# The singular locus of $q$-logarithmic foliations

Ariel Molinuevo The author was fully supported by Universidade Federal do Rio
de Janeiro, Brasil. Instituto de Matemática, Universidade Federal do Rio de
Janeiro, Brazil. Federico Quallbrunn The author was fully supported by
CONICET, Argentina. Departamento de Matemática, Universidad CAECE, Argentina.

###### Abstract

We determine the structure of the singular locus of generic codimension-$q$
logarithmic foliations and its relation with the unfoldings of these
foliations. We calculate the ideal defining the scheme of persistent
singularities.

##  1 Introduction

The study of holomorphic foliations on algebraic varieties has been an active
area of research in mathematics for several decades. We refer to the book
[Scá21] for a review of fundamental results and open problems in the subject.
Singularities of foliations are of particular interest as they provide
valuable insight into the dynamics and geometry of the foliation and how these
change under deformation.

Starting from the seminal work of Calvo-Andrade in [CA94] until recent years,
there has been significant progress in the study of foliations defined by
logarithmic forms. In [CA94], Calvo-Andrade proves that the set of codimension
$1$ foliations defined by logarithmic $1$-forms in a compact holomorphic
variety $X$ with $H^{1}(X,\mathbb{C})=0$ is stable under deformation. In the
case where $X=\mathbb{P}^{n}_{\mathbb{C}}$ this result implies that foliations
defined by forms of the type

$\omega=\left(\prod_{j=1}^{r}F_{j}\right)\sum_{i=1}^{r}\lambda_{i}\frac{dF_{i}}{F_{i}},\quad\lambda_{j}\in\mathbb{C},\
F_{j}\in\mathbb{C}[x_{0},\dots,x_{n}],$

$F_{j}$ homogeneous, $\deg F_{j}=d_{j}$, define an irreducible component
$\mathcal{L}(d_{1},\dots,d_{r})$ of the moduli space of foliations of
codimension $1$ in $\mathbb{P}^{n}$. Recently, in [CGAM19], a new proof of
this result is given using deformation theory. Also in recent years, some
progress have been made in the classification of codimension $q$ foliations.
In particular, Cerveau and Lins-Neto in [CN21] have proven that foliations
defined by logarithmic $q$-forms of the type

$\omega=\left(\prod_{j=1}^{r}F_{j}\right)\left[\sum_{\begin{subarray}{c}I\subset\\{1,\dots,r\\}\\\
|I|=q\end{subarray}}\lambda_{I}\left(\bigwedge_{i\in
I}\frac{dF_{i}}{F_{i}}\right)\right],\quad\lambda_{I}\in\mathbb{C},\
F_{j}\in\mathbb{C}[x_{0},\dots,x_{n}],$

define irreducible components of the moduli space of codimension $q$
foliations on $\mathbb{P}^{n}$, and Gargiulo in [GA20] gives an independent
proof of the case $q=2$ using deformation theory.

This article deals with foliations defined by logarithmic forms such as the
ones above. Instead of focusing on the stability of such forms under
deformation, we deal with their singular locus and their unfoldings,
generalizing the main result of [CSV06] to the case of arbitrary codimension,
and applying this results to the characterization of the unfoldings of these
foliations, as we will explain now.

Unfoldings of singular foliations, and particularly _first order unfoldings_
(see Definition 2.6 below) were introduced independently by Suwa and Mattei
(see [Suw83a, Mat91] and the references within). Although the original
definition of Suwa was stated in full generality, the focus of these earlier
articles was on germs of codimension-1 foliations around a singularity.

In later years, the work of Suwa has been expanded in several articles, see
[MMQ18, MMQ21] and the references within. In this paper we will apply the
results from [MMQ21] to study the singular locus of a $q$-logarithmic
foliation. That paper is a generalization of [MMQ18]. Since we will refer
several times to both articles, we will briefly explain here what they are
about.

In [MMQ18] we studied first order unfoldings and we defined the _Kupka scheme_
of a codimension-1 foliation on $\mathbb{P}^{n}$.

A codimension-1 foliation in $\mathbb{P}^{n}$ can be given by a global twisted
differential form $\omega\in
H^{0}(\mathbb{P}^{n},\Omega^{1}_{\mathbb{P}^{n}}(e))$, for some $e\geq 2$
defining a line bundle, such that $\omega$ verifies the _Frobenius
integrability condition_ :

$\omega\wedge d\omega=0\ ,$

see Remark 2.2.

A first order unfolding of a foliation defined by $\omega$ is a particular
type of first order deformation of $\omega$. While a (first order) deformation
of $\omega$ is given by a family of differential forms $\omega_{\varepsilon}$
parameterized by an infinitesimal parameter $\varepsilon$ such that
$\omega_{\varepsilon}$ is integrable for every fixed value of $\varepsilon$
and coincides with $\omega$ in the origin, a (first order) unfolding
$\widetilde{\omega}_{\varepsilon}$ is given by a foliation in an infinitesimal
neighbourhood of $\mathbb{P}^{n}$ such that it restricts to $\omega$ in the
central fiber.

In the codimension $1$ case, first order unfoldings define a sheaf isomorphic
to a sheaf of ideals, which is called the _unfolding ideal_ , $I(\omega)$, see
Remark 2.7. If we define as $J(\omega)$ the ideal defined by the coefficients
of $\omega$, we have that the variety defined by $J(\omega)$ is the singular
locus of $\omega$, $\mathrm{Sing}(\omega)$, see Remark 2.5. It can be also
easily seen that $J(\omega)\subset I(\omega)$. So the first order unfoldings
define a subvariety of $\mathrm{Sing}(\omega)$. We refer to [Mol16] for
details on this topic.

In the literature, see [Kup64], [DM77, Fundamental Lemma, p. 406] or [GMLN91,
Definition 2.1], the _Kupka set_ of a codimension-1 foliation $\omega$ is the
set of points such that $\omega(p)=0$ and $d\omega(p)\neq 0$. To work in an
algebraic setting first we needed to associate a scheme structure to that set.
The way for doing that was defining the Kupka scheme as the schematic support
of $d\omega$ inside $\mathrm{Sing}(\omega)$.

For a more formal definition, let us fix here
$S=\mathbb{C}[x_{0},\ldots,x_{n}]$ the homogeneous coordinate ring of
$\mathbb{P}^{n}$. We defined the Kupka scheme, $\mathpzc{Kup}(\omega)$, see
[MMQ18, Definition 4.1], as $\mathpzc{Kup}(\omega)=Proj(S/K(\omega))$ where
$K(\omega)$ is the homogeneous ideal defined as

$K(\omega)=ann(\overline{d\omega})+J(\omega),\ \
\overline{d\omega}\in\Omega^{2}_{S}\otimes S/J(\omega)\ .$

We also stated the following relation between the ideal of first order
unfoldings and the Kupka ideal, see [MMQ18, Theorem 4.12] for a complete
statement, for $\omega$ a _generic_ codimension-1 foliation in
$\mathbb{P}^{n}$ we have:

$\sqrt{I(\omega)}=\sqrt{K(\omega)}\ .$ (1)

Despite the fact that a first order unfolding of a codimension-1 foliation
identifies with an ideal, in higher codimension there was no analog to that
fact. In [MMQ21] we defined what we called the _subscheme of persistent
singularities_ , $\mathpzc{Per}(\omega)$, addressing that particular
situation, where $\omega$ defines a codimension-$q$ foliation, see Section 2
for a proper definition. We refer to Definition 2.10 for the formal definition
of $\mathpzc{Per}(\omega)$, but for now we can state an equivalent definition,
see Proposition 2.11, as:

$\mathfrak{p}\in\mathpzc{Per}(\omega)\iff\mathfrak{p}\text{ is a singular
point of every unfolding of }\omega\ .$

We also gave a definition of the Kupka scheme for a codimension-$q$ foliation,
see Definition 2.12.

In [CSV06, Theorem 3] the authors show that the singular locus of a generic
codimension-1 logarithmic foliation in $\mathbb{P}^{n}$ is given by a
codimension 2 variety (which is the Kupka variety) and a set of isolated
points. In this paper we extend that result showing that the singular locus of
a _generic_ codimension-$q$ logarithmic foliation, see Remark 4.4 for the
generic conditions, in a smooth variety $X$ is given by two varieties
$\mathpzc{Kup}(\omega)$ and $\mathcal{H}$ where $\mathpzc{Kup}(\omega)$ has
codimension $(q+1)$ and $\mathcal{H}$ has dimension $(q-1)$ or
$\mathcal{H}=\emptyset$, see Theorem 4.16:

######  Theorem 1.

Let $\omega\in H^{0}(X,\Omega^{q}_{X}\otimes\mathcal{L})$ be a generic,
locally decomposable, $q$-logarithmic foliation. Then we can decompose its
singular locus $\mathrm{Sing}(\omega)$ as the disjoint union of

$\mathrm{Sing}(\omega)=\mathpzc{Kup}(\omega)\sqcup\mathcal{H},$

where $\mathpzc{Kup}(\omega)$ is the Kupka variety of $\varpi$ of codim $q+1$
and $\mathcal{H}$ is a variety of dimension $q-1$ or $\mathcal{H}=\emptyset$.
Even more so, we have that $\mathpzc{Kup}(\omega)$ coincides with the scheme
of persistent singularities $\mathpzc{Per}(\omega)$.

This article is organized as follows:

In section Section 2 we give the definitions of first order unfolding,
persistent singularities and of the Kupka scheme all of them for a
codimension-$q$ foliation on a smooth variety $X$.

In section Section 3 we recall from [Del06] the definition of logarithmic
differential form and of the residue of such form.

Finally in Section 4 we define $q$-logarithmic foliations and we use the
relation between Kupka singularities and persistent singularities to establish
in Theorem 4.16 the dimension of the irreducible components of the singular
set of a generic logarithmic foliation on $X$, as well as determining the
equality between the schemes of Kupka singularities and persistent
singularities.

Then in Corollary 4.17 we use the result stated above to compute the ideal of
persistent singularities.

These results are of particular interest as not much is known in general about
the singular set of a higher codimension foliation, so logarithmic foliations
represent an interesting first example where the dimension of the irreducible
components of the singular set as well as the scheme of persistent
singularities are defined.

Acknowledgments We would like express our gratitude to Alicia Dickenstein and
Jorge Vitorio Pereira for their valuable suggestions and contributions.

##  2 Codimension-$q$ foliations

Along this section we give the definition of _codimension- $q$ foliation_ on a
smooth variety $X$. Then we recall the definition of unfolding of a
codimension-$q$ foliation on a variety $X$. Finally we review the definitions
of persistent singularities and of Kupka singularities for codimension-$q$
foliations, see Definition 2.10 and Definition 2.12, respectively, and we
state some general facts regarding the ideal of persistent singularities and
the Kupka scheme. We also add remarks recalling the analogous definitions for
a codimension-1 foliation in $\mathbb{P}^{n}$ for the sake of clarity.

We refer the reader to [MMQ21] for a complete treatment of the subject.

Let us fix a smooth variety as $X$. If $\Xi\in\Gamma(U,\bigwedge^{p}TX)$ is a
multivector and $\varpi\in\Gamma(U,\Omega^{q}_{X})$ a $q$-form we will denote
by $i_{\Xi}\varpi\in\Gamma(U,\Omega^{q-p}_{X})$ the contraction. Recall from
[DM00, Proposition 1.2.1] that the _Plücker relations_ for $\varpi$ are given
by

$i_{\Xi}\varpi\wedge\varpi=0$

for any $\Xi\in\bigwedge^{q-1}TX$. Which is equivalent to: if
$\varpi(\mathfrak{p})\neq 0$ for some closed point $\mathfrak{p}\in X$ then
$\varpi$ is _locally decomposable_ as a product
$\varpi=\varpi_{1}\wedge\dots\wedge\varpi_{q}$ of $q$ 1-forms at a
neighborhood of $\mathfrak{p}$. We will call a differential form $\varpi$
locally decomposable if it is locally decomposable at every closed point
$\mathfrak{p}\in X$.

######  Definition 2.1.

Let $\mathcal{L}$ be a line bundle and $\omega:\mathcal{L}\to\Omega^{q}_{X}$,
with $1\leq q\leq dim(X)-1$, be a (non trivial) morphism of sheaves, we will
say that the morphism is _integrable_ if

  * a)

For every local section $s$ of $\mathcal{L}$ and $\Xi$ of $\bigwedge^{q-1}TX$,
$\omega(s)$ verifies

$d(i_{\Xi}\omega(s))\wedge\omega(s)=0.$ (2)

  * b)

$\Omega^{q}_{X}/\mathcal{L}$ is torsion free.

  * c)

The map

$i_{\Xi}\omega\wedge\omega:\mathcal{L}\to\Omega^{q+1}_{X}\otimes\mathcal{L}^{-1}$

is zero for every local section $\Xi$ of $\bigwedge^{q-1}TX$.

We also say that $\omega$ determines a _codimension- $q$ foliation_.

Let $\omega:\mathcal{L}\to\Omega^{q}$ be a integrable morphism. Then, we can
consider two maps,

$\textstyle{\bigwedge^{q-1}TX\otimes\mathcal{L}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{(-)}\omega}$$\textstyle{\Omega^{1}_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\omega\wedge-}$$\textstyle{\Omega_{X}^{q+1}\otimes\mathcal{L}^{-1}}$

Condition c) on $\omega$ implies that this diagram is a complex and it is easy
to check that its homology is supported over the points where $\omega$ is not
decomposable.

######  Remark 2.2.

In the case of the projective space $\mathbb{P}^{n}$ a codimension-1 foliation
can be given by a global twisted differential form $\omega\in
H^{0}(\mathbb{P}^{n},\Omega^{1}_{\mathbb{P}^{n}}(e))$ such that it verifies
the Frobenius integrability condition

$\omega\wedge d\omega=0\ ,$

see [CSV06, Mol16, MMQ18]. Or, more similar to Definition 2.1, by a morphism
$\omega:\mathcal{O}_{\mathbb{P}^{n}}(-e)\to\Omega^{1}_{\mathbb{P}^{n}}$.

The integrability condition is equivalent to item a) of Definition 2.1. The
item b) of the definition is equivalent to ask the generic property of having
singular locus of codimension $\geq 2$. There is no equivalent for the item c)
in the case of codim-1.

Let us denote the ring of homogeneous coordinates of $\mathbb{P}^{n}$ by
$S=\mathbb{C}[x_{0},\ldots,x_{n}]$ and the homogeneous component of degree
$\ell$ of a graded $S$-module $M$ by $M(\ell)$. A global form, $\omega\in
H^{0}(\mathbb{P}^{n},\Omega^{1}_{\mathbb{P}^{n}}(e))$, can be written as

$\omega=\sum_{i=0}^{n}A_{i}dx_{i}\ ,$

where the $A_{i}\in S(e-1)$ are homogeneous polynomials of degree $e-1$. The
condition of descent of $\omega$ to $\mathbb{P}^{n}$ is given by the
annihilation of the contraction with the radial field
$R=\sum_{i=0}^{n}x_{i}\frac{\partial}{\partial x_{i}}$, _i.e._ ,

$i_{R}(\omega)=\sum_{i=0}^{n}A_{i}x_{i}=0\ .$

######  Definition 2.3.

We define the _conormal sheaf associated to the foliation $\omega$_, denoted
$\mathcal{E}=\mathcal{E}(\omega)$, as the kernel of
$\omega\wedge-:\Omega^{1}_{X}\to\mathcal{L}^{-1}\otimes\Omega^{q+1}_{X}$.

Notice that by [Har80, Proposition 1.1] we get that $\mathcal{E}$ is a
reflexive sheaf.

Composing a morphism $\omega:\mathcal{L}\to\Omega^{q}$ with the contraction of
forms with vector fields gives us a morphism

$\bigwedge^{q}TX\otimes\mathcal{L}\to\mathcal{O}_{X}.$

######  Definition 2.4.

The ideal sheaf $\mathscr{J}(\omega)$ is defined to be the sheaf-theoretic
image of the morphism $\bigwedge^{q}TX\otimes\mathcal{L}\to\mathcal{O}_{X}$.
The subscheme it defines is called _the singular scheme_ of $\omega$ and
denoted $\mathrm{Sing}(\omega)\subseteq X$. We will denote it just by
$\mathscr{J}$ if no confusion arises.

######  Remark 2.5.

In the case of a codimension-1 foliation in the projective space
$\mathbb{P}^{n}$ we said in Remark 2.2 that $\omega$ can be written as
$\omega=\sum_{i=0}^{n}A_{i}dx_{i}$ for some homogeneous polynomials $A_{i}\in
S(e-1)$. In this situation the definition above coincides with the ideal
$J(\omega)=(A_{0},\dots,A_{n})$ and the variety defined by it.

From [Suw83a][(4.6) Definition] we get the following definition for a
codimension-$q$ foliation:

######  Definition 2.6.

Let $T$ be a scheme, $\mathfrak{p}\in T$ a closed point, and
$\mathcal{L}\xrightarrow{\omega}\Omega^{q}_{X}$ a codimension-$q$ foliation on
$X$. An _unfolding_ of $\omega$ is a codimension-$q$ foliation
$\widetilde{\mathcal{L}}\xrightarrow{\widetilde{\omega}}\Omega^{q}_{X\times
T}$ on $X\times T$ such that
$\widetilde{\omega}|_{X\times\\{\mathfrak{p}\\}}\cong\omega$. In the case
$T=\mathrm{Spec}(\mathbb{C}[\varepsilon]/(\varepsilon^{2}))$ we will call
$\widetilde{\omega}$ a _first order unfolding_.

Let $\mathcal{D}=\mathrm{Spec}(\mathbb{C}[\varepsilon]/(\varepsilon^{2}))$ be
the scheme of _dual numbers_ over $\mathbb{C}$, $0\in\mathcal{D}$ be its
closed point, $\pi:X\times\mathcal{D}\to\mathcal{D}$ be the projection and
$\iota:X\cong X\times\\{0\\}\hookrightarrow X\times\mathcal{D}$ be the
inclusion. Then the sheaf $\Omega^{q}_{X\times\mathcal{D}}$ can be decomposed
as a direct sum of $\iota_{*}(\mathcal{O}_{X})$-modules as

$\Omega^{q}_{X\times\mathcal{D}}\cong\iota_{*}\Omega^{q}_{X}\oplus\varepsilon\cdot(\iota_{*}\Omega^{q}_{X})\oplus\iota_{*}\Omega^{q-1}_{X}\wedge
d\varepsilon.$

Given a codimension-$q$ foliation determined by an integrable morphism
$\mathcal{L}\xrightarrow{\omega}\Omega^{q}_{X}$, and a first order unfolding
$\widetilde{\omega}:\widetilde{\mathcal{L}}\to\Omega^{q}_{X\times\mathcal{D}}$
of $\omega$, we take local generators $\varpi$ of $\mathcal{L}(U)$ and
$\widetilde{\varpi}$ of $\widetilde{\mathcal{L}}(U\times\mathcal{D})$. Suppose
$\omega$ and $\widetilde{\omega}$ are locally decomposable, then we may take
$U$ small enough such that $\varpi$ and $\widetilde{\varpi}$ decompose as
products

$\varpi=\varpi_{1}\wedge\dots\wedge\varpi_{q},\qquad\widetilde{\varpi}=\widetilde{\varpi}_{1}\wedge\dots\wedge\widetilde{\varpi}_{q}.$

Then we can write
$\widetilde{\varpi}_{i}=\varpi_{i}+\varepsilon\eta_{i}+h_{i}d\varepsilon$ and
the equations $d\widetilde{\varpi}_{i}\wedge\widetilde{\varpi}=0$ for
$i=1,\dots,q$ are equivalent to the equations

$\left\\{\begin{aligned}
&d\eta_{i}\wedge\varpi+d\varpi_{i}\wedge\left(\sum_{j=1}^{q}(-1)^{j}\eta_{j}\wedge\varpi_{\widehat{j}}\right)=0,\qquad(i=1,\dots,q),\\\
&(dh_{i}-\eta_{i})\wedge\varpi+d\varpi_{i}\wedge\left(\sum_{j=1}^{q}(-1)^{j}h_{j}\varpi_{\widehat{j}}\right)=0,\qquad(i=1,\dots,q),\end{aligned}\right.$
(3)

where
$\varpi_{\widehat{j}}=\varpi_{1}\wedge\dots\wedge\varpi_{j-1}\wedge\varpi_{j+1}\wedge\dots\wedge\varpi_{q}\in\Omega^{q-1}_{X}(U)$.

As is shown in [Suw83a, proof of 6.1 Theorem] the second equation implies the
first. So we finally get that the equations
$d\widetilde{\varpi}_{i}\wedge\widetilde{\varpi}=0$ for $i=1,\dots,q$ are
equivalent to

$\displaystyle\left\\{(dh_{i}-\eta_{i})\wedge\varpi+d\varpi_{i}\wedge\left(\sum_{j=1}^{q}(-1)^{j}h_{j}\varpi_{\widehat{j}}\right)=0,\qquad(i=1,\dots,q)\
.\right.$ (4)

######  Remark 2.7.

In the case of a codimension-1 foliation in the projective space
$\mathbb{P}^{n}$ a first order unfolding of a foliation defined by an
integrable global twisted differential form $\omega\in
H^{0}(\mathbb{P}^{n},\Omega^{1}_{\mathbb{P}^{n}}(e))$ is just an integrable
twisted differential form $\widetilde{\omega}\in
H^{0}(\mathbb{P}^{n}\times\mathcal{D},\Omega^{1}_{\mathbb{P}^{n}\times\mathcal{D}|\mathbb{C}}(e))$
that can be written as

$\widetilde{\omega}=\omega+\varepsilon\eta+hd\varepsilon\ .$

The integrability condition on $\widetilde{\omega}$ gives the equations

$\left\\{\begin{aligned} &\omega\wedge d\eta+d\omega\wedge\eta=0\\\
&hd\omega=\omega\wedge(\eta-dh)\end{aligned}\right.$

in an analog way to Eq. 3. By the same argument of Eq. 4 they are equivalent
to just the second equation

$\left\\{\begin{aligned} hd\omega=\omega\wedge(\eta-dh)\
.\end{aligned}\right.$

Then, we can parameterize first order unfoldings as

$U(\omega)=\\{(h,\eta)\in
H^{0}(\mathbb{P}^{n},\mathcal{O}_{\mathbb{P}^{n}}(e))\times
H^{0}(\mathbb{P}^{n},\Omega^{1}_{\mathbb{P}^{n}}(e)):hd\omega=\omega\wedge(\eta-
dh)\ \\}\big{/}(0,\omega)\ ,$

where we are quotienting by $(0,\omega)$ to eliminate the trivial deformation
induced by $\omega$, _i.e._ ,
$\widetilde{\omega}=\omega+\varepsilon\omega=(1+\varepsilon)\ \omega$. Since
$\varepsilon^{2}=0$ we have that $1+\varepsilon$ is invertible, implying that
$\omega$ and $(1+\varepsilon)\ \omega$ define the same foliation.

Let us denote as $(\mathbb{C}^{n},0)$ the infinitesimal neighbourhood around
the origin of $\mathbb{C}^{n}$, $\mathcal{O}_{(\mathbb{C}^{n},0)}$ the ring of
germs of analytic functions and $\Omega^{1}_{(\mathbb{C}^{n},0)}$ the germs of
differential forms and consider $\omega_{0}\in\Omega^{1}_{(\mathbb{C}^{n},0)}$
a germ of an integrable differential form. Regarding the equations above we
get all the same setting, _i.e._ ,

$U_{(\mathbb{C}^{n},0)}(\omega_{0})=\\{(h,\eta)\in\mathcal{O}_{(\mathbb{C}^{n},0)}\times\Omega^{1}_{(\mathbb{C}^{n},0)}:hd\omega_{0}=\omega_{0}\wedge(\eta-
dh)\\}\big{/}(0,\omega_{0})\ ,$

the only (and main) difference is that the image of the projection to the
first coordinate

$\textstyle{U_{(\mathbb{C}^{n},0)}(\omega_{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{1}}$$\textstyle{\mathcal{O}_{(\mathbb{C}^{n},0)}}$$\textstyle{(h,\eta)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{h}$

defines an ideal of the ring of germs of analytic functions
$\mathcal{O}_{(\mathbb{C}^{n},0)}$, $Im(\pi_{1}(\omega_{0}))=I(\omega_{0})$,
instead of a finite vector space in the algebraic case. If we ask $\omega_{0}$
to have singular locus of codimension $\geq 2$, it can be easily seen that

$I(\omega_{0})\simeq U(\omega_{0})\ .$

The ideal $I(\omega_{0})$ defines the unfolding ideal on $(\mathbb{C}^{n},0)$
or, in our terms, the ideal defining the variety of persistent singularities.
For more details on the local setting we refer to [Suw83a] and for the global
algebraic setting to [Mol16].

Let $\mathcal{L}\xrightarrow{\omega}\Omega^{q}_{X}$ be an integrable morphism
inducing a morphism $\mathcal{E}\to\Omega^{1}_{X}$. Composing $\omega$ with
wedge product gives a morphism
$\Omega^{2}_{X}\xrightarrow{\omega\wedge-}\mathcal{L}^{-1}\otimes\Omega^{q+2}_{X}$.
As $\omega$ is integrable the sheaf $\mathcal{E}\otimes\Omega^{1}_{X}$ is in
the kernel of $\omega\wedge-$.

######  Definition 2.8.

Following the notation above, we define the sheaf $H^{2}(\omega)$ as

$H^{2}(\omega):=\ker(\omega\wedge-)/\mathcal{E}\otimes\Omega^{1}_{X}.$

######  Remark 2.9.

The restriction of the de Rham differential to $\mathcal{E}$ gives a sheaf map
$\mathcal{E}\to\Omega^{2}_{X}$ which is not $\mathcal{O}_{X}$-linear but whose
image is in $\ker(\omega\wedge-)$ as $\omega$ is integrable. The projection of
this map to $H^{2}(\omega)$ is however $\mathcal{O}_{X}$-linear as
$dg\varpi\cong gd\varpi\mod\mathcal{E}\otimes\Omega^{1}_{X}$ for every local
section $\varpi$ of $\mathcal{E}$.

Let us denote $\mathcal{L}\xrightarrow{\omega}\Omega^{q}_{X}$ to be an
integrable morphism determining a subsheaf $\mathcal{E}\to\Omega^{1}_{X}$.
Then we have the following definitions:

######  Definition 2.10.

The subscheme of _persistent singularities_ of $\omega$ is defined by the
ideal sheaf $\mathscr{I}(\omega)$ to be the annihilator of $d(\mathcal{E})$ in
$H^{2}(\omega)$. In other words the local sections of $\mathscr{I}(\omega)$ in
an open set $U\subseteq X$ are given by

$\mathscr{I}(\omega)(U)=\\{h\in\mathcal{O}_{X}(U)\ :\
\forall\varpi\in\mathcal{E}(U),\
hd\varpi=\sum_{j}\alpha_{j}\wedge\omega_{j}\\},$

for some local 1-forms $\alpha_{j}$ and 1-forms $\omega_{j}$ in
$\mathcal{E}(U)$. We will denote it just as $\mathscr{I}$ if no confusion
arises.

We also define $\mathpzc{Per}(\omega)$ as the subvariety (subscheme) of $X$
associated to the ideal sheaf $\mathscr{I}(\omega)$, _i.e._ ,

$\mathpzc{Per}(\omega)=\mathcal{O}_{X}/\mathscr{I}(\omega).$

For the case $q=1$ in Eq. 4, we recall the following proposition from [MMQ21,
Proposition 3.6] which gives the name to the ideal of persistent
singularities:

######  Proposition 2.11.

Let $\mathfrak{p}\in X$ be a point in $\mathrm{Sing}(\omega)$,
$\mathcal{O}_{X,\mathfrak{p}}$ the local ring around $\mathfrak{p}$, and
$X_{\mathfrak{p}}=\mathrm{Spec}(\mathcal{O}_{X,\mathfrak{p}})$. Then
$\mathfrak{p}$ is in the subscheme of persistent singularities if and only if
for any first order unfolding $\widetilde{\omega}$ of $\omega$ in
$X_{\mathfrak{p}}$, the point $(\mathfrak{p},0)\in
X_{\mathfrak{p}}\times\mathcal{D}$ is a singular point of
$\widetilde{\omega}$.

######  Definition 2.12.

The subscheme of _Kupka singularities_ of $\omega$ is the one defined by the
ideal sheaf $\mathcal{K}(\omega)=\mathrm{ann}(\\{d\omega\\})$ where
$\\{d\omega\\}$ is the class of
$d\omega\in\Omega^{q+1}_{X}\otimes\mathcal{O}_{\mathrm{Sing}(\omega)}\otimes\mathcal{L}^{-1}$.
We will denote it just as $\mathcal{K}$ if no confusion arises.

We also define as $\mathpzc{Kup}(\omega)$ the subvariety (subscheme) of $X$
associated to the variety defined by $\mathcal{K}(\omega)$, _i.e._ ,

$\mathpzc{Kup}(\omega)=\mathcal{O}_{X}/\mathcal{K}(\omega).$

Let $\mathfrak{p}\in\mathpzc{Kup}(\omega)$. We will call $\mathfrak{p}$ a
_strictly Kupka point_ if $d\omega|_{\mathfrak{p}}\neq 0$, meaning that the
evaluation of $d\omega$ at the point $\mathfrak{p}$ is different from zero.

######  Remark 2.13.

Again, in the case of a codimension-1 foliation in the projective space
$\mathbb{P}^{n}$ we can also define the ideal $\mathcal{K}(\omega)$ as the
quotient ideal defined as
$(\mathscr{J}(\omega).\Omega^{2}_{\mathbb{C}[x_{0},\ldots,x_{n}]|\mathbb{C}}:d\omega)$.
Also, the reduced structure associated to $\mathcal{K}(\omega)$ differs from
the variety defined as the closure of the strictly Kupka points, as it is
explained in [MMQ18, p. 1034].

We will need the following proposition, also from [MMQ21, Proposition 4.9],
for later use.

######  Proposition 2.14.

Given an integrable morphism $\mathcal{L}\xrightarrow{\omega}\Omega^{q}_{X}$
we have the inclusions $\mathpzc{Per}(\omega)\subset\mathrm{Sing}(\omega)$ and
$\mathpzc{Kup}(\omega)\subset\mathrm{Sing}(\omega)$. If moreover $\omega$ is
locally decomposable (_i.e._ if $\mathcal{E}$ is locally free) then we have
$\mathpzc{Kup}(\omega)\subset\mathpzc{Per}(\omega)\subset\mathrm{Sing}(\omega)$.

##  3 Differential forms with logarithmics poles

Along this section recall the definitions of $q$-logarithmic differential
forms and of residue of a differential form due to Deligne. For that we will
follow [Del06, 3. Pôles logarithmiques].

We will refer to the set $\\{1,\ldots,s\\}$ by $[s]$ and by $|I|$ to the
cardinality of the (sub) set $I\subset[s]$.

######  Definition 3.1.

Let us consider $D$ a simple simple normal crossing divisor and let
$j:X\backslash D\to X$ be the inclusion. We define the _sheaf of logarithmic
differential forms along $D$_ as the smallest sheaf $\Omega^{*}_{X}(\log D)$
of $j_{*}\Omega^{*}_{X\backslash D}$, stable by wedge products and such that
$\frac{df}{f}$ is a local section of $\Omega^{1}_{X}(\log D)$ when $f$ is a
local section of $j_{*}\mathcal{O}^{*}_{X}$, meromorphic along $D$.

By [Del06, Proposition 3.2] we know that $\Omega^{1}_{X}(\log D)$ is locally
free and

$\Omega^{q}_{X}(\log D)=\bigwedge^{q}\Omega^{1}_{X}(\log D)\ .$

Remember that given a locally free sheaf $\mathcal{E}$ and a global section
given by a morphism $\mathcal{O}_{X}\xrightarrow{s}\mathcal{E}$ the _zero
locus_ of $s$ is the subscheme defined by the sheaf of ideals defined by the
image of the dual morphism
$\mathcal{E}^{\vee}\xrightarrow{s^{\vee}}\mathcal{O}_{X}$.

######  Notation 3.2.

We denote the zero locus of a section $\omega\in H^{0}(X,\Omega^{q}_{X}(\log
D))$ by $Z(\omega)$.

Let us consider $D=\left(\bigcup_{i=1}^{s}D_{i}\right)$, where each $D_{i}$ is
an irreducible component of the divisor $D$ and
$\widetilde{D}^{r}=\bigsqcup_{\\{I\subset[s],\ |I|=r\\}}\bigcap_{i\in
I}D_{i}$, for $r\geq 1$. We will ask the $D_{i}$ to be smooth. In the case
$r=1$ we will skip the superindex in $\widetilde{D}^{1}$ and write just
$\widetilde{D}$ instead.

In [Del06, p. 77] there is the definition of _résidu de Poincaré_ as a map

$Res:Gr_{n}^{W}(\Omega^{q}_{X}(\log
D))\to\Omega^{q}_{\widetilde{D}^{n}}(\epsilon^{n})[-n]\ ,$

where the $\epsilon^{n}$ stands for choosing an ordering on the subsets of
$\widetilde{D}^{n}$, _i.e._ on the $I\subset[s]$, as it is explained in
[Del06, 3.4, p. 75]. In the following we will fix an ordering and remove the
$\epsilon^{n}$.

Taking $n=q$ we get a map

$Res:Gr_{q}^{W}(\Omega^{q}_{X}(\log
D))\to\Omega^{q}_{\widetilde{D}^{q}}[-q]=\mathcal{O}_{\widetilde{D}^{q}}\ ,$

Then we can define the _residue of a $q$-differential form_ which consists of
the extension by linearity the following application:

$\textstyle{\Omega^{q}_{X}(\log
D)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{Res}^{q}}$$\textstyle{\mathcal{O}_{\widetilde{D}^{q}}}$$\textstyle{\frac{df_{i_{1}}}{f_{i_{1}}}\wedge\dots\wedge\frac{df_{i_{q}}}{f_{i_{q}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\in\mathcal{O}_{\widetilde{D}^{q}}}$$\textstyle{\omega\
}$$\textstyle{\ 0,\text{ if }\omega\in\Omega^{q}_{X}}$

where $i_{1}<\ldots<i_{q}$. If $\omega\in H^{0}(X,\Omega^{q}_{X}(\log D))$ we
have

$\mathrm{Res}^{q}(\omega)\in
H^{0}(X,\mathcal{O}_{\widetilde{D}^{q}})=\bigoplus_{\begin{subarray}{c}I\subset[s]\\\
|I|=q,\ 1\in I\end{subarray}}H^{0}(X,\mathcal{O}_{\bigcap_{i\in I}D_{i}}).$

We can also define a residue application $R:\Omega^{q}_{X}(\log
D)\to\Omega^{q-1}_{\widetilde{D}}(\log D^{\prime})$ in the following way:
first consider $D^{\prime}$ a divisor of $\widetilde{D}$ defined by

$D^{\prime}=\bigsqcup_{i=1}^{s}\widetilde{D}|_{D_{i}}=\bigsqcup_{\begin{subarray}{c}j\neq
i\\\ j=1\end{subarray}}^{s}\left(D_{i}\bigcap D_{j}\right),$

then we define $R$ as the map obtained by extending linearly the following,

$\textstyle{\Omega^{q}_{X}(\log
D)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{R}$$\textstyle{\Omega^{q-1}_{\widetilde{D}}(\log
D^{\prime})}$$\textstyle{\frac{df_{i}}{f_{i}}\wedge\eta\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\eta|_{D_{i}},}$
(5)

where we would like to recall that we have fixed an ordering for the subsets
of $D^{\prime}$.

Notice that $\eta|_{D_{i}}$ will have logarithmic poles in the intersections
of $D_{i}$ with the other components of $D$.

######  Remark 3.3.

Consider $D_{1}$ as an irreducible component of $D$ and $\eta$ a
$q$-logarithmic form in $\Omega^{q}_{X}(\log D)$. The logarithmic $q-1$-form
$\eta|_{D_{1}}$ has as a residue

$\mathrm{Res}^{q-1}(\eta|_{D_{1}})\in\bigoplus_{\begin{subarray}{c}I\subset[s]\\\
|I|=q,\ 1\in I\end{subarray}}H^{0}(X,\mathcal{O}_{\bigcap_{i\in I}D_{i}}).$

Meaning that the residue of $\eta|_{D_{1}}$ consists of those $\lambda_{i}$
that are residues of $\eta$ and such that they correspond to the intersection
of $q$-components in which $D_{1}$ is one of them. In particular, if all the
$\lambda_{i}$ of $\eta$ are $\neq 0$, then the same is true for the residues
of the differential $q-1$-form $R(\eta)$.

######  Proposition 3.4.

Keeping the notation of the residue maps as above, lets consider $\omega\in
H^{0}(X,\Omega^{q}_{X}(\log D))$ and define $\eta$ as $\eta=R(\omega)\in
H^{0}(X,\Omega^{q-1}_{\widetilde{D}}(\log D^{\prime}))$. Then we have that

$Z(\omega)\cap D\subset Z(\eta).$

###### Proof.

Let us take $\mathfrak{p}\in\mathrm{Sing}(\omega)\cap D$, such that
$\mathfrak{p}\in D_{1}$. Locally around $\mathfrak{p}$ we can write

$\omega=\frac{df_{1}}{f_{1}}\wedge\widetilde{\eta}+\widetilde{\omega},$

where $\widetilde{\eta}$ is a $q-1$-differential form in $X$ such that
$\widetilde{\eta}|_{D_{1}}=\eta$ and $\widetilde{\omega}$ is an holomorphic
$q$-differential form. Being $\mathfrak{p}$ a singular point then

$\frac{df_{1}}{f_{1}}\wedge\widetilde{\eta}|_{\mathfrak{p}}+\widetilde{\omega}|_{\mathfrak{p}}=0,$

since $\widetilde{\omega}$ is holomorphic it is linearly independent with
$\frac{df_{1}}{f_{1}}\wedge\widetilde{\eta}$ near $\mathfrak{p}$ implying that

$\frac{df_{1}}{f_{1}}\wedge\widetilde{\eta}|_{\mathfrak{p}}=\widetilde{\omega}|_{\mathfrak{p}}=0.$

Then $\widetilde{\eta}|_{\mathfrak{p}}=0$ because $\frac{df_{1}}{f_{1}}$ is a
local generator of $\Omega^{1}_{X}(\log D)$. Then we conclude that
$\eta|_{\mathfrak{p}}=0$ as we wanted to see. ∎

##  4 Main results

Now we define our main object of study, which are $q$-logarithmic foliations
on a smooth variety $X$. For that, we first define a 1-logarithmic foliation.
Then we will state and prove our main results, which are Theorem 4.16,
Corollary 4.17.

######  Definition 4.1.

Let $\varpi\in H^{0}(X,\Omega^{1}_{X}(\log D))$ be a 1-logarithmic form, where
$D$ is a simple normal crossing divisor locally defined as
$D=(\prod_{i=1}^{s}f_{i})$, written as

$\varpi=\sum_{i=1}^{s}\lambda_{i}\frac{df_{i}}{f_{i}},\text{ with
$\lambda_{i}\in\mathbb{C}$}.$

Then we define the _1-logarithmic foliation_ associated to $\varpi$ as

$\omega=\left(\prod_{i=1}^{s}f_{i}\right)\sum_{i=1}^{s}\lambda_{i}\frac{df_{i}}{f_{i}}=\sum_{i=1}^{s}\lambda_{i}F_{\widehat{i}}\,df_{i}\
,$

where $F_{\widehat{i}}\,=\prod_{\begin{subarray}{c}j=1\\\ j\neq
i\end{subarray}}^{s}f_{j}$. This defines a global section

$F\varpi\in H^{0}(X,\Omega^{1}_{X}\otimes\mathcal{L})\ ,$

for some line bundle $\mathcal{L}$, and $F$ locally given by
$\prod_{i=1}^{s}f_{i}$.

######  Definition 4.2.

Let $\varpi\in H^{0}(X,\Omega^{q}_{X}(\log D))$ be a locally decomposable
$q$-logarithmic differential form, where $D$ is a simple normal crossing
divisor locally defined as $D=(\prod_{i=1}^{s}f_{i})$, written as

$\varpi=\sum_{\begin{subarray}{c}I\subset[s],\
|I|=q\end{subarray}}\lambda_{I}\frac{df_{I}}{f_{I}}\ .$

Then we define the _$q$ -logarithmic foliation_ associated to $\varpi$ as

$\omega=\left(\prod_{i=1}^{s}f_{i}\right)\left(\sum_{\begin{subarray}{c}I\subset[s],\
|I|=q\end{subarray}}\lambda_{I}\frac{df_{I}}{f_{I}}\right)=\sum_{\begin{subarray}{c}I\subset[s],\
|I|=q\end{subarray}}\lambda_{I}F_{\widehat{I}}df_{I}\ ,$ (6)

where $F_{\widehat{I}}$ denotes the product
$\prod_{\begin{subarray}{c}j\in[s],\ j\notin I\end{subarray}}f_{j}$ and
$df_{I}=\wedge_{i\in I}df_{i}$. This defines a global section

$F\varpi\in H^{0}(X,\Omega^{q}_{X}\otimes\mathcal{L})\ ,$

for some line bundle $\mathcal{L}$ and $F$ locally given by
$\prod_{i=1}^{s}f_{i}$. We will denote this space as $Log(X,D,\mathcal{L},q)$.

######  Remark 4.3.

If $\omega$ is a $q$-logarithmic foliation then it is indeed a foliation.
Since it is locally decomposable then it verifies condition c) of Definition
2.1. Condition b) comes from the fact that the functions $f_{i}$ are
tranversal. And condition a) can be seen because c) holds and $\omega$ is a
closed $q$-differential form multiplied by a function $F$.

Moreover, the smooth part of the divisor $D$ is invariant by the foliation
$\omega$. Indeed, using Eq. 6, we can see that the restriction of $\omega$ to
the irreducible component $D_{i}=(f_{i}=0)$ of $D$ in
$\Omega^{q}_{X}|_{D_{i}}$ is of the form

$\omega|_{D_{i}}=df_{i}|_{D_{i}}\wedge\eta$

For some form $\eta$ in $\Omega^{q-1}_{X}|_{D_{i}}$. This in turn implies that
$df_{i}|_{D_{i}}$ is in $\mathcal{E}(\omega)|_{D_{i}}$, which means that the
distribution annihilated by $\omega$ is a subsheaf of the tangent sheaf to the
divisor $D_{i}$ i.e.: the divisor $D_{i}$ is invariant by the foliation
$\omega$.

######  Remark 4.4.

Following Definition 4.2, we will refer to a _generic_ $q$-logarithmic
foliation as one where all the line bundles induced by the divisors $f_{i}$
are ample and the scalar coefficients $\lambda_{I}$ are required to be
pairwise distinct. See [Gro61, §4, 4.7].

In Corollary 4.17 we will see that under generic conditions the ideal of
persistent singularities $\mathscr{I}(\omega)$ can be given by the following
formula

$\mathscr{I}(\omega)=\bigcap_{\begin{subarray}{c}K\subset[s]\\\
|K|=q+1\end{subarray}}\sum_{i\in K}\mathcal{I}(f_{i})\ ,$

where by $\mathcal{I}(f)$ we denote the ideal generated by $f$. For that, we
would like to state now the following technical lemma, which is a
generalization of [Suw83b, Corollary 1.6].

######  Lemma 4.5.

Let us consider an open set $U\subset X$ and $f_{1},\ldots,f_{s}\in
H^{0}(U,\mathcal{O}_{X})$ sections of $X$ in $U$, for $s\in\mathbb{N}$ such
that $s>q$. Suppose that $ht(f_{i_{1}},\ldots,f_{i_{s-q}})=s-q$ if
$i_{1},\ldots,i_{s-q}\in\\{1,\ldots,s\\}$ are distinct, where ‘$ht$’ stands
for _height_. Assume that $J$ runs through every subset of cardinality $q$ of
$\\{1,\ldots,s\\}$, $q\geq 1$, then we have

$\sum_{\begin{subarray}{c}J\subset[s],\
|J|=q\end{subarray}}\mathcal{I}\left(F_{\widehat{J}}\right)=\bigcap_{\begin{subarray}{c}K\subset[s]\\\
|K|=q+1\end{subarray}}\sum_{i\in K}\mathcal{I}\left(f_{i}\right)\ .$

###### Proof.

To see the first inclusion

$\sum_{\begin{subarray}{c}J\subset[s],\
|J|=q\end{subarray}}\left(F_{\widehat{J}}\right)\subset\bigcap_{\begin{subarray}{c}K\subset[s]\\\
|K|=q+1\end{subarray}}\sum_{i\in K}\mathcal{I}\left(f_{i}\right)$

we just notice that $F_{\widehat{J}}$ has $s-q$ factors. Now, since
$s-q+q+1=s+1>s$ then for every subset $K\subset\\{1,\ldots,s\\}$ such that
$|K|=q+1$ there exists a $k_{0}\in K$ such that $k_{0}\notin J$. This implies
that $f_{k_{0}}|F_{\widehat{J}}$ and $f_{k_{0}}\in\sum_{i\in
K}\mathcal{I}(f_{i})$.

Since this happens for every $J\subset\\{1,\ldots,s\\}$, $|J|=q$, and
$K\subset\\{1,\ldots,s\\}$, $|K|=q+1$, we have that
$F_{\widehat{J}}\in\sum_{i\in K}\mathcal{I}(f_{i})$. Then

$\sum_{\begin{subarray}{c}J\subset[s],\
|J|=q\end{subarray}}F_{\widehat{J}}\subset\bigcap_{\begin{subarray}{c}K\subset[s]\\\
|K|=q+1\end{subarray}}\sum_{i\in K}\mathcal{I}(f_{i})$

as we wanted to see.

For the other inclusion we will use the transversality of the
$\\{f_{1},\ldots,f_{s}\\}$ in the following sense: we can compute the
intersection of
$\mathcal{I}(f_{i})\cap\mathcal{I}(f_{j})=\mathcal{I}(f_{i}.f_{j})$ if $i\neq
j$. Then we would like to see which is the worst choice of polynomials in the
intersection $\bigcap_{\\{K\subset[s],\ |K|=q+1\\}}\sum_{i\in
K}\mathcal{I}(f_{i})$ in the sense that we would get the least amount of
polynomials in the product. Let us suppose that we choose $f_{1}$ as many
times as possible, then we take $f_{2}$ as many times as possible and so on.
For $f_{1}$ we have $\binom{s-1}{q}$ possible choices, meaning that there are
$\binom{s-1}{q}$ ideals with $f_{1}$ in the intersection; for $f_{2}$ we have
$\binom{s-2}{q}$ possible choices, and it keeps going on like this. By the
following formula

$\sum_{i=1}^{s-q}\binom{s-i}{q}=\binom{s}{q+1}$

we are seeing that we can take, at least, $s-q$ polynomials to fill the
$\binom{s}{q+1}$ possible choices of each one of the ideals in the
intersection. Then, if $|J|=q$, we have that
$F_{\widehat{J}}\in\bigcap_{\\{K\subset[s],\ |K|=q+1\\}}\sum_{i\in
K}\mathcal{I}(f_{i})$ concluding our proof. ∎

From now on let us fix $X$ to be a smooth variety,
$D=\sum_{i=1}^{s}D_{i}=\left(\prod_{i=1}^{s}f_{i}\right)$ a simple normal
crossing divisor such that its components $D_{i}=(f_{i})$ are irreducible,
smooth, and ample.

We will also consider $\varpi\in H^{0}(X,\Omega^{q}_{X}(\log D))$ a
$q$-logarithmic differential form and $\omega=F\cdot\varpi\in
H^{0}(X,\Omega^{q}_{X}\otimes\mathcal{L})$ the associated logarithmic
$q$-differential foliation, where $F=\prod_{i=1}^{s}f_{i}$, as in Definition
4.2.

######  Proposition 4.6.

Let $\varpi\in H^{0}(X,\Omega^{q}_{X}(\log D))$ be a generic $q$-logarithmic
differential form such that every residue is not zero (_i.e._ : such that
every $\lambda_{I}$ is $\neq 0$ for every $I$), and such that $\varpi$ is
locally decomposable as a product of $q$ logarithmic $1$-forms outside its
zero locus. Then $Z(\varpi)$, the zero locus of $\varpi$, has dimension
$\dim(Z(\varpi))\leq q-1$ or $Z(\varpi)=\emptyset$. If, moreover, the $\varpi$
is _locally decomposable_ then we have that $\dim(Z(\varpi))=q-1$.

###### Proof.

The case $q=1$ is the main result of [CSV06]. Let $q\geq 2$ and suppose
$\dim(Z(\varpi))\geq q$ (in particular we are assuming that
$Z(\varpi)\neq\emptyset$). The ampleness of the components $D_{i}$ implies
that the intersection of the zeroes of $\varpi$, with $D$ has dimension
greater or equal to $q-1$. By Proposition 3.4 the intersection of $Z(\varpi)$
with $D$ is inside the zero locus of the form $R(\varpi)$. $R(\varpi)$ is a
$(q-1)$-logarithmic form on a complete intersection, the residues of
$R(\varpi)$ are a subset of the residues of $\varpi$, so they are all non-
null. Then, by induction, the set of zeros of $R(\varpi)$ must be of
$\dim(Z(R(\varpi)))=q-2$, therefore $\dim(Z(\varpi))\leq q-1$. On the other
hand, being $\varpi$ locally decomposable $\mathcal{E}$ is a locally free
sheaf of rank $n$ and $s$ is a section of $\bigwedge^{q}\mathcal{E}$ that is
locally decomposable then the expected dimension of $Z(s)$ is $q-1$, so if
$Z(s)\neq\emptyset$ we have $\dim(Z(s))\geq q-1$. Therefore
$\dim(Z(\varpi))=q-1$ concluding our statement. ∎

######  Proposition 4.7.

Let $\omega\in H^{0}(X,\Omega^{q}_{X}\otimes\mathcal{L})$ be a $q$-logarithmic
foliation, as in Eq. 6, and suppose that $\lambda_{I}\neq\lambda_{J}$ if
$I\neq J$. The Kupka scheme of $\omega$ is given by

$\mathpzc{Kup}(\omega)=\bigcup_{K\subset[s],\ |K|=q+1}\left(\bigcap_{i\in
K}D_{i}\right).$

###### Proof.

We will proceed by induction on $q$. Suppose $q=1$, then [CSV06] implies
$\mathrm{Sing}(\omega)$ consists of isolated points and a codimension-$2$
subvariety given by the intersections of the components of $D$. If
$\mathfrak{p}\in D_{i}\cap D_{j}$ is a regular point of $D_{i}\cap D_{j}$ then
locally around $\mathfrak{p}$ we have
$d\omega|_{\mathfrak{p}}=(\lambda_{i}-\lambda_{j})gdf_{i}\wedge df_{j}$ where
$g$ is a unit of $\mathcal{O}_{D_{i}\cap D_{j},\mathfrak{p}}$. Therefore
$\mathfrak{p}$ is a (strictly) Kupka point. The case $q=1$ follows form the
fact that, if $\mathrm{Sing}(\omega)$ is reduced, then $\mathpzc{Kup}(\omega)$
is the closure of the set of strictly Kupka points _i.e._ , points
$\mathfrak{p}\in\mathrm{Sing}(\omega)$ such that $d\omega|_{\mathfrak{p}}\neq
0$.

If $q>1$ then from the local expression of $\omega$ follows that
$\bigcap_{\\{i_{1},\dots,i_{q+1}\\}}D_{i_{j}}\subseteq\mathrm{Sing}(\omega)$
for any intersection of $q+1$ components of $D$. Let $\mathfrak{p}$ be a
regular point of $\bigcup_{\\{K\subset[s],|K|=q+1\\}}\left(\bigcap_{i\in
K}D_{i}\right)$ and suppose $d\omega|_{\mathfrak{p}}=0$. The logarithmic
$q-1$-form $\eta:=R(\varpi)|_{D_{j}}$ defines a logarithmic foliation on
$D_{j}$ given by the twisted form $F_{\widehat{j}}\,\eta$ whose Kupka scheme
consists of the $q+1$-wise intersections of components of $D$ containing
$D_{j}$. From the definition of $R(\varpi)$, Eq. 5, and Proposition 3.4 it
follows that if $d\omega|_{\mathfrak{p}}=0$ then
$d(F_{\widehat{j}}\,\eta)|_{\mathfrak{p}}=0$ which contradicts the inductive
hypothesis on $q$. So $\mathfrak{p}$ is a Kupka point of $\omega$ and the
result follows. ∎

For the next Proposition we will need the following technical lemmas.

The next Lemma 4.8 is, somehow, a generalization of a result from Suwa,
[Suw83b, Theorem 1.11], that says that a universal unfolding for
$\omega={f_{1}\ldots f_{s}\sum_{k=1}^{s}\lambda_{k}\frac{df_{k}}{f_{k}}}$ is
of the form

$\widetilde{\omega}=\widetilde{f}_{1}\ldots\widetilde{f}_{s}\sum_{k=1}^{s}\lambda_{k}\frac{d\widetilde{f}_{k}}{\widetilde{f}_{k}},$

where $\widetilde{f}_{k}(x,t)$ is a function germ such that
$\widetilde{f}_{k}(x,0)=f_{k}(x)$.

In case of a first order unfolding, _i.e._ in $X\times\mathcal{D}$, where
$\mathcal{D}=\mathrm{Spec}(k[\varepsilon]/(\varepsilon^{2}))$, we will have
that $\widetilde{\omega}=\omega+\varepsilon\theta+hd\varepsilon$, for some
1-logarithmic foliaton $\theta$ and some function germ $h$. Notice that the
1-logarithmic foliation $\theta$ is just (the addition of) $\omega$ with one
of the $f_{k}$’s changed by another function germ, $g$, and the function $h$
is the product of the $f_{\ell}$’s, with $\ell\neq k$, and $g$, and the scalar
$\lambda_{k}$. Without loss of generality we can suppose that $k=1$ and then
we can write an unfolding $\widetilde{\omega}$ as

$\displaystyle\widetilde{\omega}$
$\displaystyle=F_{\widehat{1}}(f_{1}+\varepsilon
g)\left(\lambda_{1}\frac{d(f_{1}+\varepsilon g)}{f_{1}+\varepsilon
g}+\sum_{\ell=2}^{s}\lambda_{\ell}\frac{df_{\ell}}{f_{\ell}}\right)\ .$ (7)

Also notice that, by Eq. 7 $\widetilde{\omega}\in
H^{0}(X\times\mathcal{D},\Omega^{1}_{X\times\mathcal{D}|\mathbb{C}}\otimes\widetilde{\mathcal{L}})$
is also a $1$-logarithmic foliation in $X\times\mathcal{D}$.

Lets consider now $\varpi\in H^{0}(X,\Omega^{q}_{X}(\log D))$ to be a locally
decomposable $q$-logarithmic differential form locally written as
$\varpi=\omega_{1}\wedge\ldots\wedge\omega_{q}$, for $\omega_{i}\in
H^{0}(X,\Omega^{1}_{X}(\log D))$, $i=1,\ldots,q$, and $\omega$ its associated
logarithmic foliation. Let us write $\omega_{i}$ as

$\omega_{i}=\lambda_{1}^{i}\ \frac{df_{1}}{f_{1}}+\omega_{i}^{+},\ \ \
i=1,\ldots,q\ .$

Then $\omega$ can be written as

$\omega=F\left(\left(\sum_{i=1}^{q}(-1)^{i+1}\lambda_{1}^{i}\
\frac{df_{1}}{f_{1}}\wedge\omega_{\widehat{i}}^{+}\right)+\omega^{+}\right)=F\
\omega_{1}\wedge\ldots\wedge\omega_{q}\ ,$ (8)

where $\omega^{+}=\omega_{1}^{+}\wedge\ldots\wedge\omega_{q}^{+}$ and we
recall from Eq. 3 that
$\omega_{\widehat{i}}^{+}=\omega_{1}^{+}\wedge\cdots\wedge\widehat{\omega_{i}^{+}}\wedge\cdots\wedge\omega_{q}^{+}$.

######  Lemma 4.8.

Following the notation above, let $\varpi\in H^{0}(X,\Omega^{q}_{X}(\log D))$
be a locally decomposable $q$-logarithmic differential form and $\omega$ its
associated logarithmic foliation.

Then, for some locally defined function $g$, there is a first order unfolding
of $\omega$ of the following form

$\displaystyle\widetilde{\omega}$
$\displaystyle=F_{\widehat{1}}(f_{1}+\varepsilon
g)\left(\left(\sum_{i=1}^{q}(-1)^{i+1}\lambda_{1}^{i}\
\frac{d(f_{1}+\varepsilon g)}{f_{1}+\varepsilon
g}\wedge\omega_{\widehat{i}}^{+}\right)+\omega^{+}\right)=$ (9)
$\displaystyle=F_{\widehat{1}}(f_{1}+\varepsilon g)\
\left[\left(\lambda_{1}^{1}\ \frac{d(f_{1}+\varepsilon g)}{f_{1}+\varepsilon
g}+\omega_{1}^{+}\right)\wedge\ldots\wedge\left(\lambda_{1}^{q}\
\frac{d(f_{1}+\varepsilon g)}{f_{1}+\varepsilon
g}+\omega_{q}^{+}\right)\right]$

###### Proof.

The proof can be established by directly computing Definition 2.1, see Remark
4.3. ∎

######  Lemma 4.9.

Let $\varpi\in H^{0}(X,\Omega^{q}_{X}(\log D))$ be a locally decomposable
$q$-logarithmic differential and $D_{1}=(f_{1}=0)$ one of the smooth
irreducible components of the divisor $D$. Then, we can write
$\varpi|_{D_{1}}$ in $\Omega^{q}_{X}(\log D)|_{D_{1}}$ as

$\varpi|_{D_{1}}=\frac{df_{1}}{f_{1}}\wedge\omega_{2}\wedge\ldots\wedge\omega_{q}\
.$

###### Proof.

By Remark 4.3 we have that $D_{1}$ is invariant by the foliation $\omega$
associated with $\varpi$, so $df_{1}|_{D_{1}}\wedge\omega|_{D_{1}}=0$ in
$\Omega^{q+1}_{X}|_{D_{1}}$. Then, as $\omega=\prod_{i}f_{i}\cdot\varpi$, we
have that, if $\omega|_{D_{1}}=df_{1}|_{D_{1}}\wedge\omega^{\prime}$ then

$\varpi|_{D_{1}}=\frac{df_{1}}{f_{1}}|_{D_{1}}\wedge\omega^{\prime},$

in $\Omega^{q+1}_{X}(\log D)|_{D_{1}}$, as $\varpi|_{D_{1}}$ is locally
decomposable and $\frac{df_{1}}{f_{1}}|_{D_{1}}$ is a local generator of
$\Omega^{q+1}_{X}(\log D)|_{D_{1}}$, the above equation implies

$\prod_{i\neq
1}f_{i}\cdot\varpi|_{D_{1}}=\left.\frac{df_{1}}{f_{1}}\right|_{D_{1}}\wedge\omega_{2}\wedge\cdots\wedge\omega_{q}\
.$

∎

As a corollary of the previous lemma we have the following statement:

######  Corollary 4.10.

$\varpi\in H^{0}(X,\Omega^{q}_{X}(\log D))$ be a locally decomposable
$q$-logarithmic differential. Then we have that the residue operation defined
in Eq. 5 as

$\textstyle{\Omega^{q}_{X}(\log
D)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{R}$$\textstyle{\Omega^{q-1}_{\widetilde{D}}(\log
D^{\prime})}$$\textstyle{\frac{df_{i}}{f_{i}}\wedge\eta\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\eta|_{D_{i}},}$

takes locally decomposable global sections of $\Omega^{q}_{X}(\log D)$ and
maps them to locally decomposable global sections of
$\Omega^{q-1}_{D_{i}}(\log D^{\prime})$ for each irreducible component $D_{i}$
of $D$, so we get map

$\textstyle{Log(X,D,\mathcal{L},q)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{R}$$\textstyle{Log(D_{i},D^{\prime},\mathcal{L},q-1).}$

###### Proof.

The proof is immediate using Lemma 4.9 and Remark 4.3. ∎

######  Remark 4.11.

Notice that taking the residue along $\\{f_{i}=0\\}$ and taking an unfolding
of the type of Lemma 4.8, by changing $f_{j}$ to $f_{j}+\varepsilon g$, with
$j\neq i$, commute.

######  Proposition 4.12.

Following the notation and assumptions above, if a point
$\mathfrak{p}\notin\mathpzc{Kup}(\omega)$ then there is a local first order
unfolding $\widetilde{\omega}$, of the type of Lemma 4.8, of $\omega$ around
$\mathfrak{p}$ such that $\widetilde{\omega}|_{\mathfrak{p}}\neq 0$.

###### Proof.

Let us first assume that $\mathfrak{p}\notin D$, then the foliation is also
defined locally by $\varpi$ which is the product of $\omega$ by a non-
vanishing local function and also is (locally) decomposable as
$\varpi=\omega_{1}\wedge\dots\wedge\omega_{q}$. To prove the existence of an
unfolding $\widetilde{\omega}$ as in the proposition we need to show that
there are $1$-forms $\alpha_{ij}$ such that
$\omega_{i}=\sum_{j}d\omega_{j}\wedge\alpha_{ij}$. In order to prove this we
will use the _generalized de Rham lemma_ of [Sai76] which states that, if the
depth of the singular ideal of the decomposable form $\varpi$ is greater than
$3$ then for every $2$-form $\beta$ such that $\beta\wedge\varpi=0$ there are
$1$-forms $\alpha_{1},\dots,\alpha_{q}$ such that
$\beta=\sum_{i}\omega_{i}\wedge\alpha_{i}$. Now, being a global logarithmic
form, $d\varpi=0$. Then, for any $1$-form $\eta$ such that
$\eta\wedge\varpi=0$, we have

$d\eta\wedge\varpi=d(\eta\wedge\varpi)=0.$

If $q<n-1$, that is, if $\omega$ is _not_ a foliation by curves, then we have
by Proposition 4.6 that $\mathrm{codim}(\mathrm{Sing}(\omega))\geq 3$, which,
as $X$ is a regular variety, in particular locally Cohen-Macaulay, implies
that $\mathrm{depth}(\mathscr{J})\geq 3$. Then we can apply the generalized de
Rham lemma which implies there are $1$-forms $\alpha_{1},\dots,\alpha_{q}$
such that

$d\eta=\sum_{i}\omega_{i}\wedge\alpha_{i}.$

So $1\in\mathscr{I}(\varpi)=\mathscr{I}(\omega)$, the definition ideal of
$\mathpzc{Per}(\omega)$, then $\mathfrak{p}$ is never in
$\mathpzc{Per}(\omega)$. Then, by [MMQ21, Proposition 4.1] there exists an
$\widetilde{\omega}$ unfolding of $\omega$ such that
$\widetilde{\omega}|_{\mathfrak{p}}\neq 0$.

For the case when $\mathfrak{p}\in D$ we can suppose that
$\mathfrak{p}\in\\{f_{1}=0\\}$. We will employ an inductive approach. The case
where $q=1$ is given by Eq. 7 using [Suw83b, Theorem 1.11] and [MMQ21,
Proposition 3.6, Theorem 3.14].

Consider a $q$-logarithmic foliation given by $\omega$ and take the residue at
$\\{f_{1}=0\\}$, $R(\frac{\omega}{F})=\eta$. Now we can consider an unfolding
of $\eta$ following Lemma 4.8 by changing $f_{2}$ to $f_{2}+\varepsilon g$,
let us call this unfolding by $\widetilde{\eta}$. Using the inductive
hypotheses we have that $\widetilde{\eta}(\mathfrak{p})\neq 0$.

By Remark 4.11, $\widetilde{\eta}$ is the residue along $\\{f_{1}=0\\}$ of an
unfolding of $\omega$ given by changing $f_{2}$ to $f_{2}+\varepsilon g$. By
virtue of the linearity of the residue operation, if the residue is nonzero at
$\mathfrak{p}$, then the differential form must also be nonzero at
$\mathfrak{p}$, thereby completing the proof. ∎

######  Corollary 4.13.

We have the inclusion of closed subvarieties

$(\mathpzc{Per}(\omega))_{\mathrm{red}}\subseteq(\mathpzc{Kup}(\omega))_{\mathrm{red}}\
.$

###### Proof.

Around a point $\mathfrak{p}\notin D$, by the same argument of the above proof
we have that $\mathfrak{p}$ is never in $\mathpzc{Per}(\omega)$. If
$\mathfrak{p}\in D$ then the above proposition together with [MMQ21,
Proposition 4.1 ] is showing that $\mathfrak{p}\notin\mathpzc{Kup}(\omega)$
implies $\mathfrak{p}\notin\mathpzc{Per}(\omega)$, from which the corollary
follows. ∎

######  Proposition 4.14.

Let $\mathfrak{p}\in\mathpzc{Kup}(\omega)$ be a strictly Kupka point. Suppose
that, when representing $\mathrm{Res}^{q}(\omega)$ as a vector in

$\mathrm{Res}^{q}(\omega)\in\bigoplus_{I\subset[s],\
|I|=q}H^{0}(X,\mathcal{O}_{\bigcap_{i\in
I}D_{i}})\simeq\mathbb{C}^{|\\{I\subset[s],\ |I|=q\\}|}\ ,$

there are two coordinates with different coefficients
$\lambda_{I}\neq\lambda_{J}$. Then locally around $\mathfrak{p}$ we have
$\mathscr{I}_{\mathfrak{p}}\subset\mathcal{K}_{\mathfrak{p}}$.

###### Proof.

Being $\mathfrak{p}$ a Kupka point, following [DM00, Proposition 1.3.1], we
can find local coordinates $(x_{1},\dots,x_{n})$ in which $\omega$ is written
completely in terms of $(x_{1},\dots,x_{q+1})$. In other words there is a
local submersion $\pi:U\to\mathbb{C}^{q+1}$ and a local form
$\alpha\in\Omega^{q}_{\mathbb{C}^{q+1},0}$ such that
$\omega|_{U}=\pi^{*}\alpha$. We can take $U$ small enough in such a way that
$0\in\mathbb{C}^{q+1}$ is the only singularity of $\alpha$. Moreover, $\alpha$
also defines a logarithmic foliation in $\mathbb{C}^{q+1}$ and its isolated
singularity is the normal crossing intersection of $q+1$ of its invariant
divisors. So we have local coordinates around $\mathfrak{p}$ such that the
logarithmic form $\frac{\omega}{F}$ is written as

$\frac{\omega}{F}=\sum_{i=1}^{q+1}f_{i}\bigwedge_{j\neq
i}\frac{dx_{j}}{x_{j}}+\xi_{1}.$

Where in this coordinates $F=x_{1}\cdots x_{q+1}$ and where $\xi_{1}\in
W^{q-1}\Omega^{q}_{X}(\log D)$, the space of germs of logarithmic $q$-forms of
weight at most $q-1$ as defined in [Del06, 3.5]. Therefore we have in this
coordinates the expression

$\omega=\sum_{i=1}^{q+1}x_{i}f_{i}\bigwedge_{j\neq i}dx_{j}+\xi_{2}.$

In this case $\xi_{2}=F\cdot\xi_{1}$ is the germ of a holomorphic $q$-form
which is actually in
$(x_{1},\dots,x_{q+1})^{2}\Omega_{X,\mathfrak{p}}^{q}=K^{2}\cdot\Omega_{X,p}^{q}$.

Now lets take $\Xi_{i,k}=\bigwedge_{j\neq i,k}\frac{\partial}{\partial
x_{j}}\in\bigwedge^{q-1}T_{X,\mathfrak{p}}$ and
$\varpi_{i,k}=i_{\Xi_{i,k}}\omega$, then

$\varpi_{i,k}=x_{i}f_{i}dx_{k}-x_{k}f_{k}dx_{i}+i_{\Xi_{i,k}}\xi_{2}.$

Notice that the $\varpi_{i,k}$ with $1\leq i<k\leq q+1$ generate the sheaf
$\mathcal{E}(\omega)$, as $i_{\frac{\partial}{\partial x_{j}}}\omega=0$ for
$q+2\leq j\leq n$. In particular every 1-form in $\mathcal{E}$ is in
$K\cdot\Omega^{1}_{X,0}$. We also have

$d\varpi_{i,k}=(f_{i}-f_{k})dx_{i}\wedge dx_{k}+\xi_{3},$

where $\xi_{3}=d\xi_{2}$ is the germ of a $2$-form in
$K\cdot\Omega^{2}_{X,0}$.

Now suppose $h\in\mathscr{I}_{\mathfrak{p}}$, we want to show $h=0$ in
$\mathcal{O}_{X,\mathfrak{p}}/K_{\mathfrak{p}}$. As
$h\in\mathscr{I}_{\mathfrak{p}}$ we have that, for every
$\varpi\in\mathcal{E}$,

$hd\varpi\in K\cdot\Omega^{2}_{X,\mathfrak{p}}.$

In particular we may take $\varpi_{i,k}$ and get

$0\equiv hd\varpi_{i,k}\equiv h(f_{i}(0)-f_{k}(0))dx_{i}\wedge dx_{k}\mod K.$

If we have two different residues in $\omega$ then we may take $i,k$ such that
$f_{i}(0)-f_{k}(0)\neq 0$ (because the residues are exactly $f_{i}(0)$ for
$i=1,\dots,q+1$). Hence we must have $h\equiv 0\mod K$. ∎

By using the two propositions above we can conclude the following.

######  Theorem 4.15.

Let $\omega$ be a codimension-$q$, locally decomposable, logarithmic foliation
on $X$ such that $\mathpzc{Kup}(\omega)$ is reduced and the singular locus be
the disjoint union of $\mathpzc{Kup}(\omega)$ and a variety $\mathcal{L}$,
_i.e._ , $\mathrm{Sing}(\omega)=\mathpzc{Kup}(\omega)\sqcup\mathcal{L}$. Then
we have

$\mathpzc{Kup}(\omega)=\mathpzc{Per}(\omega).$

###### Proof.

By Corollary 4.13 we have that
$(\mathpzc{Per}(\omega))_{red}\subset(\mathpzc{Kup}(\omega))_{red}$, then by
our hypothesis we have that
$(\mathpzc{Per}(\omega))_{red}\subset\mathpzc{Kup}(\omega)$. And by
Proposition 4.14 we know that for every $\mathfrak{p}\in\mathpzc{Kup}(\omega)$
a strictly Kupka point we have
$\mathscr{I}_{\mathfrak{p}}\subset\mathcal{K}_{\mathfrak{p}}$, meaning that
$\mathpzc{Kup}(\omega)_{\mathfrak{p}}\subset\mathpzc{Per}(\omega)_{\mathfrak{p}}$.

Since $\mathpzc{Kup}(\omega)$ is reduced then the set of strictly Kupka points
is dense in $\mathpzc{Kup}(\omega)$, implying that

$\mathpzc{Kup}(\omega)\subset\mathpzc{Per}(\omega)\ .$

By Proposition 2.14 we have that
$\mathpzc{Per}(\omega)\subset\mathrm{Sing}(\omega)$ then
$\mathpzc{Kup}(\omega)\subset\mathpzc{Per}(\omega)\subset\mathrm{Sing}(\omega)$.
And since $\mathrm{Sing}(\omega)=\mathpzc{Kup}(\omega)\sqcup\mathcal{L}$ we
have that

$\mathpzc{Kup}(\omega)_{\mathfrak{p}}\subset\mathpzc{Per}(\omega)_{\mathfrak{p}}\subset\mathrm{Sing}(\omega)_{\mathfrak{p}}=\mathpzc{Kup}(\omega)_{\mathfrak{p}}\
,\forall\mathfrak{p}\in Supp(\mathpzc{Kup}(\omega))$

implying that $\mathcal{K}(\omega)=\mathscr{I}(\omega)$. ∎

As a corollary of the theorem above we get the following decomposition of
$\mathrm{Sing}(\omega)$ for $\omega$ a $q$-logarithmic foliation.

######  Theorem 4.16.

Let $\omega\in H^{0}(X,\Omega^{q}_{X}\otimes\mathcal{L})$ be a generic,
locally decomposable, $q$-logarithmic foliation. Then we can decompose its
singular locus $\mathrm{Sing}(\omega)$ as the disjoint union of

$\mathrm{Sing}(\omega)=\mathpzc{Kup}(\omega)\sqcup\mathcal{H},$

where $\mathpzc{Kup}(\omega)$ is the Kupka variety of $\varpi$ of codim $q+1$
and $\mathcal{H}$ is a variety of dimension $q-1$ or $\mathcal{H}=\emptyset$.
Even more so, we have that $\mathpzc{Kup}(\omega)=\mathpzc{Per}(\omega)$.

###### Proof.

Since $\omega$ is a $q$-logarithmic foliation, we know that its singular locus
is made of $\mathpzc{Kup}(\omega)$ of codimension-$(q+1)$ and a variety
$\mathcal{H}$ of dimension $(q-1)$ or $\mathcal{H}=\emptyset$. Since $\omega$
is generic, we can assume that
$\mathpzc{Kup}(\omega)\cap\mathcal{H}=\emptyset$.

Finally, we can apply Theorem 4.15 and conclude
$\mathpzc{Kup}(\omega)=\mathpzc{Per}(\omega)$ as we wanted to see. ∎

Let $X$ be a smooth projective variety with an ample line bundle
$\mathcal{L}$. We will denote
$\bigoplus_{n\in\mathbb{Z}}H^{0}(X,\mathcal{F}\otimes\mathcal{L}^{n})$ by
$H^{0}_{*}(X,\mathcal{F})$.

######  Corollary 4.17.

Let $X$ a smooth projective variety with an ample line bundle and $\omega$ a
$q$-logarithmic foliation defined by a simple normal crossing divisor
$D=(\prod_{i=1}^{s}f_{i})$ as

$\omega=F\left(\sum_{I\subset[s],\
|I|=q}\lambda_{I}\frac{df_{I}}{f_{I}}\right)=\sum_{I\subset[s],\
|I|=q}\lambda_{I}F_{\widehat{I}}df_{I}=F\varpi\in
H^{0}(X,\Omega^{q}_{X}\otimes\mathcal{L})$

satisfying the hypotheses of Theorem 4.16, then we have

$H^{0}_{*}(X,\mathscr{I}(\omega))=H^{0}_{*}(X,\mathcal{K}(\omega))=\sum_{J\subset[s],\
|J|=q}\mathcal{I}\left(F_{\widehat{J}}\right)=\bigcap_{\begin{subarray}{c}K\subset[s]\\\
|K|=q+1\end{subarray}}\sum_{i\in K}\mathcal{I}\left(f_{i}\right).$

###### Proof.

For $\omega$ of the form given in the statement the Kupka scheme is, by
Proposition 4.7, given by the ideal
$\bigcap_{\begin{subarray}{c}\\{K\subset[s],|K|=q+1\\}\end{subarray}}\sum_{i\in
K}\mathcal{I}\left(f_{i}\right)$. Then applying Theorem 4.16 and Lemma 4.5 we
are done. ∎

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Ariel Molinuevo∗ |<EMAIL_ADDRESS>
---|---
Federico Quallbrunn† |<EMAIL_ADDRESS>

∗Instituto de Matemática |  † Departamento de Matemática
---|---
Universidade Federal do Rio de Janeiro | Universidad CAECE
Av. Athos da Silveira Ramos 149 | Av. de Mayo 866
Bloco C, Centro de Tecnologia, UFRJ | CP C1084AAQ Buenos Aires
Cidade Universitária, Ilha do Fundão | Argentina
CEP. 21945-970 Rio de Janeiro - RJ |
Brasil |