${\mathcal{J}}=(x^{1/w_{1}},y^{1/w_{3}},z^{1/w_{3}})$ is given by

$\operatorname{in}(f)=x^{p}+y^{p}z+z^{k}.$

Then by the similar argument as before

${\operatorname{supp}}(\operatorname{in}(f),3)=V(x,y,z)=V({\mathcal{J}})\subset
C_{{\mathcal{J}}}(X)$

The cobordant blow-up at ${\mathcal{J}}$ decreases the maximal order to $2$,
and the equation can be solved by further cobordant blow-ups.

Observe that if one considers ”characteristic zero” center
${\mathcal{J}}_{1}:=(x^{\frac{1}{p+1}},y^{\frac{1}{p}},z^{\frac{1}{p}})$
associated with maximal admissible Rees center
${\mathcal{O}}_{X}[xt^{1/p},yt^{1/(p+1)},z^{1/(p+1)})]$ then
$\operatorname{in}(f)=x^{p}+y^{p}z$, and

${\operatorname{supp}}(\operatorname{in}(f),3)=V(x,y)\neq
V({\mathcal{J}}_{1})$

In fact the full cobordant blow-up $B^{\prime}_{+}\to X$ of
${\mathcal{J}}_{1}$ transforms $f$ into

$x^{p}+y^{p}z+t^{(p+1)-k}z^{k}+t^{1-p}x^{p-1}y^{2}+t^{-3p-1}x^{p+1}yz$

On $B_{+}=B\smallsetminus V(x,y,z)$ in the chart $B\smallsetminus V(z)$, the
maximal order does not decrease:

${\operatorname{maxord}}_{B_{+}}(\sigma^{s}(f))=p.$

###  4.5. Homogenous subschemes

Let $V$ be a regular scheme and

$X:={\mathbb{A}}^{n}_{V}=\operatorname{Spec}_{V}({\mathcal{O}}_{V}[x_{1},\ldots,x_{n}]))$

be an affine space over $V$. Let
${\mathcal{J}}=(x_{1}^{1/w_{1}},\ldots,x_{k}^{1/w_{k}})$ be a center on $X$.
The center ${\mathcal{J}}$ defines the gradation on
${\mathcal{O}}_{V}[x_{1},\ldots,x_{n}]$ via the isomorphism $x_{i}\mapsto
t^{w_{i}}x_{i}$ for $i\leq k$, $x_{i}\mapsto x_{i}$:

$\phi:{\mathcal{O}}_{V}[x_{1},\ldots,x_{n}]\to{\mathcal{O}}_{V}[t^{w_{1}}x_{1},\ldots,t^{w_{k}}x_{k},x_{k+1},\ldots,x_{n}]$

It takes the monomial $x^{\alpha}=x_{1}^{a_{1}}\cdot\ldots\cdot
x_{n}^{a_{n}}\in{\mathcal{J}}^{a_{1}w_{1}+\ldots+a_{k}w_{k}}_{X}$ to
$t^{a_{1}w_{1}+\ldots+a_{k}w_{k}}\cdot x^{\alpha}$.

Let $f\in{\mathcal{J}}^{a}_{X}\smallsetminus{\mathcal{J}}^{a+1}_{X}$ be a
homogenous element in
${\mathcal{O}}_{X}={\mathcal{O}}_{V}[x_{1},\ldots,x_{n}]$ of grading $a$. Then

$\phi(f)=t^{a}f\in{\mathcal{O}}_{V}[t^{w_{1}}x_{1},\ldots,t^{w_{k}}x_{k},x_{k+1},\ldots,x_{n}].$

The latter can be interpreted as the strict transform of $f$ inside of the
cobordant algebra
${\mathcal{O}}_{B}={\mathcal{O}}_{X}[t^{-1},t^{w_{1}}x_{1},\ldots,t^{w_{k}}x_{k}]$.

######  Definition 4.5.1.

With the above notation and assumption any subscheme of $X$ defined by a
homogenous ideal ${\mathcal{I}}$ will be called a homogenous subscheme with
respect to the center ${\mathcal{J}}$.

This leads to a simple but useful result.

######  Lemma 4.5.2.

With the assumption and notation above. Let
${\mathcal{I}}\subset{\mathcal{O}}_{V}[x_{1},\ldots,x_{n}]$ be a homogenous
ideal with respect to ${\mathcal{J}}$. Let $\sigma:B\to X$ be the cobordant
blow-up of ${\mathcal{J}}$, and $\pi:Y:=X\times{\mathbb{A}}^{1}\to X$ be the
standard projection. Then

$B=\operatorname{Spec}({\mathcal{O}}_{X}[t^{-1},t^{w_{1}}x_{1},\ldots,t^{w_{k}}x_{k}]=\operatorname{Spec}({\mathcal{O}}_{V}[t^{-1},t^{w_{1}}x_{1},\ldots,t^{w_{k}}x_{k},x_{k+1},\ldots,x_{n}]$

Thus there exists an isomorphism

$B\simeq Y=X\times{\mathbb{A}}^{1}$

over $V$, defined by

$B=\operatorname{Spec}({\mathcal{O}}_{V}[t^{-1},t^{w_{1}}x_{1},\ldots,t^{w_{k}}x_{k},x_{k+1},\ldots,x_{n}]\to
Y=\operatorname{Spec}({\mathcal{O}}_{V}[x_{1},\ldots,x_{n},y])$

which takes

  * •

$x_{i}\mapsto t^{w_{i}}x_{i}$ for $i\leq k$,

  * •

$x_{i}\mapsto x_{i}$, $i>k$

  * •

$y\mapsto t^{-1}$,

  * •

${\mathcal{O}}_{Y}\cdot{\mathcal{I}}$ to the strict transform ideal
$\sigma_{B}^{s}({\mathcal{I}})$

  * •

$V(x_{1},\ldots,x_{n})$ to the vertex $V$ of $B$

∎

######  Example 4.5.3.

The lemma explains the transformation of the singularity

$f=x_{1}^{c_{1}}+\ldots+x_{k}^{c_{k}},$

from Example 3.2.2 which is taken to

$\sigma^{c}(f)=\sigma^{s}(f)=(x^{\prime}_{1})^{c_{1}}+\ldots+(x^{\prime}_{k})^{c_{k}}.$

######  Example 4.5.4.

The following example is due to Narasimhman [Nar83]. Let

$f=x^{2}+yz^{3}+zw^{3}+y^{7}w\in k[x,y,z,w],$

where $k$ is a field of characteristic $2$. Its singular locus
${\operatorname{Sing}}(f)$ coincides with

${\operatorname{supp}}(f,2)=V(D(f))=V(f,Y,Z,W),$

where

$Y:=D_{y}(f)=z^{3}+y^{6}w,\quad Z:=D_{z}(f)=yz^{2}+w^{3},\quad
W:=D_{w}(f)=zw^{2}+y^{7}.$

Note that $F:=f+yY+zZ=x^{2}+yz^{3}$ is again binomial. Hence
${\operatorname{Sing}}(f)=V(F,Y,Z,W)$ is binomial, so it is a (non-normal)
toric subvariety of ${\mathbb{A}}^{4}$ of dimension $1$. One can compute that
it coincides with the image $t\mapsto(t^{32},t^{7},t^{19},t^{15})$.

Observe that the subscheme $V(f)$ is homogenous with respect to the center

${\mathcal{J}}:=(x^{1/32},y^{1/7},z^{1/19},t^{1/15}).$

Thus, by Lemma 4.5.2, the cobordant blow-up of ${\mathcal{J}}$ transforms $X$
into $B\simeq X\times{\mathbb{A}}^{1}_{k}$ with the same equation $f$
describing $\sigma^{s}(f)$. Then $B_{+}=B\smallsetminus V(x,y,z,w)$. The
singular locus of $\sigma^{s}(f)$ on $B_{+}$ is simply the product

$\\{(t^{32},t^{7},t^{19},t^{15})\mid t\neq 0\\}\times{\mathbb{A}}^{1}.$

The functions $Y$, $Z$, and $W$ are semiinvariant on $B$. Moreover
$D_{z}(Y)=z^{2}$, $D_{w}(Z)=w^{2}$, and $D_{y}(W)=y^{6}$ are invertible on an
open subset $B_{yzw}=B\smallsetminus V(yzw)$. On the other hand
$D_{y}(Y)=D_{z}(Z)=D_{w}(W)\equiv 0$. Thus any two of $Y,Z,W$ form a partial
system of local parameters on $B_{yzw}$. Furthermore the set $B_{yzw}$
contains

${\operatorname{Sing}}(\sigma^{s}(f))\cap B_{+}=V(\sigma^{s}(f),Y,Z,W)\cap
B_{+}=V(\sigma^{s}(f),Y,Z,W)\smallsetminus V(x,y,z,w),$

as $yzw$ is invertible on $\\{(t^{32},t^{7},t^{19},t^{15})\mid t\neq
0\\}\times{\mathbb{A}}^{1}$. In addition we can write on $B_{yzw}$:

$\sigma^{s}(f)=x^{2}+\frac{1}{z^{2}}YZ+\frac{y^{6}w^{4}}{z^{2}}=X^{2}+\frac{1}{z^{2}}YZ,$

where $X:=x+\frac{y^{3}w^{2}}{z}$, $z$ is a unit, and $X,Y,Z$ form a partial
system of local parameters. The subscheme $V_{B_{+}}(\sigma^{s}(f))$ is almost
homogenous on $B_{xyz}\subset B_{+}$ with respect to the
${\mathbb{G}}_{m}$-stable smooth center described as
${\mathcal{J}}_{1}:=(X,Y,Z)$ on $B_{xyz}\subset B_{+}$. In fact,
${\mathcal{J}}_{1}$ is an ordinary ideal defining a smooth locus
${\operatorname{Sing}}(\sigma^{s}(f))$. The cobordant (or the ordinary) blow-
up of ${\mathcal{J}}_{1}$ determines a cobordant resolution. Further applying
locally toric resolution of [Wło20, Theorem 7.17.1] resolves the singularity
functorially with SNC exceptional divisor.

Alternatively, the singularity can be resolved using the characteristic zero
invariant ”${\operatorname{inv}}$,” but the process is longer and requires
three cobordant blow-ups.

######  Remark 4.5.5.

The Narasinhman example shows that the maximal contact, or even any regular
hypersurface containing equimultiple locus, that is the subset of the points
with the equal multiplicity of a function does not exist. In characteristic
zero such property follows immediately from the definition of maximal contact
$u\in{\mathcal{D}}^{a-1}({\mathcal{I}})$ at a point $p$, where
${\operatorname{ord}}_{p}({\mathcal{I}})=a$. In our setting it is a
consequence of Lemma 3.6.24(1). The equimultiple locus
${\operatorname{supp}}(f,2)={\operatorname{Sing}}(f)$ has embedding dimension
$4$ and cannot be embedded into a smooth hypersurface. In fact one can show
that the invariant ”${\operatorname{inv}}$” in positive characteristic is not
upper semicontinuous, and its locus does not have good properties.

####  4.5.6. Cobordant blow-ups vs blow-ups at smooth centers

In the cobordant resolution of the Narasinhman example, the second cobordant
blow-up was at the smooth center $(X,Y,Z)$ with weights equal to $1$. In
general, we have:

######  Lemma 4.5.7.

Let $X$ be a regular scheme. Let $B_{+}\to X$ be a cobordant blow-up of
${\mathcal{J}}$, locally given by ${\mathcal{J}}=(u_{1},\ldots,u_{k})$ with
weights $1$. Let $Bl_{{\mathcal{J}}}(X)\to X$ be the ordinary blow-ups of the
smooth center ${\mathcal{J}}$. Then the induced quotient morphsim

$B_{+}\to B_{+}/{\mathbb{G}}_{m}\simeq Bl_{{\mathcal{J}}}(X)$

is a locally trivial ${\mathbb{G}}_{m}$-bundle.

###### Proof.

The problem is local on $X$. Then
$B=\operatorname{Spec}_{X}({\mathcal{O}}_{X}[t^{-1},u_{1}t,\ldots,u_{k}t])$,
and $B_{+}$ is covered by $B_{u_{i}t^{w}}=B\smallsetminus
V(u_{i}t)=\operatorname{Spec}_{X}({\mathcal{O}}_{X}[t^{-1},u_{1}t,\ldots,u_{k}t,(u_{i}t)^{-1}])$,
and

$B_{u_{i}t}/{\mathbb{G}}_{m}=\operatorname{Spec}_{X}({\mathcal{O}}_{X}[\frac{u_{1}t}{u_{i}t},\ldots,\frac{u_{k}t}{u_{i}t}])=\operatorname{Spec}_{X}({\mathcal{O}}_{X}[\frac{u_{1}}{u_{i}},\ldots,\frac{u_{k}}{u_{i}}]).$

Thus the induced morphism $B_{+}/{\mathbb{G}}_{m}\to X$ is a weighted blow-up
which coincides with the standard blow-up of ${\mathcal{J}}$. Since we have
that $t^{-1}=(u_{i}t)^{-1}u_{i}$, we can write

${\mathcal{O}}_{B_{u_{i}t}}={\mathcal{O}}_{X}[\frac{u_{1}}{u_{i}},\ldots,\frac{u_{k}}{u_{i}}])[t^{-1},u_{i}t,(u_{i}t)^{-1}]={\mathcal{O}}_{X}[\frac{u_{1}}{u_{i}},\ldots,\frac{u_{k}}{u_{i}}])[u_{i}t,(u_{i}t)^{-1}].$

Thus locally $B_{+}\simeq(Bl_{{\mathcal{J}}})\times{\mathbb{G}}_{m}$. ∎

##  5\. Appendix

###  5.1. Cobordant blow-ups of Rees algebras and generalized weighted centers

####  5.1.1. Generalized cobordant blow-ups

######  Definition 5.1.2.

Let $R=\bigoplus_{i\in{\mathbb{Z}}_{\geq
0}}R_{a}t^{a}\subset{\mathcal{O}}_{X}[t]$ be any Rees algebra on a Noetherian
scheme $Y$, and let $R[t^{-1}]$ be an extended algebra of $R$. Then the full
cobordant blow-up of $R$ is given by $B:=Spec_{X}(R[t^{-1}])\to X$. The
cobordant blow-up is simply $B_{+}:=B\smallsetminus V(R)$.

####  5.1.3. Generalized centers

We generalize the centers accordingly.

######  Definition 5.1.4.

A Rees center on a Noetherian scheme $X$ is an extended Rees algebra
${\mathcal{A}}^{\operatorname{ext}}$ it can be locally presented as
${\mathcal{A}}^{\operatorname{ext}}={\mathcal{O}}_{X}[t^{-w},t^{1/a_{1}}f_{1},\ldots,t^{1/a_{k}}f_{k}]$,
where $V(f_{1},\ldots,f_{k})$ is a regular subscheme of $X$.

The following is a simple consequence of the definition:

######  Lemma 5.1.5.

If $f:X\to Y$ is a morphism of Noetherian schemes, and $R_{Y}$ is a Rees
algebra on $Y$ then $R_{X}={\mathcal{O}}_{X}\cdot R_{Y}$ is the induced Rees
algebra on $X$. Let $B_{Y}\to Y$ and $B_{X}\to X$ denote the full cobordant
blow-ups of $Y$ at $R_{Y}$ and $X$ at $R_{X}$ respectively. Then there are
induced morphisms $f_{B}:B_{X}\to B_{Y}$, $f_{B_{+}}:B_{X+}\to B_{Y+}$
commuting with cobordant blow-ups to $X$ and $Y$. Furtheremore, if $X\to Y$ is
a closed immersion, then $B_{X}\to B_{Y}$ and $f_{B_{+}}:B_{X+}\to B_{Y+}$ are
so.

###### Proof.

The morphism $f:X\to Y$ determines the homomorphism of graded sheaves
$R_{Y}\to f_{*}R_{X}$, so that for any open subset $U\subset X$ and $V\subset
Y$ such that $f(U)\subseteq V$, we have the induced graded homomorphisms
$R_{Y}(V)\to R_{X}(U)$ which gives rise to $B_{X}\to B_{Y}$ taking the vertex
$V_{X}$ to the vertex $V_{Y}$. If $f$ is a closed immersion then $X$ can be
thought as a closed subscheme of $Y$ and we get the induced surjective
homomorphism of graded ${\mathcal{O}}_{Y}$-modules $R_{Y}\to R_{X}$ which
defines the closed immersions $f_{B}:B_{X}\to B_{Y}$, $f_{B_{+}}:B_{X+}\to
B_{Y+}$. ∎

####  5.1.6. The maximal admissible centers on the subscheme are regular

######  Lemma 5.1.7.

Let $X$ be a smooth variety over a field $k$ of characteristic zero, and $E$
be an SNC divisor on $X$. Let $Y\subset X$ be a reduced closed subscheme of a
$X$ which is not locally contained in the divisor $E$, and let
${\mathcal{I}}_{Y}$ be its ideal on $X$. Let
${\mathcal{A}}^{{\operatorname{ext}}}$ be a maximal
${\mathcal{I}}_{Y}$-admissible center locally at a point $p\in Y$ of the form
${\mathcal{A}}^{\operatorname{ext}}={\mathcal{O}}_{X}[t^{-w},t^{1/a_{1}}u_{1}t^{1/a_{k}},\ldots,t^{1/a_{k}}u_{k}]$.
Then

${\mathcal{O}}_{Y}\cdot{\mathcal{A}}^{\operatorname{ext}}={\mathcal{O}}_{Y}[t^{-w},t^{1/a_{1}}f_{1}t^{1/a_{k}},\ldots,t^{1/a_{k}}f_{k}],$

for $f_{i}:=u_{i|Y}$, is a Rees center on $Y$.

###### Proof.

By definition ${\mathcal{I}}_{Y}t\subset{\mathcal{A}}^{{\operatorname{ext}}}$.
So

$Y=V({\mathcal{I}}_{Y})=V({\mathcal{O}}_{X}[{\mathcal{I}}_{Y}t])\supseteq
V({\mathcal{A}}^{\operatorname{ext}})=V(u_{1},\ldots,u_{k})$

Consequently

$V(f_{1},\ldots,f_{k})=V(u_{1},\ldots,u_{k})_{|Y}=V(u_{1},\ldots,u_{k})$

is regular on $Y$.

∎

###  5.2. Geometric quotient and cobordant blow-ups

In this section, we review some standard properties of the torus actions,
usually stated over algebraically closed fields, and extended here to
noetherian schemes.

######  Definition 5.2.1.

Let a torus $T={\mathbb{G}}_{m}^{l}$, where
${\mathbb{G}}_{m}=\operatorname{Spec}({\mathbb{Z}}[t,t^{-1}])$ act on a scheme
$X$. We say that an action of a torus $T$ on scheme $X$ is relatively affine
if for any point $p\in X$ admits an open $T$-stable affine neighborhood $U$ of
$p$.

Let $\pi:X\to Y$ be an invariant morphism of schemes. We say that an action of
a torus $T$ on scheme $X$ is relatively affine over $Y$ if for any point $p\in
X$ there is an open $T$-stable affine neighborhood $U$ of $p$, mapping to an
open affine neighborhood of $V$ of $q=\pi(q)\in Y$, and a $T$-equivariant
closed immersion

$\phi_{U}:U\hookrightarrow V\times{\mathbf{A}}^{n}\times{\mathbb{G}}_{m}^{r},$

with the diagonal action of $T$ on
${\mathbf{A}}^{n}\times{\mathbb{G}}_{m}^{r}$, and commuting with the invariant
morphisms

If additionally, $X$ admits geometric quotient, and $U$, and $V$ can be chosen
so that $V\times{\mathbf{A}}^{n}\times{\mathbb{G}}_{m}^{r}$ admits geometric
quotient then we say that $T$ admits relatively affine geometric quotient over
$Y$.

We shall need a well known fact for the action of the torus on any affine
schemes.

######  Lemma 5.2.2.

Let $X$ be an scheme with a relatively affine action of torus $T$, with the
group of characters $\chi(T)$. Then ${\mathcal{O}}_{X}$ admits a natural
grading and

${\mathcal{O}}_{X}=\bigoplus_{\alpha\in\chi(T)}({\mathcal{O}}_{X})_{\alpha},$

where $T$ acts on ${\mathcal{O}}(X)_{\alpha}$ by the character $t^{\alpha}$.

###### Proof.

We can reduce the situation to the case of affine $X$. The natural grading on
${\mathcal{O}}(X)$, and thus on the sheaf ${\mathcal{O}}_{X}$ is given by the
coaction

$\rho:{\mathcal{O}}(X)\to{\mathcal{O}}(X)\otimes{\mathbb{Z}}[T],$

where ${\mathbb{Z}}[T]={\mathbb{Z}}[t^{\pm}_{1},\ldots,t^{\pm}_{m}]$. For any
$f\in{\mathcal{O}}(X)$, $\rho(f)=\sum c_{\alpha}t^{\alpha}$, where
$c_{\alpha}\in{\mathcal{O}}(X)$. Moreover it follows from the associativity

$(\text{id}_{{\mathcal{O}}(X)}\otimes\mu)\circ\rho=(\rho\times\text{id}_{{\mathbb{Z}}[T]})\circ\rho$

where

$\mu:{\mathbb{Z}}[T]\to{\mathbb{Z}}[T]\otimes{\mathbb{Z}}[T],\quad\mu(t)=tt^{\prime}$

is defined by the group multiplication, such that

$\sum
c_{\alpha}(tt^{\prime})^{\alpha}=\sum\rho(c_{\alpha})(t^{\prime})^{\alpha},$

and hence $\rho(c_{\alpha})=c_{\alpha}t^{\alpha}$. From existence of the
identity map
$1_{T}:{\mathbb{Z}}[t^{\pm}_{1},\ldots,t^{\pm}_{m}]\to{\mathbb{Z}},$ such that

$(1_{T}\otimes\text{id}_{{\mathcal{O}}(X)})\circ\rho=\text{id}_{{\mathcal{O}}(X)}$

we deduce that

$f=(1_{T}\otimes\text{id}_{{\mathcal{O}}(X)})\circ\rho(f)=(1_{T}\otimes\text{id}_{{\mathcal{O}}(X)})(\sum
c_{\alpha}t^{\alpha})=\sum c_{\alpha}.$

Thus any element decomposes into a finite sum of the homogenous elements
$c_{\alpha}$ of grading $\alpha$, and

${\mathcal{O}}(X)=\bigoplus{\mathcal{O}}(X)_{\alpha},$

where $T$ acts on ${\mathcal{O}}(X)_{\alpha}$ by the character $t^{\alpha}$. ∎

######  Definition 5.2.3.

Let $X$ be a scheme with with a relatively affine action of torus $T$. By a
$T$ -stable ideal we mean a homogenous ideal ${\mathcal{I}}$ of
${\mathcal{O}}_{X}=\bigoplus_{\alpha}({\mathcal{O}}_{X})_{\alpha}$.

######  Definition 5.2.4.

Let $X$ be a normal scheme with a relatively affine action of torus $T$. A
$Q$-ideal ${\mathcal{J}}$ on a normal scheme is said to be $T$ -stable if is
of the form ${\mathcal{J}}={\mathcal{I}}^{1/n}$, where ${\mathcal{I}}$ is a
$T$-stable ideal.

######  Lemma 5.2.5.

Let ${\mathcal{J}}$ be a $T$-stable center on a regular scheme $X$ with a
relatively affine action of $T$, and a $T$-stable SNC divisor $E$. Then one
can write locally the center ${\mathcal{J}}$ as
${\mathcal{J}}=(x_{1}^{1/w_{1}},\ldots,x_{k}^{1/w_{k}})$, where
$x_{1},\ldots,x_{k}$ is a partial system of semiinvariant local parameters.

###### Proof.

We can assume that ${\mathcal{J}}$ has the form
${\mathcal{J}}=(x_{1}^{1/w_{1}},\ldots,x_{k}^{1/w_{k}})$ at a point $p\in X$.
Write $x_{1}\in({\mathcal{J}}^{w_{1}})_{X}$ as the sum of homogenous
components $x_{1}=\sum c_{\alpha}$. We conclude that
${\operatorname{ord}}_{p}(c_{\alpha})=1$ for some
$x^{\prime}_{1}:=c_{\alpha}\in({\mathcal{J}}^{w_{1}})_{X}$. So, by Lemma
3.1.29, one can replace $x_{1}$ with a semiinvariant $x^{\prime}_{1}$ in the
presentation of ${\mathcal{J}}$. By the induction on dimension, we assume that
${\mathcal{J}}_{|V(x^{\prime}_{1})}=((x^{\prime}_{2})^{1/w_{2}},\ldots,(x^{\prime}_{k})^{1/w_{k}})$
is represented by semiinvariant local parameters on $V(x_{1}^{\prime})$.
Moreover, there is a surjection of graded rings
${\mathcal{O}}_{X}\to{\mathcal{O}}_{V(x^{\prime}_{1})}={\mathcal{O}}_{X}/V(x^{\prime}_{1})$.
Then $x^{\prime}_{i}$ can be lifted to the homogenous elements in
${\mathcal{O}}_{X}$. Set

${\mathcal{J}}^{\prime}:=((x^{\prime}_{1})^{1/w_{1}},\ldots,(x^{\prime}_{k})^{1/w_{k}})=(x^{\prime}_{1})^{1/w_{1}}+((x^{\prime}_{2})^{1/w_{2}},\ldots,(x^{\prime}_{k})^{1/w_{k}}).$

Note that such a presentation is independent of the liftings of the elements
$x^{\prime}_{i}$, $i\geq 2$ on $V(x^{\prime}_{1})$ to $U$. Then
${\mathcal{J}}_{|V(x^{\prime}_{1})}={\mathcal{J}}^{\prime}_{|V(x^{\prime}_{1})}$,
and their saturated powers $(({\mathcal{J}}_{|V(x^{\prime}_{1})})^{a}))^{\rm
sat}=(({\mathcal{J}}^{\prime}_{|V(x^{\prime}_{1})})^{a})^{\rm sat}$ for
sufficiently divisible $a$ are equal as ideals. Then in the completion

$\widehat{{\mathcal{O}}_{X,p}}=x^{\prime}_{1}\cdot\widehat{{\mathcal{O}}_{X,p}}\oplus\widehat{{\mathcal{O}}_{X,p}}/(x^{\prime}_{1})$

both ideals are equal so

$\widehat{{\mathcal{J}}^{a}}:=(x^{\prime}_{1})^{a/w_{1}}+(({\mathcal{J}}_{|V(x^{\prime}_{1})})^{a}))^{\rm
sat}=(x^{\prime}_{1})^{a/w_{1}}+(({\mathcal{J}}^{\prime}_{|V(x^{\prime}_{1})})^{a}))^{\rm
sat}=\widehat{{\mathcal{J}}^{\prime a}}.$

This implies that ${\mathcal{J}}^{a}={\mathcal{J}}^{\prime a}$ as ideals and
${\mathcal{J}}={\mathcal{J}}^{\prime}$ as ${\mathbb{Q}}$-ideals. ∎

######  Lemma 5.2.6.

Let ${\mathcal{I}}_{0}$ be an ideal on a regular scheme $X_{0}$, and $X\to
X_{0}$ be a scheme obtained from $X_{0}$ by a sequence of $l$ cobordant blow-
ups of regular (possibly reducible) weighted centers $T$-stable for the
induced torus actions, and ${\mathcal{I}}=\sigma^{s}({\mathcal{I}})$ be its
controlled transform. Then $X$ is a regular scheme with the relatively affine
action of torus $T={\mathbb{G}}_{m}^{l}$ over $X_{0}$ and $X$ admits a
relatively affine geometric quotient $X\to X/T$ of $T$ over $X_{0}$. Moreover
$\sigma^{s}({\mathcal{I}})$ is a $T$-stable ideal on $X$.

###### Proof.

We use the induction on $l$. Let $X\to X_{0}$ be a sequence of $l-1$
equivariant cobordant blow-ups, and let $B_{+}\to X$ be the cobordant blow-up
of a $T$-stable center ${{\mathcal{J}}}$. Let $\pi:X\to X/T^{l-1}$ be a
$T$-invariant morphism defined by the geometric quotient. Then $B$ and $B_{+}$
admit an action of $T^{l}:=T^{l-1}\times{\mathbb{G}}_{m}$, and
$\pi\sigma_{+}:B_{+}\to X/T$ is $T^{l}$-invariant. In order to show that there
exists a geometric quotient $B_{+}/T^{l}$ it suffices to show that it exists
for $\sigma_{+}^{-1}\pi^{-1}(V_{\alpha})$ for open affine cover $V_{\alpha}$
of $X/T$. By the induction we can assume that each open affine
$U:=\pi^{-1}(V_{\alpha})$ admits a $T^{l-1}$-equivariant embedding.

$\phi:U\stackrel{{\scriptstyle\phi}}{{\hookrightarrow}}W:=U_{0}\times{\mathbb{A}}^{n}\times{\mathbb{G}}_{m}^{r}$

Furthermore the closed image $\phi(U)$ is generated by the seminvariant
parameters $v_{1},\ldots,v_{r}\in\Gamma(W,{\mathcal{O}}_{W})$.

Also we can assume that the center on $U$ is described as
${\mathcal{J}}=(x_{1}^{1/w_{1}},\ldots,x_{k}^{1/w_{k}})$. Consider the local
presentation of $B$ as in Section 2.5.2. Let us introduce the free variables
denoted by $t^{-1},x^{\prime}_{1},\ldots,x^{\prime}_{n}$ with the action

$\operatorname{Spec}({\mathcal{O}}_{U}[t^{-1},x^{\prime}_{1},\ldots,x^{\prime}_{k}])\smallsetminus
V(x^{\prime}_{1},\ldots,x^{\prime}_{k})=U\times({\mathbb{A}}^{k+1}\smallsetminus
V(x^{\prime}_{1},\ldots,x^{\prime}_{k})),$

with the action of ${\mathbb{G}}_{m}$ on
${\mathbb{A}}^{k+1}=\operatorname{Spec}({\mathbb{Z}}[t^{-1},x^{\prime}_{1},\ldots,x^{\prime}_{k}]))$,
given by

$(t^{-1}\cdot t^{-1},t^{w_{1}}x^{\prime}_{1},\ldots,t^{w_{k}}x^{\prime}_{1}),$

Then $B_{U}:=\sigma^{-1}(U)$ is a closed subscheme of
$U\times({\mathbb{A}}^{k+1}\smallsetminus
V(x^{\prime}_{1},\ldots,x^{\prime}_{k}))$:

$(B_{U})_{+}=\operatorname{Spec}({\mathcal{O}}_{U}[t^{-1},x^{\prime}_{1},\ldots,x^{\prime}_{k}])\smallsetminus
V(x^{\prime}_{1},\ldots,x^{\prime}_{k}))/(t^{-w_{i}}x^{\prime}_{i}-x_{i}).$

We consider the action of $T^{l-1}$ on
$(t^{-1},x^{\prime}_{1},\ldots,x^{\prime}_{k})$ as on the corresponding
elements of $B$, so the action of $T^{l-1}$ on $t^{-1}$ is trivial and the
action of $T^{l-1}$ on $x^{\prime}_{i}$ is the same as on $t^{w_{i}}x_{i}$.

Then for $(B_{U})_{x_{i}}=D(x_{i})=W\smallsetminus V(x_{i})\subset(B_{U})_{+}$
we obtain the $T^{l}$-equivariant closed embedding:

$(B_{U})_{i}\subset\operatorname{Spec}({\mathcal{O}}_{U}[t^{-1},x^{\prime}_{1},\ldots,x^{\prime}_{k},(x^{\prime}_{i})^{-1}])=U\times{\mathbb{A}}^{k}\times{\mathbb{G}}_{m}\hookrightarrow
U_{0}\times{\mathbb{A}}^{n+k}\times{\mathbb{G}}_{m}^{r+1},$

such that $U_{0}\times{\mathbb{A}}^{n+k}\times{\mathbb{G}}_{m}^{r+1}$ admits
geometric quotient of the group action of $T^{l}$.

∎

######  Proposition 5.2.7.

Let $Y$ be a regular affine scheme. Let $T$ act on
${\mathbf{A}}^{n}\times{\mathbb{G}}_{m}^{l}$ diagonally so that the geometric
quotient exists. Let $Z$ be a closed $T$-stable regular subscheme of
$W:=Y\times{\mathbf{A}}^{n}\times{\mathbb{G}}_{m}^{l}$ defined by a partial
system of semiinvariant local parameters.

Then for any point $p\in Z$ there is an open affine $T$-stable neigbourhood
$U\subset Z$ of $p$, over an open affine $V\subset Y$, and a $T$-equivariant
closed immersion

$U\stackrel{{\scriptstyle\phi}}{{\hookrightarrow}}W_{U}:=W=V\times{\mathbb{G}}_{m}^{a}\times({\mathbb{A}}_{k}^{s}\times{\mathbb{G}}_{m}^{b})$

for some $a,b,s$, and the induced fiber square diagram :

$\begin{array}[]{ccccc}U_{Z}&\stackrel{{\scriptstyle\phi}}{{\hookrightarrow}}&W=V\times{\mathbb{G}}_{m}^{a}\times({\mathbb{A}}_{k}^{s}\times{\mathbb{G}}_{m}^{b})&\stackrel{{\scriptstyle\pi}}{{\to}}&{\mathbb{A}}_{k}^{s}\times{\mathbb{G}}_{m}^{b}\\\
\pi_{X}\downarrow&&\downarrow&&\pi_{A}\downarrow\\\
U_{Z}/T&\stackrel{{\scriptstyle\phi/T}}{{\hookrightarrow}}&W/Z=(V\times{\mathbb{G}}_{m}^{a})\times
X_{\sigma}&\stackrel{{\scriptstyle\pi/T}}{{\to}}&X_{\sigma}=({\mathbb{A}}_{k}^{s}\times{\mathbb{G}}_{m}^{b})/T\end{array}.$

  1. (1)

$T$ acts trivially on the factor ${\mathbb{G}}_{m}^{a}$.

  2. (2)

$T$ acts diagonally on the factor
${\mathbf{A}}^{s}\times{\mathbb{G}}_{m}^{b}$, admits a geometric quotient of a
diagonal action of $T$, and the action has finite stabilizers.

  3. (3)

$\phi$, and $\phi/T$ are closed immersions, with their closed images
$\phi(U)$, and $\phi_{U/T}(U/T)$ is defined by the system of invariant
parameters $v_{1},\ldots,v_{k}\in(\Gamma(W,{\mathcal{O}}_{W})^{T})$, whose
restrictions to

$(\pi/T)^{-1}(q_{0})=(V\times{\mathbb{G}}_{m}^{a})\times\operatorname{Spec}(k(q_{0}))$

also define regular parameters.

###### Proof.

Let $x_{1},\ldots,x_{s}$, for $s\leq n$ be the coordinates on
${\mathbb{A}}^{s}$ which are noinvertible at $p$ on

$W_{0}=Y\times{\mathbb{A}}^{n}\times{\mathbb{G}}_{m}^{l}=Y\times{\mathbb{A}}^{s}\times{\mathbb{A}}^{n-s}\times{\mathbb{G}}_{m}^{l},$

and $y_{s+1},\ldots,y_{n+l-s}$ are the coordinates on
${\mathbb{A}}^{n-s}\times{\mathbb{G}}_{m}^{l}$ which are invertible at $p$.

We can replace $W$ with the open subset
$W_{0}=Y\times{\mathbb{A}}^{s}\times{\mathbb{G}}_{m}^{c}$, where $c:=n-s+l$,
and $Z$ with $U:=Z\cap W_{0}$

First, we show:

######  Lemma 5.2.8.

There is an open $T$-stable neighborhood $U$ of any point $p$ and a fiber
square:

$\begin{array}[]{ccc}U&\stackrel{{\scriptstyle\phi}}{{\hookrightarrow}}&W_{U}:=(V\times{\mathbf{A}}^{r}\times{\mathbb{G}}_{m}^{d})\\\
\pi_{U}\downarrow&&\pi_{W}\downarrow\\\
U/T=\operatorname{Spec}({\mathcal{O}}(U)^{T})&\stackrel{{\scriptstyle\phi/T}}{{\hookrightarrow}}&W_{U}/T:=V\times({\mathbf{A}}^{r}\times{\mathbb{G}}_{m}^{s})/T\end{array},$

for some $d,r\in{\mathbb{N}}$, where $\phi$, and $\phi/T$ are closed
immersions, with their closed images $\phi(U)$, and $\phi_{U/T}(U/T)$ is
defined by the system of invariant parameters
$v_{1},\ldots,v_{k}\in(\Gamma(W,{\mathcal{O}}_{W})^{T})$, whose restrictions
to $V\times\\{0\\}\times{\mathbb{G}}_{m}^{d}$ also define regular parameters.

###### Proof.

By the assumptions the image $U\subset W_{0}$, is described locally by a
partial system of semiinvariant parameters $v_{1},\ldots,v_{t}$.

Write any semiinvariant parameter $v_{j}$ as

$v_{j}=\sum c^{j}_{\beta}y^{\beta},+\sum_{\alpha\neq
0}c^{j}_{\alpha\beta}x^{\alpha}y^{\beta}=v_{0}^{j}+w_{0}^{j}$

where $c^{j}_{\alpha\beta}\in{\mathcal{O}}(Y)$, $v_{0}^{j}:=\sum
c^{j}_{\beta}y^{\beta}$, and $w_{0}^{j}:=\sum_{\alpha\neq
0}c^{j}_{\alpha\beta}x^{\alpha}y^{\beta}$, with all monomials semiinvariant of
the same weight. We will modify the sets of local parameters using the
induction on $j$.

Assume that $v_{1},\ldots,v_{j-1}$ is the set of first $j-1$ local parameters,
which are invariant and such that their images in
$\frac{m_{p}}{(x_{1},\ldots,x_{s})+m_{p}^{2}}$ are linearly independent.
Consider two cases.

Case 1.
$v_{0}^{j}\not\in(v_{1},\ldots,v_{j-1})+(x_{1},\ldots,x_{s})+m_{p}^{2}$.

Then multiplying $v_{j}$ by the inverse $y^{-\beta}$ for some $\beta$ occuring
in $v_{0}^{j}$ we reduce the situation to the invariant parameter
$v_{j}^{\prime}:=y^{-\beta}v_{j}$ on
$U_{0}\times{\mathbb{A}}^{s}\times{\mathbb{G}}_{m}^{d}$, such that
$v_{1},\ldots,v_{j}$ are invariant with their images in
$\frac{m_{p}}{(x_{1},\ldots,x_{s})+m_{p}^{2}}$ are linearly independent.

Case 2. $v_{0}^{j}\in(v_{1},\ldots,v_{j-1})+(x_{1},\ldots,x_{s})+m_{p}^{2}$.

Then at least one
$c^{j}_{\alpha\beta}x^{\alpha}y^{\beta}\in(x_{1},\ldots,x_{s})$ is a parameter
at $p$. Thus $x^{\alpha}=x_{i}$ and
$u^{j}_{\alpha\beta}:=c^{j}_{\alpha\beta}y^{\beta}$ is invertible at $p$, and
$c^{j}_{\alpha\beta}$ is such.

So, after replacing $U$ with $U_{c^{j}_{\alpha\beta}}$, and clearing the unit
$v^{\prime}_{j}=(u^{j}_{\alpha\beta})^{-1}v_{j}$ we can assume that the
semiinvariant parameter $v^{\prime}_{j}$ is of the form
$v^{\prime}_{j}=x_{i}+h_{j}$, where $h_{i}:=\sum
a_{\alpha\beta}x^{\alpha}y^{\beta}$. We can write

$v^{\prime}_{j}=x_{i}+h_{j}=x_{i}(1+\sum
b_{\alpha\beta}x^{\alpha}y^{\beta})+h^{\prime}_{j}=gx_{i}+h^{\prime}_{j},$

where $h^{\prime}_{j}:=\sum d_{\alpha\beta}x^{\alpha}y^{\beta}$, where all
$x^{\alpha}$ do not depend upon $x_{i}$, and

$g:=1+\sum b_{\alpha\beta}x^{\alpha}y^{\beta}$

is invariant invertible at $p$. Further shrinking $U$ to $U_{g}$ and
introducing new invertible variable $y_{r+1}$ on $G_{m}$ we obtain the induced
$T$-equivariant embedding

$\phi:U_{g}\hookrightarrow W^{\prime}=W\times
G_{m}=V\times{\mathbb{A}}^{s}\times{\mathbb{G}}_{m}^{r+1}$

such that $y_{r+1}\mapsto g$. This also introduces the additional invariant
local parameter $v^{\prime\prime}_{j}:=y_{r+1}-g$, such that

  * •

$v_{1},\ldots,v_{j-1},v^{\prime\prime}_{j},v^{\prime}_{j},v_{j+1},\ldots,v_{r}$
is a system of local parameters describing $\phi(U)$ on $W^{\prime}$.

  * •

$v_{1},\ldots,v_{j-1},v^{\prime\prime}_{j}$ are invariant with their images in
$m_{p}/(x_{1},\ldots,x_{s})+m_{p}^{2}$ are linearly independent.

Then $v_{j}^{\prime}$ can be written modulo $v_{j}^{\prime\prime}$ as

$v_{j}^{\prime}=y_{r+1}x_{i}+h_{i}\sim x_{i}+h^{\prime}_{j}\cdot y_{r+1}^{-1}$

Consider a $T^{l}$-equivariant automorphism of $W^{\prime}$ taking
$x_{i}+h^{\prime}_{j}\cdot y_{r+1}^{-1}\mapsto x_{i}$, and $x_{j}\mapsto
x_{j}$. Thus, after the coordinate change one can assume that the parameter
$v^{\prime}_{j}=y_{r+1}x_{j}$. This induces the embedding: into

$\phi^{\prime}:U_{g}\hookrightarrow
W^{\prime\prime}:=V(x_{i})=U_{0}\times{\mathbb{A}}^{s-1}\times{\mathbb{G}}_{m}^{r+1},$

and proves Lemma 5.2.8. ∎

Let
$\phi:U{\hookrightarrow}W_{U}:=(V\times{\mathbf{A}}^{r}\times{\mathbb{G}}_{m}^{d})$
be the closed $T$-equivariant embedding as in Lemma 5.2.8, whose image is
defined by the invariant parameters $v_{1},\ldots,v_{k}$ such that their
restrictions to closed subscheme $V\times\\{0\\}\times{\mathbb{G}}_{m}^{d}$
also define the parameters.

Consider the splitting
${\mathbb{G}}_{m}^{d}={\mathbb{G}}_{m}^{a}\times{\mathbb{G}}_{m}^{b}$, where
${\mathbb{G}}_{m}^{a}=({\mathbb{G}}_{m}^{d})^{T}$ is the maximal subtorus,
such that $T$ acts trivially on ${\mathbb{G}}_{m}^{a}$, and transitively on
${\mathbb{G}}_{m}^{b}$. Denote by $y_{1},\ldots,y_{a}$ the coordinates in
${\mathbb{G}}_{m}^{a}$ , and by $z_{1},\ldots,z_{b}$ the coordinates in
${\mathbb{G}}_{m}^{b}$.

Let

$\pi:W:=V\times{\mathbb{A}}^{s}_{k}\times{\mathbb{G}}_{m}^{d}=V\times{\mathbb{G}}_{m}^{a}\times({\mathbb{A}}_{k}^{s}\times{\mathbb{G}}_{m}^{b})\to{\mathbb{A}}^{s}_{k}\times{\mathbb{G}}_{m}^{b}.$

be the natural projection. Denote by
$q\in{\mathbb{A}}^{s}_{k}\times{\mathbb{G}}_{m}^{b}$ the image $q=\pi(p)$, and
by $p_{0}\in W/T$ the image of $p$ under the quotient $W\to W/T$.

Any invariant parameter $v_{j}$ can be written as

$v_{j}=\sum c^{j}_{\alpha\beta\gamma}x^{\alpha}y^{\beta}z^{\gamma},$

where $c^{j}_{\alpha\beta\gamma}\in{\mathcal{O}}(V)$. Then for any $\gamma\neq
0$ the action of $T$ on $z^{\gamma}$ is not trivial. Hence $\alpha\neq 0$ as
the action on $x^{\alpha}z^{\gamma}$ is trivial .

This shows that

$v_{j}=\sum c^{j}_{\beta}y^{\beta}+\sum_{\alpha\neq
0}c^{j}_{\alpha\beta\gamma}x^{\alpha}y^{\beta}z^{\gamma}$

Thus the restriction of $v_{j}$ to the smooth fiber

$F_{q}=\pi^{-1}(q)\simeq
V\times{\mathbb{G}}_{m}^{b}\times\operatorname{Spec}(k(q))\subset
V\times\\{0\\}\times{\mathbb{G}}_{m}^{d}$

determines the parameters on $F_{q}$ defined by $\sum c^{j}_{\beta}y^{\beta}$.

Let $q_{0}\in{\mathbb{A}}^{s}_{k}\times{\mathbb{G}}_{m}^{b}/T$ be the image of
$q$. The geometric quotient of $W$ by $T$ can be written as

$W/T=U_{0}\times{\mathbb{G}}_{m}^{b}\times({\mathbb{A}}_{k}^{s}\times{\mathbb{G}}_{m}^{b})/T=U_{0}\times{\mathbb{G}}_{m}^{a}\times
X_{\sigma},$

where

$X_{\sigma}:=({\mathbb{A}}_{k}^{s}\times{\mathbb{G}}_{m}^{b})/T$

is a toric variety with the abelian quotient singularities.

The composition of morphisms

$\psi:U_{Z}\stackrel{{\scriptstyle\phi}}{{\hookrightarrow}}W=U_{0}\times{\mathbb{G}}_{m}^{a}\times({\mathbb{A}}_{k}^{s}\times{\mathbb{G}}_{m}^{b})\stackrel{{\scriptstyle\pi}}{{\to}}{\mathbb{A}}_{k}^{s}\times{\mathbb{G}}_{m}^{b}$

induces

$\psi/T:U_{Z}/T\stackrel{{\scriptstyle\phi/T}}{{\hookrightarrow}}W/T=U_{0}\times{\mathbb{G}}_{m}^{a}\times
X_{\sigma}\stackrel{{\scriptstyle\pi/T}}{{\to}}X_{\sigma}=({\mathbb{A}}_{k}^{s}\times{\mathbb{G}}_{m}^{b})/T,$

for which the restrictions of
$v_{1},\ldots,v_{k}\in{\mathcal{O}}(U_{Z}/T)={\mathcal{O}}(U_{Z})^{T}$ at
$p_{0}$ to the fiber

$(\pi/T)^{-1}(q_{0})=(V\times{\mathbb{G}}_{m}^{a})\times\operatorname{Spec}(k(q_{0}))$

also define regular parameters.

∎

######  Corollary 5.2.9.

Let $X_{0}$ be a smooth variety over a field $k$. Let $X$ be obtained by a
sequence of $m$ cobordant blow-ups at smooth centers from $Y$, equivariant for
the induced torus actions. Let $Z$ be a regular $T$-stable closed subscheme of
$X$, where ${\mathbb{G}}_{m}=\operatorname{Spec}(k[t,t^{-1}])$, and
$T={\mathbb{G}}^{m}$. Then the geometric quotient $Z/T$ exists. Moreover,
locally there exists a smooth morphism

$U_{Z}/T\to X_{\sigma}:=({\mathbb{A}}_{k}^{s}\times{\mathbb{G}}_{m}^{b})/T,$

where $U_{Z}\subset Z$ is an open affine $T$-stable and $X_{\sigma}$ is a
toric variety over $k$ with abelian quotient singularities.

###### Proof.

By Lemma 5.2.6, and Proposition 5.2.7, for any point $p_{0}\in Z/T$ there is a
smooth morphism

${\pi/T}:W/T=(V\times{\mathbb{G}}_{m}^{b})\times X_{\sigma}\to X_{\sigma},$

and the closed immersion $U_{Z}/T\hookrightarrow W/T$, with the image defined
by the functions
$v_{1},\ldots,v_{r}\in{\mathcal{O}}(W)^{T}={\mathcal{O}}(W/T)$, and its
restriction to the smooth fiber

$({\pi/T})^{-1}(q_{0})\simeq
U_{0}\times{\mathbb{G}}_{m}^{b}\times\operatorname{Spec}(k(q_{0}))$

determines a partial system of local parameters. Thus, by [Sta, Lemma
10.99.3.] the morphism

$\psi/T:U_{Z}/T\to
X_{\sigma}:=({\mathbb{A}}_{k}^{s}\times{\mathbb{G}}_{m}^{b})/T$

is flat at $p_{0}$, and, by Lemma [Sta, Lemma 29.34.14(2)], it is smooth at
$p_{0}$.

∎

######  Remark 5.2.10.

The above result is not valid in the mixed characteristic. The morphism
$\pi/T$, which is a projection along regular scheme is not flat, in general,
and one needs to work with the presentation from Proposition 5.2.7 directly:

$U/T\stackrel{{\scriptstyle\phi/T}}{{\hookrightarrow}}V\times
X_{\sigma}\stackrel{{\scriptstyle\pi/T}}{{\to}}X_{\sigma},$

where $V$ is a regular scheme, ${\phi/T}$ is a closed embedding, and $\pi/T$
is the projection. Here the image ${\phi/T}(U/T)$ is defined by local
parameters at $p_{0}\in{\phi/T}(U/T)$ whose restriction to the regular fiber
$F_{q_{0}}=({\pi/T})^{-1}(q_{0})$, where $q_{0}=({\pi/T})(p_{0})$, are also
parameters.

###  5.3. Cobordant blow-ups in the logarithmic category

Let $X$ be a logarithmically regular scheme, with Zariski logarithmic
structure (or a strict toroidal variety) and assume that ${\mathcal{J}}$ has a
more general form

${\mathcal{J}}=(u_{1}^{1/w_{1}},\ldots,u_{k}^{{1/w_{k}}},m_{1}^{1/w_{k+1}},\ldots,m_{r}^{1/w_{k+r}}),$

where $u_{i}$ are free parameters and $m_{i}$ are monomials for the
logarithmic structure on $X$ then the full cobordant blow-up $B\to X$ at
${\mathcal{J}}$ can be described as

$B=\operatorname{Spec}({{\mathcal{O}}_{X}}[t^{-1},t^{w_{1}}{x_{1}},\ldots,t^{w_{k}}{x_{k}},t^{w_{k+1}}{m_{1}},\ldots,t^{w_{k+r}}{m_{r}}])^{int},$

which is again a logarithmically regular scheme,(respectively strict toroidal
variety). By functoriality, the construction extends to any logarithmically
regular schemes (or toroidal varieties).

Initially, in the papers [ATW17], [ATW20], we considered the stack-theoretic
blow-ups of the centers of the form
${\mathcal{J}}=(u_{1},\ldots,u_{k},m_{1}^{1/w_{k+1}},\ldots,m_{r}^{1/w_{k+r}})$,
in the context of Kummer étale topology on the logarithmic stacks. Then in
[ATW19], we developed the formalism of the stack-theoretic blow-ups of the
weighted centers of the form $(u_{1}^{1/w_{1}},\ldots,u_{k}^{{1/w_{k}}})$.

Soon after Quek in [Que20, Section 3.2] studied the stack-theoretic blow-ups
of the more general form

${\mathcal{J}}=(u_{1}^{1/w_{1}},\ldots,u_{k}^{{1/w_{k}}},m_{1}^{1/w_{k+1}},\ldots,m_{r}^{1/w_{k+r}})=(u_{1}^{1/w_{1}},\ldots,u_{k}^{{1/w_{k}}},Q^{1/w})$

in the context of logarithmic stacks and Kummer étale topology. Here $Q$ is
generated by $m_{i}^{w/w_{i}}$ for the corresponding $w$.

## References

  * [Abh67] Shreeram Shankar Abhyankar, _Nonsplitting of valuations in extensions of two dimensional regular local domains_ , Math. Ann. 170 (1967), 87–144. MR 207698
  * [AKMW02] Dan Abramovich, Kalle Karu, Kenji Matsuki, and Jarosław Włodarczyk, _Torification and factorization of birational maps_ , J. Amer. Math. Soc. 15 (2002), no. 3, 531–572 (electronic). MR 1896232 (2003c:14016)
  * [AQ21] Dan Abramovich and Ming Hao Quek, _Logarithmic resolution via multi-weighted blow-ups_ , 2021.
  * [ATW17] Dan Abramovich, Michael Temkin, and Jarosław Włodarczyk, _Principalization of ideals on toroidal orbifolds_ , JEMS, to appear, (2017), arXiv:1709.03185.
  * [AT19] Dan Abramovich and Michael Temkin, _Functorial factorization of birational maps for qe schemes in characteristic 0_ , Algebra Number Theory 13 (2019), no. 2, 379–424. MR 3927050
  * [ATW19] by same author, _Functorial embedded resolution via weighted blowings up_ , arXiv e-prints (2019), arXiv:1906.07106.
  * [ATW20] Dan Abramovich, Michael Temkin, and Jarosław Włodarczyk, _Relative desingularization and principalization of ideals_ , arXiv:2003.03659.
  * [BM91] Edward Bierstone and Pierre D. Milman, _A simple constructive proof of canonical resolution of singularities_ , Effective methods in algebraic geometry (Castiglioncello, 1990), Progr. Math., vol. 94, Birkhäuser Boston, Boston, MA, 1991, pp. 11–30. MR 1106412
  * [BM97] by same author, _Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant_ , Invent. Math. 128 (1997), no. 2, 207–302. MR 1440306 (98e:14010)
  * [BM08] by same author, _Functoriality in resolution of singularities_ , Publ. Res. Inst. Math. Sci. 44 (2008), no. 2, 609–639. MR 2426359
  * [BR19] Daniel Bergh and David Rydh, _Functorial destackification and weak factorization of orbifolds_ , arXiv e-prints (2019), arXiv:1905.00872.
  * [BV11] Ana Bravo and Orlando E. Villamayor, _Elimination algebras and inductive arguments in resolution of singularities_ , Asian Journal of Mathematics 15 (2011), no. 3, 321 – 356.
  * [BVU13] Angélica Benito and Orlando E. Villamayor U., _Monoidal transforms and invariants of singularities in positive characteristic_ , Compos. Math. 149 (2013), no. 8, 1267–1311. MR 3103065
  * [CP08] Vincent Cossart and Olivier Piltant, _Resolution of singularities of threefolds in positive characteristic. I. Reduction to local uniformization on Artin-Schreier and purely inseparable coverings_ , J. Algebra 320 (2008), no. 3, 1051–1082. MR 2427629
  * [CP09] by same author, _Resolution of singularities of threefolds in positive characteristic. II_ , J. Algebra 321 (2009), no. 7, 1836–1976. MR 2494751
  * [Cut11] Steven Dale Cutkosky, _A skeleton key to Abhyankar’s proof of embedded resolution of characteristic p surfaces_ , Asian J. Math. 15 (2011), no. 3, 369–416. MR 2838213
  * [Dan78] V. I. Danilov, _The geometry of toric varieties_ , Uspekhi Mat. Nauk 33 (1978), no. 2(200), 85–134, 247. MR 495499
  * [DH98] Igor V. Dolgachev and Yi Hu, _Variation of geometric invariant theory quotients_ , Inst. Hautes Études Sci. Publ. Math. (1998), no. 87, 5–56, With an appendix by Nicolas Ressayre. MR 1659282 (2000b:14060)
  * [EH02] Santiago Encinas and Herwig Hauser, _Strong resolution of singularities in characteristic zero_ , Comment. Math. Helv. 77 (2002), no. 4, 821–845. MR 1949115
  * [EV03] Santiago Encinas and Orlando Villamayor, _A new proof of desingularization over fields of characteristic zero_ , Proceedings of the International Conference on Algebraic Geometry and Singularities (Spanish) (Sevilla, 2001), vol. 19, 2003, pp. 339–353. MR 2023188
  * [EV07] by same author, _Rees algebras and resolution of singularities_ , Proceedings of the XVIth Latin American Algebra Colloquium (Spanish), Bibl. Rev. Mat. Iberoamericana, Rev. Mat. Iberoamericana, Madrid, 2007, pp. 63–85. MR 2500351
  * [Ful98] William Fulton, _Intersection theory_ , second ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 2, Springer-Verlag, Berlin, 1998. MR 1644323
  * [Hau98] Herwig Hauser, _Seventeen obstacles for resolution of singularities_ , Singularities (Oberwolfach, 1996), Progr. Math., vol. 162, Birkhäuser, Basel, 1998, pp. 289–313. MR 1652479
  * [Hir64] Heisuke Hironaka, _Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II_ , Ann. of Math. (2) 79 (1964), 109–203; ibid. (2) 79 (1964), 205–326. MR 0199184
  * [HP19] Herwig Hauser and Stefan Perlega, _Cycles of singularities appearing in the resolution problem in positive characteristic_ , J. Algebraic Geom. 28 (2019), no. 2, 391–403. MR 3912062
  * [HS06] Craig Huneke and Irena Swanson, _Integral closure of ideals, rings, and modules_ , London Mathematical Society Lecture Note Series, vol. 336, Cambridge University Press, Cambridge, 2006. MR 2266432
  * [Kaw06] Yujiro Kawamata, _Derived categories of toric varieties_ , Michigan Math. J. 54 (2006), no. 3, 517–535. MR 2280493
  * [KM16] Hiraku Kawanoue and Kenji Matsuki, _Resolution of singularities of an idealistic filtration in dimension 3 after Benito-Villamayor_ , Minimal models and extremal rays (Kyoto, 2011), Adv. Stud. Pure Math., vol. 70, Math. Soc. Japan, [Tokyo], 2016, pp. 115–214. MR 3617780
  * [Kol07] János Kollár, _Lectures on resolution of singularities_ , Annals of Mathematics Studies, vol. 166, Princeton University Press, Princeton, NJ, 2007\. MR 2289519 (2008f:14026)
  * [McQ19] Michael McQuillan, _Very fast, very functorial, and very easy resolution of singularities_ , 2019, preprint, written with the collaboration of G. Marzo.
  * [Moh87] T. T. Moh, _On a stability theorem for local uniformization in characteristic $p$_, Publ. Res. Inst. Math. Sci. 23 (1987), no. 6, 965–973. MR 935710
  * [Moh96] by same author, _On a Newton polygon approach to the uniformization of singularities of characteristic $p$_, Algebraic geometry and singularities (La Rábida, 1991), Progr. Math., vol. 134, Birkhäuser, Basel, 1996, pp. 49–93. MR 1395176
  * [Mor96] Robert Morelli, _The birational geometry of toric varieties_ , J. Algebraic Geom. 5 (1996), no. 4, 751–782. MR 1486987 (99b:14056)
  * [MP13] Michael McQuillan and Daniel Panazzolo, _Almost étale resolution of foliations_ , J. Differential Geom. 95 (2013), no. 2, 279–319. MR 3128985
  * [Nar83] R. Narasimhan, _Monomial equimultiple curves in positive characteristic_ , Proc. Amer. Math. Soc. 89 (1983), no. 3, 402–406. MR 715853
  * [Oda88] Tadao Oda, _Convex bodies and algebraic geometry—toric varieties and applications. I_ , 1988, pp. 89–94. MR 966447
  * [Pan06] Daniel Panazzolo, _Resolution of singularities of real-analytic vector fields in dimension three_ , Acta Math. 197 (2006), no. 2, 167–289. MR 2296055
  * [QR22] Ming Hao Quek and David Rydh, _Weighted blow-ups_ , posted on March, 31,2022 on Rydh’s website, in preparation.
  * [Que20] Ming Hao Quek, _Logarithmic resolution via weighted toroidal blowings up_ , arXiv e-prints (2020), arXiv:2005.05939.
  * [Rei] Miles Reid, _What is a flip?_ , Colloquium Talk, Univ. Of Utah, Dec , 1992.
  * [Reid02] Miles Reid, _Graded rings and varieties in weighted projective space_ , https://homepages.warwick.ac.uk/~masda/surf/more/grad.pdf, 2002.
  * [Rush07] David E.Rush, _Rees valuations and asymptotic primes of rational powers in Noetherian rings and lattices_ , Journal of Algebra 308 (2007), 295-320
  * [Sat13] Matthew Satriano, _Canonical Artin stacks over log smooth schemes_ , Math. Z. 274 (2013), no. 3-4, 779–804. MR 3078247
  * [Sta] The Stacks Project Authors, _Stacks Project_ , http://stacks.math.columbia.edu.
  * [Tha94a] Michael Thaddeus, _Stable pairs, linear systems and the Verlinde formula_ , Invent. Math. 117 (1994), no. 2, 317–353. MR 1273268
  * [Tha94b] by same author, _Toric quotients and flips_ , Topology, geometry and field theory, World Sci. Publ., River Edge, NJ, 1994, pp. 193–213. MR 1312182
  * [Tha96] by same author, _Geometric invariant theory and flips_ , J. Amer. Math. Soc. 9 (1996), no. 3, 691–723. MR 1333296
  * [Var76] A. N. Varčenko, _Newton polyhedra and estimates of oscillatory integrals_ , Funkcional. Anal. i Priložen. 10 (1976), no. 3, 13–38. MR 0422257
  * [Vil89] Orlando Villamayor, _Constructiveness of Hironaka’s resolution_ , Ann. Sci. École Norm. Sup. (4) 22 (1989), no. 1, 1–32. MR 985852
  * [Vil08] by same author, _Villamayor u. o.e.: Rees algebras on smooth schemes: integral closure and higher differential operators. rev. mat. iberoamericana 24(1), 213-242_ , Revista Matematica Iberoamericana - REV MAT IBEROAM 24 (2008).
  * [Wło00] Jarosław Włodarczyk, _Birational cobordisms and factorization of birational maps_ , J. Algebraic Geom. 9 (2000), no. 3, 425–449. MR 1752010 (2002d:14019)
  * [Wło03] by same author, _Toroidal varieties and the weak factorization theorem_ , Invent. Math. 154 (2003), no. 2, 223–331. MR 2013783 (2004m:14113)
  * [Wło05] by same author, _Simple Hironaka resolution in characteristic zero_ , J. Amer. Math. Soc. 18 (2005), no. 4, 779–822 (electronic). MR 2163383
  * [Wło08] by same author, _Program on resolution of singularities in characteristic p_ , Notes from lectures at RIMS, Kyoto, December 2008.
  * [Wło20] by same author,_Functorial resolution except for toroidal locus. toroidal compactification_ , arXiv e-prints (2020), arXiv:2007.13846.
  * [Wło22] by same author, _Functorial resolution by torus actions_ , Version I, March 2022, arXiv:2007.13846.
  * [Wło23] by same author, _Cox rings of morphisms and resolution of singularities_ , 2023.