# Stable rationality of hypersurfaces of mock toric variety II

Taro Yoshino  Graduate School of Mathematical Sciences, The University of
Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo, 153-8914, Japan
<EMAIL_ADDRESS>
###### Abstract.

In recent years, there has been a development in approaching rationality
problems through motivic methods (cf. [Kontsevich–Tschinkel’19],
[Nicaise–Shinder’19], [Nicaise–Ottem’21]).

This method requires the explicit construction of degeneration families of
curves with favorable properties. While the specific construction is generally
difficult, [Nicaise-–Ottem’22] combines combinatorial methods to construct
degeneration families of hypersurfaces in toric varieties and mentions the
stable rationality of a very general hypersurface in projective spaces.

In this paper, we substitute mock toric varieties for toric varieties and we
prove the following theorem from the motivic method: If a very general
hypersurface of degree $d$ in $\mathbb{P}^{2n-5}_{\mathbb{C}}$ is not stably
rational, then a very general hypersurface of degree $d$ in
$\mathrm{Gr}_{\mathbb{C}}(2,n)$ is not stably rational.

###### 2020 Mathematics Subject Classification:

14E08, 14D06, 14M25

###### Contents

  1. 1 Introduction
  2. 2 Stably birational volume of hypersurfaces of mock toric varieties
  3. 3 Mock polytopes of the mock toric variety
  4. 4 Application: Stable rationality of hypersurfaces in Grassmaniann varieties

##  1\. Introduction

###  1.1. Motivic method for the rationality problem

The rationality problem is one of the central problems in algebraic geometry.
Recently, there exists motivic approach for the rationality problem(cf. [5],
[8], [10]).

Let $K$ be a field. Two schemes X and Y of finite type over $K$ are called
stably birational if $X\times\mathbb{P}^{m}_{K}$ is birational to
$Y\times\mathbb{P}^{n}_{K}$ for some non-negative integers $m$ and $n$. If a
$K$-variety $X$ is stably birational to $\operatorname{Spec}(K)$, then we call
$X$ is stably rational. In particular, a rational $K$-variety $X$ is stably
rational. Let $\mathrm{SB}_{K}$ denote the set of stable birational
equivalence classes $\\{X\\}_{\mathrm{sb}}$ of integral $K$-schemes $X$ of
finite type, and $\mathbb{Z}[\mathrm{SB}_{K}]$ be the free abelian group on
$\mathrm{SB}_{K}$. Moreover, for any variety $X$ over $K$, it holds that $X$
is stably rational over $K$ if and only if
$\\{X\\}_{\mathrm{sb}}=\\{\operatorname{Spec}(K)\\}_{\mathrm{sb}}$ by
definition. In [8], they constructed a ring morphism as follows:

######  Proposition 1.1.

[8, Lemma.3.3.5] Let $k$ be an algebraically closed field with
$\operatorname{char}(k)=0$ and let $\mathscr{K}$ denote the field of the
Puiseux series over $k$. Then there exists a unique ring morphism
$\mathrm{Vol}_{\mathrm{sb}}\colon\mathbb{Z}[\mathrm{SB}_{\mathscr{K}}]\rightarrow\mathbb{Z}[\mathrm{SB}_{k}]$
such that for every strictly toroidal proper $\mathscr{R}$-scheme
$\mathscr{X}$ with smooth generic fiber $X=\mathscr{X}_{\mathscr{K}}$, we have

$\mathrm{Vol}_{\mathrm{sb}}(\\{X\\}_{\mathrm{sb}})=\sum_{E\in\mathcal{S}(\mathscr{X})}(-1)^{\mathrm{codim}(E)}\\{E\\}_{\mathrm{sb}}$

, where $\\{E\\}$ is a stratification of $\mathscr{X}_{k}$.

At this point, we omit the detailed definition of a strictly toroidal
model(See Definition 2.1). Still, it is worth mentioning that if $\mathscr{X}$
is flat over $\operatorname{Spec}(\mathscr{R})$ such that the generic fiber is
smooth over $\operatorname{Spec}(\mathscr{K})$ and the closed fiber is a
reduced simple normal crossing divisor of $\mathscr{X}$, then $\mathscr{X}$ is
a strictly toroidal model. We remark that such ring morphisms appeared in
these references(cf. [5], [8], [10]), although the details of the forms
differ. In application, for a $\mathscr{K}$-variety $X$, if there exists a
strictly toroidal proper $\mathscr{R}$-scheme $\mathscr{X}$ with smooth
generic fiber $X=\mathscr{X}_{\mathscr{K}}$ and
$\mathrm{Vol}_{\mathrm{sb}}(\\{X\\}_{\mathrm{sb}})\neq\\{\operatorname{Spec}(k)\\}_{\mathrm{sb}}$,
then $X$ is not stably rational over $\mathscr{K}$.

In this article, we show the following theorem by using the above ring
morphism:

######  Theorem 1.2 (See Theorem 4.4).

If a very general hypersurface of degree $d$ in
$\mathbb{P}_{\mathbb{C}}^{2n-5}$ is not stably rational, then a very general
hypersurface of degree $d$ in $\mathrm{Gr}_{\mathbb{C}}(2,n)$ is not stably
rational.

In particular, the following corollary holds from [11, Corollary 1.2]
immediately:

######  Corollary 1.3 (See Theorem 4.5).

If $n\geq 5$ and $d\geq 3+\log_{2}(n-3)$, then a very general hypersurface of
degree $d$ of $\mathrm{Gr}_{\mathbb{C}}(2,n)$ is not stably rational.

###  1.2. The difficulty of motivic method

While this method of determination is quite simple, in practice, there are
several difficult problems. For example, we have the following three problems
:

  * 1.

To construct a “good” proper model with a smooth generic fiber.

  * 2.

To enumerate strata in the stratification of the model.

  * 3.

To determine the stable birational equivalence classes (or birational
equivalence classes) for each stratum.

Previous work [7] successfully overcomes these three problems as follows(cf.
[7, Theorem. 3.14]):

  * 1.

They focused on a hypersurface $Y$ of an algebraic torus. This hypersurface
has a Tropical compactification $\overline{Y}$ (cf.[12]); in particular, they
were compactified as a hypersurface of a proper toric variety $X$. If $Y$ has
a good condition(”Newton nondegenerate”), then the toric resolution of $X$
induces the resolution of $\overline{Y}$. This property is applied to the
construction of a good model.

  * 2.

The ambient space is a toric variety, and its decomposition of torus orbits
induces the stratification of this hypersurface. Moreover, each stratum is
also a hypersurface of an algebraic torus, and we can calculate the defining
Laurant polynomial of each stratum combinatorially.

  * 3.

They construct the strictly toroidal model of a quartic 5-fold, which has a
stratum $E$ such that $E$ is birational to a very general quartic double
fourfold. In particular, it is not stably rational (cf.[2]). In addition to
this, they showed that a very general quartic 5-fold is not stably
rational([7, Cor. 5.2]).

It is natural to consider applying the method of [7] to mention the
rationality of hypersurfaces in other rational varieties. For example,
Grassmanian varieties are. Grassmanian varieties have a Schubert
decomposition; in particular, they have an affine space as a dense open
subspace. Moreover, an affine space has an algebraic torus as a dense open
subspace. Thus, it seems straightforward to construct a strictly toroidal
model of the variety, which is birational to the hypersurface of the
Grassmanian variety. However, this does not work well because the defining
Laurant polynomial of the hypersurface is not Newton nondegenerate. The
necessity of this condition arises from the fact that in this method,
singularities of this hypersurface can only be resolved through a toric
resolution of the ambient toric variety. For example, suppose this
hypersurface has a singular point in the dense open subset of the ambient
toric variety. In that case, we cannot resolve the singularity of it in this
case.

###  1.3. Mock toric varieties

In the previous subsection, we indicate that there is a limit to constructing
concretely the strict toroidal models of some varieties from toric varieties.
In this article, we substitute mock toric varieties for toric varieties and
prove the main theorem 4.4. Mock toric varieties are introduced by the author
of this paper in [14, Definition 3.1]. From [14, Section 3 & 4], mock toric
varieties inherit some same properties as toric varieties. In particular, if
some conditions hold, we can construct proper strictly toroidal models of
hypersurfaces of mock toric varieties concretely as these of toric
varieties(See [14, Proposition 7.7]). Moreover, we show that we can construct
the birational proper strict toroidal models of general hypersurfaces of
$\mathrm{Gr}_{\mathscr{K}}(2,n)$(See Proposition 4.2) from mock toric
varieties, and we show the main theorem.

Consequently, the proof strategy is the same as in [7], with the difference
that the scope has been extended from hypersurfaces of toric varieties to
those of mock toric varieties. The key point of this proof is the result of
the stable birational equivalence class of a very general hypersurface in a
projective space([7, Corollary. 4.2]). We remark that for any integer $m\geq
3$, we can’t find these models of hypersurfaces of
$\mathrm{Gr}_{\mathscr{K}}(m,n)$ related to mock toric varieties.

###  1.4. The rationality problem for hypersurfaces of Grassmanian varieties

We discuss previous work on the rationality of hypersurfaces in
$\mathrm{Gr}_{\mathbb{C}}(2,n)$. We fix a Plücker embedding
$\mathrm{Gr}_{\mathbb{C}}(2,n)\hookrightarrow\mathbb{P}^{n(n-1)/2-1}_{\mathbb{C}}$,
let $X$ denote $\mathrm{Gr}_{\mathbb{C}}(2,n)$, and $X_{(r)}$ denote the
intersection of $X$ and hypersurfaces of degree = $r$ in
$\mathbb{P}^{n(n-1)/2-1}_{\mathbb{C}}$. Because $K_{X}=\mathscr{O}_{X}(-n)$, a
general $X_{(r)}$ is a Fano variety for $1\leq r\leq n$. We itemize the
rationality of $X_{(r)}$ as follows:

  * •

$X_{(1)}$ is rational([13, Theorem 2.2.1.]).

  * •

$X_{(2)}\subset\mathrm{Gr}_{\mathbb{C}}(2,4)$ is complete intersection of 2
quadrics in $\mathbb{P}^{5}_{\mathbb{C}}$, so it is a rational.

  * •

A very general $X_{(3)}\subset\mathrm{Gr}_{\mathbb{C}}(2,4)$ is not stably
rational([3, Theorem 1.1])

  * •

$X_{(2)}\subset\mathrm{Gr}_{\mathbb{C}}(2,5)$ is a Gushel-Mukai 5-fold.
Moreover, it is rational([1, Proposition 4.2]).

  * •

Ottem showed that a very general $X_{(3)}\subset\mathrm{Gr}_{\mathbb{C}}(2,5)$
is not stably rational in his unpublished paper.

  * •

We can show that a very general $X_{(4)}\subset\mathrm{Gr}_{\mathbb{C}}(2,5)$
is not stably rational from [6, Theorem 7.1.2] and [7, Corollary 4.2.2].

###  1.5. Outline of the paper

This paper is organized as follows. In Section 2, we construct the formulation
of a stable birational volume of hypersurfaces of mock toric varieties from
[14, Proposition 7.7]. In Section 3, we define an analog of Newton polytope
for mock toric varieties and consider Newton nondegeneracy. In general,
confirming Newton nondegeneracy of a specific hypersurface takes time and
effort. Instead, we apply the formula to a general section of linear systems,
following the idea from [7]. In Section 4, we apply the formula to the stable
rationality problem of a very general hypersurface in
$\mathrm{Gr}_{\mathbb{C}}(2,n)$ of some degrees.

###  1.6. Acknowledgment

The author is grateful to his supervisor, Yoshinori Gongyo, for his
encouragement.

##  2\. Stably birational volume of hypersurfaces of mock toric varieties

In this section, we compute the stably birational volume of hypersurfaces of
mock toric varieties from the proper strictly toroidal model constructed in
[14, Proposition 7.7].

We recall some notations from [8] to compute the stable birational volume
using the results obtained from [8].

######  Definition 2.1 ([8]).

Let $\mathscr{X}$ be a flat and separated $\mathscr{R}$-scheme of finite type,
and let $x\in\mathscr{X}_{k}$ be a point. We say that $\mathscr{X}$ is
strictly toroidal at $x$ if there exist a toric monoid $S$, an open
neighbourhood $U$ of $x$ in $\mathscr{X}$, and a smooth morphism of
$\mathscr{R}$-schemes
$U\rightarrow\operatorname{Spec}(\mathscr{R}[S]/(\chi^{\omega}-t^{q}))$, where
$q$ is a positive rational number, $\omega$ is an element of $S$, and
$\chi^{\omega}$ is a torus invariant monomial in $k[S]$ associated with
$\omega$ such that $k[S]/(\chi^{\omega})$ is reduced. We call that
$\mathscr{X}$ is strictly toroidal if it is so at any $x\in\mathscr{X}_{k}$.

Let $\mathscr{X}$ be a strictly toroidal $\mathscr{R}$-scheme. Then a stratum
$E$ of $\mathscr{X}_{k}$ is a connected component of the intersection of a
non-empty set of irreducible components of $\mathscr{X}_{k}$. Let
$\mathrm{codim}(E)$ denote the codimension of $E$ in $\mathscr{X}_{k}$, and
$\mathcal{S}(\mathscr{X})$ denote the set of all strata in $\mathscr{X}_{k}$.

Now, we compute the stably birational volume of hypersurfaces of mock toric
varieties from the proper strictly toroidal model constructed in [14,
Proposition 7.7].

######  Proposition 2.2.

We keep the notation in [14, Proposition 7.7]. Then the following statements
follow:

  1. (a)

We assume that $f$ is nondegenerate for $\Delta^{\prime}$. Let
$\tau\in\Delta^{\prime}_{\operatorname{sp}}\cap{}_{f}\Delta^{\prime}$ be a
cone and $\\{E^{(1)}_{\tau},E^{(2)}_{\tau},\ldots,E^{(r_{\tau})}_{\tau}\\}$
denote all connected components of $H_{W,f}\cap W_{\tau}$. Then for any $1\leq
j\leq r_{\tau}$, $E^{(j)}_{\tau}$ is a dense open subset of
$\overline{E^{(j)}_{\tau}}$ and $\dim(E^{(j)}_{\tau})=\dim(Y)-\dim(\tau)$.
Moreover, as a closed subscheme of $\overline{W_{\tau}}$, the following
equation holds:

$H_{W,f}\cap\overline{W_{\tau}}=\coprod_{1\leq j\leq
r_{\tau}}\overline{E^{(j)}_{\tau}}$

  2. (b)

We use the notation in (a). Then for any $1\leq j\leq r_{\tau}$, the following
equations hold:

$\displaystyle\biggl{\\{}E^{(j)}_{\tau}\biggr{\\}}_{\mathrm{sb}}$
$\displaystyle=\biggl{\\{}\overline{E^{(j)}_{\tau}}\biggr{\\}}_{\mathrm{sb}}$
$\displaystyle E^{(j)}_{\tau}$ $\displaystyle=\overline{E^{(j)}_{\tau}}\cap
W_{\tau}$

  3. (c)

We use the notation in (a). We assume that the assumption of (a) holds. Let
$\mathfrak{I}$ denote a set which consists of all irreducible components of
$\mathscr{H}_{k}$. Then the following equation holds:

$\mathfrak{I}=\coprod_{\begin{subarray}{c}\gamma\in\Delta^{\prime}_{\operatorname{sp}}\cap{}_{f}\Delta^{\prime}\\\
\dim(\gamma)=1\end{subarray}}\biggl{\\{}\overline{E^{(j)}_{\gamma}}\biggr{\\}}_{1\leq
j\leq r_{\gamma}}$

  4. (d)

We assume that the assumption of [14, Proposition 7.7(k)] holds. Moreover, we
assume that $\Delta^{\prime}$ is compactly arranged. We use the notation in
(a) and [14, Proposition 7.7(j)]. Let $1\leq i\leq r$ be an integer and
$E\in\mathcal{S}(\mathscr{H}_{i})$ be a stratum. Then there unique exists
$\tau\in\Delta^{\prime}_{\operatorname{bdd}}\cap{}_{f}\Delta^{\prime}$ and
$1\leq j\leq r_{\tau}$ such that $E=\overline{E^{(j)}_{\tau}}$.

  5. (e)

We assume that the assumption in [14, Proposition 7.7(k)] holds and use the
notation in (a) and [14, Proposition 7.7(j)]. Let
$\tau\in\Delta^{\prime}_{\operatorname{bdd}}\cap{}_{f}\Delta^{\prime}$ be a
cone and $1\leq j\leq r_{\tau}$ be an integer. Then there exists $1\leq i\leq
r$ and $E\in\mathcal{S}(\mathscr{H}_{i})$ such that
$\overline{E^{(j)}_{\tau}}=E$.

  6. (f)

We assume that the assumption of [14, Proposition 7.7(k)] holds. Let
$\Delta^{\prime\prime}$ be a strongly convex rational polyhedral fan in
$(N^{\prime}\oplus\mathbb{Z})_{\mathbb{R}}$ such that $\pi^{1}$ is compatible
with the fans $\Delta^{\prime\prime}$ and $\Delta\times\Delta_{!}$, and
$\operatorname{Supp}(\Delta^{\prime\prime})=(\pi^{1}_{\mathbb{R}})^{-1}(\operatorname{Supp}(\Delta\times\Delta_{!}))$.
We use the notation in (a) and [14, Proposition 7.7(j)]. Let $W^{\prime}$ be a
mock toric variety induced by $Z^{1}$ along $\pi^{1}_{*}\colon
X(\Delta^{\prime\prime})\rightarrow X(\Delta\times\Delta_{!})$ and $s^{1}$.
Like as in (a), let $\\{E^{(j)}_{\sigma}\\}_{1\leq j\leq r_{\sigma}}$ denote
all connected components of $H_{W^{\prime},f}\cap W^{\prime}_{\sigma}$ for
$\sigma\in\Delta^{\prime\prime}$.

We assume that $\Delta^{\prime}$ is a refinement of $\Delta^{\prime\prime}$
and $f$ is fine for $\Delta^{\prime\prime}$. Then $f$ is non-degenerate for
$\Delta^{\prime\prime}$ and the following equation holds:

$\sum_{1\leq i\leq
r}\mathrm{Vol}_{\mathrm{sb}}\biggl{(}\bigl{\\{}H^{\circ}_{Y_{\mathscr{K}},f_{i}}\bigr{\\}}_{\mathrm{sb}}\biggr{)}=\sum_{\begin{subarray}{c}\sigma\in{\Delta^{\prime\prime}}_{\operatorname{bdd}}\\\
\sigma\in{}_{f}\Delta^{\prime\prime}\end{subarray}}(-1)^{\dim(\sigma)-1}\biggl{(}\sum_{1\leq
j\leq r_{\sigma}}\bigl{\\{}E^{(j)}_{\sigma}\bigr{\\}}_{\mathrm{sb}}\biggr{)}$

###### Proof.

We prove the statements from (a) to (f) in order:

  1. (a)

From [14, Proposition 6.4], there exist $\chi\in k[W_{0}]^{*}$ such that
$\operatorname{val}^{\tau}_{f}+\operatorname{val}^{\tau}_{\chi^{\prime}}\equiv
0$. Let $p^{\tau}$ denote a ring morphism
$\Gamma(W(\tau),\mathscr{O}_{W})\rightarrow\Gamma(W_{\tau},\mathscr{O}_{\overline{W_{\tau}}})$
and $f^{\tau}$ denote $p^{\tau}(\chi f)$. Then from [14, Proposition 6.11(a)],
$H_{W,f}\cap W_{\tau}=H^{\circ}_{\overline{W_{\tau}},f^{\tau}}$. Let
$f^{\tau}=g_{1}g_{2}\cdots g_{l}$ be an irreducible decomposition. We remark
that $H_{W,f}\cap W_{\tau}$ is smooth over $k$. Thus, $l=r_{\tau}$ and if it
is necessary, we may replace the index such that
$E^{(j)}_{\tau}=H^{\circ}_{\overline{W_{\tau}},g_{j}}$ for any $1\leq j\leq
r_{\tau}$. Moreover, from [14, Proposition 6.11(c)],
$H_{W,f}\cap\overline{W_{\tau}}=H_{\overline{W_{\tau}},f^{\tau}}$ and from
[14, Proposition 6.11(e)], $f^{\tau}$ is non-degenerate for
$\Delta^{\prime}[\tau]$. Thus, from [14, Proposition 6.12(c)], the following
equation is a connected decomposition of
$H_{W,f}\cap\overline{W_{\tau}}=H_{\overline{W_{\tau}},f^{\tau}}$:

$H_{\overline{W_{\tau}},f^{\tau}}=\coprod_{1\leq j\leq
r_{\tau}}H_{\overline{W_{\tau}},g_{j}}$

We remark that $\overline{E^{(j)}_{\tau}}=H_{\overline{W_{\tau}},g_{j}}$ for
any $1\leq j\leq r_{\tau}$. Because $H_{\overline{W_{\tau}},g_{j}}$ is a
scheme theoretic closure of $H^{\circ}_{\overline{W_{\tau}},g_{j}}$ in
$\overline{W_{\tau}}$ and
$H^{\circ}_{\overline{W_{\tau}},g_{j}}=H_{\overline{W_{\tau}},g_{j}}\cap
W_{\tau}$, $E^{(j)}_{\tau}$ is a dense open subset in
$\overline{E^{(j)}_{\tau}}$ and $E^{(j)}_{\tau}=\overline{E^{(j)}_{\tau}}\cap
W_{\tau}$ for any $1\leq j\leq r_{\tau}$. Furthermore, from the above
argument, $\dim(E^{(j)}_{\tau})=\dim(W_{\tau})-1$. Thus, from [14, Proposition
3.4(d)], we have $\dim(E^{(j)}_{\tau})=\dim(Y)-\dim(\tau)$.

  2. (b)

We have already shown the statement of (b) in the proof of (a).

  3. (c)

We use the notation in (a). We can identify with $\mathscr{H}_{k}$ and
$H_{W,f}\times_{\mathbb{A}^{1}}\\{0\\}$. In addition to this,
$(H_{W,f}\times_{\mathbb{A}^{1}}\\{0\\})\cap W_{\tau}\neq\emptyset$ if and
only if $\tau\in\Delta^{\prime}_{\operatorname{sp}}\cap{}_{f}\Delta^{\prime}$.
Therefore, $\mathscr{H}_{k}$ has the following stratification:

$\displaystyle\mathscr{H}_{k}$
$\displaystyle=\coprod_{\tau\in\Delta^{\prime}_{\operatorname{sp}}\cap{}_{f}\Delta^{\prime}}(\mathscr{H}_{k}\cap
W_{\tau})$
$\displaystyle=\coprod_{\tau\in\Delta^{\prime}_{\operatorname{sp}}\cap{}_{f}\Delta^{\prime}}(\coprod_{1\leq
j\leq r_{\tau}}E^{(j)}_{\tau})$

For any $\tau\in\Delta^{\prime}_{\operatorname{sp}}\cap{}_{f}\Delta^{\prime}$,
we have $\tau\not\subset N^{\prime}_{\mathbb{R}}\times\\{0\\}$. Hence, there
exists $\gamma\in\Delta^{\prime}_{\operatorname{sp}}$ such that
$\gamma\preceq\tau$ and $\dim(\gamma)=1$. From [14, Proposition 6.11(d)],
$\gamma\in{}_{f}\Delta^{\prime}$. Moreover, for any $\tau_{1}$ and
$\tau_{2}\in\Delta^{\prime}_{\operatorname{sp}}\cap{}_{f}\Delta^{\prime}$,
$\mathscr{H}_{k}\cap
W_{\tau_{1}}\subset\mathscr{H}_{k}\cap\overline{W_{\tau_{2}}}$ if and only if
$\tau_{2}\subset\tau_{1}$ from [14, Corollary 3.8]. In conclusion, the
following equation holds from (a):

$\mathscr{H}_{k}=\bigcup_{\begin{subarray}{c}\gamma\in\Delta^{\prime}_{\operatorname{sp}}\cap{}_{f}\Delta^{\prime}\\\
\dim(\gamma)=1\end{subarray}}(\bigcup_{1\leq j\leq
r_{\gamma}}\overline{E^{(j)}_{\gamma}})$

Furthermore, we remark that for any $\tau_{1}$ and
$\tau_{2}\in\Delta^{\prime}_{\operatorname{sp}}\cap{}_{f}\Delta^{\prime}$, any
$1\leq j_{1}\leq r_{\tau_{1}}$, and any $1\leq j_{2}\leq r_{\tau_{2}}$, if we
have $\overline{E^{(j_{1})}_{\tau_{1}}}=\overline{E^{(j_{2})}_{\tau_{2}}}$,
then $\tau_{1}=\tau_{2}$ and $j_{1}=j_{2}$. Indeed, because
$E^{(j_{1})}_{\tau_{1}}\cap\overline{E^{(j_{2})}_{\tau_{2}}}\neq\emptyset$, we
have $W_{\tau_{1}}\cap\overline{W_{\tau_{2}}}\neq\emptyset$. Thus, from [14,
Corollary 3.8], $\tau_{2}\subset\tau_{1}$. Similarly,
$\tau_{1}\subset\tau_{2}$ too. From (a), we have $j_{1}=j_{2}$.

Let $\gamma\in\Delta^{\prime}_{\operatorname{sp}}\cap{}_{f}\Delta^{\prime}$ be
a cone such $\dim(\gamma)=1$ for any $1\leq j\leq r_{\gamma}$. Then
$\dim(\overline{E^{(j)}_{\gamma}})=\dim(Y)-1$ from (a) for any $1\leq j\leq
r_{\gamma}$. Thus, from the above remark, the following set is equal to
$\mathfrak{I}$:

$\coprod_{\begin{subarray}{c}\gamma\in\Delta^{\prime}_{\operatorname{sp}}\cap{}_{f}\Delta^{\prime}\\\
\dim(\gamma)=1\end{subarray}}\biggl{\\{}\overline{E^{(j)}_{\gamma}}\biggr{\\}}_{1\leq
j\leq r_{\gamma}}$

  4. (d)

Let $1\leq i^{\prime}\leq r$ be an integer and
$E\in\mathcal{S}(\mathscr{H}_{i^{\prime}})$ be a stratum. From [14,
Proposition 7.7(j)], $\mathscr{H}_{k}=\coprod_{1\leq i\leq
r}(\mathscr{H}_{i})_{k}$. Thus, all irreducible components of
$(\mathscr{H}_{i})_{k}$ are in $\mathfrak{I}$. Hence, from (c), there exists
$\gamma_{1},\ldots,\gamma_{m}\in\Delta^{\prime}_{\operatorname{sp}}\cap{}_{f}\Delta^{\prime}$
and integers $\\{j_{l}\\}_{1\leq l\leq m}$, such that $\dim(\gamma_{l})=1$,
$1\leq j_{l}\leq r_{\gamma_{l}}$ for any $1\leq l\leq m$, and $E$ is a
connected component of the following closed subset:

$\bigcap_{1\leq l\leq m}\overline{E^{(j_{l})}_{\gamma_{l}}}$

In particular, from (a), $\cap_{1\leq l\leq
m}\overline{W_{\gamma_{l}}}\neq\emptyset$. Thus, from [14, Proposition
3.4(a)], there exists $\sigma\in\Delta^{\prime}$ such that
$\gamma_{l}\preceq\sigma$ for any $1\leq j\leq m$. Hence, from the assumption,
there exists $\tau\in\Delta^{\prime}_{\operatorname{bdd}}$ such that
$\gamma_{l}\preceq\tau$ for any $1\leq l\leq m$. We can take $\tau$ minimal
because $\Delta^{\prime}$ is a fan. Then the following equation holds from the
assumption of $\tau$ :

$\bigcap_{1\leq l\leq m}\overline{W_{\gamma_{l}}}=\overline{W_{\tau}}$

Thus, from (a), $\mathscr{H}_{k}\cap\overline{W_{\tau}}\neq\emptyset$. In
particular, from [14, Proposition 3.4(a)] and [14, Proposition 6.11(d)],
$\tau\in{}_{f}\Delta^{\prime}$. Moreover, the following equation holds:

$\displaystyle\mathscr{H}_{k}\cap\overline{W_{\tau}}$
$\displaystyle=\bigcap_{1\leq l\leq
m}(\mathscr{H}_{k}\cap\overline{W_{\gamma_{l}}})$
$\displaystyle=\coprod_{\\{\\{h_{l}\\}_{1\leq l\leq m}\mid 1\leq h_{l}\leq
r_{\gamma_{l}}\\}}(\bigcap_{1\leq l\leq
m}\overline{E^{(h_{l})}_{\gamma_{l}}})$

Thus, $E$ is also a connected component of
$\mathscr{H}_{k}\cap\overline{W_{\tau}}$. Therefore, from (a), there exists
$1\leq j\leq r_{\tau}$ such that $E=\overline{E^{(j)}_{\tau}}$.

We already have shown the uniqueness in the proof of (c)

  5. (e)

Let $\tau\in\Delta^{\prime}_{\operatorname{bdd}}\cap{}_{f}\Delta^{\prime}$ be
a cone and $1\leq j\leq r_{\tau}$ be an integer. Let
$\gamma_{1},\ldots,\gamma_{m}$ be all rays of $\tau$. Because
$\tau\in\Delta^{\prime}_{\operatorname{bdd}}$, we have
$\gamma_{1},\ldots,\gamma_{m}\in\Delta^{\prime}_{\operatorname{bdd}}$.
Moreover, from [14, Proposition 6.11(d)],
$\gamma_{1},\ldots,\gamma_{m}\in{}_{f}\Delta^{\prime}$. Because
$\gamma_{1},\ldots,\gamma_{m}$ are all rays of $\tau$, the following equation
holds:

$\bigcap_{1\leq l\leq m}\overline{W_{\gamma_{l}}}=\overline{W_{\tau}}$

Thus, like as the argument in proof of (d), the following equation holds:

$\displaystyle\mathscr{H}_{k}\cap\overline{W_{\tau}}$
$\displaystyle=\bigcap_{1\leq l\leq
m}(\mathscr{H}_{k}\cap\overline{W_{\gamma_{l}}})$
$\displaystyle=\coprod_{\\{\\{h_{l}\\}_{1\leq l\leq m}\mid 1\leq h_{l}\leq
r_{\gamma_{l}}\\}}(\bigcap_{1\leq l\leq
m}\overline{E^{(h_{l})}_{\gamma_{l}}})$

Hence, there exists integers $(j_{l})_{1\leq l\leq m}$ such that $1\leq
j_{l}\leq r_{\gamma_{l}}$ for any $1\leq l\leq m$ and
$\overline{E^{(j)}_{\tau}}$ is a connected component of the following closed
subset:

$\bigcap_{1\leq l\leq m}\overline{E^{(j_{l})}_{\gamma_{l}}}$

From (m), for any $1\leq l\leq m$, $\overline{E^{(j_{l})}_{\gamma_{l}}}$ is an
irreducible component of $\mathscr{H}_{k}$. Now, $\overline{E^{(j)}_{\tau}}$
is non-empty, so there unique exists $1\leq i\leq r$ such that each
$\overline{E^{(j_{l})}_{\gamma_{l}}}$ is an irreducible componenst of
$(\mathscr{H}_{i})_{k}$ from [14, Proposition 7.7(j)]. This shows that
$\overline{E^{(j)}_{\tau}}\in\mathcal{S}(\mathscr{H}_{i})$.

  6. (f)

We can regard $W$ as a mock toric variety induced by $W^{\prime}$ along
$X(\Delta^{\prime})\rightarrow X(\Delta^{\prime\prime})$. Thus, from [14,
Proposition 6.14(e)], $f$ is non-degenerate for $\Delta^{\prime\prime}$.

From (d), (e), and the remark in the proof of (c), we can check that the
following equation holds:

$\coprod_{1\leq i\leq
r}\mathcal{S}(\mathscr{H}_{i})=\coprod_{\tau\in\Delta^{\prime}_{\operatorname{bdd}}\cap{}_{f}\Delta^{\prime}}\\{\overline{E^{(j)}_{\tau}}\\}_{1\leq
j\leq r_{\tau}}$

We remark that
$\\{H_{Y_{\mathscr{K}},f_{i}}\\}_{\mathrm{sb}}=\\{H^{\circ}_{Y_{\mathscr{K}},f_{i}}\\}_{\mathrm{sb}}$
for any $1\leq i\leq r$. Thus, from (a), (b), [14, Proposition 7.7(k)], and
Proposition 1.1, the following equation holds:

$\sum_{1\leq i\leq
r}\mathrm{Vol}_{\mathrm{sb}}\biggl{(}\bigl{\\{}H^{\circ}_{Y_{\mathscr{K}},f_{i}}\bigr{\\}}_{\mathrm{sb}}\biggr{)}=\sum_{\begin{subarray}{c}\tau\in\Delta^{\prime}_{\operatorname{bdd}}\\\
\tau\in{}_{f}\Delta^{\prime}\end{subarray}}(-1)^{\dim(\tau)-1}\biggl{(}\sum_{1\leq
j\leq r_{\tau}}\bigl{\\{}E^{(j)}_{\tau}\bigr{\\}}_{\mathrm{sb}}\biggr{)}$

Let $\tau\in\Delta^{\prime}_{\operatorname{sp}}\cap{}_{f}\Delta^{\prime}$ be a
cone. Then there exists unique $\sigma\in\Delta^{\prime\prime}$ such that
$\tau^{\circ}\subset\sigma^{\circ}$ because $\Delta^{\prime}$ is a refinement
of $\Delta^{\prime\prime}$. Because
$\tau\in\Delta^{\prime}_{\operatorname{sp}}$, we have
$\sigma\in\Delta^{\prime\prime}_{\operatorname{sp}}$. Moreover, from the proof
of [14, Proposition 6.14(e)], $H_{W,f}\cap W_{\tau}$ is a trivial algebraic
torus fibration of $H_{W^{\prime},f}\cap W^{\prime}_{\sigma}$ whose dimension
of the fiber is $\dim(\sigma)-\dim(\tau)$ from (a). In particular,
$\sigma\in{}_{f}\Delta^{\prime\prime}$. Moreover, $r_{\tau}=r_{\sigma}$ and if
it is necessary, we can replace the index such that $E^{(j)}_{\tau}$ is a
trivial algebraic torus fibration of $E^{(j)}_{\sigma}$ for any $1\leq j\leq
r_{\sigma}$.

Conversely, for any
$\sigma\in\Delta^{\prime\prime}_{\operatorname{sp}}\cap{}_{f}\Delta^{\prime\prime}$
and for $\tau\in\Delta^{\prime}$ such that
$\tau^{\circ}\subset\sigma^{\circ}$, we have
$\tau\in\Delta^{\prime}_{\operatorname{sp}}\cap{}_{f}\Delta^{\prime}$ from
Proposition [14, Proposition 6.14(e)].

Thus, the following equation holds:

$\sum_{1\leq i\leq
r}\rho\biggl{(}\bigl{\\{}H^{\circ}_{Y_{\mathscr{K}},f_{i}}\bigr{\\}}_{\mathrm{sb}}\biggr{)}=\sum_{\begin{subarray}{c}\sigma\in\Delta^{\prime\prime}_{\operatorname{sp}}\\\
\sigma\in{}_{f}\Delta^{\prime\prime}\end{subarray}}\biggl{(}\sum_{\begin{subarray}{c}\tau\in\Delta^{\prime}_{\operatorname{bdd}}\\\
\tau^{\circ}\subset\sigma^{\circ}\end{subarray}}(-1)^{\dim(\tau)-1}\biggr{)}\biggl{(}\sum_{1\leq
j\leq r_{\sigma}}\bigl{\\{}E^{(j)}_{\sigma}\bigr{\\}}_{\mathrm{sb}}\biggr{)}$

Now, we use the notation in [9, $\S 3$]. Let
$\sigma\in\Delta^{\prime\prime}_{\operatorname{sp}}$ be a cone. We can
identify with $N^{\prime}_{\mathbb{Q}}$ and
$N^{\prime}_{\mathbb{Q}}\times\\{1\\}$, so we can regard $\sigma\cap
N^{\prime}_{\mathbb{Q}}\times\\{1\\}$ as a $\mathbb{Q}$-rational polyhedron in
$N^{\prime}_{\mathbb{Q}}$. Similarly, for any $\tau\in\Delta^{\prime}$ with
$\tau^{\circ}\subset\sigma^{\circ}$, we can regard $\tau\cap
N^{\prime}_{\mathbb{Q}}\times\\{1\\}$ as a $\mathbb{Q}$-rational polyhedron in
$N^{\prime}_{\mathbb{Q}}$. Moreover, there exists the following disjoint
decomposition of $\mathbb{Q}$-rational polyhedrons because $\Delta^{\prime}$
is a refinement of $\Delta^{\prime\prime}$:

$\sigma^{\circ}\cap(N^{\prime}_{\mathbb{Q}}\times\\{1\\})=\coprod_{\begin{subarray}{c}\tau\in\Delta^{\prime}\\\
\tau^{\circ}\subset\sigma^{\circ}\end{subarray}}\tau^{\circ}\cap(N^{\prime}_{\mathbb{Q}}\times\\{1\\})$

There exists Euler characteristic $\chi^{\prime}$ of definable subsets in
$N^{\prime}_{\mathbb{Q}}$ from [4, 9.6]. From [4, 9.6], $\chi^{\prime}$ has
the additivity, so the following equation holds:

$\chi^{\prime}(\sigma^{\circ}\cap(N^{\prime}_{\mathbb{Q}}\times\\{1\\}))=\sum_{\begin{subarray}{c}\tau\in\Delta^{\prime}\\\
\tau^{\circ}\subset\sigma^{\circ}\end{subarray}}\chi^{\prime}(\tau^{\circ}\cap(N^{\prime}_{\mathbb{Q}}\times\\{1\\}))$

Furthermore, for any $\tau\in\Delta^{\prime}_{\operatorname{sp}}$, the
following equation holds from [9, 3.7]:

$\chi^{\prime}(\tau^{\circ}\cap(N^{\prime}_{\mathbb{Q}}\times\\{1\\}))=\begin{cases}(-1)^{\dim(\tau)-1}&\tau\in\Delta^{\prime}_{\operatorname{bdd}}\\\
0&\tau\notin\Delta^{\prime}_{\operatorname{bdd}}\\\ \end{cases}$

Thus, for any
$\sigma\in\Delta^{\prime\prime}_{\operatorname{sp}}\cap{}_{f}\Delta^{\prime\prime}$,
there exists the following equation:

$\displaystyle\sum_{\begin{subarray}{c}\tau\in\Delta^{\prime}_{\operatorname{bdd}}\\\
\tau^{\circ}\subset\sigma^{\circ}\end{subarray}}(-1)^{\dim(\tau)-1}$
$\displaystyle=\sum_{\begin{subarray}{c}\tau\in\Delta^{\prime}_{\operatorname{bdd}}\\\
\tau^{\prime\circ}\subset\sigma^{\circ}\end{subarray}}\chi^{\prime}(\tau^{\circ}\cap(N^{\prime}_{\mathbb{Q}}\times\\{1\\}))$
$\displaystyle=\sum_{\begin{subarray}{c}\tau\in\Delta^{\prime}_{\operatorname{sp}}\\\
\tau^{\prime\circ}\subset\sigma^{\circ}\end{subarray}}\chi^{\prime}(\tau^{\circ}\cap(N^{\prime}_{\mathbb{Q}}\times\\{1\\}))$
$\displaystyle=\chi^{\prime}(\sigma^{\circ}\cap(N^{\prime}_{\mathbb{Q}}\times\\{1\\}))$

Similarly, for any $\sigma\in\Delta^{\prime\prime}_{\operatorname{sp}}$, the
following equation holds from [9, 3.7]:

$\chi^{\prime}(\sigma^{\circ}\cap(N^{\prime}_{\mathbb{Q}}\times\\{1\\}))=\begin{cases}(-1)^{\dim(\sigma)-1}&\sigma\in\Delta^{\prime\prime}_{\operatorname{bdd}}\\\
0&\sigma\notin\Delta^{\prime\prime}_{\operatorname{bdd}}\\\ \end{cases}$

Thus, the following equation holds:

$\sum_{1\leq i\leq
r}\mathrm{Vol}_{\mathrm{sb}}\biggl{(}\bigl{\\{}H^{\circ}_{Y_{\mathscr{K}},f_{i}}\bigr{\\}}_{\mathrm{sb}}\biggr{)}=\sum_{\begin{subarray}{c}\sigma\in\Delta^{\prime\prime}_{\operatorname{bdd}}\\\
\sigma\in{}_{f}\Delta^{\prime\prime}\end{subarray}}(-1)^{\dim(\sigma)-1}\biggl{(}\sum_{1\leq
j\leq r_{\sigma}}\bigl{\\{}E^{(j)}_{\sigma}\bigr{\\}}_{\mathrm{sb}}\biggr{)}$

∎

In [14, Proposition 7.7] and Proposition2.2, we assume that $f$ is
irreducible. In the following theorem, we extract and summarize the conditions
under which the above proposition holds.

######  Theorem 2.3.

We keep the notation in [14, Proposition 7.7]. Let $\Delta^{\prime\prime}$ be
a strongly convex rational polyhedral fan in
$(N^{\prime}\oplus\mathbb{Z})_{\mathbb{R}}$ such that $\pi^{1}$ is compatible
with the fans $\Delta^{\prime\prime}$ and $\Delta\times\Delta_{!}$, and
$\operatorname{Supp}(\Delta^{\prime\prime})=(\pi^{1}_{\mathbb{R}})^{-1}(\operatorname{Supp}(\Delta\times\Delta_{!}))$.
we assume that

  1. (1)

A fan $\Delta^{\prime}$ is a refinement of $\Delta^{\prime\prime}$ and $f$ is
fine for $\Delta^{\prime\prime}$

  2. (2)

A function $f$ is nondegenerate for $\Delta^{\prime}$.

  3. (3)

A fan $\Delta^{\prime}$ is compactly arranged, generically unimodular, and
specifically reduced.

  4. (4)

Every irreducible component of $H^{\circ}_{W,f}$ dominates
$\mathbb{G}_{m}^{1}$.

  5. (5)

A scheme $Z$ is proper over $k$.

Let $\\{F_{\lambda}\\}_{\lambda\in\Lambda}$ be all connected components of
$\mathscr{H}^{\circ}$, $W^{\prime}$ denote a mock toric variety induced by
$Z^{1}$ along $\pi^{1}_{*}\colon X(\Delta^{\prime\prime})\rightarrow
X(\Delta\times\Delta_{!})$ and $s^{1}$, $\\{E^{(j)}_{\sigma}\\}_{1\leq j\leq
r_{\sigma}}$ denote all connected components of $H_{W^{\prime},f}\cap
W^{\prime}_{\sigma}$ for $\sigma\in\Delta^{\prime\prime}$. Then the following
equation holds:

$\sum_{\lambda\in\Lambda}\mathrm{Vol}_{\mathrm{sb}}\biggl{(}\bigl{\\{}F_{\lambda}\bigr{\\}}_{\mathrm{sb}}\biggr{)}=\sum_{\begin{subarray}{c}\sigma\in\Delta^{\prime\prime}_{\operatorname{bdd}}\\\
\sigma\in{}_{f}\Delta^{\prime\prime}\end{subarray}}(-1)^{\dim(\sigma)-1}\biggl{(}\sum_{1\leq
j\leq r_{\sigma}}\bigl{\\{}E^{(j)}_{\sigma}\bigr{\\}}_{\mathrm{sb}}\biggr{)}$

###### Proof.

From [14, Proposition 6.14(e)], $f$ is non-degenerate for
$\Delta^{\prime\prime}$. Let $f=h_{1}\cdots h_{n}$ be an irreducible
decomposition of $f\in k[W^{\circ}]$. Because $f$ is non-degenerate for
$\Delta^{\prime}$, $h_{b}$ is non-degenerate for $\Delta^{\prime}$ too from
[14, Proposition 6.12(b)] for any $1\leq b\leq n$. Moreover, from [14,
Proposition 6.12(c)], there exist a following decomposition of $H_{W,f}$:

$H_{W,f}=\coprod_{1\leq b\leq n}H_{W,h_{b}}$

In particular, $\mathscr{H}^{\circ}$ can be decomposed as follows:

$\coprod_{1\leq b\leq
n}(H^{\circ}_{W,h_{b}}\times_{\mathbb{G}_{m}}\operatorname{Spec}(\mathscr{K}))$

We remark that $H^{\circ}_{W,f}$ is smooth over $k$. In particular, it is
reduced. Thus, for any $1\leq b\leq n$, $H^{\circ}_{W,h_{b}}$ is reduced.
Moreover, from [14, Proposition 6.12(a)], each $h_{b}$ is fine for
$\Delta^{\prime\prime}$. Therefore, for each $h_{b}$, we can compute the
summation of the stable birational volume of irreducible components of
$H^{\circ}_{W,h_{b}}\times_{\mathbb{G}_{m}^{1}}\operatorname{Spec}(\mathscr{K})$
for the refinement $\Delta^{\prime}$ of $\Delta^{\prime\prime}$ like as
Proposition 2.2(f). Moreover, for any
$\sigma\in\Delta^{\prime\prime}_{\operatorname{sp}}$, the following equation
holds from the proof of [14, Proposition 6.12(b)]:

$\displaystyle H_{W^{\prime},f}\cap W^{\prime}_{\sigma}$
$\displaystyle=\coprod_{1\leq b\leq n}(H_{W^{\prime},h_{b}}\cap
W^{\prime}_{\sigma})$

Thus, from the above disjoint decomposition, the statement holds. ∎

##  3\. Mock polytopes of the mock toric variety

Let $Z$ be a scheme over $k$ and ($N$, $\Delta$, $\iota$, $\Phi$,
$\\{N_{\varphi}\\}_{\varphi\in\Phi}$,
$\\{\Delta_{\varphi}\\}_{\varphi\in\Phi}$) be a mock toric structure of $Z$.
In the previous section, we considered the stable birational volume of
hypersurfaces of a mock toric variety. However, it isn’t easy to apply
Theorem2.3 because of the following three reasons.

  * •

To find a strongly convex rational polyhedral fan $\Delta^{\prime\prime}$ that
$f$ is fine for $\Delta^{\prime\prime}$.

  * •

To check $f$ is nondegenerate for $\Delta^{\prime}$.

  * •

To find a refinement $\Delta^{\prime}$ of $\Delta^{\prime\prime}$ that
$\Delta^{\prime}$ is compactly arranged, generically unimodular, and
specifically reduced.

For the first point, Collory[14, Corollary 6.9] might be helpful, but
calculating the fan associated with $f$ is extremely difficult. Furthermore,
for the second point, if an intersection of a hypersurface and a single
stratum is not smooth, then the calculation becomes impossible at that stage.
Similarly, for the third point, no fan satisfies ”specifically reduced” in
general.

For the first and second points, we adopt the approach already taken in [7].
In more detail, consider the finite set consisting of units in $k[Z^{\circ}]$
and focus on their linear system $D$. These linear systems correspond to what
is referred to as the Newton polytope in the argument of toric varieties. In
[7], they constructed $\Delta^{\prime\prime}$ that a general $f\in D$ is fine
for $\Delta^{\prime\prime}$ from Neton polytope associated with $D$. Moreover,
they showed that a general $f\in D$ is non-degenerate for
$\Delta^{\prime\prime}$ from Bertini’s theorem. In this section, we consider
the applicability of the method outlined above to the context of mock toric
varieties. In the latter part of the section, we explain ways of avoiding the
third issue, which geometrically corresponds to semistable reduction.

In this section, we use the notation in [14, Sections 6 & 7] and the following
definition.

######  Definition 3.1.

We itemize the notation as follows:

  * •

Let $k$ be an infinite algebraically closed field with
$\operatorname{char}(k)=0$.

  * •

Let $S$ be a finite set.

  * •

Let $\mathcal{U}_{Z,S}$ denote a set which consists of all map from $S$ to
$k[Z^{\circ}]^{*}$.

  * •

We define a relation $\sim$ of $\mathcal{U}_{Z,S}$ as follows: For
$u_{1},u_{2}\in\mathcal{U}_{Z,S}$, if there exists $\chi\in k[Z^{\circ}]^{*}$
such that $\chi u_{1}(i)=u_{2}(i)$ for any $i\in S$, then we define $u_{1}\sim
u_{2}$. We can check that this relation $\sim$ is an equivalence relation of
$\mathcal{U}_{Z,S}$.

  * •

For $u\in\mathcal{U}_{Z,S}$, let $[u]$ denote the equivalence class of $u$
along the relation $\sim$.

  * •

For $u\in\mathcal{U}_{Z,S}$, let $V_{u}$ denote a $k$-lenear subspace of
$k[Z^{\circ}]$ generated by $\\{u(i)\\}_{i\in S}$, and $d_{u}$ denote
$\dim_{k}(V_{u})$.

The following proposition is immediately evident from Definition3.1

######  Proposition 3.2.

We use the notation in Definition 3.1. Let $u_{1}$ and
$u_{2}\in\mathcal{U}_{Z,S}$ be maps. If $u_{1}\sim u_{2}$, then
$d_{u_{1}}=d_{u_{2}}$.

###### Proof.

For any $\chi$, the $k$-linear endmorphism of $k[Z^{\circ}]$ defined by
multiplication of $\chi$ is an isomorphism. Thus,
$\dim(V_{u_{1}})=\dim(V_{u_{2}})$. ∎

In the following definition, we define a mock polytope, an analog of a Newton
polytope.

######  Definition 3.3.

We use the notation in Definition 3.1. Let $u\in\mathscr{U}_{Z,S}$ be a map.
Then we call the tuple $(Z,S,u)$ a mock polytope of $Z$.

In Section 6, we construct the map
$\operatorname{val}_{f}\colon\operatorname{Supp}(\Delta)\rightarrow\mathbb{R}$
for $f\in k[Z^{\circ}]$. From now on, we construct the map
$\operatorname{val}_{u}\colon\operatorname{Supp}(\Delta)\rightarrow\mathbb{R}$
for $u\in\mathcal{U}_{Z,S}$.

######  Definition 3.4.

Let $(Z,S,u)$ be a mock polytope of $Z$. Let $\operatorname{val}_{u}$ denote a
map from $\operatorname{Supp}(\Delta)$ to $\mathbb{R}$ defined as
$\operatorname{val}_{u}(v)=\min_{i\in S}\\{\operatorname{val}_{u(i)}(v)\\}$
for $v\in\operatorname{Supp}(\Delta)$. For a subset
$A\subset\operatorname{Supp}(\Delta)$, let $\operatorname{val}^{A}_{u}$ denote
the restriction $\operatorname{val}_{u}|_{A}$.

We consider the basic properties of $\operatorname{val}_{u}$ in the following
proposition.

######  Proposition 3.5.

We use the notation in Definition 3.4. Then the following statement holds:

  1. (a)

The map $\operatorname{val}_{u}$ is continuous and is preserved by the scalar
product of non-negative real numbers.

  2. (b)

Let $\sigma\in\Delta$ be a cone and $\varphi\in\Phi$ be an element such that
$\sigma\in\Delta_{\varphi}$. Let $\sigma_{\varphi}$ denote
$(q_{\varphi})_{\mathbb{R}}(\sigma)$. Then there exists $g_{\varphi}\in
k[M_{\varphi}]$ such that the following equation holds:

$\operatorname{val}^{\sigma}_{u}=\operatorname{val}^{\sigma_{\varphi}}_{g_{\varphi}}\circ(q_{\varphi})_{\mathbb{R}}|_{\sigma}$

###### Proof.

We prove the statements from (a) to (b).

  1. (a)

For each $i\in S$, $\operatorname{val}_{u(i)}$ is a continuous map and is
preserved by the scalar product of non-negative real numbers from [14,
Proposition 6.7(a)]. Thus, $\operatorname{val}_{u}$ also has these properties
from the definition of $\operatorname{val}_{u}$.

  2. (b)

From [14, Proposition 6.3], for each $i\in S$, there exists $\omega_{i}\in
M_{\varphi}$ such that
$\operatorname{val}^{\sigma}_{u(i)}=\operatorname{val}^{\sigma}_{\iota^{*}_{\varphi}(\chi^{\omega_{i}})}$.
Let $T$ denote a subset $\\{\omega_{i}\mid i\in S\\}\subset M_{\varphi}$. Let
$g_{\varphi}$ denote $\sum_{\omega\in T}\chi^{\omega}\in k[M_{\varphi}]$. Then
for any $v\in\sigma\cap N$, we have
$q_{\varphi}(v)(g_{\varphi})=\min_{\omega\in
T}\\{q_{\varphi}(v)(\chi^{\omega})$} because the definition of the torus
invariant valuation on toric varieties. This shows that for any
$v\in\sigma\cap N$, there exists the following equation:

$\displaystyle\operatorname{val}^{\sigma_{\varphi}}_{g_{\varphi}}\circ(q_{\varphi})_{\mathbb{R}}(v)$
$\displaystyle=q_{\varphi}(v)(g_{\varphi})$ $\displaystyle=\min_{\omega\in
T}\\{q_{\varphi}(v)(\chi^{\omega})\\}$ $\displaystyle=\min_{i\in
S}\\{q_{\varphi}(v)(\chi^{\omega_{i}})\\}$ $\displaystyle=\min_{i\in
S}\\{\operatorname{val}_{Z}(v)(\iota^{*}_{\varphi}(\chi^{\omega_{i}}))\\}$
$\displaystyle=\min_{i\in
S}\\{\operatorname{val}^{\sigma}_{\iota^{*}_{\varphi}(\chi^{\omega_{i}})}(v)\\}$
$\displaystyle=\min_{i\in S}\\{\operatorname{val}^{\sigma}_{u(i)}(v)\\}$
$\displaystyle=\operatorname{val}^{\sigma}_{u}(v)$

Thus, from (a) and [14, Lemma 8.13], the statement holds.

∎

Similar to [14, Proposition 6.8], $\operatorname{val}_{u}$ provides a
refinement of $\Delta$. We define the notation in Proposition 3.6 and prove
that the set of cones is a refinement of $\Delta$ in Proposition 3.7.

######  Definition 3.6.

Let $(Z,S,u)$ be a mock polytope and $\sigma\in\Delta$ be a cone.

  * •

Let $\Omega^{\sigma}_{u}$ be a set defined as follows:

$\Omega^{\sigma}_{u}=\\{(\chi,b)\in
k[Z^{\circ}]^{*}\times\mathbb{Z}_{>0}\mid\operatorname{val}_{u}(v)\leq\frac{1}{b}\operatorname{val}_{\chi}(v)\quad(\forall
v\in\sigma)\\}$

  * •

For $(\chi,b)\in\Omega^{\sigma}_{u}$, let $C^{\sigma}_{u}(\chi,b)$ be a set
defined as follows:

$C^{\sigma}_{u}(\chi,b)=\\{v\in\sigma\mid\operatorname{val}_{u}(v)=\frac{1}{b}\operatorname{val}_{\chi}(v)\\}$

######  Proposition 3.7.

We keep the notation in Definition 3.6. Then the following statements follow:

  1. (a)

The set $\\{C^{\sigma}_{u}(\chi,b)\\}_{(\chi,b)\in\Omega^{\sigma}_{u}}$ is a
strongly convex rational polyhedral fan and a refinement of $\sigma$.

  2. (b)

Let $\sigma$ and $\tau\in\Delta$ be cones such that $\tau\preceq\sigma$. Then
$\Omega^{\sigma}_{u}\subset\Omega^{\tau}_{u}$. Moreover,
$C^{\sigma}_{u}(\chi,b)\cap\tau=C^{\tau}_{u}(\chi,b)$ for any
$(\chi,b)\in\Omega^{\sigma}_{u}$. In particular, $C^{\tau}_{u}(\chi,b)\preceq
C^{\sigma}_{u}(\chi,b)$.

  3. (c)

The following set $\Delta_{u}$ is a strongly convex rational polyhedral fan in
$N_{\mathbb{R}}$ and a refinement of $\Delta$:

$\Delta_{u}=\bigcup_{\sigma\in\Delta}\\{C^{\sigma}_{u}(\chi,b)\\}_{(\chi,b)\in\Omega^{\sigma}_{u}}$

  4. (d)

For any $(\chi,b)\in\Omega^{\sigma}_{u}$, there exists $\chi^{\prime}\in
k[Z^{\circ}]^{*}$ such that
$\operatorname{val}_{u}(v)=\operatorname{val}_{\chi^{\prime}}(v)$ for any
$v\in C^{\sigma}_{u}(\chi,b)$.

  5. (e)

Let $u_{1}$ and $u_{2}\in\mathscr{U}_{Z,S}$ be maps. If $u_{1}\sim u_{2}$,
then $\Delta_{u_{1}}=\Delta_{u_{2}}$.

###### Proof.

We prove the statements from (a) to (e) in order.

  1. (a)

From Proposition 3.5(b), there exists $f\in k[Z^{\circ}]$ such that
$\operatorname{val}^{\sigma}_{u}=\operatorname{val}^{\sigma}_{f}$. This shows
that $\Omega^{\sigma}_{u}=\Omega^{\sigma}_{f}$ and for any
$(\chi,b)\in\Omega^{\sigma}_{u}$, we have
$C^{\sigma}_{u}(\chi,b)=C^{\sigma}_{f}(\chi,b)$. Thus, from [14, Proposition
6.8(c)], the statement holds.

  2. (b)

We can check the statement likewise [14, Proposition 6.8(d)].

  3. (c)

We can check the statement likewise [14, Proposition 6.8(e)].

  4. (d)

We use the notation in the proof of (a). Then $(\chi,b)\in\Omega^{\sigma}_{f}$
and $C^{\sigma}_{u}(\chi,b)=C^{\sigma}_{f}(\chi,b)$. Thus, from [14,
Proposition 6.8(f)], there exists $\chi^{\prime}\in k[Z^{\circ}]^{*}$ such
that $\operatorname{val}_{f}(v)=\operatorname{val}_{\chi^{\prime}}(v)$ for any
$v\in C^{\sigma}_{f}(\chi,b)$. Therefore, the statement holds.

  5. (e)

Let $\eta\in k[Z^{\circ}]^{*}$ be a unit such that $u^{\prime}(i)=\eta u(i)$
for any $i\in S$. Then for any $i\in S$,
$\operatorname{val}_{u^{\prime}(i)}=\operatorname{val}_{\eta}+\operatorname{val}_{u(i)}$
from [14, Proposition 6.7(b)]. This shows that
$\operatorname{val}_{u^{\prime}}=\operatorname{val}_{\eta}+\operatorname{val}_{u}$.
Let $\sigma\in\Delta$ be a cone. Thus, for any $(\chi,b)\in
k[Z^{\circ}]^{*}\times\mathbb{Z}_{>0}$, we can check that
$(\chi,b)\in\Omega^{\sigma}_{u}$ if and only if
$(\chi\eta^{b},b)\in\Omega^{\sigma}_{u^{\prime}}$. Moreover, in this case,
$C^{\sigma}_{u^{\prime}}(\chi\eta^{b},b)=C^{\sigma}_{u}(\chi,b)$. Therefore,
the set $\\{C^{\sigma}_{u}(\chi,b)\\}_{(\chi,b)\in\Omega^{\sigma}_{u}}$ is
equal to the set
$\\{C^{\sigma}_{u^{\prime}}(\chi^{\prime},b^{\prime})\\}_{(\chi^{\prime},b^{\prime})\in\Omega^{\sigma}_{u^{\prime}}}$.

∎

From $u\in\mathscr{U}_{Z,S}$, we can define the linear system
$D\subset\Gamma(Z^{\circ},\mathscr{O}_{Z})$. In Proposition3.9, we show that
$\Delta_{f}=\Delta_{u}$ for a general $f\in D$. The following lemma is needed
to show Proposition3.9.

######  Lemma 3.8.

Let $(Z,S,u)$ be a mock polytope. Let $(a_{i})_{i\in S}\in k^{|S|}$ be a
vector. Let $f\in k[Z^{\circ}]$ be a function defined as $f=\sum_{i\in
S}a_{i}u(i)$. Then the following statements follow:

  1. (a)

Let $v\in\operatorname{Supp}(\Delta)\cap N$ be an element. Then for general
$(a_{i})_{i\in S}\in k^{|S|}$, we have
$\operatorname{val}_{f}(v)=\operatorname{val}_{u}(v)$.

  2. (b)

Let $\tau\in\Delta_{u}$ be a cone and $v_{0}\in\tau^{\circ}$ be an element.
For any $(a_{i})_{i\in S}\in k^{|S|}$, if
$\operatorname{val}_{u}(v_{0})=\operatorname{val}_{f}(v_{0})$, then
$\operatorname{val}^{\tau}_{u}=\operatorname{val}^{\tau}_{f}$.

###### Proof.

We prove the statements from (a) to (b).

  1. (a)

For $n\in\mathbb{Z}$, let $E(v,n)$ denote a subset of $k^{|S|}$ defined as
follows:

$E(v,n)=\\{(a_{i})_{i\in S}\in k^{|S|}\mid\operatorname{val}_{Z}(v)(\sum_{i\in
S}a_{i}u(i))\geq n\\}$

Because $\operatorname{val}_{Z}(v)$ is a valuation which is trivial on $k$,
$E(v,n)$ is a linear subspace of $k^{|S|}$. In particular,
$E(v,\operatorname{val}_{u}(v))=k^{|S|}$ because of the property of a
valuation and the definition of $\operatorname{val}_{u}$.

On the other hand, from the definition of $\operatorname{val}_{u}$, there
exists $j\in S$ such that
$\operatorname{val}_{u}(v)=\operatorname{val}_{u(j)}(v)$. Let $e_{j}$ denotes
$(\delta_{i,j})_{i\in S}\in k^{|S|}$, where $\delta_{\cdot,\cdot}$ is
Kronecker’s delta. Thus, $e_{j}\notin E(v,\operatorname{val}_{u}(v)+1)$.

Therefore, $E(v,\operatorname{val}_{u}(v)+1)$ is a strict linear sub space of
$k^{|S|}$. In particular, for general $(a_{i})_{i\in S}\in k^{|S|}$,
$\operatorname{val}_{f}(v)=\operatorname{val}_{u}(v)$.

  2. (b)

From the definition of $\Delta_{u}$, there exists $\sigma\in\Delta$ and
$(\chi,b)\in\Omega^{\sigma}_{u}$ such that $\tau=C^{\sigma}_{u}(\chi,b)$. From
Proposition 3.7(d), there exists $\eta\in k[Z^{\circ}]^{*}$ such that
$\operatorname{val}^{\tau}_{u}=\operatorname{val}^{\tau}_{\eta}$. In
particular, $\operatorname{val}^{\tau}_{u}$ is a linear map over $\tau$ from
[14, Proposition 6.7(d)]. For any $v\in\operatorname{Supp}(\Delta)\cap N$, we
have $\operatorname{val}_{u}(v)\leq\operatorname{val}_{f}(v)$ because of the
definition of $\operatorname{val}_{u}$ and the property of valuations. Both
functions $\operatorname{val}_{u}$ and $\operatorname{val}_{f}$ are continuous
and are preserved by the scalar product of non-negative real numbers from [14,
Proposition 6.7(a)] and Proposition 3.5(a). Thus
$\operatorname{val}^{\tau}_{u}\leq\operatorname{val}^{\tau}_{f}$ because
$\tau$ is a rational polyhedral cone. Moreover, from [14, Proposition 6.7(c)],
$\operatorname{val}^{\sigma}_{f}$ is an upper convex function. In particular,
$\operatorname{val}^{\tau}_{f}$ is an upper convex function because
$\tau\subset\sigma$. Thus,
$\operatorname{val}^{\tau}_{f}-\operatorname{val}^{\tau}_{u}$ is a non-
negative valued upper convex function. On the other hand, from the assumption,
$\operatorname{val}^{\tau}_{u}(v_{0})=\operatorname{val}^{\tau}_{f}(v_{0})$.
Therefore, $\operatorname{val}^{\tau}_{u}=\operatorname{val}^{\tau}_{f}$
because $v_{0}\in\tau^{\circ}$.

∎

######  Proposition 3.9.

We keep the notation in Lemma 3.8. Then
$\operatorname{val}_{u}=\operatorname{val}_{f}$ and $\Delta_{f}=\Delta_{u}$
for general $(a_{i})_{i\in S}\in k^{|S|}$.

###### Proof.

If $\operatorname{val}_{u}=\operatorname{val}_{f}$, then
$\Delta_{u}=\Delta_{f}$ from the definition of $\Delta_{u}$ and $\Delta_{f}$.
Thus, it is enough to show that
$\operatorname{val}_{u}=\operatorname{val}_{f}$ for general $(a_{i})_{i\in
S}\in k^{|S|}$. Because $|\Delta|$ is finite, $|\Delta_{u}|$ is also finite
from the proof of Proposition 3.7(a). Thus, it is enough to show that for any
$\tau\in\Delta_{u}$, we have
$\operatorname{val}^{\tau}_{u}=\operatorname{val}^{\tau}_{f}$ for general
$(a_{i})_{i\in S}\in k^{|S|}$. Let $\tau\in\Delta_{u}$ be a cone, and
$v_{0}\in\tau^{\circ}\cap N$ be an element. Then from Lemma 3.8(a),
$\operatorname{val}_{f}(v_{0})=\operatorname{val}_{u}(v_{0})$ for general
$(a_{i})_{i\in S}\in k^{|S|}$. Thus, from Lemma 3.8(b),
$\operatorname{val}^{\tau}_{u}=\operatorname{val}^{\tau}_{f}$ for such
$(a_{i})_{i\in S}\in k^{|S|}$. ∎

From $u\in\mathscr{U}_{Z,S}$, we construct a refinement $\Delta_{u}$ of
$\Delta$. Let $Z_{u}$ denote the mock toric variety induced by $Z$ along
$X(\Delta_{u})\rightarrow X(\Delta)$. In Definition3.10, we define new mock
polytopes for each $\tau\in\Delta_{u}$. They are important to check the
structure of the intersection of a hypersurface and a stratum of $Z_{u}$.

######  Definition 3.10.

Let $(Z,S,u)$ be a mock polytope, $Z_{u}$ be a mock toric variety induced by
$Z$ along $X(\Delta_{u})\rightarrow X(\Delta)$, $\tau\in\Delta_{u}$ be a cone,
$Z^{\tau}_{u}$ denote a mock toric variety $\overline{(Z_{u})_{\tau}}$,
$p^{\tau}$ denote a ring morphism
$\Gamma(Z_{u}(\tau),\mathscr{O}_{Z_{u}})\rightarrow\Gamma((Z_{u})_{\tau},\mathscr{O}_{Z^{\tau}_{u}})$
associated with the closed immersion $(Z_{u})_{\tau}\hookrightarrow
Z_{u}(\tau)$, and $S^{\tau}$ denote a subset $\\{i\in S\mid
val^{\tau}_{u}=\operatorname{val}^{\tau}_{u(i)}\\}$ of $S$.

From Proposition 3.7 (d), there exists $\chi\in k[Z^{\circ}]^{*}$ such that
$\operatorname{val}^{\tau}_{u}=\operatorname{val}^{\tau}_{\chi}$. Let
$u^{\tau}_{\chi}$ denote a map from $S^{\tau}$ to $k[(Z^{\tau}_{u})^{\circ}]$
defined as $u^{\tau}_{\chi}(i)=p^{\tau}(\chi^{-1}u(i))$ for $i\in S^{\tau}$.

The following proposition is immediately evident from Definition3.10.

######  Proposition 3.11.

We keep the notation in Definition 3.10. Then the following statements follow:

  1. (a)

The tuple $(Z^{\tau}_{u},S^{\tau},u^{\tau}_{\chi})$ is a mock polytope of
$Z^{\tau}_{u}$.

  2. (b)

For any $\chi^{\prime}\in k[Z^{\circ}]^{*}$ such that
$\operatorname{val}^{\tau}_{u}=\operatorname{val}^{\tau}_{\chi^{\prime}}$, we
have $u^{\tau}_{\chi}\sim u^{\tau}_{\chi^{\prime}}$.

  3. (c)

Let $u^{\prime}\in\mathscr{U}_{Z,S}$ be a map such that $u\sim u^{\prime}$.
From Proposition 3.7(e), $\Delta_{u}=\Delta_{u^{\prime}}$, in particular,
$\tau\in\Delta_{u^{\prime}}$. Then $\\{i\in
S\mid\operatorname{val}^{\tau}_{u^{\prime}}=\operatorname{val}^{\tau}_{u^{\prime}(i)}\\}$
is equal to $S^{\tau}$.

  4. (d)

We use the notation in (c). Let $\chi^{\prime}\in k[Z_{0}]^{*}$ be a unit such
that
$\operatorname{val}^{\tau}_{u^{\prime}}=\operatorname{val}^{\tau}_{\chi^{\prime}}$.
Then $u^{\tau}_{\chi}\sim u^{\prime\tau}_{\chi^{\prime}}$.

###### Proof.

we prove the statements from (a) to (d) in order.

  1. (a)

It is enough to show that for any $i\in S^{\tau}$, we have
$u^{\tau}_{\chi}(i)\in k[(Z^{\tau}_{u})^{\circ}]^{*}$. Let $i\in S^{\tau}$ be
an element. From the definition of $S^{\tau}$, we have
$\operatorname{val}^{\tau}_{\chi^{-1}u(i)}\equiv 0$. Thus, from [14,
Proposition 4.8],
$\chi^{-1}u(i)\in\Gamma(Z_{u}(\tau),\mathscr{O}_{Z_{u}})^{*}$. Hence,
$p^{\tau}(\chi^{-1}u(i))\in k[(Z^{\tau}_{u})^{\circ}]^{*}$ too.

  2. (b)

From the assumption of $\chi$ and $\chi^{\prime}$,
$\operatorname{val}^{\tau}_{\chi^{\prime-1}\chi}\equiv 0$. As the argument in
the proof of (a),
$\chi^{\prime-1}\chi\in\Gamma(Z_{u}(\tau),\mathscr{O}_{Z_{u}})^{*}$. Let
$\eta$ denote $p^{\tau}(\chi^{\prime-1}\chi)\in
k[(Z^{\tau}_{u})^{\circ}]^{*}$. Then for any $i\in S^{\tau}$, we have $\eta
u^{\tau}_{\chi}(i)=u^{\tau}_{\chi^{\prime}}(i)$. This shows that
$u^{\tau}_{\chi}\sim u^{\tau}_{\chi^{\prime}}$.

  3. (c)

Let $\chi_{0}\in k[Z^{\circ}]^{*}$ be a unit such that
$\chi_{0}u(i)=u^{\prime}(i)$ for any $i\in S$. Then
$\operatorname{val}_{u(i)}+\operatorname{val}_{\chi_{0}}=\operatorname{val}_{u^{\prime}(i)}$
from [14, Proposition 6.7(b)]. Thus,
$\operatorname{val}_{u}+\operatorname{val}_{\chi_{0}}=\operatorname{val}_{u^{\prime}}$
from the definition of $\operatorname{val}_{u}$ and
$\operatorname{val}_{u^{\prime}}$. Therefore, for any $i\in S$,
$\operatorname{val}^{\tau}_{u}=\operatorname{val}^{\tau}_{u(i)}$ if and only
if
$\operatorname{val}^{\tau}_{u^{\prime}}=\operatorname{val}^{\tau}_{u^{\prime}(i)}$.
This shows that $\\{i\in
S\mid\operatorname{val}^{\tau}_{u^{\prime}}=\operatorname{val}^{\tau}_{u^{\prime}(i)}\\}$
is equal to $S^{\tau}$.

  4. (d)

Let $\chi_{0}\in k[Z^{\circ}]^{*}$ be a unit such that
$\chi_{0}u(i)=u^{\prime}(i)$ for any $i\in S$. In the proof of (c), we have
already shown that
$\operatorname{val}_{u}+\operatorname{val}_{\chi_{0}}=\operatorname{val}_{u^{\prime}}$.
Thus,
$\operatorname{val}^{\tau}_{\chi\chi_{0}}=\operatorname{val}^{\tau}_{u^{\prime}}$
from [14, Proposition 6.7(b)]. Therefore from (b),
$u^{\prime\tau}_{\chi\chi_{0}}\sim u^{\prime\tau}_{\chi^{\prime}}$. On the
other hand, for any $i\in S$,
$(\chi\chi_{0})^{-1}u^{\prime}(i)=\chi^{-1}u(i)$. Thus,
$u^{\tau}_{\chi}=u^{\prime\tau}_{\chi\chi_{0}}$. This shows that
$u^{\tau}_{\chi}\sim u^{\prime\tau}_{\chi^{\prime}}$.

∎

In Definition3.10, we choose $\chi$ to define $S^{\tau}$ for each
$\tau\in\Delta_{u}$. However, in Proposition3.11, we showed that the
definition of $S^{\tau}$ is independent of the choice of $\chi$. Moreover, we
showed that $u^{\tau}_{\chi}$ is the same up to the equivalence class. Now, we
define a new notation for the previously established notation, which takes
into account the equivalent classes. In practice, the shifts by multiplication
of units are irrelevant when considering the linear system.

######  Definition 3.12.

Let $Z$ be a mock toric variety, $S$ be a nonempty finite set, and $[u]$ be an
equivalent class of $\mathscr{U}_{Z,S}$.

  * •

We call the tuple $(Z,S,[u])$ an abstract mock polytope of $Z$.

  * •

Let $\Delta_{[u]}$ denote $\Delta_{u}$. From Proposition 3.7(e), it is well-
defined. Let $Z_{[u]}$ denote a mock toric variety $Z_{u}$.

  * •

Let $d_{[u]}$ denote $d_{u}$. From Proposition 3.2, it is well-defined.

  * •

We use the notation in Definition 3.10. Let $\tau\in\Delta_{u}$ be a cone. We
call $(Z^{\tau}_{u},S^{\tau},[u^{\tau}_{\chi}])$ an abstract mock toric
polytope of $Z^{\tau}_{u}$ induced by $(Z,S,[u])$ associated with $\tau$. The
definition is independent of the choice of $u\in[u]$ and $\chi\in
k[Z^{\circ}]^{*}$ from Proposition 3.11(b), (c), and (d). Let $[u^{\tau}]$
denote $[u^{\tau}_{\chi}]$.

  * •

For an abstract mock polytope $(Z,S,[u])$ of $Z$, we call the following set
the decomposition of $(Z,S,[u])$:

$\\{(Z^{\tau}_{u},S^{\tau},[u^{\tau}])\\}_{\tau\in\Delta_{[u]}}$

In the following lemma, we can compute the intersection of a hypersurface and
a stratum of $Z_{u}$ for a general section in the linear system defined by
$[u]$.

######  Lemma 3.13.

We keep the notation in Definition 3.12. Let $(Z,S,[u])$ be an abstract mock
polytope of $Z$, $(a_{i})_{i\in S}\in k^{|S|}$ be a vector,
$\tau\in\Delta_{[u]}$ be a cone, $f$ denote $\sum_{i\in S}a_{i}u(i)\in
k[Z^{\circ}]$, and $f^{\tau}$ denote $\sum_{i\in S^{\tau}}a_{i}u^{\tau}(i)\in
k[(Z^{\tau}_{u})^{\circ}]$. We assume that
$\operatorname{val}_{f}=\operatorname{val}_{u}$. Then
$H_{Z_{u},f}\cap(Z_{u})_{\tau}=H^{\circ}_{Z^{\tau}_{u},f^{\tau}}$.

###### Proof.

From the assumption, $\Delta_{[u]}=\Delta_{f}$. In particular, $f$ is fine for
$\Delta_{[u]}$ from [14, Corollary 6.9]. There exists $\chi\in
k[Z^{\circ}]^{*}$ such that
$\operatorname{val}^{\tau}_{\chi}=\operatorname{val}^{\tau}_{u}$ from
Proposition 3.7(d), in particular,
$\operatorname{val}^{\tau}_{\chi^{-1}f}\equiv 0$. We can assume that
$u^{\tau}(i)=p^{\tau}(\chi^{-1}u(i))$ for any $i\in S^{\tau}$ from Proposition
3.11(b). From [14, Proposition 4.8],
$\chi^{-1}f\in\Gamma(Z_{u}(\tau),\mathscr{O}_{Z_{u}})$. Let $g$ denote
$p^{\tau}(\chi^{-1}f)$. Then from [14, Proposition 6.11(a)],
$H_{Z_{u},f}\cap(Z_{u})_{\tau}=H^{\circ}_{Z^{\tau}_{u},g}$. Thus, it is enough
to show that for any $i\notin S^{\tau}$, we have $p^{\tau}(\chi^{-1}u(i))=0$.
Let $i\in S$ be an element such that $i\notin S^{\tau}$. Then from Lemma 3.8
(b), for any $v_{0}\in\tau^{\circ}$, we have
$\operatorname{val}_{u}(v_{0})<\operatorname{val}_{u(i)}(v_{0})$. In
particular, $\operatorname{val}_{Z}(v_{0})(\chi^{-1}u(i))>0$ for any
$v_{0}\in\tau^{\circ}\cap N$. From [14, Proposition 4.7(b)], such
$\operatorname{val}_{Z}(v_{0})$ is a valuation of a valuation ring over
$Z_{[u]}$ which dominates the local ring at the generic point of
$(Z_{u})_{\tau}$. This shows that the ideal of
$\Gamma(Z_{u}(\tau),\mathscr{O}_{Z_{u}})$ associated with the closed subscheme
$(Z_{u})_{\tau}$ of $Z_{u}(\tau)$ contains $\chi^{-1}u(i)$. In particular,
$p^{\tau}(\chi^{-1}u(i))=0$. ∎

As mentioned at the beginning of this section, the following proposition
indicates that a general section in the linear system defined by $[u]$ is non-
degenerate for $\Delta_{u}$.

######  Proposition 3.14.

Let $(Z,S,[u])$ be an abstract mock polytope of $Z$, $(a_{i})_{i\in S}\in
k^{|S|}$ be a vector, and $f$ denote $\sum_{i\in S}a_{i}u(i)\in k[Z^{\circ}]$.
Then the following statements follow:

  1. (a)

If $d_{[u]}=1$, then $H^{\circ}_{Z,f}=\emptyset$ for general $(a_{i})_{i\in
S}\in k^{|S|}$.

  2. (b)

We assume that $k$ is an uncountable field and $d_{[u]}\geq 2$. Then for
general $(a_{i})_{i\in S}\in k^{|S|}$, we have $H^{\circ}_{Z,f}\neq\emptyset$.

  3. (c)

Let $\tau\in\Delta_{[u]}$ be a cone. We assume that $k$ is an uncountable
field and $d_{[u^{\tau}]}\geq 2$. Then for general $(a_{i})_{i\in S}\in
k^{|S|}$, we have $\Delta_{f}=\Delta_{[u]}$ and $H_{Z_{[u]},f}\cap
Z_{[u],\tau}\neq\emptyset$.

  4. (d)

Let $\Delta^{\geq 2}_{[u]}$ denote the following subset of $\Delta_{[u]}$:

$\Delta^{\geq 2}_{[u]}=\\{\tau\in\Delta_{[u]}\mid d_{[u^{\tau}]}\geq 2\\}$

Then $\Delta^{\geq 2}_{[u]}$ is a sub fan of $\Delta_{[u]}$.

  5. (e)

For general $(a_{i})_{i\in S}\in k^{|S|}$, $H^{\circ}_{Z,f}$ is smooth over
$k$.

  6. (f)

For general $(a_{i})_{i\in S}\in k^{|S|}$, $f$ is non-degenerate for
$\Delta_{[u]}$.

###### Proof.

We prove the statements from (a) to (f) in order.

  1. (a)

From the assumption, there exists $\chi\in k[Z^{\circ}]^{*}$ and
$(c_{i})_{i\in S}\in(k^{*})^{|S|}$ such that $\sum_{i\in
S}a_{i}u(i)=(\sum_{i\in S}a_{i}c_{i})\chi$. Because $\chi$ is a unit of
$k[Z^{\circ}]$, for general $(a_{i})_{i\in S}\in k^{|S|}$, we have
$H^{\circ}_{Z_{[u]},f}=\emptyset$.

  2. (b)

Let $h$ denote a linear morphism from $k^{|S|}$ to $k[Z^{\circ}]^{*}$ defined
as $h((a_{i})_{i\in S})=\sum_{i\in S}a_{i}u(i)$ for $(a_{i})_{i\in S}\in
k^{|S|}$. We remark that for any $\chi\in k[Z^{\circ}]^{*}$,
$h^{-1}(k\cdot\chi)$ is a strict linear subspace of $k^{|S|}$ from the
assumption of $d_{[u]}$. Let $H$ be a closed subscheme of
$\mathbb{A}^{|S|}_{k}\times Z^{\circ}_{u}$ defined by $\sum_{i\in
S}x_{i}u(i)\in k[Z^{\circ}][x_{i}]$, where $\\{x_{i}\\}_{i\in S}$ are the
coordinate functions of $\mathbb{A}^{|S|}_{k}$. Let $\theta\colon
H\rightarrow\mathbb{A}^{|S|}_{k}$ denote the first projection. From
Chevalley’s theorem, $\theta(H)$ is a constractible set of
$\mathbb{A}^{|S|}_{k}$. We show that $\theta$ is a dominant map by a
contradiction. We assume that $\theta$ is not dominant. Then there exists a
non-empty open subset $V$ of $\mathbb{A}^{|S|}_{k}$ such that
$V\cap\theta(H)=\emptyset$. This shows that for any $(a_{i})_{i\in S}\in
V(k)$, we have $\sum_{i\in S}a_{i}u(i)\in k[Z^{\circ}]^{*}$. Thus, from the
remark, $V(k)$ is covered by strict linear subspaces
$\\{h^{-1}(k\cdot\chi)\\}_{\chi\in k[Z^{\circ}]^{*}}$ of $k^{|S|}$. On the
other hand, a group $k[Z^{\circ}]^{*}/k^{*}$ is a free abelian group of finite
rank, so $|\\{h^{-1}(k\cdot\chi)\\}_{\chi\in k[Z^{\circ}]^{*}}|$ is countable.
However, $V(k)$ can not be covered by countable strict closed subsets of
$\mathbb{A}^{|S|}_{k}$ because $k$ is an uncountable field. It is a
contradiction.

  3. (c)

Let $f^{\tau}$ denote $\sum_{i\in S^{\tau}}a_{i}u^{\tau}(i)\in
k[(Z^{\tau}_{u})^{\circ}]$. From Proposition 3.9 and Lemma3.13,
$H_{Z_{u},f}\cap(Z_{u})_{\tau}=H^{\circ}_{Z^{\tau}_{u},f^{\tau}}$ for general
$(a_{i})_{i\in S}\in k^{|S|}$. Let $g$ denote $\sum_{i\in
S^{\tau}}b_{i}u^{\tau}(i)\in k[(Z^{\tau}_{u})^{\circ}]$ for $(b_{i})_{i\in
S^{\tau}}\in k^{|S^{\tau}|}$. Thus, from (b) and the assumption of
$d_{[u^{\tau}]}$, we have $H^{\circ}_{Z^{\tau}_{u},g}\neq\emptyset$ for
general $(b_{i})_{i\in S^{\tau}}\in k^{|S^{\tau}|}$. Therefore,
$H_{Z_{u},f}\cap(Z_{u})_{\tau}\neq\emptyset$ for general $(a_{i})_{i\in S}\in
k^{|S|}$.

  4. (d)

Because $|\Delta_{u}|$ is finite, for general $(a_{i})_{i\in S}\in k^{|S|}$,
$f$ is fine for $\Delta_{u}$ and $H_{Z_{u},f}\cap(Z_{u})_{\tau}\neq\emptyset$
for all $\tau\in\Delta_{u}$ such that $d_{[u^{\tau}]}\geq 2$ from Proposition
3.9 and (c). We remark that for any $\tau\in\Delta_{u}$ such that
$d_{[u^{\tau}]}=1$, we have $H_{Z_{u},f}\cap(Z_{u})_{\tau}=\emptyset$ from the
fact that $f$ is fine for $\Delta_{u}$, the argument in the proof of (a), and
Lemma3.13. Thus, $\Delta^{\geq 2}_{[u]}$ is equal to ${}_{f}(\Delta_{[u]})$
for general $(a_{i})_{i\in S}\in k^{|S|}$. Therefore, from [14, Proposition
6.11(d)], $\Delta^{\geq 2}_{[u]}$ is a sub fan of $\Delta_{[u]}$.

  5. (e)

For any $i\in S$, $u(i)$ is a unit in $k[Z^{\circ}]$. In particular,
$\\{u(i)\\}_{i\in S}\subset\Gamma(Z^{\circ},\mathscr{O}_{Z})$ has no base
locus in $Z^{\circ}$. Thus, from Bertini’s theorem, $H^{\circ}_{Z,f}$ is
smooth over $k$ for general $(a_{i})_{i\in S}\in k^{|S|}$.

  6. (f)

Let $f^{\tau}$ denote $\sum_{i\in S^{\tau}}a_{i}u^{\tau}(i)\in
k[(Z^{\tau}_{u})^{\circ}]$. From Proposition 3.9 and Lemma3.13, for general
$(a_{i})_{i\in S}\in k^{|S|}$, we have
$H_{Z_{u},f}\cap(Z_{u})_{\tau}=H^{\circ}_{Z^{\tau}_{u},f^{\tau}}$. Let $g$
denote $\sum_{i\in S^{\tau}}b_{i}u^{\tau}(i)\in k[(Z^{\tau}_{u})^{\circ}]^{*}$
for $(b_{i})_{i\in S^{\tau}}\in k^{|S^{\tau}|}$. Thus, from (e), for general
$(b_{i})_{i\in S^{\tau}}\in k^{|S^{\tau}|}$, $H^{\circ}_{Z^{\tau}_{u},g}$ is
smooth over $k$. Therefore, for general $(a_{i})_{i\in S}\in k^{|S|}$,
$H_{Z_{u},f}\cap(Z_{u})_{\tau}$ is smooth over $k$. Therefore, because
$|\Delta_{u}|$ is finite, for general $(a_{i})_{i\in S}\in k^{|S|}$, $f$ is
non-degenerate for $\Delta_{[u]}$.

∎

Until now, we have been considering mock toric varieties over
$\mathbb{A}^{1}_{k}$ and their hypersurfaces, and the same holds in this case.
In particular, we show that for a general section, its hypersurface dominates
$\mathbb{A}^{1}_{k}$ in Proposition3.17.

######  Definition 3.15.

We use the notation in [14, Proposition 4.14], that in [14, Section 7], and
the following ones:

  * •

Let $(Z_{\pi},S,u)$ be a mock polytope of $Z_{\pi}$.

  * •

Let $\kappa\colon S\rightarrow\mathbb{Z}$ be a map.

  * •

Let $u\oplus\kappa\in\mathscr{U}_{Z^{1}_{\pi^{1}},S}$ denote the map defined
as $u\oplus\kappa(i)=u(i)t^{\kappa(i)}$ for $i\in S$. We recall that
$u\oplus\kappa(i)\in k[Z^{1}_{\pi^{1}}]^{*}$ for any $i\in S$. We call a mock
polytope $(Z^{1}_{\pi^{1}},S,u\oplus\kappa)$ of $Z^{1}_{\pi^{1}}$ a model of
mock polytope of $(Z^{\pi},S,u)$ along $\kappa$.

######  Proposition 3.16.

We keep the notation in Definition 3.15. Let
$u^{\prime}\in\mathscr{U}_{Z_{\pi},S}$ be a map. If $u\sim u^{\prime}$, then
$u\oplus\kappa\sim u^{\prime}\oplus\kappa$.

###### Proof.

Let $\chi\in k[Z_{\pi}]^{*}$ be a unit such that $\chi u(i)=u^{\prime}(i)$ for
any $i\in S$. We remark that $\chi\in k[Z^{1}_{\pi^{1}}]^{*}$ too. Thus, $\chi
u(i)t^{\kappa(i)}=u^{\prime}(i)t^{\kappa(i)}$ in $k[Z^{1}_{\pi^{1}}]$ for any
$i\in S$. This equation shows that $u\oplus\kappa\sim u^{\prime}\oplus\kappa$.
∎

In Definition 3.15, we can define the equivalent class $[u\oplus\kappa]$ of
$\mathscr{U}_{Z^{1}_{\pi^{1}},S}$ from an abstract mock polytope
$(Z_{\pi},S,[u])$ by Proposition 3.16. We call that
$(Z^{1}_{\pi^{1}},S,[u\oplus\kappa])$ an abstract model mock polytope of
$(Z_{\pi},S,[u])$ along $\kappa$.

######  Proposition 3.17.

We keep the notation in Definition 3.15. Let
$(Z^{1}_{\pi^{1}},S,[u\oplus\kappa])$ be an abstract model mock polytope of
$(Z_{\pi},S,[u])$ along $\kappa$. Let $(a_{i})_{i\in S}\in k^{|S|}$ be a
vector and $f$ denote $\sum_{i\in S}a_{i}(u\oplus\kappa)(i)\in
k[Z^{1}_{\pi^{1}}]$. If $d_{[u]}\geq 2$, then every irreducible component of
$H^{\circ}_{Z^{1}_{\pi^{1}},f}$ dominates ${\mathbb{G}_{m}^{1}}_{,k}$ for
general $(a_{i})_{i\in S}\in k^{|S|}$.

###### Proof.

Let $(a_{i})_{i\in S}\in k^{|S|}$ be a vector. We assume that one irreducible
component of $H^{\circ}_{Z^{1}_{\pi^{1}},f}$ does not dominate
${\mathbb{G}_{m}^{1}}_{,k}$. Because $k[Z^{1}_{\pi^{1}}]\cong
k[t,t^{-1}][Z_{\pi}]$, there exists a non-unit polynomial $g\in k[t,t^{1}]$
such that $f$ can be divide by $g$. Because $k$ is algebraically closed, there
exists $\alpha\in k^{*}$ such that $t-\alpha|g$. In particular, $\sum_{i\in
S}\alpha^{\kappa(i)}a_{i}u(i)=0\in k[Z_{\pi}]$.

Let
$\rho_{\kappa}\colon{\mathbb{G}_{m}^{1}}_{,k}\times\mathbb{A}^{|S|}_{k}\rightarrow\mathbb{A}^{|S|}_{k}$
be an algebraic action of ${\mathbb{G}_{m}^{1}}_{,k}$ for
$\mathbb{A}^{|S|}_{k}$ defined as
$\rho_{\kappa}(t_{0})(b_{i})=(t_{0}^{-\kappa(i)}b_{i})_{i\in S}$ for $t_{0}\in
k^{*}$ and $(b_{i})_{i\in S}\in k^{|S|}$. Let $h$ denote a linear morphism
from $k^{|S|}$ to $k[Z^{1}_{\pi^{1}}]^{*}$ defined as $h((c_{i})_{i\in
S})=\sum_{i\in S}c_{i}u(i)$ for $(c_{i})_{i\in S}\in k^{|S|}$. From the
property of $g$, we have $(\alpha^{\kappa(i)}a_{i})_{i\in S}\in\ker(h)$. We
can identify with $k^{|S|}$ and $\mathbb{A}^{|S|}_{k}(k)$. Let $X$ denote a
linear subspace of $\mathbb{A}^{|S|}_{k}$ associated with $\ker(h)\subset
k^{|S|}$. On this identification, we can also check that $(a_{i})_{i\in
S}\in\rho_{\kappa}({\mathbb{G}_{m}^{1}}_{,k}(k))(X(k))$.

Because $d_{u}\geq 2$, we have $\dim(X)\leq|S|-2$. Thus, the restriction
$\rho_{\kappa}|_{{\mathbb{G}_{m}^{1}}_{,k}\times
X}\colon{\mathbb{G}_{m}^{1}}_{,k}\times X\rightarrow\mathbb{A}^{|S|}_{k}$ is
not dominant. Therefore, for general $(a_{i})_{i\in S}\in k^{|S|}$, every
irreducible component of $H^{\circ}_{Z^{1}_{\pi^{1}},f}$ dominates
${\mathbb{G}_{m}^{1}}_{,k}$ from Chevalley’s theorem. ∎

We use the notation in [14, Definition 4.11]. At the end of this article, we
compute the stable birational volume of hypersurfaces in
$\mathrm{Gr}_{\mathbb{C}}(2,n)$. In fact, we haven’t found a mock toric
structure of $\mathrm{Gr}_{\mathbb{C}}(2,n)$ yet. Alternatively, it is
necessary to construct a mock toric variety of dimension $n-3$. Unfortunately,
we were unable to find directly a proper mock variety of dimension
$2(n-2)=\dim(\mathrm{Gr}_{\mathbb{C}}(2,n))$. Instead, we consider a
$(n-1)$-dimensional algebraic torus fibration over a $(n-3)$-dimensional
proper mock toric variety. Next, by compactifying this $2(n-2)$-fold, we
obtain a proper mock toric variety of dimension $2(n-2)$. The two crucial
components for the compactification are $\operatorname{val}_{u,\pi}$ and
$\Delta_{u,\pi}$, which are constructed from
$\\{\operatorname{val}_{u(i),\pi}\\}_{i\in S}$ defined in [14, Proposition
6.16].

######  Proposition 3.18.

We keep the notation in [14, Definition 4.11]. Let $(Z_{\pi},S,u)$ be a mock
polytope of $Z_{\pi}$. We use the following notation:

  * •

Let $\operatorname{val}_{u,\pi}$ denote the function from
$\pi^{-1}_{\mathbb{R}}(\operatorname{Supp}(\Delta))$ to $\mathbb{R}$ defined
as follows for $w\in\pi^{-1}_{\mathbb{R}}(\operatorname{Supp}(\Delta))$:

$\operatorname{val}_{u,\pi}(w)=\min_{i\in
S}\\{\operatorname{val}_{u(i),\pi}(w)\\}$

We remark that $\operatorname{val}_{u,\pi}$ is continuous and is preserved by
the scalar product of non-negative real numbers. Indeed, from [14, Proposition
6.16], $\operatorname{val}_{u(i),\pi}$ is continuous and is preserved by the
scalar product of non-negative real numbers for any $i\in S$. Thus,
$\operatorname{val}_{u,\pi}$ is so from the definition of
$\operatorname{val}_{u,\pi}$.

  * •

Let $\sigma\in\Delta$ be a cone, $\sigma^{\pi}$ denote
$\pi^{-1}_{\mathbb{R}}(\sigma)$, and $\Omega^{\sigma^{\pi}}_{u,\pi}$ denote
the following set:

$\Omega^{\sigma^{\pi}}_{u,\pi}=\\{(\chi,b)\in
k[Z_{\pi}]^{*}\times\mathbb{Z}_{>0}\mid
val_{u,\pi}(v^{\prime})\leq\frac{1}{b}\operatorname{val}_{\chi,\pi}(v^{\prime})\quad(\forall
v^{\prime}\in\sigma^{\pi})\\}$

  * •

For $(\chi,b)\in\Omega^{\sigma^{\pi}}_{u}$, let
$C^{\sigma^{\pi}}_{u,\pi}(\chi,b)$ denote the following set:

$C^{\sigma^{\pi}}_{u,\pi}(\chi,b)=\\{v^{\prime}\in\sigma^{\pi}\mid
val_{u,\pi}(v^{\prime})=\frac{1}{b}\operatorname{val}_{\chi,\pi}(v^{\prime})\\}$

Then the following statements follow:

  1. (a)

Let $\varphi\in\Phi$ be an element such that $\sigma\in\Delta_{\varphi}$,
$\sigma_{\varphi}$ denote $(q_{\varphi})_{\mathbb{R}}(\sigma)$, and
$\sigma^{\pi}_{\varphi}$ denotes a rational polyhedral convex cone
$(q^{\prime}_{\varphi})_{\mathbb{R}}(\sigma^{\pi})$ in
$(N^{\prime}/s(N_{\varphi}))_{\mathbb{R}}$ . Then

     * (I)

$\sigma^{\pi}_{\varphi}=(\pi_{\varphi})^{-1}_{\mathbb{R}}(\sigma_{\varphi})$

     * (II)

$(q^{\prime}_{\varphi})_{\mathbb{R}}|_{\sigma^{\pi}}\colon\sigma^{\pi}\rightarrow\sigma^{\pi}_{\varphi}$
is bijective.

     * (III)

$q^{\prime}_{\varphi}|_{\langle\sigma^{\pi}\rangle\cap
N^{\prime}}\colon\langle\sigma^{\pi}\rangle\cap
N^{\prime}\rightarrow\langle\sigma^{\pi}_{\varphi}\rangle\cap
N^{\prime}/s(N_{\varphi})$ is isomorphic.

  2. (b)

Let $\sigma\in\Delta$ be a cone and $\varphi\in\Phi$ be an element such that
$\sigma\in\Delta_{\varphi}$. Let $\sigma^{\pi}_{\varphi}$ denotes
$(q^{\prime}_{\varphi})_{\mathbb{R}}(\sigma^{\pi})$. Then there exists
$g_{\varphi}\in k[M^{\prime}_{\varphi}]$ such that
$\operatorname{val}^{\sigma^{\pi}}_{u,\pi}=\operatorname{val}^{\sigma^{\pi}_{\varphi}}_{g_{\varphi}}\circ(q^{\prime}_{\varphi})_{\mathbb{R}}|_{\sigma^{\pi}}$.

  3. (c)

Let $g_{\varphi}\in k[M^{\prime}_{\varphi}]$ be a function that satisfies the
condition in (b). Then for any $(\chi,b)\in\Omega^{\sigma^{\pi}}_{u,\pi}$,
there exists $\omega\in\Omega^{\sigma^{\pi}_{\varphi}}_{g_{\varphi}}$ such
that the following equation holds:

$(q^{\prime}_{\varphi})_{\mathbb{R}}(C^{\sigma^{\pi}}_{u,\pi}(\chi,b))=C^{\sigma^{\pi}_{\varphi}}_{g_{\varphi}}(\omega)$

Conversely, for any $\omega\in\Omega^{\sigma^{\pi}_{\varphi}}_{g_{\varphi}}$,
there exists $(\chi,b)\in\Omega^{\sigma^{\pi}}_{u,\pi}$ such that the
following equation holds:

$C^{\sigma^{\pi}_{\varphi}}_{g_{\varphi}}(\omega)=(q^{\prime}_{\varphi})_{\mathbb{R}}(C^{\sigma^{\pi}}_{u,\pi}(\chi,b))$

  4. (d)

The set
$\\{C^{\sigma}_{u,\pi}(\chi,b)\\}_{(\chi,b)\in\Omega^{\sigma^{\pi}}_{u,\pi}}$
is a rational polyhedral convex fan and a refinement of $\sigma^{\pi}$.

  5. (e)

Let $\tau$ be a face of $\sigma$ and $\tau^{\pi}$ denote
$\pi^{-1}_{\mathbb{R}}(\tau)$. Then
$\Omega^{\sigma^{\pi}}_{u,\pi}\subset\Omega^{\tau^{\pi}}_{u,\pi}$. Moreover,
for any $(\chi,b)\in\Omega^{\sigma^{\pi}}_{u,\pi}$, we have
$C^{\sigma^{\pi}}_{u,\pi}(\chi,b)\cap\tau^{\pi}=C^{\tau^{\pi}}_{u,\pi}(\chi,b)$.
In particular, $C^{\tau^{\pi}}_{u,\pi}(\chi,b)\preceq
C^{\sigma^{\pi}}_{u,\pi}(\chi,b)$.

  6. (f)

The following set $\Delta_{u,\pi}$ is a rational polyhedral convex fan in
$N_{\mathbb{R}}$ and
$\operatorname{Supp}(\Delta_{u,\pi})=\pi^{-1}_{\mathbb{R}}(\operatorname{Supp}(\Delta))$:

$\Delta_{u,\pi}=\bigcup_{\sigma\in\Delta}\\{C^{\sigma^{\pi}}_{u,\pi}(\chi,b)\\}_{(\chi,b)\in\Omega^{\sigma^{\pi}}_{u,\pi}}$

  7. (g)

For any $(\chi,b)\in\Omega^{\sigma^{\pi}}_{u,\pi}$, there exists
$\chi^{\prime}\in k[Z_{\pi}]^{*}$ such that
$\operatorname{val}_{u,\pi}(v^{\prime})=\operatorname{val}_{\chi^{\prime},\pi}(v^{\prime})$
for any $v^{\prime}\in C^{\sigma^{\pi}}_{u,\pi}(\chi,b)$. In particular,
$\operatorname{val}_{u,\pi}$ is a linear function on
$C^{\sigma^{\pi}}_{u,\pi}$ from [14, Proposition 6.17].

  8. (h)

We use the notation in (f). We assume that $\Delta_{u,\pi}$ is a strongly
convex rational polyhedral fan in $N^{\prime}_{\mathbb{R}}$. Let
$\Delta^{\prime}$ denote $\Delta_{u,\pi}$ and $W$ denote a mock toric variety
induced by $Z$ along $\pi_{*}\colon X(\Delta^{\prime})\rightarrow X(\Delta)$
and $s$. Then for the mock polytope $(W,S,u)$, we have
$\Delta^{\prime}_{u}=\Delta^{\prime}$.

###### Proof.

We prove the statements from (a) to (h) in order.

  1. (a)

We prove the statements from (I) to (III) in order.

     * (I)

First, we prove
$\sigma^{\pi}_{\varphi}\subset(\pi_{\varphi})^{-1}_{\mathbb{R}}(\sigma_{\varphi})$.
Let $v^{\prime}\in\sigma^{\pi}$ be an element. Then
$(\pi_{\varphi})_{\mathbb{R}}\circ(q^{\prime}_{\varphi})_{\mathbb{R}}(v^{\prime})=(q_{\varphi})_{\mathbb{R}}\circ(\pi_{\varphi})_{\mathbb{R}}(v^{\prime})\in\sigma_{\varphi}$.
Thus,
$\sigma^{\pi}_{\varphi}=(q^{\prime}_{\varphi})_{\mathbb{R}}(\sigma^{\pi})\subset(\pi_{\varphi})^{-1}_{\mathbb{R}}(\sigma_{\varphi})$.
Next, we prove
$\sigma^{\pi}_{\varphi}\supset(\pi_{\varphi})^{-1}_{\mathbb{R}}(\sigma_{\varphi})$.
Let $v^{\prime\prime}\in(\pi_{\varphi})^{-1}_{\mathbb{R}}(\sigma_{\varphi})$
be an element. Then
$(\pi_{\varphi})_{\mathbb{R}}(v^{\prime\prime})\in(q_{\varphi})_{\mathbb{R}}(\sigma)$
and there exists $v\in\sigma$ such that
$(q_{\varphi})_{\mathbb{R}}(v)=(\pi_{\varphi})_{\mathbb{R}}(v^{\prime\prime})$.
Let $w\in N^{\prime}_{\mathbb{R}}$ denote $s_{\mathbb{R}}(v)$. Then
$(\pi_{\varphi})_{\mathbb{R}}(v^{\prime\prime}-(q^{\prime}_{\varphi})_{\mathbb{R}}(w))=(q_{\varphi})_{\mathbb{R}}(v-\pi_{\mathbb{R}}(w))=0$
and
$v^{\prime\prime}-(q^{\prime}_{\varphi})_{\mathbb{R}}\in\ker((\pi_{\varphi})_{\mathbb{R}})$.
We remark that $\ker(q_{\varphi}\circ\pi)=\ker(\pi)\oplus s(N_{\varphi})$, in
particular, $\ker(\pi_{\varphi})=q^{\prime}_{\varphi}(\ker(\pi))$. Similarly,
$\ker((\pi_{\varphi})_{\mathbb{R}})=(q^{\prime}_{\varphi})_{\mathbb{R}}(\ker(\pi_{\mathbb{R}}))$.
Hence, there exists $u\in\ker(\pi_{\mathbb{R}})$ such that
$v^{\prime\prime}-(q^{\prime}_{\varphi})_{\mathbb{R}}(w)=(q^{\prime}_{\varphi})_{\mathbb{R}}(u)$,
and hence $v^{\prime\prime}=(q^{\prime}_{\varphi})_{\mathbb{R}}(u+w)$.
Moreover, $\pi_{\mathbb{R}}(u+w)=\pi_{\mathbb{R}}(w)=v\in\sigma$, and hence
$u+w\in\sigma^{\pi}$. Therefore, $v^{\prime\prime}\in\sigma^{\pi}_{\varphi}$.

     * (II)

It is enough to show that
$(q^{\prime}_{\varphi})_{\mathbb{R}}|_{\sigma^{\pi}}$ is injective. Let
$v^{\prime}_{1},v^{\prime}_{2}\in\sigma^{\pi}$ such that
$(q^{\prime}_{\varphi})_{\mathbb{R}}(v^{\prime}_{1})=(q^{\prime}_{\varphi})_{\mathbb{R}}(v^{\prime}_{2})$.
Then $v^{\prime}_{1}-v^{\prime}_{2}\in
s_{\mathbb{R}}((N_{\varphi})_{\mathbb{R}})$, and hence there exists
$v\in(N_{\varphi})_{\mathbb{R}}$ such that
$v^{\prime}_{1}-v^{\prime}_{2}=s_{\mathbb{R}}(v)$ and
$v\in\langle\sigma\rangle\cap(N_{\varphi})_{\mathbb{R}}$. Because $\sigma$ is
a rational polyhedral convex cone,
$\langle\sigma\rangle\cap(N_{\varphi})_{\mathbb{R}}=0$ from [14, Proposition
3.2(b)]. Thus, $v^{\prime}_{1}=v^{\prime}_{2}$.

     * (III)

From the definition of $\sigma^{\pi}_{\varphi}$,
$(q^{\prime}_{\varphi})_{\mathbb{R}}(\langle\sigma^{\pi}\rangle)=\langle\sigma^{\pi}_{\varphi}\rangle$.
Thus, $(q^{\prime}_{\varphi})(\langle\sigma^{\pi}\rangle\cap
N^{\prime})\subset\langle\sigma^{\pi}_{\varphi}\rangle\cap
N^{\prime}/s(N_{\varphi})$. Moreover, from the proof of (II),
$q^{\prime}_{\varphi}|_{\langle\sigma^{\pi}\rangle\cap N^{\prime}}$ is
injective.

Hence, we show that $q^{\prime}_{\varphi}|_{\langle\sigma^{\pi}\rangle\cap
N^{\prime}}$ is surjective. Let
$v^{\prime\prime}\in\langle\sigma^{\pi}_{\varphi}\rangle\cap
N^{\prime}/s(N_{\varphi})$ be an element. From (I),
$(\pi_{\varphi})_{\mathbb{R}}(\langle\sigma^{\pi}_{\varphi}\rangle)\subset\langle\sigma_{\varphi}\rangle$,
and hence
$\pi_{\varphi}(v^{\prime\prime})\in\langle\sigma_{\varphi}\rangle\cap
N/N_{\varphi}$. From [14, Proposition 3.2(d)], there exist
$v\in\langle\sigma\rangle\cap N$ such that
$\pi_{\varphi}(v^{\prime\prime})=q_{\varphi}(v)$. Because
$\pi_{\varphi}(v^{\prime\prime}-q^{\prime}_{\varphi}(s(v)))=\pi_{\varphi}(v^{\prime\prime})-q_{\varphi}(\pi(s(v)))=\pi_{\varphi}(v^{\prime\prime})-q_{\varphi}(v)=0$,
$v^{\prime\prime}-q^{\prime}_{\varphi}(s(v))\in\ker(\pi_{\varphi})$. From the
remark in the proof of (I), there exists $v^{\prime}\in\ker(\pi)$ such that
$q^{\prime}_{\varphi}(v^{\prime})=v^{\prime\prime}-q^{\prime}_{\varphi}(s(v))$.
Because $v\in\langle\sigma\rangle\cap N$ and $v^{\prime}\in\ker(\pi)$,
$s(v)+v^{\prime}\in\langle\sigma^{\pi}\rangle$. Therefore,
$v^{\prime\prime}=q^{\prime}_{\varphi}(s(v)+v^{\prime})\in
q^{\prime}_{\varphi}(\langle\sigma^{\pi}\rangle\cap N^{\prime})$.

  2. (b)

From [14, Proposition 6.17], for each $i\in S$, there exists $\omega_{i}\in
M^{\prime}_{\varphi}$ such that
$\operatorname{val}^{\sigma^{\pi}}_{u(i)}=\operatorname{val}^{\sigma^{\pi}}_{\iota^{*}_{\varphi}(\chi^{\omega_{i}})}$.
Let $T$ denote a subset $\\{\omega_{i}\mid i\in S\\}$ of
$M^{\prime}_{\varphi}$ and $g_{\varphi}$ denote $\sum_{\omega\in
T}\chi^{\omega}\in k[M^{\prime}_{\varphi}]$. Then for any
$v^{\prime}\in\sigma^{\pi}\cap N^{\prime}$, we have
$q^{\prime}_{\varphi}(v)(g_{\varphi})=\min_{\omega\in
T}\\{q^{\prime}_{\varphi}(v)(\chi^{\omega})\\}$ because the definition of the
torus invariant valuation on toric varieties. This shows that for any
$v^{\prime}\in\sigma^{\pi}\cap N^{\prime}$, there exists the following
equation:

$\displaystyle\operatorname{val}_{g_{\varphi}}\circ(q^{\prime}_{\varphi})_{\mathbb{R}}(v^{\prime})$
$\displaystyle=q^{\prime}_{\varphi}(v^{\prime})(g_{\varphi})$
$\displaystyle=\min_{\omega\in
T}\\{q^{\prime}_{\varphi}(v^{\prime})(\chi^{\omega})\\}$
$\displaystyle=\min_{i\in
S}\\{q^{\prime}_{\varphi}(v^{\prime})(\chi^{\omega_{i}})\\}$
$\displaystyle=\min_{i\in
S}\\{\operatorname{val}_{Z,\pi}(v^{\prime})({\iota^{\prime}}^{*}_{\varphi}(\chi^{\omega_{i}}))\\}$
$\displaystyle=\min_{i\in
S}\\{\operatorname{val}_{{\iota^{\prime}}^{*}_{\varphi}(\chi^{\omega_{i}}),\pi}(v^{\prime})\\}$
$\displaystyle=\min_{i\in S}\\{\operatorname{val}_{u(i),\pi}(v^{\prime})\\}$
$\displaystyle=\operatorname{val}_{u,\pi}(v^{\prime})$

Thus, from [14, Lemma 8.13], the statement holds.

  3. (c)

We can check the statement likewise [14, Proposition 6.8(b)]. In the proof of
[14, Proposition 6.8(b)], we used [14, Proposition 6.3]. Instead, we use [14,
Proposition 6.17] and (a)-(III).

  4. (d)

We can check the statement likewise [14, Proposition 6.8(c)].

  5. (e)

Let $\omega\in\sigma^{\vee}\cap M$ be an element such that
$\tau=\sigma\cap\omega^{\perp}$. We remark that
$\pi^{*}(\omega)\in(\sigma^{\pi})^{\vee}\cap M^{\prime}$ and
$\tau^{\pi}=\sigma^{\pi}\cap\pi^{*}(\omega)^{\perp}$. In particular,
$\tau^{\pi}\preceq\sigma^{\pi}$. Now, we can check the statement likewise [14,
Proposition 6.8(d)].

  6. (f)

We can check the statement likewise [14, Proposition 6.8(e)].

  7. (g)

We can check the statement likewise [14, Proposition 6.8(f)].

  8. (h)

We remark that $\pi$ is compatible with the fans $\Delta^{\prime}$ and
$\Delta$ from (d) and (f). Let $\tau\in\Delta^{\prime}$ be a cone. Then there
exists $\sigma\in\Delta$ and $(\chi,b)\in\Omega^{\sigma^{\pi}}_{u,\pi}$ such
that $\tau=C^{\sigma^{\pi}}_{u,\pi}(\chi,b)$. From [14, Proposition 4.12(b)],
$\operatorname{val}_{W}=\operatorname{val}_{Z,\pi}$, in particular,
$\operatorname{val}^{\tau}_{u(i)}=\operatorname{val}^{\tau}_{u(i),\pi}$ for
any $i\in S$ and
$\operatorname{val}^{\tau}_{\chi}=\operatorname{val}^{\tau}_{\chi,\pi}$.
Hence, $\operatorname{val}^{\tau}_{u}=\operatorname{val}^{\tau}_{u,\pi}$ and
$(\chi,b)\in\Omega^{\tau}_{u}$. Moreover, from the definition of
$C^{\tau}_{u}(\chi,b)$, we have $\tau=C^{\tau}_{u}(\chi,b)$. From 3.7(c),
$\\{C^{\tau}_{u}(\chi^{\prime},b^{\prime})\\}_{(\chi^{\prime},b^{\prime})\in\Omega^{\tau}_{u}}$
is a refinment of $\tau$, and hence
$\\{C^{\tau}_{u}(\chi^{\prime},b^{\prime})\\}_{(\chi^{\prime},b^{\prime})\in\Omega^{\tau}_{u}}$
is equal to the set of all faces of $\tau$. This shows that
$\Delta^{\prime}_{u}=\Delta^{\prime}$.

∎

We construct $\Delta_{u,\pi}$ in Proposition 3.18. However, the above
construction method is not practical in the sense that it is not suitable for
computations using a programming code. The following proposition indicates
that the construction of $\Delta_{u,\pi}$ itself can be achieved through
straightforward calculations.

######  Proposition 3.19.

We keep the notation in Proposition 3.18. Let $\sigma\in\Delta$ be a cone.
From [14, Proposition 6.17], there exists $\omega_{i}\in M^{\prime}$ for any
$i\in S$ such that $\operatorname{val}_{u(i),\pi}(v^{\prime})=\langle
v^{\prime},\omega_{i}\rangle$ for any $v^{\prime}\in\sigma^{\pi}$. We fix such
$\omega_{i}\in M^{\prime}$ for any $i\in S$ and let $D^{\sigma}_{u,\pi}$ be a
rational polyhedral convex cone in $(M^{\prime}\oplus\mathbb{Z})_{\mathbb{R}}$
generated by the following set:

$\\{(\omega^{\prime},0)\in(M^{\prime}\oplus\mathbb{Z})_{\mathbb{R}}\mid\omega^{\prime}\in(\sigma^{\pi})^{\vee}\\}\cup\\{(\omega_{i},1)\in(M^{\prime}\oplus\mathbb{Z})_{\mathbb{R}}\mid
i\in S\\}$

Let $C^{\sigma}_{u,\pi}$ denote a dual cone of $D^{\sigma}_{u,\pi}$ in
$(N^{\prime}\oplus\mathbb{Z})_{\mathbb{R}}$. We remark that
$C^{\sigma}_{u,\pi}$ is also a rational polyhedral convex cone in
$(N\oplus\mathbb{Z})_{\mathbb{R}}$.

Then the following statements hold:

  1. (a)

The definition of $D^{\sigma}_{u,\pi}$ is independent of the choice of
$\\{\omega_{i}\\}_{i\in S}$.

  2. (b)

For any $(v^{\prime},a)\in C^{\sigma}_{u,\pi}$, we have that
$v^{\prime}\in\sigma^{\pi}$.

  3. (c)

Let $v^{\prime}\in\sigma^{\pi}$ be an element and $a\in\mathbb{R}$ be a real
number. Then $\operatorname{val}_{u,\pi}(v^{\prime})+a\geq 0$ if and only if
$(v^{\prime},a)\in C^{\sigma}_{u,\pi}$.

  4. (d)

Let $\tau\preceq C^{\sigma}_{u,\pi}$ be a face. We assume that
$(0,1)\notin\tau$. Then for any $(v^{\prime},a)\in\tau$, we have
$a=-\operatorname{val}_{u,\pi}(v^{\prime})$.

  5. (e)

For any $v^{\prime}\in\sigma^{\pi}$, there exists a face $\tau\preceq
C^{\sigma}_{u,\pi}$ such that
$(v^{\prime},-\operatorname{val}_{u,\pi}(v^{\prime}))\in\tau$ and
$(0,1)\notin\tau$.

  6. (f)

Let $\Delta^{\sigma,u,\pi}$ denote the following set:

$\Delta^{\sigma,u,\pi}=\\{\tau\preceq
C^{\sigma}_{u,\pi}\mid(0,1)\notin\tau\\}$

We remark that $\Delta^{\sigma,u,\pi}$ is a rational polyhedral fan in
$(N^{\prime}\oplus\mathbb{Z})_{\mathbb{R}}$. Let $\mathrm{pr}_{1}$ denote a
first projection from $N^{\prime}\oplus\mathbb{Z}$ to $N^{\prime}$. Then the
following subset $\Delta^{\sigma}_{u,\pi}$ is a rational polyhedral fan in
$N^{\prime}_{\mathbb{R}}$ with
$\operatorname{Supp}(\Delta^{\sigma}_{u,\pi})=\sigma^{\pi}$:

$\Delta^{\sigma}_{u,\pi}=\\{(\mathrm{pr}_{1})_{\mathbb{R}}(\tau)\mid\tau\in\Delta^{\sigma,u,\pi}\\}$

  7. (g)

Let $E^{\sigma}_{u,\pi}$ denote a convex cone in
$(N^{\prime}\oplus\mathbb{Z})_{\mathbb{R}}$ generated by the following set:

$\\{(v^{\prime},-\operatorname{val}_{u,\pi}(v^{\prime}))\mid
v^{\prime}\in\sigma^{\pi}\\}\cup\\{(0,1)\\}$

Then $E^{\sigma}_{u,\pi}$ is $C^{\sigma}_{u,\pi}$.

  8. (h)

The cone $C^{\sigma}_{u,\pi}$ is strongly convex if and only if
$\Delta^{\sigma}_{u,\pi}$ is a strongly convex fan in
$N^{\prime}_{\mathbb{R}}$.

  9. (i)

For any $(\chi,b)\in\Omega^{\sigma^{\pi}}_{u,\pi}$, there exists
$\tau\in\Delta^{\sigma,u,\pi}$ such that
$C^{\sigma^{\pi}}_{u,\pi}(\chi,b)=(\mathrm{pr}_{1})_{\mathbb{R}}(\tau)$.

  10. (j)

Conversely, for any $\tau\in\Delta^{\sigma,u,\pi}$, there exists
$(\chi,b)\in\Omega^{\sigma^{\pi}}_{u,\pi}$ such that
$(\mathrm{pr}_{1})_{\mathbb{R}}(\tau)=C^{\sigma^{\pi}}_{u,\pi}(\chi,b)$.
Besides the statement (i), we have that
$\\{C^{\sigma^{\pi}}_{u,\pi}(\chi,b)\\}_{(\chi,b)\in\Omega^{\sigma^{\pi}}_{u,\pi}}=\Delta^{\sigma}_{u,\pi}$.

  11. (k)

Let $\varsigma\in\Delta^{\sigma}_{u,\pi}$ be a cone. Then there exists
$\sigma\in\Delta$ and $\tau\in\Delta^{\sigma,u,\pi}$ such that
$\varsigma=(\mathrm{pr}_{1})_{\mathbb{R}}(\tau)$ from (j). Let
$(v^{\prime},a)\in\tau^{\circ}$ be an element. The following equation holds.

$\\{i\in
S\mid\operatorname{val}^{\varsigma}_{u,\pi}=\operatorname{val}^{\varsigma}_{u(i),\pi}\\}=\\{i\in
S\mid\langle(v^{\prime},a),(\omega_{i},1)\rangle=0\\}$

###### Proof.

We prove the statement from (a) to (k) in order.

  1. (a)

Let $i\in S$ be an integer and $\omega^{\prime}_{i}\in M^{\prime}$ be an
element such that $\operatorname{val}_{u(i),\pi}(v^{\prime})=\langle
v^{\prime},\omega^{\prime}_{i}\rangle$ for any $v^{\prime}\in\sigma^{\pi}$.
Then $\omega^{\prime}_{i}-\omega_{i}\in(\sigma^{\pi})^{\vee}$ because $\langle
v^{\prime},\omega^{\prime}_{i}-\omega_{i}\rangle=0$ for any
$v^{\prime}\in\sigma^{\pi}$. Thus, $(\omega^{\prime}_{i},1)\in
D^{\sigma}_{u,\pi}$ from the definition of $D^{\sigma}_{u,\pi}$. This shows
that the definition of $D^{\sigma}_{u,\pi}$ is independent of the choice of
$\\{\omega_{i}\\}_{i\in S}$.

  2. (b)

Let $(v^{\prime},a)\in C^{\sigma}_{u,\pi}$ be an element. For any
$\omega^{\prime}\in(\sigma^{\pi})^{\vee}$, we have $(\omega^{\prime},0)\in
D^{\sigma}_{u,\pi}$. Thus, $\langle
v^{\prime},\omega^{\prime}\rangle=\langle(v^{\prime},a),(\omega^{\prime},0)\rangle\geq
0$ for any $\omega^{\prime}\in(\sigma^{\pi})^{\vee}$. Therefore,
$v^{\prime}\in\sigma^{\pi}$.

  3. (c)

From the definition of $C^{\sigma}_{u,\pi}$, we can check that
$(v^{\prime},a)\in C^{\sigma}_{u,\pi}$ if and only if $\langle
v^{\prime},\omega_{i}\rangle\geq-a$ for any $i\in S$. Because
$v^{\prime}\in\sigma^{\pi}$,
$\operatorname{val}_{u(i),\pi}(v^{\prime})=\langle
v^{\prime},\omega_{i}\rangle$ for any $i\in S$, we can check the statement
follows from the definition of $\operatorname{val}_{u,\pi}$.

  4. (d)

Let $(v^{\prime},a)\in\tau$ be an element. Then $v^{\prime}\in\sigma^{\pi}$
from (b). From (c), $a\geq-\operatorname{val}_{u,\pi}(v^{\prime})$. Moreover,
$(v^{\prime},-\operatorname{val}_{u,\pi}(v^{\prime}))$ and
$(0,a+\operatorname{val}_{u,\pi}(v^{\prime}))\in C^{\sigma}_{u,\pi}$. Because
$\tau\preceq C^{\sigma}_{u,\pi}$, we have that
$(v^{\prime},-\operatorname{val}_{u,\pi}(v^{\prime}))$ and
$(0,a+\operatorname{val}_{u,\pi}(v^{\prime}))\in\tau$. Thus, from the
assumption of $\tau$, we have that
$a=-\operatorname{val}_{u,\pi}(v^{\prime})$.

  5. (e)

Let $\tau\preceq C^{\sigma}_{f}$ be a face such that
$(v^{\prime},-\operatorname{val}_{u,\pi}(v^{\prime}))\in\tau^{\circ}$. If
$\tau$ contains $(0,1)$, then there exists $\epsilon>0$ such that
$(v^{\prime},-\operatorname{val}_{u,\pi}(v^{\prime})-\epsilon)\in\tau$ because
$(v^{\prime},-\operatorname{val}_{u,\pi}(v^{\prime}))\in\tau^{\circ}$.
However, it is a contradiction to (c). Thus, we have that $(0,1)\notin\tau$.

  6. (f)

Because
$(\mathrm{pr}_{1})_{\mathbb{R}}|_{\operatorname{Supp}(\Delta^{\sigma,u,\pi})}$
is injective from (d). From [14, Lemma 8.1(c)], $\Delta^{\sigma}_{u,\pi}$ is a
polyhedral convex fan. A morphism $\mathrm{pr}_{1}$ is a lattice morphism, and
$\Sigma^{\sigma,u,\pi}$ is a rational polyhedral fan so that
$\Delta^{\sigma}_{u,\pi}$ is a rational polyhedral convex fan. Moreover,
$\operatorname{Supp}(\Delta^{\sigma}_{u,\pi})=\sigma^{\pi}$ from (a) and (e).

  7. (g)

We can check easily that $E^{\sigma}_{u,\pi}\subset C^{\sigma}_{u,\pi}$ from
(c). Let $(v^{\prime},a)\in C^{\sigma}_{u,\pi}$be an element. From (b),
$v^{\prime}\in\sigma^{\pi}$. Moreover, from (c),
$a+\operatorname{val}_{u,\pi}(v^{\prime})\geq 0$. Thus,
$(v^{\prime},a)=(v^{\prime},-\operatorname{val}_{u,\pi}(v^{\prime}))+(0,a+\operatorname{val}_{u,\pi}(v^{\prime}))\in
E^{\sigma}_{u,\pi}$.

  8. (h)

Because
$(\mathrm{pr}_{1})_{\mathbb{R}}|_{\operatorname{Supp}(\Delta^{\sigma,u,\pi})}$
is injective, the strong convexity of $\Delta^{\sigma,u,\pi}$ is equivalent to
that of $\Delta^{\sigma}_{u,\pi}$ from [14, Lemma 8.1(b)]. Let $\tau$ denote
the minimal face of $C^{\sigma}_{u,\pi}$. We remark that $\tau$ is a linear
subspace of $(N^{\prime}\oplus\mathbb{Z})_{\mathbb{R}}$. Because
$C^{\sigma}_{u,\pi}$ does not contain $(0,-1)$, $\tau$ does not contain
$(0,1)$. Thus, $\tau\in\Delta^{\sigma,u,\pi}$ and the statement follows.

  9. (i)

Let $(\chi,b)\in\Omega^{\sigma^{\pi}}_{u,\pi}$ be an element. From [14,
Proposition 6.17], there exists $\omega\in M^{\prime}$ such that
$\operatorname{val}_{\chi,\pi}(v^{\prime})=\langle v^{\prime},\omega\rangle$
for any $v^{\prime}\in\sigma^{\pi}$. Because
$(\chi,b)\in\Omega^{\sigma^{\pi}}_{u,\pi}$, we have $\langle
v^{\prime},\omega\rangle-b\operatorname{val}_{u,\pi}(v^{\prime})\geq 0$ for
any $v^{\prime}\in\sigma^{\pi}$. This indicates that $(\omega,b)\in
D^{\sigma}_{u,\pi}$ from (g). Let $\tau$ denote
$C^{\sigma}_{u,\pi}\cap(\omega,b)^{\perp}\preceq C^{\sigma}_{u,\pi}$. Then
$\tau\in\Delta^{\sigma,u,\pi}$ because $b>0$. Moreover, we can check that
$(\mathrm{pr}_{1})_{\mathbb{R}}(\tau)=C^{\sigma^{\pi}}_{u,\pi}(\chi,b)$.

  10. (j)

Let $\tau\in\Delta^{\sigma,u,\pi}$ be a cone. Then there exists $(\omega,b)\in
D^{\sigma}_{u,\pi}$ such that $\tau=C^{\sigma}_{u,\pi}\cap(\omega,b)^{\perp}$.
Because $D^{\sigma}_{u,\pi}$ is rational polyhedral cone and
$(0,1)\notin\tau$, we may assume that $\omega\in M^{\prime}$ and
$b\in\mathbb{Z}_{>0}$. Let $\varphi\in\Phi$ be an element such that
$\sigma\in\Delta_{\varphi}$. Then there exists a section $s^{\prime}$ of
$q^{\prime}_{\varphi}\colon N^{\prime}\rightarrow N^{\prime}/s(N_{\varphi})$
such that $\sigma^{\pi}\subset
s^{\prime}_{\mathbb{R}}((N^{\prime}/s(N_{\varphi}))_{\mathbb{R}})$ from [14,
Lemma 8.2] and the condition (1) and (3) in [14, Definition 3.1]. In
particular, $\ker(s^{\prime*})\subset M^{\prime}\cap(\sigma^{\pi})^{\perp}$.
Let $M^{\prime}_{\varphi}$ denote the dual lattice of
$N^{\prime}/s(N_{\varphi})$. Then there exists the direct summation
$M=\ker(s^{\prime*})\oplus(q^{\prime}_{\varphi})^{*}(M^{\prime}_{\varphi})$.
Thus, there exists
$\omega_{1}\in(q^{\prime}_{\varphi})^{*}(M^{\prime}_{\varphi})$ and
$\omega_{2}\in\ker(s^{\prime*})$ such that $\omega=\omega_{1}+\omega_{2}$.
Because $\omega_{2}\in\ker(s^{\prime*})\subset(\sigma^{\pi})^{\perp}$, we have
$(\omega_{1},b)\in D^{\sigma}_{u,\pi}$ and
$\tau=C^{\sigma}_{u,\pi}\cap(\omega_{1},b)^{\perp}$ from (a). Let $\chi$
denote $\iota^{\prime*}(\chi^{\omega_{1}})\in k[Z^{\pi}]^{*}$. Then
$\operatorname{val}_{\chi,\pi}(v^{\prime})=\langle
v^{\prime},\omega_{1}\rangle$ for any $v^{\prime}\in\sigma^{\pi}$ because
$\omega_{1}\in(q^{\prime}_{\varphi})^{*}(M^{\prime}_{\varphi})$. Thus, we can
check that $(\chi,b)\in\Omega^{\sigma^{\pi}}_{u,\pi}$ and
$(\mathrm{pr}_{1})_{\mathbb{R}}(\tau)=C^{\sigma^{\pi}}_{u,\pi}(\chi,b)$.

  11. (k)

We recall that $v^{\prime}\in\sigma^{\pi}$ from (b) and
$a=-\operatorname{val}_{u,\pi}(v^{\prime})$ from (d). Let $i\in S$ be an
integer. Thus, $\langle(v^{\prime},a),(\omega_{i},1)\rangle=0$ if and only if
$\operatorname{val}_{u,\pi}(v^{\prime})=\operatorname{val}_{u(i),\pi}(v^{\prime})$.
Moreover, Because $(\mathrm{pr}_{1})_{\mathbb{R}}|_{\tau}$ is injective,
$v^{\prime}\in\varsigma^{\circ}$. Hence,
$\operatorname{val}_{u,\pi}(v^{\prime})=\operatorname{val}_{u(i),\pi}(v^{\prime})$
if and only if
$\operatorname{val}^{\varsigma}_{u,\pi}=\operatorname{val}^{\varsigma}_{u(i),\pi}$
from [14, Proposition 6.17] and Proposition 3.18(g) and the definition of
$\operatorname{val}_{u,\pi}$.

∎

In general, the commputation of $D^{\sigma}_{u,\pi}$ is very difficult because
we have to find $\omega_{i}\in M^{\prime}$ such that
$\operatorname{val}_{u(i),\pi}(v^{\prime})=\langle
v^{\prime},\omega_{i}\rangle$ for any $v^{\prime}\in\sigma^{\pi}$ and $i\in
S$. In fact, the following lemma shows that we can compute
$D^{\sigma}_{u,\pi}$ briefly if $\\{u(i)\\}_{i\in S}$ is specific.

######  Lemma 3.20.

We keep the notation in Proposition 3.19. We assume that for any $i\in S$,
there exists $\omega_{i}\in M^{\prime}$ and $c_{i}\in k^{*}$ such that
$u(i)=c_{i}{\iota^{\prime}}^{*}(\chi^{\omega_{i}})$. Then $D^{\sigma}_{u,\pi}$
is generated by the following elements:

$\\{(\omega^{\prime},0)\in(M^{\prime}\oplus\mathbb{Z})_{\mathbb{R}}\mid\omega^{\prime}\in(\sigma^{\pi})^{\vee}\\}\cup\\{(\omega_{i},1)\in(M^{\prime}\oplus\mathbb{Z})_{\mathbb{R}}\mid
i\in S\\}$

###### Proof.

From [14, Lemma 8.13] and [14, Corollary 4.13],
$\operatorname{val}_{u(i),\pi}(v^{\prime})=\langle
v^{\prime},\omega_{i}\rangle$ for any $v^{\prime}\in\sigma^{\pi}$ and $i\in
S$. Thus, from Proposition 3.19(a), the statement holds. ∎

As mentioned at the beginning of this section, the computation of stable
birational volume of hypersurfaces of mock toric varieties assumes certain
conditions on the fan. The following proposition describes a method to
overcome these obstacles, which is geometrically referred to as the semistable
reduction.

######  Lemma 3.21.

We keep the notation in Definition 3.15. Let
$(Z^{1}_{\pi^{1}},S,u\oplus\kappa)$ be a model mock polytope of a mock
polytope $(Z_{\pi},S,u)$ along $\kappa$. Let $n$ be a positive integer. Let
$r_{n}\colon N^{\prime}\oplus\mathbb{Z}\rightarrow N^{\prime}\oplus\mathbb{Z}$
be a linear morphism defined as $r_{n}(v,c)=(nv,c)$ for $(v,c)\in
N^{\prime}\oplus\mathbb{Z}$. Then the following statements follow:

  1. (a)

$n\operatorname{val}_{u\oplus\kappa,\pi^{1}}=\operatorname{val}_{u\oplus{n\kappa},\pi^{1}}\circ(r_{n})_{\mathbb{R}}|_{(\pi^{1}_{\mathbb{R}})^{-1}(\operatorname{Supp}(\Delta\times\Delta_{!}))}$

  2. (b)

Let $\sigma\in\Delta\times\Delta_{!}$ be a cone, $\chi\in k[Z_{\pi}]^{*}$ be a
unit, $a\in\mathbb{Z}$ be an integer, and $b\in\mathbb{Z}_{>0}$ be a positive
integer. Let $\sigma^{\pi}$ denote $(\pi^{1}_{\mathbb{R}})^{-1}(\sigma)$. Then
$(\chi t^{a},b)\in\Omega^{\sigma^{\pi}}_{u\oplus\kappa,\pi^{1}}$ if and only
if $(\chi t^{an},b)\in\Omega^{\sigma^{\pi}}_{u\oplus{n\kappa},\pi^{1}}$. In
this case, $(r_{n})_{\mathbb{R}}(C^{\sigma^{\pi}}_{u\oplus\kappa,\pi^{1}}(\chi
t^{a},b))=C^{\sigma^{\pi}}_{u\oplus{n\kappa},\pi^{1}}(\chi t^{an},b)$.

  3. (c)

We use the notation in (b). Then $(\chi
t^{a},b)\in\Omega^{\sigma^{\pi}}_{u\oplus{n\kappa},\pi^{1}}$ if and only if
$(\chi^{n}t^{a},nb)\in\Omega^{\sigma^{\pi}}_{u\oplus\kappa,\pi^{1}}$. In this
case,
$(r_{n})_{\mathbb{R}}(C^{\sigma^{\pi}}_{u\oplus\kappa,\pi^{1}}(\chi^{n}t^{a},bn))=C^{\sigma^{\pi}}_{u\oplus{n\kappa},\pi^{1}}(\chi
t^{a},b)$.

  4. (d)

The map $(r_{n})_{\mathbb{R}}$ induces a one-to-one correspondence with cones
in $(\Delta\times\Delta_{!})_{u\oplus\kappa,\pi^{1}}$ and those in
$(\Delta\times\Delta_{!})_{u\oplus{n\kappa},\pi^{1}}$. Moreover, if
$(\Delta\times\Delta_{!})_{u\oplus\kappa}$ is strongly convex, then
$(\Delta\times\Delta_{!})_{u\oplus{l\kappa}}$ is strongly convex for any
$l\in\mathbb{Z}_{>0}$.

  5. (e)

There exists $m\in\mathbb{Z}_{>0}$ satisfying the following condition (*):

     * (*)

For any $l\in\mathbb{Z}_{>0}$, there exists a strongly convex rational
polyhedral fan $\Delta^{\prime}_{l}$ in
$(N^{\prime}\oplus\mathbb{Z})_{\mathbb{R}}$ such that $\Delta^{\prime}_{l}$ is
a refinement of $(\Delta\times\Delta_{!})_{u\oplus{ml\kappa},\pi^{1}}$,
generically unimodular, specifically reduced, and compactly arranged.

###### Proof.

We prove the statements from (a) to (e) in order.

  1. (a)

Let
$v^{\prime}=(v,c)\in(\pi^{1}_{\mathbb{R}})^{-1}(\operatorname{Supp}(\Delta\times\Delta_{!}))$
be an element. For any $i\in S$, the following equation folds:

$\displaystyle\operatorname{val}_{u\oplus{n\kappa}(i),\pi^{1}}\circ
r_{n}(v^{\prime})$
$\displaystyle=\operatorname{val}_{u\oplus{n\kappa}(i),\pi^{1}}\circ
r_{n}(v,c)$
$\displaystyle=\operatorname{val}_{u\oplus{n\kappa}(i),\pi^{1}}(nv,c)$
$\displaystyle=\operatorname{val}_{u(i)t^{n\kappa(i)},\pi^{1}}(nv,c)$
$\displaystyle=\operatorname{val}_{u(i),\pi}(nv)+nc\kappa(i)$
$\displaystyle=n(\operatorname{val}_{u(i),\pi}(v)+c\kappa(i))$
$\displaystyle=n(\operatorname{val}_{u(i)t^{\kappa(i)},\pi^{1}}(v,c))$
$\displaystyle=n\operatorname{val}_{u\oplus\kappa(i),\pi^{1}}(v^{\prime})$

Thus, the statement holds from the definition of $\operatorname{val}_{u\oplus
n\kappa,\pi^{1}}$ and $\operatorname{val}_{u\oplus\kappa,\pi^{1}}$.

  2. (b)

We remark that the restriction $(r_{n})_{\mathbb{R}}|_{\sigma^{\pi}}$ is an
automorphism of $\sigma^{\pi}$ from the definition of $r_{n}$. Moreover, for
any $\chi\in k[Z_{\pi}]^{*}$ and $a\in\mathbb{Z}$, we can check that
$\operatorname{val}_{\chi
t^{an},\pi^{1}}\circ(r_{n})_{\mathbb{R}}|_{(\pi^{1}_{\mathbb{R}})^{-1}(\operatorname{Supp}(\Delta\times\Delta_{!}))}=n\operatorname{val}_{\chi
t^{a},\pi^{1}}$ from the similar argument in the proof of (a). Thus, we can
check that the statement holds.

  3. (c)

We remark that for any $\chi\in k[Z_{\pi}]^{*}$ and $a\in\mathbb{Z}$, we have
$n(\frac{1}{n}\operatorname{val}_{\chi^{n}t^{a},\pi^{1}})=\operatorname{val}_{\chi
t^{a},\pi^{1}}\circ(r_{n})_{\mathbb{R}}|_{(\pi^{1}_{\mathbb{R}})^{-1}(\operatorname{Supp}(\Delta\times\Delta_{!}))}$.
Then, we can check that the statement holds from the similar argument in the
proof of (b).

  4. (d)

The restriction
$(r_{n})_{\mathbb{R}}|_{(\pi^{1})^{-1}(\operatorname{Supp}(\Delta\times\Delta_{!}))}$
is an automorphism of
$(\pi^{1})^{-1}(\operatorname{Supp}(\Delta\times\Delta_{!}))$. Thus, from (b)
and (c), $r_{n}$ induces a one-to-one correspondence with cones in
$(\Delta\times\Delta_{!})_{u\oplus\kappa,\pi^{1}}$ and those in
$(\Delta\times\Delta_{!})_{u\oplus{n\kappa},\pi^{1}}$.

For the latter part of the statement, we can check it because
$(r_{l})_{\mathbb{R}}$ is a linear isomorphism for any $l\in\mathbb{Z}_{>0}$.

  5. (e)

Let $m\in\mathbb{Z}_{>0}$ be a positive integer and $\Delta^{\prime}$ be a
strongly convex rational polyhedral fan in
$(N^{\prime}\oplus\mathbb{Z})_{\mathbb{R}}$ which is a refinement of
$(\Delta\times\Delta_{!})_{u\oplus m\kappa,\pi^{1}}$. We assume that
$\Delta^{\prime}$ is generically unimodular, specifically reduced, and
compactly arranged. For $l\in\mathbb{Z}_{>0}$, let $\Delta^{\prime}_{l}$
denote the following set:

$\Delta^{\prime}_{l}=\\{(r_{l})_{\mathbb{R}}(\tau)\mid\tau\in\Delta^{\prime}\\}$

From [14, Lemma 8.1(d)], $\Delta^{\prime}_{l}$ is a strongly convex rational
polyhedral fan in $(N^{\prime}\oplus\mathbb{Z})_{\mathbb{R}}$. Moreover, from
(d), $\Delta^{\prime}_{l}$ is a refinement of
$(\Delta\times\Delta_{!})_{u\oplus{ml\kappa},\pi^{1}}$. Now, we show that
$\Delta^{\prime}_{l}$ is also generically unimodular, specifically reduced,
and compactly arranged for any $l\in\mathbb{Z}_{>0}$.

From the definition of $r_{l}$, we can check that $\Delta^{\prime}_{l}$ is
generically unimodular.

Let $\tau\in\Delta^{\prime}$ be a cone. Then from the definition of $r_{l}$,
we can check that $\tau\in\Delta^{\prime}_{\operatorname{sp}}$ if and only if
$(r_{l})_{\mathbb{R}}(\tau)\in(\Delta^{\prime}_{l})_{\operatorname{sp}}$.
Moreover, we can check that $\tau\in\Delta^{\prime}_{\operatorname{bdd}}$ if
and only if
$(r_{l})_{\mathbb{R}}(\tau)\in(\Delta^{\prime}_{l})_{\operatorname{bdd}}$.
Therefore, from (d), $\Delta^{\prime}_{l}$ is compactly arranged.

We remark that $(r_{l})_{\mathbb{R}}$ induces a one-to-one correspondence with
$\gamma\in\Delta^{\prime}_{\operatorname{sp}}$ such that $\dim(\gamma)=1$ and
those in $\Delta^{\prime}_{l}$ from the definition of $r_{l}$. Let
$\gamma\in\Delta^{\prime}_{\operatorname{sp}}$ be a cone such that
$\dim(\gamma)=1$ and $(v,c)\in N^{\prime}\oplus\mathbb{Z}$ be a minimal
generator of $\gamma$. From the assumption, $c=1$. Thus, $(lv,1)$ is a
generator of $(r_{l})_{\mathbb{R}}(\gamma)$. Hence, $\Delta^{\prime}_{l}$ is
specifically reduced.

From the above argument, it is enough to show that the following statement
(**) holds:

    1. (**)

There exists $m\in\mathbb{Z}_{>0}$ and $\Delta^{\prime}$ be a strongly convex
rational polyhedral fan in $(N^{\prime}\oplus\mathbb{Z})_{\mathbb{R}}$ such
that $\Delta^{\prime}$ is a refinement of $(\Delta\times\Delta_{!})_{u\oplus
m\kappa,\pi^{1}}$, generically unimodular, specifically reduced, and compactly
arranged.

Let $\Delta^{\prime\prime}$ be an unimodular refinement of
$(\Delta\times\Delta_{!})_{u\oplus\kappa,\pi^{1}}$. For a positive integer
$m$, let $\Delta^{\prime\prime}_{m}$ denote the following set:

$\Delta^{\prime\prime}_{m}=\\{(r_{m})_{\mathbb{R}}(\tau)\mid\tau\in\Delta^{\prime\prime}\\}$

From [14, Proposition 7.4(b)], we can check that $\Delta^{\prime\prime}_{m}$
is a strongly convex rational polyhedral fan in
$(N^{\prime}\oplus\mathbb{Z})_{\mathbb{R}}$, a refinement of
$(\Delta\times\Delta_{!})_{u\oplus m\kappa,\pi^{1}}$, generically unimodular,
and compactly arranged for any $m\in\mathbb{Z}_{>0}$. Let $\Gamma$ denote the
following subset of $\Delta^{\prime\prime}$:

$\Gamma=\\{\gamma\in\Delta^{\prime\prime}_{\operatorname{sp}}\mid\dim(\gamma)=1\\}$

For $\gamma\in\Gamma$, let $v_{\gamma}$ denote an element in
$N^{\prime}_{\mathbb{Q}}$ such that $(v_{\gamma},1)\in\gamma$. Because
$|\Gamma|$ is finite, there exists $m_{0}\in\mathbb{Z}_{>0}$ such that
$m_{0}v_{\gamma}\in N^{\prime}$ for any $\gamma\in\Gamma$. Therefore, we can
check that $\Delta^{\prime\prime}_{m_{0}}$ is specifically reduced.

∎

From Lemma3.13, we can compute the intersection of a hypersurface and a
stratum of $Z_{u}$ for a general section in the linear system defined by
$[u]$. However, calculating this in practice is difficult in general. Indeed,
this is also a different point between toric varieties and mock toric
varieties. The following proposition indicates that, under favorable
conditions on the cone, the computation can be relatively straightforward.

######  Proposition 3.22.

We use the notation in [14, Definition 4.11] and the following notation:

  * •

Let $\sigma$ be a strongly convex rational polyhedral cone in
$N^{\prime}_{\mathbb{R}}$. We assume that $\pi_{\mathbb{R}}(\sigma)=\\{0\\}$.
Let $\tau$ denote $p_{\mathbb{R}}(\sigma)$.

  * •

Let $\Delta_{1}$ denote a fan whose cones are all faces of $\sigma$ and $W$
denote a mock toric variety induced by $Z$ along $\pi_{*}\colon
X(\Delta_{1})\rightarrow X(\Delta)$ and $s$.

Then the following statements follow:

  1. (a)

The set $\tau$ is a strongly convex rational polyhedral cone in
$(N_{1})_{\mathbb{R}}$.

  2. (b)

Let $\varphi\in\Phi$ be an element. Then, all squares in the following diagram
are Cartesian squares:

${W_{\sigma}}$${O_{\sigma}}$${O_{\sigma_{\varphi}}}$${O_{\tau}}$${W(\sigma)}$${X(\sigma)}$${X(\sigma_{\varphi})}$${X(\tau)}$${Z^{\circ}}$${T_{N}}$${T_{N/N_{\varphi}}}$${\operatorname{Spec}(k)}$$\scriptstyle{\iota^{\prime}}$$\scriptstyle{(q^{\prime}_{\varphi})_{*}}$$\scriptstyle{\pi_{*}}$$\scriptstyle{(p_{\varphi})_{*}}$$\scriptstyle{(\pi_{\varphi})_{*}}$$\scriptstyle{\iota}$$\scriptstyle{(q_{\varphi})_{*}}$

  3. (c)

Let $\epsilon^{*}$ denote the ring morphism $k[Z^{\circ}]\rightarrow
k[Z_{\pi}]$ induced by $\epsilon\colon Z_{\pi}\rightarrow Z^{\circ}$ and
$p^{*}_{0}$ denote the ring morphism $k[M_{1}]\rightarrow k[Z_{\pi}]$ induced
by $p_{0}\colon Z_{\pi}\rightarrow T_{N_{1}}$. Then
$\epsilon^{*}\otimes\pi^{*}\colon k[Z^{\circ}]\otimes k[M_{1}]\rightarrow
k[Z_{\pi}]$ is isomorphic, and all horizontal morphisms in the following
diagram are isomorphic:

${k[Z^{\circ}]\otimes_{k}k[M_{1}]}$${k[Z_{\pi}]}$${k[Z^{\circ}]\otimes_{k}k[M_{1}\cap\tau^{\vee}]}$${\Gamma(W(\sigma),\mathscr{O}_{W})}$${k[Z^{\circ}]\otimes_{k}k[M_{1}\cap\tau^{\perp}]}$${\Gamma(W_{\sigma},\mathscr{O}_{\overline{W_{\sigma}}})}$

where downward morphisms are quotient morphisms and upward morphisms are
localization morphisms.

  4. (d)

Let $(W,S,u)$ be a mock polytope of $W$. From [14, Proposition 4.12(c)], there
exists $\eta_{i}\in k[Z^{\circ}]^{*}$ and $\omega_{i}\in M_{1}$ such that
$u(i)=\epsilon^{*}(\eta_{i})p^{*}_{0}(\chi^{\omega_{i}})$ for any $i\in S$. We
assume that $\sigma\in(\Delta_{1})_{u}$. Then there exists $\omega\in M_{1}$
such that $\omega_{i}-\omega\in\tau^{\perp}$ for any $i\in S^{\sigma}$ and
$\operatorname{val}^{\sigma}_{p^{*}(\chi^{\omega})}=\operatorname{val}^{\sigma}_{u}$.

  5. (e)

We use the notation in (d). Let $u^{\prime}\colon S^{\sigma}\rightarrow
k[Z^{\circ}]\otimes_{k}k[O_{\tau}]$ be a map defined as
$u^{\prime}(i)=\eta_{i}\otimes\chi^{\omega_{i}-\omega}$ for $i\in S^{\sigma}$.
Then $u^{\sigma}\sim u^{\prime}$ on the identification with
$k[Z^{\circ}]\otimes k[O_{\tau}]$ and $k[W_{\sigma}]$ from (c).

###### Proof.

We prove the statements from (a) to (e) in order.

  1. (a)

From the assumption of $\sigma$, $p_{\mathbb{R}}|_{\sigma}$ is injective.
Thus, $\tau$ is a strongly convex cone from [14, Lemma 8.1(a)] and (d).
Because $p$ is a lattice morphism, $\tau$ is a rational polyhedral convex
cone.

  2. (b)

From the definition of the mock toric structure of $W$, two squares on the
left side of the above diagram are Cartesian squares. We can identify with
$\sigma$ and $\\{0\\}\times\tau$ by the isomorphism $\pi\times p\colon
N^{\prime}\rightarrow N\oplus N_{1}$. Thus, all squares of the following
diagram are Cartesian squares:

${O_{\sigma}}$${O_{\tau}}$${X(\sigma)}$${X(\tau)}$${T_{N}}$${\operatorname{Spec}(k)}$$\scriptstyle{p_{*}}$$\scriptstyle{\pi_{*}}$

Moreover, We can identify with $\sigma_{\varphi}$ and $\\{0\\}\times\tau$ by
the isomorphism $\pi_{\varphi}\times p_{\varphi}\colon
N^{\prime}/s(N_{\varphi})\rightarrow N/N_{\varphi}\oplus N_{1}$. Similarly,
all squares of the following diagram are Cartesian squares:

${O_{\sigma_{\varphi}}}$${O_{\tau}}$${X(\sigma_{\varphi})}$${X(\tau)}$${T_{N/N_{\varphi}}}$${\operatorname{Spec}(k)}$$\scriptstyle{{p_{\varphi}}_{*}}$$\scriptstyle{{\pi_{\varphi}}_{*}}$

  3. (c)

The statement follows from (b).

  4. (d)

Because $\sigma\in(\Delta_{1})_{u}$, there exists $\chi^{\prime}\in
k[Z_{\pi}]^{*}$ such that
$\operatorname{val}^{\sigma}_{\chi^{\prime}}=\operatorname{val}^{\sigma}_{u}$
from Proposition 3.7(d). From [14, Proposition 4.12(c)], there exists $\eta\in
k[Z^{\circ}]^{*}$ and $\omega\in M_{1}$ such that
$\chi^{\prime}=\epsilon^{*}(\eta)p^{*}_{0}(\chi^{\omega})$. From the
assumption of $\sigma$ and [14, Proposition 4.12(b)] and (d),
$\operatorname{val}^{\sigma}_{\chi^{\prime}}=\operatorname{val}^{\sigma}_{p^{*}_{0}(\chi^{\omega})}$
and
$\operatorname{val}^{\tau}_{\chi^{\omega}}=\operatorname{val}^{\tau}_{\chi^{\omega_{i}}}$.
Thus,
$\operatorname{val}^{\sigma}_{p^{*}_{0}(\chi^{\omega})}=\operatorname{val}^{\sigma}_{u}$
and $\omega_{i}-\omega\in\tau^{\perp}$ for any $i\in S^{\sigma}$.

  5. (e)

From (d) and [14, Proposition 4.8],
$\chi^{\prime-1}p^{*}_{0}(\chi^{\omega})\in\Gamma(W(\sigma),\mathscr{O}_{W})^{*}$.
Let $p^{\sigma}$ denote a ring morphism
$\Gamma(W(\sigma),\mathscr{O}_{W})\rightarrow\Gamma(W_{\sigma},\mathscr{O}_{\overline{W_{\sigma}}})$
associated with the closed immersion $W_{\sigma}\hookrightarrow W(\sigma)$.
Let $\beta$ denote $p^{\sigma}(\chi^{\prime-1}p^{*}_{0}(\chi^{\omega}))$. Then
$\beta\in\Gamma(W_{\sigma},\mathscr{O}_{\overline{W_{\sigma}}})^{*}$ and
$u^{\sigma}(i)=\beta u^{\prime}(i)$ for any $i\in S^{\sigma}$. Therefore,
$u^{\sigma}\sim u^{\prime}$.

∎

The following proposition shows that the stable birational volume of a general
section in the linear system defined by the mock polytope can be computed,
achieving the aim of this section.

######  Proposition 3.23.

We use the notation in Definition 3.12, that in Definition 3.15, and that in
Proposition 3.18. Let $(Z_{\pi},S,u)$ be a mock polytope of $Z_{\pi}$ and
$\kappa\colon S\rightarrow\mathbb{Z}$ be a map. We assume that $Z$ is proper
over $k$, $d_{u}\geq 2$, and
$(\Delta\times\Delta_{!})_{u\oplus\kappa,\pi^{1}}$ is a strongly convex
rational polyhedral fan. Let $\Delta^{\prime\prime}$ denote
$(\Delta\times\Delta_{!})_{u\oplus\kappa,\pi^{1}}$. We remark that
$\Delta^{\prime\prime}=\Delta^{\prime\prime}_{u\oplus\kappa}$ from Proposition
3.18(h) and
$\operatorname{Supp}(\Delta^{\prime\prime})=(\pi^{1}_{\mathbb{R}})^{-1}(\operatorname{Supp}(\Delta\times\Delta_{!}))$
from Proposition 3.18(f). Moreover, we assume that there exists a refinement
$\Delta^{\prime}$ of $\Delta^{\prime\prime}$ such that $\Delta^{\prime}$ is
generically unimodular, specifically reduced, and compactly arranged. Let
$W^{\prime}$ be a mock toric variety induced by $Z^{1}$ along
$\pi^{1}_{*}\colon X(\Delta^{\prime\prime})\rightarrow
X(\Delta\times\Delta_{!})$ and $s^{1}$. Let $(a_{i})_{i\in S}\in k^{|S|}$ be a
family and $f$ denote the following polynomial in $k[Z^{1}_{\pi^{1}}]$:

$f=\sum_{i\in S}a_{i}u(i)t^{\kappa(i)}$

Let $\sigma\in\Delta^{\prime\prime}$ be a cone and $f^{\sigma}$ denote the
following polynomial in $k[W^{\prime}_{\sigma}]$:

$f^{\sigma}=\sum_{i\in S^{\sigma}}a_{i}(u\oplus\kappa)^{\sigma}(i)$

Let $\\{E^{(j)}_{\sigma}\\}_{1\leq j\leq r_{\sigma}}$ be all irreducible
components of $H_{W^{\prime}_{\sigma},f^{\sigma}}$. We remark that
$r_{\sigma}$ is dependent on the choice of $f$. Let $\\{F_{i}\\}_{1\leq i\leq
r}$ be all irreducible components of
$H^{\circ}_{W^{\prime},f}\times_{{\mathbb{G}_{m}^{1}}_{,k}}\operatorname{Spec}(\mathscr{K})$.
Then for general $(a_{i})_{i\in S}\in k^{|S|}$, the following equation holds:

$\sum_{1\leq i\leq
r}\mathrm{Vol}_{\mathrm{sb}}\biggl{(}\bigl{\\{}F_{i}\bigr{\\}}_{\mathrm{sb}}\biggr{)}=\sum_{\begin{subarray}{c}\sigma\in\Delta^{\prime\prime}_{\operatorname{bdd}}\\\
\sigma\in{\Delta^{\prime\prime\geq
2}_{u\oplus\kappa}}\end{subarray}}(-1)^{\dim(\sigma)-1}\biggl{(}\sum_{1\leq
j\leq r_{\sigma}}\bigl{\\{}E^{(j)}_{\sigma}\bigr{\\}}_{\mathrm{sb}}\biggr{)}$

###### Proof.

We show that for general $(a_{i})_{i\in S}\in k^{|S|}$, $f$ satisfies the
conditions in Theorem 2.3.

First, from Proposition 3.9, for general $(a_{i})_{i\in S}\in k^{|S|}$, we
have $\operatorname{val}_{u\oplus\kappa}=\operatorname{val}_{f}$ and
$\Delta^{\prime\prime}_{u\oplus\kappa}=\Delta^{\prime\prime}_{f}$. In
particular, we remark that $\Delta^{\prime\prime}=\Delta^{\prime\prime}_{f}$.

Second, from Proposition 3.14(f), for general $(a_{i})_{i\in S}\in k^{|S|}$,
$f$ is non-degenerate for
$\Delta^{\prime\prime}=\Delta^{\prime\prime}_{u\oplus\kappa}$. In particular,
from [14, Proposition 6.14(e)], $f$ is non-degenerate for $\Delta^{\prime}$
too.

Third, from Proposition 3.17, for general $(a_{i})_{i\in S}\in k^{|S|}$, every
irreducible component of $H^{\circ}_{Z^{1}_{\pi^{1}},f}$ dominates
${\mathbb{G}_{m}^{1}}_{,k}$.

Thus, all conditions in Theorem 2.3 hold.

Moreover, We show that for general $(a_{i})_{i\in S}\in k^{|S|}$,
${}_{f}\Delta^{\prime\prime}=\Delta^{\prime\prime\geq 2}_{u\oplus\kappa}$.
Indeed, we have already shown that for general $(a_{i})_{i\in S}\in k^{|S|}$,
$\Delta^{\prime\prime}_{u\oplus\kappa}={}_{f}\Delta^{\prime\prime}$ in the
proof of Proposition 3.14(d).

We recall that $H_{W^{\prime},f}\cap
W^{\prime}_{\sigma}=H^{\circ}_{W^{\prime}_{\sigma},f^{\sigma}}$ for any
$\sigma\in\Delta^{\prime\prime}$ from 3.13. Because $f$ is non-degenerate for
$\Delta^{\prime\prime}$, any irreducible component of $H_{W^{\prime},f}\cap
W^{\prime}_{\sigma}$ is a connected component of $H_{W^{\prime},f}\cap
W^{\prime}_{\sigma}$. Therefore, the statement holds from Theorem 2.3. ∎

Now, while the formula has been obtained, grasping the stable rationality of
each stratum in the strictly toroidal model is difficult. However, when
favorable conditions are established for the shape of the cone, rationality
can be inferred. We show it in Proposition3.25. This proposition is a
generalization of [7, Example 3.8]. The following lemma is needed in the proof
of Proposition3.25.

######  Proposition 3.24.

Let $Z$ be a mock toric variety and $(Z,S,[u])$ be an abstract mock polytope
of $Z$. Let $S_{1}$ and $S_{2}$ be a non-empty subset of $S$ such that
$S_{1}\coprod S_{2}=S$. For $(a_{i})_{i\in S}\in k^{|S|}$, $f_{1}$ and $f_{2}$
denote the following functions in $k[Z^{\circ}]$:

$\displaystyle f_{1}$ $\displaystyle=\sum_{i\in S_{1}}a_{i}u(i)$
$\displaystyle f_{2}$ $\displaystyle=\sum_{i\in S_{2}}a_{i}u(i)$

Then for general $(a_{i})_{i\in S}\in k^{|S|}$, $f_{1}$ and $f_{2}$ are co-
prime.

###### Proof.

Let $(b_{i})_{i\in S_{1}}\in k^{|S_{1}|}$ be a vector such that $\sum_{i\in
S_{1}}b_{i}u(i)\neq 0$ and $\theta$ denote the following linear morphism from
$k^{|S_{2}|}$ to $k[Z^{\circ}]$:

$k^{|S_{2}|}\ni(c_{i})_{i\in S_{2}}\mapsto\sum_{i\in S_{2}}c_{i}u(i)\in
k[Z^{\circ}]$

Let $g$ be an irreducible component of $\sum_{i\in S_{1}}b_{i}u(i)$. Then the
following subset of $k^{|S_{2}|}$ is a linear subspace of $k^{|S_{2}|}$:

$\\{(c_{i})_{i\in S_{2}}\in k^{|S_{2}|}\mid\sum_{i\in
S_{2}}c_{i}u(i)\in(g)\\}$

From the definition of mock polytope, $(1,0,\ldots,0)_{i\in S_{2}}$ is not
contained in the above linear subspace. Hence, the above linear subspace is a
strict linear subspace of $k^{|S_{2}|}$. This shows that for any
$(b_{i})_{i\in S_{1}}$ such that $\sum_{i\in S_{1}}b_{i}u(i)\neq 0$,
$\sum_{i\in S_{1}}b_{i}u(i)$ and $\sum_{i\in S_{2}}c_{i}u(i)$ are co-prime for
general $(c_{i})_{i\in S_{2}}\in k^{|S_{2}|}$.

Let $X$ be a closed subscheme of $\mathbb{A}^{|S|}_{k}\times Z^{\circ}$
defined by the ideal which is generated by $f_{1}$ and $f_{2}\in
k[\\{a_{i}\\}_{i\in S}]$, where $\\{a_{i}\\}_{i\in S}$ is a family of
coordinate functions of $\mathbb{A}^{|S|}_{k}$. Let $\pi$ denote the first
projection $X\rightarrow\mathbb{A}^{|S|}_{k}$. Then from generic flatness,
there exists non-empty open subset $V$ of $\mathbb{A}^{|S|}_{k}$ such that
$\pi|_{\pi^{-1}(V)}\colon\pi^{-1}(V)\rightarrow V$ is flat. Thus, there exists
$n$ and anon-empty open subset $V_{0}$ of $V$ such that $\\{x\in
V\mid\dim_{\kappa(x)}(\pi^{-1}(x))=n\\}=V_{0}$. We claim that $n\leq\dim(Z)-2$
by a contradiction. Because, $S_{1}$ and $S_{2}$ are non-empty, we may assume
$n=\dim(Z)-1$. Let $(b_{i})_{i\in S_{1}}\in k^{|S_{1}|}$ be a vector such that
$((b_{i})_{i\in S_{1}}\times\mathbb{A}^{|S_{2}|})\cap V_{0}\neq\emptyset$ and
$\sum_{i\in S_{1}}b_{i}u(i)\neq 0$. Then for any $((b_{i})_{i\in
S_{1}},(c_{i})_{i\in S_{2}})\in V_{0}$, $\sum_{i\in S_{1}}b_{i}u(i)$ and
$\sum_{i\in S_{2}}c_{i}u(i)$ are not co-prime from the assumption of $n$.
However, it is a contradiction of the argument at first. Thus,
$n\leq\dim(Z)-2$. This inequality shows that the statement holds. ∎

######  Proposition 3.25.

We use the notation in Proposition 3.22and in the following notation:

  * •

There exists an isomorphisms $\epsilon\times p_{0}\colon Z^{\pi}\rightarrow
Z\times T_{N_{1}}$ from Proposition 3.22(c). Then $\epsilon^{*}\otimes
p^{*}_{0}\colon k[Z^{\circ}]\otimes_{k}k[M_{1}]\rightarrow k[Z_{\pi}]$ is an
isomorphism. We regard $k[Z^{\circ}]$ and $k[M_{1}]$ as subrings of
$k[Z_{\pi}]$ by $\epsilon^{*}$ and $p^{*}_{0}$, so we omit $\epsilon^{*}$ and
$p^{*}_{0}$.

  * •

Let $\omega$ denote an element of $M_{1}$ such that $M_{1}/\mathbb{Z}\omega$
is torsion-free. We fix a sub lattice $M_{2}$ of $M_{1}$ such that
$M_{2}\oplus\mathbb{Z}\omega=M_{1}$.

  * •

Let $(Z_{\pi},S,[u])$ be an abstract mock polytope of $Z_{\pi}$. We assume
that there exists the non-empty subsets $S_{1}$ and $S_{2}$ of $S$ which
satisfy the following three conditions:

    * (1)

$S=S_{1}\coprod S_{2}$

    * (2)

For any $i\in S_{1}$, there exists $v(i)\in k[Z^{\circ}]^{*}$ and $w(i)\in
k[M_{2}]^{*}$ such that $u(i)=\chi^{\omega}v(i)w(i)$.

    * (3)

For any $i\in S_{2}$, there exists $v(i)\in k[Z^{\circ}]^{*}$ and $w(i)\in
k[M_{2}]^{*}$ such that $u(i)=v(i)w(i)$.

  * •

For $(a_{i})_{i\in S}\in k^{|S|}$, let $f$ denote the following function in
$k[Z_{\pi}]$:

$f=\sum_{i\in S}a_{i}u(i)$

Then for general $(a_{i})_{i\in S}\in k^{|S|}$, $H_{Z_{\pi},f}$ is rational.

###### Proof.

Let $F_{1}$ denote the following function in $k[Z_{\pi}]$:

$F_{1}=\sum_{i\in S_{1}}a_{i}v(i)w(i)$

Let $F_{2}$ denote the following function in $k[Z_{\pi}]$:

$F_{2}=\sum_{i\in S_{2}}a_{i}v(i)w(i)$

We remark that $f=\chi^{\omega}F_{1}+F_{2}$. Hence, from Proposition 3.24,
$F_{1}$ and $F_{2}$ are co-prime for general $(a_{i})_{i\in S}\in k^{|S|}$.
Thus, $H_{Z_{\pi},f}$ is irreducible for general $(a_{i})_{i\in S}\in k^{|S|}$
because $k[Z^{\circ}][M_{2}]$ is a UFD. For such $(a_{i})_{i\in S}\in
k^{|S|}$, let $U$ denote an open subscheme
$\operatorname{Spec}(k[Z^{\circ}][M_{2}]_{F_{1}})$ of
$\operatorname{Spec}(k[Z_{0}][M_{2}])$. Then $U$ and $H_{Z_{\pi},f}$ are
birational. In particular, $\operatorname{Spec}(k[Z^{\circ}][M_{2}])$ is
rational, so that $H_{Z_{\pi},f}$ is rational. ∎

##  4\. Application: Stable rationality of hypersurfaces in Grassmaniann
varieties

As an application of the considerations thus far, we can mention the stable
rationality of some hypersurfaces in $\mathrm{Gr}_{\mathbb{C}}(2,n)$. In this
section, we use the following notation.

  * •

Let $n$ be a positive integer greater than 3. Let $k$ be an uncountable
algebraically closed field with $\operatorname{char}(k)=0$.

  * •

Let $I$ denote the following set:

$I=\\{(i,j)\in\mathbb{Z}^{2}\mid 0\leq i\leq j\leq n-3\\}$

  * •

Let $\\{e^{i,j}\\}_{(i,j)\in I}$ denote a canonical basis of
$\mathbb{Z}^{|I|}$, and $\mathbf{1}\in\mathbb{Z}^{|I|}$ be $\sum_{(i,j)\in
I}e^{i,j}$.

  * •

Let $N$ denote $\mathbb{Z}^{|I|}/\mathbb{Z}\mathbf{1}$, $\Pi$ denote the
quotient morphism $\mathbb{Z}^{|I|}\rightarrow N$, and $e_{i,j}\in N$ denote
$\Pi(e^{i,j})$ for $(i,j)\in I$.

  * •

Let $(\mathbb{Z}^{|I|})^{\vee}$ denote a dual lattice of $\mathbb{Z}^{|I|}$,
and $\\{\omega_{i,j}\\}_{(i,j)\in I}$ denote a dual basis of
$\\{e^{i,j}\\}_{(i,j)\in I}$.

  * •

Let $M$ denote a sub lattice of $(\mathbb{Z}^{|I|})^{\vee}$ as follows:

$M=\\{\sum_{(i,j)\in
I}a_{i,j}\omega_{i,j}\in(\mathbb{Z}^{|I|})^{\vee}\mid\sum_{(i,j)\in
I}a_{i,j}=0\\}$

We remark that we can regard $M$ as a dual lattice of $N$.

  * •

Let $N^{\dagger}$ be a lattice of rank $n-1$, $\\{e^{\dagger}_{j}\\}_{-1\leq
j\leq n-3}$ be a basis of $N^{\dagger}$. Let $M^{\dagger}$ be a dual lattice
of $N^{\dagger}$, $\\{\eta_{j}\\}_{-1\leq j\leq n-3}$ be a dual basis of
$\\{e^{\dagger}_{j}\\}_{-1\leq j\leq n-3}$.

  * •

Let $\\{Y_{i}\\}_{0\leq i\leq n-3}$ be a homogeneous coordinate function of
$\mathbb{P}^{n-3}_{k}$. Let
$L_{i,j}\in\Gamma(\mathbb{P}^{n-3}_{k},\mathscr{O}_{\mathbb{P}^{n-3}_{k}}(1))$
be homogeneous functions as follows for $(i,j)\in I$:

$L_{i,j}=\left\\{\begin{array}[]{ll}Y_{i}&(i=j)\\\
Y_{i}-Y_{j}&(i<j)\end{array}\right.$

  * •

Let $\iota_{0}$ denote a closed immersion from $\mathbb{P}^{n-3}_{k}$ to
$\mathbb{P}^{|I|-1}_{k}$ defined by $\\{L_{i,j}\\}_{(i,j)\in I}$.

  * •

Let $\\{X_{i,j}\\}_{(i,j)\in I}$ be a homogeneous coordinate function of
$\mathbb{P}^{|I|-1}_{k}$.

  * •

Let $\mathcal{B}$ denote $\\{L_{i,j}\\}_{(i,j)\in I}$. We remark that
$\mathcal{B}$ generates $\Gamma(\mathbb{P}^{n-3}_{k},\mathscr{O}(1))$. Let
$\mathcal{V}$, $\mathcal{C}$, $\Delta(\mathcal{B})$, $Z$,
$(N,\Delta(\mathcal{B}),\iota,\Phi,\\{N_{\varphi}\\}_{\phi\in\Phi},\\{\Delta(\mathcal{B})_{\varphi}\\}_{\phi\in\Phi})$
be data defined as those in [14, Section 5] from the above notation.

  * •

Let $H\in k[y_{1},\ldots,y_{n-3}]$ be a polynomial defined as follows:

$H=\prod_{1\leq j\leq n-3}y_{j}\prod_{1\leq j\leq n-3}(1-y_{j})\prod_{1\leq
i<j\leq n-3}(y_{j}-y_{i})$

  * •

Let $o$ denote a ringmorphism from $k[y_{1},\ldots,y_{n-3}]_{H}$ to
$k[Z^{\circ}]$ defined as $o(y_{j})=\frac{Y_{j}}{Y_{0}}$ for any $0\leq j\leq
n-3$. We can check that $o$ is isomorphic. From now, we regard $k[Z^{\circ}]$
as $k[y_{1},\ldots,y_{n-3}]_{H}$. We remark that for any $(i,j)\in I$, the
following equation holds:

$i^{*}(\frac{X_{i,j}}{X_{0.0}})=\left\\{\begin{array}[]{ll}y_{i}&(i=j)\\\
y_{i}-y_{j}&(i<j)\end{array}\right.$

where $y_{0}=1$.

  * •

Let $e_{t}$ be a generator of $\mathbb{Z}$ and $\delta$ be a dual basis of
$\mathbb{Z}^{\vee}$.

  * •

Let $\frac{X_{i,j}}{X_{l,m}}\in k[M]$ denote a torus invariant mononomial
associated with $\omega_{i,j}-\omega_{l,m}\in M$, $x_{j}\in k[M^{\dagger}]$
denote a torus invariant mononomial associated with $\eta_{j}\in M^{\dagger}$,
and $t\in k[\mathbb{Z}^{\vee}]$ is a torus invariant mononomial associated
with $\delta\in\mathbb{Z}^{\vee}$.

  * •

Let $N^{\prime}$ denote $N\oplus N^{\dagger}$ and $\pi$ denote a first
projection $N^{\prime}\rightarrow N$. Let $s$ denote the section of $\pi$ such
that $s(v)=(v,0)$ for any $v\in N$.

  * •

Let $J$ denote $\\{(i,j)\in\mathbb{Z}^{2}\mid 0\leq i<j\leq n-1\\}$.

  * •

Let $M_{k}(n,2)$ denote a set of all $n\times 2$-matrices over $k$. A set
$M_{k}(n,2)$ has a natural $\mathrm{GL}_{k}(2)$-action, and we can identify
with $M_{k}(n,2)/\mathrm{GL}_{k}(2)$ and $\mathrm{Gr}_{k}(2,n)(k)$.

  * •

For $A=(a_{i,j})_{0\leq i\leq n-1,0\leq j\leq 1}\in M_{k}(n,2)$, and integers
$(i_{1},i_{2})\in J$, let $d_{i_{1},i_{2}}(A)$ denote
$a_{i_{1},0}a_{i_{2},1}-a_{i_{1},1}a_{i_{2},0}$.

  * •

Let $\\{W_{i_{1},i_{2}}\\}_{(i_{1},i_{2})\in J}$ be a homogeneous coordinate
function of $\mathbb{P}^{|J|-1}_{k}$.

  * •

Let $\mathrm{Pl}$ denote the following Plücker embedding:

$\displaystyle\mathrm{Pl}\colon\mathrm{Gr}_{k}(2,n)\ni[A=(a_{i,j})_{0\leq
i\leq n-1,0\leq j\leq 1}]\mapsto[d_{i,j}(A)]\in\mathbb{P}^{|J|-1}_{k}$

  * •

Let $U^{i,j}$ denote the affine open subset of $\mathbb{P}^{|J|-1}_{k}$
defined by $W_{i,j}\neq 0$ for $(i,j)\in J$, $U$ denote open subset
$\mathrm{Gr}_{k}(2,n)\cap U^{0,1}$ of $\mathrm{Gr}_{k}(2,n)$, and
$\mathrm{Gr}^{\circ}_{k}(2,n)$ denote open subset
$\mathrm{Gr}_{k}(2,n)\cap\bigcap_{(i,j)\in J}U^{i,j}$ of
$\mathrm{Gr}_{k}(2,n)$. We remark that we can identify with $U$ and
$\mathbb{A}^{2(n-2)}_{k}$ as follows:

$M_{k}(n,2)/\mathrm{GL}_{k}(2)\supset U(k)\ni\biggl{[}\begin{pmatrix}1&0\\\
0&1\\\ u_{0}&v_{0}\\\ u_{1}&v_{1}\\\ \vdots&\vdots\\\ u_{n-3}&v_{n-3}\\\
\end{pmatrix}\biggr{]}\\\
\longleftrightarrow(u_{0},v_{0},u_{1},v_{1},\ldots,u_{n-3},v_{n-3})\in\mathbb{A}^{2(n-2)}_{k}(k)$

Let $\xi$ denote the closed immersion $\mathbb{A}^{2(n-2)}_{k}\hookrightarrow
U^{0,1}$ defined by the above identification with $U$ and
$\mathbb{A}^{2(n-2)}_{k}$. We regard
$\\{u_{0},v_{0},\ldots,u_{n-3},v_{n-3}\\}$ as coordinate functions of
$\mathbb{A}^{2(n-2)}_{k}$. Let $\xi^{*}$ denote a subjective ring morphism
from $k[\\{\frac{W_{i,j}}{W_{0,1}}\\}_{(i,j)\in J}]$ to
$k[u_{0},v_{0},\ldots,u_{n-3},v_{n-3}]$ induced from $\xi$.

  * •

We define $\\{f_{i,j}\\}_{(i,j)\in J}\subset
k[u_{0},v_{0},\ldots,u_{n-3},v_{n-3}]$ as follows:

$f_{i,j}=\left\\{\begin{array}[]{ll}1&((i,j)=(0,1))\\\ v_{j-2}&(i=0,j>1)\\\
-u_{j-2}&(i=1)\\\ u_{i-2}v_{j-2}-u_{j-2}v_{i-2}&(i>1)\end{array}\right.$

We remark that $\xi^{*}(\frac{W_{i,j}}{W_{0,1}})=f_{i,j}$ for any $(i,j)\in J$
from the definition of $\mathrm{Pl}$.

  * •

Let $F\in k[u_{0},v_{0},\ldots,u_{n-3},v_{n-3}]$ be a polynomial defined as
$F=\prod_{(i,j)\in J}f_{i,j}$. We remark that the inclusion morphism
$k[u_{0},v_{0},\ldots,u_{n-3},v_{n-3}]\rightarrow
k[u_{0},v_{0},\ldots,u_{n-3},v_{n-3}]_{F}$ induces an open immersion
$\mathrm{Gr}^{\circ}_{k}(2,n)\hookrightarrow U$.

  * •

Let $\zeta$ denote a ring morphism from
$k[M^{\dagger}]\otimes_{k}k[Z^{\circ}]\rightarrow
k[u_{0},v_{0},\ldots,u_{n-3},v_{n-3}]_{F}$ as follows:

$\displaystyle\zeta(x_{-1})$ $\displaystyle=v_{0}u^{-1}_{0}$
$\displaystyle\zeta(x_{j})$ $\displaystyle=u_{j}$ $\displaystyle(0\leq j\leq
n-3)$ $\displaystyle\zeta(y_{j})$
$\displaystyle=u_{0}v^{-1}_{0}u^{-1}_{j}v_{j}$ $\displaystyle(1\leq j\leq
n-3)$

We can check that $\zeta$ is well-defined and isomorphic.

  * •

For $(i,j)\in J$, let $s_{i,j}\in k[M^{\dagger}]\otimes_{k}k[Z^{\circ}]$ be
elements as follows:

$s_{i,j}=\left\\{\begin{array}[]{ll}1&((i,j)=(0,1))\\\
x_{-1}x_{0}&((i,j)=(0,2))\\\ x_{-1}x_{j-2}y_{j-2}&(i=0,j>2)\\\
-x_{j-2}&(i=1)\\\ x_{-1}x_{0}x_{j-2}(y_{j-2}-1)&(i=2)\\\
x_{-1}x_{i-2}x_{j-2}(y_{j-2}-y_{i-2})&(i>2)\\\ \end{array}\right.$

We can check that $\zeta(s_{i,j})=f_{i,j}$ and $s_{i,j}$ is a unit of
$k[M^{\dagger}]\otimes_{k}k[Z^{\circ}]$ for any $(i,j)\in J$.

  * •

Let $d\geq 2$ be a positive integer. Let $S_{J,d}$ denote the following set:

$S_{J,d}=\\{\alpha=(\alpha_{i,j})\in\mathbb{Z}^{|J|}_{\geq
0}\mid\sum_{(i,j)\in J}\alpha_{i,j}=d\\}$

  * •

For $\alpha\in S_{J,d}$, let $s_{\alpha}$ denote $\prod_{(i,j)\in
J}s^{\alpha_{i,j}}_{i,j}\in k[M^{\dagger}]\otimes k[Z^{\circ}]$.

  * •

Let $X$ be a smooth closed sub variety of $\mathbb{P}^{|J|-1}_{k}$.

  * •

For $\alpha\in S_{J,d}$, let $\mathbb{W}^{\alpha}$ denote $\prod_{(i,j)\in
J}W_{i,j}^{\alpha_{i,j}}$.

  * •

Let $\\{t_{\alpha}\\}_{\alpha\in S_{J,d}}$ denote coordinate functions of
$\mathbb{A}^{|S_{J,d}|}_{k}$, $\mathscr{H}^{n}_{d}$ denote a closed subvariety
of $\mathbb{P}^{|J|-1}_{k}\times\mathbb{A}^{|S_{J,d}|}_{k}$ defined by
$\sum_{\alpha\in S_{J,d}}t_{\alpha}\mathbb{W}^{\alpha}=0$, and
$\mathscr{X}_{d}$ denote a closed subscheme
$X\times\mathbb{A}^{|S_{J,d}|}_{k}\cap\mathscr{H}^{n}_{d}$ of
$\mathbb{P}^{|J|-1}_{k}\times\mathbb{A}^{|S_{J,d}|}_{k}$.

  * •

Let $\theta$ be a composition of a closed immersion
$\mathscr{X}_{d}\hookrightarrow\mathbb{P}^{|J|-1}_{k}\times\mathbb{A}^{|S_{J,d}|}_{k}$
and the second projection
$\mathbb{P}^{|J|-1}_{k}\times\mathbb{A}^{|S_{J,d}|}_{k}\rightarrow\mathbb{A}^{|S_{J,d}|}_{k}$.

  * •

Let $u\colon S_{J,d}\rightarrow k[M^{\dagger}]\otimes_{k}k[Z^{\circ}]$ be a
map such that $u(\alpha)=s_{\alpha}$ for $\alpha\in S_{J,d}$. Then
$(Z_{\pi},S_{J,d},u)$ be a mock polytope of $Z_{\pi}$.

  * •

Let $J_{0},J_{1}$, and $J_{2}$ denote the following subset of $J$:

$\displaystyle J_{0}$ $\displaystyle=\\{(0,1)\\}$ $\displaystyle J_{1}$
$\displaystyle=\\{(i,j)\in J\mid i<2,j>1\\}$ $\displaystyle J_{2}$
$\displaystyle=\\{(i,j)\in J\mid i>1\\}$

We remark that $J=\coprod_{0\leq i\leq 2}J_{i}$.

  * •

For $\alpha\in S_{J,d}$, we define an integer $c_{0}(\alpha),c_{1}(\alpha)$,
and $c_{2}(\alpha)$ as follows:

$\displaystyle c_{0}(\alpha)$ $\displaystyle=\alpha_{0,1}$ $\displaystyle
c_{1}(\alpha)$ $\displaystyle=\sum_{(i,j)\in J_{1}}\alpha_{i,j}$
$\displaystyle c_{2}(\alpha)$ $\displaystyle=\sum_{(i,j)\in
J_{2}}\alpha_{i,j}$

  * •

Let $\kappa$ denote a map $S_{J,d}\rightarrow\mathbb{Z}$ defined as follows:

$\kappa(\alpha)=\left\\{\begin{array}[]{ll}0&(c_{1}(\alpha)=d)\\\
2(d-c_{1}(\alpha))-1&(c_{1}(\alpha)<d)\\\ \end{array}\right.$

  * •

For non-negative integers $a,b$, and $c$, let $S_{d_{0},d_{1},d_{2}}$ denote
the following subset of $S_{J,d}$:

$S_{d_{0},d_{1},d_{2}}=\\{\alpha\in
S_{J,d}\mid(c_{0}(\alpha),c_{1}(\alpha),c_{2}(\alpha))=(d_{0},d_{1},d_{2})\\}$

  * •

Let $\pi^{1}$ denote the morphism $N^{\prime}\oplus\mathbb{Z}\rightarrow
N\oplus\mathbb{Z}$ defined by $\pi^{1}=(\pi,\mathrm{id}_{\mathbb{Z}})$. Let
$s^{1}$ denote the section of $\pi^{1}$ such that $s^{1}((v,r))=(s(v),r)$ for
$(v,r)\in N\oplus\mathbb{Z}$.

  * •

Let $j^{\circ}$ denote the section of $\mathrm{pr}_{1}\colon
N\oplus\mathbb{Z}\rightarrow N$ such that $\mathrm{pr}_{2}\circ
j^{\circ}\equiv 0$.

  * •

Let $j^{\prime}$ denote the section of $\mathrm{pr}_{1}\colon
N^{\prime}\oplus\mathbb{Z}\rightarrow N^{\prime}$ such that
$\mathrm{pr}_{2}\circ j^{\prime}\equiv 0$.

  * •

Let $Z^{1}$ be a mock toric variety induced by $Z$ along
$(\mathrm{pr}_{1})_{*}\colon X(\Delta(\mathcal{B})\times\Delta_{!})\rightarrow
X(\Delta(\mathcal{B}))$ and $j^{\circ}$.

  * •

Let $\mathscr{K}$ denote the field of the puiseüx functions over $\mathbb{C}$.

Fundamentally, the following proposition shows the reason why finding one
nonstably rational hypersurface over $\mathcal{K}$ implies the existence of a
very general nonstably rational hypersurface over $k$.

######  Proposition 4.1.

The following statements follow:

  1. (a)

There exists a non-empty open subset $U_{X,d}$ of $\mathbb{A}^{|S_{J,d}|}_{k}$
such that the restriction
$\theta|_{\theta^{-1}(U_{X,d})}\colon\theta^{-1}(U_{X,d})\rightarrow U_{X,d}$