\documentclass[ALCO,Unicode]{cedram} \usepackage{array,calrsfs,graphicx,multirow,latexsym,subfig,tikz-cd,verbatim} \newcommand\hmmax{0} \newcommand\bmmax{0} \usepackage{bm} \usepackage[cmtip, all]{xy} \title{The Grassmannian of $3$-planes in $\mathbb{C}^{8}$ is sch\"on} \author[\initial{D.} Corey]{\firstname{Daniel} \lastname{Corey}} \address{University of Nevada, Las Vegas\\ 4505 S Maryland Pkwy\\ Las Vegas, NV 89154} \email{daniel.corey@unlv.edu} \author[\initial{D.} Luber]{\firstname{Dante} \lastname{Luber}} \address{Technische Universit\"at Berlin\\ Institut f\"ur Mathematik, Sekr. MA 6-2\\ Strasse des 17 Juni 136 \\ 10623 Berlin} \email{luber@math.tu-berlin.de} \thanks{DC is supported by the SFB-TRR project ``Symbolic Tools in Mathematics and their Application'' (project-ID 286237555), and DL is supported by ``Facets of Complexity'' (GRK 2434, project-ID 385256563).} \keywords{Chow quotient, Grassmannian, matroid, tight span} \subjclass{14T90, 05E14, 14C05, 52B40} \newcommand{\A}{\mathbb{A}} \newcommand{\C}{\mathbb{C}} \newcommand{\F}{\mathbb{F}} \newcommand{\G}{\mathbb{G}} \newcommand{\K}{\mathbb{K}} \newcommand{\PP}{\mathbb{P}} \newcommand{\R}{\mathbb{R}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\pC}{\mathcal{C}} \newcommand{\pE}{\mathcal{E}} \newcommand{\pF}{\mathcal{F}} \newcommand{\pG}{\mathcal{G}} \newcommand{\pQ}{\mathcal{Q}} \newcommand{\pS}{\mathcal{S}} \newcommand{\sC}{\mathsf{C}} \newcommand{\sF}{\mathsf{F}} \newcommand{\sG}{\mathsf{G}} \newcommand{\sH}{\mathsf{H}} \newcommand{\sL}{\mathsf{L}} \newcommand{\sP}{\mathsf{P}} \newcommand{\sQ}{\mathsf{Q}} \newcommand{\sT}{\mathsf{T}} \newcommand{\sU}{\mathsf{U}} \newcommand{\sV}{\mathsf{V}} \newcommand{\sW}{\mathsf{W}} \newcommand{\sa}{\mathsf{a}} \newcommand{\su}{\mathsf{u}} \newcommand{\sv}{\mathsf{v}} \newcommand{\sw}{\mathsf{w}} \newcommand{\sx}{\mathsf{x}} \newcommand{\sy}{\mathsf{y}} \newcommand{\sz}{\mathsf{z}} \newcommand{\TS}{\mathsf{TS}} \newcommand{\init}{\mathsf{in}} \newcommand{\se}{\mathsf{e}} \newcommand{\op}{\mathsf{op}} \newcommand{\vertices}{\mathsf{vert}} \newcommand{\edge}{\mathsf{edge}} \newcommand{\col}{\mathsf{col}} \newcommand{\Leaf}{\mathsf{Leaf}} \newcommand{\matroidal}{\mathsf{mat}} \newcommand{\spMat}{\sQ_\mathsf{mk}} \newcommand{\spCone}{\pC_\mathsf{mk}} \newcommand{\spw}{\sw_\mathsf{mk}} \newcommand{\Zone}{\langle \mathbf{1} \rangle} \newcommand{\bk}[2]{\langle #1, #2 \rangle} \newcommand{\chow}[2]{#1/\!\!/#2} \newcommand{\Sn}[1]{\mathfrak{S}_{#1}} \DeclareMathOperator{\coef}{coef} \DeclareMathOperator{\Conv}{conv} \DeclareMathOperator{\Dr}{Dr} \DeclareMathOperator{\face}{face} \DeclareMathOperator{\Hom}{Hom} \DeclareMathOperator{\GL}{GL} \DeclareMathOperator{\Gr}{Gr} \DeclareMathOperator{\PGL}{PGL} \DeclareMathOperator{\semigp}{semigp} \DeclareMathOperator{\Spec}{Spec} \DeclareMathOperator{\TGr}{TGr} \DeclareMathOperator{\trop}{trop} \DeclareMathOperator{\Trop}{Trop} \DeclareMathOperator{\norm}{norm} \newcommand{\Csch}{\C\operatorname{--sch}} \begin{document} \theoremstyle{definition} \newtheorem{algm}[cdrthm]{Algorithm} \begin{abstract} We prove that the open subvariety $\operatorname{Gr}_0(3,8)$ of the Grassmannian $\operatorname{Gr}(3,8)$ determined by the nonvanishing of all Pl\"ucker coordinates is sch\"on, i.e. all of its initial degenerations are smooth. Furthermore, we find an initial degeneration that has two connected components, and show that the remaining initial degenerations, up to symmetry, are irreducible. As an application, we prove that the Chow quotient of $\operatorname{Gr}(3,8)$ by the diagonal torus of $\operatorname{PGL}(8)$ is the log canonical compactification of the moduli space of $8$ lines in $\mathbb{P}^2$, resolving a conjecture of Hacking, Keel, and Tevelev. Along the way we develop various techniques to study finite inverse limits of schemes. \end{abstract} \maketitle \section{Introduction} A closed subvariety of an algebraic torus is \textit{sch\"on} if all of its initial degenerations---flat degenerations arising via Gr\"obner theory---are smooth. This notion was introduced by Tevelev in his influential paper \cite{Tevelev}, and admits this characterization by Helm and Katz \cite{HelmKatz}. Sch\"on subvarieties of tori satisfy many desirable properties. Their tropical compactifications are sch\"on compactifications, in particular they are normal and have toroidal singularities. Notions from birational geometry, like log minimality and ampleness of the log canonical divisor, admit tropical characterizations for sch\"on subvarieties and their sch\"on compactifications, respectively \cite{HackingKeelTevelev2009}. Some notable examples include nondegenerate hypersurfaces \cite{VarchenkoNewton, VarchenkoZeta} (whose study predates the notion of sch\"onness), complements of hyperplane arrangements, open del Pezzo surfaces, and moduli spaces of marked del Pezzo surfaces \cite{HackingKeelTevelev2009}. Consider the Grassmannian $\Gr(r,n)$ of $r$-planes in $\K^{n}$, where $\K$ is an algebraically closed field, and let $\Gr_0(r,n)$ be the open locus of $\Gr(r,n)$ defined by the nonvanishing of all Pl\"ucker coordinates. Its tropicalization $\TGr_0(r,n)$ parameterizes all tropical linear spaces realizable over a valued field extension of $\K$ \cite[Theorem~4.3.17]{MaclaganSturmfels2015}, so $\TGr_0(r,n)$, and therefore the initial degenerations of $\Gr_0(r,n)$, depend on the characteristic of $\K$. For this reason, we assume that $\K=\C$ throughout. The Grassmannian $\Gr_0(r,n)$ is known to be sch\"on when $(r,n) = (2,n)$ \cite{Tevelev}, $(3,6)$ \cite{Luxton} and $(3,7)$~\cite{CoreyGrassmannians}. \begin{theo} \label{thm:schoenIntro} The Grassmannian $\Gr_0(3,8)$ is sch\"on. \end{theo} \begin{rema} The initial degenerations of Grassmannian $\Gr_0(3,8)$ are isomorphic, up to a torus factor, to those of the moduli space $X(3,8)$ parameterizing isomorphism classes of $8$ hyperplanes in $\PP^{2}$ lying in general position. See Formula \ref{eq:initStratum}. In particular, $\Gr_0(3,8)$ is sch\"on if and only if $X(3,8)$ is sch\"on. In \cite{Schock}, Schock develops a theory of quasilinear varieties. Schock proves that a quasilinear variety $X$ is sch\"on and the strata of any sch\"on compactification of $X$ are irreducible. Schock also proposes a proof that $X(3,8)$ is quasilinear, and hence sch\"on, but this argument contains a critical gap. By Theorem \ref{thm:2ConnectedComponentsInDegIntro}, $X(3,8)$ has an initial degeneration that is disconnected, and hence $X(3,8)$ is not quasilinear. \end{rema} The tropical Grassmannian $\TGr_0(r,n)$ lies in a larger space $\Dr(r,n)$ called the \textit{Dressian}, which parameterizes \textit{all} tropical linear spaces \cite{HerrmannJensenJoswigSturmfels, Speyer2008}. Initially studied by Lafforgue \cite{Lafforgue2003}, to a $\sw \in \Dr(r,n)$ is associated a finite inverse limit $\Gr(\sw)$ of thin Schubert cells parameterized by the induced matroidal subdivision of the hypersimplex. When~$\sw\in \TGr_0(r,n)$, we may also form the $\sw$--initial degeneration~$\init_{\sw}\Gr_0(r,n)$. We use in an essential way the main theorem of \cite{CoreyGrassmannians}, which asserts that, for any $\sw\in \TGr_0(r,n)$, there is a closed immersion \begin{equation*} \init_{\sw}\Gr_0(r,n) \hookrightarrow \Gr(\sw). \end{equation*} We deduce Theorem \ref{thm:schoenIntro} from a stronger result, that when $(r,n) = (3,8)$, the above maps are isomorphisms, and the limits $\Gr(\sw)$ are smooth. While the statement of this theorem is geometric, the proof relies on techniques from matroid theory, commutative algebra (computing the coordinate rings of thin Schubert cells and morphisms between them) and polyhedral complexes (regular subdivisions of the hypersimplex and their tight-spans). In a recent paper~\cite{CoreyLuber}, the authors prove that $\Gr_0(3,n)$ is not sch\"{o}n for~$n \geq 12$. \begin{figure}[h] \includegraphics[height=4cm]{figures/specialMatroid.eps} \caption{The matroid $\spMat$} \label{fig:spMatroidIntro} \end{figure} Next, we study the connectedness properties of the initial degenerations of $\Gr_{0}(3,8)$. Interestingly, they are not all connected. Let $\spMat$ be the rank-3 matroid from Figure \ref{fig:spMatroidIntro}, \ie each 3-element set not connected by a (possibly curved) line is a basis. This matroid appears frequently in the literature \cite{BokowskiSturmfels, Gropp, Grunbaum, NazirYoshinaga}, and is referred to as the \textit{M\"obius--Kantor} or \textit{MacLane} matroid. Denote by $\spCone$ the cone of $\TGr_0(3,8)$ containing the corank vector of $\spMat$ in its relative interior (see \S \ref{sec:reducibleInitDeg}). \begin{theo} \label{thm:2ConnectedComponentsInDegIntro} For any $\sw$ in the relative interior of a cone in the $\Sn{8}$--orbit of $\spCone$, the initial degeneration $\init_{\sw} \Gr_{0}(3,8)$ has 2 connected components. \end{theo} \noindent Up to $\Sn{8}$--symmetry, this is the only non-connected initial degeneration of $\Gr_0(3,8)$. \begin{theo} \label{thm:connectedInitDegIntro} If $\sw\in \TGr_0(3,8)$ is not in the relative interior of a cone in the $\Sn{8}$--orbit of $\spCone$, then $\init_{\sw}\Gr_0(3,8)$ is irreducible. \end{theo} We apply these results to study the Chow quotient compactification of $X(r,n)$, the moduli space of projective equivalence classes of $n$ marked hyperplanes of $\PP^{r-1}$ in linear general position. The diagonal torus of $\GL(n)$ acts on $\C^n$ by scaling coordinates, and this induces an action of the diagonal torus $H\subset \PGL(n)$ on $\Gr(r,n)$. This action is free on $\Gr_0(r,n)$, and $X(r,n)$ is identified with the quotient $\Gr_0(r,n)/H$ by the Gelfand--MacPherson correspondence \cite{GelfandMacPherson}. The space $X(2,n)$ is also known as $M_{0,n}$, the moduli space of smooth rational $n$-marked curves. A natural candidate to compactify $X(r,n)$ is given by the Chow quotient $\chow{\Gr(r,n)}{H}$, studied by Kapranov \cite{Kapranov1993}, where he demonstrates that the Grothendieck--Knudsen moduli space of genus 0, stable, $n$-marked curves $\overline{M}_{0,n}$ is isomorphic to $\chow{\Gr(2,n)}{H}$. This is also the log-canonical compactification of $M_{0,n}$ \cite{KeelMcKernan}. Keel and Tevelev show that $\chow{\Gr(r,n)}{H}$ is usually not log canonical, failing already for~$(r,n) = (3,9)$ \cite{KeelTevelev2006}. With Hacking, they conjecture \cite[Conjecture 1.6]{KeelTevelev2006} that it is log canonical when $(r,n) = (2,n)$, $(3,6)$, $(3,7)$, and $(3,8)$. The conjecture is known in the first 3 cases, by Keel and McKernan (mentioned above), Luxton \cite{Luxton}, and the first author \cite{CoreyGrassmannians}, respectively. Using Theorems \ref{thm:schoenIntro}, \ref{thm:2ConnectedComponentsInDegIntro} and \ref{thm:connectedInitDegIntro}, together with a convexity result on $\TGr_0(3,8)$, we prove the $(3,8)$ case. \begin{theo} \label{thm:chowQuotientIntro} The normalization of the Chow quotient $\chow{\Gr(3,8)}{H}$ is the log canonical compactification of $X(3,8)$. \end{theo} Here is an outline of the paper. In \S \ref{sec:inverseLimitsTSC}, we review the relationship between initial degenerations of $\Gr_0(r,n)$ and finite inverse limits of thin Schubert cells induced by a matroidal subdivision of the hypersimplex. We consider the general setup of diagrams of schemes in \S\ref{sec:toolsForLimits}, and develop various strategies to determine when their inverse limits are smooth and irreducible. While not all morphisms between thin Schubert cells of $\Gr(3,8)$ are smooth and dominant, we develop general criteria in \S \ref{sec:computationsTSC} to detect when they are smooth and dominant with connected fibers. The disconnected initial degeneration of Theorem \ref{thm:2ConnectedComponentsInDegIntro} is studied in \S \ref{sec:reducibleInitDeg}. Theorems \ref{thm:schoenIntro} and \ref{thm:connectedInitDegIntro} are proved in \S\ref{sec:proofThms1And3}. We use the techniques from \S\S \ref{sec:toolsForLimits}--\ref{sec:computationsTSC} to prove that each inverse limit $\Gr(\sw)$ smooth of dimension 15 and, excluding the case from Theorem \ref{thm:2ConnectedComponentsInDegIntro}, irreducible. For any individual $\sw$, once one has the regular subdivision, one may carry out this verification by hand, as we illustrate in several examples throughout that section. As there are $57\,344$ cases to consider, we must use software. Finally, we discuss the Chow quotient in \S\ref{sec:Chow} and prove Theorem \ref{thm:chowQuotientIntro}. \subsection*{Code} For Proposition \ref{prop:smoothIrredTSC} and throughout \S\ref{sec:proofThms1And3}, we use \texttt{polymake.jl} \cite{polymake,polymakeJL} and OSCAR \cite{OSCAR-book,OSCAR}, both of which run using \texttt{julia} \cite{BezansonEdelmanKarpinskiShah}. The code can be found at the following github repository: \begin{center} \url{https://github.com/dcorey2814/Gr38Schoen} \end{center} \noindent The data for $\TGr_0(3,8)$, computed in~\cite{BendleBoehmRenSchroter}, can be found at the link: \begin{center} \url{https://www.mathematik.uni-kl.de/~boehm/singulargpispace/tropical.htm} \end{center} \section{Inverse limits of thin Schubert cells} \label{sec:inverseLimitsTSC} \subsection{Matroidal subdivisions} \label{sec:matroidalSubdivisions} We assume that the reader is familiar with matroids; a good general reference is \cite{Oxley}. While there are many ways of characterizing matroids, the definition via \textit{bases} is most relevant. A matroid of rank $r$ on $[n]$, called an $(r,n)$--\textit{matroid}, is a nonempty subset $\sQ$ of $\binom{[n]}{r}$ that satisfies the basis-exchange axiom. Let $N = \Z^{n}/\Zone$ and $M = \Hom(N,\Z)$. We use the standard abbreviations $M_{A} = M \otimes_{\Z} A$ and $N_{A} = N \otimes_{\Z} A$ where $A$ is a $\Z$-module. Denote by $\epsilon_{1},\ldots,\epsilon_n \in N$ the images of the standard basis vectors under the projection $\Z^{n} \to N$, and $\epsilon_{\lambda} = \epsilon_{i_1} + \cdots + \epsilon_{i_r}$ whenever $\lambda = \{i_1,\ldots,i_r\} \in \binom{[n]}{r}$. Let $\epsilon_1^{*},\ldots,\epsilon_{n}^{*} \in M$ be the dual basis and $\epsilon_{\lambda}^{*} = \epsilon_{i_1}^{*} + \cdots + \epsilon_{i_r}^{*}$. Given $\sQ \subset \binom{[n]}{r}$, set \begin{equation*} \Delta(\sQ) = \Conv \{\epsilon_{\lambda}^{*} \, : \, \lambda \in \sQ\} \subset M_{\R}. \end{equation*} When $\sQ$ is a matroid, $\Delta(\sQ)$ is the \textit{matroid polytope} of $\sQ$. An important special case is the \textit{uniform matroid} $\sQ = \binom{[n]}{r}$, and its polytope $\Delta(\sQ)$ is the $(r,n)$--\textit{hypersimplex} which we denote $\Delta(r,n)$. Let $N(\sQ) = \Z^{\sQ}/\Zone$ and $M(\sQ) = \Hom(N(\sQ),\Z)$. Denote by $\{\se_{\lambda}\, : \, \lambda \in \sQ\}$ the images of the standard basis vectors under $\Z^{\sQ} \to N(\sQ)$. For $\sw \in N(\sQ)_{\R} := N(\sQ) \otimes_{\Z} \R $, denote by $\pQ(\sw)$ the \textit{regular subdivision} of $\Delta(\sQ)$ induced by $\sw$; this is the polyhedral complex obtained by lifting, for each $\lambda \in \sQ$, the vertex $\epsilon_{\lambda}^{*}$ of $\Delta(\sQ)$ to height $\sw_{\lambda}$ in~$M_{\R} \times \R$, then projecting the lower faces of this lifted polytope back down to $M_{\R}$. See~\cite[Chapter~2]{DeLoeraRambauSantos} for a precise treatment. If $\sQ$ is a matroid, then the subdivision $\pQ(\sw)$ is \textit{matroidal} if each cell in $\pQ(\sw)$ is a matroid polytope. The \textit{Dressian} of $\sQ$ \cite{HerrmannJensenJoswigSturmfels} is the set \begin{equation*} \Dr(\sQ) = \{\sw \in N(\sQ)_{\R} \, : \, \pQ(\sw) \; \text{ is matroidal} \}. \end{equation*} This is the support of a polyhedral fan in $N(\sQ)$. \subsection{Tropicalization} Consider a pair of finite-rank lattices $M,N$ with a perfect pairing \begin{equation*} (\su, \sv) \to \bk{\su}{\sv}. \end{equation*} Let $T = \Spec(\C[M])$ be the torus whose character and cocharacter lattices are $M$ and $N$, respectively. Denote by $x^{\su} \in \C[M]$ the monomial corresponding to $\su\in M$. Suppose $X\subset T$ is a closed subvariety defined by a prime ideal $I\subset \C[M]$. Given $\sw \in N_{\R}$, the \textit{$\sw$--initial form} of $f\in \C[M]$ is \begin{equation*} \init_{\sw} f = \sum_{\substack{\langle \su, \sw \rangle \\ \text{is minimal}} } c_{\su} x^{\su} \hspace{20pt} \text{where} \hspace{20pt} f = \sum_{\su} c_{\su} x^{\su}. \end{equation*} The \textit{$\sw$--initial ideal} of $I$ is $\init_{\sw} I = \langle \init_{\sw} f \, : \, f\in I\rangle$. The \textit{tropicalization} of $X$ is \begin{equation*} \Trop(X) = \{\sw \in N_{\R} \, : \, \init_{\sw} I \neq \langle 1 \rangle \}. \end{equation*} This set is the support of a rational polyhedral fan in $N_{\R}$. Given $\sw \in \Trop(X)$, the \textit{$\sw$--initial degeneration} of $X$, denoted $\init_{\sw} X$, is the subscheme of $T$ defined by the ideal~$\init_{\sw} I$. \subsection{The Grassmannian} \label{sec:Grassmannian} As a set, the Grassmannian $\Gr(r,n)$ consists of all $r$-dimensional linear subspaces of $\C^{n}$. It may be realized as a projective variety by the Pl\"ucker embedding: \begin{align*} \Gr(r,n) \hookrightarrow \PP(\wedge^{r}\C^n ) \cong \PP^{\binom{n}{r} - 1} \hspace{20pt} F \mapsto \wedge^r F = [\zeta_{\lambda}(F) \, : \, \lambda \in \textstyle{\binom{[n]}{r}}]. \end{align*} The homogeneous coordinates $\zeta_{\lambda}(F)$ are called the \textit{Pl\"ucker coordinates} of $F$. Concretely, if $F \subset \C^n$ is the row span of the full-rank $(r\times n)$--matrix $A$, then $\zeta_{\lambda}(F)$ is the determinant of the $(r\times r)$--submatrix of $A$ whose columns are indexed by $\lambda$. Given a matroid $\sQ \subset \binom{[n]}{r}$, its \textit{thin Schubert cell} is the scheme-theoretic intersection \begin{equation*} \Gr(\sQ) = \Gr(r,n) \cap T(\sQ) \end{equation*} where $T(\sQ) = \G_m^{\sQ} / \G_m$ is the locally-closed coordinate stratum of $\PP^{\binom{n}{r}-1}$ whose nonzero coordinates are exactly those belonging to $\sQ$. Of particular interest is the thin Schubert cell of the uniform matroid $\sQ = \binom{[n]}{r}$, which we denote by $\Gr_0(r,n)$; this is the open locus of $\Gr(r,n)$ determined by the nonvanishing of all Pl\"ucker coordinates. The decomposition of the Grassmannian into thin Schubert cells refines the decomposition by Schubert cells, see \cite[\S 1.1]{KnutsonLamSpeyer} for a nice discussion on this. Throughout, we use the abbreviations \begin{equation*} \TGr(\sQ) = \Trop(\Gr(\sQ)) \hspace{20pt} \text{and} \hspace{20pt} \TGr_0(r,n) = \Trop(\Gr_0(r,n)). \end{equation*} The set $\TGr(\sQ) \subset N(\sQ)_{\R}$ is preserved under translation by the $(n-1)$--dimensional linear subspace $L_{\R}$, where $L\subset N(\sQ)$ is the subgroup \begin{equation} \label{eq:lineality} L = \left\langle \sum_{i\in \lambda \in \sQ} \se_{\lambda}, \;\; \sum_{i\notin \lambda \in \sQ} \se_{\lambda} \, : \, i\in [n] \right\rangle. \end{equation} We have an inclusion $\TGr(\sQ) \subset \Dr(\sQ)$ by \cite[Proposition~2.2]{Speyer2008} in the uniform matroid case and \cite[Lemma~4.4.6]{MaclaganSturmfels2015} in general. Of particular importance is that \textit{$\pQ(\sw)$ is matroidal for any $\sw \in \TGr(\sQ)$.} Denote by $\pG(r,n)$ the Gr\"obner fan structure of $\TGr_0(r,n)$ and $\pS_{\matroidal}(r,n)$ the secondary fan structure on $\Dr(r,n)$. By \cite[Theorem~5.4]{Tevelev}, $\pG(r,n)$ is a subfan of a refinement of $\pS_{\matroidal}(r,n)$. For $(r,n) = (2,n)$, $(3,6)$ \cite{SpeyerSturmfels2004a}, $(3,7)$ \cite{HerrmannJensenJoswigSturmfels}, or $(3,8)$ \cite{BendleBoehmRenSchroter}, there is a subfan $\pS_{\trop}(r,n)$ of $\pS_{\matroidal}(r,n)$ whose support is $\TGr_0(r,n)$. The \textit{face order} on the set of $(r,n)$--matroids is defined by $\sP \leq \sQ$ whenever $\Delta(\sP)$ is a face of $\Delta(\sQ)$. For each pair $\sP \leq \sQ$, the coordinate projection $T(\sQ) \to T(\sP)$ induces a morphism $\varphi_{\sQ,\sP}: \Gr(\sQ) \to \Gr(\sP)$ \cite[Proposition~I.6]{Lafforgue2003}. Given $\sw \in \TGr(\sQ)$, the regular subdivision $\pQ(\sw)$ of $\Delta(\sQ)$ is matroidal in the sense of \S \ref{sec:matroidalSubdivisions}. The assignment \begin{equation*} \sP \mapsto \Gr(\sP) \hspace{20pt} \sP' \leq \sP \mapsto \varphi_{\sP,\sP'}: \Gr(\sP) \to \Gr(\sP') \end{equation*} defines a diagram of type $\pQ(\sw)$ in the category of affine $\C$-schemes, and therefore we may form its inverse limit: \begin{equation*} \Gr(\sw) := \varprojlim_{\pQ(\sw)} \Gr. \end{equation*} The following theorem \cite[Theorem~1.1]{CoreyGrassmannians} is fundamental to the proofs of the main theorems in this paper. \begin{theo} \label{thm:closedImmersion} For any $\sw\in \TGr_0(r,n)$, there is a closed immersion \begin{equation*} \init_{\sw} \Gr_0(r,n) \hookrightarrow \Gr(\sw). \end{equation*} \end{theo} \noindent We thus obtain the following. \begin{coro} \label{cor:limit2Init} Suppose $\sw\in \TGr_0(r,n)$. If $\Gr(\sw)$ is smooth and irreducible of dimension $r(n-r)$, then $\init_{\sw}\Gr_0(r,n)$ is also smooth and irreducible. \end{coro} \begin{proof} Being a limit of a flat degeneration of $\Gr_0(r,n)$, the initial degeneration $\init_{\sw} \Gr_0(r,n)$ is an affine scheme of dimension $r(n-r)$. The inverse limit~$\Gr(\sw)$ is also an affine scheme, so if it is smooth, irreducible, and has the same dimension as $\init_{\sw} \Gr(r,n)$, then the closed immersion from Theorem \ref{thm:closedImmersion} is an isomorphism \cite[Proposition~A.8]{CoreyGrassmannians}. \end{proof} In the course of proving Theorems \ref{thm:schoenIntro}, \ref{thm:2ConnectedComponentsInDegIntro}, and \ref{thm:connectedInitDegIntro}, we show that the closed immersion in Theorem \ref{thm:closedImmersion} is an isomorphism for each $\sw\in \TGr_0(3,8)$. This provides a conceptual explanation of \cite[Theorem~4.9]{BendleBoehmRenSchroter}, which states that two vectors $\sw_1,\sw_2 \in \TGr_0(3,8)$ lie in the same cone of $\pS_{\trop}(3,8)$ if and only if $\init_{\sw_1}\Gr_0(3,8) = \init_{\sw_2}\Gr_0(3,8)$. In contrast, the cones in the Gr\"obner fan parameterize the various initial ideals of the homogeneous Pl\"ucker ideal. \subsection{Limits over graphs and tight spans} \label{sec:limitsGraphsTS} Computing the inverse limit $\Gr(\sw)$ seems like an unmanageable task given the size of the full face poset of $\pQ(\sw)$. It turns out that the codimension 0 and 1 cells that meet the relative interior of $\Delta(\sQ)$ are sufficient to compute this limit. For added flexibility, we describe a collection of subposets of $\pQ(\sw)$ that carry enough information to determine the inverse limit~$\Gr(\sw)$. Let $\pQ(\sw)$ be a regular subdivision of a lattice polytope $\Delta(\sQ) \subset M_{\R}$ induced by the lifting function $\sw \in N(\sQ)$. The \textit{dual graph} of $\pQ(\sw)$, denoted by $\Gamma(\sw)$, is the graph that has a vertex $v_{\sP}$ for each codimension-0 face $\sP \in \pQ(\sw)$, and the vertices $v_{\sP}$, $v_{\sP'}$ are connected by an edge if and only if $\sP$ and $\sP'$ meet along a common codimension-$1$ face. The \textit{tight span} of $\pQ(\sw)$, denoted $\TS(\sw)$, is the polyhedral complex that has a cell of dimension $\dim \Delta(\sQ) - k$ for each $k$-dimensional cell of $\pQ(\sw)$ meeting the relative interior of $\Delta(\sQ)$, and with face identifications opposite those of $\pQ(\sw)$. See \cite{Herrmann} for a precise treatment. The dual graph is exactly the 1-skeleton of the tight-span. Given any polyhedral complex $\Sigma$ with $\Gamma(\sw) \subset \Sigma \subset \TS(\sw)$, we may view $\Sigma^{\op}$ as a subposet of $\pQ(\sw)$, and hence we may form the limits of thin Schubert cells over this diagram. We denote such a limit by $\varprojlim_{\Sigma} \Gr$. \begin{prop} \label{prop:limitTSGamma} Given any polyhedral complex $\Sigma$ with $\Gamma(\sw) \subset \Sigma \subset \TS(\sw)$, the natural morphisms \begin{equation*} \Gr(\sw) \xrightarrow{\Phi} \varprojlim_{\Sigma} \Gr \xrightarrow{\Psi} \varprojlim_{\Gamma(\sw)} \Gr \end{equation*} are isomorphisms. \end{prop} \begin{proof} The composition $\Psi \circ \Phi$ is an isomorphism by a result of Cueto \cite[Proposition~C.12]{CoreyGrassmannians}; denote by $\Theta:\varprojlim_{\Gamma(\sw)} \Gr\to\Gr(\sw)$ its inverse. One readily verifies that $\Phi \circ \Theta$ is the inverse to $\Psi$. \end{proof} \section{Tools to study finite inverse limits of schemes} \label{sec:toolsForLimits} We gather in this section a number of techniques that will allow us to study inverse limits of diagrams of schemes coming from contractible polyhedral complexes, \eg tight-spans of matroidal subdivisions of the hypersimplex. More precisely, we want techniques to prove that a finite inverse limit is smooth and irreducible just by understanding the spaces and maps in the inverse limit system. With these techniques in hand, we avoid the necessity to directly compute the inverse limit. The key facts underlying most arguments in this section are contained in the following proposition. Recall that an \textit{SDC--morphism} of $\C$-schemes is a morphism of schemes $\varphi:X\to Y$ that is smooth, dominant, and its nonempty fibers are connected. \begin{prop} \label{prop:smoothIrreducibleFiberProduct} Suppose we have morphisms of equidimensional $\C$-schemes $X\to W$ and $Y\to W$. \begin{enumerate} \item If $Y$ is smooth and irreducible, and $X\to W$ is an SDC--morphism, then $X\times_W Y$, if nonempty, is smooth and irreducible. \item If $Y$ is smooth with $k$ connected components, and $X\to W$ is smooth and surjective with connected fibers, then $X\times_W Y$ is smooth with $k$ connected components. \end{enumerate} In either case, the dimension of $X\times_W Y$ is \begin{equation*} \dim X\times_W Y = \dim X + \dim Y - \dim W. \end{equation*} \end{prop} \begin{proof} Part (1) is \cite[Proposition~A.2]{CoreyGrassmannians}, so consider (2). Smoothness, surjectivity, and connectivity of fibers are all preserved by base change, so $X\times_W Y \to Y$ has these properties. The fiber product $X\times_W Y$ is a smooth $\C$-scheme as $X\times_W Y \to \Spec \C$ is the composition of two smooth morphisms. By \cite[\href{https://stacks.math.columbia.edu/tag/0378}{Tag 0378}]{stacks-project}, the morphism $X\times_W Y \to Y$ induces a bijection between the connected components of these spaces, and hence $X\times_W Y$ has $k$ connected components. The dimension formula is standard, see, \eg \cite[\S III.9]{Hartshorne} \end{proof} Throughout this section, let $\Sigma$ be a finite connected polyhedral complex, and $\Gamma$ its 1-skeleton. We regard $\Sigma$ as a quiver induced by its face order. Thus, we have an arrow $\sC' \to \sC$ whenever $\sC'$ is a \textit{facet} of $\sC$ in $\Sigma$. A \textit{diagram} of type $\Sigma$ in $\Csch$, the category of $\C$-schemes, is simply a functor $Z:\Sigma \to \Csch$. As finite inverse limits exist in the category $\Csch$ (indeed, it is enough to have fiber products and a terminal object, see \cite[Proposition~5.21]{Awodey}), we may form the inverse limit $\varprojlim_{\Sigma} Z$. \begin{prop} \label{prop:limitTreeAllButOne} Suppose $Z: \Gamma \to \Csch$ is a diagram over a tree $\Gamma$, and that $Z(v)$ is smooth and irreducible for all $v\in V(\Gamma)$. Furthermore, suppose that $Z(v \to e)$ is an SDC--morphism for all $v\in V(\Gamma)$ and $e\in E(\Gamma)$ adjacent to $v$, except for possibly one vertex $v_0$. Then $\varprojlim_{\Gamma} Z$ is smooth and irreducible of dimension \begin{equation} \label{eq:dimTree} \dim \varprojlim_{\Gamma} Z = -\sum_{e\in E(\Gamma)} \dim Z(e) + \;\; \sum_{v\in V(\Gamma)} \dim Z(v). \end{equation} \end{prop} \begin{proof} We proceed by induction on the number of edges. The proposition is trivial if $\Gamma$ has no edges, so assume that $\Gamma$ has $n$ edges and that the proposition is true for graphs with fewer than $n$ edges. Suppose $v\neq v_0$ is connected to $v_0$ by the edge $e$. Then $Z(v)\times_{Z(e)} Z(v_0)$ is smooth and irreducible by Proposition \ref{prop:smoothIrreducibleFiberProduct}. Let $Z'$ be the diagram on $\Gamma/e$ where $Z'(v) = Z(v)\times_{Z(e)} Z(v_0)$ at the contracted vertex, and $Z' = Z$ otherwise. By \cite[Proposition~A.5]{CoreyGrassmannians} we have that $\varprojlim_{\Gamma} Z \cong \varprojlim_{\Gamma/e}Z'$. As $Z'$ is a diagram on a graph with fewer edges and satisfies the hypotheses of the proposition, the limit $\varprojlim_{\Gamma/e}Z'$ is smooth and irreducible by the inductive hypothesis. \end{proof} \subsection{Removing leaves} \label{sec:removeLeaf} A \textit{leaf} of $\Sigma$ is a vertex $v$ of $\Sigma$ that is a leaf vertex of $\Gamma$. A \textit{leaf-pair} is a pair $(v,e)$ such that $v$ is a leaf and $e$ is its adjacent edge. Let $\Sigma_{\sL} \subset \Sigma$ be the subcomplex obtained by removing all leaf-pairs from $\Sigma$. \begin{prop} \label{prop:removeLeavesLim} If $\varprojlim_{\Sigma_{\sL}} Z$ is smooth and irreducible and $Z(v\to e)$ is an SDC--morphism for each leaf pair $(v,e)$ then $\varprojlim_{\Sigma} Z$ is smooth and irreducible of dimension \begin{equation} \label{eq:dimDeLeaf} \dim \varprojlim_{\Gamma} Z = \dim \varprojlim_{\Sigma_{\sL}} Z \;\;- \sum_{\substack{e \text{ leaf} \\ \text{edge}}} \dim Z(e)\;\;+ \sum_{\substack{v \text{ leaf} \\ \text{vertex}}} \dim Z(v). \end{equation} \end{prop} \begin{proof} Let $\Gamma'$ be the star-shaped graph obtained by contracting $\Sigma_{\sL}$ to a point. Let $Z'$ be the diagram on $\Gamma'$ defined by $Z$ on all leaf-pairs, and $\varprojlim_{\Sigma_{\sL}} Z$ on the newly-formed vertex. Set \begin{equation*} Y_{\Leaf} = \prod_{\substack{v \text{ leaf} \\ \text{vertex}}} Z(v) \hspace{20pt} Y_{\Leaf}' = \prod_{\substack{e \text{ leaf} \\ \text{edge}}} Z(e) \end{equation*} and note that $Y_{\Leaf} \to Y_{\Leaf}'$ is an SDC--morphism since each $Z(v\to e)$ is an SDC--morphism. By \cite[Proposition~A.5]{CoreyGrassmannians} and Proposition \ref{prop:limitTSGamma}, we have \begin{equation*} \varprojlim_{\Sigma} Z \cong \varprojlim_{\Gamma'} Z'. \end{equation*} The proposition now follows from Proposition \ref{prop:limitTreeAllButOne}. \end{proof} Observe that $\Sigma_{\sL} \subset \Sigma$ may still have leaves. The process of iteratively removing leaf-pairs must terminate, and the result is the subcomplex $\Sigma_{\mathsf{Br}} \subset \Sigma$, which is nonempty provided $\Sigma$ is not a tree. This is similar to the \textit{split prime remainder} in the sense of~\cite{HerrmannJoswig}. A maximal connected subgraph of $\Sigma \setminus \Sigma_{\mathsf{Br}}$ is called a \textit{branch} of $\Sigma$. Any vertex in a branch is called a \textit{branch vertex} and any edge contained in a branch, or connecting a branch to the rest of $\Sigma$, is called a \textit{branch edge}. The following proposition follows from Proposition \ref{prop:removeLeavesLim} and induction. \begin{prop} \label{prop:removeBranchLim} If $\varprojlim_{\Sigma_{\mathsf{Br}}} Z$ is smooth and irreducible and $Z(v\to e)$ is an SDC--morphism for each pair $(v,e)$ of a branch vertex $v$ and adjacent edge $e$, then $\varprojlim_{\Sigma} Z$ is smooth and irreducible of dimension \begin{equation} \label{eq:dimDeBranch} \dim \varprojlim_{\Gamma} Z = \dim \varprojlim_{\Sigma_{\mathsf{Br}}} Z - \sum_{\substack{e \text{ branch} \\ \text{edge}}} \dim Z(e) + \sum_{\substack{v \text{ branch} \\ \text{vertex}}} \dim Z(v). \end{equation} \end{prop} \subsection{Removing fins}\label{sec:removeFins} Suppose $\sF$ is a closed 2-dimensional cell of $\Sigma$ with $k$ vertices ($k\geq 3$). We say that $\sF$ is a \textit{fin} if its intersection with $\overline{\Sigma \setminus \sF}$ is a path of edge-length $\ell$ with $1\leq \ell \leq k-2$; this path is called the \textit{connecting path} of $\sF$. See Figure \ref{fig:attachFin} for an illustration. Denote by $V(\sF)$ the vertices of $\sF$ and $E(\sF)$ its edges. A vertex $v\in V(\sF)$, \resp an edge $e\in E(\sF)$, is \textit{exposed} if $v\notin \overline{\Sigma\setminus \sF}$, \resp $e\not\subset \overline{\Sigma\setminus \sF}$. Denote by \begin{equation*} \pE_{\vertices}(\sF) = \{v \in V(\sF) \, : \, v \text{ is exposed}\} \hspace{20pt} \pE_{\edge}(\sF) = \{e\in E(\sF) \, : \, e \text{ is exposed}\}. \end{equation*} Order the vertices of $\sF$ cyclically $v_1,\ldots,v_{k}$ so that the first $k-\ell-1$ are the exposed vertices. Let $e_{i,i+1}$ be the edge between $v_{i}$ and $v_{i+1}$ (where the indices are taken modulo $k$). Denote by $Z_{\pE}(\sF)$ the fiber product \begin{equation} \label{eq:limitExposed} Z_{\pE}(\sF) = Z(v_{1}) \times_{Z(e_{1,2})} Z(v_{2}) \times_{Z(e_{2,3})} \cdots \times_{Z(e_{k-\ell-2,k-\ell-1})} Z(v_{k-\ell-1}) \end{equation} and let $\varphi_{\sF}:Z_{\pE}(\sF) \to Z(e_{1,k}) \times_{Z(F)} Z(e_{k-\ell-1,k-\ell})$ be the morphism induced by $Z(v_1 \to e_{1,k})$ and $Z(v_{k-\ell-1} \to e_{k-\ell-1,k-\ell})$. \begin{figure} \centering \includegraphics[height=5cm]{figures/attachFin.eps} \caption{A fin and the corresponding quiver. The vertices $v_1$, $v_2$ and edges $e_{15}$, $e_{12}$, $e_{23}$ are exposed} \label{fig:attachFin} \end{figure} \begin{prop} \label{prop:addfin} Suppose $Z:\Sigma \to \Csch$ is a diagram, $\sF$ is a fin of $\Sigma$, and $\Sigma' = \overline{\Sigma\setminus \sF}$. Then we have a pullback diagram \begin{equation*} \begin{tikzcd} \varprojlim_{\Sigma} Z \arrow[r] \arrow[d] & Z_{\pE}(\sF) \arrow[d, "\varphi_{\mathsf{F}}"] \\ \varprojlim_{\Sigma'} Z \arrow[r] & Z(e_{1,k}) \times_{Z(F)} Z(e_{k-\ell-1,k-\ell}). \end{tikzcd} \end{equation*} \end{prop} The proof amounts to showing that $\varprojlim_{\Sigma}Z$ satisfies the universal property of fiber products. It is similar to \cite[Proposition~A.5]{CoreyGrassmannians}, so we omit the proof. Given a collection of fins $\mathfrak{F}$, let $\Sigma(\mathfrak{F}) \subset \Sigma$ be the subcomplex obtained by removing, for each $\sF\in \mathfrak{F}$, the relatively open cell $\sF^{\circ}$ and the exposed vertices and edges of $\sF$. \begin{prop} \label{prop:limitFins} If $\varprojlim_{\Sigma(\mathfrak{F})} Z$ is smooth and irreducible and $\varphi_{\sF}$ is an SDC--morphism for each $\sF \in \mathfrak{F}$, then $\varprojlim_{\Sigma} Z$ is smooth and irreducible of dimension \small{\begin{equation} \label{eq:limitFin} \dim \varprojlim_{\Sigma} Z = \dim \varprojlim_{\Sigma(\mathfrak{F})} Z + \sum_{\sF \in \mathfrak{F}} \left( \dim Z(\sF) - \sum_{e\in \pE_{\edge}(\sF)} \dim Z(e) + \sum_{v\in \pE_{\vertices}(\sF)} \dim Z(v) \right). \end{equation}} \end{prop} \begin{proof} This follows from Propositions \ref{prop:smoothIrreducibleFiberProduct} and \ref{prop:addfin} applied to each fin $\sF \in \mathfrak{F}$. \end{proof} \section{Computations on thin Schubert cells} \label{sec:computationsTSC} By the Mn\"ev universality theorem (see \cite{Lafforgue2003,VakilLee}), the collection of schemes $\Gr(\sQ)$, as~$\sQ$ runs through the $\C$-realizable $(3,n)$--matroids, satisfies Murphy's law in the sense of Vakil \cite{Vakil}. Nevertheless, for small values of $n$, many such $\Gr(\sQ)$ are smooth and irreducible; in fact $\Gr(\sQ)$ is smooth and irreducible for all $(3,n)$--matroid with $n\leq 7$~\cite[Proposition~4.2]{CoreyGrassmannians}. In this section, we show that $\Gr(\sQ)$ are smooth for all $\C$-realizable $(3,8)$--matroids, but there is one $\Gr(\sQ)$ that has 2 connected components. \subsection{Affine coordinates for thin Schubert cells} \label{sec:affineCoordinates} We recall the affine coordinate ring construction of $\Gr(\sQ)$, following \cite[Construction~2.2]{CoreyGrassmannians}. Given any $r\times n$ (for $r\leq n$) matrix $A$, denote by $\col_{i}(A)$ the $i$-th column of $A$. With $\lambda = \{i_1<\cdots1} x_i^{u_i}, \hspace{10pt} \text{and} \hspace{10pt} h \in \C[x_{2},\ldots,x_b]. \end{equation*} Here, $c$ is a nonzero constant and the $u_i$ are nonnegative. Therefore, the assignment $x_1\mapsto h/g$, and $x_i \mapsto x_i$ for $i>1$ defines a ring isomorphism \begin{equation*} R \to \bar{S}^{-1} \C[x_2,\ldots,x_b] / \langle f_2,\ldots,f_b \rangle \end{equation*} where $\bar{S}$ is the image of $S$ under the above substitution. The lemma now follows from the inductive hypothesis. \end{proof} Lemma \ref{lem:upperTriangularEliminate} suffices to determine that $R_{\Sigma_{\sL}}$ is a regular integral domain for most cases in $\sG_3$. For the remaining, we use the following Algorithm. Given $f\in B$, denote by $\deg_{k}(f)$ the degree of $x_k$ in $f$, \ie the highest power of $x_k$ appearing in $f$. If $\deg_{k}(f) = 1$, then denote by $\coef(f;x_k)$ the coefficient of $x_k$ in $f$. We say that $f$ is \textit{reduced} if $f$ has no unit other than elements of $\C^*$ as a factor. Because $B$ is a unique factorization domain, if $f\in S^{-1}B$ is not a unit, then there is a unique expression (up to a factor lying in $\C^*$) $f = g\cdot \bar{f}$ such that $g\in S$ and $\bar{f}$ lie in $B$ and has no unit (other than an element of $\C^*$) as a factor. \begin{algm}\label{algm:reduceIdeal} \leavevmode \noindent \textit{Input}: The polynomial ring $B = \C[x_1,\ldots,x_b]$, $S\subset B$ a multiplicative semigroup containing all monomials, and $\pF \subset B$ a finite set of reduced generators of $I\subset S^{-1}B$. Set $R = S^{-1} B / I$. \medskip \noindent \textit{Output}: A new presentation of $R$, say $R \cong \bar{S}^{-1}\bar{B} / \bar{I}$, and a finite set $\bar{\pF}\subset \bar{B}$ of reduced generators of $\bar{I}\subset \bar{S}^{-1} \bar{B}$ such that \begin{itemize} \item[-] $\bar{B}$ is a polynomial ring on a subset of the $x_1,\ldots,x_b$, \item[-] for each $x_k\in \bar{B}$ and $f \in \bar{\pF}$, we have $\deg_{k} f \neq 1$ or $\deg_{k} f = 1$ but $\coef(f;x_k) \notin S$. \end{itemize} In particular, if $\bar{I} = \langle 0 \rangle$, then $R$ is a regular integral domain. \medskip \noindent Find $f\in \pF$ and $x_k\in B$ such that $\deg_{k} f = 1$ and $\coef(f;x_k) \in S$. This means that \begin{equation*} f = g \cdot x_k - h \end{equation*} for some $g\in S$ and $h \in B$ such that $\deg_k g = 0$ and $\deg_k h = 0$. Then perform the substitution $x_k \mapsto h/g$ followed by a reduction for all elements of $\pF$, and just the substitution for the elements of $S$; this produces new sets $\bar{\pF}$ and $\bar{S}$. Set $\bar{B} = \C[x_j \, : \, j\neq k]$. Now repeat this procedure with $\pF\gets \bar{\pF}$, $S \gets \bar{S}$ and $B \gets \bar{B}$. \medskip \noindent Once no such pair $(f,x_k)$ exists, this function returns $\pF$ and the presentation $R = S^{-1} B / I$. \end{algm} \begin{prop} \label{prop:G3} If $\sw \in \sG_3$, then $\Gr(\sw)$ is smooth and irreducible of dimension 15. \end{prop} \begin{proof} For $26\, 187$ combinatorial types $\sw\in \sG_3$, there is a presentation of the ideal $I_{\Sigma_{\sL}}$ that satisfies the conditions of Lemma \ref{lem:upperTriangularEliminate}. So $\varprojlim_{\Sigma_{\sL}}\Gr$ is smooth and irreducible. For the remaining $2040$ combinatorial types, we apply Algorithm \ref{algm:reduceIdeal}. In each case, this function returns a presentation of $R_{\Sigma_{\sL}}$ whose ideal is $\langle 0 \rangle$, whence $R_{\Sigma_{\sL}}$, and therefore $R_{\Sigma}$, is a regular integral domain. In either case, we get that $\Gr(\sw)$ is smooth and irreducible by Proposition \ref{prop:limitcvpLeaf}. Finally, we compute the dimensions of each $\Gr(\sw)$ using Formula \eqref{eq:dimLimitCVPLeaf}, verify that each dimension is $15$. \end{proof} \subsection{Case \texorpdfstring{$\sG_4$}{G4}}\label{sec:G4} There are $483$ combinatorial types belonging to the set $\sG_4$. \begin{prop} \label{prop:G4} If $\sw \in \sG_4$, then $\Gr(\sw)$ is smooth and irreducible of dimension 15. \end{prop} \begin{proof} For any $\sw\in \sG_4$, it happens to be the case that any branch of $\TS(\sw)$ is a path, and so each of its vertices has 2 or fewer neighbors. For such a vertex $v$, its dual polytope $\Delta_v$ has 2 or fewer internal facets. Thus, each such $\Gr(\sQ_v)$ is smooth and irreducible and the maps from $\Gr(\sQ_v)$ corresponding to the edges adjacent to $v$ are SDC--morphisms by Proposition \ref{prop:oneInternalFacet}. By Proposition \ref{prop:removeBranchLim}, to show that $\Gr(\sw)$ is smooth and irreducible, it suffices to show that $\varprojlim_{\Sigma_{\mathsf{Br}}} \Gr$ is smooth and irreducible. For $15$ combinatorial types, there is a presentation of the ideal $I_{\Sigma_{\mathsf{Br}}}$ that satisfies the conditions of Lemma \ref{lem:upperTriangularEliminate}. For the remaining $468$ combinatorial types, we apply Algorithm \ref{algm:reduceIdeal}, which always returns the ideal $\langle 0 \rangle$. In either case, we see that $R_{\Sigma_{\mathsf{Br}}}$ is a regular integral domain, as required. Finally, we verify that the dimensions of the $\Gr(\sw)$ are all 15 using the formula \begin{equation*} \dim \Gr(\sw) = \dim \varprojlim_{\Sigma_{\mathsf{Br}}} \Gr - \sum_{\substack{e \text{ branch } \\ \text{edge} }} \dim \Gr(\sQ_e) + \sum_{\substack{v \text{ branch } \\ \text{vertex} }} \dim \Gr(\sQ_v) \end{equation*} which follows from Formula \ref{eq:dimDeBranch}. \end{proof} \subsection{Case \texorpdfstring{$\sG_5$}{G5}} There are $14\,389$ combinatorial types belonging to the set $\sG_5$. Similar to the case $\sG_3$, given a combinatorial type $\sw$, let $\Sigma = \TS(\sw)$ and $\Sigma_{\sL} \subset \Sigma$ the subcomplex obtained by removing all leaf-vertices of $\Sigma$ together with their adjacent edges. Let $Y_{\Leaf}$ and $Y_{\Leaf}'$ be the schemes in Formula \ref{eq:leavesVE}. Formula \ref{eq:deleaf} holds, but now $\varprojlim_{\Sigma_{\sL}} \Gr$ cannot be evaluated using a presentation provided by Proposition \ref{prop:coordRingSigma} since $\Sigma_{\sL}$ is not vertex-intersecting. Recall that a fin $\sF$ is $B$--maximal if there is a $\mu \in \sQ_{\sF}$ such that $\sQ$ is $B$--maximal for all $\sQ \in \pE_{\vertices}(\sF)$. Given a collection of fins $\mathfrak{F}$ of $\Sigma_{\sL}$, let $\Sigma_{\sL}(\mathfrak{F}) \subset \Sigma_{\sL}$ be the subcomplex obtained by removing the fins $\sF \in \mathfrak{F}$. \begin{prop}\label{prop:fins} If $\varprojlim_{\Sigma_{\sL}(\mathfrak{F})}\Gr$ is smooth and irreducible and each fin $\sF \in \mathfrak{F}$ is $B$--maximal, then $\Gr(\sw)$ is smooth and irreducible of dimension \begin{align} \nonumber \dim & \Gr(\sw) = \dim \varprojlim_{\Sigma_{\sL}(\mathfrak{F})} \Gr \; - \; \sum_{\substack{e \text{ leaf} \\ \text{edge}}} \dim \Gr(\sQ_e) \; + \sum_{\substack{v \text{ leaf} \\ \text{vertex}}} \dim \Gr(\sQ_v) \; \\ & +\sum_{\sF \in \mathfrak{F}} \;\left(\dim \Gr(\sQ_{\sF}) - \sum_{e \in \pE_{\edge}(\sF)} \dim \Gr(\sQ_e) + \sum_{v \in \pE_{\vertices}(\sF)} \dim \Gr(\sQ_v) \right). \label{eq:dimDefin} \end{align} \end{prop} \begin{proof} By Propositions \ref{prop:addfin} and \ref{prop:finSDC}, we have that $\varprojlim_{\Sigma_{\sL}} \Gr$ is smooth and irreducible of dimension {\footnotesize \begin{equation*} \dim \varprojlim_{\Sigma_{\sL}} \Gr = \dim \varprojlim_{\Sigma_{\sL}(\mathfrak{F})} \Gr \; +\sum_{\sF \in \mathfrak{F}} \;\left(\dim \Gr(\sQ_{\sF}) - \sum_{e \in \pE_{\edge}(\sF)} \dim \Gr(\sQ_e) + \sum_{v \in \pE_{\vertices}(\sF)} \dim \Gr(\sQ_v) \right). \end{equation*} } Now apply Propositions \ref{prop:limitcvpLeaf}. \end{proof} \begin{prop} \label{prop:G5} If $\sw \in \sG_5$, then $\Gr(\sw)$ is smooth and irreducible of dimension 15. \end{prop} \begin{proof} Fix $\sw \in \sG_5$, $\Sigma = \TS(\sw)$, and $\Sigma_{\sL} \subset \Sigma$ the subcomplex obtained by removing all leaf vertices and edges. Let $\mathfrak{\sF}$ be the set of fins of $\Sigma_{\sL}$ that have connecting path length 1. We verify that each $\sF \in \mathfrak{F}$ is $B$--maximal. In this case, the set $\Sigma_{\sL}(\mathfrak{F})$ is vertex-connecting by Proposition \ref{prop:vertexConnectingFin} and vertex-intersecting by hypothesis, and therefore we may compute the coordinate ring of $\varprojlim_{\Sigma_{\sL}(\mathfrak{F})} \Gr$ using Proposition \ref{prop:coordRingSigma}. We verify that these rings are smooth integral domains as in the proof of Proposition \ref{prop:G3}. Finally, we verify that $\dim \Gr(\sw) = 15$ using Formula \eqref{eq:dimDefin}. The proposition now follows from Corollary \ref{cor:limit2Init}. \end{proof} \begin{exam} Let \begin{align*} \sw &= \se_{124} + \se_{125} + 2 \se_{126} + 2 \se_{134} + 3 \se_{137} + \se_{145} + \se_{146} + 2\se_{147} +\se_{148} \\ & + \se_{156} + 2 \se_{235} + 2 \se_{237} + \se_{245} + \se_{246} + \se_{256} + 2 \se_{257} +\se_{258} \\ & + 3 \se_{347} + 2 \se_{357} + 2 \se_{367} + 2 \se_{368} + 2 \se_{378} + 2 \se_{456} + 3 \se_{678}. \end{align*} The tight span $\TS(\sw)$ of $\pQ(\sw)$ is given in Figure \ref{fig:mantis}. We also see the matroid corresponding to the node $v_2$ of $\TS(\sw)$ that did not appear in Figure $\ref{fig:matroidsUVW}$. \begin{figure} \centering \includegraphics[height = 4cm]{figures/mantis_fixed.eps}\hspace{40pt} \includegraphics[height = 3cm]{figures/mantis3_new.pdf} \caption{Dual graph of $\pQ(\sw)$ and the matroid corresponding to $\sQ_3$} \label{fig:mantis} \end{figure} The matroids $\sQ_i$ corresponding to the vertices $v_i$ of $\Gamma(\sw)$ are \begin{equation*} \begin{array}{lll} \sQ_1=\sU(1,2568,37,4) &\sQ_2= \sU(1,2,34578,6) &\sQ_4 = \sW(37;1,4;256,8) \\ \sQ_5 = \sW(2568;1,4;3,7) &\sQ_6= \sW(1468;2,5;3,7) &\sQ_7 = \sW(37;146,8;2,5) \\ \sQ_8 = \sW(6;14,25;37,8) &\sQ_9 = \sU(1468,2,37,5) &\sQ_{10} = \sU(14,2568,3,7) \\ \sQ_{11} = \sW(14;25,68;3,7) &\sQ_{12}= \sW(7;1245,3;6,8) &\sQ_{13} = \sU(12345,6,7,8) \\ \sQ_{14} = \sU(12378,4,5,6) & &\\ \end{array} \end{equation*} When $v_{i}$ and $v_j$ share an edge, the matroid of the edge is denoted by $\sQ_{i,j}$; in each case, $\sQ_{i,j}$ is isomorphic to a matroid of the form $\sU'$. Then Proposition \ref{prop:TSCUVW}, and direct computation for the $i=3$ case, tells us the dimensions of the thin Schubert cells corresponding to the vertices and edges of $\TS(\sw)$ are given by \begin{equation*} \dim \Gr(\sQ_i) = 7 \text{ for } 1\leq i \leq 14 \hspace{15pt} \dim \Gr(\sQ_{i,j}) = 6\text{ for }(v_i,v_j)\in E(\Gamma(\sw)). \end{equation*} \noindent Moreover, one can check that $\dim\Gr(\sQ_{\sF}) = 5$ for the unique fin $\sF=\langle v_8,v_{11},v_{12}\rangle$ of~$\Sigma_{\sL}$. Finally, we compute $R_{\Sigma_{\sL}(\mathfrak{F})}$, to obtain the dimension of $\varprojlim_{\Sigma(\mathfrak{F})}$. We compute \begin{equation*} B_{\Sigma_{\sL}(\mathfrak{F})} = \C[x_{11},x_{12},x_{13},x_{14},x_{15},x_{21},x_{22},x_{23},x_{24},x_{25},x_{31},x_{32},x_{34},x_{35}] \end{equation*} the ideal is \begin{equation*} I_{\Sigma_{\sL}(\mathfrak{F})} = \langle x_{13}x_{25}-x_{23}x_{15},\; x_{11}x_{32}x_{25}+x_{31}x_{22}x_{15},\; x_{11}x_{32}x_{23}+x_{31}x_{22}x_{13}\rangle, \end{equation*} and $S_{\Sigma_{\sL}(\mathfrak{F})}$ is the semigroup of all monomials of $B_{\Sigma_{\sL}(\mathfrak{F})}$. One can check that the condition of Lemma \ref{lem:upperTriangularEliminate} does not hold. Thus we apply Algorithm \ref{algm:reduceIdeal}. Let $f_1,f_2,f_3$ denote the generators of $I_{\Sigma_{\sL}(\mathfrak{F})}$. The relations $f_i\equiv 0\mod I_{\Sigma_{\sL}(\mathfrak{F})}$ (for $i=1,2$) yield \begin{equation*} x_{13} \equiv \frac{x_{23}x_{15}}{x_{25}} \hspace{10pt} \text{ and } \hspace{10pt} x_{32} \equiv -\frac{x_{31}x_{22}x_{15}}{x_{11}x_{25}} \mod I_{\Sigma_{\sL}(\mathfrak{F})}. \end{equation*} Observe that with these substitutions, we have $f_3 = 0$, and that the coordinate ring of $\varprojlim_{\Sigma(\mathfrak{F})} \Gr$ reduces to $\C[x_{11},x_{12},x_{14},x_{15},x_{21},x_{22},x_{23},x_{24},x_{25},x_{31},x_{34},x_{35}]$. So we have that $\varprojlim_{\Sigma(\mathfrak{F})} \Gr$ is smooth, irreducible and of dimension 12. We also have the fin $\sF$ is $B$--maximal, in the sense that $\sQ_{12}$ is $(B,\mu)$--maximal where $\mu=\{1,7,8\} \in \sQ_{\sF}$. By Proposition \ref{prop:fins} $\Gr(\sw)$ is smooth and irreducible. One may use Formula \eqref{eq:dimDefin} to verify that $\dim \Gr(\sw) = 15$. Hence, by Corollary \ref{cor:limit2Init}, $\init_{\sw}\Gr_{0}(3,8)$ is smooth and irreducible. \end{exam} \subsection{Case \texorpdfstring{$\sG_6$}{G6}} There are $389$ combinatorial types in $\sG_6$. For each combinatorial type $\sw \in \sG_6$, the dual graph $\Gamma(\sw)$ has a star-tree subgraph $\sT$ with 4 leaves; the matroid $\sC$ of the center node is isomorphic to $\sU(12,34,56,78)$, and the matroids $\sL_1,\sL_2,\sL_3,\sL_4$ of the 4 leaves are isomorphic to either $\sV(12,34,56;7,8)$ or $\sW(12; 34, 56; 7, 8)$. The set $\sG_6$ splits further into 5 groups $\sH_0, \sH_1, \sH_2, \sH_3, \sH_4$ where $\sw \in \sH_k$ if \begin{equation*} |\{ m\, : \, \sL_{m} \cong \sV(12,34,56;7,8) \}| = k. \end{equation*} The sizes of the $\sH_k$'s are \begin{equation*} \begin{array}{lllll} |\sH_0| = 233 & |\sH_1| = 127 & |\sH_2| = 25 & |\sH_3| = 3 & |\sH_4| = 1. \end{array} \end{equation*} \begin{prop} \label{prop:dimLimitT} If $\sw \in \sH_k$, then the limit $\varprojlim_{\sT} \Gr$ is smooth and irreducible of dimension $11+k$. \end{prop} \begin{proof} The matroid $\sC_m$ corresponding to the edge between $\sC$ and $\sL_m$ is isomorphic to $\sU'(12,34,56;78)$, see Figure \ref{fig:matroidsUVW}. By Proposition \ref{prop:TSCUVW}, the thin Schubert cells $\Gr(\sC)$, $\Gr(\sC_m)$, $\Gr(\sL_m)$ are smooth, irreducible, and their dimensions are $\dim \Gr(\sC) = 7 $, $\dim \Gr(\sC_m) = 6$ and {\small \begin{align*} \dim \Gr(\sL_m) = 8 \hspace{5pt} \text{ if } \sL_m \cong \sV(12,34,56;7,8); \hspace{5pt} \dim \Gr(\sL_m) = 7 \hspace{5pt} \text{ if } \sL_m \cong \sW(12; 34, 56; 7, 8). \end{align*}} Furthermore, the morphisms $\Gr(\sL_m) \to \Gr(\sC_m)$ are smooth and dominant with connected fibers by Proposition \ref{prop:TSCUVW}. The proposition now follows from Proposition~\ref{prop:limitTreeAllButOne}. \end{proof} \begin{prop} \label{prop:G6} If $\sw \in \sG_6$, then $\Gr(\sw)$ is smooth and irreducible of dimension 15. \end{prop} \begin{proof} Suppose $\sw \in \sH_k$, and let $\Sigma = \TS(\sw)$. As usual, let $\Sigma_{\sL} \subset \Sigma$ be the subcomplex obtained by removing all leaf vertices and edges from $\Sigma$. By a direct verification, $\Sigma_{\sL}$ consists of the tree $\sT$, and 6 fins, each of which has contact-length 2. In fact, there is a fin attached to $\sT$ at each pair of edges. We verify that each fin is $B$--maximal. Thus $\Gr(\sw)$ is smooth and irreducible by Propositions \ref{prop:limitFins} and \ref{prop:finSDC}. Finally, we verify that $\dim \Gr(\sw) = 15$ using Formula \eqref{eq:dimDefin}. The proposition now follows from Corollary~\ref{cor:limit2Init}. \end{proof} \begin{exam} \label{ex:G6} The vector \begin{align*} \sw &= 2\,\se_{123} + 2\,\se_{124} + 2\,\se_{134} + 2\,\se_{135} + 2\,\se_{136} + 4\,\se_{137} + 3\,\se_{138} + \se_{156} \\ & + \se_{178} + 2\,\se_{234} + \se_{245} + \se_{246} + \se_{247} + \se_{248} + \se_{256} + \se_{278} + \se_{356} \\ & + \se_{378} + \se_{456} + \se_{478} + 2\se_{567} + 2\se_{568} + 2\se_{578} + 2\se_{678} \end{align*} \noindent represents a combinatorial type in $\sH_0$, the tight span $\TS(\sw)$ is shown in Figure \ref{fig:exampleG6}, together with the matroids corresponding to the nodes of $\TS(\sw)$. The tree $\sT$ consists of the vertices $v_{2},v_{7},v_{9},v_{10},v_{15}$, and $\varprojlim_{\sT}\Gr$ is smooth and irreducible of dimension~11 by Proposition \ref{prop:dimLimitT}. The subcomplex $\Sigma_{\sL} \subset \TS(\sw)$ obtained by removing all leaves is obtained by removing the vertex $v_3$ and its adjacent edge. The fins of $\Sigma_{\sL}$ are: \begin{equation*} \begin{array}{lll} \sF_1 = \langle v_{2}, v_{4}, v_{9}, v_{15}\rangle & \sF_2 = \langle v_{2}, v_{7}, v_{8}, v_{9}\rangle & \sF_3 = \langle v_{6}, v_{7}, v_{9}, v_{10}\rangle \\ \sF_4 = \langle v_{9}, v_{10}, v_{11}, v_{15}\rangle & \sF_5 = \langle v_{1}, v_{2}, v_{5}, v_{9}, v_{10}\rangle & \sF_6 = \langle v_{7}, v_{9}, v_{12}, v_{13}, v_{14}, v_{15}\rangle. \end{array} \end{equation*} With $\mu_1 = \{1,5,7\}$ and $\mu_2 = \{1,2,5\}$, the fins $\sF_1$, $\sF_4$, $\sF_6$ are $(B,\mu_1)$--maximal, and the fins $\sF_2$, $\sF_3$, $\sF_5$ are $(B,\mu_2)$--maximal. Therefore, $\Gr(\sw)$ is smooth and irreducible by Propositions \ref{prop:limitFins} and \ref{prop:finSDC}. Using this and Proposition \ref{prop:TSCUVW}, one may verify that $\dim \Gr(\sw) = 15$, and therefore $\init_{\sw} \Gr_0(3,8)$ is smooth and irreducible by Corollary~\ref{cor:limit2Init}. \end{exam} \begin{figure} \begin{minipage}{0.4\textwidth} \includegraphics[width=\linewidth,keepaspectratio=true]{figures/exampleG6.eps} \end{minipage} \begin{minipage}{0.6\textwidth} \small \begin{align*} \begin{array}{ll} \sQ_1 = \sW(5678;1,3;2,4) & \sQ_2 = \sW(78;1,3;24,56) \\ \sQ_3 = \sU(1,24568,3,7) & \sQ_4 = \sW(7;1,3;2456,8) \\ \sQ_5 = \sU(13,2,4,5678) & \sQ_6 = \sW(1378;2,4;5,6) \\ \sQ_7 = \sW(78;13,24;5,6) & \sQ_8 = \sW(2478;1,3;5,6) \\ \sQ_9 = \sU(13,24,56,78) & \sQ_{10} = \sW(13;2,4;56,78) \\ \sQ_{11} = \sW(1356;2,4;7,8) & \sQ_{12} = \sU(1234,56,7,8) \\ \sQ_{13} = \sU(1234,5,6,78) & \sQ_{14} = \sW(1234;5,6;7,8) \\ \sQ_{15} = \sW(56;13,24;7,8) & \end{array} \end{align*} \null \par\xdef\tpd{\the\prevdepth} \end{minipage} \caption{Left: the tight span $\TS(\sw)$; right: the matroids corresponding to the nodes of $\TS(\sw)$} \label{fig:exampleG6} \end{figure} \subsection{Completion of the proofs} The following is a compilation of Propositions~\ref{prop:specialLimit}, \ref{prop:G1}, \ref{prop:G2}, \ref{prop:G3}, \ref{prop:G4}, \ref{prop:G5}, \ref{prop:G6}, and Corollary~\ref{cor:limit2Init}. \begin{theo} \label{thm:smoothLimits} Suppose $\sw \in \TGr_0(3,8)$. \begin{enumerate} \item The inverse limit $\Gr(\sw)$ is smooth of dimension 15. \item If $\sw$ is contained in the relative interior of any cone in the $\Sn{8}$--orbit of $\spCone$, then $\Gr(\sw)$ has 2 connected components. Otherwise, $\Gr(\sw)$ is irreducible. \item The closed immersion $\init_{\sw}\Gr_0(3,8)\hookrightarrow \Gr(\sw)$ is an isomorphism. \end{enumerate} \end{theo} \begin{proof}[Proofs of Theorems \ref{thm:schoenIntro} and \ref{thm:connectedInitDegIntro}] These follow from Theorem \ref{thm:smoothLimits}. \end{proof} \section{The Chow quotient \texorpdfstring{$\chow{\Gr(3,8)}{H}$}{Gr(3,8)/H}} \label{sec:Chow} Recall that $X(r,n)$ is the moduli space of projective configurations of $n$ hyperplanes in $\PP^{r-1}$ in linear general position. In this final section, we prove Theorem \ref{thm:chowQuotientIntro}, and therefore resolve \cite[Conjecture~1.6]{KeelTevelev2006} by Hacking, Keel, and Tevelev. There are two key inputs in the proof: that $X(3,8)$ is sch\"on and that the secondary fan structure on $\Trop X(3,8)$ is \textit{convexly disjoint}, in the sense of \cite[Definition~1.15]{HackingKeelTevelev2009}. If the initial degenerations of $X(3,8)$ were also all connected, then these two facts suffice to prove Theorem \ref{thm:chowQuotientIntro}, however, $X(3,8)$ has a $\Sn{8}$--orbit of reducible initial degenerations (this follows from Theorem \ref{thm:2ConnectedComponentsInDegIntro}, see \S\ref{sec:proofChow} below). In the final subsection, we provide a work-around to this slight difficulty. Sch\"onness of $X(3,8)$ follows from Theorem \ref{thm:schoenIntro} (see \S\ref{sec:proofChow} below). Convexly disjointness of the secondary fan of $\Trop X(3,8)$ follows from \cite[Proposition~7.9]{Schock}, but we provide an independent proof of this fact in the following subsection. \subsection{Convexity properties of the Dressian} Recall that the \textit{Dressian} $\Dr(r,n)$ is the set \begin{equation*} \Dr(r,n) = \{\sw \in (\wedge^{r}\R^n)/\Zone \, : \, \pQ(\sw) \textit{ is matroidal}\} \subset (\wedge^{r}\R^n)/\Zone \end{equation*} and is the support of a subfan, denoted by $\pS_{\matroidal}(r,n)$, of the secondary fan of~$\Delta(r,n)$. The fan $\pS_{\matroidal}(r,n)$ has an $(n-1)$--dimensional lineality space $L_{\R}$ consisting of those vectors $\sw$ induced by a linear function on the vertices of $\Delta(r,n)$. The Dressian may also be realized as the tropical prevariety defined by the 3-term Pl\"ucker relations, \ie $\sw \in \wedge^{r}\R^n \cong \R^{\binom{[n]}{r}}$ lies in $\Dr(r,n)$ if and only if, for every $\eta \in \binom{[n]}{r-2}$ and distinct $i,j,k,\ell \notin \eta$, the minimum of the numbers \begin{equation*} \sw_{ij\eta} + \sw_{k\ell\eta}, \hspace{20pt} \sw_{ik\eta} + \sw_{j\ell\eta}, \hspace{20pt} \sw_{i\ell\eta} + \sw_{jk\eta} \end{equation*} is achieved at least twice. We use the standard shorthand that, \eg $ij\eta = \{i,j\} \cup \eta$. This endows $\Dr(r,n)$ with another fan structure called the \textit{Pl\"ucker fan}. By \cite[Theorem~14]{OlartePanizzutSchroter}, this coincides with the secondary fan structure on~$\Dr(r,n)$. A convex polyhedral fan $\Sigma$ is \textit{convexly disjoint} if every convex subset $S\subset \Sigma$ is contained in a single closed cone of $\Sigma$. \begin{prop} \label{prop:DrConvexlyDisjoint} The fan $\pS_{\matroidal}(r,n)$ is convexly disjoint. In particular, the secondary fan structure on $\TGr_0(r,n)$ is convexly disjoint for $(r,n) = (2,n), (3,6), (3,7), (3,8)$. \end{prop} \noindent We are indebted to Benjamin Schr\"oter for supplying the main idea in the following proof. \begin{proof} Suppose $\sx$ and $\sz$ lie in distinct closed cones of $\pS_{\matroidal}(r,n)$. By the equivalence of the Pl\"ucker and secondary fan structures on $\Dr(r,n)$ discussed above, there is a $\eta \in \binom{[n]}{r-2}$ and distinct $i,j,k,\ell \notin \eta$, such that \begin{align*} \sx_{ij\eta} + \sx_{k\ell\eta} = \sx_{ik\eta} + \sx_{j\ell\eta} < \sx_{i\ell\eta} + \sx_{jk\eta} \hspace{10pt} \text{ and } \hspace{10pt} \sz_{ij\eta} + \sz_{k\ell\eta} > \sz_{ik\eta} + \sz_{j\ell\eta} = \sz_{i\ell\eta} + \sz_{jk\eta}. \end{align*} Given a real number $t$, define $\sy^{(t)} = (1-t) \cdot \sx + t \cdot \sz$. For any $0