# Tangent cones at infinity

Luis Renato G. Dias  Instituto de Matemática e Estatística, Universidade
Federal de Uberlândia, Av. João Naves de Ávila 2121, 1F-153 - CEP: 38408-100,
Uberlândia, Brazil<EMAIL_ADDRESS>and  Nilva Rodrigues Ribeiro  Universidade
Federal de Viçosa, Campus Rio Paranaíba, Km 7, Zona Rural, MG - 230 Rodoviário
- CEP: 38810-000, Rio Paranaíba, Brazil<EMAIL_ADDRESS>
###### Abstract.

Let $X\subset{\mathbb{C}}^{m}$ be an unbounded pure $k$-dimensional algebraic
set. We define the tangent cones $C_{4,\infty}(X)$ and $C_{5,\infty}(X)$ of
$X$ at infinity. We establish some of their properties and relations. We prove
that $X$ must be an affine linear subspace of ${\mathbb{C}}^{m}$ provided that
$C_{5,\infty}(X)$ has pure dimension $k$. Also, we study the relation between
the tangent cones at infinity and representations of $X$ outside a compact set
as a branched covering. Our results can be seen as versions at infinity of
results of Whitney and Stutz.

###### Key words and phrases:

Whitney tangent cones at infinity, branched covering, algebraic set

###### 2020 Mathematics Subject Classification:

14B05, 58K30, 14R10, 32S20

##  1\. Introduction

Let $(V,0)\subset{\mathbb{C}}^{m}$ be an $n$-dimensional analytic set in some
neighborhood of the origin in ${\mathbb{C}}^{m}$. In [9, 10], Whitney
considered six tangent cones $C_{i}(V,0)$, $i=1,\ldots,6$, and, since then,
their properties and relations have been studied. The precise definitions for
the six cones can be seen in [1, p. 92], [9] and [10]. The cone $C_{3}(V,0)$
is also known as the Zariski tangent cone of $V$.

The Whitney tangent cones $C_{3}(V,0),C_{4}(V,0)$ and $C_{5}(V,0)$ play an
important role in various results. For instance, it is well known that
$C_{3}(V,0)$ determines the set of all projections $\pi$ of $(V,0)$ to
${\mathbb{C}}^{n}$ that are a branched cover when restricted to a small
neighborhood of $0\in V$. In [8], Stutz showed that if $\dim C_{4}(V,0)=n$,
then $\pi$ can be chosen such that the critical locus of $\pi$ is exactly the
singular set of $V$. Also in [8], Stutz proved that if $\dim C_{5}(V,0)=n+1$,
then a suitable choice of a projection of $(V,0)$ to ${\mathbb{C}}^{n+1}$
gives a representation of $V$ in terms of an hypersurface in
${\mathbb{C}}^{n+1}$; see also [1] and [6].

In this paper, we define the Whitney tangent cones at infinity
$C_{4,\infty}(X)$ and $C_{5,\infty}(X)$ for an unbounded pure $k$-dimensional
algebraic set $X\subset{\mathbb{C}}^{m}$ and we also consider the Whitney
tangent cone at infinity $C_{3,\infty}(X)$ defined in [4] as follows:

######  Definition 1.1.

Let $X\subset{\mathbb{C}}^{m}$ be an unbounded pure $k$-dimensional algebraic
set.

  * $\bullet$

The Whitney tangent cone $C_{3,\infty}(X)$ of $X$ at infinity is defined as
the set of the points $v\in{\mathbb{C}}^{m}$ for which there exist sequences
$(p_{j})_{j}$ in $X$ and $(t_{j})_{j}$ in ${\mathbb{C}}$ such that
$\lim_{j\to\infty}\|p_{j}\|=\infty$ and $\lim_{j\to\infty}t_{j}p_{j}=v$.

  * $\bullet$

The Whitney tangent cone $C_{4,\infty}(X)$ of $X$ at infinity is defined as
the set of the points $v\in{\mathbb{C}}^{m}$ for which there exists a sequence
$(p_{j})_{j}$ in $X\setminus{\rm Sing}X$ and, for each $j$, there is a vector
$v_{j}\in T_{p_{j}}X$ such that $\lim_{j\to\infty}\|p_{j}\|=\infty$ and
$\lim_{j\to\infty}v_{j}=v$.

  * $\bullet$

The Whitney tangent cone $C_{5,\infty}(X)$ of $X$ at infinity is defined as
the set of points $v\in{\mathbb{C}}^{m}$ for which there exist sequences
$(p_{j})_{j}$ and $(q_{j})_{j}$ in $X$ and $(t_{j})_{j}$ in ${\mathbb{C}}$
such that $\lim_{j\to\infty}\|p_{j}\|=\lim_{j\to\infty}\|q_{j}\|=\infty$ and
$\lim_{j\to\infty}t_{j}(p_{j}-q_{j})=v$.

By Definition 1.1, for $i=3,4,5$, the sets
$C_{i,\infty}(X)\subset{\mathbb{C}}^{m}$ are closed and they are complexly
homogeneous in the sense that for any $v\in C_{i,\infty}(X)$ and
$\lambda\in{\mathbb{C}}$ the vectors $\lambda v$ also belong to
$C_{i,\infty}(X)$.

In section §2, we prove basic properties of $C_{4,\infty}(X)$ and
$C_{5,\infty}(X)$. We prove that $C_{4,\infty}(X)$ and $C_{5,\infty}(X)$ are
algebraic sets and that $C_{3,\infty}(X)\subset C_{4,\infty}(X)\subset
C_{5,\infty}(X)$. Moreover, we prove that $\dim C_{5,\infty}(X)\leq 2k+1$.
Then, since $\dim C_{3,\infty}(X)=k$ (see [2, Prop. 2.10]), it follows that
$k\leq\dim C_{4,\infty}(X)\leq\dim C_{5,\infty}(X)\leq 2k+1$; see Propositions
2.1, 2.2 and 2.3.

In Section 3, we prove that any pure $k$-dimensional algebraic set
$X\subset{\mathbb{C}}^{m}$ for which $C_{5,\infty}(X)$ has pure dimension $k$
must be an affine linear subspace of ${\mathbb{C}}^{m}$; see Theorem 3.5.

The aim of Section 4 is to give versions at infinity of the above results of
Stutz for complex algebraic set in ${\mathbb{C}}^{m}$ as follows. Let
$X\subset{\mathbb{C}}^{m}$ be an unbounded pure $k$-dimensional algebraic set.
Let $W\subset{\mathbb{C}}^{m}$ be a fixed $m-k$ dimensional vector space such
that $W\cap C_{3,\infty}(X)=\\{0\\}$ (this always exists by Lemma 4.2). We may
use the coordinates $(x,y)\in V\times W={\mathbb{C}}^{m}$, where $V$ and $W$
are orthogonal. Let $\pi:X\to V$ be the restriction to $X$ of the canonical
projection $(x,y)\in V\times W\to x\in V$. We denote by $B_{R}$ the ball in
${\mathbb{C}}^{m}$ of radius $R$ centered at the origin. We have:

######  Theorem 1.2.

Suppose $C_{4,\infty}(X)\cap W=\\{0\\}$. Then, there exists a sufficiently
large $R$ such that ${\rm Sing}X\setminus\bar{B}_{R}={\rm
Sing}\,\pi\setminus\bar{B}_{R}$.

By a similar argument from the proof of Lemma 4.2, it follows that $\dim
C_{4,\infty}(X)=k$ forces $C_{4,\infty}(X)\cap W=\\{0\\}$.

We denote $y=(y_{1},\ldots,y_{m-k})$ and
$W^{i}:=\\{(0,y)\in{\mathbb{C}}^{m}\mid y_{i}=0\\},$ for $i=1,\ldots,m-k$. We
denote $\pi_{i}:{\mathbb{C}}^{m}=V\times
W\to{\mathbb{C}}^{k}\times{\mathbb{C}}={\mathbb{C}}^{k+1}$ the projection
defined by $\pi_{i}(x,y)=(x,y_{i})$. It follows that $\ker\pi_{i}=W^{i}$. We
have:

######  Theorem 1.3.

Suppose that $C_{5,\infty}(X)\cap W^{i}=\\{0\\}$. Then $\pi_{i}(X)$ is an
algebraic hypersurface in ${\mathbb{C}}^{k+1}$ and, for sufficiently large
$R$, the restriction
$\pi_{i}:X\setminus\bar{B}_{R}\to\pi_{i}(X\setminus\bar{B}_{R})$ is a
homeomorphism.

By a similar argument from the proof of Lemma 4.2, it follows that for the
hypothesis of Theorem 1.3 to be satisfied, it is necessary that $\dim
C_{5,\infty}(X)\leq k+1$.

##  2\. Properties of the tangent cones at infinity

######  Proposition 2.1.

Let $X\subset{\mathbb{C}}^{m}$ be an unbounded pure $k$-dimensional algebraic
set. Then, $C_{3,\infty}(X)\subset C_{4,\infty}(X)\subset C_{5,\infty}(X)$.

###### Proof.

Firstly, we prove that $C_{3,\infty}(X)\subset C_{4,\infty}(X)$.

Let $v\in C_{3,\infty}(X)$. By Definition 1.1, $0\in C_{3,\infty}(X)\cap
C_{4,\infty}(X)$. So it is sufficient to consider $v\neq 0$.

By Definition 1.1, there exist sequences $(p_{j})_{j}$ in $X$ and
$(t_{j})_{j}$ in ${\mathbb{C}}$ such that $\lim_{j\to\infty}\|p_{j}\|=\infty$
and $\lim_{j\to\infty}t_{j}p_{j}=v$. It is obvious that
$\lim_{j\to\infty}t_{j}=0.$

Since $X\setminus{\rm Sing}X$ is dense on $X$, there exists a sequence
$(q_{j})_{j}$ in $X\setminus{\rm Sing}X$ such that
$\|p_{j}-q_{j}\|<\frac{1}{j}$. Then, by triangle inequality, we have
$\|p_{j}\|\leq\|p_{j}-q_{j}\|+\|q_{j}\|\leq\frac{1}{j}+\|q_{j}\|.$ Since
$\lim_{j\to\infty}\|p_{j}\|=\infty$, also $\lim_{j\to\infty}\|q_{j}\|=\infty$.

Moreover, by triangle inequality, we have:

$\|t_{j}q_{j}-v\|\leq\|t_{j}q_{j}-t_{j}p_{j}\|+\|t_{j}p_{j}-v\|\leq|t_{j}|\,\|q_{j}-p_{j}\|+\|t_{j}p_{j}-v\|.$

Since both members of the right side converge to $0$, we have
$\lim_{j\to\infty}t_{j}q_{j}=v.$

Since $q_{j}\in X\setminus{\rm Sing}X$, it follows by the Curve Selection
Lemma that there exist analytic curves $\psi:(0,\epsilon)\to(X\setminus{\rm
Sing}X)$ and $t:(0,\epsilon)\to{\mathbb{C}}$ such that $\lim_{s\to
0}\|\psi(s)\|=\infty$ and $\lim_{s\to 0}t(s)\psi(s)=v$.

Now we can use the same arguments from the proof of Lemma 2.2 of [4] and we
get an analytic curve $\varphi$ which is a reparametrization of $\psi$ that
satisfies:

  1. a)

$\varphi(s)\in X\setminus{\rm Sing}X$, for any $s$.

  2. b)

$\lim_{s\to 0}\|\varphi(s)\|=\infty.$

  3. c)

$\lim_{s\to 0}\frac{\varphi(s)}{\|\varphi(s)\|}=-\lim_{s\to
0}\frac{\varphi^{\prime}(s)}{\|\varphi^{\prime}(s)\|}=\frac{v}{\|v\|}.$

It follows by a), b) and c) that $-\frac{v}{\|v\|}\in C_{4,\infty}(X)$. Since
$C_{4,\infty}(X)$ is a cone, it follows that $v\in C_{4,\infty}(X)$.

Now, we prove that $C_{4,\infty}(X)\subset C_{5,\infty}(X)$. Let $v\in
C_{4,\infty}(X)$. By the definition of $C_{4,\infty}(X)$, there exists a
sequence $(p_{j})_{j}$ in $X\setminus{\rm Sing}X$ and, for each $j$, a vector
$v_{j}\in T_{p_{j}}X$, such that $\lim_{j\to\infty}\|p_{j}\|=\infty$ and
$\lim_{j\to\infty}v_{j}=v.$

It follows from the definition of $T_{p_{j}}X$ that, for any $j$, there exists
$q_{j}$ in $X\setminus{\rm Sing}X$ and $t_{j}$ in ${\mathbb{R}}$ such that
$\|p_{j}-q_{j}\|\leq\frac{1}{j}$, $t_{j}\leq\frac{1}{j}$, and

(1) $\|\frac{p_{j}-q_{j}}{t_{j}}-v_{j}\|\leq\frac{1}{j}.$

It follows by triangle inequality that
$\|p_{j}\|\leq\|p_{j}-q_{j}\|+\|q_{j}\|\leq\frac{1}{j}+\|q_{j}\|.$ Since
$\lim_{j\to\infty}\|p_{j}\|=\infty$, we have
$\lim_{j\to\infty}\|q_{j}\|=\infty$. It follows by (1) and
$\lim_{j\to\infty}v_{j}=v$ that

$\lim_{j\to\infty}\frac{1}{t_{j}}(p_{j}-q_{j})=v,$

which shows that $v\in C_{5,\infty}(X)$. ∎

The next two propositions are motivated by the following facts: by Corollary
3.1 of [4], $C_{3,\infty}(X)$ is an algebraic set; by Proposition 2.10 of [2],
$\dim C_{3,\infty}(X)=\dim X$.

We denote the complex projective space of dimension $m$ by ${\mathbb{P}}^{m}$.
We identify ${\mathbb{C}}^{m}$ with the open set
$\\{[x_{0}:x_{1}:\ldots:x_{m}]\in{\mathbb{P}}^{m}\mid x_{0}\neq 0\\}$ of
${\mathbb{P}}^{m}$ through the map $\tau:{\mathbb{C}}^{m}\to{\mathbb{P}}^{m}$,
$\tau(x_{1},\ldots,x_{m})=[1:x_{1}:\ldots:x_{m}]$. The hyperplane at infinity
is defined by
$H_{\infty}:=\\{[x_{0}:x_{1}:\ldots:x_{m}]\in{\mathbb{P}}^{m}\mid x_{0}=0\\}$.

######  Proposition 2.2.

Let $X\subset{\mathbb{C}}^{m}$ be an unbounded pure $k$-dimensional algebraic
set. Then, $C_{5,\infty}(X)$ is an algebraic set of dimension less than or
equal to $2k+1$.

###### Proof.

Let $A=\\{(x,y,v)\in X\times X\times{\mathbb{C}}^{m}\mid(x-y)\wedge v=0\\}$.
We have that $A$ is an algebraic set of
${\mathbb{C}}^{m}\times{\mathbb{C}}^{m}\times{\mathbb{C}}^{m}$. We consider
$\sigma:A\to X\times X$, defined by $\sigma(x,y,v)=(x,y)$.

We denote by $\Delta\subset X\times X$ the subset of $X\times X$ such that
$(x,y)\in\Delta$ if and only if $x=y$. For any $(x,y)\notin\Delta$, the fiber
$\sigma^{-1}(x,y)$ has dimension equal to $1$ (in fact, $\sigma^{-1}(x,y)$ is
the set of the vectors that are complex proportional to $x-y$). Therefore,
$\dim A=2q+1$.

We consider $A$ and $\sigma^{-1}(\Delta)$ as subsets of
${\mathbb{C}}^{m}\times{\mathbb{C}}^{m}\times{\mathbb{C}}^{m}\subset{\mathbb{P}}^{m}\times{\mathbb{P}}^{m}\times{\mathbb{C}}^{m}$.
Let $\mathcal{A}:=\overline{A\setminus\sigma^{-1}(\Delta)}$ be the closure of
$A\setminus\sigma^{-1}(\Delta)$ in
${\mathbb{P}}^{m}\times{\mathbb{P}}^{m}\times{\mathbb{C}}^{m}$, which is an
algebraic set. We have $\dim\mathcal{A}=2k+1$.

Let $Z:=\mathcal{A}\cap(H_{\infty}\times H_{\infty}\times{\mathbb{C}}^{m})$
and $\tau:Z\to{\mathbb{C}}^{m}$ defined by $\tau(x,y,v)=v$. It follows that
$Z$ is an algebraic set and by definition of $C_{5,\infty}(X)$, we have
$\tau(Z)=C_{5,\infty}(X)$. This shows that $C_{5,\infty}(X)$ is a
constructible set. The proof that $C_{5,\infty}(X)$ is closed in
${\mathbb{C}}^{m}$ is straightforward (diagonal argument). Therefore,
$C_{5,\infty}(X)$ is an algebraic set.

Since $\dim\mathcal{A}=2k+1$, it follows that $\dim Z\leq 2k+1$. Therefore,
the dimension of $\tau(Z)=C_{5,\infty}(X)$ is less than or equal to $2k+1$. ∎

######  Proposition 2.3.

Let $X\subset{\mathbb{C}}^{m}$ be an unbounded pure $k$-dimensional algebraic
set. Then, $C_{4,\infty}(X)$ is an algebraic set and $k\leq\dim
C_{4,\infty}(X)\leq 2k+1$.

###### Proof.

Let $\mathcal{I}(X)$ be the vanishing ideal of $X$ and let
$\\{f_{1},\ldots,f_{l}\\}$ be a set of generators of $\mathcal{I}(X)$. We
consider $F=(f_{1},\ldots,f_{l}):{\mathbb{C}}^{m}\to{\mathbb{C}}^{l}$. For any
$p\in X\setminus{\rm Sing}X$, we have $\dim\ker d_{p}F=k$ and $\ker
d_{p}F=T_{p}X$, where $d_{p}F$ denotes the differential of $F$ at $p$; see
section 2 of [5].

Let $A=\\{(p,v)\in X\times{\mathbb{C}}^{m}\mid v\in\ker d_{p}F\\}$. Then $A$
is an algebraic set of ${\mathbb{C}}^{m}\times{\mathbb{C}}^{m}$. Since ${\rm
Sing}X$ is an algebraic set, it follows that ${\rm
Sing}X\times{\mathbb{C}}^{m}$ is an algebraic set of
${\mathbb{C}}^{m}\times{\mathbb{C}}^{m}$. We consider $A$ and ${\rm
Sing}X\times{\mathbb{C}}^{m}$ as subsets of
${\mathbb{C}}^{m}\times{\mathbb{C}}^{m}\subset{\mathbb{P}}^{m}\times{\mathbb{C}}^{m}$.
Let $\mathcal{B}:=\overline{A\setminus({\rm Sing}X\times{\mathbb{C}}^{m}})$ be
the closure of $A\setminus({\rm Sing}X\times{\mathbb{C}}^{m})$ in
${\mathbb{P}}^{m}\times{\mathbb{C}}^{m}$, which is an algebraic set.

Let $Z:=\mathcal{B}\cap(H_{\infty}\times{\mathbb{C}}^{m})$ and let
$\tau:Z\to{\mathbb{C}}^{m}$ be the restriction to $Z$ of the projection
$(x,v)\in{\mathbb{P}}^{m}\times{\mathbb{C}}^{m}\to v\in{\mathbb{C}}^{m}$. Then
$Z$ is an algebraic set and, by the definition of $C_{4,\infty}(X)$, we have
$\tau(Z)=C_{4,\infty}(X)$. This shows that $C_{4,\infty}(X)$ is a
constructible set. The proof that $C_{4,\infty}(X)$ is closed in
${\mathbb{C}}^{m}$ is straightforward (diagonal argument). Therefore,
$C_{4,\infty}(X)$ is an algebraic set.

By Proposition 2.10 of [2], $\dim C_{3,\infty}(X)=\dim X$. By Proposition 2.2,
$\dim C_{5,\infty}(X)\leq 2k+1$. It follows by Proposition 2.1 that $k\leq\dim
C_{4,\infty}(X)\leq 2k+1$, which completes the proof. ∎

The above results can be seen as versions at infinity of results of Whitney on
tangent cones at a point of an analytic variety; see [1], [9] and [10].

##  3\. On the dimension of $\dim C_{5,\infty}(X)$

The main result of this section is Theorem 3.5. We begin by recalling the
definition of an algebraic region.

######  Definition 3.1 (See p. 672 of [7]).

Let $m\geq 2$. A set $\Omega\subset{\mathbb{C}}^{m}$ is called an algebraic
region of type $(k,m)$ if there exist vector subspaces $V_{1},V_{2}$ in
${\mathbb{C}}^{m}$ and positive real numbers $A,B$ such that the following
conditions hold: $\dim V_{1}=k$, $\dim V_{2}=m-k$,
${\mathbb{C}}^{m}=V_{1}\oplus V_{2}$ and $\Omega$ consists of those
$z\in{\mathbb{C}}^{m}$ for which:

(2) $\|z^{\prime\prime}\|\leq A(1+\|z^{\prime}\|)^{B},$

where $z=z^{\prime}+z^{\prime\prime}$, with $z^{\prime}\in V_{1}$ and
$z^{\prime\prime}\in V_{2}$.

We have the following geometric property for algebraic sets:

######  Theorem 3.2 (See Theorem 2 of [7]).

A closed analytic subset $V\subset{\mathbb{C}}^{m}$ of pure dimension $k$ is
algebraic if and only if $V$ lies in some algebraic region $\Omega$ of type
$(k,m)$.

The next result is motivated by Theorem 2.9 of [2] and it is useful for the
proof of Theorem 3.5. We remember that by Proposition 2.2 the cone
$C_{5,\infty}(X)$ is an algebraic set.

######  Proposition 3.3.

Let $X\subset{\mathbb{C}}^{m}$ be a pure $k$-dimensional algebraic set.
Suppose $C_{5,\infty}(X)$ has a pure dimension $k$. Then there exists an
algebraic region $\Omega$ of type $(k,m)$ such that $X$, $C_{3,\infty}(X)$ and
$C_{5,\infty}(X)$ lie in $\Omega$.

###### Proof.

We follow the proof of Theorem 2.9 of [2].

It follows by Theorem 3.2 that there exist an algebraic region
$\tilde{\Omega}$ of type $(k,m)$ and vector spaces $V_{1}$ and $V_{2}$ in
${\mathbb{C}}^{m}$, with $\dim V_{1}=k$, $\dim V_{2}=m-k$,
${\mathbb{C}}^{m}=V_{1}\oplus V_{2}$, and positive real numbers $\tilde{A},B$
such that for any $z\in\tilde{\Omega}$ (hence, in particular, for $z\in
C_{5,\infty}(X)$) we have:

(3) $\|z^{\prime\prime}\|<\tilde{A}(1+\|z^{\prime}\|)^{B},$

where $z=z^{\prime}+z^{\prime\prime}$, with $z^{\prime}\in V_{1}$ and
$z^{\prime\prime}\in V_{2}$.

We assume in (3) that $B\geq 1$ since $(1+t)^{s}\leq(1+t)$, for any $t\geq 0$
and $0\leq s\leq 1$.

We claim that there exists a positive real number $R$ such that, for any $w\in
X$, we have:

(4) $\|w^{\prime\prime}\|<R(1+\|w^{\prime}\|)^{B},$

where $w=w^{\prime}+w^{\prime\prime}$, with $w^{\prime}\in V_{1}$ and
$w^{\prime\prime}\in V_{2}$.

If the claim is not true, then there exists a sequence $(w_{j})_{j}$ in $X$
such that $\|w_{j}^{\prime\prime}\|>j(1+\|w_{j}^{\prime}\|)^{B}$ and, up to a
subsequence, we may suppose that
$\lim_{j\to\infty}\frac{w_{j}^{\prime\prime}}{\|w_{j}^{\prime\prime}\|}=y_{0}\in
V_{2}$. We have $\lim_{j\to\infty}\|w_{j}^{\prime\prime}\|=\infty$. Since
${\mathbb{C}}^{m}$ is the direct sum of $V_{1}$ and $V_{2}$,
$\lim_{j\to\infty}\|w_{j}\|=\infty$. Then

$\frac{1}{j}>\frac{(1+\|w_{j}^{\prime}\|)^{B}}{\|w_{j}^{\prime\prime}\|}\geq\frac{(1+\|w_{j}^{\prime}\|)}{\|w_{j}^{\prime\prime}\|}\geq\frac{\|w_{j}^{\prime}\|}{\|w_{j}^{\prime\prime}\|}.$

Let $t_{j}:=(\tilde{A})^{-1}\|w_{j}^{\prime\prime}\|$. We have
$\lim_{j\to\infty}t_{j}=\infty$ and

$\lim_{j\to\infty}\frac{1}{t_{j}}w_{j}=\lim_{j\to\infty}\frac{1}{t_{j}}\left(w_{j}^{\prime}+w_{j}^{\prime\prime}\right)=\tilde{A}y_{0}.$

It follows that $\tilde{A}y_{0}\in C_{3,\infty}(X)$ by Definition 1.1. Now,
since $V_{2}$ is a linear space and $y_{0}\in V_{2}$, we have
$\tilde{A}y_{0}\in V_{2}\cap C_{3,\infty}(X)$. By Proposition 2.1, it follows
that $\tilde{A}y_{0}\in V_{2}\cap C_{5,\infty}(X)$. This is a contradiction,
because $\tilde{A}y_{0}$ does not verify (3).

Therefore, there exists a positive real number $R$ such that, for any $w\in
X$,

$\|w^{\prime\prime}\|<R(1+\|w^{\prime}\|)^{B},$

where $w=w^{\prime}+w^{\prime\prime}$, with $w^{\prime}\in V_{1}$ and
$w^{\prime\prime}\in V_{2}$. Let $A:={\rm max}\\{R,\tilde{A}\\}$, then $X$ and
$C_{5,\infty}(X)$ lie in an algebraic region $\Omega$ of type $(k,m)$, with
vector spaces $V_{1}$ and $V_{2}$ in ${\mathbb{C}}^{m}$, with $\dim V_{1}=k$,
$\dim V_{2}=m-k$ and positive real numbers $A$ and $B$. Since
$C_{3,\infty}(X)\subset C_{5,\infty}(X)$, it follows that $C_{3,\infty}(X)$
lies in $\Omega$. This finishes the proof. ∎

In the proof of Theorem 3.5, we use the definition of the degree of an
algebraic set. We follow section 11.3 of [1]. As before, the complex
projective space is denoted by ${\mathbb{P}}^{m}$ and ${\mathbb{C}}^{m}$ is
identified with the open set
$\\{[x_{0}:x_{1}:\ldots:x_{m}]\in{\mathbb{P}}^{m}\mid x_{0}\neq 0\\}$ of
${\mathbb{P}}^{m}$ through the map $\tau:{\mathbb{C}}^{m}\to{\mathbb{P}}^{m}$,
$\tau(x_{1},\ldots,x_{m})=[1:x_{1}:\ldots:x_{m}]$.

######  Definition 3.4.

Let $X\subset{\mathbb{C}}^{m}$ be an algebraic set. The degree of $X$, denoted
by $\deg X$, is the degree of its closure in ${\mathbb{P}}^{m}$.

The next one is the main result of this section:

######  Theorem 3.5.

Let $X\subset{\mathbb{C}}^{m}$ be a pure $k$-dimensional algebraic set.
Suppose $C_{5,\infty}(X)$ has a pure dimension $k$. Then $X$ is an affine
space.

###### Proof.

By Proposition 3.3, there exists an algebraic region $\Omega$ of type $(k,m)$
such that $X$, $C_{3,\infty}(X)$ and $C_{5,\infty}(X)$ lie in $\Omega$. Then,
there exist vector spaces $V_{1},V_{2}\subset{\mathbb{C}}^{m}$ and constants
$A,B$ as in Definition 3.1. We may choose linear coordinates $(x,y)$ in
${\mathbb{C}}^{m}$ such that $x\in V_{1}$ and $y\in V_{2}$. In this coordinate
system, we denote by $\pi_{\Omega}$ the canonical projection from
${\mathbb{C}}^{m}$ to $V_{1}$, i.e., $\pi_{\Omega}(x,y)=x$. Moreover, we may
assume that $V_{1}$ and $V_{2}$ are orthogonal to each other and that
$\pi_{\Omega}$ is an orthogonal projection; see Theorem 3 of page 78 of [1].

We have the following direct fact concerning the projection $\pi_{\Omega}$:

######  Lemma 3.6.

The restrictions $\pi_{\Omega}:X\to V_{1}$ and
$\pi_{\Omega}:C_{5,\infty}(X)\to V_{1}$ are proper maps.

$\square$

Then, it follows that the closures of $X$ and $V_{2}$ (respectively, the
closures of $C_{5,\infty}(X)$ and $V_{2}$) in ${\mathbb{P}}^{m}$ do not have
points at infinity in common (here, the phrase “at infinity” means the set
${\mathbb{P}}^{m}\setminus{\mathbb{C}}^{m}$). Then, it follows by Corollary 1
of [1, p. 126] the following fact:

######  Lemma 3.7.

The restriction $\pi_{\Omega}:X\to V_{1}$ $($respectively,
$\pi_{\Omega}:C_{5,\infty}(X)\to V_{1}$$)$ is a ramified covering with number
of sheets equal to $\deg X$ $($respectively, $\deg C_{5,\infty}(X))$.

$\square$

Since $C_{5,\infty}(X)$ is a cone, it follows by Lemma 3.6 that:

(5) $\ker\pi_{\Omega}\cap C_{5,\infty}(X)=\\{0\\}.$

We will show that $\deg X=1$. Let $\Sigma\subset V_{1}$ be the critical values
of $\pi_{\Omega}:X\to V_{1}$ and let $v\notin C_{3,\infty}(\Sigma)$. It
follows that $tv\notin\Sigma$, for $t>0$ big enough.

If $\deg X>1$, then there are two different liftings
$\lambda_{1}(t),\lambda_{2}(t)$ of the path $tv$ by $\pi_{\Omega}:X\to V_{1}$,
with $t>0$ big enough. So
$\pi_{\Omega}(\lambda_{1}(t))=\pi_{\Omega}(\lambda_{2}(t))=tv$.

It follows that

(6) $\left(\lambda_{1}(t)-\lambda_{2}(t)\right)\in\ker\pi_{\Omega}.$

We may assume that there is a sequence $(t_{j})_{j}$ in ${\mathbb{R}}$ such
that $\lim_{j\to\infty}t_{j}=\infty$ and

(7)
$w:=\lim_{j\to\infty}\frac{\lambda_{1}(t_{j})-\lambda_{2}(t_{j})}{\|\lambda_{1}(t_{j})-\lambda_{2}(t_{j})\|}.$

It follows that $w\in\ker\pi_{\Omega}$. We have
$\lim_{j\to\infty}\|\lambda_{1}(t_{j})\|=\lim_{j\to\infty}\|\lambda_{2}(t_{j})\|=\infty$
since $\pi_{\Omega}$ is an orthogonal projection. Hence

$w\in\ker\pi_{\Omega}\cap C_{5,\infty}(X).$

But this is a contradiction with (5) since $w\neq 0$.

Thus, $\deg X=1$. By Proposition 3.3 of [3], we have that $X$ is an affine
space. ∎

##  4\. Tangent cones and projections

Let $X\subset{\mathbb{C}}^{m}$ be an unbounded pure $k$-dimensional algebraic
set fixed. We begin with the following definition:

######  Definition 4.1.

We say that an $m-k$ dimensional vector space $W\subset{\mathbb{C}}^{m}$ is
transverse to $X$ provided that $W\cap C_{3,\infty}(X)=\\{0\\}$.

The existence of such transverse vector spaces is shown in the next result.

######  Lemma 4.2.

There exists an $m-k$ dimensional vector space transverse to $X$.

###### Proof.

By Theorem 2.9 of [2], there exists an algebraic region $\Omega$ of type
$(k,m)$ such that $X$ and $C_{3,\infty}(X)$ lie in $\Omega$. Then, there exist
vector spaces $V_{1}$ and $V_{2}$ in ${\mathbb{C}}^{m}$, with $\dim V_{1}=k$,
$\dim V_{2}=m-k$, ${\mathbb{C}}^{m}=V_{1}\times V_{2}$, and positive real
numbers $A,B$ such that for any $z\in\Omega$ (hence, in particular, for $z\in
X\cup C_{3,\infty}(X)$) we have:

(8) $\|z^{\prime\prime}\|<A(1+\|z^{\prime}\|)^{B},$

where $z=z^{\prime}+z^{\prime\prime}$, with $z^{\prime}\in V_{1}$ and
$z^{\prime\prime}\in V_{2}$.

Since $C_{3,\infty}(X)$ is complexly homogeneous in the sense that for any
$v\in C_{3,\infty}(X)$ and $\lambda\in{\mathbb{C}}$ the vector $\lambda v$
also belong to $C_{3,\infty}(X)$, it follows that $V_{2}\cap
C_{3,\infty}(X)=\\{0\\}$. So, $V_{2}$ is transverse to $X$. ∎

Let $W\subset{\mathbb{C}}^{m}$ be a fixed $m-k$ dimensional vector space
transverse to $X$. This always exists by Lemma 4.2. We may use the coordinates
$(x,y)\in V\times W={\mathbb{C}}^{m}$, where $V$ and $W$ are orthogonal.

Let $\pi:V\times W\to V$ be the canonical projection, $\pi(x,y)=x$. Then:

######  Lemma 4.3.

The restriction $\pi:X\to V$ is a proper map.

###### Proof.

We claim that there exist positive constants $C$ and $R$ such that

$X\subset\\{(x,y)\in{\mathbb{C}}^{m}=V\times W\mid\|y\|<C\|x\|\\}\cup B_{R}.$

Indeed, if the claim is not true, there exists a sequence
$\\{(x_{j},y_{j})\\}_{j}$ in $X$ such that
$\lim_{j\to\infty}\|(x_{j},y_{j})\|=\infty$ and $\|y_{j}\|\geq j\|x_{j}\|$,
for any $j$. So $\lim_{j\to\infty}\frac{x_{j}}{\|y_{j}\|}=0$. Up to a
subsequence, we may assume that
$\lim_{j\to\infty}\frac{y_{j}}{\|y_{j}\|}=y_{0}$. Then, $y_{0}\in W$ and

$\lim_{j\to\infty}\frac{1}{\|y_{j}\|}(x_{j},y_{j})=(0,y_{0}).$

By Definition 1.1, it follows that $(0,y_{0})\in C_{3,\infty}(X)$. Hence
$(0,y_{0})\in C_{3,\infty}(X)\cap W$. This is a contradiction, because $W$ is
transverse to $X$. Therefore, the claim holds.

Now, it is a direct consequence of the claim that $\pi$ restricted to $X$ is a
proper map. ∎

It follows by Lemmma 4.3 that the closures of $X$ and $W$ in
${\mathbb{P}}^{m}$ do not have points at infinity in common (here, the phrase
“at infinity” means the set ${\mathbb{P}}^{m}\setminus{\mathbb{C}}^{m}$).
Then, it follows by Corollary 1 of [1, p. 126] the following fact:

######  Lemma 4.4.

The restriction $\pi:X\to V$ is a ramified covering with number of sheets
equal to $\deg X$. $\square$

Now we are ready to give the

###### Proof of Theorem 1.2.

The inclusion ${\rm Sing}X\subset{\rm Sing}\,\pi$ is always true.

Now, assume that there is no a positive constant $R$ such that ${\rm
Sing}\,\pi\setminus\bar{B}_{R}\subset{\rm Sing}X\setminus\bar{B}_{R}$. Then,
there exists a sequence $(p_{j})_{j}$ in ${\rm Sing}\,\pi\setminus{\rm Sing}X$
such that $\lim_{j\to\infty}\|p_{j}\|=\infty$.

Then, we can take $w_{j}\in T_{p_{j}}X$, $\|w_{j}\|=1$, such that
$d_{p_{j}}\pi(w_{j})=0$. Since $\pi$ is a linear map, we have
$d_{p_{j}}\pi(w_{j})=\pi(w_{j})$. Therefore, $w_{j}\in W=\ker\pi$, for any
$j$.

Up to a subsequence, we may assume that $\lim_{j\to\infty}w_{j}=w_{0}$. Then
$w_{0}\in W\cap C_{4,\infty}(X)$ which is a contradiction. ∎

We have fixed an $m-k$ plane $W$ transverse to $X$ and coordinates $(x,y)\in
V\times W={\mathbb{C}}^{m}$, where $V$ and $W$ are orthogonal. We denote
$y=(y_{1},\ldots,y_{m-k})$ and we set $W^{i}:=\\{(0,y)\in{\mathbb{C}}^{m}\mid
y_{i}=0\\},$ for $i=1,\ldots,m-k$. We denote $\pi_{i}:{\mathbb{C}}^{m}=V\times
W\to{\mathbb{C}}^{k}\times{\mathbb{C}}={\mathbb{C}}^{k+1}$ the projection
defined by $\pi_{i}(x,y)=(x,y_{i})$. So we have $\ker\pi_{i}=W^{i}$.

###### Proof of Theorem 1.3.

Since $C_{5,\infty}(X)\cap W^{i}=\\{0\\}$, it follows by Proposition 2.1 that
$C_{3,\infty}(X)\cap W^{i}=C_{4,\infty}(X)\cap W^{i}=\\{0\\}$.

The restriction of $\pi_{i}$ to $X$ is a proper map since $C_{3,\infty}(X)\cap
W^{i}=\\{0\\}$. In particular, $\pi_{i}:X\to{\mathbb{C}}^{k+1}$ is a closed
map. It follows that $\pi_{i}(X)\subset{\mathbb{C}}^{k+1}$ is a closed
constructible set and, therefore, $\pi_{i}(X)$ is an algebraic set of
dimension $k$. It follows that $\pi_{i}(X)$ is a hypersurface in
${\mathbb{C}}^{k+1}$.

Claim 1: for sufficiently large $R$, the restriction
$\pi_{i}:X\setminus\bar{B}_{R}\to\pi(X\setminus\bar{B}_{R})$ is an injective
map.

Indeed, if the claim does not hold, then there exist sequences $(p_{j})_{j}$
and $(q_{j})_{j}$ in $X$ such that $p_{j}\neq q_{j}$,
$\lim_{j\to\infty}\|p_{j}\|=\lim_{j\to\infty}\|q_{j}\|=\infty$ and
$\pi_{i}(p_{j})=\pi_{i}(q_{j})$. In particular, $p_{j}-q_{j}\in W^{i}$. Up to
a subsequence, we may assume that
$\lim_{j\to\infty}\frac{p_{j}-q_{j}}{\|p_{j}-q_{j}\|}=w_{0}$.

It follows that $w_{0}\neq 0$. By Definition 1.1, $w_{0}\in C_{5,\infty}(X)$.
Moreover, $w_{0}\in W^{i}$ since $p_{j}-q_{j}\in W^{i}$ for any $j$.

Then, $w_{0}\in W^{i}\cap C_{5,\infty}(X)$, which is a contradiction.
Therefore Claim 1 holds.

Claim 2: for sufficiently large $R$, the restriction
$\pi_{i}:X\setminus\bar{B}_{R}\to\pi(X\setminus\bar{B}_{R})$ is a proper map.

Suppose that for sufficiently large $n\in{\mathbb{N}}$, we have that
$\pi_{i}:X\setminus\bar{B}_{n}\to\pi(X\setminus\bar{B}_{n})$ is not proper.
Then, since $\pi_{i}:X\to{\mathbb{C}}^{k+1}$ is proper, there exists a
sequence $(p_{jn})_{j}$ in $X\setminus\bar{B}_{n}$ such that
$\lim_{j\to\infty}\pi_{i}(p_{jn})=y_{n}\in\pi(X\setminus\bar{B}_{n})$ and
$\lim_{j\to\infty}p_{jn}=p_{n}$, where $p_{n}\in X$ and $\|p_{n}\|=n$. Thus,
there exists $x_{n}\in X\setminus\bar{B}_{n}$ such that
$\pi_{i}(x_{n})=y_{n}$. In particular, $x_{n}\neq p_{n}$.

Then, up to a subsequence, we may assume that
$\lim_{n\to\infty}\frac{p_{n}-x_{n}}{\|p_{n}-x_{n}\|}=w$. It follows that
$w\neq 0$. Since $p_{n}-x_{n}\in W^{i}$ for any $n$, we have $w\in W^{i}$.

By construction,
$\lim_{n\to\infty}\|p_{n}\|=\lim_{n\to\infty}\|x_{n}\|=\infty$. Hence, $w\in
C_{5,\infty}(X).$

Then $w\in W^{i}\cap C_{5,\infty}(X)$, which is a contradiction. Therefore
Claim 2 holds.

It follows directly from Claim 1 and Claim 2 that
$\pi_{i}:X\setminus\bar{B}_{R}\to\pi(X\setminus\bar{B}_{R})$ is a
homeomorphism for sufficiently large $R$. ∎

## Acknowledgments

L. R. G. Dias was partially supported by Fapemig-Brazil Grant APQ-02085-21 and
by Grants 301631/2022-0 and 403959/2023-3 of the Conselho Nacional de
Desenvolvimento Científico e Tecnológico (CNPq) of the Ministry of Science,
Technology and Innovation of Brazil.

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