B(Y^{\prime}_{U^{\prime}}/U^{\prime})_{+}$

Thus the full cobordization and cobordization of proper birational locally
toric morphisms are functorial for regular morphisms up to torus factors.

###### Proof.

This is a direct consequence of Lemma 3.2.2 and definition of locally toric
morphisms ∎

####  3.3.3. Local description of cobordization of locally toric morphisms

######  Lemma 3.3.4.

Let $\pi:Y\to X$ be a proper birational locally toric morphism over a field
$\kappa$, and $\pi_{B}:B\to X$ be its full cobordization. Then

  1. (1)

$B_{+}\subset B$ is the natural open immersion.

  2. (2)

For any point $p\in X$ there is an open neighborhood $U$ of $p$, with a toric
chart $U\to X_{\sigma}$, and the torus

$T_{B\smallsetminus B_{U}}:=\operatorname{Spec}(\,\kappa[x_{i},x_{i}^{-1}\mid
E_{i}\subset B\smallsetminus B_{U}\,\,]\,),$

and an induced regular morphism

$B_{U}=\sigma^{-1}(U)=B(Y_{U}/U)\times T_{B\smallsetminus
B_{U}}\,\,\to\,\,X_{\sigma}\times{\mathbb{A}}^{k}\times T_{B\smallsetminus
B_{U}}$

  3. (3)

If $X$ is regular then $B$ is regular.

###### Proof.

By Lemma 3.2.2, the problem reduces locally to a toric situation via toric
chart $U\to X_{\sigma}$.

(1) By Lemma 2.3.5, $B(X_{\Delta}/X_{\sigma})_{+}\hookrightarrow
B(X_{\Delta}/X_{\sigma})$ is an open inclusion. Thus, by Lemma 3.2.2,
$B_{+}\subset B$ is also such.

(2) Also locally by Lemma 1.6.1, we can write $B_{U}=B(Y_{U}/U)\times
T_{B\smallsetminus B_{U}}$. On the other hand, by Lemma 3.2.2(4)
$B(Y_{U}/U)\to B(X_{\Delta}/X_{\sigma})$ is regular. Finitely by Lemma
2.3.5(2), $B(X_{\Delta}/X_{\sigma})=X_{\sigma}\times{\mathbb{A}}^{k}$.

(3) Follows from (2).

∎

###  3.4. Local description of cobordization

####  3.4.1. Cobordization of locally monomial maps

######  Definition 3.4.2.

Let $X$ be a locally monomial scheme over a field $\kappa$. We say that
$u_{1},\ldots,u_{k}$ is a locally toric system of parameters on $X$ if there
is a chart $\phi:U\to X_{\sigma}$, and a local system of toric parameters
$x_{1},\ldots,x_{k}$ on $X_{\sigma}$, such that $u_{i}=\phi^{*}(x_{i})$.

As a Corollary from Proposition 3.3.2, and Lemma 3.3.4 we obtain the
following:

######  Theorem 3.4.3.

Let $\pi:Y\to X$ be a proper birational locally toric morphism of locally
toric schemes over a field $\kappa$ . Then locally on $X$ we can write up to
torus factors

${\mathcal{A}}_{Y/X}=\pi_{*}({\mathcal{C}}_{Y/X})={\mathcal{O}}_{X}[t_{1}^{-1},\ldots,t_{k}^{-1},u_{1}{\bf
t}^{\alpha_{1}},\ldots,u_{k}{\bf t}^{\alpha_{k}}],$

where

  1. (1)

$u_{1},\ldots,u_{k}$ is a locally toric system of parameters on an open
$U\subset X$ defining a toric chart for the morphism $\pi$,

  2. (2)

${\bf t}^{\alpha_{i}}:=t_{1}^{a_{i1}}\cdot\ldots\cdot t_{k}^{a_{ik}}$, with
$a_{ij}:=\nu_{i}(u_{j})\geq 0$.

In particular, if $X$ is regular then $B$ and $B_{+}\subset B$ are regular.∎

###### Proof.

We use the fact from Lemma 3.2.2, that the valuations are locally monomial
with respect to $u_{1},\ldots,u_{k}$, and Lemma 2.3.5(1).

∎

####  3.4.4. The cobordization of monomial morphisms

Let $Y\to X$ be a proper birational locally toric morphism over $\kappa$.

Let $x_{1},\ldots,x_{n}$ be a system of local parameters at a point $p$ on a
locally toric $X$ defining a toric chart for a $Y\to X$. Then the full
cobordization of $\pi:Y\to X$ can be represented as:

$B=\operatorname{Spec}_{X}({\mathcal{O}}_{X}[t_{1}^{-1},\ldots,t_{k}^{-1},x^{\prime}_{1},\ldots,x^{\prime}_{n}]/(x^{\prime}_{1}{\bf
t}^{-\alpha_{1}}-x_{1},\ldots,x^{\prime}_{k}{\bf t}^{-\alpha_{k}}-x_{k}).$

Thus $B=V(x^{\prime}_{1}{\bf t}^{-\alpha_{1}}-x_{1},\ldots,x^{\prime}_{k}{\bf
t}^{-\alpha_{k}}-x_{k})\subset X\times{\mathbb{A}}^{n+k}$ is locally a closed
subscheme of $X\times{\mathbb{A}}^{n+k}$ defined by a system of local
parameters. It is regular for a regular $X$. Consequently, the full
cobordization $B\to X$ can be described by a single chart up to a torus factor
with the following coordinates:

  * •

$t_{i}^{-1}$ for $i=1,\ldots,k$ is the inverse of the coordinate $t_{i}$
representing the action of torus
$T=\operatorname{Spec}(\kappa[{\operatorname{Cl}}(Y/X)]=\operatorname{Spec}(\kappa[t_{1},t_{1}^{-1},\ldots,t_{n},t_{n}^{-1}])$.

  * •

$x_{i}^{\prime}=x_{i}\cdot{\bf t}^{\alpha_{i}}$ for $1\leq i\leq k$, and

  * •

$x_{j}^{\prime}=x_{j}$ for $j>n$.

The open subsets $B_{x^{\prime}_{i}}=B\smallsetminus V(x^{\prime}_{i})$,
associated with the forms $x^{\prime}_{i}=x_{i}{\bf t}^{\alpha_{i}}$ cover the
cobordization $B_{+}=B\smallsetminus V({\mathcal{I}}_{{\operatorname{irr}}})$
producing several ”charts” similarly to the standard blow-up. These open
affine subsets can be conveniently described by using toric geometry. They
correspond to the maximal faces of the decomposition $\Delta$ of the cone
$\sigma$ associated with the local toric chart.

If $\pi:Y\to X$ is the cobordant blow-up of a locally monomial
${\mathcal{J}}$, where $\operatorname{codim}(V({\mathcal{J}})\geq 2$, then the
subset $B_{+}$, by Lemma 1.4.4, can be described as $B_{+}=B\smallsetminus
V({\mathcal{J}}{\bf t}^{\alpha})$, where $\alpha=(a_{1},\ldots,a_{k})$, and
$a_{i}$ are the coefficients of the exceptional divisor $E=\sum a_{i}E_{i}$ of
$\pi:Y\to X$. In this case, the charts of $B_{+}$ can also be interpreted by
the vertices of the Newton polytope of ${\mathcal{J}}$.(See Example 2.6.14)

######  Remark 3.4.5.

In the particular case, when considering the stack-theoretic quotients of the
blow-up of a locally monomial ideal on a regular scheme, one obtains the
definition of a multiple weighted blow-up
${\operatorname{Bl}}_{{\mathcal{J}}}=[B_{+}\sslash T]$ introduced in [AQ21] by
Abramovich-Quek via the Satriano construction in [Sat13]. The more general
definition of ${\operatorname{Bl}}_{{\mathcal{J}},b}$ is discussed in Section
5.5.

####  3.4.6. Weighted cobordant blow-ups

Recall that the weighted stack-theoretic blow-ups were considered in the
context of resolution in [McQ19] and [ATW19]. The definition of the weighted
cobordant blow-up  was introduced in [Wło22]. One can view these notions from
the more general perspective of Cox cobordant blow-ups or the multiple
weighted blow-ups of Abramovich-Quek from [AQ21].

######  Definition 3.4.7.

Let $(x_{1},\ldots,x_{k})$ be a partial system of local parameters on a
regular scheme $X$. Let ${\mathcal{J}}$ be a center of the form
$(x_{1}^{a_{1}},\ldots,x_{k}^{a_{k}})$, where $a_{1}\leq a_{2}\leq\ldots\leq
a_{k}$ are positive integers, and $k>1$. Let $\pi:Y\to X$ be the normalized
blow-up of ${\mathcal{J}}$. By the weighted cobordant blow-up of
${\mathcal{J}}$ we mean the cobordization $B_{{\mathcal{J}}+}\to X$ of
$\pi:Y\to X$.

The corresponding monomial ideal on the toric chart
${\mathbb{A}}_{\kappa}^{k}$ defines a piecewise linear function
$G:=\min_{i}(a_{i}e_{i}^{*})$ on the regular coordinate cone $\sigma=\langle
e_{1},\ldots,e_{k}\rangle$, where $e_{i}^{*}(e_{j})=\delta_{ij}$. The
functions $a_{i}e_{i}^{*}$ determine the ray

$\rho:=\\{v\in\sigma\mid a_{1}e_{1}^{*}(v)=\ldots=a_{k}e_{k}^{*}(v)\\}.$

The ray $\rho$ is generated by the primitive vector

$w=(w_{1},\ldots,w_{k})=w_{1}e_{1}+\ldots+w_{k}e_{k},$

with relatively prime components and such that

$w_{1}a_{1}=\ldots=w_{k}a_{k}.$

The normalized blow-up of ${\mathcal{J}}$ is described by the decomposition
$\Delta$ of $\sigma$ into maximal subcones where $G$ is linear.

Thus $\Delta$ is the star subdivision $\rho\cdot\langle
e_{1},\ldots,e_{k}\rangle$ at a ray $\rho$. The vector $w$ determines the
valuation $\nu_{E}$ of the unique irreducible exceptional divisor.

Then, by Lemma 2.3.5(1), the full cobordant blow-up of $X$ at the center
${\mathcal{J}}$ is defined by

$B_{\mathcal{J}}=\operatorname{Spec}_{X}({{\mathcal{O}}_{X}}[{t}^{-1},{t}^{w_{1}}{x_{1}},\ldots,{t}^{w_{k}}{x_{k}}]).$

Here we have

$w_{i}=\nu_{E}(x_{i})=(w,e_{i}^{*})$

The cobordant weighted blow-up is simply
$(B_{\mathcal{J}})_{+}=B\smallsetminus V(\sigma^{\circ}({\mathcal{J}}))$,
where, by Lemma 1.4.4, we have
$\sigma^{\circ}({\mathcal{J}})={t}^{a}\cdot{\mathcal{J}}$, where
$a=\nu({\mathcal{J}})$. Thus

$\sigma^{\circ}({\mathcal{J}})=(x_{1}^{a_{1}}{t}^{a_{1}w_{1}},\ldots,x_{k}^{a_{k}}{t}^{a_{k}w_{k}}).$

We see that the cobordant weighted blow-up is the cobordization of an ordinary
toric weighted blow-up corresponding to the star subdivision at the center
$v\in\sigma$. We will discuss this construction in the context of the blow-ups
of valuative ${\mathbb{Q}}$-ideals in Section 5.4.

Observe that both notions: the one in Definition 3.4.7, and the one given by
the formula
$B_{\mathcal{J}}=\operatorname{Spec}_{X}({{\mathcal{O}}_{X}}[{t}^{-1},{t}^{w_{1}}{x_{1}},\ldots,{t}^{w_{k}}{x_{k}}]),$
as in [Wło22], are different in the trivial case $k=1$ and the blow-up of
$(x_{1}^{a_{1}})$. Then $Y\to X$ is an isomorphism, and $B=B_{+}=B_{-}\simeq
Y\simeq X$. However the formula from [Wło22] gives us

$B=\operatorname{Spec}_{X}({{\mathcal{O}}_{X}}[{t}^{-1},{t}^{w_{1}}{x_{1}}]),$

which defines the isomorphism of the quotients:

$B/G_{m}\simeq B_{+}/G_{m}\simeq B_{-}/G_{m}\simeq Y\simeq X.$

In this case, $B_{+}$ is a locally trivial $G_{m}$-bundle. So both
constructions of $B_{+}$ differ locally by the torus factor.

###  3.5. Geometric quotients for locally toric morphisms

In general, when considering the cobordization $B_{+}$ of a locally toric
morphism $\pi:Y\to X$, one obtains the good quotient $B_{+}\sslash T\simeq Y$.
Proposition 3.5.2, below shows that if $X$ is regular then, by replacing
$B_{+}$ with an open subset $B^{s}\subseteq B_{+}$ one obtains the geometric
quotient $B^{s}/T$ with a proper birational morphism $B^{s}/T\to B_{+}\sslash
T\simeq Y$.xConsequently, $B^{s}$ has a geometric quotient $B^{s}/T$ with
abelian quotient singularities and the transformation $B^{s}\to X$ can be be
used in the resolution instead of $B_{+}\to X$.

######  Lemma 3.5.1.

Let $\pi:Y\to X$ be a locally toric morphism, with $X$ regular.

Then its cobordization $B_{+}$ determines the geometric quotient $B_{+}\to
B_{+}/T\simeq Y$ iff $Y$ has abelian quotient singularities.

###### Proof.

The problem is local and can be reduced to the toric morphism
$\pi:X_{\Delta}\to X_{\sigma}$ corresponding to the subdivision $\Delta$ of a
regular cone $\sigma$. Then, by Lemma 2.3.5, the full cobordization $B$ of
$\pi$ is a regular scheme corresponding to the cone $\Sigma_{B}$, and $B_{+}$
is its open toric subscheme. The natural projection $\sigma_{B_{+}}\to\Delta$
corresponds to the geometric quotient iff $\Delta$ is a simplicial fan, and
thus $Y$ has abelian quotient singularities.

∎

######  Proposition 3.5.2.

Let $\pi:Y\to X$ be a proper birational locally toric morphism of locally
toric schemes over a field, with $X$ regular. Then
$B_{+}=\operatorname{Cox}(Y/X)_{+}\subset B=\operatorname{Cox}(Y/X)$ contains
open maximal subsets $B^{s}\subset B_{+}$ admitting geometric quotient
$B^{s}/T_{B}$ with the projective birational morphism

$B^{s}/T_{B}\to B_{+}/T_{B}=Y.$

###### Proof.

Let $E_{1},\ldots,E_{k}$ be the irreducible exceptional divisors of $\pi:Y\to
X$, and $\nu_{1},\ldots,\nu_{k}$ be the associated exceptional valuations on
$X$. By Lemma 3.2.2(5), the valuations are locally toric on $X$. Consider the
sequence of the blow-ups at the valuations $\nu_{i}$ as in [Wło20, Proposition
8.16.6]. These are precisely the normalized blow-ups of
${\mathcal{I}}_{\nu_{i},a,X}$ for a sufficiently divisible $a$.

Locally, in the compatible toric charts, $U\to X_{\sigma}$ the sequence of the
blow-ups correspond to a sequence of the star subdivisions at the vertices
${\operatorname{Vert}}(\Delta)\smallsetminus{\operatorname{Vert}}(\sigma)$
(see [Wło20, Lemma 7.3.9]). As the result we create a new subdivision
$\Delta^{\prime}$ of $\Delta$ with
${\operatorname{Vert}}(\Delta^{\prime})={\operatorname{Vert}}(\Delta)$. This
decomposition is simplicial. Indeed, let $\delta_{0}$ be any cone in
$\Delta^{\prime}$. By the property of the star subdivisions, for any vertex
$v_{0}\in{\operatorname{Vert}}(\delta)\smallsetminus{\operatorname{Vert}}(\sigma)$
one can write $\delta_{0}=\langle v_{0}\rangle+\delta_{1}$, where $\delta_{1}$
is a face of $\delta_{0}$ of codimension one in $\delta_{0}$, and $v_{0}$ is
linearly independent of ${\operatorname{Vert}}(\delta_{1})$. We can run this
argument inductively until we can represent $\delta_{0}$ as
$\delta_{0}=\langle v_{0},\ldots,v_{r}\rangle+\delta_{r}$, where
$v_{0},\ldots,v_{r}\in{\operatorname{Vert}}(\delta_{0})$ are linearly
independent of
${\operatorname{Vert}}(\delta_{r})\subset{\operatorname{Vert}}(\sigma)$. Thus
all the vertices of $\delta_{0}$ are linearly independent.

By construction, the valuations of the exceptional divisors corresponding to
${\operatorname{Vert}}(\Delta)\smallsetminus{\operatorname{Vert}}(\sigma)$
remain unchanged. Then $B(Y/X)=B(Y^{\prime}/X)$. On the other hand, by the
description of the toric case from Lemma 2.3.5(4), we obtain the open
inclusions

$B^{s}:=B(Y^{\prime}/X)_{+}\subseteq B(Y/X)_{+}=B_{+}\subseteq B(Y/X)=B.$

∎

##  4\. Cobordant resolution of singularities

###  4.1. The dual complex of the exceptional divisor

One can extend the considerations and the results from Section 2.5.1.

####  4.1.1. The exceptional divisor

Let $\pi:Y\to X$ be a proper birational locally toric morphism, where $X$ is a
regular scheme over a field $\kappa$, and $E_{1},\ldots,E_{k}$ be the
irreducible components of the exceptional divisor $E$ of $\pi$. Let
$\pi_{B}:B\to X$ be the full cobordization of $\pi$. By Theorem 4.4.5, $B$ is
regular and there is an SNC divisor $D=V_{B}(t_{1}^{-1}\cdot\ldots\cdot
t_{k}^{-1})$ with irreducible components $D_{i}=V_{B}(t_{i}^{-1})$. So is the
divisor $D_{+}=D_{|B_{+}}$ on $B_{+}$. Moreover, the exceptional divisor $E$
of $\pi:Y\to X$ is locally toric.

We can associate with the SNC divisors $D$ on $B$, $D_{+}$ on $B_{+}$, and
with the divisor $E$ on $Y$ the divisorial stratifications $S_{D}$,
$S_{D_{+}}$, and $S_{E}$, extending the definitions from Section 2.5.1. The
strata of $S_{D}$ are defined by the irreducible components of the locally
closed sets :

$\bigcap_{i\in I}D_{i}\smallsetminus\bigcup_{j\in J}D_{j},$

where $I\cup J=\\{1,\ldots,k\\}$. Replacing $D_{i}$ with $D_{i+}$ we obtain
the definition for $S_{D+}$. Consequently, any stratum $s\in S_{D_{+}}$
extends to a stratum in $S_{D}$.

The closures of the strata $s^{E}\in S_{E}$ are defined by the irreducible
components of the intersections $\bigcap_{i\in I}E_{i}$. The strata $s_{E}$
are obtained by removing from $\overline{s}^{E}$ the proper closed subsets
$\overline{s^{\prime}}^{E}$. The stratifications $S_{D}$, $S_{D_{+}}$ and
$S_{E}$ determine the dual simplicial complexes $\Delta_{D}$,
$\Delta_{D_{+}}$, and $\Delta_{E}$. Since $D$ and $D_{+}$ are SNC, the
simplices in $\Delta_{D}$, (respectively $\Delta_{D_{+}}$) are in the
bijective correspondence with the strata of $S_{D}$ (respectively of
$S_{D_{+}}$). Moreover, by the above, $\Delta_{D_{+}}$ is a subcomplex of
$\Delta_{D}$ corresponding to the strata of $S_{D}$ which intersect
$B_{+}\subset B$. Also, under this identification
${\operatorname{Vert}}(\Delta_{D})={\operatorname{Vert}}(\Delta_{D+})$.

The divisor $E$ is usually not SNC, and the strata alone do not determine the
faces of $\Delta_{E}$. The vertices $v_{i}$ of $\Delta_{E}$ correspond to the
divisors $E_{i}\leftrightarrow v_{i}$. The simplices $\sigma=\Delta(e_{i}\mid
i\in I)$ in $\Delta_{E}$ correspond to the pairs $(s^{E}_{\sigma},E_{\sigma})$
consisting of a stratum $s_{\sigma}\in S_{E}$ and a collection of divisors
$E_{\sigma}=\\{E_{i}\mid i\in I)$, such that $\overline{s_{\sigma}}^{E}$ is an
irreducible component of $\bigcap_{i\in I}E_{i}$. Thus, in this case, the
correspondence between the faces of $\Delta_{E}$ and the strata of $S_{E}$ is
not bijective, and the closures of strata could be represented by the
intersections of components $\bigcap_{i\in I}E_{i}$ defined by different
subsets $I$. (See also Section 2.5.1.)

Summarizing we have

######  Lemma 4.1.2.

A simplex $\sigma$ in $\Delta_{B}$ (respectively $\Delta_{D_{+}}$,
$\Delta_{E}$) is represented by a pair

$(\\{D_{i}\mid i\in I\\},\quad(\bigcap D_{i})_{0})$

consisting of a collection of the irreducible divisors $D_{i}$, (respectively
$D_{i+}$, $E_{i}$) which have a nonempty intersection and an irreducible
component $(\bigcap D_{i})_{0}$ of $\bigcap D_{i}$ (respectively $\bigcap
D_{i+}$, $\bigcap E_{i}$). ∎

######  Corollary 4.1.3.

With the previous assumptions and notations:

  1. (1)

There is a bijective correspondence between the divisors $D_{i}$, $D_{i+}$,
$E_{i}$, and the valuations $\nu_{i}$.

  2. (2)

    1. (a)

If $s\in S_{D}$ then $s$ is a component of a locally closed set

$\bigcap_{D_{i}\supseteq s}D_{i}\smallsetminus\bigcap_{D_{i}\not\supseteq
s}D_{i}$

    2. (b)

The image $\pi_{B}(s)$ is closed. It is an irreducible component of the closed
set

$\pi_{B}(\bigcap_{D_{i}\supseteq
s}D_{i}\smallsetminus\bigcap_{D_{i}\not\supseteq s}D_{i})=$
$\bigcap_{D_{i}\supseteq s}Z_{X}(\nu_{i})=\bigcap_{D_{i}\supseteq
s}\pi_{B}(D_{i})=\bigcap_{D_{i}\supseteq s}\pi(E_{i})$

where $Z_{X}(\nu)$ denotes the center of a valuation $\nu$ on $X$. Moreover,
the sets $\bigcap_{D_{i}\supseteq s}D_{i}$ are locally irreducible over $X$.

    3. (c)

The morphism $\pi_{B}$ determines a bijective correspondence between the
strata defined by the irreducible components of $\bigcap_{D_{i}\supseteq
s}D_{i}\smallsetminus\bigcap_{D_{i}\not\supseteq s}D_{i}$ and the irreducible
components of $\bigcap_{D_{i}\supseteq s}Z_{X}(\nu_{i})$.

  3. (3)

The morphism $\pi_{B_{+},Y}$ determines a bijective correspondence between the
components of $\bigcap_{i\in I}D_{i+}$ and the components of $\bigcap_{i\in
I}E_{i}$. This correspondence defines the isomorphism of the dual complexes
$\Delta_{D_{+}}\simeq\Delta_{E}$.

  4. (4)

The morphism of the stratifications $S_{D_{+}}\to S_{D}$ maps a stratum
$s_{+}$ of $S_{D_{+}}$ into an open subset of a stratum $s$ of $S_{D}$ . It
determines the inclusion of the dual complexes
$\Delta_{B_{+}}\hookrightarrow\Delta_{B}$.

###### Proof.

(1) The correspondence follows from Lemmas 1.2.9, and 1.2.11.

(2)-(5) By Lemma 3.2.2 and 3.2.4, we can reduce the situation locally to the
toric case, where we use Lemmas 2.5.3, and 2.5.6, and Corollaries 2.5.5 2.5.7
and 2.5.8.

∎

####  4.1.4. Dual complex of valuations of a locally toric morphism

Let $N=\\{\nu_{1},\ldots,\nu_{k}\\}$ be the set of the exceptional valuations
of $\pi:Y\to X$. The vertices of ${\operatorname{Vert}}(\Delta_{E})$, and thus
of ${\operatorname{Vert}}(\Delta_{B})$ and
${\operatorname{Vert}}(\Delta_{B+})$ are in the bijective correspondence with
the valuations in $N$, and the exceptional divisors $E_{i}$, $D_{i+}$, and
$D_{i}$:

$\nu_{i}\leftrightarrow E_{i}=Z_{Y}(\nu_{i})\leftrightarrow
D_{i+}\leftrightarrow D_{i}.$

Consequently, one can associate with the faces of $\Delta_{E}$, $\Delta_{B+}$,
and $\Delta_{B}$ the subsets of $N$. This determines the complexes
$\Delta^{N}_{B},\Delta^{N}_{B+},\Delta^{N}_{E}$, called the dual valuation
complexes, together with natural isomorphisms of the simplicial complexes

$\Delta_{B}\to\Delta^{N}_{B},\quad,\Delta_{B_{+}}\to\Delta^{N}_{B+},\quad\Delta_{E}\to\Delta^{N}_{E}$

Then, by Lemma 4.1.3, $\Delta^{N}_{E}=\Delta_{B_{+}}$ determine the same
subcomplex of $\Delta^{N}_{B}$.

The simplices of the valuation complexes will be called the valuation faces.
The valuation faces come with natural face inclusions inherited from
$\Delta^{N}_{B},\Delta^{N}_{B+},\Delta^{N}_{E}$.

By Lemma 4.1.3 we get:

######  Lemma 4.1.5.

  1. (1)

A valuation face $\sigma$ in $\Delta^{N}_{B}$ is represented by a pair
$(\omega,Z^{0}_{X}(\omega))$ defined by the collection of valuations
$\omega=\omega_{\sigma}$ in $N$, such that

$Z_{X}(\omega):=\bigcap_{\nu\in\omega}Z_{X}(\nu)\neq\emptyset,$

and an irreducible component $Z^{0}_{X}(\omega)$ of $Z_{X}(\omega)$.

  2. (2)

A simplex $\sigma$ of $\Delta^{N}_{E}=\Delta^{N}_{B+}$ corresponds to a subset
$\omega\subset N$, such that

$Z_{Y}(\omega):=\bigcap_{\nu\in\omega}Z_{Y}(\nu)\neq\emptyset,$

and an irreducible component, denoted as $Z^{0}_{Y}(\omega)$ of the set

$Z_{Y}(\omega):=\bigcap_{\nu\in\omega}Z_{Y}(\nu).$

The face relations are given by the inclusions of the sets of valuations and
the associated components. ∎

######  Remark 4.1.6.

Thus, the dual valuation complexes could be thought of as ordinary dual
complexes of the exceptional divisors with the associated valuation structure
so that the vertices define the relevant exceptional valuations, and the faces
determine the sets of the valuations.

####  4.1.7. Dual complex associated with a locally monomial ideal

If ${\mathcal{J}}$ is locally monomial ideal on a regular scheme, such that
$\operatorname{codim}(V({\mathcal{I}}))\geq 2$, then one can associate with
${\mathcal{J}}$ the normalized blow-up $\pi:Y\to X$, and the full cobordant
blow-up $\pi_{B}:B\to X$ of ${\mathcal{J}}$. The morphism $\pi:Y\to X$ is
locally toric, and we shall call the dual complexes $\Delta_{D}$,
$\Delta_{D+}\simeq\Delta_{E}$ and the corresponding dual valuation complexes
$\Delta^{N}_{D}$, $\Delta^{N}_{D+}\simeq\Delta^{N}_{E}$ associated with
${\mathcal{J}}$.

###  4.2. Graded rings defined by the valuations

####  4.2.1. Graded rings defined by valuations

In the considerations below, let $\omega=\\{\nu_{1},\ldots,\nu_{r}\\}$ be a
set of valuations on a regular scheme $X$. We associate with each valuation
$\nu_{i}$ a dummy variable $t_{i}$ for $i=1,\ldots,r$. Set

${\bf t}:=(t_{1},\ldots,t_{k})\quad\mbox{and}\quad{\bf
t}^{-1}:=(t_{1}^{-1},\ldots,t_{k}^{-1}).$

Consider the partial componentwise order on ${\mathbb{Z}}_{\geq o}^{r}$. For
$\alpha:=(a_{1},\ldots,a_{r})\in{\mathbb{Z}}^{r}_{\geq 0}$ we define the
ideals

(5)
$\displaystyle{\mathcal{J}}_{\omega}^{\alpha}:=\bigcap_{\nu_{i}\in\omega}{\mathcal{I}}_{\nu_{i},a_{i}}\subset{\mathcal{O}}_{X},\quad\quad{\mathcal{J}}_{\omega}^{>\alpha}:=\sum_{\beta>\alpha}{\mathcal{J}}_{\omega}^{\beta}.$

This determines the ${\mathbb{Z}}^{k}_{\geq 0}$-graded Rees algebra

${\mathcal{A}}_{\omega}:=\bigoplus_{a\in{\mathbb{Z}}_{\geq
0}}{\mathcal{J}}_{\omega}^{\alpha}{\bf
t}^{\alpha}\subset{\mathcal{O}}_{X}[t],$

where ${\bf t}^{\alpha}=t_{1}^{a_{1}}\cdot\ldots\cdot t_{r}^{a_{r}}$, and the
associated gradation

(6)
$\displaystyle{\operatorname{gr}}_{\omega}({\mathcal{O}}_{X})=\bigoplus_{a\in{\mathbb{Z}}_{\geq
0}}({\mathcal{J}}_{\omega}^{\alpha}/{\mathcal{J}}_{\omega}^{>\alpha}){\bf
t}^{\alpha}={\mathcal{A}}_{\omega}/({\mathcal{A}}_{\omega}\cap{\bf
t}^{-1}{\mathcal{A}}_{\omega}])={\mathcal{A}}_{\omega}[{\bf t}^{-1}]/({\bf
t}^{-1}\cdot{\mathcal{A}}_{\omega}[{\bf t}^{-1}]).$

In particular, for $\alpha=0=(0,\ldots,0)$ we have locally on $X$:

${\mathcal{J}}_{\omega}:={\mathcal{J}}_{\omega}^{>0}={\mathcal{I}}_{Z_{X}(\omega)},$

where

$Z_{X}(\omega):=\bigcap_{i=1}^{k}Z_{X}(\nu_{i}).$

Then ${\operatorname{gr}}_{\omega}({\mathcal{O}}_{X})$ is a sheaf of graded
${\mathcal{O}}_{X}/{\mathcal{J}}_{\omega}={\mathcal{O}}_{V({\mathcal{J}}_{\omega})}$-modules.

######  Lemma 4.2.2.

Assume the valuations in the set $\omega=\\{\nu_{1},\ldots,\nu_{r}\\}$ are
monomial for a certain partial system of local parameters $u_{1},\ldots,u_{n}$
on a regular scheme $X$. Then

  1. (1)

${\mathcal{J}}_{\omega}=\sum_{i=1}^{k}{\mathcal{I}}_{\nu_{i},1}=(u_{j}\mid\nu_{i}(u_{j})>0,{\rm
for\,\,some}\,\,\nu_{i}\in\omega)$ , and

  2. (2)

${\operatorname{gr}}_{\omega}({\mathcal{O}}_{X})={\mathcal{O}}_{V({\mathcal{J}}_{\omega})}[u_{1}{\bf
t}^{\alpha_{1}},\ldots,u_{k}{\bf t}^{\alpha_{k}}],$ where
$u_{i}\in{\mathcal{J}}^{\alpha_{i}}_{\omega}\smallsetminus{\mathcal{J}}_{\omega}^{>\alpha_{i}}$,
and $\alpha_{i}=(a_{i1},\ldots,a_{in})$, with
$\nu_{i}(u_{j})=a_{ij}\in{\mathbb{Z}}_{\geq 0}$ for $i=1,\ldots,r$, and
$j=1,\ldots,n$

###### Proof.

(1) Note that ${\mathcal{I}}_{Z_{X}(\nu_{i})}=(u_{j}\mid\nu_{i}(u_{j})>0)$.
Thus

${\mathcal{J}}_{\omega}={\mathcal{I}}_{Z_{X}(\omega)}=\sum_{\nu\in\omega}{\mathcal{I}}_{Z_{X}(\nu_{i})}=(u_{j}\mid\nu_{i}(u_{j})>0,{\rm
for\,\,some}\,\,\nu_{i}\in\omega)$

(3) By definition of $A_{\omega}$, the equality (5), and the Proof of Lemma
2.3.5(1).

$\displaystyle{\mathcal{A}}_{\omega}[{\bf
t}^{-1}]=(\bigoplus_{a_{i}\in{\mathbb{Z}}_{\geq
0}}\,\,\bigcap^{r}_{i=1}\,{\mathcal{I}}_{\nu_{i},a_{i}}\,\,\cdot
t_{1}^{a_{1}}\cdot\ldots\cdot t_{r}^{a_{r}})[t_{1}^{-1},\ldots,t_{r}^{-1}]=$
$\displaystyle={\mathcal{O}}_{X}[t_{1}^{-1},\ldots,t_{r}^{-1},u_{1}{\bf
t}^{\alpha_{1}},\ldots,u_{n}{\bf t}^{\alpha_{n}}])={\mathcal{O}}_{X}[{\bf
t}^{-1},u_{1}{\bf t}^{\alpha_{1}},\ldots,u_{n}{\bf t}^{\alpha_{k}}],$

where $\alpha_{i}=(a_{i1},\ldots,a_{in})$, and
$\nu_{i}(u_{j})=a_{ij}\in{\mathbb{Z}}_{\geq 0}$ for $i=1,\ldots,r$, and
$j=1,\ldots,n$. Thus by the equality (6):

$\displaystyle{\operatorname{gr}}_{\omega}({\mathcal{O}}_{X})={\mathcal{A}}_{\omega}[{\bf
t}^{-1}]/({\bf t}^{-1}\cdot{\mathcal{A}}_{\omega}[{\bf t}^{-1}]=$
$\displaystyle=({\mathcal{O}}_{X}[{\bf t}^{-1},u_{1}{\bf
t}^{\alpha_{1}},\ldots,u_{n}{\bf t}^{\alpha_{k}}])/({\bf t}^{-1})\simeq$
$\displaystyle=({\mathcal{O}}_{X}/{\mathcal{J}}_{\omega})[u_{1}{\bf
t}^{\alpha_{1}},\ldots,u_{n}{\bf t}^{\alpha_{k}}].$

∎

We shall call the corresponding scheme

${\mathbb{N}}_{\omega}(X):=\operatorname{Spec}_{V({\mathcal{J}}_{\omega})}({\operatorname{gr}}_{\omega}({\mathcal{O}}_{X}))=\operatorname{Spec}_{Z_{X}(\omega)}({\operatorname{gr}}_{\omega}({\mathcal{O}}_{X}))$

the weighted normal bundle of $X$ at the set of valuations $\omega$.

One can extend this to any valuation face in $\Delta_{B}^{N}$, associated with
a full cobordant blow-up $B\to X$.

######  Definition 4.2.3.

By the weighted normal bundle of $X$ at the valuation face
$\omega\in\Delta_{B}^{N}$ we mean the scheme

${\mathbb{N}}_{\omega}(X):=\operatorname{Spec}_{Z^{0}_{X}(\omega)}({\operatorname{gr}}_{\omega}({\mathcal{O}}_{X}))$

over the component $Z^{0}_{X}(\omega)$ of $Z_{X}(\omega)$ associated with the
face $\omega$.

As $\Delta_{E}^{N}\subset\Delta_{B}^{N}$ the above definition is also valid
for any valuation face $\omega\in\Delta_{E}^{N}$.

####  4.2.4. The ideals of the initial forms

With any function $f\in{\mathcal{O}}_{X,p}$, regular at $p\in
V({\mathcal{J}})$, such that
$f\in{\mathcal{J}}_{\omega}^{\alpha}\smallsetminus{\mathcal{J}}_{\omega}^{>\alpha}$,
for a certain $a\in{\mathbb{N}}$ one can associate the unique homogenous
element, called the initial form

$\operatorname{in}_{\omega}(f)=(f+{\mathcal{J}}_{\omega}^{>\alpha})\in({\mathcal{J}}_{\omega}^{\alpha}/{\mathcal{J}}_{\omega}^{>\alpha}){\bf
t}_{\omega}^{\alpha}\subset{\operatorname{gr}}_{{}_{\omega}}({\mathcal{O}}_{X}).$

Similarly, we associate with an ideal sheaf ${\mathcal{I}}$, the filtration
${\mathcal{I}}_{\omega}^{\alpha}:={\mathcal{I}}\cap{\mathcal{J}}_{\omega}^{\alpha}$
and set
${\mathcal{I}}^{>\alpha}_{\omega}={\mathcal{I}}\cap{\mathcal{J}}_{\omega}^{>\alpha}$.

We define the ideal of the initial forms of ${\mathcal{I}}$ to be the ideal

$\operatorname{in}_{\omega}({\mathcal{I}})=\bigoplus_{\alpha\in{\mathbb{Z}}{\geq
0}}{\mathcal{I}}_{\omega}^{\alpha}/{\mathcal{I}}^{>\alpha}_{\omega}=\bigoplus_{\alpha\in{\mathbb{Z}}_{\geq
0}}({\mathcal{I}}^{\alpha}_{\omega}+{\mathcal{J}}_{\omega}^{>\alpha})/{\mathcal{J}}_{\omega}^{>\alpha}\subset{\operatorname{gr}}_{{\omega}}({\mathcal{O}}_{X})$

on $N_{{\mathcal{J}}}(X)$.

For the ideal sheaf ${\mathcal{I}}$, its weak ideal of the initial forms on
$N_{{\mathcal{J}}}(X)$ is given by

$\operatorname{in}^{\circ}_{\omega}({\mathcal{I}})={\operatorname{gr}}_{{\omega}}({\mathcal{O}}_{X})\cdot{\mathcal{I}}_{\omega}^{\alpha_{0}}/{\mathcal{I}}^{>\alpha_{0}}_{\omega}\subset{\operatorname{gr}}_{{\omega}}({\mathcal{O}}_{X}),$

where ${\mathcal{I}}\subset{\mathcal{J}}_{\omega}^{\alpha_{0}}$, and
${\mathcal{I}}\not\subset{\mathcal{J}}_{\omega}^{>\alpha_{0}}$.

######  Remark 4.2.5.

For any function $f\in{\mathcal{O}}_{X}$,

$\operatorname{in}_{\omega}(f)=\operatorname{in}_{\omega}({\mathcal{O}}_{X}\cdot
f)=\operatorname{in}^{\circ}_{\omega}({\mathcal{O}}_{X}\cdot f).$

####  4.2.6. Composition of gradations

######  Lemma 4.2.7.

Let $\omega=\\{\nu_{1},\ldots,\nu_{r}\\}$ be a set of valuations which are
monomial for a common partial system of local parameters $u_{1},\ldots,u_{n}$
on a regular $X$. Consider its partition into subsets
$\omega_{1}=\\{\nu_{1},\ldots,\nu_{s}\\}$, and
$\omega_{2}=\\{\nu_{s+1},\ldots,\nu_{r}\\}$. Let ${\bf
t}:=(t_{1},\ldots,t_{r})$ ( respectively ${\bf
t}_{\omega_{1}}:=(t_{1},\ldots,t_{s})$) be the set of the unknowns $t_{i}$
associated to the valuations $\nu_{i}\in\omega$, ( respectively and
$\nu_{i}\in\omega_{1}$). Let
${\mathcal{J}}_{\omega_{1}}=(u_{1},\ldots,u_{\ell})$. Then

  1. (1)

The set $\omega_{2}=\\{\nu_{s+1},\ldots,\nu_{r}\\}$ determines the set of
monomial valuations
$\overline{\omega_{2}}=\\{\overline{\nu}_{s+1},\ldots,\overline{\nu}_{r}\\}$
on the multi-graded ring

${\operatorname{gr}}_{\omega_{1}}({\mathcal{O}}_{X})={\mathcal{O}}_{V({\mathcal{J}}_{\omega_{1}})}[u_{1}{\bf
t}_{\omega_{1}}^{\alpha_{1}},\ldots,u_{\ell}{\bf
t}_{\omega_{1}}^{\alpha_{n}}]\simeq{\mathcal{O}}_{V({\mathcal{J}}_{\omega_{1}})}[u_{1},\ldots,u_{\ell}],$

with the ideals

${\mathcal{I}}_{\overline{\nu}_{i},a_{i}}=\operatorname{in}_{\omega_{1}}({\mathcal{I}}_{\nu_{i},a_{i}})$

  2. (2)

${\operatorname{gr}}_{\overline{\omega_{2}}}({\operatorname{gr}}_{\omega_{1}}({\mathcal{O}}_{X})\simeq{\operatorname{gr}}_{\omega}({\mathcal{O}}_{X}).$

  3. (3)

If ${\mathcal{I}}\subset{\mathcal{O}}_{X}$ then

    1. (a)

$\operatorname{in}_{\omega}({\mathcal{I}})=\operatorname{in}_{\overline{\omega_{2}}}(\operatorname{in}_{\omega_{1}}({\mathcal{I}})).$

    2. (b)

$\operatorname{in}^{\circ}_{\omega}({\mathcal{I}})=\operatorname{in}^{\circ}_{\overline{\omega_{2}}}(\operatorname{in}^{\circ}_{\omega_{1}}({\mathcal{I}})).$

###### Proof.

(1) For $j\leq\ell$, $\operatorname{in}_{\omega_{w_{1}}}(u_{j})$ is identified
with $u_{j}$ in
${\mathcal{O}}_{V({\mathcal{J}}_{\omega_{1}})}[u_{1},\ldots,u_{\ell}]$.
Otherwise if $j>\ell$, then $\operatorname{in}_{\omega_{w_{1}}}(u_{j})$ is a
parameter in
${\mathcal{O}}_{V({\mathcal{J}}_{\omega_{1}})}={\mathcal{O}}_{X}/{\mathcal{J}}_{\omega_{1}}={\mathcal{O}}_{X}/(u_{1},\ldots,u_{\ell})$.
Consequently $\nu_{j}$ determine the monomial valuations $\overline{\nu}_{j}$
on ${\operatorname{gr}}_{\omega}({\mathcal{O}}_{X})$ with
$\operatorname{in}_{\omega_{1}}({\mathcal{I}}_{\nu_{j},a})=({\mathcal{I}}_{\overline{\nu}_{j},a}).$

(2) and (3) For the multiindex $\alpha=(\alpha_{1},\alpha_{2})$, where
$\alpha_{i}$ correspond to $\omega_{i}$ for $i=1,2$, consider a function
$f\in{\mathcal{J}}_{\omega}^{\alpha}\smallsetminus{\mathcal{J}}_{\omega}^{>\alpha}$:

$f\in{\mathcal{J}}_{\omega}^{\alpha}\smallsetminus{\mathcal{J}}_{\omega}^{>\alpha}={\mathcal{J}}_{\omega_{1}}^{\alpha_{1}}\cap{\mathcal{J}}_{\omega_{2}}^{\alpha_{2}}\smallsetminus({\mathcal{J}}_{\omega_{1}}^{>\alpha_{1}}\cap{\mathcal{J}}_{\omega_{2}}^{\alpha_{2}}+{\mathcal{J}}_{\omega_{1}}^{\alpha_{1}}\cap{\mathcal{J}}_{\omega_{2}}^{>\alpha_{2}})$

The ideal
$\operatorname{in}_{\omega_{1}}({\mathcal{J}}_{\omega_{2}}^{\alpha_{2}})\subset{\operatorname{gr}}_{\omega_{1}}({\mathcal{O}}_{X})$
is homogenous and $\operatorname{in}_{\omega_{1}}(f)$ is in
$\alpha_{1}$-gradation of
$\operatorname{in}_{\omega_{1}}({\mathcal{J}}_{\omega_{2}}^{\alpha_{2}})_{\alpha_{1}}\subset({\operatorname{gr}}_{\omega_{1}}({\mathcal{O}}_{X}))_{\alpha_{1}}$:

$\operatorname{in}_{\omega_{1}}({\mathcal{J}}_{\omega_{2}}^{\alpha_{2}})_{\alpha_{1}}=\frac{{\mathcal{J}}_{\omega_{1}}^{\alpha_{1}}\cap{\mathcal{J}}_{\omega_{2}}^{\alpha_{2}}}{{\mathcal{J}}_{\omega_{1}}^{>\alpha_{1}}\cap{\mathcal{J}}_{\omega_{2}}^{\alpha_{2}}}\subseteq({\operatorname{gr}}_{\omega_{1}}({\mathcal{O}}_{X}))_{\alpha_{1}}=\frac{{\mathcal{J}}_{\omega_{1}}^{\alpha_{1}}}{{\mathcal{J}}_{\omega_{1}}^{>\alpha_{1}}}$

and

$\operatorname{in}_{\omega_{1}}({\mathcal{J}}_{\omega_{2}}^{>\alpha_{2}})_{\alpha_{1}}=\frac{{\mathcal{J}}_{\omega_{1}}^{\alpha_{1}}\cap{\mathcal{J}}_{\omega_{2}}^{>\alpha_{2}}+{\mathcal{J}}_{\omega_{1}}^{>\alpha_{1}}\cap{\mathcal{J}}_{\omega_{2}}^{\alpha_{2}}}{{\mathcal{J}}_{\omega_{1}}^{>\alpha_{1}}\cap{\mathcal{J}}_{\omega_{2}}^{\alpha_{2}}}$

Consequently, by the above,

$\displaystyle\operatorname{in}_{\overline{\omega_{2}}}(\operatorname{in}_{\omega_{1}}(f))\in\operatorname{in}_{\overline{\omega_{2}}}(\operatorname{in}_{\omega_{1}}({\mathcal{J}}_{\omega_{2}}^{\alpha_{2}})_{\alpha_{1}})=\frac{(\operatorname{in}_{\omega_{1}}({\mathcal{J}}_{\omega_{2}}^{\alpha_{2}}))_{\alpha_{1}}}{(\operatorname{in}_{\omega_{1}}({\mathcal{J}}_{\omega_{2}}^{>\alpha_{2}}))_{\alpha_{1}}}=$
$\displaystyle=\frac{{\mathcal{J}}_{\omega_{1}}^{\alpha_{1}}\cap{\mathcal{J}}_{\omega_{2}}^{\alpha_{2}}}{{\mathcal{J}}_{\omega_{1}}^{\alpha_{1}}\cap{\mathcal{J}}_{\omega_{2}}^{>\alpha_{2}}+{\mathcal{J}}_{\omega_{1}}^{>\alpha_{1}}\cap{\mathcal{J}}_{\omega_{2}}^{\alpha_{2}}}=\frac{{\mathcal{J}}_{\omega}^{\alpha}}{{\mathcal{J}}_{\omega}^{>\alpha}}=({\operatorname{gr}}_{\omega_{1}}({\mathcal{O}}_{X}))_{\alpha}$

On the other hand the initial form

$\operatorname{in}_{\omega}(f)\in\frac{{\mathcal{J}}_{\omega}^{\alpha}}{{\mathcal{J}}_{\omega}^{>\alpha}}=({\operatorname{gr}}_{\omega_{1}}({\mathcal{O}}_{X}))_{\alpha},$

determines the same element:

$\displaystyle\operatorname{in}_{\omega}(f)=\operatorname{in}_{\overline{\omega_{2}}}(\operatorname{in}_{\omega_{1}}(f))\in({\operatorname{gr}}_{\omega}({\mathcal{O}}_{X}))_{\alpha}={\operatorname{gr}}_{\overline{\omega_{2}}}(({\operatorname{gr}}_{\omega_{1}}({\mathcal{O}}_{X}))_{\alpha_{1}})_{\alpha_{2}},$

which implies (3).

∎

###  4.3. The weighted normal bundles at valuations

The following extends a classical result of Huneke-Swanson on extended Rees
algebras and smooth blow-ups [HS06, Definition 5.1.5], and the recent results
of Rydh in [QR19] and Włodarczyk in [Wło22, Lemma 5.1.4] on the weighted
normal cone.

######  Lemma 4.3.1.

Let $\pi:Y\to X$ be a locally toric proper birational morphism to a regular
scheme $X$ over a field $\kappa$, with the exceptional components $E_{i}$, for
$i=1,\ldots,k$ and let $\pi_{B}:B\to X$ be its full cobordization. Then for
any stratum $s=s_{\omega}\in S_{D}$ of the exceptional divisor
$D=V(t_{1}^{-1}\cdot\ldots\cdot t_{k}^{-1})$ on $B$ and the corresponding
valuation face $\omega$ in $\Delta_{B}^{N}$ there is an isomorphism:

$s\simeq{\mathbb{N}}_{\omega}(X)\times\check{\bf t}_{{\omega}}.$

where $T_{\check{\bf t}_{\omega}}:=\operatorname{Spec}(\kappa[\check{\bf
t}_{\omega},\check{\bf t}^{-1}_{\omega}])$, for the set $\check{\bf
t}_{\omega}$ of the unknowns corresponding to the remaining exceptional
valuations which are not in $\omega$.

###### Proof.

We can replace $X$ with its open subset and assume $Z_{X}(\omega)$ is
irreducible so that $\pi_{B}(s)=Z^{0}_{X}(\omega)=Z_{X}(\omega)$ .

By separating variables into ${\bf t}_{\omega}$ and $\check{\bf t}_{\omega}$
we can factor any monomial ${\bf t}^{\alpha}={\bf
t}_{\omega}^{\alpha}\cdot\check{\bf t}_{\omega}^{\alpha}$ uniquely into the
product of the relevant monomials ${\bf t}_{\omega}^{\alpha}$ and $\check{\bf
t}_{\omega}^{\alpha}$ respectively in ${\bf t}_{\omega}$ and $\check{\bf
t}_{\omega}$. Then by Corollary 4.1.3(2), we can write $\overline{s}=V({\bf
t}^{-1}_{\omega})$ in a neighborhood of $\overline{s}$, and $s=V({\bf
t}^{-1}_{\omega})\smallsetminus V(\check{\bf t}^{-1}_{\omega})$ as there is
only one component of $V({\bf t}^{-1}_{\omega})$ mapping to
$\pi_{B}(s)=Z^{0}_{X}(\omega)=Z_{X}(\omega)$.

By definition $B=\operatorname{Spec}({\mathcal{A}}[\bf{t}^{-1}])$, where

${\mathcal{A}}=\bigoplus_{\alpha\in{\mathbb{Z}}^{k}_{\geq
0}}{\mathcal{J}}^{\alpha}{\bf
t}^{\alpha},\quad{\mathcal{J}}^{\alpha}:=\bigcap^{k}_{i=1}{\mathcal{I}}_{\nu_{i},a_{i}}\subset{\mathcal{O}}_{X}.$

Let

${\mathcal{A}}_{\omega}=\bigoplus_{\alpha}{\mathcal{J}}^{\alpha}_{\omega}{\bf
t}^{\alpha}_{\omega},\quad{\mathcal{J}}^{\alpha}_{\omega}:=\bigcap_{{\nu_{i}\in\omega}}{\mathcal{I}}_{\nu_{i},a_{i}}\subset{\mathcal{O}}_{X}.$

Thus for the open subset $B_{\omega}\subset B$ where $\check{\bf
t}_{\omega}^{-1}$ are invertible we can write:

$B_{\omega}:=B_{\check{\bf
t}_{\omega}^{-1}}=\operatorname{Spec}_{X}({\mathcal{A}}[\bf{t}^{-1}][\check{\bf
t}_{\omega}])=\operatorname{Spec}_{X}({\mathcal{A}}_{\omega}[{\bf
t}_{\omega}^{-1}])[\check{\bf t}_{\omega},\check{\bf t}_{\omega}^{-1}])$

Consequently $s=V_{B_{\omega}}({\bf t}_{\omega}^{-1})\subset B_{\omega}$, by
Corollary 4.1.3(2), so we can write

$\displaystyle{\mathcal{O}}_{s}={\mathcal{A}}_{\omega}[{\bf
t}_{\omega}^{-1}][\check{\bf t}_{\omega},\check{\bf t}_{\omega}^{-1}]/({\bf
t}_{\omega}^{-1})=({\mathcal{A}}_{\omega}[{\bf t}_{\omega}^{-1}]/(({\bf
t}_{\omega}^{-1}\cdot{\mathcal{A}}_{\omega}[{\bf
t}_{\omega}^{-1}]))[\check{\bf t}_{\omega},\check{\bf t}_{\omega}^{-1}]=$
$\displaystyle=({\mathcal{A}}_{\omega}/(({\bf
t}_{\omega}^{-1}\cdot{\mathcal{A}}_{\omega})\cap{\mathcal{A}}_{\omega}))[\check{\bf
t}_{\omega},\check{\bf
t}_{\omega}^{-1}]={\operatorname{gr}}_{\omega}({\mathcal{O}}_{X})[\check{\bf
t}_{\omega},\check{\bf t}_{\omega}^{-1}].$

The latter equality follows from Section 4.2.1. ∎

####  4.3.2. The weak and the strict transforms and the ideal of the initial
forms

The identification from Lemma 4.3.1 can be extended to the strict transforms
of the ideals. The following generalizes the result from [Wło22, Lemma 5.1.4]
for the weighted blow-ups.

######  Lemma 4.3.3.

Let $X$ be a regular scheme over a field Let $B\to X$ be the full cobordant
blow-up of a locally monomial center ${\mathcal{J}}$. Let
${\mathcal{I}}\subset{\mathcal{O}}_{X}$ be an ideal sheaf on $X$. Let
$\sigma^{s}({\mathcal{I}})\subset{\mathcal{O}}_{B}$ be the strict transform of
${\mathcal{I}}$, and $\sigma^{\circ}({\mathcal{I}})\subset{\mathcal{O}}_{B}$
be its weak transform (see Definition 1.4.2). Then for any $s\in S_{D}$, the
natural isomorphism

${\mathcal{O}}_{s}\simeq{\mathcal{O}}_{B}[\check{\bf t}_{\omega},\check{\bf
t}_{\omega}^{-1}]/({\bf
t}_{\omega}^{-1})\to{\operatorname{gr}}_{\omega}({\mathcal{O}}_{X})[\check{\bf
t}_{\omega},\check{\bf t}_{\omega}^{-1}]$

takes

  1. (1)

$\sigma^{s}({\mathcal{I}})_{|s}$ onto
$\operatorname{in}_{\omega}({\mathcal{I}})[\check{\bf t}_{\omega},\check{\bf
t}_{\omega}^{-1}]\subset{\operatorname{gr}}_{\omega}({\mathcal{O}}_{X})[\check{\bf
t}_{\omega},\check{\bf t}_{\omega}^{-1}]$.

  2. (2)

$\sigma^{\circ}({\mathcal{I}})_{|s}$ onto
$\operatorname{in}^{\circ}_{\omega}({\mathcal{I}})[\check{\bf
t}_{\omega},\check{\bf
t}_{\omega}^{-1}]\subset{\operatorname{gr}}_{\omega}({\mathcal{O}}_{X})[\check{\bf
t}_{\omega},\check{\bf t}_{\omega}^{-1}]$.

###### Proof.

Let $f\in{\mathcal{I}}$ such that
$f\in{\mathcal{J}}^{\alpha}\smallsetminus{\mathcal{J}}^{>{\alpha}}$. By the
definition of

${\mathcal{O}}_{B}=(\bigoplus_{\alpha\geq 0}{\mathcal{J}}^{\alpha}{\bf
t}^{\alpha})[{\bf t}^{-1}]$

we conclude that $\sigma^{s}(f)={\bf t}^{\alpha}f.$

Then
$f\in{\mathcal{J}}^{\alpha}_{\omega}\smallsetminus{\mathcal{J}}_{\omega}^{>{\alpha}}$,
and in a neighborhood of $s$ we have that $\check{\bf t}^{-1}_{\omega}$ is
invertible. Then the strict transform

$\sigma^{s}(f)={\bf t}^{\alpha}f={\bf t}_{\omega}^{\alpha}\check{\bf
t}_{\omega}^{\alpha}f\in{\mathcal{J}}^{\alpha}{\bf
t}^{\alpha}\subset{\mathcal{J}}_{\omega}^{\alpha}{\bf
t}_{\omega}^{\alpha}[\check{\bf t}_{\omega},\check{\bf t}_{\omega}^{-1}],$

and its reduction modulo $({\bf
t}_{\omega}^{-1}{\mathcal{O}}_{B}\cap{\mathcal{J}}_{\omega}^{{\alpha}}){\bf
t}^{\alpha}={\mathcal{J}}_{\omega}^{>{\alpha}}{\bf t}^{\alpha}$ can be written
as the homogenous element

$\sigma^{s}(f)={\bf t}^{\alpha}f+{\bf
t}^{\alpha}{\mathcal{J}}_{\omega}^{{>a}}={\bf t}_{\omega}^{\alpha}\check{\bf
t}_{\omega}^{\alpha}f+{\bf t}_{\omega}^{\alpha}\check{\bf
t}_{\omega}^{\alpha}\cdot{\mathcal{J}}_{\omega}^{{>a}}$

in

${\mathcal{O}}_{B}[\check{\bf t}_{\omega},\check{\bf t}_{\omega}^{-1}]/({\bf
t}_{\omega}^{-1}\cdot{\mathcal{O}}_{B}[\check{\bf t}_{\omega},\check{\bf
t}_{\omega}^{-1}])={\operatorname{gr}}_{\omega}({\mathcal{O}}_{X})[\check{\bf
t}_{\omega},\check{\bf t}_{\omega}^{-1}]$

in the gradation

$\displaystyle{{\mathcal{J}}^{\alpha}_{\omega}{\bf
t}_{\omega}^{\alpha}[\check{\bf t}_{\omega},\check{\bf
t}_{\omega}^{-1}]}/{({\bf t}_{\omega}^{-1}{\mathcal{O}}_{B}[\check{\bf
t}_{\omega},\check{\bf t}_{\omega}^{-1}]\cap{\mathcal{O}}_{B}[\check{\bf
t}_{\omega},\check{\bf t}_{\omega}^{-1}])}$
$\displaystyle=({\mathcal{J}}^{\alpha}_{\omega}/{\mathcal{J}}_{\omega}^{{>a}}){\bf
t}_{\omega}^{\alpha}[\check{\bf t}_{\omega},\check{\bf
t}_{\omega}^{-1}]\subset{\operatorname{gr}}_{\omega}({\mathcal{O}}_{X})[\check{\bf
t}_{\omega},\check{\bf t}_{\omega}^{-1}]$

On the other hand $f$ determines its initial form

$\operatorname{in}_{\omega}(f)=(f+{\mathcal{J}}_{\omega}^{>\alpha}){\bf
t}^{\omega}\in({\mathcal{J}}_{\omega}^{\alpha}/{\mathcal{J}}_{\omega}^{>\alpha}){\bf
t}^{\alpha}_{\omega},$

and thus, by the above $\sigma^{s}(f)$ naturally and bijectively corresponds
to

$\check{\bf
t}_{\omega}^{\alpha}\operatorname{in}_{\omega}(f)\in({\mathcal{J}}^{\alpha}_{\omega}/{\mathcal{J}}_{\omega}^{>{\alpha}}){\bf
t}_{\omega}^{\alpha}[\check{\bf t}_{\omega},\check{\bf
t}_{\omega}^{-1}]\subset{\operatorname{gr}}_{\omega}({\mathcal{O}}_{X})[\check{\bf
t}_{\omega},\check{\bf t}_{\omega}^{-1}].$

The latter differs from $\operatorname{in}_{\omega}(f)$ by the unit
$\check{\bf t}_{\omega}^{\alpha}$:

$\check{\bf
t}_{\omega}^{\alpha}\operatorname{in}_{\omega}(f)\sim\operatorname{in}_{\omega}(f).$

∎

###  4.4. Cobordant resolution by locally monomial centers

####  4.4.1. Weighted normal cone

######  Definition 4.4.2.

Let $X$ be a regular scheme over a field. Let $Y\subset X$ be a closed reduced
subscheme with the ideal ${\mathcal{I}}_{Y}$. Let $\omega$ be a set of
monomial valuations for a partial system of local parameters. The subscheme
$C_{\omega}(Y)=V(\operatorname{in}_{\omega}{\mathcal{I}}_{Y})\subset{\mathbb{N}}_{\omega}(X)$
will be called the weighted normal cone of $Y$ at $\omega$.

######  Lemma 4.4.3.

Let $X$ be a regular universal catenary scheme over a field. Let $Y\subset X$
be a subscheme of pure codimension $d$. Let $\omega$ be a set of monomial
valuations for a common partial local system of parameters
$u_{1},\ldots,u_{k}$ on $X$. Then $C_{\omega}(Y)$ is of pure codimension $d$
in ${\mathbb{N}}_{\omega}(X)$.

###### Proof.

Let $\omega=\\{\nu_{1},\ldots,\nu_{r}\\}$ and
$\omega_{1}=\\{\nu_{1},\ldots,\nu_{r-1}\\}$ be its subset. Then, by Lemma
4.2.7, we can write

$\operatorname{in}_{\omega}({\mathcal{I}})=\operatorname{in}_{\nu_{r}}(\operatorname{in}_{\omega_{1}}({\mathcal{I}})),$

where $\nu_{r}$ is monomial on

${\mathbb{N}}_{\omega_{1}}(X)=\operatorname{Spec}({\operatorname{gr}}_{\omega_{1}}({\mathcal{O}}_{X}))=\operatorname{Spec}({\mathcal{O}}_{V({\mathcal{J}}_{\omega_{1}})}[u_{1},\ldots,u_{\ell}])$

is also universally catenary. Here we assume without loss of generality that
${\mathcal{J}}_{\omega_{1}}=u_{1},\ldots,u_{\ell}$ for $\ell\leq k$.

By the inductive argument for $\operatorname{in}_{\omega_{1}}({\mathcal{I}})$
on ${\mathbb{N}}_{\omega_{1}}(X)$ we can reduce the situation to a single
monomial valuation $\nu=\nu_{r}$.

Let $\nu(u_{1})=w_{1},\ldots,\nu(u_{k})=w_{k}$, and find some integers
$a_{1},\ldots,a_{k}$ such that

$a_{1}w_{1}=\ldots=a_{k}w_{k},$

Consider the full cobordant blow-up $B$ of
${\mathcal{I}}=(u_{1}^{a_{1}},\ldots,u_{k}^{a_{k}})$

$B=\operatorname{Spec}_{X}({\mathcal{O}}_{X}[{t}^{-1},u_{1}{t}^{w_{1}},\ldots,u_{k}{t}^{w_{k}}].$

We apply the argument from [Wło22, Theorem 5.2.1]. By the assumption, $B$ is
catenary. Let $d$ be the codimension of $Y$ in $X$. Then for the morphism

$\pi_{B_{-}}:B_{-}=B\smallsetminus V({t}^{-1})=X\times{\mathbb{G}}_{m}\to X,$

the inverse image $\pi_{B_{-}}^{-1}(Y)$ is of pure codimension $d$ in $B_{-}$.
So it is its scheme-theoretic closure $Y^{\prime}:=\overline{\pi_{B_{-}}(Y)}$,
which is the strict transform $V(\sigma^{s}({\mathcal{I}}))$ of $Y$.

Note that ${t}^{-1}$ is not a zero divisor in

$Y^{\prime}=V(\sigma^{s}({\mathcal{I}}))=\operatorname{Spec}({\mathcal{O}}_{B}/\sigma^{s}({\mathcal{I}})),$

since ${t}^{-1}f\in\sigma^{s}({\mathcal{I}})$ implies
$f\in\sigma^{s}({\mathcal{I}})$, by definition of the strict transform.

Then, by the Krull Hauptidealsatz, we have that each component of
$Y^{\prime}\cap V({t}^{-1})$ is of codimension $1$ in $Y^{\prime}$, and of
codimension $d+1$ in $B$. We conclude that each component of

$Y^{\prime}\cap
V({t}^{-1})=V(O_{{t}^{-1}}\cdot\sigma^{s}({\mathcal{I}}))=C_{\nu}(Y)\subset{\mathbb{N}}_{\nu}(X)$

is of codimension $d$ in $V({t}^{-1})={\mathbb{N}}_{\nu}(X)$.

∎

####  4.4.4. Cobordant resolution

For any scheme $Y$, let ${\operatorname{Sing}}(Y)$ denote its singular locus.
For any ideal ${\mathcal{I}}$ on $X$ by
${\operatorname{Sing}}(V({\mathcal{I}}))$ we mean the singular locus of the
scheme

$V({\mathcal{I}})=\operatorname{Spec}_{X}({\mathcal{O}}_{X}/{\mathcal{I}}).$

The following theorem extends [Wło22, Theorem 5.2.2].

######  Theorem 4.4.5.

Let $X$ be a regular universally catenary scheme over a field. Let $Y\subset
X$ be an integral, closed subscheme of pure codimension $d$ defined by
${\mathcal{I}}_{Y}$.

Assume there is a locally monomial ideal
${\mathcal{J}}\supset{\mathcal{I}}_{Y}$ on $X$, with the cosupport
$V({\mathcal{J}})$ of codimension $\geq 2$, and with the associated
exceptional divisor $E$ on the normalized blow-up
$\sigma:Y=bl_{{\mathcal{J}}}(X)\to X$, and the dual valuation complex
$\Delta_{E}^{N}$ such that

  1. (1)

${\operatorname{Sing}}(V({\mathcal{I}}_{Y}))\subseteq V({\mathcal{J}})$.

  2. (2)

For any valuation face $\omega\in\Delta^{N}_{E}\subset\Delta^{N}_{B}$, and the
ideal
$\operatorname{in}_{\omega}({\mathcal{I}}_{Y})\subset{\operatorname{gr}}_{{\mathcal{J}}_{\omega}}({\mathcal{O}}_{X})$111Definition
4.2.3 we have

${\operatorname{Sing}}_{{\mathbb{N}}_{\omega}(X)}(V(\operatorname{in}_{\omega}{\mathcal{I}}_{Y}))\subseteq
V(\operatorname{in}^{\circ}_{\omega}{\mathcal{J}}).$

(respectively
(2’) For any valuation face $\omega\in\Delta^{N}_{E}\subset\Delta^{N}_{B}$, we
have

${\operatorname{Sing}}_{{\mathbb{N}}_{\omega}(X)}(V(\operatorname{in}^{\circ}_{\omega}{\mathcal{I}}_{Y}))\subseteq
V(\operatorname{in}^{\circ}_{\omega}{\mathcal{J}}).)$

Then the cobordant blow-up $B_{+}\to X$ of ${\mathcal{J}}$ defines a cobordant
resolution of $Y$. That is, the strict transform
$Y^{\prime}=V(\sigma^{s}({\mathcal{I}}_{Y}))$ of $Y$ (respectively the weak
transform $Y^{\prime}=V(\sigma^{\circ}({\mathcal{I}}_{Y}))$ of $Y$) is a
regular subscheme of $B_{+}$ of the codimension equal to the codimension of
$Y$ in $X$.

###### Proof.

The problem is local on $X$. Thus, up to a torus factor, we can assume that
the full cobordant blow-up of ${\mathcal{J}}$ is given locally on $X$ by

$\sigma:B=\operatorname{Spec}({\mathcal{O}}_{X}[{\bf t}^{-1},{\bf
t}^{\alpha_{1}}u_{1},\ldots,{\bf t}^{\alpha_{k}}u_{k}])\to X.$

Then for the restriction morphism $\pi_{B-}:B_{-}=X\times T\to X$, the inverse
image $\pi_{B_{-}}^{-1}(Y)$ is irreducible of codimension $d$. So is its
closure
$Y^{\prime}:=\overline{\sigma^{-1}(Y)}=V(\sigma^{c}({\mathcal{I}}_{Y}))$,
which is the strict transform of $Y$. Since $V({\mathcal{J}})$ is of
codimension $\geq 2$, the divisor $D=V_{B}({\bf t}^{-1})$ is exceptional for
$B\to X$. Observe that

${\operatorname{Sing}}(Y^{\prime})\smallsetminus
D={\operatorname{Sing}}(Y^{\prime})\cap B_{-}\subset
V_{B_{-}}^{\circ}({\mathcal{J}})\subseteq
V_{B}(\sigma^{\circ}({\mathcal{J}}))$

On the other hand, the exceptional divisor $D_{+}=D_{|B_{+}}$ is the union of
the strata $s_{+}\in S_{D+}$. By Corollary 4.1.3(4),(5), each such a stratum
$s_{+}$ extends to $s\in S_{D}$, and corresponds to the valuation face
$\omega\in\Delta^{N}_{D_{+}}=\Delta^{N}_{E}\subset\Delta^{N}_{D}$.

Since the singular locus of $V(\operatorname{in}_{\omega}{\mathcal{I}}_{Y})$
is contained in $V(\operatorname{in}_{\omega}^{\circ}({\mathcal{J}}))$ and by
Lemmas 4.3.1, 4.3.3, we have

${\operatorname{Sing}}(Y^{\prime}\cap
s)={\operatorname{Sing}}_{{\mathbb{N}}_{\omega}(X)}(V(\operatorname{in}_{\omega}{\mathcal{I}}_{Y}))\times\check{\bf
t}_{\omega}\subseteq
V(\operatorname{in}^{\circ}_{\omega}{\mathcal{J}})\times\check{\bf
t}_{\omega}=V_{B}(\sigma^{\circ}({\mathcal{J}})_{|s}),$

Then using Lemmas 4.4.3 and 4.3.1 we conclude that the subscheme

$Y^{\prime}\cap s\simeq
V(\operatorname{in}_{\omega}{\mathcal{I}}_{Y}))\times\check{\bf t}_{\omega}$

is of pure codimension $d$ in $s\simeq{\mathbb{N}}_{\omega}(X)\times\check{\bf
t}_{\omega}$, and

$(Y^{\prime}\cap s)\cap B_{+}=((Y^{\prime}\cap s)\smallsetminus
V_{B}(\sigma^{\circ}({\mathcal{J}}))$

is regular of codimension $d$ in $s$.

Hence for $p\in((Y^{\prime}\cap s)\smallsetminus
V_{B}(\sigma^{\circ}({\mathcal{J}}))$, we can find parameters $v_{1},\ldots
v_{d}\in{\mathcal{O}}_{s,p}\cdot{\mathcal{I}}_{Y^{\prime}}=({\mathcal{O}}_{B,p}\cdot{\mathcal{I}}_{Y^{\prime}})/({\bf
t}_{\omega}^{-1})$ at $p$ which vanish on $Y^{\prime}\cap s$. But these
parameters come from local parameters in ${\mathcal{I}}_{Y^{\prime}}$ on $B$
at $p$. So they define a regular subscheme $Y^{\prime\prime}$ of $B_{+}$ of
codimension $d$, containing locally $Y^{\prime}$. Thus $Y^{\prime\prime}$
locally coincides with $Y^{\prime}$ which must be regular at $p\in
s\smallsetminus V_{B}(\sigma^{\circ}({\mathcal{J}}))$. Consequently
${\operatorname{Sing}}(Y^{\prime})$ is contained in
$V_{B}(\sigma^{\circ}({\mathcal{J}}))$, and, by Lemma 1.4.4, $Y^{\prime}$ is a
regular subscheme of $B_{+}=B\smallsetminus
V_{B}(\sigma^{\circ}({\mathcal{J}}))$ of codimension $d$.

The proof for the weak transform $\sigma^{\circ}({\mathcal{I}})$ (with
stronger assumptions in condition (2’)) is identical.

∎

As a corollary, we obtain the following:

######  Theorem 4.4.6.

Let $X$ be a smooth variety over a field $\kappa$ of any characteristic. Let
$Y\subset X$ be a closed integral subscheme of $X$ Assume there is a locally
monomial ideal ${\mathcal{J}}\supset{\mathcal{I}}_{Y}$ on $X$ , with the
cosupport $V({\mathcal{J}})$ of codimension $\geq 2$, and with the associated
exceptional divisor $E$ on the normalized blow-up
$\sigma:Y=bl_{{\mathcal{J}}}(X)\to X$, and the dual valuation complex
$\Delta_{E}^{N}$ such that

  1. (1)

${\operatorname{Sing}}(V({\mathcal{I}}_{Y}))\subseteq V({\mathcal{J}})$.

  2. (2)

For any valuation face $\omega\in\Delta^{N}_{E}$, the ideal the singular locus

${\operatorname{Sing}}_{{\mathbb{N}}_{\omega}(X)}(V(\operatorname{in}_{\omega}{\mathcal{I}}_{Y}))\subseteq
V(\operatorname{in}^{\circ}_{\omega}{\mathcal{J}}).$

Then there is a resolution of $Y$ at $Z$, that is, a projective birational
morphism $\phi:Y^{res}\to Y$ from a smooth variety $Y^{res}$ with the
exceptional locus $Z\subset Y$, such that $\phi^{-1}(Z)$ is an SNC divisor on
$Y^{\prime}$.

###### Proof.

Take the cobordant resolution $B_{+}\to X$ from Theorem 4.4.5. We use Section
3.4.4 to embed cobordant blow-up $B_{+}$ as a smooth subspace of the relative
affine space ${\mathbb{A}}^{n}_{X}$. This implies that $B_{+}\sslash T$ is
locally toric.

The locally toric singularities of $B_{+}\sslash T$ can be canonically
resolved by the combinatorial method of [Wło20, Theorem 7.17.1]. This produces
the projective birational resolution $Y^{\prime}\to Y$ of $Y$ such that the
inverse image of the singular point is an SNC divisor.

∎

###  4.5. Resolution of hypersurfaces via the Newton method

####  4.5.1. The Newton polytope of a monomial ideal

Let
$X={\mathbb{A}}^{k}_{Z}=\operatorname{Spec}({\mathcal{O}}_{Z}[x_{1},\ldots,x_{k}]),$
where $Z$ is a smooth scheme over $\kappa$, and
${\mathcal{I}}\subset{\mathcal{O}}_{Z}[x_{1},\ldots,x_{k}]$ is an ideal. One
can extend the notion of the Newton polytope of monomial ideals $I=({\bf
x}^{\alpha_{1}},\ldots,{\bf x}^{\alpha_{k}})\subset\kappa[x_{1},\ldots,x_{n}]$
considered previously in Section 2.6.1 in the case
$Z=\operatorname{Spec}(\kappa)$, where $\kappa$ is a field.

As before, by the associated Newton polytope of ${\mathcal{I}}$ we mean

${\rm P}_{\mathcal{I}}:={\operatorname{conv}}(\alpha_{1}+{\mathbb{Q}}_{\geq
0}^{n},\ldots,\alpha_{k}+{\mathbb{Q}}_{\geq 0}^{n})\subseteq{\mathbb{Q}}_{\geq
0}^{n}$

Conversely, with a polytope $P\subset{\mathbb{Q}}_{\geq 0}^{n}$ we associate
the monomial ideal

${\mathcal{I}}_{P}:=(x^{\alpha}\mid\alpha\in P\cap{\mathbb{Z}}^{n}).$

####  4.5.2. The initial forms defined by faces

######  Definition 4.5.3.

(see [AQ21]) If ${\mathcal{I}}=(x^{\alpha})_{\alpha\in
A}\subset{\mathcal{O}}_{Z}[x_{1},\ldots,x_{n}]$ is a monomial ideal and
$P_{\mathcal{I}}$ is its Newton polytope and $P\leq P_{\mathcal{I}}$ be its
face, we define the initial form with respect to a face of the $P_{I}$ to be:

${\operatorname{inv}}_{P}({\mathcal{I}}):=({\bf x}^{\alpha}\mid{\alpha\in
A\cap P}).$

The definition ${\operatorname{inv}}_{P}({\mathcal{I}})$ is a particular case
of the notion o the initial form
${\operatorname{inv}}^{\circ}_{\omega}({\mathcal{I}})$ with respect to a
valuation face $\omega\in\Delta^{N}_{E}$.

By Corollary 2.6.13 we obtain

######  Lemma 4.5.4.

Let $\Delta^{N}_{E}$ be the valuation dual complex associated with a monomial
ideal ${\mathcal{I}}$, and let $P=P_{\mathcal{I}}$ be its Newton polytope.

Any valuation face $\omega\in\Delta^{N}_{E}$ of the associated dual valuation
complex $\Delta^{N}_{E}$ defines the induced face

$P_{\omega}:=P\cap\bigcap_{\nu\in\omega}H_{\nu}$

of the Newton polytope $P$, for the supporting hyperplanes $H_{\nu}$
associated with $\nu$ and we have:

${\operatorname{inv}}^{\circ}_{\omega}({\mathcal{I}})={\operatorname{inv}}_{P_{\omega}}({\mathcal{I}}).$

Conversely for any supporting face ${P^{\prime}}$ of $P$ there is a valuation
face $\omega\in\Delta^{N}_{E}$, such that
${\operatorname{inv}}^{\circ}_{\omega}({\mathcal{I}})={\operatorname{inv}}_{P_{\omega}}({\mathcal{I}}).$
∎

######  Remark 4.5.5.

The above correspondence is not bijective. Several valuation faces $\omega$
could define the same supporting face of $P$. The information encoded in the
dual valuation complex is richer and can be applied to a more general setting.

####  4.5.6. The Newton polytopes of polynomials and ideals

By the Newton polytope of the function

$f=\sum c_{\alpha}{\bf x}^{\alpha}\in{\mathcal{O}}(Z)[x_{1},\ldots,x_{k}],$

where $c_{\alpha}\in{\mathcal{O}}(Z)$ we mean the Newton polytope of the
monomial ideal

${\mathcal{J}}_{f}:=(x^{\alpha}\mid c_{\alpha}\neq 0),$

generated by the exponents $\alpha$ occurring in the presentation of $f$ with
nonzero coefficients. Note that ${\mathcal{J}}_{f}$ is the smallest monomial
ideal which contains $f$.

This definition can be extended to any ideal
${\mathcal{I}}\subset{\mathcal{O}}_{Z}[x_{1},\ldots,x_{k}]$. We associate with
${\mathcal{I}}$ the monomial ideal ${\mathcal{J}}={\mathcal{J}}_{\mathcal{I}}$
generated by ${\mathcal{I}}_{f}$, where $f\in{\mathcal{I}}$. The Newton
polytope $P_{\mathcal{I}}$ of ${\mathcal{I}}$ is simply the Newton polytope of
the monomial ideal ${\mathcal{J}}$.

If $P\leq P_{\mathcal{I}}$ is a face of the Newton polytope $P_{\mathcal{I}}$,
and $f=\sum_{\alpha\in A}c_{\alpha}{\bf x}^{\alpha}\in{\mathcal{I}}$ we put

${\operatorname{inv}}_{P}(f):=\sum_{\alpha\in A\cap P}c_{\alpha}{\bf
x}^{\alpha}.$

Then ${\operatorname{inv}}_{P}({\mathcal{I}})$ is the ideal generated by
${\operatorname{inv}}_{P}(f)$, where $f\in{\mathcal{I}}$.

Let $\Delta^{N}_{E}$ be the dual valuation complex associated with a monomial
ideal ${\mathcal{J}}\subset{\mathcal{O}}_{Z}[x_{1},\ldots,x_{k}]$. Recall
that, by Section 4.2.4, and using identification :
${\operatorname{gr}}_{\omega}{\mathcal{O}}_{Z}[x_{1},\ldots,x_{k}]={\mathcal{O}}_{Z}[x_{1},\ldots,x_{k}]$,
for any valuation face $\omega\in\Delta^{N}_{E}$ we write

${\operatorname{inv}}_{\omega}(f):=\sum_{\alpha\in A_{\omega,f}}c_{\alpha}{\bf
x}^{\alpha}\in{\operatorname{gr}}_{\omega}({\mathcal{O}}_{Z}[x_{1},\ldots,x_{k}])={\mathcal{O}}_{Z}[x_{1},\ldots,x_{k}].$

where

$A_{\omega,f}=\\{\alpha\in A\mid\nu(x^{\alpha})=\nu(f),\nu\in\omega\\}.$

Similarly for the ideal
${\mathcal{I}}\subset{\mathcal{O}}_{Z}[x_{1},\ldots,x_{k}]$ the ideal of the
initial forms ${\operatorname{inv}}_{\omega}^{\circ}({\mathcal{I}})$ is
generated by all ${\operatorname{inv}}_{\omega}(f)$, where
$f\in{\mathcal{I}}$, and $\nu(f)=\nu({\mathcal{I}})$ for all $\nu\in\omega$.

The following is an immediate consequence of Lemma 4.5.4, and the above:

######  Lemma 4.5.7.

Let $P_{f}$ (respectively $P_{\mathcal{I}}$) be the Newton polytope of $f=\sum
c_{\alpha}x^{\alpha}$ (respectively of an ideal
${\mathcal{I}}\subset{\mathcal{O}}_{Z}[x_{1},\ldots,x_{k}]$), and let
${\mathcal{J}}_{f}$ (respectively ${\mathcal{J}}_{\mathcal{I}}$) be the
associated monomial ideal. Then for any valuation face
$\omega\in\Delta^{N}_{E}$ of the associated dual valuation complex
$\Delta^{N}_{E}$ and the corresponding face $P_{\omega}$ of $P$.

${\operatorname{inv}}_{\omega}(f)={\operatorname{inv}}_{P_{\omega}}(f).$
$\mbox{(respectively}\quad{\operatorname{inv}}_{\omega}^{\circ}({\mathcal{I}})={\operatorname{inv}}_{P_{\omega}}({\mathcal{I}})).$

∎

####  4.5.8. Resolution by the Newton polytopes

The following is a particular case of Theorem 4.4.5 for hypersurfaces, written
in a more straightforward setup.

######  Theorem 4.5.9.

$X={\mathbb{A}}^{n}_{Z}=\operatorname{Spec}{\mathcal{O}}_{Z}[x_{1},\ldots,x_{k}]$,
where $Z$ is a regular scheme over a field $\kappa$. Let

$f=\sum_{\alpha\in A_{f}}c_{\alpha}{\bf
x}^{\alpha}\in{\mathcal{O}}_{Z}[x_{1},\ldots,x_{k}]$

where $c_{\alpha}\neq 0$ for $\alpha\in A_{f}$. Let ${\mathcal{J}}=({\bf
x}^{\alpha}\mid\alpha\in A_{f})^{{\rm sat}}$ be the induced monomial ideal,
and $P_{f}=P_{\mathcal{J}}$ be its Newton polytope. Assume that

  1. (1)

The cosupport $V({\mathcal{J}})$ is of codimension $\geq 2$,

  2. (2)

${\operatorname{Sing}}(V((f))\subseteq V({\mathcal{J}}).$

  3. (3)

For any supporting face $P$ of $P_{f}$,
${\operatorname{Sing}}(V(\operatorname{in}_{P}(f))\subset
V(\operatorname{in}_{P}({\mathcal{J}}))$.

Then the cobordant blow-up $B_{+}\to X$ of ${\mathcal{J}}$ resolves the
singularity of $V(f)$. That is, the strict transform
$Y^{\prime}=V(\sigma^{s}(f))$ of $Y$ (which coincides with the weak transform
$V(\sigma^{\circ}(f))$ of $Y$) is a regular subscheme of $B_{+}$.

###### Proof.

By Lemmas 4.5.7, 4.5.4, and the assumption (3) we get that

${\operatorname{Sing}}({\operatorname{inv}}_{\omega}(f))={\operatorname{Sing}}({\operatorname{inv}}^{\circ}_{\omega}(f))\subset
V(\operatorname{in}^{\circ}_{\omega}({\mathcal{J}})),$

for any $\omega\in\Delta^{N}_{E}$ and the corollary follows from Theorem
4.4.5. ∎

######  Remark 4.5.10.

The theorem shows that in the case of hypersurface $V(f)$ the critical
combinatorial information is related to the faces $P$ of the Newton polytope
$P_{f}$. Generally, one considers the dual valuation complex $\Delta^{N}_{E}$
associated with the ideal ${\mathcal{I}}$. In such a case,
${\operatorname{inv}}_{P}(f)$ is replaced with more general
${\operatorname{inv}}_{\omega}({\mathcal{I}})$, and the role of the Newton
polytope of a monomial ideal is limited (see Theorems 4.4.5, Theorems 4.6.9).
However, it still can be used in the context of the order of the ideals in
${\mathcal{O}}_{Z}[x_{1},\ldots,x_{k}]$ (see Theorems 4.8.1, 4.8.2).

One can easily extend these results to the products of schemes:

######  Theorem 4.5.11.

Let $X=\prod_{Z}X_{j}$, where each
$X_{j}={\mathbb{A}}^{n_{j}}_{Z}=\operatorname{Spec}({\mathcal{O}}_{Z}[{\bf
x}_{j}])=\operatorname{Spec}{\mathcal{O}}_{Z}[x_{j1},\ldots,x_{jk_{j}}]$,
where $Z$ is a regular scheme over a field $\kappa$ for $j=1,\ldots,r$. Let

$f_{j}=\sum_{\alpha\in A_{f_{j}}}c_{j\alpha}{\bf
x}^{j\alpha}\in{\mathcal{O}}_{Z}[x_{j1},\ldots,x_{jk_{j}}]$

where $c_{j\alpha}\neq 0$ for $\alpha\in A_{f_{j}}$. Let

${\mathcal{J}}_{j}=({\bf x}_{j}^{\alpha}\mid\alpha\in A_{f_{j}})^{{\rm
sat}}\subset{\mathcal{O}}_{Z}[{\bf x}_{j}]$

be the induced monomial ideal, and $P_{f_{j}}:=P_{{\mathcal{J}}_{j}}$ be its
Newton polytope in ${\mathbb{Q}}^{k_{j}}$. Assume that for any $j=1,\ldots,r$

  1. (1)

The cosupport $V({\mathcal{J}}_{j})$ is of codimension $\geq 2$,

  2. (2)

${\operatorname{Sing}}(V(f_{j}))\subseteq V({\mathcal{J}}_{j}).$

  3. (3)

For any supporting face $P$ of
$P_{f_{j}}$,${\operatorname{Sing}}(V(\operatorname{in}_{P}(f))\subseteq
V(\operatorname{in}_{P}({\mathcal{J}}_{i}))$.

Then the cobordant blow-up $B_{+}\to X$ of
$\prod_{j=1}^{r}{\mathcal{O}}_{X}\cdot{\mathcal{J}}_{i}$ resolves the
singularity of $V(f_{1},\ldots,f_{k})$. That is, the strict transform
$Y^{\prime}=V(\sigma^{s}((f_{1},\ldots,f_{k}))$ of $Y$ is a regular subscheme
of $B_{+}$.

###### Proof.

The cobordant blow-up of
$\prod_{j=1}^{r}{\mathcal{O}}_{X}\cdot{\mathcal{J}}_{i}$ is equal to the
product over $Z$ of the cobordant blow-ups $B_{j+}$ of ${\mathcal{J}}_{i}$ on
$\operatorname{Spec}({\mathcal{O}}_{Z}[{\bf x}_{j}])$, each of which is smooth
over $Z$.

∎

###  4.6. The Abramovich-Quek resolution

The following result is due to Abramovich-Quek (with some minor
modifications):

######  Corollary 4.6.1.

[AQ21, Theorem 5.1.2] Let
$X={\mathbb{A}}^{n}_{Z}=\operatorname{Spec}{\mathcal{O}}_{Z}[x_{1},\ldots,x_{n}]$,
where $Z$ is a regular scheme over a field. Consider the induced SNC divisor
$D:=V(x_{1}\cdot\ldots\cdot x_{n})$. Let

$f=\sum_{\alpha\in A_{f}}c_{\alpha}{\bf
x}^{\alpha}\in{\mathcal{O}}(Z)[x_{1},\ldots,x_{k}]$

where $c_{\alpha}\neq 0$ for $\alpha\in A_{f}$. Let
${\mathcal{J}}={\mathcal{J}}_{f}:=({\bf x}^{\alpha}\mid\alpha\in A_{f})$ be
the associated monomial ideal, and $P_{f}:=P_{\mathcal{J}}$ be its Newton
polytope. Assume that the cosupport $V({\mathcal{J}})$ is of codimension $\geq
2$, and for any face $P$ of $P_{f}$, the ideal $(\operatorname{in}_{P}(f))$
determines a smooth subscheme outside of $D$.

Then the cobordant blow-up $B_{+}\to X$ of ${\mathcal{J}}$ resolves the
singularity of $V(f)$. That is, the strict transform
$Y^{\prime}=V(\sigma^{s}(f))=V(\sigma^{\circ}(f))$ of $Y$ is a regular
subscheme of $B_{+}$.

######  Remark 4.6.2.

Note that unlike in the original formulation the coefficients $c_{\alpha}$ are
not necessarily invertible. Theorem 4.6.1 is further generalized for the
ideals in the context of order. See Remark 4.8.3.

###### Proof.

Let $\sigma_{0}^{\vee}={\mathbb{Q}}_{\geq 0}^{n}=\langle
x_{1},\ldots,x_{n}\rangle$ be the cone corresponding to the ring

${\mathcal{O}}_{Z}[x_{1},\ldots,x_{n}].$

It suffices to show that conditions (2), (3) of Theorem 4.5.9 are satisfied.

To prove condition (3) let $P^{\prime}$ be any supporting face of $P_{f}$. By
Lemma 2.6.5, there is a stratification of $X$ with strata ${s_{\tau}}$, where
$\tau$ is a face of $\sigma_{0}$ . It is determined by the pull-back of the
orbit stratification on $X_{\sigma_{0}}$, via $X=X_{\sigma_{0}}\times Z\to
X_{\sigma_{0}}$.

Assume that $\overline{s_{\tau}}$ is not in
$V(\operatorname{in}_{P^{\prime}}^{\circ}({\mathcal{J}}))$. This means, by
Corollary 2.6.6, that $\tau^{*}$ intersects $P^{\prime}$ so we consider the
face $P:=P^{\prime}\cap\tau^{*}$. Moreover, by Lemma 2.6.5, we can write the
closure of the stratum $s_{\tau}$ as

$\overline{s_{\tau}}:=V(x_{i}\mid x_{i}\not\in\tau^{*}).$

On the other hand, since $P\subset\tau^{*}$, the polynomial
$\operatorname{in}_{P}(f)\in{\mathcal{O}}_{Z}[x_{i}\in\tau^{*}]\subset{\mathcal{O}}_{Z}[x_{1},\ldots,x_{n}]$
can be identified with

$\operatorname{in}_{P}(f)_{|\overline{s_{\tau}}}\in{\mathcal{O}}_{Z}[x_{1},\ldots,x_{n}]/(x_{i}\mid
x_{i}\not\in\tau^{*})\simeq{\mathcal{O}}_{Z}[x_{i}\in\tau^{*}].$

Now $\operatorname{in}_{P}(f)$ is simply equal to

$\operatorname{in}_{P}(f)_{|\overline{s_{\tau}}}=\operatorname{in}_{P^{\prime}}(f)_{|\overline{s_{\tau}}},$

By the assumption
$\operatorname{in}_{P}(f)\in{\mathcal{O}}_{Z}[x_{i}\in\tau^{*}]$ is a local
parameter on

$\displaystyle\operatorname{Spec}({\mathcal{O}}_{Z}[x_{1},\ldots,x_{n}]\smallsetminus
V(\prod{x_{i}})\quad=$
$\displaystyle(\operatorname{Spec}({\mathcal{O}}_{Z}[x_{i}\in\tau^{*}])\smallsetminus
V(\prod_{x_{i}\in\tau^{*}}x_{i}))\quad\times\quad(\operatorname{Spec}({\mathcal{O}}_{Z}[x_{i}\not\in\tau^{*}])\smallsetminus
V(\prod_{x_{i}\not\in\tau^{*}}x_{i})),$

and on

$\displaystyle(\operatorname{Spec}({\mathcal{O}}_{Z}[x_{i}\in\tau^{*}])\smallsetminus
V(\prod_{x_{i}\in\tau^{*}}x_{i}))\simeq$ $\displaystyle\simeq V(x_{i}\mid
x_{i}\not\in\tau^{*})\smallsetminus
V(\prod_{x_{i}\in\tau^{*}}x_{i}))=\overline{s_{\tau}}\smallsetminus
V(\prod_{x_{i}\in\tau^{*}}x_{i}))=s_{\tau}$

Consequently
$\operatorname{in}_{P}(f)_{|\overline{s_{\tau}}}=\operatorname{in}_{P^{\prime}}(f)_{|\overline{s_{\tau}}}$
defines a local parameter on the stratum $s_{\tau}$. This implies that
$\operatorname{in}_{P^{\prime}}(f)$ is a local parameter on all strata
$s_{\tau}$ outside of
$V(\operatorname{in}^{\circ}_{P^{\prime}}({\mathcal{J}}))$.

The proof of condition (2) is similar. Consider any face $\tau$ of
$\sigma_{0}$. If $\overline{s_{\tau}}$ is not in $V({\mathcal{J}})$, then, by
Corollary 2.6.6, $\tau$ intersects $P_{f}$ so we consider the face
$P:=P_{f}\cap\tau^{*}$. Consequently, by the assumption

$(\operatorname{in}_{P}(f))_{|\overline{s_{\tau}}}=f_{|\overline{s_{\tau}}}\in{\mathcal{O}}_{Z}[x_{i}\in\tau^{*}]$

defines a local parameter on the stratum

$s_{\tau}=\overline{s_{\tau}}\smallsetminus V(\prod_{x_{i}\in\tau^{*}}x_{i}),$

This implies that $f$ is a local parameter on all strata $s_{\tau}$ outside of
$V({\mathcal{J}})$, showing condition (2) of Theorem 4.5.9 and completing the
proof.

∎

######  Corollary 4.6.3.

Let $Z$ be a regular scheme over a field $\kappa$, and $X=\prod_{Z}X_{j}$,
where
$X_{j}:={\mathbb{A}}^{n_{j}}_{Z}=\operatorname{Spec}{\mathcal{O}}(Z)[x_{j1},\ldots,x_{jk_{j}}]$
for $j=1,\ldots,r$. Let

$f_{j}=\sum_{\alpha\in A_{f_{j}}}c_{j\alpha}{\bf
x}^{j\alpha}\in{\mathcal{O}}_{Z}[x_{j1},\ldots,x_{jk_{j}}]$

where $c_{j\alpha}\neq 0$ for $\alpha\in A_{f_{j}}$. Let
${\mathcal{J}}_{j}:=({\bf x}_{j}^{\alpha}\mid\alpha\in A_{f_{j}})$ be the
induced monomial ideal, and $P_{f_{j}}:=P_{{\mathcal{J}}_{j}}$ be its Newton
polytope in ${\mathbb{Q}}^{k_{j}}$. Assume that for any $j=1,\ldots,r$ the
cosupport $V({\mathcal{J}}_{j})$ is of codimension $\geq 2$, and for any face
$P$ of $P_{f_{j}}$, the ideal $(\operatorname{in}_{P}(f_{j}))$ determines a
smooth subscheme outside of $V(x_{j1}\cdot\ldots\cdot x_{jk_{j}})$. Then the
cobordant blow-up $B_{+}\to X$ of
$\prod_{j=1}^{r}{\mathcal{O}}_{X}\cdot{\mathcal{J}}_{i}$ resolves the
singularities of $V(f_{1},\ldots,f_{k})$. That is, the strict transform
$Y^{\prime}=V(\sigma^{s}((f_{1},\ldots,f_{k}))$ of $Y$ is a regular subscheme
of $B_{+}$.

####  4.6.4. Examples of resolution

######  Theorem 4.6.5.

Let $X=\operatorname{Spec}_{Z}({\mathcal{O}}_{Z}[x_{1},\ldots,x_{k}])$, where
$Z$ is a smooth variety over a field $\kappa$. Consider the closed subscheme
$Y$ on $X$ defined by a function $f\in H^{0}(X,{\mathcal{O}}_{X})$ of the form

$f=\sum_{i=1}^{k}c_{\alpha_{i}}(v){\bf x}^{\alpha_{i}},$

where $c_{\alpha_{i}}(v)\in{\mathcal{O}}(Z)^{*}$ are invertible.

Assume that for the presentation of $f$, one of the following holds:

  * •

${\operatorname{char}}(\kappa)=0$, and for any $\alpha_{i}$ except possibly
one, there is a variable $x_{j_{i}}$ such that a power $x_{j_{i}}^{a_{j_{i}}}$
of $x_{j_{i}}$, occurs in ${\bf x}^{\alpha_{i}}$ and $x_{j_{i}}$ does not
occur in the others ${\bf x}^{\alpha_{j}}$ for $j\neq i$.

  * •

${\operatorname{char}}(\kappa)=p$, and for any $\alpha_{i}$ except possibly
one there is a variable $x_{j_{i}}$ such that a power $x_{j_{i}}^{a_{j_{i}}}$
of $x_{j_{i}}$, occurs in ${\bf x}^{\alpha_{i}}$, with $p\nmid a_{j_{i}}$ and
$x_{j_{i}}$ does not occur in the others ${\bf x}^{\alpha_{j}}$ for $j\neq i$
except as some $k\cdot p$-th power for $k\in{\mathbb{N}}$.

Then the cobordant blow-up $B_{+}\to X$ of ${\mathcal{J}}=({\bf
x}^{\alpha_{1}},\ldots,{\bf x}^{\alpha_{k}})$ resolves singularity, so that
the strict transform $\pi_{B}^{s}(f)$ determines a regular subscheme
$\sigma^{s}(Y)$ of $B_{+}$.

###### Proof.

Let ${\mathcal{D}}(f)$ be the ideal generated by $f$, and all the derivatives
$D(f)$. At any point $p$ of ${\operatorname{Sing}}(f)$ , we have that
${\operatorname{ord}}_{p}(f)\geq 2$ which implies
${\operatorname{Sing}}(V(f))=V({\mathcal{D}}(f))$. But the ideal
${\mathcal{D}}(f)$ contains all but possibly one monomials ${\bf
x}^{\alpha_{i}}\sim x_{j(i)}D_{x_{j(i)}}(f)$ . Since
$f=\sum_{i=1}^{k}c_{\alpha_{i}}(v){\bf x}^{\alpha_{i}},$ and all but at most
one monomial ${\bf x}^{\alpha_{i}}$ are in ${\mathcal{D}}(f)$ we conclude that
${\mathcal{D}}(f)\supseteq{\mathcal{J}}=({\bf x}^{\alpha_{1}},\ldots,{\bf
x}^{\alpha_{k}}).$ So

${\operatorname{Sing}}(Y)=V({\mathcal{D}}(f))\subseteq V({\mathcal{J}}).$

Similarly

${\operatorname{Sing}}(\operatorname{in}_{P}(f))=V({\mathcal{D}}(\operatorname{in}_{P}(f))\subseteq
V((\operatorname{in}^{\circ}_{P}({\mathcal{J}})).$

Thus the conditions of Theorem 4.5.9 are satisfied.

∎

######  Example 4.6.6.

Let $f=x_{1}^{a_{1}}+\ldots+x_{k}^{a_{k}}\in\kappa[x_{1},\ldots,x_{k}]$, where
the characteristic $p$ divides at most one $a_{i}$. Then the cobordant blow-up
of $(x_{1}^{a_{1}},\ldots,x_{k}^{a_{k}})$ resolves singularity. By Example
3.4.6, it is given by

$B=\operatorname{Spec}_{X}({\mathcal{O}}_{X}[{\bf t}^{-1},x_{1}{\bf
t}^{w_{1}},\ldots,x_{k}{\bf t}^{w}_{k}])$ $B_{+}=B\smallsetminus
V_{B}(\sigma^{s}({\mathcal{J}}))=B\smallsetminus V_{B}(x_{1}{\bf
t}^{w_{1}},\ldots,x_{k}{\bf t}^{w_{k}}).$

The morphism $B_{+}\to X$ is interpreted in Section 5.4 as the cobordant blow-
up of the weighted center
${\mathcal{J}}=(x_{1}^{1/w_{1}},\ldots,x_{k}^{1/w_{k}})$, such that
${\mathcal{O}}_{B_{+}}\cdot{\mathcal{J}}={\mathcal{O}}_{B_{+}}\cdot{\bf
t}^{-1}$.

######  Example 4.6.7.

$x_{1}^{p}+ax_{2}^{p}x_{3}+bx_{1}x_{4}^{p}x_{5}^{p^{2}}\in\kappa[x_{1},x_{2},x_{3},x_{4},x_{5},],$

where $a,b\in\kappa^{*}$can be resolved by the single cobordant blow-up of

${\mathcal{J}}=(x_{1}^{p},x_{2}^{p}x_{3},x_{1}x_{4}^{p}x_{5}^{p^{2}})$

over a field $\kappa$ of characteristic $p$. Here for $x_{2}^{p}x_{3}$ the
variable $x_{3}$ does not occur in the other terms, and for
$x_{1}x_{4}^{p}x_{5}^{p^{2}}$ the coordinate $x_{1}$ occurs in the other terms
as $x_{1}^{p}$ -power or does not show at all.

######  Example 4.6.8.

$x_{1}^{2}x_{2}^{5}+7x_{4}^{7}x_{3}^{5}+25x_{1}x_{3}^{6}\in\kappa[x_{1},x_{2},x_{3},x_{4}]$

can be resolved by the cobordant blow-up of

${\mathcal{J}}=(x_{1}^{2}x_{2}^{5},x_{4}^{7}x_{3}^{5},x_{1}x_{3}^{6})$

over a field $\kappa$ of ${{\operatorname{char}}(\kappa)}\neq 5,7$. We use
$x_{2}$ for $x_{1}^{2}x_{2}^{5}$, and $x_{4}$ for $x_{4}^{7}x_{3}^{5}$.

######  Theorem 4.6.9.

Let $Z$ be a smooth variety over a field $\kappa$. Let

$X=\operatorname{Spec}_{Z}({\mathcal{O}}_{Z}[x_{1},\ldots,x_{n}])=\operatorname{Spec}_{Z}({\mathcal{O}}_{Z}[{\bf
x}_{1},\ldots,{\bf x}_{r}]),$

where

${\bf x}_{i}:=(x_{k_{i-1}},\ldots,x_{k_{i}-1}),$

for $k_{0}=1<k_{1}<\ldots<k_{r}=n+1$. Consider the closed subscheme $Y$ of $X$
defined by the set of the polynomial functions $f_{j}\in
H^{0}(X,{\mathcal{O}}_{X})$, where $j=1,\ldots,r$ of the form

$f_{j}=\sum_{i=1}^{r_{j}}c_{\alpha_{ij}}(v){\bf x}_{j}^{\alpha_{ij}},$

where $c_{\alpha_{ij}}(v)\in{\mathcal{O}}(Z)^{*}$ are invertible.

Assume that for any $j=1,\ldots,r$ and for the presentation of $f_{j}$ one of
the following holds:

  * •

${\operatorname{char}}(\kappa)=0$, and for any $\alpha_{ij}$ except possibly
one, there is a variable $x_{j_{i}}$ such that a power $x_{j_{i}}^{a_{j_{i}}}$
of $x_{j_{i}}$, occurs in ${\bf x}^{\alpha_{ij}}$ and $x_{j_{i}}$ does not
occur in the others ${\bf x}^{\alpha_{i^{\prime}j}}$ for $i^{\prime}\neq i$.

  * •

${\operatorname{char}}(\kappa)=p$, and for any $\alpha_{ij}$ except possibly
one there is a variable $x_{j_{i}}$ such that a power $x_{j_{i}}^{a_{j_{i}}}$
of $x_{j_{i}}$ occurs in ${\bf x}^{\alpha_{ij}}$ , with $p\nmid a_{j_{i}}$ and
$x_{j_{i}}$ does not occur in the others ${\bf x}^{\alpha_{i^{\prime}j}}$ for
$i^{\prime}\neq i$ except as some $k\cdot p$-th power for $k\in{\mathbb{N}}$.

Then the cobordant blow-up $B_{+}\to X$ of

${\mathcal{J}}=\prod_{j=1}^{r}({\bf x}_{j}^{\alpha_{1}},\ldots,{\bf
x}_{j}^{\alpha_{k}})$

resolves singularity, so that the strict transform
$\sigma^{s}(f_{1},\ldots,f_{r})$ determines a smooth subvariety of $B_{+}$.

###### Proof.

The space $X$ can be written as the fiber product

$X=\prod_{Z}{\mathbb{A}}_{Z}^{k_{i}-k_{i-1}}=\prod_{Z}\operatorname{Spec}_{Z}({\mathcal{O}}_{Z}[{\bf
x}_{j}]).$

The cobordant blow-up of ${\mathcal{J}}$ is equal to the product over $Z$ of
the cobordant blow-ups $B_{j+}$ of $({\bf x}_{j}^{\alpha_{1}},\ldots,{\bf
x}_{j}^{\alpha_{k}})$ on $\operatorname{Spec}_{Z}({\mathcal{O}}_{Z}[{\bf
x}_{j}])$, and each of $B_{j+}$ is smooth over $Z$ by Theorem 4.6.5.

∎

######  Example 4.6.10.

The system of equations

$\displaystyle x_{1}^{p}+ax_{1}x_{2}^{p}x_{3}+bx_{4}x_{5}^{p}x_{6}^{p^{2}}$
$\displaystyle=0$ $\displaystyle
y_{1}^{p^{3}}+cy_{2}^{p^{2}}y_{3}y_{6}+dy_{1}y_{4}^{p}y_{5}^{p^{2}}y_{6}^{2}$
$\displaystyle=0$

in

$\kappa[x_{1},\ldots,x_{6},y_{1},\ldots,y_{5}],$

where $a,b,c,d\in\kappa^{*}$, can be resolved by the single cobordant blow-up
of

${\mathcal{J}}=(x_{1}^{p},x_{2}^{p}x_{3},x_{1}x_{4}x_{5}^{p}x_{6}^{p^{2}})\cdot(y_{1}^{p^{3}},y_{2}^{p^{2}}y_{3},y_{4}y_{5}^{p}y_{6}^{p^{2}})$

in characteristic $p$.

######  Example 4.6.11.

Let $f_{j}=x_{1j}^{a_{1}}+\ldots+x_{k_{j}j}^{a_{k}}\in\kappa[x_{ij}]$, where
$j=1,\ldots,k$ and the characteristic $p$ divides at most one $a_{ij}$ for any
$j$. Then the cobordant blow-up of
$\prod_{j}(x_{1j}^{a_{1}},\ldots,x_{k_{j}j}^{a_{k}})$ resolves singularity of
$V(f_{j})_{j=1,\ldots,k}$.

###  4.7. Partial resolution by the order

The method can be linked to different invariants, particularly to the order

${\operatorname{ord}}_{p}({\mathcal{I}}):=\max\\{k\mid{\mathcal{I}}_{p}\subset{\mathfrak{m}}_{p}^{k}\\},$

where $m_{p}\subset{\mathcal{O}}_{X,p}$ is the maximal ideal of a point $p\in
X$.

######  Definition 4.7.1.

Let ${\mathcal{I}}$ be an ideal on a regular scheme $X$, and
$d\in{\mathbb{N}}$ be an integer. We define

${\operatorname{supp}}({\mathcal{I}},d):=\\{p\in
X\mid{\operatorname{ord}}_{p}({\mathcal{I}})\geq d\\}.$

The following theorem extends [Wło22, Lemma 5.3.1]:

######  Theorem 4.7.2.

Let ${\mathcal{I}}$ be an ideal on a regular scheme $X$ over a field, and let
$d\in{\mathbb{N}}$ be any natural number. Assume that there exists a locally
monomial center ${{\mathcal{J}}}$, with
$\operatorname{codim}(V({\mathcal{J}})\geq 2$, with the associated dual
valuation complex $\Delta^{N}_{E}$, and such that

  1. (1)

${\operatorname{supp}}({\mathcal{I}},d)\subseteq V({\mathcal{J}})\subset
X,\quad$

  2. (2)

${\operatorname{supp}}(\operatorname{in}_{\omega}({\mathcal{I}}),d)\subseteq
V(\operatorname{in}^{\circ}_{\omega}({\mathcal{J}}))\subset{\mathbb{N}}_{\omega}(X)$,
for any $\omega\in\Delta^{N}_{E}$.

(respectively
(2’)
${\operatorname{supp}}(\operatorname{in}^{\circ}_{\omega}({\mathcal{I}}),d)\subseteq
V(\operatorname{in}^{\circ}_{\omega}({\mathcal{J}}))\subset{\mathbb{N}}_{\omega}(X)$,
for any $\omega\in\Delta^{N}_{E}$.)

Then for the cobordant blow-up $\sigma_{+}:B_{+}\to X$ of ${\mathcal{J}}$, the
maximal order of the strict transform $\sigma^{s}({\mathcal{I}})$
(respectively the weak transform $\sigma^{\circ}({\mathcal{I}})$) on $B_{+}$
is strictly smaller than $d$.

###### Proof.

Let $q\in D_{+}=D\smallsetminus V_{B}(\sigma^{\circ}({\mathcal{J}}))$, where
$D=V(t_{1}^{-1}\cdot\ldots t_{k}^{-1})$ is the exceptional divisor of $B\to
X$. Then there is $\omega\in\Delta^{N}_{E}$, and the corresponding stratum $s$
in $S_{D}$ such that

$q\in s\smallsetminus V(\sigma^{\circ}({\mathcal{J}})).$

By Lemmas 4.3.1, 4.3.3, there is a natural isomorphism
$s\to\operatorname{Spec}({\operatorname{gr}}_{\omega}({\mathcal{O}})[\check{\bf
t}_{\omega},\check{\bf t}_{\omega}^{-1}])$, which takes
$\sigma^{s}({\mathcal{I}})_{|s}$ to
$\operatorname{in}_{\omega}({\mathcal{I}})[\check{\bf t}_{\omega},\check{\bf
t}_{\omega}^{-1}]$ (and $\sigma^{\circ}({\mathcal{J}})_{|s}$ to
$\operatorname{in}^{\circ}_{\omega}({\mathcal{J}})[\check{\bf
t}_{\omega},\check{\bf t}_{\omega}^{-1}]$). Consequently

${\operatorname{ord}}_{q}(\sigma^{s}({\mathcal{I}}))\leq{\operatorname{ord}}_{q}(\sigma^{s}({\mathcal{I}})_{|s})={\operatorname{ord}}_{q}(\operatorname{in}_{\omega}({\mathcal{I}})[\check{\bf
t}_{\omega},\check{\bf t}_{\omega}^{-1}])<d.$

If

$q\in B\smallsetminus D\smallsetminus
V(\sigma^{s}({\mathcal{J}}))=B_{-}\smallsetminus
V(\sigma^{s}({\mathcal{J}}))=(X\smallsetminus V({\mathcal{J}}))\times T,$

then since $\pi_{B}(q)\in X\smallsetminus V({\mathcal{J}})$ we conclude that

${\operatorname{ord}}_{q}(\sigma^{s}({\mathcal{I}}))={\operatorname{ord}}_{\pi_{B}(q)}({\mathcal{I}})<d.$

The proof for $\sigma^{\circ}({\mathcal{I}})$ is the same. ∎

###  4.8. The Newton method of decreasing order

As a corollary from Theorem 4.7.2, and Lemma 4.5.4 we obtain:

######  Theorem 4.8.1.

$X={\mathbb{A}}^{n}_{Z}=\operatorname{Spec}{\mathcal{O}}_{X}[x_{1},\ldots,x_{k}]$,
where $Z$ is regular over a field $\kappa$ of characteristic $p$. Let
${\mathcal{I}}\subset{\mathcal{O}}_{Z}[x_{1},\ldots,x_{k}]$ be an ideal, and
${\mathcal{J}}={\mathcal{J}}_{\mathcal{I}}$, be its associated monomial ideal
with the Newton polytope $P_{\mathcal{J}}=P_{\mathcal{I}}$ and
$d\in{\mathbb{N}}$ be any natural number such that

  1. (1)

$\operatorname{codim}(V({\mathcal{J}}))\geq 2$.

  2. (2)

${\operatorname{supp}}({\mathcal{I}},d)\subseteq V({\mathcal{J}})$.

  3. (3)

for any supporting face $P$ of $P_{\mathcal{J}}$,
${\operatorname{supp}}(\operatorname{in}_{P}({\mathcal{I}}),d)\subset
V({\operatorname{inv}}_{P}({\mathcal{J}})).$

Then the maximal order of the weak transform $\sigma^{\circ}({\mathcal{I}})$
on $B_{+}$ under cobordant blow-up of ${\mathcal{J}}$ is strictly smaller than
$d$. ∎

Thus we get

######  Theorem 4.8.2.

$X={\mathbb{A}}^{n}_{Z}=\operatorname{Spec}{\mathcal{O}}_{X}[x_{1},\ldots,x_{k}]$,
where $Z$ is regular over a field $\kappa$ of characteristic $p$. Let
${\mathcal{I}}\subset{\mathcal{O}}_{Z}[x_{1},\ldots,x_{k}]$ be an ideal, and
${\mathcal{J}}={\mathcal{J}}_{\mathcal{I}}$, be its associated monomial ideal
with with $\operatorname{codim}(V({\mathcal{J}})\geq 2$, and let
$P_{\mathcal{J}}=P_{\mathcal{I}}$ be its Newton polytope and
$d\in{\mathbb{N}}$ be any natural number such that for any face $P$ of
$P_{\mathcal{J}}$,

${\operatorname{supp}}(\operatorname{in}_{P}({\mathcal{I}}),d)\subset
D:=V(x_{1}\cdot,\ldots\cdot x_{k}).$

Then the maximal order of the weak transform $\sigma^{\circ}({\mathcal{I}})$
under cobordant blow-up $B_{+}\to X$ of ${\mathcal{J}}$ is strictly smaller
than $d$. ∎

###### Proof.

The proof uses similar arguments as the proof of Corollary 4.6.1. We need to
show that the conditions of Theorem 4.8.1 are satisfied

For condition (3) of Theorem 4.8.1, let $P^{\prime}$ be any supporting face of
$P_{\mathcal{I}}$. Consider the closure of the stratum $\overline{s_{\tau}}$,
where $\tau$ is a face of $\sigma_{0}$.

If ${s_{\tau}}$ is not in
$V(\operatorname{in}_{P^{\prime}}^{\circ}({\mathcal{J}}))$ consider the face
$P:=P^{\prime}\cap\tau^{*}$. Then

${\operatorname{supp}}(\operatorname{in}_{P^{\prime}}({\mathcal{I}}))_{|\overline{s_{\tau}}},d)={\operatorname{supp}}(\operatorname{in}_{P}(f)_{|\overline{s_{\tau}}},d)$

is contained in $\overline{s_{\tau}}\smallsetminus
s_{\tau}=V(\prod_{x_{i}\in\tau^{*}}x_{i})$ so it is not in $s_{\tau}$. This
implies that
${\operatorname{supp}}(\operatorname{in}_{P^{\prime}}({\mathcal{I}})),d)$ is
contained $V(\operatorname{in}^{\circ}_{P^{\prime}}({\mathcal{J}})$.

The proof of condition (2) of Theorem 4.8.1 is the same, except we replace
$P^{\prime}$ with $P_{\mathcal{J}}$,
$\operatorname{in}_{P^{\prime}}^{\circ}({\mathcal{J}})$ with
${\mathcal{J}}=\operatorname{in}_{P_{\mathcal{J}}}^{\circ}({\mathcal{J}})$ ,
and $P$ with $P=P_{\mathcal{J}}\cap\tau^{*}$.

Thus the corollary is a consequence of Theorem 4.8.1. ∎

######  Remark 4.8.3.

Theorems 4.8.1, 4.8.2 generalize respectively Theorem 4.5.9, and Corollary
4.6.1. We put ${\mathcal{I}}=(f)$, and $d=2$. Then
${\operatorname{Sing}}(V(f))={\operatorname{supp}}((f),2)$.

######  Example 4.8.4.

(See also Example 2.6.14)

Let $Y\subset X=\operatorname{Spec}\kappa[x,y,z]$ be described by the ideal

${\mathcal{I}}=(x^{k}+xy+y^{l},\quad
z^{kl}+x^{k-2}z^{{kl}-1}+y^{k-2}z^{{kl}-1})$

of order $2$, where $\gcd(k,l)=1$. Consider the corresponding admissible
monomial ideal

${\mathcal{J}}=(x^{k},xy,y^{l},z^{kl},x^{k-2}z^{{kl}-1},y^{k-2}z^{{kl}-1})$

and its associated Newton polytope $P$ generated by the exponents

$(k,0,0),(1,1,0),(0,l,0),(0,0,kl)\subset\sigma^{\vee}=\langle
e_{1}^{*},e_{2}^{*},e_{3}^{*}\rangle.$

This corresponds to two supporting faces $P_{1},P_{2}$ defined, respectively,
by

$(k,0,0),(1,1,0),(0,0,kl),$

and

$(1,1,0),(0,l,0),(0,0,kl).$

They intersect at the face $P_{12}=(1,1,0),(0,0,kl)$.

The faces $P_{1},P_{2}$ corresponds to the primitive vectors
$v_{1}=(a_{1},a_{2},a_{3})$ such that

$a_{1}k=a_{1}+a_{2}=a_{3}kl,$

and $v_{2}=(b_{1},b_{2},b_{3})$, where

$b_{1}+b_{2}=lb_{2}=klb_{3}$

in the dual plane. So

$v_{1}=(l,l(k-1),1),\quad v_{2}=(k(l-1),k,1).$

This defines the set of two extremal valuations $\nu_{1},\nu_{2}$.

Then

$\operatorname{in}_{P_{1}}({\mathcal{I}})=(x^{k}+xy,z^{kl}),\quad\operatorname{in}_{P_{2}}(f)=(xy+y^{l},z^{kl}),\quad\operatorname{in}_{P_{12}}(f)=(xy,z^{kl}).$

By considering the ideals of the derivatives
${\mathcal{D}}(\operatorname{in}_{P}(f))$ we see that in all cases

${\operatorname{supp}}(\operatorname{in}_{P}({\mathcal{I}}),2)=V(x,y,z)$

Similarly ${\operatorname{supp}}({\mathcal{I}},2)=V(x,y,z)$.

The cobordant blow-up of ${\mathcal{J}}=(x^{k},x^{2}y,y^{l})$ is described as

$\displaystyle
B=\operatorname{Spec}_{X}({\mathcal{O}}_{X}[t_{1}^{-1},t_{2}^{-1},\,\,\,xt_{1}^{l}t_{2}^{k(l-1)},\,\,\,yt_{1}^{l(k-1)}t_{2}^{k},\,zt_{1}t_{2}]=$
$\displaystyle\operatorname{Spec}(\kappa[t_{1}^{-1},t_{2}^{-1},\,xt_{1}^{l}t_{2}^{k(l-1)},\,\,yt_{1}^{l(k-1)}t_{2}^{k},\,\,zt_{1}t_{2}])$
$B_{+}=B\smallsetminus V(\sigma^{s}({\mathcal{J}}))=B\smallsetminus{\bf
t}^{\alpha}{\mathcal{J}},$

where $\sigma^{s}({\mathcal{J}})={\bf t}^{\alpha}{\mathcal{J}}$, and the
coefficients are given by the exceptional divisor
$E=\alpha_{1}E_{1}+\alpha_{2}E_{2}$ of the toric normalized blow-up of
${\mathcal{J}}$.

$\alpha_{1}=\nu_{1}(f)=a_{1}k=a_{1}+a_{2}=a_{3}kl=kl,$

and

$\alpha_{2}=\nu_{2}(f)=b_{1}+b_{2}=lb_{2}=klb_{3}=kl$

Thus

$B_{+}=B\smallsetminus V((x^{k},xy,y^{l},z^{kl})\cdot t_{1}^{kl}t_{2}^{kl}),$

By Lemma 4.6.9, the cobordant blow-up $B_{+}\to X$ of
${\mathcal{J}}=(x^{k},xy,y^{l},z^{kl})$ decreases the order of ${\mathcal{I}}$
to 1.

##  5\. Generalized cobordant blow-ups and ${\mathbb{Q}}$-ideals

###  5.1. Cobordization with respect to subgroups
$\Gamma\subset{\operatorname{Cl}}(Y/X)\otimes{\mathbb{Q}}$

######  Definition 5.1.1.

Let $\pi:Y\to X$ be a proper birational morphism. Let
$\Gamma\subset{\operatorname{Cl}}(Y/X)\otimes{\mathbb{Q}}$ be a finitely
generated subgroup. We define the full cobordization (resp. cobordization of
$\pi$) with respect to $\Gamma$ to be

$B=B^{\Gamma}:=\operatorname{Spec}_{X}(\pi_{*}(\bigoplus_{E\in\Gamma}{\mathcal{O}}_{X}(E))\quad
B_{+}=B^{\Gamma}_{+}=\operatorname{Spec}_{Y}(\bigoplus_{E\in\Gamma}{\mathcal{O}}_{Y}(E)).$

######  Proposition 5.1.2.

The natural morphism $B_{+}^{\Gamma}\to B^{\Gamma}$ is an open immersion if
locally on $X$ there are forms $F=fx^{-E}$, with $E\in\Gamma$ such that
$X_{F}$ are open affine and cover $X$.

###### Proof.

The proposition follows from the first part of the proof of Proposition 1.3.4.

∎

######  Definition 5.1.3.

Let $\pi:Y\to X$ be the normalized blow-up of the an ${\mathcal{I}}$ on a
normal scheme $X$. Let
$\Gamma\subset{\operatorname{Cl}}(Y/X)\otimes{\mathbb{Q}}$ be a finitely
generated subgroup. Then we define the full cobordant blow-up of
${\mathcal{I}}$ with respect to $\Gamma$ (resp. the cobordant blow-up of
${\mathcal{I}}$ with respect to $\Gamma$) to be the full cobordization (resp.
cobordization of) $\pi$ with respect to $\Gamma$.

######  Proposition 5.1.4.

$\pi:Y\to X$ be the normalized blow-up of an ideal ${\mathcal{J}}$ on a normal
scheme $X$, and let $E^{0}$ be the exceptional Cartier divisor such that
${\mathcal{O}}_{Y}(-E^{0})={\mathcal{O}}_{Y}\cdot{\mathcal{J}}$. If a finitely
generated group $\Gamma\subset{\operatorname{Cl}}(Y/X)\otimes{\mathbb{Q}}$
contains divisor $E^{0}$, then $B^{\Gamma}_{+}=B^{\Gamma}\smallsetminus
V({\mathcal{I}}{\bf t}^{-E_{0}})$.

###### Proof.

The proof is identical to the proof of Lemma 1.4.4.

∎

###  5.2. Simple cobordant blow-up of ideal ${\mathcal{I}}$

######  Definition 5.2.1.

Let $\pi:Y\to X$ be the normalized blow-up of an ideal ${\mathcal{I}}$ on a
normal scheme $X$, with the exceptional divisor $E^{0}$, such that
${\mathcal{O}}_{Y}(-E^{0})={\mathcal{O}}_{Y}\cdot{\mathcal{I}}$. By the simple
cobordant blow-up of ${\mathcal{I}}$ on $X$ we mean the cobordization
$B_{+}^{\Gamma}$ of $\pi:Y\to X$ with respect to the subgroup
$\Gamma={\mathbb{Z}}\cdot E^{0}\subset{\operatorname{Cl}}(Y/X)$ generated by
$E^{0}$.

######  Lemma 5.2.2.

The simple cobordant blow-up of ${\mathcal{I}}$ is given by

$B=\operatorname{Spec}_{X}({\mathcal{O}}_{X}[{\bf
t}^{-1},{\mathcal{I}}t])^{{\operatorname{int}}},\quad B_{+}=B\smallsetminus
V({\mathcal{I}}t).$

###### Proof.

It follows that
${\mathcal{O}}_{Y}(-nE^{0})={\mathcal{O}}_{Y}\cdot{\mathcal{I}}^{n}$. Moreover
$\pi_{*}({\mathcal{O}}_{Y}(-nE^{0}))=({\mathcal{I}}^{n})^{{\operatorname{int}}}$
is the integral closure of ${\mathcal{I}}^{n}$.

Consequently

$B=B^{\Gamma}_{\mathcal{I}}=\operatorname{Spec}_{X}(\pi_{*}(\bigoplus_{n\in{\mathbb{Z}}}{\mathcal{O}}_{Y}(nE^{0}){\bf
t}^{nE_{0}})=\operatorname{Spec}_{X}({\mathcal{O}}_{X}[{t}^{-1},{\mathcal{I}}t])^{{\operatorname{int}}},$

under the identification of ${\bf t}^{E_{0}}$ with $t^{-1}$. By Proposition
5.1.4,

$B_{+}=B\smallsetminus V(\sigma^{s}({\mathcal{I}}))=B\smallsetminus
V({\mathcal{I}}t).$

and thus is described by the standard Rees extended algebra. ∎

###  5.3. Cobordant blow-ups of ${\mathbb{Q}}$-ideals

####  5.3.1. Valuative ${\mathbb{Q}}$-ideals

The valuative ${\mathbb{Q}}$-ideals were introduced in [ATW19]. Here we
consider its particular version considered in [Wło22].

######  Definition 5.3.2.

By valuative ${\mathbb{Q}}$-ideals, or, simply, ${\mathbb{Q}}$-ideals on a
normal scheme $X$ we mean the equivalence classes of formal expressions
${\mathcal{I}}^{1/n}$, where ${\mathcal{I}}$ is the ideal on $X$, and
$n\in{\mathbb{N}}$. We say that two ${\mathbb{Q}}$-ideals
${\mathcal{I}}^{1/n}$, and ${\mathcal{J}}^{1/m}$ are equivalent if the
integral closures of ${\mathcal{I}}^{m}$, and ${\mathcal{J}}^{n}$ are the
same.

In particular, if $D$ is a Cartier effective divisor on $X$ then any
${\mathbb{Q}}$-Cartier effective divisor $\frac{1}{m}\cdot D$ determines the
${\mathbb{Q}}$-ideal ${\mathcal{O}}_{X}(-D)^{\frac{1}{m}}$.

By the vanishing locus of ${\mathcal{J}}={\mathcal{I}}^{1/n}$ we mean
$V({\mathcal{J}})=V({\mathcal{I}})$.

One can define the operation of addition and multiplication on
${\mathbb{Q}}$-ideals:

${\mathcal{I}}^{1/n}+{\mathcal{J}}^{1/m}:=({\mathcal{I}}^{m}+{\mathcal{J}}^{n})^{1/mn},\quad{\mathcal{I}}^{1/n}\cdot{\mathcal{J}}^{1/m}=({\mathcal{I}}^{m}\cdot{\mathcal{J}}^{n})^{1/mn}.$

For any valuative ${\mathbb{Q}}$-ideal ${\mathcal{J}}={\mathcal{I}}^{1/n}$ on
$X$ we define the associated ideal of sections on $X$:

${\mathcal{J}}_{X}:=\\{f\in{\mathcal{O}}_{X}\mid
f^{n}\in{\mathcal{I}}^{{\operatorname{int}}}\\},$

where ${\mathcal{I}}^{{\operatorname{int}}}$ is the integral closure of
${\mathcal{I}}$. In particular, for the effective Cartier divisor $D$, we have
the equalities

$({\mathcal{O}}_{X}(-D)^{1/m})_{X}={\mathcal{O}}_{X}(-\frac{1}{m}D)=\\{f\in{\mathcal{O}}_{X}\mid{\operatorname{div}}(f)-\frac{1}{m}D\geq
0\\}.$

With any valuative ${\mathbb{Q}}$-ideal ${\mathcal{J}}$ we associate the Rees
algebra on $X$ :

${\mathcal{O}}_{X}[{\mathcal{J}}t]_{X}=\bigoplus_{n\in{\mathbb{Z}}_{\geq
0}}({\mathcal{J}}^{n})_{X}{t}^{n}\subset{\mathcal{O}}_{X}[t],$

and the extended Rees algebra on $X$:

${\mathcal{O}}_{X}[{t}^{-1},{\mathcal{J}}t]_{X}=\bigoplus_{n\in{\mathbb{Z}}_{\geq
0}}{\mathcal{J}}^{n}_{X}{t}^{n}\oplus\bigoplus_{-n\in{\mathbb{Z}}_{<0}}{t}^{-n}\subset{\mathcal{O}}_{X}[t,{t}^{-1}].$

####  5.3.3. Cobordant blow-up of ${\mathbb{Q}}$-ideals

Let ${\mathcal{J}}={\mathcal{I}}^{1/m}$ be a ${\mathbb{Q}}$-ideal on $X$.
Consider the normalized blow-up $\pi:Y\to X$ of ${\mathcal{I}}$, with the
exceptional divisor $E^{0}$ such that
${\mathcal{O}}_{Y}(-E^{0})={\mathcal{O}}_{Y}\cdot{\mathcal{I}}$.

Then ${\mathcal{O}}_{Y}\cdot{\mathcal{I}}^{1/m}$ is the ${\mathbb{Q}}$-ideal
${\mathcal{O}}_{Y}(-E^{0})^{1/m}$, which corresponds to the
${\mathbb{Q}}$-Cartier exceptional divisor $\frac{1}{m}E^{0}$ on $Y$.

Consequently, by the blow-up of the ${\mathbb{Q}}$-ideal
${\mathcal{J}}={\mathcal{I}}^{1/m}$ we mean the the normalized blow-up
$\pi:Y\to X$ of ${\mathcal{I}}$, with the associated ${\mathbb{Q}}$-Cartier
divisor $\frac{1}{m}E^{0}$.

######  Definition 5.3.4.

By the simple cobordant blow-up/full cobordant blow-up of the
${\mathbb{Q}}$-ideal ${\mathcal{J}}={\mathcal{I}}^{1/m}$ we mean the
cobordization/full cobordization of the normalized blow-up $Y\to X$ of
${\mathcal{J}}$ with respect to the group
$\Gamma={\mathbb{Z}}\cdot\frac{1}{m}E^{0}\subset{\operatorname{Cl}}(Y/X)\otimes{\mathbb{Q}}$
generated by $\frac{1}{m}E^{0}$.

######  Lemma 5.3.5.

Let $\sigma:B\to X$ be the simple full cobordant blow-up of the
${\mathbb{Q}}$-ideal ${\mathcal{J}}={\mathcal{I}}^{1/m}$ on a normal scheme
$X$. Then

$B=\operatorname{Spec}_{X}({\mathcal{O}}_{X}[{t}^{-1},{\mathcal{J}}t])_{X}$

  1. (1)

$B_{+}=B\smallsetminus V({\mathcal{J}}t)$

  2. (2)

${\mathcal{O}}_{B_{+}}\cdot{\mathcal{J}}={t}^{-1}\cdot{\mathcal{O}}_{B_{+}}$

###### Proof.

Let $\pi:Y\to X$ be the normalized blow-up of ${\mathcal{I}}$, $E^{0}$ is the
exceptional divisor on $Y$ such
${\mathcal{O}}_{X}\cdot{\mathcal{I}}={\mathcal{O}}_{Y}(-E_{0})$. Thus, by
[Wło22, Proof of Lemma 2.1.4],

$\pi_{*}({\mathcal{O}}_{Y}(-\frac{n}{m}E^{0})=(f\in\pi_{*}({\mathcal{O}}_{Y})={\mathcal{O}}_{X}:f^{n}\in\pi_{*}({\mathcal{O}}_{Y}(-mE_{0}))=({\mathcal{I}}^{m})^{\operatorname{int}})={\mathcal{J}}^{m}_{X}.$

giving the formula for $B$:

$B=\operatorname{Spec}_{X}(\pi_{*}(\bigoplus_{n\in{\mathbb{Z}}}{\mathcal{O}}_{Y}(n\cdot(1/m)\cdot
E^{0}){t}^{n}))=\operatorname{Spec}_{X}({\mathcal{O}}_{X}[{t}^{-1},{\mathcal{J}}t])_{X}.$

By Proposition 5.1.4,

$B_{+}=B\smallsetminus V({\mathcal{I}}\cdot{\bf t}^{-E^{0}})=B\smallsetminus
V({\mathcal{J}}{\bf t}^{-(1/m)E^{0}})=B\smallsetminus V({\mathcal{J}}t),$

as $(1/m)E^{0}$ generates $\Gamma$, and ${\bf t}^{(1/m)E^{0}}$ corresponds to
${t^{-1}}$. Thus the inverse image of

${\mathcal{O}}_{B_{+}}\cdot{\mathcal{I}}^{1/m}={\mathcal{O}}_{B_{+}}\cdot{\mathcal{J}}={t}^{-1}{\mathcal{O}}_{B_{+}}\cdot{\mathcal{J}}t={\mathcal{O}}_{B_{+}}\cdot{t}^{-1}$

is a Cartier exceptional divisor. We use here the fact that the
${\mathbb{Q}}$-ideal ${\mathcal{J}}t_{|B_{+}}={\mathcal{O}}_{B_{+}}$, as
${\mathcal{J}}t=({\mathcal{I}}t^{a})^{1/a}=({\mathcal{I}}\cdot
t^{-E^{0}})^{1/a}$ is trivial on $B_{+}=B\smallsetminus
V({\mathcal{I}}t^{-E^{0}})$. ∎

###  5.4. Weighted cobordant blow-ups revisited

Let $\pi_{B}:B\to X$ be the simple cobordant blow-up of the weighted center
${\mathcal{J}}=(u_{1}^{1/w_{1}},\ldots,u_{k}^{1/w_{k}})$, where
$u_{1},\ldots,u_{k}$ is a partial system of local parameters on a regular
scheme. Assume, first, that the weights $w_{i}$ are relatively prime. The
center ${\mathcal{J}}$ can be written as ${\mathcal{J}}={\mathcal{I}}^{1/m}$,
where

${\mathcal{I}}=(u_{1}^{m/w_{1}},\ldots,u_{k}^{m/w_{k}}),$

is the ideal, and the weights $w_{i}|m$.

Let $E^{0}$ be the exceptional divisor of the blow-up $\pi:Y\to X$ of
${\mathcal{J}}$. Let $\nu_{E^{0}}$ be the associated exceptional valuation.
Using the toric chart, defined by $u_{1},\ldots,u_{k}$ one reduces the
situation to the blow-up of the toric ${\mathbb{Q}}$-ideal ${\mathcal{J}}$ on
a toric variety $X_{\sigma}$, where $\sigma=\langle e_{1},\ldots,e_{k}\rangle$
is regular. The ${\mathbb{Q}}$-ideal ${\mathcal{J}}={\mathcal{I}}^{1/m}$
defines a piecewise linear convex function

$F_{\mathcal{J}}=\min(\frac{1}{w_{1}}e_{1}^{*},\ldots,\frac{1}{w_{k}}e_{k}^{*})=1/m\cdot\min(\frac{m}{w_{1}}e_{1}^{*},\ldots,\frac{m}{w_{k}}e_{k}^{*}).$

The normalized blow-up of ${\mathcal{J}}$ defines a decomposition $\Delta$ of
$\sigma$ into the maximal subcones where $F_{\mathcal{J}}$ is linear. Let
$w:=(w_{1},\ldots,w_{k})$ , and

$F_{\mathcal{J}}(e_{i})=0,\ldots,F_{\mathcal{J}}(w)=1.$

Then $\Delta$ is the star subdivision at $\langle w\rangle$. Moreover, the
vector $mw$ corresponds to $mF_{\mathcal{J}}=F_{\mathcal{I}}$ in the sense
that they define the same Weil divisors , and $w$ and corresponds to $E^{0}$,
so the valuation $\nu_{E^{0}}$ on $Y$ is associated with $w$. In particular,

$\nu_{E^{0}}(u_{i})=e_{i}^{*}(w)=w_{i}.$

Consequently,
${\mathcal{O}}_{Y}\cdot(u_{1}^{m/w_{1}},\ldots,u_{k}^{m/w_{k}})={\mathcal{O}}_{Y}(-mE^{0})$
is the ideal of the Cartier divisor $mE^{0}$ on $Y$, associated with the
integral function $mF_{\mathcal{J}}$, and the ${\mathbb{Q}}$-ideal
${\mathcal{J}}=(u_{1}^{1/w_{1}},\ldots,u_{k}^{1/w_{k}})$ corresponds to the
${\mathbb{Q}}$-divisor $(1/m)\cdot mE^{0}=E^{0}$ which is a Weil divisor.

The cobordant blow-up associated with the group $\Gamma={\mathbb{Z}}\cdot
E^{0}={\operatorname{Cl}}(Y/X)$ is given by the standard formula from Theorem
3.4.3:

$\displaystyle
B=\operatorname{Spec}_{X}(\pi_{*}(\bigoplus_{n\in{\mathbb{Z}}}{\mathcal{O}}_{Y}(nE^{0})){t}^{n}=\operatorname{Spec}_{X}({\mathcal{O}}_{X}[{t}^{-1},{\mathcal{J}}t])_{X}=$
$\displaystyle=\operatorname{Spec}_{X}(\bigoplus_{a_{i}\in{\mathbb{Z}}}\,\,{\mathcal{I}}_{\nu,a_{i}}\,\,\cdot
t_{1}^{a_{1}}\cdot\ldots\cdot t_{k}^{a_{k}})=$
$\displaystyle=\operatorname{Spec}_{X}({{\mathcal{O}}_{X}}[{t}^{-1},{u_{1}}{t}^{w_{1}},\ldots,{u_{k}}{t}^{w_{k}}]),$

where $w_{i}=\nu_{E^{0}}(u_{i})$.

In general, for arbitrary weights, the simple cobordant blow-up of
$(u_{1}^{1/w_{1}},\ldots,u_{k}^{1/w_{k}})$ is associated with the group
$\Gamma={\mathbb{Z}}\cdot\frac{1}{w^{0}}E^{0}=\frac{1}{w^{0}}\cdot{\operatorname{Cl}}(Y/X)$,
where $w^{0}:=\gcd(w_{1},\ldots,w_{k})$, and with the valuation $w^{0}\nu$,
with

$w^{0}\nu(x_{i})=w_{i}.$

Now

${\mathcal{I}}_{w^{0}\nu,a}=(u_{1}^{b_{1}}\cdot\ldots\cdot
u_{k}^{b_{k}})\mid\sum_{j=1}^{k}{b_{j}}w_{i}\geq a).$

Comparing gradations we see

$\bigoplus_{a\in{\mathbb{Z}}}{\mathcal{I}}_{w^{0}\nu,a}t^{a}={\mathcal{O}}_{X}[t^{-1},u_{i}t^{w_{i}}].$

Then

$\displaystyle
B=\operatorname{Spec}_{X}(\pi_{*}(\bigoplus_{n\in{\mathbb{Z}}}{\mathcal{O}}_{Y}(n\cdot(1/w^{0})\cdot
E^{0})t^{n}=$
$\displaystyle=\operatorname{Spec}_{X}(\bigoplus_{a_{i}\in{\mathbb{Z}}}{\mathcal{I}}_{w_{0}\nu,a}\,\,\cdot
t^{a})=\operatorname{Spec}_{X}({{\mathcal{O}}_{X}}[t^{-1},t^{w_{1}}{x_{1}},\ldots,t^{w_{k}}{x_{k}}]).$

By the above $B_{+}=B\smallsetminus V({\mathcal{J}}t)=B\smallsetminus
V(t^{w_{1}}{x_{1}},\ldots,t^{w_{k}}{x_{k}})$, and
${\mathcal{O}}_{B_{+}}\cdot{\mathcal{J}}={\mathcal{O}}_{B_{+}}\cdot t^{-1}$.

These weighted cobordant blow-ups were studied in [Wło22] and used for the
resolution of varieties in characteristic zero and some classes of
singularities in positive and mixed characteristic. To a great extent, they
are equivalent to the stack-theoretic weighted blow-ups introduced and
considered in [McQ19], and [ATW19].

###  5.5. Multiple weighted cobordant blow-ups of Abramovich-Quek

In the paper [AQ21], the authors consider the generalization of the weighted
blow-ups, so-called, multi-weighted blow-ups ${Bl_{{\mathcal{J}},b}}$,
associated with a ${\mathbb{Q}}$-ideal ${\mathcal{J}}$ and a vector
$b=(b_{1},\ldots,b_{k})$. They are constructed locally in toric charts in the
language of fantastacks and stack-theoretic quotients via Satriano
combinatorial approach [Sat13]. The multi-weighted blow-ups are used to prove
the logarithmic resolution on smooth toroidal ambient Artin stacks in
characteristic zero.

We give here a geometric interpretation of this construction in the language
of cobordizations with respect to a subgroup. In particular, this approach
does not rely on coordinates or combinatorics.

Let $\pi:Y\to X$ be the normalized blow-up of a locally monomial center
${\mathcal{J}}$ on a regular scheme over a field. Denote by
$E_{1},\ldots,E_{k}$ the irreducible exceptional divisors. Let
$\nu_{1},\ldots,\nu_{k}$ be the associated exceptional valuations. We consider
the full cobordant blow-up of ${\mathcal{J}}$ with respect to the subgroup

$\Gamma_{b}:={\mathbb{Z}}\frac{1}{b_{1}}E_{1}\oplus\ldots\oplus{\mathbb{Z}}\frac{1}{b_{k}}E_{k}\subset{\operatorname{Cl}}(Y/X)\otimes{\mathbb{Q}},$

for any positive integers $b_{1},\ldots,b_{k}$, and $b=(b_{1},\ldots,b_{k})$.
Write

$B=\operatorname{Spec}_{X}(\pi_{*}(\bigoplus_{E\in\Gamma_{b}}{\mathcal{O}}_{X}(E)),\quad
B_{+}=\operatorname{Spec}_{Y}(\bigoplus_{E\in\Gamma_{b}}{\mathcal{O}}_{Y}(E)).$

The generators $\frac{1}{b_{1}}E_{1},\ldots,\frac{1}{b_{k}}E_{k}$ are
associated with the monomial valuations

$\nu^{b}_{1}:=b_{1}\nu_{1},\ldots,\nu^{b}_{k}:=b_{k}\nu_{k}.$

Then locally on $X$ using the Proposition 1.2.2, and the proof of Lemma
2.3.5(1) we can write

$\displaystyle
B=\operatorname{Spec}_{X}(\bigoplus_{a_{i}\in{\mathbb{Z}}}\,\,\bigcap^{k}_{i=1}\,{\mathcal{I}}_{\nu^{b}_{i},a_{i}}\,\,\cdot
t_{1}^{a_{1}}\cdot\ldots\cdot t_{k}^{a_{k}})=$
$\displaystyle=\bigcap^{k}_{i=1}{\mathcal{O}}_{X}[t_{i}^{-1},u_{j}t_{i}^{\nu^{b}_{i}(u_{j})}][\check{\bf
t}_{i},\check{\bf t}_{i}^{-1}]=$
$\displaystyle=\operatorname{Spec}_{X}({\mathcal{O}}_{X}[t_{1}^{-1},\ldots,t_{k}^{-1},u_{1}{\bf
t}^{\alpha^{b}_{1}},\ldots,u_{k}{\bf t}^{\alpha^{b}_{k}}]),$

where

  * •

$\check{\bf t}_{i}:=t_{1},\ldots,\check{t_{i}},\ldots,t_{k}$

  * •

$u_{1},\ldots,u_{k}$, is a system of coordinates on open $U\subset X$ defining
monomial generators for ${\mathcal{J}}$, and

  * •

${\bf t}^{\alpha^{b}_{i}}:=t_{1}^{a^{b}_{i1}}\cdot\ldots\cdot
t_{k}^{a^{b}_{ik}},\quad\mbox{with}\quad
a^{b}_{ij}:=\nu^{b}_{i}(u_{j})=b_{i}\nu_{i}(u_{j})\geq 0.$

Note that under this correspondence $t_{i}^{-1}\mapsto{\bf
t}^{\frac{1}{b_{i}}E_{i}}.$

Let

(7) $\displaystyle E^{0}=a_{1}E_{1}+\ldots+a_{k}E_{k}$

be the exceptional divisor of $\pi:Y\to X$, for the relevant
$a_{i}\in{\mathbb{Z}}_{\geq 0}$, such that
${\mathcal{O}}_{Y}(-E^{0})={\mathcal{O}}_{Y}\cdot{\mathcal{J}}$. By
Proposition 5.1.4,

$B_{+}=B\smallsetminus V({\mathcal{J}}{\bf t}^{-E^{0}})=B\smallsetminus
V({\mathcal{J}}{\bf t}^{\alpha^{b}}),$

where ${\bf t}^{\alpha^{b}}$ corresponds to ${\bf t}^{-E_{0}}$, under
$t_{i}^{-1}\mapsto{\bf t}^{\frac{1}{b_{i}}E_{i}}$. Thus by (7),
$\alpha^{b}=(b_{1}a_{1},\ldots,b_{k}a_{k}),$ and

$\displaystyle B_{+}=B\smallsetminus V({\mathcal{J}}{\bf
t}^{-E^{0}})=B\smallsetminus V({\mathcal{J}}t_{1}^{b_{1}a_{1}}\cdot\ldots\cdot
t_{k}^{b_{k}a_{k}}).$

In particular, if $X$ is regular over a field and ${\mathcal{J}}$ is a locally
monomial ideal on $X$, then the full cobordant blow-up $B$ of ${\mathcal{J}}$
with respect to $\Gamma_{b}$ is regular.

####  5.5.1. Multiple weighted blow-ups associated with ${\mathbb{Q}}$-ideals

Consider the normalized blow-up $\pi:Y\to X$ of a monomial
${\mathbb{Q}}$-ideal ${\mathcal{J}}$, with the associated exceptional divisor
$E^{0}=a_{1}E_{1}+\ldots+a_{k}E_{k}$ with rational, positive coefficients
$a_{i}$, as in Section 5.3.3. We choose $b=(b_{1},\ldots,b_{k})$ with
$b_{i}\in{\mathbb{Z}}_{>0}$, such that

$\Gamma_{b}={\mathbb{Z}}\frac{1}{b_{1}}E_{1}\oplus\ldots\oplus{\mathbb{Z}}\frac{1}{b_{k}}E_{k}\subset{\operatorname{Cl}}(Y/X)\otimes{\mathbb{Q}},$

is the minimal subgroup of ${\operatorname{Cl}}(Y/X)\otimes{\mathbb{Q}}$
containing $E^{0}$.

Thus any monomial ${\mathbb{Q}}$-ideal ${\mathcal{I}}$ and
$b=(b_{1},\ldots,b_{k})\in{\mathbb{Z}}^{k}_{>0}$ determines a unique
associated cobordant blow-up $B_{+}\to X$ with respect to the group
$\Gamma_{b}$. This way, taking the stack-theoretic quotient, we obtain the
Abramovich-Quek multiple weighted blow-up $[B_{+}\sslash T]\to X$ from [AQ21],
which is necessarily regular for a regular $X$.

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