# On automorphisms of flexible varieties

S. Kaliman and D. Udumyan  Department of Mathematics, University of Miami,
Coral Gables, FL 33124, USA<EMAIL_ADDRESS>Department of Mathematics,
University of Miami, Coral Gables, FL 33124, USA<EMAIL_ADDRESS>
(Date: August 27, 2024)

###### Abstract.

Let $X$ be a flexible variety and $\varphi:C_{1}\to C_{2}$ be an isomorphism
of closed one-dimensional subschemes of $X$. We develop several criteria which
guarantee that $\varphi$ extends to an automorphism of $X$.

###### Contents

  1. 1 Preliminaries
  2. 2 Properness
  3. 3 Notations and Convention
  4. 4 Normal bundles in flexible varieties
  5. 5 $\mathop{\rm SL}_{n}(A)$ over a commutative ring $A$
  6. 6 Main Theorem
  7. 7 Main Theorem. II
  8. 8 Applications
  9. 9 Holomorphic case

## Introduction

Every algebraic variety $X$ in this paper is considered over an algebraically
closed field ${\rm\bf k}$ of characteristic 0. We study the following
question:

when can an isomorphism $f:Y_{1}\to Y_{2}$ of subschemes of $X$ be extended to
an automorphism of $X$?

For $X\simeq{\mathbb{A}}^{n}$ and $Y_{i}\simeq{\mathbb{A}}^{1}$ the answer to
this question is known to be always positive in all cases but $n=3$ (see
[AMo], [Su], [Cr], [Je]). For $n=3$ the answer is unknown but if ${\rm\bf
k}=\mathbb{C}$, then $f$ can be extended to a holomorphic automorphism of
$\mathbb{C}^{3}$ [Ka92]. Furthermore, very general sufficient conditions for
extension of isomorphisms of closed subvarieties of ${\mathbb{A}}^{n}$ to
automorphisms of ${\mathbb{A}}^{n}$ where found in [Ka91] and [Sr] (in a
weaker form they were already present in [Je]). For $X\neq{\mathbb{A}}^{n}$
the first break-through is due to van Santen (formerly Stampfli) who proved
that every isomorphism of smooth polynomial curves in $X\simeq\mathop{\rm
SL}_{n}(\mathbb{C}),\,n\geq 3$ is extendable to an automorphism of $X$ [St].
Later jointly with Feller he showed that the same is true when
$Y_{i}\simeq\mathbb{C}$ and $X$ is any complex linear algebraic group
different from $\mathbb{C}^{3}$, $\mathop{\rm SL}_{2}(\mathbb{C})$ or ${\rm
PSL}_{2}(\mathbb{C})$ [FS]. Some extra cases with a positive answer to the
extension problem were found in [Ka20]. Say, the answer is positive when $X$
is a nonzero fiber of a non-degenerate quadric form in ${\mathbb{A}}^{m}$ (we
call such $X$ an $(m-1)$-sphere over ${\rm\bf k}$) and $Y_{i}$ is a smooth
closed subvariety with $\dim Y_{i}\leq\frac{m}{3}-1$.

Starting from dimension 2 affine spaces, linear algebraic groups without
nontrivial characters and spheres over ${\rm\bf k}$ are examples of so-called
flexible varieties (recall that a normal quasi-affine variety $X$ is flexible
if $\mathop{\rm SAut}(X)$ acts transitively on the smooth part $X_{\rm reg}$
of $X$ where $\mathop{\rm SAut}(X)$ is the subgroup of the group
$\operatorname{{\rm Aut}}(X)$ of algebraic automorphisms of $X$ generated by
one-parameter unipotent subgroups). For a smooth flexible variety $X$ every
isomorphism of zero-dimensional subvarieties is extendable to an automorphism
[AFKKZ]. Furthermore, recall that for a closed subvariety $Y$ of $X$ with
defining ideal $I$ in the ring $A={\rm\bf k}[X]$ of regular functions on $X$
its $k$th infinitesimal neighborhood is the subscheme of $X$ with the defining
ideal $I^{k}$ (in particular, an automorphism of such neighborhood is
essentially an automorphism of the algebra $\frac{A}{I^{k}}$). It follows from
[AFKKZ, Theorem 4.14 and Remark 4.16-4.17] that any automorphism $g:Z\to Z$ of
an $k$th infinitesimal neighborhood $Z$ of a zero dimensional subvariety $Y$
of a smooth flexible variety $X$ is extendable to an automorphism of $X$
provided that $g|_{Y}={\rm id}_{Y}$ and $g$ preserves local volume forms
modulo $I^{k-1}$ (we say that such maps of infinitesimal neighborhoods have
Jacobian 1). The second author gave a proof in [Ud] of the similar result in
the case when $Y$ is a smooth polynomial curve in a smooth flexible variety
$X$ of $\dim X\geq 4$.

In this paper we give a much stronger version of this result (Theorem 6.6) and
develop some new important technical tools for the extension problem. In
particular, instead of polynomial curves we consider smooth curves in $X$ with
trivial normal bundles. (The class of such curves is sufficiently big. It
contains, for instance, curves which have only rational irreducible
components. Furthermore, consider $X$ as a closed subvariety of an affine
space. Then by the Bertini theorem the intersection of $X$ with $n-1$ general
hyperplanes where $\dim X=n$ is a smooth curve. Every irreducible component of
this curve has a trivial normal bundle.) As an application we get the
following generalization of the Feller-van Santen theorem (Corollary 8.6).

######  Theorem 0.1.

Let $X$ be isomorphic to a complex linear algebraic group without nontrivial
characters and let $\dim X\geq 4$. Let $C_{1}$ and $C_{2}$ be smooth
polynomial curves in $X$. Then every isomorphism of $k$th infinitesimal
neighborhoods of $C_{1}$ and $C_{2}$ with Jacobian 1 extends to an
automorphism of $X$.

The absence of nontrivial characters is essential - without this assumption
this theorem does not hold (see Remark 8.7). If $C_{1}=C_{2}$, then we can
make a stronger statement in the holomorphic category without assuming that
$C_{i}$ is a polynomial curve or even that $C_{i}$ is irreducible. Namely, the
holomorphic category has the advantage of the Ivarsson-Kutzschebauch theorem
[IK] which leads, in particular, to the following (Corollary 9.11).

######  Theorem 0.2.

Let $X$ be isomorphic to a complex linear algebraic group without nontrivial
characters and $\dim X\geq 4$. Suppose that $C$ is a smooth closed curve in
$X$ with a trivial normal bundle and defining ideal ${\hat{I}}$ in the algebra
$\operatorname{{\rm Hol}}(X)$ of holomorphic functions on $X$. Let
${\hat{A}}=\frac{\operatorname{{\rm Hol}}(X)}{{\hat{I}}}$ be the algebra of
holomorphic functions on $C$. Then every ${\hat{A}}$-automorphism
$\theta:\frac{\operatorname{{\rm
Hol}}(X)}{{\hat{I}}^{k}}\to\frac{\operatorname{{\rm Hol}}(X)}{{\hat{I}}^{k}}$
with Jacobian 1 extends to a holomorphic automorphism of $X$.

In the case of $X\simeq\mathop{\rm SL}_{n}(\mathbb{C})$ we can say even more
(Corollary 9.12).

######  Theorem 0.3.

Let $X$ be isomorphic to $\mathop{\rm SL}_{n}(\mathbb{C})$ where $n\geq 3$.
Suppose that $C_{1}$ and $C_{2}$ are isomorphic smooth closed curves in $X$
such that their normal bundles in $X$ are trivial. Then every isomorphism of
$k$th infinitesimal neighborhoods of $C_{1}$ and $C_{2}$ with Jacobian 1
extends to a holomorphic automorphism of $X$.

In the case of sphere over ${\rm\bf k}$ the similar result holds in the
algebraic category (Corollary 8.11).

######  Theorem 0.4.

Let $X$ be an $(m-1)$-sphere over ${\rm\bf k}$ where $m\geq 6$. Let $C_{1}$
and $C_{2}$ be isomorphic smooth closed curves in $X$ such that their
irreducible components are rational. Then every isomorphism of $k$th
infinitesimal neighborhoods of $C_{1}$ and $C_{2}$ with Jacobian 1 extends to
an automorphism of $X$.

Let us briefly discuss the crucial step behind these theorems and the idea of
its proof. As we mentioned, we consider a more general situation when $X$ is a
flexible variety of $n:=\dim X\geq 4$, $C$ is a smooth closed curve in $X$
with a trivial normal bundle $N_{X}C$ and defining ideal $I\subset{\rm\bf
k}[X]$. The consideration can be reduced to the case of an irreducible $C$.
Then $A={\rm\bf k}[C]\simeq\frac{{\rm\bf k}[X]}{I}$ is a Dedekind domain and
$N_{X}C$ can be viewed as a free $A$-module with a basis
$v_{1},\ldots,v_{n-1}$. In particular, any $A$-automorphism of $N_{X}C$ can be
presented by an element of $\operatorname{{\rm GL}}_{n-1}(A)$ where the latter
group contains a subgroup $\mathop{\rm E}_{n-1}(A)$ generated by elementary
matrices of the form ${\rm id}+ae_{ij}$ (where ${\rm id}$ is the identity
matrix, $a\in A$, $1\leq i\neq j\leq n-1$ and $e_{ij}$ is the matrix with
entry 1 in position $(i,j)$ and zeros everywhere else). The crucial step is
that under certain mild assumptions one has the following claim.

Every $A$-automorphism of $N_{X}C$ that is presented by an element of
$\mathop{\rm E}_{n-1}(A)$ is induced by an automorphism of $X$.

Assume for simplicity that $X$ is smooth affine and that the natural action of
a one-parameter unipotent subgroup of $\operatorname{{\rm Aut}}(X)$ on $X$
(which is also called a $\mathbb{G}_{a}$-action) has a finitely generated ring
$A^{\mathbb{G}_{a}}$ of invariants. That is, there exists the algebraic
quotient morphism $\varrho:X\to Q=\operatorname{{\rm Spec}}A^{\mathbb{G}_{a}}$
of this action sending $X$ to the normal affine variety $Q$. Recall that such
a $\mathbb{G}_{a}$-action can be viewed as the flow of a so-called locally
nilpotent vector field $\delta$ on $X$. Let $C^{\prime}$ be the closure of
$\varrho(C)$ in $Q$ and $f$ be a regular function on $Q$ which vanishes on
$C^{\prime}$. Then $\varrho^{*}(f)\delta$ is again a locally nilpotent vector
field whose flow $\exp(t\varrho^{*}(f)\delta)$ leaves every point of $C$ fixed
and, in particular, it induces $A$-automorphisms of $N_{X}C$. The plan is to
show that every $A$-automorphism of $N_{X}C$ given by an element of
$\mathop{\rm E}(A)$ is induced by a composition of automorphisms of $X$ of the
form $\exp(t\varrho^{*}(f)\delta)$ with $\delta$ running over the set of
locally nilpotent vector fields on $X$. To follow through this plan we use the
fact ([Ka20, Theorem 6.1]) that $\delta$ can be chosen so that for a given
finite subset $K$ of $C$ there is an open subset $V^{\prime}$ of $C^{\prime}$
for which $V=\varrho^{-1}(C^{\prime})\cap C$ contains $K$ and
$\varrho|_{V}:V\to V^{\prime}$ is an isomorphism. Let $C\setminus V$ be the
zero locus of an ideal ${\mathfrak{q}}\subset A$ in $C$, $\mathop{\rm
F}_{n-1}(A,{\mathfrak{q}})$ be the subgroup of $\mathop{\rm E}_{n-1}(A)$
generated by elementary matrices of the form ${\rm id}+ae_{ij}$ where
$a\in{\mathfrak{q}}$ and $\mathop{\rm E}_{n-1}(A,{\mathfrak{q}})$ be the
normal closure of $\mathop{\rm F}_{n-1}(A,{\mathfrak{q}})$ in $\mathop{\rm
E}_{n-1}(A)$. Consider sections $w_{1},\ldots,w_{n-1}$ of $N_{X}C$ such that
they generate the free ${\rm\bf k}[V]$-module $N_{X}V$ and let $M$ be the free
$A$-submodule of $N_{X}C$ generated by these sections. Then abundance of
$\mathbb{G}_{a}$-action on $X$ enables us to prove that shrinking $V$ one can
suppose that for an appropriate choice of $w_{1},\ldots,w_{n-1}$ and
sufficiently large $k>0$ every $A$-automorphism of $M$ presented by a matrix
from $\mathop{\rm F}_{n-1}(A,{\mathfrak{q}}^{k})$ with respect to the basis
$w_{1},\ldots,w_{n-1}$ is induced by a composition of automorphisms of the
form $\exp(t\varrho^{*}(f)\delta)$ (see, Corollary 6.14). By [Ni, Theorem 3]
$\mathop{\rm F}_{n-1}(A,{\mathfrak{q}}^{k})\supset\mathop{\rm
E}_{n-1}(A,{\mathfrak{q}}^{2k})$. Now one needs to switch from the basis
$w_{1},\ldots,w_{n-1}$ in $M$ to the basis $v_{1},\ldots,v_{n-1}$ in $N_{X}C$
which requires some results about matrices over Dedekind domains from [BMS]
and [Ba] (see, Theorem 5.2). Under this change of bases the group $\mathop{\rm
E}_{n-1}(A,{\mathfrak{q}}^{2k})$ is transformed to a group containing
$\mathop{\rm E}_{n-1}(A,{\mathfrak{q}}^{2mk})$ for some $m>0$. Hence, every
$A$-automorphism of $N_{X}C$ presented by a matrix from $\mathop{\rm
E}_{n-1}(A,{\mathfrak{q}}^{2mk})$ with respect to the basis
$v_{1},\ldots,v_{n-1}$ is induced by a composition of automorphisms of the
form $\exp(t\varrho^{*}(f)\delta)$ (see, Corollary 6.14). Then, replacing $K$
by $C\setminus V$ in the argument above, we show that for some ideal
${\mathfrak{p}}$ in $A$ such that ${\mathfrak{q}}+{\mathfrak{p}}=A$ every
$A$-automorphism of $N_{X}C$ presented by a matrix from $\mathop{\rm
E}_{n-1}(A,{\mathfrak{p}}^{2mk})$ is again induced by an automorphism of $X$.
Observing, that $\mathop{\rm E}_{n-1}(A,{\mathfrak{q}}^{2mk})$ and
$\mathop{\rm E}_{n-1}(A,{\mathfrak{p}}^{2mk})$ generate $\mathop{\rm
E}_{n-1}(A)$ we get the desired claim.

With this claim at hand we note that for an $A$-automorphism
$\theta:\frac{{\rm\bf k}[X]}{I^{2}}\to\frac{{\rm\bf k}[X]}{I^{2}}$ of the
second infinitesimal neighborhood of $C$ the assumption that the Jacobian is 1
is equivalent to the fact that the induced automorphism of $N_{X}C$ is
contained in $\mathop{\rm SL}_{n-1}(A)$. This implies that if the claim before
is valid, then $\theta$ is induced by an automorphism of $X$ provided
$\mathop{\rm SL}_{n-1}(A)=\mathop{\rm E}_{n-1}(A)$. Say, rational curves
satisfy this property (which is the reason why such curves appear in Theorem
0.4), whereas in the holomorphic category one has always $\mathop{\rm
SL}_{n-1}({\hat{A}})=\mathop{\rm E}_{n-1}({\hat{A}})$ by virtue of the
Ivarsson-Kutzschebauch theorem (see, Proposition 9.3). To extend this result
to automorphisms $\tilde{\theta}:\frac{{\rm\bf k}[X]}{I^{k}}\to\frac{{\rm\bf
k}[X]}{I^{k}}$ of the $k$th infinitesimal neighborhood of $C$ we use induction
on $k$ with the case $k=2$ as the first step of induction. For implication
$k-1\implies k$ it suffices to show that $\tilde{\theta}$ is induced by an
automorphism of $X$ if its restriction to the $(k-1)$st infinitesimal
neighborhood of $C$ is trivial. The proof of the latter statement essentially
copies the similar step in the proof of [AFKKZ, Theorem 4.14].

The paper is organized as follows. In Section 1 we collect technical tools
developed in [Ka20] that are important for this paper. Section 2 contains some
criterion which enables us to deal with curves that are not once-punctured
(recall that a curve $C$ is once-punctured if for any completion ${\bar{C}}$
of $C$ the set ${\bar{C}}\setminus C$ is a singleton). The advantage of a
once-punctured curve (like $\mathbb{C}$) is that every morphism of this curve
to a quasi-affine algebraic variety is automatically proper. Meanwhile, the
assumptions under which we can prove the crucial step described before include
the properness of the morphisms $\varrho|_{C}:C\to Q$. The criterion we
developed guarantees that for any closed curve $Z$ in a flexible variety $X$
and a morphism $\tau:X\to P$ to an affine variety $P$ there is an automorphism
$\alpha$ of $X$ such that $\tau\circ\alpha|_{Z}:Z\to Q$ is proper. In Section
3 we fix some notations for the rest of the paper and spell out a convention
under which our criterion of properness works. In Section 4 using this
convention we describe the structure of normal bundles of smooth curves in a
large class of flexible varieties. Actually, we show later (Corollary 7.4)
that if a normal bundle of a smooth closed curve in such variety is trivial,
then every nonvanishing section of this bundle is induced by a locally
nilpotent vector field (cf., [AFKKZ, Corollary 4.3]). Section 5 contains some
facts about special linear groups and their subgroups generated by elementary
transformations over commutative rings. In Sections 6 we prove the main
theorem (Theorem 6.6) that states that under the assumption on Jacobians and
some other mild assumptions automorphisms of $k$th infinitesimal neighborhoods
of smooth closed irreducible curves in a flexible variety $X$ whose normal
bundles are trivial extend to automorphisms of $X$. In Section 7 we remove the
assumption of irreducibility of these curves in the main theorem. As
applications we prove Theorems 0.1 and 0.4 in Section 8. Section 9 contains
Theorems 0.2 and 0.3.

Acknowledgements. The authors are grateful to Leonid Vaserstein and Mikhail
Zaidenberg for useful consultations. They are also deeply indebted to the
referee for a very thorough and helpful review. In particular, the referee
corrected a mistake in the original proof of Lemma 6.21 and suggested a better
proof of Proposition 9.3 which we use in this text.

##  1\. Preliminaries

In this section we present some technical tools developed in [Ka20] which
require the following definitions.

######  Definition 1.1.

(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 $\operatorname{{\rm Aut}}(X)$.

######  Definition 1.2.

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 $\operatorname{{\rm Aut}}(X)$.

######  Definition 1.3.

(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) Note that the kernel $\operatorname{{\rm Ker}}\delta$ of $\delta$
coincides with the ring ${\rm\bf k}[X]^{\mathbb{G}_{a}}$ of
$\mathbb{G}_{a}$-invariants in ${\rm\bf k}[X]$ where ${\rm\bf
k}[X]^{\mathbb{G}_{a}}$ can be always treated as the ring of regular functions
on a quasi-affine variety $V$ [Wi1, Theorem 1]). The morphism $\tau:X\to V$
induced by the natural embedding ${\rm\bf
k}[X]^{\mathbb{G}_{a}}\hookrightarrow{\rm\bf k}[X]$ is called the categorical
quotient morphism of the $\mathbb{G}_{a}$-action. When $V$ is an affine
variety (i.e., when ${\rm\bf k}[X]^{\mathbb{G}_{a}}$ is finitely generated) we
call $\tau$ an algebraic quotient morphism. Since $V$ is not always affine by
Nagata’s example, we use sometimes instead of $\tau$ partial quotient
morphisms. Namely, a morphism $\varrho:X\to Q$ into a normal affine variety
$Q$ is called a partial quotient morphism associated with $\delta$ if the ring
$\varrho^{*}({\rm\bf k}[Q])$ is contained in $\operatorname{{\rm Ker}}\delta$
and general fibers of $\varrho$ are isomorphic to ${\mathbb{A}}^{1}$. If $X$
is normal, then partial quotient morphisms always exist by virtue of the
Rosenlicht Theorem (e.g., see [PV, Theorem 2.3] and [FKZ, Definition 2.16]).

(3) 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
$X$ as an open subset 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 $X$. As a partial quotient morphism $\varrho:X\to Q$ of
$\delta$ we consider the restriction of a partial quotient morphism of
$\tilde{\delta}$ to $X$.

######  Remark 1.4.

Definition 1.3 (2)-(3) implies that for a dense open smooth subvariety $Q_{0}$
of $Q$ and $X_{0}=\varrho^{-1}(Q_{0})$ the morphism $\varrho|_{X_{0}}:X_{0}\to
Q_{0}$ is smooth and surjectve (in fact, one can assume that $X_{0}$ is
naturally isomorphic to $Q_{0}\times{\mathbb{A}}^{1}$ by [KaMi, Theorem 2])
and $\delta$ does not vanish on $X_{0}$.

######  Definition 1.5.

(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 1.6.

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 Collective Transversality Theorem is one of the main technical
tools in [Ka20].

######  Theorem 1.7.

([Ka20, Theorem 1.4], cf. [AFKKZ, Theorem 1.15 ]) Let $X$ and $P$ be smooth
irreducible algebraic varieties and $\kappa:X\to P$ be a smooth morphism. Let
$G$ be a subgroup of the group $\operatorname{{\rm Aut}}(X/P)$ of
automorphisms of $X$ over $P$ such that $G$ is algebraically generated by a
system ${\mathcal{G}}$ of connected algebraic subgroups closed under
conjugation in $G$. Suppose that the restriction of the $G$-action to
$\kappa^{-1}(p)$ is transitive for every $p\in P$.

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

(i) $\dim(Y\cap(h_{1}\cdot\ldots\cdot h_{m}).Z)\leq\dim(Y\times_{P}Z)+\dim
P-\dim X$.

In particular, when $\dim Y\times_{P}Z\leq\dim Y+\dim Z-\dim P$ one has

(ii) $\dim(Y\cap(h_{1}\cdot\ldots\cdot h_{m}).Z)\leq\dim Y+\dim Z-\dim X$.

Furthermore, suppose that the inequality $\dim Y\times_{P}Z\leq\dim Y+\dim
Z-\dim P$ holds and also that $Z$, $Y\times_{P}Z$, and $Y\times_{P}X$ are
smooth. Then

(iii) $(h_{1}\cdot\ldots\cdot h_{m}).Z$ meets $Y$ transversally.

The crucial step in the proof of Theorem 1.7 is the follwoing.

######  Proposition 1.8.

([Ka20, Proposition 1.7]) Let the assumption of Theorem 1.7 hold. Then there
is a sequence $H_{1},\ldots,H_{m}$ in ${\mathcal{G}}$ so that for a suitable
open dense subset $U\subseteq H_{m}\times\ldots\times H_{1}$, the map

(1) $\Phi:H_{m}\times\ldots\times H_{1}\times X\longrightarrow
X\times_{P}X\quad\mbox{with}\quad(h_{m},\ldots,h_{1},x)\mapsto((h_{m}\cdot\ldots\cdot
h_{1}).x,x)$

is smooth on $U\times X$.

Namely, we have the following.

######  Proposition 1.9.

([Ka20, Proposition 1.10]) If a sequence $H_{1},\ldots,H_{m}\in{\mathcal{G}}$
satisfies the conclusions of Proposition 1.8, then it satisfies also the
conclusions of Theorem 1.7. Furthermore, for any element $H$ of
${\mathcal{G}}$ the sequence $H_{1},\ldots,H_{m},H$ (resp.
$H,H_{1},\ldots,H_{m}$) satisfies the conclusions of Proposition 1.8 and
Theorem 1.7 as well.

We shall need the following notions which, unfortunately, were not introduced
in [Ka20].

######  Definition 1.10.

Let $\kappa:X\to P$, $G$ and ${\mathcal{G}}$ satisfy the assumptions of
Theorem 1.7 and $\dim X=n$. Consider $(X\times_{P}X)\setminus\Delta$ (where
$\Delta$ is the diagonal), the complement $T^{\prime}X$ 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 $T_{x}X$ 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 $T^{\prime}X$, or ${\rm PFr}(X)$. Suppose that the $G$-action is transitive
on every fiber of $Y$ over $P$. Then we say that an algebraic $G$-family
${\mathcal{A}}$ of automorphisms of $X$ is a regular $G$-family for $Y$ over
$P$ 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

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

is smooth on $U\times Y$.

Of course, by Proposition 1.8 the assumption about transitivity implies that
regular families exist. Note also that if ${\mathcal{A}}$ is a regular
$G$-family, say, for $(X\times_{P}X)\setminus\Delta$ over $P$ and
${\mathcal{B}}$ is a regular $G$-family for $T^{\prime}X$ over $P$, then by
Proposition 1.9 ${\mathcal{A}}\times{\mathcal{B}}$ (resp.
${\mathcal{B}}\times{\mathcal{A}}$) is a regular $G$-family for both
$(X\times_{P}X)\setminus\Delta$ and $T^{\prime}X$ over $P$. This implies the
existence of algebraic $G$-families ${\mathcal{A}}$ that are regular for all
four varieties $X$, $(X\times_{P}X)\setminus\Delta$, $T^{\prime}X$ and ${\rm
PFr}(X)$ over $P$. We call such ${\mathcal{A}}$ a perfect $G$-family for $Y$
over $P$ (if $P$ is a singleton, then we just say that ${\mathcal{A}}$ is a
perfect $G$-family for $Y$).

######  Remark 1.11.

It is worth observing the following property of regular $G$-families in the
case when $P$ is a singleton. Let $K$ be a finite subset of $Y$, $Z$ be a
dense open subset of $Y\times K$ and $Z_{y}=Z\cap(Y\times y)$ where $y\in K$.
Note that $V_{y}=(\Psi|_{U\times y})^{-1}(Z_{y})$ is a nonempty open subset of
$U$ since $\Psi$ is smooth. Hence, $\Psi(\bigcap_{y\in K}V_{y}\times K)$ is a
nonempty open subset of $Y\times K$. This implies that the morphism

$U\to Y\times K,\,\alpha\mapsto(\alpha(y))_{y\in K}$

is dominant.

######  Example 1.12.

Let $X$ be a smooth $G$-flexible variety where $G$ is generated by a saturated
set ${\mathcal{N}}\subset\operatorname{{\rm LND}}(X)$. Then the $G$-action is
$m$-transitive for every $m>0$ [FKZ, Theorem 2.12]. Hence, the natural action
of $G$ on $(X\times_{P}X)\setminus\Delta$ is transitive. Similarly, treating
$X$ as an open subset of a normal affine variety one can apply [AFKKZ, Theorem
4.14 and Remark 4.16] to show that $G$ acts transitively on $T^{\prime}X$ and
on ${\rm PFr}(X)$ (but not on ${\rm Fr}(X)$). Hence, $X$ admits a perfect
$G$-family.

Theorem 1.7 was used to establish the next fact.

######  Theorem 1.13.

([Ka20, Theorem 4.2(i)(iii)]) Let $X$ and $P$ be smooth algebraic varieties
and $Q$ be a normal algebraic variety. Let $\varrho:X\to Q$ and $\tau:Q\to P$
be dominant morphisms such that $\kappa:X\to P$ is smooth for
$\kappa=\tau\circ\varrho$. Suppose that $Q_{0}$ is a non-empty open smooth
subset of $Q$ so that for $X_{0}=\varrho^{-1}(Q_{0})$ the morphism
$\varrho|_{X_{0}}:X_{0}\to Q_{0}$ is smooth. Let $G$ be an algebraically
generated subgroup of the group $\operatorname{{\rm Aut}}(X/P)$ of
automorphisms of $X$ over $P$ such that $G$ acts 2-transitively on each fiber
of $\kappa:X\to P$ and let $Z$ be a locally closed reduced subvariety in $X$.

(i) Let $\dim Z\times_{P}Z\leq 2\dim Z-\dim P$ and $\dim Q\geq\dim Z+m$ 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) Suppose that $X$ is $G$-flexible where $G$ is generated by a saturated
set ${\mathcal{N}}\subset\operatorname{{\rm LND}}(X)$ and $P_{0}=\tau(Q_{0})$.
Let $\dim Z\times_{P}Z\leq 2\dim Z-\dim P,2\dim Z+1\leq\dim Q$ and $\dim
T(Z_{0}/P_{0})\leq\dim Q-\dim P$. Then the algebraic family ${\mathcal{A}}$
from (i) can be chosen so that for a general element $\alpha\in{\mathcal{A}}$
the morphism $\varrho|_{\alpha(Z)\cap X_{0}}:\alpha(Z)\cap X_{0}\to
Z_{\alpha}^{\prime}\cap Q_{0}$ is bijective and it induces an injective map of
the tangent bundle of $\alpha(Z)\cap X_{0}$ into the tangent bundle of
$Q_{0}$.

######  Remark 1.14.

(1) We shall need below the case when $P$ is a singleton. In particular, the
assumption $\dim Z\times_{P}Z\leq 2\dim Z-\dim P$ can be omitted.

(2) If $X$ is a smooth $G$-flexible variety, then the $G$-action on $X$ is
$m$-transitive for every $m\geq 1$ [FKZ, Theorem 2.12]. In particular, when
$P$ is a singleton the assumption that $G$ acts 2-transitively on $X$ can be
also omitted in this case.

(3) It follows from the proof that any regular $G$-family for
$(X\times_{P}X)\setminus\Delta$ over $P$ can serve as ${\mathcal{A}}$ in
Theorem 1.13 (i), whereas for Theorem 1.13 (ii) any $G$-family regular for
both $(X\times_{P}X)\setminus\Delta$ and $T^{\prime}X$ over $P$ works. Thus,
whenever we apply Theorem 1.13 below we use a perfect $G$-family. In
particular, for every $H\in{\mathcal{G}}$ the families $H\times{\mathcal{A}}$
and ${\mathcal{A}}\times H$ also satisfy the conclusions of Theorem 1.13.

Another fact from [Ka20] (also based on Theorem 1.7) which we shall need later
is the following.

######  Theorem 1.15.

([Ka20, Theorem 6.1]) Let $X$ be a smooth quasi-affine algebraic variety,
${\mathcal{N}}$ be a saturated set of locally nilpotent vector fields on $X$,
and $G\subset\mathop{\rm SAut}(X)$ be the group generated by ${\mathcal{N}}$.
Suppose that $X$ is $G$-flexible. Let $\varrho:X\to Q$ be a partial quotient
morphism associated with a nontrivial $\delta\in{\mathcal{N}}$, $Z$ be a
locally closed reduced subvariety of $X$ of codimension at least 2 and $K$ be
a finite subset of $Z$ such that $\dim T_{z_{0}}Z\leq\dim Q$ for every
$z_{0}\in K$. Then there exists a connected algebraic $G$-family
${\mathcal{A}}$ of automorphisms of $X$ such that for a general element
$\alpha\in{\mathcal{A}}$ and the closure ${\bar{Z}}_{\alpha}^{\prime}$ of
$Z_{\alpha}^{\prime}=\varrho\circ\alpha(Z)$ in $Q$ one can find a neighborhood
$V_{\alpha}^{\prime}$ of $\varrho(\alpha(K))$ in ${\bar{Z}}_{\alpha}^{\prime}$
such that for $V_{\alpha}=\varrho^{-1}(V_{\alpha}^{\prime})\cap\alpha(Z)$ the
morphism $\varrho|_{V_{\alpha}}:V_{\alpha}\to V_{\alpha}^{\prime}$ is an
isomorphism.

######  Remark 1.16.

(1) The proof of Theorem 1.15 (see [Ka20, page 550]) implies that
${\mathcal{A}}$ is of the form
${\mathcal{A}}=U_{\kappa}\times{\mathcal{A}}_{1}$ where $U_{\kappa}$ is a
fixed family111This family $U_{\kappa}$ is a product of several one-parameter
unipotent subgroup depending on $\delta$ and $K$ but not on $Z$ [Ka20, Formula
(20)]., while ${\mathcal{A}}_{1}=H_{1}\times\ldots\times H_{m}$ is any regular
$G$-family for $X$ that contains an element $\alpha_{0}\in G$ with the
following property: for every $z\in\alpha_{0}(K)$ and $q=\varrho(z)$ the
induced map $\varrho_{*}:T_{z}\alpha(Z)\to T_{q}Q$ is injective. Note that by
Remark 1.11 such $\alpha_{0}$ exists automatically if one requires that
${\mathcal{A}}_{1}$ is a regular $G$-family for ${\rm PFr}(X)$. Thus, from now
on whenever we apply Theorem 1.15 we suppose that ${\mathcal{A}}_{1}$ is a
perfect family.

(2) Another important feature of Theorem 1.15 we want to emphasize is that by
construction $V_{\alpha}^{\prime}$ is contained in $Q_{0}\subset Q$ as in
Remark 1.4.

In conclusion of this section we state the following simple fact which will be
crucial in further considerations.

######  Lemma 1.17.

([AFKKZ, Lemma 4.1])222In [AFKKZ] this fact was formulated for an affine
variety $X$ but the proof goes without change in the quasi-affine case. Let
$\delta$ be a locally nilpotent vector field on a smooth quasi-affine variety
$X$, and let $p\in X$ be a point. Assume that $f\in\operatorname{{\rm
Ker}}\delta$ and $f(p)=0$. If $\Phi=\exp(f\delta)$ is the automorphism
associated with the replica $f\delta$, then

(3) ${\rm d}_{p}\Phi(w)=w+{\rm d}_{p}f(w)\delta(p)\,\,\,\text{for
all}\,\,\,w\in T_{p}X$

where $\delta(p)\in T_{p}X$ is the value of the vector field $\delta$ at $p$.

##  2\. Properness

######  Definition 2.1.

Let $\delta_{1}$ and $\delta_{2}$ be locally nilpotent vector fields on an
affine algebraic variety $X$ and $\varrho_{i}:X\to Q_{i}$ be partial quotient
morphisms associated with $\delta_{i},\,i=1,2$.

(1) The pair $\delta_{1},\delta_{2}$ will be called suitable if $\varrho_{1}$
and $\varrho_{2}$ can be chosen so that the morphism
$\tau=(\varrho_{1},\varrho_{2}):X\to Q_{1}\times Q_{2}$ is proper.

(2) The pair $\delta_{1},\delta_{2}$ will be called semi-compatible if
$\operatorname{{\rm Ker}}\delta_{1}$ and $\operatorname{{\rm
Ker}}\delta_{2}\subset{\rm\bf k}[X]$ are finitely generated (i.e.,
$\varrho_{1}$ and $\varrho_{2}$ can be chosen to be algebraic quotient
morphisms), $\tau:X\to\tau(X)$ is birational and finite where $\tau(X)$ is
closed in $Q_{1}\times Q_{2}$. In particular, every semi-compatible pair of
locally nilpotent vector fields is automatically suitable.

It is useful to keep in mind the following criterion of semi-compatibility.

######  Proposition 2.2.

([KaKu08, Proposition 3.4]) Let $\delta_{1}$ and $\delta_{2}$ be locally
nilpotent vector fields on a normal affine algebraic variety $X$ such that
their kernels $\operatorname{{\rm Ker}}\delta_{1}$ and $\operatorname{{\rm
Ker}}\delta_{2}\subset{\rm\bf k}[X]$ are finitely generated. Then these vector
fields are semi-compatible if and only if the span of $\operatorname{{\rm
Ker}}\delta_{1}\cdot\operatorname{{\rm Ker}}\delta_{2}$ contains a nontrivial
ideal of ${\rm\bf k}[X]$.

Though the next fact is not be used later in the paper, we include it as a
natural property of suitability.

######  Proposition 2.3.

(cf. [KaKu08, Proposition 3.6]) Let $\delta_{1}$ and $\delta_{2}$ be suitable
locally nilpotent vector fields on a affine algebraic variety $X$ with
finitely generated kernels $\operatorname{{\rm Ker}}\delta_{1}$ and
$\operatorname{{\rm Ker}}\delta_{2}$. Suppose that $\Gamma$ is a finite group
acting on $X$ so that each $\delta_{i}$ induces a locally nilpotent vector
field $\delta_{i}^{\prime}$ on the quotient $X^{\prime}=X/\Gamma$. Then
$\delta_{1}^{\prime}$ and $\delta_{2}^{\prime}$ are suitable on $X^{\prime}$.

###### Proof.

Let $\varrho_{i}:X\to Q_{i}$ and $\tau$ be as in Definition 2.1 and let
$K_{i}\subset\mathop{\rm SAut}(X)$ be the flow of $\delta_{i}$. Since by the
assumption the $K_{i}$-action on $X$ commutes with the $\Gamma$-action, we
have a $\Gamma$-action on $Q_{i}$ such that $\varrho_{i}$ is
$\Gamma$-equivariant. We have also the induced morphisms
$\varrho_{i}^{\prime}:X^{\prime}\to Q_{i}^{\prime}:=Q_{i}/\Gamma$. Every
morphism $\varphi^{\prime}:X^{\prime}\to Z$ that is constant on the orbits of
$K_{i}$ induces a $\Gamma$-equivariant morphism $\varphi:X\to Z$ (where $Z$ is
equipped with a trivial $\Gamma$-action). Since $\varrho_{i}$ is an algebraic
quotient morphism, $\varphi$ is the composition of $\varrho_{i}$ and a
morphism $\psi:Q_{i}\to Z$. Since $\varrho_{i}$ is $\Gamma$-equivariant, so is
$\psi$. Hence, $\psi$ factors through $Q_{i}^{\prime}$ which shows that
$\varrho_{i}^{\prime}:X^{\prime}\to Q_{i}^{\prime}$ is the algebraic quotient
morphism. Consider the following commutative diagram

$\begin{array}[]{ccc}X&\stackrel{{\scriptstyle{\tau}}}{{\rightarrow}}&Q_{1}\times
Q_{2}\\\ \,\,\,\,\downarrow{\kappa}&&\,\,\,\,\downarrow{\kappa^{\prime}}\\\
X^{\prime}&\stackrel{{\scriptstyle{\tau^{\prime}}}}{{\rightarrow}}&Q_{1}^{\prime}\times
Q_{2}^{\prime}\par\end{array}$

where $\tau^{\prime}=(\varrho_{1}^{\prime},\varrho_{2}^{\prime})$, $\kappa$ is
the quotient morphism of the $\Gamma$-action on $X$ and $\kappa^{\prime}$ is
the quotient morphism of the natural $\Gamma\times\Gamma$-action on
$Q_{1}\times Q_{2}$. Since $\tau$ and $\kappa^{\prime}$ are proper morphisms,
so is $\kappa^{\prime}\circ\tau=\tau^{\prime}\circ\kappa$. Since $\kappa$ is
surjective, this implies that $\tau^{\prime}$ is proper which is the desired
conclusion. ∎

The next simple observation is crucial.

######  Theorem 2.4.

Let $\\{\delta_{\beta}|\,\beta\in{\mathcal{B}}\\}$ (where ${\mathcal{B}}$ is
an index set) be an infinite collection of locally nilpotent vector fields on
an affine algebraic variety $X$ such that for every
$\beta\neq\gamma\in{\mathcal{B}}$ the pair $(\delta_{\beta},\delta_{\gamma})$
is suitable for appropriate partial quotient morphisms $\varrho_{\beta}:X\to
Q_{\beta}$ and $\varrho_{\gamma}:X\to Q_{\gamma}$. Suppose that $C$ is a
closed (but not necessarily irreducible) curve in $X$. Then there is a finite
subset $T$ of ${\mathcal{B}}$ such that for every
$\beta\in{\mathcal{B}}\setminus T$ the morphism $\varrho_{\beta}|_{C}:C\to
Q_{\beta}$ is proper.

###### Proof.

Suppose that ${\bar{C}}$ is a completion of $C$ such that it is smooth at
every point of ${\bar{C}}\setminus C$. By [Ha, Chapter II, Theorem 4.7] the
fact that a morphism $\theta:C\to Y$ is not proper is equivalent to the fact
that $\theta$ extends regularly to some point $c\in{\bar{C}}\setminus C$.
Assume that $\varrho_{\beta}|_{C}:C\to Q_{\beta}$ extends to
$c\in{\bar{C}}\setminus C$. Then for every $\beta\neq\gamma\in{\mathcal{B}}$
the morphism $\varrho_{\gamma}|_{C}:C\to Q_{\gamma}$ cannot be extended to
$c$. Indeed, otherwise the morphism $\tau|_{C}:C\to Q_{\beta}\times
Q_{\gamma}$ extends to $c$ where $\tau=(\varrho_{\beta},\varrho_{\gamma}):X\to
Q_{\beta}\times Q_{\gamma}$. That is, $\tau$ is not proper by [Ha, Chapter II,
Corollary 4.8] which contradicts suitability. Thus, there is at most a finite
number of elements $\beta$ of ${\mathcal{B}}$ such $\varrho_{\beta}|_{C}$
extends to some $c\in{\bar{C}}\setminus C$. This yields the desired
conclusion. ∎

######  Definition 2.5.

Let $X$ be an affine algebraic variety. We say that $X$ is parametrically
suitable if there exists a connected algebraic family ${\mathcal{B}}$ of
automorphisms of $X$ with $\dim{\mathcal{B}}\geq 1$, a locally nilpotent
vector field $\delta\in\operatorname{{\rm LND}}(X)$ and a collection
$\\{\varrho_{\beta}:X\to Q_{\beta}|\,\beta\in{\mathcal{B}}\\}$ of partial
quotient morphisms $\varrho_{\beta}$ associated with
$\delta_{\beta}:=\beta_{*}(\delta)$ such that for every
$\beta\neq\gamma\in{\mathcal{B}}$ the morphism
$(\varrho_{\beta},\varrho_{\gamma}):X\to Q_{\beta}\times Q_{\gamma}$ is proper
(i.e., the pair $(\delta_{\beta},\delta_{\gamma})$ is suitable). Suppose
additionally that $G\subset\operatorname{{\rm Aut}}(X)$ is a group generated
by a saturated set ${\mathcal{N}}\subset\operatorname{{\rm LND}}(X)$. If
$\delta$ belongs to ${\mathcal{N}}$ and ${\mathcal{B}}$ is contained in $G$,
then we say that $X$ is parametrically $G$-suitable. If we want to emphasize
that we use a specific $\delta$ in this definition we speak about parametric
$G$-suitability with respect to $\delta$.

######  Remark 2.6.

(1) For $\delta\in\operatorname{{\rm LND}}(X)$, $\alpha\in\operatorname{{\rm
Aut}}(X)$ and $\delta_{\alpha}=\alpha_{*}(\delta)$ one has by definition
$\delta_{\alpha}(f)=\delta(f\circ\alpha)$ for every $f\in{\rm\bf k}[X]$.
Hence, if $g=f\circ\alpha\in\operatorname{{\rm Ker}}\delta$, then
$f=g\circ\alpha^{-1}\in\operatorname{{\rm Ker}}\delta_{\alpha}$. This implies
that if $\varrho:X\to Q$ is a partial quotient morphism associated with
$\delta$, then $\varrho\circ\alpha^{-1}:X\to Q$ is a partial quotient morphism
associated with $\delta_{\alpha}$. In particular, we shall look for elements
of a collection $\\{\varrho_{\beta}:X\to Q_{\beta}|\,\beta\in{\mathcal{B}}\\}$
in Definition 2.5 in the form $\varrho_{\beta}=\varrho\circ\beta^{-1}$ (i.e.,
$Q_{\beta}=Q$).

(2) Note also that one can suppose that ${\mathcal{B}}$ in Definition 2.5
contains the identity automorphism. Indeed, if a pair
$(\delta_{\beta},\delta_{\gamma})$ is suitable, then the pair
$(\delta_{\alpha\circ\beta},\delta_{\alpha\circ\gamma})$ is also suitable,
since
$(\varrho_{\alpha\circ\beta},\varrho_{\alpha\circ\gamma})=(\varrho_{\beta},\varrho_{\gamma})\circ\alpha^{-1}$.
Hence, ${\mathcal{B}}$ can be replaced by $\alpha{\mathcal{B}}$. Choosing
$\alpha=\beta^{-1}$ for some $\beta\in{\mathcal{B}}$ we get the desired
statement.

(3) One can modify the notion of suitability and consider it not for pairs
but, say, for triples of locally nilpotent vector fields on $X$. That is, for
a connected nonconstant algebraic family ${\mathcal{B}}$ of automorphism of
$X$, $\delta\in\operatorname{{\rm LND}}(X)$, a collection
$\\{\varrho_{\beta}:X\to Q_{\beta}|\,\beta\in{\mathcal{B}}\\}$ of partial
quotient morphisms $\varrho_{\beta}$ associated with $\delta_{\beta}$ and
every triple $(\beta,\gamma,\chi)$ of distinct elements of ${\mathcal{B}}$ one
can require that the morphism
$(\varrho_{\beta},\varrho_{\gamma},\varrho_{\chi}):X\to Q\times Q\times Q$ is
proper. Then the straightforward adjustment of the proof of Theorem 2.4 shows
that $\varrho_{\beta}|_{C}:C\to Q$ is proper for a general
$\beta\in{\mathcal{B}}$. In this case we call $X$ weakly parametrically
suitable (we do not know if this notion is equivalent to parametric
suitability).

(4) Though we use partial quotient morphisms in the definition of (parametric)
suitability in all applications below $\delta$ (and, consequently,
$\delta_{\beta}$) admits an algebraic quotient morphism.

The following is one of the main criteria for parametric suitability.

######  Theorem 2.7.

Let $X$ be a smooth complex affine algebraic variety, $F\simeq\mathop{\rm
SL}_{2}(\mathbb{C})$ be an algebraic subgroup of $\operatorname{{\rm Aut}}(X)$
contained in a group $G\subset\operatorname{{\rm Aut}}(X)$ such that the
natural $F$-action on $X$ is fixed point free and non-degenerate333An action
of a linear algebraic group is non-degenerate if general orbits have the same
dimension as the group.. Then $X$ is parametrically $G$-suitable.

###### Proof.

Fixing some basis $\bar{u}=(u_{1},u_{2})$ in $\mathbb{C}^{2}$ with the natural
action of $\mathop{\rm SL}_{2}(\mathbb{C})$ we get a unipotent subgroup
$H_{1}$ (resp. $H_{2}$) of upper (resp. lower) triangular matrices in
$\mathop{\rm SL}_{2}(\mathbb{C})\simeq F$. The $\mathbb{G}_{a}$-action induced
by $H_{1}$ (resp. $H_{2}$) on $X$ is the flow of a locally nilpotent vector
field $\delta_{1}$ (resp. $\delta_{2}$). By [Had, Theorem 3.1] there exists an
algebraic quotient morphism $\varrho:X\to Q$ associated with $\delta_{1}$. The
central fact we use now is that $\delta_{1}$ and $\delta_{2}$ are semi-
compatible and even compatible in the terminology of [DDK, Theorem 12].
However, $H_{1}$ and $H_{2}$ (resp. $\delta_{1}$ and $\delta_{2}$) depend on
the choice of $\bar{u}$. In particular, for every $\alpha\in\mathop{\rm
SL}_{2}(\mathbb{C})$ the groups $\alpha H_{1}\alpha^{-1}$ and $\alpha
H_{2}\alpha^{-1}$ present upper and lower triangular matrices in an
appropriate basis (note that this conjugation sends $\delta_{i}$ to
$\alpha_{*}(\delta_{i})$). Consider ${\mathcal{B}}=H_{2}$ and let us show that
for general $\alpha\neq\beta\in{\mathcal{B}}$ the locally nilpotent vector
fields $\alpha_{*}(\delta_{1})$ and $\beta_{*}(\delta_{1})$ are semi-
compatible, or, equivalently, in an appropriate basis $\alpha
H_{1}\alpha^{-1}$ and $\beta H_{1}\beta^{-1}$ correspond to upper and lower
unipotent matrices respectively. The latter is equivalent to the similar claim
about the groups $\beta^{-1}\alpha H_{1}\alpha^{-1}\beta$ and $H_{1}$. We can
suppose that $\alpha:(u_{1},u_{2})\mapsto(u_{1},u_{2}+au_{1})$ and
$\beta:(u_{1},u_{2})\mapsto(u_{1},u_{2}+bu_{1})$. Then a direct computation
shows that $\beta^{-1}\alpha H_{1}\alpha^{-1}\beta$ consists of matrices of
the form

$\left[{\begin{array}[]{cc}1-(a-b)t&t\\\
-(a-b)^{2}t&1+(a-b)t\end{array}}\right].$

Another direct computation shows that for the automorphism
$\gamma:(u_{1},u_{2})\mapsto(u_{1}+(a-b)^{-1}u_{2},u_{2})$ one has $\gamma
H_{2}\gamma^{-1}=\beta^{-1}\alpha H_{1}\alpha^{-1}\beta$. Hence,
$\beta^{-1}\alpha H_{1}\alpha^{-1}\beta$ and $H_{1}$ correspond to upper and
lower unipotent matrices in an appropriate basis since $\gamma
H_{2}\gamma^{-1}$ and $H_{1}=\gamma H_{1}\gamma^{-1}$ do. Thus,
$\alpha_{*}(\delta_{1})$ and $\beta_{*}(\delta_{1})$ are suitable for general
$\alpha\neq\beta\in{\mathcal{B}}$ which yields the desired conclusion. ∎

######  Example 2.8.

Let $H$ be a complex semi-simple Lie group with Lie algebra different from
${\mathfrak{s}}{\mathfrak{l}}_{2}$. Let $R$ be a proper closed reductive
subgroup of $H$. Then the homogeneous space $H/R$ is parametrically suitable.
Indeed, by [DDK, Theorem 24 and the proof of Corollary 25] $H/R$ admits a non-
degenerate fixed point free $\mathop{\rm SL}_{2}(\mathbb{C})$-action. Hence,
we are done by Theorem 2.7.

######  Corollary 2.9.

Let $X$ be a smooth affine algebraic variety equipped with a fixed point free
non-degenerate $\mathop{\rm SL}_{2}({\rm\bf k})$-action. Let $\varrho:X\to Q$
and ${\mathcal{B}}$ be the same as in the proof of Theorem 2.7 (but with
$\mathbb{C}$ replaced by ${\rm\bf k}$). Then for every closed curve $C$ in $X$
and a general $\beta\in{\mathcal{B}}$ the morphism
$\varrho_{\beta}|_{C}=\varrho\circ\beta^{-1}|_{C}:C\to Q$ is proper.

###### Proof.

If ${\rm\bf k}$ is a universal domain (i.e., it is an algebraically closed
field of characteristic 0 with infinite transcendence degree over
$\mathbb{Q}$), then one can just refer to [Ek]. To include a finite
transcendence degree in the proof one can argue like this. Let ${\tilde{C}}$
be an open subset in a completion of $C$ consisting of $C$ and one extra point
$z$. Consider $\tau_{\beta}:{\tilde{C}}\times{\mathcal{B}}\dashrightarrow Q$
that sends each $(c,\beta)\in C\times{\mathcal{B}}$ to $\varrho_{\beta}(c)$.
Assume that $\varrho_{\beta}:C\to Q$ is not proper for a general
$\beta\in{\mathcal{B}}$. Then for some ${\tilde{C}}$ as above $\tau_{\beta}$
is a morphism. By the “finite extension principle" there exist an
algebraically closed subfield ${\rm\bf k}^{0}$ of ${\rm\bf k}$ which is of a
finite transcendence degree over of $\mathbb{Q}$, algebraic varieties
$X^{0},Q^{0},{\tilde{C}}^{0}=C^{0}\cup z^{0},{\mathcal{B}}^{0}$ over ${\rm\bf
k}^{0}$, morphisms $\varrho_{\beta}^{0}:C^{0}\to Q^{0}$,
$\tau_{\beta}^{0}:{\tilde{C}}^{0}\times{\mathcal{B}}^{0}\rightarrow Q^{0}$ and
an $\mathop{\rm SL}_{2}({\rm\bf k}_{0})$-action on $X^{0}$ such that
$\varrho_{\beta}:C\to Q$, $\tau_{\beta}:{\tilde{C}}\times{\mathcal{B}}\to Q$
and the $\mathop{\rm SL}_{2}({\rm\bf k})$-action on $X$ are obtained by the
base extension $\otimes_{\operatorname{{\rm Spec}}{\rm\bf
k}_{0}}\operatorname{{\rm Spec}}{\rm\bf k}$. By the Lefschetz principle we can
view ${\rm\bf k}^{0}$ as a subfield of $\mathbb{C}$. The extension
$\otimes_{\operatorname{{\rm Spec}}{\rm\bf k}_{0}}\operatorname{{\rm
Spec}}\mathbb{C}$ leads to morphisms
$\varrho_{\beta}^{\mathbb{C}}:C^{\mathbb{C}}\to Q^{\mathbb{C}}$,
$\tau_{\beta}^{\mathbb{C}}:{\tilde{C}}^{\mathbb{C}}\times{\mathcal{B}}^{\mathbb{C}}\rightarrow
Q^{\mathbb{C}}$ of complex varieties and a non-degenerate fixed point free
$\mathop{\rm SL}_{2}(\mathbb{C})$-action on $X^{\mathbb{C}}$. However,
Theorems 2.4 and 2.7 imply that
$\varrho_{\beta}^{\mathbb{C}}:C^{\mathbb{C}}\to Q^{\mathbb{C}}$ is proper for
general $\beta\in{\mathcal{B}}$ which contradicts the existence of the
morphism $\tau_{\beta}^{\mathbb{C}}$. Hence, we have the desired conclusion. ∎

##  3\. Notations and Convention

Let us list some notations which will remain unchanged through the rest of the
paper.

######  Setting 3.1.

We suppose further that

\- $X$ is a normal quasi-affine variety of dimension $n$;

\- $\delta$ is a locally nilpotent nonzero vector field on $X$;

\- $\varrho:X\to Q$ is a partial quotient morphism associated with $\delta$;

\- $Q_{0}$ is a smooth open dense subset of $Q$ such that
$\varrho|_{\varrho^{-1}(Q_{0})}:\varrho^{-1}(Q_{0})\to Q_{0}$ is a smooth
morphism and $\delta$ does not vanish on $\varrho^{-1}(Q_{0})$ (i.e., $Q_{0}$
is as in Remark 1.4);

\- for any $\alpha\in\operatorname{{\rm Aut}}(X)$ we let
$\delta_{\alpha}=\alpha_{*}(\delta)$ and
$\varrho_{\alpha}=\varrho\circ\alpha^{-1}:X\to Q$ (i.e., $\varrho_{\alpha}$ is
a partial quotient morphism associated with $\delta_{\alpha}$);

\- $C$ is a smooth (but not necessarily irreducible) curve in $X$ such that
$C\subset X_{\rm reg}$ and $C$ is closed in an affine variety containing $X$
as a dense open subset;

\- $I$ is the defining ideal of $C$ in ${\rm\bf k}[X]$ (except for Section 5
where $I$ is an identity matrix);

\- $\mathop{\rm pr}:TX|_{C}\to N_{X}C$ is the natural projection onto the
normal bundle $N_{X}C$ of $C$ in $X$.

In the case when $C$ contains an irreducible component which is not a once-
punctured curve we shall often use the following important convention.

######  Convention 3.2.

Suppose additionally that $X$ is a $G$-flexible variety with $\dim X=n\geq 4$
where the group $G\subset\mathop{\rm SAut}(X)$ is generated by a saturated set
${\mathcal{N}}\subset\operatorname{{\rm LND}}(X)$ such that
$0\neq\delta\in{\mathcal{N}}$ and the next two conditions are true:

($\\#$) the variety $\varrho^{-1}(Q\setminus Q_{0})$ is of codimension at
least $2$ in $X$ (in particular, the zero locus of $\delta$ is at least of
codimension 2);

($\\#\\#$) $X$ is affine and parametrically $G$-suitable with respect to
$\delta$.

######  Remark 3.3.

The condition ($\\#$) cannot be acheived for some flexible Gizatullin surfaces
but in higher dimensions the authors do no know examples of smooth affine
flexible varieties when ($\\#$) does not hold for an appropriate $\delta$. For
instance, if $X$ is a linear algebraic group without nontrivial characters,
then every nonzero nilpotent element of its Lie algebra yields a locally
nilpotent vector field on $X$ for which condition ($\\#$) is true. Another
example is a complex sphere given by $z_{1}^{2}+\ldots+z_{m}^{2}=1$ where
$\bar{z}=(z_{1},\ldots,z_{m})$ is a coordinate system on ${\mathbb{A}}^{m}$.
Then $\sigma=z_{2}\frac{\partial}{\partial
z_{1}}-z_{1}\frac{\partial}{\partial z_{2}}$ is a locally nilpotent vector
field on this sphere whose zero locus has codimension 2. The kernel of
$\sigma$ is generated by $z_{3},\ldots,z_{m}$ and, therefore, the quotient
space is smooth and ($\\#$) holds.

##  4\. Normal bundles in flexible varieties

Before describing $N_{X}C$ from Setting 3.1 we need the following.

######  Proposition 4.1.

Let $X$ be $G$-flexible where the group $G\subset\mathop{\rm SAut}(X)$ is
generated by a saturated set ${\mathcal{N}}\subset\operatorname{{\rm LND}}(X)$
such that $\delta\in{\mathcal{N}}$. Let $K$ be a finite subset of $C$. Then
there exists a perfect $G$-family ${\mathcal{A}}$ such that for general
$\alpha_{1},\ldots,\alpha_{n-1}\in{\mathcal{A}}$ there is an open neighborhood
$C^{*}$ of $K$ in $C$ for which the following is true.

(i) For every $i=1,\ldots,n-1$ the restriction of $\varrho_{\alpha_{i}}$
yields an isomorphism between $C^{*}$ and
$C^{*}_{i}:=\varrho_{\alpha_{i}}(C^{*})$ where $C_{i}^{*}$ is contained in
$Q_{0}$.

(ii) The normal bundle $N_{X}C^{*}$ is trivial and $\mathop{\rm
pr}(\delta_{\alpha_{1}}),\ldots,\mathop{\rm pr}(\delta_{\alpha_{n-1}})$
generate $N_{X}C^{*}$ as a module over ${\rm\bf k}[C^{*}]$.

(iii) The morphism ${\mathcal{A}}\to\prod_{x\in
K}T_{x}X,\,\alpha\mapsto(\delta_{\alpha}(x))_{x\in K}$ is dominant.

###### Proof.

Statement (i) follows from Theorem 1.15 and Remark 1.16 (2) applied to $X_{\rm
reg}$ and the fact that $\varrho_{\alpha}=\varrho\circ\alpha^{-1}$. Let
$T^{\prime}X_{\rm reg}$ be the complement to the zero section in the tangent
bundle $TX_{\rm reg}$. Recall that ${\mathcal{A}}$ acts transitively on
$T^{\prime}X_{\rm reg}$ (see, Example 1.12). Hence, letting
$Y=T^{\prime}X_{\rm reg}$ in Remark 1.11 we get (iii). In particular, we can
suppose that the vectors
$\delta_{\alpha_{1}}(x),\ldots,\delta_{\alpha_{n-1}}(x)$ are general in
$T_{x}X$ and, thus, linearly independent modulo $T_{x}C$. Taking, if
necessary, a smaller neighborhood $C^{*}$ of $K$ such that for every $x\in
C^{*}$ the similar fact hold we get (ii). Hence, we are done. ∎

It turns out that under Convention 3.2 we can get a much stronger version of
Proposition 4.1.

######  Proposition 4.2.

Let Convention 3.2 hold. Then one can choose a perfect $G$-family
${\mathcal{A}}$ so that for a general $\alpha\in{\mathcal{A}}$

(i) the morphism $\varrho_{\alpha}|_{C}:C\to Q$ is proper;

(ii) its restriction yields an isomorphism between $C$ and
$\varrho_{\alpha}(C)$ and

(iii) $\varrho_{\alpha}(C)\subset Q_{0}$ (in particular, $\delta_{\alpha}$
does not vanish on $C$).

###### Proof.

Theorem 1.7 implies that for a general $\alpha$ in any perfect family
$\alpha^{-1}(C)$ does not meet $\varrho^{-1}(Q\setminus Q_{0})$, since the
latter is of codimension at least 2. Since
$\varrho_{\alpha}=\varrho\circ\alpha^{-1}$, this yields (iii). Let
${\mathcal{A}}_{1}$ now be a perfect $G$-family as in Theorem 1.13(ii), i.e.
for general $\alpha\in{\mathcal{A}}_{1}$ the morphism
$\varrho_{\alpha}|_{C}:C\to Q$ is an injective immersion. Note that if $C$ is
once-punctured, then $\varrho_{\alpha}|_{C}:C\to Q$ is automatically proper
and, in particular, $\varrho_{\alpha}(C)$ is closed in $Q$. Hence, in this
case $\varrho_{\alpha}|_{C}$ yields an isomorphism between $C$ and
$\varrho_{\alpha}(C)$. If $C$ is not once-punctured consider a family
${\mathcal{B}}\subset G$ of automorphisms of $X$ as in Definition 2.5. Recall
that by Remark 1.14 the replacement of ${\mathcal{A}}_{1}$ by
${\mathcal{A}}={\mathcal{A}}_{1}\times{\mathcal{B}}$ leaves the conclusions of
Theorem 1.13 valid. Consider the morphism
${\mathcal{A}}_{1}\times{\mathcal{B}}\times
X\to{\mathcal{A}}_{1}\times{\mathcal{B}}\times
Q,\,(\alpha,\beta,x)\mapsto(\alpha,\beta,\varrho_{\alpha\beta}(x))$ and extend
it to a proper morphism
${\mathcal{X}}\to{\mathcal{A}}_{1}\times{\mathcal{B}}\times Q$ over
${\mathcal{A}}_{1}\times{\mathcal{B}}$. Let
${\mathcal{D}}={\mathcal{X}}\setminus({\mathcal{A}}_{1}\times{\mathcal{B}}\times
X)$ and ${\mathcal{C}}$ be the closure of
${\mathcal{A}}_{1}\times{\mathcal{B}}\times C$ in ${\mathcal{X}}$.

Assume that $\varrho_{\alpha\beta}|_{C}:C\to Q$ is not proper for a general
$(\alpha,\beta)$ in ${\mathcal{A}}_{1}\times{\mathcal{B}}$. Recall that the
morphism $\varrho_{\alpha\beta}|_{C}:C\to Q$ is not proper iff the closure
$C_{\alpha,\beta}$ of the curve $\alpha\times\beta\times C$ in ${\mathcal{X}}$
meets ${\mathcal{D}}$. Hence, there exists a dense subset $U$ of
${\mathcal{A}}_{1}\times{\mathcal{B}}$ such that for every $(\alpha,\beta)\in
U$ the curve $C_{\alpha,\beta}$ meets ${\mathcal{C}}\cap{\mathcal{D}}$. This
implies that there exists an irreducible component of
${\mathcal{C}}\cap{\mathcal{D}}$ whose image in
${\mathcal{A}}_{1}\times{\mathcal{B}}$ under the natural morphism
${\mathcal{D}}\to{\mathcal{A}}_{1}\times{\mathcal{B}}$ is dense. Therefore, we
can suppose that $U$ is open. For a general $\alpha\in{\mathcal{A}}_{1}$ and
the natural projection
$\theta:{\mathcal{A}}_{1}\times{\mathcal{B}}\to{\mathcal{A}}_{1}$ the
intersection of $\theta^{-1}(\alpha)\cap U$ is open and nonempty in
$\theta^{-1}(\alpha)\simeq{\mathcal{B}}$. That is, the morphism
$\varrho_{\alpha\beta}|_{C}:C\to Q$ is not proper when
$\beta\in\theta^{-1}(\alpha)\cap U$. On the other hand, this morphism factors
through the isomorphism $C\to\alpha^{-1}(C)$ and the map
$\varrho_{\beta}:\alpha^{-1}(C)\to Q$. By Theorem 2.4 and the parametric
suitability assumption the latter map is proper for a general
$\beta\in{\mathcal{B}}$. This contradiction shows that our assumption is wrong
which yields the desired conclusion. ∎

######  Lemma 4.3.

Let Convention 3.2 hold and ${\mathcal{A}}$ be a perfect $G$-family as in
Proposition 4.2. Let $Z_{\alpha}$ be the zero locus of $\delta_{\alpha}$ and
$L_{\alpha}$ be the line bundle induced by $\delta_{\alpha}$ on $X\setminus
Z_{\alpha}$. Suppose that $K$ is a finite subset of $C$ and $X_{k}$ is an open
subset of $X$ containing $C$. Let $B_{k}\subset TX_{k}$ be a vector subbundle
of rank $k\leq n-2$. For general $\alpha$ in ${\mathcal{A}}$ one has the
following.

(i) There exists a closed subvariety $Y_{k}\subset X$ of dimension at most $k$
such that $B_{k}$ and $L_{\alpha}$ generate a vector bundle $B_{k+1}$ of rank
$k+1$ on $X_{k+1}=X_{k}\setminus(Y_{k}\cup Z_{\alpha})$ and $C\subset
X_{k+1}$.

(ii) If $k\leq n-3$ and $\mathop{\rm pr}(B_{k})$ yields a vector subbundle of
$N_{X}C$ of rank $k$, then $\mathop{\rm pr}(B_{k+1})$ yields a vector
subbundle of $N_{X}C$ of rank $k+1$.

(iii) If $\mathop{\rm pr}(B_{n-2})$ is a vector subbundle of $N_{X}C$ of rank
$n-2$, then there exists a finite set $H_{\alpha}\subset C\setminus K$ such
that $\mathop{\rm pr}(B_{n-1})$ yields $N_{X}(C\setminus H_{\alpha})$.
Furthermore, $\bigcap_{\alpha\in{\mathcal{A}}}H_{\alpha}=\emptyset$.

###### Proof.

Recall that by $G$-flexibility and [AFKKZ, Theorem 4.14] $G$ acts transitively
on $T^{\prime}X$ where $T^{\prime}X$ is the complement to the zero section in
$TX$. For every subset $S$ of $TX$ let $S^{\prime}=S\cap T^{\prime}X$. Note
that $\dim B_{k}^{\prime}=n+k$, $\dim B_{k}^{\prime}|_{C}=k+1$ and $\dim
L_{\alpha}^{\prime}=n+1$. By Theorem 1.7 for general $\alpha\in{\mathcal{A}}$
the variety $L_{\alpha}^{\prime}$ meets $B_{k}^{\prime}|_{C}$ along a
subvariety $R$ of dimension $(k+1)+(n+1)-2n=k+2-n\leq 0$. However, this
dimension cannot be zero, since for each $v\in R\subset TX$ every vector
proportional to $v$ is also in $R$, i.e., $R$ is empty. Again by Theorem 1.7
$L_{\alpha}^{\prime}$ meets $B_{k}^{\prime}$ transversely along a subvariety
${\tilde{Y}}$ of dimension at most $(n+k)+(n+1)-2n=k+1$. Hence,
$Y_{k}=\mathop{\rm pr}({\tilde{Y}})$ does not meet $C$ and has dimension at
most $\dim{\tilde{Y}}-1=k$, while $B_{k}$ and $L_{\alpha}$ generate a rank
$k+1$ vector subbundle $B_{k+1}$ of the tangent bundle of $X_{k+1}$. Note that
$C\subset X_{k+1}$ since $C\cap Z_{\alpha}=\emptyset$ by Proposition 4.2
(iii). Thus, we get (i).

The claim in (ii) that $\mathop{\rm pr}(B_{k})$ is a vector subbundle of
$N_{X}C$ of rank $k$ is equivalent to the fact that the constructible set
$\bigcup_{x\in C}(B_{k}(x)+T_{x}C)$ is a bundle ${\tilde{B}}_{k}$ of rank
$k+1$ over $C$. We note that since $\dim{\tilde{B}}_{k}^{\prime}+\dim
L_{\alpha}^{\prime}-\dim T^{\prime}X=(k+2)+(n+1)-2n=k+3-n\leq 0$, by Theorem
1.7 for general $\alpha\in{\mathcal{A}}$ the variety $L_{\alpha}^{\prime}$
meets ${\tilde{B}}_{k}^{\prime}$ transversely along a variety ${\tilde{H}}$ of
dimension at most zero. However, arguing as before we see that
${\tilde{H}}=\emptyset$ which implies (ii) (to see that each $x\in K$ is still
in $X_{k+1}$ one needs to choose as before $\delta_{\alpha}(x)$ outside the
fiber $B_{k}(x)+T_{x}C$).

In (iii) the same argument shows that ${\tilde{H}}$ may have dimension 1.
Therefore, $H_{\alpha}:=\mathop{\rm pr}({\tilde{H}})$ is a finite set and
$\mathop{\rm pr}(B_{n-1})$ yields $N_{X}(C\setminus H_{\alpha})$. The
transversality argument implies also that for general $\alpha\in{\mathcal{A}}$
the variety $L_{\alpha}^{\prime}$ does not meet the fiber of
${\tilde{B}}_{n-s-1}^{\prime}$ over a fixed point $c\in C$. This implies that
$\bigcap_{\alpha\in{\mathcal{A}}}H_{\alpha}=\emptyset$ and, consequently,
$K\cap H_{\alpha}=\emptyset$ for general $\alpha$ which concludes the proof. ∎

######  Proposition 4.4.

Let Convention 3.2 hold and $K$ be a finite subset of $C$. Then there exists a
perfect $G$-family ${\mathcal{A}}$ such that for
$\bar{\alpha}=(\alpha_{1},\ldots,\alpha_{n-1})$ in a dense open subset $U$ of
${\mathcal{A}}^{n-1}$ the following is true.

(1) For every $i=1,\ldots,n-1$ the map $\varrho_{\alpha_{i}}:C\to Q$ is proper
and generates an isomorphism between $C$ and $\varrho_{\alpha_{i}}(C)\subset
Q_{0}$.

(2) The span of $\delta_{\alpha_{1}},\ldots,\delta_{\alpha_{n-2}}$ over
${\rm\bf k}[C]$ is a subbundle $B_{n-2}\subset TX|_{C}$ such that $\mathop{\rm
pr}(B_{n-2})$ is a subbundle of $N_{X}C$ whose rank is $n-2$.

(3) Let $H_{\alpha_{n}}$ be as in Lemma 4.3 and $C^{*}=C\setminus
H_{\alpha_{n-1}}$. Then $K\subset C^{*}$ and the span of $\mathop{\rm
pr}(B_{n-2})$ and $\mathop{\rm pr}(\delta_{\alpha_{n-1}})$ over ${\rm\bf
k}[C^{*}]$ is the normal bundle $N_{X}C^{*}$.

(4) The morphism ${\mathcal{A}}\to\prod_{x\in
K}T_{x}X,\,\alpha\mapsto(\delta_{\alpha}(x))_{x\in K}$ is dominant.

(5) $N_{X}C$ is naturally isomorphic to $\mathop{\rm pr}(B_{n-2})\oplus L$
where $L$ is a line subbundle of $N_{X}C$.

###### Proof.

The existence of a perfect $G$-family ${\mathcal{A}}$ satisfying (1) is
provided by Proposition 4.2. Let $B_{0}$ be the zero subbundle of $TX$. Then
applying Lemma 4.3 to $B_{0}$ and consequent vector bundles $B_{k}$ we get (2)
and (3). The same argument as in Proposition 4.1 (iii) yields (4).

Now let $v_{i}=\mathop{\rm pr}(\delta_{\alpha_{i}})$. Then each $v_{i}$ is a
non-vanishing section of $N_{X}C$ and by (3) $v_{n-1}$ meets $\mathop{\rm
pr}(B_{n-2})$ over $H_{\alpha_{n-1}}=\\{x_{1},\ldots,x_{m}\\}\subset C$.
Consider a neighborhood $U_{i}$ of $x_{i}$ in $C$ such that $U_{i}\cap
H_{\alpha_{n-1}}=x_{i}$ and the normal bundle $N_{X}U_{i}$ is trivial, i.e.,
$v_{1},\ldots,v_{n-2}$ and some section $u_{i}$ is a basis of $N_{X}U_{i}$.
Then $v_{n-1}=g_{i}u_{i}+f_{1,i}v_{1}+\ldots+f_{n-2,i}v_{n-2}$ where
$g_{i},f_{1,i},\ldots,f_{n-2,i}$ are regular functions on $U_{i}$. Consider
regular functions $f_{1},\ldots,f_{n-2}$ on $C$ such that $f_{j}-f_{j,i}$ is
divisible by $g_{i}$ for every $i$ and $j$. Consider also the projectivization
$D$ of $N_{X}C$ which is a ${\mathbb{P}}^{n-2}$-bundle over $C$. It contains a
${\mathbb{P}}^{n-3}$-bundle $D_{0}$ corresponding to $\mathop{\rm
pr}(B_{n-2})$. Note that $v_{n-1}-(f_{1}v_{1}+\ldots+f_{n-2}v_{n-2})$ defines
a section of $D$ over $C\setminus H_{\alpha_{n-1}}$ which extends to a section
over $C$ that does not meet $D_{0}$. This section yields the desired line
bundle $L$ which concludes the proof. ∎

######  Remark 4.5.

Recall that by [Se1] every vector bundle $W$ of rank $k$ over $C$ can be
presented as $W_{0}\oplus W_{1}$ where $W_{0}$ is a trivial bundle of rank
$k-1$ and $W_{1}$ is a line bundle. Proposition 4.4 (5) below tells
additionally that for $W=N_{X}C$ the bundle $W_{0}$ has a basis of sections
generated by locally nilpotent vector fields.

######  Corollary 4.6.

Let the notations and conclusions of Proposition 4.4 hold and
$v_{i}=\mathop{\rm pr}(\delta_{\alpha_{i}}),\,i=1,\ldots,n-1$. Suppose that
$N_{C}X$ and, therefore, $L$ are trivial bundles and $v_{0}$ is a nonvanishing
section of $L$. Then $v_{n-1}=qv_{0}+\sum_{i=1}^{n-2}a_{i}v_{i}$ where
$a_{1},\ldots,a_{n-2}$ and $q$ are regular functions on $C$ having simple
zeros on $C$.

###### Proof.

Consider the natural projection $N_{X}C\to L$. Let $\tilde{v}_{0}=qv_{0}$ be
the image of $v_{n-1}$. Then $v_{n-1}-\tilde{v}_{0}\in\mathop{\rm
pr}(B_{n-2})$ which yields an equality
$v_{n-1}=\tilde{v}_{0}+\sum_{i=1}^{n-2}a_{i}v_{i}$. Since $\mathop{\rm
pr}(B_{n-2})$ is a vector subbundle of $N_{X}C$ of rank $n-2$, the
constructible set $\bigcup_{x\in C}(B_{n-2}(x)+T_{x}C)$ is a bundle
${\tilde{B}}_{n-2}$ of rank $n-1$ over $C$. Let $L_{\alpha_{n-1}}$ be the line
bundle induced by $\delta_{\alpha_{n-1}}$ on $X\setminus Z_{\alpha_{n-1}}$ as
in Lemma 4.3. As we mentioned in the proof of Lemma 4.3 $L_{\alpha_{n-1}}$
meets ${\tilde{B}}_{n-2}$ transversely for a general
$\alpha_{n-1}\in{\mathcal{A}}$ by virtue of Theorem 1.7 which implies the
claim about the zeros of $q$ (which are the same as the zeros of
$\tilde{v}_{0}$). The similar argument works for the zeros of $a_{i}$. Thus,
we get the desired conclusion. ∎

##  5\. $\mathop{\rm SL}_{n}(A)$ over a commutative ring $A$

In this section we collect some facts about matrices over commutative rings
and prove Proposition 5.5 which is an essential step for our main result.

######  Notation 5.1.

Let $A$ be a commutative ring, $n\geq 3$ and $\mathop{\rm SL}_{n}(A)$ be the
special linear group with entries from $A$, whereas ${\rm Mat}_{n}(A)$ is the
ring of $(n\times n)$-matrices with entries from $A$. By $I\in\mathop{\rm
SL}_{n}(A)$ we denote the identity matrix and by $e_{ij},\,1\leq i,j\leq n$
the matrix in ${\rm Mat}_{n}(A)$ with all zero entries but the entry in
position $(i,j)$ which is 1. By $\mathop{\rm E}_{n}(A)$ we denote the subgroup
of $\mathop{\rm SL}_{n}(A)$ generated by all elementary matrices of the form
$I+ae_{ij},\,1\leq i\neq j\leq n$ where $a\in A$. Let ${\mathfrak{q}}$ be any
ideal in $A$. By $\mathop{\rm SL}_{n}(A,{\mathfrak{q}})$ we denote the kernel
of the natural group homomorphism $\mathop{\rm SL}_{n}(A)\to\mathop{\rm
SL}_{n}(A/{\mathfrak{q}})$, by $\mathop{\rm F}_{n}(A,{\mathfrak{q}})$ we
denote the subgroup of $\mathop{\rm E}_{n}(A)$ generated by all elementary
transformations of the form $I+ae_{ij}$ where $a\in{\mathfrak{q}}$ and by
$\mathop{\rm E}_{n}(A,{\mathfrak{q}})$ the normal closure of $\mathop{\rm
F}_{n}(A,{\mathfrak{q}})$ in $\mathop{\rm E}_{n}(A)$.

We need the following result of Bass, Milnor and Serre [BMS, Theorem 4.1] (see
also [Ba, Theorem 4]).

######  Theorem 5.2.

Let Notation 5.1 hold, $A$ be a Dedekind domain and $n\geq 3$. Then

(4) ${\rm E}_{n}(A,{\mathfrak{q}})=[{\mathop{\rm SL}}_{n}(A),{\mathop{\rm
SL}}_{n}(A,{\mathfrak{q}})].$

Another important fact unfortunately omitted in [Ud] is the following “Tits’
theorem" in [Ni, Theorem 3].

######  Theorem 5.3.

Let Notation 5.1 hold. Then $\mathop{\rm E}_{n}(A,{\mathfrak{q}}^{2})$ is
contained in $\mathop{\rm F}_{n}(A,{\mathfrak{q}})$.

Let us reproduce two facts from [Ud, Lemma 2.2.1 and Theorem 2.2.2].

######  Lemma 5.4.

([Ud, Theorem 2.2.2]) Let Notations 5.1 hold, ${\mathfrak{q}}=qA,\,q\in A$ be
a proper principal ideal in $A$ and $D\in{\rm Mat}_{n}(A)$ be a matrix which
is invertible in ${\rm Mat}(A[\frac{1}{q}])$.

(i) For some $m>0$ one has $D^{-1}{\mathop{\rm
SL}}_{n}(A,{\mathfrak{q}})D\supset{\mathop{\rm
SL}}_{n}(A,{\mathfrak{q}}^{m})$.

(ii) Let $A$ be a Dedekind ring. Then the group
$D^{-1}E_{n}(A,{\mathfrak{q}})D$ contains $E_{n}(A,{\mathfrak{q}}^{l})$ for
some $l>0$.

###### Proof.

Since $D^{-1}=\frac{1}{{\rm det}D}{{\rm adj}(D)}$ (where ${\rm adj}(D)$ is the
classical adjoint), one has (i) with $m$ such that $\frac{q^{m-1}}{{\rm
det}D}$ belongs to $A$.

By (i) and Formula (4) we have

$D^{-1}{\rm E}_{n}(A,{\mathfrak{q}})D=D^{-1}[{\mathop{\rm
SL}}_{n}(A),{\mathop{\rm SL}}_{n}(A,{\mathfrak{q}})]D=[D^{-1}{\mathop{\rm
SL}}_{n}(A)D,D^{-1}{\mathop{\rm SL}}_{n}(A,{\mathfrak{q}})D]\supset$
$[{\mathop{\rm SL}}_{n}(A,{\mathfrak{q}}^{m}),{\mathop{\rm
SL}}_{n}(A,{\mathfrak{q}}^{m})]\supset[{\rm E}_{n}(A,{\mathfrak{q}}^{m}),{\rm
E}_{n}(A,{\mathfrak{q}}^{m})]\supset{\rm E}_{n}(A,{\mathfrak{q}}^{2m})$

where the last inclusion follows from [BMS, Section 5]. ∎

Now we have the desired fact.

######  Proposition 5.5.

Let Notation 5.1 hold, $A$ be a Dedekind ring and let $q$ and $q^{\prime}\in
A$ generate comaximal principal ideals ${\mathfrak{q}}$ and
${\mathfrak{q}}^{\prime}\subset B$. Let a matrix $D^{\prime}$ plays the same
role for $q^{\prime}$ as $D$ for $q$ in Lemma 5.4. Suppose that $H$ is a
subgroup of $\mathop{\rm SL}_{n}(A)$ that contains all elements from
$\mathop{\rm SL}_{n}(A)\cap(D^{-1}\mathop{\rm F}_{n}(A,{\mathfrak{q}})D)$ and
$\mathop{\rm SL}_{n}(A)\cap((D^{\prime})^{-1}\mathop{\rm
F}_{n}(A,{\mathfrak{q}}^{\prime})D^{\prime})$. Then $H$ contains $\mathop{\rm
E}_{n}(A)$.

###### Proof.

By Theorem 5.3, $H$ contains the subgroups $\mathop{\rm
SL}_{n}(A)\cap(D^{-1}\mathop{\rm E}_{n}(A,{\mathfrak{q}}^{2})D)$ and
$\mathop{\rm SL}_{n}(A)\cap((D^{\prime})^{-1}\mathop{\rm
E}_{n}(A,({\mathfrak{q}}^{\prime})^{2})D^{\prime})$. Hence, by Lemma 5.4 $H$
contains $\mathop{\rm E}_{n}(A,{\mathfrak{q}}^{2l})$ and $\mathop{\rm
E}_{n}(A,({\mathfrak{q}}^{\prime})^{2l})$. Now the desired conclusion follows
from the fact that $({\mathfrak{q}}^{\prime})^{2l}+{\mathfrak{q}}^{2l}=A$. ∎

The rest of this section will be needed only for the holomorphic case in
Section 7.

######  Lemma 5.6.

Let Notation 5.1 hold, $\sigma\in{\rm Mat}_{n}(A)$ be a matrix of the form
${q}(I+\sum_{i=2}^{n}\frac{a_{i}}{q}e_{1i})$ where $q\in A\setminus\\{0\\}$
and each $a_{i}\in A$. Suppose that ${\mathfrak{p}}$ is an ideal in $A$. Let
${\mathcal{E}}$ be a subgroup of $\mathop{\rm SL}_{n}(A)$ that contains

(i) all elementary matrices of the form $\tau=I+ce_{ij}$ where $c\in A$,
$2\leq j\neq i\leq n$ and

(ii) all matrices $\sigma\tau\sigma^{-1}$ where $\tau=I+bq^{2}e_{i1}$,
$b\in{\mathfrak{p}}$ and $1\leq i\leq n$.

Then for every $2\leq k\neq l\leq n$ and every $b\in{\mathfrak{p}}$ the group
${\mathcal{E}}$ contains the following matrices

$I+a_{l}bq^{2}e_{k1}-a_{k}a_{l}bqe_{kk}-a_{l}^{2}bqe_{kl}-a_{k}bq^{2}e_{l1}+a_{k}a_{l}bqe_{ll}+a_{k}^{2}bqe_{lk}.$

###### Proof.

Note that $\sigma^{-1}={q}^{-1}(I-\sum_{i=2}^{n}\frac{a_{i}}{q}e_{1i})\in{\rm
Mat}_{n}({\rm Frac}(A))$. Let $\alpha=I+ba_{l}q^{2}e_{k1}-ba_{k}q^{2}e_{l1}$.
Then
$\sigma\alpha\sigma^{-1}=I+\sigma(ba_{l}q^{2}e_{k1}-ba_{k}q^{2}e_{l1})\sigma^{-1}=I+\sigma(ba_{l}q^{2}e_{k1})\sigma^{-1}-\sigma(ba_{k}q^{2}e_{l1})\sigma^{-1}$.
Consider

$\begin{array}[]{c}\sigma(bq^{2}e_{k1})\sigma^{-1}=bq^{2}e_{k1}+(\sum_{i=2}^{n}\frac{a_{i}}{q}e_{1i})bq^{2}e_{k1}-bq^{2}e_{k1}(\sum_{i=2}^{n}\frac{a_{i}}{q}e_{1i})\\\
-(\sum_{i=2}^{n}\frac{a_{i}}{q}e_{1i})bq^{2}e_{k1}(\sum_{i=2}^{n}\frac{a_{i}}{q}e_{1i})=\\\
bq^{2}e_{k1}+a_{k}bqe_{11}-a_{k}bqe_{kk}-a_{k}^{2}be_{1k}-\sum_{i=2,i\neq
k}^{n}a_{i}bqe_{ki}-\sum_{i=2,i\neq k}^{n}a_{k}a_{i}be_{1i}=\\\
bq^{2}e_{k1}+a_{k}bqe_{11}-a_{k}bqe_{kk}-a_{k}^{2}be_{1k}-a_{l}bqe_{kl}-a_{k}a_{l}be_{1l}\\\
-\sum_{i=2,i\neq k,l}^{n}a_{i}bqe_{ki}-\sum_{i=2,i\neq
k,l}^{n}a_{k}a_{i}be_{1i}.\end{array}$

Replacing $b$ by $a_{l}b$ we get $\sigma(ba_{l}q^{2}e_{k1})\sigma^{-1}$ and
then, exchanging the letters $k$ and $l$ we get
$\sigma(ba_{k}q^{2}e_{l1})\sigma^{-1}$. Hence, one has

$\begin{array}[]{c}\sigma\alpha\sigma^{-1}=I+a_{l}bq^{2}e_{k1}+a_{k}a_{l}bqe_{11}-a_{k}a_{l}bqe_{kk}-a_{k}^{2}a_{l}be_{1k}-a_{l}^{2}bqe_{kl}-a_{k}a_{l}^{2}be_{1l}\\\
-(a_{k}bq^{2}e_{l1}+a_{k}a_{l}bqe_{11}-a_{k}a_{l}bqe_{ll}-a_{k}a_{l}^{2}be_{1l}-a_{k}^{2}bqe_{lk}-a_{k}^{2}a_{l}be_{1k})+\beta\\\
=I+a_{l}bq^{2}e_{k1}-a_{k}a_{l}bqe_{kk}-a_{l}^{2}bqe_{kl}-a_{k}bq^{2}e_{l1}+a_{k}a_{l}bqe_{ll}+a_{k}^{2}bqe_{lk}+\beta\end{array}$

where $\beta$ is a sum of terms of the form $\sum_{i=2,i\neq
k,l}^{n}(c_{i}e_{ki}+d_{i}e_{li})$ where $c_{i},d_{i}\in A$. One can rewrite
$\sigma\alpha\sigma^{-1}$ now as

$(I+\beta)(I+a_{l}bq^{2}e_{k1}-a_{k}a_{l}bqe_{kk}-a_{l}^{2}bqe_{kl}-a_{k}bq^{2}e_{l1}+a_{k}a_{l}bqe_{ll}+a_{k}^{2}bqe_{lk}).$

Since $I+\beta\in{\mathcal{E}}$, we have the desired conclusion. ∎

######  Example 5.7.

Consdier the case of $(k,l)=(2,3)$ in the proof of Lemma 5.6. Let
$\sigma\alpha\sigma^{-1}$ and $\beta$ be as in the proof of Lemma 5.6 and
$\theta=(I+\beta)^{-1}\sigma\alpha\sigma^{-1}$. Then

$\theta=\left[{\begin{array}[]{cccc}1&0&0&\bar{0}\\\
a_{3}bq^{2}&1-a_{2}a_{3}bq&-a_{3}^{2}bq&\bar{0}\\\
-a_{2}bq^{2}&a_{2}^{2}bq&1+a_{2}a_{3}bq&\bar{0}\\\
\bar{0}&\bar{0}&\bar{0}&I_{n-3}\end{array}}\right]$

where $I_{n-3}\in\mathop{\rm SL}_{n-3}(A)$ is the identity matrix. Removing
the first row and the first column we get the matrix

$\gamma=\left[{\begin{array}[]{ccc}1-a_{2}a_{3}bq&-a_{3}^{2}bq&\bar{0}\\\
a_{2}^{2}bq&1+a_{2}a_{3}bq&\bar{0}\\\
\bar{0}&\bar{0}&I_{n-3}\end{array}}\right]$

which has determinant 1.

######  Proposition 5.8.

Let the assumptions of Lemma 5.6 hold and $\mathop{\rm
SL}_{n-1}(A)=\mathop{\rm E}_{n-1}(A)$. Then for every $b\in{\mathfrak{p}}$ the
matrix $I+a_{l}bq^{2}e_{k1}-a_{k}bq^{2}e_{l1}$ is contained in
${\mathcal{E}}$.

###### Proof.

We show only that $I+a_{3}bq^{2}e_{21}-a_{2}bq^{2}e_{31}\in{\mathcal{E}}$
since the general case of $(k,l)\neq(2,3)$ is similar. Hence, we use the
notations $\theta$ and $\gamma$ as in Example 5.7. Consider a block matrix of
the form

$\bar{\lambda}=\left[{\begin{array}[]{cc}1&\bar{0}\\\
\bar{0}&\lambda\end{array}}\right]\in{\rm SL}_{n}(A).$

Then

$\theta\bar{\lambda}=\left[{\begin{array}[]{cccc}1&\bar{0}\\\
\bar{c}&\gamma\lambda\end{array}}\right]$

where $\bar{c}$ is an (n-1)-column with the first two entries $a_{3}bq^{2}$
and $-a_{2}bq^{2}$, whereas the rest of the entries are zeros. Thus, taking
$\lambda=\gamma^{-1}\in\mathop{\rm SL}_{n-1}(A)=\mathop{\rm E}_{n-1}(A)$ we
see that $I+a_{3}bq^{2}e_{21}-a_{2}bq^{2}e_{31}\in{\mathcal{E}}$ which
concludes the proof. ∎

##  6\. Main Theorem

In this section we present a strengthened version of the main result in [Ud].
Let us start with terminology.

######  Definition 6.1.

(1) Let $C_{1}$ and $C_{2}$ be closed subvarieties of a smooth quasi-affine
variety $X$ with defining ideals $I_{1}$ and $I_{2}$ in ${\rm\bf k}[X]$. Let
$G$ be a subgroup of $\operatorname{{\rm Aut}}(X)$, i.e., $G$ acts on ${\rm\bf
k}[X]$. Suppose that for some $\varphi\in G$ one has $\varphi(I_{1})=I_{2}$.
Then $\varphi(I_{1}^{k})=I_{2}^{k}$ for every $k\in\mathbb{N}$ and, therefore,
$\varphi$ induces an isomorphism $\varphi_{k}:\frac{{\rm\bf
k}[X]}{I_{1}^{k}}\to\frac{{\rm\bf k}[X]}{I_{2}^{k}}$. We say that such a
$\varphi_{k}$ is a $G$-induced isomorphism.

(2) Let $\theta:\frac{{\rm\bf k}[X]}{I_{1}^{k}}\to\frac{{\rm\bf
k}[X]}{I_{2}^{k}}$ be a ring homomorphism. Note that one has
$\theta(I_{1})\subset{I_{2}}$ since $\theta$ sends nilpotent elements to
nilpotent ones. Hence, $\theta$ induces ring homomorphisms
$\theta^{\prime}:\frac{{\rm\bf k}[X]}{I_{1}}\to\frac{{\rm\bf k}[X]}{I_{2}}$
and $\theta^{\prime\prime}:\frac{I_{1}}{I_{1}^{2}}\to\frac{I_{2}}{I_{2}^{2}}$.
Given an isomorphism
$\lambda:\frac{I_{1}}{I_{1}^{2}}\to\frac{I_{2}}{I_{2}^{2}}$ we say that it is
$G$-induced if for some $G$-induced isomorphism $\theta:\frac{{\rm\bf
k}[X]}{I_{1}^{k}}\to\frac{{\rm\bf k}[X]}{I_{2}^{k}}$ one has
$\lambda=\theta^{\prime\prime}$.

(3) If in the previous definition $I_{1}=I_{2}$ and $\theta^{\prime}$ is the
identity map, then we say $\theta$ (resp. $\lambda$) is an
$A_{1}$-automorphism of $\frac{{\rm\bf k}[X]}{I_{1}^{k}}$ (resp.
$\frac{I_{1}}{I_{1}^{k}}$) where $A_{j}=\frac{{\rm\bf k}[X]}{I_{j}}$ is the
ring ${\rm\bf k}[C_{j}]$ of regular functions on $C_{j}$. That is, $\theta$
can be viewed as an automorphism of an $k$th infinitesimal neighborhood of
$C_{1}$ over $C_{1}$ itself.

In the case of a flexible variety the Jacobian of an automorphism must be a
constant function because of the following.

######  Lemma 6.2.

Let $X$ be a smooth quasi-affine algebraic variety equipped with a volume form
$\omega$ such that every two points of $X$ can be joined by a chain of
polynomial curves (which is the case when $X$ is flexible). Let $\alpha:X\to
X$ be an automorphism. Then the Jacobian of $\alpha$ is constant. Furthermore,
if $\alpha\in\mathop{\rm SAut}(X)$ then its Jacobian is 1.

###### Proof.

The action of $\alpha$ transforms $\omega$ into a volume form
$\omega_{\alpha}$. The Jacobian $f_{\alpha}=\frac{\omega_{\alpha}}{\omega}$ of
this automorphism is an invertible function on $X$ and $f_{e}=1$ where $e$ is
the identity automorphism. Since the restriction of $f_{\alpha}$ to every
polynomial curve is constant, we see that $f_{\alpha}$ is constant as ${\rm\bf
k}$ is an algebraically closed field. Thus, for $\alpha\in\mathop{\rm
SAut}(X)$ the Jacobian $f_{\alpha}$ can be viewed as a value of an invertible
function $f$ on $\mathop{\rm SAut}(X)$. By the definition any two elements
$\alpha_{1},\alpha_{2}\in\mathop{\rm SAut}(X)$ can be joined by a chain of
polynomial curves (corresponding to one-parameter unipotent groups). Hence,
$f$ is constant and, consequently, $f=1$ which is the desired conclusion. ∎

To see how the above condition on Jacobians can be reformulated for
automorphisms of infinitesimal neighborhoods of closed subvarieties of $X$ we
need to remind the following well-known fact.

######  Lemma 6.3.

Let $Y$ be a normal affine algebraic variety of dimension $m$ and $Z$ be a
smooth closed curve in $Y$ with defining ideal $J$ in ${\rm\bf k}[Y]$. Suppose
that $Z$ is contained in $Y_{\rm reg}$. Then any section of the conormal
bundle of $Z$ in $Y$ can be presented by a regular function on $Y$.
Furthermore, suppose that the conormal bundle is trivial. Then there exists an
open neighborhood $U$ of $Z$ in $Y$ and regular functions
$u_{1},\ldots,u_{m-1}\in{\rm\bf k}[Y]$ such that $Z$ is a strict complete
intersection in $U$ given by $u_{1}|_{U}=\ldots=u_{m-1}|_{U}=0$.

###### Proof.

Let ${\mathcal{J}}$ be the ideal sheaf on $Y$ induced by $J$. Consider the
short exact sequence
$0\to{\mathcal{J}}^{2}\to{\mathcal{J}}\to\frac{{\mathcal{J}}}{{\mathcal{J}}^{2}}\to
0$ and the long exact sequence induced by it. Since ${\mathcal{J}}^{2}$ is a
coherent sheaf on the affine variety $Y$, we have
$H^{1}(Y,{\mathcal{J}}^{2})=0$ and the long exact sequence implies that every
section of $\frac{{\mathcal{J}}}{{\mathcal{J}}^{2}}$ is induced by a section
of ${\mathcal{J}}$, i.e., by a regular function on $Y$ which belongs to $J$.
Note also that as in [Ha, Chapter II, Theorem 8.17]
$\frac{{\mathcal{J}}}{{\mathcal{J}}^{2}}$ can be viewed as a sheaf
${\mathcal{C}}$ of ${\mathcal{O}}_{Z}$-modules where ${\mathcal{O}}_{Z}$ is
the structure sheaf of $Z$. Any section of ${\mathcal{C}}$ over $Z$ can be
extended by 0 to $Y\setminus Z$ so that one gets a global section of
$\frac{{\mathcal{J}}}{{\mathcal{J}}^{2}}$ which, as we showed, is induced by
an element of $J$. Furthermore, since $Z$ is smooth ${\mathcal{C}}$ is a
locally free sheaf of rank $m-1$ which induces the conormal bundle of $Z$ in
$Y$. Since any section of this bundle can be viewed as a section of
${\mathcal{C}}$, we have the first statement. The smoothness of $Z$ implies
that $Z$ is a local complete intersection in $Y$ ([Ha, Chapter II, Theorem
8.17]). That is, for every point $z\in Z$ and any collection
$u_{1}^{z},\ldots,u_{m-1}^{z}\in{\mathcal{J}}$ which induces a local basis
$\tilde{u}_{1}^{z},\ldots,\tilde{u}_{m-1}^{z}$ of ${\mathcal{C}}$ at $z$ one
can find a neighborhood $U_{z}$ of $z$ in $Y$ such that $U_{z}\cap Z$ is a
strict complete intersection in $U_{z}$ given by the system
$u_{1}^{z}=\ldots=u_{m-1}^{z}=0$. In the case of a trivial ${\mathcal{C}}$ for
every $i=1,\ldots,m-1$ such local sections $\tilde{u}_{i}^{z}$ can be chosen
as the restrictions of a global section $\tilde{u}_{i}$ of ${\mathcal{C}}$ to
$Z\cap U_{z}$. As we saw $\tilde{u}_{i}$ is induced by a function $u_{i}\in
J$. There is also a neighborhood $U$ of $Z$ in $Y$, for which $U\cap Z$ is
given by the system $u_{1}=\ldots=u_{m-1}=0$, since for every $z\in Z$ there
is a neighborhood $W$ of $z$ in $Y$ in which $W\cap Z$ is given by this
system. This concludes the proof. ∎

######  Definition 6.4.

Let $X$ be a normal quasi-affine algebraic variety, i.e., $X$ is contained as
a dense subvariety in a normal affine variety $Y$. Let $C_{j}$ be a smooth
curve in $X_{\rm reg}$ with a trivial normal bundle such that $C_{j}$ is
closed in $Y$. By Lemma 6.3 $C_{j}$ is a strict complete intersection in some
open $U_{j}\subset X$ given by $u_{1j}=\ldots=u_{n-1,j}=0$ where
$u_{1j},\ldots,u_{n-1,j}$ are regular functions on $X$. One can see now that
the ring $\frac{{\rm\bf k}[X]}{I_{j}^{k}}$ is naturally isomorphic to
$A_{j}\oplus\bigoplus_{l=1}^{k-1}\frac{I_{j}^{l}}{I_{j}^{{l+1}}}$ where each
summand $\frac{I_{j}^{l}}{I_{j}^{l+1}}$ can be treated as the space of
homogeneous polynomials in $u_{1j},\ldots,u_{n-1,j}$ of degree $l$ over
$A_{j}$. In particular, for any homomorphism $\theta:\frac{{\rm\bf
k}[X]}{I_{1}^{k}}\to\frac{{\rm\bf k}[X]}{I_{2}^{k}}$ of algebras one can
consider the Jacobi matrix $\left[\frac{\partial\theta(u_{l1})}{\partial
u_{m2}}\right]_{l,m=1}^{n-1}$. In the case of $I_{1}=I_{2}$ and
$u_{i,1}=u_{i,2}$ the determinant of this matrix is independent of the choice
of coordinates $u_{1j},\ldots,u_{n-1,j}$ and we say that $\theta$ has Jacobian
1 if the determinant is equal to 1 modulo $I_{1}^{k-1}$.

Let $\theta:\frac{{\rm\bf k}[X]}{I_{1}^{k}}\to\frac{{\rm\bf k}[X]}{I_{1}^{k}}$
be an $A_{1}$-automorphism induced by an automorphism
$\alpha\in\operatorname{{\rm Aut}}(X)$. If $X$ possesses a volume form and the
Jacobian of $\alpha$ is 1, then the Jacobian of $\theta$ is, of course, 1.
However, one can often make the same conclusion about the Jacobian of $\theta$
without the presence of a volume form.

######  Example 6.5.

Let $0\neq\delta\in\operatorname{{\rm LND}}(X)$ and $\varrho:X\to Q$ be a
partial quotient morphism associated with $\delta$. Suppose that $x\in C_{1}$
is a general point of $X$, i.e., $\varrho(x)$ belongs to $Q_{0}$ where $Q_{0}$
is as in Remark 1.4. Let $f\in{\rm\bf k}[Q]$ vanish on $\varrho(C_{1})$ and
$\alpha\in\mathbb{G}_{a}$ be the automorphism of $X$ induced by
$\exp(\varrho^{*}(f)\delta)$. Then for every $k$ the automorphism
$\theta:\frac{{\rm\bf k}[X]}{I_{1}^{k}}\to\frac{{\rm\bf k}[X]}{I_{1}^{k}}$
induced by $\alpha$ has Jacobian 1. Indeed, consider a function $h\in{\rm\bf
k}[X]$ of degree 1 with respect to $\delta$, i.e., $h\notin\operatorname{{\rm
Ker}}\delta$ and $\delta(h)\in\operatorname{{\rm Ker}}\delta$. This is
equivalent to the fact that the restriction of $h$ to the
$\mathbb{G}_{a}$-orbit ${\mathbb{A}}^{1}$ of any general point of $X$ (and
$x$, in particular) is a polynomial of degree 1. Consider the smooth germ $S$
of the hypersuface $h^{-1}(h(x))\subset X$ at $x$. Then the
$\mathbb{G}_{a}$-orbit of $S$ is naturally isomorphic to
$S\times{\mathbb{A}}^{1}$ and one can choose a coordinate $t$ on the second
factor of $S\times{\mathbb{A}}^{1}$ so that
$\alpha|_{S\times{\mathbb{A}}^{1}}$ is given by $(s,t)\mapsto(s,t+f(s))$.
Since $S$ is a smooth germ it has a volume form $\omega_{0}$ and, hence,
$S\times{\mathbb{A}}^{1}$ has the volume form $\omega=\omega_{0}\times{\rm
d}t$. It remains to note now that $\alpha$ preserve $\omega$ which yields the
desired claim.

The main strengthened result from [Ud] is the following.

######  Theorem 6.6.

([Ud, Theorem 2.5.19]) Let $X$ be a normal $G$-flexible quasi-affine variety
of dimension $n\geq 4$ where $G\subset\mathop{\rm SAut}(X)$ is a group
generated by a saturated set ${\mathcal{N}}\subset\operatorname{{\rm
LND}}(X)$. Let $0\neq\delta\in{\mathcal{N}}$, $\varrho:X\to Q$ be an
associated partial quotient morphism of $\delta$ and $C$ be a smooth curve in
$X$ such that $C\subset X_{\rm reg}$ and $C$ is closed in an affine variety
containing $X$. Suppose also that

($\bullet$) $C$ is irreducible and either $C$ is once-punctured or Convention
3.2 is valid (in particular, $X$ is affine and parametrically suitable with
respect to $\delta$).

Let $I$ be the defining ideal of $C$ in the algebra ${\rm\bf k}[X]$ of regular
functions on $X$ and $A=\frac{{\rm\bf k}[X]}{I}\simeq{\rm\bf k}[C]$. Suppose
that the normal bundle $N_{X}C$ of $C$ in $X$ is trivial (i.e., it is a free
$A$-module of dimension $n-1$). Let $\theta:\frac{{\rm\bf
k}[X]}{I^{k}}\to\frac{{\rm\bf k}[X]}{I^{k}}$ be an $A$-automorphism with
Jacobian 1 (in particular, the automorphism
$\lambda:\frac{I}{I^{2}}\to\frac{I}{I^{2}}$ induced by $\theta$ belongs to
$\mathop{\rm SL}_{n-1}(A)$). Then one has the following

(i) If $\lambda$ belongs to $\mathop{\rm E}_{n-1}(A)$, then $\lambda$ is
$G$-induced.

(ii) If $\mathop{\rm E}_{n-1}(A)=\mathop{\rm SL}_{n-1}(A)$ (which is true in
the case when $A$ is a Euclidean domain [Lam, Chapter I, Proposition 5.4],
e.g., when $C$ is a smooth polynomial curve), then $\theta$ is $G$-induced.

Before the proof we need to consider several facts. The first of which follows
immediately from Formula (3) in Lemma 1.17.

######  Lemma 6.7.

(cf. [Ud, page 47]) Let $X,G,C$, $\delta$ and $\varrho:X\to Q$ be as in
Theorem 6.6 and $\alpha_{1},\ldots,\alpha_{n-1}\in G$. Let
$\delta_{\alpha}=\alpha_{*}(\delta)$ and
$\varrho_{\alpha}=\varrho\circ\alpha^{-1}:X\to Q$ for every
$\alpha\in\operatorname{{\rm Aut}}(X)$. Suppose that
$C_{i}:=\varrho_{\alpha_{i}}(C)$ for some $i\in\\{1,\ldots,n-1\\}$. Suppose
that $f_{j}$ and $h_{j},\,j=1,\ldots,n-1$ are regular functions on $Q$ such
that each $f_{j}$ vanishes on $C_{i}$. Let $\Phi^{ij}_{t}$ be the flow of the
locally nilpotent vector field
$\varrho_{\alpha_{i}}^{*}(h_{j}f_{j})\delta_{\alpha_{i}}$ at time $t$ and let
$\breve{\sigma}=\sigma|_{C}$ for every $\sigma\in\operatorname{{\rm LND}}(X)$
.

(i) Then for every $t$ the automorphism $\Phi^{ij}_{t}$ leaves each point of
$C$ fixed (i.e., it preserves $TC$) and it induces an automorphism of the
bundle $TX|_{C}$ (and, hence, of $N_{X}C$) such that
$\breve{\delta}_{\alpha_{i}}\mapsto\breve{\delta}_{\alpha_{i}}$ and
$\breve{\delta}_{\alpha_{k}}\mapsto\breve{\delta}_{\alpha_{k}}+t(\varrho_{\alpha_{i}}^{*}(h_{j})\delta_{\alpha_{k}}(\varrho_{\alpha_{i}}^{*}(f_{j})))|_{C}\breve{\delta}_{\alpha_{i}}$
for $1\leq k\neq i\leq n-1$.

(ii) For $j\neq l$ the locally nilpotent vector fields
$\varrho_{\alpha_{i}}^{*}(h_{j}f_{j})\delta_{\alpha_{i}}$ and
$\varrho_{\alpha_{i}}^{*}(h_{l}f_{l})\delta_{\alpha_{i}}$ commute, i.e., the
vector field $\sigma=(\sum_{j=1,j\neq
i}^{n-1}\varrho_{\alpha_{i}}^{*}(h_{j}f_{j}))\delta_{\alpha_{i}}$ is locally
nilpotent. In particular, the flow of $\sigma$ at time $t$ induces an
automorphism of $TX|_{C}$ such that
$\breve{\delta}_{\alpha_{i}}\mapsto\breve{\delta}_{\alpha_{i}}$ and

$\breve{\delta}_{\alpha_{k}}\mapsto\breve{\delta}_{\alpha_{k}}+t(\sum_{j=1,j\neq
i}^{n-1}\varrho_{\alpha_{i}}^{*}(h_{j})\delta_{\alpha_{k}}(\varrho_{\alpha_{i}}^{*}(f_{j})))|_{C}\breve{\delta}_{\alpha_{i}}$

for $1\leq k\neq i\leq n-1$.

######  Notation 6.8.

Let the notations of Lemma 6.7 hold. Let $C^{*}$ be an open subset of $C$
containing a finite set $K$ and $C_{i}^{*}=\varrho_{\alpha_{i}}(C^{*})$. Let
$Q_{0}$ be as in Remark 1.4, i.e., $Q_{0}\subset Q_{\rm reg}$,
$\varrho_{\alpha_{i}}|_{\varrho_{\alpha_{i}}^{-1}(Q_{0})}:\varrho_{\alpha_{i}}^{-1}(Q_{0})\to
Q_{0}$ is a smooth morphism and $\delta_{\alpha_{i}}$ does not vanish on
$\varrho_{\alpha_{i}}^{-1}(Q_{0})$. Suppose that for every $i=1,\ldots,n-1$

(a) $C_{i}^{*}$ is contained in $Q_{0}$;

(b) $\varrho_{\alpha_{i}}|_{C^{*}}:C^{*}\to C_{i}^{*}$ is an isomorphism and
the morphism $\varrho_{\alpha_{i}}|_{C}:C\to Q$ is proper;

(c) for every $p\in K$ the vectors
$\delta_{\alpha_{1}}(p),\ldots,\delta_{\alpha_{n-1}}(p)$ are linearly
independent in $T_{p}X$ and their span meets $T_{p}C$ at zero only.

Taking a smaller $C^{*}\supset K$ we can suppose that (c) holds for every
$p\in C^{*}$ as well (note that this implies that the vector fields
$\delta_{\alpha_{1}}|_{C^{*}},\ldots,\delta_{\alpha_{n-1}}|_{C^{*}}$ induce a
basis of $N_{X}C^{*}$). Choose functions $f_{1},\ldots,f_{i-1},f_{i+1}\ldots
f_{n-1}\in{\rm\bf k}[Q]$ such that they vanish on $C_{i}$ and at every point
of $\varrho_{\alpha_{i}}(K)$ their differential are linearly independent (this
is possible since $C_{i}^{*}$ and $Q_{0}$ are smooth). Shrinking $C^{*}$
further we can suppose that the same is true for every point of $C_{i}^{*}$.

######  Lemma 6.9.

Let Notation 6.8 hold. Then the matrix
$T^{i}=\left[\delta_{\alpha_{k}}(\varrho_{\alpha_{i}}^{*}(f_{j}))\right]_{j,k=1;j,k\neq
i}^{n-1}$ is invertible over $C^{*}$.

###### Proof.

Let $p\in C^{*}$ and $q=\varrho_{\alpha_{i}}(p)\in C_{i}^{*}$. Since
$\varrho_{\alpha_{i}}$ is smooth over $q$ by (a) in Notation 6.8,
$\delta_{i}(p)$ generates the kernel of $\varrho_{\alpha_{i}*}:T_{p}X\to
T_{q}Q$. Hence, the vectors
$\varrho_{\alpha_{i}*}(\delta_{\alpha_{j}}(p)),j\in\\{1,\ldots,n-1\\}\setminus\\{i\\}$
are linearly independent and their span meets $T_{q}C_{i}$ at zero only. In
particular, these vectors induce a basis in the fiber of the normal bundle
$N_{Q}C_{i}$ over $q$. Since each $f_{j}$ vanishes on $C_{i}$, its
differential ${\rm d}f_{j}$ vanishes on $TC_{i}$. That is, the linear
independence of ${\rm d}f_{1},\ldots,{\rm d}f_{i-1},{\rm d}f_{i+1}\ldots{\rm
d}f_{n-1}$ implies that they induce a basis in the fiber of the conormal
bundle of $C_{i}$ in $Q$ over $q$. This implies in turn that the value of the
matrix $T^{i}$ at $p$ is invertible which yields the desired conclusion. ∎

######  Lemma 6.10.

Let Notation 6.8 hold. Suppose that $a$ is a nonzero regular function on $C$
that vanishes on $C\setminus C^{*}$ only, but with a sufficiently high
multiplicity such that for every $i$ the function $\frac{a}{{\rm det}T^{i}}$
is regular on $C$. Let $\bar{b}$ be any $(n-2)$-column vector whose entries
are in ${\rm\bf k}[C]$. Then there exists another $(n-2)$-column vector
$\bar{c}_{i}$ with entries in ${\rm\bf k}[C]$ such that
$T^{i}\bar{c}_{i}=a\bar{b}$. In particular,
$T^{i}(a\bar{c}_{i})=a^{2}\bar{b}$.

###### Proof.

Indeed, the matrix $a(T^{i})^{-1}=a\frac{1}{\det T^{i}}{\rm adj}(T^{i})$
(where ${\rm adj}(T^{i})$ is the classical adjoint) is regular over $C$.
Hence, $\bar{c}_{i}=a(T^{i})^{-1}\bar{b}$ yields the desired conclusion. ∎

To use Lemma 6.10 for automorphisms of $N_{X}C$ as in Lemma 6.7 (ii) we need
the entries of $a\bar{c}_{i}$ to be pullbacks
$\varrho_{\alpha_{i}}^{*}(h_{j})|_{C}$ of regular functions $h_{j}|_{C_{i}}$
on $C_{i}$ which is guaranteed by the next fact.

######  Proposition 6.11.

([Ud, Theorem 2.3.1]) Let $\tau:C\to\bar{C}$ be a proper birational morphism
of affine curves such that its restriction to an open subset $C^{*}$ of $C$ is
an isomorphism. Then there exists $m>0$ such that every regular function
$a\in{\rm\bf k}[C]$ vanishing on $C\setminus C^{*}$ with multiplicity at least
$m$ is a pullback of a regular function on $\bar{C}$.

Recall that by Notation 6.8 $\varrho_{\alpha_{i}}|_{C}:C\to Q$ is not only
birational but also proper. Hence, Proposition 6.11 is applicable and choosing
the column $a^{2}\bar{b}$ in Lemma 6.10 so that its $k_{0}$-entry is $a^{2}$
whereas the rest of them are zeros we find now regular functions
$h_{1},\ldots,h_{i-1},h_{i+1}\ldots,h_{n-1}\in{\rm\bf k}[Q]$ for which one has
$\sum_{j=1,j\neq
i}^{n-1}(\varrho_{\alpha_{i}}^{*}(h_{j})\delta_{\alpha_{k_{0}}}(\varrho_{\alpha_{i}}^{*}(f_{j})))|_{C}=a^{2}$
and $\sum_{j=1,j\neq
i}^{n-1}(\varrho_{\alpha_{i}}^{*}(h_{j})\delta_{\alpha_{k}}(\varrho_{\alpha_{i}}^{*}(f_{j})))|_{C}=0$
for $k\neq k_{0}$. Thus, we have the following.

######  Proposition 6.12.

Let Notation 6.8 hold. For every $1\leq k_{0}\neq i\leq n-1$ a locally
nilpotent vector field $\sigma$ in Lemma 6.7 can be chosen so that its flow
induces an automorphism of $TX|_{C}$ such that
$\breve{\delta}_{\alpha_{k}}\mapsto\breve{\delta}_{\alpha_{k}}$ for $1\leq
k\neq k_{0}\leq n-1$ and
$\breve{\delta}_{\alpha_{k_{0}}}\mapsto\breve{\delta}_{\alpha_{k_{0}}}+ta^{2}d\breve{\delta}_{\alpha_{i}}$
where $t$ is a time parameter, $a$ is a fixed regular function on $C$ that
vanishes on $C\setminus C^{*}$ only, but with sufficiently high multiplicity
and $d$ is any given regular function on $C$. In particular, automorphisms of
$TX|_{C}$ that have this form are $G$-induced.

######  Corollary 6.13.

Let the notations of Proposition 6.12 hold. Let $\mathop{\rm pr}:TX|_{C}\to
N_{X}C$ be the natural projection, $A={\rm\bf k}[C]$ and $w_{j}=\mathop{\rm
pr}(\breve{\delta}_{\alpha_{j}})$ for $j=1,\ldots,n-1$. Suppose (without loss
of generality) that $C\setminus C^{*}$ is the zero locus of a function
$q\in{\rm\bf k}[C]$ (i.e., ${\rm\bf k}[C^{*}]=A[\frac{1}{q}]$). Then
$w_{1},\ldots,w_{n-1}$ generate a free $A$-module $M$ such that the module
$M[\frac{1}{q}]=M\otimes_{A}A[\frac{1}{q}]$ coincides with $N_{X}C^{*}$.
Furthermore, there is an automorphism $\alpha\in G$ of $X$ such that it
induces an automorphism of $M$ given by $w_{k}\mapsto w_{k}$ for $1\leq k\neq
k_{0}\leq n-1$ and $w_{k_{0}}\mapsto w_{k_{0}}+ta^{2}dw_{i}$.

###### Proof.

The first statement was explained in Notations 6.8. The second one follows
from an application of $\mathop{\rm pr}$ to the automorphism of $TX|_{C}$ in
Proposition 6.12. ∎

Letting the index $i$ in Proposition 6.12 run from $1$ to $n-1$ and taking a
smaller neighborhood $C^{*}$ of $K$ (where $K$ is as in Notations 6.8), if
necessary, we get the next result.

######  Corollary 6.14.

Let $M$ be as in Corollary 6.13 (i.e., every $A$-automorphism of $M$ can be
presented as a matrix from ${\rm\operatorname{{\rm Mat}}}_{n-1}(A))$. Then
$C^{*}\supset K$ can be chosen so that for some principal ideal
${\mathfrak{q}}=qA\subset A$ whose zero locus is $C\setminus C^{*}$ every
element of $\mathop{\rm F}_{n-1}(A,{\mathfrak{q}})$ is $G$-induced (where
${\rm\operatorname{{\rm Mat}}}_{n-1}(A)$ and $\mathop{\rm
F}_{n-1}(A,{\mathfrak{q}})$ are as in Notation 5.1).

###### Proof of Theorem 6.6 (i).

Recall that for general elements $\alpha_{1},\ldots,\alpha_{n-1}$ of a perfect
$G$-family ${\mathcal{A}}$ the morphism $\varrho_{\alpha_{i}}|_{C}:C\to Q$ is
proper for every $i$ because either $C$ is once-punctured or the conclusions
of Proposition 4.4 are valid by virtue of Convention 3.2. Furthermore, by
Propositions 4.1 and 4.4 $\varrho_{\alpha_{i}}|_{C^{*}}:C^{*}\to C_{i}^{*}$ is
an isomorphism and $C_{i}^{*}$ is contained in $Q_{0}$ where $C^{*}\subset C$
is a neighborhood of a given finite subset $K$ of $C$. Thus, we have the
assumptions (a) and (b) from Notations 6.8. The assumption (c) is also valid
by virtue of Propositions 4.1(iii) and 4.4(4). Hence, we can use Corollaries
6.13 and 6.14.

Suppose that $\bar{v}=(v_{1},\ldots,v_{n-1})$ is a basis of $N_{X}C$ as a free
$A$-module. Let $\bar{w}=(w_{1},\ldots,w_{n-1})$ where $w_{i}$ is as in
Corollary 6.13. That is, $\bar{w}$ is a basis of a free $A$-submodule $M$ of
$N_{X}C$ as in Corollary 6.13. Then for some matrix $D\in{\rm Mat}_{n-1}(A)$
we have $\bar{w}=D\bar{v}$. Let $q$ be as in Corollary 6.14. Since the
$A[\frac{1}{q}]$-module $M[\frac{1}{q}]$ coincides with
$N_{X}C^{*}=N_{X}C[\frac{1}{q}]$ by Corollary 6.13, the matrix $D$ is
invertible as an element of ${\rm Mat}_{n-1}(A[\frac{1}{q}])$. Let
$K^{\prime}=C\setminus C^{*}$. Then by Proposition 4.1 (resp. 4.4) and
Corollary 6.13 we can find general elements
$\alpha_{1}^{\prime},\ldots,\alpha_{n-1}^{\prime}$ of some perfect $G$-family
${\mathcal{A}}$ such that for the free $A$-module $M^{\prime}$ with the basis
$\bar{w}^{\prime}=(\mathop{\rm
pr}(\breve{\delta}_{\alpha_{1}^{\prime}}),\ldots,\mathop{\rm
pr}(\breve{\delta}_{\alpha_{n-1}^{\prime}}))$ there exists a principal ideal
${\mathfrak{q}}^{\prime}=q^{\prime}A$ such that its zero locus is disjoint
from $K^{\prime}$ and every element of $\mathop{\rm
F}_{n-1}(A,{\mathfrak{q}}^{\prime})$ is $G$-induced. Furthermore,
$\bar{w}^{\prime}=D^{\prime}\bar{v}$ for some matrix $D^{\prime}$ similar to
$D$. By construction, automorphisms of $N_{X}C$ whose matrices in basis
$\bar{v}$ belong to $\mathop{\rm SL}_{n-1}(A)\cap(D^{-1}\mathop{\rm
F}_{n-1}(A,{\mathfrak{q}})D)$ or $\mathop{\rm
SL}_{n-1}(A)\cap((D^{\prime})^{-1}\mathop{\rm
F}_{n-1}(A,{\mathfrak{q}}^{\prime})D^{\prime})$ are $G$-induced. Hence, by
Proposition 5.5 every element of $\mathop{\rm E}_{n-1}(A)$ is $G$-induced.
Switching to dual spaces we see that every $A$-automorphism
$\lambda:\frac{I}{I^{2}}\to\frac{I}{I^{2}}$ that belongs to $E_{n-1}(A)$ is
$G$-induced which concludes the proof. $\square$

######  Notation 6.15.

Let $A$ be a commutative ring, $I$ be the ideal in $A[u_{1},\ldots,u_{n-1}]$
(where each $u_{i}$ is a variable) that consists of all polynomials whose
constant terms are zero, and $\bar{u}=(u_{1},\ldots,u_{n-1})$. Consider the
space ${\mathcal{P}}\subset I$ that consists of those polynomials whose degree
is at most $k-1$ and the space $\tilde{\mathcal{F}}$ of $(n-1)$-tuples
$\bar{f}=(f_{1},\ldots,f_{n-1})$ of elements of ${\mathcal{P}}$ (note that
modulo $I^{k}$ compositions of elements of $\tilde{\mathcal{F}}$ are again
elements of $\tilde{\mathcal{F}}$). For every $f_{i}\in{\mathcal{P}}$ one has
$f_{i}=f_{i}^{1}+\ldots+f_{i}^{k-1}$ where each $f_{i}^{l}$ is a homogeneous
polynomial of degree $l$. Consider two $A$-linear maps
$\chi:\tilde{\mathcal{F}}\to\tilde{\mathcal{F}},\,\bar{f}\mapsto(f_{1}^{1},\ldots,f_{n-1}^{1})$
and
$\pi:\tilde{\mathcal{F}}\to\tilde{\mathcal{F}},\,\bar{f}\mapsto(f_{1}^{k-1},\ldots,f_{n-1}^{k-1})$,
i.e., the image of $\pi$ coincides with the space ${\mathcal{F}}$ of
$(n-1)$-tupes of $(k-1)$-homogeneous forms in $\bar{u}$ over $A$. Consider
also the subset $\tilde{\mathcal{F}}^{\prime}$ of $\tilde{\mathcal{F}}$ that
consists of all $\bar{f}\in\tilde{\mathcal{F}}$ for which
$(\bar{f}-\pi(\bar{f}))(\bar{u})=\bar{u}$.

######  Lemma 6.16.

Let $\bar{f}\in\tilde{\mathcal{F}}^{\prime}$ and
$\bar{g}\in\tilde{\mathcal{F}}$. Then

(1) $\pi(\bar{f}\circ\bar{g})=\pi(\bar{g})+\pi(\bar{f})\circ\chi(\bar{g})$ and
$\pi(\bar{g}\circ\bar{f})=\chi(\bar{g})\circ\pi(\bar{f})+\pi(\bar{g})$ (in
particular, if $\bar{g}\in\tilde{\mathcal{F}}^{\prime}$ one has
$\pi(\bar{f}\circ\bar{g})=\pi(\bar{f})+\pi(\bar{g})$);

(2)
$\bar{f}\circ\bar{g}-\pi(\bar{f}\circ\bar{g})=\bar{g}\circ\bar{f}-\pi(\bar{g}\circ\bar{f})=\bar{g}-\pi(\bar{g})$.

Furthermore, let $\bar{h}\in\tilde{\mathcal{F}}$ be such that
$\bar{h}\circ\bar{g}(\bar{u})=\bar{u}$ modulo $I^{k}$ (and, thus,
$\chi(\bar{h}\circ\bar{g})(\bar{u})=\chi(\bar{h})\circ\chi(\bar{g})(\bar{u})=\bar{u}$).
Then

(3)
$\pi(\bar{h}\circ\bar{f}\circ\bar{g})=\chi(\bar{h})\circ\pi(\bar{f})\circ\chi(\bar{g})$
and $\bar{h}\circ\bar{f}\circ\bar{g}\in\tilde{\mathcal{F}}^{\prime}$.

###### Proof.

Note that for every $\bar{e}\in{\mathcal{F}}$ one has
$\pi(\bar{g}\circ\bar{e})=\chi(\bar{g})\circ\bar{e}$ and
$\pi(\bar{e}\circ\bar{g})=\bar{e}\circ\chi(\bar{g})$. Since every element of
$\tilde{\mathcal{F}}^{\prime}$ is of the form
$\bar{f}(\bar{u})=\bar{u}+\bar{e}(\bar{u})$ where $\bar{e}=\pi(\bar{f})$, this
implies (1) and (2). Then by (2) and (1) we have
$\bar{f}\circ\bar{g}=\bar{g}\circ\bar{f}-\pi(\bar{g}\circ\bar{f})+\pi(\bar{f}\circ\bar{g})=\bar{g}\circ\bar{f}-\chi(\bar{g})\circ\pi(\bar{f})+\pi(\bar{f})\circ\chi(\bar{g})$.
Applying (1) again to the composition of $\bar{h}$ with both sides of the last
formula we get
$\pi(\bar{h}\circ\bar{f}\circ\bar{g})=\pi(\bar{f})-\chi(\bar{h})\circ\chi(\bar{g})\circ\pi(\bar{f})+\chi(\bar{h})\circ\pi(\bar{f})\circ\chi(\bar{g})=\chi(\bar{h})\circ\pi(\bar{f})\circ\chi(\bar{g})$.
Note also that $\bar{f}\circ\bar{g}=\bar{g}\circ\bar{f}+\bar{e}$ for some
$\bar{e}\in{\mathcal{F}}$. Hence, modulo $I^{k}$ one has
$\bar{h}\circ\bar{f}\circ\bar{g}=\bar{f}+\bar{h}\circ\bar{e}$. Since the last
summand belongs to ${\mathcal{F}}$, we get
$\bar{h}\circ\bar{f}\circ\bar{g}\in\tilde{\mathcal{F}}^{\prime}$ which
concludes the proof. ∎

######  Notation 6.17.

Suppose that $C$ is a smooth curve in a normal quasi-affine variety $X$ such
that $C$ is closed in an affine variety containing $X$. Suppose also that
$U\subset X_{\rm reg}$ is a neighborhood of $C$ in $X$ such that $C$ is a
strict complete intersection in $U$ given by $u_{1}=\ldots=u_{n-1}=0$ where
each $u_{i}$ belongs to the defining ideal $I$ of $C$ in ${\rm\bf k}[X]$. That
is, $\bar{u}=(u_{1},\ldots,u_{n-1})$ can be viewed as a set of generators of
the graded ring $\frac{{\rm\bf k}[X]}{I^{k}}\simeq
A\oplus\bigoplus_{l=2}^{k-1}\frac{I^{l-1}}{I^{l}}$ over $A={\rm\bf k}[C]$.
Then every $A$-automorphism $\theta$ of $\frac{{\rm\bf k}[X]}{I^{k}}$ is given
by its values on generators of $\frac{{\rm\bf k}[X]}{I^{k}}$ over $A$. That
is, in the terminology of Notation 6.15 $\theta$ is given by a map of the form
$\bar{u}\mapsto\bar{f}(\bar{u})$ where
$\bar{f}=(f_{1},\ldots,f_{n-1})\in\tilde{\mathcal{F}}$ and the automorphism
$\lambda:\frac{I}{I^{2}}\to\frac{I}{I^{2}}$ of the conormal bundle induced by
$\theta$ is given by $\bar{u}\mapsto\chi(\bar{f})(\bar{u})$. In particular, we
have the natural map $\Phi:\tilde{H}\to\tilde{\mathcal{F}}$ from the group
$\tilde{H}$ of $A$-automorphisms of $\frac{{\rm\bf k}[X]}{I^{k}}$ which
becomes a semigroup homomorphism if one equips $\tilde{\mathcal{F}}$ with the
binary operation of composition modulo $I^{k}$. Let $\tilde{\mathcal{F}}_{0}$
be the subset of $\tilde{\mathcal{F}}$ consisting of all $\bar{f}$ for which
the Jacobian $\left[\frac{\partial f_{i}}{\partial
u_{j}}\right]_{i,j=1}^{n-1}$ is equal to 1 modulo $I^{k-1}$. Note that
$\chi(\bar{f})$ can be viewed as an element of $\mathop{\rm SL}_{n-1}(A)$ for
every $\bar{f}\in\tilde{\mathcal{F}}_{0}$. For $k\geq 3$ consider
$\tilde{\mathcal{F}}_{0}^{\prime}=\tilde{\mathcal{F}}_{0}\cap\tilde{\mathcal{F}}^{\prime}$.
In particular, every
$\bar{f}=(f_{1},\ldots,f_{n-1})\in\tilde{\mathcal{F}}_{0}^{\prime}$ is of the
form $\bar{f}(\bar{u})=\bar{u}+\pi(\bar{f})(\bar{u})$. It is a straightforward
fact observed in [AFKKZ, proof of Lemma 4.13] that for such $\bar{f}$ the
assumption on the Jacobian is equivalent to the following

$\operatorname{\rm div}\pi(\bar{f})=\frac{\partial f_{1}^{k-1}}{\partial
u_{1}}+\ldots+\frac{\partial f_{n-1}^{k-1}}{\partial u_{n-1}}=0.$

That is, ${\mathcal{F}}_{0}=\pi(\tilde{\mathcal{F}}_{0}^{\prime})$ is the
space of $(n-1)$-tuples of $(k-1)$-forms with zero divergence.

######  Lemma 6.18.

Let Notations 6.17 hold, $G$ be a subgroup of $\operatorname{{\rm Aut}}(X)$
such that $G$-induced $A$-automorphisms of $\frac{{\rm\bf k}[X]}{I^{l}}$ have
Jacobian 1 (modulo $I^{l-1}$) for every $l\geq 2$ and let $H$ be the subgroup
of all $G$-induced $A$-automorphisms of $\frac{{\rm\bf k}[X]}{I^{k}}$. Suppose
that $\Phi(H)\supset\tilde{\mathcal{F}}_{0}^{\prime}$ or, equivalently,
${\mathcal{F}}^{0}:=\pi(\Phi(H)\cap\tilde{\mathcal{F}}_{0}^{\prime})$
coincides with ${\mathcal{F}}_{0}$. Then $\Phi(H)=\tilde{\mathcal{F}}_{0}$.

###### Proof.

We use induction on $k$ with the case of $k=2$ provided by Theorem 6.6(i).
Furthermore, by construction an automorphism $\alpha\in G$ whose restriction
to $C$ induces $\lambda$ in Theorem 6.6(i) is a composition of flows of
locally nilpotent vector fields of the form
$\varrho_{\alpha_{i}}^{*}(hf)\delta_{\alpha_{i}}$ from Lemma 6.7 where
$f,h\in{\rm\bf k}[Q]$ are such that $f$ vanishes on $\varrho_{\alpha_{i}}(C)$.
These vector fields satisfy the description of $\sigma$ in Example 6.5. Hence,
for every $l\geq 2$ the automorphism $\alpha$ induces an $A$-automorphism
$\frac{{\rm\bf k}[X]}{I^{l}}$ with Jacobian 1. Suppose that the statement is
true for $k-1$. In particular, for every $\bar{g}\in\tilde{\mathcal{F}}_{0}$
there exists an $\theta\in H$ such that
$\bar{g}-\pi(\bar{g})=\Phi(\theta)-\pi(\Phi(\theta))$. Then
$\bar{f}:=\Phi(\theta^{-1})\circ\bar{g}={\rm
id}+\Phi(\theta^{-1})\circ(\pi(\bar{g})-\pi(\Phi(\theta)))$ is contained in
$\tilde{\mathcal{F}}^{\prime}$ and has Jacobian 1 modulo $I^{k-1}$. That is,
$\bar{f}\in\tilde{\mathcal{F}}_{0}^{\prime}$ and, consequently,
$\Phi(\gamma)=\bar{f}$ for some $\gamma\in H$. This implies that
$\Phi(\theta\circ\gamma)=\bar{g}$ where $\theta\circ\gamma$ is induced by some
$\alpha\in G$ . This yields the desired conclusion. ∎

######  Lemma 6.19.

Let the notations of Lemma 6.18 hold.

(i) Then the map $\pi\circ\Phi:H\to{\mathcal{F}}_{0}$ is a group homomorphism
(transforming multiplication to addition).

(ii) There is a natural $\mathop{\rm SL}_{n-1}(A)$-action on ${\mathcal{F}}$
such that ${\mathcal{F}}^{0}$ is invariant with respect to this action.

###### Proof.

Statement (i) follows from Lemma 6.16 (1). Let $\theta$ belong to $H$,
$\Phi(\theta)=\bar{g}$ and $\Phi(\theta^{-1})=\bar{h}$. By Theorem 6.6 (i)
$\chi\circ\Phi(H)$ is isomorphic to $\mathop{\rm SL}_{n-1}(A)$, i.e.,
$\chi(\bar{g})$ can be any element of $\mathop{\rm SL}_{n-1}(A)$ and
$\chi(\bar{h})$ is its inverse. Hence, one has the desired natural
$\mathop{\rm SL}_{n-1}(A)$-action on ${\mathcal{F}}$ given by
$\bar{e}\mapsto\chi(\bar{h})\circ\bar{e}\circ\chi(\bar{g})$. Let
$\pi(\bar{f})\in{\mathcal{F}}^{0}$ for some
$\bar{f}\in\tilde{\mathcal{F}}_{0}^{\prime}$, i.e., there exists $\gamma\in H$
such that $\bar{f}=\Phi(\gamma)$. Since
$\chi(\bar{h})\circ\pi(\bar{f})\circ\chi(\bar{g})=\pi(\bar{h}\circ\bar{f}\circ\bar{g})=\pi(\Phi(\theta^{-1}\gamma\theta))$
and $\bar{h}\circ\bar{f}\circ\bar{g}\in\tilde{\mathcal{F}}^{\prime}$ by Lemma
6.16 (3), we have
$\chi(\bar{h})\circ\pi(\bar{f})\circ\chi(\bar{g})\in{\mathcal{F}}^{0}$.
Therefore, the $\mathop{\rm SL}_{n-1}(A)$-action sends every element of
${\mathcal{F}}^{0}$ to an element of ${\mathcal{F}}^{0}$ which yields (ii) and
the desired conclusion. ∎

We shall need also the following well-known fact from representation theory.

######  Theorem 6.20.

Let $F$ be the space of $(n-1)$-tuples of homogeneous $k$-forms in variables
$\bar{u}=(u_{1},\ldots,u_{n-1})$ over a field ${\rm\bf K}$ such that the
divergence of every $q\in F$ is zero. Then $F$ is an irreducible $\mathop{\rm
SL}_{n-1}({\rm\bf K})$-module under the $\mathop{\rm SL}_{n-1}({\rm\bf
K})$-action given by $q(\bar{u})\mapsto L^{-1}(q(L(\bar{u}))$ where
$L\in\mathop{\rm SL}_{n-1}({\rm\bf K})$ and $q\in F$.

###### Proof.

Denote by $V$ the ${\rm\bf K}$-vector space with the basis $\bar{u}$ and by
$V^{*}$ its dual. Then the space of homogeneous $k$-forms in $\bar{u}$ can be
treated as ${\rm Sym}^{k}V$ and the space $F$ as $V^{*}\otimes{\rm Sym}^{k}V$
(cf. [AFKKZ, Lemma 4.12(a) and Formula (13)]). In these notations divergence
is the contraction map $V^{*}\otimes{\rm Sym}^{k}V\to{\rm Sym}^{k-1}V$ and $F$
is its kernel which is invariant under the natural $\mathop{\rm
SL}_{n-1}({\rm\bf k})$-action.

Following [FuHa, page 216] we use the notations
$\Gamma_{a_{1},\ldots,a_{n-2}}$ (where $a_{1},\ldots,a_{n-2}$ are nonnegative
integers) to denote irreducible representations of $\mathop{\rm
SL}_{n-1}({\rm\bf k})$ classified by highest weights. In particular,
$V^{*}\simeq\bigwedge^{n-2}V$ is the natural space of the irreducible
representation $\Gamma_{0,\ldots,0,1}$ of $\mathop{\rm SL}_{n-1}({\rm\bf k})$,
while for ${\rm Sym}^{k}V$ the corresponding irreducible representation is
$\Gamma_{k,0,\ldots,0}$. By [FuHa, Proposition 15.25 ] ${\rm
Sym}^{k}V\otimes\bigwedge^{n-2}V$ is the sum of two irreducible
representations $\Gamma_{k-1,0,\ldots,0}$ and $\Gamma_{k,0,\ldots,0,1}$. The
first of them is, of course, the space ${\rm Sym}^{k-1}V$ and the second one
is, therefore, the kernel of the contraction homomorphism. Hence, $F$ is
irreducible and we are done. ∎

######  Lemma 6.21.

Let Notation 6.17 hold (i.e., every automorphism of $\frac{I}{I^{2}}$ is given
by a matrix from $\operatorname{{\rm GL}}_{n-1}(A)$) and let $G$ be generated
by a saturated set ${\mathcal{N}}\subset\operatorname{{\rm LND}}(X)$. Let all
automorphisms of $\frac{I}{I^{2}}$ given by elements of $\mathop{\rm
SL}_{n-1}(A)$ be $G$-induced. Suppose that for every $x\in C$ there exists a
neighborhood $V\subset C$ and $\sigma\in{\mathcal{N}}$ with an associated
partial quotient morphism $\tau:X\to P$ such that $\sigma(x)\neq 0$, $\tau$ is
smooth at $x$ and it induces an isomorphism $\tau|_{V}:V\to\tau(V)$ where
$\tau(V)$ is contained in the smooth part of $P$. Then every $A$-automorphism
$\theta:\frac{{\rm\bf k}[X]}{I^{k}}\to\frac{{\rm\bf k}[X]}{I^{k}}$ with
Jacobian 1 is $G$-induced.

###### Proof.

By Lemma 6.18 it suffices to prove that ${\mathcal{F}}_{0}={\mathcal{F}}^{0}$.
By Lemma 6.19 ${\mathcal{F}}^{0}$ is a subgroup of ${\mathcal{F}}_{0}$ which
is invariant under the natural $\mathop{\rm SL}_{n-1}(A)$-action. Hence,
${\mathcal{F}}^{0}$ contains a unique maximal $A$-module ${\mathcal{R}}$ which
is also invariant under the $\mathop{\rm SL}_{n-1}(A)$-action. For every
$A$-module $M$ denote by $M_{\mu}$ the localization of $M$ with respect to a
maximal ideal $\mu\subset A$. Recall that two submodules $M$ and $N$ of an
$A$-module coincide if $M_{\mu}=N_{\mu}$ for every maximal ideal $\mu$ [AM,
Proposition 3.9]. Hence, by Lemma 6.18 it suffices to prove that
$({\mathcal{F}}_{0})_{\mu}={\mathcal{R}}_{\mu}$. By [AM, Corollary 2.7] this
is true if
${\mathcal{R}}_{\mu}+\mu({\mathcal{F}}_{0})_{\mu}=({\mathcal{F}}_{0})_{\mu}$.
In turn the latter equality is equivalent to the following
$\frac{{\mathcal{R}}_{\mu}}{\mu({\mathcal{F}}_{0})_{\mu}\cap{\mathcal{R}}_{\mu}}=\frac{({\mathcal{F}}_{0})_{\mu}}{\mu({\mathcal{F}}_{0})_{\mu}}$.
Note that $\frac{({\mathcal{F}}_{0})_{\mu}}{\mu({\mathcal{F}}_{0})_{\mu}}$ is
the space of $(n-1)$-tuples of homogeneous $k$-forms in variables
$\bar{u}=(u_{1},\ldots,u_{n-1})$ over the residue field of $\mu$ such that the
divergence is zero. By Theorem 6.20 it is enough to prove that
$\frac{{\mathcal{R}}_{\mu}}{\mu({\mathcal{F}}_{0})_{\mu}\cap{\mathcal{R}}_{\mu}}$
is nonzero. By the Nullstellensatz $\mu$ is the defining ideal of a point
$x\in C$. Choose a regular function $f$ on $P$ that vanishes on $\tau(C)$ and
has a nonzero differential at $\tau(x)$. Then $\breve{f}=\tau^{*}(f)$ belongs
to the defining ideal $I$ of $C$ in ${\rm\bf k}[X]$. Furthermore, since
$\tau|_{V}:V\to\tau(V)$ is an isomorphism the assumptions on smoothness and
the differential imply that $\breve{f}$ belongs to $I\setminus I^{2}$.
Consider the flow of the locally nilpotent vector field
$\breve{\sigma}=\breve{f}^{k-1}\sigma$ with time parameter 1. It sends $u_{i}$
to
$u_{i}+\breve{\sigma}(u_{i})+\sum_{l=2}^{\infty}\frac{\breve{\sigma}^{l}(u_{i})}{l!}=u_{i}+\breve{f}^{k-1}\sigma(u_{i})+\breve{f}^{2k-3}b$
where $b$ is some regular function on $X$. Note that if $k\geq 3$, then
$\breve{f}^{2k-3}b$ belongs to $I^{k}$. On the other hand,
$\breve{f}^{k-1}\sigma(u_{i})\in I^{k-1}$ and, furthermore,
$\breve{f}^{k-1}\sigma(u_{i})\in I^{k-1}\setminus\mu I^{k-1}$ provided that
$\sigma(u_{i})(x)\neq 0$. Since $\sigma(x)\neq 0$ and $\sigma$ is not tangent
to $C$ (because of the isomorphism $\tau|_{V}:V\to\tau(V)$) we do have $i$ for
which $\sigma(u_{i})(x)\neq 0$. Hence,
$(\breve{f}^{k-1}u_{1},\ldots,\breve{f}^{k-1}u_{n-1})$ yields the desired
nonzero element of
$\frac{{\mathcal{R}}_{\mu}}{\mu({\mathcal{F}}_{0})_{\mu}\cap{\mathcal{R}}_{\mu}}$
which concludes the proof. ∎

###### Proof of Theorem 6.6 (ii).

By Propositions 4.1 and 4.4 letting $\sigma=\delta_{\alpha_{1}}$ we see that
$\sigma$ satisfies the assumptions of Lemma 6.21. Since $\mathop{\rm
E}_{n-1}(A)=\mathop{\rm SL}_{n-1}(A)$ and by Theorem 6.6(i) every element of
$\mathop{\rm E}_{n-1}(A)$ is $G$-induced, we see that all assumptions of Lemma
6.21 are valid. This implies the desired conclusion. $\square$

######  Remark 6.22.

Let us emphasize that by construction an automorphism $\alpha\in G$ whose
restriction to $C$ induces $\lambda$ (resp. $\theta$) in Theorem 6.6 is a
composition of flows of locally nilpotent vector fields of the form
$\varrho_{\alpha_{i}}^{*}(hf)\delta_{\alpha_{i}}$ from Lemma 6.7 where
$f,h\in{\rm\bf k}[Q]$ are such that $f$ vanishes on
$C_{i}=\varrho_{\alpha_{i}}(C)$. Furthermore, this construction requires only
the knowledge of $f$ and $h$ on the $k$th infinitesimal neighborhood of
$C_{i}$ in $Q$ and we do not care about the behavior of $f$ and $h$ on
$Q\setminus C_{i}$. Hence, one can require additionally that $f$ and $h$
vanish with a given multiplicity on a given closed subvariety of $Q$ disjoint
from $C_{i}$.

In conclusion of this section we want to pose the following problem.

Question. Does Theorem 6.6 remain true if the assumption $n\geq 4$ is replaced
by $n\geq 3$?

##  7\. Main Theorem. II

The main aim of this section is to show that Theorem 6.6 is valid without the
assumption that $C$ is irreducible.

######  Theorem 7.1.

(1) Theorem 6.6 is valid if assumption ($\bullet$) is replaced by one of the
following:

(i) either every irreducible component of $C$ is once-punctured, or

(ii) Convention 3.2 holds.

(2) Furthermore, let $l\geq 1$, $Z$ be a closed subvariety of $X$ disjoint
from $C$, ${\mathcal{N}}_{Z,l}$ be the subset of ${\mathcal{N}}$ that consists
of those elements of ${\mathcal{N}}$ that vanish on $Z$ with multiplicity at
least $l$ and $G_{Z,l}$ be the subgroup of $G$ generated by the flows of
elements of ${\mathcal{N}}_{Z,l}$. Suppose that in (i) (resp. (ii)) ${\rm
codim}_{X}Z\geq 2$ (resp. ${\rm codim}_{X}Z\geq 3$). Then the automorphisms
$\lambda$ and $\theta$ in Theorem 6.6 (modified as in (1)) are
$G_{Z,l}$-induced.

###### Proof of Theorem 7.1 (i)..

Let $C^{\prime}$ be an irreducible component of $C$ and
$C^{\prime\prime}=C\setminus C^{\prime}$. Assume first that $C=C^{\prime}$.
The quasi-affine variety $X\setminus Z$ is $G_{Z,l}$-flexible by [FKZ, Theorem
2.6]. Hence, the desired conclusion in this case follows from Theorem 6.6. If
$C\neq C^{\prime}$, then by induction on the number of irreducible components
of $C$ one can suppose that the desired conclusion holds for the curve
$C^{\prime\prime}$. That is, we have an automorphism $\alpha\in G_{Z,l}$ with
a prescribed restriction to the $k$-infinitesimal neighborhood of
$C^{\prime\prime}$. Replacing $Z$ by $Z\cup C^{\prime\prime}$ we get also an
automorphism $\beta\in G_{Z\cup C^{\prime\prime},l}\cap
G_{C^{\prime\prime},k}$ with a prescribed restriction to the $k$-infinitesimal
neighborhood of $C^{\prime}$. Hence, one can choose $\alpha$ and $\beta$ so
that $\alpha\circ\beta\in G_{Z,l}$ has a given restriction to the
$k$-infinitesimal neighborhood of $C$ which concludes the proof. $\square$

######  Remark 7.2.

The above argument does not work for Theorem 7.1 (ii), since $X$ must be
affine by Convention 3.2. Hence, one cannot replace $X$ by $X\setminus Z$
(resp. $X\setminus C^{\prime\prime}$) when $Z\neq\emptyset$ (resp.
$C^{\prime\prime}\neq\emptyset$).

Hence, we need the following technical result.

######  Lemma 7.3.

Let $X$ be an affine $G$-flexible variety where $G$ is generated by a
saturated set ${\mathcal{N}}\subset\operatorname{{\rm LND}}(X)$. Let
$\delta\in{\mathcal{N}}$ and its partial quotient morphism $\varrho:X\to Q$ be
such that condition ($\\#$) from Convention 3.2 is valid. Let $C$ be a closed
curve in $X$ such that $C\subset X_{\rm reg}$ and $Z$ be a closed subvariety
of $X$ with ${\rm codim}_{X}Z\geq 3$. Let ${\mathcal{A}}$ be an algebraic
$G$-family. Suppose that
$W_{\alpha}=\varrho_{\alpha}^{-1}(\overline{\varrho_{\alpha}(Z)})$ where
$\varrho_{\alpha}=\varrho\circ\alpha^{-1}$ for $\alpha\in{\mathcal{A}}$ and
$\overline{\varrho_{\alpha}(Z)}$ is the closure of ${\varrho_{\alpha}(Z)}$ in
$Q$. Then for every $l\geq 1$ there exists $\gamma\in G_{Z,l}$ such that for a
general $\alpha\in{\mathcal{A}}$ the curve $\gamma(C)$ does not meet
$W_{\alpha}$.

###### Proof.

Let ${\mathcal{X}}={\mathcal{A}}\times X$ and
${\mathcal{Z}}={\mathcal{A}}\times Z$. Consider the morphism
$\tilde{\varrho}:{\mathcal{X}}\to{\mathcal{A}}\times
Q,\,(\alpha,x)\mapsto(\alpha,\varrho_{\alpha}(x))$. Let
${\mathcal{V}}=\tilde{\varrho}({\mathcal{Z}})$, $\bar{\mathcal{V}}$ be the
closure of ${\mathcal{V}}$ in ${\mathcal{A}}\times Q$ and
${\mathcal{D}}=\bar{\mathcal{V}}\setminus{\mathcal{V}}$. By Chevalley’s
theorem ${\mathcal{V}}$ is a constructible set [Ha, Chapter II, Exercise 3.19]
and, hence, ${\mathcal{D}}$ is a subvariety of $\bar{\mathcal{V}}$. Let
$\kappa:\bar{\mathcal{V}}\to{\mathcal{A}}$ be the natural projection,
$\bar{\mathcal{V}}^{\prime}$ be an irreducible component of
$\bar{\mathcal{V}}$ and let ${\mathcal{D}}^{\prime}$ be an irreducible
component of ${\mathcal{D}}$ contained in $\bar{\mathcal{V}}^{\prime}$. By
another Chevalley’s theorem [Ha, Chapter II, Exercise 3.22] general fibers of
$\kappa|_{\bar{\mathcal{V}}^{\prime}}$ (resp.
$\kappa|_{{\mathcal{D}}^{\prime}}$) are of the same pure dimension
$\dim\bar{\mathcal{V}}^{\prime}-\dim{\mathcal{A}}$ (resp.
$\dim\bar{\mathcal{D}}^{\prime}-\dim{\mathcal{A}}$). Since
$\dim{\mathcal{D}}^{\prime}<\dim\bar{\mathcal{V}}^{\prime}$, we see that
$\kappa^{-1}(\alpha)\cap{\mathcal{D}}^{\prime}$ cannot be an irreducible
component of $\kappa^{-1}(\alpha)\cap\bar{\mathcal{V}}^{\prime}$ for a general
$\alpha\in{\mathcal{A}}$. That is,
$\kappa^{-1}(\alpha)\cap{\mathcal{D}}^{\prime}$ is contained in the closure of
$\kappa^{-1}(\alpha)\cap{\mathcal{V}}$ in the fiber $\alpha\times Q$. Hence,
for a general $\alpha\in{\mathcal{A}}$ the closure
$\overline{\varrho_{\alpha}(Z)}$ of $\varrho_{\alpha}(Z)$ in $\alpha\times Q$
coincides with $\kappa^{-1}(\alpha)\cap\bar{\mathcal{V}}$. In particular, for
${\mathcal{W}}=\tilde{\varrho}^{-1}(\bar{\mathcal{V}})=\tilde{\varrho}^{-1}(\overline{\tilde{\varrho}({\mathcal{Z}})}))$
and a general $\alpha\in{\mathcal{A}}$ one has
$\pi^{-1}(\alpha)\cap{\mathcal{W}}=W_{\alpha}$ where
$\pi:{\mathcal{X}}\to{\mathcal{A}}$ is the natural projection.

Let $G_{Z,l}$ act on ${\mathcal{X}}$ by
$\gamma.(\alpha,x)=(\alpha,\gamma(x))$. Then this action preserves
${\mathcal{Z}}$ and it is transitive on every fiber of
$\pi|_{{\mathcal{X}}\setminus{\mathcal{Z}}}$ by [FKZ, Theorem 2.6]. Let
${\mathcal{C}}={\mathcal{A}}\times C$. By Theorem 1.7 and [Ka20, Remark
1.9(1)] for a general $\gamma$ in some algebraic $G_{Z,1}$-family
$\gamma.{\mathcal{C}}$ meets ${\mathcal{W}}$ along a subvariety $R$ of
dimension at most
$\dim{\mathcal{W}}+\dim{\mathcal{C}}-\dim{\mathcal{X}}=\dim{\mathcal{A}}+\dim
W_{\alpha}+1-\dim X$ where $\alpha$ is a general element of ${\mathcal{A}}$.
Note that by condition ($\\#$) $\dim W_{\alpha}\leq\dim X-2$ since $\dim
Z\leq\dim X-3$. Hence, $\dim R<\dim{\mathcal{A}}$ and for a general $\alpha$
the set $R\cap\pi^{-1}(\alpha)=W_{\alpha}\cap\gamma(C)$ is empty which
concludes the proof. ∎

###### Proof of Theorem 7.1 (ii)..

Let ${\mathcal{A}}$ be a family as in Proposition 4.4 and
$(\alpha_{1},\ldots,\alpha_{n-1})$ be a general element of
${\mathcal{A}}^{n-1}$. That is, each $\varrho_{\alpha_{i}}|_{C}:C\to Q$ is a
closed embedding and $C_{i}=\varrho_{\alpha_{i}}(C)$ is contained in $Q_{0}$
where $Q_{0}$ is as in Remark 1.4. Replacing $C$ by $\gamma(C)$ where
$\gamma\in G_{Z,l}$ is as in Lemma 7.3 we can suppose that $C$ does not meet
$\varrho_{\alpha_{i}}^{-1}(\overline{\varrho_{\alpha_{i}}(Z)})$ for every
$i=1,\ldots,n-1$. This implies that $C_{i}$ does not meet the closed
subvariety $\overline{\varrho_{\alpha_{i}}(Z)}$ in $Q$.

Let $C^{\prime}$ be an irreducible component of $C$ and
$C^{\prime\prime}=C\setminus C^{\prime}$. Assume first that $C=C^{\prime}$.
Recall that by Remark 6.22 an automorphism $\Psi\in G$ with a prescribed
restriction $\theta$ to the $k$-infinitesimal neighborhood of $C$ can be
obtained as a composition of automorphisms of the form
$\varrho_{\alpha_{i}}^{*}(hf)\delta_{\alpha_{i}}$ where $h$ and $f$ are
regular functions on $Q$ such that $f$ vanishes on $C_{i}$. Consider
$\tilde{f}$ and $\tilde{h}\in{\rm\bf k}[Q]$ such that in the $k$th
infinitesimal neighborhood of $C_{i}$ in $Q$ they coincide with $f$ and $h$
respectively but they vanish on $\overline{\varrho_{\alpha_{i}}(Z)}$ with
multiplicity at least $l$ (we can find such functions since the closed
subvarieties $C_{i}$ and $\overline{\varrho_{\alpha_{i}}(Z)}$ of $Q$ are
disjoint). By Remark 6.22 replacing each element
$\varrho_{\alpha_{i}}^{*}(hf)\delta_{\alpha_{i}}$ in the composition with
$\varrho_{\alpha_{i}}^{*}(\tilde{h}\tilde{f})\delta_{\alpha_{i}}$ we get a new
automorphism $\tilde{\Psi}\in G$ that induces the same automorphism $\theta$
of the $k$-infinitesimal neighborhood of $C$. However, $\tilde{\Psi}$ is
already an element of $G_{Z,l}$ which concludes the case of irreducible $C$.

If $C\neq C^{\prime}$, then by induction on the number of irreducible
components of $C$ one can suppose that the desired conclusion holds for the
curve $C^{\prime\prime}$. That is, we have an automorphism
$\Phi^{\prime\prime}\in G_{Z,l}$ with a prescribed restriction to the
$k$-infinitesimal neighborhood of $C^{\prime\prime}$. Replacing $Z$ by $Z\cup
C^{\prime\prime}$ we get also an automorphism $\Psi^{\prime}\in G_{Z\cup
C^{\prime\prime},l}\cap G_{C^{\prime\prime},k}$ with a prescribed restriction
to the $k$-infinitesimal neighborhood of $C^{\prime}$. Hence,
$\Psi^{\prime}\circ\Psi^{\prime\prime}\in G_{Z,l}$ has a prescribed
restriction to the $k$-infinitesimal neighborhood of $C$ which concludes the
proof. $\square$

######  Corollary 7.4.

Let the assumptions of Theorem 7.1 hold and Convention 3.2 be valid. For every
nonvanishing section $v$ of $N_{X}C$ there exists $\sigma\in{\mathcal{N}}$
such that $v=\mathop{\rm pr}(\sigma)$ where $\mathop{\rm pr}:TX|_{C}\to
N_{X}C$ is the natural projection. In particular, $N_{X}C$ admits a basis
$v_{1}\ldots,v_{n-1}$ over $A$ such that each $v_{i}=\mathop{\rm
pr}(\sigma_{i})$ where $\sigma_{1},\ldots,\sigma_{n-1}\in{\mathcal{N}}$.

###### Proof.

The second statement is especially simple. Indeed, by Proposition 4.4(5) we
can suppose that $v_{i}=\mathop{\rm pr}(\sigma_{i})$ where
$\sigma_{i}\in{\mathcal{N}}$ for $i\leq n-2$. Then one can consider an
$A$-automorphism $\theta$ of $N_{X}C$ given by $v_{n-1}\to v_{1}$, $v_{1}\to-
v_{n-1}$ and $v_{i}\to v_{i}$ for $2\leq i\leq n-2$. Since every element of
$\mathop{\rm E}_{n-1}(A)$ is $G$-induced by Theorem 7.1 and
$\theta\in\mathop{\rm E}_{n-1}(A)$, we see that $v_{n-1}=\mathop{\rm
pr}(\sigma_{n-1})$ where $\sigma_{n-1}\in{\mathcal{N}}$.

For the first statement note that $v=\sum_{i=1}^{n-1}a_{i}v_{i}$ where
$a_{1},\ldots,a_{n-1}\in A$ have no common zeros. Since $n-1\geq 3$, by the
Bass stable range theorem [Ba, Theorem 11.1] for some $b_{i}\in A,\,i\geq 1$
the ideal generated by $a_{1}^{\prime},,\ldots,a_{n-2}^{\prime}$ is $A$ where
$a_{i}^{\prime}=a_{i}-b_{i}a_{n-1},\,i=1,\ldots,n-2$. Hence, applying
consequently elements of $\mathop{\rm E}_{n-1}(A)$ we can send $v$ first to
$\sum_{i=1}^{n-2}a_{i}^{\prime}v_{i}$, then to
$v_{n-1}+\sum_{i=1}^{n-2}a_{i}^{\prime}v_{i}$ and then to $v_{n-1}$. This
yields the desired conclusion. ∎

##  8\. Applications

######  Notation 8.1.

In this section $C_{1}$ and $C_{2}$ are smooth curves in a smooth quasi-affine
variety $X$ with defining ideals $I_{1}$ and $I_{2}$ in the algebra ${\rm\bf
k}[X]$ of regular functions on $X$. We suppose also that $C_{1}$ and $C_{2}$
are closed in an affine variety containing $X$. Let $X$ possess a volume form
$\omega$ (i.e., $\omega$ is a nonvanishing section of the canonical bundle on
$X$) and let each conormal bundle $\frac{I_{j}}{I_{j}^{2}}$ of $C_{j}$ in $X$
be trivial. Recall that by Lemma 6.3 there is a neighborhood $U_{j}$ of
$C_{j}$ in $X$ in which $C_{j}$ is a strict complete intersection given by
$u_{1,j}=\ldots=u_{n-1,j}=0$ where $u_{1,j},\ldots,u_{n-1,j}\in I_{j}$. That
is, $\frac{{\rm\bf k}[X]}{I_{j}^{k}}\simeq
A_{j}\oplus\bigoplus_{l=1}^{k-1}\frac{I_{j}^{l}}{I^{l+1}_{j}}$ is a graded
algebra where $A_{j}=\frac{{\rm\bf k}[X]}{I_{j}}$. For $k\in\mathbb{N}$ we
denote by $\theta:\frac{{\rm\bf k}[X]}{I_{1}^{k}}\to\frac{{\rm\bf
k}[X]}{I_{2}^{k}}$ an isomorphism of these algebras.

######  Definition 8.2.

Since $N_{X}C_{j}$ 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 $\tilde{\theta}^{*}\omega_{1}=\omega_{2}$ where the isomorphism
$\tilde{\theta}:C_{2}\to C_{1}$ is induced by $\theta$. Choose a section
$\mathop{\rm pr}_{j}:TX|_{C_{j}}\to TC_{j}$ of the canonical inclusion
$TC_{j}\hookrightarrow TX|_{C_{j}}$ and consider the section
$\tilde{\omega}_{j}=\omega_{j}\circ\mathop{\rm pr}_{j}$ of the dual bundle
$(TX|_{C_{j}})^{\vee}$ of $TX|_{C_{j}}$. 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 the matrix $\left[\frac{\partial\theta(u_{l,1})}{\partial
u_{m,2}}\right]_{l,m=1}^{n-1}$ 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 again that $\theta$ has Jacobian 1 if the determinant of
$\left[\frac{\partial\theta(u_{l,1})}{\partial u_{m,2}}\right]_{l,m=1}^{n-1}$
is equal to 1 modulo $I_{2}^{k-1}$.

The next fact was is a straightforward consequence of Theorem 7.1.

######  Theorem 8.3.

Let the asumptions of Theorem 7.1 be satisfied with $X$ being a smooth quasi-
affine variety equipped with a volume form. Let $\operatorname{{\rm
Aut}}^{1}(X)\subset\operatorname{{\rm Aut}}(X)$ be the subgroup of
automorphisms of $X$ that have Jacobian 1 and let $Y$ be an affine variety
containing $X$ as an open subset. Suppose that ${\mathcal{F}}$ is a family of
smooth curves in $X$ with trivial normal bundles such that each
$C\in{\mathcal{F}}$ is closed in $Y$ and every isomorphism between
$C_{1}\in{\mathcal{F}}$ and $C_{2}\in{\mathcal{F}}$ extends to an automorphism
from $\operatorname{{\rm Aut}}^{1}(X)$ (resp. $\mathop{\rm SAut}(X)$). Then
every isomorphism $\theta:\frac{{\rm\bf k}[X]}{I_{1}^{k}}\to\frac{{\rm\bf
k}[X]}{I_{2}^{k}}$ with the Jacobian 1 is $\operatorname{{\rm
Aut}}^{1}(X)$-induced (resp. $\mathop{\rm SAut}(X)$-induced).

###### Proof.

Let $\theta_{1}:\frac{{\rm\bf k}[X]}{I_{1}}\to\frac{{\rm\bf k}[X]}{I_{2}}$ be
the isomorphism generated by $\theta$. By the assumption $\theta_{1}^{-1}$ is
generated by some automorphism $\Phi\in G$ such that $\Phi(C_{2})=C_{1}$. By
Lemma 6.2 $\Phi$ generates an isomorphism
$\varphi:\frac{A}{I_{2}^{k}}\to\frac{A}{I_{1}^{k}}$ with Jacobian 1. Hence,
$\varphi\circ\theta:\frac{{\rm\bf k}[X]}{I_{1}^{k}}\to\frac{{\rm\bf
k}[X]}{I_{1}^{k}}$ has Jacobian 1 and by Theorem 7.1 it is generated by an
automorphism $\Psi\in G$. Hence, $\theta$ is induced by $\Phi^{-1}\circ\Psi$
which is the desired conclusion. ∎

######  Remark 8.4.

It is worth mentioning that in the case when $C_{1}=C_{2}$ we do not have to
require in Theorem 8.3 that $\theta$ is an $A_{1}$-automorphism as we did in
Theorem 7.1.

######  Example 8.5.

Let Notation 8.1 hold and $\varphi:C_{1}\to C_{2}$ be any isomorphism of
smooth polynomial curves.

(1) Consider the case of $X=\mathop{\rm SL}_{n}({\rm\bf k}),\,n\geq 3$. By van
Santen’s theorem [St] (see also [Ka20]) there is an automorphism
$\alpha\in\operatorname{{\rm Aut}}^{1}(X)$ such that $\alpha|_{C_{1}}=\varphi$
and also $X$ has a volume form.

(2) More generally, let $X$ be a connected complex algebraic group of
dimension at least 4 without nontrivial characters. The absence of characters
implies that the natural action of $\mathop{\rm SAut}(X)$ on $X$ is transitive
and, therefore, $X$ is a flexible variety ([AFKKZ, Theorem 0.1]). It has also
a left invariant volume form. The Feller-van Santen theorem [FS] states that