text:
\begin{document}
\thispagestyle{empty}

\title[\sc Degree of the exceptional component of foliations]
  {Degree of the exceptional component of foliations\\ of degree two and codimension one in  \mbox{$\mathbb P^ 3$}}

\author[A. Rossini]{Artur Rossini}
\author[I. Vainsencher]{Israel Vainsencher}

\subjclass[2010]{14N10, 14N15 (Primary); 37F75 (Secondary).}
\keywords{holomorphic foliations, enumerative
geometry}

\begin{nouppercase} \maketitle \end{nouppercase}
\begin{abstract}
The purpose of this work is to obtain the
degree of the exceptional component of the space
of holomorphic foliations of degree two and codimension
one in $\mathbb{P}^3$. This component is the
closure of the orbit
of the foliation defined by the differential form
$$\omega=(3fdg-2gdf)\big/x_0,
\ \mbox{ where } \ f=x_0^2x_3-x_0x_1x_2+\dfrac{x_1^3}{3},\
g=x_0x_2-\dfrac{x_1^2}{2}
$$
under the natural action of the group of automorphisms of $\mathbb{P}^3$.
Our first task is to unravel a geometric characterization
of the pair $g,f$. This leads us to
the construction of a parameter space as
an explicit fiber bundle over the
variety of complete flags.
Using tools from equivariant intersection theory,
especially Bott's formula,
the  degree is expressed as an integral over our
parameter space.
\end{abstract}

\section{Introduction}

A holomorphic foliation of codimension one and degree $d$
in the complex projective space $\mathbb{P}^n$ is given
by a differential 1-form
$\omega = A_0dx_0 + \cdots + A_ndx_n$
where $A_0,\dots,A_n$ are homogeneous polynomials of degree $d+1$, satisfying the conditions (i) $A_0x_0 + \cdots +A_nx_n = 0$ and (ii) (integrability) $\omega \wedge d\omega = 0.$

These two conditions define a closed subscheme
$\overline{\mathcal{F}(d,n)}$ of the projective
space of global sections of the twisted cotangent sheaf
$\Omega_{\mathbb{P}^n}^1(d+2)$.
Naturally, $\overline{\mathcal{F}(d,n)}$ possesses a
decomposition into irreducible components.  This
decomposition is completely known in cases $d = 0$ and $d
= 1$, thanks to the pioneering book by
\cite{Jouanolou}. For $n \geq 3$, $d=2$
a full description can be found in the celebrated work by
\cite{Alcides} where we learn that there are six
components of $\overline{\mathcal{F}(2,3)}$,
among which
is the \textit{exceptional component} $E(3)$,
our main object of interest here.

There are a few other known components for $d \geq 3$,
such as the pullback, the rational  and
the logarithmic components, see \cite{NetoeEndonvein},
\cite{CukiermanVI} and \cite{Omegar}. In
\cite{CukiermanVI} the authors managed to compute the
degree of some rational components. In particular, the
degrees of the rational components $\mathcal R(n,2,3)$
were computed for $n \leq 5$; in \cite{daniel} this
is extended to the
rational components $\mathcal R(n,2,2r+1)$ for arbitrary $n,r$.

The \textit{exceptional component} $E(3)$ is the
closure of the orbit, under the natural action of
Aut($\mathbb{P}^3$), of the foliation defined by the differential form
\begin{equation}\label{fgi}
\omega_0=\frac{3f_0dg_0-2g_0df_0}{x_0},
\ \mbox{ where } \ f_0=x_0^2x_3-x_0x_1x_2+\dfrac{x_1^3}{3},\
g_0=x_0x_2-\dfrac{x_1^2}{2}
.\end{equation}

Our goal is to describe the geometry of the family
of pairs $g,f$ in the orbit closure, enabling us to
compute the degree of $E(3)$.
Our strategy  starts with a
complete flag in $\mathbb{P}^3$
$$p \textrm{ (point)} \in \ell \textrm{ (line) }
\subset v \textrm { (plane)},
$$
over which we describe suitable cubic forms $f$ and
quadratic forms $g$ that fulfill some special
conditions. The polynomials $f,g$ will
give us the differential 1-form
$$\omega = \dfrac{3fdg-2gdf}{h},
$$
where $h$ is an equation of the plane $v$ of the
given flag.

The main technical difficulty is
to solve  the indeterminacies of the rational map
$$(g,f) \mapsto \omega.
$$ This is done by a careful analysis of the irreducible
components of the  indeterminacy locus, blown up one at a
time. Macaulay2 (\cite{M2}) is extensively used.
We get a description of a parameter
space for the exceptional component, over which we can apply appropriate tools from equivariant intersection theory to compute the desired degree.

\section{A parameter space
	for the Exceptional Component}\label{description}

The  component $E(3)$
is the orbit closure of the foliation defined by the
1-form $\omega_0$ as in (\ref{fgi}).
The dimension of  $E(3)$
is equal to 13, see  \cite{jinvariante}.

Explicitly, we have
\begin{equation}\label{omega0}
\begin{array}{rcrcr}
\omega_0&=&
(x_1x_2^2-2x_1^2x_3+x_0x_2x_3)dx_0& +&
x_0(3x_1x_3-2x_2^2)dx_1  \\
&&\! +x_0(x_1x_2-3x_0x_3)dx_2& +&\!\!
x_0(2x_0x_2-x_1^2)dx_3.
\end{array}
\end{equation}
The singular locus of the foliation (\ref{omega0}) consists
of a union of three curves:

\begin{enumerate}
	\item the conic given by ideal
	$\langle x_0,x_2^2-2x_1x_3 \rangle$
	(it lies in  the plane $x_0=0$);

	\item the line $\ell_0$ defined by
	$ x_0=x_1=0$;

	\item the twisted cubic given by $ \langle 2x_2^2-3x_1x_3, \ x_1x_2-3x_0x_3, \ x_1^2-2x_0x_2 \rangle$.
\end{enumerate} \medskip

These three components meet at the
point $p_0:=(0:0:0:1) \in \mathbb{P}^3$.
Let us examine the geometry of the  surface defined
by the cubic form
$f_0$.
We see  that it is an irreducible cubic, singular along the
line \,$\ell_0$.
Moreover, given a point $ p_t = (0: 0: t: 1) \in \ell_0
$, the tangent cone to this cubic surface at the point $
p_t $ has equation
$$ x_0^2 - tx_0x_1 = x_0 (x_0-tx_1).
$$
Thus,  the tangent cone is a pair of planes containing the double
line \,$\ell_0$, one of which is fixed and the other varies
with the point $ p_t$.

We have also perceived that at  the special
point $ p_0= (0: 0: 0: 1) $ the tangent
cone to the cubic is the double plane $ x_0^2 = 0$.

Therefore, the cubic $f_0$ comes endowed with
a companion complete flag
\begin{equation} \label{pdv}
\varphi_0:\
p_0=\{x_0=x_1=x_2=0\} \in \ell_0 = \{x_0=x_1=0\}
\subset v_0 =
\{x_0=0\} .
\end{equation}

As for the quadric
$$ g_0 = x_0x_2 - \frac{x_1^2}{2},$$
we have a cone containing the line $\ell_0$
and vertex
$p_0$. Moreover, the tangent plane to the cone at a
smooth point on $\ell_0$ is precisely the same plane $v_0$.

\subsection{\bf Remark.}	\label{recupflag}
	The flag can move under the action of an
	automorphism of $\mathbb{P}^3$, but it can be
	recovered directly from the 1-form $\omega $
	defining the exceptional foliation, just by
	looking at the singular locus. For any such
	$\omega$, the singular locus has three components
	- a conic living on the new plane, the new line
	and a twisted cubic that meets the other two
	components just at the new point; the line is
	tangent to the twisted cubic, the plane is the
	osculating plane and the conic is the osculating
	conic. By the way, the dimension 13 mentioned just
	above is the dimension of the family of pointed
	twisted cubics.\medskip

These considerations about the cubic/quadric pairs
as in (\ref{fgi}) lead us to consider the construction of a
parameter space for the family of exceptional foliations
as a fiber bundle over
the variety $ {\mathbb F}$  of complete flags on $ \mathbb{P}^3 $,
$$
p \textrm{(point)} \in \ell \textrm{(line)} \subset \pi
\textrm{(plane)}.
$$
Given such a flag we take: \medskip

$\bullet$ A cubic surface $f$ with the properties:
\begin{enumerate}
	\item $f$ is singular along $\ell$;
	\item the tangent cone to $f$ at
	each point $q \in \ell$ is
	the union of two planes $\pi \cup \alpha_q$.
	\item $\alpha_p = \pi$, that is, at the point $p$
	the tangent cone  to $f$ is the double plane.
\end{enumerate}

$\bullet$ A quadratic cone $g$ with the properties:
\begin{enumerate}
	\item the line $\ell$ is contained in the cone $g$;
	\item the plane $\pi$ is tangent to $g$ along
	the line $\ell$;
	\item the point $p$ is in the vertex of
	the cone.
\end{enumerate}

An exceptional foliation is then given by the differential form
$$\omega = \dfrac{3fdg - 2gdf}{h},$$
where $h$ is an equation of the plane $\pi$.

\subsection{\bf Definition.}
	For simplicity we will call here a cubic or a quadric that meet the requirements above as \textit{special}.\medskip

We realize that after the
construction of  special pairs
$(f, g)$, it is still necessary to
impose the condition that the differential form $
3fdg-2gdf $ be divisible by the equation of the plane. We
refer to this as  the \textit{divisibility condition}.
In order to better understand this divisibility
condition, let us fix a complete flag as in
(\ref{pdv}).

The requirements about special cubics/quadrics show that
we have

\begin{equation}\label{ff}\left\{\begin{array}l
f=\left(a_0x_0 + a_1x_1 + a_2x_2+a_3x_3\right)x_0^2 +
a_4x_0x_1^2+a_5x_0x_1x_2 + a_6x_1^3\\
g=b_0x_0^2 + b_1x_0x_1 + b_2x_0x_2 + b_3x_1^2.
\end{array}\right.
\end{equation}

We register the following invariant description.

\subsection{\bf Lemma.} \label{monsfg}
	The $7$ monomials appearing in \ $f$ \medskip
	\\\centerline{$x_{0}^{3},x_{0}^{2}x_{1},
		x_{0}^{2}x_{2}, x_{0}^{2}x_{3}, x_{0}x_{1}^{2},   x_{0}x_{1}x_{2}, x_{1}^{3} $}
	are the monomial generators of the
	subspace  of cubics,
	\\\centerline{$x_0 ^2\langle x_0,\dots,x_3\rangle+x_0
		\langle x_0,x_1\rangle \langle x_0,x_1,x_2\rangle + \langle x_0,x_1\rangle ^3\subset S_3$.}
	Likewise, the $4$ monomials \ $x_0^2,x_0x_1,x_0x_2,x_1^2$ \
	in the quadric \ $g$ \ generate the subspace
	\\\centerline{$x_0\langle x_0,x_1,x_2\rangle +  \langle x_0,x_1\rangle ^2\subset S_2
		$.}\medskip

Varying the flag, we obtain equivariant vector
subbundles
\begin{equation}
\label{AB}
\mathcal A\subset S_3 \ \text{ and } \ \mathcal B\subset S_2.
\end{equation}
with respective ranks 7 and 4.
Projectively we obtain, at this stage, the variety
$$
\mathbb{P}(\mathcal A) \times_{\mathbb F}\mathbb{P}(\mathcal B),
$$
a $\mathbb{P}^6 \times \mathbb{P}^3$ bundle
of pairs $(f,g)$ over a fixed flag.

Continuing the discussion about the divisibility
condition, we may write
$$3fdg-2gdf = x_0\omega_1 + \left[(3a_6b_1-2a_4b_3)x_1^4 + (3a_6b_2 - 2a_5b_3)x_1^3x_2\right]dx_0,$$
where $\omega_1$ is a 1-form. From this it follows that the
divisibility condition is given, on the fiber over our
fixed flag $\varphi$,  by the equations
$$\left\{
\begin{array}{c}
3a_6b_1 = 2a_4b_3,
\\3a_6b_2 = 2a_5b_3
\end{array}
\right.$$

This locus consists of two irreducible components inside
$\mathbb{P}^6 \times \mathbb{P}^3$, both of
codimension two:
\begin{equation}
\label{divcond1}
\begin{array}{lrc}
&  (\star)&
\left\{\begin{array}{lcr}
3a_6b_1 &=& 2a_4b_3\\
3a_6b_2 &=& 2a_5b_3\\
a_4b_2 &=& a_5b_1
\end{array}\right\} \end{array} \ \ \ \text{ and } \ \ \ \
\begin{array}{lcr}
& (\star \star )& a_6=b_3=0.
\end{array}
\end{equation}

For a pair $(g,f)$ satisfying $(\star \star )$,
we actually get
$$\left\{\begin{array}{l}
f=x_0\left(a_0x_0^2 + a_1x_0x_1 + a_2x_0x_2+a_3x_0x_3 +
a_4x_1^2+a_5x_1x_2\right),\\
g=x_0\left(b_0x_0 + b_1x_1 + b_2x_2\right).
\end{array}\right.$$
It means that a general element in the second component $(\star \star)$
consists of a cubic and a quadric both
divisible by the equation of the plane,
certainly not interesting for our study of the
exceptional component.

\medskip
Henceforth, we refer to the {\em divisibility condition}
as  the equations $(\star)$ in (\ref{divcond1}). \medskip

Let $\mathbb{G}=\mathbb{G}(1,3)$ be the Grassmann variety of lines in $\mathbb{P}^3$, with tautological sequence
$$0 \longrightarrow \mathcal{S} \longrightarrow \mathbb{G} \times \mathbb{C}^4 \longrightarrow \mathcal{Q} \longrightarrow 0,$$
where rank $(\mathcal S)$=2.
Denote by  $\mathcal{Q}^{\vee}$ the dual
of $\mathcal{Q}$. We also have the tautological sequence
over $\mathbb{P}^3$,
$$
0 \longrightarrow \mathcal{O}_{\mathbb{P}^3}(-1)
\longrightarrow  \mathbb{P}^3
\times \mathbb{C}^4 \longrightarrow \mathcal{P} \longrightarrow 0
$$
Then,
$$
\begin{array} {*5c}
\mathbb{P}(\mathcal{S})&=&\{(p,\ell) \,|\, p \in \ell
\} &\subset& \mathbb{P}^3 \times \mathbb{G}
\\
\mathbb{P}(\mathcal{Q}^{\vee})&=&
\{(\ell, \pi) \,|\, \ell \subset \pi \}&
\subset& \mathbb{G} \times \check{\vphantom{|}\mathbb{P}}^3
.\end{array}$$
The variety of complete flags  is just the fiber product
$$\mathbb{F}=\mathbb{P}(\mathcal{S}) \times _\mathbb{G}
\mathbb{P}(\mathcal{Q}^{\vee}) = \{(p,\ell,\pi) | \ p \in
\ell \subset \pi \}
\subset\mathbb{P}^3\times \mathbb{G}\times \check{\vphantom{|}\mathbb{P}}^3.
$$
The tautological bundles of these spaces lift
to bundles over $\mathbb{F}$ still denoted by the
same letters. They
fit together into the diagram (pullbacks omitted),
$$\begin{array}{*9c}
\mathcal O_{\mathcal Q^\vee}(-1)&=&
\mathcal O_{\check{\vphantom{|}\mathbb P}^3}(-1) & \hookrightarrow & \mathcal Q^\vee &
\hookrightarrow & \mathcal P^\vee &
\hookrightarrow & S_1:=(\mathbb{C}^4)^{\vee}\\ &\varphi_0:&
\langle x_0 \rangle &\subset& \langle x_0,x_1\rangle&\subset&\langle x_0,x_1,x_2\rangle
&\subset&\langle x_0,\dots,x_3 \rangle
\end{array}$$
where the bottom row indicates the corresponding
fibers over the flag (\ref{pdv}). \medskip

In view of Lemma \ref{monsfg}, we have  the surjections
$$\begin{array}{c}
\left(\mathcal O_{\mathcal Q^\vee}(-2)\otimes S_1\right)\oplus\left(
\mathcal O_{\mathcal Q^\vee}(-1)\otimes{\mathcal Q^\vee}\otimes\mathcal
P ^\vee\right)
\oplus{}{\textrm{Sym}}_3\mathcal Q^\vee\twoheadrightarrow
\mathcal A\subset S_3
\\
(\mathcal O_{\mathcal Q^\vee}(-1)\otimes\mathcal P^\vee)
\oplus{}\textrm{Sym}_2\mathcal Q^\vee
\twoheadrightarrow
\mathcal B\subset S_2.
\end{array}
$$

By construction, the vector bundles $\mathcal A$ and $\mathcal B$ fit into the
exact sequences
$$\begin{array} {*4cl}
S_1\otimes\mathcal{O}_{\ensuremath{\check{\vphantom{|}\mathbb P}}^3}(-2)& \hookrightarrow & \mathcal A&
\twoheadrightarrow&
\overline{\mathcal A}:=\mathcal A\big/\left(S_1\otimes\mathcal{O}_{\ensuremath{\check{\vphantom{|}\mathbb P}}^3}(-2)
\right)\\
\mathcal O_{\ensuremath{\check{\vphantom{|}\mathbb P}}^3}(-2)&\hookrightarrow&
\mathcal B &\twoheadrightarrow &\overline{\mathcal B}:=\mathcal B\big/\left(\mathcal{O}_{\ensuremath{\check{\vphantom{|}\mathbb P}}^3}(-2)
\right).
\end{array}$$

\subsection{\bf Lemma.}\label{ovAovB}
	$\overline{\mathcal A}$ is isomorphic to
	$\left(\mathcal Q^\vee/\mathcal
	O_{\ensuremath{\check{\vphantom{|}\mathbb P}}^3}(-1)\right)\otimes\overline{\mathcal B}$.

\begin{proof}
	Indeed,
	on the fiber over $\varphi_0$ we have
	$$(
	\mathcal Q^\vee/\mathcal O_{\ensuremath{\check{\vphantom{|}\mathbb P}}^3}(-1))_{\varphi_0}=\langle x_0,x_1 \rangle/ \langle x_0\rangle
	= \langle \overline{x_1}\rangle
	$$
	hence
	$$
	\left((
	\mathcal Q^\vee/\mathcal O_{\ensuremath{\check{\vphantom{|}\mathbb P}}^3}(-1))\otimes\overline{\mathcal B}\right)_{\varphi_0}=
	\langle \overline{x_1}\rangle \otimes
	\langle \overline{x_0x_1},\overline{x_0x_2},\overline{x_1^2} \rangle
	=\overline{\mathcal A}_{\varphi_0},
	$$
	where the = signs mean isomorphisms of
	representations of the stabilizer of the flag
	$\varphi_0$.
	\end{proof}

\subsection{\bf Definition.}
	We define
	$\mathbb{X}=\mathbb P(\mathcal B)$
	the corresponding projective bundle of special quadrics.
	The fiber $\mathbb{X}_{\varphi_0}
	$ is the $\mathbb{P}^3$ of special quadrics over the flag
	$\varphi_0 =(p_{0},\ell_0,v_0)$ fixed in (\ref{pdv}). \medskip

The divisibility condition $(\star)$\,(\ref{divcond1}) can be rewritten as a
system of linear equations
in variables $(\underline{a})=(a_4:a_5:a_6)$ with
coefficients $(\underline{b})=(b_0:b_1:b_2:b_3)\in \mathbb{P}^3$,
\begin{equation}\label{syslin}
\left[\begin{array}{ccc}
2b_3 & 0     & -3b_1\\
0    & -2b_3 & 3b_2\\
-b_2 & b_1   & 0
\end{array}\right] \cdot
\left[\begin{array}{c}
a_4\\
a_5\\
a_6
\end{array}\right] =
\left[\begin{array}{c}
0\\
0\\
0
\end{array}\right].
\end{equation}

The matrix of the coefficients has determinant zero and
generic rank two. This rank drops when
$b_1=b_2=b_3=0\leftrightarrow x_0^2$. Off the
point $x_0^2$  the solution space is spanned by
the vector product of two rows,
$$(-3b_1b_2:-3b_2^2:-2b_2b_3)=( 3b_1:3b_2: 2b_3)\ .$$

The idea now is to describe the locus of special cubic
forms with the divisibility condition as a
$\mathbb{P}^4$-bundle over $\mathbb{X}$ (since it has
codimension two in the $\mathbb{P}^6$ of special
cubics). But to make it possible, we need to replace
$\mathbb{X}$ by a new space, $\mathbb{X}'$,
for which the rank of the
coefficients matrix in (\ref{syslin}) is two everywhere.

Precisely, think of the fiberwise solution space to
(\ref{syslin}) as defining a rational map
$\psi:\mathbb X \rightarrow \mathbb{P}(\overline{\mathcal A})$, which on the fiber over the standard
flag $\varphi_0$ (\ref{pdv}) reads
\begin{equation}\label{b2a}
\psi:\
(\underline b)\mapsto
(a_4:a_5:a_6)=
(3b_1:3b_2:2b_3).
\end{equation}
Look at the closure $\mathbb{X}'$ of the graph of
$\psi$. On the fiber over $\varphi_0$,
this is the blowup of $\mathbb{P}^3=\mathbb{P}(\mathcal B_{\varphi_0})$
at the point $x_0^2$.
As a matter of fact, this turns out to be the
restriction to the fiber over $ \varphi_0$
of the blowup of $\mathbb{X}$ along the section
$\mathbb{P}\left(\mathcal{O}_{\ensuremath{\check{\vphantom{|}\mathbb P}}^3}(-2)\right)
\hookrightarrow \mathbb{P}(\mathcal{B})$ over $\mathbb
F$. We have the diagram
$$
\xymatrix
@R2.5pc
{&&
	\,  \mathbb X'
	\ar@{->}[d]_{\text{\small (blowup)}}
	\ar@{->}[drr]^{\ \ \ \ \psi':\text{\small\,\,($\mathbb{P}^1$-bundle)}}
	&&&
	\\
	\mathbb{P}\left(\mathcal{O}_{\ensuremath{\check{\vphantom{|}\mathbb P}}^3}(-2)\right)
	\ \ \ar@{=}[drrr]
	\!\! \ar@{^(-}[r]&

	\mathbb{P}(\mathcal B) \ar@{=}[r]& \mathbb X
	\ar@{->}[dr]
	\ar@{-->}[rr]^{\!\hskip-.31cm^\psi}&&\mathbb{P}(\overline{\mathcal A})
	\ \ar@{->}[dl] \ar@{=}[r]& \mathbb{P}(\overline{\mathcal B})
	\\&&&\ \ \ \mathbb F \ \ \ & &
}
$$
where the leftmost inclusion is defined by
taking the square of the equation of the plane in
the flag.
The equality $\mathbb{P}(\overline{\mathcal A})=\mathbb{P}(\overline{\mathcal
	B})$ comes from Lemma\,\ref{ovAovB}; under this
identification,
$$\mathcal O_{\overline{\mathcal A}}(-1)=
\mathcal O_{\overline{\mathcal B}}(-1)\otimes (
\mathcal Q^\vee/\mathcal O_{\ensuremath{\check{\vphantom{|}\mathbb P}}^3}(-1)).
$$
Pulling back the above tautological line
subbundle via $\psi'$ , we get the diagrams
\begin{equation}\label{ovAB}
\begin{array}{*5c}
S_1\otimes\mathcal{O}_{\ensuremath{\check{\vphantom{|}\mathbb P}}^3}(-2)
&\hookrightarrow&\mathcal A'&
\twoheadrightarrow&
\mathcal O_{\overline{\mathcal A}}(-1)\\
||&&\downarrow&&\downarrow\\
S_1\otimes\mathcal{O}_{\ensuremath{\check{\vphantom{|}\mathbb P}}^3}(-2) &\hookrightarrow& {\mathcal A}&
\twoheadrightarrow&
\overline{\mathcal A}\
\end{array}
\end{equation}
and
\begin{equation}\label{ovAB1}
\begin{array}{*5c}
\mathcal{O}_{\ensuremath{\check{\vphantom{|}\mathbb P}}^3}(-2)
&\hookrightarrow&\mathcal B'&
\twoheadrightarrow&
\mathcal O_{\overline{\mathcal B}}(-1)\\
||&&\downarrow&&\downarrow\\
\mathcal{O}_{\ensuremath{\check{\vphantom{|}\mathbb P}}^3}(-2) &\hookrightarrow& {\mathcal B}&
\twoheadrightarrow&
\overline{\mathcal B}\
\end{array}
\end{equation}
where rank $\mathcal A'=5$, rank $\mathcal B'=2$. We have
\begin{equation}\label{defX'}
\mathbb X'=\mathbb{P}(\mathcal B'),
\end{equation} a $\mathbb{P}^1$ bundle over the $\mathbb{P}^2$ bundle
$\mathbb{P}(\overline{\mathcal B})$ over the flag variety
$\mathbb F$.
Define
\begin{equation}\label{Y}
\mathbb Y=\mathbb{P}(\mathcal A'),
\end{equation}
an equivariant $\mathbb{P}^4$--bundle over $\mathbb X'$.

\subsection{\bf Proposition.}
	A general point in $\mathbb Y$ corresponds to a pair
	$(g,f)$ of special quadric, cubic satisfying the
	divisibility condition.
\begin{proof}
	The assertion follows from the previous
	considerations.
\end{proof}

The next step is to solve the indeterminacies of the rational map
\begin{equation}\label{mapomega}
\begin{array}{ccc}
\mathbb{Y} & \makebox{\Large$\dashrightarrow$}    &
E(3)\subset
\mathbb{P}(H^0(\Omega^1_{\mathbb{P}^3}(4)))\\
(g,g',f) & \longmapsto & \omega = \dfrac{3fdg - 2gdf}{x_0} \ .\\
\end{array}
\end{equation}
This will be accomplished by a sequence of 4 blowups,

\medskip
\medskip
$$\xymatrix
{\mathbb Y_4\,\ar@/^2.0pc/[rrrrr]\,\ar@{->}[r]
	&
	\mathbb Y_3\,\ar@{->}[r]&\mathbb Y_2\,\ar@{->}[r]&\mathbb Y_1\,\ar@{->}[r]&\mathbb Y\,\ar@{-->}[r]&E(3).
}$$

\subsection{\bf Remark.}
	Since all constructions performed so far are equivariant, we drop the reference to the fiber over $\varphi_0$, and simplify notation writing
	$$\mathbb X=\mathbb
	X_{\varphi_0},\ \mathcal A=\mathcal A_{\varphi_0},\dots,\  etc.
	$$

In view of Lemma \ref{ovAovB}, the rational map (\ref{b2a}) can also be written as the rational linear projection map
$$\begin{array}{cccc}
\psi_{\overline{\mathcal{B}}}: & \mathbb{X} &
\dashrightarrow   & \mathbb{P}(\overline{\mathcal{B}})\\
& g:=b_0x_0^2 + b_1x_0x_1 + b_2x_0x_2 + b_3x_1^2 &
\longmapsto
& g':=b_1\overline{x_0x_1} + b_2\overline{x_0x_2} + b_3\overline{x_1^2}
\end{array}$$

(class mod.\,$x_0^2$). Write for short
\begin{equation} \label{X'}
\mathbb{X}' = \{ (g,g') =
\left((b_0:b_1:b_2:b_3),(u_1:u_2:u_3)\right)
| \ b_iu_j=b_ju_i  \},
\end{equation}
where
\begin{equation} \label{defg'}
g':=u_1\overline{x_0x_1} + u_2\overline{x_0x_2} + u_3\overline{x_1^2}\ .
\end{equation}

Here, the $\underline{u}$ are
homogeneous coordinates in $\mathbb{P}^2$ = fiber of $\mathbb{P}(
\overline{\mathcal B})$ over $\varphi_0$.
The map $\psi'$ is (see  (\ref{b2a}))
$$\begin{array}{cccc}
\psi': & \mathbb{X'} & \longrightarrow    & \mathbb{P}(\overline{\mathcal{A}})\\
& (g,g')
& \mapsto & 3u_1\overline{x_0x_1^2} + 3u_2\overline{x_0x_1x_2}
+ 2u_3\overline{x_1^3} \\
\end{array}$$

Notice that the divisibility condition
(\ref{divcond1})
has now the expression
\begin{equation}\label{divcond}
\left\{\begin{array}{lcl}
3a_6u_1 &=& 2a_4u_3\\
3a_6u_2 &=& 2a_5u_3\\
a_4u_2 &=& a_5u_1
\end{array}\right.\end{equation}
It can be seen that the system above is of rank 2 for all $(u_1:u_2:u_3)\in\mathbb{P}^2$.
Equations (\ref{divcond}) enables to find all special
cubics over any point $(g,g') \in \mathbb{X}'$.

\section{Solving the Indeterminacies}\label{solvingmap}

Let us work over an affine cover of $\mathbb{X}'$ to deal with the indeterminacies of the map $(g,g',f) \mapsto \omega$.

Notice that over a fixed flag, a point in $\mathbb{Y}$ has 7 affine coordinates.

Notation as in (\ref{X'}), consider the open dense set of $\mathbb{X}'$ given by $b_0=u_1=1$. The interesting equations are
$$ \left\{\begin{array}{l}
b_2 = b_1u_2\\
b_3 = b_1u_3
\end{array}\right\} \ \textrm{ and } \ \left\{\begin{array}{l}
3a_6 = 2a_4u_3\\
a_5 = a_4u_2
\end{array}\right\},$$
where the former corresponds to the blowup of our $\mathbb{P}^3$ of
special quadrics at $x_0^2$ (\ref{X'}) and the latter
to divisibility $(\star)$\,(\ref{divcond}). So,
$$\left\{\begin{array} l
f=\left(a_0x_0 + a_1x_1 + a_2x_2
+a_3x_3\right)x_0^2 +a_4\left(
x_0x_1^2+u_2x_0x_1x_2 + \frac{2}{3}u_3x_1^3\right)
\\
g=x_0^2 + b_1x_0x_1 + b_1u_2x_0x_2 + b_1u_3x_1^2\ .
\end{array}\right.
$$

Computing $\omega$ as in (\ref{mapomega}), we find using Macaulay2
\begin{eqnarray*}
	\omega = [(3a_0b_1-2a_1)x_0^2x_1+(6a_0b_1u_3+a_1b_1-4a_4)x_0x_1^2+ \\ + (3a_0b_1u_2-2a_2)x_0^2x_2-2a_3x_0^2x_3+ \cdots ]dx_0 + \cdots
\end{eqnarray*}
For  $a_0=0$  the 1-form $\omega $ vanishes only if
$a_0=a_1=a_2=a_3=a_4=0,$ which is impossible. Thus, in order to study the indeterminacy locus we can take $a_0=1$.

Setting $a_0=1$ and collecting coefficients of $\omega$, we get a non-reduced and reducible scheme, given by an ideal $J$. Its radical $rad(J):=J_{red}$ presents two irreducible components: \medskip

\begin{enumerate}
	\item A component $C$,
	with ideal
	\begin{equation} \label{JC}
	J_C=\langle b_1-4u_3,a_1-6u_3,a_2,a_3,a_4-12u_3^2,u_2 \rangle
	\end{equation}
	The ideal $J_C$ is generated by a regular
	sequence.  The affine coordinates here are $a_1,
	a_2, a_3, a_4, b_1, u_2, u_3$.
	Therefore we have a locally complete
	intersection, which represents the curve over the
	flag $\varphi_0$ given by
	$$f=(x_0+2u_3x_1)^3, \ \ g=(x_0+2u_3x_1)^2.$$

	\item A component $E$ with ideal
	\begin{equation} \label{JEE}
	J_E=\langle a_1,a_2,a_3,a_4,b_1 \rangle,
	\end{equation}
	which is the whole $\mathbb{P}^2$--fiber of the
	exceptional divisor of $\mathbb{X}'$ over
	$g=x_0^2$, and $f=x_0^3$.
\end{enumerate} \medskip

Notice that there is precisely one  point in the
intersection of these two components, since
$$J_C + J_E = \langle a_1,a_2,a_3,a_4,b_1,u_2,u_3 \rangle .$$
This ideal represents the single point $(g,g',f)=(x_0^2,x_0x_1,x_0^3) \in \mathbb{Y} $. \medskip

Let's see what happens when we
blowup
$\mathbb{Y}$ first along $C$, followed by a blowup along
$E'$, the strict transform of $E$.

Let $\mathbb{A}^7$ be the affine neighborhood defined by $b_0=a_0=u_1=1$. The blowup of this $\mathbb{A}^7$ along $C$ is
$$\mathbb{Y}_1 = \{\left(
(\underline{a},\underline{b}),(s_0:\ldots:s_5)\right)
\ | \ s_i \cdot e_j = s_j \cdot e_i\} \subset \mathbb{A}^7 \times \mathbb{P}^5,$$
where $e_i, \ 0\leq i \leq 5$, are the equations of $J_C$, ordered as in (\ref{JC}).

There are six choices for the local equation of the first
exceptional divisor. Choose, say,
exc$_1=u_2$, {\em i.e,} look to the affine chart $s_5=1$.
Then, we make the following substitutions
$$\left\{\begin{array}{rcl}
b_1-4u_3    & = & s_0u_2 \\
a_1-6u_3    & = & s_1u_2 \\
a_2         & = & s_2u_2 \\
a_3         & = & s_3u_2 \\
a_4-12u_3^2 & = & s_4u_2
\end{array}\right.
$$

The new 7 affine coordinates on the blowup are
\begin{equation}\label{affineY1}
s_0,s_1,s_2,s_3,s_4,u_2,u_3.
\end{equation}

After the blowup $\mathbb{Y}_1 \rightarrow \mathbb{Y}$  of $\mathbb{Y}$
along $C$,
the strict transform of the radical $J_{red}$ is given by

\begin{equation} \label{KE'}
J_E'=\langle s_3,s_2,3s_0-2s_1,6u_3+s_1u_2,3s_4+s_1^2u_2 \rangle.
\end{equation}
It coincides with the strict transform $E'$ of $E$ under
the first blowup. Now, take $E'$ as our new blowup center and
denote by $\mathbb{Y}_2 \rightarrow \mathbb{Y}_1$ the blowup of $\mathbb{Y}_1$ along $E'$,
$$\mathbb{Y}_2 = \{\left(
(\underline{u},\underline{s}),(t_0:\ldots:t_4)\right)
\ | \ t_i \cdot e_j = t_j \cdot e_i\} \subset \mathbb{A}^7 \times \mathbb{P}^4,$$
where $e_i, \ 0\leq i \leq 4$, are the equations of
$J_{E'}$, ordered as in (\ref{KE'}), and
$(\underline{u},\underline{s})$ as in (\ref{affineY1}).

Choose now the equation exc$_2=s_2$ in (\ref{KE'})  as
the new local exceptional equation. Equations of
$\mathbb{Y}_2$
become
$$\left\{
\begin{array}{rcl}
s_3           & = & t_0s_2 \\
3s_0-2s_1     & = & t_2s_2 \\
6u_3+s_1u_2   & = & t_3s_2 \\
3s_4+s_1^2u_2 & = & t_4s_2
\end{array}\right.
$$
The new 7 affine coordinates on $\mathbb{Y}_2$ are
\begin{equation}\label{affineY2}
t_0,t_2,t_3,t_4,u_2,s_1,s_2.
\end{equation}

The two blowups so far performed are not enough to solve
the indeterminacies of our
map $(g,f) \mapsto \omega$. However they do make    the
new scheme of indeterminacies to become reduced.
Although reduced, the indeterminacy locus is still reducible, with two components. One of them is given by the ideal
\begin{equation} \label{JR2}
J_{R}=\langle u_2,t_3-1,t_2,t_0,3s_1-2t_4 \rangle.
\end{equation}
The ideal $J_{R}$ represents a ruled surface $R$ which is a
$\mathbb{P}^1$--subbundle of the (transform of the) exceptional divisor $\mathbb{P}(\mathcal
N_{C|\mathbb Y})$ -- note the equation exc$_1=u_2$ representing the curve $C$ and the other four linear equations on the new affine variables.

Denote by $\mathbb{Y}_3 \rightarrow \mathbb{Y}_2$ the blowup of $\mathbb{Y}_2$ along $R$. This blowup, in the affine chart $s_5=1,\ t_1=1$, is
$$\mathbb{Y}_3 = \{\left(
(\underline{u},\underline{s},\underline{t}),(v_0:\ldots:v_4)  \right)
\ | \ v_i \cdot e_j = v_j \cdot e_i\} \subset \mathbb{A}^7 \times \mathbb{P}^4,$$
where $e_i, \ 0\leq i \leq 4$, are the equations of
$J_{R}$, ordered as in (\ref{JR2}), and
$(\underline{u},\underline{s},\underline{t})$ as in (\ref{affineY2}).

Choose now the equation exc$_3=u_2$ in (\ref{JR2})  as
the new local exceptional equation, {\em i.e,} take $v_0=1$. Equations of $\mathbb{Y}_3$ become
$$\left\{
\begin{array}{rcl}
t_3-1     & = & v_1u_2 \\
t_2       & = & v_2u_2 \\
t_0       & = & v_3u_2 \\
3s_1-2t_4 & = & v_4u_2
\end{array}\right.
$$

The new 7 affine coordinates on $\mathbb{Y}_3$ are
\begin{equation}\label{affineY3}
v_1,v_2,v_3,v_4,u_2,s_2,t_4
\end{equation}
and the indeterminacy locus reduces to
\begin{equation}\label{J'''L}
J_L=\langle s_2,v_1,v_2,v_3,v_4,t_4 \rangle.
\end{equation}

The component represented by the ideal $J_L$ (\ref{J'''L}) is easy to
describe. Since the affine variables are as listed in
(\ref{affineY3}),
the six linear equations represent a
line inside the (transform of) the second blowup center $E'$ (note the presence of the equation exc$_2 = s_2$ in $J_L$), a line parametrized by the variable
$u_2$.

We call $L$ the component described by the ideal $J_L \
(\ref{J'''L})$. This is a reduced and irreducible
local complete intersection. Since it is the full indeterminacy locus,
a blowup along this subscheme will solve the indeterminacies over the
neighborhood $[b_0=1, \ u_1=1],[s_5=t_1=v_0=1]$.
That is, the map to $\omega$
becomes a morphisms,
cf. \,\cite[ex. 7.17.3, p.\,168]{hartshorne2013algebraic}.

Denote by $\mathbb{Y}_4 \rightarrow \mathbb{Y}_3$ the blowup of $\mathbb{Y}_3$ along $L$,
$$\mathbb{Y}_4 = \{\left(
(\underline{u},\underline{s},\underline{t},\underline{v}),(z_0:\ldots:z_5)\right)  \ | \ z_i \cdot e_j = z_j \cdot e_i\} \subset \mathbb{A}^7 \times \mathbb{P}^5,$$
where $e_i, \ 0\leq i \leq 5$, are the equations of $J_L$, ordered as in (\ref{J'''L}), and $(\underline{u},\underline{s},\underline{t},\underline{v})$ as
in (\ref{affineY3}).

In $\mathbb{Y}_4$ the map is solved, at least in the
affine chart [$s_5=t_1=v_0=1$].	The calculations in all
other standard neighborhoods reveal that this sequence of
four blowups, along $C$, $E$, $R$ and $L$,
will solve the map $(g,f) \mapsto \omega$ over the neighboorhod $[b_0=1, \ u_1=1]$.

There are other five standard neighborhoods to be checked to complete an affine cover of
$\mathbb{X}'$,  corresponding to
$$[b_3=1], \ [b_2=1], \ [b_1=1], \ [b_0=1 \textrm{ and }
u_2=1], \ [b_0=1 \textrm{ and } u_3=1].$$ Similar calculations show  that these four blowups described will solve the
map over each of these neighborhoods. \medskip

The whole discussion of the indeterminacy loci was made over the fixed flag (\ref{pdv}).

Since the blowup centers as described were actually fibers of
bundles  over the variety of complete flags  ${\mathbb F}$,
we have at the end a bundle which constitutes the
desired parameter space:

\begin{equation} \label{summarized}
\begin{array} c
\mathbb{Y}_4   \xrightarrow{\textrm{blowup } L} \mathbb{Y}_3 \xrightarrow{\textrm{blowup }R} \mathbb{Y}_2  \xrightarrow{\textrm{blowup }E} \mathbb{Y}_1  \xrightarrow{\textrm{blowup }C} \mathbb{Y} \\\\
\mathbb{Y} \xrightarrow{\mathbb{P}^4 \textrm{ bundle}} \mathbb{X}' \xrightarrow{\textrm{blowup }\mathbb{P}(\mathcal{O}_{\ensuremath{\check{\vphantom{|}\mathbb P}}^3}(-2))} \mathbb{X}
\xrightarrow{\mathbb{P}^3 \textrm{ bundle } } {\mathbb F}=\textrm{ all flags}
\end{array}
\end{equation}

This can be summarized as follows:

\subsection{\bf Theorem.}\label{Y4}
	Let $\mathbb{Y}_4$ be the variety obtained by the four
	blowups as described above. Then $\mathbb{Y}_4$
	is
	equipped with a morphism $\Phi$
	onto the exceptional component $E(3)$.
	\qed

\subsection{\bf Proposition.}
	The map $\Phi: \mathbb{Y}_4 \longrightarrow E(3)$ is generically injective.
	\begin{proof}\normalfont
		For a given $\omega \in E(3)$ (off the boundary) we already saw that the flag can be recovered (see Remark \ref{recupflag}). Hence, we can look at a fiber over a fixed flag.

		So fix the flag $\varphi_0=(p_{0},\ell_{0},v_0) (\ref{pdv})$. The fiber of $\mathbb{X}$ at this flag is the $\mathbb{P}^3$ of special quadrics
		$$g=b_0x_0^2 + b_1x_0x_1 + b_2x_0x_2 + b_3x_1^2.$$
		Now, we will pay attention to the dense
		open set where $b_2=1$, so
		\begin{equation} \label{gfibra}
		g=b_0x_0^2 + b_1x_0x_1 + x_0x_2 + b_3x_1^2.
		\end{equation}
		The divisibility condition (\ref{divcond1}) shows that the fiber of $\mathbb{Y}$ over a quadric as in (\ref{gfibra}) is the $\mathbb{P}^4=\{(a_0:a_1:a_2:a_3:a_5) \}$ of special cubics
		$$f=(a_0x_0+a_1x_1 +a_2x_2+a_3x_3)x_0^2
		+ a_5\left(b_1x_0x_1^2 +    x_0x_1x_2 +
		\frac{2}{3}   b_3x_1^3\right)
		.$$
		Computing $\omega = \dfrac{3fdg-2gdf}{x_0}$ one can see that there are no indeterminacies on this neighborhood $[b_2=1]$. Moreover,
		$$2g\dfrac{\partial f}{\partial x_3} - 3f\dfrac{\partial g}{\partial x_3} = 2a_3gx_0^2\ ,$$
		and the quadric $g$ can be recovered from
		$\omega $ just by looking at the coefficient of $dx_3$ (at least on the dense open set $a_3=1$).

		The coefficient of $dx_2$ in $\omega$ is

		\begin{eqnarray*}
			\dfrac{1}{x_0}\left(2g\dfrac{\partial f}{\partial x_2} -
			3f\dfrac{\partial g}{\partial x_2}\right) =
			(2a_2b_0-3a_0)x_0^3+(2a_5b_0+2a_2b_1-3a_1)x_0^2x_1+ \\ + (-a_5b_1+2a_2b_3)x_0x_1^2
			-a_2x_0^2x_2-a_5x_0x_1x_2-3a_3x_0^2x_3.
		\end{eqnarray*}

		Hence the coefficients $a_2$ and $a_5$ can be recovered from $x_0^2x_2dx_2$ and $x_0x_1x_2dx_2$,
		respectively. And since the coefficients $b_0$ and $b_1$ are also known we obtain the coefficients $a_0$ and $a_1$ from $x_0^3dx_2$ and $x_0^2x_1dx_2$
		respectively.
	\end{proof}

\section{Calculation of Degree}\label{deg}

By construction, $\mathbb{Y}_4$ is equipped with a line bundle $\mathcal{W}$, pullback of the line bundle $\mathcal{O}_{\mathbb{P}^{44}}(-1)$ over
${\mathbb{P}^{44}}=\mathbb{P}(H^0(\Omega^1_{\mathbb{P}^3}(4)))$. The fiber of $\mathcal{W}$ over a point in $\mathbb{Y}_4$ is the rank one space spanned by the computed 1-form $\omega$.

$$\begin{array}{ccccc}
\mathcal{W}=\Phi^*(\mathcal{O}(-1))&   & \mathcal{O}(-1)  & &  \\
\downarrow &  & \downarrow  & & \\
\mathbb{Y}_4 & \stackrel{\Phi}\longrightarrow    & \mathbb{P}(H^0(\Omega^1_{\mathbb{P}^3}(4))) & \supset & E(3)=\Phi(\mathbb{Y}_4)\\
\end{array}$$

\subsection{\bf Theorem.} \label{integral}
	The degree of the exceptional component of
	codimension one and degree two foliations
	in $\mathbb{P}^3$ is given by
	$$\int\limits_{\mathbb{Y}_4}-c_1^{13}(\mathcal{W}) \cap [\mathbb{Y}_4].$$
	\begin{proof}\normalfont
		We have $\dim (\mathbb{Y}_4) =13$. Since the map $\Phi$ is generically injective the required degree is (cf. definition of deg$_f\tilde{X}$ in \cite{Fulton}, page 83) $$\int\limits_{\mathbb{Y}_4}c_1(\Phi^*\mathcal{O}(1))^{13} \cap [\mathbb{Y}_4] = \int\limits_{\mathbb{Y}_4}-c_1(\Phi^* \mathcal{O}(-1))^{13} \cap [\mathbb{Y}_4]= \int\limits_{\mathbb{Y}_4}-c_1(\mathcal{W})^{13} \cap [\mathbb{Y}_4].$$
	\end{proof}

In order to obtain the value of the integral in Theorem \ref{integral}, we apply Bott's formula
\begin{equation}  \label{BOTT}
\int\limits_{\mathbb{Y}_4}-c_1^{13}(\mathcal{W}) \cap [\mathbb{Y}_4]=\sum\limits_{F}\dfrac{-c_1^T(\mathcal{W})^{13} \cap [F]_T}{c_{top}^T(\mathcal{N}_{F|\mathbb{Y}_4})},\end{equation}
where the sum runs through all fixed components $F$ under
a convenient action of the torus $\mathbb{C}^*$.  The $\mathcal{N}_{F|\mathbb{Y}_4}$ appearing in the denominator denotes
the normal bundle of a fixed component $F$ in $\mathbb Y_4$.
Fix  an action
\begin{equation}\label{actionp3}
\begin{array}{ccc}
\mathbb{C}^* \times \ensuremath{\check{\vphantom{|}\mathbb P}}^3 & \longrightarrow & \ensuremath{\check{\vphantom{|}\mathbb P}}^3 \\
(t,x_i) & \mapsto & t^{w_i}x_i
\end{array}
\end{equation}
of $\mathbb{C}^*$ in $\ensuremath{\check{\vphantom{|}\mathbb P}}^3$ with distinct weights $w_i$, $i \in \{0,1,2,3 \}$.
The only fixed flags are the 24 standard ones
$$p_{ijk}=\{x_i=x_j=x_k=0\} \in \ell_{ij}=\{x_i=x_j=0\} \subset v_i=\{x_i=0\}.$$

Over each one of these fixed flags, we can find 72 fixed isolated points and 5 fixed lines. So, there is a total of $72 \cdot 24
= 1728$ fixed points and $5 \cdot 24 = 120$ fixed lines.

A detailed exposition of the fixed points and the computations of their contributions on Bott's formula (\ref{BOTT}), including scripts for  \textit{Macaulay2} for this sum and for the resolution of singularities also in the other neighborhoods can be found in

\url{https://sites.google.com/a/ifsudestemg.edu.br/nucleo-de-matematica/artur/trabalhos/Exceptional.pdf}

\subsection{\bf Theorem.} \label{GRAUE3}
	The degree of the exceptional component of foliations of codimension one and degree two in $\mathbb{P}^3$ is \textbf{168208}. \medskip

	In the next Table \ref{tab} we list the components of
	the space of foliations of degree two and codimension one
	in $\mathbb{P}^3$ as described in  \cite{Alcides}, and
	their respective degrees.

	Notice that the degree of the logarithmic
	components have not been found yet in the literature to
	the best of our knowledge.

\begin{table}

	\caption{Components of $\overline{\mathcal{F}(2,3)}$ and their Degrees}
	\label{tab}

	\begin{tabular}{lll}
		\hline\noalign{\smallskip}
		Component & Degree & Reference  \\
		\noalign{\smallskip}\hline\noalign{\smallskip}
		Linear Pullbacks $\mathcal{S}(2,3)$ & 1320 & Ferrer and Vainsencher (to appear)  \\
		Rational $\mathcal R(2,2)$ & 1430 & \cite{CukiermanVI} \\
		Rational $\mathcal R(1,3)$ & 700 & \cite{CukiermanVI} \\
		Logarithmic $\mathcal{L}(1,1,1,1)$ & ???? &  \\
		Logarithmic $\mathcal{L}(1,1,2)$ & ???? &  \\
		Exceptional Component $E(3)$ & 168208 & Theorem \ref{GRAUE3} \\
		\noalign{\smallskip}\hline
	\end{tabular}
\end{table}

\medskip

\section{A geometric interpretation of the degree}\label{geo}

For a codimension one foliation in $\mathbb{P}^n$, given by the differential form
$$\omega = \sum_{i=0}^{n} A_idx_i,$$
the hyperplane defined by the distribution at a point $p \in \mathbb{P}^n$ is
$$H=\sum_{i=0}^{n} A_i(p)x_i = 0.$$

Thus, a tangent direction $v=(v_0: \ldots : v_n) \in \mathbb{P}(T_p\mathbb{P}^3)$ lies on this hyperplane $H$ if
$$\sum_{i=0}^{n} A_i(p) \cdot v_i = 0.$$
This can be thought of as a linear equation on the
coefficients of the $A_i$.
Hence, the point $(p,v) \in \mathbb{P}(T\mathbb{P}^3) $
defines a hyperplane in the projective space of distributions.
\begin{equation}\label{condlinear}
\omega(p) \cdot v = 0.
\end{equation}

The equation (\ref{condlinear}) shows that the degree of
a $m$--dimensional component of the space of codimension
one foliations in $\mathbb{P}^n$ can be interpreted as
the number of such foliations that are tangent  to $m$
general
directions in $\mathbb{P}^n$.

In particular, Theorem \ref{GRAUE3} means that there are 168208 exceptional foliations tangent to 13 general directions in $\mathbb{P}^3$.

\subsection*{\bf Acknowledgements}

	Our hearts and minds go with
	Flaviano Bahia P. Vieira (1984-2011). He has worked on this subject,
	especially  during a visit at the UVa. His untimely
	death has  deprived us of the company of a beloved friend.


text:
\begin{document}

\title{Uncovering Proximity of Chromosome Territories using Classical Algebraic Statistics}
\author{Javier Arsuaga}

\address{Department of Mathematics, San Francisco State University, San Francisco CA 94132}

\author{Ido Heskia}

\author{Serkan Ho\c{s}ten}

\author{Tatsiana Maskalevich}

\begin{abstract}
Exchange type chromosome aberrations (ETCAs) are rearrangements of the genome that occur when chromosomes break and the resulting fragments rejoin with other fragments from other chromosomes.
ETCAs are commonly observed in cancer cells and in cells exposed to radiation. The frequency of these chromosome rearrangements is correlated with their spatial
proximity, therefore it can be used to infer the three dimensional organization of the genome. Extracting statistical significance of spatial proximity from cancer
and radiation data  has remained somewhat elusive because of the sparsity of the data. We here propose a
new approach to study the three dimensional organization of the genome using algebraic statistics. We test our method on a published data set of irradiated human blood
lymphocyte cells.
We provide a rigorous method for testing the overall organization of the genome, and in agreement with previous results we find a random relative positioning of
chromosomes with the exception of  the chromosome  pairs \{1,22\} and \{13,14\} that have a significantly larger number of ETCAs than the rest of the chromosome pairs
suggesting their spatial proximity. We conclude that algebraic methods can successfully be used to analyze genetic data and have potential applications to larger and more complex data sets.
\end{abstract}

\maketitle

\section{Introduction}

\medskip
During the early stages of the cell cycle the mammalian genome is organized in chromosome territories \cite{Cremer82,Rabl} (for a review see \cite{Cremer10}). When DNA damaging agents, such as radiation,
cross the cell nucleus they introduce double strand breaks that produce chromosome fragments. These chromosome fragments need to be rejoined with their original
partners for the cell to survive. A small percentage of breaks however are incorrectly rejoined introducing exchange type chromosome aberrations (ETCAs).
ETCAs between non-homologue chromosomes can be detected in the laboratory by means of diverse chromosome painting and sequencing techniques (see Figure \ref{SKY_Image} for an example of ETCAs detected by the chromosome painting technique SKY). It is expected that chromosomes that are in close spatial
proximity form ETCAs more often than those that are far apart \cite{Chen96,Bickmore02,Nikiforova,Roix03,ZhangDekker12}.
Therefore the frequency of ETCAs between non-homologous chromosomes is informative of their relative position and it
can be used to reconstruct the three dimensional structure of the genome.

\begin{figure}[htbp]
\centering
\makebox{\includegraphics[height=1.75in]{SKY_Image_JuanLuis_V2}}
\caption{\label{SKY_Image} Examples of ETCAs found in a tumor cell. Pairs of homologue chromosomes were painted the same color using SKY. The arrows point to ETCAs.
Figure kindly provided by J.L. Garc\'{i}a.}
\end{figure}

Chromosome painting techniques such Fluorescence in-situ Hybridization  (FISH) or its multicolor versions   such as Spectral Karyotyping (SKY) and multiplex FISH (mFISH)
\cite{Speicher,Schrock} paint every pair of homologue chromosomes the same color (Figure \ref{SKY_Image}). These techniques have shown that the position of
chromosome territories with respect to the center of the nucleus is driven by gene density, chromosome size and/or local chromatin geometry \cite{Boyle2001,Kolbl,Parada}.
In human lymphocytes gene-rich chromosomes  such as  \{1, 19, 17, 22\}  are located near the center of the nucleus \cite{Boyle2001,Cremer10} while gene-poor chromosomes such as $\{2,4,13,18\}$ are located closer to the periphery of the nucleus \cite{Boyle2001, Cremer10}.
Studies measuring the relative position of chromosomes, using radiation induced ETCAs, on the other hand have shown that chromosomes are randomly located with respect to each other with the exception of a few chromosome clusters \cite{Corn2002,Hlatky2002,Sachs1997,Sachs2004}. Interestingly  this overall random relative organization has been corroborated by new sequencing techniques \cite{Lieberman09,ZhangDekker12}.
\medskip

Quantification of the relative position of chromosome territories commonly use tables whose entries are the number of ETCAs detected for any two non-homologue chromosomes
such as the one we use in  Table \ref{table:Lympho2004}.  Several methods have been proposed to study frequency tables of radiation induced ETCAs.
In \cite{Boei06} tables of radiation induced ETCAs detected by FISH were analyzed.
In this type of studies frequency tables were densely populated and with large entries, therefore chi-square statistics were used to find significant clusters of chromosomes.
In \cite{Ars2004,Corn2002} similar tables were generated employing mFISH data. Although more accurate in some respects, the latter data were more sparse and with small entries.
In this case the proximity of chromosome territories was tested by assigning p-values to clusters of previously reported  chromosome territories.
Here we  propose a model-based approach that builds a simple log-linear model to test the proximity of pairs of chromosome territories,
referred to as chromosome pairs from now on,  and we use a Markov Chain Monte Carlo method based on the theory of algebraic statistics to assign significance.  The mFISH data we analyze have already appeared in the literature \cite{Ars2004,Corn2002,Con2001}.
\medskip

In our study we test a no-proximity effect model and a single pair proximity model by sampling tables that have the same sufficient statistic as those observed experimentally.
Sampling of the tables is performed by running a Markov Chain Monte Carlo algorithm that uses a Markov basis \cite{Sturmfels98, LAC} for a second hypersimplex \cite{DST}.
This kind of method is one of the early contributions of algebraic statistics, hence ``classical algebraic statistics'' in the title of this work.
Our results indicate that we could not reject the hypothesis of random relative arrangement of chromosome territories when radiation induced chromosome aberrations were analyzed. This result is in qualitative agreement with previous studies
\cite{Ars2004, Corn2002, Con2001} and suggests that the specific positions of any two pairs of chromosomes do not influence the frequency of aberrations observed.
However, by assuming the existence of a proximity biasing factor we found chromosome pairs $\{1,22\}$ and $\{13,14\}$ to be significant. We conclude that methods
 develop in algebraic statistics are suitable for analyzing genetic data of moderate size in which data sparsity or low numbers of measurements are present.

\section{Data and Methods}

To test for proximity of chromosome territories we used a radiation induced exchange type chromosome aberration table published in  \cite{Ars2004,Vives05}.
 In these experiments cells from healthy donors were irradiated with sparsely ionizing $\gamma$-rays at different doses, and mFISH \cite{Speicher} was used to detect ETCAs.
This table includes a total of 3585 records of human peripheral blood lymphocytes irradiated with sparsely ionizing radiation at different doses.
\medskip

The frequency of ETCAs was summarized by recording the number of cells in which at least one exchange between two non-homologous chromosomes occurred. This
quantity is robust with respect to noise introduced by apparently incomplete aberrations (i.e. those aberrations with not all fragments accounted for) and reduces false positives. Following previous publications \cite{Ars2004,Corn2002}
we denote these values by $f(j,k)$ where $j$ and  $k$ are the chromosomes that participate in the exchange. These values are presented in Table \ref{table:Lympho2004}  as a $22\times 22 $ upper-triangular table.

\begin{table}[h]
	\begin{center}
\caption{Table of ETCAs in 3585  human lymphocytes as reported in \cite{Ars2004,Vives05}. Each entry $f(j,k)$ holds the number of cells in which at least one exchange between chromosomes $j$ and $k$ was recorded.
The total number of cells in which a given chromosome was involved in at least one exchange appears in the "sum" column.}
	{\tiny\tt
	\begin{tabular}{lllllllllllllllllllllll}

Chr&2&3&4&5&6&7&8&9&10&11&12&13&14&15&16&17&18&19&20&21&22&Sum\\
\hline1&44&38&42&29&26&29&18&39&29&25&18&15&18&34&31&22&12&14&22&9&27&541\\
2&&43&37&32&30&24&25&29&16&24&30&29&9&26&8&24&8&7&12&13&15&485\\
3&&&21&31&32&24&21&26&23&25&23&21&18&18&19&21&11&17&11&12&10&465\\
4&&&&23&27&28&24&26&20&13&19&23&22&20&16&18&11&6&12&10&7&425\\
5&&&&&17&31&26&25&24&30&25&25&15&19&8&19&13&7&16&7&4&426\\
6&&&&&&18&22&21&31&13&30&18&15&19&14&15&13&10&9&8&7&395\\
7&&&&&&&20&20&17&28&25&13&18&8&18&23&11&9&19&6&7&396\\
8&&&&&&&&13&12&24&11&25&15&16&12&16&17&4&9&7&8&345\\
9&&&&&&&&&21&25&7&23&23&27&20&15&22&8&9&7&10&416\\
10&&&&&&&&&&18&21&14&14&10&19&14&9&5&11&7&3&338\\
11&&&&&&&&&&&25&5&15&16&19&15&8&10&12&3&11&364\\
12&&&&&&&&&&&&9&16&9&12&16&8&13&10&5&5&337\\
13&&&&&&&&&&&&&29&10&10&7&16&5&6&7&9&319\\
14&&&&&&&&&&&&&&22&13&6&10&2&6&13&11&310\\
15&&&&&&&&&&&&&&&22&13&9&7&11&7&9&332\\
16&&&&&&&&&&&&&&&&12&15&12&20&8&13&321\\
17&&&&&&&&&&&&&&&&&5&4&11&5&10&291\\
18&&&&&&&&&&&&&&&&&&2&11&9&3&223\\
19&&&&&&&&&&&&&&&&&&&6&0&8&156\\
20&&&&&&&&&&&&&&&&&&&&7&10&240\\
21&&&&&&&&&&&&&&&&&&&&&6&156\\
22&&&&&&&&&&&&&&&&&&&&&&193\\
		\end{tabular}}
	\end{center}

 	 \label{table:Lympho2004}\
\end{table}

\subsection{The No-proximity and Single-pair Proximity Effects as Log-Linear Models}
ETCAs between two non-homologous chromosomes $j$ and $k$  ($j\ne k$)  were modelled by a single discrete random variable $X$ with
${22 \choose 2}=231$ values corresponding to all possible pairs of non-homologue chromosomes with a probability density function given by
$$p \, : \, \{ (j,k) \, : 1 \leq j < k \leq 22 \} \longrightarrow \Delta_{231}$$ where the set
$$\Delta_{231}  \, = \, \{ (p_{12}, p_{13}, \ldots, p_{2122}) \in \mathbb{R}^{231} \, : \, p_{jk} \geq 0 \,\,\, \sum_{j,k} p_{jk} = 1 \}$$
is the probability simplex in $\mathbb{R}^{231}$ and  $p_{jk} = p(j,k)$. In this no-proximity effect model the probability of observing an ETCA
between the chromosomes $j$ and $k$ is
$$p_{jk} \, = \, \theta_j \theta_k \,\,\, 1 \leq j < k \leq 22.$$
where $\theta_1, \theta_2, \ldots, \theta_{22}$ are positive parameters. More precisely,  the no-proximity effect model is the image of $\R_+^{22}$ in $\Delta_{231}$ under the map
$\phi(\theta_1, \ldots, \theta_{22}) = (\theta_j \theta_k \, : \, 1 \leq j < k \leq 22)$. This model can be linearized by applying logarithms to both sides of the equation,
which gives $\log(p_{jk}) = \log(\theta_j) + \log(\theta_k)=\beta_j+\beta_k$ and shows that our model is a log-linear model \cite{LLM}. Therefore the no-proximity effect model is given
by a $22 \times 231$ design matrix, denoted by $A(22)$, whose columns are $e_j + e_k$  with $1 \leq j < k \leq 22$ where $e_j$ is the $j$th standard unit vector in $\R^{22}$.
The columns of $A(22)$ are the vertices of the {\it second hypersymplex} in $\R^{22}$~\cite{DST}.

\medskip
For a fixed pair $\{r,s\}$ with $r\ne s$  we define an extended model that we call a {\it single-pair proximity effect model}. This model is given by the map
$\phi': \R_+^{23}  \to \Delta_{231}$
defined as
\begin{equation}
p_{jk} = \phi'(\theta_1,..,\theta_{22}, \mu_{rs})= \left\{ \begin{array}{ll} \theta_r \theta_s \mu_{rs} &\mbox{ if } j=r \mbox{ and } k=s,\\
   \theta_j \theta_k &\mbox{otherwise.}\end{array}\right.
\end{equation}
The parameter $\mu_{rs}$ is a bias factor for the frequency of an observed exchange between chromosomes $r$ and $s$. This bias factor corresponds to a proximity factor between
the territories of two chromosomes $r$ and $s$.
By taking the logarithm on both sides of the above equation we get the parametrization in logarithmic coordinates:
 	\begin{equation}
	 \log(p_{jk})=
	\left\{ \begin{array}{ll} \beta_r+\beta_s+\alpha_{rs}\;\;\;\; &\mbox{ if } j=r, k=s, \\
	\beta_j+\beta_k  & \hfill \mbox{ otherwise. }
         \end{array}\right. \end{equation}

The single-pair proximity effect model for any pair of chromosomes $\{r,s\}$ is also a log-linear model which extends the no-proximity effect model since
the set of probability distributions in this model are
those in the image of the map $\phi'$ with $\alpha_{rs}=0$. The design matrix $A[r,s]$ defining this model is a $23 \times 231$ matrix, identical to $A(22)$ in its first $22$ rows, and with an extra row of all zeros except a $1$
in the column corresponding to $p_{rs}$.

\medskip
In order to assign a $p$-value to the goodness-of-fit test we propose to sample tables that are similar to those observed experimentally. More specifically we sample tables with the same {\it minimal sufficient statistic} as
the data table $f$. For each chromosome  $k$ we have the {\em marginal total} $$ u_k  \, = \, \sum_{j: \, j\ne k} f(j,k) \quad \quad  \mbox{for     } k = 1, \ldots, 22.$$ This quantity is displayed in the {\tt Sum} column of Table \ref{table:Lympho2004}.
The marginal total of a given chromosome is a measure of the propensity of each individual chromosome to form ETCAs. In radiation studies
this quantity has been associated to the sensitivity of the chromosomes to radiation \cite{Ars2004,Corn2002} and to repair mechanisms \cite{Wu01}.
 For the no-proximity effect model $\phi$ the collection of marginal totals $u = (u_k: \, k=1, \ldots, 22)$
is the minimal sufficient statistic. For the single-pair proximity effect model for the chromosome pair $\{r, s\}$, the minimal sufficient statistic is the same marginal sums together with $u_{rs} = f(r,s)$.
The set of tables with the same sufficient statistic  is called {\em the fiber} of the experimentally observed table and is denoted by $\mathcal{F}(u)$. It is well-known that $\mathcal{F}(u)$ consists of lattice points in a polytope.

\subsection{Maximum Likelihood Estimation}

The maximum likelihood estimator (MLE) tables of the data with respect to the no-proximity effect and single-pair proximity models were computed by the standard numerical algorithm  {\it Iterative Proportional Scaling} \cite{IPS}. For log-linear models,
the algorithm converges to the unique MLE table $\hat f$ such that $\frac{1}{N} \hat f$ lies on the model where $N$ is the sample size and $\hat f$ has the same sufficient statistic $u$ as the data table $f$.
The existence of this unique table is guaranteed by Birch's Theorem (see \cite{LAC}).

\subsection{Hypothesis testing and Monte-Carlo simulations}
Our goodness-of-fit test for the no-proximity effect model  uses the standard chi-square statistic

\begin{equation} \label{chi-square}
 \chi^2(F)=\sum_{1 \leq j < k \leq 22}  \frac{(F(j,k)-\hat f(j,k))^2}{\hat f(j,k)}.
\end{equation}
where $\hat f(j,k)$ are the entries of the MLE table with respect to the no-proximity effect model
given the data table $f$, and $F(j,k)$ are the entries of tables $F$ drawn from the fiber $\mathcal{F}(u)$.

Typically large $\chi^2(F)$ values  would indicate that the data table $f$ is ``close'' to $\hat f$ providing no evidence for rejecting no-proximity effect model.
\medskip

The fiber $\mathcal{F}(u)$ for the no-proximity effect model is very large while at the same time the data table $f$ has some small entries including a zero entry. These observations point to using the
well-established standard Markov Chain Monte Carlo method
for running the goodness-of-fit test. One might think that computing a Markov basis for the no-proximity effect model (or equivalently the second hypersimplex $A(22)$)  is intractable. Luckily,
a Markov basis (in fact a {\it Gr\"obner basis}) for this model
is available, see \cite[Theorem 9.1]{GBCP} and \cite{DST}. Our Markov basis is defined by the following set of moves:   For each  $1 \leq i < j < k < \ell \leq 22$ one defines two moves $m[i,j; k,\ell]$
and $m[i,\ell;j,k]$. The first move is a table where the $(i,j)$ and $(k,\ell)$ entries are set to equal $1$,
the $(i,k)$ and $(j,\ell)$ entries are set to equal $-1$, and all the other entries are set to equal $0$. The second move is a table where the $(i,\ell)$ and $(j,k)$
entries are set to equal $1$, the $(i,k)$ and $(j,\ell)$ entries are set to equal $-1$, and all the other entries are set to equal $0$.
These tables  together with their negatives $-m[i,j;k,l]$ and $-m[i,\ell; j,k]$ comprise our  Markov basis $\mathcal{B}$.
This Markov basis contains $ 2 \cdot 2 \cdot {22 \choose 4} \, = \, 29,260$ moves.
\medskip

Using this MCMC we generated a set of $m$ random tables $f_1, \ldots, f_m$  using the Metropolis-Hastings algorithm and estimated the $p$-value of goodness-of-fit test
by  $$ \frac{1}{m} \left( \sum_{\chi^2(f_i) \geq \chi^2(f)} 1\right).$$
One important parameter of the Metropolis-Hastings algorithm is the number of steps it requires between each selection of tables $f_i$ and $f_{i+1}$. The rule of thumb is that one needs sufficient
number of steps  so that the Markov chain can reach any table in $\mathcal{F}(u)$ starting from an arbitrary table. We followed the method in \cite{Ido} to heuristically determine this number of steps:
It is a consequence of the Gr\"obner basis theory that there is a unique table $T_{unique}$  in ${\mathcal F}(u)$ where none of
the Gr\"obner basis moves can be applied, and every table in ${\mathcal F}(u)$ is connected to $T_{unique}$ table via the Markov basis moves. This table is the unique reduced normal form of the data
table with respect to the Gr\"obner basis from which our Markov basis $\mathcal{B}$ is constructed. Empirically, the average number of steps one needs to go from a randomly generated table in $\mathcal{F}(u)$
to $T_{unique}$ using the moves in $\mathcal{B}$ is about $15000$. Hence, we estimate that the number of steps needed to connect two tables in $\mathcal{F}(u)$ is bounded by $30000$.

\subsection {Log-Ratio Test}
Since the single-pair proximity effect model contains the no-proximity effect model  we compared the relative fit of the two models by
a \emph{ likelihood ratio test }.
The likelihood ratio test statistic is  defined as:
	\begin{equation} G^2=2\sum_{1 \leq j < k \leq 22} \hat f_{jk}^1 \log \left(\frac{\hat f_{jk}^1}{\hat f_{jk}^0}\right)
	\end{equation}
where  $\hat f_{jk}^1$ is the MLE  with respect to  the single-pair proximity effect model  and $\hat f_{jk}^0$ is the MLE with respect to the no-proximity effect model.
It is well known \cite[Theorem 10.2.8]{LLM} that, for large sample sizes $N$, if the null hypothesis is true (i.e. if the data fits the no-proximity effect model better than it fits
the single-pair proximity effect model) then $G^2$ has a $\chi^2$ distribution with
degrees of freedom equal to the difference of the ranks of the nested log-linear models. In our case, the rank of the no-proximity effect model is equal to $\mathrm{rank} \, A(22) = 21$
and the rank of the single-pair proximity effect effect model is equal to $\mathrm{rank} \,  A[r,s] = 22$.

\section{Numerical Results}

\begin{table}[h]
	\begin{center}
	\caption{ Maximum Likelihood Estimate  for experimental Table \ref{table:Lympho2004} obtained from irradiated  human lymphocytes.}
	{\tiny\tt
	\begin{tabular}{lllllllllllllllllllllll}

Chr&2&3&4&5&6&7&8&9&10&11&12&13&14&15&16&17&18&19&20&21&22&Sum\\
\hline1&47&43&38&37&33&33&27&34&26&28&25&23&22&24&23&20&14&8.9&15&8.8&11&541\\
2&&37&32&32&29&29&24&30&23&25&22&21&20&22&21&18&13&8.3&14&8.2&11&485\\
3&&&30&30&27&27&23&28&22&24&22&20&19&21&20&18&13&8.2&14&8.2&11&465\\
4&&&&27&24&24&21&26&20&22&20&18&18&19&18&16&12&7.8&13&7.8&10&425\\
5&&&&&24&24&21&26&20&22&20&19&18&19&19&17&12&8&13&8&10&426\\
6&&&&&&22&19&24&19&20&19&17&17&18&17&16&12&7.7&12&7.7&9.7&395\\
7&&&&&&&19&24&19&20&19&17&17&18&17&16&12&7.9&13&7.9&9.9&396\\
8&&&&&&&&20&16&18&16&15&15&16&15&14&11&7.2&11&7.2&9&345\\
9&&&&&&&&&20&21&20&18&18&19&18&17&13&8.6&13&8.5&11&416\\
10&&&&&&&&&&17&16&15&15&16&15&14&11&7.3&11&7.4&9.2&338\\
11&&&&&&&&&&&17&16&16&17&16&15&11&8&12&8&9.9&364\\
12&&&&&&&&&&&&15&15&16&15&14&11&7.6&12&7.6&9.4&337\\
13&&&&&&&&&&&&&14&15&15&13&10&7.4&11&7.4&9.1&319\\
14&&&&&&&&&&&&&&15&14&13&10&7.4&11&7.4&9&310\\
15&&&&&&&&&&&&&&&15&14&11&7.9&12&7.9&9.6&332\\
16&&&&&&&&&&&&&&&&14&11&7.9&11&7.9&9.5&321\\
17&&&&&&&&&&&&&&&&&10&7.4&11&7.4&8.9&291\\
18&&&&&&&&&&&&&&&&&&6&8.6&6&7.3&223\\
19&&&&&&&&&&&&&&&&&&&6.4&4.6&5.6&156\\
20&&&&&&&&&&&&&&&&&&&&6.6&7.8&240\\
21&&&&&&&&&&&&&&&&&&&&&5.7&156\\
22&&&&&&&&&&&&&&&&&&&&&&193

		\end{tabular}}
	\end{center}
 	 	 \label{MLE:Lympho2004}
\end{table}

We first computed the MLE tables for the no-proximity effect and the single-pair models as discussed in the previous section. This algorithm preserves the minimal sufficient statistic
which guarantees that the MLE table belongs to the fiber of the data table. The MLE table for the no-proximity effect model is displayed above. Note that we display just two significant digits.
The table for the no-proximity effect reveals some marked differences with the experimentally observed table. Most notably, entries for each chromosome tended to be more
homogeneous than those in the experiment. This is particularly true for small chromosomes 19 to 22.

\medskip

Using the MCMC approach explained earlier we generated $3 \times 10^{10}$ tables which were sampled every $ 3 \times 10^4$ times to reduce the intrinsic correlation in the Markov Chain.
We therefore obtained a sample size of $10^6$ tables. The chi-square statistic for each of the sampled table was computed.
Interestingly, none of the $10^6$ tables generated for Table \ref{table:Lympho2004} had a test statistic smaller than the experimentally observed.
Hence the no-proximity effect model could not be rejected.   This result is somewhat surprising but can be explained by
estimating the number of tables that are contained in the ellipsoid defined by (\ref{chi-square}). We will give the details of this estimation in the Appendix.
These results show that the data can be well fit by the no-proximity model and
that the relative positions of chromosomes are random.

\begin{table}[h!]
       \begin{adjustwidth}{-1.8cm}{}
	\begin{center}
 	\caption{Table of  deviations between observed and MLE counts. Each entry is the difference between  the observed counts and MLE for each pair of chromosomes $j$ and $k$ computed for Table \ref{table:Lympho2004}.}

	{\tiny\tt
		\begin{tabular}{rrrrrrrrrrrrrrrrrrrrrr}

Chr&2&3&4&5&6&7&8&9&10&11&12&13&14&15&16&17&18&19&20&21&22\\
\hline
1&-3.2&-5.5&4.2&-8.2&-7.2&-3.8&-9&5&3.3&-2.9&-7&-8.1&-4&10&8.4&2.2&-2&5.1&6.8&0.2&16\\
2&&6.2&4.6&-0.1&1.2&-4.6&1.1&-0.9&-6.9&-0.9&7.6&8.2&-11&4.4&-13&5.9&-4.9&-1.3&-2.1&4.8&4.3\\
3&&&-9.4&0.8&4.7&-3.2&-1.8&-2.4&1.1&1.2&1.4&0.9&-1.3&-2.8&-0.9&3.4&-1.7&8.8&-2.8&3.8&-0.5\\
4&&&&-3.9&2.5&3.6&3.3&0.3&0.0&-8.7&-0.8&4.5&4.2&0.8&-2.4&1.7&-0.9&-1.8&-0.9&2.2&-3\\
5&&&&&-7.5&6.6&5.2&-0.6&3.9&8.2&5.1&6.4&-3&-0.4&-11&2.5&0.9&-1&2.9&-1.0&-6.2\\
6&&&&&&-4.4&2.8&-2.5&12&-7.2&11&0.6&-1.8&0.9&-3.4&-0.5&1.5&2.3&-3.5&0.3&-2.7\\
7&&&&&&&0.7&-3.5&-1.7&7.7&6.4&-4.5&1.1&-10&0.5&7.3&-0.7&1.1&6.3&-1.9&-2.9\\
8&&&&&&&&-7.2&-4.3&6.4&-5.2&9.7&0.1&0.0&-3.5&2&6.5&-3.2&-2.4&-0.2&-1\\
9&&&&&&&&&1.2&3.7&-13&4.5&5.1&7.9&1.6&-1.6&9.5&-0.6&-4.5&-1.5&-0.7\\
10&&&&&&&&&&0.8&5&-1.1&-0.7&-5.8&3.7&0.1&-1.6&-2.3&-0.4&-0.4&-6.2\\
11&&&&&&&&&&&7.8&-11&-0.9&-1.0&2.6&0.1&-3.4&2&-0.3&-5&1.1\\
12&&&&&&&&&&&&-6.2&1.2&-6.8&-3.3&2&-2.8&5.4&-1.6&-2.6&-4.4\\
13&&&&&&&&&&&&&15&-5&-4.6&-6.4&5.6&-2.4&-5.2&-0.4&-0.1\\
14&&&&&&&&&&&&&&7.3&-1.3&-7.1&-0.2&-5.4&-5&5.6&2\\
15&&&&&&&&&&&&&&&6.8&-0.9&-2&-0.9&-0.7&-0.9&-0.6\\
16&&&&&&&&&&&&&&&&-1.6&4.2&4.1&8.5&0.1&3.5\\
17&&&&&&&&&&&&&&&&&-5&-3.4&0.3&-2.4&1.1\\
18&&&&&&&&&&&&&&&&&&-4&2.4&3&-4.3\\
19&&&&&&&&&&&&&&&&&&&-0.4&-4.6&2.4\\
20&&&&&&&&&&&&&&&&&&&&0.4&2.2\\
21&&&&&&&&&&&&&&&&&&&&&0.4\\

		\end{tabular}}
	\end{center}

	 	 \label{table:Lympho2004Dev}
\end{adjustwidth}
\end{table}

\begin{table}
\begin{center}
	\caption{Table of pairs of chromosomes with their corresponding Chi-squared value and $p$-value corresponding to their extended interaction model and $p$-value adjusted with the Bonferroni correction. Significant $p$-values are labeled in boldface.}
	\begin{tabular}

	  {c | c| c| c|}
	  \cline{2-4}
& \multicolumn{3}{|c|}
	{ $p$-value }\\%( with Bonferroni correction)} \\
	\hline
\textbf{\em Chromosome Pair }
	&  Chi-squared
	&  p-value before correction
	&  Bonferroni corrected p-value \\
	\hline \hline

	$\set{1,22}$ &   {\textbf{17.27}}& {\textbf{0.00005}}& \textbf{0.00138} \\
	$\set{13,14}$& {\textbf{13.66}}& {\textbf{0.00022}}& \textbf{0.01012}  \\
	$\set{3,19}$& 7.87 & 0.00502 & 0.23092 \\
	$\set{6,10}$& 7.78 & 0.00527 & 0.24242 \\
	$\set{6,12} $&  6.85 & 0.00888 & 0.40848 \\
	$\set{9,18} $&6.51 & 0.0107 & 0.4922 \\
	$\set{16, 20} $& 5.73 & 0.01671 & 0.76866  \\
	$\set{8, 13} $& 5.72 & 0.01673 & 0.76958 \\
	\hline
	\end{tabular}

	\label{table:pVal}
\end{center}
\end{table}

\medskip
Microscopy observations however have shown that some groups of chromosomes tend to be close to each other and form exchange type aberrations more frequently than what one would predict using the
no-proximity effect mode \cite{Branco2006}. These chromosomes include those that are found
 in the center of the nucleus \cite{Ars2004,Corn2002} and those that form the nucleolus \cite{Ars2004}. This observation is
further supported by the large positive deviations between entries in the experimentally observed table and the MLE table.
The differences between the observed table and the MLE table are shown in Table  \ref{table:Lympho2004Dev}.
A positive entry indicates that the observed table had more exchanges than the number predicted by the MLE table while a negative entry shows cases where the MLE table had more exchanges
than the observed table.  The largest positive entries
in the table are those for chromosome pairs \{1,22\}, \{13,14\}, \{1,15\}, and \{9,18\}. We therefore tested all chromosome pairs on the single-pair proximity effect model and  performed the log-ratio tests considering the original log-linear models  against the modified modelsfor all chromosome pairs.
We found eight pairs of chromosome pairs that were significant (Column 1 in Table \ref{table:pVal}).
However only two of them were significant after correction for multiple testing using Bonferroni  \cite{Bonf} (Column 4 in Table  \ref{table:pVal}).

\section{Discussion}

In this work we have presented a model-based approach to determine the relative positioning of chromosome territories from ETCA frequency
tables  that are sparse and with small entries.
In previous work a method for dealing with small entries was reported \cite{Corn2002},  however the assignment of p-values to specific clusters of
chromosome territories was based on groups of chromosomes previously found in the literature. Our method
builds on the techniques developed in classical algebraic statistics by Diaconis and Sturmfels \cite{Sturmfels98}.

\medskip
Our results show that the overall distribution of chromosome exchanges can be simply explained by a model in which the relative position of chromosome territories is random.
This finding does not quantitatively agree with the results reported in \cite{Ars2004,Corn2002} since a small deviation from randomness was reported in those studies.
However all studies agree upon a rather random organization of chromosome territories. Several sources can be contributing to this apparently random organization of
the genome. The first is imposed by the limitation of the data. Chromosome painting techniques are limited by the fact that homologue chromosomes are painted the same
color, and this evidently introduces unavoidable noise since territories of homologous chromosomes can be positioned in very different environments (i.e. with different neighbouring chromosome territories). It is also possible that cell to cell variation is very large or that there is a severe reorganization of the chromosome territories after the radiation insult. The fact that new sequencing analysis is consistent with this overall picture suggest that radiation has a small repositioning effect.

Our study shows that a small fraction of chromosome pairs deviate from this picture
of random ETCA formation. We identified eight pairs of chromosome territories that were significant prior to multiple testing correction (Table 4).
The first two pairs of chromosome territories (ie.  $\{13,14\}$ and $\{1,22\}$) were also significant in \cite{Ars2004,Corn2002,Vives05}. There is an easy explanation for the significance  of these pairs although their true functional significance remains to be determined \cite{Bickmore02} . The pair $\{13,14\}$ is part of the cluster of chromosomes in the nucleolus $\{13,14,15,21,22\}$, an organelle that brings chromosome territories together for specific needs of the cell.
The second pair  $\{1,22\}$ has been found to be part of a cluster of chromosome territories $ \{1,16,17,19,22\}$ located in the center of the nucleus of lymphocyte cells. We used a Bonferroni correction method for multiple testing. This method is known to be very conservative and it is possible that we rejected some informative pairs. In fact all the  pairs that were not significant after Bonferroni correction have been reported in several clinical blood malignancies suggesting that proximity of these pairs of chromosome territories may be somewhat common and furthermore may have an important
role in the development of these diseases
\cite{Bassat90,Bossi2014,Impera2011,Jarosova14,Naeem95,Zhu07}.

It is our intention to improve our results by including better outlier detection tools that help identify other chromosome pairs \cite{Rapallo, KRR} and by incorporating these results into the developent of  biophysical models.
These models are based on different properties of the genome that can obtained from basic physical priciples such as the radial organization of chromosomes
using overlapping sphere or ellipsoid packings \cite{Cremer96,Khalil,UW} , gene density \cite{Kreth04} or DNA decondensation processes \cite{Rosa08}  or through the folding of chromatin fibers \cite{Barbieri12,Blackstone11,Lieberman09}

\medskip
\textbf{Acknowledgements} This work is partially supported by NSF grant
DMS-1217324 and NIH grant RO1-GM109457 to J. Arsuaga. We want to thank J. L. Garc\'{i}a from Centro de Investigaci\'{o} del cancer de La Universidad de Salamanca (Spain) for sharing Figure 1.

\section{Appendix}
In this last section we return to our remark that none of the $10^6$ tables generated for Table \ref{table:Lympho2004} in our MCMC procedure had a test statistic smaller than $\chi^2(f)$.  In order
to give a heuristic explanation for this behavior we will estimate the size of $\mathcal{F}(u)$ and also estimate the size of the set of the tables $F$ where $\chi^2(F) \leq \chi^2(f)$.

\medskip We first give a lower bound for the cardinality of $\mathcal{F}(u)$, which is the set of lattice points in a polytope. A standard computational tool such as {\tt Latte} \cite{Latte} cannot compute
the cardinality of this immensely large set.  Instead we employ a divide-and-conquer approach where we consider the subtables consisting of the chromosomes 1 through 8 (Subtable A),
chromosomes 8 through 15 (Subtable B), chromosomes 15 through 22 (Subtable C) in Table \ref{Subtables:Lympho2004}.

\begin{table}[h]
     \caption{Subtables A-C}
	$\begin{array}{cc}
	{\tiny\tt
	\begin{tabular}{lllllllll}

Chr&2&3&4&5&6&7&8&Sum\\
\hline
1&44&38&42&29&26&29&18& 226\\
2&&43&37&32&30&24&25&235\\
3&&&21&31&32&24&21&210\\
4&&&&23&27&28&24&202\\
5&&&&&17&31&26&189\\
6&&&&&&18&22&172\\
7&&&&&&&20&174\\
8&&&&&&&&156\\
		\end{tabular}}  &
	{\tiny\tt
	\begin{tabular}{lllllllll}

Chr&9&10&11&12&13&14&15&Sum\\
\hline
8&13&12&24&11&25&15&16& 116\\
9&&21&25&7&23&23&27&139\\
10&&&18&21&14&14&10&110\\
11&&&&25&5&15&16&128\\
12&&&&&9&16&9&98\\
13&&&&&&29&10&115\\
14&&&&&&&22&134\\
15&&&&&&&&110\\
		\end{tabular}}  \\
	{\tiny\tt
	\begin{tabular}{lllllllll}

Chr&16&17&18&19&20&21&22&Sum\\
\hline
15&22&13&9&7&11&7&9& 78\\
16&&12&15&12&20&8&13&102\\
17&&&5&4&11&5&10&60\\
18&&&&2&11&9&3&54\\
19&&&&&6&0&8&39\\
20&&&&&&7&10&76\\
21&&&&&&&6&42\\
22&&&&&&&&59\\
		\end{tabular}}  &
	\end{array}$

	 	 \label{Subtables:Lympho2004}
\end{table}
\noindent
The $j$th entry in the {\it Sum} column in each subtable refers to the sum of $f(j,k)$ over all chromosomes $k\neq j$ included in the subtable. See for instance $1-8$ in Subtable A in Table \ref{Subtables:Lympho2004}.
The remaining entries of Table \ref{table:Lympho2004} were subdivided into six rectangular subtables. Each of
these subtables are indexed by two subsets of chromosomes $J$ and $K$: Subtable 1 ($J= 1-4$, $K=9-15$),   Subtable 2 ($J= 5-7$, $K=9-15$),
Subtable 3 ($J = 1-4$, $K=16-22$), Subtable 4 ($J = 5-7$, $K=16-22$), Subtable 5 ($J = 8-11$, $K=16-22$), Subtable 6 ($J = 12-14$, $K=16-22$) in Table \ref{Subtables2:Lympho2004}.

\begin{table}[h]
	\caption{ Subtables 1-6}
	$\begin{array}{cc}
	{\tiny\tt
	\begin{tabular}{llllllll|l}

Chr&9&10&11&12&13&14&15&RSum\\
\hline
1&39&29&25&18&15&18&34 & 178\\
2&29&16&24&30&29&9&26  &  163\\
3&26&23&25&23&21&18&18  & 154 \\
4&26&20&13&19&23&22&20  & 143 \\
\hline
CSum & 120 & 80 & 87 & 90 & 88 & 67 & 98 \\
		\end{tabular}}  &
	{\tiny\tt
	\begin{tabular}{llllllll|l}

Chr&9&10&11&12&13&14&15&RSum\\
\hline
5&25&24&30&25&25&15&19  &  163\\
6&21&31&13&30&18&15&19  & 147\\
7&20&17&28&25&13&18&8    & 129\\
\hline
CSum & 66 & 72 & 71 & 80 & 56 & 48 & 46 \\
		\end{tabular}}  \\
\\
	{\tiny\tt
	\begin{tabular}{llllllll|l}

Chr&16&17&18&19&20&21&22&RSum\\
\hline
1&31&22&12&14&22&9&27 & 137\\
2&8&24&8&7&12&13&15  &  87\\
3&19&21&11&17&11&12&10  & 101 \\
4&16&18&11&6&12&10&7  & 80 \\
\hline
CSum & 74 & 85 & 42 & 44 & 57 & 44 & 59\\
		\end{tabular}}  &
	{\tiny\tt
	\begin{tabular}{llllllll|l}

Chr&16&17&18&19&20&21&22&RSum\\
\hline
5&8&19&13&7&16&7&4  &  74\\
6&14&15&13&10&9&8&7  & 76\\
7&18&23&11&9&19&6&7    & 93\\
\hline
CSum & 40 & 57 & 37 & 26 & 44 & 21 & 18 \\
		\end{tabular}}  \\
\\
	{\tiny\tt
	\begin{tabular}{llllllll|l}

Chr&16&17&18&19&20&21&22&RSum\\
\hline
8&12&16&17&4&9&7&8 & 73\\
9&20&15&22&8&9&7&10  &  91\\
10&19&14&9&5&11&7&3  & 68 \\
11&19&15&8&10&12&3&11  & 78 \\
\hline
CSum & 70 & 60 & 56 & 27 & 41 & 24 & 32 \\
		\end{tabular}}  &
	{\tiny\tt
	\begin{tabular}{llllllll|l}

Chr&16&17&18&19&20&21&22&RSum\\
\hline
12&12&16&8&13&10&5&5  &  69\\
13&10&7&16&5&6&7&9  & 60\\
14&13&6&10&2&6&13&11    & 61\\
\hline
CSum & 35 & 29 & 34 & 20 & 22 & 25 & 25 \\
		\end{tabular}}  \\
	\end{array}$

	 	 \label{Subtables2:Lympho2004}
\end{table}

\medskip
In the above tables, the $j$th entry in the {\tt RSum} column refers to the sum of the numbers in the $j$th row in the corresponding table, and
the $k$th entry in the {\tt CSum} row refers to the sum of the numbers in the $k$th column in the corresponding table. Any table that has been subdivided
in a total of $9$ subtables where subtables $A-C$ have same {\tt Sum} column as in the subtables $A-C$ of the data table and where
 subtables $1-6$ have the same {\tt RSum} and {\tt CSum} columns/rows as in subtables $1-6$ of the data table is in $\mathcal{F}(u)$. So a lower bound for
the cardinality of $\mathcal{F}(u)$ can be obtained by the product of the number of subtables of type $A,B,C$ and $1, \ldots, 6$ with the given {\tt Sum} and {\tt RSum}/{\tt CSum} columns/rows.
The total number of such subtables range from $10^{34}$ for Table A to $10^{14}$ for Table \ref{Subtables2:Lympho2004}. The exact number of tables calculated by Latte \cite{Latte} are shown in
Table \ref{LATTE:Lympho2004}.

\begin{table}[h]
\caption{ Total number subtables associated to subtables A-C and 1-6  }
$$ \begin{array}{c|c}
\mbox{Subtable} & \mbox{Size} \\
\hline
A  &  2952470953799239962752797659386190 \\
B &  252762217255461089482462934497 \\
C & 242451808378958740321921 \\
1 & 384937707376563538670706387547\\
2 &  11636397863410272633 \\
3 & 51895845228141509162048464\\
4 & 5538280355961059\\
5 & 336625602844011493310899\\
6  & 777971438252448\\
\end{array}
$$

	 	 \label{LATTE:Lympho2004}
\end{table}
The product of these numbers is in the order of $10^{214}$.
This estimate however can be improved by the following arguments. Most of the $29,260$ Markov moves can be applied to Table \ref{table:Lympho2004} without changing the {\tt Sum} column.
The few moves that cannot be applied are those that make the $(19,21)$ entry  negative. In addition, only $1554$ moves will not change the {\tt Sum} or {\tt RSum}/{\tt CSum} entries of the above 9 Subtables
since the moves occur completely inside each of the subtables.
The remaining $27,706$ Markov moves will alter at least two of these subtables so that their {\tt Sum} and/or {\tt RSum}/{\tt CSum} entries will change. If we repeat the above {\tt Latte} calculations
for the nine subtables obtained after the application of each one of these $27,706$ Markov moves (in other words those that change {\tt Sum} and/or {\tt RSum}/{\tt CSum}) we will find that the number of these
new tables will be again of the order $6 \times 10^{214}$. We can repeat the same argument when two or more Markov moves are considered. For instance if we applied
exactly 30 Markov moves of the 27706 possible
moves in sequence then we would obtain a total number of new tables in ${\mathcal F}(u)$ given by $ 6 \times 10^{214}$ times
$${ 27706 \choose 30} \quad \approx \quad 7 \times 10^{100}.$$
This is approximately $4 \times 10^{315}$.
This calculation however may have some tables that are counted more than once because two distinct sequences of moves starting from Table \ref{table:Lympho2004} can lead to
 the same
table. This overcounting should be more than compensated by tables that could be reached with more or fewer than $30$ moves.
In conclusion we estimate that a lower bound of $ 4 \times 10^{315}$ possible tables is justified, but to be on the safe side   we adopt
$10^{300}$ as a very conservative estimation of the  possible size of ${\mathcal F}(u)$.

\medskip
Now we provide a very liberal upper bound on the number of tables whose $\chi^2$ value is smaller than the $\chi^2$ value of the data table ($346.63$).
The volume $V$ of the ellipsoid defined by  equation (\ref{chi-square}) is bounded above by $1.1 \times 10^{266}$ using the volume formula
for multidimensional ellipsoids.  It is known that
the number of lattice points in an $n$-dimensional ellipsoid defined by $$\sum_{1 \leq j < k \leq n} a_{jk} x_{jk} \quad \leq \quad r^2$$ is approximately equal to $ V + O(r^{n/2})$
\cite{Landau} and  in our case $r^2 = 346.63$ and $n = 231$, and we arrive at $1.8 \times 10^{293}$ as an upper bound
on the number of tables which are inside the ellipsoid defined by (\ref{chi-square}).  We note that this must be a gross estimate, since we should be counting
the tables in the ellipsoid that are also in the fiber ${\mathcal F}(u)$. For this we should be counting the lattice points  in another ellipsoid of lower dimension $209$.
We conclude that a very conservative estimate of proportion of tables $F$ where $\chi^2(F) \leq \chi^2(f) = 346.63$ in ${\mathcal F}(u)$ is extremely small:
$$ \leq \frac{1.8 \times 10^{293}}{10^{300}} \approx 1.8 \times 10^{-7}.$$
We believe that the true proportion is much smaller.

config: arxiv


TITLE: For which values of $n$ does $\mathbb{Z}/n\mathbb{Z}$ have the property that every non-zero element is either a unit or is nilpotent?
QUESTION [2 upvotes]: Question: For which values of $n$ does $\mathbb{Z}/n\mathbb{Z}$ have the property that every non-zero element is either a unit or is nilpotent?
It is straightforward to see that for $n=p$ a prime, every non-zero element in $\mathbb{Z}/n\mathbb{Z}$ is unit. We can also see that every non-zero element of the ring $\mathbb{Z}/n\mathbb{Z}$ is either a unit or a zero-divisor.
However, I do not know how to extend this to the question above. I have identified this thread but still cannot figure out which other $n$ have the property described above.

REPLY [3 votes]: Assume $n > 1$, and let $R=\mathbb{Z}/n\mathbb{Z}$.

For $x\in \mathbb{Z}$, let $\bar{x}$ denote the corresponding element of $R$.

Consider two cases . . .

Case $(1)$:$\;n$ is a power of a prime.

Let $n=p^k$, where $p$ is prime, and $k$ is a positive integer.

Suppose $\bar{x}$ is a non-unit of $R$.

Then we must have $p{\,\mid\,}x$.
\begin{align*}
\text{Then}\;\;&p{\,\mid\,}x\\[4pt]
\implies\;&p^k{\,\mid\,}x^k\\[4pt]
\implies\;&\overline{x^k}=0\\[4pt]
\implies\;&\bar{x}^k=0\\[4pt]
\end{align*}
hence $\bar{x}$ is a nilpotent element of $R$.

Case $(2)$:$\;n$ has at least two distinct prime factors.

Let $p,q$ be distinct prime factors of $n$.

Then $\bar{p}$ is not a unit of $R$, since $\gcd(p,n) = p > 1$.

Suppose $\bar{p}$ is a nilpotent element of $R$.

Then $\bar{p}^k=0$, for some positive integer $k$.
\begin{align*}
\text{Then}\;\;&\bar{p}^k=0\\[4pt]
\implies\;&\overline{p^k}=0\\[4pt]
\implies\;&n|p^k\\[4pt]
\implies\;&q|p^k\\[4pt]
\implies\;&q|p\\[4pt]
\end{align*}
contradiction.

Hence $\bar{p}$ is not a unit of $R$, and also not a nilpotent element of $R$.

Therefore, $R$ is such that every element is either a unit or nilpotent if and only if $n$ is a power of a prime.


text:
\section{Theta lifting}

In this section, we study global theta lifts for some quaternionic dual pairs. The material in this section will be needed
 in Sec. \ref{sec:construction-nonvanishing} to globalize the construction of the Kudla-Millson cohomology classes from the previous section to the group $\tilde{\mathscr{G}}_B=\GU_B(\tilde{V})^0$ and
 to show the non-vanishing of their restriction to a suitable subgroup. Moreover, we will also use it in Sec. \ref{sec:arthur-galois-hodge} to
 study their associated Galois representations and to show that they lie in the $\C$-span of the Hodge classes.

\subsection{Setup}
\label{ss:theta-setup}

Let $F$ be a number field.
Let $B$ be a quaternion \emph{division} algebra over $F$ and $*$ the main involution on $B$.
Let $E$ be a quadratic extension of $F$ which embeds into $B$.
Fix a trace zero element $\i \in E^\times$ and write $\N = \N_{E/F}$ for the norm map from $E$ to $F$.
Let $\xi_E$ be the quadratic character of $\A^\times/F^\times$ associated to $E/F$ by class field theory.

Let $V$ be an $m$-dimensional right $B$-space equipped with a skew-hermitian form $\langle \cdot, \cdot \rangle : V \times V \rightarrow B$.
Let $W = B$ be a $1$-dimensional left $B$-space equipped with a hermitian form $\langle \cdot, \cdot \rangle : W \times W \rightarrow B$ given by
\[
 \langle x, y \rangle = x \cdot y^*.
\]
Then $\GU(W) \cong B^\times$.
Put
\begin{align*}
 G & = \GU(V)^0, & G_1 & = \U(V)^0, \\
 H & = \GU(W), & H_1 & = \U(W),
\end{align*}
and
\[
 R = \{ (g,h) \in G \times H \, | \, \nu(g) = \nu(h) \},
\]
where $\nu$ denotes the similitude character.
Let $Z_G \cong F^\times$ and $Z_H \cong F^\times$ be the centers of $\GU(V)$ and $\GU(W)$, respectively.
Put
\[
 (\A^\times)^+ = \nu(G(\A)) \cap \nu(H(\A))
\]
and
\begin{align*}
 G(\A)^+ & = \{ g \in G(\A) \, | \, \nu(g) \in (\A^\times)^+ \}, &
 G(F)^+ & = G(F) \cap G(\A)^+, \\
 H(\A)^+ & = \{ h \in H(\A) \, | \, \nu(h) \in (\A^\times)^+ \}, &
 H(F)^+ & = H(F) \cap H(\A)^+.
\end{align*}

Let $\V = V \otimes_B W$ be a $4m$-dimensional $F$-space equipped with a symplectic form
\[
 \llangle \cdot, \cdot \rrangle = \frac{1}{2} \tr_{B/F} \left( \langle \cdot, \cdot \rangle \otimes \langle \cdot, \cdot \rangle^* \right).
\]
Let $\Mp(\W)_\A$ be the metaplectic group:
\[
 1 \longrightarrow \C^1 \longrightarrow \Mp(\W)_\A \longrightarrow \Sp(\W)(\A) \longrightarrow 1.
\]
Fix a complete polarization $\V = \X \oplus \Y$.
Then we can realize the Weil representation $\omega_\psi$ of $\Mp(\W)_\A$ (relative to a nontrivial additive character $\psi$ of $\A/F$) on the Schwartz space $\SS(\X(\A))$.
Assume that there exists a homomorphism $\iota : R(\A) \rightarrow \Mp(\W)_\A$ such that the diagram
\[
 \xymatrix{
 R(F) \ar@{^{(}->}[r] \ar@{->}[d] &
 R(\A) \ar@{->}[d]^-\iota \\
 \Sp(\W)(F) \ar@{->}[r]^-i &
 \Mp(\W)_\A
 }
\]
is commutative, where $i$ is the canonical splitting.
Then for any $\varphi \in \SS(\X(\A))$, we may form a theta function on $R(\A)$:
\[
 \Theta_\varphi(g,h) = \sum_{x \in \X(F)} \omega_\psi(\iota(g,h)) \varphi(x).
\]

\subsection{Theta lifts from $E^\times$ to $B^\times$}
\label{ss:theta-1}

Let $V = B$ be a $1$-dimensional right $B$-space equipped with a skew-hermitian form
\[
 \langle x, y \rangle = x^* \cdot \kappa \i \cdot y
\]
for some $\kappa \in F^\times$.
Then $\GU(V)^0 \cong E^\times$.
In Appendix \ref{sec:weil-hodge}, we define a splitting $\iota : R(\A) \rightarrow \Mp(\W)_\A$ as above.
Let $\eta$ be a character of $\A_E^\times/E^\times$.
We regard $\eta$ as an automorphic character of $G(\A)$.
For $\varphi \in \SS(\X(\A))$ and $h \in H(\A)^+$, put
\[
 \theta_\varphi(\eta)(h) = \int_{G_1(F) \backslash G_1(\A)} \Theta_\varphi(g_1 g, h) \eta(g_1 g) \, dg_1,
\]
where we choose $g \in G(\A)^+$ such that $\nu(g) = \nu(h)$.
Since $(\A^\times)^+ = \N(\A_E^\times)$ and $F^\times \cap \N(\A_E^\times) = \N(E^\times)$, this integral defines an automorphic form $\theta_\varphi(\eta)$ on $H(\A)^+$.
We may extend $\theta_\varphi(\eta)$ to an automorphic form on $H(\A)$ by the natural embedding
\[
 H(F)^+ \backslash H(\A)^+ \hookrightarrow H(F) \backslash H(\A)
\]
and extension by zero.
Let $\theta(\eta)$ be the automorphic representation of $H(\A)$ generated by $\theta_\varphi(\eta)$ for all $\varphi \in \SS(\X(\A))$.

\begin{lem}
\label{l:theta-1}
Assume that:
\begin{itemize}
 \item $B_v$ is split for all archimedean places $v$ of $F$;
 \item $\eta_v$ does not factor through the norm map for any place $v$ of $F$ such that $B_v$ is ramified.
\end{itemize}
Then we have
\[
 \theta(\eta) = \pi(\eta)_B,
\]
where $\pi(\eta)$ is the automorphic induction of $\eta$ to $\GL_2(\A)$ and $\pi(\eta)_B$ is its Jacquet--Langlands transfer to $B^\times(\A)$.
\end{lem}

\begin{proof}
Suppose that $\theta(\eta)$ is nonzero.
Let $v$ be a place of $F$.
If $B_v$ is split, then by \S \ref{ss:compatibility-hk-duke}, the splitting $\iota : R(F_v) \rightarrow \Mp(\W_v)$ agrees with the standard one for symplectic-orthogonal dual pairs.
Hence it follows from the local theta correspondence for unramified representations that the local component $\theta(\eta)_v$ is isomorphic to the automorphic induction $\pi(\eta_v)$ of $\eta_v$ to $\GL_2(F_v)$ for almost all $v$.
By the strong multiplicity one theorem, we have $\theta(\eta) = \pi(\eta)_B$.

Thus it remains to show that $\theta(\eta)$ is nonzero.
Let $\VV$ and $\WW$ be the $1$-dimensional hermitian $E$-space and the $2$-dimensional skew-hermitian $E$-space, respectively, as in \S \ref{ss:doubling-U(WW)}.
Then $\GU(V)^0 = \GU(\VV)$ and $\GU(W) \hookrightarrow \GU(\WW)$.
By \S \ref{ss:compatibility-periods2}, the splitting $\iota : R(\A) \rightarrow \Mp(\W)_\A$ agrees with the restriction of the standard one $\G(\U(\VV) \times \U(\WW))(\A) \rightarrow \Mp(\W)_\A$ for unitary dual pairs.
Hence it suffices to show that the global theta lift of $\chi := \eta|_{\A_E^1}$ (regarded as an automorphic character of $\U(\VV)(\A)$) to $\U(\WW)(\A)$ is nonzero.
By assumption, we have $\chi \ne 1$, so that the standard $L$-function $L(s, \chi)$ is nonzero and holomorphic at $s=1$.
This together with the Rallis inner product formula \cite{ichino-mathZ, gqt, yamana} implies that the non-vanishing of the global theta lift $\theta(\chi)$ to $\U(\WW)(\A)$ is equivalent to the non-vanishing of the local theta lift $\theta(\chi_v)$ to $\U(\WW_v)$ for all $v$.
If $B_v$ is split, then $\theta(\chi_v)$ is nonzero since the dual pair $(\U(\VV_v), \U(\WW_v))$ is in the stable range \cite{li-invent}.
Suppose that $B_v$ is ramified, so that $v$ is non-archimedean.
Let $r^+(\chi_v)$ and $r^-(\chi_v)$ be the first occurrence indices:
\begin{align*}
 r^+(\chi_v) & = \min \{ r \, | \, \text{the theta lift of $\chi_v$ to $\U(\mathbf{H}_v^{\oplus r})$ is nonzero} \}, \\
 r^-(\chi_v) & = \min \{ r \, | \, \text{the theta lift of $\chi_v$ to $\U(\WW_v \oplus \mathbf{H}_v^{\oplus r-1})$ is nonzero} \},
\end{align*}
where $\mathbf{H}_v$ is the hyperbolic plane over $E_v$.
Since $\chi_v \ne 1$ by assumption, we have $r^+(\chi_v) = 1$.
On the other hand, we have
\[
 r^+(\chi_v) + r^-(\chi_v) = 2
\]
by the conservation relation \cite{sun-zhu}.
Hence we have $r^-(\chi_v) = 1$, so that $\theta(\chi_v)$ is nonzero.
This completes the proof.
\end{proof}

\subsection{Theta lifts from $B_1^\times \times B_2^\times$ to $B^\times$}
\label{ss:theta-2}

Let $V = B_1 \otimes_E B_2$ be the $2$-dimensional skew-hermitian right $B$-space as in \cite[\S 2.2]{periods1}, where $B_1$ and $B_2$ are quaternion algebras over $F$ such that $E$ embeds into $B_1$ and $B_2$, and such that $B_1 \cdot B_2 = B$ in the Brauer group.
Then $\GU(V)^0 \cong (B_1^\times \times B_2^\times) / F^\times$.
In \cite[Appendix C]{periods1}, we have defined a splitting $\iota : R(\A) \rightarrow \Mp(\W)_\A$ as above.
Let $\sigma_1$ and $\sigma_2$ be irreducible unitary cuspidal automorphic representations of $B_1^\times(\A)$ and $B_2^\times(\A)$, respectively.
We assume that they have the same central character, so that we may regard $\sigma_1 \boxtimes \sigma_2$ as an automorphic representation of $G(\A)$.
For $\varphi \in \SS(\X(\A))$, $f \in \sigma_1 \boxtimes \sigma_2$, and $h \in H(\A)^+$, put
\[
 \theta_\varphi(f)(h) = \int_{G_1(F) \backslash G_1(\A)} \Theta_\varphi(g_1 g, h) f(g_1 g) \, dg_1,
\]
where we choose $g \in G(\A)^+$ such that $\nu(g) = \nu(h)$.
By Eichler's norm theorem, this integral defines an automorphic form $\theta_\varphi(f)$ on $H(\A)^+$.
Since $H(F) H(\A)^+ = H(\A)$, we may extend $\theta_\varphi(f)$ to an automorphic form on $H(\A)$.
Let $\theta(\sigma_1 \boxtimes \sigma_2)$ be the automorphic representation of $H(\A)$ generated by $\theta_\varphi(f)$ for all $\varphi \in \SS(\X(\A))$ and $f \in \sigma_1 \boxtimes \sigma_2$.

\begin{lem}
\label{l:theta-2}
Let $\pi_i$ be the Jacquet--Langlands transfer of $\sigma_i$ to $\GL_2(\A)$.
\begin{enumerate}
\item If $\pi_1 \ne \pi_2$, then $\theta(\sigma_1 \boxtimes \sigma_2) = 0$.
\item If $\pi_1 = \pi_2$, then $\theta(\sigma_1 \boxtimes \sigma_2)$ is the Jacquet--Langlands transfer of $\pi_i$ to $B^\times(\A)$ (which exists).
\end{enumerate}
\end{lem}

\begin{proof}
Let $\sigma$ be an irreducible unitary cuspidal automorphic representation of $B^\times(\A)$ and $\pi$ its Jacquet--Langlands transfer to $\GL_2(\A)$.
For $\varphi \in \SS(\X(\A))$, $f \in \sigma_1 \boxtimes \sigma_2$, and $f' \in \bar{\sigma}$ (where $\bar{\sigma}$ is the complex conjugate of $\sigma$), we have a seesaw identity
\[
 \int_{Z_H(\A) H(F) \backslash H(\A)} \theta_{\varphi}(f)(h) \cdot f'(h) \, dh
 = \int_{Z_G(\A) G(F) \backslash G(\A)} f(g) \cdot \theta_{\varphi}(f')(g) \, dg,
\]
where $\theta_\varphi(f')$ is the theta lift of $f'$ to $G(\A)$ as in \cite[\S 4]{periods1}.
Since the theta lift of $\bar{\sigma}$ to $G(\A)$ is $\bar{\pi}_{B_1} \boxtimes \bar{\pi}_{B_2}$ by \cite[Proposition 4.4]{periods1}, where $\pi_{B_i}$ is the Jacquet--Langlands transfer of $\pi$ to $B_i^\times(\A)$ (if it exists), this integral vanishes unless $\sigma_i = \pi_{B_i}$.
In particular, (i) follows.
Moreover, if $\sigma_i = \pi_{B_i}$, then we can find $\varphi$, $f$, and $f'$ such that the integral is nonzero.
This implies that
\[
 \theta(\sigma_1 \boxtimes \sigma_2) = \sigma,
\]
so that (ii) follows.
\end{proof}

\subsection{Theta lifts from $B^\times$ to $\operatorname{GSO}^*(4,2)$}
\label{ss:theta-3}

Let $V$ be the $3$-dimensional skew-hermitian right $B$-space as in \S \ref{sec:constr-gl-ex-isom}.
Then $(\A^\times)^+ = \N(\A_E^\times)$ and $G(\A)^+ = G(\A)$.
In Appendix \ref{sec:weil-hodge}, we define a splitting $\iota : R(\A) \rightarrow \Mp(\W)_\A$ as above.
Let $\tau$ be an irreducible unitary automorphic representation of $H(\A)^+$.
For $\varphi \in \SS(\X(\A))$, $f \in \tau$, and $g \in G(\A)$, put
\[
 \theta_\varphi(f)(g) = \int_{H_1(F) \backslash H_1(\A)} \Theta_\varphi(g, h_1 h) f(h_1 h) \, dh_1,
\]
where we choose $h \in H(\A)^+$ such that $\nu(h) = \nu(g)$.
This integral defines an automorphic form $\theta_\varphi(f)$ on $G(\A)$.
Let $\theta(\tau)$ be the automorphic representation of $G(\A)$ generated by $\theta_\varphi(f)$ for all $\varphi \in \SS(\X(\A))$ and $f \in \tau$.

In the rest of this section, we assume that $\theta(\tau)$ is nonzero and cuspidal.
Note that $\theta(\tau)$ is automatically cuspidal if $\U(V)$ is anisotropic.
Then:

\begin{lem}
\label{lem:irred-theta-similitude}
The global theta lift $\theta(\tau)$ is irreducible and
\[
 \theta(\tau) \cong \otimes_v \theta(\tau_v),
\]
where $\theta(\tau_v)$ is the local theta lift of $\tau_v$ (see the proof for its definition).
\end{lem}

\begin{proof}
As in \cite[Corollary 7.1.3]{kr94}, the assertion follows from the Howe duality, which we describe below.
For this, we fix a place $v$ of $F$ and suppress the subscript $v$ from the notation.
Also, we work with the category of Harish-Chandra modules if $F$ is archimedean.

Consider the compact induction
\[
 \Omega = \operatorname{ind}^{G\times H^+}_R \omega,
\]
where $\omega$ is the Weil representation of $R$ (relative to a fixed nontrivial character of $F$ and a fixed splitting over $R$).
For any irreducible representation $\tau$ of $H^+$, the maximal $\tau^\vee$-isotypic quotient of $\Omega$ is of the form
\[
 \Theta(\tau) \boxtimes \tau^\vee
\]
for some representation $\Theta(\tau)$ of $G$.
Then the Howe duality asserts that
\begin{enumerate}
 \item $\Theta(\tau)$ is of finite length;
 \item $\Theta(\tau)$ is zero or has a unique irreducible quotient $\theta(\tau)$;
 \item for any irreducible representations $\tau$ and $\tau'$ of $H^+$ which occur as quotients of $\Omega$, we have
\[
 \theta(\tau) \cong \theta(\tau') \Longrightarrow \tau \cong \tau'.
\]
\end{enumerate}
This can be deduced from the Howe duality \cite{howe-jams, waldspurger-howe-duality, gan-takeda, gan-sun} for $(\U(V), \U(W))$ as follows.

We first show that the Howe duality for $(\U(V)^0, \U(W))$ follows from the Howe duality for $(\U(V), \U(W))$.
If $B$ is ramified, then there is nothing to prove since $\U(V)^0(F) = \U(V)(F)$.
If $B$ is split, then we have $\U(V) \cong \O(V^\dagger)$ and $\U(W) \cong \Sp(W^\dagger)$, where $V^\dagger$ and $W^\dagger$ are the $6$-dimensional quadratic $F$-space and the $2$-dimensional symplectic $F$-space, respectively, associated to $V$ and $W$ by Morita theory.
For brevity, we write $\Gs = \O(V^\dagger)$ and $\Gs^0 = \SO(V^\dagger)$.
Let $\sigma_0$ be an irreducible representation of $\Gs^0$.
Then $\sigma_0$ is an irreducible component of $\sigma|_{\Gs^0}$ for some irreducible representation $\sigma$ of $\Gs$.
Note that $\sigma$ is not necessarily uniquely determined.
Namely, $\sigma_0$ is also an irreducible component of $(\sigma \otimes \sgn)|_{\Gs^0}$, where $\sgn$ denotes the unique nontrivial character of $\Gs$ trivial on $\Gs^0$.
Fix $\varepsilon \in \Gs \smallsetminus \Gs^0$ and put $\sigma_0^\varepsilon(g) = \sigma_0(\varepsilon g \varepsilon^{-1})$ for $g \in \Gs^0$.
We have
\[
 \sigma_0 \cong \sigma_0^\varepsilon \Longleftrightarrow \sigma \ncong \sigma \otimes \sgn,
\]
and
\begin{itemize}
\item if $\sigma \ncong \sigma \otimes \sgn$, then
\[
 \sigma|_{\Gs^0} = \sigma_0, \qquad
 \Ind^{\Gs}_{\Gs^0} \sigma_0 = \sigma \oplus (\sigma \otimes \sgn);
\]
\item if $\sigma \cong \sigma \otimes \sgn$, then
\[
 \sigma|_{\Gs^0} = \sigma_0 \oplus \sigma_0^\varepsilon, \qquad
 \Ind^{\Gs}_{\Gs^0} \sigma_0 = \sigma.
\]
\end{itemize}
Then, by the conservation relation \cite{sun-zhu}, we have
\begin{itemize}
\item if $\sigma \ncong \sigma \otimes \sgn$, then at most one of $\sigma$ and $\sigma \otimes \sgn$ occurs as a quotient of $\omega$;
\item if $\sigma \cong \sigma \otimes \sgn$, then $\sigma$ does not occur as a quotient of $\omega$.
\end{itemize}
This reduces the Howe duality for $(\U(V)^0, \U(W))$ to the Howe duality for $(\U(V), \U(W))$.

Finally, as in \cite{roberts}, \cite[\S 3]{gan-tantono} (noting that the projections $R \rightarrow G$ and $R\rightarrow H^+$ are surjective), the Howe duality for $(\GU(V)^0, \GU(W)^+)$ follows from the Howe duality for $(\U(V)^0, \U(W))$.
This completes the proof.
\end{proof}

Now we explicate the local theta lift $\theta(\tau_v)$ in the unramified case.
Fix a non-archimedean place $v$ of $F$ such that:
\begin{itemize}
 \item $F_v$ is of odd residual characteristic;
 \item $E_v$ is unramified over $F_v$;
 \item $B_v$ is split over $F_v$;
 \item $\det V_v \in \mathcal{O}_{F_v}^\times$;
 \item $\psi_v$ is of order zero;
 \item $\tau_v$ is unramified.
\end{itemize}
From now on, we suppress the subscript $v$ from the notation.
We may identify $G$ with the group
\[
 \{ g \in \GL_6(F) \, | \, {}^t g \mathcal{Q} g = \nu(g) \cdot \mathcal{Q}, \, \det g = \nu(g)^3 \},
\]
where
\[
 \mathcal{Q} =
 \begin{pmatrix}
  & & & & & 1 \\
  & & & & 1 & \\
  & & 1 & & & \\
  & & & -u & & \\
  & 1 & & & & \\
  1 & & & & &
 \end{pmatrix}.
\]
Let $B$ be a Borel subgroup of $G$ and $T$ a maximal torus of $G$ given by
\[
 B = \left\{
 \begin{pmatrix}
  * & * & * & * & * & * \\
  & * & * & * & * & * \\
  & & * & * & * & * \\
  & & * & * & * & * \\
  & & & & * & * \\
  & & & & & *
 \end{pmatrix}
 \right\}
 \quad \supset \quad
 T = \left\{
 \begin{pmatrix}
  * & & & & & \\
  & * & & & & \\
  & & * & * & & \\
  & & * & * & & \\
  & & & & * & \\
  & & & & & *
 \end{pmatrix}
 \right\}.
\]
Then we have an isomorphism $T \cong (F^\times)^2 \times E^\times$ defined by
\[
 (t_1, t_2, a + b\i) \longmapsto
 \begin{pmatrix}
  t_1 & & & & & \\
  & t_2 & & & & \\
  & & a & bu & & \\
  & & b & a & & \\
  & & & & \nu t_2^{-1} & \\
  & & & & & \nu t_1^{-1}
 \end{pmatrix},
\]
where $\nu = a^2 - b^2 u$.
Also, we may identify $H$ with $\GL_2(F)$, so that
\[
 H^+ = \{ h \in \GL_2(F) \, | \, \det h \in \N(E^\times) \}.
\]
Let $B'$ be the Borel subgroup of $H$ consisting of upper triangular matrices.
Recall that $\tau$ is an irreducible unramified representation of $H^+$.
Then $\tau$ is an irreducible component of
\[
 \Ind^H_{B'}(\chi_1 \otimes \chi_2)|_{H^+}
\]
for some unramified characters $\chi_1, \chi_2$ of $F^\times$.
Note that $\chi_1, \chi_2$ are not necessarily uniquely determined.
Namely, $\tau$ is also an irreducible component of $\Ind^H_{B'}(\chi_1 \xi_E \otimes \chi_2 \xi_E)|_{H^+}$.
Then:

\begin{lem}
\label{l:theta-unram}
The local theta lift $\theta(\tau)$ is an irreducible component of
\[
 \Ind^G_B(\chi_1 \chi_2^{-1} \xi_E \otimes |\cdot| \otimes (\chi_2 |\cdot|^{-\frac{1}{2}}) \circ \N ).
\]
\end{lem}

\begin{proof}
By \S \ref{ss:compatibility-hk-duke}, the splitting $\iota : R \rightarrow \Mp(\W)$ agrees with the standard one for symplectic-orthogonal dual pairs.
Hence the assertion follows from the standard unramified computation.
We omit the details.
\end{proof}