text:
\begin{document}

\title[Cohen-Macaulay Du Bois singularities with a torus action]{Cohen-Macaulay Du Bois singularities \protect\\
 with a torus action of complexity one}

\author{Antonio Laface}

\address{Departamento de Matem\'atica, Universidad de Concepci\'on,
  Casilla 160-C, Concepci\'on, Chile} \email{alaface@udec.cl}

\author{Alvaro Liendo}

\address{Instituto de Matem\'atica y F\'isica, Universidad de Talca,
  Casilla 721, Talca, Chile} \email{aliendo@inst-mat.utalca.cl}

\author{Joaqu\'\i n Moraga}

\address{Department of Mathematics, University of Utah, 155 S 1400 E,
  Salt Lake City, UT 84112} \email{moraga@math.utah.edu}

\subjclass[2010]{Primary 14B05,
Secondary 14L30, 14M25.
}

\thanks{ The first author was partially supported by Proyecto FONDECYT
  Regular N. 1150732.  The second author was partially supported by
  Proyecto FONDECYT Regular N. 1160864.}

\maketitle

\medskip

\begin{abstract}
  Using Altmann-Hausen-S{\"u}\ss\ description of
  $\mathbb{T}$-varieties via divisorial fans and
  K\'ovacs-Schwede-Smith characterization of Du Bois singularities we
  prove that any rational $\mathbb{T}$-variety of complexity one which
  is Cohen-Macaulay and Du Bois has rational singularities. In higher
  complexity, we prove an analogous result in the case where the Chow
  quotient of the $\mathbb{T}$-variety has Picard rank one and trivial
  geometric genus.
\end{abstract}

\setcounter{tocdepth}{1}
\tableofcontents

\section*{Introduction}

We study the singularities of $\mathbb{T}$-varieties, i.e. normal
algebraic varieties endowed with an effective action of an algebraic
torus $\mathbb{T}=(\kk^*)^k$, where $\kk$ is an algebraically closed
field of characteristic zero.  Given a $\mathbb{T}$-variety $X$ we
define its {\em complexity} to be $\dim(X)-\dim(\mathbb{T})$.
$\mathbb{T}$-varieties of complexity zero are called {\em toric}
varieties and they admit a well-known combinatorial description in
term of fans of polyhedral cones~\cite{KKMS73,CLS}. Such combinatorial
description can be generalized to higher complexity. The complexity
one case was considered in ~\cite{KKMS73,FZ,Tim08}. Altmann,
Hausen and S{\"u}\ss\ generalize this combinatorial description for
arbitrary complexity using the language of polyhedral divisors and
divisorial fans on normal projective algebraic varieties~\cite{AH05,
  AHH}. We will use the language of polyhedral divisors in this paper. In this context, an
affine $\TT$-variety $X$ is determined by a pair $(Y,\mathcal{D})$,
where $Y$ is the Chow quotient for the $\TT$-action on $X$ and
$\mathcal{D}$ is a p-divisor on $Y$, i.e., a formal finite sum of
prime divisors on $Y$ whose coefficients are convex polyhedra. Recall
that the Chow quotient of $X$ by the action of $\mathbb{T}$ is the
closure of the set of general $\mathbb{T}$-orbit closures seen as
points in the Chow variety (see Section~\ref{sec1} for the details).

Using toric resolution of singularities is it possible to prove that
every normal toric variety $X$ has log terminal singularities
~\cite[Proposition 11.4.24]{CLS}.  Therefore, normal toric varieties
have rational singularities and then are Cohen-Macaulay and Du Bois,
see~\cite{KM98}. However, in higher complexity none of this
statements is true if we do not impose conditions on the defining
combinatorial data. For instance, a simple computation shows that the
affine cone $X_C$ over a plane curve $C$ of genus $g$ is a normal
$\TT$-surface of complexity one which is log canonical if and only if
$g$ equals $0$ or $1$. Furthermore, in the case that $C$ is a curve of genus $g\geq 1$, the
singularity at the vertex is not rational~\cite{LS} and so by virtue
of~\cite{KK10}, $X$ is an example of a $\TT$-variety of complexity one
which is Cohen-Macaulay and Du Bois but has a singularity which is not
rational.

Therefore it is worthwhile to study characterizations of this kind of
singularities for $\TT$-varieties of higher complexity in terms of the
defining p-divisor. There are such characterizations of rational
singularities and $\qq$-Gorenstein singularities and partial
characterizations of Cohen-Macaulayness in terms of the defining
p-divisors (see~\cite{LS}). In the case of positive characteristic,
there are also characterizations of $F$-split and $F$-regular
$\mathbb{T}$-varieties of complexity one in terms of p-divisors (see~\cite{AIS}).

Cohen-Macaulay singularities are the most natural singularities which
have the same cohomological behavior than smooth varieties, for
instance vanishing theorems still hold for varieties with
Cohen-Macaulay singularities (see~\cite{KM98}).  Du Bois singularities
also play an important role in algebraic geometry since these are the
singularities appearing in the minimal model program and moduli
theory (see~\cite{Sch07,SS11}).  In this article, we study
$\mathbb{T}$-varieties with Cohen-Macaulay and Du Bois singularities.

We will restrict our attention to a certain class of
$\mathbb{T}$-varieties having a nice resolution of the rational map
into the Chow quotient. Let $X$ be a $\mathbb{T}$-variety having Chow
quotient $Y$ and let $r:X\dashrightarrow Y$ be the rational quotient
map. We say that $X$ \emph{has a rational resolution of the Chow
  quotient} if the normalization of the closure of the graph of $r$
has rational singularities. Remark that in this case $Y$ also has
rational singularities by \cite{Bou}, but $X$ may have worst
singularities.

Using Kov\'acs-Schwede-Smith characterization
of Du Bois singularities \cite[Theorem 1]{KSS} we prove the following
result:

\begin{introthm}\label{thm1}
  Let $X$ be a $\mathbb{T}$-variety having a rational resolution of
  the Chow quotient. If the Chow quotient $Y$ has Picard rank one
and $p_g(Y)=0$, then $X$ is Cohen-Macaulay and Du Bois if and
  only if $X$ has rational singularities.
\end{introthm}

Rational $\mathbb{T}$-varieties of complexity one are a very natural
generalization of toric varieties and have been extensively studied
from the point of view of topology, singularities, Cox rings, and
automorphisms, among others, see~\cite{AHHL14,LL16,BHHN16}. The
previous theorem holds for example in this case. Indeed, all
$\mathbb{T}$-varieties of complexity one have a rational resolution of
the Chow quotient \cite[Chapter~4]{KKMS73} and the Chow quotient of a
rational $\mathbb{T}$-variety of complexity one is an open set of
$\mathbb{P}^1$. Thus, we obtain the following result

\begin{introcor}\label{introcor}
  Let $X$ be a rational $\mathbb{T}$-variety of complexity one. Then
  $X$ is Cohen-Macaulay and Du Bois if and only if $X$ has rational
  singularities.
\end{introcor}

Let us remark that none of the conditions of Theorem~\ref{thm1} can be
weakened, and we devote Section~\ref{sec3} to give examples in any
dimension in which the statement of the theorem fails when we weaken
any of the hypothesis.  Moreover, in Section~\ref{sec3} we give a
characterization of complexity one $\mathbb{T}$-varieties which are
log canonical but not Cohen-Macaulay.

The paper is organized as follows: in Section~\ref{sec1} we introduce
the basic notation of $\mathbb{T}$-varieties via p-divisors and in
Section~\ref{gensing} we introduce the classes singularities needed in
the article.  In Section~\ref{sec2} we prove show some preliminary
results and in Section~\ref{sec25} we prove Theorem~\ref{thm1}.
Section~\ref{sec3} is devoted to give examples showing that the
theorem fails generically if we weaken any of the hypothesis. In
particular, in this section, we give examples in any dimension of
rational $\mathbb{T}$-varieties of complexity one which are Du Bois
and not Cohen-Macaulay and vice versa.

\subsection*{Acknowledgements}
We would like to thank Karl Schwede and Linquan Ma for useful comments
about Du Bois singularities.

\section{$\mathbb{T}$-Varieties via polyhedral divisors}\label{sec1}

We work over an algebraically closed field $\kk$ of characteristic
zero.  In this section, we briefly introduce the language of
p-divisors introduced in ~\cite{AH05} and ~\cite{AHH}.  We begin by
recalling the definition of polyhedral divisors, p-divisors and their
connection with affine $\mathbb{T}$-varieties.

Let $N$ be a finitely generated free abelian group of rank $k$ and
denote by $M$ its dual. We also let $N_\qq$ and $M_\qq$ be the
corresponding $\qq$-vector spaces obtained by tensoring $N$ and $M$
with $\qq$ over $\mathbb{Z}$, respectively.  We denote by
$\mathbb{T} : = \Spec(\kk [M] )$ the $k$-dimensional algebraic torus.
Given a polyhedron $\Delta \subset N_\qq$ we will denote its {\em
  recession cone} by
\[
\rec(\Delta) :=\{ v\in N_\qq \mid v + \Delta \subset \Delta\},
\]
where $+$ denotes the Minkowski sum,
which is defined by
\[
\Delta_1 + \Delta_2 := \{ w \in N_\qq \mid
w=v_1+v_2, v_1\in \Delta_1 \text{ and } v_2 \in \Delta_2\}.
\]
Given a pointed polyhedral cone $\sigma\subset N_\qq$
we can define a semigroup with underlying set
\[
\Pol_\qq(N,\sigma) :=\{ \Delta \subset N_\qq \mid \text{ $\Delta$ is a
polyhedron with $\rec(\Delta)=\sigma$}\}
\]
and addition being the Minkowski sum. The neutral element in the
semigroup is the cone $\sigma$. The elements in $\Pol_\qq(N,\sigma)$
are called $\sigma$-polyhedra.  In what follows we consider the
semigroup $\Pol^+_\qq(N,\sigma)$ which is the extension of the above
semigroup obtained by including the element $\emptyset$ which is an
absorbing element with respect to the addition, meaning that:
\[
  \emptyset + \Delta :=\emptyset \text{ for all
    $\Delta \in \Pol^+_\qq(N,\sigma)$}.
\]
Given a normal projective variety $Y$ we denote by
$\CaDiv_{\geq 0}(Y)$ the semigroup of effective Cartier divisors on
$Y$. We define a {\em polyhedral divisor on $(Y,N)$ with recession
  cone $\sigma$} to be an element of the semigroup
\[
\Pol_\qq^+(N,\sigma) \otimes_{\zz_{\geq 0}} \CaDiv_{\geq 0}(Y).
\]
Observe that any polyhedral divisor can be written as
\[
\mathcal{D}= \sum_{Z} \Delta_Z \cdot Z,
\]
where the sum runs over prime divisors of $Y$, the coefficients
$\Delta_Z$ are either $\sigma$-polyhedra or the empty set and all but
finitely many $\Delta_Z$ are the neutral element $\sigma$. The
recession cone of a polyhedral divisor $\mathcal{D}$ is defined as the
recession cone of any non-empty coefficient and is denoted by
$\sigma(\mathcal{D})$.  The {\em locus of $\mathcal{D}$} is the open
set
\[
\loc(\mathcal{D}) := Y \setminus \bigcup_{\Delta_Z=\emptyset} Z
\]
and we say that $\mathcal{D}$ has {\em complete locus} if the equality
$\loc(\mathcal{D})=Y$ holds, meaning that all the coefficients
$\Delta_Z$ are nonempty $\sigma(\mathcal{D})$-polyhedra.  A polyhedral
divisor $\mathcal{D}$ on $(Y,N)$ with recession cone $\sigma$, defines
a homomorphism of semigroups as follows
\[
\mathcal{D} \colon \sigma^\vee \rightarrow
\CaDiv_\qq ( \loc(\mathcal{D}))
\]
\[
u\mapsto \sum_{\Delta_Z \neq \emptyset}
\min \langle \Delta_Z, u \rangle Z|_{\loc(\mathcal{D})},
\]
where by abuse of notation we denote by $\mathcal{D}$ both the
polyhedral divisor and the homomorphism of semigroups.  Observe that
this homomorphism is well-defined since all the polyhedra $\Delta_Z$
have recession cone $\sigma(\mathcal{D})$ and then the minimum
appearing in the definition always exists.  Moreover, for any
$u\in \sigma^\vee$ we have that $\mathcal{D}(u)$ is a $\qq$-divisor in
$\loc(\mathcal{D})$ whose support contained in
\[
\supp(\mathcal{D}) :=
\loc(\mathcal{D}) \cap \bigcup_{\Delta_Z \neq \emptyset} Z.
\]

Recall that a variety $Y$ is called semiprojective if the natural map
$Y\rightarrow H^0(Y,\OO_Y)$ is projective. In particular, affine and
projective varieties are semiprojective. Furthermore, blow-ups of
semiprojective varieties are also semiprojective.

\begin{definition}
  Let $\mathcal{D}$ be a polyhedral divisor on $(Y,N)$ with recession
  cone $\sigma$.  We say that $\mathcal{D}$ is a {\em proper
    polyhedral divisor}, or {\em p-divisor} for short, if
  $\loc(\mathcal{D})$ is semiprojective, $\mathcal{D}(u)$ is semiample
  for $u \in \sigma^\vee$ and $\mathcal{D}(u)$ is big for
  $u\in \relint(\sigma^\vee)$.  Note that this condition holds
  whenever the locus of the polyhedral divisor is affine.
\end{definition}

\begin{remark}
In what follows, given a $\qq$-divisor $D$ on an algebraic variety $X$
we will write $\mathcal{O}_X(D)$ for $\mathcal{O}_X(\lfloor D \rfloor)$.
\end{remark}

Given a p-divisor $\mathcal{D}$ on $(Y,N)$
with recession cone $\sigma$
we have an induced sheaf of $\mathcal{O}_{\loc(\mathcal{D})}$-algebras
\[
\mathcal{A}(\mathcal{D}) := \bigoplus_{u \in \sigma^\vee \cap M}
\mathcal{O}_{\loc(\mathcal{D})}(\mathcal{D}(u))\chi^u.
\]
Regard that $\mathcal{A}(\mathcal{D})$ is indeed a
sheaf of $\mathcal{O}_{\loc(\mathcal{D})}$-algebras
since the inequality
\[
\mathcal{D}(u)+\mathcal{D}(u')\leq \mathcal{D}(u+u')
\]
holds for every $u,u'\in \sigma^\vee$ by convexity of the polyhedral
coefficient.  We denote by $\widetilde{X}(\mathcal{D})$ the relative
spectrum of $\mathcal{A}(\mathcal{D})$ and by $X(\mathcal{D})$ the
spectrum of the ring of global sections, both $X(\mathcal{D})$ and
$\widetilde{X}(\mathcal{D})$ comes with a $\mathbb{T}$-action induced
by the $M$-grading.  The main theorem of ~\cite{AH05} states that
every $n$-dimensional normal affine variety $X$ with an effective
action of a $k$-dimensional torus $\mathbb{T}$ is
$\mathbb{T}$-equivariantly isomorphic to $X(\mathcal{D})$ for some
$p$-divisor $\mathcal{D}$ on $(Y,N)$, where $Y$ is a projective normal
variety of dimension $n-k$ and $\rank(N)=k$.  In this situation we
have to natural morphisms
\[
  \xymatrix@C=20pt{ & \loc(\mathcal{D}) &
    \widetilde{X}(\mathcal{D})\ar[r]^-{r}
    \ar[l]_-{\pi} & X(\mathcal{D}), \\
  }
\]
where $\pi$ is the good quotient induced by the inclusion of sheaves
$\mathcal{O}_{\loc(\mathcal{D})} \hookrightarrow
\mathcal{A}(\mathcal{D})$ and $r$ is the $\mathbb{T}$-equivariant
birational map induced by taking global sections of
$\mathcal{A}(\mathcal{D})$. Moreover, every affine $\TT$-variety can be
recovered from a p-divisor on the Chow quotient by the $\TT$-action (see \cite[Definition~8.7]{AH05}).

For the proof of our main theorem, we need the following intrinsic
description of $\widetilde{X}(\mathcal{D})$ in the case where the
p-divisor $\mathcal{D}$ lies on the Chow quotient, see~\cite{Vol} for
the proof.
\begin{lemma} \label{vollmert}
  Let $X(\mathcal{D})$ be a $\mathbb{T}$-variety given by the
  p-divisor $\mathcal{D}$ on $(Y,N)$. If $Y$ is the Chow quotient for
  the $\TT$-action on $X(\mathcal{D})$, then
  $\widetilde{X}(\mathcal{D})$ is canonically isomorphic to the
  normalization of the closure of the graph of the quotient rational
  map $X\dashrightarrow Y$.
\end{lemma}

\section{Singularities of normal varieties}\label{gensing}

In this section, we define the main classes of singularities that we
will study and we recall the inclusions between these classes.  We
will proceed by introducing the smaller classes of singularities first
to then move to the wider ones.

\begin{definition}
Let $X$ be a $\qq$-Gorenstein normal algebraic variety
and $\phi \colon Y \rightarrow X$ be a log resolution,
i.e. a resolution such that the exceptional locus with
reduced scheme structure is purely divisorial and simple normal crossing.
Then we can write
\[ K_Y = \phi^*(K_X) + \sum_{i} a_iE_i,\]
where $E_i$ are pairwise different exceptional divisors over $X$.
We say that $X$ is {\em log terminal}
(resp. {\em log canonical}) if $a_i>-1$ (resp. $a_i\geq -1$) for every
$i\in I$.
\end{definition}

\begin{definition}
  Let $X$ be a normal algebraic variety and let
  $\phi \colon Y \rightarrow X$ be any resolution of singularities of
  $X$.  We say that $X$ has {\em rational singularities} if the higher
  direct image sheaves $R^{i}\phi_{*}\mathcal{O}_Y$ vanish for all
  $i>0$.
\end{definition}

Log terminal singularities are rational ~\cite[Theorem 5.22]{KM98} but
in general log canonical singularities are not rational.  Indeed, let
$X$ to be the affine cone over a planar elliptic curve $C$ in
$\mathbb{P}^2$. We can produce a log resolution
$\phi\colon Y \rightarrow X$ with only one exceptional divisor $E$ such
that $K_Y = \phi^*(K_X)-E$, so $X$ is log canonical but the stalk of
$R^1\phi_*\mathcal{O}_X$ at the vertex of the cone is isomorphic to
$H^1(C,\mathcal{O}_C)\simeq \kk$ which is non trivial and so $X$ is
not rational.

\begin{definition}
  We say that a commutative local Noetherian ring $R$ is {\em
    Cohen-Macaulay} if its depth is equal to its dimension.  An
  algebraic variety $X$ is {\em Cohen-Macaulay} if the local
  ring $\mathcal{O}_{X,x}$ is a Cohen-Macaulay for all $x\in X$.
\end{definition}

It is known that rational singularities in characteristic zero are
Cohen-Macaulay (see, e.g., ~\cite[Theorem 5.10]{KM98}), however there are many examples of log canonical
singularities which are not Cohen-Macaulay. We will construct a family
of $\mathbb{T}$-varieties of complexity two with such property in
Section~\ref{sec3}.

We will use the following characterization of Du Bois singularities
which are Cohen-Macaulay due to K\'{o}vacs-Schwede-Smith. \\

\begin{theorem}[{\cite[Theorem 1]{KSS}}]\label{caractdubois}
  Let $X$ be a normal Cohen-Macaulay algebraic variety, let
  $\phi \colon Y\rightarrow X$ be a log resolution and denote by $E$
  the exceptional divisor of $\phi$ with reduced scheme structure.
  Then $X$ has Du Bois singularities if and only if we have an
  isomorphism
  of sheaves $\phi_* \omega_Y(E)\simeq \omega_X$.
\end{theorem}

There is an analogous characterization of rational singularities in
terms of canonical sheaves: a Cohen-Macaulay normal algebraic variety
$X$ has rational singularities if and only if
the natural inclusion of sheaves $\phi_*\omega_Y \hookrightarrow \omega_X$ is an isomorphism,
where $\phi \colon Y \rightarrow X$ is
a resolution of singularities \cite{KKMS73}. From
Theorem~\ref{caractdubois} and this characterization of rational
singularities it follows that rational singularities are Du Bois.
Indeed, by ~\cite[Lemma 3.14]{KSS} we have two natural injections
$\phi_*\omega_Y \hookrightarrow \phi_* \omega_Y(E) \hookrightarrow \omega_X$,
and Theorem~\ref{caractdubois} states that $X$ is Du Bois whenever the inclusion
of sheaves $\phi_*\omega_Y(E)\hookrightarrow \omega_X$ is an isomorphism.
Moreover, it is known that log canonical singularities are Du
Bois~\cite{KK10}.

  It is known that toric singularities are log terminal
  ~\cite[Proposition 11.4.24]{CLS} and therefore they belong to all
  the above classes of singularities.  In higher complexity the
  situation becomes more complicated: rational singularities can be
  characterized in terms of divisorial fans ~\cite[Theorem 3.4]{LS},
  but there are not complete characterizations for the other classes
  of singularities defined above. Partial characterizations of
  Cohen-Macaulay and log terminal singularities with torus action are
  given in ~\cite{LS}.

\begin{remark}
The above definitions are independent of the chosen resolution
(or log resolution) of the normal variety $X$.
By ~\cite{AW97} we may assume that the resolution of singularities is
$\mathbb{T}$-equivariant.
\end{remark}

\section{Preliminary results}\label{sec2}

In this section, we collect all the ingredients we need for the proof
of our main theorem. Some are borrowed from the literature on
$\TT$-varieties while some other are proven here.

\subsection{$\mathbb{T}$-invariant divisors}\label{ss1}

We recall the description of the $\mathbb{T}$-invariant prime divisors
of the $\mathbb{T}$-varieties $\widetilde{X}(\mathcal{D})$ and
$X(\mathcal{D})$ following ~\cite[Proposition 3.13]{PS}.  Let
$\mathcal{D}$ be a p-divisor on $(Y,N)$, where $Y$ is a normal
algebraic variety, then any fiber of the good quotient $\pi$ over a
point of $Y$ which is not contained in $\supp(\mathcal{D})$ is
$\mathbb{T}$-equivariantly isomorphic to the toric variety
$X(\sigma(\mathcal{D}))$, therefore the $\mathbb{T}$-variety
$\widetilde{X}(\mathcal{D})$ admits an open subset which is isomorphic
to the product
$X(\sigma(\mathcal{D}))\times (Y - \supp(\mathcal{D}))$.

An {\em horizontal $\mathbb{T}$-divisor} is a $\mathbb{T}$-invariant
divisor of $\widetilde{X}(\mathcal{D})$ which dominates $Y$, these
divisors are in one to one correspondence with the rays $\rho$ of the
cone $\sigma(\mathcal{D})$ and they can be realized as the closure in
$\widetilde{X}(\mathcal{D})$ of the subvariety
$V(\rho)\times (Y-\supp(\mathcal{D}))$, where $V(\rho)$ is the toric
invariant divisor of $X(\sigma(\mathcal{D}))$ corresponding to the ray
$\rho$ of $\sigma(\mathcal D)$. On the other hand, we define a {\em vertical
  $\mathbb{T}$-divisor} to be a $\mathbb{T}$-invariant divisor of
$\widetilde{X}(\mathcal{D})$ which does not dominate $Y$, these
divisors are in one-to-one correspondence with pairs $(Z,v)$ where $Z$
is a prime divisor of $Y$ and $v$ is a vertex of the polyhedron
$\Delta_Z$.  We often write $\mathcal{V}(\Delta_Z)$ for the set of
vertices of the polyhedron $\Delta_Z$.  Any $\mathbb{T}$-invariant
divisor of $\widetilde{X}(\mathcal{D})$ is either horizontal or
vertical.

Now we turn to describe the horizontal and vertical
$\mathbb{T}$-divisors which are not contained in the exceptional locus
of the morphism
$r\colon \widetilde{X}(\mathcal{D})\rightarrow X(\mathcal{D})$.  We
say that a ray $\rho$ of $\sigma(\mathcal{D})$ is a {\em big ray} if
the $\qq$-divisor $\mathcal{D}(u)$ is big for every
$u\in \relint( \sigma^\vee \cap \rho^{\perp})$, the corresponding
horizontal $\mathbb{T}$-divisor will be called {\em big} as well.
Given a prime divisor
 $Z\subset Y$ and a vertex $v\in \Delta_Z$ we say
that $v$ is a {\em big vertex} if the $\qq$-divisor
$\mathcal{D}(u)|_Z$ is a big divisor for every $u$ in the interior of
the normal cone
\[
\mathcal{N}(\Delta_Z, v):=
\{
u \mid \langle u, w-v \rangle >0
\text{ for every $w\in \Delta_Z$}
\}.
\]
As before, we say that the corresponding vertical $\mathbb{T}$-divisor
is big.  Then we can say that the codimension one exceptional set of
$r$ is the union of all the $\mathbb{T}$-divisors which are not big.

Finally we turn to introduce some notation for the horizontal and
vertical $\mathbb{T}$-divisors on $\widetilde{X}(\mathcal{D})$ and
$X(\mathcal{D})$.  Given a ray $\rho$ in $\sigma(\mathcal{D})$ we may
denote by $\widetilde{D}_\rho$ the corresponding horizontal divisor of
$\widetilde{X}(\mathcal{D})$ and whenever $\rho$ is big we denote by
$D_\rho$ its image $r(\widetilde{D}_\rho)$.  Given a prime divisor
$Z\subset Y$ and a vertex $v\in \mathcal{V}(\Delta_Z)$ we write
$\widetilde{D}_{(Z,v)}$ for the corresponding vertical divisor and
whenever $v$ is big we denote by $D_{(Z,v)}$ its image in
$X(\mathcal{D})$.

\subsection{$\mathbb{T}$-orbit decomposition}\label{ss2}

Now we turn to describe the $\mathbb{T}$-orbits of the
$\mathbb{T}$-varieties $\widetilde{X}(\mathcal{D})$ and
$X(\mathcal{D})$ following ~\cite[Section 10]{AH05}.  In order to do
so, we begin by defining the toric bouquet associated to a polytope
$\Delta\subset N_\qq$.

\begin{definition}
  Given a $\sigma$-polyhedron $\Delta \subset N_\qq$ we denote by
  $\mathcal{N}(\Delta)$ the {\em normal fan} of $\Delta$, which is the
  fan in $M_{\mathbb{Q}}$ with support $\sigma^\vee$ and whose cones
  corresponds to linearity regions of the function
  $\min\langle \Delta, - \rangle \colon \sigma^\vee \rightarrow \qq$.
  Observe that the cones of $\mathcal{N}(\Delta)$ are in one-to-one
  dimension-reversing correspondence with the faces of $\Delta$.
  Given a face $F$ of $\Delta$, we will denote by $\mathcal{N}(F)$ the
  corresponding cone of $\mathcal{N}(\Delta)$.  We define the {\em
    toric bouquet} of $\Delta$ to be
\[
X(\Delta):=\Spec( \kk [ \mathcal{N}(\Delta) \cap M]),
\]
where $\kk [ \mathcal{N}(\Delta) \cap M]$ equals
$\kk[\sigma^\vee \cap M]$ as $\kk$-vector spaces, but the
multiplication rule in $\kk[\mathcal{N}(\Delta)\cap M]$ is given by
\[
\chi^u \cdot \chi^{u'}:=
\left\{
	\begin{array}{cl}
		\chi^{u+u'}  & \text{if u,u' belong to a common cone of $\mathcal{N}$}, \\
		0 &  \text{otherwise}.
	\end{array}
\right.
\]
\end{definition}

Given a point $y\in Y$, we define the {\em fiber polyhedron} to be
\[
\Delta_y :=
\left\{
	\begin{array}{cl}
		\sum_{y\in Z} \Delta_Z  & \text{if $y\in \supp(\mathcal{D})$}, \\
		\sigma(\mathcal{D}) &  \text{otherwise}.
	\end{array}
\right.
\]
In the previous section we stated that the fiber $\pi^{-1}(y)$ over a
point $y\in Y$ which is not contained in the support of $\mathcal{D}$
is $\mathbb{T}$-equivariantly isomorphic to the toric variety
$X(\sigma(\mathcal{D}))$, in general the special fibers of $\pi$ are
not irreducible, but they can still be interpreted as toric bouquets
as follows.  Given a point $y\in \supp(\mathcal{D})$, the fiber
$\pi^{-1}(y)$ is $\mathbb{T}$-equivariantly isomorphic to the toric
bouquet $X(\Delta_y)$.  Therefore, we have a dimension-reversing
bijection between between the orbits of $\pi^{-1}(y)$ and the faces of
$\Delta_y$.  Thus, the orbits of $\widetilde{X}(\mathcal{D})$ are in
one-to-one dimension-reversing correspondence with the pairs $(y,F)$
where $y\in Y$ and $F$ is a face of $\Delta_y$.

Finally, we describe the $\mathbb{T}$-orbits
of $\widetilde{X}(\mathcal{D})$ which are identified by the contraction
morphism $r$.
Given an element $u\in \sigma(\mathcal{D})^\vee\cap M$ we have an
induced morphism
\[
\phi_u \colon Y \rightarrow Y_u,
\text{ where }
Y_u=\Proj \left( \bigoplus_{n\in \zz_{\geq 0}}
H^0(Y, \mathcal{O}_Y(\mathcal{D}(nu)))\right).
\]
From ~\cite[Theorem 10.1]{AH05}
we know that two orbits of $\widetilde{X}(\mathcal{D})$
corresponding to the pairs $(y,F)$ and $(y',F')$
are identified via the $\mathbb{T}$-equivariant contraction $r$
if and only if
\[
\mathcal{N}(F)
=\mathcal{N}(F') \text{ and }
\phi_u(y)=\phi_u(y') \text{ for some $u\in \relint(\mathcal{N}(F))$}.
\]
Observe that in the case that $Y$ is $\pp^1$, then either $\phi_u$ is an isomorphism
or is the constant morphism, depending whether the divisor $\mathcal{D}(nu)$ has
positive or trivial degree, respectively.

\subsection{Sections of $\mathbb{T}$-invariant sheaves}\label{ss3}

In this subsection we describe the sections of the sheaf induced by a
$\mathbb{T}$-invariant Weil divisor on $X(\mathcal{D})$ (see, e.g., ~\cite[Section 3.3]{PS}).  Recall that
the prime $\mathbb{T}$-invariant divisors of $X(\mathcal{D})$ are of
the form $D_\rho$ or $D_{Z,v}$ for $\rho$ a big ray of
$\sigma(\mathcal{D})$ or $v\in \mathcal{V}(\Delta_Z)$ a big vertex.
Therefore, any $\mathbb{T}$-invariant Weil divisor $D$ of
$X(\mathcal{D})$ can be written as
\[
D=\sum_{\rho \text{ big }} a_\rho D_\rho
+ \sum_{\substack{(Z,v)\\ \text{ $v$ big }}}
b_{Z,v}D_{Z,v}.
\]
Observe that the field of rational functions of
$\widetilde{X}(\mathcal{D})$ and the field of rational functions of
$X(\mathcal{D})$ are isomorphic to the fraction field of $\kk(Y)[M]$,
so any quasi-homogeneous rational function of
$\widetilde{X}(\mathcal{D})$ or $X(\mathcal{D})$ can be written as
$f\chi^u$ where $u\in M$ and $f\in \kk(Y)$.  By
~\cite[Proposition 3.14]{PS} we know that the principal divisor
associated to $f\chi^u$ on $\widetilde{X}(\mathcal{D})$ is given by
\[
\divv( f\chi^u)=
\sum_{\rho} \langle \rho, u\rangle D_\rho
+\sum_{(Z,v)} \mu(v)(\langle v, u \rangle +\ord_Z f) D_{Z,v},
\]
and the associated principal divisor on $X(\mathcal{D})$ is given by
\[
\divv( f\chi^u)=
\sum_{\substack{\rho \\ \text{ $\rho$ big }}} \langle \rho, u\rangle D_\rho
+\sum_{\substack{(Z,v)\\ \text{ $v$ big }}} \mu(v)(\langle v, u \rangle +\ord_Z f) D_{Z,v},
\]
where $\mu(v)$ denotes the smaller positive integer such that $\mu(v)v$ is an integral point of $N_\qq$.
Therefore, the divisor
$\divv(f\chi^u) + D$ is effective in
$X(\mathcal{D})$ if and only if
\begin{eqnarray}
\langle \rho , u \rangle + a_\rho \geq 0, \text{ for every $\rho$ big, and }\nonumber\\
\langle v, u\rangle + \frac{b_{Z,v}}{\mu(v)} +\ord_Z f \geq 0, \text{ for every pair $(Z,v)$ with $v$ big}.\nonumber
\end{eqnarray}
We denote by $\square(D)$ the integer points of the polyhedron defined
by
\[ \{ u \in M_\qq \mid \langle \rho, u \rangle \geq -a_\rho,
\text{ for each $\rho$ big } \}.\]
Given an element $u\in \square(D)\cap M$
we denote by $\psi_D(u)$ the divisor of $Y$ given by
\[
\sum_{Z\subset Y}\left( \min_{\substack{ v\in \mathcal{V}(\Delta_Z)\\ \text{ $v$ big }}}
 \left( \langle v, u \rangle +  \frac{b_{Z,v}}{\mu(v)} \right) \right)Z .
\]
Therefore, we have that
\[
H^0 \left(X(\mathcal{D}),\mathcal{O}_{X(\mathcal{D})}(D)\right)
\simeq \bigoplus_{u\in \square(D)\cap M}
H^0 \left(X(\mathcal{D}),\mathcal{O}_{X(\mathcal{D})}(D)\right)_u,
\]
and
\[
H^0\left(X(\mathcal{D}),\mathcal{O}_{X(\mathcal{D})}(D)\right)_u
\simeq H^0(Y, \mathcal{O}_Y(\psi_D(u)))\chi^u.
\]

\subsection{Canonical sheaf}\label{ss4}

Let $K_Y$ be a canonical divisor of $Y$. Given a subvariety
$Z\subset Y$ we denote by $\coef_Z K_Y$ the coefficient of $K_Y$ along
$Z$. Then, we can describe the canonical divisor of
$\widetilde{X}(\mathcal{D})$ as
$$
K_{\widetilde{X}(\mathcal{D})}= \sum_{(Z,v)} (\mu(v)\coef_Z K_Y +
\mu(v)-1)\widetilde{D}_{Z,v} -\sum_\rho \widetilde{D}_\rho\,,
$$
and the canonical divisor of $X(\mathcal{D})$ as
$$
K_{X(\mathcal{D})} = \sum_{\substack{(Z,v)\\ \text{ $v$ big }}}
(\mu(v)\coef_Z K_Y + \mu(v)-1)D_{Z,v} -\sum_{\rho \text{ big }} D_\rho\,.
$$

\section{Proof of the main theorem}\label{sec25}

In the following lemma, we prove Theorem~\ref{thm1} in the particular
case of a 1-dimensional torus action, i.e., in the case where
$N=\mathbb{Z}$.

\begin{lemma}\label{lem1}
  Let $Y$ be a projective variety with rational singularities,
Picard rank one and $p_g(Y)=0$ and let $\mathcal{D}$ be a
  p-divisor on $(Y,\mathbb{Z})$. If $\widetilde{X}(\mathcal{D})$ has
  rational singularities, then $X(\mathcal{D})$ is Cohen-Macaulay and
  Du Bois if and only if $X(\mathcal{D})$ has rational singularities.
\end{lemma}

\begin{proof}

  If $X(\mathcal{D})$ has rational singularities, then
  $X(\mathcal{D})$ is Cohen-Macaulay and Du Bois. Hence, we only need
  to prove the ``only if'' part of the lemma.

  In the case where $\loc(\mathcal{D})$ is not complete, since the
  Picard rank is one, we have that $\loc(\mathcal{D})$ is affine.
  If $\loc(\mathcal{D})$ is affine, we have
  $X(\mathcal{D})=\widetilde{X}(\mathcal{D})$ which has rational
  singularities by assumption. In the following, we assume that
  $\loc(\mathcal{D})$ equals $Y$.

  Observe that $\mathcal{D}$ is uniquely determined by the
  $\qq$-divisor $D=\mathcal{D}(1)$, then we will just write $X(D)$ for
  the associated $\mathbb{T}$-variety. From now we will write
  \[ D=\sum_{Z\subset Y} a_Z Z,\] where the divisors $Z\subset Y$ are
  prime divisors and $a_Z$ are rational coefficients. Observe that
  since $Y$ has Picard rank one and $D$ is big we have that $D$ is
  ample. So, by Section~\ref{ss1} we conclude that the exceptional
  locus of the $\mathbb{T}$-equivariant contraction
  $r\colon \widetilde{X}(D)\rightarrow X(D)$ is the unique horizontal
  prime divisor $\widetilde{D}_\rho$. Equivalently, for any prime
  divisor $Z\subset Y$ the unique vertex of $\mathcal{V}(D_Z)$ is a
  big vertex.

  Since $X(D)$ is Cohen-Macaulay, by ~\cite[Corollary 2.4]{Wat} we
  conclude that
  \[
    H^i(Y, \mathcal{O}_Y(uD))=0, \text{ for $u\in \zz$ and
      $1\leq i <\dim(Y)$}.
  \]
  Moreover, since $\widetilde{X}(D)$ has rational singularities the
  proof of~\cite[Theorem 3.3]{LS} applies to our situation. Indeed,
  in~\cite[Theorem 3.3]{LS} the SNC hypotesis is only required to
  ensure that $\widetilde{X}(D)$ has rational singularities which is
  given in our case. We deduce then that $X(D)$ has rational
  singularities if and only if
  \begin{equation}\label{eq0}
    H^{\dim(Y)}(Y, \mathcal{O}_Y(uD))=0, \text{ for $u\geq 0$}.  \end{equation}

  We are assuming that $p_g(Y)=h^0(K_Y)=0$, so it is enough to
  prove equality~\eqref{eq0} for $u>0$, which is equivalent, by Serre
  duality, to prove that
  \[
    H^0( Y, \mathcal{O}_Y( K_Y - \lfloor uD \rfloor )) =0, \text{ for
      $u>0$}.
  \]
  Let $W\rightarrow X(D)$ be a $\TT$-equivariant log resolution of
  $X(D)$ which factors through $\widetilde{X}(D)$ giving an
  isomorphism from an open set of $W$ to the smooth locus of
  $\widetilde{X}(D)$. Since the general fiber of the morphism
  $\pi \colon \widetilde{X}(D)\rightarrow Y$ is smooth, we have that
  $W\rightarrow \widetilde{X}(D)$ does not contract horizontal
  divisors over $Y$.
Indeed, we can find an open set of $U\subsetneq Y$, such that
$\pi^{-1}(U)$ is smooth, and then in order to resolve $\widetilde{X}(D)$ it suffices to blow-up points
over the complement of $U$.  Therefore, we conclude that $W$ contains a unique $\mathbb{T}$-invariant horizontal
divisor $\widetilde{D}_{\rho}$, and we can use Section~\ref{ss4} to see that such divisor appears with coefficient zero in $K_{W}+E$,
where $E$ is the exceptional divisor of the log resolution $W\rightarrow X(D)$, with reduced scheme structure.
  Using the
  formula for $\divv(f\chi^u)$ in Section~\ref{ss3} we conclude that
  any global section $f\chi^u$ of the sheaf defined by $K_{W}+E$ has
  $u\geq 0$.

  By ~\cite[Theorem 1.1]{KSS} we have that the canonical inclusion
  $f_*\left( \omega_{X(\mathcal{D})}(E) \right) \hookrightarrow
  \omega_{X(\mathcal{D})}$ is an isomorphism and then both sheaves
  have the same global sections.  The injection is given by
  \[
    \bigoplus_{u \in \zz_{\geq 0}} H^0\left( Y, \mathcal{O}_Y\left(
        \psi_{K_{W}+E}(u)\right)\right) \hookrightarrow
    \bigoplus_{u \in \zz} H^0\left(Y,\mathcal{O}_Y\left(
        \psi_{K_{X(\mathcal{D})}}(u)\right)\right)
  \]
  and since we know that this injection is indeed surjective we
  conclude that
  \[
    H^0(Y, \mathcal{O}_Y( \psi_{K_{X(\mathcal{D})}}(-u)))=0, \text{
      for $u>0$}.
  \]
  So, it is enough to prove that
  $K_Y -\lfloor uD \rfloor \leq \lfloor
  \psi_{K_{X(\mathcal{D})}}(-u)\rfloor$ for $u>0$.  Observe that in
  this case we can use the formula in Section~\ref{ss3} for
  $\psi_{K_{X(\mathcal{D})}}(-u)$ to write
  \[
    \left\lfloor \psi_{K_{X(\mathcal{D})}}(-u) \right\rfloor = K_Y
    +\sum_{Z\subset Y} \left\lfloor \frac{\mu(a_Z)-1}{\mu(a_Z)} - ua_Z
    \right\rfloor Z \geq K_Y -\lfloor uD \rfloor
  \]
  concluding the proof.
\end{proof}

\begin{lemma}\label{lem2}
  Let $\mathcal{D}$ be a p-divisor on $(Y,N)$ with recession cone
  $\sigma \subset N_\qq$.  Assume that for some $\rho \in \sigma(1)$
  and $u\in \relint(\sigma^\vee \cap \rho^\perp)$ the divisor
  $\mathcal{D}(u)$ is principal on $Y$ and let
  \[
    f\in H^0(Y,\mathcal{O}_Y(\mathcal{D}(u))).
  \]
  Then the complement of $\divv(f\chi^u)$ in $X(\mathcal{D})$ is
  isomorphic to $X(\mathcal{D}_0)\times \kk^*$, where
$\mathcal{D}_0$ is a p-divisor on $(Y,N')$ and $\rank(N')<\rank(N)$.
\end{lemma}

\begin{proof}
Observe that $f\chi^u$ is a regular function on $X(\mathcal{D})$ which vanishes
at all the horizontal $\mathbb{T}$-invariant divisors $D_{\rho'}$ for $\rho'\neq \rho$.
Moreover, the regular function $f\chi^u$ vanishes at the vertical $\mathbb{T}$-invariant
divisor $D_{Z,v}$ if and only if $\langle v ,u \rangle + \ord_Z f >0$.

Write $\mathcal{D}=\sum_{Z\subset Y} \Delta_Z \cdot Z$
and let $N'_\qq \subset N_\qq$ be a codimension one subspace defined by $\langle u , N'_\qq \rangle =0$
and denote by $N'$ the lattice of integral points of $N$ that lies on $N'_\qq$.
Observe that for any $\Delta_Z$ the vertices $v\in \mathcal{V}(\Delta_Z)$
with $\langle v , u \rangle  = -\ord_Z f$ lie in a face of $\Delta_Z$ that we denote by $\Delta_{Z,\rho}$.
Let $\Delta'_{Z,\rho}$ the projection of $\Delta_{Z,\rho}$ in $N'$
and $\mathcal{D}' = \sum_{Z\subset Y} \Delta'_{Z,\rho}\cdot Z$ be the corresponding p-divisor on $(Y,N)$.
Then using the orbit decomposition we see that
$X(\mathcal{D}')\rightarrow X(\mathcal{D})$ defines an embedding onto $X(\mathcal{D})\setminus \divv(f\chi^u)$
and $X(\mathcal{D}')\simeq X(\mathcal{D}_0)\times \kk^*$ where $\mathcal{D}_0$ is the p-divisor $\mathcal{D}'$ restricted to $N'$.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{thm1}]
Since the statement is local on $X$, it is enough to prove it on an affine chart.
By~\cite{Sum74}, we can cover $X$ by $\mathbb{T}$-invariant affine charts
and by~\cite[Theorem 3.4]{AH05} we may asusme that such charts are $\mathbb{T}$-equivariantly isomorphic to $X(\mathcal{D})$ for some
 p-divisor $\mathcal{D}$ on its Chow quotient $Y$.  Observe that if
  $\mathcal{D}$ does not have complete locus, then its locus is affine,
  since $\rank(\Pic(Y))=1$. Therefore
  $\widetilde{X}(\mathcal{D})\simeq X(\mathcal{D})$ and
  $\widetilde{X}(\mathcal{D})$ has rational singularities since it
  equals to the normalization of the closure of the graph of the map
  $X(\mathcal{D})\dashrightarrow Y$ by Lemma~\ref{vollmert}.

  In what follows we will assume that $\mathcal{D}$ has complete
  locus, and we will denote by $n$ the dimension of $Y$.  We will
  proceed by induction on $k$, the dimension of the lattice $N$, being
  Lemma~\ref{lem1} the base case $k=1$.  From now we assume that
  $k\geq 2$.

  We claim that no vertical divisor is being contracted by the
  $\mathbb{T}$-equivariant morphism
  $r\colon \widetilde{X}(\mathcal{D})\rightarrow X(\mathcal{D})$.
  Indeed, pick some vertex $v$ of $\Delta_Z$ for some subvariety
  $Z\subset Y$, then we have that the interior of the normal cone
  $\mathcal{N}(v)=\mathcal{N}(\Delta_Z,v)$ lies inside the relative
  interior of $\sigma(\mathcal{D})^\vee$, therefore for any
  $u\in \mathcal{N}(v)$, the $\qq$-divisor $\mathcal{D}(u)$ is ample,
  so its restriction $\mathcal{D}(u)|_Z$ is ample on $Z$ as well.

  Since $\widetilde{X}(\mathcal D)$ has rational singularities the proof
  of~\cite[Theorem 3.3]{LS} applies again to our situation. By the
  proof of~\cite[Theorem 3.3]{LS} for any resolution of
  singularities $f\colon W \rightarrow X(\mathcal{D})$ we have that
  $R^if_* \mathcal{O}_W$ equals the sheaf associated to the $H^0(\mathcal{A}(\mathcal{D}))$-module
  \[
    \bigoplus_{u\in
      \sigma(\mathcal{D})^\vee \cap M} H^i(Y, \mathcal{D}(u)).
  \]
  Therefore, for any resolution $f\colon W \rightarrow X(\mathcal{D})$
  we have that $R^if_* \mathcal{O}_{W}=0$
   for
  $i>\dim(Y)$.  We claim that $R^if_*\mathcal{O}_{W}=0$
  for $1\leq i \leq \dim(Y)$.

  Using ~\cite[Lemma 3.3]{Kov},
  it is enough to prove that there
  exists a closed subset $Z\subset X(\mathcal{D})$ such that $Z$ has
  dimension at most $k-2$ and $X(\mathcal{D})\setminus Z$ has rational
  singularities.  In order to do so, we will prove that the image
  $r(E)$ of the exceptional set $E$ of
  $r\colon \widetilde{X}(\mathcal{D})\rightarrow X(\mathcal{D})$ has
  dimension at most $k-1$, and that there exists an open set of
  $X(\mathcal{D})$ with rational singularities which intersects $r(E)$
  non-trivially.

  Let $\widetilde D_\rho$ be a prime exceptional divisor of $r$,
  meaning that $\rho$ is not a big ray, then we have that for some
  $u\in \relint(\sigma^\vee \cap \rho^\perp)$ the divisor
  $\mathcal{D}(u)$ is semiample but not big on $Y$.
Therefore, $\mathcal{D}(u)$ is numerically trivial.
Since we are assuming that $Y$ has Picard rank one, then we conclude that
${\rm Pic}^0(Y)$ is torsion,  so we have that some multiple of $\mathcal{D}(u)$ is linearly
  equivalent to $\mathcal{O}_Y$.  Therefore the function $\phi_u$ is
  the constant function, and we conclude from the $\mathbb{T}$-orbit
  decomposition of $X(\mathcal{D})$ that for any two points $y$ and
  $y'$ of $Y$ not lying in $\supp(\mathcal{D})$, the orbits $(y,\rho)$
  and $(y',\rho)$ are identified by $r$, implying that
  $r(\widetilde D_\rho)$ has dimension at most $k-1$.

  Let $m\in \zz_{\geq 1}$ be such that $m\mathcal{D}(u)$ is principal and
  consider $f\in H^0(Y,\mathcal{O}_Y(m\mathcal{D}(u)))$, then by
  Lemma~\ref{lem2} we conclude that the complement of $\divv(f\chi^u)$
  in $X(\mathcal{D})$, which intersects $r(\widetilde D_\rho)$
  non-trivially, is of the form $X(\mathcal{D}')\times \kk^*$, where
  $\mathcal{D}'$ is a p-divisor on $(Y,N')$ with
  $N'\oplus \zz \simeq N$.  By the assumptions $X(\mathcal{D}')$ is
  Cohen-Macaulay and DuBois, and therefore by the induction hypothesis
  we conclude that $X(\mathcal{D}')$ has rational singularities, so
  the open set $X(\mathcal{D}')\times \kk^*$ has rational
  singularities as well.  Analogously for the other irreducible
  components of the exceptional divisor of $r$ we conclude the claim.
\end{proof}

To conclude this section, we rephrase Corollary~\ref{introcor} in
terms of p-divisors.  More precisely, by ~\cite[Theorem 3.3]{LS}, we
obtain the following combinatorial characterization of rational
$\mathbb{T}$-varieties of complexity one with Cohen-Macaulay and Du
Bois singularities.

\begin{corollary}
  Let $\mathcal{D}$ be a p-divisor on $\mathbb{P}^1$.  Then
  $X(\mathcal{D})$ is Cohen-Macaulay and Du Bois if and only if either
  \begin{itemize}
  \item $\mathcal{D}$ has affine locus, or
  \item $\mathcal{D}$ has complete locus and
    $\deg \lfloor \mathcal{D}(u)\rfloor \geq -1$, for all
    $u\in \sigma(\mathcal{D})^\vee$.
\end{itemize}
\end{corollary}

\section{Examples and corollaries}\label{sec3}

We devote this section to give examples in which the main theorem
fails when we weaken the hypotheses. We also study rational
$\mathbb{T}$-varieties of complexity one which are log canonical and
not Cohen-Macaulay.  The hypothesis of $Y$ having rational
singularities is trivially necessary since we can take products of any
$Y$ with a toric variety in order to construct a $\mathbb{T}$-variety
with arbitrary singularities.  We start by giving examples in which
the theorem fails if we drop the $p_g(Y)=0$ and Picard
rank one conditions.

\begin{example}
Consider a plane elliptic curve $C$ and denote by $X$ the cone over $C$,
then $X$ is a $\mathbb{T}$-variety of dimension two and complexity one
whose Chow quotient is $C$, which has rational singularities and one-dimensional Neron-Severi group,
but $p_g(C)\neq 0$.
Clearly, $X$ is Cohen-Macaulay and Du Bois, but the singularity at the vertex
of $X$ is not rational.
\end{example}

\begin{example}
Consider $Y$ to be the blow-up of $\pp^2$ at one point $p$,
let $H$ the pull-back of an hyperplane of $\pp^2$ not containing $p$,
and $E$ the exceptional divisor.
Observe that $Y$ is smooth, has acyclic structure sheaf and Picard rank two.
Let $\mathcal{D}$ be any p-divisor on $(Y,\zz)$, with locus $Y\setminus \supp(E+H)$
and recession cone isomorphic to $\qq_{\geq 0}$, then $X = X(\mathcal{D})$
is a $3$-fold which is Cohen-Macaulay and Du Bois
but the singularity at the vertex is not rational.
Indeed, in this case the birational contraction $\widetilde{X}(\mathcal{D})\rightarrow X$
gives a resolution of singularities, where the exceptional divisor is isomorphic to $\cc^2-\{p\}$,
therefore the stalk of $R^1 f_*\mathcal{O}_{\widetilde{X}(\mathcal{D})}$ at the vertex of $X$
is naturally isomorphic to $H^1(\cc^2-\{p\}, \mathcal{O}_{\cc^2-\{p\}})$ which is non-trivial.
Thus, the sheaf $R^1 f_*\mathcal{O}_{\widetilde{X}(\mathcal{D})}$ is non-trivial at the vertex, which means
that such singular point is not rational.
\end{example}

Now we turn to give examples of rational complexity one $\mathbb{T}$-varieties
which are Cohen-Macaulay and not Du bois and vice versa.
Since $\pp^1$ is smooth, has Picard rank one and acyclic structure sheaf,
this examples will prove that both conditions Cohen-Macaulayness and Du Bois
are necessary in Theorem~\ref{thm1}.

In what follows,
given a polytope $\Delta\subset N_\qq$,
we denote by $\mu(\Delta)$
the number $\max\{ \mu(v) \mid v\in \mathcal{V}(\Delta)\}$.
Recall that $\mu(v)$ is the smallest positive integer such that
$\mu(v)v$ is a lattice point of $N_\qq$.

\begin{proposition}\label{finprop}
Let $\mathcal{D}$ be a p-divisor on $\pp^1$.
Assume that $X(\mathcal{D})$ is $\qq$-Gorenstein.
Then $X(\mathcal{D})$ is log canonical if and only if either
\begin{itemize}
\item $\loc(\mathcal{D})$ is affine, or
\item $\loc(\mathcal{D})=\pp^1$ and
\begin{equation}\label{ineq}
\sum_{p\in \pp^1} \frac{ \mu(\mathcal{D}_p)-1}{\mu(\mathcal{D}_p)} \leq 2.
\end{equation}
\end{itemize}
\end{proposition}

\begin{proof}
Following the proof of ~\cite[Theorem 4.9]{LS} we can see
that $X(\mathcal{D})$ is log canonical if and only if the divisor
\[
-K_{\pp^1} - \sum_{p\in \pp^1}\left( \frac{ \mu(\mathcal{D}_p)-1}{\mu(\mathcal{D}_p)} \right) p
\]
is a semiample $\qq$-divisor on $\pp^1$.
\end{proof}

\begin{remark}\label{remfin}
Recall that, whenever the locus of a p-divisor $\mathcal{D}$ on $(\pp^1, N)$
is affine, then $X(\mathcal{D})$ is log terminal since it has toroidal singularities ~\cite[Proposition 2.6]{LS}.
Moreover, if $\mathcal{D}$ has complete locus and the sum
$\sum_{p\in \pp^1} \frac{ \mu(\mathcal{D}_p)-1}{\mu(\mathcal{D}_p)}$
is strictly less than $2$, then $X(\mathcal{D})$ is log terminal by ~\cite[Theorem 4.9]{LS}.
In any case $X(\mathcal{D})$ is Cohen-Macaulay and Du Bois.
Thus, from now we are interested in examples in which the equality in ~\eqref{ineq} holds.
\end{remark}

\begin{proposition}\label{log}
Let $\mathcal{D}$ be a p-divisor on $\pp^1$.
Assume that $X(\mathcal{D})$ is $\qq$-Gorenstein.
Then $X(\mathcal{D})$ is log canonical and not Cohen-Macaulay
if and only if $\mathcal{D}$
has at most four polyhedral coefficients $\mathcal{D}_{p_i}$ with
$\mu_i:=\mu(\mathcal{D}_i)\geq 2$,
which are listed in the following cases (up to permutation)
\[
(\mu_1,\dots, \mu_4) = (2,2,2,2),
\]
\[
(\mu_1,\dots, \mu_3)= (2,4,4),
\]
\[
(\mu_1,\dots, \mu_3)= (3,3,3),
\]
and $\lfloor \deg(\mathcal{D}(u)) \rfloor < -1$ for some $u\in \sigma(\mathcal{D})^\vee$.
\end{proposition}

\begin{proof}
By Proposition~\ref{finprop}
and Remark~\ref{remfin}, we know that
the locus $\mathcal{D}$ is complete
and the sum  $\sum_{p\in \pp^1} \frac{ \mu(\mathcal{D}_p)-1}{\mu(\mathcal{D}_p)}$ equals $2$.
Moreover, it is known that log canonical singularities are Du Bois,
therefore by Theorem~\ref{thm1} we know that
$X(\mathcal{D})$ is Cohen-Macaulay
if and only if it has rational singularities, concluding the proof.
\end{proof}

The following corollary shows that in order to produce an example
of a rational complexity one $\mathbb{T}$-variety with log canonical
singularities that is not Cohen-Macaulay we need to consider a threefold.
The singularities of $\kk^*$-surfaces has been extensively studied (see for example~\cite{FZ}).

\begin{corollary}
Any log canonical
rational $\kk^*$-surface
has rational singularities.
\end{corollary}

\begin{proof}
Let $X(\mathcal{D})$ be a log canonical
rational $\kk^*$-surface.
We assume that $\mathcal{D}$ has complete locus
and $X(\mathcal{D})$ is not log terminal.
If $Y$ is zero dimensional, then $X(\mathcal{D})$ is a toric variety and the result follows from the fact that toric singularities are rational,
otherwise by the rationality of the $\kk^*$-surface we conclude that $Y=\pp^1$
and $N_\qq$ is one-dimensionl.
Using the equation of ~\cite[Section 5.2]{AIPSV} that characterize p-divisors over curves, we conclude
that we can write
\[
\mathcal{D}=\sum_{i=1}^k ( q_i + \qq_{\geq 0}) \cdot p_i
\]
for some rational numbers $q_1,\dots, q_k$.
By Proposition~\ref{log}, it suffices to check the three possible cases
for $(\mu(q_1),\dots, \mu(q_k))$, up to permutation.
We will assume the first case, being the other two analogous.
Thus, we can assume that $\mu(q_1)=\dots =\mu(q_4)=2$
and $q_5,\dots,q_k$ are integers,
so we can write $q_i=n_i+\frac{1}{2}$ for $n_i$ integers
and $1\leq i \leq 4$.
We have that
\[ \deg(\mathcal{D}(u))= \sum_{i=5}^k uq_i + \sum_{i=1}^4 \left( un_i + \frac{u}{2}\right) >0,
\text{ for $u\geq 0$}.
\]
therefore $\lfloor \deg(\mathcal{D}(u))\rfloor \geq -1$, for $u\geq 0$, concluding the claim.
\end{proof}

Now we turn to give examples,
in any dimension greater or equal than $3$,
of rational $\mathbb{T}$-varieties of complexity one
that are log canonical
and not Cohen-Macaulay.

\begin{example}\label{lc}
Consider the polyhedral divisor $\mathcal{D}$ on $\pp^1$
whose recession cone $\sigma(\mathcal{D})$ is
$\sigma:=\langle e_1, \dots, e_k\rangle$, with $k\geq 2$,
where $e_1,\dots, e_k$ is a canonical basis of $M_\qq \simeq \qq^k$,
its support is $\{p_1,\dots, p_4\}\subset \pp^1$ and its polyhedral coefficients are
\[
\Delta_{p_1}=\Delta_{p_2}= \conv\left( \frac{1}{2}e + \frac{1}{2}e_1,\dots, \frac{1}{2}e+\frac{1}{2}e_k  \right) + \sigma \subsetneq N_\qq,
\]
\[
\Delta_{p_3}=\Delta_{p_4}=\conv\left( - \frac{1}{2}e + \frac{1}{2}e_1 ,\dots, -\frac{1}{2}e+\frac{1}{2}e_k  \right) + \sigma \subsetneq N_\qq,
\]
where $e=\sum_{i=1}^k e_i$
and $\conv$ denotes the convex hull.
We will denote by $q_j$, with $j\in \{1,\dots, k\}$
the vertices of the polyhedron $\Delta_{p_1}$,
and by $r_j$, with $j\in \{1,\dots, k\}$,
the vertices of the polyhedron $\Delta_{p_3}$.
Recall that a polyhedral divisor
$\mathcal{D}=\sum_{i=1}^k \Delta_{p_i}\cdot p_i$ on $(\pp^1,N)$
is a p-divisor if and only if
$\sum_{i=1}^k \Delta_{p_i}\subsetneq \sigma(\mathcal{D})$
(see, e.g.,~\cite[Section 5.2]{AIPSV}).
In this case, we have that
\[
\sum_{i=1}^4\Delta_{p_i}=
\conv(2e_1,\dots, 2e_k)+ \sigma
\subsetneq \sigma,
\]
therefore $\mathcal{D}$ is a p-divisor.
We observe that $X(\mathcal{D})$ is $\qq$-Gorenstein,
indeed we can write
\begin{equation}\label{canonicaldivisor}
K_{X(\mathcal{D})}=
- \sum_{j=1}^k D_{p_1,q_j}
- \sum_{j=1}^k D_{p_2,q_j}
+ \sum_{j=1}^k D_{p_3,r_j}
+ \sum_{j=1}^k D_{p_4,r_j}=
\divv(f),
\end{equation}
where $f$ is a rational function on $\pp^1$,
with poles on $p_1$ and $p_2$ of order one
and zeros on $p_3$ and $p_4$ of order one.
Here, we are choosing the representation $K_{\pp^1}=-p_1-p_2$ for the canonical divisor of $\pp^1$.
Also, observe that the p-divisor $\mathcal{D}$ has no big rays (see Section~\ref{ss1}), since the Chow quotient is a curve,
so the contraction morphism $\widetilde{X}(\mathcal{D})\rightarrow X(\mathcal{D})$ is contracting no vertical divisors,
therefore the equality~\eqref{canonicaldivisor} is obtained from Section~\ref{ss4}.
We claim that we can use Proposition~\ref{log} to
we conclude that $X(\mathcal{D})$ is $\qq$-Gorenstein and log canonical.
Indeed, $\mathcal{D}$ has exactly four polyhedral coefficients with $\mu_i=2$.
Moreover, we can write
\[
K_{\widetilde{X}(\mathcal{D})}= r^*( K_{X(\mathcal{D})}) - \sum_{j=1}^k \widetilde{D}_j.
\]
Moreover, $X(\mathcal{D})$ is not Cohen-Macaulay
since $\lfloor \mathcal{D}(e_1) \rfloor = -2$.
\end{example}

Finally we give an example in dimension $3$,
of rational $\mathbb{T}$-varieties
of complexity one
that are Cohen-Macaulay and not Du Bois.
By taking product with toric varieties
this variety produces such examples in any dimension.

\begin{example}\label{cm}
Consider $M_\qq \simeq \qq^2$,
with a canonical basis $e_1, e_2$
and let $\sigma=\langle e_1, e_2 \rangle$.
Choose $r$ points $p_1,\dots, p_r$ in $\pp^1$
and $r$ elements $q_1,\dots, q_r$ of $N_\qq$
such that
\begin{equation}\label{contract}
q_1+\dots + q_r = (0, Q),
\end{equation}
for some $Q \in \mathbb{Z}_{>0}$.
Consider the p-divisor
\[
\mathcal{D}_{(q_1,\dots, q_r)}:=
\sum_{i=1}^r (q_1+ \sigma)\cdot p_i.
\]
We claim that $X\left(\mathcal{D}_{(q_1,\dots, q_r)}\right)$
is Cohen-Macaulay.
Ineed, observe that $X\left(\mathcal{D}_{(q_1,\dots, q_r)}\right)$
has exactly one
 horizontal $\mathbb{T}$-invariant prime divisor,
since the birational contraction
$\widetilde{X}\left(\mathcal{D}_{(q_1,\dots, q_r)}\right) \rightarrow X\left(\mathcal{D}_{(q_1,\dots, q_r)}\right)$
contracts the $\mathbb{T}$-invariant prime divisor corresponding to the ray $e_2$, by the equality~\eqref{contract}.
Moreover, such $\mathbb{T}$-invariant prime divisor is a normal $\kk^*$-surface,
and there exists  a regular function
that only vanishes along such divisor.
Indeed, consider $m\in \zz_{\geq 1}$ big enough such that $mq_1,\dots, mq_r$ are integer numbers,
then we can pick a rational function $f$ which vanishes with order $mq_{i,1}$ at the point $p_i$, for each $i\in \{1,\dots, r\}$,
where $q_{i,1}$ is the value of the
first coordinate of the point $q_i$, with respect to the dual basis of $e_1,e_2$,
therefore $f\chi^{e_2}$ is such a regular function.
Thus, we conclude that $X=X(\mathcal{D}_{(q_1,\dots, q_r)})$ is Cohen-Macaulay,
since ${\rm div}(f\chi^{e_2})$ is Cohen-Macaulay and its complement is smooth on $X$.
From Theorem~\ref{thm1}, we know that
$X(\mathcal{D}_{(q_1,\dots,q_r)})$ is not Du Bois if and only if
it does not have rational singularities,
so it is enough to impose the condition
\[
\sum_{i=1}^r \lfloor e_1 \cdot q_i \rfloor \leq -2,
\]
to produce the desired example.
\end{example}

\end{document}


TITLE: When does strong convergence imply convergence in operator norm?
QUESTION [2 upvotes]: I have a $C_0$-semigroup $(T_t)_{t\ge0}$ and I want to show

$\lim_{t \to \infty} T_t =0 $ with respect to the operator norm.

After some effort, I was able to prove

$\lim_{t \to \infty} T_t =0 $ with respect to the strong operator topology.

Now I know that in general strong convergence doesn't imply convergence in operator norm. Therefore my question is the following:

What additional conditions on $(T_t)_{t\geq 0}$ or its generator $A$ are sufficient for strong convergence to imply convergence in operator norm?

I would be happy to be provided with some references too.

REPLY [4 votes]: Since your semigroup is $C_0$, if it satisfies some condition according to which "strong convergence implies convergence in operator norm", then it is uniformly continuous (which I suspect is not the case). Therefore, you probably should try another strategy. For this, the following result can be useful.

For a $C_0$-semigroup $(T(t))_{t\geq 0}$ on a Banach space $X$ with generator $A$, the following assertions are equivalent.

$\|T(t)\|\overset{t\to\infty}{\longrightarrow}0$ (uniform stability).
$e^{\varepsilon t}\|T(t)\|\overset{t\to\infty}{\longrightarrow}0$ for some $\varepsilon>0$ (uniform exponential stability).
$e^{\varepsilon t}\|T(t)x\|\overset{t\to\infty}{\longrightarrow}0$ for all $x\in X$ and some $\varepsilon>0$.
$\|T(t_0)\|< 1$ for some $t_0>0$.
$r(T(t_0))<1$ for some $t_0>0$, where $r$ is the spectral radius.
$\displaystyle \int_0^\infty\|T(t)x\|^p\, dt<\infty$ for some/all $p\in[1,\infty)$ and all $x\in X$ (Datko-Pazy Theorem).
If $(T(t))_{t\geq 0}$ is eventually norm continuous:
$s(A)<0$, where $s$ is the spectral bound.
If $X$ is Hilbert:
$\{\lambda\mid \operatorname{Re}\lambda>0\}\subset\rho(A)$ and $\displaystyle \sup_{\operatorname{Re}\lambda>0}\|(\lambda I-A)^{-1}\|<\infty$ (Gearhart-Prüss Theorem).
If $X$ is Hilbert and $(T(t)_{t\geq 0}$ is of contractions:
$\mathbf{i}\mathbb{R}\subset\rho(A)$ and $\displaystyle \limsup_{|\lambda|\to\infty}\|(\mathbf{i}\lambda I-A)^{-1}\|<\infty$ (Gearhart-Prüss Theorem).

Source: Engel & Nagel and Liu & Zheng. Other equivalences can be found in Neerven.


TITLE: Homeomorphism and compact spaces
QUESTION [1 upvotes]: I can't really pinpoint the idea behind the following example ( from wikipedia https://en.wikipedia.org/wiki/Homeomorphism )
"The Euclidean real line is not homeomorphic to the unit circle as a subspace of $\Bbb{R^2}$, since the unit circle is compact as a subspace of Euclidean $\Bbb{R^2}$ but the real line is not compact.
Why can't we have a homeomorphism between these spaces? The real line is considered open ( or closed and unbounded ), I believe, and thus is not compact. So we would have a mapping from a non-compact set onto a compact set ( either the mapping or its inverse, both need to be continuous ). Does this imply we can't construct a continuous mapping between these spaces?

REPLY [2 votes]: Assume that $f:C\to\mathbb R$ is a homeomorphism, where $C$ is the unit circle. Clearly
the open intervals
$$
U_n=(-n,n), \quad n\in\mathbb N,
$$
form an open cover of $\mathbb R$, i.e. $\bigcup_{n\in\mathbb N}U_n=\mathbb R$. Set
$$
V_n=f^{-1}(U_n), \quad n\in\mathbb N.
$$
Then $V_n$'s are open subsets of $C$, in its relative topology, as inverses of open sets, and they form an open cover of $C$. Since $C$ is compact, the open cover $\{V_n\}_{n\in\mathbb N}$ of $C$ contains a finite subcover $V_{n_1},\ldots,V_{n_k}$. We may assume that $n_1<n_2<\cdots<n_k$. This means that $U_{n_1}\subset\cdots\subset U_{n_k}$, and hence $V_{n_1}\subset\cdots\subset V_{n_k}$. Thus
$$
C=\bigcup _{j=1}^k V_{n_j}=V_{n_k}=f^{-1}(-n_k,n_k).
$$
Contradiction.


text:
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import analysis.inner_product_space.calculus
import analysis.inner_product_space.pi_L2

/-!
# Euclidean distance on a finite dimensional space

When we define a smooth bump function on a normed space, it is useful to have a smooth distance on
the space. Since the default distance is not guaranteed to be smooth, we define `to_euclidean` to be
an equivalence between a finite dimensional normed space and the standard Euclidean space of the
same dimension. Then we define `euclidean.dist x y = dist (to_euclidean x) (to_euclidean y)` and
provide some definitions (`euclidean.ball`, `euclidean.closed_ball`) and simple lemmas about this
distance. This way we hide the usage of `to_euclidean` behind an API.
-/

open_locale topological_space
open set

variables {E : Type*} [normed_add_comm_group E] [normed_space ℝ E] [finite_dimensional ℝ E]

noncomputable theory

/-- If `E` is a finite dimensional space over `ℝ`, then `to_euclidean` is a continuous `ℝ`-linear
equivalence between `E` and the Euclidean space of the same dimension. -/
def to_euclidean : E ≃L[ℝ] euclidean_space ℝ (fin $ finite_dimensional.finrank ℝ E) :=
continuous_linear_equiv.of_finrank_eq finrank_euclidean_space_fin.symm

namespace euclidean

/-- If `x` and `y` are two points in a finite dimensional space over `ℝ`, then `euclidean.dist x y`
is the distance between these points in the metric defined by some inner product space structure on
`E`. -/
def dist (x y : E) : ℝ := dist (to_euclidean x) (to_euclidean y)

/-- Closed ball w.r.t. the euclidean distance. -/
def closed_ball (x : E) (r : ℝ) : set E := {y | dist y x ≤ r}

/-- Open ball w.r.t. the euclidean distance. -/
def ball (x : E) (r : ℝ) : set E := {y | dist y x < r}

lemma ball_eq_preimage (x : E) (r : ℝ) :
  ball x r = to_euclidean ⁻¹' (metric.ball (to_euclidean x) r) :=
rfl

lemma closed_ball_eq_preimage (x : E) (r : ℝ) :
  closed_ball x r = to_euclidean ⁻¹' (metric.closed_ball (to_euclidean x) r) :=
rfl

lemma ball_subset_closed_ball {x : E} {r : ℝ} : ball x r ⊆ closed_ball x r :=
λ y (hy : _ < _), le_of_lt hy

lemma is_open_ball {x : E} {r : ℝ} : is_open (ball x r) :=
metric.is_open_ball.preimage to_euclidean.continuous

lemma mem_ball_self {x : E} {r : ℝ} (hr : 0 < r) : x ∈ ball x r := metric.mem_ball_self hr

lemma closed_ball_eq_image (x : E) (r : ℝ) :
  closed_ball x r = to_euclidean.symm '' metric.closed_ball (to_euclidean x) r :=
by rw [to_euclidean.image_symm_eq_preimage, closed_ball_eq_preimage]

lemma is_compact_closed_ball {x : E} {r : ℝ} : is_compact (closed_ball x r) :=
begin
  rw closed_ball_eq_image,
  exact (is_compact_closed_ball _ _).image to_euclidean.symm.continuous
end

lemma is_closed_closed_ball {x : E} {r : ℝ} : is_closed (closed_ball x r) :=
is_compact_closed_ball.is_closed

lemma closure_ball (x : E) {r : ℝ} (h : r ≠ 0) : closure (ball x r) = closed_ball x r :=
by rw [ball_eq_preimage, ← to_euclidean.preimage_closure, closure_ball (to_euclidean x) h,
  closed_ball_eq_preimage]

lemma exists_pos_lt_subset_ball {R : ℝ} {s : set E} {x : E}
  (hR : 0 < R) (hs : is_closed s) (h : s ⊆ ball x R) :
  ∃ r ∈ Ioo 0 R, s ⊆ ball x r :=
begin
  rw [ball_eq_preimage, ← image_subset_iff] at h,
  rcases exists_pos_lt_subset_ball hR (to_euclidean.is_closed_image.2 hs) h with ⟨r, hr, hsr⟩,
  exact ⟨r, hr, image_subset_iff.1 hsr⟩
end

lemma nhds_basis_closed_ball {x : E} :
  (𝓝 x).has_basis (λ r : ℝ, 0 < r) (closed_ball x) :=
begin
  rw [to_euclidean.to_homeomorph.nhds_eq_comap],
  exact metric.nhds_basis_closed_ball.comap _
end

lemma closed_ball_mem_nhds {x : E} {r : ℝ} (hr : 0 < r) : closed_ball x r ∈ 𝓝 x :=
nhds_basis_closed_ball.mem_of_mem hr

lemma nhds_basis_ball {x : E} :
  (𝓝 x).has_basis (λ r : ℝ, 0 < r) (ball x) :=
begin
  rw [to_euclidean.to_homeomorph.nhds_eq_comap],
  exact metric.nhds_basis_ball.comap _
end

lemma ball_mem_nhds {x : E} {r : ℝ} (hr : 0 < r) : ball x r ∈ 𝓝 x :=
nhds_basis_ball.mem_of_mem hr

end euclidean

variables {F : Type*} [normed_add_comm_group F] [normed_space ℝ F] {f g : F → E} {n : with_top ℕ}

lemma cont_diff.euclidean_dist (hf : cont_diff ℝ n f) (hg : cont_diff ℝ n g)
  (h : ∀ x, f x ≠ g x) :
  cont_diff ℝ n (λ x, euclidean.dist (f x) (g x)) :=
begin
  simp only [euclidean.dist],
  apply @cont_diff.dist ℝ,
  exacts [(@to_euclidean E _ _ _).cont_diff.comp hf,
    (@to_euclidean E _ _ _).cont_diff.comp hg, λ x, to_euclidean.injective.ne (h x)]
end

subset_name: curated


TITLE: Terry Tao Analysis I
QUESTION [0 upvotes]: $6 \leq 2x \leq 4$ for all $3 \leq x \leq 2$ is true but $6 \leq 2x \leq 4$ for some $3 \leq x \leq 2$ is false

Could someone help me out why the second one is false.

REPLY [2 votes]: There is no $x$ satisfying $3\leq x\leq 2$. So, any statement is true for all such $x$ (since there are none).
However, there does not exist a value of $x$ which has this property (see above), which makes the second statement false.

REPLY [1 votes]: The second one says that there is at least one value of $x$ such that $3\leq x\leq 2$ for which $6\leq 2x\leq 4$.  There are no values of $x$ satisfying $3\leq x\leq 2$, so stating that "there is at least one $x$ satisfying this" is incorrect.
If you were to suppose there was such an $x$ then you would have that $3\leq 2$ which is false... $3$ is not less than $2$ in this context.
Compare to the first which says that for every $x$ satisfying $3\leq x\leq 2$ (if any exist in the first place) you will have $6\leq 2x\leq 4$.  This is vacuously true since there are no $x$ satisfying it.

REPLY [1 votes]: Note the direction of the inequality signs. Neither inequality is ever true, for any $x$.
If you say "for any $x$ such that the second inequality is true, the first inequality is true", then that's vacuously true. That's what the first statement says. However, the second statement says that there is some $x$ which fulfills the first inequality, and we know there are none. This makes the second statement false.


TITLE: Bijection implies isomorphism for algebraic varieties
QUESTION [10 upvotes]: Let $f:X\to Y$ be a morphism of algebraic varieties over $\mathbb C$. Assume that
a) $f$ is bijective on $\mathbb C$-points
b) $X$ is connected
c) $Y$ is normal.
Does it imply that $f$ is an isomorphism? If not, are there stronger reasonable conditions under which it is true?

REPLY [11 votes]: I am just expanding the comment above.  The statement is local on $Y$, so assume that $Y$ is connected.  Since $Y$ is normal, this is equivalent to assuming that $Y$ is irreducible.  Grothendieck's formulation of Zariski's Main Theorem is EGA $\textrm{III}_2$, Théorème 4.4.3, p. 136.  According to that theorem, there exists a factorization of $f$, $f=\overline{f}\circ i$ where $i:X\hookrightarrow\overline{X}$ is a dense open immersion, and where $\overline{f}:\overline{X}\to Y$ is a finite morphism.  Identify $X$ with the image of $i$.
Since $f$ is bijective on points, there exists a unique irreducible component $\overline{X}_0$ of $\overline{X}$ that dominates $Y$.  The restriction of $\overline{f}$ to $\overline{X}_0$ is a finite, surjective morphism that is generically one-to-one.  Since the characteristic is $0$, this morphism is birational (in positive characteristic, it might be purely inseparable but not birational).  Thus, by the classical form of Zariski's Main Theorem, $\overline{f}$ restricts to an isomorphism from $\overline{X}_0$ to $Y$, cf. Mumford's Red Book of Varieties and Schemes, p. 288.
By way of contradiction, assume that $\overline{X}_0$ is a proper subset of $\overline{X}$.  Denote by $Z \subset \overline{X}$ the union of all irreducible components different from $\overline{X}_0$.  Since $X$ is a dense open in $\overline{X}$, also $X\cap Z$ is a dense open in $Z$.  Thus, also $(X\cap Z) \setminus (\overline{X}_0\cap Z)$ is a dense open subset of $Z$.  Denote by $W$ the image in $Y$ of this open subset.  Since $f$ is injective, the constructible subset $W$ is disjoint from the dense open $V=f(X\cap \overline{X}_0)$ of $Y$.  Thus, the closure of $W$ is disjoint from $V$.  Since $\overline{f}$ is finite, this closure equals the image of $Z$.  Thus, $X\cap \overline{X}_0$ is disjoint from $\overline{X}_0\cap Z$, i.e., $X\cap \overline{X}_0\cap Z$ is empty.  Thus, the open subsets $\overline{X}_0\setminus (\overline{X}_0\cap Z)$ and $Z\setminus (Z\cap \overline{X}_0)$ of $\overline{X}$ pullback to disjoint open subsets of $X$ that cover $X$.  Since $X$ is connected, this is a contradiction.
Therefore, $\overline{X}_0$ equals all of $\overline{X}$.  Since $\overline{X}_0\to Y$ is an isomorphism, it follows that $f$ is an open immersion.  Since $f$ is also surjective, $f$ is an isomorphism.


TITLE: Linearity coefficient mapping to polynomials.
QUESTION [1 upvotes]: In this Vandermonde matrix section right at the beginning of my book, the authors comment on the linearity properties of polynomials and conclude that this must mean a mapping from polynomial coefficients to sampled polynomial values is linear. Their explanation in photo:

That seems like a large jump to me. Can someone please elaborate on how the coefficients, say, $c_0,...,c_k, \, 1 < k < n-1$ can be linearly mapped as mentioned, to a corresponding polynomial of degree $k$? Importantly, because of the text leading up to that conclusion.
I think I'm just overthinking this. Thank you.

REPLY [1 votes]: Think of it this way:
Let $p(x)=c_2x^2+c_1x+c_0$. Then we want to create a map that sends $$\mathbf{c}=\begin{bmatrix}c_0\\c_1\\c_2\end{bmatrix} \to \begin{bmatrix}p(x_1)\\p(x_2)\\p(x_3)\end{bmatrix}=\begin{bmatrix}c_0+c_1x_1+c_2x_1^2\\c_0+c_1x_2+c_2x_2^2\\c_0+c_1x_3+c_2x_3^2\end{bmatrix}=\mathbf{p}.$$
But
$$\begin{bmatrix}c_0+c_1x_1+c_2x_1^2\\c_0+c_1x_2+c_2x_2^2\\c_0+c_1x_3+c_2x_3^2\end{bmatrix}=\begin{bmatrix}1&x_1&x_1^2\\1&x_2&x_2^2\\1&x_3&x_3^2\end{bmatrix}\begin{bmatrix}c_0\\c_1\\c_2\end{bmatrix}.$$
So this map is simply a multiplication by (the Vandermonde) matrix, hence linear.
Added note
Based on your comment I think this is what you might be looking for.
Let $T$ be the map from the coefficient space to the the sampled points space. Continuing with the example above, let $q(x)=d_2x^2+d_1x+d_0$, then the map $T$ will send
$$\mathbf{d}=\begin{bmatrix}d_0\\d_1\\d_2\end{bmatrix} \to \begin{bmatrix}q(x_1)\\q(x_2)\\q(x_3)\end{bmatrix}=\mathbf{q}.$$
Now consider
$$T(\mathbf{c}+\mathbf{d})=\begin{bmatrix}(p+q)(x_1)\\(p+q)(x_2)\\(p+q)(x_3)\end{bmatrix}=\begin{bmatrix}p(x_1)+q(x_1)\\p(x_2)+q(x_2)\\p(x_3)+q(x_3)\end{bmatrix}=\begin{bmatrix}p(x_1)\\p(x_2)\\p(x_3)\end{bmatrix}+\begin{bmatrix}q(x_1)\\q(x_2)\\q(x_3)\end{bmatrix}=T(\mathbf{c})+T(\mathbf{d}).$$
Similarly you can check that $T(\lambda \mathbf{c})=\lambda T(\mathbf{c})$. Hence the linearity is obtained based on the linearity of the polynomial space.


TITLE: Sequences that diverge, limit stuff
QUESTION [0 upvotes]: I have a quadratic based sequence,
say its of the general form : $y = ax^2 + bx +c$
To put a concrete example the sequence : $a(n) = n^2 +5n +6$

$\lim{a(n)}\to\infty$ when $n\to\infty$

I am trying to think of an Epsilon-N way to prove this.
I am not sure how to start to break down the quadratic, would one factor it in product form and manipulate that, or would one use some kind of Triangle Inequality on the quadratic.
Hope to get some clarification on this.

REPLY [2 votes]: We notice that for positive $n$ we have $$a(n)> n^2$$Now for each $M$ we can choose $N=\max\{|M|,1\}$ and thus for $n>N$ we get $$a(n)>n^2\ge N^2\ge |M|^2\ge |M|\ge M$$


TITLE: Is $e^{2(\cos(t)-1)}$ the characteristic function of some random variable?
QUESTION [1 upvotes]: I am asked to decide whether
$$f(t)=e^{2(\cos(t) -1)}$$
is the characteristic function of some random variable.
Attempt. I am trying to find directly a possible associated random variable (which might be a totally bad strategy). The assignment actually starts like this:

Let $X$, $Y$ be two i.i.d. random variables with characteristic function $\varphi_X$ ($=\varphi_Y$). Compute $\varphi_{-Y}$ and $\varphi_{X-Y}$ in terms of $\varphi_{X}$.

It's immediate to see that $\varphi_{X-Y}(t)=\varphi_X(t)\varphi_X(-t)$. Thus I can maybe use this observation to guess that, if $f(t)=\varphi_X(t)$ for some r.v. $X$, then $X=Y-Z$ where $\varphi_Y(t)=e^{\cos(t) -1}=e^{\cos(-t) -1}=\varphi_{-Z}(t)$.
Therefore it doesn't seem a bad idea to look for some density function $g_Y(x)$ such that
$$e^{\cos t -1} = \int_{\mathbb R} e^{itx} g_Y(x) dx = \int_{\mathbb R} \cos (tx) \ g_Y(x) dx + i \int_{\mathbb R} \sin (tx) \ g_Y(x) dx.$$
Since $e^{\cos t -1} \in \mathbb R$ for all $t \in \mathbb R$, we have that $i \int_{\mathbb R} \sin (tx) \ g_Y(x) dx$ must be $0$. Hence a good choice for $g_Y(x)$ could be an even function, defined on a symmetric domain (w.r.t. $0$).
Here I'm stuck. It seems to me that I am still quite far from a solution. "Even function on a symmetric domain" is clearly a quite small achievement (and it's even just a guess).

Please note that we didn't discuss Bochner's theorem in class. We did discuss Inversion Theorem, but apparently computations get quite messy.

I'd prefer to receive just hints for the moment, not full solutions. Thank you for your help.

REPLY [3 votes]: Hints:

If $Y$ is Bernoulli symmetric with $P(Y=1)=P(Y=-1)=\frac12$ then $\varphi_Y(t)=$ $____$.
If $N$ is Poisson with parameter $\lambda$, then the generating function of $N$ is $$g_N(s)=E(s^N)=\sum_{n\geqslant0}s^n\mathrm e^{-\lambda}\frac{\lambda^n}{n!}=\mathrm e^{-\lambda}\sum_{n\geqslant0}\frac{(\lambda s)^n}{n!}=\text{____}.$$
If $Y$ and $(Y_n)$ is i.i.d. and $N$ is Poisson with parameter $\lambda$ and independent of $(Y_n)$, then $$X=\sum\limits_{n=1}^NY_n$$ is such that, for every $n\geqslant0$, $$E(\mathrm e^{\mathrm itX}\mid N=n)=E(\mathrm e^{\mathrm it(Y_1+Y_2+\cdots+Y_n)})=E(\mathrm e^{\mathrm itY_1})^n=\varphi_Y(t)^n,$$ hence $$\varphi_X(t)=E(E(\mathrm e^{\mathrm itX}\mid N))=E(\varphi_Y(t)^n)=g_N(\varphi_Y(t)),$$
Finally, $\varphi:t\mapsto\mathrm e^{-2(1-\cos(t))}$ is the characteristic function of $X=$ $____$.


text:
In the previous sections, we have studied a simple network structure where the server communicates directly with the users via an error-free broadcast link. We here briefly describe some ways in which this aspect of the setup has been generalized in the literature.

\subsection{Network Topologies}
There have been several works in the literature which study more complex topologies for the cache network. We briefly describe two such models below:

\begin{enumerate}
\item \emph{Hierarchical networks}: In this model, the server is connected to the users via a tree network, with caches of possibly different sizes at each level of the tree and where each cache at level $i$ communicates with its' children at level $i+1$ via an error-free broadcast link. Note that the server is the root of this hierarchical caching network and the users are the leaves. \cite{KNMDhierarchical} studied the special case of a two-level hierarchical tree caching network with $N$ files of size $F$ bits each at the server, communicating via an error-free broadcast link with $K_1$ \emph{mirrors} each with a cache of memory size $M_1F$ bits, at level $1$. Each of these mirror nodes is connected to $K_2$ \emph{users} each with a cache of size $M_2F$ bits. The system operates as before in two phases: a placement phase when all caches are populated with content, and then after the user requests are revealed, a delivery phase where the server sends a common message of size $R_1F$ bits to the mirrors and each mirror sends a message of size $R_2F$ bits to its connected users. \cite{KNMDhierarchical} proposed a scheme and showed  that for any $M_1, M_2$, the required rates $R_1, R_2$ for the proposed scheme are within a constant factor (independent of all problem parameters) of the information-theoretic optimal rates. A desired feature of the proposed scheme is that the delivery phase only uses messages which involve coding across a single layer of storage at a time. Details can be found in \cite{KNMDhierarchical}. \\

\item \emph{Device-to-device networks}: In this model, there is no designated server in the network which hosts the entire catalogue of $N$ files. The system consists of $K$ co-located users, each with a cache of size $MF$ bits, which can communicate with each other over an error-free broadcast link. This setup was studied in \cite{MingyueD2D1, MingyueD2D2} where they proposed a caching and delivery scheme for this setup and analyzed the total required transmission size on the shared link, as well as compared it to information-theoretic lower bounds.

\end{enumerate}

\subsection{Interference networks and physical-layer considerations}

Most of our discussion so far has been limited to a single base station.
An interesting problem is to consider multiple base stations, each of which has access to the content library one way or another.
These base stations will then interfere with each other, and the question becomes how to manage this interference for the purpose of content distribution.

Our discussion has so far ignored the physical layer, instead treating all channels as error-free bit pipes.
In broadcast networks this is not a big issue since separating what to send from how to transmit it is natural in a broadcast setting.
However, this is no longer straightforward when several base stations are present.

This problem of caching in interference networks was first studied in \cite{MNinterference}, which had an interference channel with three transmitters that were equipped with caches and three receivers that were requesting content.
This was later extended in \cite{TSfogran} to consider an arbitrary number of transmitters and receivers.
Problems with caches both at the transmitters and at the receivers were then studied, but with restrictions on the schemes: \cite{NMAinterferencemanagement} was limited to one-shot linear schemes while \cite{XLTinterference} prohibited coding across files during the placement phase.
Furthermore, both \cite{XLTinterference} and \cite{HNDlayered} looked at a limited number of transmitters and receivers (no more than three).

The first general result was published in \cite{HNDdofcaching}, which found an approximate characterization of the information-theoretically optimal rate-memory trade-off in the high-SNR regime.
Three key insights into the problem are derived.
First, it is shown that a separation of the physical and network layers is approximately optimal: a physical-layer scheme focuses on transmitting some message set across the interference network using a technique known as interference alignment, and the network-layer scheme uses this message set as error-free bit pipes to implement a coded caching scheme.
Second, it is shown that, as long as the transmitters can collectively store exactly the entire content library, then increasing the transmitter memory has no effect on the optimal rate beyond a constant multiplicative factor.
A consequence of this is that it is not necessary for the transmitters to share information: they can all store distinct parts of the content library for most gains to be obtained.
Third, there is a trade-off between the receiver memory and the number of transmitters needed to approximately achieve maximal system performance: as the receiver memory increases, fewer transmitters are required.
In particular, when each receiver can hold a fraction of the library, then a constant number of transmitters is sufficient to achieve most benefits.
We will discuss these results in more detail below.

At the other extreme, the low-SNR regime was studied in \cite{HNDenergyefficiency}, where a similar separation of the network and physical layers is proposed.
This separation architecture is shown to be approximately optimal in some cases, namely the single-receiver and the single-transmitter cases.
Contrary to the high-SNR regime, it is shown in \cite{HNDenergyefficiency} that transmitter co-operation, by storing shared content in their caches, is crucial in the low-SNR regime.

\subsubsection{The Separation Architecture}

Although the separation architecture was studied specifically for a Gaussian interference network, we will first describe it in a very general context as the same ideas hold.
We will then show how it applies to the Gaussian network.

\paragraph{High-Level Overview of the Separation Architecture}

There is a content library containing $N$ files of size $F$ bits.
The library is separated from the users by an interference channel.
The interference channel has $K_t$ transmitters and $K_r$ receivers, who act as the users.
Each transmitter has a cache of memory $M_tF$ bits and each receiver has a cache of memory $M_rF$ bits.

During the placement phase, we place information about the files in every transmitter and receiver cache.
During the delivery phase, each transmitter $\ell\in\{1,\ldots,K_t\}$ sends a codeword $\mathbf{x}_\ell = (x_\ell(1), \ldots, x_\ell(T))$ over a block length of $T$ through the interference network.
Importantly, the codeword $\mathbf{x}_\ell$ can only depend on the file requests and the contents of transmitter $\ell$'s cache; the transmitter has no knowledge of the entire content library.
Each receiver $k\in\{1,\ldots, K_r\}$ then receives a signal $\mathbf{y}_k=(y_k(1),\ldots,y_k(T))$.
Finally, each receiver $k$ uses the received signal $\mathbf{y}_k$ in combination with the contents of its cache to recover the requested file.
The goal is to maximize the transmission rate defined as $R=F/T$.

A key aspect of this problem that has not been discussed in previous sections is that it combines caching with physical-layer delivery.
The main idea of the separation architecture is to separate the caching aspect (what to store in the caches and what to send from transmitters to receivers) from the physical-layer aspect (how to send it through the interference network).
Thus the system is split into an overlay network layer and a physical layer.
The two layers interface through a set of messages from (subsets of) transmitters to (subsets of) receivers.
This is illustrated in \figurename~\ref{coded:fig:separation}.

\begin{figure}
\centering
\includegraphics[width=\textwidth]{separation.pdf}
\caption{Illustration of the separation architecture.
The cache-aided interference channel is split into a physical layer and a network layer; the two layers interface using a message set.}
\label{coded:fig:separation}
\end{figure}

In the most general sense, we define a set of $n$ messages
\[
\mathscr{V} = \left\{ V_{\mathcal{K}_1\mathcal{L}_1}, V_{\mathcal{K}_2\mathcal{L}_2}, \ldots, V_{\mathcal{K}_n\mathcal{L}_n} \right\},
\]
where $\mathcal{K}_i\subseteq\{1,\ldots,K_r\}$ and $\mathcal{L}\subseteq\{1,\ldots,K_t\}$, and $V_{\mathcal{K}_i\mathcal{L}_i}$ is a message from the transmitters in $\mathcal{L}_i$ to the receivers in $\mathcal{K}_i$.
At the physical layer, we have the sub-problem of transmitting this message set across the interference network reliably.
It is assumed that all the transmitters in $\mathcal{L}_i$ have access to the message $V_{\mathcal{K}_i\mathcal{L}_i}$.
At the network layer, we can use these messages as orthogonal, non-interacting bit pipes through which we can pass information from the transmitters to the receivers.
The constraint is that every transmitter in $\mathcal{L}_i$ must be able to cosntruct $V_{\mathcal{K}_i\mathcal{L}_i}$ from the contents of its cache.

Suppose the physical-layer scheme can transmit every message in the message set at a rate of at least $R'$, over a block length of $T$.
Suppose also that the network-layer scheme can deliver the requested files using at most $vF$ bits through each bit pipe represented by the messages in the message set.
We thus have $vF\ge R'T$.
Then, the system would have delivered the requested files to the user at a total rate of $R = F/T \ge v/R'$.

\paragraph{The High-SNR Gaussian Interference Network}

In the particular case of the memoryless Gaussian interference network, the inputs and outputs to the channel are real-valued.
At each time step $\tau$, every output symbol $y_k(\tau)$ is a linear combination of all the input symbols $x_k(\tau)$, plus a Gaussian unit-variance random noise variable.
Furthermore, a power limit of $P$ is imposed on the input codewords, $\|\mathbf{x}_\ell\|^2\le PT$.

We are interested in the high-SNR regime, i.e., the regime in which $P$ is large.
More specifically, we will look at the degrees of freedom (DoF) of the system, which is the behavior of the rate as a scaling of the capacity of a point-to-point Gaussian channel.
Specifically, if we write the information-theoretically optimal rate for a specific power $P$ as $R^\star(P)$, then the degrees of freedom is defined as
\[
\mathsf{DoF} = \lim_{P\to\infty} \frac{R^\star(P)}{\frac12\log P}.
\]

We will next describe the approximately optimal strategy within the context of the high-SNR cache-aided Gaussian interference network.
This strategy makes use of the separation architecture in the following way.
\begin{enumerate}
\item The message set $\mathscr{V}$ that is chosen is a set of messages from every single receiver $\ell$ to every subset $\mathcal{K}$ of receivers of a fixed size $|\mathcal{K}|=\kappa+1$, where $\kappa\approx K_rM_r/N$.
Notice that this is exactly the same as the multicast size in the broadcast setup in Section~\ref{coded:sec:overview}.
Thus the message set represents a set of single-transmitter multicast channels.
\item At the network layer, we partition every file in the content library into $K_t$ parts and store each part at one transmitter.
Thus each transmiter has a content sub-library consisting of part of every file.
A standard centralized Maddah-Ali--Niesen scheme is performed on each sub-library, and each multicast message (intended for $\kappa+1$ receivers) is sent through the corresponding bit pipe.
\item At the physical layer, we apply a technique known as interference alignment in order to transmit the message set as efficiently as possible.
\footnote{In fact, interference alignment can exactly achieve the degrees of freedom of the communication problem that arises at the physical layer.
Note that this is the degrees of freedom associated with the rate $R'$ described earlier; it is not the degrees of freedom of the entire cache-aided interference network.}
\end{enumerate}

The following theorem gives the approximate degrees of freedom of the network, as determined in \cite{HNDdofcaching}.
\begin{theorem}
\label{coded:thm:dof}
The degrees of freedom of the cache-aided Gaussian interference network is approximately given by
\[
\mathsf{DoF} \approx
\frac{K_tK_r}{K_t+K_r-1}
\cdot \frac{1}{1-M_r/N}
\cdot \frac{K_rM_r/N+1}{\frac{M_r}{N}(\frac{1}{K_r}+\frac{1}{K_t-1})^{-1}+1},
\]
for all $N$, $K_t$, $K_r$, $M_r\in[0,N]$, and $M_t\ge N/K_t$.
The approximation is within a constant multiplicative factor.
\end{theorem}

Notice that the degrees of freedom can be written as the product of three gains, in a similar way to the rate expression in Section~\ref{coded:sec:overview}.
The first term is the interference alignment gain and represents the DoF when no receiver caches are present.
The second term is the local caching gain, as with the broadcast case.
The third term is the global caching gain.

Finally, some powerful insights can be gained from Theorem~\ref{coded:thm:dof}.
The first is that a separation of the network and physical layers is approximately optimal.
Second, one can see that the DoF expression does not involve the transmitter memory $M_t$, excet in the condition $K_tM_t\ge N$.
This implies that there is no benefit (no more than a constant factor) in increasing the transmitter memory.
In particular, it shows that transmitter co-operation is unnecessary: the DoF can be approximately achieved even if the transmitters share no information at all.
Third, as the receiver memory increases, the number of transmitters needed for approximately achieving the DoF decreases.
These insights together show that the cache-aided interference network problem can be solved with a system that is layered and simple to design and implement.

config: arxiv


TITLE: For continuous $f:X\rightarrow [0,1]$ is it true that $a<b\implies \overline{f^{-1}[0,a)}\subset f^{-1}[0,b)$?
QUESTION [0 upvotes]: Let $f:X\rightarrow [0,1]$ be continuous function.
Is it true that $a<b\implies \overline{f^{-1}[0,a)}\subset f^{-1}[0,b)$? I'm asking because I can't think of a counterexample. (This is not homework.)

REPLY [2 votes]: If $a < b$ then $[0,a] \subset [0,b)$ and as $f$ is continuous you have that $f^{-1}([0,a])$ is closed giving
$$\overline{f^{-1}([0,a))} \subseteq \overline{f^{-1}([0,a])} = f^{-1}([0,a]) \subseteq f^{-1}([0,b)) $$