High index Fano varieties, lattices and Batyrev–Borisov mirror symmetry

Let P be a lattice polytope in a lattice N, and let M be the dual lattice. As before, we denote the toric variety arising from the spanning fan P as $X_P$ . If P is the polytope of projective space, that is, if $X_{P}=\mathbb {P}^n$ , then $X_{P^{\vee }}$ is just $X_P/G$ , where $G=(\mathbb {Z}/(n+1)\mathbb {Z})^{n-1}$ . The group action can be seen to arise from the fact that $\mathbb {P}^n$ has Fano index $n+1$ : The vertices of $P^{\vee }$ span a sublattice $i:\overline {M} \to M$ such that $M /\overline {M} \cong G$ . The fact that $X_{P^{\vee }}=\mathbb {P}^{n}/G$ is a consequence of the fact that $i^{-1}(P^{\vee }) =P$ .

It is a simple corollary of the Batyrev–Borisov construction that if we take intermediate lattices we recover Greene–Plesser mirror symmetry for the Calabi–Yau hypersurface in projective space. Consider a lattice inclusion $\overline {M} \subset M' \subset M$ , and set $(M')^{\vee }=N'$ and $\overline {M}^{\vee }=\overline {N}$ . Then the anticanonical hypersurface in $\mathbb {P}^n/(N'/N)$ forms a mirror pair with the anticanonical hypersurface in $\mathbb {P}^n/(M'/\overline {M}).$

We show in §1 that there is an analogous picture for any Fano toric variety of index n. Let P be a reflexive Fano polytope of Fano index k. The vertices of $P^{\vee }$ span a sublattice $i:\overline {N} \to N$ . Set $G:=N/\overline {N}$ . In the projective space example above and in many other cases, $ G \cong (\mathbb {Z}/k\mathbb {Z})^{\dim X_P-1}$ . Let Q denote the polytope $i^{-1}(P)$ . Then mirror symmetry reverses the roles of P and Q:

where the arrows are Batyrev–Borisov mirror symmetry. We call Q the primitive dual of P. For polytopes whose vertices generate the ambient lattice, this is a duality. We also obtain a series of mirror pairs via considering intermediate lattices.

If $X_P$ is a sufficiently nice toric degeneration of the Grassmannian, then it is in particular a Fano variety of Fano index n. If $X_P$ is the Gelfand–Cetlin toric degeneration, then its vertices can be described via a ladder diagram ([Reference Batyrev, Ciocan-Fontanine, Kim and van Straten3]; see §2). The primitive dual Q can be described via the same diagram. We give a description of Q in this case in §2.

The series of examples of mirror pairs for projective space includes the quintic twin [Reference Doran and Morgan8], which intriguingly has the same Picard–Fuchs equations as the quintic. The perspective in this paper, allows us to generalize this ‘twin phenomenon’ to toric degenerations of the Grassmannian. See §2.4.

The Gelfand–Cetlin toric degeneration and variation of geometric invariant theory quotient

Let $P_{n,r}$ denote the reflexive Fano polytope associated to the Gelfand–Cetlin toric degeneration, and $Q_{n,r}$ its primitive dual. Let $G_{n,r}:=(\mathbb {Z}/n\mathbb {Z})^{r(n-r)-1}.$ The polytopes $P_{n,r}$ and $Q_{n,r}$ are closely related via mirror symmetry, but in fact there is another relation. The main result of §3 is Theorem 0.2: There is a weak Fano toric variety $Y_{n,r}$ that fits into the diagram

Note that, under mirror symmetry, the roles of the two maps in the diagram are reversed, suggesting the relationship is analogous to a geometric transition. The variation of GIT is a toric variation of GIT. While it is not a crepant resolution, the variation of geometric invariant theory quotient (VGIT) is a K-equivalence: It preserves the anticanonical class in the sense that the polyhedron associated to the anticanonical divisor is the same for each toric variety.

Smoothing

In the first subsection of §4, we prove Theorem 0.1 above, which we can now state more precisely. There is a natural embedding of $X_{P_{n,r}}$ into $\mathbb {P}^{\binom {n}{r}-1}$ , and it is in $\mathbb {P}^{\binom {n}{r}-1}$ that the Gelfand–Cetlin toric degeneration is seen. The embedding allows G to act not just on $X_{P_{n,r}}$ but on the whole projective space. The heart of the theorem is the remarkable characterization of $H_{n,r}$ as precisely the maximal subgroup of G for which the Grassmannian is an equivariant subvariety (see Theorem 4.9).

It is easy to construct a G-equivariant family $Z \subset \mathbb {P}^{\binom {n}{r}-1} \times {\mathbb {C}}^m$ , with coefficients $c_i$ on the second factor, such that the special fiber is $X_{P_{n,r}}$ and the general fiber is $\operatorname {{\mathrm {Gr}}}(n,r)$ . This is just the Gelfand–Cetlin toric degeneration, where we extend the action of G to the second factor in such a way that the varying Plücker relations are G-invariant. To obtain a smoothing of $X_{P_{n,r}}/G$ , we need to take the quotient of this family by G. Fibers of the quotient family are

$$\begin{align*}(Z \cap \{c_i^n=b_i\})/G \end{align*}$$

for constants $b_i$ .

We call this a smoothing, as the only singularities arise from the finite group quotient.

We also describe a partial smoothing of $Y_{n,r}$ . Using the characterization of $Y_{n,r}$ as the blow-up of the Gelfand–Cetlin degeneration, one can write down binomial equations cutting $Y_{n,r}$ out of an ambient space obtained by blowing-up $\mathbb {P}^{\binom {n}{r}-1}$ multiple times. This description allows us to reverse the Gelfand–Cetlin degeneration to (partially) smooth to the proper transform of the Grassmannian in the blow-up. We call the resulting variety $\operatorname {{\mathrm {Bl}}} \operatorname {{\mathrm {Gr}}}(n,r)$ .

Our mirror proposal

Consider the equations

(2) $$ \begin{align} \sum_{\lambda \in {\mathcal{A}(n,r)}} p_{\lambda}^n + \psi \prod_{p_{\mu} \text{ frozen}}p_{\mu}=0. \end{align} $$

The product in the second term is over frozen Plücker coordinates which are indexed by partitions $\mu $ which are maximal rectangular partitions. For details and definitions, see §5. This equation defines a family of Calabi–Yau hypersurfaces in $\operatorname {{\mathrm {Bl}}} \operatorname {{\mathrm {Gr}}}(n,r)/{H}$ . This is a candidate mirror to a Calabi–Yau hypersurface in $\operatorname {{\mathrm {Gr}}}(n,r)$ . In §5, we give evidence towards this conjecture by showing Theorem 0.3 above.

1.1 Background on toric varieties

1.1.1 Reflexive polytopes

Let P be a full-dimensional lattice polytope in a lattice M, containing the origin in its strict interior. Let $N=M^{\vee }$ be the dual lattice. The dual of P is

$$\begin{align*}P^{\vee}:=\{v \in N_{\mathbb{R}} \mid \langle v, w \rangle \geq -1, \forall w \in P\}. \end{align*}$$

Definition 1.1. A polytope P is reflexive if $P^{\vee }$ is also a lattice polytope.

Vertices of reflexive polytopes are primitive lattice vectors.

Definition 1.2. A lattice polytope $P \subset M_{\mathbb {R}}$ is vertex spanning if the set of vertices of P span the lattice M.

Reflexive polytopes may or may not be vertex spanning. For example, among the 4,319 three-dimensional reflexive polytopes, classified in [Reference Kreuzer and Skarke16], 4,075 are vertex spanning and 244 are not.

1.2 High-index Fano varieties

Throughout this section, we will fix a d-dimensional reflexive Fano polytope P giving rise (via the spanning fan) to a Fano toric variety $X_P$ . We assume that $X_P$ has Fano index n. Let N denote the ambient lattice of P and M the dual lattice.

Let $\overline {M}$ be the sublattice of M spanned by the vertices of $P_{-K_{X_P}}$ . Let Q denote the lattice polytope in $\overline {M}$ given by the preimage of $P_{-K_{X_P}}$ under the inclusion $\overline {M} \subset M$ .

The inclusion of lattices $\overline {M} \subset M$ induces an action of the finite group $M/\overline {M}$ on $X_Q$ , and

$$\begin{align*}X_{P^{\vee}}=X_Q/(M/\overline{M}). \end{align*}$$

We can extend this description of Batyrev–Borisov mirror symmetry to a series of mirror pairs, one for each intermediate lattice. The inclusion of lattices $\overline {M} \subset M$ defines an inclusion $N \subset \overline {M}^{\vee }$ . We denote $\overline {N}:=\overline {M}^{\vee }$ . Any intermediate lattice $\overline {M} \subset M_1 \subset M$ defines an intermediate dual lattice

$$\begin{align*}N \subset N_1:=M_1^{\vee} \subset \overline{N}. \end{align*}$$

Proposition 1.7. Fix a lattice $\overline {M} \subset M_1 \subset M$ , and let $N_1$ be the lattice dual to $M_1$ . Then

$$\begin{align*}X_P/(N_1/N) \text{ and } X_Q/(M_1/\overline{M}) \end{align*}$$

are a Batyrev–Borisov mirror pair.

In the following proposition, we show that primitive duality is a duality, under the assumption that P is vertex spanning.

We can be more explicit given the following assumptions on P and $X_P$ :

1. 1. There is a Cartier toric divisor $D_i$ for some ray i such that $n D_i$ is linearly equivalent to $-K_{X_P}$ .

2. 2. Both P and $P_{D_0}$ are vertex-spanning polytopes.

The two polyhedra $P_{n D_i}$ and $P_{-K_{X_P}}$ are translates of each other. Suppose that

$$\begin{align*}P_{n D_i} +h=P_{-K_{X_P}} \end{align*}$$

for some lattice element $h \in M$ .

Lemma 1.9. The lattice $\overline {M}$ is generated by h and the generators of $n M$ , and

$$\begin{align*}M/\overline{M} \cong (\mathbb{Z}/n\mathbb{Z})^{d-1}. \end{align*}$$

Proof. As noted, $P^{\vee }=P_{-K_{X_P}}$ . Fix a basis $e_1,\dots ,e_d$ of M. The vertices of $P_{n D_0}$ span the lattice given by the basis $n e_j$ . It follows the lattice spanned by the vertices of $P^{\vee }$ , that is the lattice $\overline {M}$ , is the lattice spanned by $n e_1,\dots ,n e_d,h$ . Note that, as $0$ is a vertex of $P_{D_i}$ , h is a vertex of the reflexive polytope $P^{\vee }$ and hence is a primitive lattice vector. It follows that

$$\begin{align*}M/\overline{M} \cong (\mathbb{Z}/n\mathbb{Z})^{d-1}. \end{align*}$$

2.1 Ladder diagrams

The polytope $P_{n,r}$ can be described via a ladder diagram. Fix integers $n>r>0$ , and set $k:=n-r$ . Draw an $r \times k$ grid of boxes. For example, the grid for $n=5$ , $r=2$ is

Draw vertices on the diagram as follows: at all internal crossing points and at the bottom-left and top-right corner. Continuing this example, we obtain

We consider paths in this diagram between vertices, where travel is always up and to the right. This describes a quiver: Vertices are given by the vertices in the ladder diagram, and there is an arrow from vertex a to b for every path in the ladder diagram which does not pass through any other vertex. In the example, the quiver is

A quiver moduli space is defined by assigning a vector space to each vertex of a quiver and choosing a stability condition [Reference King15]. If we assign a one-dimensional vector space to each vertex of the ladder quiver and choose a Fano stability condition, then we obtain a quiver moduli space that coincides with $X_{P_{n,r}}$ : See [Reference Kalashnikov14] for details.

The weight matrix of $X_{P_{n,r}}$ is given by the adjacency matrix of the ladder quiver. More precisely, let $\mathbb {Z}^{\overline {\operatorname {{\mathrm {LQ}}}}_0}$ be the lattice with basis $e_v, v \in \overline {\operatorname {{\mathrm {LQ}}}}_0$ . For the source vertex $0 \in \operatorname {{\mathrm {LQ}}}_0$ , set $e_0=0$ . Then the weights are given by $d_a:=-e_{s(a)}+e_{t(a)}$ .

It is often more convenient to write the weights in a different basis. We identify a basis element $f_v$ for each $v \in \overline {LQ}_0$ , as follows:

1. 1. If all arrows with $t(a)=v$ have $s(a)=0$ , then we set $f_v:=d_a=e_v$ .

2. 2. If there is a unique arrow such that both $t(a)=v$ and $s(a) \neq 0$ , then we set $f_v:=d_a$ .

3. 3. Otherwise, v is the top-right corner of a square of vertices. Label the other vertices of the square as shown:

   

   then set

   $$\begin{align*}f_v:=e_{v}-e_{v_1}-e_{v_2}+e_{v_0}. \end{align*}$$

   That is, $f_v$ is a sum $\sum d_a$ where a runs through the arrows drawn in the diagram:

   

In the basis $f_v$ , $v \in \overline {LQ}_0$ , the weight matrix

(3) $$ \begin{align} B:=[b_{v a}] \end{align} $$

has rows indexed by $v \in \overline {\operatorname {{\mathrm {LQ}}}}_0$ , columns indexed by $a \in \operatorname {{\mathrm {LQ}}}_1$ , and all entries are either $1$ or $0$ . There are precisely n positive entries in each row. It is easy to identify visually the set

$$\begin{align*}\{a \in \operatorname{{\mathrm{LQ}}}_1: b_{v a}=1\} \end{align*}$$

for each v. We describe it by way of example.

Example 2.3. Let $n=7$ and $r=3$ . For each nonsource vertex v, labeled in red, the arrows satisfying $b_{v a}=1$ are the n arrows fully contained in the red part of the ladder quiver.

As one can see from the example, for a nonsource vertex v, $b_{v a}=1$ if a is an arrow that either

• ○ has a horizontal step above and in the same column as the horizontal arrow with target v or

• ○ has a vertical step to the right and in the same row as the vertical arrow with target v.

In particular, every path from the bottom-left to top-right corner contains exactly one of the arrows in

$$\begin{align*}\{a \in \operatorname{{\mathrm{LQ}}}_1: b_{v a}=1\} \end{align*}$$

for each v.

A great deal of the geometry of the special fiber $X_{P_{n,r}}$ can be read off the ladder quiver. For example, collections of arrows give divisors. The toric variety $X_{P_{n,r}}$ has Picard rank 1, and the Picard lattice is generated by a divisor corresponding to a path from the bottom-left vertex to the top right; each such path is linearly equivalent to the ample generator for the Picard lattice. This generator is the ample line bundle that gives the embedding of $X_{P_{n,r}}$ into $\mathbb {P}^{\binom {n}{r}-1}$ , where it is the special fiber of the Gelfand–Cetlin degeneration. As such, we label it $\mathcal {O}(1)$ . The space of sections of $\mathcal {O}(1)$ has dimension $\binom {n}{r}$ and basis indexed by paths from the bottom-left vertex to the top right. Fixing such a path p, we can consider the area above p: This gives a partition $\lambda $ which fits into an $r \times k$ box. We label the corresponding section of $\mathcal {O}(1)$ by $s_{\lambda }$ . The Cox ring is a polynomial ring ${\mathbb {C}}[y_a: a \in \operatorname {{\mathrm {LQ}}}_1]$ , and $s_{\lambda } = \prod _{a \in p} y_a$ .

Partitions fitting into an $r \times k$ box also index Plücker coordinates. Given a partition $\lambda $ , there are n horizontal and vertical steps in the corresponding path p, and the location of the vertical steps gives a size-r subset of $\{1,\dots ,n\}$ .

Example 2.5. The ladder diagram for $Gr(7,3)$ is below. The path in blue describes the partition $(2,1)$ . This indexes the Plücker coordinate $p_{\{1,3,5\}}$ .

2.2 The polytope $P_{n,r}$ and its primitive dual

To describe the Gelfand–Cetlin toric degeneration via a polytope, we draw another quiver dual to the ladder quiver (this is the quiver that gives the head-over-tails mirror [Reference Eguchi, Hori and Xiong9]). There is a vertex for the dual quiver in each square of the ladder diagram, as well as two external vertices above and to the right of the ladder diagram. Arrows move down and to the left, so we obtain:

Each arrow in the ladder quiver is crossed by precisely one arrow in the dual quiver.

Notice that there are $k r$ interior vertices. Let M be an $k r$ -dimensional lattice, where we index the basis $e^{\prime }_v$ by interior vertices v in the dual quiver. Set $e^{\prime }_v=0$ for external vertices v.

Sometimes, instead of the basis $e^{\prime }_v$ it is convenient to use another basis, given by arrows. For each internal vertex v in the dual quiver, we assign an arrow $a_v$ . If there is a horizontal arrow out of vertex v, then this is $a_v$ ; otherwise, $a_v$ is the vertical arrow out of v. Then we set $b_v:=w_{a_v}$ . In the diagram below, we highlight the arrows $a_v$ in red for $n=7, r=3$ .

Let N be the lattice dual to M, with dual bases ${e'}_v^{\vee }$ and $b_v^{\vee }$ . We now construct the primitive dual of $P_{n,r}$ , which we call $Q_{n,r}$ . By definition, this is the pullback of the anticanonical polytope of $X_{P_{n,r}}$ to the lattice spanned by its vertices. We construct it in three stages:

1. 1. Construct the polytope for the divisor given by the path corresponding to the empty partition.

2. 2. Compare the n-th dilate of this polytope with anticanonical polytope of $X_{P_{n,r}}$ .

3. 3. Compute the sublattice generated by the vertices of the polytope.

Let $D_{\varnothing }$ denote the divisor corresponding to the empty partition. The path corresponding to the empty set consists of exactly one arrow, which we denote $a_{\varnothing } $ . Set

(4) $$ \begin{align} m_{\lambda}:=\sum_{\lambda \text{ crosses } a_v} b^{\vee}_v. \end{align} $$

Lemma 2.7. Let a be an arrow in the dual quiver that is not $a_{\varnothing } $ and $\lambda $ a path that does not correspond to $\varnothing $ . Then

$$\begin{align*}\langle m_{\lambda},w_a\rangle=1\end{align*}$$

if $\lambda $ crosses a and is $0$ otherwise.

Proof. To compute, we consider both vectors written in the $b_{v}$ or $b_v^{\vee }$ basis. The statement is obvious if $a=a_v$ for some v. Otherwise, we can assume that a is a vertical arrow with both source and target internal vertices. The vector $w_a$ is expanded in the $b_v$ by following the path starting at $s(a)$ , going all the way to the right, then down, and then going in reverse back to $t(a)$ . Then $w_a$ is the sum of these $b_v$ , where $b_v$ has coefficient $\pm 1$ , depending on whether it followed in the positive direction or in reverse. Note that the number of arrows followed in the positive direction is one more than the number of arrows in the reverse direction.

Suppose $\lambda $ crosses a. After crossing a, it must cross exactly one of the arrows going in the positive direction, so $\langle m_{\lambda },w_a\rangle =1$ . If $\lambda $ does not cross a, it either avoids the arrows used in the expansion of $w_a$ entirely, or it crosses one arrow in the reverse direction and one arrow in the positive, so

$$\begin{align*}\langle m_{\lambda},w_a\rangle=1-1=0. \end{align*}$$

For an illustration, consider the following example:

Corollary 2.8. The polytope $P_{D_{\varnothing }}$ corresponding to the divisor $D_{\varnothing } $ is the convex hull of the points

$$\begin{align*}\left\{m_{\lambda}:=\sum_{\lambda \text{ crosses } a_v} b^{\vee}_v \Bigg| \lambda \in {\mathcal{P}(n,r)} \right\}. \end{align*}$$

To compute $P_{n,r}^{\vee }$ , one must dilate the polyhedron $P_{D_{\varnothing } }$ by a factor of n and then perform a linear shift. That is,

$$\begin{align*}P_{n,r}^{\vee}=n P_{D_{\varnothing} }-h, \end{align*}$$

where $h=\sum _{v} b_v$ . This can be seen by comparing the homogenization procedure of $P_{n D_{\varnothing } }$ and $P_{-\sum D_a}=P_{n,r}^{\vee }$ . It follows that the vertices of $P_{n,r}^{\vee }$ are

$$\begin{align*}n \cdot m_{\lambda}-h. \end{align*}$$

Finally, we note that the lattice $\overline {M}$ generated by $h, \{n b_v: v \text { an interior vertex}\}$ is the lattice generated by the vertices of $P_{n,r}^{\vee }$ . The preimage of $P_{n,r}^{\vee }$ under the inclusion of lattice $\overline {M} \subset M$ is by definition the primitive dual $Q_{n,r}$ . This is a Fano reflexive polytope with $\binom {n}{r}$ vertices, labeled by partitions $\lambda \in {\mathcal {P}(n,r)}.$ We denote the vertices of $Q_{n,r}$ as $v_{\lambda }$ , $\lambda \in {\mathcal {P}(n,r)}$ .

2.3 Weights of $X_{Q_{n,r}}$

As $Q_{n,r}$ is a Fano reflexive polytope, we can consider the Fano toric variety $X_{Q_n,r}$ . We will describe a spanning set of the kernel of the ray sequence (a basis for the kernel would give the rows of a weight matrix). There is a relation for each m-covering of the ladder quiver $\operatorname {{\mathrm {LQ}}}$ .

The $1$ -coverings are given by crossing diagrams.

For example, the following is a crossing diagram for $\operatorname {{\mathrm {LQ}}}_{5,2}$ .

A crossing diagram defines a set of n partitions in ${\mathcal {P}(n,r)}$ . Each partition corresponds to a path in the ladder quiver. A path $\lambda $ is a path for a given crossing diagram if

• ○ For every vertex labeled X, if the path $\lambda $ contains this vertex, it goes straight through this vertex.

• ○ For every vertex labeled O, if the path $\lambda $ contains this vertex, it makes a $90^{\circ }$ turn at this vertex.

If we continue the same example as above, the paths are

. Notice that the empty partition and the maximal partition $(k^r)$ are paths for every crossing diagram. Together, the paths of a fixed crossing diagram cover every arrow in the ladder quiver precisely once, so they are a $1$ -covering.

Proof. To show the first part of the proposition, note that, though the $v_{\lambda }$ are vectors in the sublattice $\overline {M}$ spanned by the vertices of $Q_{n,r}$ , it is equivalent to show that

$$\begin{align*}\sum_{i=1}^{mn} i(v_{\lambda_i})=0, \qquad \text{that is,} \qquad \sum_{i=1}^{mn} (n m_{\lambda_i}-h) = 0. \end{align*}$$

Here, i is the inclusion $i:\overline {M} \to M$ .

Recall that $m_{\lambda }:=\sum _{\lambda \text { crosses } a_v} b^{\vee }_v$ . Since the collection of paths $\{\lambda _i\}$ together cover each arrow in $\operatorname {{\mathrm {LQ}}}$ exactly m times, it follows that

$$\begin{align*}\sum_{i=1}^{mn} m_{\lambda_i}=\sum_{v} b_v^{\vee}= m h, \end{align*}$$

which is equivalent to the desired equality.

For the second part of the proposition, suppose there is some collection $\{\mu _1,\dots ,\mu _k\}$ and coefficients $a_i$ satisfying

$$\begin{align*}\sum_{i=1}^k a_i v_{\mu_i}=0. \end{align*}$$

Note that we can assume, by adding multiples of the relations obtained from the crossing diagrams if necessary, that all the $a_i$ are nonnegative. In fact, if we allow partitions to appear multiple times, we can assume that the $a_i=1$ .

As above, we convert this to an equality in N, where we see that

$$\begin{align*}\sum_{i=1}^k (n m_{\mu_i}-h)=0. \end{align*}$$

It follows that k is a multiple of n and

$$\begin{align*}\sum_{i=1}^k m_{\mu_i}=\frac{k}{n} h. \end{align*}$$

If we reinterpret this as a statement about ladder diagrams, it says that, for each basis arrow $a_v$ in the dual quiver, precisely $k/n$ of the paths $\mu _i$ cross $a_v$ .

In fact, this is enough to show that precisely $k/n$ of the paths $\mu _i$ cross any arrow in the dual quiver. To see this, first consider a nonbasis arrow a, $a \neq a_{\varnothing } $ . It is a vertical arrow with internal source and target (in the dual quiver). There is a rectangle of arrows in the dual diagram, with vertical arrow a on the left side, and all other arrows (on the boundary) basis arrows. For example, this is such a rectangle (where a is the green arrow and the red arrows are—as before—the $a_v$ ):

In general, there will be zero or more black vertical arrows in between the green vertical arrow and the red vertical arrow. Each of the red arrows is crossed $k/n$ times, and suppose a is crossed s times. If the width of the rectangle is t arrows, this means there are $s+t k/n$ paths $\mu _i$ entering the rectangle, and $(t+1)k/n$ paths exiting the rectangle. So we can conclude that $s=k/n$ . Therefore, the collection of arrows $\mu _i$ cover the ladder diagram, omitting the arrow dual to $a_{\varnothing } $ , exactly $k/n$ times.

Finally, now suppose that $a_{\varnothing } $ is crossed r times by the paths— that is, r of the $\mu _i$ are the path corresponding to the empty set. The remaining arrows do not cross $a_{\varnothing } $ , but they cover the rest of the diagram $k/n$ times so that means that there are $(n-1)k/n$ of them. Since $r+(n-1)k/n=k$ , we conclude that $r=k/n$ as required. So the set $\{\mu _i\}$ is a $k/n$ -covering.

This proposition characterizes a (redundant form of) the weight matrix.

2.4 Grassmannian twins

One of the more interesting examples of Batyrev–Borisov mirror pairs obtained from lattice inclusions is the quintic twin [Reference Doran and Morgan8]. This is a hypersurface in a quotient of projective space whose Picard–Fuchs equations are identical to that of the quintic. The quotient is obtained by taking the action on $\mathbb {P}^4$ of the group $\mu _5$ of fifth roots of unity given by

$$\begin{align*}\zeta \cdot [z_0:\cdots: z_4]=[z_0: \zeta z_1: \zeta^2 z_2: \zeta^3 z_3: \zeta^4 z_4]. \end{align*}$$

Let $\tilde {P}$ denote the polytope of the quotient, a lattice polytope in $\tilde {M}$ , and P the polytope of projective space, a lattice polytope in M. Then the group action corresponds to a lattice inclusion $i:M \subset \tilde {M}$ . The fact that the Picard–Fuchs equations are identical follows from the fact that $i^{-1}(\tilde {P})$ is isomorphic to P.

The perspective on mirror symmetry for high-index Fano varieties described in this paper also suggests that the quintic twin can be generalized to Grassmannians, as in the following example.

4.1 Smoothing $X_{P_{n,r}}/G$

Although the group action of G on $X_{P_{n,r}}$ looks very toric and unconnected to the Grassmannian, remarkably, when smoothing, we obtain a very natural group action on $\operatorname {{\mathrm {Gr}}}(n,r)$ , which generalizes the Greene–Plesser group action for projective space. Recall that

(5) $$ \begin{align} \tilde{H}_{n,r}=\langle (\zeta_1,\dots,\zeta_{n}) \mid \zeta_i^n=1, \, (\prod \zeta_i)^r=1 \rangle \end{align} $$

acts naturally on $\operatorname {{\mathrm {Mat}}}(r \times n)$ and descends to an effective action of $H_{n,r}$ on $\operatorname {{\mathrm {Gr}}}(n,r)$ . The group $H_{n,r}$ is the quotient

$$\begin{align*}\tilde{H}_{n,r}/((\zeta,\dots,\zeta) \in \tilde{H}_{n,r}).\end{align*}$$

4.1.1 The action of G on $X_{P_{n,r}}$

It follows from Lemma 1.9 that $G \cong (\mathbb {Z}/n\mathbb {Z})^{rk-1}$ , where, as before, $k=n-r$ . The group action of G on $X_{P_{n,r}}$ can be described as a quotient of a larger group action on ${\mathbb {C}}^{|\operatorname {{\mathrm {LQ}}}_1|}$ , recalling that $X_{P_{n,r}}={\mathbb {C}}^{|LQ_1|}/\!/ T$ , $T=({\mathbb {C}}^*)^{|\overline {\operatorname {{\mathrm {LQ}}}}_0|}$ . The group

$$\begin{align*}\tilde{G}:=\{ (\psi_a)_{a \in \operatorname{{\mathrm{LQ}}}_1} \mid \psi_a^n=1, \, \prod_{a} \psi_a =1 \}\end{align*}$$

acts naturally via $\operatorname {{\mathrm {SL}}}(|\operatorname {{\mathrm {LQ}}}_1|)$ on ${\mathbb {C}}^{|\operatorname {{\mathrm {LQ}}}_1|}$ . The action descends to give an effective action of the quotient

$$\begin{align*}G=\tilde{G}/ (T \cap \tilde{G})\end{align*}$$

on the toric variety $X_{P_{n,r}}$ . To compute the intersection, we view both T and $\tilde {G}$ as diagonal subgroups of $\operatorname {{\mathrm {GL}}}(|\operatorname {{\mathrm {LQ}}}_1|)$ . Recall from equation (3) that the weight matrix of the toric variety is $B:=[b_{va}]_{v \in \overline {\operatorname {{\mathrm {LQ}}}}_0, a \in \operatorname {{\mathrm {LQ}}}_1}$ . The intersection $T \cap \tilde {G}$ is generated by

(6) $$ \begin{align} \{(\psi^{b_{va}})_a \mid \psi^n=1, v \in \overline{\operatorname{{\mathrm{LQ}}}}_0\} \subset \tilde{G}. \end{align} $$

Remark 4.1. One way to see that this group action agrees with the action described in Lemma 1.9 is to convert the fan description of $X_{Q_{n,r}^{\vee }}$ to a GIT description. Recall that the vertices of $X_{Q_{n,r}^{\vee }}$ are indexed by arrows of the ladder diagram and that $Q_{n,r}^{\vee }$ can be obtained from the polytope of a divisor of $X_{Q_{n,r}}$ , namely the divisor associated to the empty partition. Let $m_a$ , $a \in \operatorname {{\mathrm {LQ}}}_1$ , denote the vertices of the polytope of this divisor. Then the vertices of $Q_{n,r}^{\vee }$ are given by $w_a:=n m_a-h$ , where h is a (fixed) primitive lattice vector and $a \in \operatorname {{\mathrm {LQ}}}_1$ .

The rays of the fan of $X_{Q_{n,r}^{\vee }}$ do not generate the ambient lattice. To convert to the GIT description, we must add virtual rays—rays that appear in no cones of the fan—so that all the rays together generate the lattice. It suffices to add all of the $m_a$ as virtual rays. Then in addition to the weights of $X_{P_{n,r}}$ , we also obtain gradings coming from the relations

$$\begin{align*}w_a+h-n m_a=0.\end{align*}$$

Choosing an appropriate stability condition allows us to translate these gradings into quotient gradings, which exactly give the action of $\tilde {G}$ on ${\mathbb {C}}^{|\operatorname {{\mathrm {LQ}}}_1|}$ .

4.1.2 The action of G on $\mathbb {P}^{\binom {n}{r}-1}$

The embedding of $X_{P_{n,r}}$ into $\mathbb {P}^{\binom {n}{r}-1}$ allows us to extend the action of G to $\mathbb {P}^{\binom {n}{r}-1}$ . Label the coordinates of $\mathbb {P}^{\binom {n}{r}-1}$ by $p_{\lambda }$ , $\lambda \in {\mathcal {P}(n,r)}$ . These are of course the Plücker coordinates—recall that they can be indexed either by partitions in ${\mathcal {P}(n,r)}$ or by size-r subsets of $\{1,\dots ,n\}$ (Definition 2.4). For a partition $\lambda \in {\mathcal {P}(n,r)}$ , $I_{\lambda }$ denotes the corresponding subset.

The embedding is given by the $\binom {n}{r}$ sections of $\mathcal {O}(1)$ on $X_{P_{n,r}}$ , that is,

$$\begin{align*}\prod_{a \in \lambda} y_a, \: \lambda \in {\mathcal{P}(n,r)}. \end{align*}$$

The group $\tilde {G}$ acts on the coordinate $p_{\lambda }$ by multiplication by $\prod _{a \in \lambda } \psi _a.$ Either by observing that it is true by construction of the embedding or by explicitly checking, it is easy to see that the action of $\tilde {G}$ descends to an action of G on $\mathbb {P}^{\binom {n}{r}-1}$ . That is, elements of intersection $T \cap \tilde {G}$ act by scaling.

4.1.3 Degenerate Plücker relations

The toric variety $X_{P_{n,r}}$ is cut out of $\mathbb {P}^{\binom {n}{r}-1}$ by degenerate Plücker relations [Reference Batyrev, Ciocan-Fontanine, Kim and van Straten3].

Definition 4.2. Let $\prec $ be the usual partial order on partitions/Plücker coordinates: for $\lambda $ , $\sigma \in {\mathcal {P}(n,r)}$ , $\lambda \prec \sigma $ if $\lambda _i \leq \sigma _i$ for all i. Here, we use zeroes to extend any partition in ${\mathcal {P}(n,r)}$ to a length-r partition.

Following [Reference Batyrev, Ciocan-Fontanine, Kim and van Straten3], there is a relation for every incomparable pair of partitions $\lambda $ , $\sigma $ :

$$\begin{align*}p_{\sigma} p_{\lambda} -p_{\sigma \wedge \lambda} p_{\sigma \vee \lambda}.\end{align*}$$

Here, $\sigma \wedge \lambda $ is the minimal partition $\mu $ satisfying $\lambda , \sigma \prec \mu $ and $\sigma \vee \lambda $ is the maximal partition $\nu $ satisfying $\nu \prec \lambda ,\sigma .$

Example 4.3. We give an example for $Gr(7,3)$ . The partitions $\sigma =(1,1,1)$ and $\lambda =(4,2)$ are incomparable. We draw them in blue and green, respectively, below.

The partition $\sigma \vee \lambda $ is drawn in red and $\sigma \wedge \lambda $ in yellow.

Lemma 4.4. The degenerate Plücker relations are homogeneous with respect to the induced action of G on $\mathbb {P}^{\binom {n}{r}-1}$ .

Proof. Let $\sigma $ , $\lambda $ be an incomparable pair. The set of arrows contained in the union of $\sigma $ and $\lambda $ is the same as the set of arrows contained in the union of $\sigma \vee \lambda $ and $\sigma \wedge \lambda $ . Therefore, G acts with the same weights on $p_{\sigma } p_{\lambda }$ and $p_{\sigma \wedge \lambda } p_{\sigma \vee \lambda }.$

4.1.4 The group preserving the Plücker relations

Consider the Plücker relation for $\operatorname {{\mathrm {Gr}}}(4,2)$ :

which degenerates to

The second equation is homogeneous under the action of G, but the first is not. However, it is homogeneous with respect to a subgroup of G.

Definition 4.5. Let $G_h$ be the subgroup of G that preserves the Grassmannian, that is, $G_h$ is the largest subgroup of G such that all Plücker relations are homogeneous with respect to this action.

Before describing $G_h$ precisely, let’s describe it heuristically. Binomial Plücker relations are of the form:

$$\begin{align*}p_{\sigma} p_{\lambda}+\sum a_{\alpha,\beta} p_{\alpha} p_{\beta} =0. \end{align*}$$

For a complete description of these relations, see [Reference Miller and Sturmfels19]. A term $p_{\alpha } p_{\beta }$ appears with nonzero coefficient only if the multisets $I_{\sigma } \cup I_{\lambda }$ and $I_{\alpha } \cup I_{\beta }$ agree. In other words, the terms that appear in the sum must just be rearrangements of the indices of $I_{\sigma }$ and $I_{\lambda }$ . The Plücker relations are not homogeneous with respect to the action of G because this action feels the paths of the partitions, which is more precise than the location of the vertical steps (which gives the underlying subset). We are looking for a subgroup $G_h$ whose action depends only on the indices. We will show that $G_h$ is isomorphic to $H_{n,r}$ . The first step is to define a map from $H_{n,r}$ to G.

Consider the following labeling of the vertical steps of a ladder diagram:

The labeling allows assigning a subset of $\{1,\dots ,n\}$ to an arrow or path in the ladder diagram by recording the labels along the arrow or path. For example, the path corresponding to the partition

in the $\operatorname {{\mathrm {Gr}}}(2,5)$ diagram has labeling $\{1,3\}$ . Some arrows are associated to the empty set—in this example, the internal horizontal arrow has no labels.

The following remark is key to establishing the next two lemmas.

Remark 4.6. The labeling of a path $\lambda $ is $I_{\lambda }$ .

Use the labeling to define a morphism

$$\begin{align*}\tilde{\Psi}: \tilde{H}_{n,r} \to \tilde{G}, \:\: (\zeta_i)_{i=1}^n \mapsto \left(\prod_{\substack{j \text{ a label} \\ \text{of } a}} \zeta_j\right)_{a \in \operatorname{{\mathrm{LQ}}}_1}. \end{align*}$$

Lemma 4.7. The map $\tilde {\Psi }$ descends to an injective map $\psi : H_{n,r} \to G.$

Proof. Recall that G acts on $\mathbb {P}^{\binom {n}{r}-1}$ . We know that $T \cap \tilde {G}$ acts on $\mathbb {P}^{\binom {n}{r}-1}$ by scaling, so it suffices to show that, for $h \in \tilde {H}_{n,r}$ ,

$$\begin{align*}h=(\zeta,\dots,\zeta) \Leftrightarrow \tilde{\Psi}(h) \text{ acts by scaling on } \mathbb{P}^{\binom{n}{r}-1}. \end{align*}$$

Note that

$$\begin{align*}\tilde{\Psi}((\zeta_i)_{i=1}^n)\cdot \left[p_{\lambda}: \lambda \in {\mathcal{P}(n,r)}\right]=\left[\left(\prod_{j \in I_{\lambda}} \zeta_j\right) p_{\lambda}: \lambda \in {\mathcal{P}(n,r)}\right]. \end{align*}$$

Since $I_{\lambda }$ runs over all size r subsets of $\{1,\dots ,n\}$ , the only way to ensure that

$$\begin{align*}\prod_{j \in I_{\lambda}} \zeta_j =\prod_{j \in I_{\mu}} \zeta_j \end{align*}$$

for all $\lambda $ , $\mu $ is for $\zeta _i=\zeta _j$ for all i, j.

Lemma 4.8. The image of $H_{n,r}$ under $\Psi $ preserves the Grassmannian, that is,

$$\begin{align*}\Psi(H_{n,r}) \subseteq G_h. \end{align*}$$

Proof. Consider a binomial Plücker relation:

$$\begin{align*}p_{\sigma} p_{\lambda}+\sum a_{\alpha,\beta} p_{\alpha} p_{\beta} =0, \end{align*}$$

where the term $p_{\alpha } p_{\beta }$ appears with nonzero coefficient only if the multisets $I_{\sigma } \cup I_{\lambda }$ and $I_{\alpha } \cup I_{\beta }$ are equal. A group element $\tilde {\Psi }((\zeta _i))$ acts on $p_{\alpha } p_{\beta }$ by multiplication by

$$\begin{align*}\prod_{j \in I_{\alpha}} \zeta_j \prod_{j \in I_{\beta}} \zeta_j \end{align*}$$

which is equal to

$$\begin{align*}\prod_{j \in I_{\lambda}} \zeta_j \prod_{j \in I_{\sigma}} \zeta_j. \end{align*}$$

Therefore, the relation is homogeneous with respect to the $\tilde {\Psi }(\tilde {G})$ action, and hence the $\Psi (G)$ action.

The main combinatorial result of this section is the next theorem.

Theorem 4.9. There is an isomorphism $G_h \cong H_{n,r}$ .

Proof. The map $\Psi $ defines an isomorphism $H_{n,r} \to \Psi (H_{n,r})$ . We will show that $\Psi (H_{n,r})=G_h$ ; by Lemma 4.8, $\Psi (H_{n,r}) \subseteq G_h$ . We will show the other direction by demonstrating that, for every $[g] \in G_h$ with lift $g \in \tilde {G}$ , the $\tilde {G} \cap T$ -orbit of g has nonempty intersection with $\tilde {\Psi }(\tilde {H}_{n,r})$ . We will use two collections of arrows in the proof. The first collection $C_1$ is described via the following example: It is the set of $n=9$ arrows that are fully contained in the blue part of the diagram.

In general, there are n arrows in $C_1$ . We label them, starting at the bottom-left corner, by $a^1_1,\dots ,a^1_n.$ This collection is chosen so that if $(\psi _a) \in \tilde {\Psi }(\tilde {H}_{n,r})$ , then the $\psi _a$ , $a \not \in C_1$ are determined by the $\psi _a$ , $a \in C_1$ . This is because the $\psi _a$ , $a \in C_1$ , are

$$\begin{align*}\zeta_1,\zeta_1 \zeta_2, \dots, \zeta_1 \cdots \zeta_r, \zeta_{r+1}, \zeta_{r+2}, \dots, \zeta_{n-1}, \zeta_n \end{align*}$$

so that the exponent matrix of the map $\Psi $ composed with the projection $\pi :(\psi _a)_{a \in \operatorname {{\mathrm {LQ}}}_1} \mapsto (\psi _a)_{a \in C_1}$ is an invertible matrix in $\operatorname {{\mathrm {GL}}}(n,\mathbb {Z})$ . This means that, for $(\psi _a)_{a} \in \tilde {\Psi }(\tilde {H}_n,r)$ , we can express each $\psi _a$ , $a \in \operatorname {{\mathrm {LQ}}}_1$ , as a function of the $\psi _a$ , $a \in C_1$ . Denote this function by $\xi _a$ , which we view as a function on $\tilde {G}$ that depends only on the $\psi _a$ for $a \in C_1$ . Then the maps $\xi _a$ provide a way of measuring how far away an element of $\tilde {G}$ is from being in the image of $\tilde {\Psi }(\tilde {H}_{n,r})$ :

$$\begin{align*}g=(\psi_a)_{a} \in \tilde{\Psi}(\tilde{H}_{n,r}) \Leftrightarrow \xi_a(g)=\psi_a \text{ for all } a \in \operatorname{{\mathrm{LQ}}}_1. \end{align*}$$

The second collection of arrows $C_2$ is drawn in green in the example below. It is disjoint from $C_1$ .

Taking the target map from $C_2 \to \operatorname {{\mathrm {LQ}}}_0$ gives an injection.

Claim 4.10. The $T \cap \tilde {G}$ orbit of any $g \in \tilde {G}$ contains another group element $h=(\psi _a)_{a \in \operatorname {{\mathrm {LQ}}}_1}$ satisfying, for $a \in C_2$ :

(7) $$ \begin{align} \psi_a=\xi_a(h). \end{align} $$

In other words, along $a \in C_2$ , the relations we need for h to be an element of $\tilde {\Psi }(\tilde {H}_{n,r})$ are satisfied.

Proof of claim

There are two types of arrows in $C_2$ : horizontal arrows and arrows containing a vertical step, which we call vertical for simplicity. Each of the arrows in $C_2$ picks out a column in the weight matrix $[b_{va}]$ of $X_{{P_{n,r}}}$ . If we order the arrows in $C_1$ from left to right and top to bottom, the resulting submatrix of $[b_{va}]$ is the identity matrix along the vertical arrows and then upper triangular for the rest with $1$ s on the diagonal.

If a is horizontal, then the equation that h must satisfy is

$$\begin{align*}\psi_a=1.\end{align*}$$

Recall that the generators of $T \cap \tilde {G}$ are given by the rows of the weight matrix of the ladder quiver—there is a row for each vertex, and the entries are the $b_{va}$ ; see equation (6). Since the submatrix is upper triangular, we can use the generators of $T \cap \tilde {G}$ corresponding to the vertices $t(a)$ , a horizontal, to move g within its orbit to a group element satisfying the equations we need for horizontal $a \in C_2$ . Call this element $g'=(\psi ^{\prime }_a)$ .

We now focus on the vertical arrows. Note that none of the generators of $T \cap \tilde {G}$ associated to the vertical arrows act nontrivially on the $\psi _a$ , $a \in C_2$ , a horizontal. Label the vertical arrows of $C_2$ from bottom to top as $a^2_1,\dots ,a^2_{r-1}$ . Then the equations that $g'$ must satisfy are

$$\begin{align*}\psi_{a^2_i}=\psi_{a^1_{i+1}}/\psi_{a^1_{i}}. \end{align*}$$

To move g within its orbit so that it satisfies these equations, we start with the last vertical arrow, $a^2_{r-1}$ . If we act with the generator associated to $t(a^2_{r-1})$ , we can set the ${a^2_{r-1}}$ -th coordinate to

$$\begin{align*}\psi^{\prime}_{a^1_{r}}/\psi^{\prime}_{a^1_{r-1}}. \end{align*}$$

Note that that generator acts nontrivially on the $a^1_{r}$ and $a^1_{r-1}$ coordinates—however, it acts on the same weight with each, so we now have a group element which satisfies the $a^2_{r_1}$ equation and the horizontal equations. If we move one step lower, the same thing happens: Our group element acts trivially on the ratios $\psi ^{\prime }_{a^1_{r}}/\psi ^{\prime }_{a^1_{r-1}}$ and $\psi ^{\prime }_{a^1_{r-1}}/\psi ^{\prime }_{a^1_{r-2}}$ . Continuing downwards, eventually we find h as required.

Let’s rephrase what we’ve done so far: For every $[g] \in G$ , we have identified a particular lift $g \in \tilde {G}$ satisfying

(8) $$ \begin{align} \psi_a=\xi_a(g) \text{ for all } a \in C_2. \end{align} $$

To complete the proof, it is enough to show that if $[g] \in G_h$ , then the special lift g constructed above is in $\tilde {\Psi }(\tilde {H}_{n,r})$ . Since $\tilde {G} \cap T$ acts trivially on the Plücker coordinates, both g and $[g]$ preserve the Plücker relations. In particular, g preserves certain three-term Plücker relations: We will define such a relation for every internal vertex in the ladder quiver.

For every internal vertex $v \in \operatorname {{\mathrm {LQ}}}_0$ , fix an incomparable pair $\lambda _v$ , $\sigma _v$ which crosses at v and agrees outside of the $2\times 2$ box centered at v. (The argument will not depend on what happens outside the $2 \times 2$ box centered at v.) For example, here are some pairs with v highlighted in red and the box shaded in gray.

Fix an interior vertex v. Locally about vertex v, $\lambda _v$ and $\sigma _v$ look like:

(9)

By definition, $\sigma _v \wedge \lambda _v$ and $\sigma _v \vee \lambda _v$ agree with $\lambda _v, \sigma _v$ outside of the box centered at v, and locally at v they look like

The incomparable pair $\sigma _v, \lambda _v$ gives rise to a three-term Plücker relation:

$$\begin{align*}p_{\alpha_v} p_{\beta_v}-p_{\lambda_v} p_{\sigma_v}+p_{\sigma_v \wedge \lambda_v} p_{\sigma_v \vee \lambda_v}=0.\end{align*}$$

The partitions $\alpha _v$ , $\beta _v$ are characterized by agreeing with $\sigma $ , $\lambda $ outside of the box centered at v, and locally at v they look like

(10)

One can easily read off from this picture that this relation is homogeneous with respect to the $\Psi (H_{n,r})$ -action. Let $S_{v}$ be the collection of arrows of $\operatorname {{\mathrm {LQ}}}$ whose intersection with the $2\times 2$ box in equation (9) is contained in the blue or green highlighted paths in equation (9). Let $T_v$ be the collection of arrows of $\operatorname {{\mathrm {LQ}}}$ whose intersection with the $2 \times 2$ box in equation (10) is contained in the blue or green highlighted paths in equation (10). If $g=(\psi _a) \in \tilde {G}$ preserves this relationship, then

(11) $$ \begin{align} \prod_{a \in S_{v}} \psi_a= \prod_{a \in T_{v}} \psi_a. \end{align} $$

Now, fix some $[g] \in G_h$ with lift $g \in \tilde {G} $ constructed as in equation (8). The Plücker relation constructed for any v is homogeneous for $g=(\psi _a)$ , so equation (11) holds for all v.

Consider the following diagram, which highlights both $C_1$ and $C_2$ .

Consider the top-left interior vertex $v_0$ . Note that there is precisely one arrow $a_0$ in $S_{v_0} \cup T_{v_0}$ that is not contained in $C:=C_1 \cup C_2$ . Using equation (11) for $v_0$ one can see that $\psi ^{\prime }_{a_0}$ is determined by the $\psi _a$ , $a \in C$ —and so, in fact, is determined by the $\psi _a$ , $a \in C_1$ . Abusing notation, replace C by $C \cup \{a_0\}$ . Now, consider the vertex $v_1$ directly below $v_0$ , and note that there is precisely one arrow $a_1$ in $S_{v_1} \cup T_{v_1}$ that is not contained in (the new) C. Applying equation (11) for $v_1$ , we see that $\psi ^{\prime }_{a_1}$ is determined by the $\psi _a$ , $a \in C$ , and hence by $\psi _a'$ , $a \in C_1$ . Applying this argument repeatedly, moving down columns from left to right, we conclude that g is uniquely characterized by:

• ○ the values $(\psi _a)_{a \in C_1}$

• ○ equation (8)

• ○ preserving the Plücker relations.

We now use the $\psi _a$ , $a \in C_1$ , to define an element of $\tilde {\Psi }(\tilde {H}_{n,r})$ and deduce that this element agrees with g. Writing the relation $\prod _{a \in \operatorname {{\mathrm {LQ}}}_1} \psi _a=1$ in terms of $\psi _a$ , $a \in C_1$ , we see that

$$\begin{align*}\prod_{i=1}^r \left(\frac{\psi_{a^2_{i}}}{\psi_{a^2_{i-1}}}\right)^r \prod_{i=r+1}^n \psi_{a^2_{i}}^r=1. \end{align*}$$

So $(\xi _a((\psi _a)_{a \in C_1}))$ defines an element of $\tilde {\Psi }(\tilde {H}_{n,r}) \subset \tilde {G}$ which agrees with g along $a \in C_1$ and satisfies the other two conditions. It follows that $g=\xi ((\psi _a)_{a \in C_1})$ , and in particular $g \in \tilde {\Psi }(\tilde {H}_{n,r})$ . Finally, we conclude that $[g] \in \Psi (H_{n,r})$ as claimed.

To translate this combinatorial statement to geometry, we extend the G-action from $X_P$ to the family which gives the toric degeneration. The degeneration is given by the Plücker relations. To illustrate, we describe the $\operatorname {{\mathrm {Gr}}}(5,2)$ example:

Example 4.11. When $n=5$ , $r=2$ , the family is given by the five polynomials

$$\begin{align*}c p_{12} p_{34}-p_{13}p_{24}+p_{14}p_{23},\end{align*}$$

$$\begin{align*}c p_{12} p_{35}-p_{13}p_{25}+p_{15}p_{23},\end{align*}$$

$$\begin{align*}c p_{12} p_{45}-p_{14}p_{25}+p_{15}p_{24},\end{align*}$$

$$\begin{align*}c p_{13} p_{45}-p_{14}p_{35}+p_{15}p_{34},\end{align*}$$

$$\begin{align*}c p_{23} p_{45}-p_{24}p_{35}+p_{25}p_{34},\end{align*}$$

that is, this is a family in $\mathbb {P}^{9} \times {\mathbb {C}}$ , where the coordinate on the second factor is c. The fiber over $c=0$ is the degenerate toric variety $X_{P_{5,2}}$ , and the generic fiber is isomorphic to the Grassmannian $\operatorname {{\mathrm {Gr}}}(5,2)$ . To make this family G-equivariant, we need to extend the action of G to the coefficient c. Label the vertices of the ladder diagram as shown:

There are nine arrows in the ladder quiver, which we label

$$\begin{align*}a_{0,1},a_{0,1}',a_{0,2}, a_{0,3},a_{0,3}',a_{1,2},a_{1,3},a_{2,3},a_{2,3}', \end{align*}$$

where $a_{i,j}$ is an arrow from $i \to j$ , and we add a $'$ to the lower arrow if there are two such arrows. Consider the first equation above. The weight of the first monomial is $\psi _{a_{0,3}} \psi _{a_{0,2}} \psi _{a_{2,3}}$ ; the weight of both the second and third monomial is $\psi _{a_{0,1}} \psi _{a_{1,3}} \psi _{a_{0,1}'} \psi _{a_{1,2}} \psi _{a_{2,3}}$ , so for c to be equivariant in this equation, it should have weight

$$\begin{align*}(\psi_{a_{0,3}} \psi_{a_{0,2}} \psi_{a_{2,3}})^{-1} \psi_{a_{0,1}} \psi_{a_{1,3}} \psi_{a_{0,1}'} \psi_{a_{1,2}} \psi_{a_{2,3}}.\end{align*}$$

However, as we go through the remaining five relations, the required weight for c changes, so we are forced to introduce multiple coefficients. Consider the polynomials

$$\begin{align*}c_1 p_{12} p_{34}-p_{13}p_{24}+p_{14}p_{23},\end{align*}$$

$$\begin{align*}c_2 p_{12} p_{35}-p_{13}p_{25}+p_{15}p_{23},\end{align*}$$

$$\begin{align*}c_3 p_{12} p_{45}-p_{14}p_{25}+p_{15}p_{24},\end{align*}$$

$$\begin{align*}c_4 p_{13} p_{45}-p_{14}p_{35}+p_{15}p_{34},\end{align*}$$

$$\begin{align*}c_5 p_{23} p_{45}-p_{24}p_{35}+p_{25}p_{34}.\end{align*}$$

We view the toric degeneration as a family in $Z\subset \mathbb {P}^{9} \times {\mathbb {C}}^5$ , where the second factor has coordinates $(c_i)$ , and the equations cutting out Z are above. Computing the required weights of $c_1,\dots ,c_5$ , we note that $c_1$ has the same weight as $c_2$ and $c_4$ has the same weight as $c_5$ . The variable $c_3$ has the weight of the monomial $c_1 c_4$ . To get rid of the excess variables, we add the equations

$$\begin{align*}c_1=c_2, \: c_4=c_5, \: c_3= c_1 c_4.\end{align*}$$

Consider the quotient $Z/G$ . The special fiber—when $c_i=0$ —is $X_{P_{n,r}}/G$ . Now, consider a generic fiber of Z. The equation $c_i=1$ is not homogeneous for the G-action; however, we can consider the disconnected subvariety $Z_{1}$ cut out of Z by the equations $c_i^5=1$ , $i \in \{1,\ldots ,5\}$ . The subgroup of G that preserves a given component of $Z_{1}$ is precisely $H_{5,2}$ . The number of components of $Z_1$ is 25. The group $G/H_{5,2}$ has order $25$ . Comparing group sizes, it follows that $Z_{1}/G=\operatorname {{\mathrm {Gr}}}(5,2)/H_{5,2}$ .

The situation is slightly more complicated in the general case. As in the example, $X_P$ is the special fiber of a family of varieties

(12) $$ \begin{align} \tilde{Z} \subset\mathbb{P}^{\binom{n}{r}-1} \times {\mathbb{C}}^m \xrightarrow{\pi} {\mathbb{C}}^m. \end{align} $$

We denote points in ${\mathbb {C}}^m$ as $\underline {c}$ . The coordinates of $\underline {c}$ give coefficients for all monomials in the Plücker relations that disappear under the degeneration, and $\tilde {Z}_{\underline {c}}$ is $\tilde {Z} \cap \pi ^{-1}(\underline {c})$ . If $\underline {c}_1:=(1,\dots ,1)$ and $\underline {c}_0:=(0,\dots ,0)$ , then $\tilde {Z}_{\underline {c}_1}=\operatorname {{\mathrm {Gr}}}(n,r)$ and $\tilde {Z}_{\underline {c}_0}=X_{P_{n,r}}$ . The generic fiber is isomorphic to $\operatorname {{\mathrm {Gr}}}(n,r)$ .

We can extend the action of G to $\mathbb {P}^{\binom {n}{r}-1} \times {\mathbb {C}}^m \to {\mathbb {C}}^m$ by defining the action on ${\mathbb {C}}^m$ to be the one that ensures $\tilde {Z}$ is preserved by G. Note that $G_h$ is the subgroup of G that acts trivially on ${\mathbb {C}}^m$ . As in Example 4.11, in general some of these variables are redundant. In fact, the proof of Theorem 4.9 shows that we need only the special $2 \times 2$ three-term Plücker relations associated to each internal vertex in the ladder quiver. More precisely, the proof shows that if $g \in G$ preserves these relations, then it in fact preserves all Plücker relations. Let $d_v$ , where v is an internal vertex, denote the these coefficients. Then for any other coefficient c, the weight of c under the G-action can be expressed as a ratio of the $d_v$ . Whatever these ratios are, they give a collection of polynomial equations between the $c_i$ , which we call

(13) $$ \begin{align} \mathcal{P}_1. \end{align} $$

Before we can define the family we want, we need to cut down by one last equation, which is nontrivial when $\gcd (n,r)>1$ . (It did not occur in our earlier example because $5$ and $2$ are coprime.) This will be a G-equivariant equation, of course, and will take some time to describe.

Definition 4.12. Fix integers $n>r>0$ , and set $k=n-r$ and $d=\gcd (n,r)$ . Then there exist s, t such that $r=s d$ and $k=t d$ . The ladder diagram $\operatorname {{\mathrm {LQ}}}_{n,r}$ can be subdivided into an $s \times t$ grid of $d \times d$ squares. The set of d-diagonal vertices in $\operatorname {{\mathrm {LQ}}}_0$ is the set of $v \in \operatorname {{\mathrm {LQ}}}_0$ such that v lies on the diagonal of one of the $d \times d$ squares. To a d-diagonal vertex v lying in a $d \times d$ square B, we associate an incomparable pair of partitions $\sigma _{B,v}$ , $\mu _{B,v}$ . As before, we only specify these partitions locally, insisting that they agree outside of B. The partitions are specified by the statement that their union contains both the vertical and horizontal line (contained in B) passing through the vertex v and that both partitions pass through the bottom-left and top-right vertices of B. We draw the two partitions below.

We denote by $\sigma _B$ and $\mu _B$ the two partitions that agree with $\sigma _{B,v}$ and $\mu _{B,v}$ away from B and follow the border of B:

Lemma 4.13. Let v be a d-diagonal vertex in the ladder diagram $\operatorname {{\mathrm {LQ}}}_{n,r}$ . Then there is a Plücker relation that contains both $p_{\sigma _B} p_{\mu _B}$ and $p_{\sigma _{B,v}} p_{\mu _{B,v}}$ .

Proof. Suppose that v is the ath d-diagonal vertex in the box B, and suppose that the label on the bottom-left vertical step of B is i. Then $I(\sigma _{B,v})$ differs from $I(\mu _{B,v})$ only in the range $M:=\{ j \mid i \leq j \leq i+2d-1\}$ . It is easy to see that

$$\begin{align*}I(\sigma_{B,v}) \cap M = \{i,i+1,\dots,i+a-1, i+a+d,\dots, i+2d-1\},\end{align*}$$

and

$$\begin{align*}I(\mu_{B,v}) \cap M = \{i+a,\dots,i+2a-1,i+2a,\dots,i+a+d-1\}.\end{align*}$$

If $I(\sigma _{B,v})=\{s_1 < \cdots <s_r\}$ and $I(\mu _{B,v})=\{t_1<\cdots < t_r\}$ , then choose p such that $s_p=i+a-1$ . So we can obtain $I(\sigma _B)$ and $I(\mu _B)$ from $I(\sigma _{B,v})$ and $I(\mu _{B,v})$ by swapping $s_{p+1}<\cdots <s_{d-a+p}$ with $t_{p+1}<\dots <t_{d-a+p}$ . This implies that there is a Plücker relation where both terms appear. For completeness, we construct a relation using [Reference Manivel17, equation 3.1.3] in the case when $2a \geq d$ and in the case where $2a<d$ .

In each case, we will identify subsets of indices $J_s=\{s_{i_1},\dots ,s_{i_l}\}$ and $J_{t}=\{t_{j_1},\dots ,t_{j_{r-l}}\}$ . Let S be the symmetric group permuting the symbols in the union $J_s \cup J_t$ , and $S_1 \times S_2$ the subgroup of S that preserves $I(\sigma _{B,v})$ and $I(\mu _{B,v})$ . For $\omega \in S/(S_1 \times S_2)$ , set

$$\begin{align*}\omega(I(\sigma_{B,v}))=\{\omega(s_{i}): s_i \in J_s \} \cup \{s_i: s_i \not \in J_s\},\end{align*}$$

and

$$\begin{align*}\omega(I(\mu_{B,v}))=\{\mu(t_{i}): t_i \in J_t \} \cup \{t_i: t_i \not \in J_t\}.\end{align*}$$

Then for any choice of $J_s$ and $J_t$ , the following equation is a Plücker relation:

$$\begin{align*}\sum_{\omega \in S/S_1 \times S_2}\operatorname{{\mathrm{sign}}}(\omega) p_{\omega(I(\sigma_{B,v}))} p_{\omega(I(\mu_{B,v}))}.\end{align*}$$

When $2a \geq d$ , we choose

$$\begin{align*}J_s=\{i+a+d,\dots,i+2d-1\},\: J_t=\{i+k,\dots,i+2a-1\} \cup M. \end{align*}$$

When $2a < d$ , we choose

$$\begin{align*}J_s=\{i,\dots,i+a-1\}\cup M,\: J_t=\{i+2k,\dots,i+d+a-1\}. \end{align*}$$

Note that for each choice the terms $p_{\sigma _{B,v}} p_{\mu _{B,v}}$ and $p_{\sigma _{B}} p_{\mu _{B}}$ appear with nonzero coefficient as claimed.

From the proof of Theorem 4.9, the weight of the ratio $p_{\sigma _{B,v}} p_{\mu _{B,v}}/(p_{\sigma _{B}} p_{\mu _{B}})$ can be expressed as a ratio of the weights of $d_w$ , where w ranges over the internal vertices. Let $f_{B,v}(d_w)$ be this ratio.

Lemma 4.14. The group G acts trivially on the product $(\prod _{(B,v)} f_{B,v})^{\frac {n}{d}}$ , where $(B,v)$ ranges over internal diagonal vertices in the $d \times d$ boxes B of the ladder diagram.

Proof. When $d=1$ , the statement is trivial. To prove the lemma, we will show that the weight of $\prod _{(B,v)} f_{B,v}$ is an order $n/d$ element of G. We will describe this element precisely. Let $r=sd$ and $n-r=td$ . We have divided the ladder diagram into a $s \times t$ grid of $d\times d$ boxes. Consider the collection C of arrows entirely contained in the border of the $d \times d$ boxes. Then we will show that the weight of $\prod _{(B,v)} f_{B,v}$ is

(14) $$ \begin{align} \prod_{a \in C} \zeta_a^{-d}. \end{align} $$

This is an order $n/d$ element of G as desired. The easiest way to see that this weight is correct is to visualize the weights using paths in the ladder diagram. We will indicate a weight by drawing paths in the ladder diagram, possibly in reverse, and allowing multiple arrows. Each arrow appearing contributes a weight of $\zeta _a^{b^+-b^-}$ , where $b^+$ is the number of times a appears in the correct direction and $b^-$ is the number of times it appears in the reverse direction. For example, the weight of $f_{B,v}$ is pictured as

Now, consider the weight of the product of $f_{B,v}$ where B is fixed: $\prod _{v} f_{B,v}$ . Pictorially, this corresponds to overlaying the diagrams for each of internal vertices v. Note that the arrows going in a positive direction have no intersection except on the border, and every such arrow is covered exactly once. Depending on where B is located in the ladder diagram, there could be vertices along the borders of B or not, which changes how many times each arrow appears. If there are no vertices on a particular edge, then there are $d-1$ arrows in the reverse direction along that edge (one for each internal vertex). If there are border vertices, then the number of arrows follows a pattern as below (where $-k$ indicates k arrows in the reverse direction of the arrow directly below the label):

The weight of the product over all boxes B—that is, the weight of $\prod _{(B,v)} f_{B,v}$ —is read off by combining all of the $d \times d$ boxes to cover the ladder diagram. It is immediate from the above diagram that this gives us the weight

$$\begin{align*}\prod_{a \in C} \zeta_a^{-(d-1)} \prod_{a \not \in C} \zeta_a. \end{align*}$$

Since $\prod _{a} \zeta _a=1$ , this shows that the weight is equation (14) as desired.

The lemma states that the action of G on $(\prod _{(B,v)} f_{B,v})^{\frac {n}{d}}$ is trivial, so the equation $(\prod _{(B,v)} f_{B,v})^{\frac {n}{d}}=1$ is homogeneous. If we wish, since the left-hand side is a monomial, we can clear the denominator to get a polynomial equation. We call the set containing this single polynomial equation

(15) $$ \begin{align} \mathcal{P}_2. \end{align} $$

This is the last equation we need to define our family.

Definition 4.15. Let $Z \subset \mathbb {P}^{\binom {n}{r}-1} \times B \xrightarrow {\pi } B$ denote the family cut out by the following three types of the equations:

1. 1. the Plücker relations,

2. 2. the polynomial equations $\mathcal {P}_1$ from equation (13),

3. 3. the polynomial equation $\mathcal {P}_2$ from equation (15).

The last two equations cut out $B \subset {\mathbb {C}}^m$ .

By construction, Z is a G-invariant algebraic variety. We use notation as for the family $\tilde {Z}$ in equation (12). Note that the fiber $Z_{\underline {c}_0}$ is invariant under the G action, but $Z_{\underline {c}_1}$ is not. However, since the action of G on on ${\mathbb {C}}^m$ is of order n, the disconnected variety $\hat {Z}_{\underline {b}}:=Z \cap \pi ^{-1}(\{(c_1,\dots ,c_m) \in B: c_i^n=b_i\})$ is G-invariant for any $\underline {b} \in {\mathbb {C}}^m$ . We will consider the quotient family $Z/G \to B/G$ .

The next result gives the precise sense in which $\operatorname {{\mathrm {Gr}}}(n,r)/H_{n,r}$ is a smoothing of $X_{P_{n,r}}/G$ .

Theorem 4.16. Suppose b is generic (i.e., $b_i \neq 0$ for all i). Then $\hat {Z}_{\underline {b}}/G$ is a fiber of the quotient family $Z/G \to B/G$ , and $\hat {Z}_{\underline {b}}/G \cong \operatorname {{\mathrm {Gr}}}(n,r)/H_{n,r}$ .

Proof. For simplicity, we will set $b_i=1$ for all i and denote $\hat {Z}_{\underline {b}}/G$ by $\hat {Z}_1/G$ . The variety $\hat {Z}_1$ is disconnected, with components we can label $A_1 \sqcup \cdots \sqcup A_s$ . One component, which we label $A_1$ , is the Grassmannian $\operatorname {{\mathrm {Gr}}}(n,r)$ . The quotient can be studied in two steps:

$$\begin{align*}\hat{Z}_1/G=(\bigsqcup A_i/G_h)/(G/G_h). \end{align*}$$

Note that the stabilizer of any element of $\bigsqcup A_i/G_h$ under the $G/G_h$ -action is trivial, as only $G_h$ preserves the components. Thus, the size of an orbit of a point in $\bigsqcup A_i/G_h$ is

$$\begin{align*}|G/G_h|=n^{r(n-r)-1}/(n^{n-2} d)=n^{r(n-r)-n} n/d=n^{(r-1)(n-r-1)-1} n/d. \end{align*}$$

In particular, the number of components s is at least $n^{(r-1)(n-r-1)-1} n/d$ . On first glance, it may appear that $\hat {Z}_1$ has $n^m$ components, but in fact, the equations from $\mathcal {P}_1$ and $\mathcal {P}_2$ cut this down considerably. The equations from $\mathcal {P}_1$ mean that we can regard all coefficients $c_i$ as a function of the coefficients $d_v$ , where v is an internal vertex of the ladder quiver, so there are at most $n^{(r-1)(n-r-1)}$ components. The single equation in $\mathcal {P}_2$ is of the form $f^{n/d},$ which means that in fact the largest possible number of components is $n^{(r-1)(n-r-1)-1} n/d$ . So

$$\begin{align*}s=n^{(r-1)(n-r-1)-1} n/d. \end{align*}$$

This means that the residual $G/G_h$ action identifies each of the components $A_i/G_h$ , so

$$\begin{align*}\hat{Z}_1/G=A_1/G_h \cong \operatorname{{\mathrm{Gr}}}(n,r)/H_{n,r}. \end{align*}$$

This also shows that the orbit of a point in the base of the family Z is precisely the set $\{(c_1,\dots ,c_m) \in B: c_i^n=b_i\}$ .

4.2 Smoothing $Y_{n,r}$ and $Y_{n,r}/G$

Recall from Theorem 3.7 that there is a toric variety $Y_{n,r}$ characterized as a blow-up of $X_{P_{n,r}}$ obtained by adding the rays required to take $P_{n,r}$ to $Q_{n,r}$ . The variety $Y_{n,r}$ is of interest as it fits into the following diagrams:

$Y_{n,r}$ is the toric variety associated to the fan obtained from the spanning fan of $P_{n,r}$ by blowing up at the rays $v_{\lambda } \in N$ , for $\lambda \in {\mathcal {B}(n,r)}$ a redundant partition. This depends on the order in which the blow-up is completed.

Each $v_{\lambda }$ lies in the interior of an intersection of maximal cones of the spanning fan of $P_{n,r}$ . Maximal cones of this fan are also indexed by partitions, but not just those in ${\mathcal {B}(n,r)}$ : There is a maximal cone $C_{\mu }$ for each $\mu \in {\mathcal {P}(n,r)}$ corresponding to the section of $\mathcal {O}(1)$ indexed by $\mu $ .

Definition 4.17. For each $\lambda \in {\mathcal {B}(n,r)}$ , let $\operatorname {{\mathrm {Cones}}}(\lambda ):=\{\mu \in {\mathcal {P}(n,r)}: v_{\lambda } \in C_{\mu }\},$ the collection of all maximal cones of the spanning fan of $P_{n,r}$ that contain $v_{\lambda }$ . Fix an order on ${\mathcal {B}(n,r)}$ . Let $\operatorname {{\mathrm {Bl}}} \mathbb {P}$ denote the iterated blow-up of $\mathbb {P}^{\binom {n}{r}-1}$ at the linear subspaces

$$\begin{align*}Z(p_{\mu}: \mu \in \operatorname{{\mathrm{Cones}}}(\lambda)),\end{align*}$$

where we iterate over $\lambda \in {\mathcal {B}(n,r)}$ . We define $\operatorname {{\mathrm {Bl}}} \operatorname {{\mathrm {Gr}}}(n,r)$ to be the iterated blow-up of $\operatorname {{\mathrm {Gr}}}(n,r)$ at these same subspaces.

Consider the Cox ring of of $\operatorname {{\mathrm {Bl}}} \mathbb {P}$ . In addition to the $p_{\mu }$ , $\mu \in {\mathcal {P}(n,r)}$ , there is a generator $x_{\lambda }$ for each $\lambda \in {\mathcal {B}(n,r)}$ graded by the Cartier divisor corresponding to the ray added when blowing up at the $\lambda $ th step. Now, instead of blowing up $X_{P_{n,r}}$ using toric geometry, we can view it as a subvariety of $\mathbb {P}^{\binom {n}{r}-1}$ , then compute the blow-up of $X_{P_{n,r}}$ as the main component of its proper transform in the ambient blow-up.

To write down the binomial equations, we adjust the binomial equations cutting $X_{P_{n,r}}$ out of $\mathbb {P}^{\binom {n}{r}-1}$ . Let $\sigma _1$ and $\sigma _2$ be an incomparable pair, giving rise to the binomial equation

$$\begin{align*}p_{\sigma_1} p_{\sigma_2}-p_{\sigma_1 \vee \sigma_2}p_{\sigma_1 \wedge \sigma_2}.\end{align*}$$

Although this homogeneous for $\mathbb {P}^{\binom {n}{r}-1}$ , it is not when considered as an equation in the Cox ring of $\operatorname {{\mathrm {Bl}}} \mathbb {P}$ . However, there is a natural minimal way to homogenize this equation by multiplying each factor by some combination of generators $x_{\lambda }$ , $\lambda \in {\mathcal {B}(n,r)}$ . Call the subvariety cut out of $\operatorname {{\mathrm {Bl}}} \mathbb {P}$ by these adjusted equations $\operatorname {{\mathrm {Bl}}} X_{P_{n,r}}$ .

Note that we can lift the action of G on $\operatorname {{\mathrm {Gr}}}(n,r)$ and $X_{P_{n,r}}$ to their blow-ups.