TITLE: If $S$ is the cyclic subgroup of order n in the dihedral group $D_n$, show that $D_n/S$ is isomorphic to $Z_2$
QUESTION [6 upvotes]: If $S$ is the cyclic subgroup of order $n$ in the dihedral group $D_n$, show that $D_n/S$ is isomorphic to $Z_2$.
I know I'm supposed to find an epimorphism from $D_n$ to $Z_2$, such that $S$ is its Kernel, so that the quotient group $D_n/S$ will be isomorphic to $Z_2$. But I have no idea of how to find such an isomorphism. Besides, I don't even know what $n$ is, so I have no way of finding the elements of $D_n$ and defining an epimorphism on them.
Any help will be appreciated!!!!

REPLY [10 votes]: $D_n$ has order $2n$. If you have a subgroup $S$ of order $n$, then $D_n/S$ will have order 2 and hence is isomorphic to $\mathbb{Z}_2$.

REPLY [5 votes]: I'm not sure what you mean when you say you have no way of finding the elements of $D_n$. What definition have you been given for this group?
$D_n$ can be defined as the group of symmetries of a regular $n$-sided polygon. It has $2n$ elements; $n$ rotations, and $n$ "flips", or reflections. The rotations form a subgroup; that's your $S$. If you can prove that it's a normal subgroup, then you know the order of the quotient group, and you know there is, up to isomorphism, only one group of that order, and you're done.


TITLE: Lemma 2.20 of Kunen
QUESTION [0 upvotes]: I'm trying to understand the proof of a result in Kunen's Set Theory An Introduction to Independence Proofs Chapter VII, but I really can't find a way.
We want to show that if $G$ is a $\mathbb{P}$-generic filter then either $G \cap E \neq \emptyset$ or there exists $q \in G$ such that for all $r \in E$ we have that $q \perp r$ meaning that there is no common extension p such that $p \leq q$ and $p \leq r$.
We are asked to consider the dense set $D= \lbrace p \mid \exists r \in E \,\, (p \leq r) \rbrace \cup \lbrace q \mid  \forall r \in E \, (r \perp q) \rbrace$ and in the end we are told that $G\cap  D \neq \emptyset $ implies our claim.
But I can't see why this last implication should hold, and in particular why the intersection could not be included in $ \lbrace p \mid \exists r \in E \,\, (p \leq r) \rbrace \ \setminus E$. Maybe this would contradict the filter properties, but it seems that a filter as defined by Kunen (see page 53) is closed under taking elements that are "LESS THAN",  not "GREATER THAN".

REPLY [1 votes]: Take any $p\in G\cap D$. Either $\exists r\in E$ such that $p\leq r$, in which case since $G$ is a filter, $r\in G$, so $G\cap E\neq\varnothing$. Or else $\forall r\in E$ we have that $r\perp p$. But now if $r\in G\cap E$, then because $G$ is a filter, $r$ and $p$ are compatible, which is impossible.


TITLE: Nice embedding of the permutohedron of order $n$ in ${\mathbb R}^{n-1}$
QUESTION [7 upvotes]: The permutohedron $P_n$ of order $n$ ($n\geqslant 2$) is the convex hull of the points $P_\pi=(\pi(1),\dots,\pi(n))$ where $\pi$ ranges over all permutations of $\{1,2,\dots,n\}$. Obviously, since these points are all in the plane with equation $x_1+\dots+x_n=\frac{n(n+1)}{2}$, the polytope $P_n$ has dimension at most $n-1$. If I am not wrong, its dimension is exactly $n-1$.
Is it possible to find a "nice" embedding of $P_n$ in ${\mathbb R}^{n-1}$? "Nice" means here a simple description of the vertices. Since I am mostly interested in the combinatorics of polytopes, my question can be formulated as: is there a $(n-1)$-polytope in ${\mathbb R}^{n-1}$ with a description of vertices is as simple as possible and that is combinatorially equaivalent to $P_n$?

REPLY [3 votes]: There is a very simple formula. If $v(\sigma)\in\Bbb R^d$ is the vertex that corresponds to the permutation $\sigma:\{1,...,d+1\}\to\{1,...,d+1\}$, then you can choose its coordinates as
$$v_i(\sigma)=\sigma(d+1)+\sigma(i),\quad\text{for $i\in\{1,...,d\}$}.$$

If you want a standard permutahedron, centered at the origin and with all edges of the same length, then you can use
$$v_i(\sigma) = \frac{d+2}2 -\frac{\sigma(d+1)}{\sqrt{d+1}}-\sigma(i)\Big(1-\frac1{\sqrt{d+1}}\Big).$$
All edges are of length $$\sqrt2\Big(1-\frac1{\sqrt{d+1}}\Big).$$

If you use the Mathematica code with the second formula
vert[d_] := Table[
   (d+2)/2-s[[d+1]]/Sqrt[d+1]-s[[i]](1-1/Sqrt[d+1]),
{s,Permutations[Range[d+1]]}, {i,1,d}]

then vert[1] gives you the points $1/2-1/\sqrt2$ and $-1/2+1/\sqrt2$.
And if you use ConvexHullMesh to plot vert[2] and vert[3] you find


text:
\begin{document}
\date{June 23, 2014}

\title[Modular equalities for complex reflection arrangements]{
Modular equalities for complex reflection arrangements}

\author[A.D.~M\u acinic]{Daniela Anca~M\u acinic}
\address{Simion Stoilow Institute of Mathematics,
P.O. Box 1-764, RO-014700 Bucharest, Romania}
\email{Anca.Macinic@imar.ro}

\author[\c S.~Papadima]{\c Stefan Papadima$^1$}
\address{Simion Stoilow Institute of Mathematics,
P.O. Box 1-764,
RO-014700 Bucharest, Romania}
\email{Stefan.Papadima@imar.ro}
\thanks{$^1$Partially supported by the Romanian Ministry of
National Education, CNCS-UEFISCDI, grant PNII-ID-PCE-2012-4-0156}

\author[C.R.~Popescu]{Clement Radu Popescu$^2$}
\address{Simion Stoilow Institute of Mathematics,
P.O. Box 1-764, RO-014700 Bucharest, Romania}
\email{Radu.Popescu@imar.ro}
\thanks{$^2$Supported by a grant of the Romanian
National Authority for Scientific Research, CNCS-UEFISCDI,
project number PN-II-RU-TE-2012-3-0492}

\subjclass[2010]{Primary
14F35,
32S55;
Secondary
20F55,
52C35,
55N25.
}

\keywords{Milnor fibration, algebraic monodromy, hyperplane arrangement,
complex reflection group, resonance variety, characteristic variety,
modular bounds.}

\begin{abstract}
We compute the combinatorial Aomoto--Betti numbers $\beta_p(\A)$
of a complex reflection arrangement. When $\A$ has rank at least $3$,
we find that $\beta_p(\A)\le 2$, for all primes $p$. Moreover, $\beta_p(\A)=0$
if $p>3$, and $\beta_2(\A)\ne 0$ if and only if $\A$ is the Hesse arrangement.
We deduce that the multiplicity $e_d(\A)$ of an order $d$ eigenvalue of the
monodromy action on the first rational homology of the Milnor fiber is equal to
the corresponding Aomoto--Betti number, when $d$ is prime. We give a uniform
combinatorial characterization of the property $e_d(\A)\ne 0$, for $2\le d\le 4$.
We completely describe the  monodromy action for full monomial arrangements of
rank $3$ and $4$. We relate $e_d(\A)$ and $\beta_p(\A)$ to multinets, on an
arbitrary arrangement.
\end{abstract}

\maketitle
\setcounter{tocdepth}{1}
\tableofcontents

\section{Introduction and statement of results}
\label{sec:intro}

\subsection{Milnor fibration and monodromy}
\label{ss11}

The complement $M$ of a degree $n$ complex hypersurface in $\C^l$,
$\{ f=0\}$, and the associated Milnor fibration, $f:M \to \C^{\times}$,
first analysed by Milnor in his seminal book \cite{M}, attracted a lot of attention
over the years. Multiplication by $\exp (\frac{2\pi \sqrt{-1}}{n})$ induces
the geometric monodromy action on the associated {\em Milnor fiber} $F=f^{-1}(1)$,
$h:F\to F$, and the {\em algebraic monodromy action},
$\{ h_i: H_i(F, \Q) \to H_i(F, \Q) \}$.

Computing $h_i$ is a major problem in the field, when $f$ has a non-isolated singularity  at $0$.
Even for the defining polynomial of a (central) complex hyperplane arrangement $\A$ in $\C^l$ and $i=1$,
the answer is far from being clear.
This case was tackled in the recent literature by many authors,
who used a variety of tools; see for instance \cite{PS14} for a brief survey.
In this paper, we focus on {\em reflection arrangements}, associated to finite complex reflection groups.

It is well-known that, for an arbitrary arrangement $\A$, the complement $M_\A$ has the homotopy type of a connected,
finite CW-complex with torsion-free homology, whose first integral homology group, $H_1(M_\A, \Z)=\Z^n$, comes
endowed with a natural basis, given by meridian loops around the hyperplanes.

It is also well-known that $h_1$ induces a $\Q\Z$-module decomposition,

\begin{equation}
\label{eq:monintro}
H_1(F_\A, \Q)= \oplus_{d}(\Q[t] \slash \Phi_d(t))^{e_d(\A)},
\end{equation}
where $\Phi_d$ is the $d$-th cyclotomic polynomial, $e_d(\A)=0$ if $d \notdivides n$, and $e_1(\A)=n-1$.

A pleasant feature of hyperplane arrangements is the rich combinatorial structure encoded by the associated
intersection lattice, $\LL_\bullet(\A)$, whose elements are the intersections of hyperplanes from $\A$, ranked by
codimension and ordered by inclusion. In this context, the open monodromy action problem takes the following
more precise form: are the multiplicities $e_d(\A)$ combinatorial? If so, give a formula involving only $\LL_\bullet(\A)$.

\subsection{Characteristic and resonance varieties}
\label{ss12}

Our approach to decomposition \eqref{eq:monintro} is topological, based on two types of jump loci,
associated to CW-complexes
having the properties recalled for $M_\A$, and the interplay between them.

 The complex characteristic variety $\VV^i_q(M)$, sitting inside the character torus \hfill
$\TT(M) \coloneqq \Hom (H_1(M, \Z), \C^\times)=(\C^\times)^n$
is the locus of those $\rho \in \TT(M)$ for which $\dim_\C H_i(M, \C_\rho) \geq q$, where $\C_\rho$
denotes the associated rank $1$ local system on $M$.

Note that $ \Hom (H_1(M, \Z), \C)=H^1(M, \C)=\C^n$, and denote by
$\exp: H^1(M, \C) \surj \TT(M)$ the natural exponential map.
For an integer $d \geq 1$, let $\rho_d \in \TT(M)$ be the exponential
of the diagonal cohomology class equal to
$\frac{2 \pi \sqrt{-1}}{d}$, with respect to the distinguished $\Z$-basis.
When $M=M_\A$ and $d>1$, it is well-known that

\begin{equation}
\label{eq:bmult}
e_d(\A)=\dim_\C H_1(M_\A, \C_{\rho_d})
\end{equation}

The resonance variety $\RR_q^i (M, \k)$ over a field $\k$, sitting inside $H^1(M, \k)$, is the locus of those
$\sigma \in H^1(M, \k)$ for which $\dim_\k H^i (H^\bullet (M, \k), \sigma\cdot) \geq q$, where
$\sigma \cdot$ denotes the left multiplication
by $\sigma$ in the cohomology ring. When $\k$ is the prime field $\FF_p$,
denote by $\sigma_p \in H^1(M, \FF_p)$
the diagonal cohomology class equal to $1$, with respect to the distinguished $\Z$-basis,
and define the modulo $p$ {\em Aomoto-Betti number} by

\begin{equation}
\label{eq:defbeta}
\beta_p(M) \coloneqq \dim_{\FF_p} H^1(H^\bullet (M, \FF_p), \sigma_p\cdot)
\end{equation}

\noindent When $M=M_\A$, we will replace $M$ by $\A$ in the notation.

By a celebrated theorem of Orlik and Solomon \cite{OS}, the cohomology ring of $M_\A$ is combinatorial.
More precisely, $\beta_p(\A)$ may be computed from $\LL_{\leq 2}(\A)$, as well as $\RR^1_q(\A, \k)$, for all $q$ and $\k$.

\subsection{Modular bounds}
\label{ss13}

It follows from Theorem 11.3 in \cite{PS-eq} that

\begin{equation}
\label{eq:bound}
e_{p^s}(\A) \leq \beta_p(\A), \text{for all}\; s \geq 1.
\end{equation}

Actually, the above modular bound holds for all CW-complexes considered in \S\ref{ss12},
with the multiplicity replaced by the value from \eqref{eq:bmult}, and is in general strict, in the broader context.
Our first main result says that \eqref{eq:bound} becomes an equality, for reflection arrangements and $s=1$.

\begin{theorem}
\label{thm:a}
Let $\A$ be a complex reflection arrangement. Then $e_p(\A)=\beta_p(\A)$, for all primes $p$.
In particular, $e_p(\A)$ is determined by $\LL_{\leq 2}(\A)$.
\end{theorem}

\subsection{Aomoto-Betti numbers for reflection arrangements}
\label{ss14}

Reflection arrangements have a distinguished history, going back as far as Jordan's work from 1878 on the symmetry group
of the famous Hessian configuration. Another related open problem is whether the {\em Hessian
arrangement} is the only arrangement supporting a $4$-net.

Finite complex reflection groups have been classified by Shephard and Todd \cite{ST} (see also \cite{C}, \cite{OT}).
Each such group $G$ gives rise to the complex reflection arrangement $\A(G)$, consisting of the fixed points of all reflections in $G$.
Among them, we have the monomial arrangements $\A(m,m,l)$ in $\C^l \; (l \geq 2)$, with defining polynomials
$\Pi_{1 \leq i< j \leq l}\ (z_i^m-z_j^m) \; (m \geq 1)$, and full monomial arrangements
$\A(m,1,l)$ in $\C^l \; (l \geq 2)$, defined by $z_1 \dots z_l \cdot \Pi_{1 \leq i< j \leq l}\ (z_i^m-z_j^m) \; (m \geq 2)$.
We may now state our second main result.

\begin{theorem}
\label{thm:b}
For a complex reflection arrangement $\A$ of rank at least $3$, the following hold.
\begin{enumerate}
\item \label{b1} If $p>3$, then $\beta_p(\A)=0$.
\item \label{b2} $\beta_2(\A)\neq 0
\Leftrightarrow \beta_2(\A)=2 \Leftrightarrow \A$ supports a $4$-net $\Leftrightarrow \A$ is the Hessian arrangement.
\item \label{b3} The only cases when  $\beta_3(\A)\neq 0$ are: $\A(m,1,3)$ with $m \equiv 1 \;(\text{mod} \;3)$,
where $\beta_3=1$; $\A(m,m,3)$ with $m \geq 2$, where $\beta_3=1$ if $m \nequiv 0 \;(\text{mod}\; 3)$
and otherwise $\beta_3=2$; $\A(m,m,4)$, where $\beta_3=1$.
\item \label{b4} In particular, $\beta_p(\A) \leq 2$, for all primes $p$.
\end{enumerate}
\end{theorem}

We imposed the rank condition since the Aomoto-Betti numbers for arrangements of rank at most $2$ are known
(see for instance \cite{MP}). Moreover, the conclusion of Theorem \ref{thm:b}(4) may no longer hold
for $\A(m,m,2)$, when $m \equiv 0 \;(\text{mod}\; p)$.

The resonance varieties $\RR^1_q(\A, \C)$ are quite well-understood, due to work by Falk, Libgober and Yuzvinsky
(see for instance the survey \cite{Yuz01}), while the picture over $\FF_p$ largely remains a mystery.
Our Theorems \ref{thm:b} and \ref{thm:a}, together with similar results from \cite{MP}, \cite{PS14},
support the modular conjecture from \cite{PS14}.

\subsection{A combinatorial non-triviality test}
\label{ss15}

Dimca, Ibadula and M\u acinic asked in \cite{DIM} the following natural question: if $d>1$ and $e_d(\A) \neq 0$,
does this imply that $\rho_d \in \exp\ \RR_1^1(\A, \C)$? A positive answer (for all $d$) would imply that
the non-triviality of $h_1$ is combinatorial, since the converse implication is known, for all $d$.

\begin{theorem}
\label{thm:c}
If $\A$ is a complex reflection arrangement, then the above question has a positive answer, for $2 \leq d \leq 4$.
\end{theorem}

We derive Theorem \ref{thm:c} and Theorem \ref{thm:a} from Theorem \ref{thm:b} with
the aid of  a general result (proved in Theorem \ref{thm:d})
that relates the Falk-Yuzvinsky multinet structures on matroids from \cite{FY} to
the algebraic monodromy action and the
Aomoto-Betti numbers of an arrangement. The tools from our paper also enable us to give a complete,
combinatorial description in Proposition \ref{prop:full} for the monodromy action on $H_1(F_\A, \Q)$,
in the case of full monomial arrangements of rank $3$ and $4$.
Related results may be found in \cite{L} and \cite{PS14}. Proposition \ref{prop:multilie}, proved in \cite{PS14},
is of great help in Section \ref{sec:multi}.

\section{Non-exceptional reflection arrangements}
\label{sec:fam}

We first compute the Aomoto-Betti numbers of monomial and full monomial arrangements.

\subsection{The classification} (cf. \cite{ST}, \cite{C}, \cite{OT})
\label{ss21}

A finite reflection group $G$ decomposes as a product of irreducible factors of the
same kind. At the level of arrangements, $\A(G\times G')$ is the product $\A(G)\times \A(G')$, and
the corresponding complements satisfy $M(G\times G') = M(G) \times M(G')$. (Whenever convenient, we will abbreviate
notation and replace  $\A(G)$ by $G$, when speaking about associated objects.)

The irreducible reflection arrangements of rank at least 3 comprise the monomial and full monomial arrangements,
$\A(m,m,l)\ (m\geq 2, l\geq 3\ {\rm or}\ m=1, l\geq 4)$ and $\A(m,1,l)\ (l\geq 3)$, plus the exceptional arrangements
$\A(G_{23}) - \A(G_{37})$. The Hessian arrangement is $\A(G_{25})$.

\subsection{Vanishing criteria}
\label{ss22}

Given an arbitrary arrangement $\A$ and an $r$--flat $X\in \LL_r(\A)$,
set $\A_X \coloneq \{H\in\A | H\supseteq X\}$, and define
the multiplicity of $X$ to be equal to $|\A_X|$.

Using the distinguished $\Z$-basis, we identify an element $\eta\in H^1(M_\A,\FF_p)$
with the family $\{\eta_H\in \FF_p\}_{H\in\A}$.
We denote by $Z_p(\A)\subseteq H^1(M_\A,\FF_p)$ the kernel of $\sigma_p\cdot$.

Definition \eqref{eq:defbeta} implies that $\beta_p(\A)=\dim_{\FF_p} Z_p(\A)-1.$
Our computations are based on the following well-known result (see e.g. \cite{PS14}).

\begin{lemma}
\label{lem:zeq}
An element $\eta$ belongs to $Z_p(\A)$ if and only if, for any $X\in\LL_2(\A)$,
\[
 \left\{
  \begin{array}{lccr}
\sum_{H \in\A_X}\eta_H = 0 & \text{if} & |\A_X|\equiv 0 & \text{(mod p)}\\
\eta_H = \eta_{H'}, \forall\ H, H'\in\A_X & \text{if} & |\A_X|\nequiv 0 & \text{(mod p)}
  \end{array}
 \right.
\]
\end{lemma}

Clearly, $\beta_p(\A)=0$ if and only if $\eta\in Z_p(\A)$ implies that $\eta$ is constant.
A first useful vanishing criterion is due to Yuzvinsky.

\begin{lemma}
\label{lem:yuzv}(\cite{Yuz01}) If $|\A|\nequiv 0$ (mod p), then $\beta_p(\A) = 0$.
\end{lemma}

A second convenient situation is the following.

\begin{lemma}
\label{lem:prod} Assume that $\A = \A_1\times\A_2$.
 \begin{enumerate}
 \item \label{p1} $H_1(M_{\A},\C_{\rho_d}) = 0$, for all $d>1$.
 \item \label{p2} $\beta_p(\A) = 0$, for all primes p.
 \end{enumerate}
\end{lemma}

\begin{proof}
Assuming the contrary, we infer from \cite[Proposition 13.1]{PS-bns} that
$\rho_d\in\VV_1^1(M_{\A_1}\times M_{\A_2}) = \{1\}\times\VV_1^1(M_{\A_2})\cup \VV^1_1(M_{\A_1})\times\{1\}$,
respectively
$\sigma_p\in\RR_1^1(M_{\A_1}\times M_{\A_2},\FF_p) = \{0\}\times\RR_1^1(M_{\A_2},\FF_p)\cup \RR^1_1(M_{\A_1},\FF_p)\times\{0\}$,
in contradiction with the fact that all coordinates of $\rho_d$ (respectively $\sigma_p$) are different from 1 (respectively 0).
\end{proof}

By Lemma \ref{lem:prod}(\ref{p2}), we only need to compute $\beta_p(G)$ for an irreducible complex reflection group $G$.

To state the third vanishing criterion, we need to introduce certain graphs, associated to an arrangement $\A$
and an integer $k\geq 2$, with vertex set $\A$.
The edges of $\Gamma_k(\A)$ are defined by the condition $|\A_{H\cap H'}|\nequiv 0\ (\bmod\ k)$, for $k>2$,
and by $|\A_{H\cap H'}|$ is either odd or equal to 2,
for $k=2$. The defining property for $\Gamma_{(k)}(\A)$ is $|\A_{H\cap H'}| = k$. Note that
$\Gamma_{(2)}(\A)$ is a subgraph of $\Gamma_p(\A)$,
for all primes $p$. The equivalence relation on $\A$ associated to the edge paths of $\Gamma_k(\A)$,
respectively $\Gamma_{(k)}(\A)$, will be denoted by
$\sim_k$, respectively $\sim_{(k)}$.

We obtain the following immediate consequences of Lemma \ref{lem:zeq}.

\begin{lemma}
\label{lem:graphv} Each of the properties below implies that $\beta_p(\A)=0$.
 \begin{enumerate}
  \item\label{gv1} The graph $\Gamma_p(\A)$ is connected.
  \item\label{gv2} The graph $\Gamma_{(2)}(\A)$ is connected.
  \item\label{gv3} For all $X\in\LL_2(\A)$, $p\notdivides |\A_X|$.
 \end{enumerate}
\end{lemma}

\subsection{Intersection lattices}
\label{ss23}

The Aomoto-Betti numbers for $\A(1,1,l)$ were computed in \cite{MP}.
They verify all statements from Theorems \ref{thm:b} and \ref{thm:a}.
Hence, we may suppose from now on that $m\geq 2$.

We need to describe $\LL_{\leq 2}(\A)$. It will be convenient to label the various hyperplanes as follows.
Set $\omega= \exp (\frac{2\pi \sqrt{-1}}{m}),\ (H_i)\ z_i=0$ for all $1\leq i\leq l$,
and $(H_{ij^\alpha})\ z_i - \omega^\alpha z_j = 0$, for $1\leq i< j\leq l$ and $\alpha\in\Z/m\Z$.
We go on by listing the 2-flats (identified with the corresponding subarrangements $\A_X$).

\begin{description}
 \item[Case {\rm I}] $\A(m,1,l), l\geq 4:$
 \item[$\rm{I_a}$] \{$H_i, H_j, H_{ij^\alpha} (\alpha\in\Z/m\Z)$\}, with multiplicity $m+2$;
 \item[$\rm{I_b}$] \{$H_{ij^\alpha}, H_{jk^\beta}, H_{ik^{\alpha+\beta}}$\}, with multiplicity 3;
 \item[$\rm{I_c}$] \{$H_{ij^\alpha}, H_{kh^\beta}$\}, with multiplicity 2;
 \item[$\rm{I_d}$] \{$H_i, H_{jk^\alpha}$\}, with multiplicity 2.
 \item[Case {\rm II}] $\A(m,m,l), l\geq 4:$ types $\rm{I_b}$ and $\rm{I_c}$, plus \{$H_{ij^\alpha} (\alpha\in\Z/m\Z)$\},
 with multiplicity $m$.
 \item[Case {\rm III}] $\A(m,1,3):$ types $\rm{I_a}$, $\rm{I_b}$ and $\rm{I_d}$.
 \item[Case {\rm IV}] $\A(m,m,3):$ types $\rm{I_b}$ and $\rm{II}$.
\end{description}

\subsection{$\beta_p$-vanishing}
\label{ss24}

We will use Lemma \ref{lem:graphv} to treat the cases when $\beta_p(\A) = 0$
in the non-exceptional families.
To simplify things, we suppress $H$ from notation and identify the hyperplanes
with their labels, $i$ and $ij^\alpha$.

We claim that, for $\A = \A(m,1,l)$ with $l\geq 4, \Gamma_{(2)}(\A)$ is connected. Indeed, given $i<j$ we may find $h<k$
with $i,j,h,k$ distinct. Hence, $i\sim_{(2)} hk^0\sim_{(2)} j$ and $ij^\alpha\sim_{(2)} k$, which proves connectivity.
Similar arguments lead to the following conclusions. If $\A = \A(m,1,l)$ with $l=3$ and $p\neq 3$, then $\Gamma_p(\A)$
is connected; for $p = 3$ and $m\nequiv 1\ (\bmod\ 3), \Gamma_3(\A)$ is connected.
The remaining full monomial Aomoto-Betti numbers,
$\beta_3(m,1,3)$ with $m\equiv 1\ (\bmod\ 3)$, will be computed later on.

If $\A = \A(m,m,l)$ with $l\geq 5$, then $\Gamma_{(2)}(\A)$ is connected.
For $l = 3,4$ and $p\neq 3, \Gamma_p(\A)$
is connected. So, for monomial arrangements, only $\beta_3$ in ranks 3 and 4 remains to be calculated.

\subsection{The remaining non-exceptional cases}
\label{ss25}

A $\bmod\ 3$ cocycle $\eta\in Z_3(\A)$ is a family of elements of $\FF_3$, $\eta_k$ and $\eta_{ij^\alpha}$,
satisfying the equations from Lemma \ref{lem:zeq}, for any $X\in\LL_2(\A)$.

{\bf Case} $\A = \A(m,1,3)$ with $m\equiv 1 (\bmod\ 3)$.

The equations coming from 2-flats of type $\rm{I_d}$ say that
$\eta_{jk^\alpha} = \eta_i$, where $i$ is the third element of $\{1,2,3\}$.
The equations of type $\rm{I_b}$ become equivalent to $\eta_1+\eta_2+\eta_3 = 0$, while type $\rm{I_a}$
equations say that $\eta_i+\eta_j+m\eta_k = 0$, for all $i\neq j\neq k$. We infer that $\beta_3(m,1,3) = 1$,
as asserted in Theorem \ref{thm:b}.

{\bf Case} $\A = \A(m,m,4)$.

The equations of type $\rm{I_c}$ say that $\eta_{ij^\alpha} = \eta_{ij}$,
and $\eta_{34} = \eta_{12}, \eta_{24} = \eta_{13}, \eta_{23} = \eta_{14}$.
Type $\rm{I_b}$ equations reduce then to $\eta_{12}+\eta_{13}+\eta_{14} = 0$,
while type II conditions follow from
$\eta_{ij^\alpha} = \eta_{ij}$. Again, $\beta_3(m,m,4) = 1$, as claimed.

{\bf Case} $\A = \A(m,m,3)$ with $m\nequiv 0 (\bmod\ 3)$.

The equations of type II say that $\eta_{ij^\alpha} = \eta_{ij}$, and the type $\rm{I_b}$ conditions
then reduce to $\eta_{12}+\eta_{23}+\eta_{13} = 0$. This shows that $\beta_3(m,m,3) = 1$, as asserted.

{\bf Case} $\A = \A(m,m,3)$ with $m = 3n$.

Set $\eta_{12^\alpha} = a_{\alpha}, \eta_{23^\alpha} = b_{\alpha}, \eta_{13^\alpha} = c_{\alpha}$.
With this notation, the equations of type $\rm{I_b}$ are equivalent to the system
\begin{equation}
\label{eq:neteq}
a_{\alpha}+b_{\beta}+c_{\alpha + \beta} = 0,\forall\ \alpha ,\beta\in\Z/m\Z,
\end{equation}
while the conditions of type II read
\begin{equation}
\label{eq:block}
\sum a_\alpha =\sum b_\beta =\sum c_\gamma = 0.
\end{equation}

We first solve the system \eqref{eq:neteq}, as follows. It implies that
$a_\alpha + b_\beta = a_{\alpha '} + b_{\beta '}$,
if $\alpha + \beta = \alpha ' + \beta '$, in particular
$a_\alpha - a_0 = b_\alpha  - b_0 \coloneq d_\alpha$, for all $\alpha$, and
$d_\alpha + d_\beta = d_{\alpha '} + d_{\beta '}$, if $\alpha + \beta = \alpha ' + \beta '$.
We infer that $d_\alpha = \alpha d_1$,
for all $\alpha$. Hence $a_\alpha = a_0 + \alpha d_1, b_\alpha = b_0 + \alpha d_1$
and $c_\alpha = -a_0 - b_0 - \alpha d_1$,
which solves the system \eqref{eq:neteq}. In particular, its solution space is 3-dimensional.

Finally, it is an easy matter to check that \eqref{eq:neteq} $\Rightarrow$ \eqref{eq:block}, since $m = 3n$.
Therefore, $\beta_3(3n,3n,3) = 2$, as asserted.

This proves Theorem \ref{thm:b} for non-exceptional reflection arrangements.

\section{Exceptional reflection arrangements}
\label{sec:exc}

We finish the proof of Theorem \ref{thm:b}, by computing the
Aomoto-Betti numbers of the exceptional complex reflection
arrangements of rank at least 3, $G_{23} - G_{37}$.

\subsection{The groups $G_{31},\ G_{32},\ G_{33}$}
\label{ss31}

{\bf Case} $\A = \A(G_{31})$. The hyperplanes of $\A$ live in $\C^4$. Their defining equations are as follows
(see \cite{HR}). Set $\omega= \exp (\frac{2\pi \sqrt{-1}}{4})$.
The hyperplanes of $\A$ are: $(H_i)\ z_i = 0\ (1\leq i\leq 4)$;
$(H_{ij^\beta})\ z_i -\omega^\beta z_j = 0\ (1\leq i< j\leq 4,\ \beta\in\Z/4\Z)$;
$(H_{\overline\alpha})\ z_1 + \sum_{2\leq i\leq 4} \omega^{\alpha_i}z_i = 0$
$(\overline\alpha = (\alpha_2,\alpha_3,\alpha_4)\in(\Z/4\Z)^3, \alpha_2 + \alpha_3 + \alpha_4\equiv 0\ (\bmod\ 2))$.

By Lemma \ref{lem:graphv}(\ref{gv2}), it is enough to show that $\Gamma_{(2)}(G_{31})$ is connected.
This can be seen as follows. Clearly, the 2-flat $H_k\cap H_{ij^\beta}$ has multiplicity 2, when $i\neq j\neq k$.
This implies that $i\sim_{(2)} j\sim_{(2)} kh^{\beta}$, for all $1\leq i<j\leq 4,\ 1\leq k<h\leq 4$ and
$\beta\in\Z/4\Z$. Given any $H_{\overline\alpha}$, it is not hard to see that the multiplicity of
$H_{\overline\alpha}\cap H_{12^{\alpha_2}}$ is 2. This proves connectivity, as claimed.

{\bf Case} $\A = \A(G_{32})$. Set $\omega= \exp (\frac{2\pi \sqrt{-1}}{3})$. The arrangement $\A$
consists of the following hyperplanes in $\C^4$ (see \cite{Sz}): $(H_i)\ z_i = 0\ (1\leq i\leq 4)$;
$(H_{1^{\alpha\beta}})\ z_2 + \omega^\alpha z_3 + \omega^\beta z_4 = 0$;
$(H_{2^{\alpha\beta}})\ z_1 + \omega^\alpha z_3 - \omega^\beta z_4 = 0$;
$(H_{3^{\alpha\beta}})\ z_1 - \omega^\alpha z_2 + \omega^\beta z_4 = 0$;
$(H_{4^{\alpha\beta}})\ z_1 + \omega^\alpha z_2 - \omega^\beta z_3 = 0$ ($\alpha,\beta\in\Z/3\Z$).
Clearly, the 2-flats $H_i\cap H_j\ (i\neq j)$ and $H_i\cap H_{i^{\alpha\beta}}$ have multiplicity 2.
This shows that $\Gamma_{(2)}(G_{32})$ is connected and we are done.

{\bf Case} $\A = \A(G_{33})$. Here, $\omega= \exp (\frac{2\pi \sqrt{-1}}{3})$ and the hyperplanes (in $\C^6$)
are as follows (see \cite{C,R}): $(H_{ij^\beta})\ z_i - \omega^\beta z_j = 0\ (1\leq i<j\leq 4,\ \beta\in\Z/3\Z)$;
$(H_{\overline\alpha})\ \sum_{1\leq i\leq 4} \omega^{\alpha_i}z_i + z_5 + z_6 = 0$
$\ (\overline\alpha = (\alpha_1,\alpha_2,\alpha_3,\alpha_4)\in(\Z/3\Z)^4,\ \sum\alpha_i = 0)$.
Plainly, $ij^{\beta}\sim_{(2)}kh^0\sim_{(2)}ij^{\beta '}$, for all $\beta, \beta '$ (where $i\neq j\neq k\neq h$).
Like in the  Case $G = G_{31}$, it can be checked that
$H_{\overline\alpha}\cap H_{ij^\beta}$ has multiplicity 2, if $\beta = \alpha_j - \alpha_i$.
We infer that $\Gamma_{(2)}(G_{33})$ is connected, and we are done.

\subsection{More vanishing criteria}
\label{ss32}

We will no longer need defining equations to settle the remaining cases.
We will use instead a couple of new vanishing arguments.

For the beginning, let us recall from \cite[pp.~224-225]{OT} the following very useful properties of reflection
groups and arrangements, derived from a key result of Steinberg \cite{S}. For any complex reflection group $G$ and any
$X\in\LL_r(\A(G))$, the fixer subgroup $G_X \coloneq \{g\in G\ |\ gx = x,\ \forall x\in X\}$
is a reflection group, and $\A(G)_X = \A(G_X)$ is again a reflection arrangement, of rank $r$.
By the construction of $\A(G)$, the group $G$ acts on the arrangement $\A(G)$, hence on the intersection lattice
$\LL_\bullet(G)$. Let us denote by $\OO_X$ the $G$-orbit of $X\in\LL(G)$. Let $\Type (X)$ be the isomorphism type
of the reflection group $G_X$. It follows from \cite[Lemma 6.88]{OT} that the type is constant on each orbit $\OO_X$.
Moreover, Table C from \cite{OT} gives $|\A(G)|$ and the orbit partition of $\LL_\bullet(G)$ for all exceptional groups,
in terms of types of orbits.

This leads to the quick computation of the sets
\[
\pP(G) := \{ p\ \text{prime}\ |\ \exists\ X\in\LL_2(G)\ \text{such that}\ |\A(G_X)|\equiv 0\ (\bmod\ p)\}
\]

In particular, $\pP(G)\subseteq \{2,3,5\}$, for every exceptional arrangement of rank at least 3.
We infer from Lemma \ref{lem:graphv}(\ref{gv3}) that $\beta_p(G) = 0$, if $p>5$.
Hence, we may suppose from now on that $p\leq 5$.

For an arbitrary arrangement $\A, H\in\A$ and a prime $p$, we define
\begin{equation}
\label{eq:multp}
m_p(H) = 1 + \sum(|\A_X| - 1),
\end{equation}
where the sum is taken over those $X\in\LL_2(\A)$ such that $X\subseteq H$
and $|\A_X|\nequiv 0\ (\bmod\ p)$.

The numbers from \eqref{eq:multp} may be extracted from Table C in \cite{OT},
for exceptional reflection arrangements of rank at least 3. This is based on the following fact, valid for an
arbitrary reflection group $G$. For $X, Y\in\LL(G)$, let $u(X,Y)$ be the number of $Z\in\LL(G)$ such that
$Z\in\OO_Y$ and $Z\subseteq X$. Clearly, this number depends only on $\OO_X$ and $\OO_Y$.
The values $u(H,X)$ may be found in Table C, for all orbit types corresponding to $H\in\LL_1(G)$
and $X\in\LL_2(G)$.

\begin{lemma}
\label{lem:frac} For an arrangement $\A$ and a prime $p$, the following hold.
 \begin{enumerate}
 \item\label{f12} If $m_p(H)>\frac{|\A|}{2}$ for all $H\in\A$, then $\beta_p(\A) = 0.$
 \item\label{f13} If $m_p(H)>\frac{|\A|}{3}$ for all $H\in\A$ and $\A$ has no rank 2 flats of
 multiplicity $p\cdot r$ with $r>1$, then $\beta_p(\A) = 0$.
 \end{enumerate}
\end{lemma}

\begin{proof}
If $\beta_p(\A)\neq 0$, there is a non-constant function $\eta\in\FF_p^\A$
satisfying all equations from Lemma \ref{lem:zeq}. Fix $H\in\A$ and set $\eta_H = \alpha$.
We claim that $|\{\eta = \alpha\}|\geq m_p(H)$. Indeed, if $X\in\LL_2(\A)$ is contained in $H$
and $|\A_X|\nequiv 0\ (\bmod\ p)$, then $\eta$ must have the constant value $\alpha$ on $\A_X$.
An easy count of all these hyperplanes gives the claimed inequality.

In Part (\ref{f12}), this implies that $\eta$ must be constant, a contradiction. In Part (\ref{f13}),
we infer that $\eta$ has only two distinct values. By adding constant functions
and multiplying by non-zero elements in
$\FF_p$, we may assume that these values are $\eta_H = 0$ and $\eta_{H '}= 1$.
By Lemma \ref{lem:zeq}, the flat $X = H\cap H'$ has multiplicity $p\cdot r$,
imposing the condition $\sum_{K\in\A_X} \eta_K = 0$.
Since necessarily $r = 1$, we arrive again at a contradiction.
\end{proof}

\subsection{Induction on rank}
\label{ss33}

We start with a couple of general considerations. A subarrangement $\B\subseteq\A$ is called
line-closed in $\A$ if $\B_X = \A_X$, for all $X\in\LL_2(\B)$. This property implies that the
restriction map, $\FF_p^\A\longrightarrow\FF_p^\B$, sends $Z_p(\A)$ into $Z_p(\B)$.
Clearly, $\A_Y$ is line-closed in $\A$, for any $Y\in\LL (\A)$.

\begin{lemma}
\label{lem:indr} Let G be a complex reflection group. Assume that $2\leq r<\rank(\A(G))$ and
$\beta_p(\A(G')) = 0$, for all irreducible groups $G'\in \Type(\LL_r(\A(G))).$ Then $\beta_p(\A(G)) = 0$.
\end{lemma}

\begin{proof}
Assuming the contrary, there exist $\eta\in Z_p(G)$ and $H_1,\ H_2\in \A(G)$ such that
$\eta_{H_1}\neq\eta_{H_2}$. From our assumption on $r$, we may find $H_3,\dots ,\ H_r\in\A(G)$ such that
$X = H_1\cap H_2\cdots\cap H_r\in\LL_r (G)$. Set $\B = \A(G)_X = \A(G_X)$. We deduce that $\beta_p(G_X)\neq 0$.
If $G_X$ is reducible, this contradicts Lemma \ref{lem:prod}. Otherwise, our second assumption is violated.
\end{proof}

\subsection{The rank 3 case}
\label{ss34}

The rank 3 exceptional groups are $H_3, G_{24}, G_{25}, G_{26}$ and $G_{27}$. Table C from \cite{OT} provides
the following information on each group:

\begin{itemize}
 \item $\pP(G) = \{2,3,5\};\ \{2,3\};\ \{2\};\ \{2,5\};\ \{2,3,5\};$
 \item $|\A(G)|$ = 15; 21; 12; 21; 45.
\end{itemize}
By Lemma \ref{lem:yuzv} and Lemma \ref{lem:graphv}(\ref{gv3}), the only
primes $p$ which might give $\beta_p(G)\neq 0$ are as follows:

$$ 3,5\ (H_3);\ 3\ (G_{24});\ 2\ (G_{25});\ -\ (G_{26});\ 3,5\ (G_{27}).$$

Using Lemma \ref{lem:frac}(\ref{f12}), we obtain
$\beta_3(H_3) = \beta_3(G_{24}) = \beta_3(G_{27}) = \beta_5(G_{27}) =0$,
and Lemma \ref{lem:frac}(\ref{f13}) gives $\beta_5(H_3) = 0$.
Finally, $\beta_2(G_{25}) = 2$, cf. \cite{PS14}. Thus, Theorem \ref{thm:b} is proved in this case. Indeed, the Hessian arrangement
$\A(G_{25})$ supports a 4-net (see e.g. \cite{Yuz04}). This implies that $\beta_2\neq 0$, by \cite{PS14}; the other implications
from Theorem \ref{thm:b}(\ref{b2}) are obvious.

Applying Lemma \ref{lem:indr} for $r = 3$ and $p = 5$, we also infer that Theorem \ref{thm:b}(\ref{b1}) holds
for all complex reflection arrangements of rank at least $3$. Thus, we only need to show that $\beta_2(G) = \beta_3(G) = 0$,
when $G$ is exceptional of rank at least 4, in order to complete the proof of Theorem \ref{thm:b}.

\subsection{The remaining cases}
\label{ss35}

The only rank 5 exceptional arrangement is $G_{33}$, for which we know from \S\ref{ss31} that all $\beta_p$ vanish.
By the computations from Section \ref{sec:fam}, the same thing happens for non-exceptional irreducible arrangements of rank 5.
Lemma \ref{lem:indr}, applied for $r = 5$, guarantees then that we may reduce our proof to the rank 4 case. Here, the list is
$G = F_4, G_{29}, H_4, G_{31}, G_{32}$, and the last two groups were treated in \S\ref{ss31}.

{\bf Case} $G = F_4$. The irreducible rank 3 types are listed in Table C from \cite{OT}: $B_3$ and $C_3$;
in both cases, the arrangement
$\A(G')$ is $\A(2,1,3)$, for which all $\beta_p$ vanish, cf. Section \ref{sec:fam}. We may conclude by resorting to
Lemma \ref{lem:indr} for $r = 3$.

{\bf Case} $G = G_{29}$. The list of irreducible rank 3 types is: $G' = A_3, B_3, G(4,4,3)$. Taking $r = 3$ and $p = 2$
in Lemma \ref{lem:indr}, we deduce from Section \ref{sec:fam} that $\beta_2(G_{29}) = 0$. Since $|\A(G_{29})| = 40$,
$\beta_3(G_{29}) = 0$, by Lemma \ref{lem:yuzv}.

{\bf Case} $G = H_4$. The irreducible types of $\LL_3(G)$ are: $G' = A_3, H_3$. Again
by Lemma \ref{lem:indr} and previous computations,
$\beta_2(H_4) = 0$. Finally, $\beta_3(H_4) = 0$, as follows from Lemma \ref{lem:frac}(\ref{f12}).

The proof of Theorem \ref{thm:b} is complete.

\section{Multinets and jump loci}
\label{sec:multi}

In this section we prove Theorems \ref{thm:a} and \ref{thm:c}.
Along the way, we establish a useful general result that relates
multinets on an arrangement to the algebraic monodromy
of its Milnor fibration and its Aomoto-Betti numbers.

\subsection{Multinets}
\label{ss41}

The work of Falk and Yuzvinsky from \cite{FY} gives, among other things,
a description of the resonance variety of an arrangement $\A$,
$\RR^1_1(\A, \C)$, in terms of {\it multinets} on the associated matroid.
A $k-$multinet on $\LL_{\leq 2}(\A)$ is a partition $\Pi$ with $k\geq 3$
non-empty blocks, $\A= \bigsqcup _{\alpha \in [k]} \A_{\alpha}$,
together with a function, $\mathfrak{m}: \A \rightarrow \Z_{> 0}$,
satisfying certain axioms. The most important is the following:

For any $H \in \A_{\alpha}$ and $H' \in \A_{\beta}$ with $\alpha \neq \beta$, and every $\gamma \in [k]$,

\begin{equation}
\label{eq:netax}
n_X  \coloneqq \sum_{K \in \A_{\gamma} \cap \A_X} \mathfrak{m}_K
\end{equation}

is independent of $\gamma$,
where $X= H \cap H' \in \LL_2(\A)$.

 We will say that a multinet is $h-reduced$ $(h \geq 1)$ if $\mathfrak{m} \equiv 1 (\bmod\; h)$.
 The usual notion of reduced multinet corresponds to $h=1$.

 We will need a result from \cite{PS14} related to axiom \eqref{eq:netax}.
 To recollect it, we start with a few notations.
 Set $H_{\A} \coloneqq H_1(M_{\A}, \Z)$ and denote by $\{a_H\}_{H \in \A}$ the distinguished $\Z$-basis. Let $S$
be  $\C \PP^1 \setminus \{k \; \text{points}\}$ and set
$H_S \coloneqq H_1 (S, \Z)=\Z-\text{span}\langle c_\alpha | \alpha \in [k]\rangle \slash \sum _{\alpha \in [k]} c_\alpha$,
where $c_{\alpha}$ is the class of a small loop in $S$ around the point $\alpha$.

Let $\cup_{\A}: \bigwedge^2 H^1(M_{\A}, \Z) \rightarrow H^2(M_{\A}, \Z)$ be the cup product.
Recalling from \cite{OS} that $H^\bullet (M_{\A}, \Z)$ has no torsion,
we denote by $\nabla_{\A}: H_2(M_{\A}, \Z) \rightarrow \bigwedge^2 H_{\A}$ the $\Z$-dual comultiplication map.

\begin{prop}[\cite{PS14}]
\label{prop:multilie}
Axiom \eqref{eq:netax} implies  that $\bigwedge^2  \phi  \circ \nabla _{\A}=0$,
where $\phi : H_{\A} \rightarrow H_S$ sends $a_H$ to $ \mathfrak{m}_H c_{\alpha}$, for $H \in A_{\alpha}$.
Therefore, $\cup_{\A} \circ \bigwedge^2 \phi ^*=0$, by taking $\Z$-duals.
\end{prop}

\subsection{Relating multinets to jump loci}
\label{ss42}

We are now ready to state our result, keeping the previous notation.

\begin{theorem}
\label{thm:d}
Assume that $\mathcal{N}=(\Pi, \mathfrak{m})$ satisfies the $k$-multinet axiom \eqref{eq:netax},
and $\mathfrak{m} \equiv 1 (\text{mod} \; k)$. Then the following hold,
for all divisors $p, d$ of $k$ with $p$ prime and $d>1$.
\begin{enumerate}
\item \label{d1} $\rho_d(\A) \in \exp\ \RR_1^1(\A, \C)$, in particular $e_d(\A)>0$.
\item \label{d2} $\beta_p(\A) \neq 0$.
\end{enumerate}
\end{theorem}

\begin{proof}
Part \eqref{d2}. Since the multinet is $k$-reduced and $p|k, \; \phi \otimes \k$ is surjective, by construction,
for $\k=\FF_p$ and $\C$. It follows from Proposition \ref{prop:multilie} that $\im(\phi \otimes \k)^* \subseteq H^1(M_{\A}, \k)$
is a $(k-1)$-dimensional subspace, isotropic with respect to the cup product.
The linear map sending each $c_{\alpha}$ to $1 \in \FF_p$ defines an element of $H^1(S, \FF_p)$,
denoted $\sigma_p(S)$. Clearly, $\phi^* (\sigma_p(S))=\sigma_p(\A)$, since $\mathcal{N}$ is in particular  $p$-reduced.
By definition \eqref{eq:defbeta}, $\beta_p(\A) \neq 0$ as claimed, since $k \geq 3$.

Part \eqref{d1}. Note that $\phi: H_{\A} \rightarrow H_S$ induces homomorphisms $\phi^*:
\mathbb{T}(S) \rightarrow \mathbb{T} (M_{\A})$ and $\phi ^* :  H^1(S, \C) \hookrightarrow H^1(M_{\A}, \C)$,
compatible with the surjective exponential maps of $S$ and $M_{\A}$.
The map sending each $c_{\alpha}$ to $\exp (\frac{2\pi \sqrt{-1}}{k})$
defines an element of the character torus, $\rho_k(S) \in \mathbb{T}(S)$. Plainly, $\phi^* (\rho_k(S))=\rho_k(\A)$,
since $\mathcal{N}$ is $k$-reduced. We also have $(\rho_k)^{ k \slash d}= \rho_d$, for both $S$ and $\A$, since
we assumed that $d$ divides $k$. Hence,
$\phi^*(\rho_d(S))=\rho_d(\A) \in \exp \; \phi^*(H^1(S, \C)) \subseteq \exp \; \RR^1_1(\A, \C)$,
where the last inclusion follows from the argument in Part \eqref{d2}.
Indeed, we know that $\phi^*(H^1(S, \C))$ is an isotropic subspace in $H^1(M_\A, \C)$, of dimension at least $2$,
and we may simply use the definition of $\RR^1_1$. The conclusion $e_d(\A) > 0$ is a direct consequence
of \eqref{eq:bmult}, since it is well-known that $\exp \; \RR_1^1(\A, \C) \subseteq \mathcal{V}^1_1 (M_\A)$;
see e.g. \cite[Theorem D]{DP} for a more general result.
\end{proof}

\subsection{Reduced multinets on complex reflection arrangements}
\label{ss43}

We begin the proof of Theorem \ref{thm:c}.

An easy preliminary remark is that the question  from \cite{DIM}
always has a positive answer, for any arrangement $\A$ of rank at most $2$. To see this, note first that the assumption
$e_d(\A) \neq 0$ is equivalent to $\rho_d \in \mathcal{V}^1_1(M_\A)$, by  \eqref{eq:bmult}. When rank$(\A) \leq 2$,
it is known that $\mathcal{V}^1_1(M_{\A})=\exp \; \RR_1^1(\A, \C)$, so the conclusion follows trivially. Consequently,
we may also suppose that the rank is at least $3$. On the other hand, $e_d(\A)>0$ and $d=p^s$ together imply,
via  \eqref{eq:bound}, that $\beta_p(\A)>0$. Therefore, $\A$ must be either the Hessian arrangement, or one
of the arrangements from Theorem \ref{thm:b}(\ref{b3}). To apply Theorem \ref{thm:d}(\ref{d1}), we need to
describe suitable multinets on these arrangements, in each case.

The Hessian arrangement supports a reduced $4$-multinet  (actually, a $4$-net). The monomial arrangement $\A(m,m,3)$
has a reduced $3$-multinet (in fact, a $3$-net), as noted in \cite{FY}.

A (non-reduced) $3$-multinet on the full monomial arrangement $\A(m,1,3)$ was constructed in \cite{FY}. It is immediate
to check that this multinet is $3$-reduced, when $m \equiv 1 (\text{mod} \; 3)$.

The last case is $\A=\A(m,m,4)$, with hyperplanes $(H_{ij^\mu}) \; z_i - \omega^\mu z_j=0$, where
$1 \leq i<j \leq 4, \; \mu \in \Z \slash m\Z$ and $\omega=\exp (\frac{2 \pi \sqrt{-1}}{m})$. We define a partition $\Pi$
with three blocks, $\{H_{ij^\mu}, H_{kh ^\nu}| \; \mu, \nu \in \Z \slash m\Z\}$, and set $\mathfrak{m} \equiv 1$ on $\A$.
It is straightforward to verify axiom \eqref{eq:netax} by using the description of $2$-flats given in  \S\ref{ss23}.
(Actually, this is a $3$-net on $\LL_{\leq 2}(\A)$.)

\subsection{Proof of Theorem \ref{thm:c} completed}
\label{ss44}

In case $\A=\A(G_{25})$, $d$ must be $2$ or $4$. We may take $k=4$ in Theorem \ref{thm:d} to obtain the desired conclusion.
The remaining cases, described in Theorem \ref{thm:b}(\ref{b3}), lead to $d=3$. Taking $k=3$, we conclude as before.
\hfill $\Box$

\subsection{Proof of Theorem \ref{thm:a}}
\label{ss45}

We may suppose that $\rank (\A) \geq 3$, since otherwise the conclusion is known (see \cite[p. 773]{MP}). By  \eqref{eq:bound} and
Theorem \ref{thm:b}, $e_p(\A)=\beta_p (\A)$, when $\A$ is not $\A(G_{25})$ or one of
the arrangements listed in Theorem \ref{thm:b}(\ref{b3}).
Moreover, we have to verify the conclusion only for $p=2$ (in case $\A(G_{25})$) or for $p=3$ (in the remaining cases).

 The equality $e_2(G_{25})=\beta_2(G_{25})=2$ is well-known (see e.g. \cite{PS14}). When $\A$ is not $\A(m,m,3)$
with $m \equiv 0\ (\bmod \; 3)$, we know that $\beta_3(\A)=1$. In these cases, we may use the $3$-reduced $3$-multinets
from \S\ref{ss43}, for $d=k=3$, exactly as in \S\ref{ss44}, to obtain that $e_3(\A)>0$. Now we are done,
since the modular bound \eqref{eq:bound} implies that $e_3(\A) \leq 1$.

The last case, $\A=\A(m,m,3)$ with $m=3n$ and $p=3$, when $\beta_3(\A)=2$, requires a more careful treatment.

The (relabeled) hyperplanes of $\A$ are
$\{H_{12^\alpha},\; H_{23^\beta}, \; H_{13^{-\gamma}}\ |\ \alpha, \beta, \gamma \in \Z \slash m\Z\}$. We recall
from \S\ref{ss23} the two types of $2$-flats: $\{H_{ij^\alpha}\ | \; \alpha \in  \Z \slash m\Z\}$ and
$\{H_{12^\alpha},\; H_{23^\beta}, \; H_{13^{-\gamma}}\}$ with $\alpha + \beta + \gamma=0$.

 We have to show that $e_3(\A) \geq 2$, in order to finish the proof. To this end, we need two reduced $3$-multinets on $\A$.
 The first one, $\mathcal{N}$, is constructed in \cite{FY}. The blocks of the partition $\Pi$ are given by
 $\{H_{ij^\alpha}\ |\;  \alpha \in  \Z \slash m\Z\}_{1 \leq i< j \leq 3}$.
 The blocks of the second one, $\mathcal{N}'$, are defined by
$\{H_{ij^\alpha}\ | \; 1 \leq i< j \leq 3, \alpha \equiv \tau (\bmod \;3)\}_{\tau \in \FF_{3}}$,
with $\alpha$ replaced by $- \alpha$ when $\{i,j \}=\{1,3 \}$.

 Let us check  \eqref{eq:netax} for $\mathcal{N}'$. The $2$-flats $X$ appearing in the multinet axiom clearly
 coincide with those $\A_X$ that contain two hyperplanes with different colours with respect to $\Pi '$.
 For $X=\{H_{ij^\alpha}| \;\alpha \in  \Z \slash m\Z \}$ we find that $n_X=n$.
For $X=\{H_{12^\alpha}, \; H_{23^\beta}, \; H_{13^{-\gamma}}\}$ with $\alpha + \beta + \gamma=0$,
the condition on colours translates to  $\alpha \nequiv \beta \nequiv \gamma (\bmod\ 3)$,
and implies that $n_X=1$. Hence $\Pi'$ defines a reduced $3$-multinet $\mathcal{N}'$.

Consider the two (surjective)  homomorphisms from Proposition \ref{prop:multilie}, $\phi, \phi' : H_\A \rightarrow H_S$.
Clearly, $\phi^* \mathbb{T}(S)= \exp \; \phi^* H^1(S, \C)$
is a positive dimensional subtorus of $\mathbb{T} (M_\A)$, and similarly for $\phi'$. Moreover,
$\phi^* \mathbb{T}(S) \subseteq \mathcal{V}_1^1(M_\A)$, since  $\phi^* H^1(S, \C) \subseteq H^1(M_\A, \C)$ is
isotropic of dimension $2$, hence contained in $\RR_1^1(\A, \C)$, and likewise for $\phi'$. Therefore,
we may find two irreducible components of $\mathcal{V}^1_1(M_\A)$, $W$ and $W'$, such that
$\phi^*\mathbb{T}(S) \subseteq W$ and $\phi '^*\mathbb{T}(S) \subseteq W'$. On the other hand,
$\rho_3 (\A) \in W \cap W'$, by the argument from the proof of Theorem \ref{thm:d}(\ref{d1}).

In this situation, it follows from a result of Artal Bartolo, Cogolludo and Matei
\cite[Proposition 6.9]{ACM} that $\rho_3(\A) \in \mathcal{V}_2^1(M_\A)$, if $W \neq W'$.
Hence, $e_3(\A) \geq 2$, by  \eqref{eq:bmult}, and we are done.

Suppose then that $W=W'$. We know from \cite{FY} that actually $W=\phi ^*\mathbb{T}(S)$, since $\phi$ comes from a $3$-net.
Taking tangent spaces at the origin $1 \in\mathbb{T}(M_\A)$, we infer that $\phi'^* H^1(S, \C) \subseteq \phi^* H^1(S, \C)$.

We identify $H^1(M_\A, \C)$ with $\C^{\A}$, using the distinguished $\Z$-basis.
In this way, the subspace $\phi^* H^1(S, \C)$ (respectively $\phi'^* H^1(S, \C)$) is identified with
the subset of those elements $\eta \in \C^{\A}$ (respectively $\eta' \in \C^{\A}$)
taking the constant values $a,b,c$ (respectively $a', b', c'$) on the blocks of $\Pi$ (respectively $\Pi'$),
where $a+b+c=a'+b'+c'=0$.
Now, it is an easy matter to check that $\phi^* H^1(S, \C) \cap \phi'^* H^1(S, \C)=0$.
This contradiction finishes the proof of Theorem \ref{thm:a}.
\hfill $\Box$

\subsection{Full monodromy action}
\label{ss46}

It follows from  \eqref{eq:monintro} that the characteristic polynomial
$\Delta_\A(t)=(t-1)^{|\A|-1} \Pi_{1< d|n} \Phi _d(t)^{e_d(\A)}$
encodes the full monodromy action on $H_1(F_\A, \Q)$.

The approach via modular bounds works only for prime power monodromy multiplicities, $e_{p^s}(\A)$.
One way to avoid this inconvenience is to impose restrictions on multiplicities of $2$-flats,
like in \cite{PS14} for instance, to arrive at full monodromy computations.
Unfortunately, as we saw in \S\ref{ss23}, arbitrarily high flat multiplicities
may appear for non-exceptional complex reflection arrangements.

Even in this kind of situation, there is hope related to the following well-known vanishing  criterion (see e.g. \cite{MP}):
if $d \notdivides |\A_X|$ for any $X \in\LL_2(\A)$, then $e_d(\A)=0$.
It turns out that this works for full monomial arrangements of small rank.
The result below verifies in particular the strong form of the conjecture from \cite{PS14}.

\begin{prop}
\label{prop:full}
For $\A=\A(m,1,l)$, with $l=3$ or $4$, $\Delta_{\A}(t)=(t-1)^{|\A|-1}(t^2+t+1)$,
if $l=3$ and $m \equiv 1 (\text{mod} \; 3)$,
and $\Delta_{\A}(t)=(t-1)^{|\A|-1}$, otherwise.
\end{prop}

\begin{proof}
We have to compute $e_d(\A)$ for all divisors $d>1$ of $|\A|$. If $d$ is prime,
this was done in Theorem \ref{thm:a} and Theorem \ref{thm:b}.
It follows from \S\ref{ss23} that the $2$-flats of $\A$ have multiplicities
$2,3$ or $m+2, |\A| = 3(m+1)$ for $l=3$ and $|\A| = 2(3m+2)$ for $l=4$.
If $d$ is not prime and $e_d(\A) \neq 0$, the vanishing criterion forces $m \equiv -2 \; (\bmod\ d)$.
Writing that $|\A| \equiv 0 \;(\bmod \; d)$, we obtain for $l=3$ that $3 \equiv 0 \;(\bmod\;  d)$,
a contradiction, and $8 \equiv 0 \;(\bmod\;  d)$, for $l=4$.
In the second case, the modular bound implies that $\beta_2(\A)>0$, contradicting Theorem \ref{thm:b}(\ref{b2}).
\end{proof}

\begin{ack}
We are grateful to Alex Suciu for helpful discussions.
The second author thanks Universit\' e de Nice Sophia Antipolis, where
this work was completed, for support and hospitality.
\end{ack}

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{\texttt{\href{http://arxiv.org/abs/#1}{arxiv:#1}}}
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{\texttt{\href{http://arxiv.org/abs/#1}{arXiv:}}
\texttt{\href{http://arxiv.org/abs/#1}{#1}}}
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{\texttt{\href{http://dx.doi.org/#1}{doi:#1}}}

config: arxiv


TITLE: Error in Data Partitions with refinement property
QUESTION [1 upvotes]: I'm considering these data in my set:

A
B
C
D

1
a
\$
F

1
A
-
T

2
A
\$
D

2
A
\$
F

2
b
-
L

3
b
\$
O

3
C
-
F

3
C
#
R

for each attribute A,B,C, and D it is possible to define the following data partitions:
$\operatorname{Part}(A) = \{\{1, 2\}, \{3, 4, 5\}, \{6, 7, 8\}\}$
$\operatorname{Part}(B) = \{\{1\}, \{2, 3, 4\}, \{5, 6\}, \{7,8\}\}$
$\operatorname{Part}(C) = \{\{1, 3, 4, 6\}, \{2, 5, 7\}, \{8\}\}$
$\operatorname{Part}(D) = \{\{1, 4, 7\}, \{2\}, \{3\}, \{5\}, \{6\}, \{8\}\}$
$\operatorname{Part}(AB) = \{\{3, 4\}, \{1\}, \{2\}, \{5\}, \{6\}, \{7, 8\}\}$
$\operatorname{Part}(ABC) = \{\{3, 4\}, \{1\}, \{2\}, \{5\}, \{6\}, \{7\}, \{8\}\}$
$\operatorname{Part}(ABCD) = \{\{1\}, \{2\}, \{3\}, \{4\}, \{5\}, \{6\}, \{7\}, \{8\}\}$
etc.
I've this formula that calculate the error in a partition:
$err(Part(X)) = ||Part(X)|| - |Part(X)|$
where:

$||Part(X)||$ is the total number of element in a set (in the example
is 8)
$|Part(X)|$ is the number of subset in each partition (for
example in part(A) is 3 and in part(B) is 4)

and in example the error values are:
$\operatorname{err(Part(A))} = 8 - 3 = 5$
$\operatorname{err(Part(B))} = 8 - 4 = 4$
$\operatorname{err(Part(C))} = 8 - 3 = 5$
$\operatorname{err(Part(D))} = 8 - 6 = 2$
$\operatorname{err(Part(AB))} = 8 - 6 = 2$
$\operatorname{err(Part(ABC))} = 8 - 7 = 1$
$\operatorname{err(Part(ABCD))} = 8 - 8 = 0$
The main problem is that when considering the partition of multiple attributes, e.g., AB, ABC, or ABCD, for calculating $Part(X)$ it is necessary to calculate the intersection of all the partitions for the attribute involved in X.
Is there a way to define the value of $f(x)$ in:
$\operatorname{err(Part(AB))} = 8 - f(x) $
only knowing the total number of elements (i.e. 8) and the error $\operatorname{err(Part(A))}$ and $\operatorname{err(Part(B))}$?
In other words, is there a way to find a correlation between $\operatorname{err(Part(A))}$, $\operatorname{err(Part(B))}$ and  $\operatorname{err(Part(AB))}$?

REPLY [0 votes]: It is not possible to calculate $\mathrm{err}(\mathrm{Part}(AB))$ from $\mathrm{err}(\mathrm{Part}(A))$ and $\mathrm{err}(\mathrm{Part}(B))$ alone.
For example, take the date set $X_1$:

A
B

1
a

1
A

2
A

2
A

and the data set $X_2$:

A
B

1
a

1
a

2
A

2
A

You can see that, in both cases, $\mathrm{err}(\mathrm{Part}(A)) = \mathrm{err}(\mathrm{Part}(B)) = 2$, however:

For data set $X_1$, $\mathrm{err}(\mathrm{Part}(AB)) = 1$
For data set $X_2$, $\mathrm{err}(\mathrm{Part}(AB)) = 2$.


TITLE: Subset of the plane that intersects every line exactly twice
QUESTION [12 upvotes]: In a comment to this question, Tim Gowers remarked that using the axiom of choice, one can show that there exists a subset of the plane that intersects every line exactly twice (although it has yet to be shown that choice is required).
Unfortunately, I haven't been able to track down a reference, so if someone could link me to the original result (or provide a short proof) that would be great.  I have roughly worked out a proof for myself, and I'd like to check it against the literature.

REPLY [3 votes]: Here is a minor variation of the proof.
We show that there is a subset $Z$ of $\mathbb R^2$ which intersects each line exactly twice.
We define the ordinals in such a way that each ordinal $a$ is the set of those ordinals $< a$. For any set $S$ we write $|S|$ for the least ordinal equipotent to $S$, and call it the cardinality of $S$. Let $c$ (for continuum) be the cardinality of $\mathbb R$. For any subset $S$ of $\mathbb R^2$ we denote by $\langle S\rangle$ the set of lines generated by $S$ (a line being generated by $S$ if it has at least two points in common with $S$). Let $d\mapsto L_d$ be a bijection from $c$ onto the set of lines in $\mathbb R^2$.
For each $d\in c$ we define $Z_d\subset\mathbb R^2$, $f(d)\in c$, and $z_d\in\mathbb R^2$, as follows. Let $z_0$ be any point of $L_0$, put $f(0)=0$, and let $Z_0$ be the empty set. Now assume $0< d< c$, and $f(e), z_e$ already defined for $e < d$. Put $Z_d:=$ {$z_e\ |\ e< d$}. As $|\langle Z_d\rangle|< c$ (because $|Z_d|< c$), there is a least $f(d)$ in $c$ such that $L_{f(d)}\notin\langle Z_d\rangle$. Let $z_d$ be any point of the set
$$L_{f(d)}-(Z_d\cup(\cup\langle Z_d\rangle)),$$
which is easily seen to be nonempty.
Let $Z$ be the union of the $Z_d$ and $L$ any line in $\mathbb R^2$. We claim $|L \cap Z|=2$, that is, $Z$ is the sought-for subset of $\mathbb R^2$. To prove $|L\cap Z|\le2\ (*)$ we assume by contradiction that there are $g< h< i$ in $c$ such that $z_g,z_h,z_i\in L\cap Z$. We have $z_g,z_h\in Z_i$ by definition of $Z_i$, and thus $L\in\langle Z_i\rangle$, contradicting the definition of $z_i$. To prove $|L\cap Z|\ge2$ we assume by contradiction $|L\cap Z|< 2$. Put $L=L_d$ and let $g$ be in $c$. The inequality $|L_d\cap Z_g|<2$ implies $L_d\notin\langle Z_g\rangle$, and thus $f(g)\le d$ by minimality of $f(g)$. This shows that $Z$ is contained into the union of the $L_e$ such that $e\le d$. As $|L_e\cap Z|\le2$ by $(*)$, we get $|Z|\le2|d|+2< c$, contradicting the obvious equality $|Z|=c$.
A pdf version is available here (Wayback Machine).


TITLE: Mapping the reals to the naturals
QUESTION [0 upvotes]: Where does this argument fail? (my opinion; the conclusion is a non sequitur)
If you just map the reals to the naturals in a mirror image of each other you will eventually have all the reals. Like this.
1 -> 0.1
2 -> 0.2
3 -> 0.3
...
9 -> 0.9
10 -> 0.01
11 -> 0.11
12 -> 0.21
...
99 -> 0.99
100 -> 0.001
101 -> 0.101
102 -> 0.201
...
...999 -> 0.999...
If you keep going forever you will eventually have all the real numbers mapped.

REPLY [0 votes]: Consider then the interval $[0,1]$. By this argument, every natural number can be extracted from some real number in this interval. So you've shown an injection $f: \mathbb{N} \rightarrow [0,1]$, i.e. $|\mathbb{N}| \leq |[0,1]|=\mathfrak{c}$, i.e. you haven't really shown $|\mathbb{N}|> \mathfrak{c}$.