Initial degenerations of Grassmannians

Abstract

We construct closed immersions from initial degenerations of ${{\,\mathrm{Gr}\,}}_{0}(d,n)$—the open cell in the Grassmannian ${{\,\mathrm{Gr}\,}}(d,n)$ given by the nonvanishing of all Plücker coordinates—to limits of thin Schubert cells associated to diagrams induced by the face poset of the corresponding tropical linear space. These are isomorphisms when (d, n) equals (2, n), (3, 6) and (3, 7). As an application we prove ${{\,\mathrm{Gr}\,}}_0(3,7)$ is schön, and the Chow quotient of ${{\,\mathrm{Gr}\,}}(3,7)$ by the maximal torus in ${\text {PGL}}(7)$ is the log canonical compactification of the moduli space of 7 points in ${\mathbb {P}}^2$ in linear general position, making progress on a conjecture of Hacking, Keel, and Tevelev.

Appendices

Some functorial properties of SDC-morphisms

Throughout this section, all $\mathbf{k }$-schemes are of finite-type over $\mathbf{k }$. Recall from the beginning of Sect. 4 that a SDC-morphism of $\mathbf{k }$-schemes is one that is smooth and dominant with connected fibers. In this section, we will catalog properties of SDC-morphisms used throughout the paper. First, we discuss how to deduce smoothness or connectedness of a $\mathbf{k }$-scheme X from properties of a morphism $X\rightarrow Y$ and Y.

Proposition A.1

Let X, Y be $\mathbf{k }$-schemes as above.

• 1. If $f:X\rightarrow Y$ is a dominant morphism with connected fibers and Y is irreducible, then X connected.

• 2. If $f:X\rightarrow Y$ is a SDC-morphism and Y is smooth and irreducible, then so is X.

Proof

Let V be the image of f. Since f is dominant and Y is irreducible, the scheme V is also irreducible, and therefore $f:X\rightarrow V$ is a surjective morphism with connected fibers. We conclude that X is connected by [38, Tag 0378]. Finally, (2) follows easily from (1). $\square$

Next, we explore how SDC-morphisms behave under base change. This proposition is crucial in the proof of Theorem 1.2 as it will allow us to deduce smoothness and irreducibility of initial degenerations by studying thin Schubert cells and the morphisms between them.

Proposition A.2

Suppose we have a pullback diagram

and that $W\times _Z X$ is nonempty. The following properties hold:

• 1. If f is smooth and W is a smooth $\mathbf{k }$-scheme, then $W\times _Z X$ is smooth.

• 2. If f is a SDC-morphism and W is irreducible, then $f'$ is also a SDC-morphism.

• 3. If f is a SDC-morphism and W is smooth and irreducible, then $W\times _Z X$ is smooth and irreducible.

Proof

To simplify notation, set $V:=W\times _Z Y$. If f and $W \rightarrow {{\,\mathrm{{Spec}}\,}}\mathbf{k }$ are smooth morphisms, then so is $V \rightarrow {{\,\mathrm{{Spec}}\,}}\mathbf{k }$ since smoothness is preserved under composition and base-change. This proves (1).

Now suppose f is a SDC-morphism and W is irreducible. So $f': V\rightarrow W$ is smooth, in particular flat. By [17, Exercise III.9.1] $f'$ is also open. This means that $f'(V)$ is a nonempty open subscheme of W, which is dense by the irreducibility of W. For $w\in f'(V)$, the fiber $V_{w}$ is nonempty and isomorphic to $X_{h(w)}$, which is connected, hence (2). Statement (3) follows from this and Proposition A.1(2). $\square$

Proposition A.3

SDC-morphisms satisfy the following.

• 1. A dominant open immersion $U \hookrightarrow X$ is a SDC-morphism.

• 2. If $f:X\rightarrow Y$ and $g:Y\rightarrow Z$ are SDC-morphisms, then $gf:X\rightarrow Z$ is a SDC-morphism.

Proof

Statement (1) is clear, so consider (2). It is well known that smoothness and dominance are preserved under composition, so we need only show that gf has connected fibers. Let $z\in Z$, and $X_z$ (resp. $Y_z$) be the scheme-theoretic fiber of gf (resp. g) over z. Let $f_z:X_z\rightarrow Y_z$ be the morphism obtained by pulling back f along the inclusion $Y_z \rightarrow Y$. Since $Y_z$ is smooth and connected, it is irreducible. By Proposition A.2(2), the map $f_z$ is a SDC-morphism. Therefore $X_z$ is connected by Proposition A.1(1), as required. $\square$

Many of the limits that appear in this paper come from graphs in the following way. Let ${\mathcal C}$ be a category that has finite limits, e.g, the category of $\mathbf{k }$-schemes ${\mathbf{k }\text{- }{{\,\mathrm{sch}\,}}}$, and G a connected graph, possibly with loops or multiple edges. We regard each edge $e\in E(G)$ as a pair of half-edges. Let us define a quiver Q(G). The set of vertices of Q(G) is $V(G) \cup E(G)$; we write $q_{v}$ ($v\in V(G)$), resp. $q_{e}$ ($e\in E(G)$), for the corresponding vertex of Q. For every half edge h of e incident to v, there is an arrow $q_h:q_v\rightarrow q_e$. In particular, if e is a loop edge, then there are two arrows from $q_v$ to $q_e$. Viewing Q(G) as a category in the usual way, a diagram of type Q(G) in a category ${\mathcal C}$ is a functor $X:Q(G) \rightarrow {\mathcal C}$. We write $X_{v} = X(q_{v})$, $X_e = X(q_e)$, $\varphi _h = X(q_h)$ and $X_G = \varprojlim _{Q(G)}X$. For example, Fig. 3 exhibits a graph and its corresponding diagram.

Fig. 3

A graph and its associated diagram

Example A.4

Let $\Gamma _{M,w}$ be the adjacency graph to a matroid subdivision $\Delta _{M,w}$. Let $M_v$, resp. $M_e$, denote the matroid corresponding to the vertex v, resp. edge e, of $\Gamma _{M,w}$, and $\varphi _{M_v,M_e}:{{\,\mathrm{Gr}\,}}_{M_{v}} \rightarrow {{\,\mathrm{Gr}\,}}_{M_e}$ whenever e is incident to v. The data of ${{\,\mathrm{Gr}\,}}_{M_v} , {{\,\mathrm{Gr}\,}}_{M_e}$, and $\varphi _{M_v,M_e}$ defines a diagram of type $Q(\Gamma _{M,w})$ in ${\mathbf{k }\text{- }{{\,\mathrm{sch}\,}}}$.

Now, let us consider how this construction behaves with respect to contracting a connected subgraph. Let F be a connected subgraph of G, and G/F the graph obtained by contracting F to a single vertex $v_{F}$. Let $X_{F} = \varprojlim _{Q(F)} X$ and let $\xi _v:X_F \rightarrow X_v$ and $\xi _{e}:X_F \rightarrow X_e$ be the structure morphisms. Set $Y_{v_{F}} = X_F$, and $Y_{v} = X_v$ for the remaining v in V(G/F). Similarly, let $Y_e = X_e$ for the edges $e\in E(G/F)$. If h is a half edge in G/F incident $v_F$, set $\psi _h = \varphi _h\xi _v$. Otherwise, let $\psi _h = \varphi _h$. The data $(Y_{v},Y_e,\psi _h)$ defines a diagram Y of type Q(G/F).

Proposition A.5

We have an isomorphism

$$\begin{aligned} \varprojlim _{G} X \cong \varprojlim _{G/F} Y. \end{aligned}$$

Proof

To simplify notation, set $Y_{G/F} = \varprojlim _{G/F} Y$. Let $\lambda _v: Y_{G/F} \rightarrow Y_v$ and $\lambda _e: Y_{G/F} \rightarrow Y_e$ denote the structure morphisms of this limit. We show that $Y_{G/F}$ satisfies the universal property for $\varprojlim _{G} X$. First, we must define morphisms $\alpha _v: Y_{G/F} \rightarrow X_v$ and $\alpha _e: Y_{G/F} \rightarrow X_{e}$ that commute with each $\varphi _h$. This is achieved by setting $\alpha _v = \xi _v\lambda _{v_F}$, (resp. $\alpha _e = \xi _e\lambda _{v_F}$) when $v\in V(F)$ (resp. $e\in E(F)$), and $\alpha _v = \lambda _v$ (resp. $\alpha _e = \lambda _e$) otherwise. One may verify that $\varphi _h \alpha _v = \alpha _e$.

Now suppose that we have a collection of morphisms $\theta _v: M\rightarrow X_v$ and $\theta _e:M\rightarrow X_e$ such that $\varphi _h\theta _v = \theta _e$ for every $q_h:q_v\rightarrow q_e$ in Q(G). We will show that there is a unique morphism $\theta : M \rightarrow Y_{G/F}$ such that

$$\begin{aligned} \theta \alpha _{v} = \theta _v \text { and } \theta \alpha _{e} = \theta _e \end{aligned}$$

(A.1)

for all $v\in V(G)$ and $e\in E(G)$, respectively. By the universal property of $X_F$, there is a unique morphism $\theta _{v_F}: M \rightarrow Y_{v_F}$ such that $\xi _v\theta _{v_F} = \theta _v$ and $\xi _e\theta _{v_F} = \theta _e$. If h is a half edge in G/F incident to $v_F$, then

$$\begin{aligned} \psi _h \theta _{v_F} = \varphi _h\xi _v\theta _{v_F} = \varphi _h\theta _v = \theta _e. \end{aligned}$$

Otherwise, we have $\psi _h \theta _v = \theta _e$ since $\psi _h = \varphi _h$. By the universal property of $Y_{G/F}$, there is a unique morphism $\theta :M\rightarrow Y_{G/F}$ satisfying $\lambda _v \theta = \theta _v$ and $\lambda _e \theta = \theta _e$.

Now we establish the equalities in Eq. (A.1). When $v\in V(F)$,

$$\begin{aligned} \alpha _v\theta = \xi _v \lambda _{v_F}\theta = \xi _v \theta _{v_F} = \theta _v. \end{aligned}$$

A similar argument shows that $\alpha _e\theta = \theta _e$ when $e\in E(F)$. The cases where $v\in V(G){\setminus } V(F)$ or $e\in E(G){\setminus } E(F)$ follow from the identifications $\alpha _v = \lambda _v$ and $\alpha _e = \lambda _e$. Finally, the uniqueness of $\theta$ follows from the uniqueness of $\theta _F$ and the universal property of $Y_{G/F}$. $\square$

Proposition A.6

Suppose G is a tree. Let X be diagram of type Q(G) in ${\mathbf{k }\text{- }{{\,\mathrm{sch}\,}}}$ such $X_v$ and $X_e$ are smooth and irreducible $\mathbf{k }$-schemes, and for each half edge h, the map $X(h):X_v\rightarrow X_e$ is a SDC-morphism. Then $X_G$ is smooth and irreducible. Moreover,

$$\begin{aligned} \mathrm{dim}X_G = \sum _{v\in V(G)} X_v - \sum _{e\in E(G)} X_e. \end{aligned}$$

Proof

We proceed by induction on the number of vertices. When G consists of a single vertex, there is nothing to show. Now suppose that the lemma is true for all trees with fewer vertices than G. Let w be a one valent vertex of G, let e the adjacent edge, and let $G'$ the graph consisting of the remaining vertices and edges. By Proposition A.5,

$$\begin{aligned} X_G \cong X_w \times _{X_{e}} X_{G'}. \end{aligned}$$

It is smooth and irreducible by Proposition A.2 and the inductive hypothesis. Because $X_w\rightarrow X_e$ is smooth of relative dimension $\mathrm{dim}X_{w} - \mathrm{dim}X_e$, so is $X_G \rightarrow X_G'$, and therefore

$$\begin{aligned} \mathrm{dim}X_G = \mathrm{dim}X_w - \mathrm{dim}X_e + \mathrm{dim}X_{G'} \end{aligned}$$

by [17, Corollary 9.6]. By the inductive hypothesis, we get the required formula for $\mathrm{dim}X_G$. $\square$

Remark A.7

An arbitrary finite limit over a diagram of smooth and irreducible $\mathbf{k }$-schemes in which every morphism is SDC-need not be irreducible. Let $h =x^2-y^2+x+\frac{1}{4}$ and $X = {{\,\mathrm{{Spec}}\,}}((h)^{-1} \mathbf{k }[x,y])$. Define two morphisms $f,g:X \rightarrow {{\,\mathrm{{Spec}}\,}}\mathbf{k }[z]$ by:

$$\begin{aligned} f^{\#}(z) = x^2-y^2 + x&g^{\#}(z) = x. \end{aligned}$$

One may verify that f and g are SDC-morphisms between smooth and irreducible $\mathbf{k }$-schemes. However, the equalizer of f and g is

$$\begin{aligned} {{\,\mathrm{{Spec}}\,}}((h)^{-1}\mathbf{k }[x,y]/ \langle x^2-y^2 \rangle ) \end{aligned}$$

which is neither smooth nor irreducible.

We end with a proposition on when a closed immersion of affine schemes is an isomorphism.

Proposition A.8

Suppose $\varphi :X\hookrightarrow Y$ is a closed immersion of affine schemes, and Y is integral. If $\mathrm{dim}X = \mathrm{dim}Y$, then $\varphi$ is an isomorphism.

Proof

Let n be the Krull dimension of X and Y. Because $\varphi$ is a closed immersion and Y is integral, the induced morphism on rings is of the form $\varphi ^{\#}:R \rightarrow R/I$ for some integral domain R and ideal $I\subset R$. A maximal chain of prime ideals ${\mathfrak {p}}_0 \subsetneq {\mathfrak {p}}_1\subsetneq \cdots \subsetneq {\mathfrak {p}}_n$ in R/I lifts to a maximal chain of prime ideals ${\mathfrak {q}}_0 \subsetneq {\mathfrak {q}}_1\subsetneq \cdots \subsetneq {\mathfrak {q}}_n$ in R with $I\subset \mathfrak {q_0}$. Because R is an integral domain, we have ${\mathfrak {q}}_0 = \langle 0\rangle$. So $I = \langle 0 \rangle$, and therefore $\varphi ^{\#}$ is the identity. $\square$

Data for Lemma 6.4

In Table 2, we list the ideals that appear as $J_{M,w}^x$ for subdivisions of $\Delta _{M,w}$ such that M is a simple, connected $\mathbf{k }$-realizable $(3,7)$-matroid and $\Gamma _{M,w}$ has no leaves, as in the proof of Lemma 6.4. We consider all of these as ideals in the ring

$$\begin{aligned} \mathbf{k }[x_{ij}^{\pm } \ | \ 0\le i \le 2,\ 0\le j\le 3]. \end{aligned}$$

We write $\mathbf{k }[x_{ij}^{\pm }]$ for short. Many of the polynomials that appear are of the form $X_{ij,k\ell } := x_{ik}x_{j\ell } - x_{i\ell }x_{jk}$. In the second column, we list variables that may be eliminated to produce an isomorphism of $\mathbf{k }[x_{ij}^{\pm }]/J_{M,w}^x$ with a Laurent polynomial ring. For example, consider the last row. In this case,

$$\begin{aligned} J_{M,w}^x = \langle X_{01,23},X_{02,03},X_{12,12},X_{12,02},X_{12,01} \rangle . \end{aligned}$$

We use the form $X_{01,23}$ to solve for $x_{02}$, the form $X_{02,03}$ to solve for $x_{03}$, the form $X_{12,12}$ to solve for $x_{11}$, and finally the form $X_{12,02}$ to solve for $x_{22}$. This produces an isomorphism $\mathbf{k }[x_{ij}^{\pm }]/J_{M,w}^x \rightarrow \mathbf{k }[x_{ij}^{\pm } \, | \, ij\ne 02,03,11,22]$.

Table 2 Here are the unique ideals nontrivial ideals that appear in the proof of Lemma 6.4

Maps between thin Schubert cells and inverse limits (written by María Angélica Cueto)

In this appendix, we discuss how to reduce the study of geometric properties of thin Schubert cells to the case of simple and connected matroids. Because the only simple rank 2 matroid is U(2, n), this analysis gives us a complete understanding of ${{\,\mathrm{Gr}\,}}_M$ in the rank 2 case, and simplifies the study of rank 3 matroids in Sects. 4 and 5. In the following subsection, we show that the limit of thin Schubert cell ${{\,\mathrm{Gr}\,}}_{M,w}$ induced by a matroid subdivision $\Delta _{M,w}$ depends only on the adjacency graph of $\Delta _{M,w}$. This allows one to apply the results from Appendix A to study ${{\,\mathrm{Gr}\,}}_{M,w}$ as in Sect. 6.

1.1 Reduction to simple and connected matroids

The following two Lemmas demonstrate that thin Schubert cells are compatible with decomposition into connected components and removal of loops and parallel elements. Lemma C.1 appears [23, Proposition 9.4] without proof, and Lemma C.2 will appear in an upcoming paper [5]. For the reader’s convenience, we sketch their proofs.

Lemma C.1

If $M = M_1\oplus M_2$, then ${{\,\mathrm{Gr}\,}}_{M} \cong {{\,\mathrm{Gr}\,}}_{M_1}\times {{\,\mathrm{Gr}\,}}_{M_2}$. In particular, we have ${{\,\mathrm{Gr}\,}}_{M} \cong {{\,\mathrm{Gr}\,}}_{M|T}$ where $T\subset [n]$ is the set of non-loop elements.

Proof

Suppose $X_1$ and $X_2$ are matrices giving rise to the rings $R_{M_1}^{x}$ and $R_{M_2}^{x}$ as in Construction 2.2. Let X be the block matrix with $X_1$ and $X_2$ on the diagonal. Then $R_M^x \cong R_{M_1}^{x} \otimes R_{M_2}^{x}$. The second statement follow from $M \cong M|T \oplus U(0,|T|)$. $\square$

Given a matroid M, we define a simple matroid by removing loops and parallel elements in the following way. Let $\eta _1,\ldots ,\eta _k$ be the rank 1 flats of M, choose nonloop elements $s_i\in \eta _i$ and set $S = \{s_1,\ldots , s_k\}$. Then M|S is a simple matroid. Let $\ell$ be the number of loops in M.

Lemma C.2

We have an isomorphism ${{\,\mathrm{Gr}\,}}_{M} \cong {{\,\mathrm{Gr}\,}}_{M|S} \times {{\mathbb {G}}}_m^{n-k-\ell }$.

Proof

By Lemma C.1, we may assume that M has no loops. Suppose i, j are parallel in M, then ${{\,\mathrm{Gr}\,}}_{M} \cong {{\,\mathrm{Gr}\,}}_{M_{[n]{\setminus } i}} \times {{\mathbb {G}}}_m$. Suppose i and j are parallel, and let $\mu _1,\mu _2\in {[n]\atopwithdelims ()d-1}$ such that $\mu _k\cup \{i\}$ and $\mu _k\cup \{j\}$ are bases of M for $k=1,2$. The quadratic generator from Eq. (3.1) yields

$$\begin{aligned} P_{M}(\mu _1\cup \{i,j\},\mu _2) = p_{\mu _1\cup i} p_{\mu _2\cup j} - p_{\mu _1\cup j} p_{\mu _2\cup i} \end{aligned}$$

This means that $p_{\mu \cup i}/p_{\mu \cup j}$ is independent of $\mu$. At the level of rings, the desired isomorphism $R_{M_{[n]{\setminus } i}} \otimes \mathbf{k }[t^{\pm }] \cong R_{M}$ is given by

$$\begin{aligned} R_{M_{[n]{\setminus } i}} \otimes \mathbf{k }[t^{\pm }]&\longrightarrow R_{M} \\ p_{\lambda }\otimes 1&\mapsto p_{\lambda } \;\;\;\;\;\;\;\; \;\;\;\;\;\;\; \text { if } i\notin \lambda , \\ 1 \otimes t&\mapsto p_{\mu \cup i}/p_{\mu \cup j} \;\;\;\; \text { if } \lambda = \mu \cup \{i\}. \end{aligned}$$

The Lemma now follows by induction on the number of parallel elements in M. $\square$

Because the uniform matroid U(2, n) is the only simple (2, [n])-matroid, and affine coordinates realize ${{\,\mathrm{Gr}\,}}_0(2,n)$ as a open subscheme of an algebraic torus, we have the following.

Proposition C.3

If M is a rank 2 matroid then

$$\begin{aligned} {{\,\mathrm{Gr}\,}}_M \cong {{\,\mathrm{Gr}\,}}_0(2,k) \times {{\mathbb {G}}}_m^{n-k-\ell }. \end{aligned}$$

(C.1)

where k is the number of rank 1 flats and $\ell$ the number of loops. In particular, the thin Schubert cell ${{\,\mathrm{Gr}\,}}_{M}$ is smooth and irreducible.

Next, we show the morphisms $\varphi _{M,M'}:{{\,\mathrm{Gr}\,}}_{M} \rightarrow {{\,\mathrm{Gr}\,}}_{M'}$ are compatible with the following operations: decomposition of matroids into connected components, removal of loops and parallel elements, and duality. This will allow us to restrict our attention to pairs $M'\le M$ where M is simple, connected, and $d = r_M([n]) \le \lfloor n/2\rfloor$.

Lemma C.4

If $M'\le M$ and $M=M_1 \oplus M_2$, then $M' = M'_1 \oplus M'_2$ with $M'_i\le M_i$ for $i=1,2$. Furthermore, we have $\varphi _{M,M'} = \varphi _{M_1,M'_1} \times \varphi _{M_2,M'_2}$.

Proof

Recall that $Q_M= Q_{M_1} \times Q_{M_2}$ if and only if $M=M_1\oplus M_2$. Thus, a face of $Q_M$ must be of the form $Q_{M_1'}\times Q_{M_2'}$ for $M_i'\le M_i$ for $i=1,2$ so $M'=M_1'\oplus M_2'$. The statement regarding $\varphi _{M,M'}$ follows by combining this decomposition with Proposition 3.2 and Lemma C.1. $\square$

Lemma C.5

If $M' \le M$, then we have $M'|S \le M|S$ and the restrictions fit into the commutative diagram:

(C.2)

Proof

The top horizontal map in (C.2) arises from the isomorphism ${{\,\mathrm{Gr}\,}}_{M}\simeq {{\,\mathrm{Gr}\,}}_{M|S} \times {{\mathbb {G}}}_m^{n-k-\ell }$ described in Lemma C.2. Since $M'\le M$, rank-one flats in M yield rank-one flats in $M'$ we have $M'|S \le M|S$. The same lemma yields ${{\,\mathrm{Gr}\,}}_{M} \cong {{\,\mathrm{Gr}\,}}_{M'|S} \times {{\mathbb {G}}}_m^{n-k-\ell }$. This determines the bottom horizontal map. $\square$

Now we ensure the compatibility of $M'\le M$ with the duality operation and the isomorphism

$$\begin{aligned} \psi :{{\,\mathrm{Gr}\,}}(d,n)\rightarrow {{\,\mathrm{Gr}\,}}(n-d,n) \quad (p_\beta )_{\beta }\mapsto ((-1)^{{\text {sign}}(\beta ,\beta ^c)} p_{\beta ^c})_{\beta ^c} \end{aligned}$$

induced from ${\mathbb {P}}(\wedge ^d\mathbf{k }^n) \cong {\mathbb {P}}(\wedge ^{n-d}\mathbf{k }^n)$. Here, the symbol $(\beta ,\beta ^c)$ is a permutation of $S_{n}$ in one-line notation and $\beta ^c = [n]\smallsetminus \beta$. On affine patches, the correspondence for matrices is explicit: for example, a $d\times n$ matrix $(I_d|X)$ in $\{p_{[d]}\ne 0\}$ is identified with the $(n-d)\times n$ matrix $(-X^{t}|I_{n-d})$ in $\{p_{[d]^c}\ne 0\}$.

Lemma C.6

If $M'\le M$ then $(M')^*\le M^*$ and $\varphi _{M^*,(M')^*} = \psi \circ \varphi _{M,M'} \circ \psi ^{-1}$.

Proof

By definition, we have $Q_{M^*} = {{\,\mathrm{conv}\,}}(\{\mathbf{1 }-e_\beta :\beta \in {\mathcal B}(M)\}) = \mathbf{1 } - Q_M$. In particular, if $Q_{M'}$ is a face of $Q_M$, then $Q_{(M')^*} = (\mathbf{1 } - Q_{M'}) \prec (\mathbf{1 } - Q_{M}) = Q_{M^*}$, as required. The isomorphism $\psi$ identifies each $p_\beta \in \mathbf{k }[{{\,\mathrm{Gr}\,}}(d,n)]$ with $p_{\beta ^c} \in \mathbf{k }[{{\,\mathrm{Gr}\,}}(n-d,n)]$. The expression $\varphi _{M^*,(M')^*} = \psi \circ \varphi _{M,M'} \circ \psi ^{-1}$ follows from this observation. $\square$

As an application, we prove that $\varphi _{M,M'}$ is a SDC-morphism whenever $M=U(d,n)$ or M is a rank 2 matroid.

Proposition C.7

For any $M'\le M:=U(d,n)$, the map $\varphi _{M,M'}:{{\,\mathrm{Gr}\,}}_0(d,n)\rightarrow {{\,\mathrm{Gr}\,}}_{M'}$ is a SDC-morphism.

Proof

By Proposition A.3, it suffices to show that $\varphi _{M,M'}$ is a SDC morphism when $M'\lessdot M$. The nondegenerate subsets of M are of the form $\{i\}$ or $[n]{\setminus } i$ for some $i \in [n]$. If $M' = M_{\{i\}}$, then $(M')^* = M^*_{[n]{\setminus } i}$ where $M^* \cong U(n-d,n)$. By Lemma C.6, it suffices to consider just $M'=M_{[n]{\setminus } \{i\}}$. In this case, we have $R_{M'}^x = (S_{M'}^x)^{-1}B_{M'}$ and $R_{M'}^x = (S_{M}^x)^{-1}R_{M'}^x[x_{i,n-4}^{\pm } \, | \, i\in [d]\,]$. Therefore ${{\,\mathrm{Gr}\,}}_{M} \subset {{\mathbb {G}}}_m^{d\times (n-d)}$ and ${{\,\mathrm{Gr}\,}}_{M} \subset {{\mathbb {G}}}_m^{d\times (n-d-1)}$ are open subvarieties, and $\varphi _{M,M'}$ is induced by a coordinate projection ${{\mathbb {G}}}_m^{d\times (n-d)} \rightarrow {{\mathbb {G}}}_m^{d\times (n-d-1)}$, which is clearly a SDC-morphism. The result now follows from Proposition A.3. $\square$

Proposition C.8

For (2, [n])-matroids $M'\le M$, the map $\varphi _{M,M'}$ is a SDC-morphism.

Proof

Because every simple rank 2 matroid is uniform, the Proposition follows from Lemmas C.4, C.5, and C.7. $\square$

Our final result in this subsection say that the reduction to simple matroids as above is compatible taking initial degenerations and inverse limits.

Proposition C.9

Fix $w\in {{\,\mathrm{TGr}\,}}_M$ and let ${\tilde{w}}$ be the projection of w to ${{\mathbb {R}}}^{{\mathcal B}(M|S)}/{{\mathbb {R}}}\!\cdot \!\mathbf{1 }$. Then ${\tilde{w}} \in {{\,\mathrm{TGr}\,}}_{M|S}$ and

$$\begin{aligned} {\text {in}}_{w} {{\,\mathrm{Gr}\,}}_{M} \simeq {\text {in}}_{{\tilde{w}}} {{\,\mathrm{Gr}\,}}_{M|S} \times {{\mathbb {G}}}_m^{n-k-\ell }&{{\,\mathrm{Gr}\,}}_{M,w} \cong {{\,\mathrm{Gr}\,}}_{M|S,{\tilde{w}}} \times {{\mathbb {G}}}_m^{n-k-\ell } \end{aligned}$$

Proof

The assertion on initial degenerations follows from the fact that the isomorphism ${{\,\mathrm{Gr}\,}}_M \cong {{\,\mathrm{Gr}\,}}_{M|S} \times {{\mathbb {G}}}_m^{n-k-\ell }$ from Lemma C.2 is induced by a monomial map on coordinate rings. The isomorphism of limits follows from this Lemma and the description of the coordinate ring of ${{\,\mathrm{Gr}\,}}_{M,w}$ in Proposition 3.7. $\square$

1.2 Limits of thin Schubert cells via adjacency graphs

Recall that the matroid subdivision $\Delta _{M,w}$ yields a system of maps $\varphi _{M_Q,M_{Q'}}: {{\,\mathrm{Gr}\,}}_{M_{Q}}\rightarrow {{\,\mathrm{Gr}\,}}_{M_{Q'}}$ whenever $Q'\le Q$ that satisfy $\varphi _{M_{Q},M_{Q''}} = \varphi _{M_{Q'},M_{Q''}}\varphi _{M_Q,M_{Q'}}$ and $\varphi _{M_Q,M_Q} = {{\,\mathrm{id}\,}}$. This allows us to form the limit

$$\begin{aligned} {{\,\mathrm{Gr}\,}}_{M,w}:= \mathop {\lim }\nolimits _{\begin{array}{c} \longleftarrow \\ M_Q \in \Delta _{M,w} \end{array}} {{\,\mathrm{Gr}\,}}_{M_Q} \end{aligned}$$

(C.3)

Rather than keeping track of the full face poset of $\Delta _{M,w}$ it is desirable to restrict ourselves to cells of codimension 0 and 1. The following construction mimics the definition of adjacency graphs for triangulations of polytopes [6, Definition 4.5.10], so we use the same name.

Definition C.10

Given w in ${\text {Dr}}_{M}$, let $\Gamma _{M,w}$ be the adjacency graph of $\Delta _{M,w}$ defined as follows. The graph $\Gamma _{M,w}$ has a vertex $v_{Q}$ for each Q in ${\text {TC}}_{M,w}$. Two vertices $v_{Q_1}, v_{Q_2}$ are connected by an edge if $Q_1\cap Q_2$ is a facet of both cells. Similarly, given a cell F of $\Delta _{M,w}$, we let $\Gamma _{M,w}^{F}$ be the full subgraph of $\Gamma _{M,w}$ generated by those vertices $v_{Q}$ of $\Gamma _{M,w}$ with $F\le Q$.

Our next lemma shows that the graphs defined above are connected. It will play a crucial role in Proposition C.12 below.

Lemma C.11

For any $w\in {\text {Dr}}_{M}$ and any cell F of $\Delta _{M,w}$, the graphs $\Gamma _{M,w}$ and $\Gamma _{M,w}^{F}$ are connected.

Proof

The first claim follows by convexity and is valid for the adjacency graph associated to a pure-dimensional polyhedral subdivision of any polytope. We argue for $Q_M$ and $\Delta _{M,w}$. Indeed, given two vertices $v_{Q_1}, v_{Q_2}$ of $\Gamma _{M,w}$, choose two points $x_1,x_2$, with $x_i\in \text {rel int}(Q_i)$ so that the segment $[x_1,x_2]$ does not meet any cell whose codimension is 2 or greater. Since $Q_M$ is convex, the $[x_1,x_2]$ lies in $\text {rel int}(Q_M)$. All but finitely many points in $[x_1,x_2]$ lie in the relative interior of top-dimensional cells. We label the encountered cells as we move from $x_1$ towards $x_2$ by $Q_1 =:\! Q'_0, Q'_1, \ldots , Q'_k \!:= Q_2$. The collection $\{Q'_i\}_{i}$ yields a path from $v_{Q_1}$ to $v_{Q_2}$ in $\Gamma _{M,w}$.

A similar argument can be used to prove the statement for $\Gamma _{M,w}^{F}$. Let $E\subset {{\mathbb {R}}}^n$ be the affine span of $Q_M$. Given F in $\Delta _{M,w}$, write $s:=\mathrm{dim}F$ and pick a point p in its relative interior. We let H be the orthogonal complement to the linear subspace $F-p$ in $E-p$, and Q a $(m-s)$-dimensional cube in H centered at the origin with diameter $0<\varepsilon \ll 1$.

We consider the full-dimensional polytope $P':=(Q+p) \cap Q_M$ in $H+p$, and the polyhedral subdivision on $P'$ induced by $\Delta _{M,w}$. Each cell in this subdivision equals $Q'\cap Q_M$ for some $Q'$, and has dimension $(s-\mathrm{dim}Q_M +\mathrm{dim}Q)$. By construction, a matroid polytope $Q'$ yields a vertex or edge of $\Gamma _{M,w}^{F}$ if and only if $Q'\in \Delta _{M,w}$ and $Q'\cap (Q+p)\ne \emptyset$. Thus, the graph $\Gamma _{M,w}$ agrees with the adjacency graph of the subdivision of $P'$. Since the latter is connected by the discussion above, the result follows. $\square$

The adjacency graph $\Gamma _{M,w}$ encodes a subsystem of the inverse system ${{\,\mathrm{Gr}\,}}_{M,w}$ from (C.3) as in Example A.4. Our final result shows that ${{\,\mathrm{Gr}\,}}_{M,w}$ agrees with the inverse system induced by $\Gamma _{M,w}$.

Proposition C.12

Let M be a $\mathbf{k }$-realizable (d, [n])-matroid and $w \in {{\,\mathrm{TGr}\,}}_{M}$. Then,

$$\begin{aligned} {{\,\mathrm{Gr}\,}}_{M,w} \cong \varprojlim _{\Gamma _{M,w}} {{\,\mathrm{Gr}\,}}_{M'}. \end{aligned}$$

(C.4)

Proof

We write ${{\,\mathrm{Gr}\,}}_{M,w}^{\Gamma }$ for the inverse limit on the right-hand side of Eq. (C.4). Given $M'$ labeling a cell of $\Gamma _{M,w}$, we write $h_{M'}^{\Gamma }:{{\,\mathrm{Gr}\,}}_{M,w}^{\Gamma } \rightarrow {{\,\mathrm{Gr}\,}}_{M'}$ for the associated morphism. Since $\Gamma _{M,w}$ determines a subsystem of $\Delta _{M,w}$, the universal property of ${{\,\mathrm{Gr}\,}}_{M,w}^{\Gamma }$ guarantees the existence of a morphism $\psi : {{\,\mathrm{Gr}\,}}_{M,w} \rightarrow {{\,\mathrm{Gr}\,}}_{M,w}^{\Gamma }$. Next, we build a morphism $\phi : {{\,\mathrm{Gr}\,}}_{M,w}^{\Gamma } \rightarrow {{\,\mathrm{Gr}\,}}_{M,w}$.

First, we construct morphisms $g_{F}:{{\,\mathrm{Gr}\,}}_{M,w}^{\Gamma } \rightarrow {{\,\mathrm{Gr}\,}}_{F}$ for each cell $Q_{F}$ of $\Delta _{M,w}$, satisfying $g_{M''} = \varphi _{M',M''} \circ g_{M'}$ for each pair of cells in $\Delta _{M,w}$ with $M'\le M''$. The morphism $\phi$ will be unique determined once we establish the compatibility of all $g_F$’s with the subdivision $\Delta _{M,w}$.

Let ${\mathscr {V}}_F$ be the set of vertices of the graph $\Gamma _{M,w}^{F}$. Set $g_{F}:=\varphi _{M',F} \circ h_{M'}^{\Gamma }$ where $v_{M'}\in {\mathscr {V}}_F$. We must show this morphism is independent of our choice of $M'$. Suppose $v_{M''}\in V_F$ as well. Since $\Gamma _{M,w}^{F}$ is connected by Lemma C.11, we can find a collection of vertices $v_{Q_{M'}}=:\!v_{Q_0}, v_{Q_1}, \ldots , v_{Q_{k}}\!:=v_{Q_{M''}}$ where $(v_{Q_i},v_{Q_{i+1}})$ is an edge of $\Gamma _{M,w}^{F}$ for each $i=0,\ldots , k-1$. We write $M_i:=Q_{M_i}$ and $M_{i(i+1)}:=M_{Q_i\cap Q_{i+1}}$; note that $F\le Q_i\cap Q_{i+1}$ for each i. The definition of inverse limit yields k diagrams

(C.5)

where all four triangles commute. It follows that $\varphi _{M',F} \circ h_{M'}^{\Gamma } = \varphi _{M'',F} \circ h_{M''}^{\Gamma }$, so $g_F$ is well-defined.

Finally, the identity $g_{F'} = \varphi _{F,F'}\circ g_{F}$ for each pair $F'\le F$ in $\Delta _{M,w}$ follows from a similar commutative diagram argument after choosing a vertex $M'$ in $\Gamma _{M,w}^{F}$. These two properties determine $\phi$. The universal property of both schemes ${{\,\mathrm{Gr}\,}}_{M,w}$ and ${{\,\mathrm{Gr}\,}}_{M,w}^{\Gamma }$ ensures that $\phi = \psi ^{-1}$, as desired. $\square$