text:
\begin{document}

\begin{abstract}
  We give a common generalization of (1) Hassett's weighted stable
  curves, and (2) Hacking-Keel-Tevelev's stable hyperplane arrangements.
\end{abstract}
\maketitle

\tableofcontents

\section{Introduction and main statements}

The moduli space $\oM_{0,n}$ of stable $n$-pointed rational curves
has many generalizations, beginning of course with $\oM_{g,n}$. For
this paper, however, the following two generalizations will be
important:
\begin{enumerate}
\item Hassett's moduli $\oM_{0,\w}$ of weighted stable $n$-pointed
  curves \cite{Hassett_WeightedStableCurves}, and
\item Hacking-Keel-Tevelev's moduli $\oM(r,n)$ of stable hyperplane
  arrangements \cite{HackingKeelTevelev}.
\end{enumerate}

A \emph{weight data}, or simply a \emph{weight},
$\w$ is a collection of $n$ of rational (or real) numbers $0 <
b_i\le~1$.
We denote $\bo=(1,\dotsc,1)$.
A weighted stable curve of genus zero is a nodal curve
$X=\cup \bP^1$ whose dual graph is a tree, together with $n$ points
$B_1, \dotsc, B_n$ satisfying two conditions:
\begin{enumerate}
\item (on singularities) $B_i\ne$ the nodes, and whenever some points
  $\{B_i,\ i\in I\}$ coincide, one has $\sum_{i\in I}b_i\le~1$.
\item (numerical) $K_X+\sum b_i B_i$ is ample. In plain words, this
  means that for every irreducible component $E$ of $X$, one has
  \begin{math}
    |E\cap(X-E)| + \sum_{B_i\in E} b_i > 2.
  \end{math}
\end{enumerate}
The space $\oM_{0,\w}$ is the fine moduli space for flat families of
such weighted curves; it is smooth and projective.

A \emph{stable pair} $(X,B=\sum_{i=1}^n b_iB_i)$ is a natural
higher-dimensional analogue of the above notion. It consists of
a connected equidimensional projective variety $X$ together with $n$ Weil
divisors $B_i$ satisfying the following conditions
(see \cite{Alexeev_ICMtalk} for more details):
\begin{enumerate}
\item (on singularities) $X$ is reduced, and
the pair $(X,B)$ is slc (semi log canonical),
  and
\item (numerical) $K_X+B$ is ample.
\end{enumerate}

In \cite{HackingKeelTevelev} the authors construct a projective
scheme, which we will denote $\oM(r,n)$, together with a flat family
$f:(\cX,\cB_1,\dotsc,\cB_n)\to\oM(r,n)$ such that every geometric
fiber $(X,\sum B_i)$ is a stable pair in the above definition, with
all coefficients $b_i=1$.  Over an open (but not dense in general) subset
$\M(r,n) \subset \oM(r,n)$ this gives the universal family of $n$
hyperplanes $B_i$ on a projective space $X=\bP^{r-1}$ such that $B_i$
are in general position.
The construction originates in
\cite{Kapranov_QuotientsGrassmannians}, see also \cite{Lafforgue_Chirurgie}.

More generally, let $\w$ be a weight, and $B_1,\dotsc, B_n$ be $n$
hyperplanes in $\bP^{r-1}$. Then the pair $(\bP^{r-1}, \sum b_i B_i)$
is
\begin{enumerate}
\item \emph{lc (log canonical)} if for each intersection $\cap_{i\in I}B_i$
  of codimension $k$, one has $\sum_{i\in I}b_i \le k$, and
\item \emph{klt (Kawamata log terminal)} if the inequalities are
  strict, in particular all $b_i<1$.
\end{enumerate}
(This is consistent with the standard definitions of the Minimal Model
Program.)

The pair $(\bP^{r-1}, \sum b_i B_i)$ is stable in the above definition
iff it is lc (slc being an analog of lc for possibly nonnormal pairs) and
$|\w|= \sum_{i=1}^n b_i > r$.  We call such pairs \emph{weighted
  hyperplane arrangements}, or simply \emph{lc hyperplane
  arrangements}.
One easily constructs a fine moduli space
$\M_{\w}(r,n)$ for them; it is smooth, of dimension $(r-1)(n-r-1)$,
and usually not complete (but see Theorem \ref{thm:small-weights} for
the exceptions).

Throughout the paper, we work over an arbitrary commutative ring $\cA$
with identity. The main results of this paper are the three theorems
below and the detailed description of the weighted stable hyperplane
arrangements given in Section~\ref{sec:defin-moduli-univ}.

\begin{theorem}[Existence]\label{thm:existence}
  For each $r,n$ and a rational weight $\w=(b_i)$ with $|\w|=\sum
  b_i>r$, there exists a projective scheme $\oM_{\w}(r,n)$ together
  with a locally free (in particular, flat) family
  $f:(\cX,\cB_1,\dotsc,\cB_n) \to \oM\lw(r,n)$ such that:
  \begin{enumerate}
  \item Every geometric fiber of $f$ is an $(r-1)$-dimensional variety
    $X$ together with $n$ Weil divisors $B_i$ such that the pair
    $(X,\sum b_i B_i)$ is stable.
  \item For distinct geometric points of $\oM\lw(r,n)$, the fibers are
    non-isomorphic.
  \item Over an open (but not dense in general) subset
    $\M_{\w}(r,n)\subset \oM_{\w}(r,n)$, $f$ coincides with the universal
    family of weighted hyperplane arrangements.
  \end{enumerate}
  For every positive integer $m$ such that all $mb_i\in\bN$, the sheaf
  $\cO_{\cX}(m( K_{\cX}+\sum b_i\cB_i) )$ is relatively ample and
  free over $\oM\lw$.
\end{theorem}

The fibers of $f$ will be called \emph{weighted stable hyperplane
  arrangements}, or simply \emph{slc hyperplane arrangements}.
As one has $\M_{\bo}(r,n) \subset
\M_{\w}(r,n)$, in particular, each of $\oM_{\w}(r,n)$ provides a
moduli compactification of the moduli space of generic hyperplane
arrangements.

\begin{definition}\label{defn:chambers}
We define the \emph{weight domain}
\begin{displaymath}
  \cD(r,n) = \{ (b_i)\in \bQ^n \mid
  0<b_i\le 1,\quad \sum b_i > r \}
\end{displaymath}
and a subdivision of it into locally closed \emph{chambers},
denoted $\chamber(\w)$, by the
hyperplanes $\sum_{i\in I}b_i = k$ for all $1\le k \le r-1$ and
$I\subset \{1,\dotsc,n\}$, and by the faces $b_i=1$.

We introduce a partial order on the points of $\cD(r,n)$: $\w>\w'$ if
for all $1\le i \le n$ one has $b_i\ge b_i'$, with at least one
strict inequality.
\end{definition}

We will frequently assume $(r,n)$ fixed and drop it from the notation.

\begin{example}\label{exmp:3-degs}
  ($r=3$, $n=5$) Consider a 1-parameter family of 5 lines on $\bP^2$
  in general position such that in the limit $B_1,B_2,B_5$ meet at a
  point $q_1$, and $B_3,B_4,B_5$ meet at a point $q_2\ne q_1$. This is
  not allowed by the lc
  singularity condition if $b_1+b_2+b_5>2$ or
  $b_3+b_4+b_5>2$. Since the spaces $\oM_{\w}$ are proper, there is
  always a stable pair limit, but its shape depends on the weight:

  \begin{enumerate}
  \item $\w=(1,1,1,1,1-\epsilon)$. The variety is $X_{\w}=X^0\cup X^1\cup
    X^2$, where $X^0$ is the blowup of $\bP^2$ at
    $q_1$ and $q_2$, and $X^0$ is glued along the exceptional $\bP^1$'s to
    $X^1=\bP^2$, $X^2=\bP^2$.  The divisor $B_5$ has three
    irreducible components, each of $B_1,\dotsc, B_4$ has two.
    $B_1,B_2$ are contained in $X^0\cup X^1$, and $B_3,B_4$ in
    $X^0\cup X^2$. All five divisors are Cartier.
  \item $\w'=\bo=(1,\dotsc,1)$. The variety $X_{\w'}$ is obtained from
    $X_{\w}$ by contracting the $(-1)$-curve $B_5\cap X^0$. The image
    of $X^0$ is ${X'}^0=\bP^1\times\bP^1$, the divisors $B_1,B_2$
    restricted to ${X'}^0$ are fibers of a ruling, and $B_3,B_4$
    restricted to it are fibers of the second ruling. The divisor
    $B_5$ intersects ${X'}^0$ at one point, and so is not $\bQ$-Cartier.
  \item $\w''=((1+\epsilon)/2,(1+\epsilon)/2,1,1,1-\epsilon)$. The
    variety $X_{\w''}$ is obtained from $X_{\w}$ by contracting $X^1$.
    \end{enumerate}
    Note that $\w'>\w>\w''$ and $\w',\w''\in \overline{\cham(\w)}$,
    we have natural morphisms
    $X_{\w'}\gets X_{\w} \to X_{\w''}$, and the first of these
    morphisms is birational.

\end{example}

\begin{theorem}[Reduction morphisms]
\label{thm:red-moduli}
  \begin{enumerate}
  \item (Same chamber) For $\w,\w'$ lying in the same chamber, one has
    $\oM_{\w}=\oM_{\w'}$ and
    $(\cX,\cB_i)_{\w}=(\cX_,\cB_i)_{\w'}$. In particular, the divisors
    $\sum (b_i-b_i)B_i$ are $\bQ$-Cartier.
  \item For $\w'\in \overline{\chamber (\w)}$, there are
    natural reduction morphisms
    \begin{equation*}
      \xymatrix{
        (\cX,\cB_i)_{\w} \ar[r]^{\pi_{\w,\w'}} \ar[d] &
        (\cX,\cB_i)_{\w'} \ar[d] \\
        \oM_{\w} \ar[r]^{\rho_{\w,\w'}} &
        \oM_{\w'}
      }
    \end{equation*}
    One has
    \begin{displaymath}
      \pi_{\w,\w'}^* \cO_{\cX\lwp}(m( K_{\cX\lwp}+\sum b_i'\cB_i) )
      = \cO_{\cX\lw}(m( K_{\cX\lw}+\sum b_i'\cB'_i) )
    \end{displaymath}
    for any $m$ such that all $mb_i'\in\bZ$.
  \item (Specializing up) For $\w' \in \overline{\chamber (\w)}$ with
    $\w<\w'$, $\rho_{\w,\w'}$ is an isomorphism, and on the fibers
    $\pi_{\w,\w'}:X\to X'$ is a birational contraction restricting to
    an isomorphism $X\setminus\cup B_i \isoto X'\setminus\cup B'_i$.
  \item For any $\w>\w'$, there is a natural reduction morphism
    $    \rho_{\w,\w'}:\oM_{\w}\to \oM_{\w'}.    $
    On the fibers, the rational map $\pi_{\w,\w'}:X\dashrightarrow X'$ is a
    sequence of log crepant contractions and log crepant birational
    extractions. Further, $X'=\Proj \oplus_{d\ge0}H^0(X,\cO(dm(K_X+\sum
    b_i'B_i)))$ is the log canonical model for the pair $(X,\sum b_i'B_i)$.
  \end{enumerate}
\end{theorem}

\begin{theorem}[Moduli for small weights]
  \label{thm:small-weights}
  Let $\alpha=(a_i)$ be a weight with $\sum a_i=r$ (lying on the boundary of
  $\cD$) which belongs to the closure of a unique chamber $\cham(\w)$. Then
  $$\M\lw=\oM_{\w} =(\bP^{r-1})^n//PGL(r) = \Gr(r,n) //\bG_m^{n-1} $$
  is the GIT quotient for the line bundle, resp.
  linearization corresponding to $\alpha$.
\end{theorem}

For \emph{any} boundary weight (i.e. with $|\alpha|=r$), we can formally
define $\oM_{\alpha}$ to be the above GIT quotient.  Over an open
and dense subset $\M_{\alpha}$ it gives the moduli of lc hyperplane
arrangements on $\bP^{r-1}$ such that $K_{\bP^{r-1}} + \sum
a_i B_i=0$. For $\alpha$ as in the theorem, one has $\M_{\alpha}=\oM_{\alpha}
=\oM\lw$.

\begin{notations}
  We work over an arbitrary commutative base ring $\cA$ with identity,
  without the Noetherian assumption, and indeed can work over any base
  scheme. $A$ will denote an $\cA$-algebra, and $k=\bar k$ an
  $\cA$-algebra which is an algebraically closed field.  The tilde
  will be used to denote affine schemes $\wt X$, cones $\wt\Delta$,
  etc., which are cones over the corresponding projective schemes $X$,
  polytopes $\Delta$, etc.
\end{notations}

It may help the reader to grasp some combinatorial aspects of this
paper with the following general outline. The (unweighted) stable
hyperplane arrangements are described by matroid tilings of the
hypersimplex $\Delta(r,n)$. Their weighted counterparts are described
by partial tilings of $\Delta(r,n)$ as viewed through a smaller
``window'' $\Delta_{\w}(r,n)$; the window must be completely covered.

Another key idea is the GIT interpretation of the weight $\beta$
explained in Section~\ref{sec:git}.

\section{Matroid polytopes}\label{sec:matroid-polytopes}

We begin with some general definitions and then specialize them to the
case of grassmannians.

\begin{setup}\label{setup}
  We fix two lattices $\bZ^N=\oplus\bZ e_j$ and $\bZ^n$, a homomorphism
  $\phi:\bZ^N\to \bZ^n$, and a homomorphism $\deg:\bZ^n\to\bZ$, such
  that $\deg \phi(e_j)=1$ for all $j$.
  Associated to this data are affine $\bA=\bA^N$ and
  projective $\bP=\bP^{N-1}$ spaces over $\cA$
  and linear actions of split tori $\wT=\bG_m^n$ on $\bA$ and of
  $T=\wT/\diag\bG_m$ on $\bP$.

  Let $\Delta$ be the lattice polytope that is the convex hull of
  $\phi(e_i)$, and $\wt\Delta$ be the corresponding cone in $\bR^n$.
  We also fix a $\bZ^n$-graded ideal
  $I[\wZ]\subset\cA[z_1,\dotsc,z_N]$ such that the quotient is a locally
  free (i.e. projective) $\cA$-module.
  Hence, $\wZ\subset\bA$ is a $\wT$-invariant closed subscheme. Let
  $Z\subset \bP$ be the corresponding $T$-invariant closed subscheme.
\end{setup}

\begin{definition}
For a geometric point $p\in Z(k)$, the closure of the orbit
$\overline{T.p}$ is a possibly nonnormal toric subvariety of
$Z_k=Z\times_{\cA}k$.
It corresponds to a
lattice polytope $P$ which we will call the \emph{$Z$-polytope} or the
\emph{moment polytope of $p$}.
(Indeed, when $k=\bC$, $P$ is  the moment polytope of
$\overline{T.p}$, as defined in symplectic geometry.) A character
$\chi\in\bZ^n$ is in the cone $\wP$ iff there exists a monomial
$z^m=\prod_i^N z_i^{m_i}$ such that $\phi(m)=d\chi$ and $z^m(p)\ne0$.

A \emph{$Z$-tiling} $\uP$ is a face-fitting subdivision of $\Delta$
into $Z$-polytopes.
\end{definition}

We fix several faces $F_i$, $i=1,\dotsc,n'$, of $\Delta$. Each of them
is defined by the inequality $l_i\le1$ for a unique $\bZ$-primitive
linear function $l_i(x_1,\dotsc,x_n)$. In a completely parallel
fashion with our grassmannian setup, an element $\w=(b_i)\in\bQ^{n'}$,
$b_i\le1$, is called a weight.
For each weight we define a subpolytope
\begin{displaymath}
  \Delta\supset\Delta\lw = \{ l_i\le b_i \}
\end{displaymath}
The \emph{weight domain} $\cD\subset\bR^{n'}$
is the set of the weights for which $\Delta\lw$ is nonempty and
maximal-dimensional.

\begin{definition}
  A \emph{weighted $Z$-polytope} is a polytope of the form $P\lw=
  P\cap \Delta_{\w}$ for some $Z$-polytope $P$, called
  \emph{the parent} of $P\lw$, such that $\Int(P)\cap
  \Delta_{\w}\ne \emptyset$.

  A \emph{weighted $Z$-tiling} $\uP\lw$ is a face-fitting
  tiling of $\Delta_{\w}(r,n)$ by weighed $Z$-polytopes.
  The \emph{partial} cover $\uP$ of $\Delta$ by the parent
  polytopes is called \emph{the parent cover} of $\uP\lw$.
\end{definition}

\begin{definition}\label{defn:Z-chamber-decomp}
  The $Z$-chamber decomposition of $\cD$ is defined as follows: $\w,\w'$
  lie in the same chamber if for every $Z$-polytope $P$, one has
  $P\cap\Delta\lw\ne\emptyset \iff P\cap\Delta\lwp\ne\emptyset$.
  Consequently, weighted $Z$-tilings of $\Delta\lw$ and $\Delta\lwp$
  are in a bijection.
\end{definition}

We now specialize these definitions to the case of the
grassmannians. The polytope $\Delta$ in this case is called the
\emph{hypersimplex} and the $Z$-polytopes are called \emph{matroid
  polytopes}. For the unweighted version, these notions were
introduced in \cite{GelfandGoreskyMacPhersonSerganova}.

Let $\Gr(r,n)$ be the grassmannian of $r$-planes in a fixed affine
space $\bA^n$, together with its Pl\"ucker embedding into
$\bP(\wedge^r \bA^n)=\bP^{N-1}$, where   $N={n\choose r}$.
Let $\wgr(r,n)\subset \bA^N$  be the affine cone. It is defined by the
classical quadratic Pl\"ucker relations.

For $I=(i_1,\dotsc,i_r)$, the Plu\"cker coordinate $p_I$ has character
$$\weight(p_I)=(1,\dotsc,1,0\dotsc,0)$$ with $r$ ones in the places
$i_1,\dotsc,i_r$ and with $(n-r)$ zeros elsewhere.

\begin{definition}
  The convex hull of these $N$ points is called the
  \emph{hypersimplex} $\Delta(r,n)$. Alternatively, it can be
  described as follows:
  \begin{displaymath}
    \Delta(r,n) = \left\{ (x_i)\in \bR^n \mid 0\le x_i \le 1, \quad
    \sum_{i=1}^n x_i = r \right\}
  \end{displaymath}
It has $2n$
faces $F_i^+= \{x_i=1\}$ and $F_i^-=\{x_i=0\}$, isomorphic to
$\Delta(r-1,n)$ and $\Delta(r,n-1)$ respectively.

We fix the lattice $\Lambda\simeq\bZ^n$ in $\bR^n$ consisting
of the vectors $(x_i)\in\bZ^n$ such that $\sum x_i$ is divisible by
$r$ and a homomorphism $\deg:\Lambda\to\bZ$ so that the characters of the
Pl\"ucker coordinates $p_I$ generate $\Lambda$ and have degree 1.
\end{definition}

\begin{definition}
  A \emph{matroid polytope} $P_V\subset \Delta(r,n)$ is the polytope
  corresponding to the toric variety $\overline{T.V}$ for some
  geometric point $[V\subset\bA^n]\in\Gr(r,n)(k)$.
  (Theorem~\ref{thm:matroid-polytopes}(1) implies that this projective
  toric variety and the corresponding affine variety are normal,
  unlike the case of general $Z$).
\end{definition}

The equations of the coordinate hyperplanes restricted to $V$ give
$n$ vectors $z_1, \dotsc, z_n \in V^*$, what is called a
\emph{realizable matroid}.
Then $\weight(p_{i_1,\dotsc,i_r})$ is a vertex of $P_V$ iff
$z_{i_1}, \dotsc, z_{i_r}$ form a basis of $V^*$. Alternatively,
$P_V$ can be described inside $\Delta(r,n)$ by the inequalities
$\sum_{i\in I}x_i \le \dim\ \Span(z_i,\ i\in I)$, for all $I\subset
\{1,\dotsc,n\}$.

One can also describe the matroid polytopes in terms of hyperplane
arrangements. Let $\bP V\simeq\bP^{r-1}$ be the corresponding
projective space and assume that it is not contained in any of the $n$
coordinate hyperplanes $H_i$ (i.e. all $z_i\ne0$ on $\bP V$); let
$B_1,\dotsc,B_n\subset \bP V$ be $H_i\cap\bP V$. Then
$\weight(p_{i_1,\dotsc, i_r})$ is a vertex of $P_V$ iff
$B_{i_1}\cap\dotsc \cap B_{i_r}$ is a point. Alternatively,
$P_V$ can be
described inside $\Delta(r,n)$ by the inequalities $\sum_{i\in I}x_i
\le \codim\cap_{i\in I}B_i$ for all $I\subset \{1,\dotsc,n\}$. Note
that the matroid polytope in this case is not contained in any of the faces
$F_i^-=\{x_i=0\}$.

\begin{definition}
  A \emph{matroid tiling} $\uP$ of $\Delta(r,n)$ is a face-fitting
  subdivision $\cup P(V_{s})$  of $\Delta(r,n)$ into matroid polytopes.
\end{definition}

Matroid polytopes form a very particular class of lattice polytopes,
with many properties not shared by general lattice polytopes. Some of their
properties can be summarized as follows:

\begin{theorem}\label{thm:matroid-polytopes}
  \begin{enumerate}
  \item Every matroid polytope is generating, i.e. its integral points
    generate the group of integral points of $\bR P$. Moreover, the
    semigroup of integral points in $\wP$ is generated by the vertices
    of $P$.
  \item A matroid polytope of codimension $c$ is in a canonical way
    the product of $c+1$ maximal-dimensional matroid polytopes for smaller
    $r,n$. So, one has $r=r_0+\dotsc+ r_c$ and
    $\{1,\dotsc,n\}=I_0\sqcup \dotsb \sqcup I_c$, and $P=\prod P_j$,
    where $P_j\subset \Delta(r_j,|I_j|)$ is a maximal-dimensional
    matroid polytope.
  \end{enumerate}
\end{theorem}

We now introduce the weighted versions of these notions.

\begin{definition}
  Let $\w=(b_1,\dotsc,b_n)$ be a weight. A \emph{weighted
    hypersimplex} is the polytope given by
  \begin{displaymath}
    \Delta_{\w}(r,n) = \left\{ (x_i)\in \bR^n \mid 0\le x_i \le b_i, \quad
    \sum_{i=1}^n x_i = r \right\}.
  \end{displaymath}
  Similarly, we also have definitions of a \emph{weighted matroid
  polytope}, a \emph{weighted matroid tiling $\uP\lw$}, and \emph{the
  parent cover} of $\uP\lw$.
\end{definition}

\begin{question}\label{qu:extend-to-parent}
  Can every parent cover be extended to a \emph{complete cover} of
  $\Delta(r,n)$? For $r=2$ the answer is easily seen to be ``yes''.
  For $r\ge3$ we expect the answer to be ``no'', following the general
  philosophy of ``Mnev's universality theorem'' (cf.
  \cite[Thm.I.14]{Lafforgue_Chirurgie} which shows that matroid geometry
  can be arbitrarily complicated.
\end{question}

\begin{theorem}[Chamber decomposition]
  \label{thm:chamber-decs-coincide}
  The chamber decomposition of $\cD(r,n)$ defined in
  \ref{defn:Z-chamber-decomp} coincides with that of
  Definition \ref{defn:chambers}.
\end{theorem}
\begin{proof}
  Starting with $\w=\w'$ and then varying the weight $\w$, the matroid
  decompositions of $\Delta\lw$ and $\Delta\lwp$ may possibly change if for
  some matroid polytope $P\subset\Delta(r,n)$, a vertex of $\Delta\lw$
  would lie on a face of $P$ that $\w'$ did not belong to.

  Every vertex of $\Delta\lw$ is of the following form: $x_i=0$ or
  $b_i$, for all but possibly one ${i_0}$.  Every face of a matroid
  polytope lies in the intersection of hyperplanes $\sum_{i\in I}x_i =
  k$ for some $1\le k\le n-1$ and $I\subset\{1,\dotsc,n\}$.
  Possibly after replacing
  $I$ by its complement, we can assume that $i_0\not\in I$. Then for
  some $J\subset I$ we get $\sum_{i\in J}b_i=k$. If $\w$ belongs to a
  face of $P$ that $\w'$ did not belong to, then we get a new equation
  of this form. So $\w$ lies in a different, smaller chamber.
\end{proof}

\cite{GelfandGoreskyMacPhersonSerganova} gives three different
interpretations of matroid polytopes. Here, we add another one.

\begin{theorem}\label{thm:matroid-lc}
  The matroid polytope $P_V$ is the set of points $(x_i)\in \bR^n$
  such that the pair $(\bP V,\sum x_iB_i)$ is lc and
  $K_{\bP V}+\sum x_iB_i=0$;
  the interior $\Int P_V$ is the set of points such that
  $(\bP V,\sum x_iB_i)$ is klt and   $K_{\bP V}+\sum x_iB_i=0$.
\end{theorem}
\begin{proof}
  Indeed, the defining inequalities $\sum_{i\in I}x_i\le \codim \cap_{i\in
    I}B_i$ of $P_V$ also happen to be the conditions for the   pair $(\bP
  V,\sum x_iB_i)$ to be lc. Similarly with the strict inequalities and klt.
\end{proof}

We don't even have to assume that $\bP V\not\subset H_i=\{z_{i_0}=0\}$
for this theorem: the pair $(\bP V,\sum x_iB_i)$ can only be lc if
$x_{i_0}=0$, otherwise $\sum x_iB_i$ is not a divisor. And indeed if
$\bP V\subset H_i$ then $P_V\subset \{x_i=0\}$, so the theorem
still holds.

\section{Moduli spaces for varieties with torus action}
\label{sec:two-moduli-spaces}

Let $Z\subset \bP$ be a projective scheme locally free over $\cA$ and
invariant under the $T$-action, and $\wt Z\subset\bA$ be its affine cone,
with the $\wT$-action, as in our general setup \ref{setup}.
Two moduli spaces of varieties with torus
action will be relevant for this paper.

\begin{enumerate}
\item The toric Hilbert scheme $\Hilb^{\wT}(\wZ,\wt\Delta)$, constructed
  in \cite{PeevaStillman,HaimanSturmfels}.
\item The moduli space $\M^T(Z,\Delta)$ of finite $T$-equivariant maps
  $Y\to Z$ of stable toric varieties $Y$ over $Z$,
  constructed in \cite{AlexeevBrion_SphericalModuli}, see also
  \cite{AlexeevBrion_Affine,AlexeevBrion_Projective,Alexeev_CMAV}.
  This is the equivariant multiplicity-free version of the moduli
  space of branchvarieties   \cite{AlexeevKnutson}.
\end{enumerate}
Both of these moduli spaces are projective schemes. Both
are available in much more general
settings; we will only need the simplest versions.

For an $\cA$-algebra $A$, $\Hilb^{\wT}(\wZ,\wt\Delta)(A)$ is the set
of closed $\wT$-invariant subschemes $\wt Y \subset
\wZ_A=\wZ\times_{\cA} A$ which are
\emph{multiplicity-free:} for
every $x\in\bZ^n$ the graded piece $A[\wY]_x$ is a locally free rank-1
$A$-module if $x\in\wt\Delta$ and is $0$ otherwise. A geometric fiber
$\wY_k$ need not be reduced. $(\wY_k)_{\rm red}$ is a
union of possibly non-normal toric varieties glued along torus orbits
in a fairly complicated way.

In contrast, $\M^T(Z,\Delta)(A)$ is the set of locally free proper families
$Y$ over $\Spec A$ together with a finite $T$-equivariant morphism
$f:Y\to Z_A$ such that every geometric fiber $Y_k$ is a projective
\emph{stable toric variety.} In the ring $\oplus_{d\ge0}
H^0(Y,f^*\cO(d))$, for each $x\in\bZ^n$, the $x$-graded piece is a
locally free rank-1 $A$-module if $x\in\wt\Delta$ and is $0$
otherwise.  A stable toric variety is a \emph{reduced} variety glued
from \emph{normal} toric varieties along orbits in a very simple way,
so that the result is seminormal.  The price for such niceness is that $Y_k\to
Z_k$ is a finite morphism rather than a closed embedding.

The projective stable toric variety $Y_k$ comes with the polarization
$L=f^*\cO_{Z}(1)$. For each irreducible component of $Y_k$, this
gives a lattice polytope $P^{s}$, and together they give a
tiling $\Delta=\cup_{s\in \uP} P^{s}$ describing the gluing
in a rather precise way.
If the cone semigroups $\wP^{s}\cap\bZ^n$ are generated in degree
1 then $Y_k\to Z_k$ is a closed embedding and gives a point of
$\Hilb^{\wT}(\wZ,\wt\Delta)$.

In general, $\M^T(Z,\Delta)$ is only a coarse moduli space since a
finite map $Y_k\to Z_k$ may have automorphisms (deck
transformations). However, it is a fine moduli space over an open
subscheme where $Y_k\to Z_k$ is birational to its image (on every
irreducible component).

Finally, we note how these moduli spaces change if we replace $\bP$ by
a $d$-tuple Veronese embedding, which means replacing the ring
$\cA[z_1,\dotsc,z_N]$ by the subring generated by monomials of degrees
divisible by $d$. The answer for $\Hilb^{\wT}(\wZ)$ is very
non-obvious, and sometimes they indeed change. The moduli space
$\M^T(Z)$, however, does not change. Indeed, the scheme $Z$ does not
change, and neither does the $T$-action.  Therefore, $\M^T(Z)$ can be
defined as easily for a \emph{rational polytope} $\Delta$: it can
always be rescaled to make it integral.

\section{Review of the unweighted case}
\label{sec:unweighted-case}

We now apply the general theory of the previous section to the
grassmannians. Let $\gr=\gr(r,n)$ be the grassmannian with its
Pl\"ucker embedding into $\bP^N$, $N={n \choose r}$, and $\wgr$ be its
affine cone. Hence, $\cA[\wgr]$ is generated by the $N$ Pl\"ucker
coordinates $p_{i_1,\dotsc,i_r}$, modulo the usual quadratic
relations. The corresponding polytope is precisely the hypersimplex
$\Delta(r,n)$, and the polytopes $P^{s}$ appearing in the
constructions of the previous section are the matroid polytopes.

\begin{definition}[\cite{HackingKeelTevelev}]
  $\oM(r,n) = \Hilb^{\wt  T}(\wgr,\wt\Delta).$
\end{definition}

A closed $\wT$-invariant multiplicity-free subscheme $Y_k\subset
\gr_k$ gives a matroid subdivision of $\Delta(r,n)$.
\ref{thm:matroid-polytopes}(1) implies that $Y_k$ is reduced and
is a stable toric variety, so we are in the ``nice case''.  Note that $\dim
Y=n-1\ne r-1$, so $Y$ is not the required stable variety $X$. Instead,
it should be thought of as the \emph{log Albanese variety} $\logAlb(X,B)$.

\begin{definition}[\cite{HackingKeelTevelev}]
  $X\subset Y$ is the intersection of $Y$ with the subvariety
  in $\Gr(r,n)$ defined by
  \begin{displaymath}
    \gr^e = \{ V\subset \bA^n \mid (1,1,\dotsc,1) \in V \}.
  \end{displaymath}
  $\gr^e$ is a Schubert variety, isomorphic
  to $\Gr(r-1,n-1)$.
\end{definition}

One easily shows that $\gr^e \into \Gr$ is a regular codimension $n-r$
embedding (not $\bG_m^n$-equivariant), the zero set of a section of the
tautological bundle $Q$ on $\Gr$.  $X$ does not contain any
$T$-orbits. This implies that $X\subset Y$ is a regular codimension
$n-r$ embedding as well, the zero section of the bundle $Q|_Y$ with
$c_1(Q|_Y)=L$.

On the other hand, by \cite{Alexeev_CMAV}, the pair $(Y,\sum
B_i^{\pm})$ has semi log canonical singularities and $K_Y+\sum
B_i^{\pm}=0$, where $B_i^{\pm}$ are the divisors corresponding the
faces $F_i^{\pm}$ of $\Delta(r,n)$. In addition, $B_i^-\cap
X=\emptyset$. Denoting $B_i^+|_X= B_i$ and combining this together
gives the following:

\begin{theorem}[\cite{HackingKeelTevelev}]
  \label{thm:HKT-main}
  \begin{enumerate}
  \item There exists a smooth morphism $X\times T \to Y$ whose image
    is the open subset $Y\setminus \cup B_i^-$ swept by the $T$-orbits
    of $X\subset Y$; this is compatible with the divisors $B_i$ and $B_i^+$.
  \item Consequently, $(X,\sum B_i)$ and $(Y\setminus \cup B_i^-, \sum
    B_i^+)$ are isomorphic locally in smooth topology; in particular,
    the pair $(X,\sum B_i)$ is slc.
  \item $K_X + \sum B_i = (K_Y + \sum B_i^{\pm} + L)|_X = L|_X$; and
    so is ample.
  \item The poset of the stratification of $X$ defined by the
    irreducible components, their intersections, and the divisors
    $B_i$ coincides with the poset of the stratification on
    $\Delta(r,n)\setminus \cup F_i^-$ defined by the subdivision $\uP$.
  \end{enumerate}
\end{theorem}

\section{Weighted grassmannians}
\label{sec:weight-grassm}

Here we define certain projective schemes $\gr\lw$ and describe their
basic properties. We begin with the elementary case which already
contains the pertinent combinatorics of the general situation.

Let $P'$ be a lattice polytope, and $Y$ be the correponding projective
toric scheme over $\cA$ (a toric variety when working over a field $k$),
together with an ample $T$-linearized ample invertible sheaf $L'$.
Let $m$ be a positive integer, $P=P'/m$ a rational polytope, and
$L=L'/m \in \Pic(Y)\otimes\bQ$
be the corresponding $\bQ$-polarization.

Let us fix several faces $F_i$ of this polytope. Each of
them is given by a linear equation $x_i=b_i\in \bQ$, where $x_i$ is an
integral primitive linear function, so that $P$ lies in the half space
$x_i\le b_i$. Each of these faces corresponds to a
divisor $B_i$ on $Y$.

Now consider the polytope $P_{\w'}$ obtained by replacing the inequalities
$x_i\le b_i$ with  $x_i\le b_i'$ for some $b_i'\in\bQ$.
Here, we denote $\w=(b_i)$ and $\w'=(b_i')$, so that $P=P_{\w}$.
Note that for some $\w'$ one may have $\dim P_{\w'}< \dim P_{\w}$.

One says that
two polytopes are \emph{normally equivalent} if their normal fans
coincide, in other words, they define the same toric variety (with
possibly different $\bQ$-polarizations). The following elementary lemma is
well-known, and we omit the proof.

\begin{lemma}\label{lem:toric-ex}
  \begin{enumerate}
  \item $P_{\w}$ and $P_{\w'}$ are normally equivalent iff $\w'$
    belongs to the interior $\cham(\w)$ of a certain rational
    polytope.
  \item If $\w'\in \overline{\cham(\w)}$ then there exists a natural
    morphism $\pi_{\w,\w'}:Y_{\w}\to Y_{\w'}$; it is birational if $\w'>\w$.
  \item One has $\pi^*(L_{\w'})= L_{\w} + \sum (b_i'-b_i)B_i$. Thus,
    any positive multiple of this $\bQ$-line bundle that is integral,
    is semiample.
  \end{enumerate}
\end{lemma}

We note that this is precisely the kind of data that appears in
Theorem~\ref{thm:red-moduli}.
Now consider a projective scheme $Z\subset\bP$ as in the general setup
\ref{setup}, $Z=\Proj \cA[\wZ]$.

\begin{definition}
  For each weight $\wei$, let $\cA[\wZ\lw]\subset \cA[\wZ]$ be the
  subalgebra generated by the monomials whose characters lie in
  $\wt\Delta\lw$. We define $Z\lw:=\Proj \cA[\wZ\lw]$.
\end{definition}

\begin{theorem}\label{thm:properties-of-Zbeta}
  \begin{enumerate}
  \item Every $Z\lw$-polytope is the intersection of a $Z$-polytope
    with~$\Delta\lw$.
  \item There exists a rational map $Z\dashrightarrow Z\lw$. It is
    regular on the open subset of points of $Z$ whose moment polytopes
    intersect $\Delta\lw$.
  \item $Z$ and $Z\lw$ share an open subset $Z\lw^0$ whose points are
    the points with moment polytopes intersecting $\Int\Delta\lw$.
  \item There exists a chamber decomposition of the weight domain into
    finitely many interiors of polytopes, with the following properties:
    \begin{enumerate}
    \item If $\wei,\wei'$ belong to the same chamber, then $Z\lw=Z\lwp$.
    \item If $\wei'\in\overline{\cham(\wei)}$ then there exists a
      proper morphism $Z\lw\to Z\lwp$.
    \end{enumerate}
  \item Further, assume that for any $\cA$-field $k$, the corresponding
    variety $Z_k$ is integral, normal, and its monomials span the
    whole $\wt\Delta$. Then $Z\dashrightarrow Z\lw$ is a birational
    map, and in the
    previous statement one can take the $Z$-chamber decomposition
    defined in \ref{defn:Z-chamber-decomp}.
  \end{enumerate}
\end{theorem}
\begin{proof}
  We reduce the proof to the elementary case \ref{lem:toric-ex} of
  toric schemes, as follows. The homomorphism $\phi:\bZ^N\to \bZ^n$ in
  the setup~\ref{setup} gives a surjective map of polytopes $\sigma\to
  \Delta$, where $\sigma$ is a simplex with $N$ vertices. The preimage
  of $\Delta\lw$ is a certain subpolytope $\sigma\lw \subset \sigma$.
  Monomials of high enough degree $d$ generate the subalgebra
  $\cA[\wZ\lw]^{(d)}$, and this gives the embedding of $Z\lw$ into the
  toric scheme corresponding to the polytope $\sigma\lw$.

  Now the properties (1--4) are elementary for the ambient toric
  schemes, and hence they also hold for the subschemes $Z\lw$.

  To prove (5), note that that (2) and (3) together imply that
  $Z\dashrightarrow Z\lw$ is a birational map. Let $\wei,\wei'$
  belong to the same $Z$-chamber. Then on every geometric fiber we get a
  birational morphism $\varphi_k:(Z\lw)_k \to (Z\lwp)_k$. Since the
  $Z\lw$-polytopes and $Z\lwp$-polytopes are the same,
  the $T$-orbits of $Z\lw,Z\lwp$ are in a dimension-preserving
  bijection, and so $\varphi_k$ is
  finite. Since $(Z\lwp)_k$ is normal, $\varphi_k$ is an isomorphism by
  the Main Zariski theorem. Since $Z\lw$ and $Z\lwp$ are free over
  $\cA$, and  $\varphi:Z\lw \to Z\lwp$ is an isomorphism fiberwise, it
  is an isomorphism.
\end{proof}

We now specialize to the case of grassmanians. Thus,
for every weight $\wei\in\cD(r,n)$ we get a projective scheme
$\gr\lw$, which we will call the \emph{weighted grassmannian}, and the
collection $\{\gr\lw\}$ satisfies the conclusions of
Theorem~\ref{thm:properties-of-Zbeta}, where the chamber decomposition
is the one defined in Definition~\ref{defn:chambers}.

\section{GIT theory of the universal family over the grassmanian}
\label{sec:git}

A key role in our definition of weighted stable hyperplane
arrangements will be played by the Geometric Invariant Theory of the
universal family $U\to \gr(r,n)$. Let us first review the basics
relevant to our case.

Let $Z\subset \bP$ be as in the setup~\ref{setup}. Then we have an
action of $T=\wT/\diag(\bG_m)$ on $Z$ and an action of $\wT$ on each
$\cO_Z(d)$, $d\in\bN$. The character group of $T$ is $\chi(T)=\{
(x_i)\in\bZ^n \mid \sum x_i=0 \}$.

A $T$-linearization of $\cO_Z(d)$ is an extension of the $T$-action
from $Z$ to $\cO_Z(d)$. It can be given by assigning to a monomial
$z^m=z_1^{m_1}\dots z_n^{m_n}$ of degree $\sum m_i = d$ an element
$\phi(m)\in \chi(T)$ so that $\phi(z^m)-\phi(z^{m'}) = m-m'$. This is
equivalent to choosing an element $\wei\in\bZ^n$ of degree $d$, so
that $\phi(z^m)=m-\wei$.
We can also take $\wei$ to be of arbitrary positive degree, and
subtract the unique element of degree $d$ on the line $\bQ\wei$.
Then for every $d'$ that is a multiple of $d$, the
element of degree $d'$ on this ray also describes the induced linearization
of $\cO_Z(d')$.

Given a $T$-linearized ample sheaf $L=\cO_Z(d)$, one considers the ring of
sections $R=R(Z,L)=\oplus_{d\ge0} \Gamma(Z,L^d)$. This ring was
already graded by $\bZ^n$ by the setup. The linearization provides a
new grading by $\chi(T)=\bZ^{n-1}\subset \chi(\wT)=\bZ^n$.

The GIT quotient $Z\lwq T$ is defined to be $\Proj R\lw$, where
the latter denotes the elements of degree 0 in the
$\chi(T)$-grading. In the original $\bZ^n$-grading, this means that we
consider the elements spanned by the monomials whose character in
$\bZ^n$ lies on the line $\bQ\wei$. Note as well that replacing $L$ by
a positive power does not change $Z\lwq T$. Hence, the input for this
construction is a weight $\wei$ up to a multiple, and an ample
invertible sheaf $L$ up to a multiple.

Applied to the grassmannian $\gr(r,n)$ and the Pl\"ucker line bundle
$\cO_{\gr}(1)$, this means that every weight $\wei\in\cD(r,n)$ gives a
linearization and a GIT quotient $\gr\lwq T$. The quotients do not
respect the chamber structure of $\cD(r,n)$, however.

\emph{Our key observation now is that the chamber structure describes
  not the GIT quotients of $\gr$ but those of the universal family $U$
  over it.}

Let $U\subset \bP^{n-1}\times \gr(r,n)$ be the universal family of linear
spaces $\bP V \subset \bP^{n-1}$. Each of the $n$ hyperplanes
$H_i=\{z_i=0\}$ in $\bP^{n-1}$ defines a hyperplane $B_i \subset \bP V$,
unless $\bP V\subset H_i$.

The natural ample invertible sheaves on $U$ are
$L_{a,b}=p_1^*\cO_{\bP^{n-1}}(a) \otimes p_2^*\cO_{\gr}(b)$ for
$a,b\in\bN$. The total ring of global sections of all of them is
\begin{displaymath}
  \oplus_{a,b\ge0}H^0(U,L_{a,b}) = \frac{\cA[z_i,p_I]}
  {( \text{Pl\"ucker relations on }p_I,\ r_J )}
\end{displaymath}
where for each $J=\{j_0,\dotsc,j_r\}$, $r_J = \sum (-1)^k z_{j_k}
p_{J\setminus j_k} $. The $\bZ^n$-character of each $z_i$ is $e_i$, the
$i$-th coordinate vector in $\bZ^n$, and for $p_I$ it is $\sum_{i\in
  I}e_i$. Hence, the $\bZ$-degrees of $z_i$ and $p_I$ are $1$ and $r$
respectively, differing slightly from \ref{setup}.

\begin{definition}
  We choose:
\begin{enumerate}
\item The ample $\bQ$-line bundle $L_{a,b}$ with $(a,b)=(|\wei|-r,1)$,
  or any actual ample invertible sheaf for a multiple $(ma,mb)$ such
  that $m\wei$ is integral.

  Note that if $\bP V\not\subset \cup H_i$ then $\cO(K_{\bP V}+ \sum
  b_i B_i) = \cO_{\bP V}(|\wei|-r)$.
\item The $T$-linearization corresponding to $\wei$.
\end{enumerate}
We denote the corresponding GIT quotient $U//\lw T$ by $\gr\lw^e$.
\end{definition}

\begin{lemma}\label{lem:gr-e}
  $\gr\lw^e$ is a closed subscheme of $\gr\lw$.
\end{lemma}
\begin{proof}
  Suppose that $m\wei$ is integral, and restrict to the subalgebra
  $\cA[\wgr\lw]^{(m)}$ whose homogeneous elements have degrees
  divisible by $m$.
  A monomial in $p_I$ can be complemented to a monomial in $p_I,z_i$
  whose character is proportional to $\wei$ exactly when its
  character, divided by the number of $p_I$'s, lies in $\Delta(r,n)$ and
  in the cone $\wei-\bR^n_{\ge0}$. The intersection of these two sets
  is precisely $\Delta\lw$.

  Hence, we have a surjective homomorphism $\cA[\wgr\lw]^{(m)}\to
  R^{(m)}\lw$ sending a monomial $\prod p_I$ to its complement
  $z^s\prod p_I$. This gives the closed embedding.
\end{proof}

\begin{example}
If $\wei=\bo$ then every monomial in $p_I$ can be
complemented, and $\gr^e\subset\gr$ is the zero set of the equations
$r_J(p_I,\bo)$. Thus, $\gr^e_{\bo}$ is the same as $\gr^e$ that appeared in
Section~\ref{sec:unweighted-case}.
\end{example}

GIT gives the description of $U\lwq T$ in terms of orbits for each
geometric fiber. To recall,
there are two open subsets in $U$:
\begin{enumerate}
\item The set $U\uss\lw$ of semistable points $p$ for which there
  exists a section
  $$
  s\in R\lw = \oplus_{(a,b)\in\bQ\wei} H^0(U,L_{a,b})
  \text{ such that }  s(p)\ne 0.
  $$
\item The set $U\us\lw$ of (properly) stable points whose orbit in
  $U\uss\lw$ is closed and the stabilizer is finite; in our case
  trivial. (This set was denoted by $U\us_{(0)}$ in
  \cite{Mumford_GIT}. We use the currently prevalent notation.)
\end{enumerate}

Then we have a surjective morphism $U\lw\uss\to U\lwq T$,
the action is free on $U\us\lw$ and $U\lw\us/T$ is a geometric
quotient. Points of $U\uss\lw$ have the same image iff the
closures of their orbits intersect. Among such orbits, there
exists a unique closed one.

For the torus action, the Hilbert-Mumford's criterion for
(semi)stability takes an especially simple form. The following criterion
is well-known (e.g., cf. \cite{BrionProcesi}):
\begin{enumerate}
\item $p\in U\lw\uss \iff$ $\wei$ belongs to the moment polytope of $p$.
\item $p\in U\lw\us \iff$ $\wei$ lies in the interior of the moment
  polytope of $p$ and the latter is maximal-dimensional.
\end{enumerate}

The moment polytope here lies in $\bR^{n-1}$ which we shift so that it
lies in the hyperplane $\sum x_i = |\wei|$ in $\bR^n$. For our choice
$L_{|\wei|-r,1}$ of an ample $\bQ$-line bundle, the moment polytope of
the point $[p\in \bP V\subset \bP^{n-1}]\in U$ is:
\begin{displaymath}
  P_V + (|\wei|-r) \sigma_p,\quad \text{where}
\end{displaymath}
\begin{enumerate}
\item $P_V$, as before, is the matroid polytope of
  $[V\subset\bA^n]$, and
\item Denoting $I(p)=\{i \mid z_i(p)=0\}$,
  $$
  \sigma_p = \left\{ (x_i)\in \bR^n \mid x_i\ge0, \ \sum x_i =1,
  \text{ and } x_i=0 \text{ for } i\in I(p) \right\}
  $$
\end{enumerate}

\begin{definition}
  $\Delta\lw^p$ is the face of $\Delta\lw$ where $x_i=b_i$ for $i\in I(p)$.
\end{definition}

\begin{lemma}
  \begin{enumerate}
  \item $p\in U\uss\lw \iff P_V \cap \Delta\lw^p \ne\emptyset.$
  \item $p\in U\us\lw \iff \Int P_V \cap \Int\Delta\lw^p \ne\emptyset$ and
    $P_V+\Delta\lw^p$ spans $\bR^{n-1}$.
  \end{enumerate}
\end{lemma}
\begin{proof}
  (1) $   \wei\in P_V + (|\wei|-r) \sigma_p \iff
  P_V \cap \{\wei - (|\wei|-r)\sigma_p\} \ne\emptyset $. The intersection
  of the latter polytope with $\Delta$ is $\Delta\lw^p$.

  (2) The point is stable iff we can replace $\wei$ with any nearby
  $\wei'$. This means that $\Int P_V \cap \Int\Delta\lw^p
  \ne\emptyset$ and $P_V+\Delta\lw^p$ spans $\bR^{n-1}$.
\end{proof}

\begin{theorem}\label{thm:git-lc-klt}
  \begin{enumerate}
  \item If $P_V\cap\Delta\lw=\emptyset$ or $V\subset \{z_i=0\}$ for
    some $i$ then no $p\in V$ is $\wei$-semistable.
  \item Suppose $P_V\cap\Delta\lw\ne\emptyset$. Then
    $p\in U\uss\lw \iff (\bP V,\sum b_iB_i)$ is lc at $p$.
  \item Suppose $P_V\cap\Int\Delta\lw\ne\emptyset$. Then
    $p\in U\us\lw \iff (\bP V,\sum b_iB_i)$ is klt at $p$.
  \end{enumerate}
\end{theorem}
\begin{proof}
  (1) If $P_V\cap\Delta\lw=\emptyset$ then $p\not\in U\uss\lw$ by the
  lemma. If $V\subset \{z_i=0\}$ then $P_V\subset\{x_i=0\}$, and
  $\Delta\lw^p \subset \{x_i=b_i\}$, so they do not
  intersect.

  (2) Suppose $P_V \cap \Delta\lw^p \ne\emptyset.$ Take $\alpha=(a_i)$
  in this intersection. By Theorem~\ref{thm:matroid-lc} the pair $(\bP
  V, \sum a_iB_i)$ is lc. Since one has $\sum a_i B_i = \sum b_i B_i$
  near $p$, the latter divisor is lc as well.

  Vice versa, assume that $(\bP V, \sum b_iB_i)$ is lc at $p$.
  By assumption, there exists
  $\alpha\in P_V\cap\Delta\lw$. If $\alpha\not\in\Delta\lw^p$ then
  we are going to construct another $\alpha'=(a_i')\in
  P_V\cap\Delta\lw^p$.

  If $P_V$ is maximal-dimensional and $\alpha\in\Int P_V$ then begin
  by increasing $x_i$ for $i\in I(p)$ until we get to $x_i=b_i$ while
  decreasing $x_i$ with $i\not\in I(p)$ and keeping $x_i\ge0$. This
  is possible to do since $\sum_{i\in I(p)} b_i \le \codim \cap_{i\in
    I(p)} B_i\le r-1$. By doing this, we either achieve the required
  $\alpha'$ or get to a lower-dimensional matroid polytope $P_{V'}$.
  But by Theorem~\ref{thm:matroid-polytopes}

  $P_{V'}$ is the product of maximal-dimensional polytopes for lower
  $(r_j,n_j)$. We finish by induction on $r$.

  (3) is proved by the same argument using the second part of
  Theorem~\ref{thm:matroid-lc}.
\end{proof}

\begin{definition}
  Denote by $\bP\lw$ the projective toric scheme over $\cA$ (a toric
  variety when working over $k$) corresponding to the polytope $\Delta\lw$.
\end{definition}

In particular, $\bP_{\bo}$ is the toric variety corresponding to the
hypersimplex $\Delta(r,n)$.

\begin{theorem}
  The morphism $U\uss\lw \to U\lwq T$ factors through $\bP\lw\times\gr\lw$.
\end{theorem}
\begin{proof}
  Consider $\bP^{n-1}\times\gr$ with the very ample sheaf $L_{a,b}$,
  $(a,b)\in\bQ\wei$. The rational map $\bP^{n-1}\times\gr
  \dashrightarrow \bP\lw \times \gr\lw$ is given by the monomials
  $\prod p_I z^m$ whose character is proportional to $\wei$ and, when
  normalized, the $\prod p_I$-part belongs to $\Delta\lw$ and the
  $z^m$-part belongs to $\beta-\Delta\lw$, which is just another copy
  of $\Delta\lw$, reflected.

  This rational map is regular on the open subset where at least one
  of these monomials, considered as a section of $L^d$, is
  nonzero. But the ring generated by these monomials contains $R\lw$,
  so this open subset contains $U\uss\lw$.
\end{proof}

Recall from Section~\ref{sec:weight-grassm} that we denoted by
$\gr\lw^0$ an open subset of $\gr\lw$ and
$\gr$ that corresponds to $\Int\Delta\lw$.

\begin{definition}
  $U\lw^0 \to \gr\lw^0$ will denote the pullback of $U\uss\to
  \gr\lw$ under the open inclusion $\gr\lw^0\to\gr$.
\end{definition}

\begin{theorem}\label{thm:free-git-quot}
  \begin{enumerate}
  \item The $T$-action on $U\lw^0$ is free.
  \item The geometric quotient $\hat\gr\lw^e = U\lw^0/T$ is projective
    and comes with a semiample invertible sheaf defining a proper
    birational morphism $\hat\gr\lw^e\to\gr\lw^e$, that is an
    isomorphism over $\gr\lw^e \cap \gr\lw^0$.
  \end{enumerate}
\end{theorem}
\begin{proof}
  (1) A point $V\in \gr\lw^0$ corresponds to a linear space
  $\bP V\subset\bP^{n-1}$ whose matroid polytope intersects
  $\Int\Delta\lw$. Take $p\in V$ such that $p\in U\uss\lw$. Then by
  Theorem~\ref{thm:git-lc-klt}(1) the pair $(\bP V,\sum b_iB_i)$ is lc
  at $p$. If we take a nearby $\wei'=(b_i')$ with $b_i'<b_i$ then
  $(\bP V,\sum b_iB_i)$ will be klt. Then by \ref{thm:git-lc-klt}(2)
  we have $p\in U\us\lwp$. Hence, $U^0\lw = U\us\lwp$, the action
  is free, and the quotient is projective.

  By removing $(U\uss\lw\setminus U^0\lw)$, we changed the
  equivalence relation on $U\lw^0$: for some of the orbits in $U\lw^0$
  their closures in $U\uss\lw$ intersect, and so they map to the same
  point of $U\lwq T$. The criterion of Theorem~\ref{thm:git-lc-klt}
  implies that every closed orbit in $U\uss\lw$ is contained in the
  closure of an
  orbit of $U\lw^0$. Hence, $\hat \gr\lw^e \to \gr\lw^e$ is
  surjective. It is given by the pullback of an ample invertible sheaf
  $\cO(m)$ on $\Proj R\lw$. This morphism is an isomorphism on the
  open subset $U\us\lw/T$ which contains $\gr\lw^e\cap \gr\lw^0$.
\end{proof}

\begin{remark}
  In the case of $\wei=\bo$ our construction is different from that of
  \cite{HackingKeelTevelev}. To explain it succinctly,
  \cite{HackingKeelTevelev} proceeds ``horizontally'', while we proceed
  ``vertically''. The points in $U\uss_{\bo}\setminus U\us_{\bo}$ are:
  the points $p\in \cup B_i$
  and the points $p\in V$ such that $P_V\subset \{x_i=1\}$
  for some $i$, satisfying the conditions of Theorem~\ref{thm:git-lc-klt}.
  So the action of $T$ on $U\uss_{\bo}$ is not
  free. There are several ways to restrict it to a subset with a free
  action:
  \begin{enumerate}
  \item ``Horizontally'', by removing the points $p\in \cup B_i$. The
    remaining set then is $U\cap(\bG_m^{n-1}\times\gr)$, where
    $\bG_m^{n-1}=\bP^{n-1}\setminus \cup H_i$. This is the choice of
    \cite{HackingKeelTevelev}.
  \item ``Vertically'', by removing points with $P_V\subset \{x_i=1\}$
    -- our choice.
  \end{enumerate}
\end{remark}

\section{Definitions of the moduli space and the family}
\label{sec:defin-moduli-univ}

\begin{definition}
  (over $k=\bar k$)
  For each stable toric variety $Y\to \gr\lw$ over $\gr\lw$, we define
  the corresponding \emph{weighted stable hyperplane arrangement} as
  \begin{displaymath}
    X := (Y\times_{\gr\lw}U\uss\lw)\lwq T.
  \end{displaymath}
  We also define divisors $\bar B_i = (H_i\times\gr\lw) \cap U\uss\lw$ and
  then
  \begin{math}
    B_i := (Y\times_{\gr\lw} \bar B_i)\lwq T.
  \end{math}
\end{definition}

\begin{theorem}\label{thm:X-Xhat-Y}
  \begin{enumerate}
  \item The $T$-action on the restriction to $Y\times_{\gr\lw^0}U^0\lw$ is
    free. The geometric quotient $(\hat X, \hat B_i)$
    by this free action is projective and comes
    with a semiample invertible sheaf defining a proper birational morphism
    $\hat X\to X$, an isomorphism on the complements of $\hat B_i$, $B_i$.
  \item $X=Y\cap \gr\lw^e$.
  \item $X$ is reduced and $B_i$ are reduced Weil divisors on $X$.
  \end{enumerate}
\end{theorem}
\begin{proof}
  (1) follows immediately from Theorem~\ref{thm:free-git-quot} and (2) from
  Lemma~\ref{lem:gr-e}  by functoriality of GIT quotients. (3) follows
  since GIT quotients of reduced schemes are reduced.
\end{proof}

\begin{example}\label{exmp:XY-for-ha}
  Let $(\bP V,\sum b_iB_i)$ be an lc hyperplane arrangement. Then
  $\Delta\lw \subset P_V$. The weighted moment polytope of $(\bP
  V,\sum b_iB_i)$ is therefore $\Delta\lw$ itself. The
  normalization $Y$ of the closure of the orbit $T.V$ is a toric variety
  and it comes with a finite morphism to $\gr\lw$.

  The pullback of $\gr\lw^0$ under $Y\to \gr\lw$ is simply the orbit $T.V$,
  isomorphic to~$T$. Then $Y\times_{\gr\lw}U^0\lw = \bP V\times T$, and the
  quotient $\hat X$ is $\bP V$ itself, together with the original divisors
  $B_i$. Since $\hat X\simeq\bP^{r-1}$,  the morphism $\hat X\to X$
  must be an isomorphism. Hence, every lc hyperplane arrangement
  appears as a particular case of our construction.
\end{example}

\begin{theorem}
  \label{thm:wsha-sings}
  \begin{enumerate}
  \item   Any weighted stable hyperplane arrangement $(X,\sum b_i B_i)$ is a
  stable pair, i.e. it has slc singularities and $K_X+\sum b_iB_i$ is
  an ample $\bQ$-divisor.
\item $\hat X$   is Gorenstein.
  \item $X$ is Cohen-Macaulay, and $X\setminus \cup B_i$ is Gorenstein.
  \item Assume $mb_i\in\bZ$. Then
    $m(K_X+\sum b_iB_i)$ is the
  restriction under $Y\subset X$ of the ample invertible sheaf $F_Y$ on $Y$
  corresponding to the polytope $m\Delta\lw$.
  \end{enumerate}
\end{theorem}
\begin{proof}
(1)  By Theorems~\ref{thm:matroid-lc} and \ref{thm:git-lc-klt},
  $(U\lw^0,\sum b_i \bar B^0_i)$ is a family of open lc subsets of
  hyperplane arrangements. Hence, $U^0$ is smooth, and there exists a finite
  sequence of blowups of $\bP^{n-1}$ giving a simultaneous resolution
  of singularities of $(U\lw^0,\sum b_i \wB^0_i)$.

    On the other hand, a stable toric variety $Y$ together with its
    boundary is slc by \cite{Alexeev_CMAV}; and the boundary is
    contained in $\gr\lw\setminus\gr\lw^0$ so can be omitted.
    The stable toric variety is Cohen-Macaulay, and its interior is
    Gorenstein by the Stanley-Reisner theory because the topological
    space $\Delta\lw$ is a smooth manifold with boundary.
    Therefore, the pullback $V:=Y\times_{\gr\lw}U^0\lw$, together with
    the boundary, has slc singularities, and it is Gorenstein.

    Then the geometric quotient $\hat X=V/T$ by the free $T$-action is
    Gorenstein, giving (2).

    Now let $m\in\bN$ be such that $m\wei$ is integral, and let $F_V$,
    $F_{\hat X}$, $F_X$ be the invertible sheaves on $V$, $\hat X$,
    $X$ given by the GIT construction: $F_V$ is the pullback of
    $L_{m\wei-mr,m}$, sections of $F_V$ descend to
    sections of $F_{\hat X}$ and $F_X$, $F_{\hat X}$ is semiample and
    defines the contraction $\hat X\to X$, $F_X$ is ample.

    We observe that by construction one has $F_{\hat X}=\cO_{\hat
      X}(m(K_{\hat X}+\sum b_i\hat B_i))$. This implies that
    $F_{X}=\cO_{X}(m(K_{X}+\sum b_iB_i))$ and that $(X,\sum b_iB_i)$
    is slc.

    Since $X\setminus\cup B_i = \hat X\setminus\cup \hat B_i$,
    $X_i\setminus \cup B_i$ is Gorenstein. $X$ is Cohen-Macaulay
    because it is the result of a log crepant contraction isomorphic
    outside of $\cup B_i$ and there exists a positive combination of
    $B_i$ which is Cartier.

    Let $F_Y$ be the (integral) ample invertible sheaf corresponding
    to the polytope $m\Delta\lw$. Then by the same argument as in
    Theorem~\ref{lem:gr-e} sections of $F_Y$ restrict to sections of
    $F_X$. This gives (3).
\end{proof}

Let $Y\to\gr\lw$ be a stable toric variety over $\gr\lw$, and
$\uP\lw=\{P_{V,\wei}\}$ be the corresponding cover of $\Int\Delta\lw$.
Each of these polytopes has a parent, so that $P_{V,\wei}=P_V\cap
\Delta\lw$.

We denote by $Y[P_{V,\wei}]$ the corresponding projective toric
variety. We also denote by $\sigma_n$ the simplex $\{(x_i)\in \bR^n
\mid x_i\ge0, \ \sum x_i=r\}$. The corresponding to it toric variety is
$\bP^{n-1}$. If $\pwei$ is maximal-dimensional then toric geometry
gives a natural birational map
$\psi[\pwei]:\bP^{n-1}\dashrightarrow Y[P_{V,\wei}]$, an isomorphism
on the torus $\bG_m^{n-1}$.

Now let $\pwei$ be a weighted matroid polytope of codimension $c$, and
let $P_V$ be its parent, a matroid polytope. Recall from
Theorem~\ref{thm:matroid-polytopes} that $P_V= \prod P_j$, the product
of maximal-dimensional polytopes for a subdivision $\{1,\dotsc,n\}=
\sqcup_{j=0}^c I_j$. Then we have a natural rational map
$\bP^{n-1}\dashrightarrow \prod \bP^{|I_j|-1}$ which on an open subset
is the quotient by a free action by $\bG_m^{c}$.

In this case, we denote by $\psi[\pwei]:\bP^{n-1}\dashrightarrow
Y[P_{V,\wei}]$ the latter rational map followed by the birational map
$\prod \bP^{|I_j|-1} \to Y[\pwei]$ corresponding to polytopes $\prod
\sigma_{|I_j|}$ and $\pwei$.

\begin{theorem}\label{thm:wsha-detailed-desc}
  The stratification of $X$ into irreducible components and their
  intersections is in a dimension-preserving,
  with a shift by $n-r$,  bijection with the tiling
  $\cup P_{V,\wei}$ of $\Int\Delta\lw$. Moreover, the closure of the
  stratum corresponding to $\pwei$ is the closure of the image of $\bP
  V$ under $\psi[\pwei]:\bP^{n-1}\dashrightarrow Y[\pwei]$. This
  rational map is regular on the open subset of $\bP V$ where $(\bP V,
  \sum b_iB_i)$ is lc. The image of this regular set gives a locally
  closed stratum in $X$ corresponding to $\pwei$.
\end{theorem}
\begin{proof}
  Each of the polytopes corresponds to an arrangement $\bP V\subset
  \bP^{n-1}$ so that the moment polytope $P_V$ intersects
  $\Int\Delta\lw$. Then we simply follow the $T$-orbit
  of $\bP V$ in $\gr\lw^0$, to the pullback in $U^0\lw$, to the quotient
  in $X$, then back to the irreducible component of $Y[\pwei]$ under
  the inclusion $X\subset Y$.

  When $\pwei$ is maximal-dimensional, the orbit $T.V$ in $\gr\lw^0$
  has trivial stabilizer. Hence, under the quotient by the free action
  by $T$, the open subset of lc points on $\bP_V$ is preserved; then
  part is contracted by a birational morphism. If $\bP_V$ has codimension
  $c$ then the stabilizer of $T.V$ in $\gr\lw^0$ is $\bG_m^{c}$. Then
  the quotient factors through the quotient of an open subset of
  $\bP^{n-1}$ by $\bG_m^c$.
\end{proof}

We are now ready to define the moduli space and the universal family
of pairs over it.

\begin{definition}\label{defn:my-moduli}
  $\oM\lw(r,n) = \M^{T}(\gr\lw,\Delta\lw).$
\end{definition}

\begin{lemma}
  $\M^{T}(\gr\lw,\Delta\lw)$ is a fine moduli space.
\end{lemma}
\begin{proof}
  Indeed, every $T$-orbit in $\gr\lw^0$ is also an orbit in $\gr$. If
  its stabilizer is finite then it is in fact trivial. Therefore,
  every irreducible component of a stable toric variety $Y\to \gr\lw$
  maps to its image birationally, and the automorphism group of $Y\to
  \gr\lw$ is trivial. Then $M^T$ is a fine moduli space.
\end{proof}

\begin{definition}\label{defn:my-family}
  The family of
  weighted stable hyperplanes arrangements $(\cX,\cB_i)\to
  \oM\lw(r,n)$ is the GIT quotient of the pullback of $U\lw^0\to
  \gr\lw$ by the universal family of stable toric varieties $\cY\to
  \M^{T}(\gr\lw,\Delta\lw).$
\end{definition}

\begin{theorem}
  $(\cX,\cB_i)\to\oM\lw(r,n)$ is a locally free (in particular, flat)
  morphism.
\end{theorem}
\begin{proof}
  The families $U\to \gr$ and $\cY\to \oM\lw(r,n)$ are locally free,
  i.e. locally they are given by locally free modules. This implies
  that the pullback $\cY \times_{\oM} U\lw\uss$ is locally free.
  Algebraically, the GIT quotient is constructed by taking the
  degree-0 component in an algebra. Thus, this subalgebra is a direct
  summand, and a direct summand of a locally free module is locally
  free (by Kaplansky's theorem \cite{Kaplansky_Projective}, over any
  ring a module is locally free iff it is projective).
\end{proof}

\section{Completing the proofs of main theorems}
\label{sec:completing-proofs}

\begin{proof}[Proof of Theorem \ref{thm:existence} (Existence)]
  The parts (1) and (3) were established in the previous section. The
  subset $\M\lw(r,n) \subset \oM\lw(r,n)$ is the open subset of
  $\M^{T}(\gr\lw,\Delta\lw)$ where the stable toric varieties are
  irreducible, cf. Example~\ref{exmp:XY-for-ha}. The sheaf
  \linebreak
  $\cO_{\cX}(m(K_{\cX} + \sum b_iB_i))$ is free over $\oM\lw$ because
  by Theorem~\ref{thm:wsha-sings} it is the restriction of the
  invertible ample sheaf $F_{\cY}$ from the universal family of
  stable toric varieties that corresponds to the lattice polytope
  $m\Delta\lw$, and $F_{\cY}$ is free: it's sections give the finite
  morphism to $\gr\lw$.

  The remaining part (2) is
  proved in the Reconstruction Theorem~\ref{thm:reconstruction} below.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm:red-moduli} (Reduction morphisms)]
(1) For $\wei,\wei'$ in the same chamber, we have $\gr\lw=\gr\lwp$ by
  Theorem~\ref{thm:properties-of-Zbeta}, applied to
  grassmannians. Also the conditions for GIT (semi)stability are the
  same. So the moduli and the families are the same.

(2) If $\wei'\in \overline{\cham(\wei)}$, we have a reduction morphism
  $\gr\lw\to\gr\lwp$ again by
  Theorem~\ref{thm:properties-of-Zbeta}. The third application of the
  same theorem gives the reduction morphism between the stable toric
  varieties $Y\lw,Y\lwp$ over $\gr\lw,\gr\lwp$. Finally, this gives in a
  canonical way the reduction morphisms between the pullbacks of the
  universal families and their GIT quotients.

  Each $\pi_{\wei,\wei'}$ is log crepant. That is because the
  morphism on the ambient stable toric varieties is given by pullback
  of $L_{\wei}+\sum(b_i'-b_i)B_i$ (cf.  \ref{lem:toric-ex},
  \ref{thm:properties-of-Zbeta}), and $L_{\wei}$ on $Y$ restricts to
  $K_X+\sum b_i B_i$ on $X$ by Theorem~\ref{thm:wsha-sings}.

(3)  When specializing up, the morphism $\oM\lw\to \oM\lwp$ is an
  isomorphism. Indeed, a stable toric variety $Y\to\gr\lw$ is uniquely
  determined by its restriction $Y^0$ to $\gr\lw^0$: $Y$ is the partial
  normalization at the boundary of the closure of $Y^0$. But
  $\gr\lw^0=\gr\lwp^0$ in this case.

  Additionally, when specializing up, the morphism $X\lw\to X\lwp$ is
  simply our morphism $\hat X\to X$, so by Theorem~\ref{thm:X-Xhat-Y}
  it is birational and an isomorphism outside $\hat B_i,B_i$.

  (4) is an immediate consequence of the parts (1,2,3).
\end{proof}

Note that if the source of the GIT quotient were fixed, with only the
line bundle and the polarization changing, the statement would be an
application of the well-known theory of variation of GIT
quotients \cite{BrionProcesi, DolgachevHu}.

\begin{proof}[Proof of Theorem~\ref{thm:small-weights} (Moduli for
  small weights)]
  In this case $\gr\lw^0=\gr\uss\lw=\gr\us\lw$, the $T$-action on it is
  free, and every stable toric variety over $\gr\lw$ is the closure of
  a unique $T$-orbit. Hence, $M^T(\gr\lw,\Delta\lw)=\gr\us\lw/T
  = \gr\lwq T$.

  The equivalence of the GIT quotients
  $(\bP^{r-1})^n//\PGL(r)$ and $\gr(r,n)//T$ is well-known, see,
  e.g., \cite[2.4.7]{Kapranov_QuotientsGrassmannians}.
\end{proof}

  Intuitively, the contribution $p_1^* \cO_{\bP^{n-1}}(|\wei|-r)$ to
  the polarization $L_{|\wei|-r,1}$ in
  this case approaches zero and only the quotient of $\gr\lw$ remains.

\begin{theorem}[Reconstruction Theorem]
  \label{thm:reconstruction}
  The stable toric variety $Y\to\gr\lw$ can be uniquely reconstructed
  from $(X,\sum b_iB_i)$.
\end{theorem}
\begin{proof}
  Let $Y[\pwei]$ be an irreducible component of a stable toric variety
  $Y\to \gr\lw$, as in Theorem~\ref{thm:wsha-detailed-desc}, and
  $X[\pwei] \subset Y[\pwei]$ be the corresponding irreducible
  component of $X$. We first show that $Y[\pwei]$ can be reconstructed from
  $X[\pwei]$ intrinsically.

  Indeed, the boundary of $X[\pwei]$ in $X$ is labelled by the
  divisors $B_i$, some of them coinciding. Then
  the defining inequalities of $\pwei$ can be read off this
  configuration: every missing intersection $\cap_{i\in I} B_i$ of
  codimension $k$ gives the inequality $\sum_{i\in I}x_i\le k$.
  This recovers the polytope $\pwei$.

  Then the embedding $X[\pwei]\to Y[\pwei]$ is recovered as
  follows. For every $m$ such that $m\wei$ is integral, every integral
  point $(x_i)\in m\pwei$ gives a section of
  the sheaf $\cO_{X[\pwei]}( m(K_X+\sum b_iB_i) )$.
   Namely, it is a unique up to a constant section vanishing
  at $B_i$ to the order $x_i$. The collection of these sections gives
  the embedding $X[\pwei]\to Y[\pwei]$ defined up to $n$ choices of
  multiplicative constants, one for each $B_i$, i.e. up to the action
  of $\wT$.

  Finally, $\bP V$ is recovered from the image of $X[\pwei]\to
  Y[\pwei]$ by applying the inverse of the rational map $\psi[\pwei]$
  of Theorem~\ref{thm:wsha-detailed-desc}. Then the orbit $T.V$ in
  $\gr\lw^0$ gives the morphism $Y[\pwei]\to \gr\lw$. The whole
  stable toric variety  $Y\to \gr\lw$ is recovered this way by looking
  at all maximal-dimensional polytopes $\pwei$.
\end{proof}

We note that for $\wei=\bo$ this proof is very different from the one
given in \cite{HackingKeelTevelev}, which does not extend to the
weighted case.

\section{Some simple examples}
\label{sec:some-examples}

\begin{example}
  $(r,n)=(2,4)$, $\wei=\bo$. Consider the subdivision of $\Delta(2,4)$,
  the octahedron on the ${4 \choose 2}$ vertices $ij$,
  into two pyramids: $P_1$ missing the vertex 34, and $P_2$, missing
  the vertex $12$.

  $P_1$ corresponds to the configuration of 4 points in $\bP^1$ for
  which the Pl\"ucker coordinate
  $p_{34}=0$, i.e. $B_3=B_4$. This polytope is given by the
  inequality $x_3+x_4\le 1$, which is precisely the lc condition for
  this configuration.
  Similarly, for $P_2$ one has
  $B_1=B_2$. On the intersection $P_1\cap P_2$ one has $B_1=B_2$,
  $B_3=B_4$, and the defining inequalities become $x_1+x_2=1$,
  $x_3+x_4=1$, i.e. $P_1\cap P_2 = \Delta(1,2) \times \Delta(1,2)$.

  The irreducible component $X_1$ then is the closure of the image of
  $\bP^1\setminus B_3$, so isomorphic to $\bP^1$; and similarly for
  $X_2$. The intersection $X_1\cap X_2$ is the quotient
  $(\bP^1\setminus \{B_1,B_3\})/\bG_m$, so a point. So $X$ is a union
  of two $\bP^1$'s intersecting at a point.
\end{example}

\begin{example}
  $(r,n)=(2,4)$, $\wei=(1/2,1/2,1,1)$. Consider the trivial
  subdivision of $\Delta\lw$, with just the polytope itself.
  The points $B_1$ and $B_2$ may or may not coincide,
  depending on whether the parent polytope
  is the pyramid $P_2$ from the previous example, or $\Delta(2,4)$,
  otherwise the points are
  pairwise distinct. $X=\bP V=\bP^1$.

  Say, $B_1=B_2$. Then $X\subset Y$ intersects the stratum corresponding
  to the edge $x_1=x_2=1/2$ of $\Delta\lw$, at a point $q$. In this case, the
  $T$-translates of $X$ do not sweep out an open subset of $Y$, and this is
  very different from the unweighted situation of
  Section~\ref{sec:unweighted-case}.

  If we consider the GIT quotient of the pullback family over the
  whole $Y$ (not just $Y\cap\gr\lw^0$ as in our construction), then on
  the boundary some fibers to the GIT quotient are modelled on the
  curve $\bA^1\cup_q \bA^1$, which is a transversal slice of $Y$
  at the point $q$.
\end{example}

In all cases with  $r=2$ the considerations are quite similar, and
produce a tree of $\bP^1$'s.

\begin{example}
  $(r,n)=(3,5)$, this will correspond to
  Example~\ref{exmp:3-degs}. Begin with $\wei=\bo$, and consider the
  subdivion of $\Delta(3,5)$ into 3 polytopes:
  $P_0=\{x_1+x_2+x_5\le2, \ x_3+x_4+x_5\le2\}$, $P_1=\{x_1+x_2\le1\}$,
  and $P_2=\{x_3+x_4\le 1\}$.

  Then $P_0$ corresponds to the configuration of $5$ lines such that
  $B_1\cap B_2\cap B_5$ is a point, $B_3\cap B_4\cap B_5$ is a point,
  and otherwise generic. The matroid polytope $P_V$ is obtained from
  $\Delta(3,5)$ by cutting two corners, and the intersection
  $P_V\cap \{x_5=~1\}$ has codimension 2, not 1 as might be expected: it is
  $\{x_5=1,x_1+x_2\le 1,x_3+x_4\le 1\}$, so the corresponding face gets
  contracted.

  As in Theorem~\ref{thm:wsha-detailed-desc}, the irreducible
  component $X^0$ is the image of $\bP^2$ under $\bP^4\dashrightarrow
  Y[\pwei]$. On $\bP^2$ it blows up two points and contracts the
  strict preimage of~$B_5$.
  The configuration $(\bP V,\sum B_i)$ is lc outside of two points, so
  the divisor $B_5$ is present on $\hat X$; it is contracted by the
  log crepant morphism $\hat X\to X$.

  For the weight $\wei=(1,1,1,1,1-\epsilon)$, the face $P_V\cap
  \{x_5=1-\epsilon \}$ has codimension 1, and the curve $B_5$ is not
  contracted.
\end{example}

\begin{example}
  Consider the subdivision of $\Delta\lw$
  by a single hyperplane $x_1
  + \dotsb + x_{n_1} = r_1$, equivalently $x_{n_1+1}+\dotsb +
  x_n=r_2$, with $r_1+r_2=r$, $n_1+n_2=n$. Then $X$ is the union of
  $\Bl_{\bP^{r_1-1}}\bP^{r-1}$  and $\Bl_{\bP^{r_2-1}}\bP^{r-1}$ glued
  along   $\bP^{r_1-1} \times \bP^{r_2-1}$.
\end{example}

\begin{example}
  Let $(a_1,\dotsc,a_{n-r+1})\in\cD(1,n-r+1)$ be a weight such that
  $\sum a_i>1$ but $\sum_{i\in I} a_i\le 1$ for any proper subset
  $I$. Let $\wei\in\cD(r,n)$ be the weight consisting of $\alpha$
  preceded by $(r-1)$ $1$'s.

  Then $\M\lw=\oM\lw= (\bP^{n-r-1})^{r-1}$. For $r=2$ this was
  established in \cite[4.5]{AlexeevGuy}. For the general case, we
  first observe that Theorem~\ref{thm:small-weights} applies in this
  case after replacing $1$ with $1-\epsilon$ for some $0<\epsilon\ll1$,
  and so $\oM\lw$ a moduli space of lc hyperplane
  arrangements. The lc condition implies that the $(r-1)$ hyperplanes
  with weight $1$ must intersect normally. Restricting to an
  intersection to any $(r-2)$ of these hyperplanes, a line, gives the $r=2$
  situation, for the weight $(1,a_i)$, and the moduli space for this
  is $\bP^{n-r-1}$. Each of the hyperplanes with weight $a_i$ is
  uniquely determined by the intersections with these $(r-1)$ lines,
  and all of these configurations are lc. So $\M\lw= (\bP^{n-r-1})^{r-1}$.
\end{example}

\begin{example}
  Let $\wei=(1,\dotsc,1,\epsilon,\dotsc,\epsilon)$,
  $|\wei|=r+(n-r)\epsilon$. The case of $r=2$ was introduced in
  \cite{LosevManin}, and $\oM\lw(2,n)$ is the toric variety for the
  permutohedron, see also \cite[2.11(4)]{AlexeevGuy}.

  For any $r$, the closure of $\M\lw$ in $\oM\lw$ is the toric variety
  for the fiber polytope $\Sigma(\sigma_r^{\oplus (n-r)} \to
  (n-r)\sigma_r)$, where $\sigma_r$ is the simplex with $r$ vertices
  and side 1, and $(n-r)\sigma_r$ is $\sigma_r$ dilated by $(n-r)$.

  This moduli space also has an interpretation as the moduli space of
  stable toric pairs $(X,D_1,\dotsc,D_{n-r})$, as in
  \cite{Alexeev_CMAV} but with $(n-r)$ divisors instead of
  one. Explaining this in detail would take quite some space, and is
  better done elsewhere.
\end{example}

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text:
\section{Numerical Results} \label{sec:Numericals}

In this section, we empirically evaluate ZDPG

and compare its performance to vanilla Policy Gradient (PG) \cite{sutton2000policy} and Policy Gradient with Baseline (PG-B) \cite{greensmith2004variance} with varying levels of system noise and MC variance reduction. First, we formally describe the Markov Decision Process we choose to solve, then we present our results and discuss the key takeaways.

\subsection{Problem Setting: Navigating Around an Obstacle with Unknown Agent Dynamics}
For our empirical evaluation of ZDPG, we consider a two-dimensional continuous state-action navigation problem, where a point agent aims

to avoid a spherical (for simplicity) obstacle and converge fast to some fixed target while satisfying its motion control dynamics. This goal is achieved by appropriately shaping the associated reward function.

Let $s^* \in \mathbb{R}^2$ represent the target the point agent wishes to reach. Let $s^c \in \mathbb{R}^2$ and $r\in \mathbb{R}_+$ denote the center and radius of the spherical obstacle the agent wishes to avoid. Then, we define the reward $R: \mathbb{R}^2 \to \mathbb{R}$ as a combination of two artificial potential functions: An attractive potential $\phi^{att}: \mathbb{R}^2\to \mathbb{R}_+$ with a maximum at the target, and a repulsive artificial potential $\phi^{rep}:\mathbb{R}^2 \to \mathbb{R}_+$ with a minimum at the center of the object. Specifically, we set the attractive potential to be the negative Euclidean distance to the goal location $s^*$ squared, that is, $\phi^{att}(s) :=- \|s - s^*\|_2^2$, where $s$ is the location (state) of the agent. Similar to \cite{kalogerias2014mobi}, we define the repulsive potential as $\phi^{rep}(s) := 1 - 1/\beta(s)$, where

\begin{equation*}
    \beta(s) := \left( 1 - \left(\frac{1 + r^4}{r^4}\right) \cdot \frac{\left(\|s - s^c\|^2 - r^2\right)^2}{1+\left(\|s - s^c\|^2 - r^2\right)^2} \right)^{\frac{1}{2}\left(1 - \textrm{sign}\left(\|s - s^c\| - r\right)\right)}.
\end{equation*}

Completely characterized by the state of the agent, the reward is equal to the sum of these attractive and repulsive potentials, that is, $R(s) = \phi^{att}(s) + \phi^{rep}(s)$. Also, the motion dynamics of the point agent are described by the controlled random walk \begin{equation}\label{state-model}
    s_{t+1} = A s_{t} + \eta \dfrac{\pi_\theta(s_t)}{\Vert \pi_\theta(s_t) \Vert} + n_t,
\end{equation}
where $A\in\mathbb{R}^{2\times2}$ is the motion state transition matrix of the agent, $\{n_t\}$ is a random process modeling state process noise, and $\eta>0$ (together with policy normalization) imposes implicit resource constraints on the agent motion. Here, we assume that, except from the agent control vector (second term on the right-hand side of \eqref{state-model}), the dynamics of the agent are \textit{unknown}, an assumption which conforms with the standard policy gradient setting.
Lastly, we consider a linear, deterministic policy to determine optimal directions that the point agent will follow given its location to its prescribed destination. In particular, we adopt

a Radial-Basis-Function (RBF) expansion as our (universal) policy parameterization, i.e.,

\begin{equation} \label{equ:det_policy}
    \pi_\theta(s) = \sum_{i = 1}^d \theta_i \kappa(s, \bar s_i),

\end{equation}

where $\kappa(s, s') = \textrm{exp}(-\|s-s'\|^2_2/2\sigma_{\kappa}^2)$ defines the RBF kernel, $\theta_i \in \mathbb{R}^2, i = 1,\dots, d$ are the parameter vectors to be learned, $\sigma_{\kappa}^2$ determines the variance of the kernel, and $\bar s_i, i = 1,\dots, d$ are the associated RBF centers.

For all simulations, we select the target of the agent to be located at $s^* = (-5,-5)$, and the obstacle to be centered at $s^c = (0,0)$ with a radius of $r = 2.5$. We define the potential with a slightly larger radius ($r = 3$) to ensure that the agent does not collide with the boundary of the obstacle. We select the centers of the RBF kernels to be spaced $0.25$ units from each other populating the $[-10,10]\times [-10,10]$ grid. The initial state distribution is uniform on $[-10,10]\times [-10,10]$.

\subsection{Results}
\vspace{-6pt}

\begin{figure*}[!t]
	\centering
	\begin{tabular}{ccc}
		\hspace{3mm}\includegraphics[trim = {5cm, 10cm, 5cm, 10cm}, width=.274\linewidth]{fig1a_final1}   &\hspace{3.5mm}
		\includegraphics[trim = {5cm, 10cm, 5cm, 10cm},width=.274\linewidth]{Corrected_fig1b2} &\hspace{3.5mm}
		\includegraphics[trim = {5cm, 10cm, 5cm, 10cm},width=.274\linewidth]{fig1b_final1} \\ \vspace{.4cm}
		~&~&~\\
		\small (a) & \small (b) & \small (c)
	\end{tabular}
	\caption{Average reward per episode with confidence bounds over 50 trials. Learning rate set to $\alpha = 10^{-7}$ for all simulations. (a) Noiseless case with no MC variance reduction ($N = 1$), ZDPG and ZDPG-S smoothing parameters set to $0.05$ and $0.5$ respectively. Action noise for PG and PG-B set to $\sigma_a^2 = 0.025$. (b) Noisy case ($\omega = 0.01$) with no MC reduction ($N = 1$), ZDPG and ZDPG-S smoothing parameters set to $0.25$ and $0.5$ respectively. Action noise for PG and PG-B set to $\sigma_a^2 = 0.025$. (c) Noisy case (same as (b)) with MC reduction ($N = 10$).}\vspace{2mm}
	\label{fig:LC}
\end{figure*}

The learning curves shown in Figure \ref{fig:LC} illustrate the performance of ZDPG and its symmetric counterpart ZDPG-S (based on \eqref{equ:symm_diff}), compared with PG and PG with a \textit{stochastic} baseline (PG-B), with discount factor $\gamma = 0.8$ in the case with and without system noise ($n_t = \mathbf{0}$ and $n_t \sim \mathcal{N}(\mathbf{0},\omega \cdot I_2)$, respectively). Also, we set $A = I_2$. For PG and PG-B, we let $\tilde \pi$ denote the stochastic policy where the action is selected by $a_t \sim \tilde \pi_\theta(s_t)$. In particular, we let $\tilde \pi$ be a Gaussian policy with mean equal to $\pi_\theta(s)$ [c.f. \eqref{equ:det_policy}] and a standard choice of covariance matrix $\Sigma = \sigma_a^2 \cdot I_2$.

For PG-B, the baseline is fairly chosen and is analogous as compared to ZDPG in that it scales the update direction $\nabla_\theta \log \tilde \pi_\theta(s)$ by a two-point difference ($\hat Q(s,a) - \tilde{V}(s)$), where $\tilde{V}(s)$ is a single-rollout estimate of the associated value function at state $s$.

After each policy update, we evaluate the system by generating a rollout of fixed length $T = 20$ from a fixed starting point $s^0 = (5,5)$. As done commonly in practice (as a heuristic), we evaluate the stochastic policies by setting the covariance matrix equal to zero thereby making the evaluation policy deterministic.

\begin{figure*}[!t]
	\centering
	\begin{tabular}{ccc}
		\hspace{.8mm}
		\includegraphics[trim = {7cm, 10cm, 5cm, 10cm}, width=.278\linewidth]{no_noise_fig2a}   &\hspace{6mm}
		\includegraphics[trim = {7cm, 10cm, 5cm, 10cm}, width=.278\linewidth]{noise_fig2b}
		&\hspace{6mm}
		\includegraphics[trim = {7cm, 10cm, 5cm, 10cm}, width=.278\linewidth]{Quiver2} \\
		~&~&\\
		\small (a) & \small (b) & \small (c)
	\end{tabular}
	\caption{Sample trajectories with fixed length $T = 100$ of learned policies corresponding to noiseless ((a): $N = 1$, $\omega = 0$) and noisy ((b): $N = 10$, $\omega =0.1$) settings (c) Learned field plot for ZDPG-S with noise level $\omega = 0.1$ shows that the variance in Figure \ref{fig:LC} comes only from the system noise.}
	\label{fig:traj}
	\vspace{-12pt}
\end{figure*}
Depending on the level of the system noise, ZDPG(-S) consistently outperforms PG-B (and also PG), both in mean performance and the associated variance.
Interestingly, the improvement of ZDPG over PG-B is sharper when the agent dynamics are noisy (also see our Supplementary Material, Section \ref{supp:results}).

In fact, we observe that, either with \textit{or} without system noise, the perturbation of the initial state with a deterministic policy indeed provides more meaningful information for policy improvement than in the setting of a stochastic policy (also see Figure \ref{fig:traj}).

One should also note that PG-B performs better in the noisy setting with MC variance reduction (\ref{fig:LC} (c)) than the noiseless setting (\ref{fig:LC} (a)). Indeed, variance reduction improves the performance of all methods, except for vanilla policy gradient. We emphasize though that we evaluate stochastic policies using only their means. Our empirical results corroborate the claim that randomness in the policy is \emph{not} necessary to solve MDPs within the PG-based RL framework.

\section{Conclusions and Future Work}
In this work, we introduced the ZDPG algorithm, which is based on an efficient two-point zeroth-order approximation of the deterministic policy gradient, for the infinite horizon Markov decision problem. Using random horizons to evaluate the zeroth-order estimates of the state-action function, we showed that low-dimensional perturbations in the action space can effectively replace the need for randomized policies in the classical policy gradient setting. We also presented finite sample complexity results for ZDPG, which improve upon the state of the art by up to two orders of magnitude. Finally, we corroborated our results with a numerical evaluation on a navigation problem which showcased the advantages of ZDPG over both standard PG and PG with a baseline. For future work, variance reduced ZDPG could be realized with more sample efficient variance reduction techniques, such as gradient averaging or using a critic network to evaluate zeroth order state action function.

config: arxiv