# On the Poincaré Problem for foliations on Compact Toric Orbifolds

Miguel Rodríguez Peña  A. M. Rodríguez
DEMAT - UFSJ
Departamento de Matemática e Estadística
São João del-Rei MG, Brazil<EMAIL_ADDRESS>
(Date: August 27, 2024)

###### Abstract.

We give an optimal upper bound of the degree of quasi-smooth hypersurfaces
which are invariant by a one-dimensional holomorphic foliation on a compact
toric orbifold, i.e. on a complete simplicial toric variety. This bound
depends only on the degree of the foliation and of the degrees of the toric
homogeneous coordinates.

##  1\. Introduction

Henri Poincaré studied in [23] the problem to decide whether a holomorphic
foliation $\mathcal{F}$ on the complex projective plane $\mathbb{P}^{2}$
admits a rational first integral. Poincaré observed that in order to solve
this problem is sufficient to find a bound for the degree of the generic curve
invariant by $\mathcal{F}$. Find such bound is now known as the Poincaré
problem. Although it is well-known that such a bound does not exist in
general, under certain hypotheses there are several works about Poincaré
problem and its generalizations; see for instance [25], [7], [5], [6], [4],
[15], [22] and [18].

Toric varieties form an important class of examples in algebraic geometry.
Furthermore, its geometry is fully determined by the combinatorial of its
associated fan, which often makes computations far more tractable. Recently,
one-dimensional holomorphic foliations on toric varieties has been studied in
[11] and [10], here the author presents a result of Darboux-Jouanolou-Ghys’s
type for these foliations. In this paper, we consider one-dimensional
holomorphic foliations on compact toric orbifold, i.e., a compact toric
variety with at most quotient singularities. Note that similarly to the
complex projective space, one can consider the Poincaré problem for
holomorphic one-dimensional foliations on a compact toric orbifold, it is
possible since that one can find homogenoeus coordinates in a toric variety
and therefore we can define the notion of degree of a foliation and of an
invariant hypersurface. In order to provide a solution to Poincaré problem we
give a normal form for quasi-homogeneous vector fields with a quasi-smooth
hypersurface invariant on a compact toric orbifold; see for instance theorems
4.1 and 4.3. As a consequence of theorems 4.1 and 4.3, we will see that the
solution of Poincaré problem on compact toric orbifolds is optimal. It is
important point out here that our main results improve and generalize the
results obtained in [8] and [3] about the Poincaré problem in weighted
projective spaces, also we generalize the results of [9], where the authors
studied the Poincaré problem for foliations on multiprojective spaces.
Moreover, we give an upper bound for Poincaré problem for foliations on
rational scrools, and an upper bound for Poincaré problem for foliations on a
compact toric orbifold surface with Weil divisor class group having torsion.
Finally, in section 5, we will present families of examples of one-dimensional
holomorphic foliations to show that our results are optimals and an example
that show that our hypotheses are necessary.

Let $\mathbb{P}_{\Delta}$ be an $n$-dimensional compact toric orbifold,
determined by a fan $\Delta$ in $N\simeq\mathbb{Z}^{n}$. As $\Delta(1)$ spans
$N_{\mathbb{R}}=N\otimes_{\mathbb{Z}}\mathbb{R}\simeq\mathbb{R}^{n}$, we have
that $\mathbb{P}_{\Delta}$ is a geometric quotient
$(\mathbb{C}^{n+r}-{\mathcal{Z}})/G$, where the group
$G=Hom_{\mathbb{Z}}(\mathcal{A}_{n-1}(\mathbb{P}_{\Delta}),\mathbb{C}^{*})$
acts on $\mathbb{C}^{n+r}$, $\mathcal{A}_{n-1}(\mathbb{P}_{\Delta})$ is the
group of Weil divisor classes of $\mathbb{P}_{\Delta}$ and $\mathcal{Z}$ is an
algebraic set of codimension at least two in $\mathbb{C}^{n+r}$; see [14,
Cox]. Consider
$\mathcal{T}\mathbb{P}_{\Delta}=\mathcal{H}om(\Omega_{\mathbb{P}_{\Delta}}^{1},\mathcal{O}_{\mathbb{P}_{\Delta}})$
the Zariski tangent sheaf of $\mathbb{P}_{\Delta}$. Since
$\mathbb{P}_{\Delta}$ is a complex orbifold then
$\mathcal{T}\mathbb{P}_{\Delta}\simeq
i_{\ast}\Theta_{\mathbb{P}_{\Delta,reg}}$, where
$i:\mathbb{P}_{\Delta,reg}\rightarrow\mathbb{P}_{\Delta}$ is the inclusion of
regular part
$\mathbb{P}_{\Delta,reg}=\mathbb{P}_{\Delta}-Sing(\mathbb{P}_{\Delta})$ and
$\Theta_{\mathbb{P}_{\Delta,reg}}$ is the tangent sheaf of
$\mathbb{P}_{\Delta,reg}$; see [12, Appendix A.2]. A singular one-dimensional
holomorphic foliation $\mathcal{F}$ on $\mathbb{P}_{\Delta}$ is a global
section of $\mathcal{T}\mathbb{P}_{\Delta}\otimes\mathcal{L}$, where
$\mathcal{L}$ is an invertible sheaf on $\mathbb{P}_{\Delta}$. We say that
$V=\left\\{f=0\right\\}\subset\mathbb{P}_{\Delta}$ is a quasi-smooth
hypersurface if its tangent cone $\left\\{f=0\right\\}$ on
$\mathbb{C}^{n+r}-\mathcal{Z}$ is smooth and it is said to be strongly quasi-
smooth hypersurface if its tangent cone $\left\\{f=0\right\\}$ on
$\mathbb{C}^{n+r}-\left\\{0\right\\}$ is smooth. Note that $V$ is quasi-smooth
if, and only if, $V$ is a suborbifold of $\mathbb{P}_{\Delta}$ and both
definitions coincide when the rank of $\mathcal{A}_{n-1}(\mathbb{P}_{\Delta})$
is one; see for instance [14] and [1]. Let $X$ be a quasi-homogeneous vector
field which defines $\mathcal{F}$ in homogeneous coordinates, we say that
$V=\left\\{f=0\right\\}\subset\mathbb{P}_{\Delta}$ is invariant by
$\mathcal{F}$ if $X(f)=g\cdot f$ for some polynomial $g$.

The group of Weil divisor classes $\mathcal{A}_{n-1}(\mathbb{P}_{\Delta})$ is
a finitely generated abelian group of rank $r$. By the fundamental theorem of
finitely generated abelian groups, we have

$\mathcal{A}_{n-1}(\mathbb{P}_{\Delta})\simeq\mathbb{Z}^{\,r}\oplus\mathbb{Z}_{(p_{1})^{\lambda_{1}}}\oplus\cdots\oplus\mathbb{Z}_{(p_{m})^{\lambda_{m}}},$

where the $p_{i}$ are primes, not necessarily distinct, and the $\lambda_{i}$
are positive integers. The direct sum is unique except for possible
rearrangement of the factors. Suppose that the homogeneous coordinate ring
$\mathrm{S}$ of $\mathbb{P}_{\Delta}$ has the following property; there is a
positive integer number $1\leq k\leq r$ such that

(1) $\begin{array}[]{ccccccccc}(\deg\mathrm{P})_{k}\geq 0\,\,\textup{for
all}\,\,\mathrm{P}\in\mathrm{S},\end{array}$

where the bottom index $k$ denotes the $k$-th coordinate of
$\deg\mathrm{P}\in\mathcal{A}_{n-1}(\mathbb{P}_{\Delta})$ i.e. the $k$-th
component of the multidegree $\deg\mathrm{P}$. First, we consider the question
of bounding the degree of strongly quasi-smooth hypersurfaces which are
invariant by a one-dimensional holomorphic foliation on a compact toric
orbifold:

######  Theorem 1.1 (Poincaré problem $1$).

Let $\mathbb{P}_{\Delta}$ be a complete simplicial toric varity of dimension
$n$. Let $\mathcal{F}$ be a one-dimensional holomorphic foliation on
$\mathbb{P}_{\Delta}$ and $X$ the quasi-homogeneous vector field which defines
$\mathcal{F}$ in homogeneous coordinates. Let
$V=\left\\{f=0\right\\}\subset\mathbb{P}_{\Delta}$ be a strongly quasi-smooth
hypersurface, invariant by $\mathcal{F}$. Suppose that $X\notin Lie(G)$, then

$\deg(V)_{k}\leq\deg(\mathcal{F})_{k}+\max_{1\leq i<j\leq
n+r}\big{\\{}deg(z_{i})_{k}+deg(z_{j})_{k}\big{\\}},$

for each $1\leq k\leq r$ as in the hypothesis $(\ref{eq0})$. Here, $Lie(G)$ is
defined in subsection 2.3.

There is a situation in which the above theorem is valid for a quasi-smooth
hypersurface $V=\left\\{f=0\right\\}\subset\mathbb{P}_{\Delta}$ invariant by
$\mathcal{F}$ with

$\left\\{0\right\\}\subsetneq Sing(V)\subset\mathcal{Z}.$

Suppose that there exist integer numbers $1\leq i_{1}<\cdots<i_{k}\leq n+r$
such that

  1. (i)

there is a regular subsequence $\left\\{\frac{\partial f}{\partial
z_{i_{1}}},\dots,\frac{\partial f}{\partial
z_{i_{k}}}\right\\}\subset\left\\{\frac{\partial f}{\partial
z_{1}},\dots,\frac{\partial f}{\partial z_{n+r}}\right\\}$ and

  2. (ii)

there is a radial vector field given by
$R_{i_{1},\ldots,i_{k}}=\sum_{j=1}^{k}a_{i_{j}}z_{i_{j}}\frac{\partial}{\partial
z_{i_{j}}}$.

Note that the property $($i$)$ is equivalent to

$codim\left(\left\\{\frac{\partial f}{\partial
z_{i_{1}}}=\cdots=\frac{\partial f}{\partial z_{i_{k}}}=0\right\\}\right)=k;$

for more details see [19]. Now, we consider the question of bounding the
degree of quasi-smooth hypersurfaces which are invariant by a one-dimensional
holomorphic foliation on a compact toric orbifold:

######  Theorem 1.2 (Poincaré problem $2$).

Let $\mathcal{F}$ be a one-dimensional holomorphic foliation on
$\mathbb{P}_{\Delta}$ and $X$ the quasi-homogeneous vector field which defines
$\mathcal{F}$ in homogeneous coordinates. Let
$V=\left\\{f=0\right\\}\subset\mathbb{P}_{\Delta}$ be a quasi-smooth
hypersurface. Suppose there are integers numbers $1\leq i_{1}<\cdots<i_{k}\leq
n+r$ with the properties $($_i_ $)$ and $($_ii_ $)$. Suppose that in
homogeneous coordinates $X=X_{1}+X_{2}$, where
$X_{1}=\sum_{j=1}^{k}P_{i_{j}}\frac{\partial}{\partial z_{i_{j}}}$ leaves $V$
invariant and such that $X_{1}\notin Lie(G)$. Then

$\deg(V)_{k}\leq\deg(\mathcal{F})_{k}+\max_{1\leq i_{j_{1}}<i_{j_{2}}\leq
k}\big{\\{}deg(z_{i_{j_{1}}})_{k}+deg(z_{i_{j_{2}}})_{k}\big{\\}},$

for each $1\leq k\leq r$ as in the hypothesis $(\ref{eq0})$.

##  2\. Generalities on toric varieties

In this section we recall some basic definitions and results about simplicial
complete toric varieties. For more details, we refer reader to the literature
(e.g. see [13], [17]).

Let $N$ be a free $\mathbb{Z}$-module of rank $n$ and
$M=\mathrm{Hom}(N,\mathbb{Z})$ be its dual. A subset $\sigma\subset
N\otimes_{\mathbb{Z}}\mathbb{R}\simeq\mathbb{R}^{n}$ is called a strongly
convex rational polyhedral cone if there exists a finite number of elements
$v_{1},\dots,v_{k}$ in the lattice $N$ such that

$\sigma=\\{a_{1}v_{1}+\cdots+a_{k}v_{k}:a_{i}\in\mathbb{R},a_{i}\geq 0\\},$

and $\sigma$ does not contain any straight line going through the origin i.e.
$\sigma\cap(-\sigma)=\left\\{0\right\\}$. We say that a subset $\tau$ of
$\sigma$ given by some $a_{i}$ being equal to zero is a proper face of
$\sigma$, and we write $\tau\prec\sigma$. A cone $\sigma$ is called simplicial
if its generators can be chosen to be linearly independent over $\mathbb{R}$.
The dimension of a cone $\sigma$ is, by definition, the dimension of a minimal
subspace of $\mathbb{R}^{n}$ containing $\sigma$.

######  Definition 2.1.

A non-empty collection $\Delta=\\{\sigma_{1},\dots,\sigma_{s}\\}$ of
$k$-dimensionais strongly convex rational polyhedral cones in
$N\otimes_{\mathbb{Z}}\mathbb{R}\simeq\mathbb{R}^{n}$ is called a fan if
satisfies:

  * (i)

if $\sigma\in\Delta$ and $\tau\prec\sigma$, then $\tau\in\Delta;$

  * (ii)

if $\sigma_{i},\sigma_{j}\in\Delta$, then
$\sigma_{i}\cap\sigma_{j}\prec\sigma_{i}$ and
$\sigma_{i}\cap\sigma_{j}\prec\sigma_{j};$

The Fan $\Delta$ is called complete if
$N\otimes_{\mathbb{Z}}\mathbb{R}=\sigma_{1}\cup\cdots\cup\sigma_{s}$. The
dimension of a fan is the maximal dimension of its cones. An $n$-dimensional
complete fan is simplicial if all its $n$-dimensional cones are simplicial.

Let $\Delta$ be a fan in $N\otimes_{\mathbb{Z}}\mathbb{R}$. It follows from
Gordan’s Lemma that each $k$-dimensional cone $\sigma$ in $\Delta$ (let say us
generated by $v_{i}$) defines a finitely generated semigroup $\sigma\cap N$.
The dual $(n-k)$-dimensional cone

$\check{\sigma}=\\{m\in M\otimes_{\mathbb{Z}}\mathbb{R},\langle
m,v_{i}\rangle\geq 0\\}$

is then a rational polyhedral cone in $M\otimes_{\mathbb{Z}}\mathbb{R}$ and
$\check{\sigma}\cap M$ is also a finitely generated semigroup. An affine
$n$-dimensional toric variety corresponding to $\sigma$ is the variety

$\mathcal{U}_{\sigma}=\mathrm{Spec}\,\mathbb{C}[\check{\sigma}\cap M].$

If a cone $\tau$ is a face of $\sigma$ then $\check{\tau}\cap M$ is a
subsemigroup of $\check{\sigma}\cap M$, hence $\mathcal{U}_{\tau}$ is embedded
into $\mathcal{U}_{\sigma}$ as an open subset. The affine varieties
corresponding to all cones of the fan $\Delta$ are glued together according to
this rule into the toric variety $\mathbb{P}_{\Delta}$ associated with
$\Delta$. Is possible to show that a toric variety $\mathbb{P}_{\Delta}$
contain a complex torus $\mathbb{T}^{n}=(\mathbb{C}^{*})^{n}$ as a Zariski
open subset such that the action of $\mathbb{T}^{n}$ on itself extends to an
action of $\mathbb{T}^{n}$ on $\mathbb{P}_{\Delta}$. A toric variety
$\mathbb{P}_{\Delta}$ determined by a complete simplicial fan $\Delta$ is a
compact complex orbifold i.e. a compact complex variety with at most quotient
singularities. Note that $\mathbb{T}^{n}$, $\mathbb{C}^{n}$ and
$\mathbb{P}^{n}$ are toric varieties. For more details see [13], [17].

###  2.1. The homogeneous coordinate ring

Let $\mathbb{P}_{\Delta}$ be the toric variety determined by a fan $\Delta$ in
$N\simeq\mathbb{Z}^{n}$. As usual, $M$ will denote the $\mathbb{Z}$-dual of
$N$, and cones in $\Delta$ will be denoted by $\sigma$. The one-dimensional
cones of $\Delta$ form the set $\Delta(1)$, and given $\rho\in\Delta(1)$, we
set $n_{\rho}$ the unique generator of $\rho\,\cap N$. If $\sigma$ is any cone
in $\Delta$, then $\sigma(1)=\\{\rho\in\Delta(1)\,:\,\rho\subset\sigma\\}$ is
the set of one-dimensional faces of $\sigma$. We will assume that $\Delta(1)$
spans $N_{\mathbb{R}}=N\otimes_{\mathbb{Z}}\mathbb{R}\simeq\mathbb{R}^{n}$.

Each $\rho\in\Delta(1)$ corresponds to an irreducible $\mathbb{T}$-invariant
Weil divisor $D_{\rho}$ in $\mathbb{P}_{\Delta}$, where
$\mathbb{T}=N\otimes_{\mathbb{Z}}\mathbb{C}^{*}\simeq
Hom_{\mathbb{Z}}(M,\mathbb{C}^{*})$ is the torus acting on
$\mathbb{P}_{\Delta}$. The free abelian group of $\mathbb{T}$ -invariant Weil
divisors on $\mathbb{P}_{\Delta}$ form a free abelian group of rank
$|{\Delta}(1)|$, that will be denoted $\mathbb{Z}^{{\Delta}(1)}$. Thus an
element $D\in\mathbb{Z}^{{\Delta}(1)}$ is a sum
$D=\sum_{\rho}a_{\rho}D_{\rho}$. The $\mathbb{T}$-invariant Cartier divisors
form a subgroup
$\mathrm{Div}_{\mathbb{T}}(\mathbb{P}_{\Delta})\subset\mathbb{Z}^{\Delta(1)}$.

Each $m\in M$ gives a character
$\chi^{m}:\mathbb{T}\rightarrow\mathbb{C}^{*}$, and hence $\chi^{m}$ is a
rational function on $\mathbb{P}_{\Delta}$. As is well-known, $\chi^{m}$ gives
the $\mathbb{T}$-invariant Cartier divisor $div(\chi^{m})=-\sum_{\rho}\langle
m,n_{\rho}\rangle D_{\rho}$. We will consider the map

$\begin{array}[]{ccc}M&\longrightarrow&\mathbb{Z}^{\Delta(1)}\\\
m&\longmapsto&D_{m}=\sum_{\rho}\langle m,n_{\rho}\rangle D_{\rho}.\end{array}$

This map is injective since $\Delta(1)$ spans $N_{\mathbb{R}}$. By [17], we
have a commutative diagram

(2)
$\begin{array}[]{ccccccccc}0&\rightarrow&M&\rightarrow&\mathrm{Div}_{\mathbb{T}}(\mathbb{P}_{\Delta})&\rightarrow&\mathrm{Pic}(\mathbb{P}_{\Delta})&\rightarrow&0\\\
&&\shortparallel&&\downarrow&&\downarrow&&\\\
0&\rightarrow&M&\rightarrow&\mathbb{Z}^{\Delta(1)}&\rightarrow&\mathcal{A}_{n-1}(\mathbb{P}_{\Delta})&\rightarrow&0\end{array}$

For each $\rho\in\Delta(1)$, introduce a variable $z_{\rho}$, and consider the
polynomial ring

$\mathrm{S}=\mathbb{C}[z_{\rho}]=\mathbb{C}[z_{\rho}:\rho\in\Delta(1)].$

Note that a monomial $\prod_{\rho}z_{\rho}^{a_{\rho}}$ determines a divisor
$D=\sum_{\rho}a_{\rho}D_{\rho}$ and to emphasize this relationship, we will
write the monomial as $z^{D}$. We will grade $\mathrm{S}$ as follows, the
degree of a monomial $z^{D}\in\mathrm{S}$ is
$\deg(z^{D})=[D]\in\mathcal{A}_{n-1}(\mathbb{P}_{\Delta})$. Using the exact
sequence (2), it follows that two monomials $\prod_{\rho}z_{\rho}^{a_{\rho}}$
and $\prod_{\rho}z_{\rho}^{b_{\rho}}$ in $\mathrm{S}$ have the same degree if
and only if there is some $m\in M$ such that $a_{\rho}=\langle
m,n_{\rho}\rangle+b_{\rho}$ for each $\rho$. Then

$\mathrm{S}=\bigoplus_{\alpha\in
A_{n-1}(\mathbb{P}_{\Delta})}\mathrm{S}_{\alpha},$

where $\mathrm{S}_{\alpha}=\bigoplus_{\deg(z^{D})=\alpha}\mathbb{C}\cdot
z^{D}$. Note also that
$\mathrm{S}_{\alpha}\cdot\mathrm{S}_{\beta}=\mathrm{S}_{\alpha+\beta}$. The
polynomial ring $\mathrm{S}$ is called homogeneous coordinate ring of the
toric variety $\mathbb{P}_{\Delta}$.

Denote by $\mathcal{O}_{\mathbb{P}_{\Delta}}$ the structure sheaf of
$\mathbb{P}_{\Delta}$. Let $\mathcal{O}_{\mathbb{P}_{\Delta}}(D)$ be the
coherent sheaf on $\mathbb{P}_{\Delta}$ determined by a Weil divisor $D$. If
$\alpha=[D]\in\mathcal{A}_{n-1}(\mathbb{P}_{\Delta})$, then it follows from
[14] that

$\mathrm{S}_{\alpha}\simeq\mathrm{H}^{0}(\mathbb{P}_{\Delta},\mathcal{O}_{\mathbb{P}_{\Delta}}(D)),$

moreover, if $\alpha=[D_{1}]$ and $\beta=[D_{2}]$, there is a commutative
diagram

$\begin{array}[]{ccc}\mathrm{S}_{\alpha}\otimes\mathrm{S}_{\beta}&\longrightarrow&\mathrm{S}_{\alpha+\beta}\\\
\downarrow&&\downarrow\\\
\mathrm{H}^{0}(\mathbb{P}_{\Delta},\mathcal{O}_{\mathbb{P}_{\Delta}}(D_{1}))\otimes\mathrm{H}^{0}(\mathbb{P}_{\Delta},\mathcal{O}_{\mathbb{P}_{\Delta}}(D_{2}))&\longrightarrow&\mathrm{H}^{0}(\mathbb{P}_{\Delta},\mathcal{O}_{\mathbb{P}_{\Delta}}(D_{1}+D_{2}))\end{array}$

where the top arrows is the polynomial multiplication. If
$\mathbb{P}_{\Delta}$ is a complete toric variety, then it follows from [14]
that

  * (i)

$\mathrm{S}_{\alpha}$ is finite dimensional for every $\alpha$, and in
particular, $\mathrm{S}_{0}=\mathbb{C}$.

  * (ii)

If $\alpha=[D]$ for an effective divisor $D=\sum_{\rho}a_{\rho}D_{\rho}$ ,
then $dim_{\mathbb{C}}(\mathrm{S}_{\alpha})=\\#(P_{D}\cap M)$, where
$P_{D}=\\{m\in M_{\mathbb{R}}:\langle m,n_{\rho}\rangle\geq-a_{\rho}$ for all
$\rho\\}$.

###  2.2. The toric homogeneous coordinates

Given a toric variety $\mathbb{P}_{\Delta}$, the group of Weil divisor classes
$\mathcal{A}_{n-1}(\mathbb{P}_{\Delta})$ is a finitely generated abelian group
of rank $r=k-n$, where $k=|\Delta(1)|$. If we apply
$Hom_{\mathbb{Z}}(-,\mathbb{C}^{*})$ to the bottom exact sequence of (2), then
we get the exact sequence

$1\longrightarrow
G\longrightarrow(\mathbb{C}^{*})^{\Delta(1)}\longrightarrow\mathbb{T}\longrightarrow
1\,,$

where
$G=Hom_{\mathbb{Z}}(\mathcal{A}_{n-1}(\mathbb{P}_{\Delta}),\mathbb{C}^{*})$.
Since $(\mathbb{C}^{*})^{\Delta(1)}$ acts naturally on
$\mathbb{C}^{\Delta(1)}$, the subgroup $G\subset(\mathbb{C}^{*})^{\Delta(1)}$
acts on $\mathbb{C}^{\Delta(1)}$ by

$g\cdot t=\Big{(}g\big{(}[D_{\rho}]\big{)}\,t_{\rho}\Big{)}\,,$

for $g:\mathcal{A}_{n-1}(\mathbb{P}_{\Delta})\rightarrow\mathbb{C}^{*}$ in
$G$, and $t=(t_{\rho})\in\mathbb{C}^{\Delta(1)}$. The explicit equations for
$G$ as a subgroup of the torus $(\mathbb{C^{*}})^{\Delta(1)}$ is given by

$G=\left\\{(t_{\rho})\in(\mathbb{C}^{*})^{\Delta(1)}\,\middle|\,\prod_{\rho}t_{\rho}^{\left\langle\,m_{i},\,n_{\rho}\right\rangle}=1,\,\,1\leq
i\leq n\right\\},$

where $m_{1},\dots,m_{n}$ is a basis of $M$; see [13].

For each cone $\sigma\in\Delta$, define the monomial

$z^{\widehat{\sigma}}=\prod_{\rho\notin\sigma(1)}z_{\rho}\,,$

which is the product of the variables corresponding to rays not in $\sigma$.
Then define

$\mathcal{Z}=V(z^{\widehat{\sigma}}\,:\,\sigma\in\Delta)\subset\mathbb{C}^{\Delta(1)}.$

We have that $\mathcal{Z}\subset\mathbb{C}^{\Delta(1)}$ has codimension at
least two, and $\mathcal{Z}=\left\\{0\right\\}$ when $r=1$; see [14].

######  Theorem 2.1.

[14] If $\mathbb{P}_{\Delta}$ is a $n$-dimensional toric variety where
$\Delta(1)$ spans $N_{\mathbb{R}}$, then:

  1. (i)

The set $\mathbb{C}^{\Delta(1)}-{\mathcal{Z}}$ is invariant under the action
of the group $G$.

  2. (ii)

$\mathbb{P}_{\Delta}$ is naturally isomorphic to the categorical quotient
$(\mathbb{C}^{\Delta(1)}-{\mathcal{Z}})/G$.

  3. (iii)

$\mathbb{P}_{\Delta}$ is the geometric quotient
$(\mathbb{C}^{\Delta(1)}-{\mathcal{Z}})/G$ if and only if
$\mathbb{P}_{\Delta}$ is an orbifold.

###  2.3. Quasi-smooth hypersurfaces

Let $\mathbb{P}_{\Delta}$ be a complex orbifold. An element
$\alpha\in\mathcal{A}_{n-1}(\mathbb{P}_{\Delta})$ gives the character
$\chi^{\alpha}:G\rightarrow\mathbb{C}^{*}$ that sends $g\in
G=Hom_{\mathbb{Z}}(\mathcal{A}_{n-1}(\mathbb{P}_{\Delta}),\mathbb{C}^{*})$ to
$g(\alpha)\in\mathbb{C}^{*}$. The action of $G$ on $\mathbb{C}^{\Delta(1)}$
induces an action on $\mathrm{S}$ with the property that given
$f\in\mathrm{S}$, we have

$f\in\mathrm{S}_{\alpha}\Leftrightarrow f(g\cdot z)=\chi^{\alpha}(g)f(z),\ \
\forall\ g\in G,\ z\in\mathbb{C}^{\Delta(1)}.$

We say that $f\in\mathrm{S}_{\alpha}$ is quasi-homogeneous of degree $\alpha$.
It follows that the equation $V=\left\\{f=0\right\\}$ is well-defined in
$\mathbb{P}_{\Delta}$ and it defines a hypersurface, we say that
$V=\left\\{f=0\right\\}$ is a quasi-homogeneous hypersurface of degree
$\alpha$. We say that $V=\left\\{f=0\right\\}$ is quasi-smooth if its tangent
cone $\left\\{f=0\right\\}$ on $\mathbb{C}^{\Delta(1)}-{\mathcal{Z}}$ is
smooth. We have the following theorem.

######  Theorem 2.2.

[1] Let $V=\left\\{f=0\right\\}\subset\mathbb{P}_{\Delta}$ be a quasi-
homogeneous hypersurface, then $V$ is quasi-smooth if and only if $V$ is a
suborbifold of $\mathbb{P}_{\Delta}$.

Suppose there is a complex number $a_{\rho}$ for each $\rho\in\Delta(1)$ with
the property that $\sum_{\rho}a_{\rho}n_{\rho}=0$ in $\mathbb{C}^{\Delta(1)}$.
Then, for any class $\alpha\in\mathcal{A}_{n-1}(\mathbb{P}_{\Delta})$, there
is a constant $\theta(\alpha)$ with the property that for any quasi-
homogeneous polynomial $f\in\mathrm{S}$ of degree $\alpha$, we have

(3) $\begin{array}[]{ccccccccc}i_{R}(df)=\theta(\alpha)\cdot f,\end{array}$

where $R=\sum_{\rho}a_{\rho}z_{\rho}\frac{\partial}{\partial z_{\rho}}$ is the
radial vector field $($it is tangent to the orbits of $G$$)$. The identity (3)
is called the Euler formula determined by $\left\\{a_{\rho}\right\\}$ and $R$.
Moreover, considering the $r=k-n$ linearly independent over $\mathbb{Z}$
relations between the $n_{\rho}$, we have $r$ radial vector fields $R_{i}$ and
$Lie(G)=\langle R_{1},\dots,R_{r}\rangle$; see [1].

We shall consider the following subfield of
$\mathbb{C}(z_{\rho})=Frac(\mathbb{C}[z_{\rho}])$ given by

$\widetilde{K}(\mathbb{P}_{\Delta})=\left\\{\frac{P}{Q}\in\mathbb{C}(z_{\rho}):P\in\mathrm{S}_{\alpha},Q\in\mathrm{S}_{\beta}\right\\}.$

Thus, the field of rational functions on $\mathbb{P}_{\Delta}$, denoted by
$K(\mathbb{P}_{\Delta})$, is the subfield of
$\widetilde{K}(\mathbb{P}_{\Delta})$ such that $\deg(P)=\deg(Q)$. It is
follows that the polynomials $P,Q\in\mathrm{S}_{\alpha}$ define a rational
function $\frac{P}{Q}:\mathbb{P}_{\Delta}\dashrightarrow\mathbb{P}^{1}$.

###  2.4. Examples

Let $\mathbb{P}_{\Delta}$ be a $n$-dimensional toric variety where $\Delta(1)$
spans $N_{\mathbb{R}}$. We know that $|\Delta(1)|=n+r$, where $r$ is the rank
of finitely generated abelian group $\mathcal{A}_{n-1}(\mathbb{P}_{\Delta})$.
We will denote $\Delta(1)=\left\\{\rho_{1},\dots,\rho_{n+r}\right\\}$,
$\mathrm{S}=\mathbb{C}\left[z_{1},\dots,z_{n+r}\right]$ and
$D_{i}=D_{\rho_{i}}$ for all $i=1,\dots,n+r$.

  1. (1)

Weighted projective spaces. Let $\omega_{0},\dots,\omega_{n}$ be positive
integers with $gcd(\omega_{0},\dots,\omega_{n})=1$. Set
$\omega=(\omega_{0},\dots,\omega_{n})$. Consider the lattice
$N=\mathbb{Z}^{n+1}/\mathbb{Z}\cdot\omega$. The dual lattice is

$M=\left\\{(a_{0},\dots,a_{n})\in\mathbb{Z}^{n+1}\,|\,a_{0}\omega_{0}+\cdots+a_{n}\omega_{n}=0\right\\}.$

Denote by $e_{0},\dots,e_{n}$ the standard basis of $\mathbb{Z}^{n+1}$. We
have the exact sequence

$0\longrightarrow
M\stackrel{{\scriptstyle\alpha}}{{\longrightarrow}}\mathbb{Z}^{n+1}\stackrel{{\scriptstyle\beta}}{{\longrightarrow}}\mathbb{Z}\longrightarrow
0,$

where $\alpha(m)=\left(\left\langle
m,\bar{e}_{0}\right\rangle,\dots,\left\langle
m,\bar{e}_{n}\right\rangle\right)$ and
$\beta(a_{0},\dots,a_{n})=a_{0}\omega_{0}+\cdots+a_{n}\omega_{n}$. Let
$\Delta$ be the fan made up of the cones generated by all the proper subsets
of $\\{\bar{e}_{0},\dots,\bar{e}_{n}\\}$. Then, $G$ is given by
$G=\left\\{(t^{\omega_{0}},\dots,t^{\omega_{n}})\,|\,t\in\mathbb{C}^{*}\right\\}\simeq\mathbb{C}^{*}$
and its action on $\mathbb{C}^{n+1}$ is given by

$t\cdot(z_{0},\dots,z_{n})=(t^{\omega_{0}}z_{0},\dots,t^{\omega_{n}}z_{n}).$

Since $\Delta$ is simplicial and has $n+1$ rays, we have
$\mathcal{Z}=\left\\{0\right\\}$ and

$\mathbb{P}(\omega):=\mathbb{P}_{\Delta}=\left(\mathbb{C}^{n+1}-\\{0\\}\right)/\mathbb{C}^{*},$

the usual representation of weighted projective spaces as a quotient. If
$\omega_{0}=\cdots=\omega_{n}=1$, then $\mathbb{P}(\omega)=\mathbb{P}^{n}$ and
when $\omega_{0},\dots,\omega_{n}$ are two by two co-primes, we have

$Sing(\mathbb{P}(\omega))=\left\\{\bar{e}_{i}\,|\,\omega_{i}>1\right\\}.$

Moreover, we have $\mathcal{A}_{n-1}(\mathbb{P}(\omega))\simeq\mathbb{Z}$ and
$\deg(z_{i})=\omega_{i}$ for all $0\leq i\leq n+1$. Consequently the
homogeneous coordinate ring of $\mathbb{P}(\omega)$ is given by
$\mathrm{S}=\oplus_{\alpha\geq 0}\mathrm{S}_{\alpha}$, where

$\mathrm{S}_{\alpha}=\bigoplus_{p_{0}\omega_{0}+\cdots+p_{n}\omega_{n}=\alpha}\mathbb{C}\cdot{z_{0}}^{p_{0}}\dots\,{z_{n}}^{p_{n}}.$

  2. (2)

Multiprojective spaces. Let $e_{1,1},\dots,e_{1,n}$ be a basis of
$N_{1}\simeq\mathbb{Z}^{n}$, and set $e_{1,0}=-e_{1,1}-\cdots-e_{1,n}$. Let
$\Delta_{1}$ be the fan in $(N_{1})_{\mathbb{R}}$ made up of the cones
generated by all the proper subsets of
$\left\\{e_{1,0},\dots,e_{1,n}\right\\}$, then
$\mathbb{P}_{\Delta_{1}}=\mathbb{P}^{n}$. Similarly, let
$e_{2,1},\dots,e_{2,m}$ be a basis of $N_{2}\simeq\mathbb{Z}^{m}$ and set
$e_{2,0}=-e_{2,1}-\cdots-e_{2,m}$, then we have
$\mathbb{P}_{\Delta_{2}}=\mathbb{P}^{m}$. If $N=N_{1}\oplus N_{2}$, then
$\Delta=\Delta_{1}\times\Delta_{2}$ is a fan in $N_{\mathbb{R}}$ and
$\mathbb{P}_{\Delta}=\mathbb{P}_{\Delta_{1}}\times\mathbb{P}_{\Delta_{2}}=\mathbb{P}^{n}\times\mathbb{P}^{m}$.
Denote $a=(a_{1},\dots,a_{n})\in\mathbb{Z}^{n}$ and
$b=(b_{1},\dots,b_{m})\in\mathbb{Z}^{m}$, we have the exact sequence

$0\longrightarrow\mathbb{Z}^{n}\oplus\mathbb{Z}^{m}\stackrel{{\scriptstyle\alpha}}{{\longrightarrow}}\mathbb{Z}^{n+1}\oplus\mathbb{Z}^{m+1}\stackrel{{\scriptstyle\beta}}{{\longrightarrow}}\mathbb{Z}\oplus\mathbb{Z}\longrightarrow
0,$

where $\alpha(a,b)=(-a_{1}-\cdots-a_{n},a,-b_{1}-\cdots-b_{m},b)$ and
$\beta(a_{0},a,b_{0},b)=(a_{0}+\cdots+a_{n},b_{0}+\cdots+b_{m})$. Then, $G$ is
given by
$G=\left\\{(\mu,\dots,\mu,\lambda,\dots,\lambda)\in(\mathbb{C}^{*})^{n+1}\times(\mathbb{C}^{*})^{m+1}\right\\}\simeq\mathbb{C}^{*}\times\mathbb{C}^{*}$
and its action on $\mathbb{C}^{n+1}\times\mathbb{C}^{m+1}$ is given by

$(\mu,\lambda)\cdot(z_{1},z_{2})=(\mu\cdot z_{1},\lambda\cdot z_{2})=(\mu
z_{1,0},\dots,\mu z_{1,n},\lambda z_{2,0},\dots,\lambda z_{2,m}).$

It is possible to show that
$\mathcal{Z}=\left(\left\\{0\right\\}\times\mathbb{C}^{m+1}\right)\cup\left(\mathbb{C}^{n+1}\times\left\\{0\right\\}\right)$.
So, we have

$\mathbb{P}^{n}\times\mathbb{P}^{m}=\left(\mathbb{C}^{n+1}\times\mathbb{C}^{m+1}-\mathcal{Z}\right)\,/\,(\mathbb{C}^{*})^{2},$

the usual representation of $\mathbb{P}^{n}\times\mathbb{P}^{m}$ as a quotient
space.

Moreover, we have
$\mathcal{A}_{n+m-1}(\mathbb{P}^{n}\times\mathbb{P}^{m})\simeq\mathbb{Z}^{2}$
and $\deg(z_{1,i})=(1,0)$ for all $0\leq i\leq n$, $\deg(z_{2,j})=(0,1)$ for
all $0\leq j\leq m$. Consequently the homogeneous coordinate ring of
$\mathbb{P}^{n}\times\mathbb{P}^{m}$ is given by
$\mathrm{S}=\oplus_{\alpha,\,\beta\geq 0}\mathrm{S}_{(\alpha,\beta)}$, where

$\mathrm{S}_{(\alpha,\beta)}=\bigoplus_{p_{0}+\cdots+p_{n}=\alpha\,;\,q_{0}+\cdots+q_{m}=\beta}\mathbb{C}\cdot{z_{1,0}}^{p_{0}}\dots\,{z_{1,n}}^{p_{n}}{z_{2,0}}^{q_{0}}\dots\,{z_{2,m}}^{q_{m}},$

the ring of bihomogeneous polynomials of bidegree $(\alpha,\beta)$.

  3. (3)

Rational scrolls. Let $N\simeq\mathbb{Z}^{2}$ and $1\leq a\leq b$ integers.
Consider the polygon

$P_{a,b}=Conv(0,ae_{1},e_{2},be_{1}+e_{2})\subset
M_{\mathbb{R}}\simeq\mathbb{R}^{2}.$

The polygon $P_{a,b}$ has $a+b+2$ lattice points. The rational normal scroll
$\mathbb{F}(a,b)$ is the toric surface associated to the normal fan of
$P_{a,b}$. The rational normal scroll is a smooth projective surface because
$P_{a,b}$ is full dimensional smooth lattice polytope.

Consider the map
$\varphi:\mathbb{C}^{*}\times\mathbb{C}^{*}\rightarrow\mathbb{P}^{a+b+2}$

$\varphi(s,t)=(1:s:s^{2}:\dots:s^{a}:t:st:s^{2}t:\dots:s^{b}t),$

then $P_{a,b}$ is the Zariski closure of the image. Rewriting the map as
$\tilde{\varphi}:\mathbb{C}\times\mathbb{P}^{1}\rightarrow\mathbb{P}^{a+b+2}$

$\tilde{\varphi}(s,(t_{1}:t_{2}))=(t_{1}:st_{1}:s^{2}t_{1}:\dots:s^{a}t_{1}:t_{2}:st_{2}:s^{2}t_{2}:\dots:s^{b}t_{2}),$

we have that $s\mapsto\tilde{\varphi}(s,(1:0))$ and
$s\mapsto\tilde{\varphi}(s,(0:1))$ are the rational normal curves
$\mathcal{C}_{a}\subset\mathbb{P}^{a+1}\subset\mathbb{P}^{a+b+2}$ and
$\mathcal{C}_{b}\subset\mathbb{P}^{b+1}\subset\mathbb{P}^{a+b+2}$.

The rational normals scroll are Hirzebruch surfaces because the normal fan of
$P_{a,b}$ defines a Hirzebruch surface $\mathcal{H}_{b-a}$, so
$\mathbb{F}(a,b)\simeq\mathcal{H}_{b-a}$. Analogously, for a $n$-dimensional
rational scroll, consider $1\leq a_{1}\leq a_{2}\leq\dots\leq a_{n}$ integers,
we have a full dimensional smooth lattice polytope
$P_{a_{1},\dots,a_{n}}\subset\mathbb{R}^{n}$ having $2n$ lattice points as
vertices. The rational normal scroll $\mathbb{F}(a_{1},\dots,a_{n})$ is the
smooth projective toric variety associated to the normal fan of
$P_{a_{1},\dots,a_{n}}$. It is possible to show that
$\mathbb{F}(a_{1},\dots,a_{n})\simeq\mathbb{P}(\mathcal{O}_{\mathbb{P}^{1}}(a_{1})\oplus\dots\oplus\mathcal{O}_{\mathbb{P}^{1}}(a_{n}))$.
For more details see [13].

In general, let $a_{1},\ldots,a_{n}$ be integers. Consider the
$(\mathbb{C}^{*})^{2}$ action on $\mathbb{C}^{2}\times\mathbb{C}^{n}$ given as
follows

$(\lambda,\mu)(z_{1,1},z_{1,2},z_{2,1},\dots,z_{2,n})=(\lambda z_{1,1},\lambda
z_{1,2},\mu\lambda^{-a_{1}}z_{2,1},\dots\mu\lambda^{-a_{n}}z_{2,n}).$

Then

$\mathbb{F}(a_{1},\dots,a_{n})=(\mathbb{C}^{2}\times\mathbb{C}^{n}-\mathcal{Z})\,/\,(\mathbb{C}^{*})^{2},$

where
$\mathcal{Z}=\left(\left\\{0\right\\}\times\mathbb{C}^{n}\right)\cup\left(\mathbb{C}^{2}\times\left\\{0\right\\}\right)$.

Moreover, we have
$\mathcal{A}_{n-1}(\mathbb{F}(a_{1},\dots,a_{n}))\simeq\mathbb{Z}^{2}$ and the
homogeneous coordinate ring associated to $\mathbb{F}(a_{1},\dots,a_{n})$ is
given by $\mathrm{S}=\oplus_{\alpha\in\mathbb{Z},\,\beta\geq
0}\mathrm{S}_{(\alpha,\beta)}$, where

$\mathrm{S}_{(\alpha,\beta)}=\bigoplus_{\alpha=p_{1}+p_{2}-\sum_{i}q_{i}a_{i}\,;\,\beta=\sum_{i}q_{i}}\mathbb{C}\cdot{z_{1,1}}^{p_{1}}{z_{1,2}}^{p_{2}}{z_{2,1}}^{q_{1}}\dots\,{z_{2,n}}^{q_{n}}.$

In particular $\deg(z_{1,1})=\deg(z_{1,2})=(1,0)$ and
$\deg(z_{2,i})=(-a_{i},1)$. Thus the total coordinate rings can have some
elements with effective degree and other elements without. Finally we have

$\mathbb{F}(a_{1},\dots,a_{n})=\mathbb{F}(b_{1},\dots,b_{n})\Longleftrightarrow\left\\{a_{1},\dots,a_{n}\right\\}=\left\\{b_{1}+c,\dots,b_{n}+c\right\\},$

for some $c\in\mathbb{Z}$. For more details see [24, pp. 14].

  4. (4)

A toric surface. Let us consider an example where $G$ and
$\mathcal{A}_{n-1}(\mathbb{P}_{\Delta})$ have torsion. Let $\Delta$ be a
complete simplicial fan in $\mathbb{Z}^{2}$ with edges along
$v_{1}=2e_{1}-e_{2}$, $v_{2}=-e_{1}+2e_{2}$ and $v_{3}=-e_{1}-e_{2}$. Then
$\mathbb{P}_{\Delta}:=\mathbb{P}_{\Delta(0,2,1)}$ is a compact orbifold toric
surface. We have the exact sequence

$0\longrightarrow
M\stackrel{{\scriptstyle\iota}}{{\longrightarrow}}\mathbb{Z}^{3}\stackrel{{\scriptstyle\pi}}{{\longrightarrow}}\frac{\,\,\,\mathbb{Z}^{3}}{Im(\iota)}\longrightarrow
0,$

where $\iota(a,b)=(2a-b)D_{1}+(-a+2b)D_{2}+(-a-b)D_{3}$. We have

$\mathcal{A}_{1}(\mathbb{P}_{\Delta(0,2,1)})=\frac{\,\,\,\mathbb{Z}^{3}}{Im(\iota)}=\frac{\mathbb{Z}D_{1}+\mathbb{Z}D_{2}+\mathbb{Z}D_{3}}{\mathbb{Z}(2D_{1}-D_{2}-D_{3})+\mathbb{Z}(-D_{1}+2D_{2}-D_{3})}$

therefore

$\mathcal{A}_{1}(\mathbb{P}_{\Delta(0,2,1)})\simeq\frac{\mathbb{Z}D_{1}+\mathbb{Z}D_{2}}{\mathbb{Z}(3D_{1}-3D_{2})}\simeq\frac{\mathbb{Z}D_{1}+\mathbb{Z}(D_{1}-D_{2})}{\mathbb{Z}3(D_{1}-D_{2})}\simeq\mathbb{Z}\oplus\mathbb{Z}_{3},$

and

$\pi(aD_{1}+bD_{2}+cD_{3})=\big{(}a+b+c,\left[2b+c\right]\big{)}.$

Note that $G\subset(\mathbb{C}^{*})^{3}$ is given by

$G=\left\\{(t_{1},t_{2},t_{3})\,|\,t_{1}^{2}=t_{2}t_{3}\,,\,t_{2}^{2}=t_{1}t_{3}\right\\}=\left\\{(\omega
t,t,\omega^{2}t)\,|\,t\in\mathbb{C}^{*}\,,\,\omega^{3}=1\right\\},$

that is $G\simeq\mathbb{C}^{*}\times\mu_{3}$ and its action on
$\mathbb{C}^{3}$ is given by

$(t,\omega)\cdot(z_{1},z_{2},z_{3})=(t\omega z_{1},tz_{2},t\omega^{2}z_{3}).$

Since $\Delta$ is simplicial and $\mathcal{A}_{1}(\mathbb{P}_{\Delta(0,2,1)})$
has rank $1$, we have $\mathcal{Z}=\left\\{0\right\\}$ and

$\mathbb{P}_{\Delta(0,2,1)}=\left(\mathbb{C}^{3}-\\{0\\}\right)/\left(\mathbb{C}^{*}\times\mu_{3}\right).$

The singular set of $\mathbb{P}_{\Delta(0,2,1)}$ is

$Sing(\mathbb{P}_{\Delta(0,2,1)})=\left\\{\left[1,0,0\right],\left[0,1,0\right],\left[0,0,1\right]\right\\}.$

Moreover, we have $\deg(z_{1})=(1,\left[0\right])$,
$\deg(z_{2})=(1,\left[2\right])$ and $\deg(z_{3})=(1,\left[1\right])$.
Consequently the homogeneous coordinates ring of $\mathbb{P}_{\Delta(0,2,1)}$
is given by $\mathrm{S}=\oplus_{\alpha\geq
0,\,\beta\in\mathbb{Z}_{3}}\mathrm{S}_{(\alpha,\beta)}$, where

$\mathrm{S}_{(\alpha,\beta)}=\bigoplus_{\alpha=m_{1}+m_{2}+m_{3}\,;\,\beta=\left[2m_{2}+m_{3}\right]}\mathbb{C}\cdot{z_{1}}^{m_{1}}{z_{2}}^{m_{2}}{z_{3}}^{m_{3}}.$

For another interesting example where $\mathcal{A}_{n-1}(\mathbb{P}_{\Delta})$
have torsion, consider the toric variety $\mathbb{P}_{\Delta}$ associated to
the normal fan of octahedron $P=Conv(\pm e_{1},\pm e_{2},\pm e_{3})\subset
M_{\mathbb{R}}\simeq\mathbb{R}^{3}$, then $\mathbb{P}_{\Delta}$ is a
projective toric threefold $($is not an orbifold$)$ and
$\mathcal{A}_{2}(\mathbb{P}_{\Delta})=\mathbb{Z}^{5}\oplus\mathbb{Z}_{2}\oplus\mathbb{Z}_{2}$.

##  3\. One-dimensional foliations

Let $\mathbb{P}_{\Delta}$ be an complete simplicial toric varity of dimension
$n$. Let $r$ be the rank of finitely generated abelian group
$\mathcal{A}_{n-1}(\mathbb{P}_{\Delta})$. There exist an exact sequence known
by generalized Euler’s sequence [1]

$0\rightarrow\mathcal{O}_{\mathbb{P}_{\Delta}}^{\oplus
r}\rightarrow\bigoplus_{i=1}^{n+r}\mathcal{O}_{\mathbb{P}_{\Delta}}(D_{i})\rightarrow\mathcal{T}\mathbb{P}_{\Delta}\rightarrow
0,$

where
$\mathcal{T}\mathbb{P}_{\Delta}=\mathcal{H}om(\Omega_{\mathbb{P}_{\Delta}}^{1},\mathcal{O}_{\mathbb{P}_{\Delta}})$
is the so-called Zariski tangent sheaf of $\mathbb{P}_{\Delta}$. Let
$i:\mathbb{P}_{\Delta,reg}\rightarrow\mathbb{P}_{\Delta}$ be the inclusion of
regular part
$\mathbb{P}_{\Delta,reg}=\mathbb{P}_{\Delta}-Sing(\mathbb{P}_{\Delta})$. Since
$\mathbb{P}_{\Delta}$ is a complex orbifold then
$\mathcal{T}\mathbb{P}_{\Delta}\simeq
i_{\ast}\Theta_{\mathbb{P}_{\Delta,reg}}$, where
$\Theta_{\mathbb{P}_{\Delta,reg}}$ is the tangent sheaf of
$\mathbb{P}_{\Delta,reg}$; see [12, Appendix A.2].

Let
$\mathcal{O}_{\mathbb{P}_{\Delta}}(d_{1},\dots,d_{n+r})=\mathcal{O}_{\mathbb{P}_{\Delta}}(\sum_{i=1}^{n+r}d_{i}D_{i})$
be, where $\sum_{i=1}^{n+r}d_{i}D_{i}$ is a Weil divisor. Tensorizing the
Euler’s sequence by $\mathcal{O}_{\mathbb{P}_{\Delta}}(d_{1},\dots,d_{n+r})$
we get

$0\rightarrow\mathcal{O}_{\mathbb{P}_{\Delta}}(d_{1},\dots,d_{n+r})^{\oplus
r}\rightarrow\bigoplus_{i=1}^{n+r}\mathcal{O}_{\mathbb{P}_{\Delta}}(d_{1},\dots,d_{i}+1,\dots,d_{n+r})\rightarrow\mathcal{T}\mathbb{P}_{\Delta}(d_{1},\dots,d_{n+r})\rightarrow
0,$

where
$\mathcal{T}\mathbb{P}_{\Delta}(d_{1},\dots,d_{n+r})=\mathcal{T}\mathbb{P}_{\Delta}\otimes\mathcal{O}_{\mathbb{P}_{\Delta}}(d_{1},\dots,d_{n+r})$.

######  Definition 3.1.

A one-dimensional holomorphic foliation $\mathcal{F}$ on $\mathbb{P}_{\Delta}$
of degree
$\left[\sum_{i=1}^{n+r}d_{i}D_{i}\right]\in\mathcal{A}_{n-1}(\mathbb{P}_{\Delta})$
is a global section of
$\mathcal{T}\mathbb{P}_{\Delta}\otimes\mathcal{O}_{\mathbb{P}_{\Delta}}(d_{1},\dots,d_{n+r})$.
For simplicity of notation we say that $\mathcal{F}$ has degree
$(d_{1},\dots,d_{n+r})$.

From above exact sequence, we conclude that a foliation on
$\mathbb{P}_{\Delta}$ of degree $(d_{1},\dots,d_{n+r})$ is given by a
polynomial vector in homogeneous coordinates of the form

$X=\sum_{i=1}^{n+r}P_{i}\frac{\partial}{\partial z_{i}},$

where $P_{i}$ is a polynomial of degree $(d_{1},\dots,d_{i}+1,\dots,d_{n+r})$
for all $i=1,\dots,n+r$, modulo addition of a vector field of the form
$\sum_{i=1}^{r}g_{i}R_{i}$, where the $R_{1},\dots,R_{r}$ are the radial
vector field. We say that $X$ is a quasi-homogeneous vector field. Moreover we
have

$Sing(\mathcal{F})=\pi\left(\\{p\in\mathbb{C}^{n+r}:R_{1}\wedge\cdots\wedge
R_{r}\wedge X(p)=0\\}\right),$

where $\pi:(\mathbb{C}^{n+r}-{\mathcal{Z}})/G\rightarrow\mathbb{P}_{\Delta}$
is the canonical projection.

Let $\mathcal{F}$ be a foliation on $\mathbb{P}_{\Delta}$ and $V=\\{f=0\\}$ a
quasi-homogeneous hypersurface. We recall that $V$ is invariant by
$\mathcal{F}$ if and only if $X(f)=g\cdot f$, where $X$ is a quasi-homogeneous
vector field which defines $\mathcal{F}$ in homogeneous coordinates.

###  3.1. Examples

  1. (1)

Weighted projective spaces. The Euler’s sequence on $\mathbb{P}(\omega)$ is an
exact sequence of orbibundle

$0\longrightarrow\underline{\mathbb{C}}\longrightarrow\bigoplus_{i=0}^{n}\mathcal{O}_{\mathbb{P}(\omega)}(\omega_{i})\longrightarrow
T\mathbb{P}(\omega)\longrightarrow 0,$

where $\underline{\mathbb{C}}$ is the trivial line orbibundle on
$\mathbb{P}(\omega)$, see [20]. Then, a one-dimensional holomorphic foliation
$\mathcal{F}$ on $\mathbb{P}(\omega)$ of degree $d$ is a global section of
$\mathcal{T}\mathbb{P}(\omega)\otimes\mathcal{O}_{\mathbb{P}(\omega)}(d)$.
Here, the radial vector field is given by
$R=\omega_{0}z_{0}\frac{\partial}{\partial
z_{0}}+\cdots+\omega_{n}z_{n}\frac{\partial}{\partial z_{n}}$.

  2. (2)

Multiprojective spaces. The Euler’s sequence over $\mathbb{P}^{n}$

$0\longrightarrow\underline{\mathbb{C}}\;\;{\longrightarrow}\;\;\mathcal{O}_{\mathbb{P}^{n}}(1)^{\oplus
n+1}\;\;{\longrightarrow}\;\;T\mathbb{P}^{n}\longrightarrow 0,$

gives, by direct summation, the exact sequence

$0\longrightarrow\underline{\mathbb{C}^{2}}\,{\longrightarrow}\mathcal{O}_{\mathbb{P}^{n}\times\mathbb{P}^{m}}(1,0)^{\oplus
n+1}\oplus\mathcal{O}_{\mathbb{P}^{n}\times\mathbb{P}^{m}}(0,1)^{\oplus
m+1}{\longrightarrow}\,T(\mathbb{P}^{n}\times\mathbb{P}^{m})\longrightarrow
0.$

Then, a one-dimensional holomorphic foliation $\mathcal{F}$ on
$\mathbb{P}^{n}\times\mathbb{P}^{m}$ of bidegree $(\alpha,\beta)$ is a global
section of
$T(\mathbb{P}^{n}\times\mathbb{P}^{m})\otimes\mathcal{O}_{\mathbb{P}^{n}\times\mathbb{P}^{m}}(\alpha,\beta)$.
Here, the radial vector fields are given by
$R_{1}=z_{1,0}\frac{\partial}{\partial
z_{1,0}}+\cdots+z_{1,n}\frac{\partial}{\partial z_{1,n}}$ and
$R_{2}=z_{2,0}\frac{\partial}{\partial
z_{2,0}}+\cdots+z_{2,m}\frac{\partial}{\partial z_{2,m}}$.

  3. (3)

Rational scrolls. The Euler’s sequence on
$\mathbb{F}(a):=\mathbb{F}(a_{1},\dots,a_{n})$ is

$0\rightarrow\mathcal{O}_{\mathbb{F}(a)}^{\oplus
2}\rightarrow\mathcal{O}_{\mathbb{F}(a)}(1,0)^{\oplus
2}\oplus\bigoplus_{i=1}^{n}\mathcal{O}_{\mathbb{F}(a)}(-a_{i},1)\rightarrow\mathcal{T}\mathbb{F}(a)\rightarrow
0.$

Then, a one-dimensional holomorphic foliation $\mathcal{F}$ on $\mathbb{F}(a)$
of bidegree $(d_{1},d_{2})$ is a global section of
$T\mathbb{F}(a)\otimes\mathcal{O}_{\mathbb{F}(a)}(d_{1},d_{2})$. Here, the
radial vector fields are given by
$R_{1}=z_{1,1}\frac{\partial}{\partial
z_{1,1}}+z_{1,2}\frac{\partial}{\partial
z_{1,2}}+\sum_{i=1}^{n}-a_{i}z_{2,i}\frac{\partial}{\partial z_{2,i}}$ and
$R_{2}=\sum_{i=1}^{n}z_{2,i}\frac{\partial}{\partial z_{2,i}}$.

  4. (4)

A toric surface. The Euler’s sequence on $\mathbb{P}_{\Delta(0,2,1)}$ is given
by

$0\rightarrow\mathcal{O}_{\mathbb{P}_{\Delta(0,2,1)}}\rightarrow\mathcal{O}_{\mathbb{P}_{\Delta(0,2,1)}}(1,\left[0\right])\oplus\mathcal{O}_{\mathbb{P}_{\Delta(0,2,1)}}(1,\left[2\right])\oplus\mathcal{O}_{\mathbb{P}_{\Delta(0,2,1)}}(1,\left[1\right])\rightarrow\mathcal{T}\mathbb{P}_{\Delta(0,2,1)}\rightarrow
0.$

Then, a one-dimensional holomorphic foliation $\mathcal{F}$ on
$\mathbb{P}_{\Delta(0,2,1)}$ of bidegree $(\alpha,\beta)$ is a global section
of
$\mathcal{T}\mathbb{P}_{\Delta(0,2,1)}\otimes\mathcal{O}_{\mathbb{P}_{\Delta(0,2,1)}}(\alpha,\beta)$.
Here, the radial vector field is given by $R=z_{1}\frac{\partial}{\partial
z_{1}}+z_{2}\frac{\partial}{\partial z_{2}}+z_{3}\frac{\partial}{\partial
z_{3}}$.

##  4\. Poincaré problem

Let $\mathbb{P}_{\Delta}=(\mathbb{C}^{n+r}-{\mathcal{Z}})/G$ be a complete
simplicial toric varity of dimension $n$.

######  Definition 4.1.

Let $f\in\mathbb{C}[z_{1},\dots,z_{n+r}]$ be a quasi-homogeneous polynomial.
We say that $V=\left\\{f=0\right\\}\subset\mathbb{P}_{\Delta}$ is strongly
quasi-smooth if its tangent cone $\left\\{f=0\right\\}$ on
$\mathbb{C}^{n+r}-\\{0\\}$ is smooth. Note that strongly quasi-smooth implies
quasi-smooth.

Consider the $r$ linearly independent over $\mathbb{Z}$ relations between the
$n_{\rho_{1}},\dots,n_{\rho_{n+r}}$

$\sum_{j=1}^{n+r}a_{i,j}n_{\rho_{j}}=0,\,\,\,i=1,\dots,r.$

######  Theorem 4.1 (Normal form 1).

Let $V=\left\\{f=0\right\\}\subset\mathbb{P}_{\Delta}$ be a strongly quasi-
smooth hypersurface of degree
$\alpha\in\mathcal{A}_{n-1}(\mathbb{P}_{\Delta})$. If $X$ is a quasi-
homogeneous vector field that leaves $V$ invariant, then

$X=\sum_{j<k}P_{j,k}^{\,i}\left(\frac{\partial f}{\partial
z_{j}}\frac{\partial}{\partial z_{k}}-\frac{\partial f}{\partial
z_{k}}\frac{\partial}{\partial
z_{j}}\right)+\frac{g}{\theta_{i}(\alpha)}\sum_{j=1}^{n+r}a_{i,j}z_{j}\frac{\partial}{\partial
z_{j}}\,,\,i=1,\dots,r.$

where $P_{j,k}^{\,i},\,g\in\mathbb{C}[z_{1},\dots,z_{n+r}]$ are quasi-
homogeneous polynomials. Here, $\theta_{i}(\alpha)$ is a complex number
defined in subsection 2.3.

###### Proof.

We follow the ideas of Zariski-Esteves using the Koszul complex in [16]. Then,
we have that $X=\sum_{j=1}^{n+r}P_{j}\frac{\partial}{\partial z_{j}}$. Euler’s
generalized formula implies that $i_{R_{i}}(df)=\theta_{i}(\alpha)\cdot f$ for
all $i=1,\dots,r$. That is

$\theta_{i}(\alpha)\cdot f=\sum_{j=1}^{n+r}a_{i,j}z_{j}\frac{\partial
f}{\partial z_{j}}\,,\,i=1,\dots,r.$

The invariance of $V$ implies that

$g\cdot f=\sum_{j=1}^{n+r}P_{j}\frac{\partial f}{\partial z_{j}},$

for some polynomial $g\in\mathbb{C}[z_{1},\dots,z_{n+r}]$. From these two
equations we obtain the following polynomial relationship

$\sum_{j=1}^{n+r}\Big{(}P_{j}-\frac{g}{\theta_{i}(\alpha)}\,a_{i,j}z_{j}\Big{)}\frac{\partial
f}{\partial z_{j}}=0\,,\,i=1,\dots,r.$

This identity says that the vector fields

$X^{\prime}_{i}=X-\frac{g}{\theta_{i}(\alpha)}\sum_{j=1}^{n+r}a_{i,j}z_{j}\frac{\partial}{\partial
z_{j}}\,,\,i=1,\dots,r$

satisfies $\partial_{1}(X^{\prime}_{i})=0$, that is, $X^{\prime}_{i}\in
Ker(\partial_{1})$, where $\partial_{1}$ is the operator of the Koszul complex
associated with $(\frac{\partial f}{\partial z_{1}},\dots,\frac{\partial
f}{\partial z_{n+r}})$; see [19, pp. 688]. By hypothesis, the singular set of
the hypersurface consists of

$\Big{\\{}\frac{\partial f}{\partial z_{1}}=\cdots=\frac{\partial f}{\partial
z_{n+r}}=0\Big{\\}}=\\{0\\},$

we have that $(\frac{\partial f}{\partial z_{1}},\dots,\frac{\partial
f}{\partial z_{n+r}})$ is a regular sequence and consequently
$Ker(\partial_{1})=Im(\partial_{2})$. Therefore

$X^{\prime}_{i}=X-\frac{g}{\theta_{i}(\alpha)}\sum_{j=1}^{n+r}a_{i,j}z_{j}\frac{\partial}{\partial
z_{j}}=\sum_{j<k}P_{j,k}^{\,i}\left(\frac{\partial f}{\partial
z_{j}}\frac{\partial}{\partial z_{k}}-\frac{\partial f}{\partial
z_{k}}\frac{\partial}{\partial z_{j}}\right),\,i=1,\dots,r.$

Hence

$X=\sum_{j<k}P_{j,k}^{\,i}\left(\frac{\partial f}{\partial
z_{j}}\frac{\partial}{\partial z_{k}}-\frac{\partial f}{\partial
z_{k}}\frac{\partial}{\partial
z_{j}}\right)+\frac{g}{\theta_{i}(\alpha)}\sum_{j=1}^{n+r}a_{i,j}z_{j}\frac{\partial}{\partial
z_{j}}\,,\,i=1,\dots,r.$

∎

The group of Weil divisor classes group
$\mathcal{A}_{n-1}(\mathbb{P}_{\Delta})$ is a finitely generated abelian group
of rank $r$. By the fundamental theorem of finitely generated abelian groups,
we have

$\mathcal{A}_{n-1}(\mathbb{P}_{\Delta})\simeq\mathbb{Z}^{\,r}\oplus\mathbb{Z}_{(p_{1})^{\lambda_{1}}}\oplus\cdots\oplus\mathbb{Z}_{(p_{m})^{\lambda_{m}}},$

where the $p_{i}$ are primes, not necessarily distinct, and the $\lambda_{i}$
are positive integers. The direct product is unique except for possible
rearrangement of the factors.

Suppose that the homogeneous coordinate ring $\mathrm{S}$ of
$\mathbb{P}_{\Delta}$ has the following property; there is a positive integer
number $1\leq k\leq r$ such that

(4) $\begin{array}[]{ccccccccc}(\deg\mathrm{P})_{k}\geq 0\,\,\textup{for
all}\,\,\mathrm{P}\in\mathrm{S},\end{array}$

where the bottom index $k$ denotes the $k$-th coordinate in
$\mathcal{A}_{n-1}(\mathbb{P}_{\Delta})$ i.e. the $k$-th component of the
multidegree. The first result is the following:

######  Theorem 4.2 (Poincaré problem $1$).

Let $\mathbb{P}_{\Delta}$ be a complete simplicial toric varity of dimension
$n$. Let $\mathcal{F}$ be a one-dimensional holomorphic foliation on
$\mathbb{P}_{\Delta}$ and $X$ the quasi-homogeneous vector field which defines
$\mathcal{F}$ in homogeneous coordinates. Let
$V=\left\\{f=0\right\\}\subset\mathbb{P}_{\Delta}$ be a strongly quasi-smooth
hypersurface, invariant by $\mathcal{F}$. Suppose that $X\notin Lie(G)$, then

$\deg(V)_{k}\leq\deg(\mathcal{F})_{k}+\max_{1\leq i<j\leq
n+r}\big{\\{}deg(z_{i})_{k}+deg(z_{j})_{k}\big{\\}},$

for each $1\leq k\leq r$ as in the hypothesis $(\ref{eq0})$.

###### Proof.

By Theorem 4.1, the quasi-homogeneous vector field $X$ that defines
$\mathcal{F}$ is given by

$X=\sum_{j<k}P_{j,k}^{\,i}\left(\frac{\partial f}{\partial
z_{j}}\underline{\frac{\partial}{\partial z_{k}}}-\frac{\partial f}{\partial
z_{k}}\frac{\partial}{\partial
z_{j}}\right)+\frac{g}{\theta_{i}(\alpha)}\sum_{j=1}^{n+r}a_{i,j}z_{j}\frac{\partial}{\partial
z_{j}}\,,\,i=1,\dots,r.$

where $P_{j,k}^{\,i},\,g\in\mathbb{C}[z_{1},\dots,z_{n+r}]$ are quasi-
homogeneous polynomials. As $X\notin Lie(G)=\langle R_{1},\dots,R_{r}\rangle$,
we have $P_{j,k}^{\,i}\neq 0$ for some $j<k$. Then

$deg\Big{(}P_{j,k}^{\,i}\cdot\frac{\partial f}{\partial
z_{j}}\Big{)}=(d_{k}+1)\left[D_{k}\right]+\sum_{i\neq
k}d_{i}\left[D_{i}\right].$

Therefore

$\displaystyle\deg(P_{j,k}^{\,i})$ $\displaystyle=$
$\displaystyle(d_{k}+1)\left[D_{k}\right]+\sum_{i\neq
k}d_{i}\left[D_{i}\right]-\deg\big{(}\frac{\partial f}{\partial z_{j}}\big{)}$
$\displaystyle=$ $\displaystyle(d_{k}+1)\left[D_{k}\right]+\sum_{i\neq
k}d_{i}\left[D_{i}\right]-\big{(}\deg(V)-\left[D_{j}\right]\big{)}$
$\displaystyle=$
$\displaystyle\sum_{i}d_{i}\left[D_{i}\right]+\left[D_{k}\right]+\left[D_{j}\right]-\deg(V)$
$\displaystyle=$
$\displaystyle\deg(\mathcal{F})+\left[D_{j}\right]+\left[D_{k}\right]-\deg(V)$
$\displaystyle=$
$\displaystyle\deg(\mathcal{F})+\deg(z_{j})+\deg(z_{k})-\deg(V).$

Let $1\leq\ell\leq r$ be as in the hypothesis $(\ref{eq3})$. Then

$0\leq\deg(P_{j,k}^{\,i})_{\ell}=\deg(\mathcal{F})_{\ell}+\deg(z_{j})_{\ell}+\deg(z_{k})_{\ell}-\deg(V)_{\ell}.$

Finally we have

$\deg(V)_{\ell}\leq\deg(\mathcal{F})_{\ell}+\deg(z_{j})_{\ell}+\deg(z_{k})_{\ell}\leq\deg(\mathcal{F})_{\ell}+\max_{1\leq
j<k\leq n+r}\big{\\{}deg(z_{j})_{\ell}+deg(z_{k})_{\ell}\big{\\}}.$

∎

######  Remark 4.1.

In Theorem 4.2

  1. (i)

A more precise conclusion is
$\deg(V)_{k}\leq\deg(\mathcal{F})_{k}+deg(z_{i})_{k}+deg(z_{j})_{k}$, for some
$i<j$ and for each $1\leq k\leq r$ as in the hypothesis $(\ref{eq3})$.

  2. (ii)

Suppose that the homogeneous coordinate ring $\mathrm{S}$ of
$\mathbb{P}_{\Delta}$ has elements with arbitrary degree, then we only have
$\deg(P_{j,k}^{\,i})=\deg(\mathcal{F})+\deg(z_{j})+\deg(z_{k})-\deg(V)$, for
some $j<k$.

There is a situation in which the above Theorem 4.2 is valid for a quasi-
smooth hypersurface $V=\left\\{f=0\right\\}\subset\mathbb{P}_{\Delta}$
invariant by $\mathcal{F}$ with

$\left\\{0\right\\}\subsetneq Sing(V)\subset\mathcal{Z}.$

Suppose that there exist integer numbers $1\leq i_{1}<\cdots<i_{k}\leq n+r$
such that

  1. (i)

there is a regular subsequence $\left\\{\frac{\partial f}{\partial
z_{i_{1}}},\dots,\frac{\partial f}{\partial
z_{i_{k}}}\right\\}\subset\left\\{\frac{\partial f}{\partial
z_{1}},\dots,\frac{\partial f}{\partial z_{n+r}}\right\\}$ and

  2. (ii)

there is a radial vector field given by
$R_{i_{1},\ldots,i_{k}}=\sum_{j=1}^{k}a_{i_{j}}z_{i_{j}}\frac{\partial}{\partial
z_{i_{j}}}$.

The property $($i$)$ is equivalent to

$codim\left(\left\\{\frac{\partial f}{\partial
z_{i_{1}}}=\cdots=\frac{\partial f}{\partial z_{i_{k}}}=0\right\\}\right)=k;$

for more details see [19]. It is easy to understand that the same arguments in
the proof of Theorems 4.1 and 4.2 can be applied here to obtain the following
theorems:

######  Theorem 4.3 (Normal form 2).

Let $V=\left\\{f=0\right\\}\subset\mathbb{P}_{\Delta}$ be a quasi-smooth
hypersurface of degree $\alpha\in\mathcal{A}_{n-1}(\mathbb{P}_{\Delta})$.
Suppose there are integers numbers $1\leq i_{1}<\cdots<i_{k}\leq n+r$ with the
properties $($_i_ $)$ and $($_ii_ $)$. Let $X$ be a quasi-homogeneous vector
field in $\mathbb{P}_{\Delta}$. Suppose that in homogeneous coordinates
$X=X_{1}+X_{2}$, where $X_{1}=\sum_{j=1}^{k}P_{i_{j}}\frac{\partial}{\partial
z_{i_{j}}}$ leaves $V$ invariant. Then

$X_{1}=\sum_{j_{1}<j_{2}}P_{j_{1},j_{2}}\left(\frac{\partial f}{\partial
z_{i_{j_{1}}}}\frac{\partial}{\partial z_{i_{j_{2}}}}-\frac{\partial
f}{\partial z_{i_{j_{2}}}}\frac{\partial}{\partial
z_{i_{j_{1}}}}\right)+\frac{g}{\theta(\alpha)}\sum_{j=1}^{k}a_{i_{j}}z_{i_{j}}\frac{\partial}{\partial
z_{i_{j}}},$

where $P_{j_{1},j_{2}},\,g\in\mathbb{C}[z_{1},\dots,z_{n+r}]$ are quasi-
homogeneous polynomials.

######  Theorem 4.4 (Poincaré problem $2$).

Let $\mathcal{F}$ be a one-dimensional holomorphic foliation on
$\mathbb{P}_{\Delta}$ and $X$ the quasi-homogeneous vector field which defines
$\mathcal{F}$ in homogeneous coordinates. Let
$V=\left\\{f=0\right\\}\subset\mathbb{P}_{\Delta}$ be a quasi-smooth
hypersurface. Suppose there are integers numbers $1\leq i_{1}<\cdots<i_{k}\leq
n+r$ with the properties $($_i_ $)$ and $($_ii_ $)$. Suppose that in
homogeneous coordinates $X=X_{1}+X_{2}$, where
$X_{1}=\sum_{j=1}^{k}P_{i_{j}}\frac{\partial}{\partial z_{i_{j}}}$ leaves $V$
invariant and such that $X_{1}\notin Lie(G)$. Then

$\deg(V)_{k}\leq\deg(\mathcal{F})_{k}+\max_{1\leq i_{j_{1}}<i_{j_{2}}\leq
k}\big{\\{}deg(z_{i_{j_{1}}})_{k}+deg(z_{i_{j_{2}}})_{k}\big{\\}},$

for each $1\leq k\leq r$ as in the hypothesis $(\ref{eq3})$.

It is worth mentioning that an analogous remark as remark 4.1 is valid here.

##  5\. Applications

In this section we use Theorem 4.2 to give an upper bound for Poincaré problem
for strongly quasi-smooth invariant hypersurfaces on weighted projective
spaces, multiprojective spaces, rational scroll and on a compact toric
orbifold surface with Weil divisor class group having torsion
$\mathbb{P}_{\Delta(0,2,1)}$. Acoording to Theorem 4.4 it is clear that
analogous results can be obtained for invariant quasi-smooth hypersurfaces. We
build several families of one-dimensional foliations where the upper bound is
sharp; see examples 5.1, 5.2, 5.3, 5.4 and 5.5.

######  Corollary 5.1 (Weighted projective spaces).

Let $\mathcal{F}$ be a one-dimensional holomorphic foliation on
$\mathbb{P}(\omega)$ and $X$ the quasi-homogeneous vector field which defines
$\mathcal{F}$ in homogeneous coordinate. Let $V\subset\mathbb{P}(\omega)$ be a
quasi-smooth hypersurface invariant by $\mathcal{F}$. Suppose that $X\notin
Lie(G)$, then

$\deg(V)\leq\deg(\mathcal{F})+\max_{0\leq i<j\leq
n}\\{\omega_{i}+\omega_{j}\\}.$

###### Proof.

Follow directly from Theorem 4.2, because
$\mathcal{A}_{n-1}(\mathbb{P}_{\omega})\simeq\mathbb{Z}$,
$\deg(z_{i})=\omega_{i}$ for all $0\leq i\leq n$ and the one-dimensional
holomorphic foliation $\mathcal{F}$ on $\mathbb{P}(\omega)$ is a global
section of
$\mathcal{T}\mathbb{P}(\omega)\otimes\mathcal{O}_{\mathbb{P}(\omega)}(d)$. ∎

Now, let us consider the family of examples of one-dimensional holomorphic
foliations in [3]

######  Example 5.1.

Let $\omega_{0},\omega_{1},\dots,\omega_{2m+1}$ be positive integers with
$gcd(\omega_{0},\dots,\omega_{2m+1})=1$ and $\xi$ such that

$\xi=\omega_{2j}+\omega_{2j+1}\,\,\textup{for all}\,\,j=0,1,\dots,m.$

Let us consider the weighted projective space
$\mathbb{P}^{n}(\omega_{0},\dots,\omega_{n})$, where $n=2m+1$ and
$\mathcal{F}$ a holomorphic foliation on
$\mathbb{P}^{n}(\omega_{0}\dots,\omega_{n})$ induced by the quasi-homogeneous
vector field with isolated singularities given by

$X=\sum_{k=0}^{m}\Big{(}d_{2k+1}z_{2k+1}^{d_{2k+1}-1}\frac{\partial}{\partial
z_{2k}}-d_{2k}z_{2k}^{d_{2k}-1}\frac{\partial}{\partial z_{2k+1}}\Big{)},$

where $d_{0},\dots,d_{n}\in\mathbb{N}$ and $\zeta$ satisfy the relation

$\zeta=\omega_{k}d_{k}\,\,\textup{for all}\,\,k=0,\dots,n.$

For each $c=(c_{0}:\cdots:c_{m})\in\mathbb{P}^{m}$, $V_{c}$ is the quasi-
smooth hypersurface on $\mathbb{P}^{n}(\omega_{0},\dots,\omega_{n})$ of degree
$\zeta$ given by

$V_{c}=\left\\{\sum_{k=0}^{m}c_{k}\big{(}z_{2k}^{d_{2k}}+z_{2k+1}^{d_{2k+1}}\big{)}=0\right\\}.$

We can see that $V_{c}$ is invariant by $\mathcal{F}$ and
$\deg(\mathcal{F})=\zeta-\xi$. Then

$\deg(V_{c})-\deg(\mathcal{F})=\xi\leq\max_{0\leq i<j\leq
n}\\{\omega_{i}+\omega_{j}\\}.$

Consider a similar foliation on even dimensional weighted projective spaces
$\mathbb{P}^{n}(\omega_{0},\dots,\omega_{n})$ where $n=2m+2$ and
$\xi=\omega_{2k}+\omega_{2k+1}\,\,\textup{for all}\,\,k=0,\dots,m$. Suppose
that $\zeta=\omega_{k}d_{k}$ for all $\,k=0,\dots,n$ and consider the vector
field $X$ of the previous example. Thus, for each
$c=(c_{0}:\cdots:c_{m+1})\in\mathbb{P}^{m+1}$, the quasi-smooth hypersurface
on $\mathbb{P}^{n}(\omega_{0},\dots,\omega_{n})$ of degree $\zeta$ given by

$V_{c}=\left\\{\sum_{k=0}^{m}c_{k}\big{(}z_{2k}^{d_{2k}}+z_{2k+1}^{d_{2k+1}}\big{)}+c_{m+1}z_{n}^{d_{n}}=0\right\\}$

is invariant by $X$ and therefore we obtain the same conclusions

$\deg(V_{c})-\deg(\mathcal{F})=\xi\leq\max_{0\leq i<j\leq
n}\\{\omega_{i}+\omega_{j}\\}.$

We see that the upper bound for Poincaré problem is optimal. Compare with the
remark 4.1-(i).

######  Corollary 5.2 (Multiprojective spaces).

Let $\mathcal{F}$ be a one-dimensional holomorphic foliation on
$\mathbb{P}^{n_{1}}\times\cdots\times\mathbb{P}^{n_{r}}$ $(r>1)$ and $X$ the
quasi-homogeneous vector field which defines $\mathcal{F}$ in homogeneous
coordinates. Let
$V\subset\mathbb{P}^{n_{1}}\times\cdots\times\mathbb{P}^{n_{r}}$ be a strongly
quasi-smooth hypersurface invariant by $\mathcal{F}$. Suppose that $X\notin
Lie(G)$, then

$\deg(V)_{k}\leq\deg(\mathcal{F})_{k}+2,\,\,1\leq k\leq r.$

###### Proof.

In homogeneous coordinates
$z=(z_{1},\dots,z_{r})\in\mathbb{C}^{(n_{1}+\cdots+n_{r})+r}$ with
$z_{i}=(z_{i,0},\dots,z_{i,n_{i}})\in\mathbb{C}^{n_{i}+1}$, we have
$\deg(z_{i,j})=\left(0,\dots,1_{i},\dots,0\right)$, for all $0\leq j\leq
n_{i}$. Then the corollary follows directly from Theorem 4.2, because
$\mathcal{A}_{n_{1}+\cdots+n_{r}-1}(\mathbb{P}^{n_{1}}\times\cdots\times\mathbb{P}^{n_{r}})\simeq\mathbb{Z}^{r}$
and one-dimensional holomorphic foliation $\mathcal{F}$ on
$\mathbb{P}^{n_{1}}\times\cdots\times\mathbb{P}^{n_{r}}$ of multidegree
$(\alpha_{1},\dots,\alpha_{r})$ is a global section of
$T(\mathbb{P}^{n_{1}}\times\cdots\times\mathbb{P}^{n_{r}})\otimes\mathcal{O}_{\mathbb{P}^{n_{1}}\times\cdots\times\mathbb{P}^{n_{r}}}(\alpha_{1},\dots,\alpha_{r})$.
∎

Now, let us consider a family of examples of one-dimensional holomorphic
foliations in $\mathbb{P}^{n}\times\mathbb{P}^{n}$

######  Example 5.2.

In homogeneous coordinates $z_{1,0},z_{1,1},\ldots
z_{1,n},z_{2,0},z_{2,1},\ldots z_{2,n}$ with $n=2m+1$, let us consider the
multiprojective space $\mathbb{P}^{n}\times\mathbb{P}^{n}$ and $\mathcal{F}$ a
holomorphic foliation on $\mathbb{P}^{n}\times\mathbb{P}^{n}$ induced by the
quasi-homogeneous vector field given by

$X=\sum_{k=0}^{m}a_{k}z_{1,2k}^{2}\Big{(}z_{2,2k+1}\frac{\partial}{\partial
z_{1,2k}}-z_{2,2k}\frac{\partial}{\partial
z_{1,2k+1}}\Big{)}+\sum_{k=0}^{m}b_{k}z_{2,2k}^{2}\Big{(}z_{1,2k+1}\frac{\partial}{\partial
z_{2,2k}}-z_{1,2k}\frac{\partial}{\partial z_{2,2k+1}}\Big{)},$

where $a=(a_{0}:\ldots:a_{m}),b=(b_{0}:\ldots:b_{m})\in\mathbb{P}^{m}$.
Consider the strongly quasi-smooth hypersurface on
$\mathbb{P}^{n}\times\mathbb{P}^{n}$ invariant by $\mathcal{F}$ given by

$V=\left\\{\sum_{k=0}^{n}z_{1,k}z_{2,k}=0\right\\}.$

It is easy seen that $\deg(V)=\deg(\mathcal{F})=(1,1)$ and so

$\deg(V)_{k}\leq\deg(\mathcal{F})_{k}+2,\,\,k=1,2.$

######  Corollary 5.3 (Rational scrolls).

Let $\mathcal{F}$ be a one-dimensional holomorphic foliation on
$\mathbb{F}(a_{1},\dots,a_{n})$ and $X$ the quasi-homogeneous vector field
which defines $\mathcal{F}$ in homogeneous coordinates. Let
$V\subset\mathbb{F}(a_{1},\dots,a_{n})$ be a strongly quasi-smooth
hypersurface invariant by $\mathcal{F}$. Suppose that $X\notin Lie(G)$, then

  1. (i)

if $n=1$, then $\deg(V)_{2}\leq\deg(\mathcal{F})_{2}+1,$

  2. (ii)

if $n>1$, then $\deg(V)_{2}\leq\deg(\mathcal{F})_{2}+2.$

Moreover, suppose that $a_{1},\dots,a_{n}$ are non-positive integers, we have

  1. (i)

if $a_{1}=a_{2}=\cdots=a_{n}=0$, then
$\deg(V)_{1}\leq\deg(\mathcal{F})_{1}+2,$

  2. (ii)

if $a_{2}=\cdots=a_{n}=0$ and $a_{1}$ is negative, then
$\deg(V)_{1}\leq\deg(\mathcal{F})_{1}+1-a_{1},$

  3. (iii)

if at least two of the $a_{i}$ are negatives, then
$\deg(V)_{1}\leq\deg(\mathcal{F})_{1}-\min_{1\leq i<j\leq
n}\left\\{a_{i}+a_{j}\right\\}.$

###### Proof.

Here $\mathcal{A}_{n-1}(\mathbb{F}(a_{1},\dots,a_{n}))=\mathbb{Z}^{2}$ and
every bihomogeneous polynomial of bidegree $(\alpha,\beta)$ has $\beta\geq 0$.
Then, the corollary follows directly from Theorem 4.2 and example $3.4$-(3),
because

$\deg(V)_{2}\leq\deg(\mathcal{F})_{2}+\max_{1\leq i<j\leq
n+2}\left\\{\deg(z_{i})_{2}+\deg(z_{j})_{2}\right\\}.$

Moreover, if $a_{1},\dots,a_{n}$ are non-positive integers, the corollary
follows directly from Theorem 4.2 and Example $3.4$-(3), because $\alpha\geq
0$ and

$\deg(V)_{1}\leq\deg(\mathcal{F})_{1}+\max_{1\leq i<j\leq
n+2}\left\\{\deg(z_{i})_{1}+\deg(z_{j})_{1}\right\\}.$

∎

######  Corollary 5.4 (A toric surface).

Let $\mathcal{F}$ be a one-dimensional holomorphic foliation on
$\mathbb{P}_{\Delta(0,2,1)}$ and $X$ the quasi-homogeneous vector field which
defines $\mathcal{F}$ in homogeneous coordinates. Let
$V\subset\mathbb{P}_{\Delta(0,2,1)}$ be a quasi-smooth hypersurface invariant
by $\mathcal{F}$. Suppose that $X\notin Lie(G)$, then

$\deg(V)_{1}\leq\deg(\mathcal{F})_{1}+2.$

###### Proof.

Here
$\mathcal{A}_{1}(\mathbb{P}_{\Delta(0,2,1)})=\mathbb{Z}\oplus\mathbb{Z}_{3}$
and every bihomogeneous polynomial of bidegree $(\alpha,\beta)$ has
$\alpha\geq 0$. Then, the corollary follows directly from Theorem 4.2. ∎

Now, let us consider a family of examples of one-dimensional holomorphic
foliations in $\mathbb{P}_{\Delta(0,2,1)}$

######  Example 5.3.

Consider a toric surface $\mathbb{P}_{\Delta(0,2,1)}$. Let $m$ be positive
integers with $m\equiv 0\,\textup{mod}\,3$ and $\mathcal{F}$ a holomorphic
foliation on $\mathbb{P}_{\Delta(0,2,1)}$ induced by the quasi-homogeneous
vector field given by

$X=z_{2}^{m}\frac{\partial}{\partial
z_{1}}+z_{1}(z_{3}^{m-1}-z_{1}^{m-2}z_{2})\frac{\partial}{\partial
z_{2}}-z_{1}z_{2}^{m-1}\frac{\partial}{\partial z_{3}}.$

We have that
$Sing(\mathcal{F})=\left\\{(1:a:-a^{-1})\in(\mathbb{C^{*}})^{3}\,|\,a^{2m}+a^{m}-1=0\right\\}$.
Consider the quasi-smooth hypersurface on $\mathbb{P}_{\Delta(0,2,1)}$
invariant by $\mathcal{F}$ given by

$V=\left\\{z_{1}^{m}+z_{2}^{m}+z_{3}^{m}=0\right\\}.$

It is easy seen that $\deg(\mathcal{F})=(m-1,\bar{0})$ and
$\deg(V)=(m,\bar{0})$ and so
$\deg(V)_{1}=m=\deg(\mathcal{F})_{1}+1<\deg(\mathcal{F})_{1}+2$. On the other
hand, according to Theorem 4.1 we have $P_{1,2}=-\frac{1}{m}z_{2}$,
$P_{1,3}=0$ and $P_{2,3}=-\frac{1}{m}z_{1}$. Hence

$\deg(P_{1,2})=(1,\bar{2})=(m-1,\bar{0})+(1,\bar{0})+(1,\bar{2})-(m,\bar{0})=\deg(\mathcal{F})+\deg(z_{1})+\deg(z_{2})+\deg(V),$

and

$\deg(P_{2,3})=(1,\bar{0})=(m-1,\bar{0})+(1,\bar{2})+(1,\bar{1})-(m,\bar{0})=\deg(\mathcal{F})+\deg(z_{2})+\deg(z_{3})+\deg(V).$

Compare with the Remark 4.1-(ii).

The next example is an application of Theorems 4.3 and 4.4

######  Example 5.4.

In homogeneous coordinates $z_{1,0},z_{1,1},z_{2,0},z_{2,1}$, let us consider
the multiprojective space

$\mathbb{P}^{1}\times\mathbb{P}^{1}=\left(\mathbb{C}^{2}\times\mathbb{C}^{2}-\mathcal{Z}\right)\,/\,(\mathbb{C}^{*})^{2},$

where
$\mathcal{Z}=\left(\left\\{0\right\\}\times\mathbb{C}^{2}\right)\cup\left(\mathbb{C}^{2}\times\left\\{0\right\\}\right)$.
Let $\alpha_{1}$, $\alpha_{2}$, $\alpha$ be positive integers with
$\alpha=\alpha_{1}+\alpha_{2}$ and $c=(c_{1}:c_{2})\in\mathbb{P}^{1}$. Let
$\mathcal{F}$ be a holomorphic foliation on
$\mathbb{P}^{1}\times\mathbb{P}^{1}$ induced by the quasi-homogeneous vector
field $X=X_{1}+X_{2}$, where

$X_{1}=c_{1}z_{1,0}^{2}z_{2,1}^{\alpha}\frac{\partial}{\partial
z_{1,0}}-c_{1}z_{1,0}^{2}z_{2,0}^{\alpha_{1}}(z_{2,0}^{\alpha_{2}}+z_{2,1}^{\alpha_{2}})\frac{\partial}{\partial
z_{1,1}}$

and

$X_{2}=c_{2}z_{2,0}^{2}z_{2,1}^{\alpha_{2}-1}(\alpha_{2}z_{1,0}z_{2,0}^{\alpha_{1}}+\alpha
z_{1,1}z_{2,1}^{\alpha_{1}})\frac{\partial}{\partial
z_{2,0}}-c_{2}z_{2,0}^{\alpha_{1}+1}(\alpha
z_{1,0}z_{2,0}^{\alpha_{2}}+\alpha_{1}z_{1,0}z_{2,1}^{\alpha_{2}})\frac{\partial}{\partial
z_{2,1}}.$

Consider the quasi-smooth hypersurface on $\mathbb{P}^{1}\times\mathbb{P}^{1}$
invariant by $\mathcal{F}$ given by

$V=\left\\{z_{1,0}z_{2,0}^{\alpha}+z_{1,1}z_{2,1}^{\alpha}+z_{1,0}z_{2,0}^{\alpha_{1}}z_{2,1}^{\alpha_{2}}=0\right\\}.$

Then $V$ is invariante by $X_{1}$, $X_{2}$ and $\left\\{0\right\\}\subsetneq
Sing(V)=\mathbb{C}^{2}\times\left\\{0\right\\}\subset\mathcal{Z}$. We have
$\deg(V)=\deg(\mathcal{F})=(1,\alpha)$ and so

$\deg(V)_{k}\leq\deg(\mathcal{F})_{k}+2,\,\,k=1,2.$

Finally, the next example tells us that our hypotheses are necessary

######  Example 5.5.

In homogeneous coordinates $z_{1,0},z_{1,1},z_{2,0},z_{2,1}$, let us consider
the multiprojective space

$\mathbb{P}^{1}\times\mathbb{P}^{1}=\left(\mathbb{C}^{2}\times\mathbb{C}^{2}-\mathcal{Z}\right)\,/\,(\mathbb{C}^{*})^{2},$

where
$\mathcal{Z}=\left(\left\\{0\right\\}\times\mathbb{C}^{2}\right)\cup\left(\mathbb{C}^{2}\times\left\\{0\right\\}\right)$.
Let $\mathcal{F}$ be a holomorphic foliation on
$\mathbb{P}^{1}\times\mathbb{P}^{1}$ induced by the quasi-homogeneous vector
field given by

$X=z_{1,0}z_{2,1}^{2}\frac{\partial}{\partial
z_{1,0}}+z_{2,0}z_{1,1}^{2}\frac{\partial}{\partial z_{2,0}}.$

Consider the hypersurface on $\mathbb{P}^{1}\times\mathbb{P}^{1}$ invariant by
$\mathcal{F}$ given by

$V=\left\\{z_{1,0}^{\alpha}z_{2,0}^{\beta}=0\right\\}.$

Then $\mathcal{Z}\subset
Sing(V)=\left(\left\\{0\right\\}\times\mathbb{C}\times\mathbb{C}\times\mathbb{C}\right)\cup\left(\mathbb{C}\times\mathbb{C}\times\left\\{0\right\\}\times\mathbb{C}\right)$.
We have $\deg(V)=(\alpha,\beta)$ and $\deg(\mathcal{F})=(1,1)$. Depending on
$\alpha,\beta$, it is easy to see that the Poincaré problem could be or not
could be valid.

Acknowlegments. I am grateful to Mauricio Corrêa for helpful comments and
suggestions during the preparation of this paper and for interesting
conversations about toric varieties. I also thank Arturo Fernández Pérez for
useful suggestions which improved the readability of this paper.

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  * [23] Poincaré M. H.: _Sur l’intégration algébrique des équations différentielles du premier ordre et du premier degré_. Rend. Circ. Mat. Palermo. Vol. 5, 161-191 (1891).
  * [24] Reid M.: _Chapters on algebraic surfaces_. AMS (1996).
  * [25] Soares M. G.: _The Poincaré problem for hypersurfaces invariant by one-dimensional foliations_. Inventiones Mathematicae. Vol. 128, 495-500 (1997).


# EMMT: A simultaneous eye-tracking, 4-electrode EEG and audio corpus for
multi-modal reading and translation scenarios

Sunit Bhattacharya  Věra Kloudová  Vilém Zouhar  Ondřej Bojar

###### Abstract

We present the Eyetracked Multi-Modal Translation (EMMT) corpus, a dataset
containing monocular eye movement recordings, audio and 4-electrode
electroencephalogram (EEG) data of 43 participants. The objective was to
collect cognitive signals as responses of participants engaged in a number of
language intensive tasks involving different text-image stimuli settings when
translating from English to Czech.

Each participant was exposed to 32 text-image stimuli pairs and asked to (1)
read the English sentence, (2) translate it into Czech, (3) consult the image,
(4) translate again, either updating or repeating the previous translation.
The text stimuli consisted of 200 unique sentences with 616 unique words
coupled with 200 unique images as the visual stimuli.

The recordings were collected over a two week period and all the participants
included in the study were Czech natives with strong English skills. Due to
the nature of the tasks involved in the study and the relatively large number
of participants involved, the corpus is well suited for research in
Translation Process Studies, Cognitive Sciences among other disciplines.

##  1 Background & Summary

Rapid advances in Artificial Intelligence over the last few years have
translated to significant advances in Natural Language Processing (NLP). The
extraordinary effectiveness of the neural network approach for artificial
intelligence is seen in the performance of the state-of-the-art language
models[1] and in the ways that they are being fine-tuned for different NLP
applications. The state-of-the-art machine translation systems have been
reported to have achieved news translation quality comparable to human
professionals.[2] Inspired by these successes, machine translation research in
the last few years has also attempted to target the problem of multimodal
translation.[3]

While traditional translation is thought of as being about printed words,[4]
the rise of the Internet and new media have made multimodal texts ubiquitous.
This implies that there is a need for a systematic understanding of
translation processes for multimodal texts. This understanding, we posit,
should arise from learning how both human and machine “brains” function. The
discipline of artificial intelligence has been historically ‘conceptualized in
anthropomorphic terms’. [5] And although there are many risks and incorrect
assumptions when anthropomorphizing modern deep neural networks,[6] there has
been a growing body of work analyzing this very aspect of such systems. One
reasoning behind such an analysis is that a comparative analysis of human and
machine intelligence should lead to a better understanding of intelligence.
There is a view that language is a key to understanding human intelligence.[7]
It is thus fair to assume that cognitive signal data from a participant
performing a set of tasks involving language use would be a great resource to
understand how the brain works with language. If the tasks are designed
carefully such that they can also be presented to trained deep neural network
algorithms, they can be used to compare the human and machine perspective of
handling the same problem, see e.g. the recent comparison of interpreting
occluded scenes.[8] A similar approach in computer vision has led to simpler
and yet equally effective models for object detection.[9]

Our objective is to collect cognitive data in the form of gaze and EEG signals
from participants while they translate sentences in multimodal conditions. We
hope that this dataset would help researchers study how behavioural patterns
(in terms of cognitive signals) change with stimuli of differing modalities.
The tasks that the participants are subjected to have been designed so that
similar experiments can be performed on the state-of-the-art algorithms for
natural language processing.[3]

Whether the observed eye-tracking and EEG data are sufficient to relate human
and machine processing of text accompanied by an image is an empirical
question. Over the years, many eye-tracking studies have been successfully
done for translation process research, leading to key insights.[10, 11, 12,
13] In recent years, many studies involving EEG signals, too, have been
conducted to understand the neural underpinnings of language.[14] Our dataset
will hopefully foster further research in that direction.

##  2 Methods

###  2.1 Equipment Used

An EyeLink 1000 Plus by SR-Research was used for recording the gaze data. The
eye-tracker was kept at a distance of 45 cm from the participants. The screen
where the stimuli were presented was kept at 65 cm from the participants. The
experimental setup from the perspective of the participants is shown in Figure
1. The Muse 2111choosemuse.com/muse-2 headband was used to record the EEG data
from the experiments. Low-cost portable headbands like Muse have been recently
used for a number of ERP-based (Event-Related Potentials[15, 16]) studies with
reasonable results.[17] Along with this, the audio of the entire session was
recorded using a Samsung Galaxy M21 device.

Figure 1: Experiment setup with eyetracker (EEG and audio recorder not shown).

###  2.2 Participants

Data were recorded from 44 people who volunteered to be a part of the study.
Requests for participants to take part was circulated via flyers, emails,
tweets and word of mouth. Out of the 44 people who volunteered, data from 43
people were used in the dataset. All of the participants were native speakers
of Czech (in the age group between 17 and 51) who also spoke English as a
second or third language. The distribution of the age of participants is shown
in Table 1.

The participants were asked to fill out a self-reporting questionnaire to
gather additional data. That data included facts like the level of proficiency
in English and if the participant offered translation services professionally.
Information about the distribution of the level of proficiency in English for
the participants (on the CEFR222coe.int/en/web/common-european-framework-
reference-languages/level-descriptions scale) is given in Table 2.

Age | 17-20 | 21-25 | 26-30 | 31-35 | 36-40 | 41-45 | 45-50 | 51-55
---|---|---|---|---|---|---|---|---
Participants | 5 | 21 | 8 | 2 | 4 | 1 | 1 | 1
Table 1: Distribution of the participants’ age. CEFR Level | B1 | B2 | C1 | C2
---|---|---|---|---
Participants | 2 | 15 | 16 | 8
Table 2: Level of proficiency in English of the participants on the CEFR
scale.

| Professional | Amateur
---|---|---
Participants | 16 | 27
Table 3: Distribution of participants providing language services (translation
or interpretation) professionally.

Service | Occasionally | 1-5 years | >10 years
---|---|---|---
Translation | 11 | 0 | 1
Interpretation | 10 | 3 | 1
Table 4: Self-reported experience of participants in providing either
translation or interpretation services.

Table 4 shows the number of people self-reporting to be amateurs and
professionals (those who provide translation and interpretation services
professionally). Table 4 shows the distribution of people self-reporting their
experience in providing translation and interpretation services respectively.
The cognitive signals from participants with such diversity of professional
experience in providing language services could provide interesting insights
into how language processing strategies vary across people.

The experiments were supervised across a span of 12 by two proctors (authors
of this paper). The experiments were designed keeping in mind the code of
ethics of Charles University.333Available in Czech: cuni.cz/UK-9490.html. The
experiment is further GDPR-compliant and approved by a qualified university
staff Libuše Kaprálová. All the participants were duly informed about the
tasks involved in the experiment before the beginning of the experiment. The
participants in the study also signed a consent form agreeing to the recording
of the gaze, audio EEG data together with the matching questionnaire answers
and releasing it for research purposes. The collected data were carefully
stripped of all personal information.444Internal records were pseudo-
anonymized, actual data do not contain any identifiable personal information.

###  2.3 Materials

The dataset that we present is composed of sentences drawn from three well-
known datasets used in various machine learning tasks: Ambiguous COCO,[18]
LAVA[19] and Hindi Visual Genome (HVG).[20] All three corpora are multimodal
corpora with an image associated with every sentence. Furthermore, all three
corpora are publicly available. The original datasets have all been released
with licenses that allow reuse of the images for research purposes.

The sentence selection process initially involved annotators identifying
ambiguous sentences. Accordingly, 176 sentences were marked as being
‘ambiguous’.555Here we do not explicitly differentiate between ambiguity and
vagueness. Many ‘ambiguous’ sentences in the dataset might be considered
vague. Two different annotators then further ranked the 176 sentences for the
‘level of ambiguity’ in the sentence on a scale of 1 to 5. The top 100
sentences from the ranked list were selected for the experiment. Another 100
sentences from the three corpora (i.e. avoiding the 176 initially chosen by
annotators) were then chosen to serve as contrastive, ‘unambiguous’,
sentences. It was however found that the chosen sentences from the original
datasets had many grammatical errors. Thus, another annotator (a native
English speaker) proofread and corrected all the selected sentences.

The sentences are complemented with images as described in the Stimuli section
below. All images always come from the same corpus as the respective sentence.

###  2.4 Stimuli & Experiment Design

For each participant, the experiment consisted of the presentation of 30
different stimuli, each based on an image-text probe. Each probe had four
sequential stages which we denote as READ, TRANSLATE, SEE and UPDATE. The
description of the stages is provided later in the section. While the text was
presented to the participants in all the stages, the image in each stimulus
was presented only for the SEE and UPDATE stages.

In total, we prepared 600 different stimuli across 2$\times$3 conditions and
divided them randomly into 20 probes (constrained by not showing the same
sentence more than once), each containing 30 stimuli. With 43 participants,
the objective was to have at least two sets of observations for each probe.

The 2$\times$3 conditions were: 100 ambiguous (A) and 100 unambiguous (U)
sentences, as described above, each complemented with a congruent (C),
incongruent (I) and missing (M) image. Congruent images are those where the
image corresponds to the original sentence from the given source corpus.
Incongruent images were selected randomly from the same corpus as the
sentence, avoiding the already selected congruent images. The image thus
matches the style of the corpus but it is unrelated to the sentence. Finally,
100+100 stimuli contained the “missing” image, i.e. an image with the fixed
text No visual clue for this case!.

The text was presented with a black font of font size 28 (Free Sans Bold) on a
light grey background. This resulted in a letter size of approximately 9.9 mm,
i.e. 0.015∘ (the distance between the screen and the participants was 65 cm).
The longest sentence presented contained 97 letters (21 words) while the
shortest contained 19 letters (4 words). All the sentences were presented in a
single line. The participants used the spacebar key to proceed with the
experiment. All the stages were self-paced. Each press of the spacebar was
confirmed with a short audio signal (a beep), thus indicating the progress in
the experiment. There were 4 example stimuli before the start of the
experiment to get the participants acquainted with the format of the
experiment. The description of the stages involved in the experiment is given
below and schemed in Figure 3.

(a) READ and TRANSLATE stages.

(b) SEE and UPDATE stages.

Figure 2: Example screens (cropped) for two different stimuli.

####  2.4.1 Stage 1: Read

In the READ stage, the participants were shown a sentence on the display
screen. The stage involved loud reading of the sentence (in English). An
example of the screen during this stage is shown in Figure 2(a). The
participants were instructed to press the space bar once they were ready to
move on to the next stage.

####  2.4.2 Stage 2: Translate

The TRANSLATE stage used the same screen as READ. The participants were
expected to translate the English sentence into Czech, speaking it out loud.
Upon completion of the translation, the participants pressed the space bar,
indicating the end of the stage.

####  2.4.3 Stage 3: See

Upon entering the SEE stage, the image (or the “missing” image) was presented
below the sentence. The participant was expected to look at the image and
decide if they would like to change the translation they produced in the
previous stage. When decided, the participant verbally indicated that they
would either like to change the translation or keep the same translation.
Pressing the space bar then led the participant to the final stage. An example
screen for this stage is shown in Figure 2(b).

####  2.4.4 Stage 4: Update

This stage involved the participants either repeating the translation that
they made during TRANSLATE or updating their translation based on the visual
stimuli during SEE. The idea is that the participants may decide to integrate
more information or correct an incorrectly guessed ambiguity based on the
image (especially for congruent images) in their translation. The collected
cognitive data would hopefully reflect how that integration manifests during
SEE and UPDATE. Inevitably, the participants will not have remembered their
original translation exactly, so we expect many “unchanged” translations to
actually differ.

Figure 3: Visualization of the four experiment stages.

###  2.5 Procedure

Data were recorded from each participant in a single session. Each experiment
started with the calibration and validation of the equipment involved (eye-
tracker and EEG recorder). Each participant was then led through a practice
round with four dummy stimuli, to get them acquainted with the procedures of
the experiment. Being a self-paced experiment design, the participants were
given an option to temporarily pause the experiment after completing the four
stages of a stimulus to take a small break. If the participants opted to
pause, the experiment would resume again with calibration and validation
before starting from where it was stopped.

##  3 Data Records

The Eyetracked Multi-Modal Translation (EMMT) corpus is hosted on the Center
for Open Science (OSF) repository: osf.io/hxymj. It is distributed under the
Creative Commons License CC0 version 4.0. All data has been anonymized in
accordance with the consent provided by participants and abiding by the GDPR
policies. We segmented the recordings based on stimuli, leading to a total of
1169 audio files. The dataset also includes EEG recordings from 28
participants and log files and gaze data corresponding to 43 participants. No
data were excluded but the EEG failed for some participants, resulting in only
702 fully valid EEG recordings.

##  4 Technical Validation

In this section, we provide a small-scale visualization to better understand
the data structure and its quality. We also list several experiment
applications for this data.

###  4.1 Audio

Each session audio is recorded together with stage timestamps. The audio
analysis can serve as a probe to translation difficulty or surprisal, e.g
pauses in hard-to-translate parts or prosody. It can also help in
understanding human text to speech translation characteristics.

###  4.2 Eyetracking

The data were sampled at 20kHz.666Line format: timestamp, x-location,
y-location, pupil dilation. Following the annotator gaze is important in
determining the attention level and location. This is crucial not only for
determining which parts of the image were important for the translation but
also helps us in determining the sequence in which the screen was processed
(e.g. Was the sentence read linearly or with saccades? Did the annotators
prefer to first read the whole sentence again and then look at the image or
did they jump between the two areas?).

Figure 4(a) shows the heatmap of annotators’ gaze (averaged over multiple
annotators) which reveals which parts of the sentence were important/on what
parts the annotators focused. Figure 4(b) shows the whole gaze sequence for a
single participant with colour-coded time progression. It shows, for example,
which parts of the sentence had to be consulted with the image.

(a) Heatmap of gaze frequency (averaged over multiple annotators).

(b) Gaze progression of a single participant for a specific probe. Old gaze
points are occluded by newer.

Figure 4: Eyetracking visualization.

###  4.3 EEG

The data contains the recorded EEG signal of 4 nodes: TP9, AF7, AF8 and TP10
(see precise description of the placement[21]) sampled at 256Hz. A more
convenient view of the data is its dominant frequencies (using the fast
Fourier transform). The bands are established as: Delta: 1-4Hz, Theta: 4-8Hz,
Alpha: 7.5-13Hz, Beta: 13-30Hz, Gamma: 30-44Hz. The headband also records
accelerometer data and certain muscle movements (blinks and jaw
clenches).777Line format:
web.archive.org/web/20181105231756/http://developer.choosemuse.com/tools/available-
data The two figures in Figure 5 show the EEG of the first stage of a specific
sentence & participant combination either as the power spectrum or as the raw
EEG voltages.

(a) Raw EEG signal for all four nodes, measured in uV, of the whole first
stage. They are separated for visibility (otherwise overlapping) and the
y-axis is hidden.

(b) Four precomputed power bands for two nodes (others hidden) of the whole
first stage.

Figure 5: EEG recordings of the first stage of a single participant.

Many psycholinguistic experiments make use of Event-Related Potentials: a
brain response in a specific place after exposure to a stimuli (after a
specific time period).[22, 23, 24, 25, 26] This EEG data allows researchers to
explore the brain response to ambiguity in the task of translation with and
without multimodal (image) cues. Future experiments may include analysing the
signals and seeing if it contains enough information to determine whether the
sentence was ambiguous or whether the image was related.

##  5 Usage Notes

The provided data is stored in the following structure (two directories). The
probes/Sentences.csv file maps sentences ids to the sentence texts. The
probes/Participants.csv file maps participant ids to the pseudo-anonymous
names used by the participants for the experiment. The *.eeg, *.et and *.mp3
files contain the recorded EEG, eyetracking and audio data, respectively. The
*.events file contains timestamps of when the text was presented, the
recording was started, the image was presented etc. They all belong to a
single sentence + ambiguity configuration + annotator tuple. The sentence and
ambiguity for all the runs in the probe* folder is listed in probe*/probes.

probes/

...

probe05/

...

P10-00-S182-A-C-ia018.eeg

P10-00-S182-A-C-ia018.et

P10-00-S182-A-C-ia018.events

P10-00-S182-A-C-ia018.mp3

...

probes

...

Sentences.csv

Participants.csv

images/

...

a006.jpg

a007.jpg

...

This storage format was chosen to be easily globbed (e.g. probe*/*-S112-*.eeg
yields the sentence 112 across all annotators). Working directly with so many
files will be slow on most filesystems and we, therefore, recommend
preprocessing and saving only the relevant view of the data (e.g. only the EEG
of the first stage of ambiguous sentences) and saving it in a binary format,
such as pickle,888docs.python.org/3/library/pickle.html which is easier and
faster to load.999We avoided using this format for distribution for forward
compatibility and to not publish a large binary blob.

##  6 Code availability

The source code for the recording of the experiment using PyGaze is available
open-source online github.com/NEUREM3/recording-code-for-eyetracked-multi-
modal-translation. The data itself is also accessible on GitHub
github.com/ufal/eyetracked-multi-modal-translation.

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##  7 Acknowledgements

This work has been funded from the grants 19-26934X (NEUREM3) of the Czech
Science Foundation and H2020-ICT-2018-2-825303 (Bergamot) of the European
Union. The work has also been supported by the Ministry of Education, Youth
and Sports of the Czech Republic, Project No. LM2018101 LINDAT/CLARIAH-CZ.

##  8 Author contributions statement

S.B. and O.B conceived the experiment and all authors contributed to the
experiment design. S.B, V.Z and O.B designed and wrote the code. S.B. and V.K.
conducted the experiment. All authors wrote and reviewed the manuscript.

##  9 Competing interests

I declare that the authors have no competing interests as defined by Nature
Research, or other interests that might be perceived to influence the results
and/or discussion reported in this paper.