Smooth points of the space of plane foliations with a center

Lubomir Gavrilov 111Institut de Mathématiques de Toulouse, UMR 5219,
Université de Toulouse, 31062 Toulouse, France<EMAIL_ADDRESS>toulouse.fr Hossein Movasati 222 Instituto de Matemática Pura e Aplicada,
IMPA, Estrada Dona Castorina, 110, 22460-320, Rio de Janeiro, RJ, Brazil,
<EMAIL_ADDRESS>
###### Abstract

We prove that a logarithmic foliation corresponding to a generic line
arrangement of $d+1\geq 3$ lines in the complex plane, with pairwise natural
and co-prime residues, is a smooth point of the center set of plane foliations
(vector fields) of degree $d$.

##  1 Introduction

The present paper is a contribution to the classical center-focus problem (the
problem of distinguishing between a center and a focus of a plane vector
field). We consider the set of complex polynomial plane vector fields of
degree at most $d$, or equivalently, affine polynomial degree $d$ foliations
in $\mathbb{C}^{2}$:

$\displaystyle{\mathcal{F}}(d)=\left\\{{\mathcal{F}}(P(x,y)dy-Q(x,y)dx)\mid
P,Q\in\mathbb{C}[x,y],\ \deg(P),\deg(Q)\leq d\right\\}.$

We identify ${\mathcal{F}}(d)$ to the set of coefficients of the polynomials
$P,Q$ (that is to say to $\mathbb{C}^{(d+1)(d+2)}$. We say that a given
foliation (a point in ${\mathcal{F}}(d)$ ) has a Morse center at
$p\in\mathbb{C}^{2}$, or simply a center, if it allows a local analytic first
integral which has a Morse critical point at $p$. It is well-known that the
Zariski closure of the set of foliations with a Morse center, the so called
_center set_ , is an algebraic set, see [Lin14, Mov04a]. We denote this center
set by ${\mathcal{M}}(d)\subset{\mathcal{F}}(d)$. It has a canonical
decomposition (up to a permutation)

$\displaystyle{\mathcal{M}}(d)=\cup_{i}\mathcal{\bar{L}}_{i},\;\;\mathcal{\bar{L}}_{i}\not\subseteq\mathcal{\bar{L}}_{j},i\neq
j$ (1)

into closed irreducible algebraic varieties $\mathcal{\bar{L}}_{i}$. _The
center-focus problem in this setting is to describe the irreducible components
$\mathcal{\bar{L}}_{i}$ of the center set ${\mathcal{M}}(d)$. _ The problem is
largely open, except in the quadratic case ($d=2$). It follows from the
Dulac’s computation of quadratic systems with a center [Dul23], that
${\mathcal{M}}(2)$ has four irreducible components, parameterised via their
explicit first integrals. In the case $d>2$ only some irreducible components
of ${\mathcal{M}}(d)$ are known. For a conjecturally complete list of cubic
systems with a center we refer the reader to [Zol94b, Zol94a, BK10, Bot07].

Suppose that $\mathcal{L}\subset{\mathcal{M}}(d)$ is an irreducible algebraic
set (algebraic variety) formed by foliations with a center. To show that its
Zariski closure $\mathcal{\bar{L}}$ is also an irreducible component of
${\mathcal{M}}(d)$, like in (1), is a local problem. Therefore we may choose a
suitable point ${\mathcal{F}}_{0}\in\mathcal{L}\subset{\mathcal{M}}(d)$ and
compare the tangent space of $\mathcal{L}$ at ${\mathcal{F}}_{0}$ and the
tangent space of ${\mathcal{M}}(d)$ at ${\mathcal{F}}_{0}$. If the dimension
of these spaces are the same, then the condition
$\mathcal{L}\not\subseteq\mathcal{L}_{j}$ (1) is certainly satisfied and
therefore $\mathcal{L}$ is an irreducible component of the center set
${\mathcal{M}}(d)$.

The computation of the tangent cone of ${\mathcal{M}}(d)$ (even if
${\mathcal{M}}(d)$ is not known!) turns out to be possible by making use of
the machinery of Melnikov functions, as shown by Ilyashenko [Ily69] (in the
Hamiltonian case), Movasati [Mov04a, Mov04b] (the case of logarithmic
foliations), Zare [Zar19] (pull back foliations), Gavrilov [FGX20] (centers of
Abel equations). In all these cases it has been shown, that the corresponding
irreducible algebraic set of systems with a center is indeed an irreducible
component of ${\mathcal{M}}(d)$.

In the present paper we focus our attention to logarithmic foliations of the
form

${\mathcal{F}}_{0}:\ \ l_{1}l_{2}\dots
l_{d+1}\left(\sum_{i=1}^{d+1}\lambda_{i}\frac{dl_{i}}{l_{i}}\right)=0,\ \
d\geq 2,$ (2)

where $\lambda_{i}\in\mathbb{C}^{*}$ and $l_{i}=l_{i}(x,y)$ are complex
bivariate polynomials of degree one. Obviously the foliation
${\mathcal{F}}_{0}$ has a first integral of the form

$\displaystyle l_{1}^{\lambda_{1}}l_{2}^{\lambda_{2}}\dots
l_{d+1}^{\lambda_{d+1}}.$ (3)

In what follows we suppose that the polynomials $l_{i}$ define a line
arrangement without triple intersection points (a general line arrangement),
and that $\lambda_{i}\neq 0$. The set of such foliations is denoted by
${\mathcal{L}}(1^{d+1})={\mathcal{L}}(1,1,\ldots,1)$. The Zariski closure
${\bar{\mathcal{L}}}(1^{d+1})\subset{\mathcal{F}}(d)$ is an irreducible
component of the corresponding center set ${\mathcal{M}}(d)$ [Mov04b]. If
another irreducible component of ${\mathcal{M}}(d)$ is of dimension at least
equal to the co-dimension of ${\mathcal{L}}(1^{d+1})$, then it certainly
intersects ${\bar{\mathcal{L}}}(1^{d+1})$. Therefore, the study of the
structure of the center set in a small neighbourhood of
${\bar{\mathcal{L}}}(1^{d+1})$ implies also a global information on
${\mathcal{M}}(d)$. Note that if the foliation ${\mathcal{F}}_{0}$ belongs to
the intersection of ${\mathcal{L}}(1^{d+1})$ with another irreducible
component of the center set, then ${\mathcal{F}}_{0}$ is a non smooth point of
${\mathcal{M}}(d)$. This motivates the following problem, which is partially
solved in the paper : _Classify the smooth points of ${\mathcal{M}}(d)$ along
the irreducible component ${\mathcal{L}}(1^{d+1})$._ We prove the following

######  Theorem 1.

Let $\lambda_{i}$, $i=1,\dots,d+1$, be mutually prime distinct natural
numbers. Let $l_{i}=l_{i}(x,y)$ , $i=1,\dots,d+1$, be linear bivariate
polynomials defining a generic line arrangement (generic means that there are
no triple points). Then the logarithmic foliation ${\mathcal{F}}_{0}$ defined
by (2) is a smooth point of the center set ${\mathcal{M}}(d)$.

If ${\mathcal{F}}_{0}$ is a general logarithmic foliation of the form (2) such
that ${\mathcal{F}}_{0}$ is a smooth point of the center set
${\mathcal{M}}(d)$, then obviously every small degree $d$ deformation with a
persistent center is also a deformation by logarithmic foliations. Therefore
the above theorem is close to another classical result which we recall now.
Consider the set $\mathcal{L}(d+1)\subset{\mathcal{M}}(d)$ formed by
Hamiltonian foliations ${\mathcal{F}}:dH=0$ where $H$ is an arbitrary degree
$d+1$ bivariate polynomial. Suppose in addition that $H$ is a ”Morse plus”
polynomial (has only Morse cticical points with distinct critical values). It
is proved by Ilyashenko [Ily69], that if in a deformation of the Morse plus
Hamiltonian foliation $dH$ the center persists, then this deformation is
Hamiltonian too. The proof implies also that ${\mathcal{F}}:dH=0$ is a smooth
point of ${\mathcal{M}}(d)$.

It is clear that when two irreducible components of ${\mathcal{M}}(d)$
intersect at ${\mathcal{F}}_{0}$, then ${\mathcal{F}}_{0}$ is a non-smooth
point of ${\mathcal{M}}(d)$. It is less known that even when
${\mathcal{F}}_{0}$ does not belong to different irreducible components of
${\mathcal{M}}(d)$, it can still be a non-smooth point of ${\mathcal{M}}(d)$.
This happens even in the quadratic case (d=2), for an example see the last
section of the paper.

Our final remark is that it follows from the computation of the tangent cone
(which turns out to be a tangent space) that ${\mathcal{L}}(1^{d+1})$ is an
irreducible component of the center set ${\mathcal{M}}(d)$. This proof is
quite different compared to the original proof [Mov04b], as the tangent cone
to ${\mathcal{M}}(d)$ is computed at a smooth pont ${\mathcal{F}}_{0}$ (like
in [Ily69]) .

The article is organised in the following way. In Section 2 we develop the
Picard-Lefschetz theory of the fibration of the polynomial $x^{n}y^{m}$ where
$n,m$ are natural numbers (not necessarily coprime). In Section 3 we study the
topology of the fibers of $f$

$f=l_{1}^{n_{1}}l_{2}^{n_{2}}\dots l_{d+1}^{n_{d+1}}.$

where $l_{i}$ are lines in a general position, and $n_{i}$ are positive
integers without common divisors. (we do not suppose that $n_{i},n_{j}$ are
relatively prime). As a by-product we get a genus formula for the fibers of
$f$. In Section 4 we generalise a theorem due to A’Campo and Gusein-Zade
[A’C75, GZ74] in the context of a logarithmic foliation defined by the
polynomial $f$. In Section 5 we compute the orbit of a vanishing cycle under
the action of the monodromy in the homology bundle of $f$. As a by product,
this implies that the orbit of this vanishing cycle contains the homology of
the compactified fiber. This is the only place where we use the fact that
$n_{i}$’s are pairwise coprime. Summing up all these results leads to the
proof of Theorem 1, given in the last Section 6.

The article was written while the first author was visiting the University of
Toulouse III. He would like to thank for the stimulating research atmosphere,
as well for the financial support of CIMI and CNRS.

##  2 The Picard-Lefschetz formula of a plane non-isolated singularity

The first attempt to describe Picard-Lefschetz theory of fibrations with non-
reduced fibers is done in [Cle69], however, the main result of this paper
Theorem 4.4 is not applicable in our context, and so, we elaborate Theorem 2
which explicitly describe a kind of Picard-Lefschetz formula.

In this section we consider the local fibration
$f:(\mathbb{C}^{2},0)\to(\mathbb{C},0)$ given by $f=x^{m}y^{n}$, where $m,n$
are two positive integers. We will use the notation:

$e:=(m,n),\ p:=\frac{m}{e},q:=\frac{n}{e}$

where $(m,n)={\rm gcd}(m,n)$ means the greatest common divisor of $m$ and $n$.
It might be easier to follow the content of the present section for the case
$e=1$. For $t\in\mathbb{R}^{+}$ let

$\Gamma:=\left\\{(r,s)\in\mathbb{R}^{+}\times\mathbb{R}^{+},\Big{|}r^{m}s^{n}=t\right\\}.$

We consider it as an oriented path in $f^{-1}(t)$ for increasing $s$ for which
we use the letter $\gamma$. We consider the following parameterization of the
fiber $f^{-1}(t)$ for $t\in\mathbb{R}^{+}$

$\displaystyle\mathbb{R}\times\Gamma\to f^{-1}(t),\ \ h=0,1,2,\ldots,e-1$ (4)
$\displaystyle(\theta,r,s)\mapsto(x,y)=(re^{2\pi i(\theta
q+\frac{h}{m})},se^{-2\pi i\theta p}).$

The fiber $f^{-1}(t)$ consists of $e$ cylinders indexed by $h$ and the above
parametrization is periodic in $\theta$ with period $1$.

######  Definition 1.

By a straight path in $f^{-1}(t)$ we mean a path which is the image of a path
$\alpha$ in $\mathbb{R}\times\Gamma$ under the parameterization (4) and with
the following property: $\alpha$ maps bijectively to its image under the
projection $\mathbb{R}\times\Gamma\to\mathbb{R}$.

For simplicity, we consider the parameters with $|t|<1$ and define
$L_{t}:=f^{-1}(t)\cap B$, where $B$ is the complex square
$\\{(x,y)\in\mathbb{C}^{2}\mid|x|\leq 1,|y|\leq 1\\}$. In this way, $L_{t}$ is
a union of $e$ compact cylinders, let us say $L_{t}=\cup_{h=0}^{e-1}L_{t,h}$.
A circle in each cylinder $L_{t,h}$ is parameterized with fixed $(r,s)$ and
for $(r,s)=(1,|t|^{\frac{1}{n}})$ and $(|t|^{\frac{1}{m}},1)$ we get two
circles of its boundary and denote them by $\delta_{1,h}$ and $\delta_{2,h}$,
respectively, and give them a natural orientation coming from $\theta\in[0,1]$
running from $0$ to $1$. We denote by $\delta_{h}:[0,1]\to L_{t,h}$ the closed
path given by the parameterization (4) and fixed $(r,s)$. This is homotopic to
$\delta_{1,h}$ and $\delta_{2,h}$. We also denote by $\gamma_{h}$ the non
closed path in $L_{t,h}$ given by the parameterization (4) and $\theta=0$.
Note that $\gamma:=\gamma_{0}$ is the only path from $(1,t^{\frac{1}{n}})$ to
$(t^{\frac{1}{m}},1)$ in the real plane $\mathbb{R}^{2}$.

We consider in $B$ two transversal sections
$\Sigma_{1}:=\\{x=1\\},\Sigma_{2}:=\\{y=1\\}$ to the $x$ and $y$-axis,
respectively, and define $\Sigma:=\Sigma_{1}\cup\Sigma_{2}$. The intersections
$\\{|x|=1\\}\cap L_{t}$ and $\\{|y|=1\\}\cap L_{t}$ are the union of circles
$\cup_{h=0}^{e-1}\delta_{i,h}$ for $i=1,2$ respectively, and they have the
following finite subsets

$\displaystyle\Sigma_{1}\cap L_{t}$ $\displaystyle=$
$\displaystyle\left\\{\zeta_{k,h},\ k=0,1\cdots,q-1,\
h=0,1,\ldots,e-1\right\\},$ $\displaystyle\Sigma_{2}\cap L_{t}$
$\displaystyle=$ $\displaystyle\left\\{\xi_{l,h},\ l=0,1,\cdots,p-1,\
h=0,1,\ldots,e-1\right\\},$

where

$\zeta_{k,h}:=(1,t^{\frac{1}{n}}e^{-2\pi i(\frac{km-h}{n})}),\ \
\xi_{l,h}:=(t^{\frac{1}{m}}e^{2\pi i(\frac{ln+h}{m})},1).$ (5)

For $\Sigma_{1}\cap L_{t}$ we have set $\theta=\frac{mk-h}{[m,n]}$ and for
$\Sigma_{2}\cap L_{t}$ we have set $\theta=\frac{nl}{[m,n]}$. We have a
natural action of the multiplicative group of $n$-th (resp. $m$-th) roots of
unity on the set $\Sigma_{1}\cap L_{t}$ (resp. $\Sigma_{2}\cap L_{t}$) which
is given by multiplication in the second coordinate.

Figure 1: A cylinder and straight path

.

######  Proposition 1.

The relative homology group $H_{1}(L_{t},L_{t}\cap\Sigma;\mathbb{Z})$ is
freely generated $\mathbb{Z}$-module of rank $n+m$.

###### Proof.

This follows from the long exact sequence in homology of the pair
$L_{t},L_{t}\cap\Sigma$:

$\begin{array}[]{cccccccccccc}0&\to&H_{1}(L_{t})&\to&H_{1}(L_{t},\Sigma\cap
L_{t})&\to&H_{0}(\Sigma\cap L_{t})&\to&H_{0}(L_{t})&\to&0&\\\
&&\parallel&&\parallel&&\parallel&&\parallel&&&\\\
&&\mathbb{Z}^{e}&&\mathbb{Z}^{m+n}&&\mathbb{Z}^{m+n}&&\mathbb{Z}^{e}&&&\\\
\end{array}$

∎

Since $p$ and $q$ are coprime positive integers, we can find
$a,b\in\mathbb{Z}$ such that

$ap-bq=1,\ \ 0\leq a\leq q-1,\ \ 0\leq b\leq p-1,$

for $p,q\geq 2$. Equivalently, $am-bn=e$. We also consider the cases:

$\left\\{\begin{array}[]{ll}a=1,\ b=0&\hbox{ if }p=1\\\ a=1,b=p-1&\hbox{ if
}q=1\end{array}\right..$

If we change the order of $p$ and $q$ we only need to replace $a$ and $b$ with
$q-a$ and $p-b$, respectively.

######  Theorem 2.

Let $\gamma$ be a straight path which connects $\zeta_{k,h}\in
L_{t}\cap\Sigma_{1}$ to $\zeta_{l,h}\in L_{t}\cap\Sigma_{2}$.The anticlockwise
monodromy $h(\gamma)$ of $\gamma$ around $t=0$ is a straight path which
connects

$\left\\{\begin{array}[]{cc}\zeta_{k,h+1}\hbox{ to }\xi_{l,h+1}&\hbox{ if }\ \
\ h+1<e,\\\ \zeta_{k-a,0}\hbox{ to }\xi_{l-b,0}&\hbox{ if }\ \ \
h+1=e.\end{array}\right.\ \ \ $

In particular, we have the classical Picard-Lefschetz formula

$h^{[m,n]}(\gamma)=\gamma+\delta,$ (6)

where $[m,n]$ is the lowest common multiple of $m$ and $n$.

###### Proof.

We consider the differential form $\omega:=m\frac{dx}{x}=-n\frac{dy}{y}$ in
$L_{t}$, where the last equality is written restricted to $L_{t}$. We observe
that

$\displaystyle\int_{\gamma_{h}}\omega$ $\displaystyle=$ $\displaystyle\ln(t),$
$\displaystyle\int_{\delta(\theta)}^{\delta(\theta+\alpha)}\omega$
$\displaystyle=$ $\displaystyle 2\pi i[m,n]\alpha,\ \ \hbox{ and hence
}\int_{\delta}\omega=2\pi i[m,n].$

Actually, in the first formula $\gamma$ can be any path with parametrization
in (4) with fixed $\theta$. We have

$\zeta_{k,h}e^{2\pi i\frac{1}{n}}=(1,e^{-2\pi
i(\frac{km-(h+1)}{n})})=\left\\{\begin{array}[]{cc}\zeta_{k,h+1}&\hbox{ if }\
\ \ h+1<e\\\ (1,e^{-2\pi i\frac{(k-a)m}{n}})=\zeta_{k-a,0}&\hbox{ if }\ \ \
h+1=e\end{array}\right.$ (7)

and

$\xi_{l,h}e^{2\pi i\frac{1}{m}}=(e^{-2\pi
i(\frac{ln+(h+1)}{m})},1)=\left\\{\begin{array}[]{cc}\xi_{l,h+1}&\hbox{ if }\
\ \ h+1<e\\\ (1,e^{-2\pi i\frac{(l-b)n}{m}})=\xi_{l-b,0}&\hbox{ if }\ \ \
h+1=e\end{array}\right.$ (8)

For the equalities in the case $h+1=e$ we have used $e=am-bn$. The above
equalities imply that $h(\gamma)$ has the right starting and end points as
announced in the theorem. By Cauchy’s theorem we have

$\int_{\gamma}\omega=\int_{\gamma_{h}}\omega+2\pi i[m,n]\frac{nl}{[m,n]}-2\pi
i[m,n]\frac{mk-h}{[m,n]}=\int_{\gamma_{h}}\omega+2\pi i(nl-mk+h).$

Now, we consider a straight path $\tilde{\gamma}$ in $L_{t,h+1}$ which
connects (7) to (8). A similar formula for $\tilde{\gamma}$ as above, and
knowing that $\int_{\gamma_{h}}\omega=\int_{\gamma_{h+1}}\omega=\ln(t)$ give
us

$\int_{\tilde{\gamma}}\omega=\int_{\gamma}\omega+2\pi i$

which implies $h(\gamma)=\tilde{\gamma}$ for $h+1<e$. For $h+1=e$ this follows
from

$\int_{\tilde{\gamma}}\omega=\int_{\gamma_{0}}\omega+2\pi
i[m,n]\frac{n(l-b)}{[m,n]}-2\pi
i[m,n]\frac{m(k-a)}{[m,n]}=\int_{\gamma_{0}}\omega+2\pi i(nl-mk+e).$

As a corollary we can get the classical formula for the monodromy
$h^{[m,n]}(\gamma)$ in (6). We know that $h^{e}(\gamma)$ is the straight path
connecting $\zeta_{k-a,0}$ to $\xi_{l-b,0}$, and so its $pq$ times iteration
is the straight path connecting $\zeta_{k-pqa,0}$ to $\xi_{l-pqb,0}$. Since
$ap-bq=1$ we get (6). ∎

Figure 2: A passage from one transversal section to another: $n=9,m=6$

.

In order to make the content of this section more accessible for applications
we have made Figure 3, and an example of it in Figure 2, which shows the
deformation retract of $L_{t}$ for which one can describe the action of
monodromy. The points in $\Sigma_{i}\cap L_{t},\ \ i=1,2$ are ordered
according to the usual order of roots of unity and we identify them with
$\Sigma_{1}:=\\{0,1,2,\ldots,n-1\\}$ and $\Sigma_{2}:=\\{0,1,2,\ldots,m-1\\}$,
respectively. In this way

$\displaystyle\Sigma_{1}\cap L_{t,h}$ $\displaystyle=$
$\displaystyle\\{h,e+h,2e+h,\ldots,(q-1)e+h\\},$ $\displaystyle\Sigma_{2}\cap
L_{t,h}$ $\displaystyle=$ $\displaystyle\\{h,e+h,2e+h,\ldots,(p-1)e+h\\}.$

In $\Sigma_{1}\cap L_{t,h}$ and $\Sigma_{2}\cap L_{t,h}$ we take minus $h$ and
divide by $e$ and connect them to $\Sigma_{1,h}:=\\{0,1,2,\ldots,q-1\\}$ and
$\Sigma_{2,h}:=\\{0,1,\ldots,p-1\\}$, respectively. We consider another copy
$\Sigma_{1,h}^{\prime}$ of $\Sigma_{1,h}$ and connect $x\in\Sigma_{1,h}$ to
$x(-p)^{-1}\in\Sigma_{1,h}^{\prime}$ modulo $q$ and another copy
$\Sigma_{2,h}^{\prime}$ of $\Sigma_{2,h}$ connecting $x\in\Sigma_{2,h}$ to
$xq^{-1}\in\Sigma_{2,h}^{\prime}$ modulo $p$. Now, all the points of
$\Sigma_{i,h}^{\prime},\ i=1,2$ are connected to a single point $p_{h}$ for
which we also consider a loop $\delta_{h}$ at $p_{h}$ with orientation. We now
describe the monodromy. Consider a path $\gamma$ from $ie+h\in\Sigma_{1}$ to
$je+h$ which turns in $\delta_{h}$, $s_{\gamma}\in\mathbb{Z}$ times. If
$h<e-1$ the monodromy $h(\gamma)$ of $\gamma$ is a similar path starting from
$ie+h+1$ and $je+h+1$ and turning in the loop $\delta_{h+1}$, $s_{\gamma}$
times. If $h=e-1$ then $h(\gamma)$ starts from $ie$ and ends in $jb$. If
$\gamma$ passes through $k\in\Sigma_{1,e-1}^{\prime}$ and
$l\in\Sigma_{2,e-1}^{\prime}$ then $h(\gamma)$ passes through
$k-a\in\Sigma_{1,0}^{\prime}$ and $l-b\in\Sigma_{2,0}^{\prime}$. The number of
turns in $\delta_{0}$ of $h(\gamma)$ is
$s_{\gamma}+\left[\frac{k-a}{q}\right]+\left[\frac{l-b}{p}\right]$.

Figure 3: The passage from $\Sigma_{1}$ to $\Sigma_{2}$.

##  3 Product of $d+1$ lines in general position

We consider the polynomial

$f=l_{1}^{n_{1}}l_{2}^{n_{2}}\dots l_{d+1}^{n_{d+1}}.$ (9)

where $l_{i}$ are lines in a general position, and $n_{i}$ are positive
integers without common divisors. We do not suppose that $n_{i},n_{j}$ are
relatively prime. Let

$L_{t}=\\{(x,y)\in\mathbb{C}:f(x,y)=t\\}.$

######  Theorem 3.

If $t$ is a regular value of $f$, then

$\displaystyle
H_{1}(L_{t},\mathbb{Z})\cong\mathbb{Z}^{(d-1)(n_{1}+n_{2}+\cdots+n_{d+1})+1}.$
(10)

###### Proof.

We fix a fiber $X:=L_{t}$ with $t$ near to zero, consider the projection in
$x$ coordinate $\pi:X\to\mathbb{C},\ (x,y)\to x$ and assume that the parallel
lines $x=$constant are transversal to lines $l_{i}$ and any two intersection
points of $l_{i}$’s have not the same $x$-coordinate. It turns out that the
set of critical points of $\pi$ is a union of $\frac{d(d+1)}{2}$ sets $P_{ij}$
which is near to $l_{i}\cap l_{j}$. Let $C_{ij}=\pi(P_{ij})$ and consider a
regular point $b\in\mathbb{C}$ for $\pi$ and $\Sigma:=\pi^{-1}(b)$. This is a
union of $\sum_{i=1}^{d+1}n_{i}$ distinct points. Let also $D_{ij}$ be a small
disc around $C_{ij}$ and $b_{ij}$ be a point in its boundary. A classical
argument in the topology of algebraic varieties involving deformation retracts
and excision theorem, see for instance [Lam81, 5.4.1],[Mov21, Section 6.7],
gives us:

$\displaystyle
H_{1}(X,\Sigma,\mathbb{Z})=\oplus_{ij}H_{1}(\pi^{-1}(D_{ij}),\pi^{-1}(b_{ij}),\mathbb{Z}).$
(11)

Now $\pi^{-1}(D_{ij})$ is a union of ${}(n_{i},n_{j})$ cylinders with
$n_{i}+n_{j}$ points from $\pi^{-1}(b_{ij})$ in its boundary (as in Section 2)
and $(\sum_{k=1}^{d+1}n_{k})-n_{i}-n_{j}$ discs, each one with one point from
$\pi^{-1}(b_{ij})$ in its boundary. Using Proposition 1 we conclude that

$H_{1}(X,\Sigma)\cong\mathbb{Z}^{d(n_{1}+n_{2}+\cdots+n_{d+1})}.$

The long exact sequence of the pair $X,\Sigma$ finishes the proof.

$\begin{array}[]{cccccccccc}0&\to&H_{1}(L_{t})&\to&H_{1}(L_{t},\Sigma)&\to&H_{0}(\Sigma)\to
H_{0}(L_{t})&\to&0&\\\ &&&&\parallel&&\parallel\ \ \ \ \ \ \ \ \ \ \ \ \ \
\parallel&&&\\\
&&&&\mathbb{Z}^{d(n_{1}+n_{2}+\cdots+n_{d+1})}&&\mathbb{Z}^{n_{1}+n_{2}+\cdots+n_{d+1}}\mathbb{Z}&&&\\\
\end{array}$

∎

######  Corollary 1.

The genus of the curve $L_{t}$ equals

$\frac{1}{2}\left((d-1)n+2-\sum_{i=1}^{d+1}{}(n_{i},n)\right),\ \ \
n:=\sum_{j=1}^{d+1}n_{j},$

where $f:=l_{1}^{n_{1}}l_{2}^{n_{2}}\cdots l_{d+1}^{n_{d+1}}$ and $t$ is a
regular value of $f$.

###### Proof.

By genus of $L_{t}$ we mean the genus of the compactified and desingularized
curve. The hypothesis ${\rm gcd}(n_{1},n_{2},\ldots,n_{d+1})=1$ and $t$ is not
a critical value of $f$, together imply that the curve $f=t$ is an irreducible
polynomial. The curve $f=t$ intersects the line at infinity $\mathbb{P}^{1}$
at the intersection $p_{i}$ of the lines $l_{i}=0$ with $\mathbb{P}^{1}$. Near
$p_{i}$ our curve has $(n_{i},n)$ local irreducible components because

$f-t=\frac{(a_{i}x+by_{i}+c_{i}z)^{n_{i}}}{z^{n}}g_{i}$

where $l_{i}=a_{i}x+by_{i}+c_{i}$ and $g_{i}$ is a holomorphic function near
$p_{i}$ with $g(p_{i})\not=0$. ∎

The genus of the degree three curve $\\{xy(x+y-1)=1\\}$ is one. The genus of
the degree six curve $\\{xy^{2}(x+y-1)^{3}=1\\}$ is also one.

Figure 4: Deformation retract of $L_{t}$

.

We are going to define a graph with
$\sum_{i\not=j}(n_{i},n_{j})+\sum_{i=1}^{d+1}n_{i}$ vertices. The
$(n_{i},n_{j})$ vertices corresponds to the intersection points $l_{i}\cap
l_{j}$. The $n_{i}$ vertices corresponds to the intersection of the line
$\Sigma$ in the proof of Theorem 3 with the fiber $L_{t}$, $t$ near to zero.
Each group of $(n_{i},n_{j})$ vertices are connected with $n_{i}$ edges, each
one with $\frac{n_{i}}{(n_{i},n_{j})}$ edges, to $n_{i}$ vertices in the
second group corresponding the intersection of $\Sigma$ with $L_{t}$ near
$l_{i}=0$. This description is trivially unique for $(n_{i}.n_{j})=1$. If
$(n_{i},n_{j})\not=1$ we have to determine the decomposition of $n_{i}$
vertices into $\frac{n_{i}}{(n_{i},n_{j})}$ sets of cardinality
$(n_{i},n_{j})$. This might be done using the description of the deformation
retract at the end of Section 2. Moreover, we consider a loop for each
$(n_{i},n_{j})$ vertices. This will correspond to the saddle vanishing cycles.
From the proof of Theorem 3 it follows that

######  Proposition 2.

The graph $G$ is a deformation retract of $L_{t}$.

Figure 5: Getting the graph $\check{G}$ from $G$

.

Figure 6: Deformation retract of $L_{t}$ for four lines with multiplicities
1,2,3,4

.

For $f$ and lines $l_{i}$ defined over real numbers, there is another way to
describe a simpler graph $\check{G}$ which shows the homotopy type of $L_{t}$.
Each vertex in the $n_{i}$ group is connected with $d$ edges to $d$ vertices
corresponding to the intersection of $l_{i}$ with other lines. We order them
as they meet $l_{i}$. We replace this with the one in Figure 5 and we get a
graph $\check{G}$ with $\sum_{i\not=j}(n_{i},n_{j})$ vertices which can be
described easily using the real geometry of lines as follows. We cut out
infinite segments of the union of lines
$\cup_{i=1}^{d+1}l_{i}\subset\mathbb{R}^{2}$, replace each intersection point
$l_{i}\cap l_{j}$ with $(n_{i},n_{j})$ vertices and replace each finite
segment which connects $l_{i}\cap l_{j}$ to $l_{i}\cap l_{k}$ (and does not
intersects other lines in its interior) with $n_{i}$ edges connecting
$(n_{i},n_{j})$ vertices with $(n_{i},n_{k})$ vertices, provided that each
vertex in the first and second group has only $\frac{n_{i}}{(n_{i},n_{j})}$
and $\frac{n_{i}}{(n_{i},n_{k})}$ edges respectively. Moreover, we consider a
loop in each $(n_{i},n_{j})$ vertices. We obtain the new graph $\check{G}$.

######  Remark 1.

The the deformation retracts above appeared first in the study of the topology
and the monodromy of the logarithmic foliation with first integral
$f=x^{p}y^{p}(1+x+y)$ in [BG11] in relation with the classical paper [vGH86].

##  4 Computation of intersection indices

The computation of intersection indices between vanishing cycles is an
important ingredient in the study of deformations of singularities. By analogy
we define intersection index for paths in the leafs of a holomorphic
foliation. Our main result Theorem 4 in this section is a generalization of a
theorem by S. Gusein-Zade and N. A’Campo, see [Mov04b, Section 2].

Let us consider a holmorphic foliation ${\mathcal{F}}(\omega)$ in
$\mathbb{R}^{2}$ given by a polynomial 1-form $\omega$ with real coefficients.
We consider an open subset $U\subset\mathbb{R}^{2}$ with exactly two saddle
singularities $O_{1}$ and $O_{2}$ of ${\mathcal{F}}$ and assume that $O_{1}$
and $O_{2}$ have a common separatrix. We assume that the 1-forms $\omega$ near
$O_{i},\ i=1,2$ in local coordinates $(x_{i},y_{i})$ is given by the linear
equation $x_{i}dy_{i}+\alpha_{i}y_{i}dx_{i},\ \alpha_{i}>0$, and so, it has
the meromorphic local first integral $y_{i}x_{i}^{\alpha_{i}}$. In a
neighborhood of $O_{i}$ the foliation has two separatrices $x_{i}=0$ and
$y_{i}=0$. The common separatrix is given by $y_{i}=0$. We consider
transversal sections to ${\mathcal{F}}$ at the points $b_{0},b_{1},b_{2}$
respectively in the common separatrix, $x_{1}=0$ and $x_{2}=0$. Let
$\gamma_{0}$ be the real trajectory of ${\mathcal{F}}$ which connects a point
$p_{1}\in\Sigma_{1}$ to $p_{2}\in\Sigma_{2}$ crossing the point
$p_{0}\in\Sigma_{0}$, see Figure 7.

We consider now the complex foliation ${\mathcal{F}}$ in $\mathbb{C}^{2}$ and
use the same notation for complexified objects. We consider a path
$\lambda:\mathbb{R}\to\Sigma_{0}$ which has period one and restricted to
$[0,1]$ turns once around $b_{0}$ anticlockwise. The path $\gamma_{0}$ from
$p_{1}$ to $p_{2}$ can be lifted to a unique path $\gamma_{t}$ in a leaf of
${\mathcal{F}}$ which crosses $\lambda(t)\in\Sigma_{0}$ and connects
$q_{1}(t)\in\Sigma_{1}$ to $q_{2}(t)\in\Sigma_{2}$. This lifting in general is
not possible, however, in our situation this follows from the fact that
$O_{1}$ and $O_{2}$ are linearizable and $\alpha_{i}\in\mathbb{R}$. Since
$\alpha_{i}>0$, the trace of $q_{i}(t)$ in $\Sigma_{i}$ will give us paths
$\lambda_{i}$ in $\Sigma_{i}$ turning around $b_{i}$ anticlockwise. If we
assume that $\lambda(\frac{1}{2})$ is again in the real domain
$\mathbb{R}^{2}$ and it lies in a real leaf $\gamma$ of ${\mathcal{F}}$ in the
other side of the common separatrix, then we have the main result of this
section.

######  Theorem 4.

With the notations as above

$\langle\gamma_{\frac{1}{2}},\gamma\rangle=+1,$ (12)

where we have oriented $\gamma$ from $O_{1}$ to $O_{2}$.

###### Proof.

We look at the projection $\pi(\gamma_{t})$ of the path $\gamma_{t}$ in the
common separatrix and we see Figure 8. The projection can be constructed in a
$C^{\infty}$ context by gluing the local transversal foliations $dx_{i}=0,\
i=1,2$. The intersection number is not changed under this projection and (12)
follows. ∎

Figure 7: Two saddles

.

Figure 8: Projection into the common separatrix

.

######  Remark 2.

The above proof uses arguments close to the one used by Gusein-Zade [GZ74] for
germs of isolated singularities. A second proof can be produced by making use
of more elaborated arguments of A’Campo [A’C75, p.23-24] as follows. Consider
the complex-conjugate path
$\overline{\gamma_{\frac{1}{2}}}=\gamma_{-\frac{1}{2}}$. As the complex
conjugation inverses the orientation then
$\langle\gamma_{-\frac{1}{2}},\gamma\rangle=-\langle\gamma_{\frac{1}{2}},\gamma\rangle$.
On the other hand the class $\gamma_{\frac{1}{2}}-\gamma_{-\frac{1}{2}}$ can
be represented by two disjoint paths $\alpha_{1},\alpha_{2}$ connecting
$\Sigma_{1}$ to $\Sigma_{1}$ and $\Sigma_{2}$ to $\Sigma_{2}$ respectively.
These paths define geometrically the holonomy of the two vertical
separatrices. It remains to compute the intersection index of $\alpha_{1}$
(representing holonomy) and the class of $\gamma_{\frac{1}{2}}$ (representing
the Dulac map near $O_{1}$). This is of course a local computation in a
neighborhood of $O_{1}$ and it follows from the local description of a complex
saddle that their intersection index equals one. Similar computation holds for
$\alpha_{2}$ from which the result follows.

A third proof can be obtained by deformation. Namely, it suffices to note that
the intersection index depends continuously on parameters, hence it is a
constant. Such a deformation is possible in any compact interval for the
parameter, provided that the initial and end points of the path on the cross-
section $\Sigma_{1}$ and $\Sigma_{2}$ are sufficiently close to the vertical
separatrix. Therefore, it is enough to check the claim of the theorem for some
toy example, like $df=0$ with $f=(x^{2}-1)y$, in which an explicit computation
of different paths and their deformations is possible.

######  Remark 3.

Theorem 4 holds true without the assumption that the saddle are linearizable
(with similar proof). We only need to know the asymptotic behavior of the
Dulac map.

##  5 The orbit of a vanishing cycle of center type

Let us consider the polynomial $f$ in $\mathbb{C}^{2}$ given by (9). The map
$f:\mathbb{C}^{2}\to\mathbb{C}$ defines a locally trivial fibration over the
$\mathbb{C}\backslash C$, where $C$ consists of $\frac{d(d-1)}{2}+1$ critical
values of $f$. It is the union of the values of critical points of center type
(which we assume that such critical values are distinct) and the critical
value $0$ over the $\frac{d(d+1)}{2}$ saddles points which are intersection of
lines. We choose a point $b\in\mathbb{C}$ with ${\rm Im}(b)>0$ and fix
straight paths $\gamma_{c},\ c\in C$ joining $b$ to the critical values of $f$
(a distinguished set of paths). Let also $h_{c}:H_{1}(L_{b},\mathbb{Z})\to
H_{1}(L_{b},\mathbb{Z})$ be the monodromy along $\gamma_{c}$ until getting
near to $c$, turning around $c$ anticlockwise and returning to $b$ along
$\gamma_{c}^{-1}$. Let $\delta_{c}\in H_{1}(L_{b},\mathbb{Z})$ be the center
vanishing cycle along $\lambda_{c},\ c\not=0$. Along $\gamma_{0}$ we get
$\sum_{i\not=j}{}(n_{i},n_{j})$ saddle vanishing cycles in
$H_{1}(L_{b},\mathbb{Z})$. We denote by $\overline{L_{b}}$ the curve obtained
by a smooth compactification of $L_{b}$.

######  Theorem 5.

Assume that $n_{i}$’s are pairwise coprime. The $\mathbb{Q}$-vector space
$O_{\delta}\subset H_{1}(L_{b},\mathbb{Q})$ generated by the action of
monodromy on a fixed center vanishing cycle $\delta$ has codimension $d$ in
$H_{1}(L_{b},\mathbb{Q})$. Moreover,

$O_{\delta}=\left\\{\gamma\in
H_{1}(L_{b},\mathbb{Q})\Big{|}\int_{\gamma}\frac{dl_{i}}{l_{i}}=0,i=1,\dots,d+1\right\\}$
(13)

and the restriction of the map $H_{1}(L_{b},\mathbb{Q})\to
H_{1}(\overline{L_{b}},\mathbb{Q})$ induced by inclusion, to $O_{\delta}$ is
surjective.

###### Proof.

Let $S\subset H_{1}(L_{b},\mathbb{Q})$ be the $\mathbb{Q}$-vector space
generated by saddle vanishing cycles. We first compute the action of monodromy
in $H_{1}(L_{b},\mathbb{Q})/S$. For this we prove that all center vanishing
cycles are in $O_{\delta}$. Consider center vanishing cycles
$\delta=\delta_{c_{1}}$, $\delta_{c_{2}}$, the critical points $p_{1},p_{2}$
with $f(p_{i})=c_{i}$ which are inside two adjacent polygons $P_{1}$ and
$P_{2}$ formed by the lines $l_{i}=0$. Let $l_{1}$ be the line of the common
edge which has multiplicity $n_{1}$. We are in the situation of Theorem 4. The
restriction of the map $f$ to $\Sigma_{0}$ in a local coordinate $z$ in
$\Sigma_{0}$ is given by $z\mapsto z^{n_{1}}$. Let $p_{0},\tilde{p}_{0}$ be
two points in the real transversal section $\Sigma_{0}$ in $\mathbb{R}^{2}$,
$p_{0}$ above and $\tilde{p}_{0}$ under the line $l_{1}$, and $\lambda$ as in
before Theorem 4. The image of $\lambda$ under $f|_{\Sigma_{0}}$ is a path
which starts at $f(p_{0})$ and turns $\frac{n_{1}}{2}$ times around
$0\in\mathbb{C}$. The conclusion is that
$h_{0}^{[\frac{n_{1}}{2}]+\epsilon}(\delta_{c_{1}})$ has a non-zero
intersection with $\delta_{c_{2}}$, where $\epsilon=0$ if $f(p_{0})>0$ and
$\epsilon=1$ if $f(p_{0})<0$. Using the classical Picard-Lefschetz formula,
see Theorem 2, we conclude that $\delta_{c_{2}}\in O_{\delta}$. Further
applications of Picard-Lefschetz formula will imply that all center vanishing
cycles are in $O_{\delta}$.

Our hypothesis on $n_{i}$’s implies that over the point $l_{i}\cap l_{j}$ we
have exactly one saddle vanishing cycle. For a finite polygon in the
complement of $\cup_{i=1}^{d+1}l_{i}$ in $\mathbb{R}^{2}$, let
$a_{1},a_{2},\ldots,a_{s}$ be the multiplicity of its edges formed by the
lines $l_{1},l_{2},\ldots,l_{s}$. Let also $\delta$ be the center vanishing
cycle inside this polygon. We look $\delta$ in the deformation retract $G$ of
$L_{b}$ in Proposition 2. The monodromy $h^{a_{1}a_{2}\cdots\hat{a_{i}}\cdots
a_{s}}(\delta)$, where $\hat{a_{i}}$ means $a_{i}$ is removed, fixes all the
edges of $\delta_{i}$ except for the $i$-the edge, and its iteration will
replace its $i$-th edge with any other $a_{i}$ paths in the deformation
retract of $L_{b}$. Moreover, any path in $H_{1}(G,\mathbb{Z})/S$ is a linear
combination of center vanishing cycles. The conclusion is that the action of
monodromy on $H_{1}(L_{b},\mathbb{Q})/S$ generates the whole space. By the
classical Picard-Lefschetz formula we have

$h^{a}(\delta)=\delta+\sum_{i=1}^{s}\frac{a}{a_{i}a_{i+1}}\delta_{i,i+1},\ \ \
a:=a_{1}a_{2}\cdots a_{s},\ \ \ s+1:=1,$ (14)

where $\delta_{i,i+1}$ is the saddle vanishing cycle over $l_{i}\cap l_{i+1}$.
It follows that by the action of monodromy we can generate a sum of saddle
vanishing cycles as above. It is easy to see that these elements are linearly
independent in $H_{1}(L_{b},\mathbb{Q})$. The codimesnion in $S$ of the
$\mathbb{Q}$-vector space generated by these elements is exactly $d$.

Let $\delta_{i,h},\ i=1,2,\ldots,d+1,\ h=1,2,\ldots,(n,n_{i})$ be the closed
cycles around the points at infinity $p_{i,h}$ of $L_{b}$ corresponding to the
intersection of $l_{i}$ with the line at infinity. An easy residue calculation
shows that

$\int_{\delta_{j,h}}\frac{dl_{i}}{l_{i}}=\left\\{\begin{array}[]{cc}\frac{n-n_{j}}{(n,n_{j})}&i=j\\\
\frac{-n_{j}}{(n,n_{j})}&i\not=j\end{array}.\right.$ (15)

This shows that cohomology classes of the $d+1$ logarithmic one-forms
$\frac{dl_{i}}{l_{i}}$ in $H^{1}_{dR}(L_{b})$ generate a vector space of
dimension $d$ (there is one linear relation between these forms restricted on
$L_{b}$). The equality (13) follows, as both sides of the equality are of
codimension $d$ and $\subseteq$ is trivially true. Moreover, by (15) we have
$\delta_{i,h}-\delta_{i,0}\in O_{\delta},\ \ i=1,2,\ldots,(n,n_{i})-1$ and
$H_{1}(L_{b},\mathbb{Q})$ is a direct sum of $O_{\delta}$ with the the
$\mathbb{Q}$-vector space generated by $\delta_{i,0},\ i=1,2,\ldots,d$. ∎

######  Remark 4.

A purely topological argument for the last part of the proof of Theorem 5 can
be formulated following [Mov04b, Section 2] and it is as follows. Let us
orient the center vanishing cycles using the anticlockwise orientation of
$\mathbb{R}^{2}$. We can orient the saddle vanishing cycle $\delta$ attached
to $l_{i}\cap l_{j}$ in such a way that it intersects positively the center
vanishing cycles in the finite polygons with $l_{i}\cap l_{j}$ vertex. For any
line $l_{i}$, let $\delta^{i}$ be the alternative sum of saddle vanishing
cycles in the order which $l_{i}$ intersects others. It turns out that the
intersection of $\delta^{i}$’s with center vanishing cycles is zero. Since it
is invariant under monodromy $h_{0}$ around $0$, its intersection with all
$h^{k}(\delta)$, $\delta$ center vanishing cycle, is also zero. The conclusion
is that the intersection of $\delta^{i}$ with all the elements in
$H_{1}(L_{b},\mathbb{Z})$ is zero and hence it is in the kernel of
$H_{1}(L_{b},\mathbb{Q})\to H_{1}(\overline{L_{b}},\mathbb{Q})$. After taking
a proper sign for $\delta^{i}$, we know that $\sum_{i=1}^{d+1}\delta^{i}=0$.
Now, it is an elementary problem to check that $\delta^{i},\ i=1,2,\ldots,d$
and $\frac{d(d-1)}{2}$ elements (14) attached to each polygon are linearly
independent and form a basis for the vector space generated by saddle
vanishing cycles.

##  6 Proof of Theorem 1

Let $\mathbb{C}[x,y]_{\leq 1}$ be the complex vector space of bi-variate
complex polynomials of degree at most one. The set ${\mathcal{L}}(1^{d+1})$ of
logarithmic foliations (2), is parameterized by the map

$\tau:\mathbb{C}^{d+1}\times\mathbb{C}[x,y]_{\leq
1}^{d+1}\rightarrow{\mathcal{F}}(d)$ (16)
$\tau(\lambda_{1},\ldots,\lambda_{d+1},l_{1},\ldots,l_{d+1})=l_{1}\cdots
l_{d+1}\sum_{i=1}^{d+1}\lambda_{i}\frac{dl_{i}}{l_{i}}$ (17)

and hence it is an irreducible algebraic set. The differential

$\displaystyle D\tau(n_{1},\dots,n_{d+1},l_{1},\ldots,l_{d+1})$ (18)

of $\tau$ at the point

$\displaystyle\omega_{0}=l_{1}l_{2}\cdots
l_{d+1}(\sum_{i=1}^{d+1}n_{i}\frac{dl_{i}}{l_{i}})\in{\mathcal{L}}(1^{d+1})$
(19)

applied to the vector
$(\lambda_{1},\ldots,\lambda_{d+1},p_{1},\ldots,p_{d+1})$ is

$l_{1}l_{2}\cdots
l_{d+1}\left\\{\sum_{i=1}^{d+1}\lambda_{i}\frac{dl_{i}}{l_{i}}+\left(\sum_{i=1}^{d+1}\frac{p_{i}}{l_{i}}\right)\left(\sum_{i=1}^{d+1}n_{i}\frac{dl_{i}}{l_{i}}\right)+d\left(\sum_{i=1}^{d+1}n_{i}\frac{p_{i}}{l_{i}}\right)\right\\}.$
(20)

It is easy to check that if a vector
$(\lambda_{1},\ldots,\lambda_{d+1},p_{1},\ldots,p_{d+1})$ is in the kernel of
$D\tau$ then for every $i$ the polynomial $p_{i}$ is colinear to $l_{i}$. It
follows from this that the dimension of the kernel is $d+1$, respectively the
dimension of the image (20) is $3(d+1)$.

Let $\omega_{0}$ be the polynomial one-form defined by (19) which we assume to
be with real coefficients. We denote by $\delta_{t}\subset\\{f=t\\}$ a
continuous family of real vanishing cycles around a real center of
$\omega_{0}$, where the parameter $t$ is the restriction of the first integral
$f=\Pi_{i=1}^{d+1}l_{i}^{n_{i}}$ to a cross-section to $\\{f=t\\}$. Let

${\mathcal{F}}_{\epsilon}:\ \
\omega_{\epsilon}:=\omega_{0}+\epsilon\omega_{1}+\cdots$ (21)

be an arbitrary degree $d$ deformation of ${\mathcal{F}}_{0}$. The return map
associated to the family $\delta_{t}$ and the deformation
${\mathcal{F}}_{\epsilon}$ takes the form

$t+\varepsilon\int_{\delta_{t}}\tilde{\omega}_{1}+O(\varepsilon),\;\;\tilde{\omega}_{1}=\frac{\omega_{1}}{l_{1}\cdots
l_{d+1}}$

######  Proposition 3.

The Melnikov integral
$M_{1}(t):=\int_{\delta_{t}}\frac{\omega_{1}}{l_{1}\cdots l_{d+1}}$ vanishes
identically if and only if $\omega_{1}$ is of the form (20).

###### Proof.

It is trivially seen that if $\omega_{1}$ belongs to the image of $D\tau$
(20), then $\int_{\delta_{t}}\tilde{\omega}_{1}$ vanishes identically. We
shall prove now the converse. If $\int_{\delta_{t}}\tilde{\omega}_{1}$
vanishes then it vanishes on every other family of cycles which are in the
orbit of $\delta_{t}$, and hence on the vector space $O_{\delta_{t}}$ spanned
by the orbit. By Theorem 5 the dual of $O_{\delta}$ in $H_{dR}^{1}(L_{t})$ has
a basis given by $\frac{dl_{i}}{l_{i}},\ i=1,2,\ldots,d$. Therefore, there are
unique constants $\lambda_{i},i=1,2,\ldots,d$ (depending on $t$) such that the
cohomology class of

$\displaystyle\tilde{\omega}_{1}-\sum_{i=1}^{d}\lambda_{i}\frac{dl_{i}}{l_{i}}$
(22)

in $H^{1}_{DR}(L_{t})$ is zero. Using (15), we can see that $\lambda_{i}$ is a
linear combination of integrals $\frac{1}{2\pi
i}\int_{\delta_{i,0}}\tilde{\omega}_{1},\ \ i=1,2,\ldots,d$ with rational, and
hence constant, coefficients. We assume that $n_{1},n_{2},\ldots,n_{d}>1$.
Multiplying $\int_{\delta_{i,0}}\tilde{\omega}_{1}$ with $t^{\frac{1}{n_{i}}}$
and $t^{\frac{-1}{n}}$ we can see that it grows like $t^{-\frac{1}{n_{i}}}$ as
$t\to 0$ and like $t^{\frac{1}{n_{i}}}$ as $t^{-1}\to 0$. Since
$\lambda_{i}$’s are uni-valued, it follows that they are meromorphic function
in $t$ with pole order $<1$ at $t=0,\infty$ and hence they are constant.

With the same arguments as in [Mov04a, Theorem 4.1] we deduce that if a one-
form on $L_{t}$ is co-homologically zero, then it is relatively exact, that is
to say

$\tilde{\omega}_{1}-\sum_{i=1}^{d+1}\lambda_{i}\frac{dl_{i}}{l_{i}}=d\tilde{P}+\tilde{Q}\tilde{\omega}_{0},$
(23)

where $\tilde{P}$ and $\tilde{Q}$ have only poles of order $\leq 1$ along the
lines $l_{i}=0$ and the line at infinity and
$\tilde{\omega}_{0}=\frac{\omega_{0}}{l_{1}l_{2}\cdots l_{d+1}}$. The crucial
observation is that the one-form (23) is logarithmic along the line at
infinity (after compactifying $\mathbb{C}^{2}$ to $\mathbb{P}^{2}$). Namely,
$\omega_{1}$ is of (affine) degree $\leq d$ which implies $\tilde{\omega}_{1}$
and $d\tilde{\omega}_{1}$ have a pole of order at most one along the infinite
line of $\mathbb{P}^{2}$. This implies that
$d\tilde{Q}\wedge\tilde{\omega}_{0}$ has pole order $\leq 1$ at infinity, and
hence $\tilde{Q}$ is holomorphic at infinity, and by the equality (23),
$\tilde{P}$ is also holomorphic at infinity. The conclusion is that we can
write

$\tilde{P}=\frac{P}{l_{1}l_{2}\cdots l_{d+1}},\ \
\tilde{Q}:=\frac{Q}{l_{1}l_{2}\cdots l_{f+1}},$

where $P,Q\in\mathbb{C}[x,y]$ are polynomials of degree $\leq d+1$.
Multiplying the equality (23) with $l_{1}l_{2}\cdots l_{d+1}$ and considering
it modulo $l_{i}=0$, we get $l_{i}|P-Qn_{i}$. If $n_{i}\not=n_{j}$ this
implies that $P$ and $Q$ vanishes in the intersection points $l_{i}\cap
l_{j}$. Knowing the degree of $P$ and $Q$, we conclude that both $P$ and $Q$
are of the form $l_{1}l_{2}\cdots
l_{d+1}(\sum_{i=1}^{d+1}a_{i}\frac{p_{i}}{l_{i}})$, where $\deg(p_{i})\leq 1$
and $a_{i}\in\mathbb{C}$ depend on $P,Q$. Substituting this ansatz for $P$ and
$Q$ in (23) we get the desired form of $\omega_{1}$ in (20). ∎

The geometric meaning of the above Proposition is that the tangent space of
${\mathcal{L}}(1^{d+1})$ and ${\mathcal{M}}(d)$ at the point
${\mathcal{F}}_{0}$ are the same, and that they are given by (20). By
dimension count the dimension of this tangent space equals the dimension of
${\mathcal{L}}(1^{d+1})$. Therefore ${\mathcal{F}}_{0}$ is a smooth point on
${\mathcal{M}}(d)$ and moreover ${\mathcal{L}}(1^{d+1})$ is an irreducible
component of the center set ${\mathcal{M}}(d)$. Note that there are no other
components of ${\mathcal{M}}(d)$ containing ${\mathcal{F}}_{0}$ and tangent to
${\mathcal{L}}(1^{d+1})$, otherwise the Zariski tangent space would be bigger.

######  Remark 5.

Assuming Corollary 3 we may complete the proof of Theorem 1 by general
arguments which we sketch in what follows, see [FGX20]. Let
$\mathcal{B}=(b_{1},b_{2},\dots,b_{N})$ be generators of the Bautin ideal near
the point ${\mathcal{F}}_{0}\in{\mathcal{F}}(d)$ defined in Theorem 1. Here
$\mathcal{B}$ is seen as an ideal of the local ring of convergent power
series. Following [FGX20, section 3.2] we divide the return map $P(t)$
associated to the family of vanishing cycles $\delta_{t}$ of
${\mathcal{F}}_{0}$ under a general (multi-parameter) perturbation in
${\mathcal{F}}(d)$ of the form $\omega_{0}+\omega_{1}$ to obtain

$P(t)-t=\sum b_{i}(\Phi_{i}(t)+\dots).$

Here the dots replace some convergent power series in the parameters and $t$,
which vanish at ${\mathcal{F}}_{0}$. Every Melnikov function (of arbitrary
order) is an appropriate linear combination of $\Phi_{i}(t)$, although the
converse is not necessarily true. In particular we may suppose that the vector
space of first order (or linear) Melnikov function has a basis formed by
$\Phi_{i}(t),i=1,\dots,k$. The linear independence of the first order Melnikov
function is equivalent then to the linear independence of the differentials
$Db_{i},i=1,\dots,k$ computed at ${\mathcal{F}}_{0}$, see [FGX20, Corollary
2]. Thus, performing a local bi-analytic change of variables in the parameter
space, we may assume that $b_{1},b_{2},\dots,b_{k}$ are new parameters
defining a plane which contains ${\mathcal{L}}(1^{d+1})$ near
${\mathcal{F}}_{0}$. By comparison of dimensions we conclude that
${\mathcal{L}}(1^{d+1})$ coincides to the plane $\\{b_{1}=\dots=b_{k}=0\\}$
near ${\mathcal{F}}_{0}$. As for the other generators $b_{i},i>k$ they already
belong to the ideal $(b_{1},\dots,b_{k})$, because they vanish identically
along ${\mathcal{L}}(1^{d+1})$. Thus ${\mathcal{M}}(d)$ coincides near
${\mathcal{F}}_{0}$ with ${\mathcal{L}}(1^{d+1})$.

######  Remark 6.

Our hypothesis in Theorem 1 suggests to study the subset of
$(\mathbb{Q}^{+})^{d}$ given by points
$(\frac{n_{1}}{n_{d+1}},\frac{n_{2}}{n_{d+1}},\ldots,\frac{n_{d}}{n_{d+1}})$,
where $n_{i}$’s are pairwise relatively prime positive integers. For instance,
it is not clear whether this set is dense in $(\mathbb{Q}^{+})^{d}$ or not.
Note that its projection in each coordinate is dense and the fibers of this
projection are finite sets.

##  7 Quadratic foliations

For quadratic foliations, that is the case $d=2$, the classification of
components of ${\mathcal{M}}(2)$ follows from the computations of H. Dulac in
[Dul23], see [FGX20, Appendix A], [Lin14, Theorem 1.1] and [IY08, Section
13.9]. The algebraic set ${\mathcal{M}}(2)$ has four components

  1. 1.

${\mathcal{L}}(1^{3})$,

  2. 2.

the set ${\mathcal{L}}(1,2)$ of logarithmic foliations of the form

$f_{1}f_{2}(\lambda_{1}\frac{df_{1}}{f_{1}}+\lambda_{2}\frac{df_{2}}{f_{2}}),\
\deg(f_{1})=1,\deg(f_{2})=2,\lambda_{1},\lambda_{2}\in\mathbb{C}-\\{0\\}$

  3. 3.

the set ${\mathcal{L}}(3)$ of Hamiltonian foliations ${\mathcal{F}}(df),\
\deg(f)=3$

  4. 4.

an exceptional component obtained by the action of yhe affine group ${\rm
Aff}(\mathbb{C}^{2})$ on the foliation with the first integral
$\frac{(x^{2}+2y+\alpha)^{3}}{(x^{3}+3xy+1)^{3}},\ \alpha\in\mathbb{P}^{1}$

see [Gav20, Proposition 4.7]. Using this, one may prove the following: the
only singular points of ${\mathcal{M}}(2)$ in ${\mathcal{L}}(1^{3})$ are
${\mathcal{L}}(1^{3})\cap{\mathcal{L}}(1,2)$ and
${\mathcal{L}}(1^{3})\cap{\mathcal{L}}(3)$, that is,

${\rm
Sing}\;{\mathcal{M}}(2)\cap{\mathcal{L}}(1^{3})=\left({\mathcal{L}}(1,2)\cap{\mathcal{L}}(1^{3})\right)\cup\left({\mathcal{L}}(3)\cap{\mathcal{L}}(1^{3})\right).$
(24)

A finer result is the classification of the components of the Bautin scheme
which is done by many authors and for many subspaces of ${\mathcal{F}}(2)$,
see [Zol94] and references therein for an overview of this. Following [IY08]
we consider the following normal form of quadratic systems with a Morse center
at the origin

$\left\\{\begin{array}[]{ccc}\dot{x}&=&-ix+Ax^{2}+Bxy+Cy^{2},\\\
\dot{y}&=&iy+C^{\prime}y^{2}+B^{\prime}xy+A^{\prime}x^{2},\end{array}\
A,B,C,A^{\prime},B^{\prime},C^{\prime}\in\mathbb{C}.\right.$ (25)

The Bautin ideal of the above system has been extensively studied in the
literature, see [Zol94, IY08]. The Bautin ideal associated is generated by
$g_{2},g_{3},g_{4}$, where

$\displaystyle g_{2}$ $\displaystyle:=$ $\displaystyle
AB-A^{\prime}B^{\prime},$ $\displaystyle g_{3}$ $\displaystyle:=$
$\displaystyle(2A+B^{\prime})(A-2B^{\prime})CB^{\prime}-(2A^{\prime}+B)(A^{\prime}-2B)C^{\prime}B,$
$\displaystyle g_{4}$ $\displaystyle:=$
$\displaystyle(BB^{\prime}-CC^{\prime})((2A+B^{\prime})B^{\prime
2}C-(2A^{\prime}+B)B^{2}C^{\prime}).$

The computation of the primary decomposition of this ideal implies four
reduced components which are explicitly written in [Zol94, Theorem 1]:

  1. 1.

Lotdka-Volterra component ${\mathcal{L}}(1^{3})$: $B=B^{\prime}=0$,

  2. 2.

Hamiltonian ${\mathcal{L}}(3)$: $2A+B^{\prime}=2A^{\prime}+B=0$,

  3. 3.

Reversible ${\mathcal{L}}(1,2)$: $AB-A^{\prime}B^{\prime}=B^{\prime
3}C-B^{3}C^{\prime}=AB^{\prime
2}C-A^{\prime}B^{2}C^{\prime}=A^{2}B^{\prime}C-A^{\prime
2}BC^{\prime}=A^{3}C-A^{\prime 3}C^{\prime}$,

  4. 4.

Exceptional: $A-2B^{\prime}=A^{\prime}-2B^{\prime}=CC^{\prime}-BB^{\prime}=0$.

Note that the ideal of the reversible component is radical and is written in a
Groebner basis (in contrast to [IY08, section 13] where the corresponding
”symmetric” component turns out to be reducible). We can also compute the
ideal of its singular set. It is clear that the Hamiltonian and Lotka-Volterra
components are smooth and the exceptional component has an isolated
singularity at $A=\cdots=C^{\prime}=0$. The reversible component has more
interesting singularities:

${\rm
Sing}({\mathcal{L}}(1,2))=\\{B=B^{\prime}=A=A^{\prime}=0\\}={\mathcal{L}}(3)\cap{\mathcal{L}}(1^{3}).$

The foliation ${\mathcal{F}}$ with
$A=B=A^{\prime}=B^{\prime}=0,C=C^{\prime}=1$ has the first integral
$f:=(\frac{1}{2}-x)(y^{2}-\frac{1}{3}(x+1)^{2})$, see [Ili98, page 159]. For
the computer codes used in this computation see the latex file of the present
article in arxiv. It must be noted that ${\mathcal{L}}(1^{3})$ itself is not
smooth, for instance, it has a nodal singularity at the foliation with the
first integral $\frac{x^{2}+y^{2}}{2y-1}$ which has been studied in [FG20].

######  Proposition 4.

The singular set of ${\mathcal{L}}(1^{3})$ is the orbit of the affine group
${\rm Aff}(\mathbb{C}^{2})$ on the foliation with the first integral
$\frac{(x+1)(y-1)}{xy}$.

For an illustration of the above phenomenon see [FG20, Figure 2].

###### Proof.

We know that the kernel of the derivation of the parametrization $\tau$ in
(16) has constant dimension. This implies that all singularities of
${\mathcal{L}}(1^{3})$ are due to the noninjectivity of $\tau$. For a
foliation ${\mathcal{F}}={\mathcal{F}}(\omega)\in{\rm
Sing}({\mathcal{L}}(1^{3}))$ we get $f=l_{1}l_{2}l_{3}$ and
$g=\tilde{l}_{1}\tilde{l}_{2}\tilde{l}_{3}$, where $\\{l_{i}=0\\}$’s (resp.
$\\{\tilde{l}_{i}=0\\}$’s) are distinct lines, such that
$d(\frac{\omega}{f})=d(\frac{\omega}{g})=0$, and hence, $F:=\frac{f}{g}$ is a
first integral of ${\mathcal{F}}$. It turns out that one of the lines
$\\{l_{i}=0\\}$’s must be equal to one of $\\{\tilde{l}_{i}=0\\}$’s, and since
${\mathcal{F}}$ is of degree $2$, $F$ is the quotient of two lines by another
two lines. Further, $F-1$ is the quotient of a line with another two lines. We
conlude that up to the action of ${\rm Aff}(\mathbb{C}^{2})$, the foliation
${\mathcal{F}}$ has the first integral $F:=\frac{(x+1)(y-1)}{xy}$. Note that
two branches of ${\mathcal{L}}(1^{3})$ near ${\mathcal{F}}(\omega)$ correspond
to $F-1=\frac{-x+y-1}{xy}$ and $\frac{F-1}{F}=\frac{-x+y-1}{(x+1)(y-1)}$. ∎

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