Notation 2.1. Given a fan $\Sigma $ supported in $\mathbb {R}^E/\mathbb {R}e_E$ , we denote its corresponding toric variety by $X(\Sigma )$ .

Proof of Lemma 3.2.

Let $\langle \mathcal {K}_{\mathbf {w}}^{\mathbf {a}}\rangle $ be the set of all possible scheme-theoretic intersections of the elements $\mathcal {K}_{\mathbf {w}}^{\mathbf {a}}$ . Since $\Delta _I^{\mathbf {a}}\cap \Delta ^{\mathbf {a}}_J=\Delta ^{\mathbf {a}}_{I\cup J}$ if and only if $I\cap J\neq \emptyset $ , we have the following decomposition:

$$\begin{align*}\langle\mathcal{K}_{\mathbf{w}}^{\mathbf{a}}\rangle= \left\{ \Delta_{I_1}^{\mathbf{a}}\cap\cdots\cap\Delta_{I_r}^{\mathbf{a}}\;\big|\; r\geq 2,\, \Delta_{I_j}^{\mathbf{a}}\in\mathcal{K}_{\mathbf{w}}^{\mathbf{a}},\text{ and }I_j\cap I_k=\emptyset\text{ for all }j\neq k \right\} \cup \mathcal{K}_{\mathbf{w}}^{\mathbf{a}}. \end{align*}$$

Notice that all elements in $\langle \mathcal {K}_{\mathbf {w}}^{\mathbf {a}}\rangle $ are smooth varieties. Moreover, the set $\langle \mathcal {K}_{\mathbf {w}}^{\mathbf {a}}\rangle $ is an arrangement of subvarieties in the sense of Li (see Definition 2.11), that is, any two elements intersect cleanly in $\mathbb {C}^{\mathbf {a}}$ . To see this, recall that two nonsingular subvarieties $A,B\subseteq Y$ of a variety Y are said to intersect cleanly if their intersection is smooth and their tangent spaces satisfy $T_{A,y}\cap T_{B,y}=T_{A\cap B,y}$ for all $y\in A\cap B$ . In the case at hand every element of $\langle \mathcal {K}_{\mathbf {w}}^{\mathbf {a}}\rangle $ is a linear subspace of $\mathbb {C}^{\mathbf {a}}$ , in which case this condition is immediate, since each one of the subvarieties can be identified with its tangent space.

It only remains to prove that $\mathcal {K}_{\mathbf {w}}^{\mathbf {a}}$ is a building set for the arrangement $\langle \mathcal {K}_{\mathbf {w}}^{\mathbf {a}}\rangle $ . By definition, this amounts to proving that given any $S\in \langle \mathcal {K}_{\mathbf {w}}^{\mathbf {a}}\rangle \setminus \mathcal {K}_{\mathbf {w}}^{\mathbf {a}}$ , the minimal elements of $\{\Delta _I^{\mathbf {a}}\in \mathcal {K}_{\mathbf {w}}^{\mathbf {a}}:\Delta _I^{\mathbf {a}}\supseteq S\}$ intersect transversely and their intersection is S. This is a direct dimension count.

Let us once again identify each one of the diagonals with their tangent spaces. By [Reference LiLi09, Section 5.1.2] a collection $\Delta _{I_1}^{\mathbf {a}},\dots ,\Delta _{I_r}^{\mathbf {a}}$ of diagonals with $\{I_1,\ldots ,I_r\}$ pairwise disjoint intersects transversely if and only if for each $p \in \mathbb {C}^{\mathbf {a}}$

(3.1) $$ \begin{align} \sum_{j=1}^r\operatorname{codim}_p\left(\Delta_{I_j}^{\mathbf{a}},\mathbb{C}^{\mathbf{a}}\right) = \operatorname{codim}_p\left(\bigcap_{j=1}^r \Delta_{I_j}^{\mathbf{a}},\mathbb{C}^{\mathbf{a}}\right). \end{align} $$

Since a subcollection of $\Delta _{I_1}^{\mathbf {a}},\dots ,\Delta _{I_r}^{\mathbf {a}}$ is also a collection of diagonals with disjoint indices, we may assume without loss of generality that $p \in \bigcap _{j=1}^r \Delta _{I_j}^{\mathbf {a}}$ . Define $i_j$ to be the largest element of $I_j$ . Then, a direct dimension count shows that

$$\begin{align*}\operatorname{codim}(\Delta_{I_j}^{\mathbf{a}},\mathbb{C}^{\mathbf{a}}) = \sum_{k\in I_j}a_k - a_{i_j}, \end{align*}$$

so that the left hand side of Equation (3.1) becomes $\sum _{k\in \cup I_j} a_k - a_{i_1} - \cdots - a_{i_r}$ . On the other hand, computing the right-hand side by directly counting the codimension of $\Delta _{I_1}^{\mathbf {a}}\cap \cdots \cap \Delta _{I_r}^{\mathbf {a}}$ in $\mathbb {C}^{\mathbf {a}}$ shows that it is also equal to $\sum _{k\in \cup I_j} a_k - a_{i_1} - \cdots - a_{i_r}$ .

Proof. Fix any subvariety Z in $\mathcal {G}$ . By [Reference HartshorneHar13, Corollary 7.15], the closed immersion $i:Y \hookrightarrow X$ induces a closed immersion $i_Z: \operatorname {Bl}_{Z \cap Y} Y \hookrightarrow \operatorname {Bl}_{Z} X$ such that the following diagram commutes

Then, if we let Z vary and we take the product over all $Z \in \mathcal {G}$ , then we get a closed immersion $\widetilde {Y} \hookrightarrow \widetilde {X}$ where

$$\begin{align*}\widetilde{Y} := \prod_{Z \in \mathcal{G}} \operatorname{Bl}_{Z \cap Y} Y \longrightarrow \widetilde{X} := \prod_{Z \in \mathcal{G}} \operatorname{Bl}_{Z} X. \end{align*}$$

Let $U_X = X \setminus \bigcup _{Z \in \mathcal {G}} Z$ and $U_Y = Y \setminus \bigcup _{Z \in \mathcal {G}} (Z \cap Y)$ . From the construction of the wonderful compactifications, the morphisms f and g are isomorphisms over $U_X$ and $U_Y$ . We notice that $U_Y = Y \cap U_X$ and hence also $U_Y = \widetilde {Y} \cap U_X$ . Therefore, we get a commutative diagram

where the vertical maps are open immersions and the horizontal maps are closed immersions. This diagram induces an identification between the closure of $U_Y$ in $\widetilde {Y}$ , that is $Y_{\mathcal {H}}$ , and the closure of $U_Y$ in $\widetilde {X}$ . The diagram also induces a closed immersion from the closure of $U_Y$ in $\widetilde {X}$ into the closure of $U_X$ in $\widetilde {X}$ , that is $X_{\mathcal {G}}$ . Then we obtain a closed immersion $j:Y_{\mathcal {H}} \hookrightarrow X_{\mathcal {G}}$ .

If we consider the diagram in the statement using the closed immersion j that we constructed as the top morphism, the diagram commutes because by construction it commutes over the dense open subset $U_Y$ of $Y_{\mathcal {H}}$ .

Now, notice that there can be at most one morphism $Y_{\mathcal {H}} \hookrightarrow X_{\mathcal {G}}$ making the diagram in the statement commute because such a morphism has prescribed values over the dense open subset $U_Y$ of $Y_{\mathcal {G}}$ . Indeed, by the commutativity of the diagram, when the morphism $Y_{\mathcal {H}} \hookrightarrow X_{\mathcal {G}}$ is restricted to $U_Y$ it agrees with the composition

$$\begin{align*}(f|_{U_X})^{-1} \circ i \circ (g|_{U_Y}). \end{align*}$$

Then, the existence and uniqueness of the closed immersion j making the diagram commute follows. Finally, the image of $Y_{\mathcal {H}}$ under j is a subvariety of $X_{\mathcal {G}}$ with the same dimension as Y and which maps birationally onto Y under f, but this characterizes the strict transform of Y under f, and this completes the proof.

Proof. This result follows as a corollary of Lemma 3.5. Indeed, let $X = \mathbb {C}^{\mathbf {d}}$ , $Y = \mathbb {C}^{\mathbf {a}}$ , and let $\mathcal {G}=\mathcal {K}_{\mathbf {w}}^{\mathbf {d}}$ be the set of all diagonals $\Delta _I^{\mathbf {d}}$ associated with $\mathbf {w}$ . Consider any $I_0 \subseteq [n]$ such that $|I_0|\geq 2$ . Notice that the $I_0$ -diagonal $\Delta _{I_0}^{\mathbf {{a}}}$ is in $\mathcal {K}_{\mathbf {w}}^{\mathbf {a}}$ if and only if the $I_0$ -diagonal $\Delta _{I_0}^{\mathbf {{d}}}$ is in $\mathcal {K}_{\mathbf {w}}^{\mathbf {d}}$ . Also, the $I_0$ -diagonal $\Delta _{I_0} = \Delta _{I_0}^{\mathbf {{d}}} \subseteq \mathbb {C}^{\mathbf {d}}$ intersects $\mathbb {C}^{\mathbf {a}}$ , since even the deepest diagonal $\Delta _{[n]} \subseteq \Delta _{I_0}$ clearly intersects $\mathbb {C}^{\mathbf {a}}$ . Moreover, $\Delta _{I_0}^{\mathbf {{a}}} = \mathbb {C}^{\mathbf {a}}\cap \Delta _{_0}^{\mathbf {{d}}}$ scheme-theoretically. Then, the intersections of the diagonals in $\mathcal {G}=\mathcal {K}_{\mathbf {w}}^{\mathbf {d}}$ with $Y = \mathbb {C}^{\mathbf {a}}$ form the corresponding building set $\mathcal {H}=\mathcal {K}_{\mathbf {w}}^{\mathbf {a}}$ of all diagonals $\Delta _I^{\mathbf {a}}$ in $\mathbb {C}^{\mathbf {a}}$ associated with $\mathbf {w}$ . By applying Lemma 3.5 in this setting, we obtain the existence and uniqueness of the desired closed immersion $j_{\mathbf {a}}: \mathbb {C}^{[\mathbf {a}]}_{\mathbf {w}} \hookrightarrow \mathbb {C}^{[\mathbf {d}]}_{\mathbf {w}}$ , making the diagram commute.

Proof. From Definition 3.9 we get the following commutative diagram consisting of Cartesian squares:

(3.7)

where $N:= a_1+\cdots + a_{(n-1)}-1$ . Let us recall that $\mathbb {C}_{\mathbf {w}}^{[\mathbf {a}]}$ is the wonderful compactification of the building set $\mathcal {G}:=\mathcal {K}_{\mathbf {w}}^{\mathbf {a}}$ in $\mathbb {C}^{\mathbf {a}}$ . Since $\Delta $ is minimal in $\mathcal {G}$ by [Reference LiLi09, Proposition 2.8] the dominant transforms to $\operatorname {Bl}_{\Delta } \mathbb {C}^{\mathbf {a}}$ of the elements in $\mathcal {G}$ form a building set $\widetilde {\mathcal {G}}$ in $\operatorname {Bl}_{\Delta } \mathbb {C}^{\mathbf {a}}$ . From the inductive construction of wonderful compactifications in [Reference LiLi09, Definition 2.12], we know that $\mathbb {C}_{\mathbf {w}}^{[\mathbf {a}]}$ is the wonderful compactification of the building set $\widetilde {\mathcal {G}}$ in $\operatorname {Bl}_{\Delta } \mathbb {C}^{\mathbf {a}}$ .

Moreover, the morphism $\mathbb {C}_{\mathbf {w}}^{[\mathbf {a}]} \rightarrow \operatorname {Bl}_{\Delta } \mathbb {C}^{\mathbf {a}}$ is the iterated blow-up of the dominant transforms of the elements in $\widetilde {\mathcal {G}}$ in any order compatible with inclusion of subvarieties. Since the exceptional divisor $E^{\mathbf {a}}_{[n]}$ of the blow-up $\operatorname {Bl}_{\Delta } \mathbb {C}^{\mathbf {a}} \rightarrow \mathbb {C}^{\mathbf {a}}$ is maximal in $\widetilde {\mathcal {G}}$ , we can order the iterated blow-up process so that the last blow-up step is that of a dominant transform of $E^{\mathbf {a}}_{[n]}$ . Such dominant transform is a Cartier divisor, and then that last blow-up step can be omitted without changing the resulting variety $\mathbb {C}_{\mathbf {w}}^{[\mathbf {a}]}$ . Hence we can consider the building set for the wonderful compactification $\mathbb {C}_{\mathbf {w}}^{[\mathbf {a}]} \rightarrow \operatorname {Bl}_{\Delta } \mathbb {C}^{\mathbf {a}}$ to be $\widetilde {\mathcal {G}}' = \widetilde {\mathcal {G}} \setminus \{ E^{\mathbf {a}}_{[n]} \}$ . Moreover, since $\Delta $ is contained in all elements in $\mathcal {G}$ , then all the dominant transforms that yield the elements in $\widetilde {\mathcal {G}}'$ from elements in $\mathcal {G} \setminus \{\Delta \}$ are actually strict transforms.

Since $\mathbb {C}^{\mathbf {a}}$ is an affine space and all elements in $\mathcal {G} \setminus \{\Delta \}$ are linear subspaces containing $\Delta $ , then Lemma 3.13 computes their intersections with $\mathbb {P}((\mathbb {C}^{\mathbf {a}}/\Delta )^\vee ) \cong \mathbb {P}^N$ seen as the fiber over $0 \in \Delta $ of the exceptional divisor of $\operatorname {Bl}_{\Delta } \mathbb {C}^{\mathbf {a}} \rightarrow \mathbb {C}^{\mathbf {a}}$ . For any $I \subsetneq [n]$ with $|I| \geq 2$ such that $\Delta _I^{\mathbf {a}}$ is in $\mathcal {G} \setminus \{\Delta \} = \mathcal {K}_{\mathbf {w}}^{\mathbf {a}} \setminus \{\Delta \}$ , the intersection of the strict transform $\widetilde {\Delta _I^{\mathbf {a}}}$ of $\Delta _I^{\mathbf {a}}$ in $\operatorname {Bl}_{\Delta } \mathbb {C}^{\mathbf {a}}$ with $\mathbb {P}((\mathbb {C}^{\mathbf {a}}/\Delta )^\vee ) \cong \mathbb {P}^N$ is precisely $\delta _I^{\mathbf {a}}$ by Lemma 3.13.

Therefore, the set of intersections of the elements of the building set $\widetilde {\mathcal {G}}'$ with $\mathbb {P}^N=\mathbb {P}^{a_1+\cdots +a_{(n-1)}-1}$ is precisely the building set $\mathcal {H}_{\mathbf {w}}^{\mathbf {a}}=\{\delta _I^{\mathbf {a}}\,\vert \, I\subsetneq [n],\, \mathbf {w}_I>1\}$ in $\mathbb {P}^N$ . From the diagram (3.7) with Cartesian squares, we have that $T_{\mathbf {w}}^{\mathbf {a}}$ is the strict transform of $\mathbb {P}^N$ in $\mathbb {C}_{\mathbf {w}}^{[\mathbf {a}]}$ under the iterated blow-up $\mathbb {C}_{\mathbf {w}}^{[\mathbf {a}]} \rightarrow \operatorname {Bl}_{\Delta } \mathbb {C}^{\mathbf {a}}$ . Therefore by Lemma 3.5, $T_{\mathbf {w}}^{\mathbf {a}}$ is equal to the wonderful compactification of the building set $\mathcal {H}_{\mathbf {w}}^{\mathbf {a}}=\{\delta _I^{\mathbf {a}}\,\vert \, I\subsetneq [n],\, \mathbf {w}_I>1\}$ in $\mathbb {P}^N=\mathbb {P}^{a_1+\cdots +a_{(n-1)}-1}$ , and this completes the proof.

Proof. By Proposition 3.14, we know that $T_{\mathbf {w}}^{\mathbf {a}}$ is a wonderful compactification. So, the claim about the iterated blow-up follows from the general theory of Li [Reference LiLi09, Theorem 1.3]. The claim about the case $\mathbf {w}_{[n-1]}\leq 1$ follows from the fact that such condition implies $n\in I$ if $\delta _I^{\mathbf {a}} \in \mathcal {H}_{\mathbf {w}}^{\mathbf {a}}$ . By (3.6), such locus $\delta _I^{\mathbf {a}}$ is a torus-invariant subvariety of $\mathbb {P}^{a_1+\cdots +a_{(n-1)}-1}$ under the standard toric structure of projective space.

Proof. We define $N:=a_1+\cdots +a_{(n-1)}-1$ and recall our convention that $a_1 \geq \cdots \geq a_{n}$ . Let us recall the description of the inclusion $g_{\mathbf {a}}:\mathbb {P}^{N} \hookrightarrow \mathbb {P}^{a_1(n-1)-1}$ . We have inclusions $\mathbb {C}^{a_i} \hookrightarrow \mathbb {C}^{a_1}$ as the span of the first $a_i$ coordinates, for each i. By taking the product, we get an inclusion of affine spaces $\mathbb {C}^{a_1+\cdots +a_{n-1}} \hookrightarrow \mathbb {C}^{a_1(n-1)}$ , which induces the closed immersion $g_{\mathbf {a}}:\mathbb {P}^N \hookrightarrow \mathbb {P}^{a_1(n-1)-1}$ between their projectivizations.

Consider any $I_0 \subsetneq [n]$ . Notice that $\delta _{I_0}^{\mathbf {{a}}}$ is in $\mathcal {H}_{\mathbf {w}}^{\mathbf {a}}$ if and only if $\delta _{I_0}^{\mathbf {{d}}}$ is in $\mathcal {H}_{\mathbf {w}}^{\mathbf {d}}$ . Also, $\delta _{I_0} = \delta _{I_0}^{\mathbf {{d}}} \subseteq \mathbb {P}^{a_1(n-1)-1}$ intersects $\mathbb {P}^{N}$ , since even $\delta _{[n]} \subseteq \delta _{I_0}$ clearly intersects $\mathbb {P}^{N}$ . Moreover, $\delta _{I_0}^{\mathbf {{a}}} = \mathbb {P}^{N} \cap \delta _{_0}^{\mathbf {{d}}}$ scheme-theoretically.

Then, the intersections with $Y = \mathbb {P}^{N}$ of the varieties $\delta _I^{\mathbf {a}}$ in the building set $\mathcal {G}=\mathcal {H}_{\mathbf {w}}^{\mathbf {d}}$ in $X=\mathbb {P}^{a_1(n-1)-1}$ form the building set $\mathcal {H}=\mathcal {H}_{\mathbf {w}}^{\mathbf {a}}$ in $\mathbb {P}^{N}$ . By applying Lemma 3.5 in this setting, we obtain the existence and uniqueness of the desired closed immersion $f_{\mathbf {a}}:T^{\mathbf {a}}_{\mathbf {w}} \hookrightarrow T^{\mathbf {d}}_{\mathbf {w}}$ , making the diagram commute.

Construction 3.26 [Reference Maria Feichtner and YuzvinskyFY04, Theorem 4].

Start with the fan $\Sigma _{E}$ in $\mathbb {R}^E/\mathbb {R}e_{E}$ whose rays have been labeled by the singletons i for $i\in E$ . Given a nonsingleton set $I\in \mathcal {B}$ , define the cone $\sigma _I:=\operatorname {cone}(\overline {e}_i\,\vert \, i\in I)\in \Sigma _{E}$ . Finally, define a total order of $\mathcal {B}=\{I_1,\dots ,I_k\}$ compatible with inclusion, so that larger sets come before smaller sets. Then, define the fan $\Sigma (\mathcal {B})$ as that one obtained from $\Sigma _{E}$ after performing iterated star subdivisions along the cones $\sigma _I$ as I ranges through the elements of $\mathcal {B}$ in the order we defined.

Notation 3.29. Consider a building set $\mathcal {B}$ over E, we denote by $X(\mathcal {B})$ the toric variety constructed from the nested fan $\Sigma (\mathcal {B})$ .

Proof. Consider a nested set M of $\pi ^*\mathcal {B}$ . If $|M|=1$ , then the result is direct, so let us consider the case $|M|\geq 2$ . Then, by definition, $M\subseteq 2^A\setminus \emptyset $ consists of a collection of singletons and sets of the form $\pi ^{-1}(S)$ for subsets $S\in \mathcal {B}$ . Let I denote the collection of singletons in M that are not full fibers of $\pi $ , in other words, all singletons excluding those that are a full fiber. If I were to contain any fiber $\pi ^{-1}(i)$ for some $i\in E$ , this fiber would have size at least two, the collection of all singletons making up this fiber would be disjoint, and their union $\pi ^{-1}(i)$ would be an element of $\pi ^*\mathcal {B}$ with at least two elements, which contradicts M being a nested set. Therefore, I cannot contain any full fibers of size at least two.

On the other hand, consider any $k\geq 2$ elements $\pi ^{-1}(S_1),\dots ,\pi ^{-1}(S_k)\in M$ , so that in particular $S_1,\dots ,S_k\in \mathcal {B}$ . We remark that, in particular, it is possible for some $\pi ^{-1}(S_i)$ to be a singleton. Then, $\pi ^{-1}(S_i)\subseteq \pi ^{-1}(S_j)$ if and only if $S_i\subseteq S_j$ . Similarly, $\pi ^{-1}(S_1)\cup \cdots \cup \pi ^{-1}(S_k)\in \pi ^*\mathcal {B}$ if and only if $S_1\cup \cdots \cup S_k\in \mathcal {B}$ . Therefore, the collection of all subsets $S\in \mathcal {B}$ such that $\pi ^{-1}(S)\in M$ forms a nested set N of $\mathcal {B}$ .

Conversely, given a $\pi $ -pair $(I,N)$ , it is a direct verification to check that $N_{(I,N)}:=\{\{i\}\,\vert \, i\in I\}\cup \{ \pi ^{-1}(S)\,\vert \, S\in N\}$ defines a nested set of $\pi ^*\mathcal {B}$ .

Proof. It is well-known that nested sets of the boolean building set $\mathcal {B}_{LM}$ over $[n]$ are in bijection with flags of nonempty subsets $S_1\subsetneq S_2\subsetneq \cdots \subsetneq S_k\subsetneq [n]$ . Indeed, a nested set of $\mathcal {B}_{LM}$ cannot contain two incomparable sets, as their union is also an element of $\mathcal {B}_{LM}$ . Therefore, in light of Corollary 3.34, the cones of $\pi ^*\mathcal {B}_{LM}$ are

$$\begin{align*}\sigma_{(I,N)} = \sigma_{(I,\emptyset)}+\sigma_{(\emptyset, N)}, \end{align*}$$

where $N=\{S_1\subsetneq S_2\subsetneq \cdots \subsetneq S_k\subsetneq [n]\}$ . This is precisely the Bergman fan of the boolean polymatroid with cage $\pi $ , as introduced in [Reference Crowley, Huh, Larson, Simpson and WangCHL⁺22, Definition 1.2]. It is proved in [Reference Crowley, Huh, Larson, Simpson and WangCHL⁺22, Appendix A] that this is the inner normal fan of the polypermutohedron.

Proof of Theorem 3.22.

Recall that the compactification $T^{\mathbf {a}}_{\mathbf {w}}$ is the iterated blow-up of the projective space $\mathbb {P}^{a_1+\cdots +a_{(n-1)}-1}$ along the dominant transforms of the varieties in the geometric building set

$$\begin{align*}\mathcal{H}_{\mathbf{w}}^{\mathbf{a}} = \left\{ \delta_I^{\mathbf{a}}\,\vert\,\mathbf{w}_I>1\right\}. \end{align*}$$

When $\mathbf {w}$ is such that $\mathbf {w}_{[n-1]}\leq 1$ and $w_i+w_n>1$ for all $i\in [n-1]$ , every subvariety in $\mathcal {H}_{\mathbf {w}}^{\mathbf {a}}$ is of the form

$$\begin{align*}\delta_I^{\mathbf{a}} = \bigcap_{i\in I\setminus\{n\}} V(x_{i1},\dots,x_{ia_i}), \end{align*}$$

for some $I\subsetneq [n]$ such that $n\in I$ and $|I|\geq 2$ ; see Equation (3.6). These subvarieties are invariant under the standard torus action on $\mathbb {P}^{a_1+\cdots +a_{(n-1)}-1}$ .

On the other hand, let $\mathcal {B}_{LM}=2^{[n-1]}\setminus \emptyset $ be the boolean building set over $[n-1]$ , and $\rho :A^-\to [n-1]$ be a caging with cage $\mathbf {a}^-$ . Then, for every I as above, the cone $\sigma _{\rho ^{-1}(I\setminus \{n\})}\in \Sigma (\rho ^*\mathcal {B}_{LM})$ corresponds to the subvariety $\delta _I^{\mathbf {a}}$ under the toric orbit-cone correspondence. Moreover, every cone in $\Sigma (\rho ^*\mathcal {B}_{LM})$ is of this form by definition. Therefore, both constructions can be obtained as iterated blow-ups of $\mathbb {P}^{a_1+\cdots +a_{(n-1)}-1}$ along the same subvarieties, in an order that can be assumed to be the same, so they are the same.

Proof of Theorem 3.38.

By Lemma 3.34, every cone of $\Sigma (\pi ^*\mathcal {B})$ corresponds to a $\pi $ -pair $(I,N)$ and has the form

$$\begin{align*}\sigma_{(I, N)} = \sigma_{(I,\emptyset)}+\sigma_{(\emptyset, N)}, \end{align*}$$

where

$$\begin{align*}\sigma_{(I,\emptyset)}=\operatorname{cone}(\overline{e}_i\,\vert\, i\in I)\quad\text{and}\quad\sigma_{(\emptyset, N)}=\operatorname{cone}(\overline{e}_{\pi^{-1}(S)}\,\vert\, S\in N)\subseteq\mathbb{R}^A/\mathbb{R}e_A, \end{align*}$$

and, by the definition of a $\pi $ -pair, $I\subsetneq A$ does not contain any fiber of $\pi $ and N is a nested set of $\mathcal {B}$ .

Consider the short exact sequence of vector spaces

Here, the maps in the sequence are defined as $\iota (\overline {e}_i)=\overline {e}_{A_i}$ for all $i\in [n]$ , and if $i\in A_i\subseteq A$ , then $\varphi (\overline {e}_i)=(0,\dots ,\overline {e}_i,\dots ,0)$ . One can check directly that these maps are well-defined and that the sequence is exact.

Let us show that $\iota $ defines an inclusion of fans. Consider a cone $\sigma _N\in \Sigma (\mathcal {B})$ corresponding to a nested set N of $\mathcal {B}$ . Then, $\sigma _N=\operatorname {cone}(\overline {e}_S\,\vert \,S\in N)$ , and its image under $\iota $ is precisely the cone $\sigma _{(\emptyset , N)}$ in $\mathbb {R}^A/\mathbb {R}e_A$ . In particular, $\iota $ is an inclusion of fans.

Next, let us see that $\varphi $ defines a surjective map of fans. Consider a cone $\sigma _{(I,N)}=\sigma _{(I,\emptyset )}+\sigma _{(\emptyset , N)}\in \Sigma (\pi ^*\mathcal {B})$ . By the exactness of the sequence, $\sigma _{(\emptyset , N)}$ is in the kernel of $\varphi $ . On the other hand, the cone $\sigma _{(I,\emptyset )}$ maps to the cone $\sigma _{I\cap A_1}\times \cdots \times \sigma _{I\cap A_n}$ in $\Sigma _{A_1}\times \cdots \times \Sigma _{A_n}$ . To show surjectivity note that every cone in the product fan is of the form $\sigma _{J_1}\times \cdots \times \sigma _{J_n}$ for some subsets $\emptyset \subseteq J_i\subsetneq A_i$ (the empty set corresponding to the zero cone). Then, $(I:=J_1\cup \cdots \cup J_n,\emptyset )$ is a $\pi $ -pair because I does not contain any fiber of $\pi $ . Moreover, we have that $\varphi (\sigma _{(I,\emptyset )})=\sigma _{J_1}\times \cdots \times \sigma _{J_n}$ .

Let us proceed to show that $\Sigma (\pi ^*\mathcal {B})$ is split by $\Sigma (\mathcal {B})$ and $\Sigma _{A_1}\times \cdots \times \Sigma _{A_n}$ . Condition (1) in Definition 3.39 is an immediate consequence of Corollary 3.34 saying that every cone $\sigma _{(I, N)}$ can be decomposed as the sum $\sigma _{(I,\emptyset )}+\sigma _{(\emptyset , N)}$ . To verify condition (2) we define the subfan $\widehat {\Sigma }$ of $\Sigma (\pi ^*\mathcal {B})$ to be the fan consisting of all cones of the form $\sigma _{(I,\emptyset )}$ as I ranges over all the subsets of A not containing any fiber of $\pi $ . Then, it follows directly from our discussion on the surjectivity of $\varphi $ that $\varphi $ defines a bijection between the cones of $\widehat {\Sigma }$ and those of $\Sigma _{A_1}\times \cdots \times \Sigma _{A_n}$ . Moreover, since $\varphi $ restricts to a map of lattices, it follows that $\varphi (\sigma _{(I,\emptyset )}\cap \mathbb {Z}^A/\mathbb {Z}e_A)=\varphi (\sigma _{(I,\emptyset )})\cap \mathbb {Z}^{A_1}/\mathbb {Z}e_{A_1}\times \cdots \times \mathbb {Z}^{A_n}/\mathbb {Z}e_{A_n}$ .

To conclude, let us prove that the fibration is not trivial. Consider any (vector space) splitting $\nu :\mathbb {R}^{A_1}/\mathbb {R}e_{A_1}\times \cdots \times \mathbb {R}^{A_n}/\mathbb {R}e_{A_n}\to \mathbb {R}^A/\mathbb {R}e_A$ . Then, for the fibration to be trivial we must have that $\nu (\sigma _I)=\sigma _{(I,\emptyset )}$ for all $I=J_1\cup \cdots \cup J_n$ with $J_i\subsetneq A_i$ . However, note that the one-dimensional cones of the form $\sigma _{(I,\emptyset )}$ span all of $\mathbb {R}^A/\mathbb {R}e_A$ . This would imply that the image of $\nu $ is all of $\mathbb {R}^A/\mathbb {R}e_A$ , which is impossible since the image of $\nu $ is the complement of $\mathbb {R}^{[n]}/\mathbb {R}e_{[n]}\hookrightarrow \mathbb {R}^A/\mathbb {R}e_A$ .

Proof. Recall from Corollary 3.23 that $\overline {M}^{LM}_{0,n+1}$ is the toric variety corresponding to the $(n-2)$ -dimensional permutohedron. This permutohedron is the nestohedron defined by the building set $\mathcal {B}_{LM}=2^{[n-1]}\setminus \emptyset $ , so that $\overline {M}^{LM}_{0,n+1}\cong X(\mathcal {B}_{LM})$ .

On the other hand, by Theorem 3.22, we know that $T^{\mathbf {a}}_{LM}\cong PP_{(\mathbf {a}^{-})}$ , where $\mathbf {a}^-=(a_1,\dots ,a_{n-1})$ . Therefore, $T^{\mathbf {a}}_{LM}$ is isomorphic to the toric variety $X(\rho ^*\mathcal {B}_{LM})$ , where $\rho :A^{-}\to [n-1]$ is any caging with cage $\mathbf {a}^-$ . The result then follows from Theorem 3.38.

Proof. After constructing the compactifications $\mathbb {C}^{\mathbf {a}}_{\mathbf {w}}$ in Definition 3.3, the moduli spaces $T^{\mathbf {a}}_{\mathbf {w}}$ are defined in Subsection 3.2; see also Theorem 3.9. The claim that $T^{\mathbf {a}}_{\mathbf {w}}$ is smooth, and its boundary divisors are normal crossings is a direct consequence of Li’s general theory of wonderful compactifications; see Proposition 2.15.

The proof of part (i) of the Theorem is given in Theorem 3.22. Part (iii) is Corollary 3.41. Let us proceed to prove part (ii).

Define $N=a_1+\cdots +a_{(n-1)}-1$ . Let $\mathbf {w}_{LM}$ be any weight vector such that $w_{[n-1]}<1$ and $w_i+w_n>1$ for all $i\in [n-1]$ , and let $\mathbf {w}=(1,\dots ,1)$ . Then, $T_{LM}^{\mathbf {a}}$ and $T^{\mathbf {a}}$ are the wonderful compactifications of $\mathbb {P}^{N}$ with respect to the geometric building sets $\mathcal {H}_{\mathbf {w}_{LM}}^{\mathbf {a}}$ and $\mathcal {H}_{\mathbf {w}}^{\mathbf {a}}$ , respectively. In particular, since $\mathbf {w}_{LM}\leq \mathbf {w}$ , we have that $\mathcal {H}_{\mathbf {w}_{LM}}^{\mathbf {a}}\subseteq \mathcal {H}_{\mathbf {w}}^{\mathbf {a}}$ . For simplicity let us denote these building sets as $\mathcal {H}_{LM}$ and $\mathcal {H}$ . Suppose that

$$\begin{align*}\mathcal{H} = \{U_1,\dots,U_p,V_1,\dots,V_q\} \end{align*}$$

is such that $\mathcal {H}_{LM}=\{U_1,\dots ,U_p\}$ and $\mathcal {H} \setminus \mathcal {H}_{LM}=\{V_1,\dots ,V_q\}$ . Each of the subvarieties $U_k$ equals a $\delta _{I_k}^{\mathbf {a}}$ for some $I_k\subsetneq [n]$ such that $|I_k|\geq 2$ and $n\in I_k$ . Similarly, each $V_k$ is of the form $\delta _{J_k}^{\mathbf {a}}$ for some $J_k\subsetneq [n]$ such that $|J_k|\geq 2$ and $n\not \in J_k$ ; see Equation (3.6). We refer to the sets $I_k,J_k$ as the index sets of $U_k$ and $V_k$ , respectively.

Let us order $\mathcal {H}_{LM}$ so that the subvarieties with larger index sets appear before those with smaller index sets. Similarly, order $\mathcal {H}$ so that the elements of $\mathcal {H}_{LM}\subseteq \mathcal {H}$ come first – appearing in the order we just fixed – and are followed by the subvarieties $V_k$ ordered so that those subvarieties with larger index sets appear before those with smaller index sets.

We claim that these orders for $\mathcal {H}_{LM}$ and $\mathcal {H}$ both satisfy the conditions of Proposition 2.16, so that $T^{\mathbf {a}}_{LM}$ and $T^{\mathbf {a}}$ are both the iterated blow-ups of $\mathbb {P}^{N}$ along the building sets in the orders we just prescribed for their building sets. This would imply the result, as the pth step of the iterated blow-up process of $T^{\mathbf {a}}$ is precisely that one in the construction of $T_{LM}^{\mathbf {a}}$ .

The fact that $\mathcal {H}_{LM}$ in this order satisfies the conditions of Proposition 2.16 is a direct consequence of the fact that, for any $1\leq r\leq p$ , the subset $\{U_1,\dots ,U_r\}\subseteq \mathcal {H}_{LM}$ is closed under intersections of its own elements. In particular, every such subset is a building set so the claim follows.

Consider a subset $\mathcal {H}_r:=\{U_1,\dots ,U_p,V_1,\dots ,V_r\}\subseteq \mathcal {H}$ for some $1\leq r\leq q$ , and let $\mathcal {S}_r$ be the induced arrangement of subvarieties of $\mathcal {H}_r$ in $\mathbb {P}^{N}$ . Every element $S\in \mathcal {S}_r$ can be written as an intersection of the form $S= U_I\cap V_{J_1}\cap \cdots \cap V_{J_k}$ for some $0\leq k\leq r$ , $I\subsetneq [n]$ such that $n\in I$ and $|I|\geq 2$ , and subsets $J_i\subsetneq [n]$ such that $|J_i|\geq 2$ and $n\not \in J_i$ . This follows from the fact that $\mathcal {H}_{LM}$ is closed under intersection. Moreover, as in the proof of Lemma 3.2, we can assume that all the subsets $I,J_1,\dots ,J_k$ are disjoint.

In order to show $\mathcal {H}_r$ is a building set we prove that the intersection $U_I\cap V_{J_1}\cap \cdots \cap V_{J_k}$ is transverse. As in Lemma 3.2, this is a direct dimension count. Since all the subvarieties $U_I, V_{J_1},\cdots , V_{J_k}$ are linear, we can identify them with their tangent spaces. Then, transversality will follow from the equality

$$\begin{align*}\operatorname{codim}(U_I\cap V_{J_1}\cap\cdots\cap V_{J_k}, \mathbb{P}^{N}) = \operatorname{codim}(U_I,\mathbb{P}^N)+\sum_{i=1}^k\operatorname{codim}(V_{J_i},\mathbb{P}^N). \end{align*}$$

This verification is left to the reader, as the dimension count is very similar to that one in the proof of Lemma 3.2, using the fact that all the subsets $I,J_1,\dots ,J_k$ are disjoint.

Since the previous argument works for all elements S of the arrangement $\mathcal {S}_r$ , we obtain that $\mathcal {H}_r$ is a building set. Since the argument holds for all values of r, the conditions of Proposition 2.16 are met, so $T^{\mathbf {a}}$ is the iterated blow-up of $\mathbb {P}^N$ along the subvarieties in $\mathcal {H}$ in the order we described previously. In particular, we have a sequence of blow-up maps $T^{\mathbf {a}}\to T^{\mathbf {a}}_{LM}\to \mathbb {P}^N$ , as desired.

Proof. The nonempty poset $\mathcal {L}=2^{A}$ is finite and there exists a greatest lower bound for any nonempty subset. Hence, $\mathcal {L}$ is an atomic meet-semilattice (see [Reference Feichtner and MüllerFM05, Section 2] for relevant definitions). Since $\pi \succeq \pi ^{\prime }$ , for every element $S'\in \mathcal {B}^{\prime }_{LM}$ the subset $S=\pi (\pi ^{\prime -1}(S'))\subseteq [n]$ is such that $\pi ^{-1}(S) = \pi ^{\prime -1}(S')$ . Therefore, $\pi ^{\prime \ast }(\mathcal {B}^{\prime }_{LM})\subseteq \pi ^{\ast }(\mathcal {B}_{LM})$ . Then, the result follows from [Reference Feichtner and MüllerFM05, Theorem 4.2]. The fact that the resulting fan of these subdivisions is unimodular follows from [Reference Maria Feichtner and YuzvinskyFY04, Proposition 2].

Proof. Define $A^{-}=\pi ^{-1}([n-1])$ and $\rho =\pi |_{A^-}$ . By Lemma 3.43, there is a reduction map $X(PP_{\mathbf {1}^{m}})\to X(PP_{\mathbf {a}^-})$ . By Theorem 3.22 and Corollary 3.23, we have that $X(PP_{\mathbf {1}^{m}})\cong T_{LM}^{\mathbf {1}^{(m+1)}}\cong \overline {M}_{0,m+2}^{LM}$ . On the other hand, $X(PP_{\mathbf {a}^-})\cong T_{LM}^{\mathbf {a}}$ . The result follows.

Proof. The building set $\mathcal {G}_{\mathbb {P}^d,H}$ is closed under intersection, so $\mathcal {G}_{\mathbf {a}}$ is as well. Therefore, it is enough to show that the intersection of its elements is clean.

Since $H_{i+1}=\mathbb {P}^{a_{i+1}}\cap H_{i}$ , we obtain that $H_{i+1} = \mathbb {P}^{a_{i+1}}\cap H$ . Therefore, the subvariety $H_S\cap \mathbb {P}^{\mathbf {a}}$ is the product subvariety of $\mathbb {P}^{\mathbf {a}}$ obtained by taking $\mathbb {P}^{a_i}$ in the components labeled by $i\in [n]\setminus S$ , and $H_i$ in those labeled by $i\in S$ . The fact that the elements of $\mathcal {G}_{\mathbf {a}}$ intersect cleanly is a direct consequence of the fact that $H_S\cap \mathbb {P}^{\mathbf {a}}$ is a multilinear subspace. Indeed, by identifying $H_S\cap \mathbb {P}^{\mathbf {a}}$ with its embedded tangent space one obtains that

$$\begin{align*}T_{H_{S_1}\cap\mathbb{P}^{\mathbf{a}},y}\cap T_{ H_{S_2}\cap\mathbb{P}^{\mathbf{a}},y} = H_{S_1}\cap\mathbb{P}^{\mathbf{a}}\cap H_{S_2}\cap\mathbb{P}^{\mathbf{a}} =H_{S_1}\cap H_{S_2}\cap\mathbb{P}^{\mathbf{a}} = T_{H_{S_1}\cap H_{S_2}\cap\mathbb{P}^{\mathbf{a}}, y} \end{align*}$$

for all $y\in H_{S_1}\cap H_{S_2}\cap \mathbb {P}^{\mathbf {a}}$ and $S_1,S_2\subseteq [n]$ .

Proof. Let $X = \mathbb {P}^{\mathbf {d}}$ , $Y = \mathbb {P}^{\mathbf {a}}$ , and let $\mathcal {G}=\mathcal {G}_{\mathbf {d}}$ and $\mathcal {H}=\mathcal {G}_{\mathbf {a}}$ . Lemma 4.3 shows that the intersections with Y of the elements of the building set $\mathcal {G}$ on X are precisely the elements of the building set $\mathcal {H}$ on Y, and therefore the conclusion follows by Lemma 3.5.

Notation 4.7. Let us denote the canonical basis of $\mathbb {R}^{A}$ by $\{ e_{i} : i \in A \}$ and for each $I \subseteq A$ let $e_I=\sum _{i \in I} e_i$ . We let $\{ e^{*}_{i} : i \in A \}$ denote the associated dual basis of the dual vector space $\left (\mathbb {R}^{A}\right )^*$ . We denote the classes of $e_i$ and $e_I$ in the quotient vector space $\mathbb {R}^{A}/\langle e_A \rangle $ by $\overline {e}_i$ and $\overline {e}_I$ , for any $e_i$ and $e_I$ .

Convention 4.11 (When is the flag $\mathcal {F}$ considered to be empty).

The conditions in Definition 4.10 are interpreted in analogy to the case of compatible pairs, as we now explain. The sets $F_l$ are allowed to be empty. Just like in the case of compatible pairs, the chain $\mathcal {F}$ is considered to be empty when there are no sets $F_l$ for $1 \leq l \leq k$ . Hence, the flag $\mathcal {F}$ is empty and we write $\mathcal {F} = \emptyset $ when there is only $F_{k+1}$ . This convention becomes visible in Definition 4.21 below, and it plays an essential role in the proof of Proposition 4.28.