2.1 Localization theorems

Let $K_T(X_{\Sigma })$ be the T-equivariant K-ring of $X_{\Sigma }$ , the Grothendieck ring of T-equivariant vector bundles on $X_{\Sigma }$ . Let $K(X_{\Sigma })$ denote the K-ring of $X_{\Sigma }$ . By forgetting the equivariant structure, one has a surjective map $K_T(X_{\Sigma }) \to K(X_{\Sigma })$ . By taking the T-equivariant sheaf Euler characteristic, one has a $K_T(\operatorname {pt})$ -module homomorphism $\chi ^T \colon K_T(X_{\Sigma }) \to K_T(\operatorname {pt})$ . We identify $K_T(\operatorname {pt}) = {\mathbb {Z}}[\operatorname {Char}(T)]$ with the Laurent polynomial ring $\mathbb {Z}[T_1^{\pm 1}, \ldots , T_n^{\pm 1}]$ , where $T_i$ is the standard character of $i\in E$ under the identification $\operatorname {Char}(T) = {\mathbb {Z}}^E$ .

Let $A_T^{\bullet }(X_{\Sigma })$ denote the equivariant Chow ring of $X_{\Sigma }$ , as defined in [Reference Edidin and GrahamEG98], and let $A^{\bullet }(X_{\Sigma })$ denote the Chow ring of $X_{\Sigma }$ . Similar to the K-rings, one has a surjective map $A_T^{\bullet }(X_{\Sigma }) \to A^{\bullet }(X_{\Sigma })$ and a $A_T^{\bullet }(\operatorname {pt})$ -module homomorphism $\int ^T \colon A_T^{\bullet }(X_{\Sigma }) \to A_T^{\bullet }(\operatorname {pt})$ . We identify $A^\bullet _T(\operatorname {pt})$ with the polynomial ring ${\mathbb {Z}}[t_1, \ldots , t_n]$ . Let $\int \colon A^\bullet (X_\Sigma ) \to {\mathbb {Z}}$ be the (nonequivariant) degree map.

Let $\Sigma (k)$ denote the set of cones of dimension k of $\Sigma $ . For each maximal cone $\sigma $ of $\Sigma $ , we have a map $K_T(X_{\Sigma }) \to K_T(\operatorname {pt}_\sigma ) = \mathbb {Z}[T_1^{\pm 1}, \ldots , T_n^{\pm 1}]$ given by pulling back to or localizing at the corresponding fixed point $\operatorname {pt}_\sigma $ . Similarly, we have a map $A_T^{\bullet }(X_{\Sigma }) \to A_T^\bullet (\operatorname {pt}_\sigma ) = \mathbb {Z}[t_1, \ldots , t_n]$ . These maps can be combined into maps $K_T(X_{\Sigma }) \to K_T(X_{\Sigma }^T) = \prod _{\sigma \in \Sigma (n)} K_T(\operatorname {pt})$ and $A^{\bullet }_T(X_{\Sigma }) \to A_{T}^{\bullet }(X_{\Sigma }^T) = \prod _{\sigma \in \Sigma (n)} A_T^{\bullet }(\operatorname {pt})$ , where $X_\Sigma ^T$ denotes the set of T-fixed points of $X_\Sigma $ . For a character $v = (v_1, \ldots , v_n) \in {\mathbb {Z}}^E$ , we denote $T^v = T_1^{v_1} \dotsb T_n^{v_n}$ and $t_v = v_1 t_1+ \cdots + v_nt_n$ . Then we have the following localization theorem.

2.2 Duality, rank, symmetric powers, exterior powers, Chern classes and Segre classes

We now recall the description of several operations on the equivariant K-ring of a toric variety in terms of localization at fixed points. Let $[\mathcal {E}] \in K_T(X_{\Sigma })$ be an equivariant K-class, localizing to $[\mathcal {E}]_{\sigma } = \sum _{i=1}^{k_{\sigma }} a_{\sigma , i}T^{m_{\sigma , i}}$ at a torus-fixed point corresponding to a maximal cone $\sigma \in \Sigma (n)$ .

There is a ring involution $D_K$ on $K_T(X_{\Sigma })$ defined by sending the class of an equivariant vector bundle to the class of the dual vector bundle. The dual class $D_K([\mathcal {E}]) := [\mathcal {E}]^{\vee }$ has

$$ \begin{align*}D_K([\mathcal{E}])_{\sigma} = \sum_{i=1}^{k_{\sigma}} a_{\sigma, i}T^{-m_{\sigma, i}}.\end{align*} $$

There is a corresponding ring involution, denoted $D_A$ , on $A_T^{\bullet }(X_{\Sigma })$ , defined by $D_A(t_i) \mapsto -t_i$ at each torus-fixed point. This multiplies by $(-1)^k$ on $A^k_T(X_{\Sigma })$ . These involutions descend to $K(X_{\Sigma })$ and $A^{\bullet }(X_{\Sigma })$ .

As toric varieties are integral, every coherent sheaf on a toric variety has a rank. As the rank is additive in short exact sequences, this defines a ring homomorphism $\operatorname {rk} \colon K_T(X_{\Sigma }) \to \mathbb {Z}$ , which descends to $K(X_{\Sigma }) \to \mathbb {Z}$ . The rank of $[\mathcal {E}]$ is $\sum _{i=1}^{k_{\sigma }} a_{\sigma , i}$ , which is independent of the choice of $\sigma $ .

The operation that assigns to each equivariant vector bundle its j-th symmetric or exterior power extends naturally to $K(X_{\Sigma })$ and $K_T(X_{\Sigma })$ . Explicitly, with u a formal variable, we have that

$$ \begin{align*}\sum_{j=0}^{\infty} \textstyle\bigwedge^j \displaystyle[\mathcal{E}]_{\sigma} u^j = \prod_{i=1}^{k_{\sigma}}(1 + T^{m_{\sigma, i}}u)^{a_{\sigma, i}}, \text{ and } \sum_{j=0}^{\infty} \operatorname{Sym}^j[\mathcal{E}]_{\sigma}u^j = \prod_{i=1}^{k_{\sigma}} \left(\frac{1}{1 - T^{m_{\sigma, i}}u}\right )^{a_{\sigma, i}}.\end{align*} $$

The function that sends a vector bundle to its equivariant total Chern class extends to a function $c^T \colon K_T(X_{\Sigma }) \to A_T^{\bullet }(X_{\Sigma })$ , which is multiplicative in the sense that $c^T({\mathcal {E}} + {\mathcal {F}}) = c^T({\mathcal {E}}) \cdot c^T({\mathcal {F}})$ . The equivariant Chern polynomial $c^T(\mathcal {E}, u)$ is the polynomial $c_0^T(\mathcal {E}) + c_1^T(\mathcal {E})u + c_2^T(\mathcal {E})u^2 + \dotsb $ , where u is a formal variable. Define similarly the Chern polynomial $c(\mathcal {E}, u) \in A^{\bullet }(X_{\Sigma })[u]$ . The equivariant total Chern class localizes to

$$ \begin{align*}c^T(\mathcal{E}, u)_{\sigma} = \sum_{j=0}^{\infty} c_j^T(\mathcal{E})_{\sigma} u^j = \prod_{i=1}^{k_{\sigma}}(1 + ut_{m_{\sigma, i}})^{a_{\sigma, i}},\end{align*} $$

where u is a formal variable.

If $\mathcal {E}$ is a vector bundle on $X_{\Sigma }$ , then $\mathcal {E}$ has a Segre class in $A^{\bullet }(X_{\Sigma })$ , characterized by the property that $c(\mathcal {E}) s(\mathcal {E}) = 1$ . We define the equivariant Segre class to be the inverse of $c^T(\mathcal {E})$ in $A^{\bullet }_T(X_{\Sigma })[c^T(\mathcal {E})^{-1}]$ . Because $c(\mathcal {E})$ is a unit in $A^{\bullet }(X_{\Sigma })$ , there is a natural map $A^{\bullet }_T(X_{\Sigma })[c^T(\mathcal {E})^{-1}] \to A^{\bullet }(X_{\Sigma })$ , and the image of $s^T(\mathcal {E})$ is $s(\mathcal {E})$ . Define the (equivariant) Segre polynomial in the same way as the (equivariant) Chern polynomial.

3.1 The stellahedral fan via compatible pairs

We describe the stellahedral fan in terms of its cones. We start by describing the closely related permutohedral fan, which both serves as a motivation for and appears as a substructure in the stellahedral fan.

Definition 3.1. The permutohedral fan ${\underline \Sigma }_E$ is a fan in ${\mathbb {R}}^E/{\mathbb {R}}{\mathbf {e}}_E$ that consists of cones $\underline \sigma _{\mathscr F}$ for each chain $\mathscr F: F_1 \subsetneq \cdots \subsetneq F_k$ of nonempty proper subsets of E, where

$$\begin{align*}\underline{\sigma}_{\mathscr F} = \operatorname{cone}\{\overline{{\mathbf{e}}}_{F_1}, \ldots, \overline{{\mathbf{e}}}_{F_k}\}. \end{align*}$$

Here, we denoted $\overline u$ for the image of $u\in {\mathbb {R}}^E$ in ${\mathbb {R}}^E/{\mathbb {R}}{\mathbf {e}}_E$ .

That this definition of ${\underline \Sigma }_E$ is equivalent to its description as the normal fan of the permutohedron ${\underline \Pi }_E =\operatorname {conv}\{ w \cdot (1,2,\ldots , n) \mid w \text { is a permutation of }E\} \subseteq {\mathbb {R}}^E$ is a standard fact about Coxeter reflection groups; see for instance [Reference Björner and BrentiBB05]. We now give a similar description of the stellahedral fan $\Sigma _E$ in terms of ‘compatible pairs’ as given in [Reference Braden, Huh, Matherne, Proudfoot and WangBHM⁺22, §2].

Both the subset I and the chain $\mathscr F$ are allowed to be empty. In contrast to the permutohedral case, the empty set is allowed to be an element in the chain $\mathscr F$ . Make the following a definition.

Proposition 3.3. [Reference Braden, Huh, Matherne, Proudfoot and WangBHM⁺22, Proposition 2.6]

The stellahedral fan $\Sigma _E$ is a simplicial fan that consists of cones $\sigma _{I\leq \mathscr F}$ for each compatible pair $I\leq \mathscr F$ , where

$$\begin{align*}\sigma_{I\leq \mathscr F} = \operatorname{cone}\{{\mathbf{e}}_i \mid i\in I\} + \operatorname{cone}\{-{\mathbf{e}}_{E\setminus F} \mid F\in \mathscr F\}. \end{align*}$$

We denote the rays of the fan $\Sigma _E$ by

$$\begin{align*}\rho_i = \sigma_{\{i\}\leq \emptyset} = \operatorname{cone}({\mathbf{e}}_i) \text{ for each}\ i\in E \quad\text{and}\quad \rho_S = \sigma_{\emptyset \leq \{S\}} = \operatorname{cone}(-{\mathbf{e}}_{E\setminus S}) \text{ for each }S\subsetneq E. \end{align*}$$

The proposition gives the following corollary concerning the stars of the stellahedral fan. Recall that for a fan $\Sigma $ in ${\mathbb {R}}^E$ , the star of a cone $\sigma \in \Sigma $ is a fan, denoted $\operatorname {star}_\sigma \Sigma $ , in ${\mathbb {R}}^E/{\mathbb {R}}\sigma $ whose cones are the images of the cones in $\Sigma $ containing $\sigma $ .

Corollary 3.4. [Reference Braden, Huh, Matherne, Proudfoot and WangBHM⁺22, Proposition 2.7]

Let $I = \{i_1, \ldots , i_j\} \leq \mathscr F: F_1 \subsetneq \cdots \subsetneq F_k$ be a compatible pair, and by convention set $F_{k+1} = E$ (so $F_1 = E$ if $\mathscr F$ is an empty chain). Then, the isomorphism

$$\begin{align*}{\mathbb{R}}^E / {\mathbb{R}}\sigma_{I\leq \mathscr F} = {\mathbb{R}}^E/{\mathbb{R}}\{{\mathbf{e}}_{i_1},\ldots, {\mathbf{e}}_{i_j}, -{\mathbf{e}}_{E\setminus F_1}, \ldots, -{\mathbf{e}}_{E\setminus F_k}\} \simeq {\mathbb{R}}^{F_1\setminus I} \times \prod_{i=1}^k {\mathbb{R}}^{F_{i+1}\setminus F_i}/{\mathbb{R}}{\mathbf{e}}_{F_{i+1}\setminus F_i}\end{align*}$$

induces an isomorphism of fans

$$\begin{align*}\operatorname{star}_{\sigma_{I\leq \mathscr F}}\Sigma_E \simeq \Sigma_{{F_1\setminus I}} \times \prod_{i = 1}^k \underline\Sigma_{{F_{i+1}\setminus F_i}}. \end{align*}$$

We will often use Example 3.5 to recover or relate the ‘augmented’ structures on stellahedral varieties to the ‘nonaugmented’ versions on permutohedral varieties. We will use the more general star structures of the stellahedral fan in §4.2, where we study the restriction of augmented tautological bundles to various torus-invariant subvarieties of the stellahedral variety.

3.2 Refinements and coarsenings

We record how the stellahedral fan $\Sigma _E$ arises as either a refinement or a coarsening of certain fans. First, we note that $\Sigma _E$ is an iterated stellar subdivision of coarser fans in two distinguished ways. Both statements can be verified via Proposition 3.3.

Since the toric varieties of $\Sigma _E$ and $(\Sigma _1)^E$ are ${\mathbb {P}}^E$ and $({\mathbb {P}}^1)^E$ , respectively, the above two descriptions of $\Sigma _E$ can be rephrased to say that the stellahedral variety $X_E$ is an iterated blow-up along smooth centers from ${\mathbb {P}}^E$ and from $({\mathbb {P}}^1)^E$ . The two maps $\pi _E\colon X_E \to {\mathbb {P}}^E$ and $\pi _{1^E}\colon X_E \to ({\mathbb {P}}^1)^E$ are the blow-down maps. For $i\in E$ , let $\pi _i\colon X_E \to {\mathbb {P}}^1$ be the composition of $\pi _{1^E}$ with the projection to the i-th ${\mathbb {P}}^1$ . These maps from $X_E$ to projective spaces give the following distinguished divisor classes on $X_E$ .

Definition 3.7. With notations as above, we denote

$$\begin{align*}\alpha = \pi_E^*(\text{hyperplane class of}\ {\mathbb{P}}^E) \quad\text{and}\quad y_i = \pi_i^*(\text{hyperplane class of}\ {\mathbb{P}}^1). \end{align*}$$

We now describe the stellahedral fan $\Sigma _E$ as a coarsening of a permutohedral fan. This description of $\Sigma _E$ will be useful for our discussion of the tropical geometry of augmented wonderful varieties in §5.3 and for producing a basis for $\Sigma _E$ in §7.2.

Denote by $\widetilde E = E\sqcup \{0\}$ . Let p be the isomorphism of lattices

$$\begin{align*}p\colon {\mathbb{Z}}^{\widetilde E} /{\mathbb{Z}} {\mathbf{e}}_{\widetilde E} \to {\mathbb{Z}}^E \text{ given by } (a_0, a_1, \ldots, a_n) \mapsto (a_1 - a_0, \ldots, a_n - a_0). \end{align*}$$

That is, for $S\subseteq \widetilde E$ we have $\overline {{\mathbf {e}}}_S\mapsto {\mathbf {e}}_S$ if $0\notin S$ and $\overline {{\mathbf {e}}}_S \mapsto -{\mathbf {e}}_{E\setminus S}$ if $0\in S$ . To show that the stellahedral fan $\Sigma _E$ of E is the image under p of a coarsening of the permutohedral fan $\underline {\Sigma }_{{\widetilde E}}$ of $\widetilde E$ , we use the following notions from [Reference De Concini and ProcesiDCP95; Reference Feichtner and YuzvinskyFY04] in an equivalent formulation given in [Reference PostnikovPos09, §7]. A building set is a collection ${\mathcal {G}}$ of subsets of $\widetilde E$ such that $\{i\} \in {\mathcal {G}}$ for any $i\in \widetilde E$ , and if S and $S'$ are in ${\mathcal {G}}$ with $S\cap S' \neq \emptyset $ , then so is $S\cup S'$ . The nested complex ${\mathcal {N}}$ of a building set ${\mathcal {G}}$ is a simplicial complex on vertices ${\mathcal {G}}$ whose faces are collections $\{X_1, \ldots , X_k\} \subseteq {\mathcal {G}}$ such that for every subcollection $\{X_{i_1}, \ldots , X_{i_\ell }\}$ with $\ell \geq 2$ consisting only of pairwise incomparable elements, one has $\bigcup _{j=1}^\ell X_{i_j} \notin {\mathcal {G}}$ . When $\widetilde E\in {\mathcal {G}}$ , the set of cones

$$\begin{align*}\big\{\operatorname{cone}\{\overline{{\mathbf{e}}}_{X_1}, \ldots, \overline{{\mathbf{e}}}_{X_k}\}\subseteq {\mathbb{R}}^{\widetilde E}/{\mathbb{R}}{\mathbf{e}}_{\widetilde E} \mid \{X_1, \ldots, X_k\}\subseteq {\mathcal{G}}\setminus\{\emptyset, \widetilde E\} \text{ a face of}\ {\mathcal{N}} \big\} \end{align*}$$

is a smooth fan in ${\mathbb {R}}^{\widetilde E}/{\mathbb {R}}{\mathbf {e}}_{\widetilde E}$ that coarsens the permutohedral fan $\underline {\Sigma }_{{\widetilde E}}$ .

3.3 Polymatroids

A standard correspondence between polyhedra and divisors on toric varieties [Reference Cox, Little and SchenckCLS11, §6.2] (see also [Reference Ardila, Castillo, Eur and PostnikovACEP20, §2.4]) states the following: For a lattice polytope Q and the toric variety $X_Q$ defined by its normal fan $\Sigma _Q$ , the base-point-free torus-invariant divisors on $X_Q$ are in bijection with deformations of Q, which are lattice polytopes whose normal fans coarsen $\Sigma _Q$ . We show that specializing this to the stellahedral variety $X_E$ gives a correspondence between the set of base-point-free divisor classes on $X_E$ and a family of polytopes called ‘polymatroids’ introduced in [Reference EdmondsEdm70].

We will use the following ‘strong normality’ of integral polymatroids in the proof of Proposition 3.16.

This property of polymatroids implies that the closure of the image of the map

$$\begin{align*}T \to {\mathbb{P}}^{|P_1 \cap {\mathbb{Z}}^E|-1} \times \cdots \times {\mathbb{P}}^{|P_k \cap {\mathbb{Z}}^E|-1} \text{ defined by } t\mapsto ([t^m]_{m\in P_1\cap {\mathbb{Z}}^E}, \dots, [t^m]_{m\in P_k\cap {\mathbb{Z}}^E}) \end{align*}$$

is isomorphic to the toric variety of the normal fan of $P_1 + \cdots + P_k$ . For a general discussion of normal polytopes in toric geometry, see [Reference Cox, Little and SchenckCLS11, Chapter 2].

To relate polymatroids to base-point-free divisor classes on $X_E$ , we will need the following equivalent description of (integral) polymatroids. A function $f \colon 2^E \to {\mathbb {R}}$ with $f(\emptyset ) = 0$ is said to be nondecreasing and submodular if

• (nondecreasing) $f(S)\leq f(S')$ whenever $S\subseteq S' \subseteq E$ , and

• (submodular) $f(S\cup S') + f(S\cap S') \leq f(S) + f(S')$ for all $S,S'\subseteq E$ .

Theorem 3.11. [Reference EdmondsEdm70, (8)]

Polymatroids on E are in bijection with nondecreasing and submodular functions $f \colon 2^E \to {\mathbb {R}}$ with $f(\emptyset ) = 0$ . The bijection is given by

$$ \begin{align*} \text{a polytope}\ P \quad&\mapsto\quad f \colon 2^E \to {\mathbb{R}} \text{ where } f_P(S) = \max\{ \langle u, {\mathbf{e}}_S\rangle \mid u\in P\} \text{ for}\ S\subseteq E\\ \text{a function } f \colon 2^E\to {\mathbb{R}} \quad&\mapsto\quad P = \{u\in {\mathbb{R}}_{\geq 0}^E \mid \langle {\mathbf{e}}_S, u \rangle \leq f(S) \text{ for all }S\subseteq E\}. \end{align*} $$

A polymatroid P is integral if and only if the function f is ${\mathbb {Z}}$ -valued.Footnote ²

The following proposition implies that, up to translation, polymatroids are exactly the deformations of the stellahedron.

Proposition 3.13. For a proper subset $\emptyset \subseteq S\subsetneq E$ , let $D_S$ be the torus-invariant divisor on $X_E$ corresponding to the ray $\sigma _{\emptyset \leq \{S\}} = \operatorname {cone}(-{\mathbf {e}}_{E\setminus S})$ of $\Sigma _E$ . Let $[D_S]$ be its divisor class in $A^1(X_E)$ . Then the map defined by

$$\begin{align*}(\text{integral polymatroid}\ P\ \text{defined by}\ f \colon 2^E \to {\mathbb{Z}}) \mapsto \sum_{\emptyset\subseteq S \subsetneq E} f(E\setminus S)[D_S] \in A^1(X_E) \end{align*}$$

is a bijection between the set of integral polymatroids on E and the set of base-point-free divisor classes on $X_E$ .

For the proof, we will need the following consequence of Proposition 3.3, which follows from [Reference Cox, Little and SchenckCLS11, Theorem 6.1.7].

Corollary 3.14. (cf. [Reference Braden, Huh, Matherne, Proudfoot and WangBHM⁺22, Proposition 2.10]) A collection of rays in $\Sigma _E$ is a minimal collection of rays that do not form a cone in $\Sigma _E$ if and only if the collection is either

$$\begin{align*}\{\rho_i, \rho_S\} \text{ for}\ i\notin S \subsetneq E \qquad\text{or}\qquad \{\rho_S, \rho_{S'}\} \text{ for incomparable}\ S,S'\subsetneq E. \end{align*}$$

For an integral polymatroid P, let $D_P = \sum _{S\subsetneq E} f(E\setminus S)D_S$ be the corresponding divisor on $X_E$ . Let $X_P$ be the toric variety of the normal fan of P, considered as a fan in ${\mathbb {R}}^E$ so that $X_P$ is considered as a T-variety. Note that $X_P$ may have dimension less than n, so the action of T on $X_P$ may have a nontrivial kernel.

3.4 Orbit-closure in a flag variety and additive-equivariance

We have so far described the structure of $X_E$ as a toric variety, that is, in terms of the T-action. Here, we show that $X_E$ admits an action by a larger group that contains the additive group $\mathbb {G}_a^E$ . Let us begin with the one-dimensional case.

The multiplicative group $\mathbb {G}_m$ acts on the additive group $\mathbb {G}_a$ via $t \cdot b = tb$ for $t \in \mathbb {G}_m$ and $b \in \mathbb {G}_a$ . Let $\mathbb {G} = \mathbb {G}_m \ltimes \mathbb {G}_a$ be semidirect product. Concretely, the groups $\mathbb {G}_m$ , $\mathbb {G}_a$ , and $\mathbb {G}$ embed into $GL_2$ as follows.

$$ \begin{align*} \mathbb{G}_m, \mathbb{G}_a, \mathbb{G} \hookrightarrow GL_2 \quad\text{via} \quad t \mapsto \begin{pmatrix}t & 0 \\ 0 & 1\end{pmatrix},\ b \mapsto \begin{pmatrix} 1 & b \\ 0 & 1\end{pmatrix},\ (t,b) \mapsto \begin{pmatrix} t & b \\ 0 & 1\end{pmatrix}. \end{align*} $$

We denote by $V = {\mathbb {k}}^2$ the resulting ${\mathbb {G}}$ -representation. The group $\mathbb {G}$ thus acts on ${\mathbb {P}}(V) = \mathbb {P}^1$ by

$$\begin{align*}(t,b) \cdot [x : y] = [tx + by : y] \end{align*}$$

with two orbits $\{[x:1] \mid b\in {\mathbb {k}}\}\simeq {\mathbb {A}}^1$ and $\{[1:0]\}$ , denoted $\{\infty \}$ . When we treat ${\mathbb {P}}^1$ as the toric variety of the fan in ${\mathbb {R}}^1$ consisting of the three cones $\{{\mathbb {R}}_{\geq 0}, {\mathbb {R}}_{\leq 0}, \{0\}\}$ , the orbit $\mathbb {A}^1_o$ is identified with the toric affine chart of $\mathbb {P}^1$ corresponding to ${\mathbb {R}}_{\geq 0}$ . In particular, letting $D_{[0,1]}$ be the toric divisor on ${\mathbb {P}}^1$ corresponding to the interval $[0,1]\subset {\mathbb {R}}^1$ , we may identify $V = H^0(\mathbb {P}^1, {\mathcal {O}}_{{\mathbb {P}}^1}(1))^\vee $ by giving T-linearization of ${\mathcal {O}}_{{\mathbb {P}}^1}(1)$ as ${\mathcal {O}}_{{\mathbb {P}}^1}(\infty ) = {\mathcal {O}}_{{\mathbb {P}}^1}(D_{[0,1]})$ .

Let us now show that the stellahedral variety $X_E$ admits a ${\mathbb {G}}^E$ -action. We do this by realizing $X_E$ as a ${\mathbb {G}}^E$ -orbit closure in a flag variety. While there are several alternate ways to exhibit the ${\mathbb {G}}^E$ -action on $X_E$ , as listed in Remark 3.18, the orbit closure description will be useful for defining the augmented tautological bundles in the next section.

From the ${\mathbb {G}}$ -action on $V = {\mathbb {k}}^2$ , we endow $V^E \simeq {\mathbb {k}}^E \oplus {\mathbb {k}}^E$ with the ${\mathbb {G}}^E$ -action given by $(\mathbf {t}, \mathbf {b}) \mapsto \begin {pmatrix} \text {diag}(\mathbf {t}) & \text {diag}(\mathbf {b}) \\ 0 & I \end {pmatrix}$ . Let $\Delta \colon {\mathbb {k}}^E \to V^E$ be the diagonal embedding.

Proof. We first consider the case when $\ell = 1$ , so we are taking the $\mathbb {G}^E$ -orbit closure of $[\Delta (L_1)]$ in $Gr(\dim (L_1); V^E)$ . Let A be a matrix whose rows form a basis for $L_1$ , so the rows of $\begin {pmatrix} A & A \end {pmatrix}$ form a basis for $\Delta (L_1)$ . Then the $\mathbb {G}^E$ -action on $Gr( \dim (L_1); V^E)$ is given by

$$ \begin{align*}(\mathbf{t}, \mathbf{b}) \cdot \left[ \begin{pmatrix} A & A \end{pmatrix} \right] = \left[ \begin{pmatrix} A & A \end{pmatrix} \begin{pmatrix} \text{diag}(\mathbf{t}) & \text{diag}(\mathbf{b}) \\ 0 & I \end{pmatrix}^t \right] = \left[ \begin{pmatrix} (\mathbf{t} + \mathbf{b})A & A\end{pmatrix} \right].\end{align*} $$

This implies that the T-orbit closure coincides with the $\mathbb {G}^E$ -orbit closure.

The normalization of $\overline {T \cdot [\Delta (L_1)]}$ is a toric variety, so it is defined over $\operatorname {Spec} \mathbb {Z}$ . We may therefore consider the moment polytope of its complexification, which is given a polarization via the Plücker embedding of the Grassmannian. The vertices of the moment polytope are given by the T-weights of the nonzero maximal minors of $\begin {pmatrix} A & A \end {pmatrix}$ , where T acts by scaling the first n columns. Every nonzero maximal minor of $\begin {pmatrix} A & A\ \end {pmatrix}$ is given by a subset $S_1$ of the first n rows and a subset $S_2$ of the second n rows such that $S_1 \sqcup S_2$ is a basis for ${\mathrm {M}}_1$ . The T-weight of this minor is ${\mathbf {e}}_{S_1}$ , so the moment polytope is $I({\mathrm {M}}_1)$ .

Let S be the set of nonloops of ${\mathrm {M}}_1$ . The vertices of $I({\mathrm {M}}_1)$ generate the lattice $\mathbb {Z}^S$ , which implies that the character lattice of the embedded torus in the normalization of $\overline {T \cdot [\Delta (L_1)]}$ is $\mathbb {Z}^S$ . Every lattice point in $I({\mathrm {M}}_1)$ is a vertex, so the restriction map $H^0(Gr(\dim (L_1); V^E); \mathcal {O}(1)) \to H^0(\overline {T \cdot [\Delta (L_1)]}, \mathcal {O}(1))$ is surjective. By Proposition 3.10, $\overline {T \cdot [\Delta (L_1)]}$ is projectively normal and therefore normal, so $\overline {T \cdot [\Delta (L_1)]}$ is isomorphic to $X_{I({\mathrm {M}}_1)}$ .

We now treat the general case. There is an embedding $Fl(\dim (L_{1}), \dotsc , \dim (L_{\ell }); V^E) \hookrightarrow \prod _{i=1}^{\ell } Gr(\dim (L_i); V^E)$ and the computation above implies that the T-orbit closure of $[\Delta (\mathscr {L})]$ is also the $\mathbb {G}^E$ -orbit closure. By Proposition 3.10, the Segre embedding of $\overline {T \cdot [\Delta (\mathcal {L})]}$ corresponds to the Minkowski sum of polytopes (with the complete linear series), which implies that the moment polytope of $\overline {T \cdot [\Delta (\mathscr {L})]}$ is P. Using that P is a normal polytope, we get that $\overline {T \cdot [\Delta (\mathscr {L})]}$ is isomorphic to $X_{P}$ .

The flag of matroids realized by a general full flag $\mathscr {L} = \{L_1 \subsetneq L_2 \subsetneq \cdots \subsetneq L_n = {\mathbb {k}}^E\}$ over an infinite field ${\mathbb {k}}$ are exactly the uniform matroids ${\mathrm {U}}_{1,E}, \ldots , {\mathrm {U}}_{n,E}$ . Since the stellahedron $\Pi _E$ is the Minkowski sum $I({\mathrm {U}}_{1, E}) + \dotsb + I({\mathrm {U}}_{n, E})$ , we have the following corollary.

4.1 Well-definedness

We now construct the augmented tautological bundles and augmented tautological classes. Recall the notation $V^E = {\mathbb {k}}^E \oplus {\mathbb {k}}^E$ . Recall that for any polymatroid P (such as an independence polytope), one has a T-equivariant map $X_E \to X_P$ because the normal fan $\Sigma _P$ coarsens $\Sigma _E$ . Let us prepare with the following trivial case.

Given a linear subspace $L\subseteq {\mathbb {k}}^E$ , we now construct vector bundles fitting into a short exact sequence that is modeled after $0\to L \to {\mathbb {k}}^E \to {\mathbb {k}}^E/L \to 0$ . Because we would like at least one of the vector bundles to be globally generated, the vector bundles ${\mathcal {S}}_L$ and ${\mathcal {Q}}_L$ will be defined so that they fit into the short exact sequence $ 0 \to {\mathcal {S}}_L \to \bigoplus _{i\in E} \pi _i^*\mathcal {O}_{\mathbb {P}^1}(1) \to {\mathcal {Q}}_L \to 0 $ with $\bigoplus _{i\in E} \pi _i^*\mathcal {O}_{\mathbb {P}^1}(1)$ in the middle instead of $\bigoplus _{i\in E} \pi _i^*\mathcal {O}_{\mathbb {P}^1}(-1)$ . As a result, when we define the dual bundle ${\mathcal {Q}}_L^\vee $ , we are led to consider the orthogonal dual $L^\perp = ({\mathbb {k}}^E/L)^\vee \subseteq {\mathbb {k}}^E$ of the realization $L\subseteq {\mathbb {k}}^E$ of a matroid ${\mathrm {M}}$ , which realizes the dual matroid ${\mathrm {M}}^\perp $ .

Definition 4.2. Let $L\subseteq {\mathbb {k}}^E$ be a realization of a rank r matroid ${\mathrm {M}}$ on E. Setting $\ell =2$ and $L_1 = L^{\perp } \subseteq L_2 = {\mathbb {k}}^E$ in Proposition 3.16 supplies us with a map

$$\begin{align*}X_E \to X_{I({\mathrm{M}}^\perp) + I({\mathrm{U}}_{n,E})}\to Fl(n-r,n;V^E). \end{align*}$$

Define the augmented tautological bundles ${\mathcal {S}}_L$ and ${\mathcal {Q}}_L$ by

$$ \begin{align*} {\mathcal{Q}}_L &= \text{the dual of the pullback to}\ X_E\ \text{of the tautological rank}\ n-r\ \text{subbundle of}\ Fl(n-r,n;V^E)\\ {\mathcal{S}}_L &= \text{the dual of the quotient bundle}\ \bigoplus_{i \in E} \pi_i^* \mathcal{O}(-1)/\mathcal{Q}_L^{\vee}. \end{align*} $$

That ${\mathcal {Q}}_L^\vee $ is a subbundle of $\bigoplus _{i \in E} \pi _i^* \mathcal {O}(-1)$ follows from Lemma 4.1 and the fact that Proposition 3.16 supplies us with a commuting diagram

Remark 4.3. By construction, we have a short exact sequence of $\mathbb {G}^E$ -equivariant vector bundles

$$ \begin{align*}0 \to \mathcal{S}_L \to \bigoplus_{i \in E} \pi_i^* \mathcal{O}_{\mathbb{P}^1}(1) \to \mathcal{Q}_L \to 0,\end{align*} $$

which, when restricted to the ${\mathbb {G}}^E$ -orbit $\mathbb {A}^E$ , is canonically identified with

$$ \begin{align*}0 \to \mathcal{O}_{\mathbb{A}^E} \otimes L \to \mathcal{O}_{\mathbb{A}^E} \otimes {\mathbb{k}}^E \to \mathcal{O}_{\mathbb{A}^E} \otimes {\mathbb{k}}^E/L \to 0.\end{align*} $$

For arbitrary matroids ${\mathrm {M}}$ , we construct (T-equivariant) K-classes $[{\mathcal {S}}_{\mathrm {M}}]$ and $[{\mathcal {Q}}_{\mathrm {M}}]$ on $X_E$ . By Theorem 2.1.(1), the T-equivariant K-ring of $X_E$ is identified with a subring of the product ring $\prod _{\Sigma _E(n)} {\mathbb {Z}}[T_1^{\pm 1}, \dotsc , T_n^{\pm 1}]$ . So we will specify these classes by specifying their localization values at each torus-fixed point indexed by a maximal cone of $\Sigma _E$ .

By Proposition 3.3, the maximal cones of $\Sigma _E$ are in bijection with compatible pairs $I \leq \mathscr F$ , where $\emptyset \subseteq I\subseteq E$ and $\mathscr F$ is a (possibly empty) maximal chain of proper subsets of E containing I. For a chain $\mathscr F$ containing I, write $\mathscr F/I$ for the new chain of subsets of $E\setminus I$ obtained by removing I from each subset in the original chain. A maximal chain $\mathscr F: \emptyset \subsetneq F_1 \subsetneq \cdots \subsetneq F_{n-1}$ orders the ground set by $F_1 < F_2\setminus F_1 < \cdots < E\setminus F_{n-1}$ , and for each matroid ${\mathrm {M}}$ on E we denote:

• • $B_{\mathscr F}({\mathrm {M}})$ the minimal basis of ${\mathrm {M}}$ under the lexicographic ordering, and

• • $B^c_{\mathscr F}({\mathrm {M}})$ the complement of $B_{\mathscr F}({\mathrm {M}})$ in the ground set of ${\mathrm {M}}$ .

Proposition 4.4. For a matroid ${\mathrm {M}}$ on E, the augmented tautological classes defined as

$$ \begin{align*} [{\mathcal{S}}_{\mathrm{M}}]_{I \leq \mathscr F} &= \operatorname{rk}_{\mathrm{M}}(I)+ \sum_{i\in B_{\mathscr F/I}({\mathrm{M}}/I)} T_i^{-1} \quad\text{and}\\ [{\mathcal{Q}}_{\mathrm{M}}]_{I \leq \mathscr F} &= |I| - \operatorname{rk}_{\mathrm{M}}(I) + \sum_{i\in B^c_{\mathscr F/I}({\mathrm{M}}/I)} T_i^{-1} \end{align*} $$

are well-defined T-equivariant K-classes on $X_E$ . Moreover, if L is a realization of ${\mathrm {M}}$ , then $[{\mathcal {S}}_L] = [{\mathcal {S}}_{\mathrm {M}}]$ and $[{\mathcal {Q}}_L] = [{\mathcal {Q}}_{\mathrm {M}}$ ].

Proof. First, we check that $[\mathcal {Q}_L] = [\mathcal {Q}_{\mathrm {M}}]$ . Then taking the case $L = \{0\}$ gives that

$$ \begin{align*}[\bigoplus_{i \in E} \pi_i^* \mathcal{O}_{\mathbb{P}^1}(1)]_{I \le \mathscr{F}} = |I| + \sum_{i \in E \setminus I} T_i^{-1}.\end{align*} $$

As $[\mathcal {S}_L] + [\mathcal {Q}_L] = [\bigoplus _{i \in E} \pi _i^* \mathcal {O}_{\mathbb {P}^1}(1)]$ , this implies that $[\mathcal {S}_L] = [\mathcal {S}_{\mathrm {M}}]$ .

Let $L \subseteq {\mathbb {k}}^E$ be a subspace of dimension r. Note that the rank $n - r$ tautological subbundle $\mathcal {\mathcal {S}}$ on $Fl(n-r, n; V^E)$ is pulled back from the forgetful map $Fl(n-r, n; V^E) \to Gr(n-r; V^E)$ . The image of the T-fixed point on $X_E$ corresponding to a maximal compatible pair $I \le \mathscr {F}$ is a T-fixed point $\mathrm {p}$ of $Gr(n-r; V^E)$ such that every nonzero Plücker has weight equal to the vertex of $I({\mathrm {M}}^{\perp })$ on which any functional in the interior of $\sigma _{I \le \mathscr {F}}$ attains its minimum, which is ${\mathbf {e}}_{B^c_{\mathscr {F}/I}({\mathrm {M}}/I)}$ . Then

$$ \begin{align*}[\mathcal{S}]_{\mathrm{p}} = |I| - \operatorname{rk}_{\mathrm{M}}(I) + \sum_{i \in B^c_{\mathscr{F}/I}({\mathrm{M}}/I)} T_i \in K_T(\mathrm{p}).\end{align*} $$

As pullbacks commute with each other, this implies that $[\mathcal {Q}_L^{\vee }]_{I \le \mathscr {F}} = [\mathcal {S}]_{\mathrm {p}} = |I| - \operatorname {rk}_{\mathrm {M}}(I) + \sum _{i \in B^c_{\mathscr {F}/I}({\mathrm {M}})} T_i$ , so applying $D_K$ gives that $[\mathcal {Q}_L] = [\mathcal {Q}_{\mathrm {M}}]$ . In particular, it gives the claimed formula for $[\bigoplus _{i \in E} \pi _i^* \mathcal {O}_{\mathbb {P}^1}(1)] = [\mathcal {Q}_{\{0\}}]$ .

Now, we check well-definedness. As $[\mathcal {S}_{\mathrm {M}}] + [\mathcal {Q}_{\mathrm {M}}] = [\bigoplus _{i \in E} \pi _i^* \mathcal {O}_{\mathbb {P}^1}(1)]$ , it suffices to check that $[\mathcal {S}_{\mathrm {M}}]$ is well-defined. There are two types of codimension $1$ cones in $\Sigma _E$ . The first type is given by a compatible pair $I \le \mathscr {F}$ where $I = {F}_1$ and there is some $\ell $ such that ${F}_{\ell + 1} \setminus F_{\ell } = \{i, j\}$ . This cone is contained in the kernel of the functional ${\mathbf {e}}_i - {\mathbf {e}}_j$ . Let $\sigma _{I \le \mathscr {F}_1}$ and $\sigma _{I \le \mathscr {F}_2}$ be the two maximal cones containing $\sigma _{I \le \mathscr {F}}$ ; they are obtained by inserting either $F_{\ell } \cup i$ or $F_{\ell } \cup j$ into $\mathscr {F}$ . Because the normal fan of $I({\mathrm {M}}^{\perp })$ coarsens $\Sigma _E$ , the vertices of $I({\mathrm {M}}^{\perp })$ that functionals in the interiors of $\sigma _{I \le \mathscr {F}_1}$ and $\sigma _{I \le \mathscr {F}_2}$ attain their minimum on are either identical or differ by an edge. Because $\sigma _{I \le \mathscr {F}_1}$ and $\sigma _{I \le \mathscr {F}_2}$ have the same ‘I’, this edge must be parallel to ${\mathbf {e}}_i - {\mathbf {e}}_j$ , and so the symmetric difference of $B_{\mathscr {F}_1/I}({\mathrm {M}}/I)$ and $B_{\mathscr {F}_2/I}({\mathrm {M}}/I)$ is either $\{i, j\}$ or $\emptyset $ . This implies that, along $\sigma _{I \le \mathscr {F}}$ , $[\mathcal {S}_{\mathrm {M}}]$ satisfies the condition of Theorem 2.1.

The second type of codimension $1$ cone is given by a compatible pair $I \le \mathscr {F}$ when $I \cup j = F_1$ , which is contained in the kernel of ${\mathbf {e}}_j$ . Then the maximal cones containing $\sigma _{I \le \mathscr {F}}$ are $\sigma _{I \cup j \le \mathscr {F}}$ and $\sigma _{I \le \tilde {\mathscr {F}}}$ , where $\tilde {\mathscr {F}}$ is obtained by adding I to $\mathscr {F}$ . Then a similar argument to the first case shows that $B_{\mathscr {F}/I \cup j}({\mathrm {M}}/I \cup j)$ and $B_{\tilde {\mathscr {F}}/I}({\mathrm {M}}/I)$ either coincide or differ by $\{j\}$ .

These augmented tautological bundles and classes are related to the nonaugmented tautological bundles and classes introduced in [Reference Berget, Eur, Spink and TsengBEST23] as follows. Endow $\mathcal {O}_{\underline {X}_E}^{\oplus E}$ with the inverse T-equivariant structure, that is, $(t_1, \dotsc , t_n) \cdot (x_1, \dotsc , x_n) = (t_1^{-1} x_1, \dotsc , t_n^{-1} x_n)$ .

Definition 4.5. Let $L \subseteq {\mathbb {k}}^E$ be a realization of a matroid ${\mathrm {M}}$ . Then the (nonaugmented) tautological bundles $\underline {\mathcal {S}}_L$ and $\underline {\mathcal {Q}}_L$ are the unique T-equivariant vector bundles on $\underline {X}_E$ that fit into a short exact sequence

$$ \begin{align*}0 \to \underline{\mathcal{S}}_L \to \mathcal{O}_{\underline{X}_E}^{\oplus E} \to \underline{\mathcal{Q}}_L \to 0,\end{align*} $$

where the fiber over the identity is identified with

$$ \begin{align*}0 \to L \to {\mathbb{k}}^E \to {\mathbb{k}}^E/L \to 0.\end{align*} $$

One can show that the short exact sequence in the above definition is the restriction to $\underline X_E$ of the short exact sequence $0 \to {\mathcal {S}}_L \to \bigoplus _{i\in E} \pi _i^* \mathcal {O}_{\mathbb {P}^1}(1) \to {\mathcal {Q}}_L \to 0$ .

For each matroid ${\mathrm {M}}$ , the authors of [Reference Berget, Eur, Spink and TsengBEST23] define classes $[\underline {\mathcal {S}}_{\mathrm {M}}]$ and $[\underline {\mathcal {Q}}_{\mathrm {M}}]$ in $K_T(\underline {X}_E)$ . The T-fixed points on $\underline {X}_E$ are in bijection with complete flags $\mathscr {F}$ of subsets of E. The tautological classes are described by

$$ \begin{align*}[\underline{\mathcal{S}}_{\mathrm{M}}]_{\mathscr{F}} = \sum_{i \in B_{\mathscr{F}}({\mathrm{M}})} T_i^{-1} \quad \text{ and } \quad[\underline{\mathcal{Q}}_{\mathrm{M}}]_{\mathscr{F}} = \sum_{i \in B_{\mathscr{F}}^c({\mathrm{M}})} T_i^{-1}.\end{align*} $$

In particular, these are restrictions to $\underline X_E$ of the augmented tautological classes $[{\mathcal {S}}_{\mathrm {M}}]$ and $[{\mathcal {Q}}_{\mathrm {M}}]$ .

4.2 Basic properties

We now develop some basic properties of augmented tautological classes. These properties and their proofs are similar to those considered in [Reference Berget, Eur, Spink and TsengBEST23, Section 5].

Proof. As a T-equivariant K-class, we have from Proposition 4.4 that

$$\begin{align*}[\det {\mathcal{Q}}_{\mathrm{M}}]_{I\leq \mathscr F} = \prod_{i\in B_{\mathscr F/I}^c({\mathrm{M}}/I)}T_i^{-1} \end{align*}$$

for a maximal cone $\sigma _{I\leq \mathscr F}$ of $\Sigma _E$ . Since the vertex of $I({\mathrm {M}}^\perp )$ that minimizes the pairing with a vector in the interior of $\sigma _{I\leq \mathscr F}$ is ${\mathbf {e}}_{B_{\mathscr F/I}^c({\mathrm {M}}/I)}$ , the result follows.

Alternatively, by appealing to Proposition 4.7 one can reduce to the case where ${\mathrm {M}}$ admits a realization L, in which case the diagram above Remark 4.3 implies that $\det {\mathcal {Q}}_L$ defines the map $X_E \to X_{I({\mathrm {M}}^\perp )}$ given by the line bundle ${\mathcal {O}}_{X_E}(D_{I({\mathrm {M}}^\perp )})$ .

For instance, the proposition implies that the assignments ${\mathrm {M}}\mapsto c({\mathcal {Q}}_{\mathrm {M}})$ and ${\mathrm {M}} \mapsto s({\mathcal {Q}}_{\mathrm {M}}^\vee )$ are valuative.

Proof. Let ${\mathbb {Z}}^{2^E}$ be the free abelian group with the standard basis indexed by the subsets of E. Consider the function

$$\begin{align*}\operatorname{Mat}(E) \to \bigoplus_{\Sigma_E(n)} \mathbb{Z}^{2^{E}} \text{ given by } {\mathrm{M}} \mapsto \sum_{\sigma_{I \le \mathscr{F}} \in \Sigma_E(n)} {\mathbf{e}}_{B_{\mathscr{F}/I}({\mathrm{M}}/I)}. \end{align*}$$

By Proposition A.4, this function is valuative; see also [Reference Ardila, Fink and RincónAFR10, Theorem 5.4]. Any fixed polynomial expression in the augmented tautological classes or their Chern classes factors through this map and is therefore valuative.

We now consider how augmented tautological classes restrict to T-invariant subvarieties of $X_E$ . By Corollary 3.4, for a (not necessarily maximal) compatible pair $I \leq \mathscr F: F_1 \subsetneq \cdots \subsetneq F_k$ , the corresponding T-invariant subvariety $Z_{I \leq \mathscr F} \subseteq X_E$ corresponding to the cone $\sigma _{I\leq \mathscr F}$ is naturally identified with

$$\begin{align*}Z_{I\leq\mathscr F} \simeq X_{F_1 \setminus I} \times \prod_{i = 1}^k \underline{X}_{{F_{i+1}\setminus F_i}}. \end{align*}$$

This identification then induces isomorphisms

$$ \begin{align*}K_T(Z_{I\leq\mathscr F}) \stackrel{\sim}{\to} K_T(X_{F_1 \setminus I}) \otimes \bigotimes_{i=1}^{k} K_T(\underline{X}_{{F_{i+1}\setminus F_i}}) \text{ and } A_T^{\bullet}(Z_{I\leq\mathscr F}) \stackrel{\sim}{\to} A_T^{\bullet}(X_{F_1 \setminus I}) \otimes \bigotimes_{i=1}^{k} A_T^{\bullet}(\underline{X}_{{F_{i+1}\setminus F_i}}).\end{align*} $$

Proposition 4.8. Under the above identification, we have that

$$ \begin{align*} [{\mathcal{S}}_{\mathrm{M}}]|_{Z_{I\leq \mathscr F}} &= \operatorname{rk}_{\mathrm{M}}(I)[{\mathcal{O}}_{Z_{I\leq \mathscr F}}] + [{\mathcal{S}}_{{\mathrm{M}}|F_1/I}] \otimes 1^{\otimes k} + \sum_{i = 1}^k 1^{\otimes (i-1)} \otimes [\underline {{\mathcal{S}}}_{{\mathrm{M}}|F_{i+1}/F_i}] \otimes 1^{\otimes (k-i)},\quad\text{and}\\[5pt] [{\mathcal{Q}}_{\mathrm{M}}]|_{Z_{I\leq \mathscr F}} &= (|I| -\operatorname{rk}_{\mathrm{M}}(I))[{\mathcal{O}}_{Z_{I\leq \mathscr F}}] + [{\mathcal{Q}}_{{\mathrm{M}}|F_1/I}]\otimes 1^{\otimes k} + \sum_{i = 1}^k 1^{\otimes (i-1)} \otimes [\underline {{\mathcal{Q}}}_{{\mathrm{M}}|F_{i+1}/F_i}] \otimes 1^{\otimes (k-i)}. \end{align*} $$

In particular, when ${\mathcal {F}} = \emptyset $ , we have that $c({\mathcal {S}}_{\mathrm {M}})|_{Z_I} \simeq c({\mathcal {S}}_{{\mathrm {M}}/I})$ as a class in $A^\bullet (Z_I) \simeq A^\bullet (X_{{E\setminus I}})$ , and similarly for ${\mathcal {Q}}_{\mathrm {M}}$ .

5.1 Augmented wonderful varieties

We note an equivalent description of the augmented wonderful variety, which can be deduced from Proposition 3.6. For a flat $F\subseteq E$ of ${\mathrm {M}}$ , let $L_F = L \cap ({\mathbb {k}}^{E\setminus F} \oplus 0^F)$ . The projective completion $\mathbb {P}(L \oplus {\mathbb {k}})$ of L contains a copy of $\mathbb {P}(L)$ as the hyperplane at infinity, and so it contains a subspace identified with $\mathbb {P}(L_F)$ for every flat F of ${\mathrm {M}}$ . Under the iterated blow-up $\pi _E\colon X_E \to {\mathbb {P}}^E$ , the augmented wonderful variety $W_L$ is the strict transform of ${\mathbb {P}}(L \oplus {\mathbb {k}}) \subseteq {\mathbb {P}}({\mathbb {k}}^E \oplus {\mathbb {k}}) = {\mathbb {P}}^E$ , fitting into the diagram

This makes $W_L$ equal to the variety obtained by blowing up $\mathbb {P}(L \oplus k)$ at the linear spaces ${\mathbb {P}}(L_F)$ corresponding to corank $1$ flats of ${\mathrm {M}}$ , then blowing up at the strict transforms of linear spaces corresponding to corank $2$ flats of ${\mathrm {M}}$ and so on.

We relate augmented wonderful varieties to augmented tautological bundles as follows.

We prepare to prove Theorem 5.2 with the following lemma.

Proof of Theorem 5.2.

Take the vector $v = (1, \ldots , 1, 0, \ldots , 0) \in {\mathbb {k}}^{E} \oplus {\mathbb {k}}^E$ . Let us identify ${\mathbb {k}}^{E} \oplus {\mathbb {k}}^E = H^0(X_E, \bigoplus _{i\in E} \pi ^*_i{\mathcal {O}}(1)) = (V^E)^\vee $ . The vector v then defines a global section of $\bigoplus _{i\in E}\pi ^*_i{\mathcal {O}}(1)$ , and hence a global section of ${\mathcal {Q}}_L$ via the surjection $\bigoplus _{i\in E}\pi ^*_i{\mathcal {O}}(1) \twoheadrightarrow {\mathcal {Q}}_L$ . On the ${\mathbb {G}}^E$ -orbit $\mathbb {A}^E$ of $X_E$ , Remark 4.3 identifies the restriction of v with the section

$$\begin{align*}(x_1, \dotsc, x_n) \in \big({\mathbb{k}}[x_1, \dotsc, x_n]\big)^E = H^0({\mathbb{A}}^E, \mathcal{O}_{{\mathbb{A}}^E} \otimes {\mathbb{k}}^E). \end{align*}$$

So the image of v in $H^0({\mathbb {A}}^E, \mathcal {O}_{{\mathbb {A}}^E} \otimes {\mathbb {k}}^E/L)$ vanishes exactly on L. The $\mathbb {G}^E$ -orbit of v is dense in ${\mathbb {k}}^{E} \oplus {\mathbb {k}}^E$ . Hence, by $\mathbb {G}^E$ -equivariance, the $\mathbb {G}^E$ -orbit of the image of v in $H^0(X_E, {\mathcal {Q}}_L)$ is dense in a subspace of $H^0(X_E, {\mathcal {Q}}_L)$ that globally generates ${\mathcal {Q}}_L$ . In other words, the section v is a sufficiently general section satisfying the conclusion of the above lemma, from which the theorem now follows.

Proof. As $W_L$ is a smooth subvariety of $X_E$ of dimension r, that $W_L$ is the vanishing locus of a global section of ${\mathcal {Q}}_L$ implies that the Koszul complex

$$\begin{align*}0 \to \textstyle{\bigwedge^{n- r} {\mathcal{Q}}_L^\vee}\to \cdots \to \textstyle{\bigwedge^2 {\mathcal{Q}}_L^\vee} \to {\mathcal{Q}}_L^\vee \to {\mathcal{O}}_{X_E} \end{align*}$$

is a resolution of ${\mathcal {O}}_{W_L}$ . Both statements now follow.

5.2 Augmented Bergman classes

We describe the Chern classes of augmented tautological classes and recover the augmented Bergman class as the top Chern class. We use the language of Minkowski weights, defined as follows.

Definition 5.5. A d-dimensional Minkowski weight on a unimodular fan $\Sigma $ is a function $w \colon \Sigma (d) \to {\mathbb {Z}}$ such that the following balancing condition is satisfied: for every cone $\tau '\in \Sigma (d-1)$

$$\begin{align*}\sum_{\tau\succ \tau'} w(\tau)u_{\tau'\setminus\tau} \in \operatorname{span}(\tau'), \end{align*}$$

where the summation is over all cones $\tau \in \Sigma (d)$ containing $\tau '$ , and $u_{\tau '\setminus \tau }$ denotes the primitive generator of the unique ray of $\tau $ that is not in $\tau '$ . Write $\operatorname {MW}_d(\Sigma )$ for the set of d-dimensional Minkowski weights on $\Sigma $ .

Minkowski weights play the role of homology classes on smooth complete toric varieties in the following sense.

Theorem 5.6. [Reference Fulton and SturmfelsFS97, Theorem 3.1]

Let $\Sigma $ be a complete unimodular fan of dimension m, and let $X_\Sigma $ be its toric variety. Then, for every $0\leq d \leq m$ , one has an isomorphism

$$\begin{align*}A^{m-d}(X_\Sigma) \overset\sim\to \operatorname{MW}_d(\Sigma) \quad\text{defined by}\quad \xi \mapsto \left( \tau \mapsto \int_X \xi \cdot [Z_\tau]\right). \end{align*}$$

For a smooth complete toric variety $X_\Sigma $ , when a Chow class $\xi \in A^\bullet (X_\Sigma )$ maps to a Minkowski weight $w \in \operatorname {MW}_\bullet (\Sigma )$ by the isomorphism in Theorem 5.6, we say that w and $\xi $ are Poincaré duals of each other, which is notated by writing

$$\begin{align*}\xi \cap [X_{\Sigma}] = w. \end{align*}$$

We compute the Chern classes of the augmented tautological classes in terms of Minkowski weights on $\Sigma _E$ . By Theorem 5.6, this amounts to computing how they intersect with the various torus-invariant strata of $X_E$ , for which we use Proposition 4.8 to reduce to understanding the Chern classes in the top degrees. We hence begin by computing what happens in the top degrees.

Lemma 5.7. We have that

$$ \begin{align*} \int_{X_E} c(\mathcal{Q}_{\mathrm{M}}) &= \begin{cases} 1 & {\mathrm{M}} = {\mathrm{U}}_{0,E} \\ 0 & \text{otherwise,} \end{cases} \qquad\text{ and}\\ \int_{X_E} c(\mathcal{S}_{\mathrm{M}}) &= \begin{cases} 1 & {\mathrm{M}} = {\mathrm{U}}_{n,E} \\ 0 & \text{otherwise}. \end{cases} \end{align*} $$

We will also need the analogous statement for tautological bundles.

Lemma 5.8. [Reference Berget, Eur, Spink and TsengBEST23, Lemma 7.3]

We have that

$$ \begin{align*}\int_{\underline{X}_{E}} c(\underline{\mathcal{Q}}_{{\mathrm{M}}}) &= \begin{cases} 1 & {\mathrm{M}} = {\mathrm{U}}_{1,E} \text{ or }{\mathrm{M}} = {\mathrm{U}}_{0,1} \\ 0 & \text{otherwise}, \end{cases} \text{ and}\\\int_{\underline{X}_{E}} c(\underline{\mathcal{S}}_{{\mathrm{M}}}) &= \begin{cases} (-1)^{n-1} & {\mathrm{M}} = {\mathrm{U}}_{n-1,E} \text{ or } {\mathrm{M}} = {\mathrm{U}}_{1,1} \\ 0 & \text{otherwise}. \end{cases} \end{align*} $$

We now compute the intersection numbers of the Chern classes of $[\mathcal {S}_{\mathrm {M}}]$ and $[\mathcal {Q}_{\mathrm {M}}]$ with the boundary stata. When the minimal element of $\mathscr {F}$ is the empty set, we recover [Reference Berget, Eur, Spink and TsengBEST23, Proposition 7.4].

Proposition 5.9. Let $I \le \mathscr {F}: F_1 \subsetneq F_2 \subsetneq \dotsc \subsetneq F_k$ be a compatible pair, and set $\ell = \operatorname {codim} Z_{I \le \mathscr {F}}$ . As before, we set $F_{k + 1} = E$ , and when $\mathscr {F}$ is empty we interpret $F_1$ as E. Let $[Z_{I\leq {\mathcal {F}}}] \in A^\bullet (X_\Sigma )$ be the Chow class of the T-invariant subvariety $Z_{I\leq {\mathcal {F}}}$ . Then

$$\begin{align*}\int_{X_E} c_{n - \ell}(\mathcal{Q}_{{\mathrm{M}}}) \cdot [Z_{I \le \mathscr{F}}] &= \begin{cases}1 & \begin{aligned} & F_1 \subseteq \operatorname{cl}_{{\mathrm{M}}}(I),\ \text{and for}\ i = 1, \ldots, k,\ \text{exactly}\ k + \operatorname{rk}_{{\mathrm{M}}}(I) - \operatorname{rk}_{{\mathrm{M}}}({\mathrm{M}})\ \text{of} \\ & \text{the minors}\ {\mathrm{M}}|F_{i+1}/F_{i}\ \text{are loops, and the rest are}\ U_{1,F_{i+1}\setminus F_{i}},\end{aligned}\\ \\ 0 & \text{otherwise, and}\end{cases}\\\int_{X_E} c_{n - \ell}(\mathcal{S}_{{\mathrm{M}}}) \cdot [Z_{I \le \mathscr{F}}] &= \begin{cases}(-1)^{\epsilon} & \begin{aligned} &\operatorname{rk}_{{\mathrm{M}}}(F_1) - \operatorname{rk}_{{\mathrm{M}}}(I) = |F_1| - |I|\text{, and for}\ i = 1, \ldots, k,\ \text{exactly} \\ & k + \operatorname{rk}_{{\mathrm{M}}}({\mathrm{M}}) - \operatorname{rk}_{{\mathrm{M}}}(I) - n\ \text{of the minors}\ {\mathrm{M}}|F_{i+1}/F_{i}\ \text{are coloops,} \\ & \text{and the rest are}\ U_{|F_{i+1}\setminus F_{i}| - 1 ,F_{i+1}\setminus F_{i}},\end{aligned}\\ \\ 0 & \text{otherwise,}\end{cases} \end{align*}$$

where $\epsilon = n - k - |F_1|$ .

Proof. We do the case of $\mathcal {S}_{{\mathrm {M}}}$ , the case of $\mathcal {Q}_{{\mathrm {M}}}$ is similar. By Proposition 4.8, we have that

$$ \begin{align*}c(\mathcal{S}_{{\mathrm{M}}}, u)|_{Z_{I \le \mathscr{F}}} = c(\mathcal{S}_{{\mathrm{M}}|F_1/I}, u) \otimes \bigotimes_{i=1}^{k} c(\underline{\mathcal{S}}_{{\mathrm{M}}|_{F_{i+1}}/F_{i}}, u) \in A^{\bullet}(X_{{F_1 \setminus I}}) \otimes \bigotimes_{i=1}^{k} A^{\bullet}(\underline X_{{F_{i+1} \setminus F_{i}}}).\end{align*} $$

Then Lemma 5.7 implies that the intersection number vanishes unless ${\mathrm {M}}|F_1/I$ is Boolean, and each ${\mathrm {M}}|F_{i+1}/F_i$ is either a coloop or is a corank $1$ uniform matroid. Note that ${\mathrm {M}}|F_1/I$ is Boolean if and only if $\operatorname {rk}_{{\mathrm {M}}}(F_1) - \operatorname {rk}_{{\mathrm {M}}}(I) = |F_1| - |I|$ , and the fact that $\operatorname {rk}_{{\mathrm {M}}}({\mathrm {M}}) = \operatorname {rk}_{\mathrm {M}}(I) + \operatorname {rk}_{{\mathrm {M}}}({\mathrm {M}}|F_1/I) + \dotsb + \sum \operatorname {rk}_{{\mathrm {M}}}({\mathrm {M}}|F_{i+1}/F_i)$ implies that, if the intersection number is nonzero, then exactly $k + \operatorname {rk}_{{\mathrm {M}}}({\mathrm {M}}) - \operatorname {rk}_{{\mathrm {M}}}(I) - n$ of the minors ${\mathrm {M}}|F_{i+1}/F_{i}$ are coloops. In this case, the intersection number is $(-1)^{\epsilon }$ , where

$$ \begin{align*}\epsilon = \sum\left(|F_{i+1}/F_i| - 1 \right),\end{align*} $$

where the sum is over the minors such that ${\mathrm {M}}|F_{i+1}/F_i$ is not a coloop. The set E decomposes into a disjoint union of elements where the corresponding minor is a coloop, is in I, is in a noncoloop minor, or is in $F_1 \setminus I$ , so

$$ \begin{align*}n = (k + \operatorname{rk}_{{\mathrm{M}}}({\mathrm{M}}) - \operatorname{rk}_{{\mathrm{M}}}(I) - n) + |I| + (\sum |F_{i+1}/F_i|) + (|F_1| - |I|).\end{align*} $$

We also have that the number of noncoloops is $n + \operatorname {rk}_{{\mathrm {M}}}(I) - \operatorname {rk}_{{\mathrm {M}}}({\mathrm {M}})$ . Substituting, we see that $\epsilon = n - k - |F_1|$ .

We now define and derive certain properties of augmented Bergman fans and augmented Bergman classes.

Definition 5.10. For a matroid ${\mathrm {M}}$ of rank r on E, the augmented Bergman fan, denoted $\Sigma _{\mathrm {M}}$ , is the subfan of $\Sigma _E$ consisting of cones $\sigma _{I \leq \mathscr {F}}$ , where the subset $I\subseteq E$ is independent in ${\mathrm {M}}$ and the flag $\mathscr F$ consists of proper flats of ${\mathrm {M}}$ . The augmented Bergman class $[\Sigma _{\mathrm {M}}]$ of ${\mathrm {M}}$ is the weight

$$\begin{align*}[\Sigma_{\mathrm{M}}] \colon \Sigma_E(r) \to {\mathbb{Z}} \quad\text{defined by}\quad \sigma \mapsto \begin{cases} 1 & \text{if}\ \sigma\in \Sigma_{\mathrm{M}}\\ 0 & \text{otherwise.} \end{cases} \end{align*}$$

[Reference Braden, Huh, Matherne, Proudfoot and WangBHM⁺22, Proposition 2.8] states that, up to scaling, the augmented Bergman class is the unique way to assign weights to the cones of the augmented Bergman fan that results in a Minkowski weight.

By restricting to the permutohedral variety, we recover properties of ‘nonaugmented’ Bergman fans and classes as follows. Note that for a loopless matroid ${\mathrm {M}}$ , the augmented Bergman fan $\Sigma _{\mathrm {M}}$ contains the ray $\rho _\emptyset $ .

The Bergman class of a matroid with a loop is defined to be zero. Since $[{\mathcal {Q}}_{\mathrm {M}}]$ restricts to $[\underline {{\mathcal {Q}}}_{\mathrm {M}}]$ on $\underline {X}_E$ and $[\Sigma _{\mathrm {M}}]$ restricts to $[\underline {\Sigma }_{\mathrm {M}}]$ , Corollary 5.11 recovers the properties of Bergman classes stated in [Reference Berget, Eur, Spink and TsengBEST23, Corollary 7.11].

5.3 Tropical geometry of augmented Bergman fans

The contents of this subsection are not logically necessary for the rest of the paper but will be useful elsewhere. We explain how augmented Bergman fans are related to tropicalizations. We point to [Reference Maclagan and SturmfelsMS15] for a background in tropical geometry.

Proof. Let $\widetilde E = E\sqcup \{0\}$ , and let $p \colon {\mathbb {Z}}^{\widetilde E}/{\mathbb {Z}}{\mathbf {e}}_{\widetilde E} \to {\mathbb {Z}}^E$ be the isomorphism described in §3.2. Under the isomorphism p, we may identify T with the projectivization ${\mathbb {P}} \widetilde T$ of the torus $\widetilde T = ({\mathbb {k}}^*)^{\widetilde E}$ . We show that the tropicalization of $\mathring {L}_b \subseteq {\mathbb {P}} \widetilde T$ is the support of a subfan in $\underline {\Sigma }_{\widetilde E}$ that maps isomorphically under p onto the augmented Bergman fan $\Sigma _{\mathrm {M}}$ .

Let $L = \{{\mathbf {x}} \in {\mathbb {k}}^E \mid A^\perp {\mathbf {x}} = 0\}$ for an $(n-r)\times n$ matrix $A^\perp $ . For an element $b\in \mathbb {G}_a^E$ , let $b'\in \mathbb {G}_a^E$ be such that $L+b = \{{\mathbf {x}} \in {\mathbb {k}}^E \mid A^\perp {\mathbf {x}} = b'\}$ . In other words, the closure of $L+b$ in the projective completion ${\mathbb {P}}({\mathbb {k}}^E \oplus {\mathbb {k}}) = {\mathbb {P}}({\mathbb {k}}^{\widetilde E})$ is the projectivization of the linear subspace $\{({\mathbf {x}}, x_0) \in {\mathbb {k}}^{\widetilde E} \mid A^\perp {\mathbf {x}} - b' x_0 = 0\}$ . Since $b'$ is general because b was, this linear subspace is a realization of the matroid $\widetilde {{\mathrm {M}}} = {\mathrm {M}} \times 0$ on $\widetilde {E}$ called the free coextension of ${\mathrm {M}}$ , whose set of bases is defined as

$$\begin{align*}\{B\cup 0 \mid B \text{ a basis of}\ {\mathrm{M}}\} \cup \{S \subseteq E \mid S\ \text{contains a basis of}\ {\mathrm{M}}\ \text{and}\ |S| = r+1\}. \end{align*}$$

It is a classical statement [Reference SturmfelsStu02; Reference Ardila and KlivansAK06] that the tropicalization of a linear subspace is the support of the Bergman fan of the corresponding matroid. Thus, it suffices now to show that the support of the Bergman fan of the free coextension is equal to that of the augmented Bergman fan under the isomorphism p. This follows from the lemma below, which is a restatement of the discussion in [Reference Mastroeni and McCulloughMM, §5.1].

Lemma 5.14. Let ${\mathrm {M}}$ be a matroid on E, and $\widetilde {{\mathrm {M}}}$ its free coextension matroid on $\widetilde {E}$ . The collection

$$\begin{align*}{\mathcal{G}} = \{F\cup 0 \mid F\subseteq E\text{ a flat of}\ {\mathrm{M}}\} \cup \{i \in E \mid i \text{ not a loop in}\ {\mathrm{M}}\} \end{align*}$$

is a building set on the lattice of flats of $\widetilde {{\mathrm {M}}}$ that induces the fan structure on the support $|\underline {\Sigma }_{\widetilde {{\mathrm {M}}}}| \subseteq {\mathbb {R}}^{\widetilde {E}}/{\mathbb {R}}{\mathbf {e}}_{\widetilde E}$ of the Bergman fan of $\widetilde {{\mathrm {M}}}$ consisting of cones

$$\begin{align*}\operatorname{cone}\{\overline{{\mathbf{e}}}_i \mid i\in I\} + \operatorname{cone}\{\overline{{\mathbf{e}}}_{F\cup 0} \mid F\in \mathscr F\} \end{align*}$$

for each compatible pair $I\leq \mathscr F$ with $I\subseteq E$ independent in ${\mathrm {M}}$ and $\mathscr F$ a flag of nonempty proper flats of ${\mathrm {M}}$ .

We remark that the tropicalization of $(L+b) \cap T$ for a nongeneral b can differ from the support of $\Sigma _{\mathrm {M}}$ . Nonetheless, by ${\mathbb {G}}^E$ -equivariance, the homology class of the closure $W_{L+b}$ of $L+b$ in the stellahedral variety $X_E$ is independent of $b\in {\mathbb {k}}^E$ . Taking b to be general, Proposition 5.13 gives an alternate proof that $[W_L] = [\Sigma _{\mathrm {M}}]$ , for instance by [Reference KatzKat09, Proposition 9.4].