#  $K$-rings of wonderful varieties and matroids

Matt Larson  Department of Mathematics, Stanford University, Stanford, CA
94305<EMAIL_ADDRESS>,  Shiyue Li  Department of Mathematics, Brown
University, Providence, RI 02906<EMAIL_ADDRESS>,  Sam Payne  Department
of Mathematics, University of Texas, Austin, TX 78712<EMAIL_ADDRESS>and
Nicholas Proudfoot  Department of Mathematics, University of Oregon, Eugene,
OR 97403<EMAIL_ADDRESS>
###### Abstract.

We study the $K$-ring of the wonderful variety of a hyperplane arrangement and
give a combinatorial presentation that depends only on the underlying matroid.
We use this combinatorial presentation to define the $K$-ring of an arbitrary
loopless matroid. We construct an exceptional isomorphism, with integer
coefficients, to the Chow ring of the matroid that satisfies a
Hirzebruch–Riemann–Roch-type formula, generalizing a recent construction of
Berget, Eur, Spink, and Tseng for the permutohedral variety (the wonderful
variety of a Boolean arrangement). As an application, we give combinatorial
formulas for Euler characteristics of arbitrary line bundles on wonderful
varieties. We give analogous constructions and results for augmented wonderful
varieties, and for Deligne–Mumford–Knudsen moduli spaces of stable rational
curves with marked points.

###### Contents

  1. 1 Introduction
  2. 2 Feichtner–Yuzvinsky presentations
  3. 3 Simplicial presentations
  4. 4 The $K$-ring of $\overline{\mathcal{M}}_{0,n}$
  5. 5 Matroid $K$-rings
  6. 6 Adams operations and Serre duality
  7. 7 Euler characteristics and simplicial Snapper polynomials
  8. 8 Snapper polynomials in the Feichtner–Yuzvinsky generators
  9. 9 Euler characteristics and Snapper polynomials for $\overline{\mathcal{M}}_{0,n}$
  10. A The simplicial presentation of the augmented Chow ring

##  1\. Introduction

Let $L$ be a finite dimensional vector space over a field $\mathbb{F}$, and
let $\mathcal{A}=\\{H_{e}\mid e\in E\\}$ be a finite multiset of hyperplanes
in $L$ intersecting only at the origin. The wonderful variety
$W_{\\!\mathcal{A}}$ is a smooth compactification of
$\mathbb{P}(L)\setminus\bigcup_{H\in\mathcal{A}}\mathbb{P}(H)$, originally
studied by De Concini and Procesi [DCP95]. The augmented wonderful variety
$W_{\\!\mathcal{A}}^{{\operatorname{aug}}}$ is a smooth compactification of
$L$, introduced in [BHM+22a], that contains $W_{\\!\mathcal{A}}$ as a divisor.
The Chow rings of these spaces have been extensively studied and have
combinatorial presentations that depend only on the underlying matroid. As a
result, such rings are naturally defined for arbitrary, not necessarily
realizable, matroids. Presentations for the Chow ring of $W_{\\!\mathcal{A}}$
appear in [DCP95, FY04, BES21], and this ring is used to prove log concavity
of the characteristic polynomial of $\mathcal{A}$ (and more generally of any
matroid) in [AHK18]. A presentation of the Chow ring of
$W^{{\operatorname{aug}}}_{\\!\mathcal{A}}$ appears in [BHM+22a], and this
ring is used to prove the top-heavy conjecture and nonnegativity of the
Kazhdan–Lusztig polynomials of matroids [BHM+22b].

Our goal is to study the $K$-rings of these spaces. We first give
presentations for the $K$-rings using generators analogous to the
Feichtner–Yuzvinsky generators for the Chow rings. Despite the fact that the
(non-homogeneous) relations among the Feichtner–Yuzvinsky generators in the
$K$-ring are different from the (homogeneous) relations among the
Feichtner–Yuzvinsky generators in the Chow ring, we construct an integral
isomorphism from the $K$-ring to the Chow ring which satisfies a
Hirzebruch–Riemann–Roch-type formula, generalizing recent results for Boolean
arrangements [BEST22, EHL22]. The $K$-rings admit additional structures, such
as Adams operations, an Euler characteristic map, and Serre duality, which
leads us to new results and new questions. Most of our results can be extended
to the moduli space $\overline{\mathcal{M}}_{0,n}$ of stable rational curves
with $n$ marked points, which is closely related to the wonderful variety for
the braid arrangement $\mathcal{B}_{n-1}$.

###  1.1. Definitions of the varieties

For any $S\subset E$, let

$L_{S}:=\bigcap_{e\in S}H_{e}L^{S}:=L/L_{S}.$

The dimension of $L^{S}$ is called the rank of $S$, and the dimension of
$L_{S}$ is called the corank. A subset $F\subset E$ is called a flat if it is
maximal within its rank, or, equivalently, if $L_{F}\subset H_{e}\Rightarrow
e\in F$.

The wonderful variety $W_{\\!\mathcal{A}}$ is defined as the closure of the
image of the rational map

$\mathbb{P}(L)\dasharrow\prod_{F}\mathbb{P}(L^{F}),$

and the augmented wonderful variety
$W^{{\operatorname{aug}}}_{\\!\mathcal{A}}$ is defined as the closure of the
image of the rational map

$\mathbb{P}(L\oplus\mathbb{F})\dasharrow\prod_{F}\mathbb{P}(L^{F}\oplus\mathbb{F}),$

where both products are over the set of nonempty flats. For any nonempty flat
$F$, let

$\pi_{F}:W_{\\!\mathcal{A}}\to\mathbb{P}(L^{F})\pi^{\operatorname{aug}}_{F}:W^{{\operatorname{aug}}}_{\\!\mathcal{A}}\to\mathbb{P}(L^{F}\oplus\mathbb{F})$

be the natural projections. We will write

$\mathcal{L}_{F}:=\pi_{F}^{*}\mathcal{O}_{\mathbb{P}(L^{F})}(1)\mathcal{L}_{F}^{\operatorname{aug}}:=(\pi_{F}^{\operatorname{aug}})^{*}\mathcal{O}_{\mathbb{P}(L^{F}\oplus\,\mathbb{F})}(1).$

The map $\pi_{E}:W_{\\!\mathcal{A}}\to\mathbb{P}(L)$ is an iterated blow-up.
It is obtained by blowing up first the points $\mathbb{P}(L_{F})$ for all
corank 1 flats $F$, then the strict transforms of the lines
$\mathbb{P}(L_{F})$ for all corank 2 flats $F$, and so on. For any nonempty
proper flat $F$, let $D_{F}\subset W_{\\!\mathcal{A}}$ be the strict transform
of $\mathbb{P}(L_{F})$. Similarly, the map
$\pi^{\operatorname{aug}}_{E}:W^{{\operatorname{aug}}}_{\\!\mathcal{A}}\to\mathbb{P}(L\oplus\mathbb{F})$
is an iterated blow-up, obtained by blowing up first the points
$\mathbb{P}(L_{F}\oplus\\{0\\})$ for all corank 1 flats $F$, then the strict
transforms of the lines $\mathbb{P}(L_{F}\oplus\\{0\\})$ for all corank 2
flats $F$, and so on. For any proper flat $F$, let
$D_{F}^{\operatorname{aug}}\subset W^{{\operatorname{aug}}}_{\\!\mathcal{A}}$
be the strict transform of $\mathbb{P}(L_{F}\oplus\\{0\\})$. In addition, for
any $e\in E$, let $D_{e}^{\operatorname{aug}}\subset
W^{{\operatorname{aug}}}_{\\!\mathcal{A}}$ be the strict transform of
$\mathbb{P}(H_{e}\oplus\mathbb{F})$. We have a canonical isomorphism
$W_{\\!\mathcal{A}}\cong D_{\varnothing}^{\operatorname{aug}}\subset
W^{{\operatorname{aug}}}_{\\!\mathcal{A}}$, which induces identifications
$\mathcal{L}_{F}=\mathcal{L}_{F}^{\operatorname{aug}}|_{W_{\\!\mathcal{A}}}$
and $D_{F}=D_{F}^{\operatorname{aug}}\cap W_{\\!\mathcal{A}}$ for all nonempty
proper $F$.

###  1.2. Feichtner–Yuzvinsky presentations

Let $A(W_{\\!\mathcal{A}})$ and $A(W^{{\operatorname{aug}}}_{\\!\mathcal{A}})$
denote the Chow rings of $W_{\\!\mathcal{A}}$ and
$W^{{\operatorname{aug}}}_{\\!\mathcal{A}}$, respectively. Let
$T_{\mathcal{A}}:=\mathbb{Z}[x_{F}\mid\text{$F$ a
flat}]\otimes\mathbb{Z}[y_{e}\mid e\in E].$ Consider the map

$\Phi_{\operatorname{FY}}^{\operatorname{aug}}:T_{\mathcal{A}}\to
A(W^{{\operatorname{aug}}}_{\\!\mathcal{A}})$

that sends $x_{F}$ to $[D_{F}^{\operatorname{aug}}]$ for all proper $F$,
$x_{E}$ to $-c_{1}(\mathcal{L}_{E}^{\operatorname{aug}})$, and $y_{e}$ to
$[D_{e}^{\operatorname{aug}}]$ for all $e\in E$. Composing with the pullback
along the inclusion $W_{\\!\mathcal{A}}\subset
W^{{\operatorname{aug}}}_{\\!\mathcal{A}}$, we obtain a map

$\Phi_{\operatorname{FY}}:T_{\mathcal{A}}\to A(W_{\\!\mathcal{A}})$

that sends $x_{F}$ to $[D_{F}]$ for all nonempty proper $F$ and $y_{e}$ to
zero.

We define the following ideals in $T_{\mathcal{A}}$:

(1) $\displaystyle\mathcal{I}_{1}$
$\displaystyle:=\Big{\langle}\sum_{F}x_{F}\Big{\rangle}$
$\displaystyle\mathcal{I}_{2}$
$\displaystyle:=\Big{\langle}y_{e}-\sum_{e\notin F}x_{F}\;\Big{|}\;e\in
E\Big{\rangle}$ $\displaystyle\mathcal{I}_{3}$ $\displaystyle:=\langle
x_{F}x_{G}\mid\text{$F$ and $G$ incomparable}\rangle$
$\displaystyle\mathcal{I}_{4}$ $\displaystyle:=\langle y_{e}\mid e\in
E\rangle$ $\displaystyle\mathcal{I}_{4}^{\operatorname{aug}}$
$\displaystyle:=\langle y_{e}x_{F}\mid e\notin F\rangle.$

######  Theorem 1.1.

Let $\mathcal{A}$ be a hyperplane arrangement.

  * (1)

The map $\Phi_{\operatorname{FY}}$ is surjective with kernel
$\mathcal{I}_{1}+\mathcal{I}_{2}+\mathcal{I}_{3}+\mathcal{I}_{4}$ [FY04,
Corollary 2].

  * (2)

The map $\Phi_{\operatorname{FY}}^{\operatorname{aug}}$ is surjective with
kernel
$\mathcal{I}_{1}+\mathcal{I}_{2}+\mathcal{I}_{3}+\mathcal{I}_{4}^{\operatorname{aug}}$
[BHM+22a, Remark 2.13].

######  Remark 1.2.

The original Feichtner–Yuzvinsky presentation of $A(W_{\\!\mathcal{A}})$ does
not include the variable $x_{\varnothing}$. Many later sources omit both
$x_{\varnothing}$ and $x_{E}$, since the remaining $x_{F}$ are sufficient to
generate and correspond to boundary divisors on $W_{\\!\mathcal{A}}$ (or to
rays of the Bergman fan). In [AHK18], $-\Phi_{\operatorname{FY}}(x_{E})$ is
called $\alpha$ and $-\Phi_{\operatorname{FY}}(x_{\varnothing})$ is called
$\beta$. Similarly, the presentation of
$A(W^{{\operatorname{aug}}}_{\\!\mathcal{A}})$ in [BHM+22a] does not include
the variable $x_{E}$.

Next, consider the map

$\Psi_{\operatorname{FY}}^{\operatorname{aug}}:T_{\mathcal{A}}\to
K(W^{{\operatorname{aug}}}_{\\!\mathcal{A}})$

that sends $x_{F}$ to $[\mathcal{O}_{D_{F}^{\operatorname{aug}}}]$ for all
proper $F$, $x_{E}$ to $1-[\mathcal{L}_{E}^{\operatorname{aug}}]$, and $y_{e}$
to $[\mathcal{O}_{D_{e}^{\operatorname{aug}}}]$ for all $e\in E$. Composing
with the pullback along the inclusion $W_{\\!\mathcal{A}}\subset
W^{{\operatorname{aug}}}_{\\!\mathcal{A}}$, we obtain a map

$\Psi_{\operatorname{FY}}:T_{\mathcal{A}}\to K(W_{\\!\mathcal{A}})$

that sends $x_{F}$ to $[\mathcal{O}_{D_{F}}]$ for all nonempty proper $F$ and
$y_{e}$ to zero.

We define the following ideals in $T_{\mathcal{A}}$:

(2) $\displaystyle\mathcal{I}_{1}^{\prime}$
$\displaystyle:=\Big{\langle}1-\prod_{F}(1-x_{F})\Big{\rangle}$
$\displaystyle\mathcal{I}_{2}^{\prime}$
$\displaystyle:=\Big{\langle}-(1-y_{e})+\prod_{e\notin
F}(1-x_{F})\;\Big{|}\;e\in E\Big{\rangle}.$

Note that the generators of $\mathcal{I}_{1}^{\prime}$ and
$\mathcal{I}_{2}^{\prime}$ are inhomogeneous, and their lowest order terms
coincide with the generators of $\mathcal{I}_{1}$ and $\mathcal{I}_{2}$. Our
first main result is the following $K$-theoretic analogue of Theorem 1.1.

######  Theorem 1.3.

Let $\mathcal{A}$ be a hyperplane arrangement.

  * (1)

The map $\Psi_{\operatorname{FY}}$ is surjective with kernel
$\mathcal{I}_{1}^{\prime}+\mathcal{I}^{\prime}_{2}+\mathcal{I}_{3}+\mathcal{I}_{4}$.

  * (2)

The map $\Psi_{\operatorname{FY}}^{\operatorname{aug}}$ is surjective with
kernel
$\mathcal{I}_{1}^{\prime}+\mathcal{I}^{\prime}_{2}+\mathcal{I}_{3}+\mathcal{I}_{4}^{\operatorname{aug}}$.

######  Remark 1.4.

The difference between the presentations in Theorems 1.1 and 1.3 is that the
homogeneous ideals $\mathcal{I}_{1}$ and $\mathcal{I}_{2}$ are replaced by the
inhomogeneous ideals $\mathcal{I}_{1}^{\prime}$ and
$\mathcal{I}_{2}^{\prime}$. Indeed, the Chow rings of these varieties are
isomorphic to the associated graded of the coniveau filtrations on the
$K$-rings.

###  1.3. Exceptional isomorphisms and simplicial presentations

For any nonempty flat $F$, we define the following Chow and $K$-classes:

$\displaystyle h_{F}$ $\displaystyle:=c_{1}(\mathcal{L}_{F})\in
A(W_{\\!\mathcal{A}})$ $\displaystyle\quad\quad h_{F}^{\operatorname{aug}}$
$\displaystyle:=c_{1}(\mathcal{L}_{F}^{\operatorname{aug}})\in
A(W^{{\operatorname{aug}}}_{\\!\mathcal{A}})$ $\displaystyle\eta_{F}$
$\displaystyle:=1-[\mathcal{L}_{F}^{-1}]\in K(W_{\\!\mathcal{A}})$
$\displaystyle\eta_{F}^{\operatorname{aug}}$
$\displaystyle:=1-[(\mathcal{L}_{F}^{\operatorname{aug}})^{-1}]\in
K(W^{{\operatorname{aug}}}_{\\!\mathcal{A}}).$

######  Remark 1.5.

The Chow classes $\\{h_{F}\\}$ and $\\{h_{F}^{\operatorname{aug}}\\}$ may be
interpreted in terms of divisors on the permutohedral or stellahedral variety
coming from simplices and are therefore known as simplicial generators. See
[BES21, Section 3.2] for more details. We will similarly refer to
$\\{\eta_{F}\\}$ and $\\{\eta_{F}^{\operatorname{aug}}\\}$ as the simplicial
generators of the $K$-ring. The statement that they generate their respective
$K$-rings is true but not obvious (Corollary 1.9).

Our next result, which is somewhat surprising from the vantage point of
Theorems 1.1 and 1.3, says that there exist isomorphisms from the $K$-ring to
the Chow ring of the varieties $W_{\\!\mathcal{A}}$ and
$W^{{\operatorname{aug}}}_{\\!\mathcal{A}}$ that satisfy analogues of the
Hirzebruch–Riemann–Roch formula, in which $\frac{1}{1-h_{E}}$ and
$\frac{1}{1-h_{E}^{\operatorname{aug}}}$ play the roles of the respective Todd
classes. The special case in which $\mathcal{A}$ is the Boolean arrangement
(i.e., the number of hyperplanes is equal to the dimension of $L$) appeared in
[BEST22, Theorem D] in the non-augmented setting and in [EHL22, Theorem 1.8]
in the augmented setting. Our techniques give new proofs of these results.

######  Theorem 1.6.

There are isomorphisms $\zeta_{\mathcal{A}}:K(W_{\\!\mathcal{A}})\to
A(W_{\\!\mathcal{A}})$ and
$\zeta_{\mathcal{A}}^{\operatorname{aug}}:K(W^{{\operatorname{aug}}}_{\\!\mathcal{A}})\to
A(W^{{\operatorname{aug}}}_{\\!\mathcal{A}})$ characterized by the property
that $\zeta_{\mathcal{A}}(\eta_{F})=h_{F}$ and
$\zeta_{\mathcal{A}}^{\operatorname{aug}}(\eta^{\operatorname{aug}}_{F})=h^{\operatorname{aug}}_{F}.$
For any classes $\xi\in K(W_{\\!\mathcal{A}})$ and
$\xi^{\operatorname{aug}}\in K(W^{{\operatorname{aug}}}_{\\!\mathcal{A}})$, we
have

$\chi(W_{\\!\mathcal{A}},\xi)=\deg_{W_{\\!\mathcal{A}}}\left(\frac{\zeta_{\mathcal{A}}(\xi)}{1-h_{E}}\right)\chi(W^{{\operatorname{aug}}}_{\\!\mathcal{A}},\xi^{\operatorname{aug}})=\deg_{W^{{\operatorname{aug}}}_{\\!\mathcal{A}}}\left(\frac{\zeta_{\mathcal{A}}^{\operatorname{aug}}(\xi^{\operatorname{aug}})}{1-h_{E}^{\operatorname{aug}}}\right).$

######  Remark 1.7.

The isomorphisms of Theorem 1.6 are not related to the Chern character
homomorphisms; in particular, they do not coincide with the respective Chern
characters after tensoring with the rational numbers. If we replaced
$\zeta_{\mathcal{A}}$ (respectively
$\zeta_{\mathcal{A}}^{\operatorname{aug}}$) with the Chern character, a
similar formula would hold with $\frac{1}{1-h_{E}}$ (respectively
$\frac{1}{1-h_{E}^{\operatorname{aug}}}$) replaced by the Todd class.

Let $S_{\mathcal{A}}:=\mathbb{Z}[u_{F}\mid\text{$F$ a nonempty flat}\\},$ and
consider the map

$\Phi_{\nabla}^{\operatorname{aug}}:S_{\mathcal{A}}\to
A(W^{{\operatorname{aug}}}_{\\!\mathcal{A}})$

that sends $u_{F}$ to $h^{\operatorname{aug}}_{F}$ for all $F$. Composing with
the pullback along the inclusion $W_{\\!\mathcal{A}}\subset
W^{{\operatorname{aug}}}_{\\!\mathcal{A}}$, we obtain a map

$\Phi_{\nabla}:S_{\mathcal{A}}\to A(W_{\\!\mathcal{A}})$

that sends $u_{F}$ to $h_{F}$ for all $F$. For any pair of flats $F$ and $G$,
let $F\vee G$ denote the smallest flat containing both $F$ and $G$. Consider
the following ideals in $S_{\mathcal{A}}$:

(3) $\displaystyle\mathcal{J}_{1}$ $\displaystyle:=\langle(u_{F}-u_{F\vee
G})(u_{G}-u_{F\vee G})\mid\textrm{$F$, $G$ arbitrary}\rangle$
$\displaystyle\mathcal{J}_{2}$ $\displaystyle:=\langle
u_{F}\mid\operatorname{rk}F=1\rangle$
$\displaystyle\mathcal{J}_{2}^{\operatorname{aug}}$ $\displaystyle:=\langle
u_{F}^{2}\mid\text{$\operatorname{rk}F=1$}\rangle+\langle u_{F}(u_{G}-u_{F\vee
G})\mid\text{$\operatorname{rk}F=1$, $G$ arbitrary}\rangle.$

######  Theorem 1.8.

Let $\mathcal{A}$ be a hyperplane arrangement. Then

  * (1)

The map $\Phi_{\nabla}$ is surjective with kernel
$\mathcal{J}_{1}+\mathcal{J}_{2}$ [BES21, Remark 3.2.6].

  * (2)

The map $\Phi_{\nabla}^{\operatorname{aug}}$ is surjective with kernel
$\mathcal{J}_{1}+\mathcal{J}_{2}^{\operatorname{aug}}$ (Theorem A.2).

For the $K$-theoretic analogue of Theorem 1.8, consider the map

$\Psi_{\nabla}^{\operatorname{aug}}:S_{\mathcal{A}}\to
K(W^{{\operatorname{aug}}}_{\\!\mathcal{A}})$

that sends $u_{F}$ to $\eta^{\operatorname{aug}}_{F}$ for all $F$. Composing
with the pullback along the inclusion $W_{\\!\mathcal{A}}\subset
W^{{\operatorname{aug}}}_{\\!\mathcal{A}}$, we obtain a map

$\Psi_{\nabla}:S_{\mathcal{A}}\to K(W_{\\!\mathcal{A}})$

that sends $u_{F}$ to $\eta_{F}$ for all $F$. Theorems 1.6 and 1.8 immediately
imply the following, which we refer to as the simplicial presentations of
$K(W_{\\!\mathcal{A}})$ and $K(W^{{\operatorname{aug}}}_{\\!\mathcal{A}})$.

######  Corollary 1.9.

Let $\mathcal{A}$ be a hyperplane arrangement.

  * (1)

The map $\Psi_{\nabla}$ is surjective with kernel
$\mathcal{J}_{1}+\mathcal{J}_{2}$.

  * (2)

The map $\Psi_{\nabla}^{\operatorname{aug}}$ is surjective with kernel
$\mathcal{J}_{1}+\mathcal{J}_{2}^{\operatorname{aug}}$.

###  1.4. The $K$-ring of $\overline{\mathcal{M}}_{0,n}$

Our results generalize to $\overline{\mathcal{M}}_{0,n}$, the
Deligne–Mumford–Knudsen compactification of the moduli space of stable
rational curves with $n$ marked points, for $n\geq 3$. For
$i\in\\{1,\dotsc,n\\}$, there is a line bundle $\mathbb{L}_{i}$ on
$\overline{\mathcal{M}}_{0,n}$ whose fiber over a point is the cotangent space
of the $i$th marked point in the corresponding curve. The first Chern class of
$\mathbb{L}_{i}$ is denoted $\psi_{i}$.

Kapranov [Kap93] showed that each $\mathbb{L}_{i}$ is a base-point-free line
bundle whose complete linear system induces a birational map
$\overline{\mathcal{M}}_{0,n}\to\mathbb{P}^{n-3}$. For every subset $S$ of
$[n]:=\\{1,\dotsc,n\\}$, we have a forgetful map
$f_{S}\colon\overline{\mathcal{M}}_{0,n}\to\overline{\mathcal{M}}_{0,S}=\overline{\mathcal{M}}_{0,|S|}$.
We therefore obtain a map

$\overline{\mathcal{M}}_{0,n}\to\prod_{S\subset[n-1]}\mathbb{P}^{|S|-2}$

by composing $f_{S\cup n}$ with the map induced by the complete linear system
of $\mathbb{L}_{n}$ on $\overline{\mathcal{M}}_{0,S\cup n}$. This map is a
closed embedding.

Consider the braid arrangement $\mathcal{B}_{n-1}$ in
$\mathbb{F}^{n-1}/\mathbb{F}\cdot(1,\dotsc,1)$, whose hyperplanes are normal
to $e_{i}-e_{j}$ for $i<j\in[n-1]$. The lattice of flats of
$\mathcal{B}_{n-1}$ may be identified with the collection of partitions of the
set $[n-1]$. For any subset $S\subset[n-1]$ of cardinality at least 2, let
$F_{S}$ denote the flat corresponding to the partition of $[n-1]$ into $S$
along with a bunch of singletons. Forgetting the flats of $\mathcal{B}_{n-1}$
not of the form $F_{S}$ for some $S$, we have a map

$W_{\mathcal{B}_{n-1}}\to\prod_{S}\mathbb{P}(L^{F_{S}})$

whose image is $\overline{\mathcal{M}}_{0,n}$ under the embedding described
previously [DCP95, Section 4.3]. The relation between
$\overline{\mathcal{M}}_{0,n}$ and $W_{\mathcal{B}_{n-1}}$ allows us to deduce
results about $\overline{\mathcal{M}}_{0,n}$ from our study of wonderful
varieties.

Let $\mathcal{L}_{S}=f_{S\cup n}^{*}\mathbb{L}_{n}$. This bundle is trivial
when $|S|=2$, and the bundles corresponding to sets of cardinality at least 3
form a basis for the Picard group of $\overline{\mathcal{M}}_{0,n}$. Let

$c_{S}:=c_{1}(\mathcal{L}_{S})=f_{S\cup n}^{*}\psi_{n}\in
A^{1}(\overline{\mathcal{M}}_{0,n}).$

######  Theorem 1.10.

There is an isomorphism $\zeta_{n}:K(\overline{\mathcal{M}}_{0,n})\to
A(\overline{\mathcal{M}}_{0,n})$ that sends $1-[\mathcal{L}_{S}^{-1}]$ to
$c_{S}$. For any $\xi\in K(\overline{\mathcal{M}}_{0,n})$, the Euler
characteristic of $\xi$ is equal to the degree of
$\frac{\zeta_{n}(\xi)}{1-c_{[n-1]}}$.

###  1.5. Matroids

The presentations of $A(W_{\\!\mathcal{A}})$ and
$A(W^{{\operatorname{aug}}}_{\\!\mathcal{A}})$ depend only on the matroid
associated with $\mathcal{A}$, which led to the definitions of Chow rings and
augmented Chow rings of arbitrary loopless matroids [FY04, BHM+22a].
Similarly, our descriptions of $K(W_{\\!\mathcal{A}})$ and
$K(W^{{\operatorname{aug}}}_{\\!\mathcal{A}})$, along with their Euler
characteristic functionals, only depend on the matroid associated with
$\mathcal{A}$. With the exception of the latter part of Section 8, we will
always assume that all matroids are loopless.

Let $M$ be a matroid on the ground set $E$. Let $\Sigma_{M}$ be the Bergman
fan of $M$, which has rays $\\{\rho_{F}\mid\text{$F$ a nonempty proper
flat}\\}$. Let $\Sigma^{\operatorname{aug}}_{M}$ be the augmented Bergman fan
[BHM+22a] of $M$, which has rays
$\\{\rho_{F}^{\operatorname{aug}}\mid\text{$F$ a proper
flat}\\}\cup\\{\rho_{e}^{\operatorname{aug}}\mid e\in E\\}$. These two fans
are related by the fact that $\Sigma_{M}$ is isomorphic to the star of the ray
$\rho_{\varnothing}$ in $\Sigma_{M}^{\operatorname{aug}}$ [BHM+22a,
Proposition 2.7(2)]. The rings $A(M)$ and $A^{\operatorname{aug}}(M)$ are
defined to be the Chow rings of the toric varieties $X_{\Sigma_{M}}$ and
$X_{\Sigma^{\operatorname{aug}}_{M}}$, respectively.

If $M$ is the matroid associated with a hyperplane arrangement $\mathcal{A}$,
then, after choosing a linear functional defining each hyperplane in
$\mathcal{A}$, we obtain canonical inclusions $W_{\\!\mathcal{A}}\subset
X_{\Sigma_{M}}$ and $W^{{\operatorname{aug}}}_{\\!\mathcal{A}}\subset
X_{\Sigma^{\operatorname{aug}}_{M}}$ with the properties that
$D_{F}=W_{\\!\mathcal{A}}\cap D_{\rho_{F}}$ for any nonempty proper flat $F$,
$D_{F}^{\operatorname{aug}}=W^{{\operatorname{aug}}}_{\\!\mathcal{A}}\cap
D_{\rho_{F}^{\operatorname{aug}}}$ for any proper flat $F$, and
$D_{e}^{\operatorname{aug}}=W^{{\operatorname{aug}}}_{\\!\mathcal{A}}\cap
D_{\rho_{e}^{\operatorname{aug}}}$ for any $e\in E$. The restriction maps
$A(M)\to A(W_{\\!\mathcal{A}})$ and $A^{\operatorname{aug}}(M)\to
A(W^{{\operatorname{aug}}}_{\\!\mathcal{A}})$ are both isomorphisms [FY04,
BHM+22a].

Consider the polynomial ring $T_{M}:=\mathbb{Z}[x_{F}\mid\text{$F$ a
flat}]\otimes\mathbb{Z}[y_{e}\mid e\in E]$ along with the homomorphism
$\Phi_{\operatorname{FY}}^{\operatorname{aug}}:T_{M}\to
A^{\operatorname{aug}}(M)$ taking $x_{F}$ to
$[D_{\rho_{F}^{\operatorname{aug}}}]$ for all proper flats $F$, $x_{E}$ to
$-\sum_{F\neq E}[D_{\rho_{F}^{\operatorname{aug}}}]$, and $y_{e}$ to
$[D_{\rho_{e}^{\operatorname{aug}}}]$. Composing with the pullback along the
inclusion $X_{\Sigma_{M}}\cong
D_{\rho_{\varnothing}^{\operatorname{aug}}}\subset
X_{\Sigma_{M}^{\operatorname{aug}}}$, we obtain a homomorphism
$\Phi_{\operatorname{FY}}:T_{M}\to A(M)$ that sends $x_{F}$ to $D_{\rho_{F}}$
for all nonempty proper $F$ and $y_{e}$ to zero. Consider also the polynomial
ring $S_{M}:=\mathbb{Z}[u_{F}\mid\text{$F$ a nonempty flat}].$ We define
$\Phi_{\nabla}:S_{M}\to A(M)$ and $\Phi_{\nabla}^{\operatorname{aug}}:S_{M}\to
A^{\operatorname{aug}}(M)$ via the formulas

(4) $\Phi_{\nabla}(u_{F}):=-\sum_{F\subset
G}\Phi_{\operatorname{FY}}(x_{G})\Phi_{\nabla}^{\operatorname{aug}}(u_{F}):=-\sum_{F\subset
G}\Phi_{\operatorname{FY}}^{\operatorname{aug}}(x_{G}).$

######  Remark 1.11.

When $M$ is the matroid associated with the arrangement $\mathcal{A}$, the
following diagrams (along with their augmented analogues) commute, thus
justifying our repeated use of the notation $\Phi_{\operatorname{FY}}$ and
$\Phi_{\nabla}$:

${T_{M}}$${T_{\mathcal{A}}}$${S_{M}}$${S_{\mathcal{A}}}$${A(M)}$${A(W_{\\!\mathcal{A}})}$${A(M)}$${A(W_{\\!\mathcal{A}}).}$$\scriptstyle{=}$$\scriptstyle{\Phi_{\operatorname{FY}}}$$\scriptstyle{\Phi_{\operatorname{FY}}}$$\scriptstyle{=}$$\scriptstyle{\Phi_{\nabla}}$$\scriptstyle{\Phi_{\nabla}}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$

Commutativity of the first diagram is immediate from the definitions, while
commutativity of the second diagram is proved by reducing to the case of the
Boolean matroid [BES21, Section 3.2].

Define the ideals
$\mathcal{I}_{1},\mathcal{I}_{1}^{\prime},\mathcal{I}_{2},\mathcal{I}_{2}^{\prime},\mathcal{I}_{3},\mathcal{I}_{4},\mathcal{I}_{4}^{\operatorname{aug}}\subset
T_{M}$ and
$\mathcal{J}_{1},\mathcal{J}_{2},\mathcal{J}_{2}^{\operatorname{aug}}\subset
S_{M}$ as in Equations (1), (2), and (3).

######  Theorem 1.12.

Let $M$ be a loopless matroid.

  * (1)

The map $\Phi_{\operatorname{FY}}$ is surjective with kernel
$\mathcal{I}_{1}+\mathcal{I}_{2}+\mathcal{I}_{3}+\mathcal{I}_{4}$ [FY04].

  * (2)

The map $\Phi_{\operatorname{FY}}^{\operatorname{aug}}$ is surjective with
kernel
$\mathcal{I}_{1}+\mathcal{I}_{2}+\mathcal{I}_{3}+\mathcal{I}_{4}^{\operatorname{aug}}$
[BHM+22a].

  * (3)

The map $\Phi_{\nabla}$ is surjective with kernel
$\mathcal{J}_{1}+\mathcal{J}_{2}$ [BES21].

  * (4)

The map $\Phi_{\nabla}^{\operatorname{aug}}$ is surjective with kernel
$\mathcal{J}_{1}+\mathcal{J}_{2}^{\operatorname{aug}}$ (Theorem A.2).

Similarly, we define $K(M)$ and $K^{\operatorname{aug}}(M)$ to be the
$K$-rings of $X_{\Sigma_{M}}$ and $X_{\Sigma_{M}^{\operatorname{aug}}}$,
respectively.

######  Proposition 1.13.

If $M$ is the matroid associated with $\mathcal{A}$, then the inclusions
$W_{\\!\mathcal{A}}\subset X_{\Sigma_{M}}$ and
$W^{{\operatorname{aug}}}_{\\!\mathcal{A}}\subset
X_{\Sigma_{M}^{\operatorname{aug}}}$ induce isomorphisms $K(M)\cong
K(W_{\\!\mathcal{A}})$ and $K^{\operatorname{aug}}(M)\cong
K(W^{{\operatorname{aug}}}_{\\!\mathcal{A}})$.

Let $\mathcal{L}_{E}^{\operatorname{aug}}$ be the line bundle on
$X_{\Sigma_{M}^{\operatorname{aug}}}$ whose first Chern class is equal to
$\sum_{F\neq E}[D_{\rho_{F}^{\operatorname{aug}}}]$. We define the map
$\Psi_{\operatorname{FY}}^{\operatorname{aug}}:T_{M}\to
K^{\operatorname{aug}}(M)$ by sending $x_{F}$ to
$[\mathcal{O}_{D_{\rho_{F}^{\operatorname{aug}}}}]$ for all proper $F$,
$x_{E}$ to $1-[\mathcal{L}_{E}^{\operatorname{aug}}]$, and $y_{e}$ to
$[\mathcal{O}_{D_{\rho_{e}^{\operatorname{aug}}}}]$ for all $e\in E$.
Composing with the pullback along the inclusion $X_{\Sigma_{M}}\cong
D_{\rho_{\varnothing}^{\operatorname{aug}}}\subset
X_{\Sigma_{M}^{\operatorname{aug}}}$, we obtain a homomorphism
$\Psi_{\operatorname{FY}}:T_{M}\to K(M)$ that sends $x_{F}$ to
$[\mathcal{O}_{D_{\rho_{F}}}]$ for all nonempty proper $F$ and $y_{e}$ to
zero.

######  Remark 1.14.

When $M$ is the matroid associated with the arrangement $\mathcal{A}$, the
following diagram commutes (as does its augmented analogue), thus justifying
the repeated use of the notation $\Psi_{\operatorname{FY}}$:

${T_{M}}$${T_{\mathcal{A}}}$${K(M)}$${K(W_{\\!\mathcal{A}}).}$$\scriptstyle{=}$$\scriptstyle{\Psi_{\operatorname{FY}}}$$\scriptstyle{\Psi_{\operatorname{FY}}}$$\scriptstyle{\cong}$

We have the following generalization of Theorem 1.3 to matroids.

######  Theorem 1.15.

Let $M$ be a loopless matroid.

  * (1)

The map $\Psi_{\operatorname{FY}}$ is surjective with kernel
$\mathcal{I}^{\prime}_{1}+\mathcal{I}^{\prime}_{2}+\mathcal{I}_{3}+\mathcal{I}_{4}$.

  * (2)

The map $\Psi_{\operatorname{FY}}^{\operatorname{aug}}$ is surjective with
kernel
$\mathcal{I}^{\prime}_{1}+\mathcal{I}^{\prime}_{2}+\mathcal{I}_{3}+\mathcal{I}_{4}^{\operatorname{aug}}$.

We also have the following extension of Theorem 1.6 to matroids.

######  Theorem 1.16.

There exist isomorphisms $\zeta_{M}\colon K(M)\to A(M)$ and
$\zeta^{{\operatorname{aug}}}_{M}\colon K^{{\operatorname{aug}}}(M)\to
A^{{\operatorname{aug}}}(M)$, characterized by the property that

$\zeta_{M}^{-1}(h_{F})=1-\prod_{F\subset
G}\left(1-\Psi_{\operatorname{FY}}(x_{F})\right)^{-1}(\zeta_{M}^{{\operatorname{aug}}})^{-1}(h_{F}^{{\operatorname{aug}}})=1-\prod_{F\subset
G}\left(1-\Psi_{\operatorname{FY}}^{\operatorname{aug}}(x_{F})\right)^{-1}.$

Motivated by the above characterization of $\zeta_{\mathcal{A}}$ and
$\zeta_{\mathcal{A}}^{{\operatorname{aug}}}$, we define maps
$\Psi_{\nabla}:S_{M}\to K(M)$ and $\Psi_{\nabla}^{\operatorname{aug}}:S_{M}\to
K^{\operatorname{aug}}(M)$ by the formulas

(5) $\Psi_{\nabla}(u_{F})=1-\prod_{F\subset
G}\left(1-\Psi_{\operatorname{FY}}(x_{G})\right)^{-1}\Psi_{\nabla}^{\operatorname{aug}}(u_{F})=1-\prod_{F\subset
G}\left(1-\Psi_{\operatorname{FY}}^{\operatorname{aug}}(x_{G})\right)^{-1}.$

######  Remark 1.17.

Let $M$ be the matroid associated with an arrangement $\mathcal{A}$.
Identifying $K(M)$ with $K(W_{\\!\mathcal{A}})$ via the isomorphism in
Proposition 1.13, we have constructed two isomorphisms
$\zeta_{M},\zeta_{\mathcal{A}}\colon K(M)\to A(M)$. Proposition 5.1 implies
that these two isomorphisms coincide, as do the analogous pair of isomorphisms
in the augmented setting. Similarly, we have constructed two maps
$\Psi_{\nabla}\colon S_{M}\to K(M)$, one defined geometrically in Section 1.3,
and the other defined algebraically in Equation (5). Proposition 5.1 also
implies that these maps coincide, as do the analogous pair of maps in the
augmented setting.

Theorem 1.12 and Theorem 1.16 immediately give the following presentations of
$K(M)$ and $K^{{\operatorname{aug}}}(M)$, extending Corollary 1.9 to matroids.

######  Corollary 1.18.

Let $M$ be a loopless matroid.

  * (1)

The map $\Psi_{\nabla}$ is surjective with kernel
$\mathcal{J}_{1}+\mathcal{J}_{2}$.

  * (2)

The map $\Psi_{\nabla}^{\operatorname{aug}}$ is surjective with kernel
$\mathcal{J}_{1}+\mathcal{J}_{2}^{\operatorname{aug}}$.

The Chow rings $A(M)$ and $A^{\operatorname{aug}}(M)$ satisfy Poincaré
duality, meaning that there exist maps

$\deg_{M}:A(M)\to\mathbb{Z}\deg^{\operatorname{aug}}_{M}:A^{\operatorname{aug}}(M)\to\mathbb{Z}$

such the pairing $A(M)\otimes A(M)\to\mathbb{Z}$ taking $f\otimes g$ to
$\deg_{M}(fg)$ is perfect [AHK18, Theorem 6.19], and similarly in the
augmented setting [BHM+22a, Theorem 1.3(4)]. Define maps

$\chi:K(M)\to\mathbb{Z}\chi^{\operatorname{aug}}:K(M)\to\mathbb{Z}$

by putting

$\chi(M,\xi):=\deg_{M}\left(\frac{\zeta_{M}(\xi)}{1-h_{E}}\right)\chi^{\operatorname{aug}}(M,\xi^{\operatorname{aug}}):=\deg_{M}^{\operatorname{aug}}\left(\frac{\zeta_{M}^{\operatorname{aug}}(\xi^{\operatorname{aug}})}{1-h_{E}^{\operatorname{aug}}}\right).$

Then the analogous pairing $K(M)\otimes K(M)\to\mathbb{Z}$ taking
$\xi\otimes\eta$ to $\chi(M,\xi\eta)$ is perfect, and similarly in the
augmented setting. For realizable matroids, this is a general property of the
Euler pairing on the $K$-theory of a smooth proper linear variety [AP15,
Theorem 1.3].

######  Remark 1.19.

When $M$ is not realizable, the maps $\chi$ and $\chi^{\operatorname{aug}}$
still have a natural geometric interpretation as Euler characteristics on the
wonderful variety (respectively augmented wonderful variety) of the Boolean
arrangement. See Proposition 5.3.

Although the $K$-rings $K(M)$ and $K^{\operatorname{aug}}(M)$ are isomorphic
to their respective Chow rings, the fact that they are $K$-rings endow them
with several additional structures, including the structure of augmented
$\lambda$-rings (see Section 6 for details). Furthermore, when $M$ is
realizable, Serre duality gives a nontrivial identity satisfied by $\chi$ and
$\chi^{\operatorname{aug}}$. In Theorem 6.2, we show that that these
identities extend to matroids.

###  1.6. Applications to Euler characteristics

Much is known about intersection theory on (augmented) wonderful varieties.
The Hirzebruch–Riemann–Roch-type formulas in Theorem 1.6 allow us to transfer
intersection-theoretic computations to $K$-theory. As applications, we give
formulas for the Euler characteristic of the class of any line bundle in
$K(M)$ expressed in terms of the bundles $\mathcal{L}_{F}$ (Theorem 7.2 and
Corollary 7.5), and similarly in the augmented setting. We also give a formula
for the Euler characteristic of any line bundle expressed in terms of the
Feichtner–Yuzvinsky generators of $K(M)$ (Theorem 8.2), extending as a
corollary Eur’s formula [Eur20] for the degrees of monomials in the
Feichtner–Yuzvinsky generators of $A(M)$ (Corollary 8.4). Likewise, we give
formulas for the Euler characteristic of any line bundle on
$\overline{\mathcal{M}}_{0,n}$ expressed in terms the line bundles
$\\{\mathcal{L}_{S}\\}$ (Theorem 9.1) or $\\{\mathbb{L}_{i}\\}$ (Theorem 9.2).

###  1.7. Structure of the paper

In Sections 2 and 3, we prove Theorems 1.3 and 1.6, assuming the following
technical lemma:

######  Lemma 1.20.

The $K$-rings of $W_{\\!\mathcal{A}}$ and
$W^{{\operatorname{aug}}}_{\\!\mathcal{A}}$ are generated by the classes of
line bundles.

In Section 4, we discuss the presentation of $K(\overline{\mathcal{M}}_{0,n})$
and the exceptional isomorphism to $A(\overline{\mathcal{M}}_{0,n})$ in
Theorem 1.10. In Section 5, we prove Theorem 1.15, and we derive Lemma 1.20 as
a corollary. We have chosen to order our presentation in this way, rather than
beginning with arbitrary matroids, in order better to highlight the geometric
ideas underlying the proofs of our main results.

Section 6 is devoted to a description of the Adams operations in $K(M)$ and
$K^{\operatorname{aug}}(M)$ along with a combinatorial Serre duality theorem
(Theorem 6.2). Section 7 is dedicated to computing Euler characteristics in
the simplicial generators, Section 8 to Euler characteristics in the
Feichtner–Yuzvinsky generators, and Section 9 to Euler characteristics on
$\overline{\mathcal{M}}_{0,n}$. Finally, we include an appendix with a proof
of the simplicial presentation for the augmented Chow ring of a matroid, which
has not previously appeared in the literature.

We note that some of the proofs in Sections 6 and 8 proceed by first giving
geometric proofs for realizable matroids, and then using the notion of
valuativity to extend the results to all matroids. Thus, in these sections,
separating the statements for hyperplane arrangements from the statements for
matroids is absolutely essential.

###  1.8. Acknowledgements

We thank Chris Eur for explaining the simplicial generators of the augmented
Chow ring to us, for suggesting that the simplicial generators should behave
well with respect to the exceptional isomorphisms, and for other helpful
conversations. We thank June Huh for helpful conversations. This collaboration
is supported by the NSF FRG grant DMS–2053261. The work of ML is also
supported by an NDSEG fellowship, the work of SL is also supported by NSF
DMS–1844768 and a Coline M. Makepeace Fellowship from Brown University, the
work of SP is also supported by NSF DMS–2001502, and the work of NP is also
supported by NSF DMS–1954050.

##  2\. Feichtner–Yuzvinsky presentations

In this section, we prove Theorem 1.3. We begin with some general lemmas about
a smooth variety $X$, which we will also apply later in the proofs of Theorems
1.6 and 1.15.

######  Lemma 2.1.

If $A(X)$ is a free abelian group of finite rank $p$, then so is $K(X)$, and
the Chern character $\textup{ch}:K(X)\to A(X)_{\mathbb{Q}}$ is injective.

###### Proof.

The Chern character becomes an isomorphism after tensoring with $\mathbb{Q}$
[Ful98, Example 15.2.16], so $K(X)$ has rank $p$. There is a surjective map
from $A(X)$ to the associated graded of $K(X)$ with respect to the coniveau
filtration [Ful98, Example 15.1.5]. Since $A(X)$ is free of rank $p$ and
$K(X)$ also has rank $p$, this implies that that $K(X)$ is free. Finally, the
Chern character factors as $K(X)\to K(X)_{\mathbb{Q}}\to A(X)_{\mathbb{Q}}$,
with the first map being injective by freeness of $K(X)$ and the second map
being an isomorphism, so the Chern character is injective. ∎

######  Remark 2.2.

Lemma 2.1 will be applied to the varieties $W_{\\!\mathcal{A}}$,
$X_{\Sigma_{M}}$ [FY04, Corollary 1] and $\overline{\mathcal{M}}_{0,n}$ [FY04,
Corollary 2], as well as to $W^{{\operatorname{aug}}}_{\\!\mathcal{A}}$ and
$X_{\Sigma_{M}^{\operatorname{aug}}}$ [AHK18, Theorem 6.19].

######  Lemma 2.3.

Let $X$ be a smooth variety, and suppose that $K(X)$ is generated as a ring by
the classes of line bundles. Let $D_{1},\ldots,D_{k}$ be divisors on $X$. If
$A^{1}(X)$ is generated as an abelian group by $[D_{1}],\ldots,[D_{k}]$, then
$K(X)$ is generated as a ring by
$[\mathcal{O}_{D_{1}}],\ldots,[\mathcal{O}_{D_{k}}]$.

###### Proof.

Let $R$ be the subring of $K(X)$ generated by
$[\mathcal{O}_{D_{1}}],\ldots,[\mathcal{O}_{D_{k}}]$. We need to show that the
class of every line bundle is contained in $R$. Since $[D_{1}],\ldots,[D_{k}]$
generate $A^{1}(X)$ as an abelian group, the line bundles $\mathcal{O}(\pm
D_{1}),\ldots,\mathcal{O}(\pm D_{k})$ generate the Picard group of $X$ under
multiplication, so it will be sufficient to show that $[\mathcal{O}(\pm
D_{i})]\in R$ for all $i$.

For any divisor $D$, we have an exact sequence

$0\to\mathcal{O}(-D)\to\mathcal{O}\to\mathcal{O}_{D}\to 0,$

which implies that

$[\mathcal{O}(-D)]=[\mathcal{O}]-[\mathcal{O}_{D}]=1-[\mathcal{O}_{D}].$

We also have

(6)
$[\mathcal{O}(D)]=[\mathcal{O}(-D)]^{-1}=\frac{1}{1-[\mathcal{O}_{D}]}=1+[\mathcal{O}_{D}]+[\mathcal{O}_{D}]^{2}+\cdots.$

Since $[\mathcal{O}_{D}]$ lives in the first piece of the coniveau filtration
on $K(X)$, it is nilpotent, so the sum terminates. This allows us to conclude
that both $[\mathcal{O}(-D_{i})]$ and $[\mathcal{O}(D_{i})]$ live in the ring
$R$. ∎

###### Proof of Theorem 1.3..

We begin with statement (2). Surjectivity of the map
$\Phi_{\operatorname{FY}}^{\operatorname{aug}}$ follows from Theorem 1.1,
Lemma 1.20, and Lemma 2.3. Next, we show that
$\mathcal{I}_{1}^{\prime}+\mathcal{I}_{2}^{\prime}+\mathcal{I}_{3}+\mathcal{I}_{4}^{\operatorname{aug}}$
is contained in the kernel of $\Phi_{\operatorname{FY}}^{\operatorname{aug}}$.
The ideal $\mathcal{I}_{3}$ is contained in the kernel because $D_{F}$ and
$D_{G}$ are disjoint when $F$ and $G$ are incomparable. Similarly,
$\mathcal{I}_{4}^{\operatorname{aug}}$ is contained in the kernel because
$D_{e}$ is disjoint from $D_{F}$ whenever $e\notin F$. To prove that
$\mathcal{I}_{1}^{\prime}$ is contained in the kernel, we observe that its
generator maps to

$1-[\mathcal{L}_{E}^{\operatorname{aug}}]\prod_{F\neq
E}\Big{(}1-[\mathcal{O}_{D_{F}^{\operatorname{aug}}}]\Big{)}=1-[\mathcal{L}_{E}^{\operatorname{aug}}]\prod_{F\neq
E}[\mathcal{O}(-D_{F}^{\operatorname{aug}})].$

Since $\mathcal{I}_{1}$ is contained in the kernel of
$\Phi_{\operatorname{FY}}^{\operatorname{aug}}$, the line bundle

$\mathcal{L}_{E}^{\operatorname{aug}}\otimes\bigotimes_{F\neq
E}\mathcal{O}(-D_{F}^{\operatorname{aug}})$

is trivial, thus the generator of $\mathcal{I}_{1}^{\prime}$ is contained in
the kernel of $\Psi_{\operatorname{FY}}^{\operatorname{aug}}$.

To prove that $\mathcal{I}_{2}^{\prime}$ is contained in the kernel, Lemma 2.1
tells us that it is sufficient to prove that, for all $e\in E$, we have

$\textup{ch}\Big{(}1-[\mathcal{O}_{D_{e}^{\operatorname{aug}}}]\Big{)}=\prod_{e\notin
F}\textup{ch}\Big{(}1-[\mathcal{O}_{D_{F}^{\operatorname{aug}}}]\Big{)}.$

We have

$\textup{ch}\Big{(}1-[\mathcal{O}_{D_{e}^{\operatorname{aug}}}]\Big{)}=\textup{ch}([\mathcal{O}(-D_{e}^{\operatorname{aug}})])=\exp\Big{(}-[D_{e}^{\operatorname{aug}}]\Big{)}$

and

$\prod_{e\notin
F}\textup{ch}\Big{(}1-[\mathcal{O}_{D_{F}^{\operatorname{aug}}}]\Big{)}=\prod_{e\notin
F}\exp\Big{(}-[D_{F}^{\operatorname{aug}}]\Big{)}=\exp\Big{(}-\sum_{e\notin
F}[D_{F}^{\operatorname{aug}}]\Big{)}.$

The fact that these are equal follows from the fact that $\mathcal{I}_{2}$ is
contained in the kernel of $\Phi_{\operatorname{FY}}^{\operatorname{aug}}$.

Let
$R:=T_{\mathcal{A}}\,\big{/}\,\mathcal{I}_{1}^{\prime}+\mathcal{I}_{2}^{\prime}+\mathcal{I}_{3}+\mathcal{I}_{4}^{\operatorname{aug}}$.
We have now shown that $R$ surjects onto
$K(W^{{\operatorname{aug}}}_{\\!\mathcal{A}})$, and we need to prove that the
map is injective. Let $p$ be the rank of the free abelian group
$A(W^{{\operatorname{aug}}}_{\\!\mathcal{A}})$. Consider the decreasing
filtration

$R=F_{0}\supset F_{1}\supset\cdots,$

where $F_{i}$ is the span of all monomials of total degree $\geq i$. Since the
leading terms of the generators $\mathcal{I}_{1}^{\prime}$ and
$\mathcal{I}_{2}^{\prime}$ coincide with the generators of $\mathcal{I}_{1}$
and $\mathcal{I}_{2}$, we have a surjection

$A(W^{{\operatorname{aug}}}_{\\!\mathcal{A}})\cong
T_{\mathcal{A}}\,\big{/}\,\mathcal{I}_{1}+\mathcal{I}_{2}+\mathcal{I}_{3}+\mathcal{I}_{4}^{\operatorname{aug}}\to\operatorname{gr}R.$

In particular, this implies that the abelian group $\operatorname{gr}R$ can be
generated by $p$ elements, and so the same is true of $R$. Lemma 2.1 tells us
that $K(W^{{\operatorname{aug}}}_{\\!\mathcal{A}})$ is also free abelian of
rank $p$, so any surjection from $R$ to
$K(W^{{\operatorname{aug}}}_{\\!\mathcal{A}})$ must be an isomorphism.

The proof of statement (1) is nearly identical. The only extra ingredient is
the argument that $\mathcal{I}_{4}$ is contained in the kernel of the map
$\Psi_{\operatorname{FY}}$, which follows from the fact that
$D_{e}^{\operatorname{aug}}$ is disjoint from
$D_{\varnothing}^{\operatorname{aug}}\cong W_{\\!\mathcal{A}}$ inside of
$W^{{\operatorname{aug}}}_{\\!\mathcal{A}}$. ∎

##  3\. Simplicial presentations

The purpose of this section is to prove Theorem 1.6. We begin by observing
that we have an inclusion

$W_{\\!\mathcal{A}}\;\;\;\subset\prod_{\begin{subarray}{c}\text{$F\subset
E$}\\\ \text{a nonempty
flat}\end{subarray}}\mathbb{P}(L^{F})\;\;\;\subset\prod_{\begin{subarray}{c}\text{$S\subset
E$}\\\ \text{a nonempty subset}\end{subarray}}\mathbb{P}(L^{S}).$

The first inclusion comes from the definition of $W_{\\!\mathcal{A}}$, and the
second from the diagonal embedding of $\mathbb{P}(L^{F})$ into the product of
$\mathbb{P}(L^{S})$ for all $S$ such that $L^{S}=L^{F}$.111In matroid
theoretic language, this is the collection of subsets $S$ whose closure is
$F$. Similarly, we have

$W^{{\operatorname{aug}}}_{\\!\mathcal{A}}\;\;\;\subset\prod_{\begin{subarray}{c}\text{$F\subset
E$}\\\ \text{a nonempty
flat}\end{subarray}}\mathbb{P}(L^{F}\oplus\mathbb{F})\;\;\;\subset\prod_{\begin{subarray}{c}\text{$S\subset
E$}\\\ \text{a nonempty
subset}\end{subarray}}\mathbb{P}(L^{S}\oplus\mathbb{F}).$

Suppose that $L^{\prime}\subset L$ is a linear subspace. Let
$E^{\prime}:=\\{e\in E\mid L^{\prime}\not\subset H_{e}\\}$, and define a new
hyperplane arrangement

$\mathcal{A}_{L^{\prime}}:=\\{H_{e}\cap L^{\prime}\mid e\in E^{\prime}\\}.$

If $E^{\prime}=E$, then we have inclusions
$W_{\\!\mathcal{A}_{L^{\prime}}}\subset W_{\\!\mathcal{A}}$ and
$W^{\operatorname{aug}}_{\\!\mathcal{A}_{L^{\prime}}}\subset
W^{{\operatorname{aug}}}_{\\!\mathcal{A}}$, each inside of the corresponding
product of projective spaces indexed by subsets of $E$. If $L^{\prime}$ is
contained in one or more hyperplane, then we still have an inclusion
$W^{\operatorname{aug}}_{\\!\mathcal{A}_{L^{\prime}}}\subset
W^{{\operatorname{aug}}}_{\\!\mathcal{A}}$, where
$W^{\operatorname{aug}}_{\\!\mathcal{A}_{L^{\prime}}}$ sits inside the product
indexed by subsets of $E$ by setting the $S$ coordinate to $0\in
L^{S}\subset\mathbb{P}(L^{S}\oplus\mathbb{F})$ unless $S\subset E^{\prime}$.

For any flat $F$ of $\mathcal{A}$, let ${F}^{\prime}:=\\{e\in E^{\prime}\mid
L_{F}\cap L^{\prime}\subset H_{e}\\}$ be the corresponding flat of
$L^{\prime}$. Then the line bundle $\mathcal{L}_{F}^{{\operatorname{aug}}}$ on
$W^{{\operatorname{aug}}}_{\\!\mathcal{A}}$ restricts to the line bundle
$\mathcal{L}_{F^{\prime}}$ on
$W_{\\!\mathcal{A}_{L^{\prime}}}^{{\operatorname{aug}}}$. If $E=E^{\prime}$,
then the line bundle $\mathcal{L}_{F}$ on $W_{\\!\mathcal{A}}$ restricts to
the line bundle $\mathcal{L}_{F^{\prime}}$ on
$W_{\\!\mathcal{A}_{L^{\prime}}}$.

######  Lemma 3.1.

Fix a flat $F$, and suppose that $H\subset L$ is a hyperplane with the
property that $L_{F}\subset H$ and $L_{G}\not\subset H$ for all flats
$G\subsetneq F$. Then

$h^{\operatorname{aug}}_{F}=[W^{\operatorname{aug}}_{\\!\mathcal{A}_{H}}]\eta^{\operatorname{aug}}_{F}=[\mathcal{O}_{W^{\operatorname{aug}}_{\\!\mathcal{A}_{H}}}].$

If the rank of $F$ is greater than 1, then

$h_{F}=\left[W_{\\!\mathcal{A}_{H}}\right]\eta_{F}=\left[\mathcal{O}_{W_{\\!\mathcal{A}_{H}}}\right].$

###### Proof.

We prove only the first statement; the second is similar. Let $g$ be a section
of $\mathcal{O}_{\mathbb{P}(L^{F}\oplus\mathbb{F})}(1)$ with vanishing locus
equal to
$\mathbb{P}(H/L_{F}\oplus\mathbb{F})\subset\mathbb{P}(L^{F}\oplus\mathbb{F})$.
Then $(\pi_{F}^{\operatorname{aug}})^{*}g$ is a regular section of
$\mathcal{L}_{F}^{\operatorname{aug}}$ with vanishing locus
$W^{\operatorname{aug}}_{\\!\mathcal{A}_{H}}$, which shows that
$h_{F}^{\operatorname{aug}}=c_{1}(\mathcal{L}_{F}^{\operatorname{aug}})=[W^{\operatorname{aug}}_{\\!\mathcal{A}_{H}}]$.
We interpret $(\pi_{F}^{\operatorname{aug}})^{*}g$ as an element of
$\operatorname{Hom}(\mathcal{O}_{W^{{\operatorname{aug}}}_{\\!\mathcal{A}}},\mathcal{L}_{F}^{\operatorname{aug}})\cong\operatorname{Hom}\big{(}(\mathcal{L}_{F}^{\operatorname{aug}})^{-1},\mathcal{O}_{W^{{\operatorname{aug}}}_{\\!\mathcal{A}}}\big{)}$
to obtain an exact sequence

$0\to(\mathcal{L}_{F}^{\operatorname{aug}})^{-1}\overset{(\pi^{{\operatorname{aug}}}_{F})^{*}g}{\longrightarrow}\mathcal{O}_{W^{{\operatorname{aug}}}_{\\!\mathcal{A}}}\to\mathcal{O}_{W^{\operatorname{aug}}_{\\!\mathcal{A}_{H}}}\to
0,$

which shows that
$\eta^{\operatorname{aug}}_{F}=1-[(\mathcal{L}_{F}^{\operatorname{aug}})^{-1}]=[\mathcal{O}_{W^{{\operatorname{aug}}}_{\\!\mathcal{A}}}]-[(\mathcal{L}_{F}^{\operatorname{aug}})^{-1}]=[\mathcal{O}_{W^{\operatorname{aug}}_{\\!\mathcal{A}_{H}}}]$.
∎

For any tuple of natural numbers $\textbf{m}=(m_{F}\mid F\text{ a nonempty
flat})$, we define the monomials

$\displaystyle h^{\textbf{m}}$ $\displaystyle:=\prod_{F}h_{F}^{m_{F}}\in
A(W_{\\!\mathcal{A}})$
$\displaystyle\quad\quad(h^{\operatorname{aug}})^{\textbf{m}}$
$\displaystyle:=\prod_{F}(h_{F}^{\operatorname{aug}})^{m_{F}}\in
A(W^{{\operatorname{aug}}}_{\\!\mathcal{A}})$ $\displaystyle\eta^{\textbf{m}}$
$\displaystyle:=\prod_{F}\eta_{F}^{m_{F}}\in K(W_{\\!\mathcal{A}})$
$\displaystyle(\eta^{\operatorname{aug}})^{\textbf{m}}$
$\displaystyle:=\prod_{F}(\eta_{F}^{\operatorname{aug}})^{m_{F}}\in
K(W^{{\operatorname{aug}}}_{\\!\mathcal{A}}).$

######  Lemma 3.2.

Suppose that $\mathbb{F}$ is infinite. For any m, one of the following two
statements holds:

  * •

$\eta^{\textbf{m}}=0$ and $h^{\textbf{m}}=0$.

  * •

There exists a linear subspace $L^{\prime}\subset L$, not contained in any of
the hyperplanes in $\mathcal{A}$, such that
$\eta^{\textbf{m}}=[\mathcal{O}_{W_{\\!\mathcal{A}_{L^{\prime}}}}]$ and
$h^{\textbf{m}}=[W_{\\!\mathcal{A}_{L^{\prime}}}]$.

In addition, one of the following two statements holds:

  * •

$(\eta^{\operatorname{aug}})^{\textbf{m}}=0$ and
$(h^{\operatorname{aug}})^{\textbf{m}}=0$.

  * •

There exists a linear subspace $L^{\prime}\subset L$ such that
$(\eta^{\operatorname{aug}})^{\textbf{m}}=[\mathcal{O}_{W^{\operatorname{aug}}_{\\!\mathcal{A}_{L^{\prime}}}}]$
and
$(h^{\operatorname{aug}})^{\textbf{m}}=[W^{\operatorname{aug}}_{\\!\mathcal{A}_{L^{\prime}}}]$.

###### Proof.

We will prove only the first claim; the augmented case is similar. We proceed
by induction on $\sum m_{F}$. When $\textbf{m}=0$, we can take $L^{\prime}=L$.
For the inductive step, assume that the second statement holds for
$\textbf{m}^{\prime}$, and take m such that $m_{F}=m^{\prime}_{F}+1$ and m
agrees with $\textbf{m}^{\prime}$ in every other coordinate. Then

$\eta^{\textbf{m}}=\eta_{F}\eta^{\textbf{m}^{\prime}}=\eta_{F}[\mathcal{O}_{W_{\\!\mathcal{A}_{L^{\prime}}}}]=\big{(}1-[\mathcal{L}_{F}^{-1}]\big{)}[\mathcal{O}_{W_{\\!\mathcal{A}_{L^{\prime}}}}]=[\mathcal{O}_{W_{\\!\mathcal{A}_{L^{\prime}}}}]-[\mathcal{L}_{F^{\prime}}^{-1}],$

where $F^{\prime}=\\{e\mid L_{F}\cap L^{\prime}\subset H_{e}\\}$ and
$\mathcal{L}_{F^{\prime}}$ is the corresponding line bundle on
$W_{\\!\mathcal{A}_{L^{\prime}}}$. If $F^{\prime}$ has rank 1, then
$\mathcal{L}_{F^{\prime}}$ is trivial, and we get zero. If $F^{\prime}$ has
rank greater than 1, then we may choose a hyperplane222This is where we use
the fact that $\mathbb{F}$ is infinite. $H^{\prime}\subset L^{\prime}$ such
that $H^{\prime}\notin\mathcal{A}^{\prime}$, $L_{F^{\prime}}\subset
H^{\prime}$, and $L_{G^{\prime}}\not\subset H^{\prime}$ for all flats
$G^{\prime}\subsetneq F^{\prime}$. Then by Lemma 3.1,
$\eta^{\textbf{m}}=[\mathcal{O}_{W_{\\!\mathcal{A}_{L^{\prime}}}}]-[\mathcal{L}_{F^{\prime}}^{-1}]=[\mathcal{O}_{W_{\\!\mathcal{A}_{H^{\prime}}}}]$.
Similarly, we have

$h^{\textbf{m}}=h_{F}h^{\textbf{m}^{\prime}}=h_{F}[W_{\\!\mathcal{A}_{L^{\prime}}}]=c_{1}(\mathcal{L}_{F})[W_{\\!\mathcal{A}_{L^{\prime}}}].$

By the projection formula, this is equal to the pushforward of
$c_{1}(\mathcal{L}_{F^{\prime}})$ from $W_{\\!\mathcal{A}_{L^{\prime}}}$ to
$W_{\\!\mathcal{A}}$. If $F^{\prime}$ has rank 1, then
$\mathcal{L}_{F^{\prime}}$ is trivial, and we get zero. If $F^{\prime}$ has
rank greater than 1, then we may choose $H^{\prime}$ as above, and we get
$[W_{\\!\mathcal{A}_{H^{\prime}}}]$ by Lemma 3.1. ∎

######  Remark 3.3.

It is useful to think about the special case when $F=E$ in the proof of Lemma
3.2. In this case, $F^{\prime}=E$, and the rank of $F^{\prime}$ is equal to
the dimension of $L^{\prime}$. This means that, if
$h^{\textbf{m}}=[W_{\\!\mathcal{A}_{L^{\prime}}}]$ for some $L^{\prime}$ of
dimension greater than 1, then
$h^{\textbf{m}}h_{E}=[W_{\\!\mathcal{A}_{H^{\prime}}}]$ for some generic
hyperplane $H^{\prime}\subset L^{\prime}$. Iterating this observation, we see
that, if $k=\dim L-1-\sum m_{F}$, then $h^{\textbf{m}}h_{E}^{k}\in A^{\dim
L-1}(W_{\\!\mathcal{A}})$ is the class of a point. Similarly,
$(h^{\operatorname{aug}})^{\textbf{m}}(h_{E}^{\operatorname{aug}})^{k+1}\in
A^{\dim L}(W^{{\operatorname{aug}}}_{\\!\mathcal{A}})$ is the class of a
point.

######  Proposition 3.4.

We have

(7)
$\chi(W_{\\!\mathcal{A}},\eta^{\textbf{m}})=\deg_{W_{\\!\mathcal{A}}}\left(\frac{h^{\textbf{m}}}{1-h_{E}}\right)\quad
and\quad\chi(W^{{\operatorname{aug}}}_{\\!\mathcal{A}},(\eta^{{\operatorname{aug}}})^{\textbf{m}})=\deg_{W^{{\operatorname{aug}}}_{\\!\mathcal{A}}}\left(\frac{(h^{{\operatorname{aug}}})^{\textbf{m}}}{1-h_{E}^{{\operatorname{aug}}}}\right).$

###### Proof.

We will prove only the non-augmented case; the augmented case is identical.
The Euler characteristic can be computed after extension of scalars, so we can
assume that $\mathbb{F}$ is infinite. By Lemma 3.2, either
$\eta^{\textbf{m}}=0$ and $h^{\textbf{m}}=0$, in which case Equation (7) holds
trivially, or there exists some $L^{\prime}\subset L$ such that
$\eta^{\textbf{m}}=[\mathcal{O}_{W_{\\!\mathcal{A}_{L^{\prime}}}}]$ and
$h^{\textbf{m}}=[W_{\\!\mathcal{A}_{L^{\prime}}}]$. Then
$\chi(W_{\\!\mathcal{A}},\eta^{\textbf{m}})=1$ because
$W_{\\!\mathcal{A}_{L^{\prime}}}$ is a smooth iterated blow-up of projective
space, and
$\deg_{W_{\\!\mathcal{A}}}\left(\frac{h^{\textbf{m}}}{1-h_{E}}\right)=1$ by
Remark 3.3. ∎

We are now ready to prove the existence of exceptional isomorphisms from
$K$-rings to Chow rings for wonderful and augmented wonderful varieties.

###### Proof of Theorem 1.6.

Again, we prove only the non-augmented case. Note that, once we prove the
first sentence in the theorem, the second sentence will follow from
Proposition 3.4.

For each nonempty flat $F$, we have an isomorphism

$\zeta_{F}:K(\mathbb{P}(L^{F}))\to A(\mathbb{P}(L^{F}))$

sending the structure sheaf of a hyperplane (which we will denote by
$\sigma_{F}$) to the Chow class of a hyperplane (which we will denote by
$s_{F}$). This isomorphism has the property that, for any $\xi_{F}\in
K(\mathbb{P}(L^{F}))$,

$\chi(\mathbb{P}(L^{F}),\xi_{F})=\deg_{\mathbb{P}(L^{F})}\left(\frac{\zeta_{F}(\xi_{F})}{1-s_{F}}\right).$

By the Künneth formula [AP15, Proposition 6.4], we have an isomorphism

$\zeta=\otimes_{F}\zeta_{F}:K\bigg{(}\prod_{F}\mathbb{P}(L^{F})\bigg{)}\to
A\bigg{(}\prod_{F}\mathbb{P}(L^{F})\bigg{)}$

that takes $\sigma_{F}$ to $s_{F}$ for every $F$, and has the property that,
for any $\xi\in K\left(\prod_{F}\mathbb{P}(L^{F})\right)$,

(8)
$\chi\Big{(}\prod_{F}\mathbb{P}(L^{F}),\xi\Big{)}=\deg_{\prod_{F}\mathbb{P}(L^{F})}\left(\frac{\zeta(\xi)}{\prod_{F}(1-s_{F})}\right).$

By Theorem 1.8, the restriction map

$\displaystyle A\Big{(}\prod_{F}\mathbb{P}(L^{F})\Big{)}\to
A(W_{\\!\mathcal{A}})\qquad\text{given by}\qquad s_{F}\mapsto h_{F}$

is surjective. By Poincaré duality, its kernel is equal to the annihilator of
$[W_{\\!\mathcal{A}}]$. By Lemma 1.20 and Lemma 2.3, the analogous restriction
map

$\displaystyle K\Big{(}\prod_{F}\mathbb{P}(L^{F})\Big{)}\to
K(W_{\\!\mathcal{A}})\qquad\text{given by}\qquad\sigma_{F}$
$\displaystyle\mapsto$ $\displaystyle\eta_{F}$

is also surjective. By the nondegeneracy of the Euler pairing [AP15,
Proposition 6.3], the kernel is equal to the annihilator of
$[\mathcal{O}_{W_{\\!\mathcal{A}}}]$. We will prove that

(9) $\zeta([\mathcal{O}_{W_{\\!\mathcal{A}}}])=\prod_{F\neq
E}(1-s_{F})\cdot[W_{\\!\mathcal{A}}].$

Since $s_{F}$ is nilpotent, the above product is a unit, which implies that
$\zeta$ takes the annihilator of $[\mathcal{O}_{W_{\\!\mathcal{A}}}]$ to the
annihilator of $[W_{\\!\mathcal{A}}]$. This in turn shows that $\zeta$
descends to an isomorphism $\zeta_{\mathcal{A}}:K(W_{\\!\mathcal{A}})\to
A(W_{\\!\mathcal{A}})$.

We now prove Equation (9). For any $\textbf{m}=(m_{F}\mid F\text{ a nonempty
flat})$, let

$s^{\textbf{m}}:=\prod_{F}s_{F}^{m_{F}}\sigma^{\textbf{m}}:=\prod_{F}\sigma_{F}^{m_{F}},$

so that $s^{\textbf{m}}\mapsto h^{\textbf{m}}$ and
$\sigma^{\textbf{m}}\mapsto\eta^{\textbf{m}}$ under restriction. We have

$\displaystyle\deg_{\prod_{F}\mathbb{P}(L^{F})}\Big{(}s^{\textbf{m}}\cdot\zeta([\mathcal{O}_{W_{\\!\mathcal{A}}}])\Big{)}$
$\displaystyle=$
$\displaystyle\chi\Big{(}\prod_{F}\mathbb{P}(L^{F}),\sigma^{\textbf{m}}\cdot\prod_{F}(1-\sigma_{F})\cdot[\mathcal{O}_{W_{\\!\mathcal{A}}}]\Big{)}\qquad\text{by
Equation \eqref{eq-euler-deg}}$ $\displaystyle=$
$\displaystyle\chi\Big{(}W_{\\!\mathcal{A}},\eta^{\textbf{m}}\cdot\prod_{F}(1-\eta_{F})\Big{)}$
$\displaystyle=$
$\displaystyle\deg_{W_{\\!\mathcal{A}}}\Big{(}h^{\textbf{m}}\cdot\prod_{F\neq
E}(1-h_{F})\Big{)}\qquad\text{by Proposition \eqref{prop:HRRtype}}$
$\displaystyle=$
$\displaystyle\deg_{\prod_{F}\mathbb{P}(L^{F})}\Big{(}s^{\textbf{m}}\cdot\prod_{F\neq
E}(1-s_{F})\cdot[W_{\\!\mathcal{A}}]\Big{)}.$

Equation (9) then follows from Poincaré duality. ∎

##  4\. The $K$-ring of $\overline{\mathcal{M}}_{0,n}$

We now apply our results to study the $K$-ring of
$\overline{\mathcal{M}}_{0,n}$, and in particular prove the exceptional
isomorphism of Theorem 1.10. From Section 1.4, we observe that the following
diagram commutes:

${W_{\mathcal{B}_{n-1}}}$${\displaystyle\prod_{F}\mathbb{P}(L^{F})}$${\overline{\mathcal{M}}_{0,n}}$${\displaystyle\prod_{S\subset[n-1]}\mathbb{P}(L^{F_{S}}),}$$\scriptstyle{p}$$\scriptstyle{pr}$

where $pr$ is the projection onto the factors indexed by flats of the form
$F_{S}$, and $p$ is the restriction of $pr$. By [FM05, Theorem 4.2], $p$ is a
composition of blow-ups at smooth centers. By [Ful98, Proposition 6.7(b)], the
pullback maps

$\displaystyle p^{*}:A(\overline{\mathcal{M}}_{0,n})\to
A(W_{\mathcal{B}_{n-1}})\qquad\text{given by}\qquad c_{S}\mapsto
h_{F_{S}},\qquad\text{and}$ $\displaystyle
p^{*}:K(\overline{\mathcal{M}}_{0,n})\to
K(W_{\mathcal{B}_{n-1}})\qquad\text{give by}\qquad
1-[\mathcal{L}_{S}^{-1}]\mapsto\eta_{F_{S}}$

are both injective.

Consider the ring $R_{n}:=\mathbb{Z}[u_{S}\mid S\subset[n-1],|S|\geq 2]\subset
S_{\mathcal{B}_{n-1}},$ where the inclusion sends $u_{S}$ to $u_{F_{S}}$. Then
the map $S_{\mathcal{B}_{n-1}}\to A(W_{\mathcal{B}_{n-1}})$ of Theorem 1.8
restricts to a map $R_{n}\to A(\overline{\mathcal{M}}_{0,n})$. Consider the
following ideals in $R_{n}$:

$\displaystyle\mathcal{K}_{1}$ $\displaystyle:=\left\langle(u_{S}-u_{S\cup
T})(u_{T}-u_{S\cup T})\mid S\cap T\neq\varnothing\right\rangle$
$\displaystyle\mathcal{K}_{2}$ $\displaystyle:=\langle u_{S}\mid|S|=2\rangle.$

######  Theorem 4.1.

[Sin04] [EHKR10, Theorem 5.5] The map $R_{n}\to
A(\overline{\mathcal{M}}_{0,n})$ is surjective with kernel
$\mathcal{K}_{1}+\mathcal{K}_{2}$.

The surjectivity statement in Theorem 4.1 allows us to prove Theorem 1.10.

###### Proof of Theorem 1.10..

It is known to experts that $K(\overline{\mathcal{M}}_{0,n})$ is generated by
line bundles; see the discussion in the introduction of [CT20]. This can also
be proved in the same way as Lemma 1.20. Therefore, Theorem 4.1 and Lemma 2.3
imply that $K(\overline{\mathcal{M}}_{0,n})$ is generated by
$1-[\mathcal{L}_{S}^{-1}]$ for all subsets $S\subset[n-1]$ of cardinality at
least 2. By Theorems 1.6 and 4.1, restriction of the isomorphism
$\zeta_{\mathcal{B}_{n-1}}:K(W_{\mathcal{B}_{n-1}})\to
A(W_{\mathcal{B}_{n-1}})$ to

$K(\overline{\mathcal{M}}_{0,n})\subset K(W_{\mathcal{B}_{n-1}})$

takes $K(\overline{\mathcal{M}}_{0,n})$ isomorphically to
$A(\overline{\mathcal{M}}_{0,n})\subset A(W_{\mathcal{B}_{n-1}}).$ ∎

######  Remark 4.2.

In [CT21], Castravet and Tevelev studied $K(\overline{\mathcal{M}}_{0,n})$ as
a representation of the symmetric group $S_{n}$; in particular, they show that
the $S_{n}$-action on $K(\overline{\mathcal{M}}_{0,n})$ is a permutation
representation over $\mathbb{Z}$. The action of $S_{n}$ on
$A(\overline{\mathcal{M}}_{0,n})$ has also been studied, beginning with
[Get95]. Our isomorphism in Theorem 1.10 is not $S_{n}$-equivariant, but it is
equivariant with respect to the action of the subgroup $S_{n-1}\subset S_{n}$
that fixes $n$. Note that $K(\overline{\mathcal{M}}_{0,n})\otimes\mathbb{Q}$
and $A(\overline{\mathcal{M}}_{0,n})\otimes\mathbb{Q}$ are
$S_{n}$-equivariantly isomorphic via the Chern character, but one can check
that $A(\overline{\mathcal{M}}_{0,5})$ is not a permutation representation of
$S_{5}$ over $\mathbb{Z}$, and therefore cannot be $S_{5}$-equivariantly
isomorphic to $K(\overline{\mathcal{M}}_{0,5})$.

Theorems 1.10 and 4.1 combine to give us the following corollary.

######  Corollary 4.3.

The homomorphism $R_{n}\to K(\overline{\mathcal{M}}_{0,n})$ taking $u_{S}$ to
$1-[\mathcal{L}_{S}^{-1}]$ is surjective with kernel
$\mathcal{K}_{1}+\mathcal{K}_{2}$.

######  Remark 4.4.

The variety $\overline{\mathcal{M}}_{0,n}$ is a special case of a wonderful
variety with a _building set_. The proof of Theorem 1.10 generalizes to any
wonderful variety with a building set, as the surjectivity part of Theorem 4.1
can be proved for any wonderful variety with a building set along of the lines
of [BES21, Section 3.2]. Therefore, the application in this section can be
extended to the Hassett compactification of heavy/light weighted rational
stable curves [Has03]. It is also a wonderful variety with a certain building
set [CHMR16], and its Chow ring has been studied in [KKL21].

##  5\. Matroid $K$-rings

In this section, we prove Proposition 1.13, Theorem 1.15, Theorem 1.16,
Proposition 5.1, and Lemma 1.20. We also prove Proposition 5.3, which provides
a geometric interpretation of the Euler characteristic map on the $K$-ring and
augmented $K$-ring of an arbitrary matroid. We begin by introducing the
following pieces of notation

$\displaystyle t_{F}$ $\displaystyle:=\Phi_{\operatorname{FY}}(x_{F})\in A(M)$
$\displaystyle\quad\quad t_{F}^{\operatorname{aug}}$
$\displaystyle:=\Phi_{\operatorname{FY}}^{\operatorname{aug}}(x_{F})\in
A^{\operatorname{aug}}(M)$ $\displaystyle\tau_{F}$
$\displaystyle:=\Psi_{\operatorname{FY}}(x_{F})\in K(M)$
$\displaystyle\tau_{F}^{\operatorname{aug}}$
$\displaystyle:=\Psi_{\operatorname{FY}}^{\operatorname{aug}}(x_{F})\in
K^{\operatorname{aug}}(M)$

By Equation (4), we have

(10) $h_{F}=-\sum_{F\subset G}t_{G}h_{F}^{\operatorname{aug}}=-\sum_{F\subset
G}t_{G}^{\operatorname{aug}}.$

###### Proof of Proposition 1.13..

Since $A(M)=A(X_{\Sigma_{M}})$ is torsion-free, it is isomorphic to the
associated graded of $K(M)=A(X_{\Sigma_{M}})$ with respect to the coniveau
filtration [Ful93, Example 15.2.16]. Since we know that the associated graded
map $A(M)\to A(W_{\\!\mathcal{A}})$ is an isomorphism, the filtered map
$K(M)\to K(W_{\\!\mathcal{A}})$ is also an isomorphism [Wei94, Theorem
5.2.12]. The augmented case is identical. ∎

###### Proof of Lemma 1.20..

By Proposition 1.13, this is equivalent to the statement that the rings $K(M)$
and $K^{\operatorname{aug}}(M)$ are generated by line bundles on the toric
varieties $X_{\Sigma_{M}}$ and $X_{\Sigma_{M}^{\operatorname{aug}}}$. This is
a general property of smooth toric varieties, see [AP15, Lemma 2.2]. ∎

###### Proof of Theorem 1.15.

The proof of statements (1) and (2) is essentially identical to the proof of
Theorem 1.3, except that we use Theorem 1.12 in place of Theorem 1.1, and we
use the fact that the $K$-ring of a smooth toric variety is generated by
classes of line bundles (see [AP15, Lemma 2.2]) in place of Lemma 1.20. ∎

###### Proof of Theorem 1.16.

We prove only the augmented case; the non-augmented case is similar. Define a
map $\kappa_{M}:S_{M}\to K^{\operatorname{aug}}(M)$ by putting

$\kappa_{M}(u_{F})=1-\prod_{F\subset
G}(1-\tau^{\operatorname{aug}}_{G})^{-1}=1-\prod_{F\subset
G}(1+\tau^{\operatorname{aug}}_{G}+(\tau^{\operatorname{aug}}_{G})^{2}+\cdots).$

Our first task will be to show that $\kappa$ vanishes on
$\mathcal{J}_{1}+\mathcal{J}_{2}^{\operatorname{aug}}$, and therefore descends
to a map from $A^{\operatorname{aug}}(M)$ to $K^{\operatorname{aug}}(M)$. We
will make use of Theorem 1.15(2), which says that
$\tau^{\operatorname{aug}}_{F}\tau^{\operatorname{aug}}_{G}=0$ for any
incomparable $F$ and $G$,

$\prod_{F}(1-\tau^{\operatorname{aug}}_{F})=1,\qquad\text{and}\qquad\Psi_{FY}^{\operatorname{aug}}(y_{e})=1-\prod_{e\notin
G}(1-\tau_{G}^{\operatorname{aug}}).$

The later equation implies that $K^{{\operatorname{aug}}}(M)$ is generated by
the $\tau_{G}^{{\operatorname{aug}}}$. Using the relations in
$\mathcal{I}_{4}^{{\operatorname{aug}}}$, we get that

(11) $\tau^{\operatorname{aug}}_{F}\left(1-\prod_{e\notin
G}(1-\tau^{\operatorname{aug}}_{G})\right)=0$

for any $e\notin F$.

We begin by checking that $\kappa_{M}$ vanishes on a generator of
$\mathcal{J}_{1}$. We have

$\displaystyle\kappa_{M}\Big{(}(u_{F}-u_{F\vee G})(u_{G}-u_{F\vee G})\Big{)}$
$\displaystyle=$ $\displaystyle\left(\prod_{F\subset
H}\left(1-\tau^{\operatorname{aug}}_{H}\right)^{-1}-\prod_{F\vee G\subset
I}\left(1-\tau^{\operatorname{aug}}_{I}\right)^{-1}\right)\left(\prod_{G\subset
J}\left(1-\tau^{\operatorname{aug}}_{J}\right)^{-1}-\prod_{F\vee G\subset
I}\left(1-\tau^{\operatorname{aug}}_{I}\right)^{-1}\right)$ $\displaystyle=$
$\displaystyle\prod_{F\vee G\subset
I}\left(1-\tau^{\operatorname{aug}}_{I}\right)^{-2}\left(\prod_{F\subset
H\subsetneq F\vee
G}\left(1-\tau^{\operatorname{aug}}_{H}\right)^{-1}-1\right)\left(\prod_{G\subset
J\subsetneq F\vee
G}\left(1-\tau^{\operatorname{aug}}_{J}\right)^{-1}-1\right),$

which vanishes because $H$ and $J$ are incomparable for any $H$ appearing in
the second product and $J$ appearing in the third product.

Next, we check that $\kappa_{M}$ vanishes on a generator of
$\mathcal{J}_{2}^{\operatorname{aug}}$. Fix a flat $F$ of rank 1 and an
element $e\in F$, so that

$\kappa_{M}(u_{F})=1-\prod_{F\subset
H}\left(1-\tau^{\operatorname{aug}}_{H}\right)^{-1}=1-\prod_{e\in
H}\left(1-\tau^{\operatorname{aug}}_{H}\right)^{-1}=1-\prod_{e\notin
K}\left(1-\tau^{\operatorname{aug}}_{K}\right).$

Then

$\kappa_{M}(u_{F}^{2})=\left(1-\prod_{e\in
H}\left(1-\tau^{\operatorname{aug}}_{H}\right)^{-1}\right)\left(1-\prod_{e\notin
K}\left(1-\tau^{\operatorname{aug}}_{K}\right)\right),$

which vanishes by Equation (11). Now for any flat $G$, we have

$\displaystyle\kappa_{M}\Big{(}u_{F}(u_{G}-u_{F\vee G})\Big{)}$
$\displaystyle=$ $\displaystyle\left(1-\prod_{e\notin
K}\left(1-\tau^{\operatorname{aug}}_{K}\right)\right)\prod_{F\vee G\subset
I}\left(1-\tau^{\operatorname{aug}}_{I}\right)^{-1}\left(\prod_{G\subset
J\subsetneq F\vee
G}\left(1-\tau^{\operatorname{aug}}_{J}\right)^{-1}-1\right),$

which also vanishes by Equation (11) because $e\notin J$ for any $J$ appearing
in the last product.

We have now proved that $\kappa_{M}$ descends to a homomorphism
$\overline{\kappa}_{M}:A^{\operatorname{aug}}(M)\to
K^{\operatorname{aug}}(M)$. Möbius inversion tells us that

$1-\tau_{F}^{\operatorname{aug}}=\prod_{F\subset
G}\Big{(}1-\overline{\kappa}_{M}(h_{G})\Big{)}^{-\mu(F,G)},$

where $\mu$ is the Möbius function on the lattice of flats of $M$. This, along
with the surjectivity statement of Theorem 1.15(2), implies that
$\overline{\kappa}_{M}$ is surjective. By Lemma 2.1 applied to the variety
$X_{\Sigma_{M}^{\operatorname{aug}}}$, $A^{\operatorname{aug}}(M)$ and
$K^{\operatorname{aug}}(M)$ are free abelian groups of the same rank, thus
$\overline{\kappa}_{M}$ is an isomorphism. We may then take
$\zeta_{M}^{\operatorname{aug}}$ to be the inverse of $\overline{\kappa}_{M}$.
∎

######  Proposition 5.1.

Let $M$ be the matroid associated with a hyperplane arrangement $\mathcal{A}$.
The following diagrams commute:

${K(M)}$${A(M)}$${K^{{\operatorname{aug}}}(M)}$${A^{{\operatorname{aug}}}(M)}$${K(W_{\\!\mathcal{A}})}$${A(W_{\\!\mathcal{A}})}$${K(W_{\\!\mathcal{A}}^{{\operatorname{aug}}})}$${A(W_{\\!\mathcal{A}}^{{\operatorname{aug}}}).}$$\scriptstyle{\zeta_{M}}$$\scriptstyle{\zeta_{M}^{{\operatorname{aug}}}}$$\scriptstyle{\zeta_{\mathcal{A}}}$$\scriptstyle{\zeta_{\mathcal{A}}^{{\operatorname{aug}}}}$

###### Proof.

We do the non-augmented case. Since $\zeta_{\mathcal{A}}$ takes $\eta_{F}$ to
$h_{F}$ and $\zeta_{M}$ is characterized in the statement of Theorem 1.16, we
need to show that

$\eta_{F}=1-\prod_{F\subset G}(1-\tau_{G})^{-1}.$

By considering first Chern classes, we have

$\mathcal{L}_{F}\cong\mathcal{L}_{E}\otimes\bigotimes_{F\subset G\subsetneq
E}\mathcal{O}(-D_{G})$

of line bundles on the wonderful variety $W_{\\!\mathcal{A}}$. Since
$\tau_{E}=1-[\mathcal{L}_{E}]$, $\tau_{G}=1-[\mathcal{O}(-D_{G})]$ for
$G\not=E$, and $\eta_{F}=1-[\mathcal{L}_{F}^{-1}]$, the result follows. ∎

We now discuss an analogue of the projection formula that will allow us to
interpret the Euler characteristic maps on matroid $K$-rings using geometry.
Let $\mathcal{U}_{E}$ denote the Boolean arrangement on the ground set $E$,
consisting of the coordinate hyperplanes in $\mathbb{F}^{E}$, realizing the
Boolean matroid $U_{E}$. The wonderful variety of $\mathcal{U}_{E}$ is a toric
variety, called the permutohedral variety, which we denote $X_{E}$. The
augmented wonderful variety is also a toric variety, called the stellahedral
variety, which we denote $X_{E}^{{\operatorname{aug}}}$. For any matroid $M$,
there is an open inclusion $\iota\colon X_{\Sigma_{M}}\hookrightarrow X_{E}$,
so we have restriction maps $\iota^{*}\colon K(X_{E})\to K(M)$ and
$\iota^{*}\colon A(X_{E})\to A(M)$. Similarly, there is an open inclusion
$\iota^{{\operatorname{aug}}}\colon X_{\Sigma_{M}^{{\operatorname{aug}}}}\to
X_{E}^{{\operatorname{aug}}}$ and corresponding restriction maps on $K$-rings
and Chow rings. These maps are characterized by the property that

$\iota^{*}t_{S}=\begin{cases}t_{S},&S\text{ is a flat of }M\\\
0,&\text{otherwise}\end{cases}\iota^{*}\tau_{S}=\begin{cases}\tau_{S},&S\text{
is a flat of }M\\\ 0,&\text{otherwise,}\end{cases}$

and similarly in the augmented setting.

######  Lemma 5.2.

For any subset $S\subset E$, let $\bar{S}$ be the closure of $S$ in $M$. We
have

$\displaystyle\iota^{*}(\eta_{S})$ $\displaystyle=\eta_{\bar{S}}\in K(M)$
$\displaystyle\quad\quad(\iota^{\operatorname{aug}})^{*}(\eta^{\operatorname{aug}}_{S})$
$\displaystyle=\eta^{\operatorname{aug}}_{\bar{S}}\in
K^{\operatorname{aug}}(M)$ $\displaystyle\iota^{*}(h_{S})$
$\displaystyle=h_{\bar{S}}\in A(M)$
$\displaystyle(\iota^{\operatorname{aug}})^{*}(h^{\operatorname{aug}}_{S})$
$\displaystyle=h^{\operatorname{aug}}_{\bar{S}}\in A^{\operatorname{aug}}(M).$

###### Proof.

We have that

$\iota^{*}h_{S}=-\iota^{*}\sum_{S\subset
S^{\prime}}t_{S^{\prime}}=-\sum_{\begin{subarray}{c}S\subset F\\\ \text{$F$ a
flat}\end{subarray}}t_{F}=h_{\bar{S}}.$

The other statements are similar. ∎

There exist unique Chow classes

$\Delta_{M}\in A(U_{E})\Delta_{M}^{\operatorname{aug}}\in
A^{\operatorname{aug}}(U_{E}),$

called the Bergman class and augmented Bergman class, respectively, with the
properties that, for any $\xi\in A(U_{E})$ and $\xi^{\operatorname{aug}}\in
A^{\operatorname{aug}}(U_{E})$, we have

$\deg_{M}(\iota^{*}\xi)=\deg_{U_{E}}(\xi\cdot\Delta_{M})\deg_{M}((\iota^{\operatorname{aug}})^{*}\xi^{\operatorname{aug}})=\deg_{U_{E}}(\xi^{\operatorname{aug}}\cdot\Delta_{M}^{\operatorname{aug}}).$

In the non-augmented setting, this is proved in [BES21, Theorem 4.2.1]. The
same argument works in the augmented setting, using [EHL22, Theorem 1.11].

######  Proposition 5.3.

The following diagrams commute:

${K(U_{E})}$${A(U_{E})}$${K^{\operatorname{aug}}(U_{E})}$${A^{\operatorname{aug}}(U_{E})}$${K(M)}$${A(M)}$${K^{{\operatorname{aug}}}(M)}$${A^{{\operatorname{aug}}}(M).}$$\scriptstyle{\zeta_{U_{E}}}$$\scriptstyle{\iota^{*}}$$\scriptstyle{\iota^{*}}$$\scriptstyle{\zeta_{U_{E}}^{{\operatorname{aug}}}}$$\scriptstyle{(\iota^{\operatorname{aug}})^{*}}$$\scriptstyle{(\iota^{\operatorname{aug}})^{*}}$$\scriptstyle{\zeta_{M}}$$\scriptstyle{\zeta^{{\operatorname{aug}}}_{M}}$

Additionally, for any $\xi\in K(U_{E})$ and $\xi^{{\operatorname{aug}}}\in
K^{\operatorname{aug}}(U_{E})$, we have

$\chi\big{(}M,\iota^{*}(\xi)\big{)}=\chi\left(U_{E},\xi\cdot\zeta_{U_{E}}^{-1}\left(\Delta_{M}\right)\right),\qquad\text{and}$
$\chi^{{\operatorname{aug}}}\big{(}M,(\iota^{\operatorname{aug}})^{*}\left(\xi^{{\operatorname{aug}}}\right)\big{)}=\chi\left(U_{E},\xi^{{\operatorname{aug}}}\cdot(\zeta^{{\operatorname{aug}}}_{U_{E}})^{-1}\left(\Delta_{M}^{{\operatorname{aug}}}\right)\right).$

###### Proof of Proposition 5.3.

Commutativity of the diagrams follows from Lemma 5.2. For the second
statement, we do the non-augmented case. For any $\xi\in K(U_{E})$, we have

$\displaystyle\chi\big{(}M,\iota^{*}(\xi)\big{)}$
$\displaystyle=\deg_{M}\left(\frac{\zeta_{M}\circ\iota^{*}(\xi)}{1-h_{E}}\right)$
$\displaystyle=\deg_{M}\left(\frac{\iota^{*}\circ\zeta_{U_{E}}(\xi)}{1-h_{E}}\right)$
$\displaystyle=\deg_{U_{E}}\left(\frac{\zeta_{U_{E}}(\xi)\cdot\Delta_{M}}{1-h_{E}}\right)$
$\displaystyle=\chi\left(U_{E},\xi\cdot\zeta^{-1}_{U_{E}}(\Delta_{M})\right).$

The augmented case is similar. ∎

As a corollary, we are to prove the following characterizing properties of
$\zeta_{U_{E}}$ and $\zeta_{U_{E}}^{\operatorname{aug}}$, which demonstrate
that they agree with the maps denoted $\zeta$ in [BEST22, Corollary 10.6] and
[EHL22, Theorem 8b].

######  Corollary 5.4.

Let $\mathcal{A}$ be a hyperplane arrangement equipped with a choice of linear
function cutting out each hyperplane, which induces embeddings
$W_{\\!\mathcal{A}}\subset X_{E}$ and
$W_{\\!\mathcal{A}}^{{\operatorname{aug}}}\subset X_{E}^{\operatorname{aug}}$.
Then

$\zeta_{U_{E}}\big{(}[\mathcal{O}_{W_{\\!\mathcal{A}}}]\big{)}=[W_{\\!\mathcal{A}}]\zeta_{U_{E}}^{\operatorname{aug}}\big{(}[\mathcal{O}_{W^{{\operatorname{aug}}}_{\\!\mathcal{A}}}]\big{)}=[W^{{\operatorname{aug}}}_{\\!\mathcal{A}}].$

###### Proof.

Let $M$ be the matroid associated with $\mathcal{A}$. For any $\xi\in
K(X_{E})=K(U_{E})$, we have

$\chi\left(X_{E},\xi\cdot\zeta_{U_{E}}^{-1}(\Delta_{M})\right)=\chi\left(U_{E},\xi\cdot\zeta_{U_{E}}^{-1}(\Delta_{M})\right)=\chi(M,\iota^{*}(\xi))=\chi(W_{\\!\mathcal{A}},\iota^{*}(\xi))=\chi(X_{E},\xi\cdot[\mathcal{O}_{W_{\\!\mathcal{A}}}]),$

where the second equality comes from Proposition 5.3 and the fourth comes from
the projection formula on the permutohedral variety. The nondegeneracy of the
pairing $(x,y)\mapsto\chi(X_{E},xy)$ [AP15, Theorem 1.3] implies that
$\zeta_{U_{E}}^{-1}(\Delta_{M})=[\mathcal{O}_{W_{\\!\mathcal{A}}}]$. Applying
$\zeta_{U_{E}}$, we find that

$[W_{\\!\mathcal{A}}]=\Delta_{M}=\zeta_{U_{E}}\big{(}[\mathcal{O}_{W_{\\!\mathcal{A}}}]\big{)}.$

The augmented case is similar. ∎

Motivated by Corollary 5.4, we define

$[\mathcal{O}_{M}]:=\zeta_{U_{E}}^{-1}(\Delta_{M})\in
K(U_{E})[\mathcal{O}_{M}^{{\operatorname{aug}}}]:=(\zeta_{U_{E}}^{{\operatorname{aug}}})^{-1}(\Delta_{M}^{{\operatorname{aug}}})\in
K^{\operatorname{aug}}(U_{E})$

for any loopless matroid $M$. Proposition 5.3 may then be interpreted as a
purely combinatorial projection formula.

We conclude this section with the following lemma, which we will need in
Section 8.

######  Lemma 5.5.

The exceptional isomorphism $\zeta_{M}:K(M)\to A(M)$ has the following
properties:

  * •

$\zeta_{M}(\tau_{\varnothing})=t_{\varnothing}$

  * •

$\zeta_{M}(\tau_{E})=\frac{t_{E}}{1+t_{E}}$

  * •

For any flat $F$, $\zeta_{M}(\tau_{F})=t_{F}+\text{terms of degree at least
2}$.

###### Proof.

By Proposition 5.3, it is sufficient to prove these statements when $M=U_{E}$
is the Boolean matroid. The first two statements for the Boolean matroid are
proved in the course of the proof of [BEST22, Theorem 10.11].

For the last statement, let $F$ be a nonempty flat. After restricting from
$X_{\Sigma^{\operatorname{aug}}_{M}}$ to $X_{\Sigma_{M}}$, we have that

$\tau_{F}=-\sum_{F\subset G}\mu(F,G)\eta_{G}+\text{terms of degree at least 2
in $\\{\eta_{G}\mid F\subset G\\}$}.$

Applying $\zeta_{M}$, we find that

$\displaystyle\zeta_{M}(\tau_{F})$ $\displaystyle=$
$\displaystyle-\sum_{F\subset G}\mu(F,G)h_{G}+\text{terms of degree at least 2
in $\\{h_{G}\mid F\subset G\\}$}$ $\displaystyle=$ $\displaystyle
t_{F}+\text{terms of degree at least 2},$

which completes the proof. ∎

##  6\. Adams operations and Serre duality

We now discuss some additional structures on matroid $K$-rings. The fact that
$K(M)$ and $K^{{\operatorname{aug}}}(M)$ are (by definition) $K$-rings of
varieties endows them with the structure of augmented $\lambda$-rings [BGI71,
Exposé V, Exemple 3.9.1]. This means that we have a rank function $\epsilon$
that takes values in $\mathbb{Z}$, and for each natural number $k$, we have
operations $\lambda^{k}$ and $\Psi^{k}$ (the latter called Adams operations)
characterized by the property that

$\lambda^{k}([\mathcal{E}])=[\wedge^{k}\mathcal{E}]\Psi^{k}([\mathcal{E}])=[\otimes^{k}\mathcal{E}]$

for any vector bundle $\mathcal{E}$. Since our simplicial generators
$\eta_{F}$ are all of the form $1-[\mathcal{E}]$ for some line bundle
$\mathcal{E}$, we have $\epsilon(\eta_{F})=0$, and

$\Psi^{k}(\eta_{F})=\sum_{i=1}^{k}(-1)^{i+1}\binom{k}{i}\eta_{F}^{i}.$

The formula for augmented simplicial generators is identical. Note that the
Adams operations are ring homomorphisms, which is not at all combinatorially
obvious from the above formula. The Adams operations become simultaneously
diagonalizable after tensoring with $\mathbb{Q}$, and their eigenspaces (which
are independent of $k>1$) map isomorphically to the graded pieces of the Chow
ring via the Chern character. We also have a duality automorphism $D$,
characterized by the property that $D([\mathcal{E}])=[\mathcal{E}^{\vee}]$ for
any vector bundle $\mathcal{E}$. In terms of the simplicial generators, this
takes the form

$D(\eta_{F})=\frac{-\eta_{F}}{1-\eta_{F}}=-\eta_{F}-\eta_{F}^{2}-\dotsb,$

and similarly in the augmented setting. We note that the operations
$\epsilon$, $\lambda^{k}$, $\Psi^{k}$, and $D$ all commute with the maps
$\iota^{*}$ and $(\iota^{\operatorname{aug}})^{*}$ introduced in Section 5
because they are compatible with functorial maps between $K$-rings of
varieties.

On a $d$-dimensional smooth projective variety $X$ with canonical bundle
$\omega_{X}$, Serre duality implies that, for any vector bundle $\mathcal{E}$,

$\chi(X,\mathcal{E})=(-1)^{d}\chi(X,\omega_{X}\otimes\mathcal{E}^{\vee}).$

In particular, this holds for $X=W_{\\!\mathcal{A}}$ or
$W^{{\operatorname{aug}}}_{\\!\mathcal{A}}$. We will show that a similar
formula holds on $K(M)$ for any matroid $M$, even if $M$ is not realizable.
Our first task is to define classes $\omega_{M}\in K(M)$ and
$\omega_{M}^{{\operatorname{aug}}}\in K^{{\operatorname{aug}}}(M)$ that will
play the roles of the canonical bundles.

For any matroid on the ground set $E$, the matroid polytope
$P(M)\subset\mathbb{R}^{E}$ is defined to be the convex hull of the indicator
functions of bases of $M$, and the independence polytope
$I\\!P(M)\subset\mathbb{R}^{E}$ is defined to be the convex hull of indicator
functions of independent subsets of $M$.

Recall that, on a smooth projective toric variety with fan $\Sigma$, there is
a correspondence between torus equivariant nef line bundles and lattice
polytopes whose normal fans coarsen $\Sigma$. For any matroid on the ground
set $E$, $P(M)$ coarsens $\Sigma_{U_{E}}$ and $I\\!P(M)$ coarsens
$\Sigma_{U_{E}}^{\operatorname{aug}}$, so we obtain line bundles $[P(M)]$ and
$[I\\!P(M)]$ on $X_{E}$ and $X_{E}^{{\operatorname{aug}}}$, respectively.

Let $M^{\perp}$ be the matroid dual to $M$, characterized by the property that
the bases of $M^{\perp}$ are the complements of the bases of $M$. We define
the classes

$\omega_{M}:=\iota^{*}\left(\omega_{X_{E}}\cdot[P(M^{\perp})]\right)\in K(M)$

and

$\omega_{M}^{\operatorname{aug}}:=(\iota^{\operatorname{aug}})^{*}\left(\omega^{{\operatorname{aug}}}_{X_{E}}\cdot[I\\!P(M^{\perp})]\right)\in
K^{\operatorname{aug}}(M).$

These definitions are motivated by the following proposition.

######  Proposition 6.1.

Let $\mathcal{A}$ be a hyperplane arrangement with associated matroid $M$. The
isomorphisms $K(M)\cong K(W_{\\!\mathcal{A}})$ and
$K^{\operatorname{aug}}(M)\cong K(W^{{\operatorname{aug}}}_{\\!\mathcal{A}})$
take $\omega_{M}$ and $\omega_{M}^{\operatorname{aug}}$ to the canonical
bundles of $W_{\\!\mathcal{A}}$ and
$W^{{\operatorname{aug}}}_{\\!\mathcal{A}}$, respectively.

###### Proof.

The adjunction formula states that the canonical class of
$W^{{\operatorname{aug}}}_{\\!\mathcal{A}}$ is equal to the restriction of the
canonical class of $X_{E}^{{\operatorname{aug}}}$ tensored with the
determinant of the normal bundle to
$W^{{\operatorname{aug}}}_{\\!\mathcal{A}}$ inside of
$X_{E}^{{\operatorname{aug}}}$. The determinant of the normal bundle is equal
to the restriction of $[I\\!P(M^{\perp})]$ by [EHL22, Proposition 4.6 and
Corollary 5.4]. The non-augmented case can be prove similarly, using [BEST22,
Theorem 7.10]. ∎

The following theorem is a combinatorial version of Serre duality for
$K$-rings of matroids.

######  Theorem 6.2.

For any $\xi\in K(M)$ and $\xi^{{\operatorname{aug}}}\in
K^{{\operatorname{aug}}}(M)$, we have

$\chi(M,\xi)=(-1)^{\operatorname{rk}M-1}\chi(M,\omega_{M}\cdot
D(\xi))\chi^{{\operatorname{aug}}}(M,\xi^{{\operatorname{aug}}})=(-1)^{\operatorname{rk}M}\chi^{{\operatorname{aug}}}(M,D(\xi^{{\operatorname{aug}}})).$

In the realizable case, Theorem 6.2 follows from Proposition 6.1 and Serre
duality for the varieties $W_{\\!\mathcal{A}}$ and
$W^{{\operatorname{aug}}}_{\\!\mathcal{A}}$. Our strategy is to deduce the
general case using the concept of valuativity. Given an abelian group $A$, a
function $f\colon\\{\text{matroids on }E\\}\to A$ is called valuative if it
factors through the map that takes $M$ to the indicator function of its
matroid polytope $P(M)\subset\mathbb{R}^{E}$. That is, for any matroids
$M_{1},\ldots,M_{k}$ and integers $a_{1},\ldots,a_{k}$ such that $\sum
a_{i}1_{P(M_{i})}=0$, we require that $\sum a_{i}f(M_{i})=0$.

######  Remark 6.3.

The notion of valuativity is typically defined on functions that take values
on the set of all matroids on $E$, whereas we are only considering loopless
matroids in this paper. This is not an important distinction, as the valuative
group of loopless matroids on $E$ is a direct summand of the valuative group
of all matroids on $E$. That is, a function on the set of loopless matroids on
$E$ is valuative if and only if it extends to a valuative function on the set
of all matroids on $E$, if and only if it extends by zero to a valuative
function on the set of all matroids on $E$.

######  Lemma 6.4.

Fix a pair of classes $\xi\in K(U_{E})$ and $\xi^{\operatorname{aug}}\in
K^{\operatorname{aug}}(U_{E})$. The following four $\mathbb{Z}$-valued
functions are valuative:

$\displaystyle M$ $\displaystyle\mapsto\chi(M,\iota^{*}\xi)$ $\displaystyle M$
$\displaystyle\mapsto\chi^{\operatorname{aug}}(M,(\iota^{\operatorname{aug}})^{*}\xi^{\operatorname{aug}})$
$\displaystyle M$
$\displaystyle\mapsto(-1)^{\operatorname{rk}M-1}\chi\big{(}M,\omega_{M}\cdot
D(\iota^{*}\xi)\big{)}$ $\displaystyle M$
$\displaystyle\mapsto(-1)^{\operatorname{rk}M}\chi^{\operatorname{aug}}\big{(}M,\omega_{M}^{\operatorname{aug}}\cdot
D((\iota^{\operatorname{aug}})^{*}\xi^{\operatorname{aug}})\big{)}.$

###### Proof.

We begin with the first function. By Proposition 5.3,
$\chi(M,\iota^{*}\xi)=\chi(X_{E},[\mathcal{O}_{M}]\cdot\xi)$. Recall that
$[\mathcal{O}_{M}]=\zeta_{U_{E}}^{-1}(\Delta_{M})$, and the function
$M\mapsto\Delta_{M}$ is valuative by [BEST22, Corollary 7.11]. Therefore the
function $M\mapsto[\mathcal{O}_{M}]$ is valuative, and the result follows by
linearity of the Euler characteristic map on $K(X_{E})$. The proof of
valuativity of the second function is similar, except that we now use [EHL22,
Proposition 4.7] to establish the valuativity of the function
$M\mapsto[\mathcal{O}_{M}^{{\operatorname{aug}}}]$.

After applying $\zeta_{U_{E}}^{\operatorname{aug}}$, it follows from [EHL22,
Proposition 4.7 and Corollary 6.5] that the function
$M\mapsto[\mathcal{O}_{M}^{\operatorname{aug}}]\cdot[I\\!P(M^{\perp})]$ is
valuative. This implies the valuativity of the fourth function. One can argue
similarly for the third function. ∎

###### Proof of Theorem 6.2.

We have already used Proposition 6.1 to prove the theorem for realizable
matroids. We have shown that each side of both claimed equalities is
valuative. The full theorem now follows from a result of Derksen and Fink
[DF10] (see also [BEST22, Lemma 5.9]), which says that the matroid polytope of
any matroid can be expressed as a linear combination of indicator functions of
matroid polytopes of realizable matroids. ∎

######  Question 6.5.

Can Theorem 6.2 be applied to give interesting identities for matroids? For
example, let $f(\ell):=\chi(M,\omega_{M}^{\ell})$; then Theorem 6.2 implies
that $f(\ell)=(-1)^{r-1}f(1-\ell)$. Does this statement admit an elementary
proof, without using valuativity and Serre duality for wonderful varieties?

##  7\. Euler characteristics and simplicial Snapper polynomials

In this section, we give purely combinatorial formulas for the Euler
characteristic of a monomial in the simplicial generators on $K(M)$ or
$K^{\operatorname{aug}}(M)$. Given a pair of tuples

$\textbf{m}=(m_{F}\mid F\text{ a nonempty
flat})\textbf{m}^{\prime}=(m^{\prime}_{F}\mid F\text{ a nonempty flat}),$

we say that $\textbf{m}^{\prime}\leq\textbf{m}$ if $m^{\prime}_{F}\leq m_{F}$
for all $F$. We say that m satisfies the Hall–Rado condition if, for all
$\textbf{m}^{\prime}\leq\textbf{m}$, we have

$\operatorname{rank}\bigcup_{m^{\prime}_{F}>0}F\;\geq\;\sum_{F}m^{\prime}_{F}.$

If the inequality is always strict, we say that m satisfies the dragon
Hall–Rado condition. The following theorem computes the degrees of the
monomials $h^{\textbf{m}}$ and $(h^{\operatorname{aug}})^{\textbf{m}}$.

######  Theorem 7.1.

If $\sum m_{F}=\operatorname{rk}M-1$, then [BES21, Theorem 5.2.4]

$\deg_{M}(h^{\textbf{m}})=\begin{cases}1&\text{if $\textbf{m}$ satisfies the
dragon Hall--Rado condition}\\\ 0&\text{otherwise.}\end{cases}$

If $\sum m_{F}=\operatorname{rk}M$, then [EL]

$\deg^{\operatorname{aug}}_{M}((h^{\operatorname{aug}})^{\textbf{m}})=\begin{cases}1&\text{if
$\textbf{m}$ satisfies the Hall--Rado condition}\\\
0&\text{otherwise.}\end{cases}$

The $K$-theoretic analogue of Theorem 7.1 is almost exactly the same, except
without the condition on $\sum m_{F}$.

######  Theorem 7.2.

For any $m$, we have

$\chi(M,\eta^{\textbf{m}})=\begin{cases}1&\text{if $\textbf{m}$ satisfies the
dragon Hall--Rado condition}\\\ 0&\text{otherwise}\end{cases}$

and

$\chi^{\operatorname{aug}}(M,(\eta^{\operatorname{aug}})^{\textbf{m}})=\begin{cases}1&\text{if
$\textbf{m}$ satisfies the Hall--Rado condition}\\\
0&\text{otherwise.}\end{cases}$

###### Proof.

We prove the first statement; the augmented case is similar. If $\sum
m_{F}\geq\operatorname{rk}M$, then $\eta^{\textbf{m}}=0$ and m fails to
satisfy the dragon Hall–Rado condition. If $\sum m_{F}<\operatorname{rk}M$, we
let

$k=\operatorname{rk}M-1-\sum m_{F}\geq 0,$

and define $\tilde{\textbf{m}}$ by putting $\tilde{m}_{E}=m_{E}+k$ and
$\tilde{m}_{F}=m_{F}$ for all $F\neq E$. Then

$\chi(M,\eta^{\textbf{m}})=\deg_{M}\left(\frac{\zeta_{M}(\eta^{\textbf{m}})}{1-h_{E}}\right)=\deg_{M}\left(\frac{h^{\textbf{m}}}{1-h_{E}}\right)=\deg_{M}\left(h^{\tilde{\textbf{m}}}\right).$

We observe that m satisfies dragon Hall–Rado if and only if
$\tilde{\textbf{m}}$ does, thus our formula follows from Theorem 7.1. ∎

Given a proper variety $X$ and a finite tuple of line bundles
$\mathcal{L}=(\mathcal{L}_{1},\ldots,\mathcal{L}_{k})$ on $X$, the function
from $\mathbb{Z}^{k}$ to $\mathbb{Z}$ given by the formula

$\textup{Snap}_{\mathcal{L}}(\textbf{a}):=\chi\left(X,\mathcal{L}_{1}^{a_{1}}\otimes\cdots\otimes\mathcal{L}_{k}^{a_{k}}\right)$

is a polynomial of degree at most $\dim X$ [Sna59, Theorem 9.1], which we call
the Snapper polynomial. Given an integer $x$ and a natural number $d$, let

$x^{(d)}=\frac{x(x+1)\cdots(x+d-1)}{d!}=\binom{d+k-1}{d}=(-1)^{d}\binom{-x}{d}.$

If $\textbf{a}\in\mathbb{Z}^{k}$ and $\textbf{d}\in\mathbb{N}^{k}$, let

$\textbf{a}^{(\textbf{d})}:=a_{1}^{(d_{1})}\cdots a_{k}^{(d_{k})}.$

######  Lemma 7.3.

If $\sigma_{i}=1-[\mathcal{L}_{i}^{-1}]$ for each $i\in\\{1,\ldots,k\\}$, then

$\textup{Snap}_{\mathcal{L}}(\textbf{a})=\sum_{\textbf{d}\in\mathbb{N}^{k}}\chi\left(X,\sigma_{1}^{d_{1}}\cdots\sigma_{k}^{d_{k}}\right)\,\textbf{a}^{(\textbf{d})}.$

###### Proof.

We have

$\mathcal{L}_{i}^{a_{i}}=\frac{1}{(1-\sigma_{i})^{a_{i}}}=\sum_{d_{i}=0}^{\infty}a_{i}^{(d_{i})}\sigma_{i}^{d_{i}}.$

The lemma follows. ∎

Let $M$ be a matroid, and consider a tuple $\textbf{a}=(a_{F}\mid\text{$F$ a
nonempty flat})$. Let

$\textup{Snap}_{M}(\textbf{a}):=\sum_{m}\chi(M,\eta^{\textbf{m}})\textbf{a}^{(\textbf{m})}\textup{Snap}_{M}^{\operatorname{aug}}(\textbf{a}):=\sum_{m}\chi^{\operatorname{aug}}(M,(\eta^{\operatorname{aug}})^{\textbf{m}})\textbf{a}^{(\textbf{m})}.$

This definition is motivated by the following observation.

######  Lemma 7.4.

Let $\mathcal{A}$ be a hyperplane arrangement with associated matroid $M$.
Then $\textup{Snap}_{M}$ and $\textup{Snap}_{M}^{\operatorname{aug}}$ coincide
with the geometrically defined Snapper polynomials for the varieties
$W_{\\!\mathcal{A}}$ and $W^{{\operatorname{aug}}}_{\\!\mathcal{A}}$ with
respect to the line bundles $(\mathcal{L}_{F})$ and
$(\mathcal{L}_{F}^{\operatorname{aug}})$.

###### Proof.

By definition, $\eta_{F}=1-[\mathcal{L}_{F}^{-1}]$. The fact that
$\textup{Snap}_{M}$ coincides with the Snapper polynomial for
$W_{\\!\mathcal{A}}$ with respect to $(\mathcal{L}_{F})$ then follows from
Lemma 7.3. The augmented case is identical. ∎

We can now rephrase Theorem 7.2 as a statement about Snapper polynomials.

######  Corollary 7.5.

We have

$\textup{Snap}_{M}(\textbf{a})\;\;=\sum_{\begin{subarray}{c}\text{$\textbf{m}$
satisfies}\\\ \text{dragon Hall--
Rado}\end{subarray}}\textbf{a}^{(\textbf{m})}\textup{Snap}_{M}^{\operatorname{aug}}(\textbf{a})\;\;=\sum_{\begin{subarray}{c}\text{$\textbf{m}$
satisfies}\\\ \text{Hall--Rado}\end{subarray}}\textbf{a}^{(\textbf{m})}.$

######  Remark 7.6.

In the special case where $M=U_{E}$ is Boolean, the first equality in
Corollary 7.5 appears in [Pos09, Theorem 11.3].

##  8\. Snapper polynomials in the Feichtner–Yuzvinsky generators

Section 7 was about the Snapper polynomial of the wonderful variety of an
arrangement with respect to the line bundles $\\{\mathcal{L}_{F}\mid\text{$F$
a nonempty flat}\\}$, or of an arbitrary matroid with respect to the
corresponding $K$-classes. This was fundamentally a “simplicial” construction,
since the simplicial generators for $K$-theory were defined in terms of these
line bundles. In this section, we consider the Snapper polynomial with respect
to a collection of line bundles related to the Feichtner–Yuzvinsky generators.
For simplicity, we work only in the non-augmented setting.

Recall that, for all flats $F$, we have defined

$t_{F}:=\Phi_{\operatorname{FY}}(x_{F})\in
A(M)\tau_{F}:=\Psi_{\operatorname{FY}}(x_{F})\in K(M).$

For any tuple of natural numbers $\textbf{m}=(m_{F}\mid\text{$F$ a flat})$,
let

$\mathbf{\tau}^{\textbf{m}}:=\prod_{F}\tau_{F}^{m_{F}}.$

For any tuple of integers $\textbf{a}=(a_{F}\mid\text{$F$ a flat})$, we define

$\textup{Snap}_{M}^{\operatorname{FY}}(\textbf{a}):=\sum_{\textbf{m}}\chi(M,\mathbf{\tau}^{\textbf{m}})\textbf{a}^{(\textbf{m})}.$

If $M$ is the matroid associated with a hyperplane arrangement $\mathcal{A}$,
let $D_{\emptyset}$ and $D_{E}$ be the divisor classes on $W_{\\!\mathcal{A}}$
with first Chern classes $t_{\emptyset}$ and $t_{E}$ respectively, so that we
have $\tau_{F}=1-[\mathcal{O}(-D_{F})]$ for all flats $F$. Lemma 7.3
immediately implies that $\textup{Snap}_{M}^{\operatorname{FY}}$ is the
Snapper polynomial of $W_{\\!\mathcal{A}}$ with respect to the tuple of line
bundles $\left(\mathcal{O}(D_{F})\mid\text{$F$ a flat}\right)$.

We now provide an explicit formula for the polynomial
$\textup{Snap}_{M}^{\operatorname{FY}}(\textbf{a})$. For any natural number
$k$, let $\operatorname{Flag}_{k}$ be the collection of flags of flats of the
form

$\mathcal{F}=\\{\varnothing=F_{0}\subsetneq F_{1}\subsetneq\cdots\subsetneq
F_{k}=E\\}.$

If $F$ and $G$ are incomparable flats, then $\tau_{F}\tau_{G}=0\in K(M)$,
therefore $\textup{Snap}_{M}^{\operatorname{FY}}(\textbf{a})$ does not contain
any monomials that are multiples of $a_{F}a_{G}$. This implies that we may
write

$\textup{Snap}_{M}^{\operatorname{FY}}(\textbf{a})=\sum_{k\geq
1}\sum_{\mathcal{F}\in\operatorname{Flag}_{k}}\sum_{\textbf{m}}c(\mathcal{F},\textbf{m})\,a_{F_{0}}^{(m_{0})}a_{F_{1}}^{(m_{1}+1)}a_{F_{2}}^{(m_{2}+1)}\cdots
a_{F_{k-1}}^{(m_{k-1}+1)}a_{F_{k}}^{(m_{k})},$

where m ranges over all tuples $(m_{0},\ldots,m_{k})$ of natural numbers. It
remains only to compute the constants $c(\mathcal{F},\textbf{m})$.

######  Remark 8.1.

It may be slightly confusing that the first and last exponents are $(m_{0})$
and $(m_{k})$, whereas the middle exponents are $(m_{i}+1)$ for $1\leq i<k$.
The point is that a particular “monomial” appears in the summand indexed by
$\mathcal{F}$ if and only if the nonempty proper flats in its support are
precisely $\\{F_{1},\ldots,F_{k-1}\\}$. Also, this convention leads to a
tidier formula for $c(\mathcal{F},\textbf{m})$.

Given $\mathcal{F}\in\operatorname{Flag}_{k}$ and $i\in\\{1,\ldots,k\\}$,
consider the matroid $M^{F_{i}}_{F_{i-1}}$ obtained by contracting $F_{i-1}$
and deleting the complement of $F_{i}$, whose poset of flats is canonically
isomorphic to the interval $[F_{i-1},F_{i}]$ in the poset of flats of $M$. Let
$d_{i}=\operatorname{rk}M^{F_{i}}_{F_{i-1}}-1$. Let $\mu^{j}(M)$ denote the
absolute value of the coefficient of $t^{\operatorname{rk}M-1-j}$ in the
reduced characteristic polynomial of $M$.

######  Theorem 8.2.

For any $k\geq 1$, $\mathcal{F}\in\operatorname{Flag}_{k}$, and
$\textbf{m}=(m_{0},\ldots,m_{k})$, we have

$c(\mathcal{F},\textbf{m})=\sum_{\begin{subarray}{c}\textbf{e},\textbf{f}\in\mathbb{N}^{k}\\\
e_{1}=m_{0}\end{subarray}}(-1)^{|\textbf{e}|+|\textbf{f}|}\prod_{i=1}^{k}\binom{d_{i}-e_{i}}{f_{i}}\mu^{e_{i}}(M^{F_{i}}_{F_{i-1}})\binom{m_{i}}{e_{i+1}+f_{i}-m_{i},m_{i}-f_{i},m_{i}-e_{i+1}},$

where we adopt the convention that $e_{k+1}=0$.

######  Corollary 8.3.

Fix an element $\mathcal{F}\in\operatorname{Flag}_{k}$ and a tuple of natural
numbers $\textbf{m}=(m_{0},\ldots,m_{k})$ with $\sum
m_{i}=\operatorname{rk}M-k$. Then the degree

$\deg_{M}\left(t_{F_{0}}^{m_{0}}t_{F_{1}}^{m_{1}+1}\cdots
t_{F_{k-1}}^{m_{k-1}+1}t_{F_{k}}^{m_{k}}\right)$

is equal to

$(-1)^{\operatorname{rk}M-k}\sum_{\begin{subarray}{c}\textbf{e}\in\mathbb{N}^{k}\\\
e_{1}=m_{0}\end{subarray}}\prod_{i=1}^{k}\mu^{e_{i}}(M^{F_{i}}_{F_{i-1}})\binom{m_{i}}{e_{i+1}-e_{i}+d_{i}-m_{i},m_{i}-d_{i}+e_{i},m_{i}-e_{i+1}},$

again with the convention that $e_{k+1}=0$.

###### Proof.

By the definition of $\chi$ and the third item of Lemma 5.5, we have

$\deg_{M}\left(t_{F_{0}}^{m_{0}}t_{F_{1}}^{m_{1}+1}\cdots
t_{F_{k-1}}^{m_{k-1}+1}t_{E}^{m_{k}}\right)=\chi\left(M,\tau_{\varnothing}^{m_{0}}\tau_{F_{1}}^{m_{1}+1}\cdots\tau_{F_{k-1}}^{m_{k-1}+1}\tau_{F_{k}}^{m_{k}}\right)=c(\mathcal{F},\textbf{m}).$

In the formula for $c(\mathcal{F},\textbf{m})$ in Theorem 8.2, we see that a
particular term vanishes unless $e_{i}+f_{i}\leq d_{i}$ for all
$i\in\\{1,\ldots,k\\}$. Taking the sum over all $i$, this implies that

$|e|+|f|\leq|d|=\operatorname{rk}M-k=m_{0}+\cdots+m_{k}.$

On the other hand, a term also vanishes unless $m_{i}\leq e_{i+1}+f_{i}$ for
all $i\in\\{1,\ldots,k\\}$. Taking the sum over all $i$ and including the
equality $e_{0}=m_{0}$, this implies that

$m_{0}+\cdots+m_{k}\leq|e|+|f|.$

Thus the aforementioned inequalities are all equalities, and we can simplify
our formula for $c(\mathcal{F},\textbf{m})$ by setting $f_{i}=d_{i}-e_{i}$ for
all $i$. The result follows. ∎

If we take $m_{0}=0=m_{k}$ in Corollary 8.3, we recover a result of Eur
[Eur20, Theorem 3.2].

######  Corollary 8.4.

Fix an element $\mathcal{F}\in\operatorname{Flag}_{k}$ and a tuple of natural
numbers $(m_{1},\ldots,m_{k-1})$ with
$(m_{1}+1)+\cdots+(m_{k-1}+1)=\operatorname{rk}M-1$. For all
$i\in\\{1,\ldots,k\\}$, let $e_{i}=(m_{1}-d_{1})+\cdots+(m_{i-1}-d_{i-1})$,
and let $e_{k+1}=0$. Then we have

$\deg_{M}\left(t_{F_{1}}^{m_{1}+1}\cdots
t_{F_{k-1}}^{m_{k-1}+1}\right)=(-1)^{\operatorname{rk}M-k}\prod_{i=1}^{k}\mu^{e_{i}}(M^{F_{i}}_{F_{i-1}})\binom{m_{i}}{e_{i+1}}.$

###### Proof.

The fact that $m_{0}=0=m_{k}$ implies that the only nonzero term in Corollary
8.3 is the one in which $e_{i}=(m_{1}-d_{1})+\cdots+(m_{i-1}-d_{i-1})$ for all
$i$. ∎

The remainder of this section is devoted to the proof of Theorem 8.2. We begin
by analyzing a particular specialization of
$\textup{Snap}_{M}^{\operatorname{FY}}$, where we set all variables except
$a_{\emptyset}$ and $a_{E}$ to $0$.

######  Lemma 8.5.

We have

$\textup{Snap}_{M}^{\operatorname{FY}}(a_{\emptyset},\mathbf{0},a_{E})=\sum_{e,f\in\mathbb{N}}(-1)^{e+f}\binom{\operatorname{rk}M-1-e}{f}\mu^{e}(M)a_{\emptyset}^{(e)}a_{E}^{(f)}.$

###### Proof.

By definition, we have

$\textup{Snap}_{M}^{\operatorname{FY}}(a_{\emptyset},\mathbf{0},a_{E})=\sum_{e,f\in\mathbb{N}}\chi(M,\tau_{\varnothing}^{e}\tau_{E}^{f})a_{\emptyset}^{(e)}a_{E}^{(f)}.$

We also have

$\chi(M,\tau_{\varnothing}^{e}\tau_{E}^{f})=\deg_{M}\left(\frac{\zeta_{M}(\tau_{\varnothing}^{e}\tau_{E}^{f})}{1-h_{E}}\right)=\deg_{M}\left(\frac{\zeta_{M}(\tau_{\varnothing}^{e}\tau_{E}^{f})}{1+t_{E}}\right).$

By Lemma 5.5, this is equal to

$\deg_{M}\left(\frac{t_{\varnothing}^{e}t_{E}^{f}}{(1+t_{E})^{f+1}}\right)=\sum_{\ell}(-1)^{\ell-f}\binom{\ell}{f}\deg_{M}(t_{\varnothing}^{e}t_{E}^{\ell}).$

The degree of $t_{\varnothing}^{e}t_{E}^{\ell}$ vanishes unless
$e+\ell=\operatorname{rk}M-1$, in which case it is equal to
$(-1)^{\operatorname{rk}M-1}\mu^{e}(M)$ [AHK18, Proposition 9.5], thus

$\chi(M,\tau_{\varnothing}^{e}\tau_{E}^{f})=(-1)^{e+f}\binom{\operatorname{rk}M-1-e}{f}\mu^{e}(M).$

This completes the proof. ∎

For the statement and proof of the next proposition, it will be convenient to
regard $\textup{Snap}_{M}^{\operatorname{FY}}(\textbf{a})$ as a function that
takes inputs $a_{S}$ for all subsets $S\subset E$, with the property that the
coefficient of any monomial involving $a_{S}$ is zero if $S$ is not a flat of
$M$. It will also be convenient to allow $M$ to have loops, with the
convention that $\textup{Snap}_{M}^{\operatorname{FY}}(\textbf{a})=0$ whenever
$M$ has a loop. This allows us to define the matroid $M_{F}^{G}$ on the ground
set $G\setminus F$ with respect to an arbitrary pair of sets $F\subset G$;
note that this matroid is loopless if and only if $F$ is a flat.

Fix a subset $G\subset E$, and for any $\textbf{a}=(a_{F})$, define
$\textbf{a}^{\prime}$ by putting $a^{\prime}_{G}=a_{G}-1$ and
$a^{\prime}_{F}=a_{F}$ for all $F\neq G$. For any polynomial $P$ in a, define
$\partial_{G}P$ by putting

$\partial_{G}P(\textbf{a}):=P(\textbf{a})-P(\textbf{a}^{\prime}).$

Note that if $P$ does not depend on $a_{G}$, then $\partial_{G}P=0$. For any
pair of subsets $F\subset G$, let $\textbf{a}|_{[F,G]}$ be the restriction of
a to this interval, which we identify with the collection of subsets of
$G\setminus F$.

######  Proposition 8.6.

For any proper nonempty subset $G\subset E$, we have

$\partial_{G}\textup{Snap}_{M}^{\operatorname{FY}}(\textbf{a})=\textup{Snap}^{\operatorname{FY}}_{M_{\varnothing}^{G}}(\textbf{a}|_{[\varnothing,G]})\,\textup{Snap}^{\operatorname{FY}}_{M_{G}^{E}}(\textbf{a}|_{[G,E]}).$

###### Proof.

We will first prove the statement when $M$ is the matroid associated with a
hyperplane arrangement $\mathcal{A}$, and then deduce the general case from
valuativity. Note that if $G$ is not a flat of $M$, then $M_{\emptyset}^{G}$
has a loop and both sides vanish. We therefore assume that $G$ is a flat.

Consider the hyperplane arrangements

$\mathcal{A}_{\varnothing}^{G}=\\{H_{e}/L_{G}\mid e\in
G\\}\mathcal{A}_{G}^{E}:=\\{H_{e}\cap L_{G}\mid e\notin G\\}.$

We observe that $\mathcal{A}_{\varnothing}^{G}$ is an arrangement in the
vector space $L^{G}$ with associated matroid $M_{\varnothing}^{G}$, and
$\mathcal{A}_{G}^{E}$ is an arrangement in the vector space $L_{G}$ with
associated matroid $M_{G}^{E}$.

There is a short exact sequence

$0\to\mathcal{O}\left((t_{G}-1)D_{G}+\sum_{F\neq
G}t_{F}D_{F}\right)\to\mathcal{O}\left(\sum
t_{F}D_{F}\right)\to\mathcal{O}_{D_{G}}\left(\sum t_{F}D_{F}\right)\to 0.$

Taking Euler characteristics, we find that

$\partial_{G}\textup{Snap}_{M}^{\operatorname{FY}}(\textbf{a})=\textup{Snap}_{M}^{\operatorname{FY}}(\textbf{a})-\textup{Snap}_{M}^{\operatorname{FY}}(\textbf{a}^{\prime})=\chi\left(D_{G},\mathcal{O}_{D_{G}}\left(\sum
t_{F}D_{F}\right)\right).$

By [DCP95, Page 482], the divisor $D_{G}$ is isomorphic to
$W_{\\!\mathcal{A}_{\varnothing}^{G}}\times W_{\\!\mathcal{A}_{G}^{E}}$.
Furthermore, the restriction of $\mathcal{O}_{D_{G}}(\sum t_{F}D_{F})$ to
$D_{G}$ is isomorphic to the pullback from
$W_{\\!\mathcal{A}_{\varnothing}^{G}}$ of $\mathcal{O}_{D_{G}}(\sum_{F\leq
G}t_{F}D_{F})$ tensored with the pullback from $W_{\\!\mathcal{A}_{G}^{E}}$ of
$\mathcal{O}_{D_{G}}(\sum_{G\leq F}t_{F}D_{F})$ [BHM+22a, Proposition 2.20].
Then by the Künneth formula, we have

$\begin{split}\chi\left(D_{G},\mathcal{O}_{D_{G}}\left(\sum
t_{F}D_{F}\right)\right)&=\chi\left(W_{\\!\mathcal{A}_{\varnothing}^{G}},\mathcal{O}_{D_{G}}\left(\sum_{F\leq
G}t_{F}D_{F}\right)\right)\cdot\chi\left(W_{\\!\mathcal{A}_{G}^{E}},\mathcal{O}_{D_{G}}\left(\sum_{G\leq
F}t_{F}D_{F}\right)\right)\\\
&=\textup{Snap}^{\operatorname{FY}}_{M_{\varnothing}^{G}}(\textbf{a}|_{[\varnothing,G]})\,\textup{Snap}^{\operatorname{FY}}_{M_{G}^{E}}(\textbf{a}|_{[G,E]}).\end{split}$

This completes the proof in the realizable case. For the general case, it will
suffice to show that both sides of the equation in the statement of the
proposition are valuative invariants of $M$. It follows from Lemma 6.4 that
$\textup{Snap}_{M}^{\operatorname{FY}}(\textbf{a})$ is valutive, and therefore
that $\partial_{G}\textup{Snap}_{M}^{\operatorname{FY}}(\textbf{a})$ is
valuative. The valuativity of
$\textup{Snap}^{\operatorname{FY}}_{M_{\varnothing}^{G}}(\textbf{a}|_{[\varnothing,G]})\,\textup{Snap}^{\operatorname{FY}}_{M_{G}^{E}}(\textbf{a}|_{[G,E]})$
follows from the valuativity of
$\textup{Snap}_{M}^{\operatorname{FY}}(\textbf{a})$ and general properties of
valuativity [McM09, Theorem 4.6] (see also [AS21, Theorem A]). ∎

######  Remark 8.7.

Our reason for regarding $\textup{Snap}_{M}^{\operatorname{FY}}(\textbf{a})$
as a polynomial with variables indexed by arbitrary subsets rather than flats,
and for allowing matroids with loops, is that it would not otherwise make
sense to assert that the two sides of the equation in the statement of
Proposition 8.6 are valuative invariants of the matroid $M$. When considering
all matroids at once, we cannot know in advance whether or not $G$ is a flat.

Iterating Proposition 8.6, we obtain the following corollary.

######  Corollary 8.8.

For any $\mathcal{F}\in\operatorname{Flag}_{k}$, we have

$\partial_{F_{1}}\cdots\partial_{F_{k-1}}\textup{Snap}_{M}^{\operatorname{FY}}(\textbf{a})=\prod_{i=1}^{k}\textup{Snap}_{M^{F_{i}}_{F_{i-1}}}^{\operatorname{FY}}(\textbf{a}|_{[F_{i-1},F_{i}]}).$

###### Proof of Theorem 8.2..

We use the fact that

$\partial_{F_{1}}\cdots\partial_{F_{k-1}}\left(a_{F_{0}}^{(m_{0})}a_{F_{1}}^{(m_{1}+1)}a_{F_{2}}^{(m_{2}+1)}\cdots
a_{F_{k-1}}^{(m_{k-1}+1)}a_{F_{k}}^{(m_{k})}\right)=a_{F_{0}}^{(m_{0})}\cdots
a_{F_{k}}^{(m_{k})}$

to interpret $c(\mathcal{F},\textbf{m})$ as the coefficient of
$a_{F_{0}}^{(m_{0})}\cdots a_{F_{k}}^{(m_{k})}$ in the polynomial
$\partial_{F_{1}}\cdots\partial_{F_{k-1}}\textup{Snap}_{M}^{\operatorname{FY}}(\textbf{a})$.
Corollary 8.8 provides us with a formula for this polynomial. Setting
$a_{F}=0$ for all $F$ not appearing in $\mathcal{F}$, we obtain the polynomial

$\prod_{i=1}^{k}\textup{Snap}_{M^{F_{i}}_{F_{i-1}}}^{\operatorname{FY}}(a_{F_{i-1}},\mathbf{0},a_{F_{i}}).$

By Lemma 8.5, we may rewrite this as

(12)
$\sum_{\textbf{e},\textbf{f}\in\mathbb{N}^{k}}(-1)^{|\textbf{e}|+|\textbf{f}|}\prod_{i=1}^{k}\binom{d_{i}-e_{i}}{f_{i}}\mu^{e_{i}}(M^{F_{i}}_{F_{i-1}})a_{F_{i-1}}^{(e_{i})}a_{F_{i}}^{(f_{i})}.$

To compute the coefficient of $a_{F_{0}}^{(m_{0})}\cdots a_{F_{k}}^{(m_{k})}$,
we make use of the identity

$x^{({m})}x^{(n)}=\sum_{\ell=0}^{m}\binom{m+n-\ell}{\ell,m-\ell,n-\ell}x^{(m+n-\ell)}$

to rewrite (12) as

$\sum_{\textbf{e},\textbf{f}\in\mathbb{N}^{k}}(-1)^{|\textbf{e}|+|\textbf{f}|}a_{\varnothing}^{(e_{1})}\prod_{i=1}^{k}\binom{d_{i}-e_{i}}{f_{i}}\mu^{e_{i}}(M^{F_{i}}_{F_{i-1}})\sum_{\ell_{i}=0}^{e_{i+1}}\binom{e_{i+1}+f_{i}-\ell_{i}}{\ell_{i},e_{i+1}-\ell_{i},f_{i}-\ell_{i}}a_{F_{i}}^{(e_{i+1}+f_{i}-\ell_{i})},$

with the convention that $e_{k+1}=0$. The coefficient of
$a_{F_{0}}^{(m_{0})}\cdots a_{F_{k}}^{(m_{k})}$ consists of those terms for
which $e_{1}=m_{0}$ and $e_{i+1}+f_{i}-\ell_{i}=m_{i}$ for all $0\leq i\leq
k$. Thus

$c(\mathcal{F},\textbf{m})=\sum_{\begin{subarray}{c}\textbf{e},\textbf{f}\in\mathbb{N}^{k}\\\
e_{1}=m_{0}\end{subarray}}(-1)^{|\textbf{e}|+|\textbf{f}|}\prod_{i=1}^{k}\binom{d_{i}-e_{i}}{f_{i}}\mu^{e_{i}}(M^{F_{i}}_{F_{i-1}})\binom{m_{i}}{e_{i+1}+f_{i}-m_{i},m_{i}-f_{i},m_{i}-e_{i+1}}.\qed$

##  9\. Euler characteristics and Snapper polynomials for
$\overline{\mathcal{M}}_{0,n}$

Last, we turn our attention to the moduli space
$\overline{\mathcal{M}}_{0,n}$. Recall that we have defined a tuple of line
bundles

$\mathcal{L}:=(\mathcal{L}_{S}\mid S\subset[n-1],|S|\geq 3).$

We begin this section by computing the Snapper polynomial
$\textup{Snap}_{\mathcal{L}}(\textbf{a})$ with respect to these line bundles.
Given a tuple of natural numbers $\textbf{m}=(m_{S}\mid
S\subset[n-1]\mid|S|\geq 3)$, we say that m satisfies the Cerberus condition
if, for every $\textbf{m}^{\prime}\leq\textbf{m}$, we have

$\Big{|}\bigcup_{m^{\prime}_{S}>0}S\cup\\{n\\}\,\,\Big{|}-3\;\geq\;\sum_{S}m^{\prime}_{S}.$

######  Theorem 9.1.

We have

$\textup{Snap}_{\mathcal{L}}(\textbf{a})\;\;=\sum_{\begin{subarray}{c}\text{$\textbf{m}$
satisfies}\\\ \text{Cerberus}\end{subarray}}\textbf{a}^{(\textbf{m})}.$

###### Proof.

Let m be given, and define

$\tilde{\textbf{m}}=(\tilde{m}_{F}\mid\text{$F$ a nonempty flat of
$\mathcal{B}_{n-1}$})$

by putting $\tilde{m}_{F_{S}}=m_{S}$ and $\tilde{m}_{F}=0$ for all flats $F$
not of the form $F_{S}$. Because the map
$p:W_{\mathcal{B}_{n-1}}\to\overline{\mathcal{M}}_{0,n}$ is a composition of
blowups at smooth centers, the Euler characteristic of a line bundle on
$\overline{\mathcal{M}}_{0,n}$ is equal to the Euler characteristic of its
pullback to $W_{\mathcal{B}_{n-1}}$. Thus, if we can show that the Cerberus
condition for m is equivalent to the dragon Hall–Rado condition for
$\tilde{\textbf{m}}$, our result will follow from Theorem 7.2. Note that
$\tilde{\textbf{m}}$ fails the dragon Hall–Rado condition if and only if there
is $\tilde{\textbf{m}}^{\prime}\leq\tilde{\textbf{m}}$ with

$\operatorname{rank}\bigcup_{\tilde{m}^{\prime}(F)>0}F\;\geq\;\sum_{F}\tilde{m}^{\prime}(F)\text{
and }\bigcup_{\tilde{m}^{\prime}(F)>0}F\text{ connected}.$

The remainder of the proof follows by direct calculation, using the facts that
$\operatorname{rk}F_{S}=|S|-1$ and $\operatorname{rk}F_{S}\cup
F_{T}=\operatorname{rk}F_{S\cup T}$ if $S$ and $T$ are not disjoint. ∎

We conclude by computing the Snapper polynomial with respect to a different
tuple of line bundles on $\overline{\mathcal{M}}_{0,n}$. Recall that
$\mathbb{L}_{i}$ is the $i^{\text{th}}$ cotangent line bundle, whose first
Chern class is equal to $\psi_{i}$, and let
$\mathbb{L}:=(\mathbb{L}_{1},\ldots,\mathbb{L}_{n}).$

######  Theorem 9.2.

We have

$\textup{Snap}_{\mathbb{L}}(\textbf{a})=\sum_{|\textbf{d}|\leq
n-3}\binom{a_{1}}{d_{1}}\cdots\binom{a_{n}}{d_{n}}\binom{n-3}{n-3-\left\lvert\textbf{d}\right\rvert,d_{1},\ldots,d_{n}}.$

In order to prove Theorem 9.2, we begin with the following lemma. Recall the
exceptional isomorphism $\zeta_{n}:K(\overline{\mathcal{M}}_{0,n})\to
A(\overline{\mathcal{M}}_{0,n})$ of Theorem 1.10. Let
$z_{i}:=1-[\mathbb{L}_{i}]\in K(\overline{\mathcal{M}}_{0,n})$.

######  Lemma 9.3.

We have $\zeta_{n}(z_{i})=-\psi_{i}$ for all $i<n$, and
$\zeta_{n}(z_{n})=-\psi_{n}/(1-\psi_{n})$.

###### Proof.

We will consider a family of closely related moduli spaces of curves, called
the Losev–Manin spaces [LM00]. The Losev–Manin space
$\overline{\mathcal{M}}_{0,w_{k,\ell}}$ is a moduli space of stable rational
curves with $n$ weighted marked points [Has03], with weights 1 for the the
$k^{\text{th}}$ and $\ell^{\text{th}}$ points and
$\varepsilon\in\mathbb{Q}\cap(0,\frac{1}{n-2})$ for the remaining $n-2$
points. Reduction of weights from $(1,\ldots,1)$ to $w_{k,\ell}$ induces a
morphism [Has03, Theorem 4.1]
$\rho_{k,\ell}:\overline{\mathcal{M}}_{0,n}\to\overline{\mathcal{M}}_{0,w_{k,\ell}}.$
Furthermore, $\overline{\mathcal{M}}_{0,w_{k,\ell}}$ is isomorphic to the
permutohedral variety $X_{A_{[n]\smallsetminus\\{k,\ell\\}}}$ [LM00], so we
have an exceptional isomorphism

$\zeta_{[n]\smallsetminus\\{k,\ell\\}}\colon
K(\overline{\mathcal{M}}_{0,w_{k,\ell}})\to
A(\overline{\mathcal{M}}_{0,w_{k,\ell}}).$

For $i\not=n$, consider the classes $t_{E},t_{\emptyset}\in
A(\overline{\mathcal{M}}_{0,w_{i,n}})\cong
A(X_{A_{[n]\smallsetminus\\{i,n\\}}})$ and $\tau_{E},\tau_{\emptyset}\in
K(\overline{\mathcal{M}}_{0,w_{i,n}})\cong
K(X_{A_{[n]\smallsetminus\\{i,n\\}}})$. Then we have the following identities:

$\displaystyle\rho_{n,i}^{\ast}t_{E}$ $\displaystyle=-\psi_{n}$
$\displaystyle\quad\quad\rho_{n,i}^{\ast}\tau_{E}$
$\displaystyle=1-[\mathbb{L}_{n}]$
$\displaystyle\rho_{n,i}^{\ast}t_{\emptyset}$ $\displaystyle=-\psi_{i}$
$\displaystyle\quad\quad\rho_{n,i}^{\ast}\tau_{\emptyset}$
$\displaystyle=1-[\mathbb{L}_{i}],$

see, for example, [DR22, Section 2]. For each $i\neq n$, we have the following
diagram

${K(\overline{\mathcal{M}}_{0,w_{n,i}})}$${A(\overline{\mathcal{M}}_{0,w_{n,i}})}$${K(\overline{\mathcal{M}}_{0,n})}$${A(\overline{\mathcal{M}}_{0,n}),}$$\scriptstyle{\zeta_{[n]\smallsetminus\\{n,i\\}}}$$\scriptstyle{\rho_{n,i}^{\ast}}$$\scriptstyle{\rho_{n,i}^{\ast}}$$\scriptstyle{\zeta_{n}}$

whose commutativity can be checked by using that
$\rho_{n,i}^{*}\tau_{E}=1-[\mathbb{L}_{n}^{-1}]$, and that similar formulas
hold for pullbacks of the $n$th cotangent line bundle under forgetful maps.

The Lemma 5.5 gives that

$\zeta_{n}(z_{i})=\zeta_{n}(1-[\mathbb{L}_{i}])=\zeta_{n}\circ\rho_{n,i}^{\ast}(\tau_{\emptyset})=\rho_{n,i}^{\ast}\circ\zeta_{[n]\smallsetminus\\{n,i\\}}(\tau_{\emptyset})=\rho_{n,i}^{\ast}(t_{\emptyset})=-\psi_{i},$

and similarly
$\zeta_{n}(z_{n})=\rho_{n,i}^{\ast}(\frac{t_{E}}{1+t_{E}})=\frac{-\psi_{n}}{1-\psi_{n}}.$
∎

###### Proof of Theorem 9.2..

By Lemma 7.3, we have

$\textup{Snap}_{\mathbb{L}}(-\textbf{a})=\sum_{\textbf{d}\in\mathbb{N}^{n}}\chi\left(\overline{\mathcal{M}}_{0,n},\,z_{1}^{d_{1}}\cdots
z_{n}^{d_{n}}\right)\,\textbf{a}^{(\textbf{d})}.$

(Note that the minus sign comes from the fact that $z_{i}=1-[\mathbb{L}_{i}]$
rather than $1-[\mathbb{L}_{i}^{-1}]$.) By Theorem 1.10 and Lemma 9.3, we have

$\chi\left(\overline{\mathcal{M}}_{0,n},\,z_{1}^{d_{1}}\cdots
z_{n}^{d_{n}}\right)=\deg\left(\frac{\zeta_{n}(z_{1}^{d_{1}}\cdots
z_{n}^{d_{n}})}{1-c_{[n-1]}}\right)=(-1)^{|\textbf{d}|}\deg\left(\frac{\psi_{1}^{d_{1}}\cdots\psi_{n}^{d_{n}}}{(1-\psi_{n})^{d_{n}+1}}\right).$

Next, we observe that

$\frac{\psi_{1}^{d_{1}}\cdots\psi_{n}^{d_{n}}}{(1-\psi_{n})^{d_{n}+1}}=\psi_{1}^{d_{1}}\cdots\psi_{n}^{d_{n}}\sum_{k=0}^{\infty}(d_{n}+1)^{(k)}\psi_{n}^{k},$

and therefore the part of this sum in degree $n-3$ is equal to

$\psi_{1}^{d_{1}}\cdots\psi_{n-1}^{d_{n-1}}\psi_{n}^{n-3-d_{1}-\cdots-
d_{n-1}}(d_{n}+1)^{(n-3-|\textbf{d}|)}$

if $|\textbf{d}|\leq n-3$, and zero otherwise. By work of Witten [Wit91], the
degree of this term is equal to

$\binom{n-3}{d_{1},\ldots,d_{n-1},n-3-d_{1}-\cdots-
d_{n-1}}(d_{n}+1)^{(n-3-|\textbf{d}|)}.$

Putting it all together, we have

$\displaystyle\textup{Snap}_{\mathbb{L}}(-\textbf{a})$ $\displaystyle=$
$\displaystyle\sum_{d\in\mathbb{N}^{n}}\chi\left(\overline{\mathcal{M}}_{0,n},\,z_{1}^{d_{1}}\cdots
z_{n}^{d_{n}}\right)\,\textbf{a}^{(\textbf{d})}$ $\displaystyle=$
$\displaystyle\sum_{|\textbf{d}|\leq
n-3}(-1)^{|\textbf{d}|}\textbf{a}^{(\textbf{d})}\deg\left(\frac{\psi_{1}^{d_{1}}\cdots\psi_{n}^{d_{n}}}{(1-\psi_{n})^{d_{n}+1}}\right)$
$\displaystyle=$ $\displaystyle\sum_{|\textbf{d}|\leq
n-3}(-1)^{|\textbf{d}|}\textbf{a}^{(\textbf{d})}(d_{n}+1)^{(n-3-|\textbf{d}|)}\binom{n-3}{d_{1},\ldots,d_{n-1},n-3-d_{1}-\cdots-
d_{n-1}}$ $\displaystyle=$ $\displaystyle\sum_{|\textbf{d}|\leq
n-3}(-1)^{|\textbf{d}|}\textbf{a}^{(\textbf{d})}\binom{d_{n}+n-3-|\textbf{d}|}{n-3-|\textbf{d}|}\binom{n-3}{d_{1},\ldots,d_{n-1},n-3-d_{1}-\cdots-
d_{n-1}}$ $\displaystyle=$ $\displaystyle\sum_{|\textbf{d}|\leq
n-3}(-1)^{|\textbf{d}|}\textbf{a}^{(\textbf{d})}\binom{n-3}{d_{1},\ldots,d_{n},n-3-|\textbf{d}|}.$

Finally, this implies that

$\textup{Snap}_{\mathbb{L}}(\textbf{a})=\sum_{|\textbf{d}|\leq
n-3}\binom{a_{1}}{d_{1}}\cdots\binom{a_{n}}{d_{n}}\binom{n-3}{d_{1},\ldots,d_{n},n-3-|\textbf{d}|},$

which completes the proof. ∎

##  Appendix A The simplicial presentation of the augmented Chow ring

This appendix is devoted to proving the simplicial presentation for the
augmented Chow ring $A^{\operatorname{aug}}(M)$. Theorem A.2 is closely
analogous to its non-augmented counterparts in [BES21], and it generalizes a
result proved in the special case of a Boolean matroid [EHL22, Section 7.2].

For any pair of flats $F$ and $G$, let

$z_{F,G}:=\sum_{\begin{subarray}{c}F\subset F^{\prime}\subsetneq F\vee G\\\
G\subset G^{\prime}\subsetneq F\vee
G\end{subarray}}x_{F^{\prime}}x_{G^{\prime}}\in T_{M}.$

For any element $e\in E$ and flat $F\neq\emptyset$, let

$w_{e,F}:=\sum_{F\subset G\not\ni e}y_{e}x_{G}\in T_{M}.$

Recall the definitions of the ideals
$\mathcal{I}_{1},\mathcal{I}_{2},\mathcal{I}_{3},\mathcal{I}_{4}^{\operatorname{aug}}\subset
T_{M}$ in Equation (1).

######  Lemma A.1.

We have the following equalities of ideals in $T_{M}$:

$\displaystyle\mathcal{I}_{3}$ $\displaystyle=$ $\displaystyle\left\langle
z_{F,G}\mid\text{$F,G$ {\em arbitrary}}\right\rangle$
$\displaystyle\mathcal{I}_{2}+\mathcal{I}_{4}^{\operatorname{aug}}$
$\displaystyle=$ $\displaystyle\mathcal{I}_{2}+\langle y_{e}^{2}\mid e\in
E\rangle+\left\langle w_{e,F}\mid e\in E,F\neq\varnothing\right\rangle.$

###### Proof.

We begin with the first statement. Every term in $z_{F,G}$ is a product of
incomparable flats, thus $z_{F,G}$ is contained in $\mathcal{I}_{3}$. The
opposite inclusion follows by induction on sum of the coranks of $F$ and $G$,
using the observation that, whenever $F$ and $G$ are incomparable,

$z_{F,G}=x_{F}x_{G}+\text{terms with lower sum of coranks.}$

For the second statement, we write

$\mathcal{I}_{4}^{\operatorname{aug}}=\langle y_{e}x_{\varnothing}\mid e\in
E\rangle+\langle y_{e}x_{F}\mid e\notin F\neq\varnothing\rangle.$

A similar argument to the one above shows that

$\langle y_{e}x_{F}\mid e\notin F\neq\varnothing\rangle=\left\langle
w_{e,F}\mid e\in E,F\neq\varnothing\right\rangle,$

so it remains only to show that $y_{e}x_{\varnothing}$ is congruent to
$y_{e}^{2}$ modulo the ideal $\mathcal{I}_{2}+\left\langle w_{e,F}\mid e\in
E,F\neq\varnothing\right\rangle$. Indeed, we have

$y_{e}^{2}-x_{\varnothing}y_{e}=y_{e}\left(y_{e}-\sum_{e\notin
F}x_{F}\right)+\sum_{e\notin F\neq\varnothing}y_{e}x_{F},$

which completes the proof. ∎

Recall that we have defined the homomorphism
$\Phi_{\nabla}^{\operatorname{aug}}:S_{M}\to A^{\operatorname{aug}}(M)$ by the
formula

$\Phi_{\nabla}^{\operatorname{aug}}(u_{F}):=-\sum_{F\subset
G}\Phi_{\operatorname{FY}}(x_{G}).$

######  Theorem A.2.

The map $\Phi_{\nabla}^{\operatorname{aug}}$ is surjective with kernel
$\mathcal{J}_{1}+\mathcal{J}_{2}^{{\operatorname{aug}}}$.

###### Proof.

Consider the homomorphism $\Theta:S_{M}\to T_{M}$ defined by the formula

$\Theta(u_{F}):=-\sum_{F\subset G}x_{G}$

for all nonempty flats $F$, so that
$\Phi_{\nabla}^{\operatorname{aug}}=\Phi_{\operatorname{FY}}^{\operatorname{aug}}\circ\Theta$.
The image of $\Theta$ is equal to the subring $\mathbb{Z}[x_{F}\mid\text{$F$ a
nonempty flat}]\subset T_{M}$. In particular, the composition

$\bar{\Theta}:S_{M}\to T_{M}\to T_{M}/(\mathcal{I}_{1}+\mathcal{I}_{2})$

is an isomorphism. This implies that $\Phi_{\nabla}^{\operatorname{aug}}$ is
surjective, and its kernel is equal to

$\Theta^{-1}(\mathcal{I}_{1}+\mathcal{I}_{2}+\mathcal{I}_{3}+\mathcal{I}_{4}^{\operatorname{aug}}).$

For the remainder of the proof, we put a bar over an element of $T_{M}$ or an
ideal in $T_{M}$ to denote its image in
$T_{M}/(\mathcal{I}_{1}+\mathcal{I}_{2})$. Then we need to compute
$\bar{\Theta}^{-1}(\bar{\mathcal{I}}_{3}+\bar{\mathcal{I}}_{4}^{\operatorname{aug}})$.
By Lemma A.1, we may rewrite this as

$\bar{\Theta}^{-1}\Big{(}\left\langle\bar{z}_{F,G}\mid\text{$F,G$
arbitrary}\right\rangle+\langle\bar{y}_{e}^{2}\mid e\in E\rangle+\left\langle
w_{e,F}\mid e\in E,F\neq\varnothing\right\rangle\Big{)}.$

We have the following equalities:

$\displaystyle\bar{z}_{F,G}$ $\displaystyle=$
$\displaystyle\bar{\Theta}\Big{(}(u_{F}-u_{F\vee G})(u_{G}-u_{F\vee
G})\Big{)},$ $\displaystyle\bar{y}_{e}^{2}$ $\displaystyle=$
$\displaystyle\bar{\Theta}(u_{\bar{e}}^{2}),$ $\displaystyle\bar{w}_{e,F}$
$\displaystyle=$ $\displaystyle\bar{\Theta}\Big{(}u_{\bar{e}}(u_{\bar{e}\vee
F}-u_{F})\Big{)},$

which together imply that the kernel of $\Phi_{\nabla}^{\operatorname{aug}}$
is equal to

$\mathcal{J}_{1}+\langle u_{\bar{e}}^{2}\mid e\in E\rangle+\langle
u_{\bar{e}}(u_{\bar{e}\vee F}-u_{F})\mid e\in E,F\neq\varnothing\rangle.$

The theorem now follows from the fact that the second and third summand above
add to $\mathcal{J}_{2}^{\operatorname{aug}}$. ∎

## References

  * [AHK18] Karim Adiprasito, June Huh, and Eric Katz.  Hodge theory for combinatorial geometries.  Ann. Math., 188(2):381–452, 2018.
  * [AP15] Dave Anderson and Sam Payne.  Operational $K$-theory.  Doc. Math., 20:357–399, 2015.
  * [AS21] Federico Ardila and Mario Sanchez.  Valuations and the Hopf monoid of generalized permutahedra.  Preprint, arXiv:2010.11178v3, to appear in Int. Math. Res. Not., 2021\.
  * [BES21] Spencer Backman, Christopher Eur, and Connor Simpson.  Simplicial generation of Chow rings of matroids.  Preprint, arXiv:1905.07114v4, to appear in J. Eur. Math. Soc., 2021.
  * [BEST22] Andrew Berget, Christopher Eur, Hunter Spink, and Dennis Tseng.  Tautological classes of matroids.  Preprint, arXiv:2103.08021v3, 2022.
  * [BGI71] Pierre Berthelot, Alexandre Grothendieck, and Luc Illusie.  Théorie des intersections et théorème de Riemann-Roch.  Lecture Notes in Mathematics, Vol. 225. Springer-Verlag, Berlin, 1971\.  Séminaire de Géométrie Algébrique du Bois-Marie 1966–1967 (SGA 6).
  * [BHM+22a] Tom Braden, June Huh, Jacob Matherne, Nicholas Proudfoot, and Botong Wang.  A semi-small decomposition of the Chow ring of a matroid.  Adv. Math., 409:Paper No. 108646, 2022.
  * [BHM+22b] Tom Braden, June Huh, Jacob Matherne, Nicholas Proudfoot, and Botong Wang.  Singular Hodge theory for combinatorial geometries.  Preprint, arXiv:2010.06088v3, 2022.
  * [CHMR16] Renzo Cavalieri, Simon Hampe, Hannah Markwig, and Dhruv Ranganathan.  Moduli spaces of rational weighted stable curves and tropical geometry.  Forum Math. Sigma, 4:Paper No. e9, 35, 2016.
  * [CT20] Ana-Maria Castravet and Jenia Tevelev.  Derived category of moduli of pointed curves. I.  Algebraic Geometry, 7(6):722–757, 2020.
  * [CT21] Ana-Maria Castravet and Jenia Tevelev.  Derived category of moduli of pointed curves. II.  Preprint, arXiv:2002.02889v3, 2021.
  * [DCP95] Corrado De Concini and Claudio Procesi.  Wonderful models of subspace arrangements.  Selecta Math. (N.S.), 1(3):459–494, 1995.
  * [DF10] Harm Derksen and Alex Fink.  Valuative invariants for polymatroids.  Adv. Math., 225(4):1840–1892, 2010.
  * [DR22] Jeshu Dastidar and Dustin Ross.  Matroid psi classes.  Selecta Math. (N.S.), 28(3):Paper No. 55, 38, 2022.
  * [EHKR10] Pavel Etingof, André Henriques, Joel Kamnitzer, and Eric Rains.  The cohomology ring of the real locus of the moduli space of stable curves of genus 0 with marked points.  Ann. Math., 171(2):731–777, 2010.
  * [EHL22] Christopher Eur, June Huh, and Matt Larson.  Stellahedral geometry of matroids.  Preprint, arXiv:2207.10605v1, 2022.
  * [EL] Christopher Eur and Matt Larson.  $K$-rings and polymatroids.  In preparation.
  * [Eur20] Christopher Eur.  Divisors on matroids and their volumes.  J. Combin. Theory Ser. A, 169:105135, 31, 2020.
  * [FM05] Eva Maria Feichtner and Irene Müller.  On the topology of nested set complexes.  Proc. Amer. Math. Soc., 133(4):999–1006, 2005.
  * [Ful93] William Fulton.  Introduction to toric varieties, volume 131 of Annals of Mathematics Studies.  Princeton University Press, Princeton, NJ, 1993.
  * [Ful98] William Fulton.  Intersection theory, volume 2 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics.  Springer-Verlag, Berlin, second edition, 1998.
  * [FY04] Eva Maria Feichtner and Sergey Yuzvinsky.  Chow rings of toric varieties defined by atomic lattices.  Invent. Math., 155(3):515–536, 2004.
  * [Get95] Ezra Getzler.  Operads and moduli spaces of genus $0$ Riemann surfaces.  In The moduli space of curves (Texel Island, 1994), volume 129 of Progr. Math., pages 199–230. Birkhäuser, Boston, MA, 1995.
  * [Has03] Brendan Hassett.  Moduli spaces of weighted pointed stable curves.  Adv. Math., 173(2):316–352, 2003.
  * [Kap93] Mikhail Kapranov.  Chow quotients of Grassmannians. I.  In I. M. Gelfand Seminar, volume 16 of Adv. Soviet Math., pages 29–110. Amer. Math. Soc., Providence, RI, 1993.
  * [KKL21] Siddarth Kannan, Dagan Karp, and Shiyue Li.  Chow rings of heavy/light Hassett spaces via tropical geometry.  J. Combin. Theory Ser. A, 178(1):Paper No. 105348, 2021.
  * [LM00] Andrey Losev and Yuri Manin.  New moduli spaces of pointed curves and pencils of flat connections.  Michigan Math. J., 48:443–472, 2000.
  * [McM09] Peter McMullen.  Valuations on lattice polytopes.  Adv. Math., 220(1):303–323, 2009.
  * [Pos09] Alexander Postnikov.  Permutohedra, associahedra, and beyond.  Int. Math. Res. Not. IMRN, 2009(6):1026–1106, 2009.
  * [Sin04] Daniel Singh.  The moduli space of stable $N$-pointed curves of genus zero.  PhD thesis, University of Sheffield, 2004.
  * [Sna59] Ernst Snapper.  Multiples of divisors.  J. Math. Mech., 8:967–992, 1959.
  * [Wei94] Charles Weibel.  An introduction to homological algebra, volume 38 of Cambridge Studies in Advanced Mathematics.  Cambridge University Press, Cambridge, 1994.
  * [Wit91] Edward Witten.  Two-dimensional gravity and intersection theory on moduli space.  In Surveys in differential geometry (Cambridge, MA, 1990), pages 243–310. Lehigh Univ., Bethlehem, PA, 1991.