setminus O(\theta)$ is
that for $w\in R\setminus O(\theta)$ there exists
$j\in\\{1,\ldots,r\\}\setminus\\{i_{1},\ldots,i_{k}\\}$ such that $w$ is also
fixed under the $H_{j}$-action (because, $w$ is contained in a
$\mathbb{T}$-orbit of a smaller dimension corresponding to a cone in $\sigma$
containing $\theta$ and some $\varrho_{j}$), whereas for any point in
$O(\theta)$ such $j$ does not exist.

Let $\bar{x}\in{\tilde{T}}=E\setminus\bigcup_{i=1}^{r}{\tilde{D}}_{i}$. Since
$\pi$ is ${\tilde{T}}$-equivariant $G.\bar{x}$ is a general orbit, i.e.,
$\pi(\bar{x})\in U_{0}$ by Lemma 2.1 (i). Since $E\cap{\tilde{D}}_{i_{1}}$ is
a ${\tilde{T}}$-orbit dense in ${\tilde{D}}_{i_{1}}$ and
$\pi({\tilde{D}}_{i_{1}})=D_{i_{1}}$ by Lemma 2.1 (v) we see that
$\pi(E\cap{\tilde{D}}_{i_{1}})=O_{i_{1}}$. For every $\bar{x}\in
E\cap{\tilde{D}}_{i_{1}}$ its ${\tilde{T}}$-orbit $Q$ is naturally isomorphic
to $G$ by Lemma 2.1 (ii) and $\pi(x)=u\in O_{i_{1}}$. Note that if $Q$ is not
closed, then its closure contain a point with some coordinates $x_{j}=0$ where
$j\neq i_{1}$. However, this implies that $u$ is a fixed point under both
$H_{i_{1}}$-action and $H_{j}$-action contrary to the argument before. Hence,
$Q$ is closed. Furthermore, $Q$ is a unique closed $G$-orbit in $\pi^{-1}(u)$
by [CLS, Theorem 5.0.7]. If there exists another orbit $Q^{\prime}$ in
$\pi^{-1}(u)$, then the closure of $Q^{\prime}$ contains $Q$ (e.g., see [PV,
Theorem 4.7 and Corollary]). However, this is impossible since $\dim
Q^{\prime}\leq\dim Q=\dim G$. Hence, $\pi^{-1}(u)=Q$ and $u\in U_{0}$ which is
(2).

Consider $u$ in the smooth part of $X_{\sigma}$. Then $u$ is contained in some
$O(\theta)$ as before where $\theta$ must be regular by [CLS, Theorem 1.3.12
and Example 1.2.20]. Let ${\tilde{O}}(\theta)\subset{\mathbb{A}}^{r}$ be the
${\tilde{T}}$-orbit consisting of all points $\bar{x}$ whose zero coordinates
are exactly $x_{i_{1}},\ldots,x_{i_{k}}$. Let $\theta^{\prime}$ be a cone in
$\sigma$ properly contained in $\theta$, i.e., $O(\theta)$ is contained in the
closure of $O(\theta^{\prime})$. Let us, say, that $\theta^{\prime}$ is
generated by extremal rays $\varrho_{i_{2}},\ldots,\varrho_{i_{k}}$. Then we
can suppose by induction that such $\theta^{\prime}$ is regular and that
$\pi({\tilde{O}}(\theta^{\prime}))=O(\theta^{\prime})\subset U_{0}$. In
particular, $\pi^{-1}(u)$ belongs to the closure of
${\tilde{O}}(\theta^{\prime})$ and
$\pi^{-1}(u)\cap{\tilde{O}}(\theta^{\prime})=\emptyset$. This implies that
every $\bar{x}\in\pi^{-1}(u)$ cannot have a nonzero coordinate $x_{i_{1}}$.
Hence, $\bar{x}$ must be contained in ${\tilde{O}}(\theta)$ (indeed, if
$\bar{x}$ has a zero coordinate $x_{j}$ with
$j\notin\\{i_{1},\ldots,i_{k}\\}$, then $u$ is fixed under the $H_{j}$-action
contrary to the argument before). This implies that the ${\tilde{T}}$-orbit
$Q$ of $\bar{x}$ is closed since otherwise its closure contains a point with
an undesirable zero coordinate. By Lemma 2.1(iii) $Q$ is naturally isomorphic
to $G$ and arguing as before we see that $\pi^{-1}(u)=Q$. Hence, $u\in U_{0}$
which yields (3) and concludes the proof. ∎

######  Corollary 2.3.

Let $Y$ be a open subset of $X_{\sigma}$ such that ${\rm
codim}_{X_{\sigma}}X_{\sigma}\setminus Y\geq 2$. Then
$\pi^{-1}(X_{\sigma}\setminus Y)$ has codimension at least 2 in
${\mathbb{A}}^{r}$.

###### Proof.

Note that $\pi^{-1}(Y)\subset\pi^{-1}(U\cap Y)\cup({\mathbb{A}}^{r}\setminus
E)$. The definition of $U$ implies that $\pi^{-1}(U\setminus Y)$ has
codimension at least 2 in ${\mathbb{A}}^{r}$ and the same is true for
$A^{r}\setminus E$. Hence, we have the desired conclusion. ∎

######  Corollary 2.4.

Let $C$ be a closed curve in $X_{\sigma}$ contained in $U_{0}$. Suppose that
either

(1) $C$ is isomorphic to the line ${\mathbb{A}}^{1}$ or

(2) $G$ is connected and $C$ is a smooth rational curve.

Then there exists a closed curve ${\tilde{C}}\subset{\mathbb{A}}^{r}$ such
that $\pi|_{\tilde{C}}:{\tilde{C}}\to C$ is an isomorphism.

###### Proof.

By Proposition 2.2 $\pi^{-1}(C)$ is a locally trivial principal $G$-bundle.
Statement (1) now follows from [FS, Theorem A.1] which states that for each
affine algebraic group $F$ every principal $F$-bundle over a line admits a
section.

In (2) by Proposition 2.2(1) we can find an open cover $\\{V_{i}\\}$ of $C$
for which $\pi^{-1}(V_{i})$ is naturally isomorphic to $V_{i}\times G$. In
particular, one has sections $s_{i}:V_{i}\to\pi^{-1}(V_{i})$ and
$s_{j}|_{V_{i}\cap V_{j}}=g_{ij}s_{i}$ where $g_{ij}:V_{i}\cap V_{j}\to G$ is
a morphism. Since $G\simeq\mathbb{G}_{m}^{r-n}$ we see that $g_{ij}$ can be
presented as a collection of $r-n$ sections of ${\mathcal{O}}_{C}^{*}$ over
$V_{i}\cap V_{j}$. Hence, $H^{1}(C,G)$ is the direct sum of $r-n$ samples of
$H^{1}(C,{\mathcal{O}}_{C}^{*})$. Since $C$ is a smooth rational curve we have
$H^{1}(C,{\mathcal{O}}_{C}^{*})=\mathop{\rm Pic}C=0$ and, hence,
$H^{1}(C,G)=0$. Thus, we can suppose that every pair of sections $s_{i}$ and
$s_{j}$ agree on $V_{i}\cap V_{j}$. Consequently, we have a global section of
$\pi|_{\pi^{-1}(C)}:\pi^{-1}(C)\to C$ which yields the desired conclusion. ∎

##  3 Embedding Theorem

######  Lemma 3.1.

Let $\pi:{\mathbb{A}}^{r}\to X_{\sigma}$ be as in Formula (1) and $A$ be the
algebra of polynomials on ${\mathbb{A}}^{r}$ invariant under the $G$-action
(i.e., $A$ can be viewed as the algebra of regular functions on $X_{\sigma}$),
let $\bar{0}$ be the origin in ${\mathbb{A}}^{r}$ and $o=\pi(\bar{0})$. Then
one can choose a collection of monomials as generators of $A$ and the set $V$
of common zeros of this collection is contained in $\pi^{-1}(o)$. In
particular, unless $X_{\sigma}=U$ (which is the case of a simplicial $\sigma$
by [CLS, Theorem 5.1.10]) $V\subset\pi^{-1}(X_{\sigma}\setminus U)$.

###### Proof.

Since the natural ${\tilde{T}}$-action respects monomials the same is true for
the $G$-action. Thus, any $G$-invariant polynomial is the sum of $G$-invariant
monomials which yields the first claim. Since $V\subset{\mathbb{A}}^{r}$ is
closed and $G$-invariant $Z=\pi(V)$ is closed in $X_{\sigma}$ and for every
$z\in Z$ the only closed orbit in $\pi^{-1}(z)$ is contained also in $V$.
Assume that $Z$ contains two distinct points $z_{1}$ and $z_{2}$ and $L_{i}$
is the closed orbit of $\pi^{-1}(z_{i})$. Note that the restriction of every
polynomial from $A$ to $V$ is constant. Hence, elements of $A$ do not separate
$L_{1}$ and $L_{2}$ contrary to the fact that the regular functions on
$X_{\sigma}$ separate $z_{1}$ and $z_{2}$. Thus, $Z$ is at most a singleton.
Since $\bar{0}\in V$ and $o\in Z$ we see that $V\subset\pi^{-1}(o)$. Let
$\dim\pi^{-1}(z)>r-n$ for some $z\in X_{\sigma}\setminus o$. Since $\pi$ is
${\tilde{T}}$-equivarinat the same is true for all points in the
$\mathbb{T}$-orbit $P$ of $z$ in $X_{\sigma}$. The closure of $P$ contains a
$\mathbb{T}$-orbit $Q$ of a smaller dimension and for every $w\in Q$ one has
$\dim\pi^{-1}(w)>r-n$ by Chevalley’s theorem. Reducing the dimension of such
$\mathbb{T}$-orbits further we see that $\dim\pi^{-1}(o)>r-n$, i.e., $o\notin
U$. This concludes the proof. ∎

######  Notation 3.2.

Let $Z$ be an affine algebraic variety and $TZ$ be its Zariski tangent bundle.
Then we let $\operatorname{{\rm ED}}(Z)=\max(2\dim Z+1,\dim TZ)$.

By [Hol, Theorem 7.4] (later rediscovered in [Ka91] and [Sr]) for every affine
algebraic variety $Z$ there exists a closed embedding of $Z$ into
${\mathbb{A}}^{\operatorname{{\rm ED}}(Z)}$.

######  Theorem 3.3.

Let $Z$ be an affine algebraic variety and $X_{\sigma}$ be a normal affine
toric variety without torus factors. Let $\pi:{\mathbb{A}}^{r}\to X_{\sigma}$
be as in Formula (1). Let $l={\rm
codim}_{{\mathbb{A}}^{r}}\pi^{-1}(X_{\sigma}\setminus U_{0})$. Suppose that
$\operatorname{{\rm ED}}(Z)\leq\dim X_{\sigma}$ and $\dim Z<l$. Then there
exists a closed embedding $\iota:Z\hookrightarrow X_{\sigma}$ such that
$\iota(Z)$ is contained in the regular part $U_{0}$ of $X_{\sigma}$.
Furthermore, $l\geq 2$ and, in particular, for every affine curve $C$ with
$\operatorname{{\rm ED}}(C)\leq\dim X_{\sigma}$ there exists a closed
embedding of $C$ in $X_{\sigma}$ with the image in $U_{0}$.

###### Proof.

By [Hol, Theorem 7.4] $Z$ can be treated as a closed subvariety of
${\mathbb{A}}^{r}$. Since $\dim Z<l$ Theorem 6.2 implies that there exists an
algebraic family ${\mathcal{A}}$ of automorphisms of ${\mathbb{A}}^{r}$ such
that for a general $\alpha\in{\mathcal{A}}$ the variety $\alpha(Z)$ does not
meet $\pi^{-1}(X_{\sigma}\setminus U_{0})$. Thus, we suppose further that
$\pi(Z)\subset U_{0}$. Note that $l\geq 2$ by Corollary 2.3. In particular, by
[FKZ, Theorem 2.6], $\pi^{-1}(U_{0})\subset{\mathbb{A}}^{r}$ is a flexible
variety. By Proposition 2.2(1) the morphism
$\pi|_{\pi^{-1}(U_{0})}:\pi^{-1}(U_{0})\to U_{0}$ is smooth. Hence, by Theorem
6.6 one can suppose that $\pi|_{\alpha(Z)}:\alpha(Z)\to U_{0}$ is an injective
immersion since $\operatorname{{\rm ED}}(Z)\leq\dim X_{\sigma}$.

Furthermore, consider the natural embedding
${\mathbb{A}}^{r}\hookrightarrow{\mathbb{P}}^{r}$,
$D={\mathbb{P}}^{r}\setminus{\mathbb{A}}^{r}\simeq{\mathbb{P}}^{r-1}$ and
$H=\operatorname{{\rm GL}}_{r}({\rm\bf k})$. Then we have the natural
$H$-action on ${\mathbb{P}}^{r}$ such that $D$ is invariant under it. By
Proposition 6.8 (3) we can replace ${\mathcal{A}}$ by the family
$H\times{\mathcal{A}}$. That is, for a general $h$ in $H$ and a general
$\alpha$ in ${\mathcal{A}}$ the morphism
$\pi|_{h\circ\alpha(Z)}:h\circ\alpha(Z)\to U_{0}$ is still an injective
immersion.

By Lemma 3.1 we can find generators $g_{1},\ldots,g_{s}$ of ${\rm\bf
k}[X_{\sigma}]$ such that the polynomials $f_{i}=g_{i}\circ\pi$ are monomials
and the codimension (in ${\mathbb{A}}^{r}$) of the variety given by
$f_{1}=\ldots=f_{s}=0$ is at least $l$. Note also that $f_{1},\ldots,f_{s}$
can be viewed as coordinate functions of $\pi:{\mathbb{A}}^{r}\to
X_{\sigma}\subset{\mathbb{A}}^{s}$ and they can be extended to rational
functions on ${\mathbb{P}}^{r}$. Since each $f_{i}$ is homogeneous with
respect to the standard degree function the intersection $R$ of the
indeterminacy sets of these extensions is given by the common zeros of
$f_{1},\ldots,f_{s}$ in $D$. In particular, $R$ has codimension at least $l$
in $D$. Let $P$ be the intersection of $D$ with the closure of
$h\circ\alpha(Z)$ in ${\mathbb{P}}^{r}$, i.e., $\dim P\leq\dim Z-1<l-1$. Since
the restriction of the $H$-action to $D$ is transitive $P$ does not meet $R$
for general $h\in H$ and $\alpha\in{\mathcal{A}}$ by Theorem 6.2. Hence,
$\pi|_{h\circ\alpha(Z)}:h\circ\alpha(Z)\to X_{\sigma}$ is a proper morphism by
[Ka20, Corollary 5.4]. Consequently, it is a closed embedding which concludes
the proof. ∎

######  Remark 3.4.

The proof actually implies that we can suppose that $\iota(Z)\subset V$ where
$V$ is any dense open subset of $U_{0}$ such that ${\rm
codim}_{{\mathbb{A}}^{r}}\pi^{-1}(X_{\sigma}\setminus V)\geq l$.

######  Corollary 3.5.

Let $X_{\sigma}$ be a simplicial toric variety. Let $Z$ be an affine algebraic
variety such that $\operatorname{{\rm ED}}(Z)\leq\dim X_{\sigma}$ and $\dim Z$
is less than the codimension of the singularities of $X_{\sigma}$ in
$X_{\sigma}$. Then there is a closed embedding of $Z$ into $X_{\sigma}$ with
the image in $U_{0}$.

##  4 Locally nilpotent vector fields on affine toric varieties

Locally nilpotent vector fields on $X_{\sigma}$ were described by Demazure
[De] and later by Liendo [Li]. Recall that a Demazure root associated with
some $\varrho_{i}$ is any element $e$ of $M$ such that $<e,\varrho_{i}>=-1$
and $<e,\varrho_{j}>$ is nonnegative for every $j\neq i$. The vector field on
$X_{\sigma}$ defined by

$\partial_{\varrho_{i},e}(\chi^{m})=<m,\varrho_{i}>\chi^{m+e}$ (4)

is locally nilpotent and up to a constant factor every homogeneous locally
nilpotent vector field is of this form. For a Demazure root $e\in M$
associated with $\varrho_{i}$ one has
$\tilde{e}:=\pi^{*}(\chi^{e})=(\tilde{e}_{1},\ldots,\tilde{e}_{r})$ where by
Formula (2) the $i$-th coordinate $\tilde{e}_{i}$ is equal to $-1$. Let
$\tilde{e}^{\prime}=(\tilde{e}_{1}^{\prime},\ldots,\tilde{e}_{r}^{\prime})$
where the $i$-th coordinate $\tilde{e}_{i}^{\prime}$ is equal to zero, whereas
$\tilde{e}_{l}^{\prime}=\tilde{e}_{l}$ for $l\neq i$. Formulas (2) and (4)
imply now the following fact which was first discovered in [AKuZ19]).

######  Lemma 4.1.

The polynomial $\pi^{*}(\partial_{\varrho_{i},e}(\chi^{m}))$ coincides with
$\tilde{\partial}_{\varrho_{i},e}(\pi^{*}(\chi^{m}))$ where the locally
nilpotent vector field $\tilde{\partial}_{\varrho_{i},e}$ on
${\mathbb{A}}^{r}$ is given by
$\bar{x}^{\tilde{e}^{\prime}}\frac{\partial}{\partial x_{i}}$ with
$\bar{x}^{\tilde{e}^{\prime}}=\prod_{l=1}^{r}x_{l}^{\tilde{e}_{l}^{\prime}}$,
i.e., the flow of $\tilde{\partial}_{\varrho_{i},e}$ is given by

$\bar{x}=(x_{1},\ldots,x_{r})\mapsto(x_{1},\ldots,x_{i-1},x_{i}+t\bar{x}^{\tilde{e}^{\prime}},x_{i+1},\ldots,x_{r})$
(5)

where $t$ is the time parameter.

The algebra ${\rm\bf k}[D_{i}]$ of regular functions on $D_{i}$ can be viewed
as the semigroup algebra of $\tau_{i}=\varrho_{i}^{\bot}\cap\sigma^{\vee}\cap
M$. Note that ${\rm\bf k}[D_{i}]$ is the kernel of $\partial_{\varrho_{i},e}$
viewed as a derivation on ${\rm\bf k}[X_{\sigma}]$. The natural embedding
${\rm\bf k}[D_{i}]\hookrightarrow{\rm\bf k}[X_{\sigma}]$ yields a dominant
$\mathbb{T}$-equivariant morphism $\kappa_{i}:X_{\sigma}\to D_{i}$ that is the
categorical quotient of the $\mathbb{G}_{a}$-action associated with
$\partial_{\varrho_{i},e}$. Note that it is also the categorical quotient of
the natural $H_{i}$-action on $X_{\sigma}$ since ${\rm\bf k}[D_{i}]$ is the
subring of $H_{i}$-invariants of ${\rm\bf k}[X_{\sigma}]$. Furthermore, as we
mentioned before in the proof of Proposition 2.2 $D_{i}$ is the fixed point
set of the $H_{i}$-action on $X_{\sigma}$.

######  Notation 4.2.

Similarly, consider a cone $\theta\in\sigma(2)$ containing two extremal rays
$\varrho_{i}$ and $\varrho_{j}$ and the subgroup $H_{ij}$ of $\mathbb{T}$
generated by $H_{i}$ and $H_{j}$. The dual cone of $\theta$ meets $M$ along
$\tau_{i}\cap\tau_{j}$. The semigroup algebra of $\tau_{i}\cap\tau_{j}$ can be
viewed as the algebra of regular functions on $D_{ij}=D_{i}\cap D_{j}$. As
before, one can see that $D_{ij}=X_{\sigma}//H_{ij}$ and $D_{ij}$ is the fixed
point set of the $H_{ij}$-action on $X_{\sigma}$. Since ${\rm\bf k}[D_{ij}]$
has no zero-divisors and its transcendence degree is $n-2$ one can see that
$D_{ij}$ is an irreducible $\mathbb{T}$-invariant Weil divisor in $D_{i}$. In
particular, $D_{ij}$ contains a dense $\mathbb{T}$-orbit $O(\theta)$ (which is
associated with $\theta$ via the Orbit-Cone correspondence).

######  Lemma 4.3.

Let Notation 4.2 hold and $\theta$ be regular. Then $\kappa_{i}$ is smooth
over $O(\theta)$ and $\kappa_{i}^{-1}(u)$ is isomorphic to ${\mathbb{A}}^{1}$
for every $u\in O(\theta)$.

###### Proof.

Since $\theta$ is regular $O(\theta)$ is contained in the regular part $U_{0}$
of $X_{\sigma}$ by [CLS, Theorem 1.3.12 and Example 1.2.20]. Let $u\in
O(\theta)$. Then $T_{u}X_{\sigma}$ is equipped with the induced linear
$H_{ij}$-action. By the Luna slice étale theorem for smooth points (e.g., see
[PV, Theorem 6.4]) there exists an $H_{ij}$-equivariant étale morphism
$\varphi:Y\to T_{u}X_{\sigma}$ from a dense open $H_{ij}$-invariant subset $Y$
of $X_{\sigma}$ containing $u$. Hence, since the map $T_{u}X_{\sigma}\to
T_{u}X_{\sigma}//H_{i}$ is smooth so is $\kappa_{i}|_{Y}:Y\to Y//H_{i}$. For
every point $w\in\kappa_{i}^{-1}(u)$ the closure of its $H_{i}$-orbit contains
the fixed point $u$ (e.g., [PV, Theorem 4.7 and Corollary]), i.e., this orbit
is contained in $Y$. Hence, $\kappa_{i}^{-1}(u)\subset Y$ (and, consequently,
$\kappa_{i}^{-1}(u)$ is isomorphic via $\varphi$ to a line through the origin
in $T_{u}X_{\sigma}$). This yields the desired conclusion. ∎

######  Lemma 4.4.

Suppose that for some $\varrho_{i}$ every $\theta\in\sigma(2)$ containing
$\varrho_{i}$ is regular. Then there exists an open subset $V_{i}$ of
$D_{i}\cap U_{0}$ such that $\mathop{\rm codim}_{D_{i}}D_{i}\setminus
V_{i}\geq 2$ and for every $v\in V_{i}$ one can find a locally nilpotent
vector field $\delta$ of the form $g\partial_{\varrho_{i},e}$ where
$g\in{\rm\bf k}[D_{i}]\subset{\rm\bf k}[X_{\sigma}]$ which does not vanish on
$\kappa_{i}^{-1}(v)$.

###### Proof.

Let $\varrho_{j_{1}},\ldots,\varrho_{j_{k}}$ be the collection of all extremal
rays distinct from $\varrho_{i}$ such that for every $s=1,\ldots,k$ there
exists $\theta\in\sigma(2)$ containing $\varrho_{j_{s}}$ and $\varrho_{i}$.
Formula (4) implies that $\partial_{\varrho_{i},e}$ does not vanish over
$O_{i}=D_{i}\setminus\bigcup_{s=1}^{k}D_{j_{s}}$. Thus, we have to consider
the case when $v$ is a general point of some $D_{j_{s}}$. Choose a rational
function $f_{s}$ on $D_{i}$ with poles on $D_{i}\cap D_{j_{s}}$ only such that
these poles are simple at general points of $D_{i}\cap D_{j_{s}}$. Let $l_{s}$
be the zero multiplicity of $\partial_{\varrho_{i},e}$ at general points of
$D_{j_{s}}$. Then the vector field
$\delta=f_{s}^{l_{s}}\partial_{\varrho_{i},e}$ is regular, locally nilpotent,
tangent to the fibers of $\kappa_{i}$ and it does not vanish at general points
of $D_{j_{s}}$. By Lemma 4.3 $\kappa_{i}^{-1}(v)$ is isomorphic to a line and
since $v\in D_{i}\cap D_{j_{s}}$ one has $\kappa_{i}^{-1}(v)\subset
D_{j_{s}}$. Thus, $\delta|_{\kappa_{i}^{-1}(v)}$ does not vanish since it is
tangent to the line $\kappa_{i}^{-1}(v)$ and nonzero at a general point of
$\kappa_{i}^{-1}(v)$. This yields the desired conclusion. ∎

######  Proposition 4.5.

Let every $\theta\in\sigma(2)$ containing $\varrho_{i}$ be regular and let
$V_{i}\subset D_{i}$ be as in Lemma 4.4. Let $Z$ be a closed subvariety of
$D_{i}\subset X_{\sigma}$ which is contained in $V_{i}$. Let
$s:Z\to{\mathbb{A}}^{r}$ be a section of $\pi:{\mathbb{A}}^{r}\to X_{\sigma}$
over $Z$ for which ${\tilde{Z}}=s(Z)$ is closed in ${\mathbb{A}}^{r}$. Then
one can find a locally nilpotent vector field $\delta$ equivalent to
$\partial_{\varrho_{i},e}$ 111Two locally nilpotent derivations are equivalent
if they have the same kernels. and such that $\delta$ does not vanish on
$\kappa_{i}^{-1}(Z)$.

###### Proof.

Lemma 4.4 implies that for every $u\in Z$ one can find a locally nilpotent
vector field $\delta_{z}$ of the form
$g_{z}\delta_{\varrho_{i},e},\,g_{z}\in{\rm\bf k}[D_{i}]\subset{\rm\bf
k}[X_{\sigma}]$ which does not vanish on $\kappa_{i}^{-1}(u)$. Recall that
$g_{z}$ as an element of ${\rm\bf k}[X_{\sigma}]$ is invariant under the
$H_{i}$-action. Hence, by Lemma 4.1 $\delta_{z}=\pi_{*}(\tilde{\delta}_{z})$
where $\tilde{\delta}_{z}$ is of the form
$\tilde{f}_{z}\frac{\partial}{\partial x_{i}}$ with $\tilde{f}_{z}$ being a
polynomial independent of $x_{i}$ since $x_{i}$ is not invariant under the
${\tilde{H}}_{i}$-action. By assumption $\tilde{\delta}_{z}$ and, therefore,
$\tilde{f}_{z}$ do not vanish at $\pi^{-1}(u)\cap{\tilde{Z}}$. Hence, by the
Nullstellensatz one can find polynomials $\tilde{h}_{z}$ such that only finite
number of them are nonzero and
$\sum_{z}\tilde{h}_{z}\tilde{f}_{z}|_{\tilde{Z}}=1$. By the assumption every
regular function on ${\tilde{Z}}$ is a lift of a regular function on $Z$ which
extends to an element ${\rm\bf k}[D_{i}]=\operatorname{{\rm
Ker}}\partial_{\varrho_{i},e}\subset{\rm\bf k}[X_{\sigma}]$. In particular,
one can suppose that $\tilde{f}_{z}=\pi^{*}f_{z}$ and
$\tilde{h}_{z}=\pi^{*}h_{z}$ where $f_{z},h_{z}\in{\rm\bf k}[D_{i}]$. Hence,
$\delta=\sum_{z}h_{z}\delta_{z}$ (resp.
$\tilde{\delta}=\sum_{z}\tilde{h}_{z}\tilde{\delta}_{z}$) is a locally
nilpotent non-vanishing vector field on $\kappa_{i}^{-1}(Z)$ (resp.
$\pi^{-1}(\kappa_{i}^{-1}(Z))$) which yields the desired conclusion. ∎

######  Corollary 4.6.

Let the assumptions of Proposition 4.5 hold and $Z$ (and, therefore,
${\tilde{Z}}$) be isomorphic to a line ${\mathbb{A}}^{1}$ equipped with a
coordinate $t$. Let $\tilde{\delta}$ be the locally nilpotent derivation on
${\mathbb{A}}^{r}$ as in the proof of Proposition 4.5, i.e.,
$\pi_{*}(\tilde{\delta})=\delta$. Then for every polynomial $h(t)$ there
exists a function $g\in{\rm\bf k}[D_{i}]\subset{\rm\bf k}[X_{\sigma}]$ such
for the flow $\tilde{\beta}_{h}^{i}$ of the locally nilpotent vector field
$\pi^{*}(g)\tilde{\delta}$ a time 1 one has
$x_{i}\circ\tilde{\beta}_{h}^{i}(t)=h(t),\,t\in{\tilde{Z}}$.

###### Proof.

Since $\delta$ is equivalent to $\partial_{\varrho_{i},e}$ and does not vanish
on $Z$ one can suppose (by Lemma 4.1) that the restriction of $\tilde{\delta}$
to ${\tilde{Z}}$ coincides with $\frac{\partial}{\partial x_{i}}$. Let
$\check{g}(t)=h(t)-x_{i}(t)$. Note that $\check{g}(t)$ (as a function on $Z$)
admits an extension to a function $g\in{\rm\bf k}[D_{i}]=\operatorname{{\rm
Ker}}\delta$. This extension yields the desired function. ∎

##  5 Affine simplicial toric varieties

Recall that an affine toric variety $X_{\sigma}$ is simplicial if every face
of $\sigma$ is simplex, i.e., $n=r$ and $G$ is a finite group. This implies,
in particular, that for every $j\neq l$ the extremal rays $\varrho_{l}$ and
$\varrho_{j}$ are contained in some $\theta\in\sigma(2)$ and $D_{lj}=D_{l}\cap
D_{j}$ is always a Weil divisor in $D_{l}$.

######  Lemma 5.1.

Let $X_{\sigma}$ a simplicial toric variety of dimension at least 4 which is
smooth in codimension 2 and $C$ be a smooth polynomial curve in the regular
part of $X_{\sigma}$. Let $V_{i}$ be as in Lemma 4.4,
$V=\bigcap_{i=1}^{r}\kappa_{i}^{-1}(V_{i})$,
$W_{l}=\kappa_{l}^{-1}(V_{l})\setminus\bigcap_{j\neq
l}\kappa_{j}^{-1}(V_{j}))$ and $W_{l}^{\prime}=\kappa_{l}(W_{l})$. For every
$\theta\in\sigma(2)$ containing extremal rays $\varrho_{l}$ and $\varrho_{j}$
let $\psi_{\theta}:X_{\sigma}\to D_{lj}$ be the morphism induced by the
homomorphism of the semigroup algebras associated with the natural embedding
$\tau_{l}\cap\tau_{j}\hookrightarrow\sigma$. Then replacing $C$ with its image
under an automorphism of $X_{\sigma}$ one can suppose that

(i) $C$ is contained in $V$;

(ii) $C_{l}=\kappa_{l}(C)$ meets $W_{l}^{\prime}$ at a finite set for every
$l=1,\ldots,r$;

(iii) $\kappa_{l}|_{C}:C\to D_{l}$ is a closed embedding for every
$l=1,\ldots,r$;

(iv) $\psi_{\theta}:C\to D_{lj}$ is a birational morphism for every
$\theta\in\sigma(2)$ containing $\varrho_{l}$ and $\varrho_{j}$.

###### Proof.

Since $D_{lj}$ is a divisor in $D_{l}$ and $D_{l}\setminus V_{l}$ has
codimesion at least 2 in $D_{l}$ we see that $V_{l}$ contains an open subset
of $D_{lj}$ and $\kappa_{l}^{-1}(V_{l})$ contains an open part of $D_{j}$.
Hence, $X_{\sigma}\setminus\kappa_{l}^{-1}(V_{l})$ does not contain Weil
divisors in $X_{\sigma}$, i.e., it is of codimension at least 2. Consequently,
${\rm codim}_{X_{\sigma}}X_{\sigma}\setminus V\geq 2$. Recall that
$X_{\sigma}$ is flexible by [AKuZ] and, therefore, $U_{0}$ and $V$ are
flexible by [FKZ, Theorem 2.6]. By Theorem 6.2 for a general $\alpha$ in any
perfect family ${\mathcal{A}}$ of automorphisms of $U_{0}$ (which are
extendable to automorphisms of $X_{\sigma}$ by the Hartogs’ theorem)
$\alpha(C)$ is contained in $V$ and $\alpha(C)$ meets every
$\kappa_{l}^{-1}(W_{l}^{\prime})$ at a finite set which yields (i) and (ii).
Lemma 4.3 implies that every $\kappa_{l}$ is smooth over $\kappa_{l}(V)$.
Thus, by Theorem 6.6 and Proposition 6.8(5) for a general
$\alpha\in{\mathcal{A}}$ each morphism $\kappa_{l}:\alpha(C)\to D_{l}$ is a
closed embedding and each morphism $\psi_{\theta}:\alpha(C)\to D_{lj}$ is
birational which yields (iii)-(iv) and the desired conclusion. ∎

######  Lemma 5.2.

Let the assumptions of Lemma 5.1 hold and $C$ satisfy conditions (i)-(iv).
Suppose that $i,\delta$, $h$ and $g$ are as in Corollary 4.6 and
$\beta_{h}^{i}$ is flow of the locally nilpotent vector field $g\delta$ a time
1. Suppose further that $h(t)=ct+d$ where $c$ and $d$ are general constants.
Then the curve $\beta_{h}^{i}(C)$ also satisfies conditions (i)-(iv).

###### Proof.

Let ${\tilde{V}}=\pi^{-1}(V)$. By Corollary 2.4 there exists a curve
${\tilde{C}}\subset{\tilde{V}}$ such $\pi|_{\tilde{C}}:{\tilde{C}}\to C$ is an
isomorphism. Note that $\pi\circ\tilde{\beta}_{h}^{i}=\beta_{h}^{i}\circ\pi$
where $\tilde{\beta}_{h}^{i}$ is as in Corollary 4.6. Hence, besides
conditions (i) and (ii) for $\beta_{h}^{i}(C)$ it suffices to prove that

(iii′) for $\tilde{\kappa}_{l}=\kappa_{l}\circ\pi$ the morphism
$\tilde{\kappa}_{l}|_{\tilde{\beta}_{h}^{i}({\tilde{C}})}:\tilde{\beta}_{h}^{i}({\tilde{C}})\to
D_{l}$ is a closed embedding for every $l=1,\ldots,r$;

(iv′) for $\tilde{\psi}_{\theta}=\psi_{\theta}\circ\pi$ the morphism
$\tilde{\psi}_{\theta}|_{\tilde{\beta}_{h}^{i}({\tilde{C}})}:\tilde{\beta}_{h}^{i}({\tilde{C}})\to
D_{lj}$ is birational for every $\theta\in\sigma(2)$ containing $\varrho_{l}$
and $\varrho_{j}$.

Let us start with (iv′). One can choose coordinate functions of
$\tilde{\psi}_{\theta}$ in the form $\pi^{*}(\chi^{m})$ where
$m\in\tau_{l}\cap\tau_{j}$. By Formula(2) $\pi^{*}(\chi^{m})$ is of the form
$x_{i}^{k_{m}}y_{m}$ where $y_{m}$ is a monomial independent of $x_{i}$.
Condition (iv) implies that there exist
$m^{\prime},m^{\prime\prime}\in\tau_{l}\cap\tau_{j}$ such that for
$t\in{\mathbb{A}}^{1}\simeq{\tilde{C}}$ the functions
$x_{i}^{k_{m^{\prime}}}(t)y_{m^{\prime}}(t)$ and
$x_{i}^{k_{m^{\prime\prime}}}(t)y_{m^{\prime\prime}}(t)$ are not proportional
and, in particular, $\frac{y_{m^{\prime}}(t)}{y_{m^{\prime\prime}}(t)}$ is a
nonzero rational function. Hence, for general $c$ and $d$ the functions
$(ct+d)^{k_{m^{\prime}}}y_{m^{\prime}}(t)$ and
$(ct+d)^{k_{m^{\prime\prime}}}(t)y_{m^{\prime\prime}}(t)$ are not proportional
and Corollary 4.6 implies that the morphism
$\tilde{\psi}_{\theta}|_{\tilde{\beta}_{h}^{i}({\tilde{C}})}:\tilde{\beta}_{h}^{i}({\tilde{C}})\to
D_{lj}$ is birational which is (iv′).

Let $S_{lj}$ be a finite subset of ${\tilde{C}}$ for which
$\tilde{\psi}_{\theta}|_{\tilde{\beta}_{h}^{i}({\tilde{C}}\setminus
S_{lj})}:\tilde{\beta}_{h}^{i}({\tilde{C}}\setminus S_{lj})\to D_{lj}$ is an
embedding. Then condition (iii) implies that for every $t_{0}\in S_{lj}$ there
exists $m\in\tau_{l}\cap\tau_{j}$ such that $\frac{{\rm d}}{{\rm
d}t}x_{i}^{k_{m}}y_{m}|_{t=t_{0}}$ is nonzero. This implies that either
$y_{m}(t_{0})\neq 0$ or $\frac{{\rm d}}{{\rm d}t}y_{m}|_{t=t_{0}}\neq 0$.
Consequently, $\frac{{\rm d}}{{\rm d}t}(ct+d)^{k_{m}}y_{m}|_{t=t_{0}}$ is
nonzero for general $c$ and $d$. Hence, we can suppose that
$\tilde{\kappa}_{l}|_{\tilde{\beta}_{h}^{i}({\tilde{C}})}:\tilde{\beta}_{h}^{i}({\tilde{C}})\to
D_{l}$ is an immersion. Furthermore, for every $t_{0}\neq t_{1}\in S_{lj}$
there exists $m\in\tau_{l}\cap\tau_{j}$ such that
$x_{i}^{k_{m}}(t_{0})y_{m}(t_{0})\neq x_{i}^{k_{m}}(t_{1})y_{m}(t_{1})$. Again
for general $c$ and $d$ this implies that
$(ct_{0}+d)^{k_{m}}y_{m}(t_{0})\neq(ct_{1}+d)^{k_{m}}y_{m}(t_{1})$. Hence,
$\tilde{\kappa}_{l}|_{\tilde{\beta}_{h}^{i}({\tilde{C}})}:\tilde{\beta}_{h}^{i}({\tilde{C}})\to
D_{l}$ is a closed embedding which is (iii′).

By (ii) for every $l$ the curve $C_{l}$ meets $W_{l}^{\prime}$ at a finite set
$Q_{l}^{\prime}$ or, equivalently, for a general point $z\in C$ one has
$\kappa_{l}(z)\notin W_{l}^{\prime}$. The same remains true for a general
point $\beta_{h}^{i}(z)$ of the curve $\beta_{h}^{i}(C)$. Indeed, it suffices
to show that it is true for some point of $\beta_{h}^{i}(C)$. Let
$t\in{\tilde{C}}\simeq{\mathbb{A}}^{1}$ be the preimage of $z$ in
${\tilde{C}}$. Note that $\beta_{h}^{i}(C)$ meets $C$ at the points where
$x_{i}(t)=ct+d$. Since $c$ and $d$ are general the solutions of the latter
equation yield general points of $C$ and, hence, condition (ii) for the curve
$\beta_{h}^{i}(C)$.

Recall that $\beta_{h}^{i}$ is the flow at time 1 of a locally nilpotent
vector field $g\delta$ as in Corollary 4.6 which is equivalent to a vector
field $\partial_{\varrho_{i},e}$ and, therefore, which is tangent to the
fibers of $\kappa_{i}$. In particular, $\beta_{h}^{i}(C)$ is contained in
$\kappa_{i}^{-1}(C_{i})$ where $C_{i}=\kappa_{i}(C)$. By construction
$\kappa_{i}^{-1}(C_{i}\setminus Q_{i}^{\prime})\subset V$. By Lemma 4.3 every
fiber $L$ of
$\kappa_{l}|_{\kappa_{l}^{-1}(Q_{i}^{\prime})}:\kappa^{-1}(Q_{i}^{\prime})\to
Q_{i}^{\prime}$ is a line and such $L$ meets $V$ since $C$ does at some point
$z_{0}\in C$ (where this $z_{0}$ is unique since $\kappa_{l}:C\to D_{l}$ is a
closed embedding). Let $t$ be a coordinate on ${\tilde{C}}\simeq
C\simeq{\mathbb{A}}^{1}$. Recall that by construction in Corollary 4.6
$g\subset{\rm\bf k}[D_{i}]$ is an extension of the function $ct+d-x_{i}(t)$.
Hence, $g\delta=\delta_{1}+d\delta$ where the locally nilpotent vector field
$\delta_{1}$ commutes with $\delta$. In particular, the flow of $g\delta$ at
time 1 and the composition of the flows of $\delta_{1}$ at time 1 and $\delta$
at time $d$. Since by Proposition 4.5 $\delta$ does not vanish on $L$ and $d$
is general we see that $\beta_{h}^{i}(z_{0})$ is a general point of $L$ and,
therefore, it belongs to $V$. This yields condition (i) for $\beta_{h}^{i}(C)$
and the desired conclusion. ∎

######  Theorem 5.3.

Let $\varphi:C\to C^{\prime}$ be an isomorphism of two smooth polynomial curve
contained in the regular part of an affine simplicial toric variety
$X_{\sigma}$ of dimension at least 4 such that $X_{\sigma}$ is smooth in
codimension 2. Then $\varphi$ extends to an automorphism of $X_{\sigma}$.

###### Proof.

Let ${\tilde{C}}$ (resp. ${\tilde{C}}^{\prime}$) be a curve in
${\mathbb{A}}^{r}$ such that $\pi|_{{\tilde{C}}}:{\tilde{C}}\to C$ (resp.
$\pi|_{{\tilde{C}}^{\prime}}:{\tilde{C}}^{\prime}\to C^{\prime}$) is an
isomorphism. Let $t^{\prime}$ be a coordinate on
$C^{\prime}\simeq{\tilde{C}}^{\prime}$ and $t=\varphi^{*}(t^{\prime})$ be the
coordinate on $C\simeq{\tilde{C}}$. Applying consequently automorphisms
$\beta_{h}^{i}$ as in Lemma 5.2 with $i$ running over $\\{1,\ldots,r\\}$ one
can suppose that ${\tilde{C}}$ is a curve such that $x_{i}(t)=c_{i}t+d_{i}$
for every $i$ where $(c_{1},d_{1},\ldots,c_{r},d_{r})$ is a general point in
${\mathbb{A}}^{2r}$. Similarly, one can suppose that ${\tilde{C}}^{\prime}$ is
a curve such that $x_{i}(t^{\prime})=c_{i}^{\prime}t^{\prime}+d_{i}^{\prime}$
for every $i$ where
$(c_{1}^{\prime},d_{1}^{\prime},\ldots,c_{r}^{\prime},d_{r}^{\prime})$ is a
general point in ${\mathbb{A}}^{2r}$. Choosing these two general points equal
we get the desired conclusion. ∎

We need to remind the following [KaUd, Definition 8.3].

######  Definition 5.4.

Let $C_{1}$ and $C_{2}$ be smooth curves in a smooth quasi-affine variety $Y$
with defining ideals $I_{1}$ and $I_{2}$ in ${\rm\bf k}[Y]$. We suppose also
that $C_{1}$ and $C_{2}$ are closed in an affine variety containing $Y$. Let
$Y$ possess a volume form $\omega$ (i.e., $\omega$ is a nonvanishing section
of the canonical bundle on $Y$) and let each conormal bundle
$\frac{I_{j}}{I_{j}^{2}}$ of $C_{j}$ in $Y$ be trivial. By [KaUd, Lemma 6.3]
there is a neighborhood $W_{j}$ of $C_{j}$ in $Y$ in which $C_{j}$ is a strict
complete intersection given by $u_{1,j}=\ldots=u_{n-1,j}=0$ where
$u_{1,j},\ldots,u_{n-1,j}\in I_{j}$ and $n=\dim Y$. That is, for
$A_{j}=\frac{{\rm\bf k}[Y]}{I_{j}}$ we have the graded algebra $\frac{{\rm\bf
k}[Y]}{I_{j}^{k}}\simeq
A_{j}\oplus\bigoplus_{l=1}^{k-1}\frac{I_{j}^{l}}{I^{l+1}_{j}}$ which can be
viewed as the algebra of polynomials in $u_{1,j},\ldots,u_{n-1,j}$ over
$A_{j}$ of degree at most $k-1$. Consider an isomorphism
$\varphi:\frac{{\rm\bf k}[Y]}{I_{1}^{k}}\to\frac{{\rm\bf k}[Y]}{I_{2}^{k}}$ of
these algebras for a natural $k$. Up to the induced isomorphism $A_{1}\simeq
A_{2}$ this isomorphism $\varphi$ is determined by its values
$\varphi(u_{i,1}),\,i=1,\ldots,n-1$. These values can be viewed as polynomials
in $u_{1,2},\ldots,u_{n-1,2}$ over $A_{2}$, i.e., one has the matrix
$\left[\frac{\partial\varphi(u_{l,1})}{\partial
u_{s,2}}\right]_{l,s=1}^{n-1}$. Since the normal bundle $N_{Y}C_{j}$ is
trivial, the existence of $\omega$ implies the existence of a volume form on
$C_{j}$. Fix volume forms $\omega_{j}$ on $C_{j}$ such that
$\tilde{\varphi}^{*}\omega_{1}=\omega_{2}$ where the isomorphism
$\tilde{\varphi}:C_{2}\to C_{1}$ is induced by $\varphi$. Choose a section
$\mathop{\rm pr}_{j}:TY|_{C_{j}}\to TC_{j}$ of the canonical inclusion
$TC_{j}\hookrightarrow TY|_{C_{j}}$ and consider the section
$\tilde{\omega}_{j}=\omega_{j}\circ\mathop{\rm pr}_{j}$ of the dual bundle
$(TY|_{C_{j}})^{\vee}$ of $TY|_{C_{j}}$. Then one can require that
$\omega|_{C_{j}}$ coincides with $\tilde{\omega}_{j}\wedge{\rm
d}u_{1,j}\wedge\ldots\wedge{\rm d}u_{n-1,j}$. Under this requirement the
determinant of $\left[\frac{\partial\varphi(u_{l,1})}{\partial
u_{s,2}}\right]_{l,s=1}^{n-1}$ is well-defined modulo $I_{2}^{k-1}$ (i.e., it
is independent of the choice of coordinates $u_{1,j},\ldots,u_{n-1,j}$).
Hence, we say that $\varphi$ has Jacobian $a\in{\rm\bf k}\setminus\\{0\\}$ if
the determinant of $\left[\frac{\partial\varphi(u_{l,1})}{\partial
u_{s,2}}\right]_{l,s=1}^{n-1}$ is equal to $a$ modulo $I_{2}^{k-1}$.

Note that Definition 5.4 is applicable in the case when $C_{1}$ and $C_{2}$
are smooth polynomial curves in a simplicial toric variety $X_{\sigma}$
contained in its regular part. Indeed, $U_{0}$ as the regular part of
$X_{\sigma}$ is flexible [AKuZ]. Recall that ${\mathbb{A}}^{r}$ admits a
volume form invariant under the natural $\mathop{\rm SL}_{r}({\rm\bf
k})$-action. Hence, we can push this volume form down to a volume form
$\omega$ on $U_{0}$ since $\pi|_{\pi^{-1}(U_{0})}:\pi^{-1}(U_{0})\to U_{0}$ is
an unramified covering by Proposition 2.2. Furthermore, the normal bundles of
smooth polynomial curves are always trivial, i.e., we are under the
assumptions of Definition 5.4.

######  Corollary 5.5.

Let $\varphi:{\mathcal{C}}_{1}\to{\mathcal{C}}_{2}$ be an isomorphism of
$k$-th infinitesimal neighborhoods of two smooth polynomial curve $C_{1}$ and
$C_{2}$ contained in the regular part $U_{0}$ of an affine simplicial toric
variety $X_{\sigma}$ of dimension at least 4 which is smooth in codimension 2.
Suppose that the Jacobian of $\varphi$ is a nonzero constant $a$. Then
$\varphi$ extends to an automorphism of $X_{\sigma}$.

###### Proof.

Recall that by [KaUd, Lemma 6.2] every automorphism $\alpha$ of $U_{0}$ has a
constant Jacobian where the Jacobian is computed as
$\frac{\alpha^{*}(\omega)}{\omega}$ [KaUd, Lemma 6.2] and if $\alpha$ is a
composition of elements of flows of locally nilpotent vector fields, then its
Jacobian is 1. Let $\psi:C_{2}\to C_{1}$ be an isomorphism. By Theorem 5.3
$\psi$ extends to an automorphism $\Psi$ of $X_{\sigma}$ which in turn induces
an automorphism ${\mathcal{C}}_{2}\to{\mathcal{C}}_{1}$ also denoted by
$\psi$. By construction $\Psi$ is a composition of elements of flows of
locally nilpotent vector fields. Hence, its Jacobian is 1. Taking a
composition of $\varphi$ with the action of an appropriate element of
$\mathbb{T}$ and replacing $C_{2}$ with its image under this action we can
suppose that the Jacobian of $\varphi$ is also 1 nodulo $I_{2}^{k-1}$ in the
sense of Definition 5.4 where $I_{j}$ is the defining ideal of $C_{j}$ in
${\rm\bf k}[X_{\sigma}]$. Then the automorphism
$\lambda:=\psi\circ\varphi:{\mathcal{C}}_{1}\to{\mathcal{C}}_{1}$ has Jacobian
1 modulo $I_{1}^{k-1}$. By [KaUd, Theorem 6.5] $\lambda$ extends to an
automorphism $\Lambda$ of $X_{\sigma}$. It remains to note that
$\Psi^{-1}\circ\Lambda$ is the desired extension of $\varphi$ and we are done.
∎

##  6 Appendix

In this section we present some technical tools developed in [Ka20] with later
clarifications in [KaUd] which we use in this paper.

######  Definition 6.1.

(1) Given an irreducible algebraic variety ${\mathcal{A}}$ and a map
$\varphi:{\mathcal{A}}\to\operatorname{{\rm Aut}}(X)$ (where
$\operatorname{{\rm Aut}}(X)$ is the group of algebraic automorphisms of $X$)
we say that $({\mathcal{A}},\varphi)$ is an algebraic family of automorphisms
of $X$ if the induced map ${\mathcal{A}}\times X\to X$,
$(\alpha,x)\mapsto\varphi(\alpha).x$ is a morphism (see [Ra]).

(2) If we want to emphasize additionally that $\varphi({\mathcal{A}})$ is
contained in a subgroup $G$ of $\operatorname{{\rm Aut}}(X)$, then we say that
${\mathcal{A}}$ is an algebraic $G$-family of automorphisms of $X$.

(3) In the case when ${\mathcal{A}}$ is a connected algebraic group and the
induced map ${\mathcal{A}}\times X\to X$ is not only a morphism but also an
action of ${\mathcal{A}}$ on $X$ we call this family a connected algebraic
subgroup of $\operatorname{{\rm Aut}}(X)$.

(4) Following [AFKKZ, Definition 1.1] we call a subgroup $G$ of
$\operatorname{{\rm Aut}}(X)$ algebraically generated if it is generated as an
abstract group by a family ${\mathcal{G}}$ of connected algebraic subgroups of
$\operatorname{{\rm Aut}}(X)$.

We have the following important fact [AFKKZ, Theorem 1.15] (which is the
analogue of the Kleiman transversality theorem [Kl] for algebraically
generated groups).

######  Theorem 6.2.

(Transversality Theorem) Let a subgroup $G\subseteq\operatorname{{\rm
Aut}}(X)$ be algebraically generated by a system ${\mathcal{G}}$ of connected
algebraic subgroups closed under conjugation in $G$. Suppose that $G$ acts
with an open orbit $O\subseteq X$.

Then there exist subgroups $H_{1},\ldots,H_{m}\in{\mathcal{G}}$ such that for
any locally closed reduced subschemes $Y$ and $Z$ in $O$ one can find a
Zariski dense open subset $U=U(Y,Z)\subseteq H_{1}\times\ldots\times H_{m}$
such that every element $(h_{1},\ldots,h_{m})\in U$ satisfies the following:

  1. (a)

The translate $(h_{1}\cdot\ldots\cdot h_{m}).Z_{\mathop{\rm reg}}$ meets
$Y_{\mathop{\rm reg}}$ transversally.

  2. (b)

$\dim(Y\cap(h_{1}\cdot\ldots\cdot h_{m}).Z)\leq\dim Y+\dim Z-\dim X$. 222We
put the dimension of empty sets equal to $-\infty$.
In particular $Y\cap(h_{1}\cdot\ldots\cdot h_{m}).Z=\emptyset$ if $\dim Y+\dim
Z<\dim X$.

######  Definition 6.3.

(1) A nonzero derivation $\delta$ on the ring $A$ of regular functions on an
affine algebraic variety $X$ is called locally nilpotent if for every $0\neq
a\in A$ there exists a natural $n$ for which $\delta^{n}(a)=0$. This
derivation can be viewed as a vector field on $X$ which we also call locally
nilpotent. The set of all locally nilpotent vector fields on $X$ will be
denoted by $\operatorname{{\rm LND}}(X)$. The flow of
$\delta\in\operatorname{{\rm LND}}(X)$ is an algebraic $\mathbb{G}_{a}$-action
on $X$, i.e., the action of the group $({\rm\bf k},+)$ which can be viewed as
a one-parameter unipotent group $U$ in the group $\operatorname{{\rm Aut}}(X)$
of all algebraic automorphisms of $X$. In fact, every $\mathbb{G}_{a}$-action
is a flow of a locally nilpotent vector field (e.g, see [Fr, Proposition
1.28]).

(2) If $X$ is a quasi-affine variety, then an algebraic vector field $\delta$
on $X$ is called locally nilpotent if $\delta$ extends to a locally nilpotent
vector field $\tilde{\delta}$ on some affine algebraic variety $Y$ containing
$X$ such that $\tilde{\delta}$ vanishes on $Y\setminus X$ where ${\rm
codim}_{Y}(Y\setminus X)\geq 2$. Note that under this assumption $\delta$
generates a $\mathbb{G}_{a}$-action on $X$ and we use again the notation
$\operatorname{{\rm LND}}(X)$ for the set of all locally nilpotent vector
fields on $X$.

######  Definition 6.4.

(1) For every locally nilpotent vector fields $\delta$ and each function
$f\in\operatorname{{\rm Ker}}\delta$ from its kernel the field $f\delta$ is
called a replica of $\delta$. Recall that such replica is automatically
locally nilpotent.

(2) Let ${\mathcal{N}}$ be a set of locally nilpotent vector fields on $X$ and
$G_{\mathcal{N}}\subset\operatorname{{\rm Aut}}(X)$ denotes the group
generated by all flows of elements of ${\mathcal{N}}$. We say that
$G_{\mathcal{N}}$ is generated by ${\mathcal{N}}$.

(3) A collection of locally nilpotent vector fields ${\mathcal{N}}$ is called
saturated if ${\mathcal{N}}$ is closed under conjugation by elements in
$G_{\mathcal{N}}$ and for every $\delta\in{\mathcal{N}}$ each replica of
$\delta$ is also contained in ${\mathcal{N}}$.

######  Definition 6.5.

Let $X$ be a normal quasi-affine algebraic variety of dimension at least 2,
${\mathcal{N}}$ be a saturated set of locally nilpotent vector fields on $X$
and $G=G_{\mathcal{N}}$ be the group generated by ${\mathcal{N}}$. Then $X$ is
called $G$-flexible if for any point $x$ in the smooth part $X_{\rm reg}$ of
$X$ the vector space $T_{x}X$ is generated by the values of locally nilpotent
vector fields from ${\mathcal{N}}$ at $x$ (which is equivalent to the fact
that $G$ acts transitively on $X_{\rm reg}$ [FKZ, Theorem 2.12]). In the case
of $G=\mathop{\rm SAut}(X)$ we call $X$ flexible without referring to
$\mathop{\rm SAut}(X)$ (recall that $\mathop{\rm SAut}(X)$ is the subgroup of
$\operatorname{{\rm Aut}}X$ generated by all one-parameter unipotent
subgroups).

The following is a simplified version of [Ka20, Theorem 4.2].

######  Theorem 6.6.

Let $X$ be a smooth algebraic variety and $Q$ be a normal algebraic variety.
Let $\varrho:X\to Q$ be a dominant morphism. Suppose that $Q_{0}$ is a smooth
open dense subset of $Q$ such that for $X_{0}=\varrho^{-1}(Q_{0})$ the
morphism $\varrho|_{X_{0}}:X_{0}\to Q_{0}$ is smooth. Let
$G\subset\operatorname{{\rm Aut}}(X)$ be an algebraically generated group
acting 2-transitively on $X$ and $Z$ be a locally closed reduced subvariety in
$X$.

(i) Let $\dim Q\geq\dim Z+m$ where $m\geq 1$. Then there exists an algebraic
$G$-family ${\mathcal{A}}$ of automorphisms of $X$ such that for a general
element $\alpha\in{\mathcal{A}}$ one can find a constructible subset $R$ of
$\alpha(Z)\cap X_{0}$ of dimension $\dim R\leq\dim Z-m$ for which $\varrho(R)$
and $\varrho(\alpha(Z)\setminus R)$ are disjoint and the restriction
$\varrho|_{(\alpha(Z)\cap X_{0})\setminus R}:(\alpha(Z)\cap X_{0})\setminus
R\to Q_{0}$ of $\varrho$ is injective. In particular, if $\dim Q\geq 2\dim
Z+1$ and ${\bar{Z}}_{\alpha}^{\prime}$ is the closure of
$Z_{\alpha}^{\prime}=\varrho\circ\alpha(Z)$ in $Q$, then for a general element
$\alpha\in{\mathcal{A}}$ the map $\varrho|_{\alpha(Z)\cap X_{0}}:\alpha(Z)\cap
X_{0}\to Z_{\alpha}^{\prime}\cap Q_{0}$ is a bijection, while in the case of a
pure-dimensional $Z$ and $\dim Q\geq\dim Z+1$ the morphism
$\varrho|_{\alpha(Z)\cap X_{0}}:\alpha(Z)\cap
X_{0}\to{\bar{Z}}_{\alpha}^{\prime}\cap Q_{0}$ is birational.

(ii) Let $G$ be generated by a saturated set ${\mathcal{N}}$ of locally
nilpotent vector fields on $X$ (in particular, $X$ is $G$-flexible). Let $\dim
TZ\leq\dim Q$ and $2\dim Z+1\leq\dim Q$. Then there exists an algebraic family
${\mathcal{A}}$ of $G$-automorphisms of $X$ such that for a general element
$\alpha\in{\mathcal{A}}$, every $z\in\alpha(Z)\cap X_{0}$ the induced map
$\varrho_{*}:T_{z}\alpha(Z_{0})\to T_{\varrho(z)}Q$ of the tangent spaces is
injective.

Let us describe some $G$-families ${\mathcal{A}}$ satisfying the conclusions
of Theorem 6.6.

######  Definition 6.7.

Let $X$ be a smooth algebraic variety and $G$ be a subgroup of $X$. Consider
$(X\times X)\setminus\Delta$ (where $\Delta$ is the diagonal), the complement
$(TX)^{*}$ to the zero section in the tangent bundle of $X$ and the frame
bundle ${\rm Fr}(X)$ bundle of $TX$ (i.e., the fiber of ${\rm Fr}(X)$ over
$x\in X$ consists of all bases of $T_{x}X$). Projectivization of $TX$ replaces
${\rm Fr}(X)$ with a bundle ${\rm PFr}(X)$ whose fiber over $x$ consists of
all ordered $n$-tuples of points in the proectivization ${\mathbb{P}}^{n}$ of
$T_{x}X$ (where $n=\dim X$) that are not contained in the same hyperplane of
${\mathbb{P}}^{n}$. Then we have natural $G$-actions on all these objects. Let
$Y$ be either $X$, or $(X\times_{P}X)\setminus\Delta$, or $(TX)^{*}$, or ${\rm
PFr}(X)$. Suppose that the $G$-action is transitive on $Y$. Then we say that
an algebraic $G$-family ${\mathcal{A}}$ of automorphisms of $X$ is a regular
$G$-family for $Y$ if

(i) ${\mathcal{A}}=H_{m}\times\ldots\times H_{1}$ where each $H_{i}$ belongs
to ${\mathcal{G}}$;

(ii) for a suitable open dense subset $U\subseteq H_{m}\times\ldots\times
H_{1}$, the map

$\Psi:H_{m}\times\ldots\times H_{1}\times Y\longrightarrow Y\times
Y\quad\mbox{with}\quad(h_{m},\ldots,h_{1},y)\mapsto((h_{m}\cdot\ldots\cdot
h_{1}).y,y)$ (6)

is smooth on $U\times Y$.

An algebraic algebraic $G$-families ${\mathcal{A}}$ that are regular for all
four varieties $X$, $(X\times_{P}X)\setminus\Delta$, $(TX)^{*}$ and ${\rm
PFr}(X)$ will be called a perfect $G$-family for $Y$.

######  Proposition 6.8.

Let $X$ be a smooth algebraic variety and $G\subset\operatorname{{\rm
Aut}}(X)$ be a group algebraically generated by a family ${\mathcal{G}}$ of
algebraic connected subgroups of $\operatorname{{\rm Aut}}(X)$. Suppose that
$G$ acts transitively on $X$.

(1) Then there exists a regular $G$-family for $X$ (which is of the form
${\mathcal{A}}=H_{1}\times\ldots\times H_{m}$ where each $H_{i}$ is an element
of ${\mathcal{G}}$).

(2) Every regular $G$-family for $X$ satisfies the conclusions of Theorem 6.2.

(3) If ${\mathcal{A}}$ is a regular (resp. perfect) $G$-family for $X$ and $H$
is an element of ${\mathcal{G}}$ then $H\times{\mathcal{A}}$ and
${\mathcal{A}}\times H$ are also regular (resp. perfect) $G$-families for $X$.

(4) In particular, if $X$ is $G$-flexible, then there exists a perfect
$G$-family.

(5) Let $X$ be $G$-flexible. Every $G$-family regular for $X$ (resp. for
$(TX)^{*}$) satisfies the conclusion of Theorem 6.6 (i) (resp. (ii)). In
particular, every perfect $G$-family satisfies the conclusion Theorem 6.6
(i)-(ii).

###### Proof.

Statement (1) is proven in [AFKKZ, Proposition 1.15]. Statements (2) and (3)
are proven in [Ka20, Proposition 1.10]. The fourth statement follows from the
fact that in the flexible case for every $m>0$ the group $G$ acts
$m$-transitively on $X$ and also transitively on $(TX)^{*}$ and ${\rm PFr}(X)$
[AFKKZ, Theorem 4.11 and Remark 4.16]. Statement (5) is essentially the
content of the proof of [Ka20, Theorem 4.2]. ∎

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