text:
\section{Introduction}

Let $X$ be a complex surface with a rational singularity at a point $O\in X$
and ${\mathcal O}_{X,O}$ its corresponding local ring.  Let $\mathfrak{a} \subseteq {\mathcal O}_{X,O}$ be an ${\mathfrak m}$-primary
ideal where ${\mathfrak m}={\mathfrak m}_{X,O}$ is the maximal ideal of ${\mathcal O}_{X,O}$. Then, for any real exponent $c>0$,
we may consider its corresponding \emph{multiplier ideal} $\mathcal{J}(\mathfrak{a}^c)$. Indeed, the multiplier ideals form a discrete nested sequence
\begin{equation} \label{sequence_mult}
\mathcal{O}_{X,O}\varsupsetneq\mathcal{J}(\mathfrak{a}^{\lambda_1})\varsupsetneq\mathcal{J}(\mathfrak{a}^{\lambda_2})\varsupsetneq\ldots\varsupsetneq\mathcal{J}(\mathfrak{a}^{\lambda_i})\varsupsetneq\ldots
\end{equation}
indexed by an increasing sequence of rational numbers $0 < \lambda_1 < \lambda_2 < \ldots$ such that
$\mathcal{J}(\mathfrak{a}^{\lambda_i})=\mathcal{J}(\mathfrak{a}^c)\varsupsetneq\mathcal{J}(\mathfrak{a}^{\lambda_{i+1}})$ for any $c \in \left[\lambda_i,\lambda_{i+1}\right)$. The
$\lambda_i$ are the so-called {\em jumping numbers} of the ideal $\mathfrak{a}$.
Ein, Lazarsfeld, Smith and Varolin \cite{ELSV04}, using the fact that the multiplier ideals are ${\mathfrak m}$-primary as well,
defined the {\em multiplicity} of a point $c \in \mathbb{R}$ as
\begin{equation} \label{eq-def-mult-general}
m\left(c\right) := \dim_{\mathbb{C}}\frac{\mathcal{J}\left(\mathfrak{a}^{c-\varepsilon}\right)}{\mathcal{J}\left(\mathfrak{a}^c\right)}
\end{equation}
for $\varepsilon>0$ small enough. With this definition, it is clear that $c$ is a jumping number if and only if $m\left(c\right)>0$.
A way to encode the information provided by the filtration of ideals (\ref{sequence_mult})
is by means of its {\em Poincar\'e series of multiplier ideals}
\begin{equation} \label{eq-def-Poinc}
P_\mathfrak{a} (t)= \sum_{c\in \mathbb{R}_{>0}} m(c) \hskip 1mm t^{c}
\end{equation}
introduced by Galindo and Montserrat in \cite{GM10}. Actually, they proved that this series is a rational function
when $X$ is smooth and $\mathfrak{a}$ is simple. In \cite{ACAMDCGA13} we gave a systematic study of multiplicities and
proved the rationality of the Poincar\'e series for any ideal $\mathfrak{a}$ in a complex surface with a rational singularity at $O$.

\vskip 2mm

Whenever we extend to the case of {\it mixed multiplier ideals}
$\mathcal{J}\left(\pmb{\mathfrak{a}}^{\mathbf{c}}\right):=\mathcal{J}\left(\mathfrak{a}_1^{c_1}\cdots\mathfrak{a}_r^{c_r}\right)$ associated to a tuple of
${\mathfrak m}$-primary ideals $\pmb{\mathfrak{a}}:=(\mathfrak{a}_1,\ldots,\mathfrak{a}_r )\subseteq ({\mathcal O}_{X,O})^r$ and a point $\mathbf{c}:=(c_1,\ldots,c_r)$  in the positive
orthant  $\mathbb{R}_{\geqslant0}^r$, things become a little bit more trickier. Instead of having a partition of the positive real line into
intervals defined by the jumping numbers  where the multiplier ideals are constant, we get a partition of the positive real orthant
into {\it constancy regions} whose boundary is described by the so-called {\it jumping walls}.  We point out that only a few results on mixed
 multiplier ideals are available in the literature.
Libgober and Musta\c{t}\v{a} \cite{LM11} studied  properties of the {\it log-canonical wall}, i.e. the jumping wall
associated to $\pmb{\lambda}_0=(0,\ldots,0)$.
Naie in \cite{Nai13}  describes a nice property that jumping walls must satisfy.
Cassou-Nogu\`es and Libgober study in \cite{CNL11,CNL14}  mixed multiplier ideals and jumping
walls associated to germs of plane curves, under the analogous notion of {\it ideals of quasi-adjunction} and {\it faces of quasi-adjunction} (see \cite{Lib02}).

\vskip 2mm

We may define the multiplicity
of any given point ${\pmb{c}} \in \mathbb{R}_{\geqslant0}^r$  and we say that it is a {\it jumping point}, if and only if $m\left({\pmb{c}}\right)>0$.
In particular, jumping points  lie on  jumping walls. The most natural generalization of a Poincar\'e series for mixed multiplier ideals is to consider
a filtration of ideals $\mathcal{J}\left(\pmb{\mathfrak{a}}^{\mathbf{c}}\right)$ indexed by points over a ray $L$ in $\mathbb{R}_{\geqslant0}^r$ with direction vector in ${\mathbb{Q}}^r_{\geqslant0}$ and set
\begin{equation} \label{eq-def-Poinc2}
P_{\pmb{\mathfrak{a}}} (\underline{t}; L)= \sum_{{\pmb{c}} \in  L} m({\pmb{c}}) \hskip 1mm \underline{t}^{{\pmb{c}}}
\end{equation}
where $\underline{t}^{{\pmb{c}}}:=t_1^{c_1}\cdots t_r^{c_r}$.
We have to mention that the multiplicity
is sensitive to small perturbation of a given point. Indeed, if we consider a sequence of mixed multiplier ideals
 $$\mathcal{J} \left(\pmb{\mathfrak{a}}^{{{\pmb{c}}_0}}\right) \supsetneq \cdots \supsetneq \mathcal{J} \left(\pmb{\mathfrak{a}}^{{\pmb{c}}_{i-1}}\right) \supsetneq
 \mathcal{J} \left(\pmb{\mathfrak{a}}^{{\pmb{c}}_{i}}\right)  \supsetneq   \mathcal{J} \left(\pmb{\mathfrak{a}}^{{\pmb{c}}_{i+1}}\right) \supsetneq  \cdots $$
 indexed by jumping points $\{{\pmb{c}}_i\}_{i\geqslant 0}$ over a ray $L$ and perturb minimally this ray, for example taking a
 parallel ray $L'$ that is close enough,
 then the sequence of mixed multiplier ideals indexed
 by jumping points in $L'$ may vary (see Example \ref{rays}) and thus, the corresponding Poicar\'e series also varies.

\vskip 2mm

The organization of the paper is as follows. In Section \ref{Sec1} we recall all the basics on mixed multiplier ideals.
In Section \ref{multiplicities} we extend the results of \cite{ACAMDCGA13} to this setting. Namely, we make a systematic study of
the multiplicities of points in the positive orthant. The main result is Theorem \ref{thm:multiplicity1_cMMI} where we
give a precise formula for the multiplicity. We also prove in Theorem \ref{thm:rational} that the Poincar\'e series associated to a ray is a
rational function. In Section \ref{perturbations} we study the variation of the multiplicity of a jumping point by
small perturbations. We prove that this multiplicity does not vary for points in the interior (with the Euclidean
topology) of a ${\mathcal C}$-facet (see Proposition \ref{interior}) but it does so in a controlled way at the intersection of ${\mathcal C}$-facets
(see Theorem \ref{intersection}). In Section \ref{Sec4} we study the exceptional divisors that contribute to the log-canonical wall.
Our main result is Theorem \ref{thm:Pi} where we establish (except for a very particular case considered in Proposition \ref{lem:2rdiv}) a one-to-one correspondence between the facets of the log-canonical
wall and the exceptional divisors of the so-called {\it Newton nest} (see Definition \ref{Newton}) generalizing
a result of Cassou-Nogu\`es and Libgober \cite[Theorem 4.22]{CNL14}.

\section{Mixed multiplier ideals} \label{Sec1}

Let $X$ be a complex surface with at most a rational singularity at a point $O \in X$ (see Artin
\cite{Art66} and  Lipman \cite{Lip69} for details) and ${\mathfrak m} ={\mathfrak m}_{X,O} $ be the maximal ideal of the local ring ${\mathcal O}_{X,O}$ at $O$.
Given a tuple of ${\mathfrak m}$-primary ideals $\pmb{\mathfrak{a}}=(\mathfrak{a}_1,\ldots,\mathfrak{a}_r )\subseteq ({\mathcal O}_{X,O})^r$ we will
consider a common {\em log-resolution}, that is, a birational
morphism $\pi: X' \rightarrow X$ such that $X'$ is smooth, $\mathfrak{a}_i\cdot{\mathcal O}_{X'} = {\mathcal O}_{X'}\left(-F_i\right)$ for some effective
Cartier divisors $F_i$,  $i=1,\dots , r$ and $\sum_{i=1}^r F_i+E$ is a divisor with simple normal crossings, where $E = Exc\left(\pi\right)$ is the exceptional locus. Actually, the divisors $F_i$ are supported on the exceptional locus since the ideals are ${\mathfrak m}$-primary.
The {\it fundamental cycle} is the
unique smallest non-zero effective divisor $Z$
(with exceptional support) such that
$Z\cdot E_i \leqslant 0$ {for every }  $i=1,\dots, r.$
The fundamental cycle satisfies
${\mathfrak m}\cdot{\mathcal O}_{X'} = {\mathcal O}_{X'}\left(-Z\right)$ (see \cite[Theorem 4]{Art66}). We point out that any effective divisor with integer coefficients
$\widetilde{D}$ is called
{\it antinef}  if $\widetilde{D}\cdot E_i \leqslant 0$ for every $i=1,\dots, r.$ Indeed, for any effective divisor
$D$ there exists a unique minimal antinef divisor $\widetilde{D}$ satisfying $D\leqslant \widetilde{D}$
that is called the {\it antinef closure} of $D$.
It can be computed using an inductive procedure called {\it unloading} (see \cite{Reg96} and \cite{ACAMDC16} for details).

\vskip 2mm

Since the point $O$ has a rational singularity, the
exceptional locus $E$ is a tree of smooth rational curves
$E_1,\dots,E_s$. Moreover,  the matrix of
intersections $\left(E_i\cdot E_j\right)_{1\leqslant i,j \leqslant
s}$ is negative-definite. For any exceptional component $E_j$, we define the {\em excess} of
$\mathfrak{a}_i$ at $E_j$ as $\rho_{i,j} = - F_i \cdot E_j$.  We also recall the following notions:
\begin{enumerate}
\item[$\cdot$] A component $E_j$ of $E$ is a {\em rupture} component if
it intersects at least three more components of $E$ (different from
$E_j$).

\item[$\cdot$] We say that $E_j$ is {\em dicritical} if $\rho_{i,j} > 0$ for some $i$.
Such components  correspond to {\it Rees valuations}
(see \cite{Lip69}).
\end{enumerate}

\vskip 2mm

We define the {\it mixed multiplier ideal}  at a point
$\mathbf{c}:=(c_1,..,c_r) \in \mathbb{R}_{\geqslant0}^r$
as\footnote{By an abuse of notation, we will also denote
$\mathcal{J}\left(\pmb{\mathfrak{a}}^{{\pmb{c}}}\right)$ its stalk at $O$ so we will omit
the word ''sheaf'' if no confusion arises.}
\begin{equation} \label{eq:mmi}
\mathcal{J}\left(\pmb{\mathfrak{a}}^{\mathbf{c}}\right):=\mathcal{J}\left(\mathfrak{a}_1^{c_1}\cdots\mathfrak{a}_r^{c_r}\right)
= \pi_*{\mathcal O}_{X'}\left(\left\lceil K_{\pi} - c_1 F_1- \cdots -
c_r F_r \right\rceil\right)
\end{equation}

\noindent where  $\lceil \cdot \rceil$ denotes the
 {\em round-up} and the {\it relative canonical divisor} $$ K_{\pi}=
\sum_{i=1}^s k_j E_j $$ is the ${\mathbb{Q}}$-divisor on $X'$ supported on the
exceptional locus $E$  characterized by the property $\left(K_{\pi}+E_i\right)\cdot E_i  = -2$
for every exceptional component $E_j$, $j=1,\dots,s$. We say that $X$ is log-canonical (resp. log-terminal)
at $O$ if $k_j\geqslant -1$ ( resp. $k_j > -1$)  $ \forall j$.

\vskip 2mm

Associated to any point ${\pmb{c}}\in \mathbb{R}_{\geqslant0}^r$, we consider:

\vskip 2mm

$\cdot$ The {\it region} of ${\pmb{c}} $: \hskip 21mm
$\mathcal{R}_{\pmb{\mathfrak{a}}}\left({\pmb{c}}\right)=\left\{{\pmb{c}}' \in
\mathbb{R}_{\geqslant 0}^r \hskip 2mm \left|
 \hskip 2mm \mathcal{J}\left(\pmb{\mathfrak{a}}^{{\pmb{c}}'}\right)\supseteq\mathcal{J}\left(\pmb{\mathfrak{a}}^{{\pmb{c}}}\right)\right\}\,\right.$

 \vskip 2mm

$\cdot$ The {\it constancy region} of ${\pmb{c}}$: \hskip 3mm
$\mathcal{C}_{\pmb{\mathfrak{a}}}\left({\pmb{c}}\right)=\left\{{\pmb{c}}'  \in
\mathbb{R}_{\geqslant 0}^r \hskip 2mm \left| \hskip 2mm
\mathcal{J}\left(\pmb{\mathfrak{a}}^{{\pmb{c}}'}\right)=\mathcal{J}\left(\pmb{\mathfrak{a}}^{{\pmb{c}}}\right)\right\}\,\right.$

\vskip 2mm

\noindent The boundary of the region ${\mathcal R}_{\pmb{\mathfrak{a}}}({\pmb{c}})$ is what we call  the {\it jumping wall} associated to ${\pmb{c}}$. One usually
refers to the jumping wall of the origin as the {\it log-canonical wall}. It follows from the definition of mixed multiplier ideals that the
jumping walls must lie on {\it supporting hyperplanes} of the form
\begin{equation} \label{eq:hyperplanes}
 V_{j,\ell}: \hskip 1mm e_{1,j} z_1+ \cdots +  e_{r,j} z_r= \ell + k_j \,,
 \hskip 8mm j=1,\dots,s
\end{equation} for a suitable $\ell \in {\mathbb{Z}}_{>0}$. Here we assume that the effective divisors $F_i$ such that
$\mathfrak{a}_i\cdot{\mathcal O}_{X'} = {\mathcal O}_{X'}\left(-F_i\right)$, for  $ i=1,\dots,r$, are of the form  $F_i=\sum_{j=1}^s e_{i,j} E_j$.
Notice that each supporting hyperplane $V_{j,\ell}$ is associated to an exceptional component $E_j$. Indeed, we may find other
exceptional components associated to the same hyperplane, that is, we may find  $E_i$  and $\ell' \in {\mathbb{Z}}_{>0}$ such that
$V_{j,\ell} = V_{i,\ell'}$.

\vskip 2mm

It is proved in  \cite[Theorem 3.3]{ACAMDC16}  that
the region ${\mathcal R}_{\pmb{\mathfrak{a}}}({\pmb{c}})$ is  (the interior of) a {\it rational convex polytope} defined  by  the inequalities
\[e_{1,j} z_1 + \cdots +  e_{r,j} z_r < k_j + 1 + e_j^{{{\pmb{c}}}}\,,  \hskip 8mm j=1,\dots,s\]
corresponding to either rupture or dicritical divisors $E_j$ and  $D_{{{\pmb{c}}}}=\sum e_j^{{{\pmb{c}}}} E_j$ is the
{ antinef closure} of $\left\lfloor c_1 F_1 + \cdots +
c_rF_r  - K_{\pi} \right\rfloor$.

The intersection of the boundary of a connected component of a constancy region ${\mathcal C}_{\pmb{\mathfrak{a}}}({\pmb{c}})$ with a supporting hyperplane of ${\mathcal R}_{\pmb{\mathfrak{a}}}({\pmb{c}})$ is what we call  a ${\mathcal C}$-facet of ${\mathcal C}_{\pmb{\mathfrak{a}}}({\pmb{c}})$. Every facet of a jumping wall decomposes into
several ${\mathcal C}$-facets associated to different mixed multiplier ideals.

\vskip 2mm

The main result of \cite{ACAMDC16} is an algorithm to compute all the constancy regions, and their corresponding mixed multiplier ideals, in any desired range of the positive orthant $\mathbb{R}_{\geqslant0}^r$. In particular the set of jumping walls of $\pmb{\mathfrak{a}}$, that we will
denote from now on as {\rm $ {\bf JW}_{\pmb{\mathfrak{a}}}$}, is precisely described.
The points on the jumping walls,
which we will denote with  $\pmb{\lambda}$ when we want to emphasize this fact, satisfy the property  $\mathcal{J}\left(\pmb{\mathfrak{a}}^{{\pmb{c}}}\right)\varsupsetneq\mathcal{J}\left(\pmb{\mathfrak{a}}^{\pmb{\lambda}}\right)$
for all ${\pmb{c}} \in \{\pmb{\lambda} - \mathbb{R}_{\geqslant0}^r\} \cap B_{\varepsilon}(\pmb{\lambda})$ and $\varepsilon >0$ small enough.
In the sequel, we will refer to these points as  the {\it jumping points} of  the tuple of ideals $\pmb{\mathfrak{a}}$.

\section{Multiplicities of jumping points} \label{multiplicities}
In this section we are going to provide a systematic study of the
multiplicity of any point ${\pmb{c}}\in\mathbb{R}^r_{\geqslant0}$. The results
that we present are a natural generalization of the ones we
obtained in  \cite{ACAMDCGA13}.

\begin{definition}
Let $\pmb{\mathfrak{a}}=\left(\mathfrak{a}_1,\ldots,\mathfrak{a}_r\right)\subseteq (\mathcal{O}_{X,O})^r$ be a tuple of ${\mathfrak m}$-primary ideals. We define the
multiplicity attached to a point ${\pmb{c}}\in\mathbb{R}^r_{\geqslant0}$ as the codimension
of $\mathcal{J} \left(\pmb{\mathfrak{a}}^{{\pmb{c}}}\right)$ in $\mathcal{J} \left(\pmb{\mathfrak{a}}^{(1-\varepsilon){\pmb{c}}}\right)$ for
$\varepsilon>0$ small enough, i.e.

\[m({\pmb{c}}):= \dim_{\mathbb{C}} \frac{\mathcal{J} \left(\pmb{\mathfrak{a}}^{(1-\varepsilon){\pmb{c}}}\right)}{\mathcal{J} \left(\pmb{\mathfrak{a}}^{{\pmb{c}}}\right)}\,.\]
\end{definition}

Our first goal is to compute explicitly these multiplicities using the theory of  {\it jumping divisors}  in this mixed multiplier ideals
setting as considered in \cite{ACAMDC16}. Since we are dealing with any general point, it will be more convenient
to consider  the  notion of {\it maximal jumping divisors} as opposed to the minimal jumping divisors, which are only meaningful for
jumping points.

\begin{definition}
Let $\pmb{\mathfrak{a}}=\left(\mathfrak{a}_1,\ldots,\mathfrak{a}_r\right)\subseteq
\left({\mathcal O}_{X,O}\right)^r$ be a tuple of ideals. Given any point ${\pmb{c}} \in \mathbb{R}^r_{\geqslant0}$, we define its {\it maximal jumping divisor} as the reduced divisor $H_{{\pmb{c}}} \leqslant
\sum_{i=1}^r F_i$ supported on those components $E_j$ such that
\[c_1 e_{1,j}  + \cdots +  c_ r e_{r,j} - k_j \in {\mathbb{Z}}_{>0}\,.\]
Equivalently, for a sufficiently small $\varepsilon>0$,
\[H_{{\pmb{c}}}= \lceil K_\pi- (1-\varepsilon)c_1 F_1-\cdots- (1-\varepsilon)c_r  F_r\rceil -
\lceil K_\pi- c_1F_1-\cdots-c_rF_r\rceil\,.\]
In particular, we have
\[\mathcal{J}({\pmb{\mathfrak{a}}}^{(1-\varepsilon){\pmb{c}} })= \pi_{*}\mathcal{O}_{X'}(\lceil K_\pi-  c_1 F_1 - \cdots - c_r F_r \rceil + H_{{\pmb{c}}})\,.\]
\end{definition}

\vskip 2mm

The following numerical properties for  maximal jumping divisors will be useful for our purposes.   We will skip the details of the proof just because
it is a natural generalization of \cite[Proposition 3.6]{ACAMDCGA13} and  the same proof holds mutatis mutandi.

\begin{proposition} \label{prop:Num_condition_cMMI}
Fix any $\pmb{c}\in \mathbb{R}_{\geqslant0}^r$, and let $H_{\pmb{c}}$ be its associated maximal jumping divisor. Then the following inequalities hold:
\begin{itemize}
\item $\left(\left\lceil K_{\pi}- c_1 F_1-\cdots- c_r F_r\right\rceil + H_{\pmb{c}}\right)\cdot E_i \geqslant -1$ for all $E_i\leqslant H_{\pmb{c}}$, and
\item $\left(\left\lceil K_{\pi}- c_1 F_1-\cdots- c_r F_r\right\rceil + H_{\pmb{c}}\right)\cdot H \geqslant -1$ for any connected component $H\leqslant H_{\pmb{c}}$.
\end{itemize}
\end{proposition}

The main result of this section is the following:

\begin{theorem}\label{thm:multiplicity1_cMMI}
Let $\pmb{\mathfrak{a}}=\left(\mathfrak{a}_1,\ldots,\mathfrak{a}_r\right) \subseteq \left({\mathcal O}_{X,O}\right)^r$ be a tuple of ${\mathfrak m}$-primary ideals and $H_{{\pmb{c}}}$
the maximal jumping divisor associated to some ${\pmb{c}} \in \mathbb{R}_{\geqslant0}^r$. Then,
\begin{align*}
m\left({\pmb{c}}\right) & = \left(\left\lceil K_{\pi}-c_1F_1-\cdots-c_rF_r\right\rceil +
H_{\pmb{c}}\right)\cdot H_{\pmb{c}} +\#\left\{\text{connected components of }
H_{\pmb{c}}\right\}.
\end{align*}
\end{theorem}

\begin{proof}
Given the short exact sequence
\begin{align*}
 &0 \longrightarrow \mathcal{O}_{X'}\left(\left\lceil K_\pi-c_1F_1-\cdots -c_rF_r\right\rceil\right)
\longrightarrow \mathcal{O}_{X'}\left(\left\lceil K_\pi-c_1F_1-\cdots -c_rF_r\right\rceil + H_{{\pmb{c}}}\right)\longrightarrow\\
 &\qquad\qquad \longrightarrow \mathcal{O}_{H_{{\pmb{c}}}}\left(\left\lceil K_\pi-c_1F_1-\cdots -c_rF_r\right\rceil+H_{{\pmb{c}}}\right) \longrightarrow 0
\end{align*}
we have, after pushing it forward to $X$ and applying local vanishing \cite{Laz04} for the case of mixed multiplier ideals
\begin{multline*}
0 \longrightarrow\mathcal{J}(\pmb{\mathfrak{a}}^{\pmb{c}}) \longrightarrow\mathcal{J}(\pmb{\mathfrak{a}}^{(1-\varepsilon) {\pmb{c}}})\longrightarrow\\ \longrightarrow H^0\left(H_{{\pmb{c}}},\mathcal{O}_{H_{{\pmb{c}}}}\left(\left\lceil K_\pi-c_1F_1-\cdots -c_rF_r\right\rceil+H_{{\pmb{c}}}\right)\right) \otimes \mathbb{C}_O \longrightarrow 0
\end{multline*}
for $\varepsilon$ small enough.   Therefore the multiplicity of ${\pmb{c}}$ is just
\begin{align*}
m\left({\pmb{c}}\right) & = h^0\left(H_{{\pmb{c}}},\mathcal{O}_{H_{{\pmb{c}}}}\left(\left\lceil K_\pi-c_1F_1-\cdots -c_rF_r\right\rceil+H_{{\pmb{c}}}\right)\right) \\
& = \sum_{E_i \leqslant H_{{\pmb{c}}}} h^0\left(E_i,\mathcal{O}_{E_i}\left(\left\lceil
K_\pi-c_1F_1-\cdots -c_rF_r\right\rceil+H_{{\pmb{c}}}\right)\right) -  a_{H_{{\pmb{c}}}},
\end{align*}
where in the second equality we have used that $H_{{\pmb{c}}}$ has simple
normal crossings, and hence the sections of the line bundle
$\mathcal{O}_{H_{{\pmb{c}}}}\left(\left\lceil K_\pi-c_1F_1-\cdots -c_rF_r\right\rceil+H_{{\pmb{c}}}\right)$
correspond to sections over each component that agree on the
$a_{H_{{\pmb{c}}}}$ intersections, where $a_{H_{{\pmb{c}}}}$ denotes the number of edges  of $H_{{\pmb{c}}}$ in the dual graph.
Then, since we have
\begin{align*}
 &\deg \mathcal{O}_{E_i}\left(\left\lceil K_\pi-c_1F_1-\cdots -c_rF_r\right\rceil+H_{{\pmb{c}}}\right) =\left(\left\lceil K_\pi-c_1F_1-\cdots -c_rF_r\right\rceil+H_{{\pmb{c}}}\right)\cdot E_i \geqslant -1
\end{align*}
by Proposition \ref{prop:Num_condition_cMMI}, we get
\begin{align*}
m\left({\pmb{c}}\right) & = \sum_{E_i \leqslant H_{{\pmb{c}}}}\left(\left(\left\lceil K_\pi-c_1F_1-\cdots -c_rF_r\right\rceil+H_{{\pmb{c}}}\right)\cdot E_i + 1\right) -  a_{H_{{\pmb{c}}}} \\
& = \left(\left\lceil K_\pi-c_1F_1-\cdots -c_rF_r\right\rceil+H_{{\pmb{c}}}\right)\cdot H_{{\pmb{c}}} + v_{H_{{\pmb{c}}}} -  a_{H_{{\pmb{c}}}} \\
& = \left(\left\lceil K_\pi-c_1F_1-\cdots -c_rF_r\right\rceil+H_{{\pmb{c}}}\right)\cdot H_{{\pmb{c}}} +\#\left\{\text{connected components of } H_{{\pmb{c}}}\right\}.
\end{align*}
\end{proof}

The above formula can be rephrased as follows

\begin{corollary} \label{cor:multiplicity2MMI}
Let $\pmb{\mathfrak{a}}=\left(\mathfrak{a}_1,\ldots,\mathfrak{a}_r\right) \subseteq \left({\mathcal O}_{X,O}\right)^r$ be a tuple of ${\mathfrak m}$-primary ideals and $H_{\pmb{c}}$
the maximal jumping divisor associated to some ${\pmb{c}} \in \mathbb{R}_{\geqslant0}^r$. Then,
\begin{align*}
&m\left({\pmb{c}}\right)  = \sum_{E_i \leqslant H_{\pmb{c}}} \left( \sum_{ E_j \in
\mathrm{Adj}(E_i)} \left\{c_1 e_{1,j}+ \cdots+c_r e_{r,j} -
k_j\right\}+c_1\rho_{1,i}+\cdots +c_r\rho_{r,i}\right)\\&\qquad\qquad-\#\left\{\text{connected components of }
H_{{\pmb{c}}}\right\}.
\end{align*}
\end{corollary}

We may also provide a very simple numerical criterion to
detect whether a given point ${\pmb{c}} \in \mathbb{R}^r_{>0}$ is a jumping point.

\begin{theorem} \label{cond_suf}
Let $\pmb{\mathfrak{a}}=\left(\mathfrak{a}_1,\ldots,\mathfrak{a}_r\right)\subseteq \left({\mathcal O}_{X,O}\right)^r$ be a tuple of ${\mathfrak m}$-primary ideals and ${\pmb{c}} \in
\mathbb{R}^r_{\geqslant0}$. Then, ${\pmb{c}}$ is a jumping point if and only if $m({\pmb{c}}) > 0$ or equivalently,  there exists a connected  component $H\leqslant
H_{{\pmb{c}}}$ such that
$$\left(\left\lceil K_\pi-c_1F_1-\cdots -c_rF_r\right\rceil + H_{{\pmb{c}}}\right)\cdot H \geq 0.$$
\end{theorem}

\begin{proof}
We have
\begin{align*}
m\left({\pmb{c}}\right) & = \left(\left\lceil K_{\pi}-c_1F_1-\cdots-c_rF_r\right\rceil +
H_{\pmb{c}}\right)\cdot H_{\pmb{c}} +\#\left\{\text{connected components of }
H_{\pmb{c}}\right\} \\  & = \sum_{H\leqslant
H_{{\pmb{c}}}} \left( \left(\left\lceil K_{\pi}-c_1F_1-\cdots-c_rF_r\right\rceil + H_{{\pmb{c}}}\right)\cdot H +1\right),
\end{align*}
where the sum is taken over all the connected components $H\leqslant
H_{{\pmb{c}}}$. The result follows since  we have $\left(\left\lceil K_{\pi}-c_1F_1-\cdots-c_rF_r\right\rceil + H_c\right)\cdot H \geqslant -1$
by Proposition \ref{prop:Num_condition_cMMI}.
\end{proof}

\subsection{Poincar\'e series of mixed multiplier ideals}
Given a $\mathfrak{m}$-primary ideal $\mathfrak{a} \subseteq {\mathcal O}_{X,O}$ we consider its {\em Poincar\'e series}
\begin{equation} \label{eq-def-Poinc11}
P_\mathfrak{a} (t)= \sum_{c\in \mathbb{R}_{\geqslant0}} m(c) \hskip 1mm t^{c}.
\end{equation} which was first considered, in the case that $X$ is smooth and $\mathfrak{a}$ is simple,
by Galindo and Montserrat \cite{GM10} and extended in  \cite{ACAMDCGA13} to the case where $X$ has a rational singularity
and $\mathfrak{a}$ is any $\mathfrak{m}$-primary  ideal.

\vskip 2mm

For a tuple of $\mathfrak{m}$-primary ideals $\pmb{\mathfrak{a}}=\{\mathfrak{a}_1,\dots,\mathfrak{a}_r\} \subseteq \left({\mathcal O}_{X,O}\right)^r$ we are going
to give a generalization of this series by considering a sequence of mixed multiplier ideals indexed by points in a ray
$L: {\pmb{c}}_0 + \mu {\bf u}$ in the positive orthant $\mathbb{R}^r_{\geqslant0} $ with  ${\bf u}=(u_1,\dots , u_r) \in {\mathbb{Z}}^r_{\geqslant 0}$, ${\bf u}\neq {\bf 0}$ and ${\pmb{c}}_0\in\mathbb{Q}^r_{\geqslant0}$.  Here we are considering, for simplicity, a point ${\pmb{c}}_0$
belonging to a coordinate hyperplane but not necessarily being the origin and $\mu \in \mathbb{R}_{\geqslant0}$. Namely, we consider the sequence of mixed multiplier ideals
$$\mathcal{J} \left(\pmb{\mathfrak{a}}^{{\pmb{c}}_0}\right) \supsetneq \mathcal{J} \left(\pmb{\mathfrak{a}}^{{\pmb{c}}_1}\right) \supsetneq \mathcal{J} \left(\pmb{\mathfrak{a}}^{{\pmb{c}}_2}\right)  \supsetneq  \cdots  \supsetneq  \mathcal{J} \left(\pmb{\mathfrak{a}}^{{\pmb{c}}_i}\right) \supsetneq  \cdots $$ where $\{{{\pmb{c}}_i}\}_{i>0}=L \cap {\rm \bf JW}_{\pmb{\mathfrak{a}}} $
 or equivalently $\{{{\pmb{c}}_i}\}_{i>0}$ is the set of jumping points of this sequence. Then we define the
 {\it Poincar\'e series of $\pmb{\mathfrak{a}}$ alongside the ray $L$} as

\begin{equation} \label{eq-def-Poinc22}
P_{\pmb{\mathfrak{a}}} (\underline{t}; L)= \sum_{{\pmb{c}} \in  L} m({\pmb{c}}) \hskip 1mm \underline{t}^{{\pmb{c}}}.
\end{equation}
where $\underline{t}^{{\pmb{c}}}:=t_1^{c_1}\cdots t_r^{c_r}$.  Notice that we have
$$P_{\pmb{\mathfrak{a}}} (\underline{t}; L)= \sum_{{\pmb{c}} \in  L} m({\pmb{c}}) \hskip 1mm \underline{t}^{{\pmb{c}}} =
\sum_{{\pmb{c}} \in [{\pmb{c}}_0,{\pmb{c}}_0 +{\bf u})} \sum_{k\in {\mathbb{N}}} m({\pmb{c}}+k {\bf u}) \hskip 1mm \underline{t}^{{\pmb{c}} +k{\bf u}}
=\underline{t}^{{\pmb{c}}_0}\sum_{\mu  \in [0,1)} \sum_{k\in {\mathbb{N}}} m({\pmb{c}}+k {\bf u}) \hskip 1mm \underline{t}^{(\mu +k){\bf u}}
$$ where the last equality follows from the fact that we are considering points of the form  ${\pmb{c}}={\pmb{c}}_0+\mu {\bf u}$ with $\mu\in [0,1)$.
Our goal is to prove that this Poincar\'e series  is rational
in the sense that it belongs to the field of fractional functions $\mathbb{C}(z_1,\dots , z_r)$, where the indeterminate $z_i$
corresponds to a fractional power $t_i^{1/e}$ for $e\in {\mathbb{N}}_{>0}$ being the least common multiple of
the denominators of the coordinates of all jumping points.  To do so  we need to prove a linear recurrence among the
coefficients of the series. A key ingredient will be a periodicity property  of the maximal jumping divisor which
 follows from its definition.

\begin{lemma}\label{lem:sameMJD}
For any ${\pmb{c}} \in \mathbb{R}^r_{>0}$ and ${\pmb{\alpha}}=(\alpha_1,\ldots,\alpha_r)\in\mathbb{Z}^r_{\geqslant0}$ we have $H_{{\pmb{c}}}= H_{{\pmb{c}} + {\pmb{\alpha}}}$.
\end{lemma}

The linear recurrence that the multiplicities satisfy is described in terms of the excesses at dicritical components.

\begin{proposition} \label{prop:growthMMI}
Let $\pmb{\mathfrak{a}}=\left(\mathfrak{a}_1,\ldots,\mathfrak{a}_r\right)\subseteq \left({\mathcal O}_{X,O}\right)^r$ be a tuple of ${\mathfrak m}$-primary ideals and
${\pmb{\alpha}}=(\alpha_1,\ldots,\alpha_r)\in\mathbb{Z}^r_{\geqslant0}$.
Then,
$$m\left({\pmb{c}}+{\pmb{\alpha}}\right)- m\left({\pmb{c}}\right) = \sum_{E_i \leqslant H_{{\pmb{c}}}} \sum_{j=1}^{r} \alpha_j\rho_{j,i}\,.$$
\end{proposition}
 \begin{proof}
Lemma \ref{lem:sameMJD} above states that ${\pmb{c}}$ and ${\pmb{c}} + {\pmb{\alpha}}$ have the same maximal jumping divisor, say $H_{{\pmb{c}}}$. Therefore, by Theorem \ref{thm:multiplicity1_cMMI}, we have
 $$m\left({\pmb{c}}+{\pmb{\alpha}}\right)- m\left({\pmb{c}}\right)= -(\alpha_1F_1+\cdots +\alpha_rF_r)\cdot H_{{\pmb{c}}}= \sum_{E_i \leqslant H_{{\pmb{c}}}} \sum_{1\leqslant j\leqslant r} \alpha_j\rho_{j,i}.$$
 \end{proof}

In the sequel, we will just denote
$\rho_{{\pmb{c}}, {\pmb{\alpha}}}:= \sum_{E_i \leqslant H_{{\pmb{c}}}} \sum_{j=1}^{r} \alpha_j\rho_{j,i}\,.$ The formula for the Poincar\'e series that we obtain  is the following:

\begin{theorem} \label{thm:rational}
 Let $\pmb{\mathfrak{a}}=\{\mathfrak{a}_1,\dots,\mathfrak{a}_r\} \subseteq \left({\mathcal O}_{X,O}\right)^r$ be a tuple of $\mathfrak{m}$-primary ideals and let
$L: {\pmb{c}}_0 + \mu {\bf u}$  be a ray in the positive orthant $\mathbb{R}^r_{\geqslant0} $ with ${\bf u}\in {\mathbb{Z}}_{\geqslant 0}$, ${\bf u}\neq {\bf 0}$. The Poincar\'e series of $\pmb{\mathfrak{a}}$ alongside $L$ can be expressed as
$$P_{\pmb{\mathfrak{a}}} (\underline{t}; L)=  \underline{t}^{{\pmb{c}}_0} \sum_{\mu  \in [0,1)} \left( \frac{m({\pmb{c}}_0+\mu{\bf u})}{1-\underline{t}^{\bf u}} + \rho_{{\pmb{c}}_0+\mu{\bf u}, {\bf u}}
\frac{\underline{t}^{\bf u}}{(1-\underline{t}^{\bf u})^2} \right) \underline{t}^{\mu {\bf u }} $$

\end{theorem}

\begin{proof}
Given a point  ${\pmb{c}}= {\pmb{c}}_0 + \mu {\bf u}$, with $\mu \in [0,1)$ we have, using Proposition \ref{prop:growthMMI}, that
\[m({\pmb{c}}+k{\bf u})= m({\pmb{c}}) + k\sum_{E_i\leqslant H_{{\pmb{c}}}}\sum_{j=1}^{ r}u_j\rho_{j,i} = m({\pmb{c}}) + k \rho_{{\pmb{c}}, {\bf u}} \]

Therefore
\begin{eqnarray*}
\sum_{k\geqslant 0}m({\pmb{c}}+k {\bf u})\hskip 1mm \underline{t}^{{\pmb{c}}+k {\bf u}} & = & m({\pmb{c}} ) \hskip 1mm  \underline{t}^{{\pmb{c}}}+
(m({\pmb{c}}) +  \rho_{{\pmb{c}}, {\bf u}}) \hskip 1mm \underline{t}^{{\pmb{c}}+ {\bf u}} + (m({\pmb{c}}) +  2\rho_{{\pmb{c}}, {\bf u}}) \hskip 1mm  \underline{t}^{{\pmb{c}}+2 {\bf u}} + \cdots   \\
&= &  \left(\frac{m({\pmb{c}})}{1-\underline{t}^{\bf u}}+ \rho_{{\pmb{c}}, {\bf u}}\frac{\underline{t}^{\bf u}}{(1-\underline{t}^{\bf u})^2}\right) \underline{t}^{{\pmb{c}}}
\end{eqnarray*}
and the result follows.
\end{proof}

\begin{remark}
In the case that $ L $ is the $i$-th axis of  the positive orthant $\mathbb{R}^r_{\geqslant0} $, in particular if ${\pmb{c}}_0$ is the origin,  we obtain the Poincar\'e series   of the ideal $\mathfrak{a}_i$.
\end{remark}

\section{Multiplicities of jumping points after small perturbations} \label{perturbations}

Let $\pmb{\mathfrak{a}}=\left(\mathfrak{a}_1,\ldots,\mathfrak{a}_r\right) \subseteq \left({\mathcal O}_{X,O}\right)^r$ be a tuple of $\mathfrak{m}$-primary ideals
and consider two parallel rays  $L: {\pmb{c}}_0 + \mu {\bf u}$ and $L': {\pmb{c}}'_0 + \mu {\bf u}$ as those considered in the previous section
that are close enough. Our aim is to compare the sequences of mixed multiplier ideals indexed by points in both rays and see how
the multiplicity of a jumping point varies with a small perturbation. To illustrate this phenomenon we start with
the following example.

\begin{example} \label{rays}
Consider the tuple of ideals $\pmb{\mathfrak{a}}=(\mathfrak{a}_1,\mathfrak{a}_2)$ on a smooth surface $X$ given by:

\vskip 2mm

$\cdot$ $\mathfrak{a}_1=\left((x+y)^4,\right.$ $x^9(x+y),$ $x^{11},$ $x^6(x+y)^2,$ $\left.x^3(x+y)^3\right)$,

$\cdot$ $\mathfrak{a}_2=\left(y^3,x^7,x^5y,x^3y^2\right)$.

\vskip 2mm

\noindent The dual graph of the log-resolution of $\pmb{\mathfrak{a}}$ is as follows:

\vskip 2mm

\begin{center}
\begin{tabular}{c}
   \begin{tikzpicture
   \draw  (-5,0) -- (4,0);
   \draw [dashed,->,thick,red] (2,0) -- (3,1);
   \draw [dashed,->,thick] (-4,0) -- (-5,1);
   \draw (-0.2,-0.3) node {{ $E_1$}};
   \draw (0.8,-0.3) node {{ $E_2$}};
   \draw (3.8,-0.3) node {{ $E_3$}};
   \draw (2.8,-0.3) node {{ $E_4$}};
   \draw (1.8,-0.3) node {{ $E_5$}};
   \draw (-1.2,-0.3) node {{ $E_{6}$}};
   \draw (-5.2,-0.3) node {{ $E_{7}$}};
   \draw (-2.2,-0.3) node {{ $E_{8}$}};
   \draw (-3.2,-0.3) node {{ $E_{9}$}};
   \draw (-4.2,-0.3) node {{ $E_{10}$}};
   \filldraw  (0,0) circle (2pt)
              (1,0) circle (2pt)
              (3,0) circle (2pt)
              (4,0) circle (2pt)
              (-1,0) circle (2pt)
              (-2,0) circle (2pt)
              (-3,0) circle (2pt)
              (-5,0) circle (2pt);
   \filldraw  [fill=white]  (2,0) circle (3pt)
                            (-4,0) circle (3pt);
   \end{tikzpicture}
\\
\end{tabular}
\end{center}

\noindent where the blank dots correspond to dicritical divisors and their excesses are represented by broken arrows.
The divisors associated with this resolution are

\vskip 2mm

$\cdot$ $F_1=4E_1+4E_2+4E_3+8E_4+12E_5+8E_6+11E_7+20 E_8+32E_9+44E_{10}$,

$\cdot$ $F_2=3E_1+6E_2+7E_3+14E_4+21E_5+3E_6+3E_7+6E_8+9E_9+12E_{10}$

\vskip 2mm

\noindent and the relative canonical divisor is:

\vskip 2mm

$\cdot$ $K_{\pi}=E_1+2E_2+3E_3+6E_4+9E_5+2E_6+3E_7+6E_8+10E_9+14E_{10}$.

\vskip 2mm

\noindent In Figure 2, we present the constancy regions of the corresponding mixed multiplier ideals, those regions
are computed using the algorithm in \cite{ACAMDC16}. The chains of mixed multiplier ideals over the parallel rays $L: \left(0,\frac{101}{780}\right)  + \mu (1,1)$ and $L': \left(0,\frac{37}{390}\right) + \mu (1,1)$ are given in Table \ref{table:T1}. The sets of generators for these  ideals are computed using the
algorithm in \cite{AAB17} (see also \cite{BD17} ).

{ \begin{figure}[ht!!!]\label{fig:EX4.1}
  \begin{center}
  \medskip
    \includegraphics[width=.40\textwidth]{Ex_Ray.pdf}

  \end{center}
\caption{ Constancy regions of the mixed multiplier ideals of $\pmb{\mathfrak{a}}$ and the rays $L, L'$,  in blue and Bordeaux respectively.}
\end{figure}}

\def1.3{1.3}
\begin{table}[!ht]
\centering
\begin{tabular}{|m{23mm}|m{49mm}|m{23mm}|m{49mm}|}
\hline
Jumping\newline points in $L$ &  \(\mathcal{J}(\pmb{\mathfrak{a}}^{\pmb{\lambda}})\)& Jumping\newline points in $L'$&  \(\mathcal{J}(\pmb{\mathfrak{a}}^{\pmb{\lambda}})\) \\
[1.5pt] \hline
\hline
$\left(\frac {81}{260}, \frac {86}{195} \right)$ 	& {\small $(x,y)$} & $\left(\frac {67}{130},\frac {119}{195} \right)$ 	 & {\small $(x,y)$} \\[1.5pt] \hline
\multicolumn{2}{| c |}{}& $\left(\frac {203}{520},\frac {757}{1560} \right)$ 	 & {\small $(x+y,x^2)$} \\[1.5pt] \hline
$\mathbf{ \left(\frac {631}{2860}, \frac {751}{2145} \right)}$ 	& {\small $(x^2,y^2,xy)$} & $\left(\frac {267}{455},\frac {1861}{2730} \right)$ 	 & {\small $(x^2,y^2,xy)$} \\[1.5pt] \hline
$\left(\frac {697}{1820}, \frac {1399}{2730} \right)$ 	 & {\small $(xy + x^2, x^3, y^2 + 2xy + x^2)$}& $\left(\frac {347}{1430},\frac {724}{2145} \right)$ 	 & {\small $(xy + x^2, x^3, y^2 + 2xy + x^2)$} \\[1.5pt] \hline
$\left(\frac {827}{1820}, \frac {797}{1365} \right)$ 	& {\small $(y^2 + xy, x^2y, x^2y + x^3)$}& $\left(\frac {477}{1430},\frac {919}{2145} \right)$ 	 & {\small $(y^2 + xy, x^2y, x^2y + x^3)$} \\[1.5pt] \hline
$\left(\frac {957}{1820}, \frac {1789}{2730} \right)$ 	& {\small $(y^2 + xy, x^2y, x^2y + x^3)$}& $\left(\frac {607}{1430},\frac {1114}{2145} \right)$ 	 & {\small $(y^2 + xy, x^2y, x^2y + x^3)$} \\[1.5pt] \hline
$\left(\frac {1151}{2860}, \frac {1141}{2145} \right)$ 	& {\small $( y^2 + xy, x^3y, x^3y + x^4)$}& \multicolumn{2}{ c |}{}  \\[1.5pt] \hline
$\left(\frac {1411}{2860}, \frac {1336}{2145} \right)$ 	& {\small $(y^3 + xy^2, x^3y, x^3y + x^4, x^2y^2,\newline xy^2 + x^2y)$} & $\mathbf{ \left(\frac {1161}{3640},\frac {4519}{10920} \right)}$ & {\small $(y^3 + xy^2, x^3y, x^3y + x^4, x^2y^2,\newline xy^2 + x^2y)$} \\[1.5pt] \hline
$\left(\frac {1849}{3640}, \frac {6961}{10920} \right)$ & {\small$(y^3 + 2xy^2 + x^2y, x^4y, xy^2 + x^2y,\newline x^2y^2 + 2x^3y + x^4, x^3y + x^4)$}& $\left(\frac {1681}{3640},\frac {6079}{10920} \right)$ & {\small$(y^3 + 2xy^2 + x^2y, x^4y, xy^2 + x^2y,\newline x^2y^2 + 2x^3y + x^4, x^3y + x^4)$} \\[1.5pt] \hline
$\left(\frac {2109}{3640},\frac {7741}{10920} \right)$ & {\small $( y^3 + 2xy^2 + x^2y, x^2y^2 + x^3y,\newline x^4y, x^2y^2 + 2x^3y + x^4)$} & $\left(\frac {1941}{3640},\frac {6859}{10920} \right)$ & {\small $( y^3 + 2xy^2 + x^2y, x^2y^2 + x^3y,\newline x^4y, x^2y^2 + 2x^3y + x^4)$} \\[1.5pt] \hline
$\left(\frac {2369}{3640},\frac {8521}{10920}\right)$ & {\small $( y^3 + 2xy^2 + x^2y, x^2y^2 + x^3y,\newline x^5y, x^2y^2 + 2x^3y + x^4)$}& $\left(\frac {2201}{3640},\frac {7639}{10920} \right)$ & {\small $( y^3 + 2xy^2 + x^2y, x^2y^2 + x^3y,\newline x^5y, x^2y^2 + 2x^3y + x^4)$} \\[1.5pt] \hline

\end{tabular}
\vspace{5pt}
\caption{  Chains of mixed multiplier ideals of $\pmb{\mathfrak{a}}$ over the rays $L$ and $L'$. In bold type we present the jumping points with multiplicity $2$.
} \label{table:T1}
\end{table}

\end{example}

In the previous example, we  observe that the chains of mixed multiplier ideals  differ whenever the
corresponding ray crosses the intersection of ${\mathcal C}$-facets. Indeed, the multiplicity of a jumping point at the intersection of ${\mathcal C}$-facets
is bigger than the multiplicities of jumping points in its neighborhood. The aim of this section is to provide an explanation to this phenomenon. We start with the fact that the multiplicity
does not increase in the interior of ${\mathcal C}$-facets.

\begin{proposition}\label{interior}
Let $\pmb{\mathfrak{a}}=\left(\mathfrak{a}_1,\ldots,\mathfrak{a}_r\right)\subseteq \left({\mathcal O}_{X,O}\right)^r$ be a tuple of $\mathfrak{m}$-primary ideals
and let $\pmb{\lambda}, \pmb{\lambda'}$ be two jumping points in the interior of a $\mathcal{C}$-facet.
Then $m(\pmb{\lambda})=m( \pmb{\lambda'})$.
\end{proposition}

To prove this result it is more convenient to compute the multiplicity
of a jumping point by using the so-called {\it minimal jumping divisor} instead of
the maximal jumping divisor as we did in Section \ref{multiplicities}. This minimal jumping
divisor is closely related to the algorithm developed in \cite{ACAMDC16} to compute
the constancy regions of mixed multiplier ideals.  We give its definition below but we refer
to \cite{ACAMDC16} for details.

\begin{definition} \label{def:minimal}
Let $\pmb{\mathfrak{a}}=\left(\mathfrak{a}_1,\ldots,\mathfrak{a}_r\right)\subseteq
\left({\mathcal O}_{X,O}\right)^r$ be a tuple of ideals. Given a jumping point $\pmb{\lambda}=(\lambda_1,\dots , \lambda_r) \in \mathbb{R}^r_{\geqslant0}$, its corresponding
{\it minimal jumping divisor} is the reduced divisor $G_{\lambda} \leqslant
\sum_{i=1}^r F_i$ supported on those components $E_j$ for which the
point $\pmb{\lambda}$ satisfies
$$ \lambda_1  e_{1,j}+ \cdots +   \lambda_r e_{r,j} = k_j + 1 +
e_j^{(1-\varepsilon){\pmb{\lambda}}},$$ where, for a sufficiently small $\varepsilon>0$,
$D_{(1-\varepsilon)\pmb{\lambda}} = \sum e_j^{(1-\varepsilon){\pmb{\lambda}}}
\hskip 1mm E_j$ is the antinef closure of $$\lfloor
(1-\varepsilon){\lambda_1} F_1 + \cdots + (1-\varepsilon){\lambda_r}
F_r - K_\pi\rfloor.$$
\end{definition}

Using the same arguments that we used in the proof of Theorem \ref{thm:multiplicity1_cMMI}
we may provide the following formula for the multiplicity of a jumping point in terms of the minimal
jumping divisor.

\begin{proposition}\label{eq:multiplicity1_MMI}
Let $\pmb{\mathfrak{a}}=\left(\mathfrak{a}_1,\ldots,\mathfrak{a}_r\right) \subseteq \left({\mathcal O}_{X,O}\right)^r$ be a tuple of ${\mathfrak m}$-primary ideals and $G_{\pmb{\lambda}}$
the maximal jumping divisor associated to some jumping point $\pmb{\lambda} \in \mathbb{R}_{>0}^r$. Then,
\begin{align*}
m(\pmb{\lambda}) & = \left(\left\lceil K_\pi-\lambda_1 F_1-\cdots-\lambda_r F_r\right\rceil +
G_{\pmb{\lambda}}\right)\cdot G_{\pmb{\lambda}} +\#\{\text{connected components
of } G_{\pmb{\lambda}}\}
\end{align*}
\end{proposition}

\vskip 2mm

It was proved in \cite[Lemma 4.6]{ACAMDC16} that two interior points of a $\mathcal{C}$-facet have the same minimal jumping divisor, which we refer to as {\it the minimal jumping divisor associated to} the $\mathcal{C}$-facet. Therefore by applying \ref{eq:multiplicity1_MMI}, the multiplicity is constant along the interior points of a $\mathcal{C}$-facet and thus proving Proposition \ref{interior}.
We point out that two interior points of a $\mathcal{C}$-facet may have  different maximal jumping divisor

\vskip 2mm

This constancy property for the multiplicities is no longer true when considering jumping  points at the intersection of $\mathcal{C}$-facets.
However we can control the multiplicity depending on the number of $\mathcal{C}$-facets that contain this jumping point.

\vskip 2mm

\begin{theorem} \label{intersection}

Let $\pmb{\mathfrak{a}}=\left(\mathfrak{a}_1,\ldots,\mathfrak{a}_r\right) \subseteq \left({\mathcal O}_{X,O}\right)^r$ be a tuple of $\mathfrak{m}$-primary ideals
and let  $L: {\pmb{c}}_0 + \mu {\bf u}$ and $L': {\pmb{c}}'_0 + \mu {\bf u}$ be two parallel rays that are close enough.
Let $\pmb{\lambda} \in L$ be a jumping point and  $B_\varepsilon(\pmb{\lambda})$ be a ball centered at $\pmb{\lambda}$ of a sufficiently small radius $\varepsilon>0$ such that
$L\cap {\rm \bf JW}_{\pmb{\mathfrak{a}}} \cap B_\varepsilon(\pmb{\lambda}) =\{\pmb{\lambda}\}$. If
$L'\cap {\rm \bf JW}_{\pmb{\mathfrak{a}}} \cap B_\varepsilon(\pmb{\lambda}) =\{\pmb{\lambda}_1, \dots , \pmb{\lambda}_n\}$ then
\[m(\pmb{\lambda})=m(\pmb{\lambda}_1)+m(\pmb{\lambda}_2)+\cdots+m(\pmb{\lambda}_n)\,.\]

\end{theorem}

\begin{proof}

Let $V_1,\dots, V_k$ be all the hyperplanes associated to exceptional divisors that contain the jumping point $\pmb{\lambda}$.
For each hyperplane $V_i$ we consider the divisor
$H_i = \sum_j  E_j$ where the sum is taken over the exceptional divisors that support the hyperplane $V_i$, i.e.  for all
$E_j \leq H_i$ there exists some $\ell_j \in {\mathbb{Z}}_{>0}$ such that
the hyperplane $V_i$ is of the form $e_{1,j} z_1+ \cdots +  e_{r,j} z_r= \ell_j + k_j$.
Notice that, even though it is possible that not all of these hyperplanes support a jumping wall, we have a decomposition of the
maximal jumping divisor as $H_{\pmb{\lambda}}=H_1 + \cdots + H_k$. Let $\{{\bf c}_1, \dots , {\bf c}_k\}$ be the ordered\footnote{The order
on the set of points $\{{\bf c}_1, \dots , {\bf c}_k\}$ is given by their distance to the origin. We order the hyperplanes $V_1,\dots, V_k$
accordingly.} set of points
resulting from the intersection of the ray $L'$ with the  hyperplanes $V_1,\dots, V_k$. Notice that we have
$L'\cap {\rm \bf JW}_{\pmb{\mathfrak{a}}} \cap B_\varepsilon(\pmb{\lambda}) =\{\pmb{\lambda}_1, \dots , \pmb{\lambda}_n\} \subseteq
\{{\bf c}_1, \dots , {\bf c}_k\}$.

\vskip 2mm

For each ${\bf c}_i=({ c}_{i,1}, \dots , { c}_{i,r})$  we may find
a point $(1-\varepsilon'){\bf c}_i := ((1-\varepsilon'_1){c}_{i,1}, \dots , (1-\varepsilon'_r){ c}_{i,r})$ over the ray $L'$ that is close enough but smaller than ${\bf c}_1$ and a point  over the ray $L$ and smaller than $\pmb{\lambda} $  that we will denote as
$(1-\varepsilon)\pmb{\lambda} := ((1-\varepsilon_1)\lambda_1, \dots , (1-\varepsilon_r)\lambda_r)$ satisfying
$$\lceil K_{\pi} - (1-\varepsilon_1)\lambda_1 F_1 - \cdots - (1-\varepsilon_r)\lambda_rF_r\rceil = \lceil K_{\pi} - (1-\varepsilon'_1){ c}_{i,1} F_1 - \cdots - (1-\varepsilon'_r){ c}_{i,r}  F_r\rceil.$$

From the  construction of the hyperplanes $V_i$ we have:

\vskip 2mm

$\cdot$  $\lceil K_{\pi} - \lambda_1 F_1 - \cdots - \lambda_rF_r\rceil = \lceil K_{\pi} - (1-\varepsilon_1)\lambda_1 F_1 - \cdots - (1-\varepsilon_r)\lambda_rF_r\rceil + H_{\pmb{\lambda}}.$

\vskip 2mm

$\cdot$ $\lceil K_{\pi} - { c}_{i,1} F_1 - \cdots - {c}_{i,r}  F_r\rceil = \lceil K_{\pi} - (1-\varepsilon'_1){c}_{i,1} F_1 - \cdots - (1-\varepsilon'_r){c}_{i,r}  F_r\rceil + H_1 + \cdots + H_i.$

\vskip 2mm

Therefore
\begin{equation} \label{epsilon}
\lceil K_{\pi} - \lambda_1 F_1 - \cdots - \lambda_rF_r\rceil - \lceil K_{\pi} - {c}_{i,1} F_1 - \cdots - { c}_{i,r}  F_r\rceil = H_{i+1} + \cdots H_k.
\end{equation}

\vskip 2mm

By Theorem \ref{thm:multiplicity1_cMMI}, one has
\begin{multline*}
m(\pmb{\lambda})  = \left(\left\lceil K_\pi-\lambda_1 F_1-\cdots-\lambda_r F_r\right\rceil +
H_{\pmb{\lambda}}\right)\cdot H_{\pmb{\lambda}} +\#\{\text{connected components
of } H_{\pmb{\lambda}}\}\,.
\end{multline*}
Thus, we can rewrite this formula as
\begin{align*}
 m(\pmb{\lambda})&= \sum_{i=1}^k (\lceil K_{\pi} - \lambda_1 F_1 - \cdots - \lambda_rF_r\rceil+H_{i})\cdot H_i\\
 &\qquad+\sum_{i=1}^k\sum_{\substack{j=1 \\ j\neq i}}^k H_iH_j+\#\{\text{connected components of }H_{\pmb{\lambda}}\} \\
 &= \sum_{i=1}^k(\lceil K_{\pi} - { c}_{i,1} F_1 - \cdots - { c}_{i,r}F_r\rceil+H_{i})\cdot H_i\\
 &\qquad+\sum_{i=1}^k\sum_{j>i}^k H_iH_j+\#\{\text{connected components of }H_{\pmb{\lambda}}\}\,
\end{align*}

\noindent where the last equality follows from Equation \ref{epsilon}.  Now, recall that for any divisor $D$ with exceptional support
$\#\{\text{connected components of }D\}=v_{D}-a_{D},$
where $v_{D}$ and $a_{D}$ denote the number of vertices and edges of $D$
in the dual graph. Since $v_{H_{\pmb{\lambda}}}=v_{H_1}+\cdots+v_{H_k}$ and $a_{H_{\pmb{\lambda}}}=a_{H_1}+\cdots+a_{H_k}+\sum_{i=1}^k\sum_{j>i}^k H_iH_j$ we deduce
\begin{align*}
     \#\{\text{\small{connected components of }}H_{\pmb{\lambda}}\
      =\sum_{i=1}^k\#\{\text{\small{connected components of }}H_i\
      -\sum_{i=1}^k\sum_{j>i}^k H_iH_j .
\end{align*}
Therefore
\begin{align*}
m(\pmb{\lambda}) & = \sum_{i=1}^k \left[ (\lceil K_{\pi} - {c}_{i,1} F_1 - \cdots - {c}_{i,r}F_r\rceil+H_{i})\cdot H_i  + \#\{\text{connected components of }H_i\} \right] \\
&= m({\bf c}_1)+ \cdots + m({\bf c}_k) .
\end{align*}

The only points with non zero multiplicity are those over a jumping wall, namely the jumping points $\{\pmb{\lambda}_1, \dots , \pmb{\lambda}_n\}$
and thus we get the desired result.
\end{proof}

\section{Contribution to the log-canonical wall} \label{Sec4}

Let $X$ be a smooth complex surface and  $\mathfrak{a}\subseteq \mathcal{O}_{X,O}$ an ideal. A common theme in the
study of multiplier ideals is to check which exceptional divisors contribute to the jumping numbers of $\mathfrak{a}$.
In the case of the  log-canonical threshold we know that it is  described by the formula
$${\rm lct}(\mathfrak{a})= \min_i\left\{\frac{k_i+1}{e_i}\right\}.$$
In the case that $\mathfrak{a}$ is $\mathfrak{m}$-primary and simple, this minimum is achieved at  the first
rupture or dicritical  exceptional component, starting from the origin, in the dual graph of the log-resolution of $\mathfrak{a}$
(see \cite{Jar11}, \cite{Tuc10}). For non simple ideals we may find
some analogous statements in  \cite{Kuw99}, \cite{GHM12}, \cite{ACNLM08}, \cite{AN10}.

\vskip 2mm

For the case of mixed multiplier ideals, Cassou-Nogu\`es and Libgober \cite[Theorem 4.22]{CNL14} studied the contribution
of exceptional divisors to the log-canonical wall for the case where the tuple of ideals corresponds to the branches
of a plane curve.
In this section we will give a generalization of their result that works for general
tuples of ${\mathfrak m}$-primary ideals  $\pmb{\mathfrak{a}}=\left(\mathfrak{a}_1,\ldots,\mathfrak{a}_r \right)\subseteq \left(\mathcal{O}_{X,O}\right)^r$,
where $X$ is a complex surface with a rational singularity at $O$ and the points in the log-canonical wall
have multiplicity one.

\vskip 2mm

Their result is described in terms of the so-called {\it Newton nest} introduced in \cite[Definition 4.19]{CNL14}. In order to give a generalization to our setup of the Newton nest we will need to fix some notation.
When $X$ has a rational singularity  we may have an strict inclusion
$\mathcal{O}_{X,O}\varsupsetneq\mathcal{J}(\pmb{\mathfrak{a}}^{\bf 0})$ where ${\bf 0}=(0,\dots, 0)$ is the origin of the positive orthant $\mathbb{R}^r_{\geqslant 0}$.
Indeed,  the mixed multiplier ideal $\mathcal{J}(\pmb{\mathfrak{a}}^{\bf 0})$  is described by a divisor $D_{\bf 0}=\sum e_j^{\bf 0} E_j$
which is the antinef closure of $\lfloor  - K_\pi \rfloor$ that can be computed using the {\it unloading procedure}
described in \cite{ACAMDC13}. Therefore, the log-canonical wall is supported on hyperplanes of the form
\[  e_{1,j} z_1 + \cdots +  e_{r,j} z_r = k_j + 1 + e_j^{{\bf 0}}\,,  \hskip 8mm j=1,\dots,s.\]
For each point $\pmb{z}_i=(0,\ldots,0,{\rm lct}(\mathfrak{a}_i),0,\ldots,0)$  in the
$i$-th coordinate axis corresponding to the log-canonical threshold of the ideal $\mathfrak{a}_i$, $i=1,\dots, r$, we
consider the reduced divisor $G'_{\pmb{z}_i}= \sum E_j$,  where the sum is taken over those exceptional divisors
associated to the supporting hyperplanes  of the log-canonical wall which contain the point ${\pmb{z}_i}$. Notice that this divisor
is contained in the minimal jumping divisor of ${\pmb{z}_i}$, that is $G'_{\pmb{z}_i}\leqslant G_{\pmb{z}_i}$.

\vskip 2mm

\begin{definition} \label{Newton}
 Consider the minimal connected subgraph $\Gamma_{\pmb{\mathfrak{a}}}'$ of the dual graph $\Gamma_{\pmb{\mathfrak{a}}}$
 containing the divisors $G'_{\pmb{z}_i}$, for $i=1\dots, r$. The {\it Newton nest} of $\Gamma_{\pmb{\mathfrak{a}}}$
 is the set of rupture or dicritical divisors belonging to $\Gamma'_{\pmb{\mathfrak{a}}}$.
\end{definition}

\begin{remark}
 In the case that $X$ is smooth and the ideals $\pmb{\mathfrak{a}}_i$  are simple,  this definition coincides with the one
given  by Cassou-Nogu\`es and Libgober in \cite[Definition 4.19]{CNL14} since in this case
we have that $G'_{\pmb{z}_i}= G_{\pmb{z}_i}=E_{j_i}$, where $E_{j_i}$ is the rupture divisor in the dual graph
$\Gamma_{\mathfrak{a}_i}$ which is closest to its root.
 \end{remark}

\vskip 2mm

Cassou-Nogu\`es and Libgober  \cite[Theorem 4.22]{CNL14} established a one-to-one correspondence between the divisors of the Newton nest and the ${\mathcal C}$-facets of the log-canonical wall in the case where $X$ is smooth and the tuple of ideals correspond to the branches of a plane curve. The only restriction that we are going to impose in our generalization is that the multiplicity of all the points in the log-canonical wall have multiplicity one. This condition is achieved, for example, in the case that $X$ has a log-terminal singularity at $O\in X$.

\begin{lemma}
Let
$\pmb{\mathfrak{a}}=\left(\mathfrak{a}_1,\ldots,\mathfrak{a}_r\right)\subseteq \left({\mathcal O}_{X,O}\right)^r$ be a tuple of simple ${\mathfrak m}$-primary ideals
and $X$ is a complex surface with a log-terminal singularity. Then, all the points in the log-canonical wall
have multiplicity one.
\end{lemma}

\begin{proof}
 From the definition of log-terminal singularity, it follows that the antinef closure of $\lfloor -K_{\pi}\rfloor$ is $0$
 because all the coefficients of $\lfloor -K_{\pi}\rfloor$ are strictly smaller than one. Therefore the ideal associated
 to the point ${\pmb 0}$ is the whole ring.

 \vskip 2mm

 Let $\pmb{\lambda}$ be a jumping point in  the log-canonical wall. All the coefficients of the divisor
 $\lfloor \pmb{\lambda} F -K_{\pi} \rfloor$ must be smaller or equal to one so we have
 $\lfloor \pmb{\lambda} F -K_{\pi} \rfloor \leqslant Z$ where $Z$ is the fundamental cycle. Therefore we have
 ${\mathfrak m}=\pi_*\mathcal{O}_{X'}(-Z)\subseteq \mathcal{J}\left(\pmb{\mathfrak{a}}^{\pmb{\lambda}}\right) \subsetneq {\mathcal O}_{X,O}$. So $ \mathcal{J}\left(\pmb{\mathfrak{a}}^{\pmb{\lambda}}\right)={\mathfrak m}$,
 and consequently $m(\pmb{\lambda})= 1$ for all points in the log-canonical wall.
\end{proof}

\vskip 2mm

Before stating the main result of this section  we will present some properties concerning  jumping points of multiplicity one.
This is a very restrictive condition on the corresponding minimal jumping divisors. To such purpose we have to introduce
some technical notation. Given any exceptional component $E_i$ and a reduced  divisor
$D \leqslant E =Exc(\pi) $, we define the set of components adjacent to $E_i$ {\em inside} $D$ and its number as:
$$\mathrm{Adj}_D\left(E_i\right) = \left\{E_j \leqslant D \hskip 2mm | \hskip 2mm E_i \cdot E_j = 1\right\} \quad \text{and} \quad a_D\left(E_i\right) = \#\mathrm{Adj}_D\left(E_i\right)$$

\begin{lemma}\label{lem:1divisor}
 Let $\pmb{\mathfrak{a}}=\left(\mathfrak{a}_1,\ldots,\mathfrak{a}_r\right) \subseteq \left({\mathcal O}_{X,O}\right)^r$ be a tuple of ${\mathfrak m}$-primary ideals and $\pmb{\lambda}$ a jumping point such that $m(\pmb{\lambda})=1$. Then, the minimal jumping divisor $G_{\pmb{\lambda}}$  has only one connected component and no rupture or dicritical divisor $E_i$ such that $a_{G_{\pmb{\lambda}}}(E_i)>1$.
\end{lemma}

\begin{proof}
Using Proposition \ref{eq:multiplicity1_MMI}  we have  the following formula.
$$
m(\pmb{\lambda})  = \left(\left\lceil K_\pi-\lambda_1 F_1-\cdots-\lambda_r F_r\right\rceil +
G_{\pmb{\lambda}}\right)\cdot G_{\pmb{\lambda}}+\#\{\text{connected components
of } G_{\pmb{\lambda}}\}\,.
$$
In the case that $m(\pmb{\lambda})=1$
we can deduce that
$\#\{\text{connected components of } G_{\pmb{\lambda}}\}=1$ and
\begin{align*}
      \left(\left\lceil K_\pi-\lambda_1 F_1-\cdots-\lambda_r F_r\right\rceil +
G_{\pmb{\lambda}}\right)\cdot G_{\pmb{\lambda}}=\sum_{E_i\leqslant G_{\pmb{\lambda}}} \left(\left\lceil K_\pi-\lambda_1 F_1-\cdots-\lambda_r F_r\right\rceil +
G_{\pmb{\lambda}}\right)\cdot E_i=0
\end{align*}
since we already had $\left(\left\lceil K_\pi-\lambda_1 F_1-\cdots-\lambda_r F_r\right\rceil +
G_{\pmb{\lambda}}\right)\cdot G_{\pmb{\lambda}}\geqslant 0$  by   \cite[Proposition 4.13]{ACAMDC16}.
Indeed, using again  this result we have
\[\left(\left\lceil K_\pi-\lambda_1 F_1-\cdots-\lambda_r F_r\right\rceil +
G_{\pmb{\lambda}}\right)\cdot E_i=0\] for all $E_i\leqslant G_{\pmb{\lambda}}$.
We may provide a more explicit description of this equation using  \cite[Lemma 4.11]{ACAMDC16}. Namely
we have
\begin{multline*}
\left(\left\lceil K_\pi-  \lambda_1  F_1-\cdots-\lambda_r F_r\right\rceil  + G_{\pmb{\lambda}}\right)\cdot E_i  =  \\
 =  -2 + \lambda_1 \rho_{1,i}+\cdots+ \lambda_r \rho_{r,i} +a_{G_{\pmb{\lambda}}}\left(E_i\right) + \sum_{ E_j \in \mathrm{Adj}_E(E_i)} \left\{\lambda_1 e_{1,j}+\cdots+\lambda_r e_{r,j}-k_j\right\}  \,.
\end{multline*}
Thus, if $E_i$ is a rupture or dicritical component with $a_{G_{\pmb{\lambda}}}\left(E_i\right)>1$, then we have \[\left(\left\lceil K_\pi-\lambda_1  F_1-\cdots-\lambda_r F_r\right\rceil + G_{\pmb{\lambda}}\right)\cdot E_i>0\,\]  so we get a contradiction and the result follows.
\end{proof}

\begin{corollary}\label{cor:Cfacet}
Let $\pmb{\lambda} \in \mathbb{R}^r_{\geqslant 0}$ be a jumping point not contained in any coordinate hyperplane such that  $m(\pmb{\lambda})=1$. Then:

 \begin{itemize}
 \item[i)] If  $\pmb{\lambda}$ is an interior  point of a ${\mathcal C}$-facet which does not intersect any other ${\mathcal C}$-facet,
  the minimal jumping divisor $G_{\pmb{\lambda}}$ contains at most two dicritical or rupture divisors.
   \item[ii)] If  $\pmb{\lambda}$ is an interior  point of a ${\mathcal C}$-facet which intersects, at least, another ${\mathcal C}$-facet,
  the minimal jumping divisor $G_{\pmb{\lambda}}$ is a dicritical or rupture divisor.
  \item[iii)] If $\pmb{\lambda}$ is at the intersection of two  ${\mathcal C}$-facets,  the minimal jumping divisor $G_{\pmb{\lambda}}$
  is connected and contains exactly two dicritical or rupture divisors, which are its two ends.

 \end{itemize}

\end{corollary}

\begin{proof}
Let $\pmb{\lambda} \in \mathbb{R}^r_{\geqslant 0}$ be a jumping point. By \cite[Theorem 4.14]{ACAMDC16},  the ends of the connected
components of the minimal jumping divisor $G_{\pmb{\lambda}}$ over the dual graph are either rupture or dicritical divisors.
If we assume $m(\pmb{\lambda})=1$, then, using Lemma \ref{lem:1divisor}, we have that  $a_{G_{\pmb{\lambda}}}(E_j)\leqslant 1$
for any rupture or dicritical divisor $E_j$. Therefore either $G_{\pmb{\lambda}}$ is just one exceptional component or it
is connected with just two ends which are rupture or dicritical divisors in the dual graph.
In particular,  {\rm i)} follows.

\vskip 2mm

Now assume that $\pmb{\lambda}$ is at the intersection of two ${\mathcal C}$-facets ${\mathcal C}_1$ and ${\mathcal C}_2$
with associated minimal jumping divisors $G_1$ and $G_2$ respectively. We
have $G_{\pmb{\lambda}}=G_1+G_2+G'$ for some divisor $G'$ with exceptional support. Moreover,
the jumping points in the interior of ${\mathcal C}_1$ and ${\mathcal C}_2$ have multiplicity $1$ so
the same properties considered above also apply for $G_1$ and $G_2$. The two
${\mathcal C}$-facets are supported on different hyperplanes with different slope, so  $G_1$ and $G_2$ do not share
any exceptional divisor. By Lemma \ref{lem:1divisor}, this forces $G_1$ and $G_2$ to be just one exceptional component
being a rupture or dicritical divisor and the minimal jumping divisor
$G_{\pmb{\lambda}}$ contains exactly two dicritical or rupture divisors. Thus,  {\rm ii)} and {\rm iii)} follow.
\end{proof}

\begin{remark}
Given a tuple of ${\mathfrak m}$-primary ideals $\pmb{\mathfrak{a}}=(\mathfrak{a}_1,\ldots,\mathfrak{a}_r )\subseteq ({\mathcal O}_{X,O})^r$ we may pick  a subfamily
$\mathfrak{a}'=\{\mathfrak{a}_{i_1}, \dots , \mathfrak{a}_{i_k} \hskip 2mm | \hskip 2mm 1\leq i_1 < \cdots < i_k \leq r  \}$ and, if no confusion arise, we may view
it either as a tuple $ ({\mathcal O}_{X,O})^s$ or a subtuple  of $\pmb{\mathfrak{a}}$ in $({\mathcal O}_{X,O})^r$ in the obvious way. Notice for example that
the Newton nest of $\pmb{\mathfrak{a}}'$   is a subset of the Newton nest of $\pmb{\mathfrak{a}}$.
 In the case that $\pmb{\lambda}=(\lambda_1, \dots , \lambda_r) \in \mathbb{R}^r_{\geqslant 0}$ is a jumping point contained in a coordinate hyperplane, we may consider the tuple  $\pmb{\mathfrak{a}}'=(\mathfrak{a}_i \hskip 2mm | \hskip 2mm \lambda_i \neq 0)$. Corollary \ref{cor:Cfacet}
 holds whenever we consider $\pmb{\lambda}$ as a jumping point for $\pmb{\mathfrak{a}}'$ and thus, a point not in the coordinate
 hyperplanes of the lower dimensional positive orthant.

\end{remark}

\vskip 2mm

Notice that Corollary \ref{cor:Cfacet} already singles out a very particular case where we may not have our desired one-to-one correspondence.
Namely, assume that the log-canonical wall has a unique ${\mathcal C}$-facet with points of multiplicity one. Part ${\rm i)}$ of Corollary \ref{cor:Cfacet}
says that the Newton nest contains either one or two divisors. Therefore the desired one-to-one correspondence fails when we have exactly two divisors and this case can indeed be achieved. Recall that the effective divisors $F_i$ such that
$\mathfrak{a}_i\cdot{\mathcal O}_{X'} = {\mathcal O}_{X'}\left(-F_i\right)$  are of the form  $F_i=\sum_{j=1}^s e_{i,j} E_j$   for  $ i=1,\dots,r$ and the relative canonical divisor is $ K_{\pi}=\sum_{i=1}^s k_j E_j $. Let $E_j$ and $E_{\ell}$ be the divisors in the Newton nest and
$V_{j,1}:\hskip 1mm e_{1,j} z_1+ \cdots +  e_{r,j} z_r=  k_j +1 $ and  $  V_{\ell,1}: \hskip 1mm e_{1,\ell} z_1+ \cdots +  e_{r,\ell} z_r=  k_\ell +1$ be their associated hyperplanes. The numerical conditions for which these hyperplanes support the unique ${\mathcal C}$-facet of the  log-canonical wall are
\[\frac{e_{1,\ell}}{e_{1,j}}=\cdots=\frac{e_{r, \ell}}{e_{r,j}}=\frac{k_\ell+1}{k_j+1}\,.\]
This result can be reformulated in the following

\vskip 2mm

\begin{lemma}\label{lem:2rdiv}

Let
$\pmb{\mathfrak{a}}=\left(\mathfrak{a}_1,\ldots,\mathfrak{a}_r\right)\subseteq \left({\mathcal O}_{X,O}\right)^r$ be a tuple of ${\mathfrak m}$-primary ideals
where $X$ is a complex surface with a rational singularity at $O \in X$. Assume that all the points in the log-canonical wall have multiplicity one and the Newton nest contains two divisors $E_j$ and $E_{\ell}$.
Then, the log-canonical wall has a unique ${\mathcal C}$-facet if and only if
\[\frac{e_{1,\ell}}{e_{1,j}}=\cdots=\frac{e_{r, \ell}}{e_{r,j}}=\frac{k_\ell+1}{k_j+1}\,.\]
\end{lemma}

\vskip 2mm

We illustrate this case with the following

\begin{example}

Consider a smooth surface $X$ and a tuple of ideals $\pmb{\mathfrak{a}}=\{\mathfrak{a}_1,\mathfrak{a}_2\}$ such that they have a minimal log-resolution with the following vertex ordering

\begin{center}

   \begin{tikzpicture}[scale=0.9]
   \draw  (-1,0) -- (3,0);
   \draw  (2,0) -- (3,1);
   \draw  (2,0) -- (3,-1);
   \draw  (3,1) -- (5,1);
   \draw  (3,-1) -- (5,-1);
   \draw  (-1,0) -- (-2,-1);
   \draw  (-1,0) -- (-2,1);
   \draw  (-2,-1) -- (-3,-1);
   \draw  (-2,1) -- (-4,1);
   \draw [dashed,->,thick,red] (4,1) -- (5,0);
   \draw [dashed,->,thick,blue] (3,-1) -- (4,0);
   \draw [dashed,->,thick,red] (-2,1) -- (-3,0.33);
   \draw [dashed,->,thick,blue] (-2,1) -- (-3,0);
   \draw (-.2,-0.3) node {{\small $E_1$}};
   \draw (2.8,-0.3) node {{\small $E_2$}};
   \draw (0.8,-0.3) node {{\small $E_3$}};
   \draw (1.8,-0.3) node {{\small $E_4$}};
   \draw (4.8,-1.3) node {{\small $E_5$}};
   \draw (3.8,-1.3) node {{\small $E_6$}};
   \draw (2.8,-1.3) node {{\small $E_7$}};
   \draw (4.8,1.3) node {{\small $E_8$}};
   \draw (2.8,1.3) node {{\small $E_9$}};
   \draw (3.8,1.3) node {{\small $E_{10}$}};
   \draw (-3.2,-1.3) node {{\small $E_{11}$}};
   \draw (-2.2,-1.3) node {{\small $E_{12}$}};
   \draw (-1.5,0) node {{\small $E_{13}$}};
   \draw (-4.2,1.3) node {{\small $E_{14}$}};
   \draw (-3.2,1.3) node {{\small $E_{15}$}};
   \draw (-2.2,1.3) node {{\small $E_{16}$}};
   \filldraw  (0,0) circle (2pt)
              (1,0) circle (2pt)
              (2,0) circle (2pt)
              (3,0) circle (2pt)
              (3,1) circle (2pt)
              (5,1) circle (2pt)
              (4,-1) circle (2pt)
              (5,-1) circle (2pt)
              (-1,0) circle (2pt)
              (-2,-1) circle (2pt)
              (-3,-1) circle (2pt)
              (-3,1) circle (2pt)
              (-4,1) circle (2pt);
   \filldraw  [fill=white]  (-2,1) circle (3pt)
                            (4,1) circle (3pt)
                            (3,-1) circle (3pt);
   \end{tikzpicture

   \end{center}

\noindent and the divisors given by the ideals are:

\vskip 2mm

$\cdot$ $F_1= (18,24,45,72,73,146,218,72,144,216,21,42,63,64,128,194)$, and

$\cdot$ $F_2= (18,24,45,72,72,144,216,74,147,222,21,42,63,64,128,194)$.

\vskip 2mm

\noindent with $K_\pi=(1,2,4,7,8,16,24,8,16,25,2,4,6,7,14,21)$.

\vskip 2mm

The divisors in the Newton nest are $E_4$ and $E_{13}$ and the log-canonical wall only has a unique ${\mathcal C}$-facet whose
supporting hyperplane has the following equation: \[ V_{4,1}: 72x_1+72x_2=8 \text{, or equivalently,  }  V_{13,1}: 63x_1+63x_2=7\,.\]
\end{example}

The main result of this section is that the one-to-one correspondence established by Cassou-Nogu\`es and Libgober  in \cite[Theorem 4.22]{CNL14} still holds in our setup except for this very particular case where we may have two divisors in the Newton nest and just a unique ${\mathcal C}$-facet.

\begin{theorem} \label{thm:Pi} Let
$\pmb{\mathfrak{a}}=\left(\mathfrak{a}_1,\ldots,\mathfrak{a}_r\right)\subseteq \left({\mathcal O}_{X,O}\right)^r$ be a tuple of ${\mathfrak m}$-primary ideals
where $X$ is a complex surface with a rational singularity at $O \in X$. Assume that the log-canonical wall has at least two ${\mathcal C}$-facets and all its points
have multiplicity one.
Then, there is a one-to-one correspondence between the exceptional divisors
 in the Newton nest of $\pmb{\mathfrak{a}}$  and the ${\mathcal C}$-facets of the log-canonical wall.
\end{theorem}

\begin{proof}
We start with the case $r=2$, that is $\pmb{\mathfrak{a}}=\left(\mathfrak{a}_1,\mathfrak{a}_2\right)$.
In order to construct the Newton nest of $\pmb{\mathfrak{a}}$ we start considering
two points $\pmb{z}_1$ and $\pmb{z}_2$ in the coordinate axes corresponding to the log-canonical
thresholds of the ideals $\mathfrak{a}_1$ and $\mathfrak{a}_2$ respectively.
Then we order the ${\mathcal C}$-facets of the log-canonical wall ${\mathcal C}_1, {\mathcal C}_2, \dots , {\mathcal C}_l$
in such a way that $\pmb{z}_1 \in {\mathcal C}_1$, $\pmb{z}_2 \in {\mathcal C}_l$ and each ${\mathcal C}$-facet ${\mathcal C}_i$
intersects ${\mathcal C}_{i-1}$ and ${\mathcal C}_{i+1}$. Roughly speaking, we are considering a path from $\pmb{z}_1$ to $\pmb{z}_2$
over the log-canonical wall.
By Corollary \ref{cor:Cfacet}, the exceptional divisors associated to the supporting hyperplanes of
${\mathcal C}_1, {\mathcal C}_2, \dots , {\mathcal C}_l$ are unique, but they are also different because the hyperplanes
have different slopes. Therefore,
we can order these dicritical or rupture divisors
$E_{j_1}, E_{j_2}, \dots, E_{j_l}$ accordingly to their corresponding ${\mathcal C}$-facets. Moreover,
Corollary \ref{cor:Cfacet}  implies that they form a path in the dual graph $\Gamma_{\pmb{\mathfrak{a}}}$ of the
log-resolution of $\pmb{\mathfrak{a}}$  with no dicritical or rupture divisors in between two consecutive $E_{j_i}$'s.

\vskip 2mm

Now, let $\Gamma_{\pmb{\mathfrak{a}}}' \subset \Gamma_{\pmb{\mathfrak{a}}}$ be the minimal connected subgraph containing $E_{j_1}$ and $E_{j_s}$.
Then we have that $E_{j_2}, \dots, E_{j_{s-1}}$  belong to $\Gamma_{\pmb{\mathfrak{a}}}'$
and the result follows because the Newton nest is the set  $\{E_{j_1}, E_{j_2}, \dots, E_{j_s}\}$ by construction.

\vskip 2mm

For the case $r>2$, that is $\pmb{\mathfrak{a}}=\left(\mathfrak{a}_1,\ldots,\mathfrak{a}_r\right)$, we  have to consider
points $\pmb{z}_1, \dots , \pmb{z}_r$ in the coordinate axes corresponding to the log-canonical
thresholds. We are going to pick one of them, say $\pmb{z}_1$, and a  ${\mathcal C}$-facet ${\mathcal C}_1$ containing this point $\pmb{z}_1$.
Let ${\bf c} \in {\mathcal C}_1$ be an interior point with rational coordinates and ${\bf q}:=(0,q_2,\dots,q_r)$ a point in the coordinate hyperplane
$\{x_1=0 \}$ with rational coordinates. Notice that ${\bf q}$ corresponds to the mixed multiplier ideal
$\mathcal{J}(\mathfrak{a}_2^{q_2} \cdots \mathfrak{a}_r^{q_r})=\mathcal{J}\left((\mathfrak{a}_2^{dq_i}\cdots \mathfrak{a}_r^{dq_r})^{\frac{1}{d}}\right)$ for some $d\in\mathbb{Z}$ such that $dq_i\in\mathbb{Z}$ for $i=2,\ldots,r$.

\vskip 2mm

The mixed multiplier ideals appearing in the restriction of the positive orthant $\mathbb{R}^r_{\geqslant 0}$
to the plane containing the points ${\bf c}$, ${\bf q}$ and the origin ${\bf 0}$  are the mixed
multiplier ideals of the duple $\pmb{\mathfrak{a}}'= ( \mathfrak{a}_1, \mathfrak{a}_2^{dq_2}\cdots \mathfrak{a}_r^{dq_r})$ so the facets of
the corresponding log-canonical wall are in one-to-one correspondence with the exceptional divisors
in the Newton nest of $\pmb{\mathfrak{a}}'$ which is contained in the Newton nest of $\pmb{\mathfrak{a}}$. Just moving the points
${\bf c}$ and ${\bf q}$ conveniently allows us to cover the whole log-canonical wall of $\pmb{\mathfrak{a}}$ and the result
follows. We point out that a log-resolution of $\pmb{\mathfrak{a}}$ is also a
log-resolution of $\pmb{\mathfrak{a}}'$.
\end{proof}

In the following example we show that with our definition of the Newton nest we may also
consider the case of non-simple ideals in a smooth surface which was not considered in \cite{CNL14}.

\begin{example} \label{ex:nest3}

Consider the tuple of ideals $\pmb{\mathfrak{a}}=(\mathfrak{a}_1,\mathfrak{a}_2,\mathfrak{a}_3)$  on a smooth surface $X$:

\vskip 2mm

$\cdot$ $\mathfrak{a}_1=( {y}^{3},{x}^{6}y,{x}^{8},{x}^{3}{y}^{2})$,

$\cdot$ $\mathfrak{a}_2=(x\left(x^2+x+y\right)^4,  \left( x^2+x+y \right)^2\left( x^2-x-y \right)^2, x^5 \left( x^2-x-y \right) ^2, x^5 \left( x^2+x+y \right) ^2,$

\hskip 1.3cm $x^3\left(x^2+x+y\right)\left( x^2-x-y \right)^2,$
$x^3 \left( x^2+x+y\right) ^3),$

$\cdot$ $\mathfrak{a}_3=($ $-x^6 \left( x-y \right),$ $x^7,$ $x^5
 \left( x-y \right) ^2,$ $-x^3 \left( x-y \right) ^3, x^
5 \left( x-y \right) ^2,$ $- \left( x-y \right) ^5,x^2 \left( x-y \right) ^4).$

\vskip 2mm

\noindent The dual graph of the log-resolution of $\pmb{\mathfrak{a}}$ is:

\hspace{1mm}
\begin{center}

\begin{tabular}{c}
\begin{tikzpicture}[scale=0.8]
   \draw  (0,0) -- (-1,0);
   \draw  (0,0) -- (1,1);
   \draw  (0,0) -- (1,-1);
   \draw  (4,1) -- (1,1);
   \draw  (4,-1) -- (1,-1);
   \draw  (-1,0) -- (-2,1);
   \draw  (-1,0) -- (-2,-1);
   \draw  (-2,1) -- (-3,1);
   \draw  (-2,-1) -- (-3,-1);
   \draw [dashed,->,thick,red] (3,1) -- (4,.5);
   \draw [dashed,->,thick,blue] (2,-1) -- (3,-.5);
   \draw [dashed,->,thick] (-2,-1) -- (-3,-.5);
   \draw [dashed,->,thick] (-2,1) -- (-3,.5);
   \draw (0.4,0) node {{ $E_1$}};
   \draw (0.7,1.4) node {{ $E_2$}};
   \draw (3.7,1.4) node {{ $E_3$}};
   \draw (1.7,1.4) node {{ $E_4$}};
   \draw (2.7,1.4) node {{ $E_5$}};
   \draw (-1.6,0) node {{ $E_{6}$}};
   \draw (-2.7,1.3) node {{ $E_{7}$}};
   \draw (-1.7,1.3) node {{ $E_{8}$}};
   \draw (-2.7,-1.3) node {{ $E_{9}$}};
   \draw (-1.7,-1.3) node {{ $E_{10}$}};
   \draw (3.7,-1.3) node {{ $E_{11}$}};
   \draw (2.7,-1.3) node {{ $E_{12}$}};
   \draw (.7,-1.3) node {{ $E_{13}$}};
   \draw (1.7,-1.3) node {{ $E_{14}$}};
   \filldraw  (1,1) circle (2pt)
              (2,1) circle (2pt)
              (4,1) circle (2pt)
              (1,-1) circle (2pt)
              (3,-1) circle (2pt)
              (4,-1) circle (2pt)
              (-3,1) circle (2pt)
              (-3,-1) circle (2pt);
   \filldraw  [fill=white]  (-2,1) circle (3pt)
                            (-2,-1) circle (3pt);
   \filldraw  [fill=magenta]  (3,1) circle (3pt);
   \filldraw  [fill=grey]  (0,0) circle (3pt);
   \filldraw  [fill=green]  (2,-1) circle (3pt);
   \filldraw  [fill=red]  (-1,0) circle (3pt);
   \end{tikzpicture}
\end{tabular}

\end{center}

\noindent For simplicity we denote the divisors associated with this resolution as

\vskip 2mm

$\cdot$ $F_1= (3,6,8,15,24,3,3,6,3,6,3,6,9,15)$,

$\cdot$ $F_2=(4,4,4,8,12,8,9,18,9,18,4,8,12,20)$,

$\cdot$ $F_3=(5,5,5,10,15,5,5,10,5,10,7,14,20,35)$.

\vskip 2mm

\noindent In the same manner, the relative canonical divisor is $K_{\pi}=(1,2,3,6,10,2,3,6,3,6,2,4,6,10)$.
The Newton nest of $\pmb{\mathfrak{a}}$ consists of the exceptional divisors $E_1, E_{5}, E_{6}$ and $E_{14}$ and they correspond, matching the colors, to the ${\mathcal C}$-facets
of the log-canonical wall which is:

\begin{center}

\begin{tikzpicture}
\begin{axis}[view/h=120
\addplot3 [patch, magenta]
table {
x y z

0.45833333333333 0 0
0.41025641025641 0.346153846153846 0
0.33333333333333 0 0.2
};
\addplot3 [patch, grey]
table {
x y z

0.41025641025641 0.346153846153846 0
0.33333333333333 0 0.2
0 .25 .2
};
\addplot3 [patch, red]
table {
x y z

0 0.375 0
0.41025641025641 0.346153846153846 0
0 .25 .2
};
\addplot3 [patch, green]
table {
x y z

0 0 0.342857142857143
0.33333333333333 0 0.2
0 .25 .2
};
\end{axis}
\end{tikzpicture}
\end{center}

\end{example}

In the following example we present a case where $X$ has a log-canonical singularity
and Theorem \ref{thm:Pi} does not hold. This shows how sharp is the condition of having
points in the log-canonical wall with multiplicity one.

\begin{example}\label{ex:Sing_racional_2}

Consider a surface $X$  with a rational singularity at $O$ whose
minimal resolution $\pi:X' {\longrightarrow} X$ has six exceptional components $E_1,\dots, E_6$
with the following dual graph and intersection matrix:

\begin{center}

   \begin{tikzpicture}[scale=0.9]
   \draw  (0,0) -- (3,0);
   \draw  (1,0) -- (0,1);
   \draw  (2,0) -- (3,1);
   \draw [dashed,->,thick] (0,0) -- (-1,1);
   \draw [dashed,->,thick] (0,0) -- (-1.2,0.67);
   \draw [dashed,->,thick] (0,0) -- (-1.3,0.33);
   \draw [dashed,->,thick] (0,0) -- (-1.4,0);
   \draw [dashed,->,thick] (3,0) -- (4,1);
   \draw [dashed,->,thick] (3,0) -- (4.4,0);
   \draw [dashed,->,thick] (0,1) -- (-1.2,1.67);
   \draw [dashed,->,thick] (1,0) -- (0.5,1.1);
   \draw [dotted,thick] (3.75,0) arc [radius=.75, start angle=0, end angle= 45];
   \draw (1.8,-0.3) node {{\small $E_1$}};
   \draw (0.8,-0.3) node {{\small $E_2$}};
   \draw (2.8,1.3) node {{\small $E_3$}};
   \draw (2.8,-0.3) node {{\small $E_4$}};
   \draw (0,1.3) node {{\small $E_5$}};
   \draw (-.2,-0.3) node {{\small $E_6$}};
   \draw (3.9,0.3) node {{\tiny $12$}};
   \filldraw  (3,1) circle (2pt);
   \filldraw  [fill=white]  (0,0) circle (3pt)
                            (1,0) circle (3pt)
                            (2,0) circle (3pt)
                            (0,1) circle (3pt)
                (3,0) circle (3pt);
   \end{tikzpicture}\\ \parbox{30mm}{\begin{center} Vertex ordering\end{center}}\\

   \end{center}

  $$M=(E_i\cdot E_j)_{1\leqslant i,j\leqslant 6}=
  \left( \begin {array}{cccccc} -2&1&1&1&0&0\\ \noalign{\medskip}1&-3&0&0&1&1\\ \noalign{\medskip}1&0&-1&0&0&0\\
\noalign{\medskip}1&0&0&-3&0 &0\\ \noalign{\medskip}0&1&0&0&-3&0\\
\noalign{\medskip}0&1&0&0&0&-6
\end {array} \right)}.$$

\vskip 2mm

The fundamental cycle is the divisor $Z = (3, 2, 3, 1, 1, 1)$ and the relative canonical divisor is
$K_\pi=\left(-\frac{1}{2},-1,\frac{1}{2},-\frac{1}{2},-\frac{2}{3},-\frac{5}{6}\right)$ so the singularity is log-canonical. Then we consider a
duple of ideals $\pmb{\mathfrak{a}}=(\mathfrak{a}_1,\mathfrak{a}_2)$, with $\mathfrak{a}_1$ non singular
and $\mathfrak{a}_2=\mathfrak{m}$ given by the  divisors  $F_1=(15, 6, 15, 9, 2, 1)$ and
$F_2=Z = (3, 2, 3, 1, 1, 1)$. The log-canonical wall has two ${\mathcal C}$-facets and the corresponding mixed multiplier ideals are different (see
Figure 2).
In particular we have jumping points on the log-canonical wall with multiplicity bigger that $1$. In this case the Newton nest consists of the exceptional divisors
$E_1, E_2$ and $E_4$ so we no longer have
the bijection  given in Theorem \ref{thm:Pi}.

\begin{figure}[ht!!!]
  \begin{center}
  \medskip
    \includegraphics[width=.40\textwidth]{Ex_Sing_racional2.pdf}

  \end{center}
\caption{Constancy regions of the the mixed multiplier ideals of $\pmb{\mathfrak{a}}$. }
\end{figure}
\end{example}

redpajama_set_name: RedPajamaArXiv


text:
. . . And to All Some Good Nights
illustration: Gina Triplett and Matt Curtius
Jaded New Yorkers bored with the usual holiday fare take heed. No need to ooh and aah at big trees and sleep through dancing fairies in order to indulge the family. We're here to help. Our list of seasonal entertainment is filled with gay Santas, boozing reindeer, and other offbeat fun. And if you decide not to venture out until 2006, we've got some long DVDs to keep you busy till then. Enjoy!
'A Very Bette Christmas'
Welcome to bitchery beneath the mistletoe. Novelist Elizabeth Fuller, who once played host to Bette Davis during an impromptu month-long stay, imagines the difficult Ms. Davis as star of a 1962 televised special. Through song, story, and hissed aside, famed female impersonator Tommy Femia will play Bette as she cows co-host Joan Crawford and spars with the show's director. SOLOSKI
Through Jan 8, Don't Tell Mama, 343 W 46th, 212-757-0788
'The Eight: Reindeer Monologues'
Apparently Dasher, Dancer, Prancer, Vixen, et al. get up to some fairly kinky reindeer games. In this series of monologues by Jeff Goode, Santa's eight-deer team gathers on-stage and dashes away eggnog-curdling confessions of boozing, bestiality, and the indignities suffered at the hands of Santa Claus. Come buck the Christmas spirit. SOLOSKI
Through Dec 17, Red Room, 85 E 4th, 212-868-4444
Dyker Heights Christmas Lights
Dreamy and even a little grotesque, the Christmas lights in Dyker Heights seem to get increasingly elaborate every year. By mid December, the lawns on Tenth Avenue are littered with high-voltage dioramas, 20-foot toys, and life-size reindeer. Approximately 100,000 people visit each season (many in SUVs)—it's best to walk, not drive. AVIV
83rd & 84th sts, btwn Tenth & 13th aves, Bklyn
'The Baby Jesus One-Act Jubilee'
If exchanging shiny commodities and drinking too much at the office party are popular ways of celebrating the birth of the Christ, then commissioning a one-act-play festival to honor his natal day isn't such an odd choice. Williamsburg's Brick Theatre presents 12 15-minute plays in two programs: "Shepherds" and "Magi." SOLOSKI
Dec 1 through 17, Brick Theatre, 575 Metropolitan Ave, Bklyn, 718-907-3457
'Balletto Stiletto'
Mary Fulham, an original member of the '80s all-girl comedy act the High Heeled Women, directs a multimedia play about nine daughters who can't stop doing ballet. It's like the Grimm fairy tale "The Twelve Dancing Princesses," except the sisters are "drugged out," fond of beer, and live in New Jersey. AVIV
Dec 1 through 18, La MaMa E.T.C., 74A E 4th, 212-475-7710
'Celebrating Fifty Years of Jim Henson & the Muppets'
Whether you're five or 55, the odds are good you've been charmed by one of Henson's colorful, furry oddballs. (Even our government's given them accolades with their own postage stamp!) This two-month retrospective celebrates Henson's creations—Kermit, Miss Piggy, Bert and Ernie, Cookie Monster—with rare screenings of specials (many holiday related, such as The Great Santa Claus Switch), episodes, clips, and gems from various TV appearances. BOSLER
Dec 2 through Jan 26, Museum of Television & Radio, 25 W 52nd, 212-621-6800, mtr.org
Holiday Cooking Courses
Caught in a teary rush of sentimentality, many people with absolutely no culinary talents volunteer to cook massive Christmas feasts every year. Looking out for the guests, the Institute of Culinary Education offers several one-day courses on various kinds of holiday food, including ham, cakes, stews, puddings, biscuits, and eggnog. AVIV
Dec 3 through 31, Institute of Culinary Education, 50 W 23rd, 212-847-0700
'4th Annual Gingerbread on Broadway'
After spending hundreds of hours creating new candy forms, students from various culinary schools across the East Coast will showcase their best attempt at a gingerbread home. In previous years, the chefs have created more colorful versions of the Brooklyn Bridge, Rockefeller Center, and Central Park carousel. AVIV
Dec 5, New York Marriott Marquis, 1535 Bway, 212-398-1900
Holiday Poetry Book Sale
The largest poetry library in the country offers a "maddeningly various" selection of obscure journals, recordings, and chapbooks, some of them written at the beginning of the century. Many of these books (which sell for $7 or less) are long out of print and impossible to purchase elsewhere. AVIV
Dec 9 & 10, Poets House, 72 Spring, 212-431-7920
What with Tiny Tim's crippling illness, Jacob Marley's rattling chains, and the spooky Ghost of Christmas Future, we've always found Dickens's story a rather creepy holiday classic. But the family-oriented Theatreworks/NYC has tempered the tale into a musical appropriate for children five and up. With a live band and a cast of Broadway vets, audiences won't likely be saying "Bah, humbug." SOLOSKI
Dec 10 through 31, Lucille Lortel Theatre, 121 Christopher, 212-279-4200
'Original Holiday Doo-Wop Extravaganza'
Ever wonder where all the boybands of a few years back stole their moves and vocal harmony ideas from? Your ole uncle Barney knows, and b'gosh, instead of getting him another bad tie or shaving kit this season, treat him to a time-machine ride to his past via Dick Fox's doo-wop holiday show. Featuring (the probably not-so-young) Frankie Lymon's Teenagers, Johnny Maestro & the Brooklyn Bridge, and Kenny Vance & the Planotones. BOSLER
Dec 10, 8 pm, North Fork Theatre, Westbury Music Fair, 960 Brush Hollow Rd, Westbury, NY, 516-334-0800
Early Music New York
Dressed in black and holding various excit- ing medieval instruments like shawm and sackbuts, the men's vocal ensemble will sing 14th-century motets in front of the Metropolitan Museum of Art's 20-foot Christmas tree. Many of their songs honor the Good King Wenceslas, the Duke of Bohemia, born in 907 and murdered on his way to church. AVIV
Dec 11, Medieval Sculpture Hall at the Metropolitan Museum of Art, 1000 Fifth Ave, 212-570-3949
'Red Hot Holiday Stomp'
Wynton's Crescent City Christmas card always makes the cut at my house during holiday time. This show looks to be a live update of the same. Second-lining through "Sleigh Ride" could be a thrill, and I bet the team—which includes Wycliffe Gordon, Joe Temperly, Victor Goines, and banjo player Don Vappie—will put a little Preservation Hall polyphony on the book. They also plan to romp through pieces by that "Spanish Tinge" Santa, Jellyroll Morton. MACNIE
Dec 15 through 17, 8pm; Dec 18, 3pm; Jazz at Lincoln Center, Time Warner Center, Rose Theater, Bway & 60th, 212-258-9800
'Twas the Night Before the Twelve Days of a Nutcracker Christmas Carol'
In Ken Nintzel's last piece, Lapse, he used a Hallmark's worth of crepe centerpieces and paper chains to depict a year in the life of a coma patient. Though he's now concentrating exclusively on Christmas, this show recombines several seasons worth of holiday song and lore. SOLOSKI
Dec 15 through 25, PS122, 150 First Ave, 212-352-3101
'Jewish Soul Celebration'
With popular songs like "Go Down Moses" and "Observant Jew," Joshua Nelson, the self-described Prince of Kosher Gospel Music, hopes to "spearhead a new revolution in Jewish liturgical" tunes. In a womanly voice, he performs a combination of soul, klezmer, and prayer—it's the sound of "suffering speckled with joy." AVIV
Dec 17, Symphony Space, 2537 Bway, 212-864-5400
'Santacon'
Sometime this December (the organizers are very secretive), more than 400 men and women, dressed as Santas, will wander drunkenly through the streets "for absolutely no reason whatsoever." Go to santacon.com to keep track of this "not-for-profit, non-political, non-religious, and non-logical Santa Claus convention." AVIV
TBA, various venues, santacon.com
'The Trachtenburg Family Slideshow Players . . . on Ice!'
Those few who believe slide shows are the province of sleepy, high school biology classes or glassy-eyed evenings at Grandpa's have clearly never seen a Trachtenburg Family show. In these incomparably indie evenings, Dad sings, Mom runs the slide carousel, and nine-year-old Rachel drums and narrates. Occasionally, there are cupcakes. This latest incarnation features holiday-themed visuals and live skating. SOLOSKI
Dec 18 through 23, P.S.122, 150 First Ave, 212-352-3101
"You will love it—it's totally your kind of music, that hard metal stuff . . . but you know, for Christmas!" says Mom. She may be way off on what you like, but there could be far worse ways to experience a holiday musical. Created by heavy-metal producer Paul O'Neill, TSO is the Polar Express of rock operas, tailor-made for prog-metal lovers, aging rockers, kids, and any/all fans of showy Broadway spectacles. Expect exuberantly electrified Christmas songs aplenty, especially tunes from their most recent, The Lost Christmas Eve. BOSLER
Dec 19, Madison Square Garden, Seventh Ave & 32nd, 212-465-MSG1
'The Beat Goes On: A Rock 'n' Glitter Xmas'
Favorite local musicians including Willie Nile ( Beautiful Wreck of a World), Don Fleming, actor-singer Carlton Smith, folk duo the Kennedys, and surprise guests take the stage to perform some classic holiday ditties from the Yule log—oops, catalog—of rockers, including the Ramones, the Pretenders, Paul McCartney, XTC, James Brown, and much more. ABER
Dec 21, B.B King Blues Club & Grill, 237 W 42nd, 212-997-4144, bbkingblues.com
Santa Claus Is Coming Out, or How the Gay Agenda Came Down My Chimney'
Historically, Santa makes lists of naughty and nice, rather than straight and gay. But Jeffrey Solomon puts the "ho, ho, ho" in homosexuality in this one-man show, in which a boy's wish for a doll somehow outs Santa. (What will Mrs. Claus say?) SOLOSKI
Dec 21 through 23, Queens Theatre in the Park, Flushing Meadow–Corona Park, Qns, 718-760-0064
Sledding at Riverside Park
Sledding is rarely as fun as it's supposed to be—snow mysteriously sneaks up your sleeves, or down your boots, and the walk up the hill is too long. If you insist on trying, the slopes at Riverside Park are the steepest (far better than Central Park), and some even have what enthusiasts call "moguls" (e.g., bumps, sticks, rocks). Enter at 91st or 108th for the best ride. AVIV
Riverside Park at 91st or 108th
Matzo Ball Party
Small talk and desperate jokes come more easily when you're a Jew on Christmas Eve. Hosted by the Society of Young Jewish Professionals, this party is good for '80s tunes, kosher sandwiches, and drunken, desperate coupling. Gentiles can join too. AVIV
Dec 24, Capitale, 130 Bowery, 888-633-5326
Wave Hill Conservatory
Like a giant sauna with flowers, the Wave Hill Conservatory is a comforting place to go when the weather begins to dip below freezing. With temperatures around 80 degrees, you can simultaneously walk, talk, and look at nature without having to constantly adjust all your baggy winter clothes. AVIV
Wave Hill Conservatory, 675 W 252nd, Bx 718-549-3200
'Chinese Food and a Movie on Christmas Day: Born to Be Wilder'
Watching movies (doing anything) is always better while sporadically and thoughtlessly snacking. Making good on the most beloved of Jewish clichés, Makor hosts a Christmas evening of unlimited Chinese food in the movie theater. The center will screen Young Frankenstein and Silver Streak, starring Gene Wilder (born Jerome Silberman), and director Billy Wilder's Some Like It Hot and The Apartment—perfect pick-me-ups for an otherwise lonely day. AVIV
Dec 25, Steinhardt Building, 35 W 67th, 212-601-1000, makor.com
Coney Island Polar Bear Club's Annual New Year's Day Swim
To understand the joys of diving repeatedly into freezing cold water, join the country's oldest winter-swimming organization on Coney Island beach every Sunday. Wearing nothing but a bathing suit and neoprene surf boots, members of the group dip into the Atlantic while onlookers cheer. On January 1, hundreds of people will join the club for a painful yearly ritual, featuring plenty of screaming. AVIV
Every Sun through May & Jan 1, Coney Island Beach, 718-356-7741, polarbearclub.org
More:Bette ChristmasBrooklyn (New York City)Coney IslandCultureHoliday GuideSanta Claus

redpajama_set_name: RedPajamaCommonCrawl


text:
Il Capo Passero è l'estrema punta sud-orientale dell'isola di Sicilia e anche il nome dell'isolotto sito a pochi metri di distanza. Si trova nel territorio del comune di Portopalo, in provincia di Siracusa. Il capo è dominato da un faro.

Nel XVI secolo, il Capo Passero fu oggetto di intense attività piratesche turco-barbaresche. I pirati, infatti, al capo si rifornivano di acqua e saccheggiavano la zona, prendendo talvolta in schiavitù gli europei in cui si imbattevano. Nel 1607, fu completato il Forte di Capo Passero, che ebbe una funzione difensiva almeno fino al 1830.

La località è famosa per la battaglia navale, avvenuta nel 1718, nel corso della quale la flotta spagnola venne sconfitta dalla flotta inglese comandata dall'ammiraglio sir George Byrn, facendo l'Inghilterra parte della Quadruplice Alleanza (Austria, Gran Bretagna, Francia e Paesi Bassi).

La tonnara

Di particolare impatto visivo, la tonnara di Capo Passero ricoprì un ruolo più che mai fondamentale nell'economia della zona, lungo tutta la sua esistenza.
Situata appena fuori dal paese in direzione Marzamemi, ha origini antichissime risalenti fino al Medioevo, ma è solo dalla seconda metà del Settecento che la tonnara moderna comincia a prendere la forma attuale, grazie all'intervento del principe di Villadorata, Corradino Nicolaci.

L'attività di pesca continuò fino a fine secolo, ma - in seguito a una cessione di proprietà - s'interruppe ai primi anni dell'Ottocento. La tonnara venne riattivata nel 1895 per opera di don Pietro Bruno di Belmonte, che all'epoca possedeva i titoli di esercizio dell'attività ittica, e continuò la sua attività annualmente fino al 1969. Da allora, sino al 2000, si passò a una cadenza quinquennale. Oggi è completamente inattiva, ma svolge, insieme con il castello Tafuri, un importante ruolo folcloristico e ambientale per i residenti e i turisti della zona.

Mario Soldati, nel suo viaggio alla scoperta dei vini genuini, nel 1968, che poi riportò sulla pubblicazione 'Vino al Vino' scrisse in riferimento a Capo Passero: "Per arrivarci, abbiamo attraversato chilometri e chilometri di vigneti: e qui è la tonnara. Non vi accennerei neanche, se non fosse uno dei luoghi più meravigliosi che ho visto, in tutto il mondo, America e Africa comprese". Nel 1981 vennero scoperte dieci vasche in muratura, nella spiaggia "del Collo", vicino Scalo Mandrie. All'interno, sono state trovate vertebre di tonno e numerose monete nonché resti di lavorazione del pesce, conservato sotto sale.

Note

Bibliografia
 Salvo Sorbello, La pesca del tonno nel capolinea del sud. Le tonnare di Vendicari, Marzamemi e Portopalo di Capo Passero. ISBN 978-88-7428-093-3.

Voci correlate
 Battaglia di Capo Passero (1718)
 Battaglia di Capo Passero (1940)
 Isola di Capo Passero
 Forte di Capo Passero
 Portopalo di Capo Passero

Collegamenti esterni

Passero, capo

redpajama_set_name: RedPajamaWikipedia


text:
Q: Bind 2 dataframes with dataframes elements I need to bind 2 dataframes that contain dataframe as element. The problem - inlayed dataframes column names could be different, and number of columns could be different. That's the result I need:

Here is my data:

issue_desc_1 <- data.frame(
                   MinAvailableMb = 50,
                   Threshold = 100
                   )

issues.df_1 <- data.frame(
    IssueId = 1,
    IssueSubId = 1,
    Categories = "Cat1",
    Solution = "Sol1",
    Level = 'Critical',
    AffectedObjects = "comp1.domain.local",
    Arguments = I(issue_desc_1)
)

issue_desc_2 <- data.frame(
                     MaxCommitedGB = 82,
                     RamSize = 64,
                     Threshold = 64
)

issues.df_2 <- data.frame(
    IssueId = 1,
    IssueSubId = 2,
    Categories = "Cat1",
    Solution = "Sol1",
    Level = 'Critical',
    AffectedObjects = "comp2.domain.local",
    Arguments = I(issue_desc_2)
)

Arguments element is dataframe here.
I am trying different ways to bind dataframes but I'm getting errors.
rbind(issues.df_1, issues.df_2) and rbind(issues.df_1, issues.df_2, fill = T)
Warning in [<-.data.frame(*tmp*, ri, , value = list(MaxCommitedGB = 82,  :
provided 3 variables to replace 2 variables
Error in dim(rvec) <- dim(x) :
dims [product 6] do not match the length of object [2]
plyr::rbind.fill(issues.df_1, issues.df_2)
Error in allocate_column(df[[var]], nrows, dfs, var) :
Data frame column 'Arguments' not supported by rbind.fill
bind_rows(issues.df_1, issues.df_2)
Error in dim(rvec) <- dim(x) :
dims [product 8] do not match the length of object [4]
How to create dataframe I need from 2 dataframes I have?

A: The solution is - to use tibble instead of data.frame. That case command dplyr::bind_rows(issues.df_1, issues.df_2) gives correct result.
library(jsonlite)
library(dplyr)

issue_desc_1 <- tibble(
                   MinAvailableMb = 50,
                   Threshold = 100
                   )

issues.df_1 <- tibble(
    IssueId = 1,
    IssueSubId = 1,
    Categories = "Cat1",
    Solution = "Sol1",
    Level = 'Critical',
    AffectedObjects = "comp1.domain.local",
    Arguments = (issue_desc_1)
)

issue_desc_2 <- tibble(
                     MaxCommitedGB = 82,
                     RamSize = 64,
                     Threshold = 64
)

issues.df_2 <- tibble(
    IssueId = 1,
    IssueSubId = 2,
    Categories = "Cat1",
    Solution = "Sol1",
    Level = 'Critical',
    AffectedObjects = "comp2.domain.local",
    Arguments = (issue_desc_2)
)

dplyr::bind_rows(issues.df_1, issues.df_2)

The result:
# A tibble: 2 x 7
  IssueId IssueSubId Categories Solution Level    AffectedObjects    Arguments$MinAvailableMb $Threshold $MaxCommitedGB $RamSize
    <dbl>      <dbl> <chr>      <chr>    <chr>    <chr>                                 <dbl>      <dbl>          <dbl>    <dbl>
1       1          1 Cat1       Sol1     Critical comp1.domain.local                       50        100             NA       NA
2       1          2 Cat1       Sol1     Critical comp2.domain.local                       NA         64             82       64

redpajama_set_name: RedPajamaStackExchange


text:
THIS ARTICLE/PRESS RELEASE IS PAID FOR AND PRESENTED BY NIBIO - Norwegian Institute of Bioeconomy Research - read more
Sardinia holds almost half of the sheep in Italy.
(Photo: Colourbox)
Norwegian researchers exchange knowledge about greenhouse gases from sheep with colleagues in Sardinia
Despite major climatic and operational differences, there are some similarities that provide a basis for important exchange of expertise.
Liv Jorunn Hind Communications Adviser
Presented by: NIBIO - Norwegian Institute of Bioeconomy Research
saturday 21. January 2023 - 04:30
With its 24,000 km2, Sardinia is the second largest island in the Mediterranean. For thousands of years Sardinia has specialised in sheep breeding, and today the island has one of the highest sheep densities in the world.
More than three million sheep graze here, almost half of all the sheep in Italy. By comparison, Norway has around two million sheep on pasture.
Sardinia is a mountainous island with great grazing potential, especially in the highlands. The lowlands, on the other hand, offer fertile soils well suited both for grazing and for intensive farming. However, with so many sheep in a relatively limited area, the animals must be fed with both roughage and concentrated feed.
Northern Norway and Sardinia represent different climate zones and different production systems. The sheep in Norway are primarily used to produce meat and wool, while the sheep in Sardinia are primarily used for milk production. Despite these differences, the researchers have found learning opportunities and value in knowledge exchange.
"Our hypothesis is that grazing animals emit less greenhouse gases when they graze than when they are fed in the barn," researcher Vibeke Lind says.
(Photo: Vibeke Lind)
Knowledge exchange on greenhouse gases from grazing animals
In 2022, project manager Vibeke Lind and research technician Jon Asbjørn Johnsen, both from NIBIO, travelled to Sardinia. The aim was to exchange experiences with Italian researchers about more climate-friendly sheep farming, and to provide direct guidance in greenhouse gas measurements from sheep at pasture.
Local researchers received training in the SF6 method for measuring gas emissions on pasture.
"We have gained many useful experiences that are worth sharing," Lind says.
For the past years, she has strengthened NIBIO's expertise in measuring greenhouse gas emissions from sheep. From NIBIO Tjøtta in Northern Norway, she led the first research project in Norway with a respiration chamber for greenhouse gas measurements from sheep. In addition, the researchers measured greenhouse gas from sheep on pasture using the SF6 tracer gas technique, a method where the animals wear a halter with a canister that collects gas emitted from the sheep's burp.
"The SF6 method on grazing animals has been shown to give measurements that correspond well with controlled measurements in respiration chambers," Lind says. "The various methods have their advantages and disadvantages. Measurements made in respiration chambers limit the sheep's natural grazing behaviour. On the other hand, we have poorer control of feed intake when sheep are grazing. During the experiments at Tjøtta, we collected feed samples from the pastures. This provided a basis for viewing feed analyses in connection with measured methane gas emissions. Our hypothesis is that grazing animals emit less greenhouse gases than when they are fed roughage and concentrate in the barn."
There has been a gradual development in the grazing industry towards a more efficient operation, both in Sardinia and in Norway. The sheep are often kept in the lowlands instead of grazing in the mountain areas. This causes a lot of overgrowth in the highlands.
Development in the grazing industry
In Sardinia, as in Norway, there has been a gradual development in the grazing industry towards a more efficient production. The sheep are often kept in the lowlands instead of grazing in the mountain areas. This leads to overgrowth in the highlands, like what we see in Norway.
Pecorino is a type of Italian cheese made from sheep's milk. The cheese is compared to parmesan, but has a sharper and clearer sheep's milk taste. The different regions have different variants. In Sardinia they make pecorino sardo.
"Traditionally, in Sardinia the soil is turned over every year for grass production. This means more use of fertiliser, more processing, and more machineries on the land. All this contribute to higher greenhouse gas emissions from the soil, compared to the use of perennial plants and turning the soil every three years. In this type of production system, mixtures of grasses and meadow legumes such as clover are important," Lind says.
On their visit to Sardinia in 2022, Lind and Johnsen brought equipment for the SF6 technique with them. Through the collaboration, local researchers received training in the method for measuring emissions from sheep grazing on annual and perennial pastures.
"Measuring greenhouse gases from animals in mountainous areas is difficult. We want to continue the collaboration with the researchers in Sardinia. We believe that this will give us important knowledge about greenhouse gas measurements from sheep under different operating systems to benefit both nationally and internationally," Lind concludes.
This article/press release is paid for and presented by NIBIO - Norwegian Institute of Bioeconomy Research
This content is created by NIBIO's communication staff, who use this platform to communicate science and share results from research with the public. NIBIO is one of more than 80 owners of ScienceNorway.no. Read more here.
See more content from NIBIO:
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The sense of apples as part of cultural identity
40 years of service to plant virology
Antarctic sea - ice plays an important role in regulating Earth's energy budget
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climate change partner farming sheep greenhouse gas emissions nibio greenhouse gas


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Gatehouse Chambers goes live and moves premises
Following our press release last month, Hardwicke officially changes its name to Gatehouse Chambers.
The change in name also coincides with Gatehouse Chambers' move to state of the art and impressive new premises at 1 Lady Hale Gate, Gray's Inn.
Although sad to leave Lincoln's Inn after many happy years, our move is an exciting development for the barristers and staff at Gatehouse Chambers who have enjoyed growing commercial success over recent years, involvement in some of the highest profile litigation work, and are the authors of a number of leading market texts.
Both the move and the name change are in line with Gatehouse Chambers' commitment to attract and support talented people to have rewarding and fulfilling careers. Gatehouse Chambers is a modern, forward-thinking organisation which prides itself on its values, leadership and strategic management in underpinning its commercial success. It is proud to launch the Gatehouse Chambers Charter as an organisational pledge in respect of its responsibilities to equality, diversity and inclusion.
Amanda Illing, Chief Executive says: "We have launched as Gatehouse Chambers today after a prolonged period of extraordinary success and growth. We have made bold decisions based on our values, to enable us to continue to provide a first-class service and environment for clients, in an amazing and collaborative place to work for barristers and staff. The future looks very exciting!"
See further details of Gatehouse Chambers being joined today by leading personal injury/clinical negligence practitioners Emma Zeb and Vanessa McKinlay.
Gatehouse Chambers can be located at: 1 Lady Hale Gate, Gray's Inn, London, WC1X 8BS
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If you would like further information, please contact Amanda Illing, Chief Executive (020 7242 2523/amanda.illing@gatehouselaw.co.uk).