# Foliations on smooth algebraic surfaces over positive characteristic

Wodson Mendson

00footnotetext: 2010 _Mathematics Subject Classification._ 32S65; 13N15;
13A3500footnotetext: _Keywords._ Foliations; Foliations over positive
characteristic; Frobenius morphism; $p$-divisor00footnotetext: This work was
developed at IMPA. W. Mendson acknowledges support of CNPq and Faperj

######  Abstract.

We investigate the notion of the $p$-divisor for foliations on a smooth
algebraic surface defined over a field of positive characteristic $p$ and we
study some of their properties. We present a structure theorem for the
$p$-divisor of foliations in the projective plane and the Hirzebruch surfaces
where we show that, under certain conditions, such $p$-divisors are reduced.

###### Contents

  1. 1 Introduction
  2. 2 Notation
  3. 3 Foliations on surfaces and the $p$-divisor
  4. 4 The $p$-divisor for foliations on $\operatorname{\mathbb{P}_{k}^{2}}$
  5. 5 The $p$-divisor for foliations on $\operatorname{\mathbb{P}_{k}^{1}\times\mathbb{P}_{k}^{1}}$
  6. 6 The $p$-divisor for foliations on Hirzebruch surfaces

##  1 Introduction

In his monograph [7] Jouanolou showed that a very generic foliation of degree
$d\geq 2$ in the projective plane has no algebraic solutions. The crucial
point of his argument consists in constructing examples, for each degree
$d\geq 2$, of foliations with no algebraic solutions. The term very generic
means that there exists a countable union $F$ of closed sets in the space of
holomorphic foliations of degree $d$ such that for any foliations
$\mathcal{F}$ lying outside $F$ has no algebraic solutions.

In order to prove his theorem, Jouanolou showed the following result.

######  Theorem (Jouanolou).

For every $d\in\mathbb{Z}_{\geq 2}$ the foliation in
$\mathbb{P}_{\mathbb{C}}^{2}$ defined by the vector field

$v_{d}=(xy^{d}-1)\frac{\partial}{\partial
x}-(x^{d}-y^{d+1})\frac{\partial}{\partial y}$

has no algebraic solutions.

Recently, a similar problem was considered in [11] for Hirzebruch surfaces. By
assuming certain conditions in the normal bundle it is shown that a foliation
in the Hirzebruch surfaces has no algebraic solutions (see [11, Theorem C]).

Interestingly, the analogue of the Jouanolou’s theorem over a field of
positive characteristic is completely false (see [15]). The results
established in [15] imply the following proposition.

######  Proposition A.

Let $\operatorname{k}$ be an algebraically closed field of characteristic
$p>0$. Let $\mathcal{F}$ be a foliation on $\mathbb{P}_{\operatorname{k}}^{2}$
and suppose that $\deg(\mathcal{F})<p-2$. Then $\mathcal{F}$ has an invariant
algebraic curve.

In positive characteristic $p$ there are two classes of foliations: the
foliations that are $p$-closed and the foliations that are not. Given a
foliation $\mathcal{F}$ in a smooth algebraic surface $X$ defined over a field
$\operatorname{k}$ of characteristic $p>0$, we say that it is $p$-closed if
their tangent sheaf $T_{\mathcal{F}}$ is closed by $p$-powers. This is
equivalent to say that the $\mathcal{O}_{X}$-morphism

$\psi_{\mathcal{F}}\colon F_{X}^{*}T_{\mathcal{F}}\longrightarrow
N_{\mathcal{F}}\qquad v\mapsto v^{p}\mod T_{\mathcal{F}}$

is the zero morphism where $F_{X}$ is the absolute Frobenius morphism and
$N_{\mathcal{F}}$ is the normal bundle of $\mathcal{F}$. The morphism
$\psi_{\mathcal{F}}$ is called the $p$-curvature of the foliation
$\mathcal{F}$.

When the foliation is $p$-closed it follows from [2, Théorème 1] that there
are infinitely many algebraic solutions. On the other hand, if the foliation
$\mathcal{F}$ is not $p$-closed then there is a divisor
$\Delta_{\mathcal{F}}$, the $p$-divisor, which is defined as the degeneracy
locus of the $p$-curvature morphism $\psi_{\mathcal{F}}$. An interesting
property of the $p$-divisor is that every irreducible algebraic solution of
the foliations is contained in the support of the $p$-divisor.

Since the set of foliations that are not $p$-closed is open we can ask about
the structure of the $p$-divisor for generic foliations.

######  Problem 1.

Let $X$ be a smooth algebraic surface defined over a field $\operatorname{k}$
of characteristic $p>0$. What can we say about the $p$-divisor
$\Delta_{\mathcal{F}}$ of a generic foliation $\mathcal{F}$?

We present results towards the solution of this problem in the case where $X$
is the projective plane or a Hirzebruch surface. More precisely, we show the
following theorems.

######  Theorem A.

Let $\operatorname{k}$ be an algebraically closed field of characteristic
$p>0$. A generic foliation in the projective plane
$\mathbb{P}_{\operatorname{k}}^{2}$ of degree $d\geq 1$ with $p\nmid d-1$ has
reduced $p$-divisor.

######  Theorem B.

Let $\operatorname{k}$ be an algebraically closed field of characteristic
$p>0$. Let $\Sigma_{d}$ be the $d$-Hirzebruch surface defined over
$\operatorname{k}$ and let $d_{1},d_{2}\in\mathbb{Z}_{\geq 0}$ such that
$p\nmid d_{i}$, if $d_{i}\neq 0$. Let $F$ be a fiber of the natural projection
$\pi\colon\Sigma_{d}\longrightarrow\mathbb{P}_{\operatorname{k}}^{1}$ and
$M_{d}$ be a section which satisfies $F\cdot M_{d}=1$ and $M_{d}^{2}=d$. Then
a generic foliation in $\Sigma_{d}$ with normal bundle $N$ which is
numerically equivalent to $(d_{1}-d+2)F+(d_{2}+2)M_{d}$ has reduced
$p$-divisor.

###  1.1 Organization of the paper

In Section 2 we fix the notations that will be used in the paper. In Section
3, we introduce the definition of $p$-divisor of foliations on algebraic
surfaces defined over a field of characteristic $p>0$. We recall the global
representation of foliations in the projective plane and in the Hirzebruch
surface. We present the definition of the $p$-divisor of a foliation and we
investigate some of their properties, in particular, some applications to
algebraicity of foliations on complex projective plane. In Section 4 we
consider the problem of the structure of the $p$-divisor for foliations in the
projective plane and in the Section 5 we study the problem for
$\mathbb{P}_{\operatorname{k}}^{1}\times\mathbb{P}_{\operatorname{k}}^{1}$. In
Section 6 we finalize by considering the Hirzebruch surface of type $d>0$.

##  2 Notation

  * •

$\operatorname{k}=$ algebraically closed field of characteristic $p>0$.

  * •

$X=$ smooth algebraic surface defined over $\operatorname{k}$.

  * •

Curve in $X$ $=$ effective divisor in $X$.

  * •

$d$-Hirzebruch surface over $\operatorname{k}$ ($d\geq 0$) $=$
$\Sigma_{d}=\mathbb{P}(\mathcal{O}_{\mathbb{P}_{\operatorname{k}}^{1}}\oplus\mathcal{O}_{\mathbb{P}_{\operatorname{k}}^{1}}(d))$.

  * •

$M=$ the curve in $\Sigma_{d}$ ($d>0$) such that $M^{2}=-d$.

  * •

$\equiv$ $=$ numerical equivalence in $\operatorname{Div}(X)$.

  * •

$\operatorname{Num}_{\mathbb{Q}}(X)=(\operatorname{Div}(X)/\equiv)\otimes\mathbb{Q}$.

  * •

If $(X,H)$ is a polarized surface and $D$ is a divisor in $X$, $\deg(D)=D\cdot
H$.

  * •

$U(X,H,N),V(X,H,\mathcal{N})=$ open sets of Lemma 3.12.

  * •

$m_{Q}(C)=$ algebraic multiplicity of the curve $C\subset X$ at $Q\in X$.

  * •

If $\pi_{Q}\colon Bl_{Q}(X)\longrightarrow X$ is the blow up at $Q$,
$\mathcal{F}$ is a foliation in $X$, $\mathcal{G}=\pi_{Q}^{*}\mathcal{F}$ and
$E$ is the exceptional divisor,
$l(Q)=\operatorname{ord}_{E}(N_{\mathcal{G}}^{*}-\pi_{Q}^{*}N_{\mathcal{F}}^{*})$.

  * •

$O(2)=$ terms of the order at least $2$.

##  3 Foliations on surfaces and the $p$-divisor

Let $\operatorname{k}$ be an algebraically closed field and $X$ be a smooth
algebraic surface defined over $\operatorname{k}$. A foliation $\mathcal{F}$
in $X$ is a coherent subsheaf $T_{\mathcal{F}}\subset T_{X}$ of rank one which
satisfies the following properties:

  * •

The sheaf $T_{\mathcal{F}}$ is closed by Lie brackets;

  * •

The quotient $T_{X}/T_{\mathcal{F}}$ is torsion free, that is,
$T_{\mathcal{F}}$ is saturated in $T_{X}$.

We can define a foliation in $X$ in more explicitly terms. In this terms, a
foliation in $X$ consists of a system $\\{(U_{i},\omega_{i},v_{i})\\}_{i\in
I}$ such that:

  * •

The collection $\\{U_{i}\\}_{i\in I}$ is an open cover of $X$;

  * •

For each $i\in I$ we have $v_{i}\in T_{X}(U_{i})$,
$\omega_{i}\in\Omega_{X/k}^{1}(U_{i})$ such that $i_{v_{i}}\omega_{i}=0$;

  * •

In $U_{i}\cap U_{j}$ we have $\omega_{i}=f_{ij}\omega_{j}$ and
$v_{i}=g_{ij}v_{j}$ for some functions
$f_{ij},g_{ij}\in\mathcal{O}_{X}(U_{ij})^{*}$;

  * •

For each $i\in I$ we have
$\operatorname{codim}\operatorname{sing}(\omega_{i})\geq 2$ and
$\operatorname{codim}\operatorname{sing}(v_{i})\geq 2$.

Note that the second definition is an alternative version of the first.
Indeed, given a foliation $\\{(U_{i},\omega_{i},v_{i})\\}_{i\in I}$ in $X$ we
can construct a saturated subsheaf of $T_{X}$ in the following way: For each
open $U$ of $X$ we define $T_{\mathcal{F}}(U)$ by

$T_{\mathcal{F}}(U)=\\{v\in T_{X}(U)\mid i_{v}\omega_{i}|_{U_{i}}=0\mbox{ in
$U\cap U_{i}$ for all $i\in I$}\\}.$

We ensure that $T_{\mathcal{F}}$ is a saturated subsheaf of $T_{X}$ by using
the condition imposed in the singular set of $\omega_{i}$. Reciprocally, let
$T_{\mathcal{F}}$ be a foliation in $X$ and consider the global section
$\omega\in\operatorname{H^{0}}(X,\Omega_{X}^{1}\otimes N_{\mathcal{F}})$
induced by the morphism $T_{X}\longrightarrow N_{\mathcal{F}}$. Then, by
definition we obtains a open cover $\\{U_{i}\\}_{i\in I}$ of $X$, $1$-forms
$\omega_{i}\in\Omega_{X}^{1}(U_{i})$ and functions $\\{f_{ij}\\}_{i,j}$ with
$f_{ij}\in\mathcal{O}_{X}^{*}(U_{i}\cap U_{j})$ representing
$N_{\mathcal{F}}^{*}$ such that $\omega_{i}=f_{ij}\omega_{j}$. Since
$T_{\mathcal{F}}$ is a saturated subsheaf of $T_{X}$ we ensure that
$\operatorname{codim}\operatorname{sing}(\omega_{i})\geq 2$ for every $i$.
Note that the vector fields $v_{i}$ are obtained in a similar way by
considering a global section $v\in\operatorname{H^{0}}(X,T_{X}\otimes
T_{\mathcal{F}}^{*})$ induced by the inclusion $T_{\mathcal{F}}\subset T_{X}$.
By construction, we obtain vector fields $\\{v_{i}\\}_{i\in I}$ in
$\\{U_{i}\\}_{i\in I}$ such that for every $i$ the vector field $v_{i}$ is
tangent to the $1$-form $\omega_{i}$ that defines $\mathcal{F}$ in the open
set $U_{i}$.

Let $\\{(U_{i},\omega_{i},v_{i})\\}_{i\in I}$ be a foliation in $X$. The
collection $\\{f_{ij}^{-1}\\},\\{g_{ij}\\}$ determines elements of
$\operatorname{H^{0}}(X,\mathcal{O}_{X}^{*})=\operatorname{Pic}(X)$ and the
line bundles associated are the conormal bundle $\Omega^{1}_{X/\mathcal{F}}$
and the cotangent bundle $\Omega^{1}_{\mathcal{F}}$ associated to
$\mathcal{F}$. Any divisor in the correspondent linear classes to
$\Omega_{\mathcal{F}}^{1}$ and $(\Omega^{1}_{X/\mathcal{F}})^{*}$ will be
called the canonical divisor and the normal divisor associated to
$\mathcal{F}$ and will be denoted by $K_{\mathcal{F}}$ and $N_{\mathcal{F}}$.

###  3.1 The $p$-divisor for foliations on smooth algebraic surfaces

We start this section recalling the following basic lemma about derivations
over fields of positive characteristic.

######  Lemma 3.1.

Let $\operatorname{k}$ be an field of characteristic $p>0$ and $R$ be a
$\operatorname{k}$-domain. Let $D\in\operatorname{Der}_{k}(R)$ be a
$\operatorname{k}$-derivation. Then
$D^{p}\in\operatorname{Der}_{\operatorname{k}}(R)$. If $f\in R$ then

$(fD)^{p}=f^{p}D^{p}-fD^{p-1}(f^{p-1})D.$

###### Proof.

[8, Proposition (5.3)]. ∎

Let $X$ be an algebraic surface defined over a algebraically closed field
$\operatorname{k}$ of characteristic $p>0$. Let
$\mathcal{F}=\\{(U_{i},\omega_{i},v_{i})\\}$ be a foliation on $X$.

######  Definition 3.2.

We say that $\mathcal{F}$ is $p$-closed if for some $i$ we have $v_{i}\wedge
v_{i}^{p}=0$.

######  Remark 3.3.

By Lemma 3.1 we conclude that $v_{i}\wedge v_{i}^{p}=0$ for some $i$ if and
only if $v_{j}\wedge v_{j}^{p}=0$ for every $j$.

Suppose that $\mathcal{F}$ is not $p$-closed. In the open $U_{ij}=U_{i}\cap
U_{j}$ we have $\omega_{i}=f_{ij}\omega_{j}$ and $v_{i}=g_{ij}v_{j}$. Since we
are assuming that $\mathcal{F}$ is not $p$-closed we have for each $i,j\in I$

$0\neq
i_{v_{i}^{p}}\omega_{i}=i_{(g_{ij}v_{j})^{p}}f_{ij}\omega_{j}=i_{(g_{ij}^{p}v_{j}^{p}+g_{ij}v_{j}^{p-1}(g_{ij}^{p-1})v_{j})}f_{ij}\omega_{j}=g_{ij}^{p}f_{ij}i_{v_{j}^{p}}\omega_{j}\neq
0.$

So, the collection $\\{i_{v_{i}^{p}}\omega_{i}\\}_{i\in I}$ determines a
global section $0\neq s_{\mathcal{F}}\in
H^{0}(X,(\Omega_{\mathcal{F}}^{1})^{\otimes p}\otimes N_{\mathcal{F}})$.

######  Definition 3.4.

Let $\mathcal{F}$ be a foliation in $X$ that is not $p$-closed. The
$p$-divisor associated to $\mathcal{F}$ is the zero divisor of the section
$s_{\mathcal{F}}$, that
is,$\Delta_{\mathcal{F}}=(s_{\mathcal{F}})_{0}\in\operatorname{Div}(X).$

######  Remark 3.5.

Let $\mathcal{F}$ be a foliation in $\operatorname{\mathbb{P}_{k}^{2}}$ of
degree $d>0$. Then, we have the formulas:
$\Omega_{\mathcal{F}}^{1}=\mathcal{O}_{\operatorname{\mathbb{P}_{k}^{2}}}(d-1)$
and $N_{\mathcal{F}}=\mathcal{O}_{\operatorname{\mathbb{P}_{k}^{2}}}(d+2)$. In
particular, if $\mathcal{F}$ is not $p$-closed then the $p$-divisor is a
divisor of degree $\deg(\Delta_{\mathcal{F}})=p(d-1)+d+2$.

######  Example 3.6.

Let $X=\mathbb{A}_{\operatorname{k}}^{2}$ and $\alpha\in\operatorname{k}$. Let
$\mathcal{F}$ be a foliation defined by the vector field
$v=x\partial_{x}+\alpha y\partial_{y}.$ Then, $\mathcal{F}$ is $p$-closed if
and only if $\alpha\in\mathbb{F}_{p}$ and if $\alpha\not\in\mathbb{F}_{p}$ we
have $s_{\mathcal{F}}=(\alpha^{p}-\alpha)xy$.

######  Proposition 3.7.

Let $X$ be smooth algebraic surface defined over $\operatorname{k}$ and
$\mathcal{F}$ be a foliation in $X$ that is not $p$-closed. Let $C$ be an
irreducible algebraic curve on $X$. If $C$ is $\mathcal{F}$-invariant then
$\operatorname{ord}_{C}(\Delta_{\mathcal{F}})>0$. Reciprocally, if $p$ does
not divides $\operatorname{ord}_{C}(\Delta_{\mathcal{F}})$ then $C$ is
$\mathcal{F}$-invariant.

###### Proof.

Suppose that $C$ is $\mathcal{F}$-invariant and let $R=\mathcal{O}_{X,C}$ be
the ring of regular functions of $X$ along $C$. Let $U$ be an affine open set
such that $T_{\mathcal{F}}$ if given by a regular vector field $v$ and
$N_{\mathcal{F}}^{*}$ is given by a regular $1$-form $\omega$. Let $\\{f=0\\}$
be the local equation for $C$ in $U$ and note that $f$ is an uniformizer
parameter to the ring $R$. We need to show that
$\operatorname{ord}_{f}(\Delta_{\mathcal{F}})>0$. Now, we have $v(f)=fH$ and
$\omega\wedge df=f\sigma$ for some $H\in R$ and
$\sigma\in\Omega_{R/\operatorname{k}}^{2}$. Contraction with the vector field
$v^{p}$ gives $fi_{v^{p}}(\sigma)=i_{v^{p}}(\omega\wedge df)=i_{v^{p}}\omega
df-i_{v^{p}}(df)\omega.$ By using the equality $v(f)=fH$ we conclude that
$v^{p}(f)=fH_{p}$ for some regular function $H_{p}$. So,
$fi_{v^{p}}\sigma+v^{p}(f)\omega=i_{v^{p}}\omega df$ and we conclude that
$i_{v^{p}}\omega\in\langle f\rangle$. It follows that
$\operatorname{ord}_{C}(\Delta_{\mathcal{F}})>0$.

Reciprocally, suppose that
$\operatorname{ord}_{C}(\Delta_{\mathcal{F}})=\alpha\not\equiv 0\mod p$ and
write $\Delta_{\mathcal{F}}=f^{\alpha}g$ with $g\in R^{*}$. By [4, Theorem
6.2] we know that $d(\Delta_{\mathcal{F}}^{p-1}\omega)=0$ and expanding this
formula we obtain $\alpha\cdot gdf\wedge\omega=f(gd\omega-dg\wedge\omega),$
which implies $f|df\wedge\omega$. So, $C$ is $\mathcal{F}$-invariant. ∎

The Proposition 3.7 has the following consequence (compare with Proposition
A).

######  Corollary 3.8.

Let $\operatorname{k}$ be an algebraically closed field of characteristic
$p>0$. Let $\mathcal{F}$ be a foliation on $\mathbb{P}_{\operatorname{k}}^{2}$
and suppose that $p\nmid d+2$. Then $\mathcal{F}$ has an invariant algebraic
curve.

###### Proof.

If $\mathcal{F}$ is $p$-closed then $\mathcal{F}$ admits infinitely many
invariant curves by [2, Théorème 1]. We can assume that $\mathcal{F}$ is not
$p$-closed. Since $p$ does not divide $d+2$ the degree formula:
$\deg(\Delta_{\mathcal{F}})=p(d-1)+d+2$ shows that
$\deg(\Delta_{\mathcal{F}})\not\equiv 0\mod p$. In particular, there exists a
prime divisor $\mathfrak{P}$ in the support of $\Delta_{\mathcal{F}}$ such
that $\operatorname{ord}_{\mathfrak{P}}(\Delta_{\mathcal{F}})\not\equiv 0\mod
p$. By Proposition 3.7, the divisor $\mathfrak{P}$ defines a
$\mathcal{F}$-invariant irreducible curve. ∎

Let $\mathcal{F}$ be a foliation in a smooth algebraic surface $X$ and
$Q\in\operatorname{sing}(\mathcal{F})$. We say that $Q$ is not degenerated
with eigenvalue $\alpha\in\operatorname{k}$ if there exists an affine open
subset $U\subset\mathbb{A}_{\operatorname{k}}^{2}$ which contains $Q$ such
that $\mathcal{F}|_{U}$ can be represent by a polynomial vector field
$v=v_{1}+O(2)$ with $v_{1}=x\partial_{x}+\alpha y\partial_{y}$ and $\alpha\neq
0$.

######  Definition 3.9.

Let $\mathcal{F}$ be a foliation on $X$ and
$Q\in\operatorname{sing}(\mathcal{F})$. We say that $Q$ is $p$-reduced if $Q$
is not degenerated and has type $\alpha(Q)$ satisfying the condition:
$\alpha(Q)\not\in\mathbb{F}_{p}$.

######  Lemma 3.10.

Let $\mathcal{F}$ be a foliation in a algebraic surface $X$ and
$Q\in\operatorname{sing}(\mathcal{F})$ be a $p$-reduced singularity of
$\mathcal{F}$. Then, $\mathcal{F}$ is not $p$-closed.

###### Proof.

Let $U$ be an affine open set that contains $Q$ and $x,y\in\mathcal{O}_{X,Q}$
local parameters system at $Q$ in $U$. In the open $U$ the foliation is given
by a vector field $v=v_{1}+\tilde{v}$ where $v_{1}=x\partial_{x}+\alpha
y\partial_{y}$ with $\alpha\not\in\mathbb{F}_{p}$ and $\tilde{v}$ consists of
terms that has order at least two. Then, $v^{p}=v_{1}^{p}+\tilde{v}_{p}$ where
$\tilde{v}_{p}$ contains only homogeneous terms with order at least two.
Observe that $v_{1}^{p}=x\partial_{x}+\alpha^{p}y\partial_{y}$ and
$v_{1}\wedge v_{1}^{p}$ is the homogeneous component of the smallest degree
that occurs in $v\wedge v^{p}$. Since $\alpha\not\in\mathbb{F}_{p}$ we ensure
that $v_{1}\wedge
v_{1}^{p}=(\alpha^{p}-\alpha)\partial_{x}\wedge\partial_{y}\neq 0$ and so
$v\wedge v^{p}\neq 0$. ∎

######  Lemma 3.11.

Let $X$ be a smooth projective surface defined over $\operatorname{k}$ and
$\mathcal{F}$ be a foliation in $X$ that is not $p$-closed. Let
$Q\in\operatorname{sing}(\mathcal{F})$ be a $p$-reduced singular point of
$\mathcal{F}$. Suppose that $\Delta_{\mathcal{F}}$ is a reduced divisor and
let $\pi_{Q}:Bl_{Q}(X)\longrightarrow X$ the blowup with center at $Q$. Then,
$\pi^{*}\mathcal{F}$ defines a foliation in $Bl_{Q}(X)$ with reduced
$p$-divisor.

###### Proof.

Let $\mathcal{G}=\pi_{Q}^{*}\mathcal{F}$ be a foliation induced in
$Bl_{Q}(X)$. Since $Q$ is $p$-reduced we have
$K_{\mathcal{G}}-\pi_{Q}^{*}K_{\mathcal{F}}=0$ and
$N_{\mathcal{G}}-\pi_{Q}^{*}N_{\mathcal{F}}=-E.$ Indeed, in an affine open set
$U\subset\mathbb{A}_{\operatorname{k}}^{2}$ the foliation is represented by a
$1$-form $\omega=\omega_{1}+O(2)$ where $\omega_{1}=ydx-\alpha xdy$ with
$\alpha\not\in\mathbb{F}_{p}$ and $O(2)$ containing only terms with order at
least two. In a convenient coordinates system the map
$\pi_{Q}:Bl_{Q}(X)\longrightarrow X$ associate $(x,t)\mapsto(x,xt)$. Since
$\pi_{Q}^{*}\omega$ is a local section of $N^{*}_{\pi_{Q}^{*}\mathcal{F}}$ we
have $[\pi_{Q}^{*}\omega]_{0}=N^{*}_{\pi_{Q}^{*}\mathcal{F}}$. In the other
hand, since $Q$ is $p$-reduced we have

$\pi_{Q}^{*}\omega=\pi_{Q}^{*}\omega_{1}+O(2)=(xtdx-\alpha
x(xdt+tdx))+O(2)=x(t(1-\alpha)dx+xdt+O(2)).$

Denote $\tilde{\omega}=t(1-\alpha)dx+xdt+O(2)$ and note that $\tilde{\omega}$
is a local section of $\pi_{Q}^{*}N_{\mathcal{F}}$. So,
$N_{\mathcal{G}}^{*}=N^{*}_{\pi_{Q}^{*}\mathcal{F}}=[\pi_{Q}^{*}\omega]_{0}=[x]_{0}+[\tilde{\omega}]_{0}=E+\pi_{Q}^{*}N_{\mathcal{F}}^{*}$.
The formula that compares $\pi_{Q}^{*}K_{\mathcal{F}}$ and
$K_{\pi_{Q}^{*}\mathcal{F}}$ follows by the adjunction formula:
$K_{X}=K_{\mathcal{F}}-N_{\mathcal{F}}$. So, we have

$[\Delta_{\mathcal{G}}]=pK_{\mathcal{G}}+N_{\mathcal{G}}=p\pi_{Q}^{*}K_{\mathcal{F}}+\pi_{Q}^{*}N_{\mathcal{F}}-E=\pi_{Q}^{*}[\Delta_{\mathcal{F}}]-E=\tilde{\Delta}_{\mathcal{F}}+(m_{Q}(\Delta_{\mathcal{F}})-1)E$

where $\tilde{\Delta}_{\mathcal{F}}$ denotes the strict transform of the
divisor $\Delta_{\mathcal{F}}$ and $m_{Q}(\Delta_{\mathcal{F}})$ is the
algebraic multiplicity of $\Delta_{\mathcal{F}}$ at $Q$. Since $Q$ is
$p$-reduced, by using [13, Fact 2.8] we conclude that
$m_{Q}(\Delta_{\mathcal{F}})=2$ so that
$[\Delta_{\mathcal{G}}]=\tilde{\Delta}_{\mathcal{F}}+E$. ∎

######  Lemma 3.12.

Let $(X,H)$ be a polarized smooth projective surface defined over
algebraically closed field $\operatorname{k}$ with characteristic $p>2$. Let
$\mathcal{N}$ be an invertible sheaf and let
$\mathbb{F}ol_{\mathcal{N}}(X)=\mathbb{P}(H^{0}(X,\Omega_{X}^{1}\otimes\mathcal{N}))$
the space of foliations on $X$ that has normal bundle $\mathcal{N}$. Suppose
that that space is not empty and consider the following sets

$U=\\{\mathcal{F}\in\mathbb{F}ol_{\mathcal{N}}(X)\mid\Delta_{\mathcal{F}}\mbox{
is reduced }\\}\quad\mbox{and}\quad
V=\\{\mathcal{F}\in\mathbb{F}ol_{\mathcal{N}}(X)\mid\Delta_{\mathcal{F}}\mbox{
is prime}\\}.$

Then, $U$ and $V$ are open sets.

###### Proof.

First, note that $\deg(\Delta_{\mathcal{F}})$ depends only of $X$ and
$\mathcal{N}$. Indeed, we have the following formula:

$\deg(\Delta_{\mathcal{F}})=pK_{\mathcal{F}}\cdot H+\mathcal{N}\cdot
H=p(K_{X}+\mathcal{N})\cdot H+\mathcal{N}\cdot H=pK_{X}\cdot
H+(p+1)\mathcal{N}\cdot H.$

Given $e\in\mathbb{Z}_{\geq 1}$ let $Z_{e}(X)$ the space that consists of all
curves in $X$ that has degree $e$. In the following, we will use the fact that
$Z_{e}(X)$ is a projective algebraic variety over $\operatorname{k}$ (see [9,
Theorem 1.4]). Define the following sets

$S_{e}=\\{(C,\mathcal{F})\in
Z_{e}(X)\times\mathbb{F}ol_{\mathcal{N}}(X)\mid\mbox{$2C\leq\Delta_{\mathcal{F}}$}\\},$
$\tilde{S}_{e}=\\{(C,\mathcal{F})\in
Z_{e}(X)\times\mathbb{F}ol_{\mathcal{N}}(X)\mid\mbox{$C\leq\Delta_{\mathcal{F}}$}\\}.$

Since the conditions $2C\leq\Delta_{\mathcal{F}}$ and
$C\leq\Delta_{\mathcal{F}}$ are closed relations we have that $S_{e}$ and
$\tilde{S}_{e}$ are closed sets in
$Z_{e}(X)\times\mathbb{F}ol_{\mathcal{N}}(X)$. Let
$\pi_{e}:Z_{e}(X)\times\mathbb{F}ol_{\mathcal{N}}(X)\longrightarrow\mathbb{F}ol_{\mathcal{N}}(X)$
the natural projection. Since $\pi_{e}$ is a proper morphism we ensure that
$\pi_{e}(S_{e})$ and $\pi_{e}(\tilde{S}_{e})$ are closed sets in
$\mathbb{F}ol_{\mathcal{N}}(X)$. Denote by
$\mathbb{F}\subset\mathbb{F}ol_{\mathcal{N}}(X)$ the closed set consisting of
$p$-closed foliations and consider the following sets:

$T_{1}=\mathbb{F}ol_{\mathcal{N}}(X)-(\mathbb{F}\cup\bigcup_{j=1}^{\lfloor\frac{\deg(\Delta_{\mathcal{F}})}{2}\rfloor}\pi_{j}(S_{j}))\qquad\mbox{and}\qquad
T_{2}=\mathbb{F}ol_{\mathcal{N}}(X)-(\mathbb{F}\cup\bigcup_{j=1}^{\deg(\Delta_{\mathcal{F}})-1}\pi_{j}(\tilde{S}_{j})).$

We claim that the following identities holds: $U=T_{1}$ and $V=T_{2}$. Indeed,
the inclusions $U\subset T_{1}$ and $V\subset T_{2}$ are trivial. Now, let
$\mathcal{G}_{1}\in T_{1}$ and $\mathcal{G}_{2}\in T_{2}$. Suppose, by
contradiction, that $\Delta_{\mathcal{G}_{1}}$ is not reduced and that
$\Delta_{\mathcal{G}_{2}}$ is not a prime divisor. In particular, there are
curves $C_{1}$ and $C_{2}$ in $X$ such that
$2C_{1}\leq\Delta_{\mathcal{G}_{1}}$ and $C_{2}\leq\Delta_{\mathcal{G}_{2}}$
with $\deg(C_{2})<\deg(\Delta_{\mathcal{G}_{2}})$. Since
$\deg(\Delta_{\mathcal{G}_{i}})$ depends only on $K_{X}$ and $\mathcal{N}$ by
computing the degrees we have:

$2\deg(C_{1})\leq\deg(\Delta_{\mathcal{G}_{1}})=\deg(\Delta_{\mathcal{F}})\qquad\mbox{and}\qquad\deg(C_{2})\leq\deg(\Delta_{\mathcal{G}_{2}})-1=\deg(\Delta_{\mathcal{F}})-1$

and this implies

$\mathcal{G}_{1}\in\mathbb{F}\cup\bigcup_{j=1}^{\lfloor\frac{\deg(\Delta_{\mathcal{F}})}{2}\rfloor}\pi_{j}(S_{j})\qquad\mbox{and}\qquad\mathcal{G}_{2}\in\mathbb{F}\cup\bigcup_{j=1}^{\deg(\Delta_{\mathcal{F}})-1}\pi_{j}(\tilde{S}_{j})$

a contradiction. ∎

In the following we will denote the open sets above by $U(X,H,\mathcal{N})$ e
$V(X,H,\mathcal{N})$.

###  3.2 Global equations for foliations on
$\operatorname{\mathbb{P}_{k}^{2}}$

Let $e\in\mathbb{Z}_{\geq 0}$ and $\operatorname{k}$ be an algebraically
closed field. A foliation in $\operatorname{\mathbb{P}_{k}^{2}}$ of degree $e$
is determined by a global section of
$\Omega_{\operatorname{\mathbb{P}_{k}^{2}}}^{1}(e+2)$. By using the Euler
exact sequence (see [6, Theorem 8.13]) it follows that a foliation in
$\operatorname{\mathbb{P}_{k}^{2}}$ is given, module elements of
$\operatorname{k}^{*}$, by a $1$-form $\Omega=Adx+Bdy+Cdz$ where
$A,B,C\in\operatorname{k}[x,y,z]_{e+1}$ with
$\mathcal{Z}(A,B,C)\subset\operatorname{\mathbb{P}_{k}^{2}}$ finite set and
such that $i_{R}\Omega=0$, where $R$ is the radial vector
field:$R=x\partial_{x}+y\partial_{y}+z\partial_{z}.$

Suppose that $\operatorname{k}$ has characteristic $p>0$ and let $\mathcal{F}$
be a foliation in $\operatorname{\mathbb{P}_{k}^{2}}$ with normal bundle
$N=\mathcal{O}_{\operatorname{\mathbb{P}_{k}^{2}}}(d+2)$ and suppose that
$p\nmid\deg(N)$. Suppose that $\mathcal{F}$ is defined by the homogeneous
$1$-form: $\omega=Adx+Bdy+Cdz$ and put: $d\omega=(d+2)(Ldy\wedge dz-Mdx\wedge
dz+Ndx\wedge dy).$ Let
$v\in\mathfrak{X}_{d}(\mathbb{A}_{\operatorname{k}}^{3})$ be the homogeneous
vector field defined by:
$v_{\omega}=L\partial_{x}+M\partial_{y}+N\partial_{z}.$ By the [7, Proposition
1.1.4] the association $\omega\mapsto v_{\omega}$ defines a bijection between
the set of projective $1$-forms of degree $d+2$ and the set of homogeneous
vector fields in $\mathbb{A}_{\operatorname{k}}^{3}$ of degree $d$ that has
zero divergent $div(v_{\omega})=L_{x}+M_{y}+N_{z}=0.$ The $p$-divisor is
explicitly given by
$\Delta_{\mathcal{F}}=[i_{v^{p}}\omega]\in\operatorname{Div}(\operatorname{\mathbb{P}_{k}^{2}}).$

######  Example 3.13.

Let $\mathcal{F}$ be a foliation of degree two in
$\mathbb{P}_{\mathbb{C}}^{2}$ defined by the projective $1$-form

$\omega=yz^{2}dx-z(4yz+2xz+2y^{2})dy+(xyz+4y^{2}z+2y^{3})dz.$

Given a prime number $p\in\mathbb{Z}_{>3}$ consider $\mathcal{F}_{p}$ the
foliation obtained by reducing modulo $p$ the $1$-form that defines
$\mathcal{F}$. Then, $\mathcal{F}_{p}$ is not $p$-closed and
$\Delta_{\mathcal{F}_{p}}=3\\{y=0\\}+(p+1)\\{z=0\\}.$

###### Proof.

We will show first that $\mathcal{F}_{p}$ is not $p$-closed for every prime
$p>3$. Fix a prime number $p>3$ and consider the foliation $\mathcal{F}_{p}$
defined in $\operatorname{\mathbb{P}_{k}^{2}}$ by reduction modulo $p$ of the
coefficients of the $1$-form $\omega.$ Since this problem is local, we can
restrict the foliation to the open set
$U=D_{+}(z)\cong\mathbb{A}_{\operatorname{k}}^{2}$. In $U$, the foliation is
given by the vector field $v=(4y+2x+2y^{2})\partial_{x}+y\partial_{y}$.
Observe that $v(x)=4y+2x+2y^{2}$. An inductive argument shows that for every
$k\geq 3$ we have

$v^{k}(x)=2^{2}(2(2^{k-2}+2^{k-3}+\cdots+2+1)+1)y+k2^{k}y^{2}+2^{k}x.$

So, we conclude $v^{p}(x)=4y+2^{p}x=4y+2x$, since
$0=2^{p-1}-1=(2^{p-2}+\cdots+1)$ in $\mathbb{F}_{p}$. In particular, we have

$v^{p}(x)v(y)-v^{p}(y)v(x)=y(v^{p}(x)-4y-2y^{2}-2x)=y(-2y^{2})=-2y^{3}\neq 0.$

So, the foliation defined by $v$ is not $p$-closed, if $p>3$. This implies
that the $p$-divisor of $\mathcal{F}_{p}$ is given by
$\Delta_{\mathcal{F}_{p}}=3\\{y=0\\}+(p+1)\\{z=0\\}$. ∎

###  3.3 Applications: foliations on $\mathbb{P}_{\mathbb{C}}^{2}$ without
algebraic invariant curves

The next two propositions show that, in some cases, the irreducibility of the
$p$-divisor for foliations in $\operatorname{\mathbb{P}_{k}^{2}}$ is relate
with holomorphic foliations in $\mathbb{P}_{\mathbb{C}}^{2}$ without algebraic
invariant curves.

######  Proposition 3.14.

Let $\mathcal{F}$ be a non-dicritical foliation in
$\mathbb{P}_{\mathbb{C}}^{2}$ defined by a projective $1$-form
$\Omega=Adx+Bdy+Cdz$. Let $K$ be a number field and suppose that
$A,B,C\in\mathcal{O}_{K}[x,y,z]_{d+1}$, where $\mathcal{O}_{K}$ is the integer
ring of $K$. Let $\mathfrak{m}\in\operatorname{Spm}(\mathcal{O}_{K})$ of
characteristic $p$ and suppose that $p$ does not divides $d+2$. Let
$\mathcal{F}_{p}$ be a foliation in $\mathbb{P}_{k(\mathfrak{m})}^{2}$
obtained by reduction modulo $\mathfrak{m}$ of the coefficients of $\Omega$.
If $\Delta_{\mathcal{F}_{p}}$ is irreducible then $\mathcal{F}$ has no
algebraic solutions. This can be used to give a simple proof that the
Jouanolou foliations of degree two and tree has no algebraic solutions.

###### Proof.

Suppose, by contradiction, that $\mathcal{F}$ has an algebraic solution $C$.
By using Galois automorphism, we can assume that $C$ is defined by an
irreducible polynomial over $\mathcal{O}_{K}$. In particular, $C$ is reduced
as a curve in $\mathbb{P}_{\mathbb{C}}^{2}$. Let $F\in\mathcal{O}_{K}[x,y,z]$
be the irreducible polynomial defining $C$. By [3, theorem] we know that
$\deg(F)\leq d+2$. Let $F\otimes k(\mathfrak{m})$ be the polynomial obtained
by reduction modulo $\mathfrak{m}$ of $F$. Note that the reduction modulo
$\mathfrak{m}$ preserves the invariance in the sense that the curve describe
by $F\otimes k(\mathfrak{m})$ is invariant by the foliation $\mathcal{F}_{p}$.
Let $G\in k(\mathfrak{m})[x,y,z]$ be an irreducible factor of $F$. We have
that the curve $\\{G=0\\}\subset\mathbb{P}_{k(\mathfrak{m})}^{2}$ is
$\mathcal{F}_{p}$-invariant and by the Proposition 3.7 we conclude that
$\\{G=0\\}\leq\Delta_{\mathcal{F}_{p}}.$ Since $\Delta_{\mathcal{F}}$ is
irreducible we have $\\{G=0\\}=\Delta_{\mathcal{F}}$. But, this is a
contraction by comparison of degrees, since
$p(d-1)+d+2=\deg(\Delta_{\mathcal{F}_{p}})>d+2\geq\deg(G)$. ∎

Let $\mathcal{F}$ be a foliation in $\mathbb{P}_{\mathbb{C}}^{2}$ and $Q$ a
reduced singularity of $\mathcal{F}$. Suppose that $Q$ is not degenerated. In
this case, we know that if $\alpha$ is the eigenvalue of $Q$ then
$\alpha\not\in\mathbb{Q}_{+}$. By [12, Appendice II] we know that there is an
analytic coordinate system such that the foliation is given by the $1$-form
$\omega=-\alpha y(1+b(x,y))dx+x(1+a(x,y))dy$ with $a,b\in\langle x,y\rangle$.
In particular, if $C$ is a reduced algebraic curve that is
$\mathcal{F}$-invariant with $Q\in C$ then in the analytic coordinate system
above the $C$ is given by $\\{x=0\\}$, $\\{y=0\\}$ or $\\{xy=0\\}$. So,
computing the Camacho-Sad index we conclude that

$CS(\mathcal{F},C,Q)=\begin{cases}1/\alpha&\text{if}\ C=\\{x=0\\},\\\
\alpha&\text{if}\ C=\\{y=0\\},\\\ \alpha+\alpha^{-1}+2&\text{if}\
C=\\{xy=0\\}.\end{cases}$

In particular, we have that the norm of the index is bounded by a constant
which depends only on eigenvalue $\alpha$ of $Q$. More precisely, we have
$|CS(\mathcal{F},C,Q)|\leq|\alpha|+|\alpha|^{-1}+2$. We will use this in the
following example.

######  Proposition 3.15.

Let $\mathcal{F}$ be a non-degenerated foliation in
$\mathbb{P}_{\mathbb{C}}^{2}$ defined by a projective $1$-form $\omega$ with
coefficients in $\mathbb{Z}$. Suppose that $\omega$ is primitive, that is, the
greatest common divisor of the coefficients of $\omega$ is equal to $1$.
Suppose that $\mathcal{F}$ is reduced and define

$\alpha_{\mathcal{F}}:=\sup\\{|\alpha(Q)|\mid
Q\in\operatorname{sing}(\mathcal{F})\\}\quad\mbox{and}\quad\overline{\alpha}_{\mathcal{F}}:=\sup\\{|\alpha(Q)|^{-1}\mid
Q\in\operatorname{sing}(\mathcal{F})\\}.$

Let
$\beta_{\mathcal{F}}=\alpha_{\mathcal{F}}+\overline{\alpha}_{\mathcal{F}}+2$
and $p$ be a prime number such that
$p>(d_{\mathcal{F}}+1)\beta_{\mathcal{F}}^{1/2}$ and suppose that
$\Delta_{\mathcal{F}_{p}}$ is a prime divisor. Then, $\mathcal{F}$ has no
algebraic solutions.

###### Proof.

Suppose by contraction that $\mathcal{F}$ has an algebraic solution $C$. By
using Galois automorphism we can assume that $C$ is given by an irreducible
polynomial defined over $\mathbb{Z}$. Since $\mathcal{F}$ is not degenerated
and reduced, using the Camacho-Sad formula (see [1, Theorem 3.2]) we have

$d_{C}^{2}=C^{2}\leq\sum_{Q\in\operatorname{sing}(\mathcal{F})\cap
C}|CS(\mathcal{F},C;Q)|\leq\\#\operatorname{sing}(\mathcal{F})\beta_{\mathcal{F}}=(d_{\mathcal{F}}^{2}+d_{\mathcal{F}}+1)\beta_{\mathcal{F}}\leq(d_{\mathcal{F}}+1)^{2}\beta_{\mathcal{F}}$

The formula above implies that
$d_{C}\leq(d_{\mathcal{F}}+1)\beta_{\mathcal{F}}^{1/2}<p$. So, reducing $C$
modulo $p$ we ensure that the reduction $C\otimes\mathbb{F}_{p}$ has no
$p$-factor, that is, there is no an effective divisor, $D$, such that $pD\leq
C\otimes\mathbb{F}_{p}$. Let $E$ be an irreducible factor of
$C\otimes\mathbb{F}_{p}$. By Proposition 3.7 we know that $E$ is a
$\mathcal{F}_{p}$-invariant irreducible curve. Since the $p$-divisor of
$\mathcal{F}_{p}$ is irreducible we conclude that
$\Delta_{\mathcal{F}_{p}}=E$. But, this is a contradiction by degree
comparison. ∎

The interesting fact about Proposition 3.14 and Proposition 3.15 is that we
get information about the algebraicity of foliations on
$\mathbb{P}_{\mathbb{C}}^{2}$ by using only a maximal ideal.

###  3.4 Global equations for foliations on $\Sigma_{d}$

In this subsection we recall the construction of the Hirzebruch surfaces and
how to represent globally foliations in that surfaces. The reference for this
section is [5].

Let $\mathbb{G}_{m}=\operatorname{k}^{*}$ the multiplicative group of
$\operatorname{k}$ and $d\in\mathbb{Z}_{\geq 0}$. Let $\mu_{d}$ the action of
$\mathbb{G}_{m}^{2}$ in
$X=(\mathbb{A}_{\operatorname{k}}^{2}-0)\times(\mathbb{A}_{\operatorname{k}}^{2}-0)$
defined by the morphism:

$\displaystyle\mu_{d}:\mathbb{G}_{m}^{2}\times X$
$\displaystyle\longrightarrow X$
$\displaystyle((a,b),(x_{0},x_{1};y_{0},y_{1}))$
$\displaystyle\mapsto(ax_{0},ax_{1},by_{0},\frac{b}{a^{d}}y_{1}).$

The quotient $\Sigma_{d}=X/\mu_{d}$ is a smooth surface defined over
$\operatorname{k}$ and is it isomorphic to the $d$-Hirzebruch surface, that
is, the surface
$\mathbb{P}(\mathcal{O}_{\mathbb{P}_{\operatorname{k}}^{1}}\oplus\mathcal{O}_{\mathbb{P}_{\operatorname{k}}^{1}}(d))$.
It is a ruled surface with structural morphism to
$\operatorname{\mathbb{P}_{k}^{1}}$ defined by

$\displaystyle\pi\colon\Sigma_{d}$
$\displaystyle\longrightarrow\operatorname{\mathbb{P}_{k}^{1}}$
$\displaystyle\overline{(x_{0},x_{1};y_{0},y_{1})}$
$\displaystyle\mapsto[x_{0}:x_{1}].$

Let $d_{1},d_{2}\in\mathbb{Z}$ and
$G\in\operatorname{k}[x_{0},x_{1},y_{0},y_{1}]$. We say that $G$ is bi-
homogeneous of the bi-degree $(d_{1},d_{2})$ if for any monomial
$x_{0}^{a_{0}}x_{1}^{a_{1}}y_{0}^{b_{0}}y_{1}^{b_{1}}$in the support of $G$ we
have $d_{1}=a_{0}+a_{1}-db_{1}$ and $d_{2}=b_{0}+b_{1}$. Let $F$ and $M_{d}$
curves in $\Sigma_{d}$ such that $F$ is a fiber of the structural projection
$\pi:\Sigma_{d}\longrightarrow\operatorname{\mathbb{P}_{k}^{1}}$ and $M_{d}$
is a section of $\pi$ which satisfies the conditions $M_{d}\cdot F=1$ and
$M_{d}^{2}=d$. Then $\\{F,M_{d}\\}$ forms a base for the vector space
$\operatorname{Num}_{\mathbb{Q}}(\Sigma)$. If
$D\in\operatorname{Div}(\Sigma_{d})$ is a divisor such that $D\equiv
d_{1}F+d_{2}M_{d}$ then the global sections of the
$\mathcal{O}_{\Sigma_{d}}(D)$ correspond to bi-homogeneous polynomials of bi-
degree $(d_{1},d_{2})$ in $\Sigma_{d}$. In the following, we will use the
following description of foliations in Hirzebruch surfaces (see [5,
Proposition 3.2]).

######  Proposition 3.16.

Let $d\in\mathbb{Z}_{\geq 0}$, $d_{1},d_{2}\in\mathbb{Z}$ and
$N=\mathcal{O}_{\Sigma_{d}}(d_{1}-d+2,d_{2}+2)$. Then, any foliation
$\mathcal{F}$ in $\Sigma_{d}$ with normal bundle $N$ is uniquely determined,
module $\operatorname{k}^{*}$, by a differential $1$-form of the type
$\Omega=A_{0}dx_{0}+A_{1}dx_{1}+B_{0}dy_{0}+B_{1}dy_{1}$ where $A_{0},A_{1}\in
H^{0}(\Sigma_{d},\mathcal{O}(d_{1}-d+1,d_{2}+2))$, $B_{0}\in
H^{0}(\Sigma_{d},\mathcal{O}_{\Sigma_{d}}(d_{1}-d+2,d_{2}+1))$ and $B_{1}\in
H^{0}(\Sigma_{d},\mathcal{O}_{\Sigma_{d}}(d_{1}+2,d_{2}+1))$ are bi-
homogeneous and satisfies the following conditions:
$x_{0}A_{0}+x_{1}A_{1}-dy_{1}B_{1}=0$ and $y_{0}B_{0}+y_{1}B_{1}=0.$

##  4 The $p$-divisor for foliations on $\operatorname{\mathbb{P}_{k}^{2}}$

In this section, we investigate the structure of the $p$-divisor for generic
foliations in the projective plane. In the next section, $\operatorname{k}$
will be denote an algebraically closed field that has characteristic $p>0$.
The following lemma will be important to the next sections.

######  Lemma 4.1.

Let $F\in\operatorname{k}[x_{0},\ldots,x_{n}]$ be a reduced polynomial and
$l\in\mathbb{Z}$ positive integer with $\operatorname{gcd}(l,p)=1$. Suppose
that $x_{0}\nmid F$. Then, $F(x_{0}^{l},x_{1},\ldots,x_{n})$ is reduced.

###### Proof.

First, note that if $x_{0}$ does not occurs in $F$ then is nothing to prove.
So, we can suppose that $x_{0}$ occurs in $F$. We first consider the case
where $F$ is irreducible. Note that it is sufficient to consider the case
where $n=0$. Indeed, if $R=\operatorname{k}[x_{1},\ldots,x_{n}]$ and $K$ is
their fraction field then by the Gauss Lemma (see [10, Theorem 2.1]) we know
that $F\in R[x_{0}]$ is irreducible if and only if
$F=a_{0}+a_{1}x_{0}+\cdots+a_{d}x_{0}^{d}\in K[x_{0}]$ is irreducible and has
content $1$, that is, the greatest common divisor of the coefficients is equal
to $1$. Since we are assuming that $F$ is irreducible we have, in particular,
that $F$ is irreducible over $K[x_{0}]$ where
$K=\operatorname{k}(x_{1},\ldots,x_{n})$. Let $g(x_{0})=F(x_{0}^{l})\in
K[x_{0}]$. Note that the content of $g$ is equal to $1$, since the
coefficients of $g$ are equal to the coefficients of $F$. In the other hand,
the irreducibility of $F$ implies that $g$ is reduced: indeed, to see this
note that it is sufficient to proof that
$\operatorname{gcd}(g(x_{0}),\frac{dg}{dx_{0}})=1$. Now, by taking derivatives
we have $\frac{dg}{dx_{0}}=lF^{{}^{\prime}}(x_{0}^{l})x_{0}^{l-1}\neq 0.$
Since $F$ is irreducible we ensure that $F^{{}^{\prime}}(x_{0}^{l})$ and
$F(x_{0}^{l})$ are coprime. In particular, we have
$\operatorname{gcd}(g(x_{0}),g^{{}^{\prime}}(x_{0}))=1$. So, $g(x_{0})$ is
reduced in $R[x_{0}]=\operatorname{k}[x_{0},x_{1},\ldots,x_{n}]$.

Now we will consider the general case, where $F$ is reduced. Let
$R=\operatorname{k}[x_{1},\ldots,x_{n}]$ and $K$ their fraction field. Note
that, without loss of generality, we can assume that $F\in R[x_{0}]$ has
content equal to $1$. Let $G$ and $H$ two irreducible factors of $F$. We will
show that $\tilde{G}=G(x_{0}^{l},x_{1}\ldots,x_{n})$ and
$\tilde{H}=H(x_{0}^{l},x_{1}\ldots,x_{n})$ do not have common irreducible
factors. By the Gauss Lemma (see [10, Theorem 2.1]) is it sufficient to show
that $\tilde{G}$ and $\tilde{H}$ do not have common irreducible factors over
$K[x_{0}]$, where $K=\operatorname{k}(x_{1},\ldots,x_{n})$. In the other hand,
since $G$ and $H$ has not common factors over
$\operatorname{k}[x_{0},\ldots,x_{n}]$ we have, in particular, that there is
no common irreducible factors over $K[x_{0}]$. So, there exists
$A(x_{0}),B(x_{0})\in K[x_{0}]$ such that $A(x_{0})G+B(x_{0})H=1.$
Specializing the identity above to $x_{0}^{l}$ we conclude that
$A(x_{0}^{l})\tilde{G}+B(x_{0}^{l})\tilde{H}=1$ and so $\tilde{G}$,$\tilde{H}$
have no common irreducible factors over $K[x_{0}]$. So, we consider only the
case where the polynomial $F$ is irreducible by considering their
decomposition in irreducible factors. ∎

######  Remark 4.2.

Let $l\in\mathbb{Z}_{>0}$ with $\operatorname{gcd}(l,p)=1$. In general, we do
not ensure that if $f\in\operatorname{k}[x]$ is irreducible then
$g(x)=f(x^{l})$ is irreducible. Indeed, let $p>2$ and consider $f(x)=x-1$. If
$l\in 2\mathbb{Z}$ then we have
$g(x)=f(x^{l})=(x^{\frac{l}{2}}-1)(x^{\frac{l}{2}}+1).$

In the following, we present the proof of the Theorem A. Recall the statement.

######  Theorem A.

Let $\operatorname{k}$ be an algebraically closed field of characteristic
$p>0$. A generic foliation in $\operatorname{\mathbb{P}_{k}^{2}}$ of degree
$d\geq 1$ with $p\nmid d-1$ has reduced $p$-divisor. More precisely, if $N$ is
a fixed bundle then 111Section 2 for
notations$U(\mathbb{P}_{\operatorname{k}}^{2},\mathcal{O}_{\mathbb{P}_{\operatorname{k}}^{2}}(1),N)\neq\emptyset$,
if $\deg(N)=3$ or if $\deg(N)>3$ with $p\nmid\deg(N)-3$.

We will consider first the case where $d\in\\{1,2\\}$ and after that we will
use the case $d=2$ to show the general case. We will divide the proof in
propositions.

######  Proposition 4.3.

A generic foliation in $\operatorname{\mathbb{P}_{k}^{2}}$ of degree
$d\in\\{1,2\\}$ has reduced $p$-divisor. More precisely, we have
$U(\mathbb{P}_{\operatorname{k}}^{2},\mathcal{O}_{\mathbb{P}_{\operatorname{k}}^{2}}(1),N)\neq\emptyset$,
if $\deg(N)=\\{3,4\\}$.

###### Proof.

If $\deg(N)=3$, that is, $d=1$ then there are examples of foliations with
reduced $p$-divisor. Indeed, if $\alpha\in\operatorname{k}-\mathbb{F}_{p}$
consider the foliation $\mathcal{F}_{\alpha}$ in
$\operatorname{\mathbb{P}_{k}^{2}}$ given by the $1$-form
$\Omega_{\alpha}=-yzdx+\alpha xzdy+(1-\alpha)xydz$. The condition
$\alpha\not\in\mathbb{F}_{p}$ implies that $\mathcal{F}_{\alpha}$ is not
$p$-closed (see Lemma 3.10) with
$\Delta_{\mathcal{F}_{\alpha}}=\\{x=0\\}+\\{y=0\\}+\\{z=0\\}$. Now, consider
the case where $\deg(N)=4$ and let
$D_{+}(z)=\\{[x:y:z]\in\operatorname{\mathbb{P}_{k}^{2}}\mid z\neq 0\\}$. We
will get the example by compactification via the isomorphism

$\displaystyle\Phi\colon D_{+}(z)$
$\displaystyle\longrightarrow\mathbb{A}_{\operatorname{k}}^{2}$
$\displaystyle[x:y:z]$
$\displaystyle\mapsto\left(\frac{x}{z},\frac{y}{z}\right)$

of a foliation $\mathcal{G}$ in $\mathbb{A}_{\operatorname{k}}^{2}$ given by
the $1$-form: $\omega=ydx-xdy+\omega_{2}$ where $\omega_{2}=a(x,y)dx+b(x,y)dy$
for $a,b\in\operatorname{k}[x,y]$ generic homogeneous polynomial of degree
two. Note that $\mathcal{G}$ has tree $\mathcal{G}$-invariant lines which
contains the point $(0,0)$. Indeed, the lines are given explicitly by the
polynomial $i_{R}\omega_{2}=l_{1}l_{2}l_{3}$, where $R$ is the radial vector
field in $\mathbb{A}_{\operatorname{k}}^{2}$: $R=x\partial_{x}+y\partial_{y}$.
Since $a$ and $b$ are generic, we can assume that $l_{1},l_{2}$ and $l_{3}$
has multiplicity $1$ along $\Delta_{\mathcal{G}}$. Indeed, we choose the lines
$l_{1},l_{2},l_{3}$ such that for each $i$ we ensure that $l_{i}\cap
l_{\infty}=\\{P_{i}\\}$ is a $p$-reduced singularity of $\mathcal{G}$. In this
case, by [13, Fact 2.8] we ensure that $l_{i}$ occurs with multiplicity $1$ in
the $p$-divisor $\Delta_{\mathcal{G}}$. Let $\mathcal{F}$ be a foliation
obtained via compactification of $\mathcal{G}$ in
$\operatorname{\mathbb{P}_{k}^{2}}$, via $\Phi$. By Lemma 3.10 we know that
$\mathcal{F}$ is not $p$-closed with four invariant lines, $l_{1},l_{2},l_{3}$
and $l_{\infty}=\\{z=0\\}$, and so
$\Delta_{\mathcal{F}}=l_{1}+l_{2}+l_{3}+l_{\infty}+C$ for some curve $C$ of
degree $p$.

We will show that a generic choice of $a,b$ implies that $C$ is irreducible
and has $Q=[0:0:1]$ as a singularity of multiplicity $m_{Q}(C)=p-1$. For this,
let
$\pi:Bl_{Q}(\operatorname{\mathbb{P}_{k}^{2}})\longrightarrow\operatorname{\mathbb{P}_{k}^{2}}$
the blowup at $Q$ fix $E$ the exceptional divisor and $F$ a fiber of the
natural projection
$\pi:Bl_{Q}(\operatorname{\mathbb{P}_{k}^{2}})\longrightarrow\operatorname{\mathbb{P}_{k}^{1}}$.
Note that $\\{F,E\\}$ forms a base to the vector space
$\operatorname{Num}_{\mathbb{Q}}(Bl_{Q}(\operatorname{\mathbb{P}_{k}^{2}}))$
and satisfies the following conditions $E^{2}=-1$, $E\cdot F=1$ and $F^{2}=0$.
Let $\mathcal{H}=\pi^{*}\mathcal{F}$ the induced foliation. Since $Q$ is a
radial singularity we have (see [1, Chapter 2, Section 3])
$N_{\mathcal{H}}^{*}=\pi^{*}N_{\mathcal{F}}^{*}+2E$ and
$K_{\mathcal{H}}=\pi^{*}K_{\mathcal{F}}-E$. Let $H$ be a line containing the
point $Q$. Since
$\pi^{*}N_{\mathcal{F}}^{*}=\pi^{*}(-4H)\equiv-4\pi^{*}H\equiv-4(F+E)$ and
$\pi^{*}K_{\mathcal{F}}\equiv\pi^{*}H\equiv F+E$, we conclude

$\Delta_{\mathcal{H}}\equiv pK_{\mathcal{H}}+N_{\mathcal{H}}\equiv
2E+\deg(\Delta_{\mathcal{F}})F=(p+4)F+2E.$

Let $\tilde{l}_{1}$, $\tilde{l}_{2},\tilde{l}_{3}$ and $\tilde{l}_{\infty}$ be
the strict transform of $l_{1},l_{2},l_{3}$ and $l_{\infty}$ respectively.
Since $\pi\colon
Bl_{Q}(\operatorname{\mathbb{P}_{k}^{2}})-E\longrightarrow\mathbb{P}_{\operatorname{k}}^{2}-\\{Q\\}$
is an isomorphism we verify that $\tilde{l}_{1}$,
$\tilde{l}_{2},\tilde{l}_{3}$ and $\tilde{l}_{\infty}$ occurs in
$\Delta_{\mathcal{H}}$, that is
$\Delta_{\mathcal{H}}-\tilde{l}_{1}-\tilde{l}_{2}-\tilde{l}_{3}-\tilde{l}_{\infty}\geq
0$. Moreover, since $a,b$ are generic we can assume that
$\operatorname{ord}_{l_{i}}(\Delta_{\mathcal{H}})\geq 0$ for
$i\in\\{1,2,3,\infty\\}$. As
$(\Delta_{\mathcal{H}}-\tilde{l}_{1}-\tilde{l}_{2}-\tilde{l}_{3}-\tilde{l}_{\infty})\cdot
F=1$ there exist an irreducible curve $C\subset Bl_{Q}(\mathbb{P}^{2})$ that
is $\mathcal{H}$-invariant and such that $C\cdot F=1$. In particular, we have
$C\equiv E+\alpha F$. Note that $\alpha>0$, since $Q$ is a radial singularity
and, in particular, $E$ is not $\mathcal{H}$-invariant. Writing
$\Delta_{\mathcal{H}}-\tilde{l}_{1}-\tilde{l}_{2}-\tilde{l}_{3}-\tilde{l}_{\infty}=C+R$
for some divisor $R$ we conclude (using $C\cdot F=1$) that $R\equiv bF$, for
some $b\in\mathbb{Z}_{\geq 0}$. Now, since $\omega_{2}$ is generic we may
assume $b=0$. Indeed, if $b\neq 0$, we see that there exist a fiber of the
natural projection $p\colon
Bl_{Q}(\operatorname{\mathbb{P}_{k}^{2}})\longrightarrow\operatorname{\mathbb{P}_{k}^{1}}$
that is $\mathcal{H}$-invariant (see Proposition 3.7). By projecting this
fiber in $\operatorname{\mathbb{P}_{k}^{2}}$, we obtain a line which contain
$Q$ that is $\mathcal{F}$-invariant. Since $l_{1}$,$l_{2}$,$l_{3}$ and
$l_{\infty}$ occurs in $\Delta_{\mathcal{F}}$ with multiplicity $1$ and since
they are the unique invariant lines of $\mathcal{F}$, we obtain a contraction.
So, we conclude that $R=0$ and $C$ is an irreducible curve that is
$\mathcal{H}$-invariant with $C\equiv E+pF$. Projecting $C$ via the map $\pi$
we obtain in $\operatorname{\mathbb{P}_{k}^{2}}$ an irreducible algebraic
curve of degree $p$ that has $Q$ as a singularity and with multiplicity $p-1$.
indeed, $\pi_{*}C$ is irreducible and the degree is given by

$\deg(\pi_{*}(C))=\pi_{*}C\cdot
H=\pi^{*}\pi_{*}C\cdot\pi^{*}H=(E+pF)\cdot(E+F)=-1+p+1=p.$

In other hand, the multiplicity can be computed in the following way:

$\pi^{*}\pi_{*}C=\tilde{C}+m_{Q}(C)E\equiv(E+pF)+m_{Q}(C)E\Longrightarrow
0=\pi^{*}\pi_{*}C\cdot E=(E+pF)\cdot E-m_{Q}(C)$

and so $0=-1+p-m_{Q}(C)$ which implies $m_{Q}(C)=p-1$. This concludes the
proof for $d\in\\{1,2\\}$. ∎

######  Proposition 4.4.

A generic foliation in $\operatorname{\mathbb{P}_{k}^{2}}$ with normal bundle
$N$ and of degree $d\geq 3$ with $p\nmid d-1$ has reduced $p$-divisor. More
precisely,
$U(\mathbb{P}_{\operatorname{k}}^{2},\mathcal{O}_{\mathbb{P}_{\operatorname{k}}^{2}}(1),N)\neq\emptyset$,
if $\deg(N)\geq 5$ with $p\nmid\deg(N)-3$.

###### Proof.

We use the case $d=2$ to show the general case. Let $e\in\mathbb{Z}_{>0}$ such
that $\operatorname{gcd}(e,p)=1$ and $\mathcal{F}$ be a foliation in
$\operatorname{\mathbb{P}_{k}^{2}}$ of degree $2$ that has reduced
$p$-divisor. By the precedent proposition, we can assume that there exists a
foliations that leaves invariant an irreducible algebraic curve $C$ of degree
$p$ and four lines: $l_{1},l_{2},l_{3}$ and $l_{\infty}$ where $l_{1}\cap
l_{2}=l_{0}\cap l_{1}=l_{0}\cap l_{2}=\\{Q\\}$. For simplicity, we suppose
that $l_{1}=\\{x=0\\}$, $l_{2}=\\{y=0\\}$ and $l_{3}=\\{ux+vy=0\\}$ for some
constants $u,v\in\operatorname{k}^{*}$. In this case, $\mathcal{F}$ is defined
by a $1$-form of the type $\Omega_{1}=yzA_{0}dx+xzA_{1}dy+xyA_{2}dz$ for
certain $A_{0},A_{1},A_{2}\in\operatorname{k}[x,y,z]$ homogeneous of degree
$1$ and such that $A_{0}+A_{1}+A_{2}=0$. Consider the finite morphism

$\displaystyle\Phi_{e}\colon\operatorname{\mathbb{P}_{k}^{2}}$
$\displaystyle\longrightarrow\operatorname{\mathbb{P}_{k}^{2}}$
$\displaystyle[x_{0}:x_{1}:x_{2}]$
$\displaystyle\mapsto[x_{0}^{e}:x_{1}^{e}:x_{2}^{e}].$

Let $\mathcal{H}$ the foliation in $\operatorname{\mathbb{P}_{k}^{2}}$ defined
by the saturation of the $1$-form

$\Omega=\Phi_{e}^{*}\Omega_{1}=e(xyz)^{e-1}[yzA_{0}(x^{e},y^{e},z^{e})dx+xzA_{1}(x^{e},y^{e},z^{e})dy+xyA_{2}(x^{e},y^{e},z^{e})dz].$

Observe that $\mathcal{H}$ is a foliation of degree $e+1$ which is not
$p$-closed. Since $\operatorname{gcd}(e,p)=1$, we can use the Lemma 4.1 to
ensure that $\Phi_{e}^{*}l_{3}$ and $\Phi_{e}^{*}C$ are reduced curves with
irreducible distinct components. In particular, by the Proposition 3.7 it
follows that $\Phi_{e}^{*}l_{3}$ and $\Phi_{e}^{*}C$ are
$\mathcal{H}$-invariant curves. We claim that $\mathcal{H}$ has $p$-divisor
$\Delta_{\mathcal{H}}=\\{x=0\\}+\\{y=0\\}+\\{z=0\\}+\Phi_{e}^{*}l_{3}+\Phi_{e}^{*}C.$
Indeed, note that $\Delta_{\mathcal{H}}$ is a divisor with degree given by the
formula

$\deg(\Delta_{\mathcal{H}})=pK_{\mathcal{H}}\cdot\mathcal{O}_{\mathbb{P}_{\operatorname{k}}^{2}}(1)+N_{\mathcal{H}}\cdot\mathcal{O}_{\mathbb{P}_{\operatorname{k}}^{2}}(1)=pe+e+3.$

In the other hand, by construction we know that the curves
$\\{x=0\\}$,$\\{y=0\\}$,$\\{z=0\\}$, $\Phi_{e}^{*}l_{3}$ and $\Phi_{e}^{*}C$
are $\mathcal{H}$-invariant. In particular,
$\Delta_{\mathcal{H}}\geq\\{x=0\\}+\\{y=0\\}+\\{z=0\\}+\Phi_{e}^{*}l_{3}+\Phi_{e}^{*}C.$
By comparison of the degrees we conclude the equality
$\Delta_{\mathcal{H}}=\\{x=0\\}+\\{y=0\\}+\\{z=0\\}+\Phi_{e}^{*}l_{3}+\Phi_{e}^{*}C.$
So, it follows that $\mathcal{H}$ is a foliation of degree $e+1$ in
$\operatorname{\mathbb{P}_{k}^{2}}$ with reduced $p$-divisor. This finishes
the proof of the proposition. ∎

######  Remark 4.5.

In the proof of the Theorem A we saw that is possible to find a foliation
$\mathcal{F}$ of degree $2$ in $\operatorname{\mathbb{P}_{k}^{2}}$ that is not
$p$-closed with $p$-divisor in the form
$\Delta_{\mathcal{F}}=l_{1}+l_{2}+l_{3}+l_{4}+C$ where $C$ is an irreducible
algebraic curve of degree $p$ and $l_{i}\neq l_{j}$ if $i\neq j$. Let
$P,Q\in\operatorname{\mathbb{P}_{k}^{2}}$ and fix
$\Phi:\operatorname{\mathbb{P}_{k}^{2}}\longrightarrow\operatorname{\mathbb{P}_{k}^{2}}$
an automorphism such that $\Phi(P),\Phi(Q)\ \not\in C$. Then,
$\Phi^{*}\mathcal{F}$ is foliation that is not $p$-closed with $p$-divisor
$\Delta_{\Phi^{*}\mathcal{F}}=\Phi^{*}l_{1}+\Phi^{*}l_{2}+\Phi^{*}l_{3}+\Phi^{*}l_{4}+\Phi^{*}C$
where $\Phi^{*}C$ is an irreducible algebraic curve of degree $p$ such that
$P,Q\not\in\Phi^{*}C$.

##  5 The $p$-divisor for foliations on
$\operatorname{\mathbb{P}_{k}^{1}\times\mathbb{P}_{k}^{1}}$

In the following, we will fix $x_{0},x_{1}$ (resp. $y_{0},y_{1}$) as the
coordinates functions of the first factor (resp. second factor) of
$\operatorname{\mathbb{P}_{k}^{1}\times\mathbb{P}_{k}^{1}}$. Let $\pi_{1}$ the
projection of $\operatorname{\mathbb{P}_{k}^{1}\times\mathbb{P}_{k}^{1}}$ over
the first factor and $\pi_{2}$ the projection of
$\operatorname{\mathbb{P}_{k}^{1}\times\mathbb{P}_{k}^{1}}$ over the second
factor. Let $F$ and $M$ fibers of $\pi_{1}$ and $\pi_{2}$ respectively. Recall
that $\\{F,M\\}$ is a base for the vector space
$\operatorname{Num}_{\mathbb{Q}}(\operatorname{\mathbb{P}_{k}^{1}\times\mathbb{P}_{k}^{1}})$
which satisfies the following numerical conditions $F^{2}=0$, $F\cdot M=1$,
and $M^{2}=0.$ Observe that for any fibers
$F_{0}=\pi_{1}^{-1}(\operatorname{pt})$ and
$M_{0}=\pi_{2}^{-1}(\operatorname{pt})$ we have $F_{0}\equiv F$ and
$M_{0}\equiv M$.

######  Lemma 5.1.

Let $C=\mathcal{Z}(f)\subset\mathbb{A}_{\operatorname{k}}^{2}$ be an
irreducible algebraic curve of degree $d$ and $f=f_{k}+f_{k+1}+\cdots+f_{d}$
the decomposition of $f$ in homogeneous terms as element of
$\operatorname{k}[x,y]$. Suppose that $x^{d}$ and $y^{d}$ effectively occurs
in $f_{d}$ and consider $\overline{C}$ the bi-projectivization of $C$ in
$\operatorname{\mathbb{P}_{k}^{1}\times\mathbb{P}_{k}^{1}}$ via the
isomorphism:

$\displaystyle\Phi\colon U_{11}$
$\displaystyle\longrightarrow\mathbb{A}_{\operatorname{k}}^{2}$
$\displaystyle([x_{0}:x_{1}],[y_{0}:y_{1}])$
$\displaystyle\mapsto\left(\frac{x_{0}}{x_{1}},\frac{y_{0}}{y_{1}}\right).$

Then, $\overline{C}$ is irreducible and if $\\{F,M\\}$ is a base for
$\operatorname{Num}_{\mathbb{Q}}(\operatorname{\mathbb{P}_{k}^{1}\times\mathbb{P}_{k}^{1}})$,
with $F^{2}=0$, $F\cdot M=1$ and $M^{2}=0$ then
$\overline{C}\equiv\deg(C)(F+M).$

###### Proof.

The compactification $\overline{C}$ is defined by the following polynomial

$F=(x_{1}y_{1})^{d-k}f_{k}(x_{0}y_{1},x_{1}y_{0})+(x_{1}y_{1})^{d-k-1}f_{k+1}(x_{0}y_{1},x_{1}y_{0})+\cdots+f_{d}(x_{0}y_{1},x_{1}y_{0})$

It is easy to see that $F$ is a homogeneous polynomial of bi-degree $(d,d)$,
so that we need only to check their irreducibility. In the other hand, note
that if exists an irreducible factor $H$ of $F$ then necessarily we have
$H\not\in\langle x_{1}y_{1}\rangle$ since the hypothesis in $f_{d}$ implies
that $f_{d}(x_{0}y_{1},x_{1}y_{0})\not\in\langle x_{1}y_{1}\rangle$. But,
specializing to $x_{1}=1,y_{1}=1$ we conclude that $f$ is reducible.
Contradiction! ∎

######  Example 5.2.

Let $C$ be an affine curve in $\mathbb{A}_{\operatorname{k}}^{2}$ defined by
the equation $f=x+y+xy$. Using the map

$\displaystyle\Phi\colon U_{11}$
$\displaystyle\longrightarrow\mathbb{A}_{\operatorname{k}}^{2}$
$\displaystyle([x_{0}:x_{1}],[y_{0}:y_{1}])$
$\displaystyle\mapsto\left(\frac{x_{0}}{x_{1}},\frac{y_{0}}{y_{1}}\right)$

and projectivizing $C$ in
$\operatorname{\mathbb{P}_{k}^{1}\times\mathbb{P}_{k}^{1}}$ via $\Phi$ we
obtain the curve $\overline{C}=\mathcal{Z}(x_{0}y_{1}+x_{1}y_{0}+x_{0}y_{0})$
of bi-degree $(1,1)\neq(2,2)$. This shows that the condition in Lemma 5.1 is
necessary.

######  Theorem 5.3.

Let $d_{1},d_{2}\in\mathbb{Z}_{\geq 0}$ such that $p\nmid d_{i}$, if
$d_{i}\neq 0$. Then, a generic foliation in
$\operatorname{\mathbb{P}_{k}^{1}\times\mathbb{P}_{k}^{1}}$ with normal bundle
$N\equiv(d_{1}+2)F+(d_{2}+2)M$ has reduced $p$-divisor. More precisely,
$U(\operatorname{\mathbb{P}_{k}^{1}\times\mathbb{P}_{k}^{1}},\mathcal{O}_{\operatorname{\mathbb{P}_{k}^{1}\times\mathbb{P}_{k}^{1}}}(F+M),N)\neq\emptyset$
if

  * •

$N\cdot F-2\geq 0$ and $p\nmid N\cdot F-2$ (if nonzero);

  * •

$\deg(N)-N\cdot F+d-2\geq 0$ and $p\nmid\deg(N)-N\cdot F-2$ (if nonzero).

The proof of the Theorem 5.3 is divided in propositions. We show that given
$d_{1},d_{2}\in\mathbb{Z}_{\geq 0}$ such that $p\nmid d_{i}$ (if $d_{i}\neq
0$) we can find a foliation $\mathcal{G}$ in
$\Sigma_{0}=\operatorname{\mathbb{P}_{k}^{1}\times\mathbb{P}_{k}^{1}}$ with
the following properties:

  1. (i)

$K_{\mathcal{G}}\equiv d_{1}F_{0}+d_{2}M_{0}$ where $F_{0}=\\{x_{0}=0\\}$ and
$M_{0}=\\{y_{0}=0\\}$.

  2. (ii)

$F_{0}$ and $M=\\{y_{1}=0\\}$ are $\mathcal{G}$-invariant curves with
$\\{Q\\}=F_{0}\cap M$ a $p$-reduced singularity of $\mathcal{G}$.

  3. (iii)

The $p$-divisor $\Delta_{\mathcal{G}}$ is reduced.

We will consider two cases: A and B.

###  5.1 Case A

In this subsection, we proof, in particular, that a generic foliation
$\mathcal{G}$ in $\operatorname{\mathbb{P}_{k}^{1}\times\mathbb{P}_{k}^{1}}$
with cotangent divisor $K_{\mathcal{G}}\equiv d_{1}F_{0}+d_{2}M_{0}$ has
reduced $p$-divisor when

$(d_{1},d_{2})\in\\{(0,0)\\}\cup\\{(l,0)\in\mathbb{Z}^{2}\mid l>0\mbox{ and
}p\nmid l\\}\cup\\{(0,l)\in\mathbb{Z}^{2}\mid l>0\mbox{ and }p\nmid l\\}.$

We start with the case $d_{1}=d_{2}=0$.

######  Proposition 5.4.

A generic foliation in
$\operatorname{\mathbb{P}_{k}^{1}\times\mathbb{P}_{k}^{1}}$ with cotangent
divisor trivial $K_{\mathcal{G}}\equiv 0$ satisfies (i),(ii) e (iii).

###### Proof.

Let $\omega$ be a $1$-form in $\mathbb{A}_{\operatorname{k}}^{2}$ given by
$\omega=\alpha ydx+xdy\quad\mbox{with}\quad\alpha\not\in\mathbb{F}_{p}.$
Consider $\mathbb{A}_{\operatorname{k}}^{2}$ as an open subset of
$\operatorname{\mathbb{P}_{k}^{1}\times\mathbb{P}_{k}^{1}}$ via the
isomorphism

$\Phi\colon
U_{11}\longrightarrow\mathbb{A}_{\operatorname{k}}^{2}\qquad([x_{0},x_{1}],[y_{0},y_{1}])\mapsto\left(\frac{x_{0}}{x_{1}},\frac{y_{0}}{y_{1}}\right)$

where $U_{11}=\\{y_{1}\neq 0\\}\cap\\{x_{1}\neq
0\\}\subset\operatorname{\mathbb{P}_{k}^{1}\times\mathbb{P}_{k}^{1}}$.
Projectivizing the $1$-form $\omega$ in
$\operatorname{\mathbb{P}_{k}^{1}\times\mathbb{P}_{k}^{1}}$ by the map $\Phi$
we obtain:

$\Phi^{*}\omega=\frac{\alpha y_{0}x_{1}dx_{0}-\alpha
y_{0}x_{0}dx_{1}}{x_{1}^{2}y_{1}}+\frac{x_{0}y_{1}dy_{0}-y_{0}x_{0}dy_{1}}{x_{1}y_{1}^{2}}$

and considering their saturation it follows that $\Omega=\alpha
x_{1}y_{0}y_{1}dx_{0}-\alpha
x_{0}y_{0}y_{1}dx_{1}+x_{0}x_{1}y_{1}dy_{0}-x_{0}x_{1}y_{0}dy_{1}$ a
projective $1$-form that is bi-homogeneous which defines a foliation
$\mathcal{F}_{\Omega}$ with $K_{\mathcal{F}_{\Omega}}\equiv 0$. In particular,
satisfies (i). Note that $F_{0}$, $F_{1}=\\{x_{1}=0\\}$, $M=\\{y_{1}=0\\}$ and
$M_{0}$ are $\mathcal{F}_{\Omega}$-invariant curves, so that
$\Delta_{\mathcal{F}_{\Omega}}\geq M_{0}+M+F_{1}+F_{2}.$ In the other hand,
$\Delta_{\mathcal{F}_{\Omega}}\equiv pK_{\mathcal{F}}+N_{\mathcal{F}}\equiv
N_{\mathcal{F}}\equiv 2F_{0}+2M_{0}$ and by degree comparison we conclude the
equality $\Delta_{\mathcal{F}_{\Omega}}=M_{0}+M+F_{1}+F_{2}.$ In particular,
$\mathcal{F}_{\Omega}$ satisfies (iii). Now, let $U_{10}$ be the open set
given by $\\{y_{0}\neq 0\\}\cap\\{x_{1}\neq 0\\}$. Observe that there exists
an isomorphism

$\displaystyle\psi\colon U_{10}$
$\displaystyle\longrightarrow\mathbb{A}_{\operatorname{k}}^{2}$
$\displaystyle\left([x_{0}:x_{1}],[y_{0}:y_{1}]\right)$
$\displaystyle\mapsto\left(\frac{x_{0}}{x_{1}},\frac{y_{1}}{y_{0}}\right)=(x,y).$

Restricting the foliation $\mathcal{F}_{\Omega}$ to the open
$U_{10}\cong\mathbb{A}_{\operatorname{k}}^{2}$ we obtain a foliation given by
the $1$-form: $\sigma=\alpha ydx-xdy$ and so $M\cap
F_{0}=\\{y=0\\}\cap\\{x=0\\}=\\{(0,0)\\}$ is a $p$-reduced singularity. In
this way, we conclude that $\mathcal{F}_{\Omega}$ satisfies (ii). This
finishes the proof for $d_{2}=d_{1}=0$. ∎

We pass now to the a more general case.

######  Proposition 5.5.

Let $d_{1},d_{2}\in\mathbb{Z}_{\geq 0}$ such that $p\nmid d_{i}$, if
$d_{i}\neq 0$. Suppose that

$(d_{1},d_{2})\in\\{(0,0)\\}\cup\\{(l,0)\in\mathbb{Z}^{2}\mid l>0\mbox{ and
}p\nmid l\\}\cup\\{(0,l)\in\mathbb{Z}^{2}\mid l>0\mbox{ and }p\nmid l\\}.$

Then, a generic foliation in
$\operatorname{\mathbb{P}_{k}^{1}\times\mathbb{P}_{k}^{1}}$ with cotangent
divisor $K_{\mathcal{G}}\equiv d_{1}F+d_{2}M$ satisfies (i),(ii) and (iii).

###### Proof.

The case $d_{1}=d_{2}=0$ was considered in the precedent proposition. We will
consider the case where $d_{1}=l$ and $d_{2}=0$ for $l>0$ with $p\nmid l$. By
the symmetry of the the problem we will automatically consider the case where
$d_{1}=0$ and $d_{2}=l$ for $l>0$ with $p\nmid l$.

Case $l=1$: Let $\mathcal{F}$ be a Riccati foliation with respect the first
projection
$\pi_{1}\colon\operatorname{\mathbb{P}_{k}^{1}\times\mathbb{P}_{k}^{1}}\longrightarrow\operatorname{\mathbb{P}_{k}^{1}}.$
Recall (see [1, Chapter 4, Section 1]) that a Riccati foliation is defined as
a foliation in $\operatorname{\mathbb{P}_{k}^{1}\times\mathbb{P}_{k}^{1}}$
whose the general fiber $F$ of $\pi_{1}$ is transverse to $\mathcal{F}$.
Suppose that

  1. (a)

The foliation $\mathcal{F}$ leaves invariant only tree fibers of the first
projection $\pi_{1}$. Denote that fibers by $F_{0},F_{1},F_{2}.$

  2. (b)

The foliation $\mathcal{F}$ leaves invariant only one fiber of the second
projection $\pi_{2}$. Denote that fiber by $M$.

  3. (c)

The intersections $\\{Q_{i}\\}=F_{i}\cap M$ are $p$-reduced singularities of
$\mathcal{F}$.

We show that that foliation satisfies (i),(ii) and (iii). Note that by (a)
(see [1, Section 1, Chapter 4])

$K_{\mathcal{F}}=\pi_{1}^{*}K_{\operatorname{\mathbb{P}_{k}^{1}}}+F_{0}+F_{1}+F_{2}\equiv-2F+3F=F.$

So, $\mathcal{F}$ satisfies (i). Note that the item (c) ensures that
$\mathcal{F}$ is not $p$-closed (see Lemma 3.10) and that $\mathcal{F}$
satisfies (ii), module a change of coordinates. We will show that
$\mathcal{F}$ satisfies (iii). Observe that

$D=\Delta_{\mathcal{F}}-F_{0}-F_{1}-F_{3}-M\geq 0$

and by (b) it follows that
$\operatorname{ord}_{M}(\Delta_{\mathcal{F}})=\operatorname{ord}_{F_{i}}(\Delta_{\mathcal{F}})=1$
for all $i$ (see [13, Fact 2.8]). Since
$\Delta_{\mathcal{F}}=pK_{\mathcal{F}}+N_{\mathcal{F}}\equiv
pF_{0}+3F_{0}+2M_{0}\equiv(p+2)F_{0}+2M_{0}$, the following equality holds

$D\cdot F=\Delta_{\mathcal{F}}\cdot F-M\cdot F=2-1=1.$

So, there exists an irreducible curve $C\leq D$ such that $C\cdot F=1$. Write
$D=C+R$ and note that we have $C\equiv uF_{0}+M_{0}$ and $R\equiv vF_{0}$, for
some $u,v\in\mathbb{Z}_{\geq 0}$. We will show that $v\equiv 0\mod p$.
Suppose, by contradiction, that $v\neq 0$. Then, since $R\equiv vF_{0}$ we
conclude by the Proposition 3.7 that there exists a fiber of the first
projection, $F_{4}\leq R$, that is $\mathcal{F}$-invariant. In the other hand,
by construction of $\mathcal{F}$, we know that $F_{0}$,$F_{1}$ and $F_{2}$ are
the complete list of $\mathcal{F}$-invariant fibers. Moreover, all that fibers
has multiplicity $1$ along the $p$-divisor $\Delta_{\mathcal{F}}$. So, the
fiber $F_{4}$ can not exist and we conclude that $v\equiv 0\mod p$. In the
other hand,

$u+v=C\cdot M_{0}+R\cdot M_{0}=D\cdot M_{0}=\Delta_{\mathcal{F}}\cdot
M_{0}-3=p+3-3=p.$

So, if $v\neq 0$ the condition $v\equiv 0\mod p$ implies that $v=p$ and by
consequence it follows $u=0$. But this implies $C\equiv M$ so that $C$ is a
fiber of the second projection which is $\mathcal{F}$-invariant. This is a
contradiction, since by construction we know that $M$ is the unique fiber of
the second projection that is $\mathcal{F}$-invariant and satisfies
$\operatorname{ord}_{M}(\Delta_{\mathcal{F}})=1$. So, $v=0$ and $C$ is an
irreducible curve $\mathcal{F}$-invariant with $C\equiv pF_{0}+M_{0}$ and
$\mathcal{F}$ has $p$-divisor given by

$\Delta_{\mathcal{F}}=F_{0}+F_{1}+F_{2}+M+C$

which is reduced. So, $\mathcal{F}$ satisfies (i),(ii) and (iii) and we
conclude the argument to the case $(d_{1},d_{2})=(1,0)$.

Case $l>0$ and $p\nmid l$: We use the precedent case ($l=1$) to study the
present case. Let $\mathcal{F}$ be a foliation in
$\operatorname{\mathbb{P}_{k}^{1}\times\mathbb{P}_{k}^{1}}$ as in the
precedent case. As we had proved, a generic foliation which satisfies (a),(b)
and (c) has $p$-divisor in the form

$\Delta_{\mathcal{F}}=F_{0}+F_{1}+F_{3}+M+C$

where $C$ is an irreducible curve with $C\equiv pF_{0}+M_{0}$, $F_{0},F_{1}$
and $F_{2}$ are fibers of the first projection $\pi_{1}$ and $M$ is a fiber of
the second projection $\pi_{2}$.

Let
$\Phi\colon\operatorname{\mathbb{P}_{k}^{1}\times\mathbb{P}_{k}^{1}}\longrightarrow\operatorname{\mathbb{P}_{k}^{1}\times\mathbb{P}_{k}^{1}}$
be a finite ramified map of degree $l$ with ramification divisor $R$. Suppose
that $\Phi$ ramifies only along the curves $F_{0}$ and $F_{1}$. Let
$\mathcal{G}=\Phi^{*}\mathcal{F}$ the foliation defined by the pull-back of
$\mathcal{F}$ by $\Phi$. Explicitly, if $\Omega$ is the bi-homogeneous
$1$-form which defines $\mathcal{F}$ then $\mathcal{G}$ is the foliation
defined by the saturation of $\Phi^{*}\Omega$. Observe that since $F_{0}$ and
$F_{1}$ are $\mathcal{F}$-invariant $K_{\mathcal{G}}=\Phi^{*}K_{\mathcal{F}}$.
So, $\Phi^{*}K_{\mathcal{F}}=\Phi^{*}F=lF$ and we conclude that $\mathcal{G}$
satisfies (i) and $N_{\mathcal{G}}\equiv(l+2)F+2M$ and
$\Delta_{\mathcal{G}}\equiv
pK_{\mathcal{G}}+N_{\mathcal{G}}\equiv(pl+l+2)F+2M$. By Lemma 4.1 we know that
$\Phi^{*}C$ and $\Phi^{*}F_{2}$ are reduced curves that has no irreducible
components in common. In particular,

$\Delta_{\mathcal{G}}\geq\Phi^{*}C+\Phi^{*}F_{2}+F_{1}+F_{0}+M.$

in the other hand, $\Phi^{*}C\equiv\Phi^{*}(pF+M)\equiv plF+M$ and
$\Phi^{*}F_{2}=lF$ and so

$\Phi^{*}C+\Phi^{*}F_{2}+F_{0}+F_{1}+M\equiv(pl+l+2)F+2M$

and by comparison degree we obtain the following equality

$\Delta_{\mathcal{G}}=\Phi^{*}C+\Phi^{*}F_{2}+F_{1}+F_{0}+M.$

In particular, $\Delta_{\mathcal{G}}$ is a reduced divisor and $\mathcal{G}$
satisfies (iii). We will show that $\mathcal{G}$ satisfies (ii). For that, let
$Q$ the point in $F_{0}\cap M$. By the precedent case, more precisely, by (c)
we know that $Q$ is a $p$-reduced singularity. Fix $U$ an affine open set
around $Q=(0,0)$ such that $\Phi|_{\Phi^{-1}(U)}:\Phi^{-1}(U)\longrightarrow
U$ is locally defined by
$\Phi\colon(x,y)\mapsto(x^{l},y)=(\tilde{x},\tilde{y})$ and $\mathcal{F}$ is
given by the $1$-form
$\omega=\alpha\tilde{y}d\tilde{x}-\tilde{x}d\tilde{y}+O(2)$. Then,
$\Phi^{*}\mathcal{F}$ is given on $\Phi^{-1}(U)$ by the $1$-form
$\sigma=l\alpha ydx-xdy+O(2).$ Since we are assuming that $Q$ is a $p$-reduced
singularity $\alpha\not\in\mathbb{F}_{p}$ and so
$l\alpha\not\in\mathbb{F}_{p}$. We conclude that $\mathcal{G}$ has
$\\{Q\\}=F\cap M$ as $p$-reduced singularity. In particular, satisfies (ii).
This finishes the proof of the case $(d_{1},d_{2})=(l,0)$ with $l>1$ and
$p\nmid l$. The cases above finish the proof of the proposition. ∎

###  5.2 Case B

In this subsection, we show that a generic foliation $\mathcal{G}$ in
$\operatorname{\mathbb{P}_{k}^{1}\times\mathbb{P}_{k}^{1}}$ with cotangent
divisor $K_{\mathcal{G}}\equiv d_{1}F+d_{2}M$ satisfies the following

  1. (i)

$K_{\mathcal{G}}\equiv d_{1}F_{0}+d_{2}M_{0}$ where $F_{0}=\\{x_{0}=0\\}$ and
$M_{0}=\\{y_{0}=0\\}$,

  2. (ii)

$F_{0}$ and $M=\\{y_{1}=0\\}$ are $\mathcal{G}$-invariant curves with
$\\{Q\\}=F_{0}\cap M$ a $p$-reduced singularity of $\mathcal{G}$,

  3. (iii)

The $p$-divisor $\Delta_{\mathcal{G}}$ is reduced

when $(d_{1},d_{2})\in\\{(l_{1},l_{2})\in\mathbb{Z}^{2}\mid
l_{1},l_{2}>0\mbox{ and }p\nmid l_{1}l_{2}\\}$. We start with the case
$d_{1}=d_{1}=1$.

######  Proposition 5.6.

A generic foliation in
$\operatorname{\mathbb{P}_{k}^{1}\times\mathbb{P}_{k}^{1}}$ with cotangent
divisor $K_{\mathcal{G}}\equiv F+M$ satisfies (i),(ii) and (iii).

###### Proof.

Indeed, in the open set $U_{11}$ consider the foliation defined by the
$1$-form: $\omega=ydx-xdy+\tilde{b}(x,y)dx+\tilde{a}(x,y)dy.$ In the proof of
Theorem A, we saw that for a generic choice of
$\tilde{a},\tilde{b}\in\operatorname{k}[x,y]_{2}$ the $1$-form $\omega$
defines a foliation with reduced $p$-divisor in
$\mathbb{A}_{\operatorname{k}}^{2}$ which is explicitly given by
$\Delta=l_{1}+l_{2}+l_{3}+C$ where $l_{1},l_{2},l_{3}$ are distinct lines that
pass through point $(0,0)$ and $C$ is an irreducible curve that passes through
point $(0,0)$ of degree $p$ and with multiplicity $p-1$ over $(0,0)$.
Moreover, by the Remark 4.5 we can assume that $C$ does not passes through
points $\\{[0:1:0],[1:0:0]\\}$. For simplicity, we will assume that
$l_{1}=\\{x=0\\}$, $l_{2}=\\{y=0\\}$ and $l_{3}=\\{ux+vy=0\\}$ for some non
zero constants $u,v\in\operatorname{k}$. In this case, we have
$\tilde{a}(x,y)=xa(x,y)$ and $\tilde{b}(x,y)=yb(x,y)$ for some
$a,b\in\operatorname{k}[x,y]_{1}$.

Using the isomorphism $\phi\colon
U_{11}\longrightarrow\mathbb{A}_{\operatorname{k}}^{2}$ which associates
$([x_{0}:x_{1}],[y_{0}:y_{1}])\mapsto\left(\frac{x_{0}}{x_{1}},\frac{y_{0}}{y_{1}}\right)$
we can compactify $\omega$ to a foliation in
$\operatorname{\mathbb{P}_{k}^{1}\times\mathbb{P}_{k}^{1}}$. In this case, the
foliation obtained is explicitly given by the $1$-form:
$\Omega=A_{0}dx_{0}+A_{1}dx_{1}+B_{0}dy_{0}+B_{1}dy_{1}$ where

$A_{0}=x_{1}y_{0}y_{1}(x_{1}y_{1}+b(x_{0}y_{1},x_{1}y_{0})),\qquad
A_{1}=-x_{0}y_{0}y_{1}(x_{1}y_{1}+b(x_{0}y_{1},x_{1}y_{0})),$
$B_{0}=x_{0}x_{1}y_{1}(-x_{1}y_{1}+a(x_{0}y_{1},x_{1}y_{0})),\quad\mbox{and}\quad
B_{1}=-x_{0}x_{1}y_{0}(-x_{1}y_{1}+a(x_{0}y_{1},x_{1}y_{0})).$

The projective $1$-form $\Omega$ defines a foliation $\mathcal{F}_{\Omega}$ in
$\operatorname{\mathbb{P}_{k}^{1}\times\mathbb{P}_{k}^{1}}$ such that
$K_{\mathcal{F}_{\Omega}}\equiv F+M$ and so $\mathcal{F}_{\Omega}$ satisfies
(i). We claim that $\Delta_{\mathcal{F}_{\Omega}}$ is reduced. Indeed, note
that

$\Delta_{\mathcal{F}_{\Omega}}\equiv
pK_{\mathcal{F}_{\Omega}}+N_{\mathcal{F}_{\Omega}}\equiv
p(F+M)+(3F+3M)=(p+3)(F+M).$

By the homogeneous equations of $\mathcal{F}_{\Omega}$, we see that
$F_{0}=\\{x_{0}=0\\}$, $F_{1}=\\{x_{1}=0\\}$, $M_{0}=\\{y_{0}=0\\}$ and
$M=\\{y_{1}=0\\}$ are $\mathcal{F}_{\Omega}$-invariant. Moreover, bi-
projetivizing the line $L$ via $\Phi$ in
$\operatorname{\mathbb{P}_{k}^{1}\times\mathbb{P}_{k}^{1}}$ we know by the
Lemma 5.1 that the obtained curve ,$\overline{L}$, is irreducible
$\mathcal{F}_{\Omega}$-invariant with $\overline{L}\equiv F_{0}+M_{0}.$ Since
$C$ does not pass through the point $\\{[0:1:0],[1:0:0]\\}$ we can apply the
Lemma 5.1 to conclude that the irreducible curve of degree $p$ in
$\mathbb{A}_{\operatorname{k}}^{2}$ projectivize to an irreducible curve,
$\overline{C}$, in $\operatorname{\mathbb{P}_{k}^{1}\times\mathbb{P}_{k}^{1}}$
that is $\mathcal{F}_{\Omega}$-invariant and such that: $\overline{C}\equiv
p(F_{0}+M_{0})$. So, we obtain
$\Delta_{\mathcal{F}_{\Omega}}=F_{0}+F_{1}+M_{0}+M+\overline{L}+\overline{C}$
and we ensure that $\Delta_{\mathcal{F}_{\Omega}}$ is a reduced divisor. This
shows that $\mathcal{F}$ satisfies (iii).

Observe that restricting the foliation to the open set $U_{10}=\\{y_{0}\neq
0\\}\cap\\{x_{1}\neq 0\\}$ the foliation in
$\mathbb{A}_{\operatorname{k}}^{2}$ is given by the $1$-form
$\sigma=y_{1}(y_{1}+b(x_{0}y_{1},1))dx_{0}-x_{0}(-y_{1}+a(x_{0}y_{1},1))dy_{1}.$
Since $a,b\in\operatorname{k}[x,y]_{1}$ are generic, we can assume that
$a(x_{0}y_{1},1)=a_{0}+O(2)$ and $b(x_{0}y_{1},1)=b_{0}+O(2)$ where
$a_{0}/b_{0}\not\in\mathbb{F}_{p}$, so that we obtain a foliation in
$\Sigma_{0}$ which satisfies (i), (ii) and (iii) with $d_{1}=d_{2}=1$. ∎

Using the precedent proposition we will consider the general case.

######  Proposition 5.7.

Let $d_{1},d_{2}\in\mathbb{Z}_{\geq 0}$ such that $p\nmid d_{i}$,if $d_{i}\neq
0$. Suppose that

$(d_{1},d_{2})\in\\{(l_{1},l_{2})\in\mathbb{Z}^{2}\mid l_{1},l_{2}>0\mbox{ and
}p\nmid l_{1}l_{2}\\}.$

Then, a generic foliation in
$\operatorname{\mathbb{P}_{k}^{1}\times\mathbb{P}_{k}^{1}}$ with cotangent
divisor $K_{\mathcal{G}}\equiv d_{1}F+d_{2}M$ satisfies (i),(ii) and (iii).

###### Proof.

Let $l\in\mathbb{N}$ be a positive integer coprime to $p$ and consider the
finite map:

$\displaystyle\Phi_{l}\colon\operatorname{\mathbb{P}_{k}^{1}\times\mathbb{P}_{k}^{1}}$
$\displaystyle\longrightarrow\operatorname{\mathbb{P}_{k}^{1}\times\mathbb{P}_{k}^{1}}$
$\displaystyle([x_{0},x_{1}],[y_{0},y_{1}])$
$\displaystyle\mapsto([x_{0}^{l}:x_{1}^{l}],[y_{0}:y_{1}]).$

Let $R_{\Phi_{l}}$ be the ramification divisor of $\Phi_{l}$ and $\mathcal{G}$
the foliation in $\operatorname{\mathbb{P}_{k}^{1}\times\mathbb{P}_{k}^{1}}$
described by the saturation of the $1$-form $\Phi_{l}^{*}\Omega$, where
$\Omega$ is the projectivization of the $1$-form in
$\mathbb{A}_{\operatorname{k}}^{2}$ given by
$\sigma=y_{1}(y_{1}+b(x_{0}y_{1},1))dx_{0}-x_{0}(-y_{1}+a(x_{0}y_{1},1))dy_{1}$
where $a,b\in\operatorname{k}[x,y]_{1}$ are generic with
$a(x_{0}y_{1},1)=a_{0}+O(2)$, $b(x_{0}y_{1},1)=b_{0}+O(2)$ and
$a_{0}/b_{0}\not\in\mathbb{F}_{p}$. Recall that by Proposition 5.6 we can
assume that the foliation defined by $\Omega$, $\mathcal{F}_{\Omega}$, is not
$p$-closed with $p$-divisor given by
$\Delta_{\mathcal{F}_{\Omega}}=F_{0}+F_{1}+M_{0}+M+\overline{L}+\overline{C}$
where $F_{0}=\\{x_{0}=0\\}$, $F_{1}=\\{x_{1}=0\\}$, $M_{0}=\\{y_{0}=0\\}$,
$M=\\{y_{1}=0\\}$ and with $\overline{L}$, $\overline{C}$ irreducible curves
such that $\overline{C}\equiv
p(F_{0}+M_{0})\quad\mbox{and}\quad\overline{L}\equiv F_{0}+M_{0}.$ We have
$R_{\Phi_{l}}=(l-1)F_{0}+(l-1)F_{1}\equiv 2(l-1)F_{0}$ and since $F_{0}$ and
$F_{1}$ are $\mathcal{F}_{\Omega}$-invariant we conclude that
$K_{\mathcal{G}}=\Phi^{*}K_{\mathcal{F}}=\Phi^{*}(F+M)=lF+M$. In explicit
terms, the foliation $\mathcal{G}$ is defined by the $1$-form:
$\gamma=\tilde{A}_{0}dx_{0}+\tilde{A}_{1}dx_{1}+\tilde{B}_{0}dy_{0}+\tilde{B}_{1}dy_{1}$
where

$\tilde{A}_{0}=lx_{1}y_{0}y_{1}(x_{1}^{l}y_{1}+b(x_{0}^{l}y_{1},x_{1}^{l}y_{0})),\qquad\tilde{A}_{1}=-lx_{0}y_{0}y_{1}(x_{1}^{l}y_{1}+b(x_{0}^{l}y_{1},x_{1}^{l}y_{0})),$
$\tilde{B}_{0}=x_{0}x_{1}y_{1}(-x_{1}^{l}y_{1}+a(x_{0}^{l}y_{1},x_{1}^{l}y_{0})),\quad\mbox{and}\quad\tilde{B}_{1}=-x_{0}x_{1}y_{0}(-x_{1}^{l}y_{1}+a(x_{0}^{l}y_{1},x_{1}^{l}y_{0})).$

We will show the following equality:
$\Delta_{\mathcal{G}}=F_{0}+F_{1}+M_{0}+M+\Phi_{l}^{*}\overline{L}+\Phi_{l}^{*}\overline{C}.$
Note that the curves $\Phi_{l}^{*}\overline{L}$ and $\Phi_{l}^{*}\overline{C}$
are $\mathcal{G}$-invariant. Indeed, since $\operatorname{gcd}(l,p)=1$ and
$x_{0},x_{1}$ do not divide the equations which define $\overline{L}$ and
$\overline{C}$, using the Lemma 4.1 we conclude that
$\Phi_{l}^{*}\overline{L}$ and $\Phi_{l}^{*}\overline{C}$ are reduced and have
distinct irreducible components. In particular, follows that
$\Phi_{l}^{*}\overline{L}$ and $\Phi_{l}^{*}\overline{C}$ are
$\mathcal{G}$-invariant (see Proposition 3.7). Since $F_{0}$, $F_{1}$,
$M_{0}$, $M$ are $\mathcal{G}$-invariant $\Delta_{\mathcal{G}}\geq
F_{0}+F_{1}+M_{0}+M+\Phi_{l}^{*}\overline{L}+\Phi_{l}^{*}\overline{C}.$ Note
that $\Delta_{\mathcal{G}}$ is a divisor that has bi-degree given by the
formula:

$pK_{\mathcal{G}}+N_{\mathcal{G}}\equiv
p(lF_{0}+M_{0})+((l+2)F_{0}+3M_{0})\equiv(pl+l+2)F_{0}+(p+3)M_{0}.$

In the other hand

$F_{0}+F_{1}+M_{0}+M+\Phi_{l}^{*}\overline{L}+\Phi_{l}^{*}\overline{C}\equiv
F_{0}+F_{0}+M_{0}+M_{0}+(lF_{0}+M_{0})+(plF_{0}+pM_{0})$
$=(pl+l+2)F_{0}+(p+3)M_{0}.$

So, by bi-degree comparison, we conclude the following equality
$\Delta_{\mathcal{G}}=F_{0}+F_{1}+M_{0}+M+\Phi_{l}^{*}\overline{L}+\Phi_{l}^{*}\overline{C}.$
This shows that $\mathcal{G}$ satisfies (iii).

We need to show that $F_{0}\cap M=\\{x_{0}=0\\}\cap\\{y_{1}=0\\}$ is a
$p$-reduced singularity of $\mathcal{G}$. Indeed, restricting the foliation
$\mathcal{G}$ to the open set $U_{10}$ we obtain a foliation in
$\mathbb{A}_{\operatorname{k}}^{2}$ given by the $1$-form
$\omega=ly(y+b(x^{l}y,1))dx-x(-y+a(x^{l}y,1))dy$ and we conclude that $(0,0)$
is $p$-reduced since we are assuming that $a(x_{0}y_{1},1)=a_{0}+O(2)$ and
$b(x_{0}y_{1},1)=b_{0}+O(2)$ with $a_{0}/b_{0}\not\in\mathbb{F}_{p}$. By
symmetry, that is, replacing $F_{0}$ by $M_{0}$, follows that given
$d_{1},d_{2}\in\mathbb{Z}_{\geq 0}$ with $p\nmid d_{i}$ (if $d_{i}\neq 0$) we
can construct foliations in
$\operatorname{\mathbb{P}_{k}^{1}\times\mathbb{P}_{k}^{1}}$ with the following
properties:

  1. (i)

The canonical divisor $K_{\mathcal{G}}$ is numerically equivalent to the
divisor $d_{1}F_{0}+d_{2}M_{0}$ where $F_{0}$ and $M_{0}$ are fibers of the
first and second projection, respectively;

  2. (ii)

The curves $F_{0}$ and $M_{0}$ are $\mathcal{G}$-invariant and
$\\{Q\\}=F_{0}\cap M_{0}$ is a $p$-reduced singularity of $\mathcal{G}$;

  3. (iii)

The $p$-divisor $\Delta_{\mathcal{G}}$ is reduced.

This finishes the proof of the proposition. ∎

Considering the junction of the case A and B we obtain the complete proof of
the Theorem 5.3.

##  6 The $p$-divisor for foliations on Hirzebruch surfaces

Let $d\in\mathbb{Z}_{\geq 0}$ and $M_{d}$ be a section of the natural
projection $\pi:\Sigma_{d}\longrightarrow\operatorname{\mathbb{P}_{k}^{1}}$
and $F$ be a fiber of $\pi$ such that $M_{d}^{2}=d$ and $M_{d}\cdot F=1$. The
curves $M_{d}$ and $F$ form a base of the vector space
$\operatorname{Num}_{\mathbb{Q}}(\Sigma_{d})$. Note that for any other divisor
$D\in\operatorname{Div}(\Sigma_{d})$ such that $\langle
D,F\rangle=\operatorname{Num}_{\mathbb{Q}}(\Sigma_{d})$ and which satisfies
$D^{2}=d$ and $D\cdot F=1$ we have $D\equiv M_{d}$. Indeed, write
$D=aM_{d}+bF$ for some $a,b\in\mathbb{Q}$ and observe that $1=D\cdot F=a$ and
$d=D^{2}=a^{2}d+2ab=d+2b$ and so $b=0$.

Fix $\\{F,M_{d}\\}$ as a basis for
$\operatorname{Num}_{\mathbb{Q}}(\Sigma_{d})$ and let $Q$ be the point in the
intersection $F\cap M$, where $M=\\{y_{1}=0\\}$ is the curve in $\Sigma_{d}$
with negative self-intersection, that is, $M^{2}=-d$. Let $Bl_{Q}\colon
Bl_{Q}(\Sigma_{d})\longrightarrow\Sigma_{d}$ the blowup of $\Sigma_{d}$ at $Q$
and $c_{\tilde{F}}\colon Bl_{Q}(\Sigma_{d})\longrightarrow\Sigma_{d+1}$ the
map that consists in the contraction of the strict transform of $F$ in
$Bl_{Q}(\Sigma_{d})$. Let $\overline{E}$ the exceptional divisor associated to
$Bl_{Q}$ and $\Phi_{d}^{d+1}\colon\Sigma_{d+1}-\\!\\!\rightarrow\Sigma_{d}$
the rational map which consist in the composition of birational maps:

$\Phi_{d}^{d+1}=Bl_{Q}\circ c_{\tilde{F}}^{-1}:\Sigma_{d+1}-\\!\\!\rightarrow
Bl_{Q}(\Sigma_{d})\longrightarrow\Sigma_{d}.$

Let $\tilde{M}_{d+1}$ the strict transform of $M_{d}$ in $Bl_{Q}(\Sigma_{d})$
and consider $E=(c_{\tilde{F}})_{*}\overline{E}$ and
$M_{d+1}=(c_{\tilde{F}})_{*}\tilde{M}_{d+1}$ the induced curves by the
contraction. Observe that $\\{E,M_{d+1}\\}$ forms a basis for
$\operatorname{Num}_{\mathbb{Q}}(\Sigma_{d})$ and satisfies the following
conditions: $E^{2}=0$, $E\cdot M_{d+1}=1$ and $M_{d+1}^{2}=d+1$.

######  Lemma 6.1.

If $\mathcal{F}$ is a foliation in $\Sigma_{d}$ with $K_{\mathcal{F}}\equiv
d_{1}F+d_{2}M_{d}$ then $\mathcal{G}=(\Phi_{d}^{d+1})^{*}\mathcal{F}$ is a
foliation in $\Sigma_{d+1}$ with
$K_{\mathcal{G}}\equiv(d_{1}-l(Q)+1)E+d_{2}M_{d+1}.$

###### Proof.

Let $\mathcal{H}=Bl_{Q}^{*}\mathcal{F}$ be the induced foliation in
$Bl_{Q}(\Sigma_{d})$. Then,
$K_{\mathcal{G}}=(c_{\tilde{F}})_{*}K_{\mathcal{H}}$ and

$K_{\mathcal{H}}=Bl_{Q}^{*}K_{\mathcal{F}}+(1-l(Q))E\equiv
Bl_{Q}^{*}(d_{1}F+d_{2}M_{d})+(1-l(Q))\overline{E}.$

Since $Bl_{Q}^{*}F=\tilde{F}+\overline{E}$ and
$Bl_{Q}^{*}M_{d}=\tilde{M}_{d+1}$, we conclude that $K_{\mathcal{H}}\equiv
d_{1}\tilde{F}+d_{2}\tilde{M}_{d+1}+(d_{1}-l(Q)+1)\overline{E}$ and it follows

$K_{\mathcal{G}}=(c_{\tilde{F}})_{*}K_{\mathcal{H}}\equiv(d_{1}-l(Q)+1)(c_{\tilde{F}})_{*}\overline{E}+d_{2}(c_{\tilde{F}})_{*}\tilde{M}_{d+1}\equiv(d_{1}-l(Q)+1)E+d_{2}M_{d+1}$

which finishes the proof of the lemma. ∎

######  Theorem B.

Let $\Sigma_{d}$ the $d$-Hirzebruch surface over $\operatorname{k}$ and
$d_{1},d_{2}\in\mathbb{Z}_{\geq 0}$ such that $p\nmid d_{i}$, if $d_{i}\neq
0$. Then, a generic foliation in $\Sigma_{d}$ with normal bundle
$N\equiv(d_{1}-d+2)F+(d_{2}+2)M_{d}$ has reduced $p$-divisor. More precisely,
$U(\Sigma_{d},\mathcal{O}_{\Sigma_{d}}(F+M_{d}),N)\neq\emptyset$ if

  * •

$N\cdot F-2\geq 0$ and $p\nmid N\cdot F-2$ (if nonzero);

  * •

$\deg(N)-(1+d)N\cdot F+d-2\geq 0$ and $p\nmid\deg(N)-(1+d)N\cdot F+d-2$ (if
nonzero).

###### Proof.

The proof will be done by induction on $d$. The case $d=0$ was considerate in
the Theorem 5.3. Suppose that the result is true for all $j$-Hirzebruch
surfaces with $j\leq d$, that is, suppose that for each $j\in\\{0,\ldots,d\\}$
and for each $d_{1},d_{2}\in\mathbb{Z}_{\geq 0}$ with $p\nmid d_{i}$ (if
$d_{i}\neq 0$) we can find a foliation $\mathcal{H}$ such that

  1. (i)

$K_{\mathcal{H}}\equiv d_{1}F_{0}+d_{2}M_{j}$, where $F_{0}$ is a fiber of the
natural projection
$\pi\colon\Sigma_{j}\longrightarrow\operatorname{\mathbb{P}_{k}^{1}}$ and
$M_{j}$ is a section of $\pi$ such that $M_{j}^{2}=j$ and $M_{j}\cdot F=1$;

  2. (ii)

If $M$ is the section of negative self-intersection of $\Sigma_{j}$ then
$F_{0}$ and $M$ are $\mathcal{H}$-invariant and $\\{Q\\}=F_{0}\cap M$ is a
$p$-reduced singularity of $\mathcal{H}$ where $M^{2}=-j$;

  3. (iii)

The $p$-divisor $\Delta_{\mathcal{H}}$ is reduced.

We will show that the same holds over $\Sigma_{d+1}$. Fix
$d_{1},d_{2}\in\mathbb{Z}_{\geq 0}$ such that $p\nmid d_{i}$, if $d_{i}\neq
0$. Let $\mathcal{G}$ a foliation in $\Sigma_{d}$ which satisfies (i),(ii) and
(iii) for $j=d$.

Let $Q$ the $p$-reduced singularity in $F_{0}\cap M$ and $\pi_{Q}\colon
Bl_{Q}(\Sigma_{d})\longrightarrow\Sigma_{d}$ the blowup with center at $Q$. By
the Lemma 3.11, we know that $Bl_{Q}^{*}\mathcal{F}$ is a foliation in
$Bl_{Q}(\Sigma_{d})$ with reduced $p$-divisor. More precisely, if
$\overline{E}$ denotes the exceptional divisor and
$\mathcal{G}=\pi_{Q}^{*}\mathcal{F}$, we know that
$\Delta_{\mathcal{G}}=\tilde{\Delta}_{\mathcal{F}}+\overline{E}.$ We claim
that since $Q$ is $p$-reduced we ensure that over $\overline{E}$ there exists
at least a $p$-reduced singularity of $\mathcal{G}$.

Indeed, in an open neighborhood of $Q$ the foliation is given by a $1$-form of
the type$\omega=\alpha ydx+xdy+O(2)$ and considering the chart
$\pi_{Q}\colon(x,t)\mapsto(x,xt)$ we see that in a neighborhood of
$\overline{E}$ the foliation $\mathcal{G}$ is given by the
$1$-form:$\sigma=(\alpha+1)tdx+xdt+O(2).$ In particular, $Q$ is $p$-reduced.
Denote by $c_{F}\colon Bl_{Q}(\Sigma_{d})\longrightarrow Y$ the contraction of
the line $\tilde{F}$ that is the strict transform of $F_{0}$. Since $Q\in
F_{0}$, the line $\tilde{F}$ has self-intersection $-1$ so that there exists
that contraction. The out surface, $Y$, is the Hirzebruch surface of type
$d+1$ ($\Sigma_{d+1}$). We will show that $c_{F_{*}}\mathcal{G}$ is a
foliation in $\Sigma_{d+1}$ that satisfies (i), (ii) and (iii). Let
$\overline{E}$ the exceptional divisor associated with the blowup $Bl_{Q}$ and
$\tilde{M}_{d+1}$ the strict transform of the curve $M_{d}$. Denote by
$E=c_{F_{*}}\overline{E}$ and $M_{d+1}=c_{F_{*}}\tilde{M_{d+1}}$ the induced
curves by the contraction $c_{F}$. Observe that we have the following
formulas: $E^{2}=0$,$E\cdot M_{d+1}=1$ and $M_{d+1}^{2}=d+1.$ If
$\\{P\\}=c_{\tilde{F}}(\tilde{F})$ then $c_{\tilde{F}}\colon
Bl_{Q}(\Sigma_{d})-\\{\tilde{F}\\}\longrightarrow\Sigma_{d+1}-\\{P\\}$ is an
isomorphism. In particular, as $\Delta_{\mathcal{F}}$ is reduced we have
$\Delta_{c_{\tilde{F}_{*}}\mathcal{G}}$ is reduced and so that we have (iii).
The local verification done above shows that $E\cap M=\\{Q_{2}\\}$ is a
$p$-reduced singularity of $c_{\tilde{F}_{*}}\mathcal{G}$ and the Lemma 6.1
ensures that $K_{\mathcal{G}}\equiv d_{1}E+d_{2}M_{d+1}$. So, we obtains (i).
This finishes the proof of the theorem. ∎

######  Remark 6.2.

Let $d\in\mathbb{Z}_{\geq 0}$ and $F$ be a section of the natural projection
$\pi\colon\Sigma_{d}\longrightarrow\operatorname{\mathbb{P}_{k}^{1}}$ and
$M_{d}$ a section of $\pi$ such that $F\cdot M_{d}=1$ and $M_{d}^{2}=d.$
Define

$S_{d}=\\{(d_{1},d_{2})\in\mathbb{Z}^{2}\mid\mbox{there exists a foliation in
$\Sigma_{d}$ with canonical divisor $K\equiv d_{1}F+d_{2}M_{d}$}\\}.$

Follows from [5, Proposition 3.6] that:

  * •

$S_{0}=\\{(d_{1},d_{2})\in\mathbb{Z}^{2}\mid d_{1},d_{2}\geq
0\\}\cup\\{(-2,0)\\}\cup\\{(0,-2)\\}$,

  * •

If $d>0$: $S_{d}=\\{(d_{1},d_{2})\in\mathbb{Z}^{2}\mid d_{1}\geq-1,d_{2}\geq
0\\}\cup\\{(d,-2)\\}.$

Now, define

$R_{d}=\\{(d_{1},d_{2})\in\mathbb{Z}^{2}\mid\mbox{there exists a foliation in
$\Sigma_{d}$ with reduced $p$-divisor and $K\equiv d_{1}F+d_{2}M_{d}$}\\}.$

Theorem B can be formulated in the following way:

  * •

$R_{0}=S_{0}-S_{0}(p)\cup\\{(-2,0)\\}\cup\\{(0,-2)\\},$

  * •

Se $d>0$: $R_{d}=S_{d}-S_{d}(p)\cup\\{(-1,d)\mid d\in\mathbb{Z}_{\geq
0}\\}\cup\\{(d,-2)\\}$

where $S_{d}(p)=\\{(d_{1},d_{2})\in S_{d}\mid p\mid d_{1}\mbox{ or }p\mid
d_{2}\\}-\\{(0,0)\\}$.

The following proposition shows that for each point $P$ on
$l=\\{(-1,e)\in\mathbb{Z}^{2}\mid e\in\mathbb{Z}_{\geq 0}\\}$ we can find an
example of foliation such that their $p$-divisor has a $p$-factor.

######  Proposition 6.3.

Let $\mathcal{F}$ be a foliation that is not $p$-closed of degree $d$ in
$\operatorname{\mathbb{P}_{k}^{2}}$ and
$Q\in\operatorname{\mathbb{P}_{k}^{2}}-\operatorname{sing}(\mathcal{F})\cup\mathcal{Z}(\Delta_{\mathcal{F}})$.
Let $\mathcal{G}=Bl_{Q}^{*}\mathcal{F}$ the induced foliation in
$Bl_{Q}(\operatorname{\mathbb{P}_{k}^{2}})$. Then, $\mathcal{G}$ is not
$p$-closed and $\Delta_{\mathcal{G}}$ has a $p$-factor. Moreover,
$K_{\mathcal{G}}\equiv-F+dM_{1}.$

###### Proof.

The condition $Q\not\in\operatorname{sing}(\mathcal{F})$ implies that
$l(Q)=0$. So, we have $N_{\mathcal{G}}=Bl_{Q}^{*}N_{\mathcal{F}}$ and
$K_{\mathcal{G}}=Bl_{Q}^{*}K_{\mathcal{F}}+(1-l(Q))E=Bl_{Q}^{*}K_{\mathcal{F}}+E.$
So,
$[\Delta_{\mathcal{G}}]=pK_{\mathcal{G}}+N_{\mathcal{G}}=Bl_{Q}^{*}\Delta_{\mathcal{F}}+pE=\tilde{\Delta}_{\mathcal{F}}+pE$
where the last equality follows from the condition
$Q\not\in\mathcal{Z}(\Delta_{\mathcal{F}})$. In particular,
$\operatorname{ord}_{E}(\Delta_{\mathcal{G}})=p.$ Now, we have
$K_{\mathcal{G}}=Bl_{Q}^{*}K_{\mathcal{F}}+E\equiv(d-1)(E+F)+E\equiv(d-1)F+dE.$
Write $M_{1}\equiv aF+bE$. Since $1=M_{1}\cdot F=b$ we have $M_{1}\equiv aF+E$
and since $1=M_{1}^{2}=(aF+E)^{2}=2a-1$ we obtain $a=1$. So, $M_{1}=F+E$ and
this implies $E=M_{1}-F$. So,
$K_{\mathcal{G}}\equiv(d-1)F+dE\equiv(d-1)F+d(M_{1}-F)\equiv-F+dM_{1}$ which
ends the proof of the proposition. ∎

Acknowledgements. W.Mendson thanks J.V. Pereira for discussions about the
results of this paper and for their remarks about this work. The author also
thanks Eduardo Vital for suggestions and comments on the manuscript and also
thanks Cesar Hilario for comments on the introduction. The author acknowledge
support of CNPq, Faperj, and the Instituto de Matemática Pura e Aplica (IMPA).

## References

  * [1] M. Brunella, Birational geometry of foliations, vol. 1 of IMPA Monographs, Springer, 2015.
  * [2] M. Brunella and M. Nicolau, Sur les hypersurfaces solutions des équations de pfaff, Comptes Rendus De L Academie Des Sciences Serie I-mathematique, 329 (1999), pp. 793–795.
  * [3] M. M. Carnicer, The Poincaré problem in the nondicritical case, Ann. of Math. (2), 140 (1994), pp. 289–294.
  * [4] D. Cerveau, A. Lins-Neto, F. Loray, J. V. Pereira, and F. Touzet, Complex codimension one singular foliations and Godbillon-Vey sequences, Mosc. Math. J., 7 (2007), pp. 21–54, 166.
  * [5] C. Galindo, F. Monserrat, and J. Olivares, Foliations with isolated singularities on Hirzebruch surfaces, arXiv preprint arXiv:2007.10071, (2020).
  * [6] R. Hartshorne, Algebraic geometry, vol. 52, Springer Science & Business Media, 1977.
  * [7] J. P. Jouanolou, Équations de Pfaff algébriques, vol. 708 of Lecture Notes in Mathematics, Springer, Berlin, 1979.
  * [8] N. M. Katz, Nilpotent connections and the monodromy theorem: Applications of a result of Turrittin, Publications mathématiques de l’IHES, 39 (1970), pp. 175–232.
  * [9] J. Kollár, Rational curves on algebraic varieties, vol. 32 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., Springer-Verlag, Berlin, 1996.
  * [10] S. Lang, Algebra, vol. 211 of Graduate Texts in Mathematics, Springer-Verlag, New York, third ed., 2002.
  * [11] R. Lizarbe, On the density of foliations without algebraic solutions on weighted projective planes, arXiv preprint arXiv:2106.12494, (2021).
  * [12] J.-F. Mattei and R. Moussu, Holonomie et intégrales premieres, Annales Scientifiques de l’École Normale Supérieure, 13 (1980), pp. 469–523.
  * [13] M. McQuillan, Curves on surfaces of mixed characteristic, Eur. J. Math., 3 (2017), pp. 433–470.
  * [14] W. Mendson, Folheações de codimensão um em característica positiva e aplicações, PhD thesis, Instituto de Matemática Pura e Aplicada, 2022.
  * [15] J. V. Pereira, Invariant hypersurfaces for positive characteristic vector fields, J. Pure Appl. Algebra, 171 (2002), pp. 295–301.

Instituto de Matemática Pura e Aplicada, Estrada Dona Castorina 110, 22460-320
Rio de Janeiro RJ, Brazil

E-mail address<EMAIL_ADDRESS>