# On foliations by curves with singularities of positive dimension

Arturo Fernández-Pérez and Gilcione Nonato Costa  Departamento de Matemática -
ICEX, Universidade Federal de Minas Gerais, UFMG Av. Antônio Carlos 6627,
31270-901, Belo Horizonte-MG, Brasil<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We present enumerative results for holomorphic foliations by curves on
$\mathbb{P}^{n}$, $n\geq 3$, with singularities of positive dimension. Some of
the results presented improve previous ones due to Corrêa–Fernández-
Pérez–Nonato Costa–Vidal Martins [2] and Nonato Costa [11]. We also present an
enumerative result bounding the number of isolated singularities in a
projective subvariety invariant by a holomorphic foliation by curves on
$\mathbb{P}^{n}$ with a singularity of positive dimension.

###### Key words and phrases:

Foliations by curves, non-isolated singularities, Milnor number

###### 2010 Mathematics Subject Classification:

Primary 32S65; Secondary 58K45

##  1\. Introduction and statement of results

The purpose of the present paper is to establish numerical relations between
the invariants of a foliation by curves ${\mathcal{F}}$ on the complex
projective space $\mathbb{P}^{n}$ and the characteristic classes of the
irreducible components of its singular set ${\rm Sing}({\mathcal{F}})$. By
numerical invariants of a foliation ${\mathcal{F}}$, we refer to Chern numbers
of its tangent sheaf, the order of vanishing and the multiplicity of
${\mathcal{F}}$ along its singularities. Since any foliation by curves on
$\mathbb{P}^{n}$ with singular set of positive dimension can be deformed in to
a foliation by curves with isolated singularities [16], we address the problem
of determining, in terms of Milnor number, the numerical contribution that
each irreducible component of the singular set offers after being deformed.

More precisely, a holomorphic foliation by curves ${\mathcal{F}}$ of
$\mathbb{P}^{n}$, $n\geq 2$, of degree $k\geq 1$, is defined by a morphism

$\Phi:\mathcal{O}_{\mathbb{P}^{n}}(1-k)\to T\mathbb{P}^{n}$

whose singular set ${\rm Sing}({\mathcal{F}}):=\\{p:\Phi(p)=0\\}$ has
codimension at least two. Outside the singular set of ${\mathcal{F}}$, the
morphism $\Phi$ induces an integrable distribution $\mathcal{D}$ of tangent
directions. By definition, the leaves of ${\mathcal{F}}$ are the leaves
induced by $\mathcal{D}$ on $\mathbb{P}^{n}\setminus{\rm
Sing}({\mathcal{F}})$. We call
$\mathcal{L}_{{\mathcal{F}}}:=\mathcal{O}_{\mathbb{P}^{n}}(1-k)$ the tangent
sheaf of ${\mathcal{F}}$. It follows from the definition that ${\mathcal{F}}$
can be represented in affine coordinates $(z_{1},...,z_{n})$ by a vector field
of the form

$X(z)=\sum^{k}_{j=0}X_{j}(z)+g(z)\cdot R(z)$

where $R(z)=\displaystyle\sum^{n}_{i=1}z_{i}\frac{\partial}{\partial z_{i}}$,
$g\in\mathbb{C}[z_{1},\ldots,z_{n}]$ is homogeneous of degree $k$ and $X_{j}$,
$0\leq j\leq k$, is a vector field whose components are homogenous polynomials
of degree $j$.

Let $W$ be a projective subvariety of $\mathbb{P}^{n}$ of codimension $d\geq
2$. Let $\widetilde{\mathbb{P}}^{n}$ be the blowup of $\mathbb{P}^{n}$ along
$W$, and $\pi:\widetilde{\mathbb{P}}^{n}\rightarrow\mathbb{P}^{n}$ the blowup
morphism with exceptional divisor $E:=\pi^{-1}(W)$. Here and subsequently
$\widetilde{{\mathcal{F}}}:=\pi^{*}{\mathcal{F}}$ denotes the strict
transformed of ${\mathcal{F}}$ by $\pi$ and $\ell$ denotes the order of
vanishing of $\widetilde{{\mathcal{F}}}$ along $E$. It is easy to check that
tangent sheaf of $\widetilde{{\mathcal{F}}}$ is

$\mathcal{L}_{\tilde{{\mathcal{F}}}}=\pi^{*}(\mathcal{L}_{{\mathcal{F}}})\otimes\mathcal{O}_{\widetilde{\mathbb{P}}^{n}}(\ell
E).$

A holomorphic foliation by curves ${\mathcal{F}}$ on $\mathbb{P}^{n}$ it said
to be special  along $W\subset{\rm Sing}({\mathcal{F}})$, if $E$ is an
invariant set of $\widetilde{{\mathcal{F}}}$, and ${\rm
Sing}(\widetilde{{\mathcal{F}}})$ meets $E$ in at most isolated singularities
. Special foliation along curves has been studied for $n=3$ in [11] and for
$n>3$ in [2] and [12]. As we will see below, the notion of speciality of a
foliation is fundamental to the study of the irreducible components of the
singular set of a foliation by curves.

In what follows $\mathcal{W}^{(d)}_{\delta}$ and $\sigma_{i}^{(d)}$ are the
Wronski (or complete symmetric) and elementary symmetric functions of degree
$\delta$ in $d$ variables, respectively. That is

$\mathcal{W}^{(d)}_{\delta}:=\mathcal{W}^{(d)}_{\delta}(k_{1},\ldots,k_{d})=\sum_{i_{1}+\ldots+i_{d}=\delta}k^{i_{1}}_{1}\ldots
k^{i_{d}}_{d}$

and

$\sigma^{(d)}_{i}:=\sigma_{i}^{(d)}(k_{1},\ldots,k_{d})=\displaystyle\sum_{1\leq
j_{1}<\ldots<j_{i}\leq d}k_{j_{1}}\cdots k_{j_{i}}.$

When $W$ is a smooth intersection complete on $\mathbb{P}^{n}$ of multidegree
$(k_{1},\ldots,k_{d})$, we will use the symbol $\nu({\mathcal{F}},W)$ to
denote

(1)
$\displaystyle\nu({\mathcal{F}},W)=-\deg(W)\sum_{|a|=0}^{n-d}\sum_{m=0}^{n-d-|a|}(-1)^{\delta_{|a|}^{m}}\frac{\varphi_{a}^{(m)}(\ell)}{m!}(\deg({\mathcal{F}}-1)^{m}\sigma_{a_{1}}^{(d)}\tau_{a_{2}}^{(d)}\mathcal{W}_{\delta_{|a|}^{m}}^{(d)},$

where $a=(a_{1},a_{2})$, $0\leq a_{1}\leq d$, $0\leq a_{2}\leq n-d$,
$|a|=a_{1}+a_{2}$, $\delta_{|a|}^{m}:=n-d-|a|-m$,

$\displaystyle\varphi_{a}(x)=x^{n-d-
a_{2}}(1+x)^{d-a_{1}}\,\,\,\text{and}\,\,\,\,\tau_{i}^{(d)}=\displaystyle\sum_{j=0}^{i}(-1)^{j}{{n+1}\choose{i-j}}\mathcal{W}_{j}^{(d)}(k_{1},\ldots,k_{d}).$

We will show that $\nu({\mathcal{F}},W)$ is strongly related with the
numerical contribution that $W$ offers after a small deformation of
${\mathcal{F}}$. To complete our notation, we introduce the notion of embedded
close points of an irreducible component of $W$ of ${\rm
Sing}({\mathcal{F}})$. More precisely, for an irreducible component $W$ of
${\rm Sing}({\mathcal{F}})$, we will prove that it is possible to deform
${\mathcal{F}}$ into a foliation by curves ${\mathcal{F}}_{t}$ (for some
$0<t<\epsilon$) on $\mathbb{P}^{n}$ such that either $W$ is invariant by
${\mathcal{F}}_{t}$ or ${\mathcal{F}}_{t}$ is special along $W$ (see Lemma
3.1). The set of isolated points $\\{p_{i}^{t},i=1,\ldots,s_{t}\\}\subset{\rm
Sing}({\mathcal{F}}_{t})$ such that $p_{i}^{t}\notin W$ but
$\displaystyle\lim_{t\to 0}p_{i}^{t}\in W$ will be called embedded close
points associated to $W$ and we will be denoted by $\mathcal{A}_{W}$.

With the above notation, we state the main result.

######  Theorem 1.

Let ${\mathcal{F}}$ be a holomorphic foliation by curves on $\mathbb{P}^{n}$,
$n\geq 3$, of degree $k$. Suppose that its singular set ${\rm
Sing}({\mathcal{F}})$ is the disjoint union of smooth scheme-theoretic
complete intersection subvarieties $W_{1},\ldots,W_{r}$ of pure codimension
$d_{j}\geq 2$, respectively, and points $p_{1},\ldots,p_{s}$. Then

$\displaystyle\sum_{i=1}^{s}\mu({\mathcal{F}},p_{i})=\displaystyle\sum_{i=0}^{n}k^{i}+\displaystyle\sum_{i=1}^{r}\nu({\mathcal{F}},W_{i})-\sum_{i=1}^{r}N({\mathcal{F}},\mathcal{A}_{W_{i}}),$

where $N({\mathcal{F}},\mathcal{A}_{W_{i}})$ denotes the number (counted with
multiplicity) of embedding closed points associated to each $W_{i}$.

It follows easily that the numerical contribution of each $W_{i}$ is exactly
$N({\mathcal{F}},\mathcal{A}_{W_{i}})-\nu({\mathcal{F}},W_{i})$. In this
sense, it can be interpret it as the Milnor number of ${\mathcal{F}}$ at
$W_{i}$,

(2)
$\mu({\mathcal{F}},W_{i}):=N({\mathcal{F}},\mathcal{A}_{W_{i}})-\nu({\mathcal{F}},W_{i}).$

In general, for an arbitrary foliation with non-isolated singularities on
$\mathbb{P}^{n}$ it seems to be difficult to determine effectively
$N({\mathcal{F}},\mathcal{A}_{W_{i}})$. In the final section of the paper, we
present a family of foliations by curves on $\mathbb{P}^{n}$ where it is
possible to obtain an upper bound to this number.

On the other hand, to prove Theorem 1 we need prove the following result.

######  Theorem 2.

Let ${\mathcal{F}}$ be a holomorphic foliation by curves on $\mathbb{P}^{n}$,
$n\geq 3$, of degree $k$. Suppose that its singular set ${\rm
Sing}({\mathcal{F}})$ is the disjoint union of smooth scheme-theoretic
complete intersection subvarieties $W_{1},\ldots,W_{r}$ of pure codimension
$d_{j}\geq 2$, respectively, and points $p_{1},\ldots,p_{s}$. If
${\mathcal{F}}$ is special along each $W_{i}$. Then

$\displaystyle\sum_{i=1}^{s}\mu({\mathcal{F}},p_{i})=\displaystyle\sum_{i=0}^{n}k^{i}+\displaystyle\sum_{i=1}^{r}\nu({\mathcal{F}},W_{i}).$

In particular,

$\mu({\mathcal{F}},W_{i})=-\nu({\mathcal{F}},W_{i}).$

Note that the speciality condition implies that the set of embedding closed
points of subvarieties $W_{i}$ is empty, for all $i=1,\ldots,r$. Moreover, in
this case, $\mu({\mathcal{F}},W_{i})$ is explicitly determined and depends of
the invariants of ${\mathcal{F}}$ and $W$. Since we work with irreducible
components of ${\rm Sing}({\mathcal{F}})$ of any dimension, it is clear that
Theorem 2 generalizes Theorem 1 of [2] and the main result of [11].

Now, in the same spirit, the paper also addresses the following question:
given a foliation by curves ${\mathcal{F}}$ on $\mathbb{P}^{n}$, $n\geq 3$,
and a smooth algebraic subvariety $V$ of $\mathbb{P}^{n}$ invariant by
${\mathcal{F}}$. Assuming that the restriction foliation ${\mathcal{F}}|_{V}$
to $V$ has only an irreducible component $W$ of positive dimension and some
isolated points of ${\rm Sing}(\mathcal{F})$ contained in $V$. How many
isolated singularities of ${\mathcal{F}}$, counted with multiplicity, are
there on $V$?

The above question was completely solved by Marcio Soares [14] in the case
that $W$ is empty, that is, when ${\mathcal{F}}$ only admits isolated
singularities on $V$. It is known from [14] that the number of isolated
singularities of ${\mathcal{F}}$ in $V$ only depends of invariants of
${\mathcal{F}}$ and $V$. The results of [14] were applied to solve an
important question, originally raised by Henri Poincaré [8], of bounding the
degree of an irreducible plane algebraic curve invariant by a holomorphic
foliation ${\mathcal{F}}$ of $\mathbb{P}^{2}$, in terms of the degree of the
foliation. For more details about this important problem we refer to a [15]
and the references within. Of course, in the case $W\neq\emptyset$, our
expectation is proving a similar result.

Let $V=Z(f_{1},\ldots,f_{m})$ be a smooth complete intersection on
$\mathbb{P}^{n}$. Let $\widetilde{{\mathcal{F}}}$ be the strict transform of
${\mathcal{F}}$ by the blowup $\pi$ at $W$ with exceptional divisor $E$.
Assume that $W\subset V$. Let $\widetilde{V}=\overline{\pi^{-1}(V\setminus
W)}$ and $V_{E}=\widetilde{V}\cap E$. Set $N:=N_{W/\mathbb{P}^{n}}$,
$\zeta:=c_{1}(\mathcal{O}_{N}(-1))$ and $\mathbf{h}$ the class of a hyperplane
of $\mathbb{P}^{n}$. If $W=Z(h_{1},\ldots,h_{d})$ is a smooth complete
intersection on $\mathbb{P}^{n}$ of multidegree $(k_{1},\ldots,k_{d})$, we
define

$\tau_{i}^{(m)}:=\tau_{i}^{(m)}(d_{1},\ldots,d_{m})=\displaystyle\sum_{j=0}^{i}(-1)^{j}{{n+1}\choose{i-j}}\mathcal{W}_{j}^{(m)}(d_{1},\ldots,d_{m}).$

with $d_{j}=\deg(f_{j})$ and also define the following

$\displaystyle\nu({\mathcal{F}},V,W)$
$\displaystyle=\displaystyle\sum_{i=0}^{n-d}\sum_{j=0}^{n-d-i}\frac{\Omega^{(j)}(\ell)}{j!}(-1)^{\gamma_{i}^{j}}\tau_{i}^{(m)}(k-1)^{j}\alpha_{\widetilde{V}_{E}}^{\gamma_{i}^{j}-1}$

where $\gamma_{i}^{j}=n-m-i-j$,
$\alpha_{{V}_{E}}^{(i)}:=\displaystyle\int_{\widetilde{V}_{E}}\pi^{*}(\mathbf{h})^{n-m-1-i}\cdot\zeta^{i}$
and $\Omega(x)=\displaystyle\sum_{p=i}^{n-m-j}x^{n-m-p}$.

Let
$\mathcal{I}(W):=\\{g\in\mathbb{C}[x_{0},\ldots,x_{n}]:g=\lambda_{1}h_{1}+\ldots+\lambda_{d}h_{d},\quad\text{for
some}\quad(\lambda_{1},\ldots,\lambda_{d})\in\mathbb{C}^{d}\\}.$ With the
above notation, we give a partial solution to the above question in a very
particular case.

######  Theorem 3.

Let ${\mathcal{F}}$ be a foliation by curves on $\mathbb{P}^{n}$, $n\geq 3$,
of degree $k$, such that

$\mbox{Sing}({\mathcal{F}})=W\cup\\{p_{1},\ldots,p_{s}\\},$

where $W=Z(h_{1},\ldots,h_{d})$ is a smooth complete intersection of
$\mathbb{P}^{n}$ and $p_{j}$, $1\leq j\leq s$, are isolated points disjoint to
$W$. Suppose that $V=Z(f_{1},\ldots,f_{m})$ is a smooth irreducible complete
intersection of $\mathbb{P}^{n}$, invariant by ${\mathcal{F}}$ and such that
$f_{i}\in\mathcal{I}(W)$ for all $1\leq i\leq m$. Then $W\subset V$ and

$\sum_{p\in V\setminus W}\mu({\mathcal{F}},p)\leq
N({\mathcal{F}},V)+\nu({\mathcal{F}},V,W)+N({\mathcal{F}},A_{V\setminus W}),$

where $N({\mathcal{F}},A_{V\setminus W})$ is the number, counted with
multiplicity, of embedding closed points of $V\setminus W$.

The proof of Theorem 3 is similar to the proof of Theorem 1. More precisely,
we construct a foliation by curves ${\mathcal{F}}_{t}$ of $\mathbb{P}^{n}$
that is a deformation of ${\mathcal{F}}$ such that $V$ is invariant and either
$W$ is invariant by ${\mathcal{F}}_{t}$ or ${\mathcal{F}}_{t}$ is special
along $W$, with some isolated singularities. Furthermore, we will show that
the total sum of Milnor numbers of ${\mathcal{F}}_{t}$ associated to each
singularity is independent of the parameter $t$.

##  2\. Preliminaries

In this section we focus on some basic definitions and known results of
holomorphic foliations by curves on $\mathbb{P}^{n}$.

###  2.1. Milnor number of an isolated singularity

Let ${\mathcal{F}}$ be a holomorphic foliation by curves on $\mathbb{P}^{n}$
and $p$ a singular point of ${\mathcal{F}}$. Without loss of generality we can
suppose that $p=0\in\mathbb{C}^{n}$ and ${\mathcal{F}}$ is defined, in a
neighborhood of $p$, by a polynomial vector field
$X_{p}=\displaystyle\sum^{n}_{j=1}P_{j}(z)\frac{\partial}{\partial{z_{n}}},$
where $P_{i}\in\mathbb{C}[z_{1},\ldots,z_{n}]$. The Milnor number of
${\mathcal{F}}$ at $p$, denoted by $\mu({\mathcal{F}},p)$, is the
$\mathbb{C}$-dimension of the local algebra associated to $X_{p}$,

$\mu({\mathcal{F}},p)={\rm
dim}_{\mathbb{C}}\frac{\mathcal{O}_{n,0}}{\displaystyle\langle
P_{1},\ldots,P_{n}\rangle}$

where $\mathcal{O}_{n,0}$ is the ring of germs of holomorphic functions at
$0\in\mathbb{C}^{n}$. It is well-known that this definition is independent of
the vector field representing ${\mathcal{F}}$ at $p$.

We say that ${\mathcal{F}}$ is non-degenerate on $\mathbb{P}^{n}$ if
${\mathcal{F}}$ has only isolated singularities and the Milnor number of a
vector field defining the foliation around each singularity is 1. The
Jouanolou’s foliation $\mathcal{J}_{k}$ of degree $k\geq 1$, given by the
vector field in affine coordinates $(z_{1},\ldots,z_{n})$

(3)
$\sum_{i=1}^{n}\Big{(}\sum^{k}_{j=1}z_{i}z_{1}^{k}-z^{k}_{i+1}\Big{)}\frac{\partial}{\partial{z}_{i}}+(z_{n}z_{1}^{k}-1)\frac{\partial}{\partial{z}_{n}}$

is an example of a non-degenerate foliation on $\mathbb{P}^{n}$. Now if
${\mathcal{F}}$ is non-degenerate foliation on $\mathbb{P}^{n}$ then it
follows from the classical Baum-Bott formula [1] that

$\sum_{p\in\mathbb{P}^{n}}\mu({\mathcal{F}},p)=k^{n}+k^{n-1}+\ldots+k+1.$

###  2.2. Multiplicity along subvarieties

We describe the multiplicity of ${\mathcal{F}}$ along an irreducible variety
$W$ of pure codimension ${\rm cod}(W)=d$, with $d\geq 2$. From now on we make
the assumption $W\subset{\rm Sing}(\mathcal{F})$. By a holomorphic change of
coordinates, $W$ can be locally given as $\\{z_{1}=\ldots=z_{d}=0\\}$. In this
neighborhood, ${\mathcal{F}}$ is described by the vector field

(4) $\mathcal{D}_{{\mathcal{F}}}=\displaystyle
P_{1}(z)\frac{\partial}{\partial z_{1}}+\ldots+\displaystyle
P_{n}(z)\frac{\partial}{\partial z_{n}}.$

Therefore, one may write the local sections as

(5) $P_{i}(z)=\displaystyle\sum_{|a|=m_{i}}z_{1}^{a_{1}}\cdots
z_{d}^{a_{d}}P_{i,a}(z)$

where $a:=(a_{1},\ldots,a_{d})\in{\mathbb{Z}}^{d}$ with
$|a|:=a_{1}+\ldots+a_{d}$, $a_{i}\geq 0$ and at least one among the
$P_{i,a}(z)$ does not vanish at $\\{z_{1}=\ldots=z_{d}=0\\}$. We define the
multiplicity of each function $P_{i}$ along $W$ as $m_{W}(P_{i})=m_{i}$. As a
consequence, the multiplicity of ${\mathcal{F}}$ along $W$ is defined as

(6) $m_{W}({\mathcal{F}})={\rm min}\\{m_{1},\ldots,m_{n}\\}.$

In order to simplify our analysis, we use the following Lemma:

######  Lemma 2.1.

Let ${\mathcal{F}}$ be a holomorphic foliation by curves, defined in the
neighborhood of $p$ by the polynomial vector field

$X_{p}=\displaystyle\sum^{n}_{j=1}P_{j}(z)\frac{\partial}{\partial{z_{j}}}$

where $m_{i}=m_{W}(P_{i})$. Then, by a linear change of coordinates,
${\mathcal{F}}$ may be described by the polynomial vector field

$Y_{p}=\displaystyle\sum^{n}_{j=1}Q_{j}(w)\frac{\partial}{\partial{z_{j}}}$

with

$m_{W}(Q_{j})=\left\\{\begin{array}[]{ll}m^{\prime}_{W}({\mathcal{F}}),&{\rm
for}\quad j=1,\ldots,d\cr m_{W}({\mathcal{F}}),&{\rm for}\quad
j=d+1,\ldots,n\end{array}\right.$

where $m^{\prime}_{W}({\mathcal{F}})={\rm min}\\{m_{1},\ldots,m_{d}\\}$.

###### Proof.

In fact, it is enough to consider the linear transformation $w=Az$ where
$A=(a_{ij})\in GL(n,\mathbb{C})$ with $a_{ij}=0$ for $i=1,\ldots,d$ and
$j=d+1,\ldots,n$. In this way, $B=(b_{ij})=A^{-1}$ has the same properties,
i.e.; the subspace $W$ given by $\\{z_{1}=\ldots=z_{d}=0\\}$ is an invariant
set under this transformation. Adjusting the coefficients $a_{ij}$, if
necessary, we can admit that

$\dot{w}_{i}=\left\\{\begin{array}[]{ll}\displaystyle\sum_{j=1}^{d}a_{ij}\dot{z}_{j}=\displaystyle\sum_{j=1}^{d}a_{ij}P_{j}(Bw)=Q_{i}(w),&\mathrm{for}\quad
i=1,\ldots,d\\\
\displaystyle\sum_{j=1}^{n}a_{ij}\dot{z}_{j}=\displaystyle\sum_{j=1}^{n}a_{ij}P_{j}(Bw)=Q_{i}(w),&\mathrm{for}\quad
i=d+1,\ldots,n\end{array}\right.$

having the required properties. ∎

###  2.3. Blowup of subvarieties

We start by recalling the blowup procedures of $\mathbb{P}^{n}$ along $W$ and
describe the behavior of ${\mathcal{F}}$ under this transformation. See [5]
for more details. Let $\Delta$ be an $n$-dimensional polydisc with holomorphic
coordinates $z_{1},\ldots,z_{n}$ and $\Gamma\subset\Delta$ the locus
$z_{1}=\ldots=z_{d}=0$. Take $[y_{1}:\ldots:y_{d}]$ to be homogeneous
coordinates on $\mathbb{P}^{d-1}$. The blowup of $\Delta$ along $\Gamma$ is
the smooth variety

$\widetilde{\Delta}=\\{(z,[y])\in\Delta\times\mathbb{P}^{d-1}\,|\,z_{i}y_{j}=z_{j}y_{i}\
{\rm for}\ 1\leq i,j\leq d\\}.$

The projection $\pi:\widetilde{\Delta}\rightarrow\Delta$ on the first factor
is an isomorphism away from $\Gamma$, while the inverse image of a point
$z\in\Gamma$ is at projective space $\mathbb{P}^{d-1}.$ The inverse image
$E=\pi^{-1}(\Gamma)$ is the exceptional divisor of the blowup.

The standard open cover $U_{j}=\\{[y_{1}:\ldots:y_{d}]\,|\,y_{j}\neq 0\\}$,
with $1\leq j\leq d$, of $\mathbb{P}^{d-1}$ yields a cover of
$\widetilde{\Delta}$ where each open set, for $1\leq j\leq d$, is defined by

(7) ${\widetilde{U}_{j}}=\\{(z,[y])\in\widetilde{\Delta}\ \,|\ \,[y]\in
U_{j}\\}$

with holomorphic coordinates
$\sigma_{j}(u_{1},\ldots,u_{n})=(z_{1},\ldots,z_{n})$ given by

$z_{i}=\begin{cases}u_{i}&{\rm if}\ i=j\ \mbox{or}\ i=d+1,\ldots,n\\\
u_{i}u_{j}&{\rm if}\ i=1,\ldots,{\widehat{j}},\ldots,d.\end{cases}$

The coordinates $u\in\mathbb{C}^{n}$ are affine coordinates on each fiber
$\pi^{-1}(p)\cong\mathbb{P}^{d-1}$ of $E$.

Now consider the algebraic subvariety $W\subset\mathbb{P}^{n}$. Let
$\\{\phi_{\lambda},U_{\lambda}\\}$ be a collection of local charts covering
$W$ and $\phi_{\lambda}:U_{\lambda}\to\Delta_{\lambda},$ where
$\Delta_{\lambda}$ is an $n$-dimensional polydisc. One may suppose that
$\Gamma_{\lambda}=\phi_{\lambda}(W)\cap U_{\lambda}$ is given by
$z_{1}=\ldots=z_{d}=0.$ Let
$\pi_{\lambda}:\widetilde{\Delta}_{\lambda}\to\Delta_{\lambda}$ be the blowup
of $\Delta_{\lambda}$ along $\Gamma_{\lambda}$. One can patch the
$\pi_{\lambda}$ together and use the chart maps $\phi_{\lambda}$ to get a
blowup $\widetilde{\mathbb{P}}_{n}$ of $\mathbb{P}^{n}$ along $W$ and a blowup
morphism $\pi:\widetilde{\mathbb{P}}_{n}\to\mathbb{P}^{n}$. The exceptional
divisor $E$ is a fibre bundle over $W$ with fiber $\mathbb{P}^{d-1}$, which is
naturally identified with the projectivization
$\mathbb{P}(N_{W/_{\mathbb{P}^{n}}})$ of the normal bundle
$N_{W/_{\mathbb{P}^{n}}}$.

###  2.4. Nondicritical and Dicritical singularities

Let us denote by $\widetilde{{\mathcal{F}}}$ the strict transform of
${\mathcal{F}}$ under $\pi$, that is, $\widetilde{{\mathcal{F}}}$ is a
foliation by curves on $\widetilde{\mathbb{P}^{n}}$ whose leaves are projected
via $\pi$ in to leaves of ${\mathcal{F}}$. We say that $W$ is nondicritical if
the exceptional divisor $E$ is invariant by $\widetilde{{\mathcal{F}}}$,
otherwise $W$ is dicritical.

Now, by Lemma 2.1, we can assume that the foliation ${\mathcal{F}}$ is
described by vector field
$\mathcal{D}_{{\mathcal{F}}}=\displaystyle\sum^{n}_{j=1}P_{j}(z)\frac{\partial}{\partial{z_{j}}}$
such that

$q_{j}=m_{W}(P_{j})=\begin{cases}m_{1}&{\rm for}\ j=1,\ldots,d\\\ m_{n}&{\rm
for}\ j=d+1,\ldots,n\end{cases}$

with $m_{1}\geq m_{n}$. In order to simplify our notation, we only consider on
the chart $\widetilde{U}_{1}$ with the coordinate system given by
$\sigma_{1}(u)=(u_{1},u_{1}u_{2},\ldots,u_{1}u_{d},u_{d+1},\ldots,u_{n})=z\in\mathbb{C}^{n}$.
Thus, for $j\in\\{1,d+1,\ldots,n\\}$, we have that

$\displaystyle\dot{u}_{j}$
$\displaystyle=P_{j}(\sigma_{1}(u))=\displaystyle\sum_{|a|=q_{j}}u_{1}^{a_{1}}(u_{1}u_{2})^{a_{2}}\cdots(u_{1}u_{d})^{a_{d}}P_{j,a}(\sigma_{1}(u))$
$\displaystyle=u_{1}^{m_{j}}\displaystyle\sum_{|a|=q_{j}}u_{2}^{a_{2}}\cdots
u_{d}^{a_{d}}P_{j,a}(\sigma_{1}(u)).$

But,

$P_{j,a}(\sigma_{1}(u))=P_{j,a}(0,\ldots,0,u_{d+1},\ldots,u_{n})+u_{1}{\widetilde{P}}_{j,a}(u)=p_{j,a}(u_{d+1},\ldots,u_{n})+u_{1}\widetilde{P}_{j,a}(u)$

and hence

(8)
$\dot{u}_{j}=u_{1}^{q_{j}}\left(\displaystyle\sum_{|a|=q_{j}}u_{2}^{a_{2}}\cdots
u_{d}^{a_{d}}p_{j,a}(u_{d+1},\ldots,u_{n})+u_{1}\widetilde{P_{j}}(u)\right):=u_{1}^{q_{j}}\left(Q_{j}(u)+u_{1}\widetilde{P_{j}}(u)\right)$

for some functions $\widetilde{P_{j}}$.

Finally, for $j\in\\{2,\ldots,d\\}$, since $z_{j}=u_{1}u_{j}$ we obtain that

(9)
$\dot{u}_{j}=P_{j}(\sigma_{1}(u))=u_{1}^{m_{1}-1}\left(G_{j}(u)+u_{1}\widetilde{P_{j}}(u)\right)$

where

(10) $G_{j}(u):=\displaystyle\sum_{|a|=m_{1}}u_{2}^{a_{2}}\cdots
u_{d}^{a_{d}}\big{(}p_{j,a}(u_{d+1},\ldots,u_{n})-u_{j}\cdot
p_{1,a}(u_{d+1},\ldots,u_{n})\big{)}:=Q_{j}(u)-u_{j}Q_{1}(u)$

for some functions $\widetilde{P_{j}}$. With the equations (8), (9) and (10),
we obtain the expressions of $\pi^{*}{\mathcal{F}}$ (in a similar way with the
others charts $\widetilde{U}_{j}$),

(11) $\displaystyle\mathcal{D}_{\pi^{*}{\mathcal{F}}}=$
$\displaystyle\displaystyle
u_{1}^{m_{1}}[Q_{1}(u)+u_{1}\widetilde{P_{1}}(u)]\frac{\partial}{\partial
u_{1}}+u_{1}^{m_{1}-1}\sum_{j=2}^{d}[G_{j}(u)+u_{1}\widetilde{P_{j}}(u)]\frac{\partial}{\partial
u_{j}}+$ (12) $\displaystyle
u_{1}^{m_{n}}\sum_{j=d+1}^{n}[Q_{j}(u)+u_{1}\widetilde{P_{j}}(u)]\frac{\partial}{\partial
u_{j}}.$

The vector field that generates $\widetilde{{\mathcal{F}}}$ is obtained from
(11), after a division by an adequate power of $u_{1}$. The situation where
$W$ is a nondicritical component is described in the following cases

(i) $m_{n}+1=m_{1}$ and $G_{j_{0}}\not\equiv 0$ for some
$j_{0}\in\\{2,\ldots,d\\}$.

In this condition, dividing (11) by $u_{1}^{m_{n}}$ we get

(13) $\displaystyle\mathcal{D}_{\widetilde{{\mathcal{F}}}}=$
$\displaystyle\displaystyle
u_{1}\big{(}Q_{1}(u)+u_{1}\widetilde{P_{1}}(u)\big{)}\frac{\partial}{\partial
u_{1}}+\sum_{j=2}^{d}\big{(}G_{j}(u)+u_{1}\widetilde{P_{j}}(u)\big{)}\frac{\partial}{\partial
u_{j}}+$ (14)
$\displaystyle\sum_{j=d+1}^{n}\big{(}Q_{j}(u)+u_{1}\widetilde{P_{j}}(u)\big{)}\frac{\partial}{\partial
u_{j}}.$

If there is some $G_{j_{i}}\equiv 0$ then the set ${\rm
Sing}(\widetilde{{\mathcal{F}}})\cap E$ contains some component of positive
dimension. Furthermore, in order to ${\mathcal{F}}$ be special along $W$ then
necessary we have that $G_{j}\not\equiv 0$, $\forall j$.

(ii) $m_{n}+1<m_{1}$.

As in the case before, dividing (11) by $u_{1}^{m_{n}}$ we get

(15) $\displaystyle\mathcal{D}_{\widetilde{{\mathcal{F}}}}=$
$\displaystyle\displaystyle
u_{1}^{m_{1}-m_{n}}\big{(}Q_{1}(u)+u_{1}\widetilde{P_{1}}(u)\big{)}\frac{\partial}{\partial
u_{1}}+u_{1}^{m_{1}-m_{n}-1}\sum_{j=2}^{d}\big{(}G_{j}(u)+u_{1}\widetilde{P_{j}}(u)\big{)}\frac{\partial}{\partial
u_{j}}+$ (16)
$\displaystyle\sum_{j=d+1}^{n}\big{(}Q_{j}(u)+u_{1}\widetilde{P_{j}}(u)\big{)}\frac{\partial}{\partial
u_{j}}.$

In this situation, the leaves of $\widetilde{{\mathcal{F}}}$ when restricted
on the exceptional divisor $E$ are contained in the hyperplane $u_{j}=k_{j}$,
for $j\in\\{2,\ldots,d\\}$. Furthermore, always there exist non-isolated
singularities of $\widetilde{{\mathcal{F}}}$ on $E$.

(iii) $m_{n}=m_{1}$ and $G_{j_{0}}\not\equiv 0$ for some
$j_{0}\in\\{2,\ldots,d\\}$.

We may divide (11) by $u_{1}^{m_{1}-1}$. As consequence, we obtain

(17) $\displaystyle\mathcal{D}_{\widetilde{{\mathcal{F}}}}=$
$\displaystyle\displaystyle
u_{1}\big{(}Q_{1}(u)+u_{1}\widetilde{P_{1}}(u)\big{)}\frac{\partial}{\partial
u_{1}}+\sum_{j=2}^{d}\big{(}G_{j}(u)+u_{1}\widetilde{P_{j}}(u)\big{)}\frac{\partial}{\partial
u_{j}}+$ (18) $\displaystyle
u_{1}^{m_{n}-m_{1}+1}\sum_{j=d+1}^{n}\big{(}Q_{j}(u)+u_{1}\widetilde{P_{j}}(u)\big{)}\frac{\partial}{\partial
u_{j}}.$

Now, the leaves of $\widetilde{{\mathcal{F}}}$ on $E$ are contained in the
hyperplane $u_{j}=k_{j}$ for $j\in\\{d+1,\ldots,n\\}$.

On the other hand, when $W$ is dicritical, we have the following condition:
(i) $m_{n}=m_{1}$ and $G_{j}\equiv 0$ for $j\in\\{2,\ldots,d\\}$.

Therefore, after the division of (11) by $u_{1}^{m_{1}}$ we get

(19)
$\displaystyle\mathcal{D}_{\pi^{*}{\mathcal{F}}}=\displaystyle\big{(}Q_{1}(u)+u_{1}\widetilde{P_{1}}(u)\big{)}\frac{\partial}{\partial
u_{1}}+\sum_{j=2}^{d}\big{(}\widetilde{P_{j}}(u)\big{)}\frac{\partial}{\partial
u_{j}}+\sum_{j=d+1}^{n}\big{(}Q_{j}(u)+u_{1}\widetilde{P_{j}}(u)\big{)}\frac{\partial}{\partial
u_{j}}.$

Note that the exceptional divisor $E$ is not an invariant set by
$\widetilde{{\mathcal{F}}}$.

######  Lemma 2.2.

With the notation given in the Lemma 2.1, we get

  1. (1)

$m_{W}({\mathcal{F}})={\rm min}\\{m_{1},m_{n}\\}$;

  2. (2)

Let $\mathcal{L}_{{\mathcal{F}}}$ and
$\mathcal{L}_{\widetilde{{\mathcal{F}}}}$ be tangent sheaves of the foliations
${\mathcal{F}}$ and $\widetilde{{\mathcal{F}}}$, respectively. If $\ell$ is
the integer such that

$\mathcal{L}_{\widetilde{{\mathcal{F}}}}\cong\pi^{*}\mathcal{L}_{{\mathcal{F}}}\otimes\mathcal{O}_{\widetilde{Y}}(\ell
E)$

then

$\ell=\begin{cases}{\rm min}\\{m_{1}-1,m_{n}\\}&\mbox{ if }{W\mbox{ is
nondicritical}}\\\ m_{1}&\mbox{ if }{W\mbox{ is dicritical. }}\end{cases}$

In particular, if $m_{1}=m_{n}$ then

$\ell=\begin{cases}m_{W}({\mathcal{F}})-1&\mbox{ if }{W\mbox{ is
nondicritical}}\\\ m_{W}({\mathcal{F}})&\mbox{ if }{W\mbox{ is dicritical.
}}\end{cases}$

The number $\ell:=m_{E}(\widetilde{\mathcal{F}},W)$ will be called the order
of vanishing of $\widetilde{\mathcal{F}}$ at $E$.

###  2.5. Chern Class

Let $\pi:\widetilde{\mathbb{P}}^{n}\to\mathbb{P}^{n}$, $n\geq 3$, be the
blowup of $\mathbb{P}^{n}$ centered at $W$, with exceptional divisor denoted
by $E$. Set $N:=N_{W/\mathbb{P}^{n}}$ and $\rho:=\pi|_{E}$. Since
$E\cong\mathbb{P}(N)$, recall that $A(E)$ is generated as an
$A({\mathcal{C}})$-algebra by the Chern class

$\zeta:=c_{1}(\mathcal{O}_{N}(-1))$

with the single relation

(20)
$\zeta^{d}-\rho^{*}c_{1}(N)\zeta^{d-1}+\ldots+(-1)^{d-1}\rho^{*}c_{d-1}(N)\zeta+(-1)^{d}\rho^{*}c_{d}(N)=0.$

The normal bundle $N_{E/\widetilde{\mathbb{P}}^{n}}$ agrees with the
tautological bundle $\mathcal{O}_{N}(-1)$, and hence

(21) $\zeta=c_{1}(N_{E/\widetilde{\mathbb{P}}^{n}}).$

If $\iota:E\hookrightarrow\widetilde{\mathbb{P}}^{n}$ is the inclusion map, we
also get

(22) $\iota_{*}(\zeta^{i})=(-1)^{i}E^{i+1}.$

Now, from Porteous Theorem (cf. [9], [3]), writing
$c_{j}(\widetilde{\mathbb{P}}^{n})$ and $c_{j}(\mathbb{P}^{n})$ for
$c_{j}(T_{\widetilde{\mathbb{P}}^{n}})$ and $c_{j}(T_{\mathbb{P}^{n}})$,
respectively, we have

(23)
$c(\widetilde{\mathbb{P}^{n}})-\pi^{*}c(\mathbb{P}^{n})=\iota_{*}(\rho^{*}C(T_{W})\cdot\alpha)$

where

$\alpha=\frac{1}{\zeta}\displaystyle\sum_{i=0}^{d}\big{[}1-(1-\zeta)(1+\zeta)^{i}\big{]}\rho^{*}c_{d-i}(N).$

In this way, we can rewrite $\alpha$,

(24)
$\alpha=\displaystyle\sum_{m=0}^{d}\sum_{j=0}^{d-m}\bigg{[}{{d-m}\choose{j}}-{{d-m}\choose{j+1}}\bigg{]}\zeta^{j}\rho^{*}c_{m}(N)=\displaystyle\sum_{i=0}^{d}\alpha_{i}$

where

(25)
$\alpha_{i}=\sum_{m=0}^{i}\bigg{[}{{d-m}\choose{i-m}}-{{d-m}\choose{i-m+1}}\bigg{]}\zeta^{i-m}\rho^{*}c_{m}(N).$

Writing $(\rho^{*}C(W)\cdot\alpha)=\displaystyle\sum_{j=0}^{n}\beta_{j}$ where

$\beta_{j}=\sum_{i=0}^{j}\alpha_{i}\rho^{*}c_{j-i}(T_{W}),$

we get,

(26)
$c_{j}(\widetilde{\mathbb{P}}^{n})=\pi^{*}c_{j}(\mathbb{P}^{n})+\iota_{*}(\beta_{j-1}).$

Now, defining $a=(a_{1},a_{2})\in{\mathbb{Z}}^{2}$, with $0\leq a_{1}:=m\leq
d$, $0\leq a_{2}=j-1-i\leq n-d$ and $|a|=a_{1}+a_{2}$, we can rewrite (26) in
the following way

(27)
$c_{j}(\widetilde{\mathbb{P}^{n}})=\pi^{*}c_{j}(\mathbb{P}^{n})+\displaystyle\sum_{|a|=0}^{j-1}(-1)^{j-|a|-1}\Gamma^{j}_{a}\cdot\rho^{*}c_{a_{1}}(N)\rho^{*}c_{a_{2}}(T_{W})E^{j-|a|}$

where

(28) $\Gamma^{j}_{a}={{d-a_{1}}\choose{j-|a|-1}}-{{d-a_{1}}\choose{j-|a|}}.$

Here we are assuming that $\displaystyle{{p}\choose{q}}=0$ if $p<q$ or $q<0$.

Now, let $i:W\to\mathbb{P}^{n}$ be a closed embedding of $W$ in
$\mathbb{P}^{n}$. From the exact sequence

$0\to T_{W}\to i^{*}T_{\mathbb{P}^{n}}\to N\to 0$

one has

$c(T_{W})=c(i^{*}T_{\mathbb{P}^{n}})/c(N).$

If $W$ is an intersection of divisors $D_{1},\ldots,D_{d}$, then

$c(N)=\displaystyle\prod_{j=1}^{d}(1+c_{1}(i^{*}\mathcal{O}_{\mathbb{P}^{n}}(D_{i})))=\prod_{j=1}^{d}(1+k_{j}\mathbf{h})=\sum
c_{i}(N)$

where $\mathbf{h}$ is the class of a hyperplane of $\mathbb{P}^{n}$ and
$k_{j}$ is the $deg(D_{j})$. Therefore,

(29) $c_{i}(N)=\sigma_{i}^{(d)}\cdot{\mathbf{h}}^{i}.$

From the projection formula one deduces that

$c(T_{W})=\displaystyle\frac{(1+\mathbf{h})^{n+1}}{\displaystyle\prod_{j=1}^{d}(1+k_{j}\mathbf{h})}=\sum
c_{i}(T_{W}).$

Consequently,

(30) $c_{i}(T_{W})=\tau_{i}^{(d)}\cdot\mathbf{h}^{i}.$

Repeatedly applying the equation (20), we get

(31)
$\zeta^{m}=\displaystyle\sum_{i=1}^{d}\rho^{*}(\beta_{m,i}\cdot\mathbf{h}^{m-d+i})\zeta^{d-i}$

where

$\beta_{m,i}=\displaystyle\sum_{j=0}^{i-1}(-1)^{j}\sigma_{j}^{(d)}\cdot\mathcal{W}_{m-d+i-j}^{(d)}$

for $m\geq d$. In fact, it is clear for $m=d$ because

$\displaystyle\sum_{j=0}^{m}(-1)^{j}\sigma_{j}^{(d)}\cdot\mathcal{W}_{m-j}^{(d)}=0,\quad
m>0$

and $\beta_{d,i}=(-1)^{i-1}\sigma_{i}^{(d)}$. Assuming that the induction
hypothesis is true for $m\geq d$, we get

$\displaystyle\zeta^{m+1}$
$\displaystyle=\zeta\cdot\zeta^{m}=\displaystyle\sum_{i=1}^{d}\rho^{*}(\beta_{m,i}\cdot\mathbf{h}^{m-d+i})\zeta^{d-i+1}$
$\displaystyle=\rho^{*}(\beta_{m,1}\cdot\mathbf{h}^{m-d+1})\zeta^{d}+\displaystyle\sum_{i=1}^{d-1}\rho^{*}(\beta_{m,i+1}\cdot\mathbf{h}^{m-d+i+1})\zeta^{d-i}.$

Again, from (20), we yields

$\displaystyle\beta_{m+1,i}$
$\displaystyle=(-1)^{i-1}\sigma_{i}^{(d)}\cdot\mathcal{W}_{m-d+1}^{(d)}+\beta_{m,i+1}$
$\displaystyle=\displaystyle\sum_{j=1}^{i-1}(-1)^{j}\sigma_{j}^{(d)}\cdot\mathcal{W}_{m+1-d+i-j}^{(d)}.$

In particular, $\beta_{m,1}=\mathcal{W}_{m-d+1}^{(d)}$ which results

(32)
$\displaystyle\int_{E}\rho^{*}c_{i}(N)\zeta^{n-i-1}=\left\\{\begin{array}[]{ll}(-1)^{d-1}\sigma_{i}^{(d)}\mathcal{W}_{n-d-i}^{(d)}\deg(W),&0\leq
i\leq n-d\cr 0,&i>n-d.\end{array}\right.$

###  2.6. Embedded closed points

Let ${\mathcal{F}}ol(\mathbb{P}^{n};k)$ be the space of holomorphic foliations
by curves of degree $k\geq 1$ of $\mathbb{P}^{n}$. Denote by
$N(\mathbb{P}^{n};k)\subset{\mathcal{F}}ol(\mathbb{P}^{n};k)$ the set of non-
degenerate foliations of degree $k$. Since the Jouanolou’s foliation given in
(3) is non-degenerated, we have that $N(\mathbb{P}^{n};k)$ is not empty. The
following result guarantees that a foliation by curves with singularities of
positive dimension can be deformed to a non-degenerated foliation of
$\mathbb{P}^{n}$.

######  Theorem 2.3 (Soares [16]).

$N(\mathbb{P}^{n};k)$ is open, dense and connected in
${\mathcal{F}}ol(\mathbb{P}^{n};k)$.

We note that this result can not be used to determine the Milnor numbers of an
irreducible component of ${\rm Sing}({\mathcal{F}})$, because the deformation
is not explicit. In order to prove Theorem 1, we use the notion of embedded
closed points associated to $W$. In fact, we will see that it is possible to
deform ${\mathcal{F}}$ to a foliation by curves ${\mathcal{F}}_{t}$ on
$\mathbb{P}^{n}$ (for some small $0<|t|<\epsilon$) such that, either $W$ is
${\mathcal{F}}_{t}$-invariant or else ${\mathcal{F}}_{t}$ is special along $W$
(see for instance Lemma 3.1). Therefore, these isolated singularities that
appear after this deformation and do not lie in $W$ will be called embedded
closed points associated to $W$. More precisely, the set of embedded closed
points of $W$ is

(33) $A_{W}:=\\{p^{t}_{j}\in{\rm
Sing}({\mathcal{F}}_{t}):\displaystyle\lim_{t\to 0}p^{t}_{j}\in
W,\quad\text{such that}\quad p^{t}_{j}\not\in W\quad\forall t\neq 0\\}.$

Let $X\subset\mathbb{P}^{n}$ be a non-empty subset. Here and subsequently
$N({\mathcal{F}},X)$ denotes the number of isolated singularities of
${\mathcal{F}}$ on $X$ (counted with multiplicity). Hence,

$N({\mathcal{F}},A_{W})=\sum_{q^{t}_{j}\in
A_{W}}\mu({\mathcal{F}}_{t},q^{t}_{j}).$

Now, suppose that $V\subset\mathbb{P}^{n}$ is a smooth irreducible complete
intersection defined by

$V=Z(F_{1},\ldots,F_{n-m})$

where $F_{l}\in\mathbb{C}[z_{0},\ldots,z_{n}]$ is homogeneous of degree
$d_{l}$, $1\leq l\leq n-m$. In the proof of Theorem 3, we will use the
following result due to Soares in [14].

######  Proposition 2.4.

Assume that $V$ is invariant by ${\mathcal{F}}$, where ${\mathcal{F}}$ is a
foliation by curves of $\mathbb{P}^{n}$ of degree $k$, non-degenerate along
$V$. Then the number of singularities of ${\mathcal{F}}$ in $V$ is

$N({\mathcal{F}},V)=(d_{1}\ldots
d_{n-m})\sum^{m}_{j=0}\left[\sum^{j}_{\delta=0}(-1)^{\delta}\mathcal{W}^{(n-m)}_{\delta}(d_{1}-1,\ldots,d_{n-m}-1)\right]k^{m-j},$

where $\mathcal{W}^{(n-m)}_{\delta}$ is the Wronski of degree $\delta$ and
$n-m$ variables.

##  3\. Deformation of a foliation by curves adapted to singular set

We use the following fundamental lemma.

######  Lemma 3.1.

Let ${\mathcal{F}}$ be a holomorphic foliation by curves on $\mathbb{P}^{n}$,
$n\geq 3$, of degree $k$. Suppose that

$\mbox{Sing}({\mathcal{F}})=W\cup\\{p_{1},\ldots,p_{s}\\}$

is a disjoint union where $W=Z(h_{1},\ldots,h_{d})$ is a smooth complete
intersection of $\mathbb{P}^{n}$ and $p_{j}$ are isolated points. Let
$\ell=m_{E}({\mathcal{F}},W)$. Suppose that

$V=Z(f_{1},\ldots,f_{m})$

is ${\mathcal{F}}$-invariant such that $f_{i}\in\mathcal{I}(W)$ for all $1\leq
i\leq m$. Then, $W\subset V$ and there exists a one-parameter family of
holomorphic foliations by curves on $\mathbb{P}^{n}$, given by
$\\{{\mathcal{F}}_{t}\\}_{t\in D}$ where
$D=\\{t\in\mathbb{C}\,:\,|t|<\epsilon\\}$ satisfying:

  1. (i)

${\mathcal{F}}_{0}={\mathcal{F}}$;

  2. (ii)

$\deg({\mathcal{F}}_{t})=\deg({\mathcal{F}})$;

  3. (iii)

If $\ell=0$ then ${\rm
Sing}({\mathcal{F}}_{t})=\\{p_{1}^{t},\ldots,p_{s_{t}}^{t}\\}$ , if $\ell>0$
then ${\rm
Sing}({\mathcal{F}}_{t})=W\cup\\{p_{1}^{t},\ldots,p_{s_{t}}^{t}\\}$, disjoint
union where $p_{i}^{t}$ are isolated points, for any $t\in D\setminus\\{0\\}$;

  4. (iv)

If $\ell=0$ then $W$ is ${\mathcal{F}}_{t}$-invariant, for any $t\in
D\setminus\\{0\\}$;

  5. (v)

If $\ell>0$ then $W\subset{\rm Sing}({\mathcal{F}}_{t})$,
$\ell=m_{E}({\mathcal{F}}_{t},W)$ and ${\mathcal{F}}_{t}$ is special along $W$
for any $t\in D\setminus\\{0\\}$;

  6. (vi)

$V$ is ${\mathcal{F}}_{t}$-invariant, for any $t\in D$.

###### Proof.

For simplicity we assume that $W$ is given in the affine coordinates
$(z_{1},\ldots,z_{n})$ by the zeros of the polynomials $h_{1},\ldots,h_{d}$
and $V$ is given by the zeros of the polynomials $f_{1},\ldots,f_{m}$, with
$f_{i}\in\mathcal{I}(W)$. Now let us consider the holomorphic function

$\displaystyle\begin{matrix}F\ :&\mathbb{C}^{n}\
&\longrightarrow&\mathbb{C}^{n}\\\
&z=(z_{1},\ldots,z_{n})&\longmapsto&\bigg{(}h_{1}(z)\,,\,\ldots\,,\,h_{d}(z)\,,z_{d+1},\ldots,\,z_{n}\bigg{)}.\end{matrix}$

By hypotheses we get

$\begin{array}[]{cccc}f_{1}=&\lambda_{11}h_{1}&+\ldots+&\lambda_{1d}h_{d}\\\
\vdots&\vdots&\vdots&\vdots\\\
f_{m}=&\lambda_{m1}h_{1}&+\ldots+&\lambda_{md}h_{d},\end{array}$

where $\lambda_{ij}\in\mathbb{C}$ for all $1\leq i\leq m$ and $1\leq j\leq d$.
Without loss of generality, since $W$ is smooth, we can admit that
$Det(M(z))\neq 0$ where $M(z)=\left(\frac{\partial h_{i}}{\partial
z_{j}}\right)$ is the $d\times d$-minor of the Jacobian matrix $D_{z}F$ for
$z$ in some open set $U\subset\mathbb{C}^{n}$. Thus, note that $F|_{U}$ is a
local biholomorphism onto an open set $\widetilde{U}\subset\mathbb{C}^{n}$ and
restricted to $\widetilde{U}$ so one may fix coordinates $w=F(z)$.

Then the image of $W\cap U$ by $F$ is the hyperplane of codimension $d$,

$H_{d}:=\\{(w_{1},\ldots,w_{n})\in\mathbb{C}^{n}:w_{1}=\ldots=w_{d}=0\\}$

while the image of $V\cap U$ by $F$ is given by

$H_{m}:=\\{(w_{1},\ldots,w_{n})\in\mathbb{C}^{n}:\displaystyle\sum_{j=1}^{d}\lambda_{ij}w_{j}=0,\quad
1\leq i\leq m\\}.$

To continue we describe the push-forward $(F)_{*}{\mathcal{F}}$. We write

$\mathcal{D}_{(F)_{*}{\mathcal{F}}}=P_{1}(w)\,\frac{\partial}{\partial
w_{1}}+\ldots+P_{n}(w)\,\frac{\partial}{\partial w_{n}}$

where

(34) $P_{i}(w)=\sum_{|a|=m_{i}}w_{1}^{a_{1}}\cdots w_{d}^{a_{d}}P_{i,a}(w)$

with at least one $P_{i,a}(z)$ not vanishing in $H_{d}$. Therefore we define
${\mathcal{G}}_{t}$ by the vector field

(35)
$\mathcal{D}_{{\mathcal{G}}_{t}}=\mathcal{D}_{(F)_{*}{\mathcal{F}}}+t\cdot Y$

where

$Y=Q_{1}\,\frac{\partial}{\partial
w_{1}}+\ldots+Q_{n}\,\frac{\partial}{\partial w_{n}}$

with

$Q_{j}(w)=\begin{cases}\displaystyle\sum_{|I|=q_{j}}a_{I,j}w_{1}^{i_{1}}\cdots
w_{d}^{i_{d}}&\text{if}\qquad 1\leq j\leq d\\\
\displaystyle\sum_{|I|=q_{j}}R_{I,j}(w)w_{1}^{i_{1}}\cdots
w_{d}^{i_{d}}&\text{if}\qquad d+1\leq j\leq n.\end{cases}$

where $a_{I,j}\in\mathbb{C}$, $R_{I,j}$ are affine linear functions and

(36) $q_{1}=\ldots=q_{d}=q_{d+1}+1=\ldots=q_{n}+1=\ell+1.$

One can choose $a_{I,j}$ in way that $H_{m}$ are invariants by $Y$( rank-
nullity theorem). If $\ell=0$ then follows from (35) that $H_{d}$ and $H_{m}$
are invariant by ${\mathcal{G}}_{t}$ for any $t\in D\setminus\\{0\\}$, indeed

$\mathcal{D}_{{\mathcal{G}}_{t}}|_{H_{d}}=\mathcal{D}_{(F)_{*}{\mathcal{F}}}|_{H_{d}}+t\cdot
Y|_{H_{d}}=t\cdot
Y|_{H_{d}}=t\sum_{j=d+1}^{n}R_{0,j}(w)\frac{\partial}{\partial w_{j}},$

because $q_{d+1}=\ldots=q_{n}=0$.

Now, if $\ell>0$ then $W\subset{\rm Sing}({\mathcal{G}}_{t})$ and
$\ell=m_{E}({\mathcal{G}}_{t},W)$ for all $t\in D\setminus\\{0\\}$. Again,
$H_{m}$ is also invariant by ${\mathcal{G}}_{t}$.

Denoting by ${\mathcal{F}}_{t}$ the pull-back foliation of ${\mathcal{G}}_{t}$
by $F$, we have that ${\mathcal{F}}_{t}$ define a one-dimensional singular
holomorphic foliation in $U$. Indeed, this foliation is given by a vector
field

$\mathcal{D}_{{\mathcal{F}}_{t}}=P^{t}_{1}\frac{\partial}{\partial
z_{1}}+\ldots+P^{t}_{n}\,\frac{\partial}{\partial z_{n}}$

where $P^{t}(z)=(P^{t}_{1}(z),\ldots,P^{t}_{n}(z))$ are obtained by the system

$Q^{t}\circ F(z)=D_{z}F(z)\cdot P^{t}(z),$

with $Q^{t}=(P_{1}+tQ_{1},\ldots,P_{n}+tQ_{n})$. Solving this system by
Cramer’s rule, we get

$P^{t}_{i}(z)=\frac{\det(A^{t}_{i})}{\det(M)}$

where one gets $A^{t}_{i}$ replacing the $i$-th column of $DF$ by the vector
column $Q^{t}\circ F(z)$. In particular,

$P^{t}_{i}(z)=\frac{Q^{t}_{i}\circ F(z)\cdot\det(M)}{\det(M)}=Q^{t}_{i}\circ
F(z)\cdot\det(M)\qquad\text{for all}\qquad 1\leq i\leq d.$

Therefore, after normalizing by the factor $\det(M)$, one may write
${\mathcal{F}}_{t}$ in $U$ as

$\displaystyle\mathcal{D}_{{\mathcal{F}}_{t}}$ $\displaystyle=$
$\displaystyle\det(A^{t}_{1})\frac{\partial}{\partial{z_{1}}}+\ldots+\det(A^{t}_{d})\frac{\partial}{\partial{z_{d}}}+$
$\displaystyle+Q^{t}_{d+1}\circ
F(z)\cdot\det(M)\frac{\partial}{\partial{z_{d+1}}}+\ldots+Q^{t}_{n}\circ
F(z)\cdot\det(M)\frac{\partial}{\partial{z_{n}}}.$

Since the components of $\mathcal{D}_{{\mathcal{F}}_{t}}$ are polynomials, we
can consider ${\mathcal{F}}_{t}$ defined in $\mathbb{C}^{n}$ by Hartogs
extension Theorem. Moreover, $V$ is invariant by ${\mathcal{F}}_{t}$ (for any
$t\in D\setminus\\{0\\}$), hence it proved $(iv)$.

By construction, $F$ is a local biholomorphism and adjusting the functions
$R_{I,j}$ and the constant $a_{I,j}$ and shrinking $\epsilon$, if necessary,
we can admit that ${\mathcal{F}}_{t}$ is special along $W$ and ${\rm
Sing}\big{(}{\mathcal{F}}_{t}\big{)}=W\cup\\{p_{1}^{t},\ldots,p_{s_{t}}^{t}\\}$,
if $\ell>0$ and ${\rm
Sing}\big{(}{\mathcal{F}}_{t}\big{)}=\\{p_{1}^{t},\ldots,p_{s_{t}}^{t}\\}$, if
$\ell=0$, where each $p_{i}^{t}$ is an isolated point, proving $(iii)-(v)$.
Also by construction, ${\mathcal{F}}_{t}$ satisfies the (vi)-condition. It is
immediate that ${\mathcal{F}}_{0}={\mathcal{F}}$ proving $(i)$. The
coefficients of $R_{I,j}$ are chosen so that we have
$\deg({\mathcal{F}}_{t})=\deg({\mathcal{F}})$ proving $(ii)$. ∎

It follows from Lemma 3.1 that we need to describe two situations: the case in
that ${\mathcal{F}}$ is special along $W$ and the case in that $W$ is
invariant by ${\mathcal{F}}$. First we assume that ${\mathcal{F}}$ is special
along $W$. In particular, the following result gives a proof of Theorem 2.

######  Theorem 3.2.

Let ${\mathcal{F}}$ be a holomorphic foliation by curves of degree $k\geq 1$
defined in $\mathbb{P}^{n}$ such that

${\rm Sing}{{\mathcal{F}}}=W\cup\\{p_{1},\ldots,p_{s}\\}$

is a disjoint union where $p_{i}$ are isolated singularities. Assume
${\mathcal{F}}$ special along $W=Z(h_{1},\ldots,h_{d})$ with
$k_{j}=\mathrm{deg}(h_{j})$. Then

$\sum_{i=1}^{s}\mu({\mathcal{F}},p_{i})=\displaystyle\sum_{i=1}^{n}k^{i}+\nu({\mathcal{F}},W),$

where $\nu({\mathcal{F}},W)$ is given as in (1).

###### Proof.

Let $\pi:\widetilde{\mathbb{P}}^{n}\to\mathbb{P}^{n}$ be the blowup of
$\mathbb{P}^{n}$ along $W$, with exceptional divisor $E$. Given that
${\mathcal{F}}$ is special along $W$, the singular set of
$\widetilde{\mathcal{F}}$ consists only of closed points. Then

$\sum_{j=1}^{s}\mu({\mathcal{F}},p_{j})=N(\widetilde{{\mathcal{F}}},\widetilde{\mathbb{P}}^{n})-N(\widetilde{{\mathcal{F}}},E),$

because $\pi$ is an isomorphism away from $W$. By [1], we have that

$N(\widetilde{\mathcal{F}},\widetilde{\mathbb{P}}^{n})=\displaystyle\int_{\widetilde{\mathbb{P}}^{n}}c_{n}\big{(}T_{\widetilde{\mathbb{P}}^{n}}\otimes\mathcal{L}_{\widetilde{{\mathcal{F}}}}^{*}\big{)}$

where

(37)
$c_{n}(T_{\widetilde{\mathbb{P}}^{n}}\otimes\mathcal{L}_{\widetilde{{\mathcal{F}}}}^{*})=\displaystyle\sum_{j=0}^{n}c_{j}(T_{\widetilde{\mathbb{P}}^{n}})\cdot
c_{1}^{n-j}(\mathcal{L}_{\widetilde{{\mathcal{F}}}}^{*})$

and by Lemma 2.2

(38)
$c_{1}^{n-j}(\mathcal{L}_{\widetilde{{\mathcal{F}}}}^{*})=\displaystyle\sum_{m=0}^{n-j}{{n-j}\choose{m}}\pi^{*}c_{1}^{m}(\mathcal{L}_{{\mathcal{F}}}^{*})(-\ell
E)^{n-j-m}.$

In order to simplify our notation, each term of the expression
$\displaystyle\int_{\widetilde{\mathbb{P}}^{n}}c_{j}(T_{\widetilde{\mathbb{P}}^{n}})\cdot
c_{1}^{n-j}(\mathcal{L}_{\widetilde{{\mathcal{F}}}}^{*})$ will be computed as
the sum of two terms separately. Thus, from (27) its first term is

$\displaystyle\int_{\widetilde{\mathbb{P}}^{n}}\pi^{*}c_{j}(T_{\mathbb{P}^{n}})c_{1}^{n-j}(\mathcal{L}_{\widetilde{{\mathcal{F}}}}^{*})=\sum_{m=0}^{n-j}\int_{\widetilde{\mathbb{P}}^{n}}{{n-j}\choose{m}}\pi^{*}c_{j}(T_{\mathbb{P}^{n}})\pi^{*}c_{1}^{m}(\mathcal{L}_{{\mathcal{F}}}^{*})(-\ell
E)^{n-j-m}.$

Therefore, in this last expression, it is enough to focus only for $m=n-j$ and
$m+j\leq n-d$. Consequently,

$\displaystyle\displaystyle\int_{\widetilde{\mathbb{P}}^{n}}\pi^{*}c_{j}(T_{\mathbb{P}^{n}})c_{1}^{m}(\mathcal{L}_{\widetilde{{\mathcal{F}}}}^{*})$
$\displaystyle=\int_{\widetilde{\mathbb{P}}^{n}}\pi^{*}c_{j}(\mathbb{P}^{n})\pi^{*}c_{1}^{n-j}(\mathcal{L}_{{\mathcal{F}}}^{*})+$
$\displaystyle\displaystyle\sum_{m=0}^{n-d-j}\int_{\widetilde{\mathbb{P}}^{n}}{{n-j}\choose{m}}\rho^{*}c_{j}(\mathbb{P}^{n})\rho^{*}c_{1}^{m}(\mathcal{L}_{{\mathcal{F}}}^{*})(-\ell
E)^{n-j-m}.$

Therefore, the first term results in

(39)
$\begin{array}[]{ll}\displaystyle\int_{\widetilde{\mathbb{P}}^{n}}&\pi^{*}c_{j}(T_{\mathbb{P}^{n}})\cdot
c_{1}^{n-j}(\mathcal{L}_{\widetilde{{\mathcal{F}}}}^{*})=\displaystyle{{n+1}\choose{j}}(k-1)^{n-j}+\\\
&\displaystyle\sum_{m=0}^{n-d-j}\int_{E}{{n-j}\choose{m}}\rho^{*}c_{j}(\mathbb{P}^{n})\rho^{*}c_{1}^{m}(\mathcal{L}_{{\mathcal{F}}}^{*})(-\ell)^{n-j-m}\zeta^{n-j-m-1}.\end{array}$

In a similar way as computed in (32), we obtain the following sum

(40)
$\begin{array}[]{ll}\displaystyle\sum_{j=0}^{n}\int_{\widetilde{\mathbb{P}}^{n}}&\pi^{*}c_{j}(T_{\mathbb{P}^{n}})\cdot
c_{1}^{n-j}(\mathcal{L}_{\widetilde{{\mathcal{F}}}}^{*})=1+k+k^{2}+k^{3}+\ldots+k^{n-1}+k^{n}+\\\
&\displaystyle\sum_{j=0}^{n-d}\sum_{m=0}^{n-d-j}(-1)^{\delta_{j}^{m}-1}{{n+1}\choose{j}}{{n-j}\choose{m}}(k-1)^{m}\ell^{n-j-m}\mathcal{W}_{\delta_{j}^{m}}^{(d)}\deg(W)\end{array}$

where $\delta_{j}^{m}=n-d-j-m$. Now, the second term of (37) is

$\displaystyle\int_{\widetilde{\mathbb{P}}^{n}}\sum_{|a|=0}^{j-1}(-1)^{j-|a|-1}\Gamma_{a}^{j}\rho^{*}c_{a_{1}}(N)\rho^{*}c_{a_{2}}(T_{W})c_{1}^{n-j}(\mathcal{L}_{\widetilde{{\mathcal{F}}}}^{*})E^{j-|a|}.$

In this time, given that $|a|+m\leq j-1+n-j=n-1$, it is enough to consider
$m+|a|\leq n-d$. In the same way as done in the case before, the second term
of (37) results

$\displaystyle\sum_{|a|=0}^{j-1}\sum_{m=0}^{n-d-|a|}(-1)^{\delta_{|a|}^{m}}{{n-j}\choose{m}}\Gamma_{a}^{j}\ell^{n-j-m}(k-1)^{m}\sigma_{a_{1}}^{(d)}\tau_{a_{2}}^{(d)}\mathcal{W}_{\delta_{|a|}^{m}}^{(d)}\deg(W).$

Consequently, after adding these terms,, we get

$\begin{array}[]{rl}&N(\widetilde{{\mathcal{F}}},\widetilde{\mathbb{P}}^{n})=1+k+k^{2}+k^{3}+\ldots+k^{n-1}+k^{n}+\cr&\deg(W)\displaystyle\sum_{j=0}^{n-d}\sum_{m=0}^{n-d-j}(-1)^{\delta_{j}^{m}-1}{{n+1}\choose{j}}{{n-j}\choose{m}}(k-1)^{m}\ell^{n-j-m}\mathcal{W}_{\delta_{j}^{m}}^{(d)}+\cr&\deg(W)\displaystyle\sum_{j=1}^{n}\sum_{|a|=0}^{j-1}\sum_{m=0}^{n-d-|a|}(-1)^{\delta_{|a|}^{m}}{{n-j}\choose{m}}\Gamma_{a}^{j}\ell^{n-j-m}(k-1)^{m}\sigma_{a_{1}}^{(d)}\tau_{a_{2}}^{(d)}\mathcal{W}_{\delta_{|a|}^{m}}^{(d)}.\end{array}$

By Whitney sum, we know that $c(T_{\mathbb{P}^{n}})|_{W}=c(N)|_{W}\cdot
c(T_{W})$, i.e.,

$\displaystyle{{n+1}\choose{j}}=\sum_{a_{1}+a_{2}=j}\sigma_{a_{1}}^{(d)}\tau_{a_{2}}^{(d)},$

we get

(41)
$\begin{array}[]{ll}&N(\widetilde{{\mathcal{F}}},\widetilde{\mathbb{P}}^{n})=1+k+k^{2}+k^{3}+\ldots+k^{n-1}+k^{n}+\cr&\deg(W)\displaystyle\sum_{|a|=0}^{n-d}\sum_{j=|a|}^{n}\sum_{m=0}^{n-d-|a|}(-1)^{\delta_{|a|}^{m}}{{n-j}\choose{m}}\Gamma_{a}^{j}\ell^{n-j-m}(k-1)^{m}\sigma_{a_{1}}^{(d)}\tau_{a_{2}}^{(d)}\mathcal{W}_{\delta_{|a|}^{m}}^{(d)}.\end{array}$

Now, we will compute the number of isolated singularities, counted with
multiplicity, of $\widetilde{{\mathcal{F}}}$ on $E$. Again, by Baum-Bott’s
formula, we have that

$N(\widetilde{{\mathcal{F}}},E)=\int_{E}c_{n-1}(\mathcal{T}_{E}\otimes\mathcal{L}_{\widetilde{{\mathcal{F}}}}^{*})$

with

$c_{n-1}(\mathcal{T}_{E}\otimes\mathcal{L}_{\widetilde{{\mathcal{F}}}}^{*})=\sum_{i=0}^{n-1}c_{i}(E)\cdot
c_{1}(\mathcal{L}_{\widetilde{{\mathcal{F}}}}^{*})^{n-i-1}.$

On the one hand,

(42)
$c_{i}(E)=c_{i}(\mathcal{T}_{\widetilde{\mathbb{P}}^{n}}\otimes\mathcal{O}_{E})-c_{i-1}(E)\zeta$

and reapplying (42) recursively we obtain

$c_{i}(E)=\displaystyle\sum_{j=0}^{i}(-1)^{j}c_{i-j}(\mathcal{T}_{\widetilde{\mathbb{P}}^{n}}\otimes\mathcal{O}_{E})\zeta^{j}.$

Then, using also (26), we get

$\displaystyle c_{i}(E)$
$\displaystyle=\sum_{j=0}^{i}(-1)^{i-j}\pi^{*}c_{j}(\mathbb{P}^{n})\zeta^{i-j}+$
$\displaystyle\ \ \
+\displaystyle\sum_{j=0}^{i}\sum_{|a|=0}^{j-1}(-1)^{i-|a|-1}\Gamma_{a}^{j}\rho^{*}c_{a_{1}}(N)c_{a_{2}}(T_{W})\zeta^{i-|a|}$
$\displaystyle=\sum_{|a|=0}^{i}(-1)^{i-|a|}\rho^{*}c_{a_{1}}(N)c_{a_{2}}(T_{W})\zeta^{i-|a|}+$
$\displaystyle\ \ \
+\displaystyle\sum_{|a|=0}^{i-1}\sum_{j=|a|+1}^{i}(-1)^{i-|a|-1}\Gamma_{a}^{j}\rho^{*}c_{a_{1}}(N)c_{a_{2}}(T_{W})\zeta^{i-|a|}$

Given that

$\sum_{j=|a|+1}^{i}\Gamma_{a}^{j}=\displaystyle 1-{{d-a_{1}}\choose{i-|a|}}$

follows that

(43)
$c_{i}(E)=\displaystyle\sum_{|a|=0}^{i}(-1)^{i-|a|}{{d-a_{1}}\choose{i-|a|}}\rho^{*}c_{a_{1}}(N)c_{a_{2}}(T_{W})\zeta^{i-|a|}$

On the other hand, as
$c_{1}(\mathcal{L}_{\widetilde{{\mathcal{F}}}}^{*})=\pi^{*}c_{1}(\mathcal{L}_{{\mathcal{F}}}^{*})-\ell\zeta$,
we have

(44)
$c_{1}(\mathcal{L}_{\widetilde{{\mathcal{F}}}}^{*})^{n-i-1}=\displaystyle\sum_{m=0}^{n-i-1}{{n-i-1}\choose{m}}\pi^{*}c_{1}(\mathcal{L}_{{{\mathcal{F}}}}^{*})^{m}(-\ell\zeta)^{n-i-m-1}.$

Again, as done in the case before, we obtain that
$N(\widetilde{{\mathcal{F}}},E)$ is equal to

$\displaystyle\deg(W)\sum_{i=0}^{n-1}\sum_{|a|=0}^{i}\sum_{m=0}^{n-d-|a|}(-1)^{\delta_{|a|}^{m}}{{n-i-1}\choose{m}}{{d-a_{1}}\choose{i-|a|}}\frac{(k-1)^{m}}{\ell^{m}}\ell^{n-1-i}\sigma_{a_{1}}^{(d)}\tau_{a_{2}}^{(d)}\mathcal{W}_{\delta_{|a|}^{m}},$

in which can be rewritten as

$\deg(W)\sum_{|a|=0}^{n-d}\sum_{j=|a|+1}^{n}\sum_{m=0}^{n-d-|a|}(-1)^{\delta_{|a|}^{m}}{{n-j}\choose{m}}{{d-a_{1}}\choose{j-|a|}-1}\frac{(k-1)^{m}}{\ell^{m}}\ell^{n-j}\sigma_{a_{1}}^{(d)}\tau_{a_{2}}^{(d)}\mathcal{W}_{\delta_{|a|}^{m}}.$

However,

${\psi}_{a}(x)=(1+x)^{d-a_{1}}x^{n-d-
a_{2}-1}=\sum_{i=0}^{n-|a|-1}{{d-a_{1}}\choose{i}}x^{n-|a|-i-1}$

which yields

$\sum_{j=|a|+1}^{n}{{n-j}\choose{m}}{{d-a_{1}}\choose{j-|a|}-1}\ell^{n-j-m}=\frac{{\psi}_{a}^{(m)}(x)}{m!}.$

Therefore,

(45)
$N(\widetilde{{\mathcal{F}}},E)=\sum_{|a|=0}^{n-d}\sum_{m=0}^{n-d-|a|}(-1)^{\delta_{|a|}^{m}}\frac{{\psi}_{a}^{(m)}(\ell)}{m!}(k-1)^{m}\sigma_{a_{1}}^{(d)}\tau_{a_{2}}^{(d)}\mathcal{W}_{\delta_{|a|}^{m}}.$

Now, the formula stated in the theorem is obtained making

$\sum_{j=1}^{s}\mu({\mathcal{F}},p_{j})=N(\widetilde{{\mathcal{F}}},\widetilde{\mathbb{P}}^{n})-N(\widetilde{{\mathcal{F}}},E),$

that is,

$\displaystyle\displaystyle\sum_{j=1}^{s}\mu({\mathcal{F}},p_{j})=1+k+k^{2}+k^{3}+\ldots+k^{n-1}+k^{n}+$
$\displaystyle\deg(W)\sum_{|a|=0}^{n-d}\sum_{j=|a|}^{n}\sum_{m=0}^{n-d-|a|}(-1)^{\delta_{|a|}^{m}-1}{{n-j}\choose{m}}{{d-a_{1}}\choose{j-|a|}}(k-1)^{m}\ell^{n-i-m}\sigma_{a_{1}}^{(d)}\tau_{a_{2}}^{(d)}\mathcal{W}_{\delta_{|a|}^{m}}^{(d)}.$

But, given that

$\displaystyle\sum_{j=|a|}^{n}{{n-j}\choose{m}}{{d-a_{1}}\choose{j-|a|}}\ell^{n-j-m}=\frac{\varphi_{a}^{(m)}(\ell)}{m!}$

with $\varphi_{a}(x)=x^{n-d-a_{2}}(1+x)^{d-a_{1}}$, we get the final
expression given in the Theorem 1. ∎

Now we consider the case where $W$ is invariant by ${\mathcal{F}}$.

######  Proposition 3.3.

Let ${\mathcal{F}}$ be a holomorphic foliation by curves on $\mathbb{P}^{n}$
of degree $k$. Assume that $W\subset\mathbb{P}^{n}$ is smooth subvariety of
codimension $d\geq 2$ invariant by ${\mathcal{F}}$. Then, the number of
isolated singularities (counted with multiplicity) of ${\mathcal{F}}$ on $W$
is

$N({\mathcal{F}},W)=-\nu({\mathcal{F}},W)|_{\ell=0}.$

###### Proof.

It is enough to observe that

$\varphi_{a}(x)=x^{n-d-a_{2}}(1+x)^{d-a_{1}}=x^{n-d-
a_{2}}+(d-a_{1})x^{n-a_{1}-a_{2}}+\ldots,$

thus, $\varphi_{a}^{(m)}(0)=0$ for $m=0,\ldots,n-d-a_{2}-1$ and
$\varphi_{a}^{(m)}(0)=m!$ for $m=n-d-a_{2}$. Now, if $a=(a_{1},a_{2})$ with
$a_{1}>0$ then $\varphi_{a}^{(m)}(0)=0$ for $m=0,\ldots,n-d-|a|$, because
$m\leq n-d-|a|=n-d-a_{1}-a_{2}<n-d-a_{2}$. Consequently,
$\varphi_{a}^{(m)}(0)\neq 0$ if, and only if, $a=(a_{1},a_{2})$ with $a_{1}=0$
and $m=n-d-|a|$. Writing $|a|=a_{2}=i$, we get

$\displaystyle\displaystyle\nu({\mathcal{F}},W)|_{\ell=0}$
$\displaystyle=-\displaystyle\sum_{i=1}^{n-d}\deg(W)\tau_{i}^{(d)}(k-1)^{n-d-i}$
$\displaystyle=-\displaystyle\sum_{i=0}^{n-d}\int_{W}c_{i}(T_{W})\cdot
c_{1}(\mathcal{L}_{{\mathcal{F}}}^{*})^{n-d-i}$
$\displaystyle=-\displaystyle\int_{W}c_{n-d}(T_{W}\otimes\mathcal{L}_{{\mathcal{F}}}^{*})$
$\displaystyle=-N({\mathcal{F}},W).$

∎

##  4\. Proof of Theorem 1

It suffices to prove Theorem 1 for a holomorphic foliation by curves
${\mathcal{F}}$ with only an irreducible component $W$ (of positive dimension)
of ${\rm Sing}({\mathcal{F}})$. In this sense, let ${\mathcal{F}}$ be a
foliation by curves on $\mathbb{P}^{n}$ of degree $k$ such that

$\mbox{Sing}({\mathcal{F}})=W\cup\\{p_{1},\ldots,p_{s}\\},$

where $W=Z(h_{1},\ldots,h_{d})$. There exists a one-parameter family of
foliations by curves on $\mathbb{P}^{n}$, given by
$\\{{\mathcal{F}}_{t}\\}_{t\in D}$ where
$D=\\{t\in\mathbb{C}\,:\,|t|<\epsilon\\}$ satisfying conditions $(i)-(v)$ of
Lemma 3.1. When $\ell>0$, we have

$\displaystyle\sum_{j=1}^{s}\mu({\mathcal{F}},p_{j})=\lim_{t\to
0}\sum_{\lim\widetilde{p}_{j}^{t}\notin
E}\mu(\widetilde{{\mathcal{F}}}_{t},\widetilde{p}_{j}^{t}),$

where $\widetilde{{\mathcal{F}}}_{t}$ is the induced foliation by
${\mathcal{F}}_{t}$ via the blowup
$\pi:\widetilde{\mathbb{P}}^{n}\to\mathbb{P}^{n}$ centered at $W$ with
exceptional divisor $E$ and ${\rm
Sing}(\widetilde{{\mathcal{F}}}_{t})=\\{\widetilde{p}_{j}^{t}:\,\,j=1,\ldots,\widetilde{s}_{t}\\}$.
On the other hand,

$\displaystyle\sum_{j=1}^{s}\mu({\mathcal{F}},p_{j})=N(\widetilde{{\mathcal{F}}_{t}},\widetilde{\mathbb{P}}^{n})-\lim_{t\to
0}\sum_{\lim\widetilde{p}_{j}^{t}\in
E}\mu(\widetilde{{\mathcal{F}}_{t}},\widetilde{p}_{j}^{t}).$

Since ${\mathcal{F}}_{t}$ is special along $W$, by Theorem 3.2, we get

$\displaystyle\sum_{j=1}^{s}\mu({\mathcal{F}},p_{j})=N(\widetilde{{\mathcal{F}}}_{t},\widetilde{\mathbb{P}}^{n})-N(\widetilde{{\mathcal{F}}}_{t},E)-N({\mathcal{F}},A_{W}).$

Then,

$\displaystyle\sum_{j=1}^{s}\mu({\mathcal{F}},p_{j})=\displaystyle\sum_{i=1}^{n}k^{i}+\nu({\mathcal{F}},W)-N({\mathcal{F}},A_{W}).$

We finish the proof by considering $\ell=0$. According to Lemma 3.1, we can
suppose that $W$ is invariant by ${\mathcal{F}}_{t}$. Note that

$\displaystyle\sum_{j=1}^{s}\mu({\mathcal{F}},p_{j})=\lim_{t\to 0}\sum_{\lim
p_{j}^{t}\notin W}\mu({\mathcal{F}}_{t},p_{j}^{t}).$

Therefore,

$\displaystyle\sum_{j=1}^{s}\mu({\mathcal{F}},p_{j})=N({\mathcal{F}}_{t},\mathbb{P}^{n})-\lim_{t\to
0}\sum_{\lim p_{j}^{t}\in W}\mu({\mathcal{F}}_{t},p_{j}^{t}).$

Thus,

$\displaystyle\sum_{j=1}^{s}\mu({\mathcal{F}},p_{j})=N({\mathcal{F}}_{t},\mathbb{P}^{n})-N({\mathcal{F}}_{t},W)-N({\mathcal{F}},A_{W}).$

By Proposition 3.3, we get

$\displaystyle\sum_{j=1}^{s}\mu({\mathcal{F}},p_{j})=\displaystyle\sum_{i=1}^{n}k^{i}+\nu({\mathcal{F}},W)|_{\ell=0}-N({\mathcal{F}},A_{W}).$

##  5\. Deformation of a foliation by curves in presence of an invariant
subvariety

The aim of this section is to prove an adaption of Lemma 3.1 for foliations by
curves which admit an invariant subvariety of $\mathbb{P}^{n}$.

######  Proposition 5.1.

Let ${\mathcal{F}}$ be a holomorphic foliation by curves on $\mathbb{P}^{n}$,
$n\geq 3$, of degree $k$ such that $W=Z(h_{1},\ldots,h_{d})\subset{\rm
Sing}({\mathcal{F}})$. Assume that the smooth variety
$V=Z(f_{1},\ldots,f_{m})$ is invariant by ${\mathcal{F}}$ such that
$f_{j}\in\mathcal{I}(W)$ and $1\leq m<d$. If ${\mathcal{F}}$ is special along
$W$ then

  1. (i)

$N(\widetilde{{\mathcal{F}}},\widetilde{V})=N({\mathcal{F}},V)+\displaystyle\sum_{i=0}^{n-d}\sum_{j=0}^{n-d-i}{{n-m-i}\choose{j}}\tau_{i}^{(m)}(k-1)^{j}(-\ell)^{\gamma_{i}^{j}}\alpha_{\widetilde{V}_{E}}^{(\gamma_{i}^{j}-1)},$

  2. (ii)

$N(\widetilde{{\mathcal{F}}},\widetilde{V}_{E})=\displaystyle\sum_{i=0}^{n-d}\sum_{j=0}^{n-d-i}\sum_{p=i+1}^{n-m-j}{{n-m-p}\choose{j}}(-1)^{\gamma_{i}^{j}-1}\ell^{\gamma_{p}^{j}}\tau_{i}^{(m)}(k-1)^{j}\alpha_{\widetilde{V}_{E}}^{\gamma_{i}^{j}-1},$

  3. (iii)

$\nu({\mathcal{F}},V,W)=N({\mathcal{F}},V)+\displaystyle\sum_{i=0}^{n-d}\sum_{j=0}^{n-d-i}\frac{\Omega^{(j)}(\ell)}{j!}(-1)^{\gamma_{i}^{j}}\ell^{\gamma_{p}^{j}}\tau_{i}^{(m)}(k-1)^{j}\alpha_{\widetilde{V}_{E}}^{\gamma_{i}^{j}-1},$

where $\tau_{j}^{(m)}:=\tau_{j}^{(m)}(d_{1},\ldots,d_{q})$, as (30),
$d_{j}=\deg(f_{j})$, $\gamma_{j}^{i}=n-m-i-j$, $\widetilde{{\mathcal{F}}}$ is
the strict transform of ${\mathcal{F}}$ by the blowup $\pi$ at $W$,
$\widetilde{V}=\overline{\pi^{-1}(V\setminus W)}$,
$\widetilde{V}_{E}=\widetilde{V}\cap E$,
$\alpha_{\widetilde{V}_{E}}^{(i)}:=\displaystyle\int_{{V}_{E}}\pi^{*}(\mathbf{h})^{n-m-1-i}\cdot\zeta^{i}$
and $\Omega(x)=(x^{n-m-i}+x^{n-m-i-1}+\ldots+x^{j})$.

###### Proof.

Let
$\alpha_{V}^{(i)}:=\displaystyle\int_{\widetilde{V}}\pi^{*}(\mathbf{h})^{n-m-i}\cdot\zeta^{i}$
for $i\geq 0$. It is not difficult to see that $\alpha_{V}^{(0)}=\deg(V)$ and
$\alpha_{V}^{(i)}=\alpha_{V_{E}}^{(i-1)}$, for $i\geq 1$. Now, given that
$\alpha_{V_{E}}^{(i)}=0$ for $i<d-m-1$, then

$\alpha_{V_{E}}^{(d-m-1)}=(-1)^{d-m-1}\deg(W).$

Assume that we have defined $\alpha_{V_{E}}^{q}$ for $q=0,\ldots,i-1<d-1$. By
(43), we have the following relation

$(-1)^{i}{{d}\choose{i}}\zeta^{i}=c_{i}(E)-\displaystyle\sum_{|a|=1}^{i}(-1)^{i-|a|}{{d-a_{1}}\choose{i-|a|}}\rho^{*}c_{a_{1}}(N)c_{a_{2}}(T_{W})\zeta^{i-|a|}.$

As a consequence

(46)
$(-1)^{i}{{d}\choose{i}}\alpha_{V_{E}}^{(i)}={{d}\choose{\varrho_{m}}}\tau_{i-\varrho_{m}}^{(d)}\deg(W)-\displaystyle\sum_{|a|=1}^{i}(-1)^{i-|a|}{{d-a_{1}}\choose{i-|a|}}\sigma_{a_{1}}^{(d)}\tau_{a_{2}}^{(d)}\alpha_{V_{E}}^{(i-|a|)}.$

where $\varrho_{m}=d-m-1$. Furthermore, in order to compute
$\alpha_{{V}_{E}}^{(i)}$ for $i\geq d$, we will use (31). Again, the
computation of $N(\widetilde{{\mathcal{F}}},\widetilde{V})$ follows directly
from the Baum-Bott formula. Thus,

$N(\widetilde{{\mathcal{F}}},\widetilde{V})=\displaystyle\int_{\widetilde{V}}c_{n-m}(T_{\widetilde{V}}\otimes\mathcal{L}_{\widetilde{{\mathcal{F}}}}^{*})=\displaystyle\int_{\widetilde{V}}\sum_{i=0}^{n-m}c_{i}(T_{\widetilde{V}})c_{1}(\mathcal{L}_{\widetilde{{\mathcal{F}}}}^{*})^{n-m-i}$

where $c_{i}(T_{\widetilde{V}})=\pi^{*}c_{i}(T_{V})$. As before, we obtain

$\displaystyle
N(\widetilde{{\mathcal{F}}},\widetilde{V})=\sum_{i=0}^{n-m}\sum_{j=0}^{n-m-i}\int_{\widetilde{V}}{{n-m-i}\choose{j}}\pi^{*}c_{i}(T_{V})\pi^{*}c_{1}(\mathcal{L}_{{{\mathcal{F}}}}^{*})^{j}(-\ell\zeta)^{\gamma_{j}^{i}}.$

For $j=n-m-i$ and $i+j\leq n-d$ in the last sum, we have concluded the proof
of (i).

To prove $(ii)$, it is enough to observe that $\widetilde{V}$ and the
exceptional divisor $E$ are nonsingular subvarieties of a nonsingular complex
variety ${\widetilde{\mathbb{P}}^{n}}$. Thus, given that $\widetilde{V}$ and
$E$ intersect properly and transversally, the intersection $\widetilde{V}_{E}$
is nonsingular and an elementary Chern class computation proves that

$c(T_{E})\cdot c(\widetilde{V})=c(T_{\widetilde{\mathbb{P}}^{n}})\cdot
c(T_{\widetilde{V}_{E}}).$

As consequence, we obtain that

$c_{q}(T_{\widetilde{V}_{E}})=\displaystyle\sum_{i=0}^{q}(-1)^{q-i}c_{i}(T_{\widetilde{V}})\cdot(\zeta)^{q-i}.$

To conclude the proof of (2), it is enough to consider

$N(\widetilde{{\mathcal{F}}},\widetilde{V}_{E})=\displaystyle\int_{\widetilde{V}_{E}}c_{n-m-1}(T_{\widetilde{V}_{E}}\otimes\mathcal{L}_{\widetilde{{\mathcal{F}}}}^{*})=\int_{\widetilde{V}_{E}}\sum_{q=0}^{n-m-1}c_{q}(T_{\widetilde{V}_{E}})c_{1}(\mathcal{L}_{\widetilde{{\mathcal{F}}}}^{*})^{n-m-1-q}.$

Now, we need to repeat the same steps,

$N(\widetilde{{\mathcal{F}}},\widetilde{V}_{E})=\displaystyle\sum_{q=0}^{n-m-1}\sum_{i=0}^{q}\sum_{j=0}^{n-m-1-q}{{n-m-1-q}\choose{j}}(-1)^{\gamma_{i}^{j}-1}\ell^{\gamma_{q}^{j}-1}\tau_{i}^{(m)}(k-1)^{j}\alpha_{\widetilde{V}_{E}}^{\gamma_{i}^{j}-1}.$

Thus,

$N(\widetilde{{\mathcal{F}}},\widetilde{V}_{E})=\displaystyle\sum_{i=0}^{n-m-1}\sum_{j=0}^{n-m-1-i}\sum_{q=i}^{n-m-1-j}{{n-m-1-q}\choose{j}}(-1)^{\gamma_{i}^{j}-1}\ell^{\gamma_{q}^{j}-1}\tau_{i}^{(m)}(k-1)^{j}\alpha_{\widetilde{V}_{E}}^{\gamma_{i}^{j}-1}.$

Making $q:=p-1$ and considering $i+j\leq n-d$, we conclude the proof of (ii).
Finally, to proof (iii) it is enough to observe that
$\nu({\mathcal{F}},V,W)=N({\mathcal{F}},V\setminus
W)=N(\widetilde{{\mathcal{F}}},\widetilde{V})-N(\widetilde{{\mathcal{F}}},\widetilde{V}_{E}).$
Therefore,
$\nu({\mathcal{F}},V,W)=N({\mathcal{F}},V)+\displaystyle\sum_{i=0}^{n-d}\sum_{j=0}^{n-d-i}\sum_{p=i}^{n-d-j}{{n-m-p}\choose{j}}(-1)^{\gamma_{i}^{j}}\ell^{\gamma_{p}^{j}}\tau_{i}^{(m)}(k-1)^{j}\alpha_{\widetilde{V}_{E}}^{\gamma_{i}^{j}-1}$.
But,

$\sum_{p=i}^{n-m-j}{{n-m-p}\choose{j}}\ell^{\gamma_{p}^{j}}=\displaystyle\frac{1}{j!}\frac{d^{j}}{dx^{j}}(x^{n-m-i}+x^{n-m-i-1}+\ldots+x^{j})\bigg{|}_{x=\ell}$

which concludes the proof of (iii). ∎

######  Remark 5.2.

The proof of Proposition 5.1 may be obtained by Theorem 1 replacing
$\mathbb{P}^{n}$ by $V$. Indeed, let $\pi_{V}:\widetilde{V}\to V$ be the
blowup of $V$ centered at $W$ with $E_{V}$ the exceptional divisor. Naturally,
$E_{V}=E\cap V=V_{E}$, $\zeta_{V}=\zeta|_{V}:=c_{1}(N_{V_{E}/\widetilde{V}})$
and $\pi_{V}=\pi|_{V}$. Furthermore, if we consider $\ell=0$ then we obtain
$\nu({\mathcal{F}},V,W)=N({\mathcal{F}},V)-N({\mathcal{F}},W)$. In fact, for
the case where $d-m=1$ we have that $\zeta_{V}-\pi^{*}(c_{1}(N_{W/V})=0$. See
(20). Therefore,

$N(\widetilde{{\mathcal{F}}},\widetilde{V}_{E})=\displaystyle\sum_{i=0}^{n-d}\int_{\widetilde{V}_{E}}\pi_{V}^{*}c_{i}(T_{\widetilde{V}_{E}})\pi_{V}^{*}c_{1}(\mathcal{L}_{{{\mathcal{F}}}}^{*})^{n-d-i}.$

By induction, we can prove that

$c_{i}(T_{\widetilde{V}_{E}})=\sum_{j=0}^{i}=(-1)^{i-j}\zeta_{V}^{i-j}\pi_{V}^{*}c_{j}(T_{V})=\pi_{V}^{*}c_{i}(T_{W})$

because $c_{i}(T_{V})|_{W}=c_{1}(N_{W/V})|_{W}c_{i-1}(T_{W})+c_{i}(T_{W})$. As
a consequence, if $d-m=1$ then

$N(\widetilde{{\mathcal{F}}},\widetilde{V}_{E})=\sum_{i=0}^{n-d}\int_{W}c_{i}(T_{W})\cdot\pi_{V}^{*}c_{1}(\mathcal{L}_{{\mathcal{F}}}^{*})^{n-d-i}=N({\mathcal{F}},W).$

For the general case where $q=d-m\geq 2$, again by (20) we have that

$\zeta_{V}^{q}-\pi_{V}^{*}c_{1}(N_{W/V})+\ldots+(-1)^{q}\pi_{V}^{*}c_{q}(N_{W/V})=0,$

$\alpha_{\widetilde{V}_{E}}^{(i)}=\int_{\widetilde{V}_{E}}\pi_{V}^{*}(\mathbf{h})^{n-m-1-i}\zeta_{V}^{i}=0$
for $i=0,\ldots,q-2$ and
$\alpha_{\widetilde{V}_{E}}^{(q-1)}=(-1)^{q-1}\deg(W)$ which yields

$N(\widetilde{{\mathcal{F}}},\widetilde{V}_{E})=\int_{\widetilde{V}_{E}}\sum_{i=q-1}^{n-m-1}\sum_{j=0}^{i-q+1}(-1)^{i-j}\zeta_{V}^{i-j}\pi_{V}^{*}c_{j}(T_{V})\cdot\pi_{V}^{*}c_{1}(\mathcal{L}_{{{\mathcal{F}}}}^{*})^{n-m-1-i}.$

Fixing $i\in\\{q-1,\ldots,n-m-1\\}$, by induction we prove that

$\sum_{j=0}^{i-q+1}(-1)^{i-j}\zeta_{V}^{i-j}\pi_{V}^{*}c_{j}(T_{V})=(-1)^{q-1}\pi_{V}^{*}c_{i-q+1}(T_{W})\zeta_{V}^{q-1}+\ldots$

which yields

$N(\widetilde{{\mathcal{F}}},\widetilde{V}_{E})=\sum_{i=q-1}^{n-m-1}\tau_{i-q+1}^{(d)}(k-1)^{n-d-i+q-1}deg(W)=N({\mathcal{F}},W)$

for $\ell=0$.

##  6\. Proof of Theorem 3

Let ${\mathcal{F}}$ be a foliation by curves on $\mathbb{P}^{n}$ of degree $k$
such that

$\mbox{Sing}({\mathcal{F}})=W\cup\\{p_{1},\ldots,p_{s}\\},$

where $W=Z(h_{1},\ldots,h_{d})$ and

$V=Z(f_{1},\ldots,f_{m})$

is invariant by ${\mathcal{F}}$ such that $f_{i}\in\mathcal{I}(W)$ for all
$1\leq i\leq m$. It is clear that $W\subset V$. By Lemma 3.1, there exists a
one-parameter family of holomorphic foliations by curves on $\mathbb{P}^{n}$
given by $\\{{\mathcal{F}}_{t}\\}_{t\in D}$ where
$D=\\{t\in\mathbb{C}\,:\,|t|<\epsilon\\}$ satisfying conditions $(i)-(vi)$.

We will consider two cases, $\ell=0$ and $\ell>0$. If $\ell=0$, then $W$ and
$V$ are ${\mathcal{F}}_{t}$-invariant for all $t\in D\setminus\\{0\\}$. As
consequence,

$\displaystyle\sum_{p_{j}\in V}\mu({\mathcal{F}},p_{j})=\lim_{t\to
0}\sum_{\lim p_{i}^{t}\in V}\mu({\mathcal{F}}_{t},p_{i}^{t})-\lim_{t\to
0}\sum_{\lim p_{i}^{t}\in W}\mu({\mathcal{F}}_{t},p_{i}^{t}).$

Thus,

$\displaystyle\sum_{p_{j}\in
V}\mu({\mathcal{F}},p_{j})=N({\mathcal{F}}_{t},V)+N({\mathcal{F}},A_{V})-N({\mathcal{F}}_{t},W)-N({\mathcal{F}},A_{W}).$

Therefore,

$\displaystyle\sum_{p_{j}\in
V}\mu({\mathcal{F}},p_{j})=N({\mathcal{F}}_{t},V)-N({\mathcal{F}},W)+N({\mathcal{F}},A_{V\setminus
W})$

with $N({\mathcal{F}},A_{V\setminus W})\geq 0.$ Now, in the case where
$\ell>0$ we get

$\displaystyle\sum_{p_{j}\in V}\mu({\mathcal{F}},p_{j})=\lim_{t\to
0}\sum_{\lim\widetilde{p}_{i}^{t}\in\widetilde{V}}\mu(\widetilde{{\mathcal{F}}_{t}},\widetilde{p}_{i}^{t})-\lim_{t\to
0}\sum_{\lim\widetilde{p}_{j}^{t}\in\widetilde{V}_{E}}\mu(\widetilde{{\mathcal{F}}_{t}},\widetilde{p}_{j}^{t})$

where $\widetilde{V}$ is the strict transform of $V$ via $\pi$ and
$\widetilde{V}_{E}=\widetilde{V}\cap E$. Then,

$\displaystyle\sum_{p_{j}\in
V}\mu({\mathcal{F}},p_{j})=N(\widetilde{{\mathcal{F}}_{t}},\widetilde{V})+N(\widetilde{{\mathcal{F}}},A_{\widetilde{V}})-N(\widetilde{{\mathcal{F}}_{t}},\widetilde{V}_{E})-N(\widetilde{{\mathcal{F}}},A_{\widetilde{V}_{E}}).$

So, given that $N({\mathcal{F}},A_{W})\geq
N(\widetilde{{\mathcal{F}}},A_{\widetilde{V}_{E}})$, we get

$\displaystyle\sum_{p_{j}\in V}\mu({\mathcal{F}},p_{j})\leq
N(\widetilde{{\mathcal{F}}},\widetilde{V})-N(\widetilde{{\mathcal{F}}},\widetilde{V}_{E})+N(\widetilde{{\mathcal{F}}},A_{V\setminus
W}),$

where $N(\widetilde{{\mathcal{F}}},\widetilde{V})$ and
$N(\widetilde{{\mathcal{F}}},\widetilde{V}_{E})$ are given in Proposition 5.1.
Since

$N(\tilde{{\mathcal{F}}},\tilde{V})-N(\tilde{{\mathcal{F}}},\tilde{V}_{E})=N({\mathcal{F}},V)+\nu({\mathcal{F}},V,W),$

we conclude

$\sum_{p\in V\setminus W}\mu({\mathcal{F}},p)\leq
N({\mathcal{F}},V)+\nu({\mathcal{F}},V,W)+N({\mathcal{F}},A_{V\setminus W}).$

Finally, note that as in Theorem 1, these expressions coincide if $\ell=0$.

##  7\. Explicit computations for a family of foliations by curves

######  Lemma 7.1.

Let ${\mathcal{F}}$ be a foliation by curves on $\mathbb{P}^{n}$ of degree $k$
such that $W\subset{\rm Sing}({\mathcal{F}})$ with $W=V(f_{1},\ldots,f_{d})$
and $k_{j}=\deg(f_{j})=1$ for $i=1,\ldots,d$. Assume that
$\ell=mult_{E}({\mathcal{F}})=k-1$. Then

  1. (i)

$\nu({\mathcal{F}},W)=-(k^{d}+\ldots+k^{n})$,

  2. (ii)

$N(\widetilde{{\mathcal{F}}},E)=(n+1-d)(1+k+\ldots+k^{d-1}).$

###### Proof.

Given that $k_{1}=\ldots=k_{d}=1$,
$c(N_{W/\mathbb{P}^{n}})=\sum\sigma_{i}^{(d)}\mathbf{h}^{i}$,
$\sigma_{i}^{(d)}={{d}\choose{i}}$,
$c(T_{W})=\sum\tau_{i}^{(d)}\mathbf{h}^{i}$,
$\tau_{i}^{(d)}={{n+1-d}\choose{i}}$ and
$\mathcal{W}^{(d)}_{\delta}:=\mathcal{W}^{(d)}_{\delta}(k_{1},\ldots,k_{d})={{\delta+d-1}\choose{d-1}}$.
To prove (i) it is enough to observe that

$\sum_{a_{1}+a_{2}=j}\varphi_{a}(x)\sigma_{a_{1}}^{(d)}\tau_{a_{2}}^{(d)}=\frac{\Psi_{n}^{(j)}(x)}{x\cdot
j!}$

where $\Psi_{n}(x)=(1+x)^{d}\cdot x^{n+1-d}$. Then,

$\nu_{n}({\mathcal{F}},W)=-\sum_{m=0}^{n-d}\sum_{j=0}^{n-d-m}\sum_{i=0}^{m}(-1)^{\delta_{i}^{j}}\frac{\ell^{i-1}\Psi_{n}^{(i+j)}(\ell)}{i!j!}{{n-m-j-1}\choose{d-1}},$

with $\delta_{i}^{j}=n-d-i-j$. For fixed $d$, we obtain the relation

$\nu_{n+1}({\mathcal{F}},W)=\nu_{n}({\mathcal{F}},W)-\sum_{i=0}^{n+1-d}\sum_{j=0}^{n+1-d-i}\frac{\ell^{i}\Psi_{n}^{(i+j)}(\ell)}{i!j!}{{n-i-j}\choose{d-1}}.$

Result (i) follows because $\nu_{d}({\mathcal{F}},W)=k^{d}$ and

$\sum_{i=0}^{n+1-d}\sum_{j=0}^{n+1-d-i}\frac{\ell^{i}\Psi_{n}^{(i+j)}(\ell)}{i!j!}{{n-i-j}\choose{d-1}}=(1+\ell)^{n+1}=k^{n+1}.$

The proof of (ii) is done in a similar way using (45). ∎

######  Example 7.2.

Let ${\mathcal{F}}_{k}$ be the holomorphic foliation by curves of degree $k$
defined in $\mathbb{P}^{n}$, $n\geq 3$, such that in the affine open
$U_{n}=\\{[\xi_{0}:\cdots:\xi_{n}]\in\mathbb{P}^{n},\xi_{n}\neq 0\\}$ is
described by the following vector field

$X_{{\mathcal{F}}_{k}}=\sum_{i=1}^{n}\sum_{|a|=m_{i}}\vartheta_{a,i}(x)x_{1}^{a_{1}}\cdots
x_{d}^{a_{d}}\frac{\partial}{\partial
x_{i}}:=\sum_{i=1}^{n}P_{i}(x)\frac{\partial}{\partial x_{i}},\quad
x_{i}=\frac{\xi_{i-1}}{\xi_{n}}$

where $2\leq d\leq n-1$, $a=(a_{1},\ldots,a_{d})$,
$m_{1}=\ldots=m_{d}=m_{d+1}+1=\ldots=m_{n}+1=k$, $\vartheta_{a,i}$ is a
constant for $i=1,\ldots,d$ and $\vartheta_{a,i}$ is an affine linear function
for $i=d+1,\ldots,n$. If $k\geq 2$ then the set $W_{0}\cong\mathbb{P}^{n-d}$
given by $\xi_{0}=\ldots=\xi_{d-1}=0$ is contained in the singular locus of
${\mathcal{F}}_{k}$. Otherwise, if $k=1$ then $W_{0}$ is an invariant set by
${\mathcal{F}}_{k}$ since $m_{i}=0$, for $d<i\leq n$. Let us assume that

$W_{0}=Z(P_{1}(\xi),\ldots,P_{d}(\xi)).$

This hypothesis is essential in order that $W_{0}$ be an irreducible component
of ${\rm Sing}({\mathcal{F}}_{k})$. As before, for $k\geq 2$, let
$\pi:\widetilde{\mathbb{P}}^{n}\to\mathbb{P}^{n}$ be the blowup of
$\mathbb{P}^{n}$ at $W_{0}$ with $E$ is the exceptional divisor. It is not
difficult to see that $\deg(W_{0})=1$ and
$\ell=m_{E}({\mathcal{F}}_{k},W_{0})=k-1$.

If ${\rm Sing}({\mathcal{F}}_{k})=W_{0}\cup\\{p_{1},\ldots,p_{r}\\}$ then

(47) $\sum_{j=1}^{r}\mu({\mathcal{F}}_{k},p_{j})=1+k+\ldots+k^{d-1}$

for all $k\geq 2$. In fact, for $k\geq 2$ and $n=d+1$, i.e.; $W_{0}$ is a
curve, we will consider $\mathbb{P}^{n}=U_{n}\cup H_{n}^{\infty}$ where
$H_{n}^{\infty}$ is the hyperplane at infinity. Let us define
${\mathcal{F}}_{k}^{(1)}={\mathcal{F}}_{k}|_{H_{n}^{\infty}}$ and
$p_{\infty}=W_{0}\cap H_{n}^{\infty}$. It is not difficult to see that
$H_{n}^{\infty}\cong\mathbb{P}^{n-1}$ is an invariant set of
${\mathcal{F}}_{k}$ and ${\mathcal{F}}_{k}^{(1)}$ is a holomorphic foliation
in $H_{n}^{\infty}$ of degree $k$ too. Furthermore, given that
$\mu({\mathcal{F}}_{k}^{(1)},p_{\infty})=k^{n-1}$,

$\sum_{j=1}^{r}\mu({\mathcal{F}}_{k},p_{j})=\sum_{j=1}^{r}\mu({\mathcal{F}}_{k}^{(1)},p_{j})=\sum_{j=0}^{n-1}k^{j}-k^{n-1}=\sum_{j=0}^{d-1}k^{j}.$

Therefore, (47) is true for $n=d+1$. Now, by the hypothesis of induction let
us assume that the result is true for $n>d$ where $d$ is fixed. Let us
consider ${\mathcal{F}}_{k}$ defined in $\mathbb{P}^{n+1}$ and special along
$W_{0}\subset\mathbb{P}^{n+1}$ with $cod(W_{0})=d$. Thus,
$\mathbb{P}^{n+1}=U_{n+1}\cup H_{n+1}^{\infty}$. Again,
$H_{n+1}^{\infty}\cong\mathbb{P}^{n}$ is an invariant set of
${\mathcal{F}}_{k}$. Defining
${\mathcal{F}}_{k}^{(1)}={\mathcal{F}}_{k}|_{H_{n+1}^{\infty}}$ and
$W_{1}=W_{0}\cap H_{n+1}^{\infty}$ we obtain a holomorphic foliation by curves
in $H_{n+1}^{\infty}$ of degree $k$ and special along $W_{1}$ too. Given that
$W_{1}$ in $H_{n+1}^{\infty}$ also has the codimension $d$, we get

$\sum_{j=1}^{r}\mu({\mathcal{F}}_{k},p_{j})=\sum_{j=1}^{r}\mu({\mathcal{F}}_{k}^{(1)},p_{j})=1+k+\ldots+k^{d-1}$

by hypothesis of induction. This number agrees with the Theorem 1 taking
$\ell=k-1$ and $\deg(W_{0})=1$. See the Lemma 7.1. For $k=1$, the equation
(47) may be interpreted as follows: let us consider the singular points
$p_{j}\notin W_{0}$ then

$\sum_{p_{j}\in\mathbb{P}^{n}\setminus
W_{0}}\mu({\mathcal{F}}_{k},p_{j})=\sum_{p_{j}\in\mathbb{P}^{n}}\mu({\mathcal{F}}_{k},p_{j})-\sum_{p_{j}\in
W_{0}}\mu({\mathcal{F}}_{k},p_{i}).$

Given that $W_{0}\cong\mathbb{P}^{n-d}$ is invariant by ${\mathcal{F}}_{k}$,
we get

$\sum_{p_{j}\notin W_{0}}\mu({\mathcal{F}}_{k},p_{j})=n+1-(n-d+1)=d.$

Again for $k\geq 2$, by the Lemma (7.1), we obtain that

(48)
$N(\widetilde{{\mathcal{F}}}_{k},\widetilde{\mathbb{P}}^{n})=\sum_{j=1}^{r}\mu({\mathcal{F}}_{k},p_{j})+N(\widetilde{{\mathcal{F}}}_{k},E)=(n+2-d)\sum_{j=0}^{d-1}k^{j}.$

Now, let us consider $V_{\lambda}\subset\mathbb{P}^{n}$ a $(n-d+1)$-plane
defined by $V_{\lambda}=\\{\xi_{j}=\lambda_{j}\xi_{0},j=1,\ldots,d\\}$,
$\lambda=(\lambda_{1},\ldots,\lambda_{d})\in\mathbb{C}^{d-1}$. Thus,
$W_{0}\subset V_{\lambda}$ and $cod(V_{\lambda})=d-1$. In order to
$V_{\lambda}$ be invariant by ${\mathcal{F}}_{k}$ the parameter $\lambda$ must
be a singular point of a holomorphic foliation by curves of degree $k$ on
$\mathbb{P}^{d-1}$. Consequently, the number of $(n-d+1)$-plane $V_{\lambda}$
invariant by ${\mathcal{F}}_{k}$ is $1+k+\ldots+k^{d-1}$. Let $V$ one of them.
By (48), we obtain that
$N(\widetilde{{\mathcal{F}}}_{k},\widetilde{V})=(n+2-d)$ where $\widetilde{V}$
is the strict transform of $V$ by $\pi^{-1}$. Furthermore, by Lemma (3.1) we
get

$N(\widetilde{{\mathcal{F}}}_{k},\widetilde{V}\cap E)=n+1-d.$

These numbers agree with Proposition (5.1) taking $\deg(V)=1$ and $\ell=k-1$
since
$\alpha_{V}^{(0)}:=\displaystyle\int_{\widetilde{V}}\pi^{*}(\mathbf{h})^{n-d}=1$
and
$\alpha_{V_{E}}^{(i)}:=\int_{V_{E}}\pi^{*}(\mathbf{h})^{n-d-i-1}\cdot\zeta^{i}=1$
for $i\geq 0$.

Now, let $\mathcal{G}_{k}$ be an one-dimensional holomorphic foliation of
degree $k$ in $\mathbb{P}^{n}$ also described in $U_{n}$ by the vector field

(49) $X_{\mathcal{G}_{k}}=\sum_{i=1}^{n}\sum_{|a|=k}c_{a,i}x_{1}^{a_{1}}\cdots
x_{d}^{a_{d}}\frac{\partial}{\partial
x_{i}}=\sum_{i=1}^{n}Q_{i}(x)\frac{\partial}{\partial x_{i}}$

where $x_{i}=\xi_{i-1}/\xi_{n}$, $a=(a_{1},\ldots,a_{d})$ and $c_{a,i}$ a
constant. Thus, $W_{0}\subset{\rm Sing}(\mathcal{G}_{k})$ and
$\ell=m_{E}(\pi^{*}\mathcal{G}_{k})=k-1$. We have two different situations
according to $\ell=0$ or $\ell\geq 1$. We start our analysis considering
$\ell=0$, i.e.; $\mathcal{G}_{k}$ has degree one. Let ${\mathcal{F}}_{t}$ be a
holomorphic foliation on $\mathbb{P}^{n}$ given in the Lemma 3.1, for $t\in
D\setminus\\{0\\}$, $D=D(0,\epsilon)$, with $\epsilon$ sufficiently small,

$X_{{\mathcal{F}}_{t}}=\sum_{i=1}^{n}\sum_{j=1}^{d}c_{i,j}x_{j}\frac{\partial}{\partial
x_{i}}+t\sum_{i=1}^{n}Y_{i}(x)\frac{\partial}{\partial x_{i}}$

where

$Y_{i}=\left\\{\begin{array}[]{ll}\sum_{j=1}^{d}b_{ij}x_{j},&\mathrm{for}\quad
1\leq i\leq d\cr b_{i0}+\sum_{j=1}^{n}b_{ij}x_{j},&\mathrm{for}\quad d+1\leq
i\leq n\end{array}\right.$

with $b_{ij}\in\mathbb{C}$. Let $A_{t}=(a_{ij}^{t})_{1\leq i,j\leq d}\in
M_{\mathbb{C}}(d,d)$ be $d\times d$ matrix given by
$a_{ij}^{t}=c_{ij}+tb_{ij}$. We can admit $\mathrm{Det}(A_{t})\neq 0$ and
$A_{t}$ has $d$ distinct eigenvalues for all $t\in D\setminus\\{0\\}$ which
yields that the only singular point $p_{n}\in{\rm Sing}({\mathcal{F}}_{t})\cap
U_{n}$ is given by $p_{n}=(0,\ldots,0,p_{d+1,n},\ldots,p_{n,n})$ where
$(p_{d+1,n},\ldots,p_{n,n})$ is the solution of linear system
$b_{i0}+\sum_{j=d+1}^{n}b_{ij}x_{j}=0$ for $d+1\leq i\leq n$, for all $t\in
D\setminus\\{0\\}$. In the affine open $U_{n-1}\subset\mathbb{P}^{n}$ with
coordinate $(y_{i})$, $y_{i}={\xi_{i-1}}/{\xi_{n-1}}$ for $1\leq i\leq n-1$
and $y_{n}={\xi_{n}}/{\xi_{n-1}}$, ${\mathcal{F}}_{t}$ is described by the
vector field

$X_{{\mathcal{F}}_{t}}=\sum_{i=1}^{n}Q_{i}^{t}(y)\frac{\partial}{\partial
y_{i}}$

where

$Q_{i}^{t}(y)=\left\\{\begin{array}[]{ll}\sum_{j=1}^{d}(c_{ij}+tb_{ij})y_{j}-y_{i}g_{n}(y),&\mathrm{for}\quad
1\leq i\leq d\cr g_{i}(y)-y_{i}g_{n}(y),&\mathrm{for}\quad d+1\leq i\leq
n-1\cr-y_{n}g_{n}(y),&\mathrm{for}\quad i=n\end{array}\right.$

with
$g_{i}(y)=\sum_{j=1}^{d}c_{ij}y_{j}+t(b_{i0}y_{n}+b_{in}+\sum_{j=1}^{n-1}b_{ij}y_{j}$.
In order to determine the singular points of ${\mathcal{F}}_{t}$ we need to
consider $y_{n}=0$ or $g_{n}(y)=0$. It is not possible to have these two
conditions satisfied simultaneously because they would result in an
inconsistent linear system. Namely, $g_{i}(y)=0$, for $d+1\leq i\leq n$ and
$y_{1}=\ldots=y_{d}=y_{n}=0$, i.e.; the number of equations is greater than
the number of unknown variables. Note that condition $g_{n}(y)=0$ results in
the singular point $p_{n}\in U_{n}\cap U_{n-1}$. Therefore, we can only
consider $y_{n}=0$ and $g_{n}(y)\neq 0$. For all $t\in D\setminus\\{0\\}$, the
point $p_{n-1}\in{\rm Sing}({\mathcal{F}}_{t})\cap U_{n-1}$ where
$p_{n-1}=(0,\ldots,0,p_{n-1,d+1},\ldots,p_{n-1,n-1},0)$ such that
$(p_{n-1,d+1},\ldots,p_{n-1,n-1})$ is the solution of the linear system
$b_{in}+\sum_{j=d+1}^{n-1}b_{ij}y_{j}=0$ for $d+1\leq i\leq n-1$. This way, in
each an affine open $U_{i}$, for $i=d,\ldots,n$ we get one singular point
$p_{i}\in W_{0}\cap U_{i}$. Therefore, $\mu({\mathcal{F}}_{k},W_{0})\geq
n-d+1$. The other singular points $p_{m}^{t}=(y_{1}^{t},\ldots,y_{n-1}^{t},0)$
of ${\mathcal{F}}_{t}$ are obtained as follows: $(y_{1}^{t},\ldots,y_{d}^{t})$
is the eigenvector of $A_{t}$ associated to eigenvalue $\lambda_{m}^{t}$.
Thus, $Q_{i}^{t}(p_{m}^{t})=y_{i}^{t}(\lambda_{m}^{t}-g_{n}(p_{m}^{t}))=0$ for
$i=1,\ldots,d$. which gives $\lambda_{m}^{t}=g_{n}(p_{m}^{t})$. The other
components of $p_{m}^{t}$ are obtained from the linear system
$\sum_{j=1}^{d}c_{ij}y_{j}^{t}+t(b_{in}+\sum_{j=1}^{n-1}b_{ij}y_{j}^{t})=0$,
for $d+1\leq i\leq n-1$. Without loss of generality, we can suppose that
$y_{1}^{t}\neq 0$ which results in

$y_{1}^{t}=\frac{\lambda_{m}^{t}-tb_{nn}}{\sum_{j=1}^{d}c_{nj}u_{j}^{t}+t(\sum_{j=1}^{n-1}b_{nj}u_{j}^{t})}$

where $u_{j}^{t}=y_{j}^{t}/y_{1}^{t}$.

Consequently, we obtain $d$ points on $p_{m}^{t}\in\mathbb{P}^{n}\setminus
W_{0}$, counted with multiplicity. However,

$\lim_{t\to
0}y_{1}^{t}=\frac{\lambda_{m}^{0}}{c_{n1}+\sum_{j=2}^{d}c_{nj}u_{j}^{0}}$

where $\lambda_{m}^{0}$ is a eigenvalue of $A_{0}$. Therefore,
$p_{n-1}=\lim_{t\to 0}p_{m}^{t}$ is an embedded point of $W_{0}$ if only if
$\lambda_{m}^{0}=0$, i.e.; $\mathrm{det}(A_{0})=0$. Let
$P_{A_{0}}(\lambda)=Det(A_{0}-\lambda I)=\lambda^{q}p_{0}(\lambda)$ with
$q\geq 0$ and $p_{0}(0)\neq 0$. It follows that $W_{0}$ has $q$ embedded
closed points, counted with multiplicity. However, $q\leq d-1$, we have that

(50) $n-d+1\leq\mu(\mathcal{G}_{k},W_{0})=n-d+1+q\leq n$

for $k=1$. At this point, we will analyze the case where $\ell\geq 1$. At now,
$W_{0}$ is a nondicritical of type (iii) component of ${\rm
Sing}(\mathcal{G}_{k})$. Let $\mathcal{G}_{t}$ be a holomorphic foliation by
curves as given in Lemma 3.1. Again, in the affine open $U_{n}$,
$\mathcal{G}_{t}$ is described by the following vector field

$X_{\mathcal{G}_{t}}=\sum_{i=1}^{n}Q_{i}(x)\frac{\partial}{\partial
x_{i}}+t\sum_{i=1}^{n}G_{i}(x)\frac{\partial}{\partial
x_{i}}=\sum_{i=1}^{n}G_{i}^{t}(x)\frac{\partial}{\partial x_{i}}$

where

$G_{i}(x)=\left\\{\begin{array}[]{ll}\displaystyle\sum_{|a|=k}b_{a,i}x_{1}^{a_{1}}\cdots
x_{d}^{a_{d}},&\mathrm{for}\quad 1\leq i\leq
d\cr\displaystyle\sum_{|a|=k-1}\upsilon_{a,i}(x)x_{1}^{a_{1}}\cdots
x_{d}^{a_{d}},&\mathrm{for}\quad d+1\leq i\leq n\end{array}\right.$

with $\upsilon_{a,i}(x)=\varrho_{a,i,0}+\sum_{j=1}^{n}\varrho_{a,i,j}\cdot
x_{j}$ is an affine linear function and $a=(a_{1},\ldots,a_{d})$.

Even if the foliation $\mathcal{G}_{k}$ has an invariant variety
$V=Z(f_{1},\ldots,f_{d})$ such that $f_{i}\in\mathcal{I}(W)$, we can choose
the parameters $b_{a,i}$ in order that $\mathcal{G}_{t}$ also has $V$ as
invariant set and $W_{0}=Z(G_{1}^{t}(\xi),\ldots,G_{d}^{t}(\xi))$ for $t\in
D\setminus\\{0\\}$.

In the chart $\widetilde{U}_{1}$ with coordinate $\varsigma\in\mathbb{C}^{n}$
such that $\sigma_{1}(u)=x$, we obtained the induced foliation
$\widetilde{\mathcal{G}}_{t}$ by $\mathcal{G}_{k}$ via blowup $\pi$ as follows

$X_{\widetilde{\mathcal{G}}_{t}}=u_{1}\widetilde{G}_{1}^{t}(u)\frac{\partial}{\partial
u_{1}}+\sum_{i=2}^{d}[\widetilde{G}_{i}^{t}(u)-u_{i}\widetilde{G}_{1}^{t}(u)]\frac{\partial}{\partial
u_{i}}+\sum_{i=d+1}^{n}\widetilde{G}_{i}^{t}(u)\frac{\partial}{\partial
u_{i}}$

where

$\widetilde{G}_{i}^{t}(u)=\left\\{\begin{array}[]{ll}\displaystyle\sum_{|a|=k}(c_{a,i}+tb_{a,i})u_{2}^{a_{2}}\cdots
u_{d}^{a_{d}},&\mathrm{for}\quad i=1,\ldots,d\cr
t\displaystyle\sum_{|a|=k-1}\widetilde{\upsilon}_{a,i}(u)u_{2}^{a_{2}}\cdots
u_{d}^{a_{d}}+u_{1}\widetilde{Q}_{i}(u),&\mathrm{for}\quad
i=d+1,\ldots,n,\end{array}\right.$

with
$\widetilde{\upsilon}_{a,i}(u)=\varrho_{a,i,0}+\sum_{j=d+1}^{n}\varrho_{a,i,j}u_{j}$
and some $\widetilde{Q}_{i}$. For $t\in D\setminus\\{0\\}$, it is not
difficult to see that there exist $1+k+\ldots+k^{d-1}$ singular points,
counted with multiplicities, on $\widetilde{U}_{1}\cap E$. Note that the
exceptional divisor $E$ is defined in $\widetilde{U}_{1}$ by $u_{1}=0$ and the
system $\widetilde{G}_{i}^{t}(u)-u_{i}\widetilde{G}_{1}^{t}(u)=0$, for
$i=2,\ldots,d$ has $1+k+\ldots+k^{d-1}$ solutions, counted with
multiplicities. Furthermore, ${\rm
Sing}(\widetilde{\mathcal{G}}_{t})\cap\widetilde{U}_{1}\setminus E=\emptyset$
because $W_{0}=Z(Q_{i}(\xi)+tY_{i}(\xi))$, $i=1,\ldots,d$; $t\neq 0$.

In the the affine open $U_{n-1}\subset\mathbb{P}^{n}$, with coordinate
$y=(y_{i})\in\mathbb{C}^{n}$, with $y_{i}=x_{i}/x_{n}$ for $1\leq i\leq n-1$
and $y_{n}=1/x_{n}$, we get

$X_{\mathcal{G}_{t}}=\sum_{i=1}^{d}\big{(}G_{i}^{t}(y)-y_{i}H_{n}(y)\big{)}\frac{\partial}{\partial
y_{i}}+\sum_{i=d+1}^{n-1}\big{(}H_{i}(y)-y_{i}H_{n}(y)\big{)}\frac{\partial}{\partial
y_{i}}-y_{n}H_{n}(y)\frac{\partial}{\partial y_{n}}$

where

$H_{i}(y)=t\sum_{|a|=k-1}\widetilde{\kappa}_{a,i}(y)y_{1}^{a_{1}}\cdots
y_{d}^{a_{d}}+\widetilde{Q}_{i}(y),$

and
$\widetilde{\kappa}_{a,i}(y)=\varrho_{a,i,0}y_{n}+\varrho_{a,i,n}+\sum_{j=1}^{n-1}\varrho_{a,i,j}\cdot
y_{j}$. Again, in the chart $\widetilde{U}_{1}$ with coordinate
$u\in\mathbb{C}^{n}$ such that $y=\sigma_{1}(u)$ we get the vector field which
describes the induced foliation $\widetilde{\mathcal{G}}_{t}$ in
$\widetilde{U}_{1}$ as follows

$\begin{array}[]{ll}\displaystyle
X_{\widetilde{\mathcal{G}}_{t}}=&u_{1}\big{(}\widetilde{G}_{1}^{t}(u)-\widetilde{H}_{n}(u)\big{)}\frac{\partial}{\partial
u_{1}}+\displaystyle\sum_{i=2}^{d}\big{(}\widetilde{G}_{i}^{t}(u)-u_{i}\widetilde{G}_{1}^{t}(u)\big{)}\frac{\partial}{\partial
u_{i}}+\cr&\displaystyle\sum_{i=d+1}^{n-1}\big{(}\widetilde{H}_{i}^{t}(u)-u_{i}\widetilde{H}_{1}^{t}(u)\big{)}\frac{\partial}{\partial
u_{i}}-u_{n}\widetilde{H}_{n}(u)\frac{\partial}{\partial u_{n}}\end{array}$

where

$\widetilde{G}_{i}^{t}(u)=\sum_{|a|=k}(c_{a,i}+tb_{a,i})u_{2}^{a_{2}}\ldots
u_{d}^{a_{d}}$

and

$\widetilde{H}_{i}^{t}(u)=u_{1}\sum_{|a|=k}(c_{a,i})u_{2}^{a_{2}}\ldots
u_{d}^{a_{d}}+t\sum_{|a|=k-1}\widetilde{\kappa}_{a,i}(\sigma_{1}(u))u_{2}^{a_{2}}\ldots
u_{d}^{a_{d}}.$

In order to determine the singular points of $\widetilde{G}_{t}$ in
$\widetilde{U}_{1}$, we need to analyze two conditions $u_{n}=0$ and
$\widetilde{H}_{n}(u)=0$. If $u_{n}=0$ and $\widetilde{H}_{n}(u)=0$ then we
would have a inconsistent system, a namely, $\widetilde{H}_{i}(u)=0$ for
$i=d+1,\ldots,n$ i.e.; more equations than unknown variables. If $u_{n}\neq 0$
and $\widetilde{H}_{n}(u)=0$ we would have the same singularities obtained
when we analysis the foliation $\widetilde{G}_{t}$ in $U_{n}$. Therefore, it
is enough to consider only the case where $u_{n}=0$.

On the exceptional divisor $E$ which on $\widetilde{U}_{1}$ is defined by
$u_{1}=0$ we obtain $1+k+\ldots+k^{d-1}$ more singular points, counted with
multiplicities. In fact, the system
$\widetilde{G}_{i}^{t}(u)-u_{i}\widetilde{G}_{1}^{t}(u)=0$, for $i=2,\ldots,d$
admits $1+k+\ldots+k^{d-1}$. Thus, with each solution obtained, we will solve
the system $\widetilde{H}_{i}(u)-u_{i}\widetilde{H}_{n}(u)=0$ for
$i=d+1,\ldots,n-1$. Consequently, after we consider each affine open $U_{i}$,
for $i=d,\ldots,n$, we obtain the total number $(n-d+1)(1+k\ldots+k^{d-1})$ of
singularities on $E$ which agrees with Theorem 1. Again, see Lemma 7.1. The
singular points $\widetilde{p}_{m}^{t}$ that are not contained in the
exceptional divisor $E$ are obtained as follows:
$\widetilde{p}_{m}^{t}=(u_{1}^{t},\ldots,u_{n-1},0)$ where
$(u_{2}^{t},\ldots,u_{d}^{t})$ is a solution of the system
$\widetilde{G}_{i}^{t}(u)-u_{i}\widetilde{G}_{1}^{t}(u)=0$, for $i=2,\ldots,d$
and

$u_{1}^{t}=\frac{\displaystyle\widetilde{G}_{1}^{t}(u)-t\sum_{|a|=k-1}\widetilde{\kappa}_{a,i}(\sigma_{1}(u))u_{2}^{a_{2}}\ldots
u_{d}^{a_{d}}}{\displaystyle\sum_{|a|=k}c_{a,n}u_{2}^{a_{2}}\ldots
u_{d}^{a_{d}}}.$

Thus, after we obtain $(u_{i}^{t})$, for $i=1,\ldots,d$ we get the others
components of $\widetilde{p}_{m}^{t}$ solving the system
$H_{i}^{t}-u_{i}H_{n}(u)=0$ for $i=d+1\ldots,n-1$.

Therefore, $p_{m}^{t}=\sigma_{1}(\widetilde{p}_{m}^{t})$ is a singular point
of $\widetilde{G}_{t}$. By the same argument, if $t\in D\setminus\\{0\\}$ then
the foliation $\widetilde{\mathcal{G}}_{t}$ admits $1+k+\ldots+k^{d-1}$
singular isolated points, counted with multiplicities.

$\mu(\mathcal{G},W_{0})\geq\displaystyle\sum_{i=0}^{n}k^{i}-\displaystyle\sum_{i=0}^{d}k^{i}=\displaystyle\sum_{i=d}^{n}k^{i}.$

However, $\displaystyle\lim_{t\to 0}p_{m}^{t}\in W_{0}$ if only if
$\widetilde{G}_{i}^{0}=0$ for $i=1,\ldots,d$. That is,
$Q_{1}(\xi)=\ldots=Q_{d}(\xi)=0$ admits a nontrivial solution. Thus, if the
multivariate resultant $Res(Q_{1},\ldots,Q_{d})=0$ then $W_{0}$ has an
embedded closed point. Given that all functions $Q_{i}$ are not identically
null, the upper bound of number of embedded points $W_{0}$ is
$k+\ldots+k^{d-1}$. Comparing with (50), we obtain that

$\displaystyle\sum_{i=d}^{n}k^{i}\leq\mu(\mathcal{G},W_{0})=\displaystyle\sum_{i=d}^{n}k^{i}+q\leq\displaystyle\sum_{i=1}^{n}k^{i}$

where $q$ is the number of embedded closed points of $W_{0}$, for all $k\geq
1$.

#### Acknowledgments.

We would like to acknowledge to Renato Vidal Martins (UFMG) and John
MacQuarrie (UFMG) for the suggestions and corrections to improve the paper.

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