# Functorial resolution by torus actions

Jarosław Włodarczyk  Department of Mathematics, Purdue University
150 N. University Street,
West Lafayette, IN 47907-2067<EMAIL_ADDRESS>
###### Abstract.

We show a simple and fast embedded resolution of varieties and
principalization of ideals in the language of torus actions on ambient, smooth
varieties with SNC divisors. The canonical functorial resolution of varieties
in characteristic zero is given by the introduced here operations of cobordant
blow-ups with smooth weighted centers. The centers are defined by the
geometric invariant measuring the singularities on smooth schemes with SNC
divisors.

As the result of the procedure, we obtain a smooth variety with a torus action
and the exceptional divisor having simple normal crossings. Moreover, its
geometric quotient is birational to the resolved variety, has abelian quotient
singularities, and can be desingularized directly by combinatorial methods.
The paper is based upon the ideas of the joint work with Abramovich and Temkin
[ATW19] and a similar result by McQuillan [McQ19] on resolution in
characteristic zero via stack-theoretic weighted blow-ups.

As an application of the method, we show the resolution of certain classes of
singularities in positive and mixed characteristic.

This research is supported by BSF grant 2014365

###### Contents

  1. 1 Introduction
  2. 2 Cobordant blow-ups
  3. 3 Resolution of singularities in characteristic zero by weighted centers
  4. 4 Free characteristic resolution of almost homogenous singularities
  5. 5 Appendix

##  1\. Introduction

The significance of $G_{m}$-actions in birational geometry and their
connection with Mori theory has already been identified by Reid, Thaddeus, and
various other scholars (refer to [Tha94a], [Tha94b], [Tha96], [Rei], [DH98]).
Particularly, Reid [Reid02] highlighted the role of weighted blowings up and
flips in birational geometry utilizing $G_{m}$-actions. This notion was also
reflected in the proof of the Weak Factorization theorem, which relied on the
concept of birational cobordism and a pivotal role of $G_{m}$-action ([Wło00],
[Wło03], [AKMW02]).

In the present paper, we propose approaching embedded resolution problems and
the existing resolution algorithms from the point of view of torus actions.
This is an alternative and, to a great extent, equivalent approach to the one
pursued in a paper [McQ19] of McQuillan-Marzo, and a recent paper of Quek
[Que20], and most notably, in the series of papers by Abramovich-Temkin-
Włodarczyk [ATW17], [ATW20],[ATW19], where all kind of stack-theoretic blow-
ups were considered to simplify resolution procedure or to adapt the algorithm
to the relative situation of morphisms. The introduced here definition of
cobordant blow-up interprets the stack-theoretic weighted blow-ups,
logarithmic Kummer blow-ups, and more general birational transformations of
stacks in terms of torus actions. Such interpretation leads to the
presentation of the weighed blow-ups and other transformations as smooth
birational cobordisms. The operation is done in the language of smooth schemes
with torus actions in terms of simple transformation rules. In fact, only one
chart is needed to describe the cobordant blow-up, which makes it a very
simple tool for algorithmic computations and possible theoretical
implementations. The method can be applied to many existing and potential
embedded resolution algorithms originally relying on smooth centers,
regardless of characteristic. It also avoids many problems associated with
smooth centers and studied intensely in [Abh67, Moh87, Moh96, Hau98, Wło08,
CP08, CP09, Cut11, BVU13, KM16, HP19]. Moreover, it is significantly faster
and eliminates unnecessary blow-ups. Furthermore, the algorithm in
characteristic zero has a very simple structure with the centers defined by a
geometric invariant related to the weighted normal cone (see Section 4.1.1).
The smooth centers are replaced with more general smooth weighted centers. In
fact, the present paper, similarly to [ATW19], contains a highly simplified
version of the strong Hironaka algorithm in characteristic zero. Recall that
the standard Hironaka algorithm uses a rather complicated invariant where the
geometry of singularities is mixed with the combinatorics of the monomials. In
the present paper, similarly to [ATW19] and [McQ19], the algorithm is purely
geometric, with the combinatorial part hidden in the torus action or
respectively in the stack-theoretic structure as, in the two papers above.

###  1.1. Cobordant blow-ups

Unlike the formalism in [ATW19], or in [McQ19], the resolution here is given
in the language of schemes with torus actions and does not use stack-theoretic
language, although, if needed, it can be converted into such. The algorithm is
carried in the language of cobordant blow-ups of weighted centers on regular
schemes. A similar operation in the language of stacks was independently
studied in the paper of Quek-Rydh [QR22].

The cobordant blow-ups of regular varieties have many advantages even when
compared to the standard blow-ups of smooth blow-ups. They are described by
particularly simple transformation rules with one chart only defined by the
affine morphism. They carry important additional information about
singularities which is contained in the vertex of the transformation. The
singularities along the vertex after the blow-up are to a great extent
equivalent to the singularities at the center, and they are removed as
typically the worst singularities in the resolution process. Moreover the
torus action contains additional additional important information for the
resolution process.

All of this is especially beneficial for all kinds of generalizations of the
resolution to classes of singularities in positive and mixed characteristic,
foliations and others. In particular in Section 4 we show how to apply
cobordant blow-ups with weighted centers to introduced here, class of almost
homogenous singularities in any characteristic (see Theorems 4.2.2, 4.2.3).
This class includes, in particular, the isolated singularities whose
generalized weighted tangent cone also has an isolated singularity. Moreover,
the approach can be generalized to the case where the equimultiple locus of
the subscheme and its weighted tangent cone coincide (see Lemma 4.3.1).

The language and the formalism of the cobordant blow-ups are related to the
notion of birational cobordism introduced as the main tool in the proof of the
Weak factorization theorem ([Wło00], [Wło03], [AKMW02]). On the other hand
when looking from a more general perspective of Cox rings of morphisms in the
subsequent paper [Wło23] one shows that any proper birational morphism of
normal noetherian schemes has such a cobordant presentation called
cobordization, where the considered here cobordant blow-up of weighted center
is the cobordization of the standard weighted blow-up. Moreover the theorems
on the resolution from Chapter 4 are extended in [Wło23] to generalized
cobordant blow-ups of locally monomial ideals.

###  1.2. Rees algebras

Another novel of the paper is the use of rational Rees algebras, giving a fast
algorithm with automatic uniqueness. One shall mention that the rational Rees
algebras was also briefly used in the first version of [ATW19] to interpret
the algebra of the center.

Recall that there are presently two approaches to the uniqueness and glueing
of the embedded resolutions. The first method is known as ”Hironaka’s trick”
which uses an equivalence relation defined by permissible blow-ups. It was
used in the original Hironaka’s approach, and in the subsequent papers of
Bierstone-Milman and Villamayor and others (see [Hir64],[Vil89],[BM97],[EH02],
[EV03]). The other method is known as ”homogenization-tuning” trick was
introduced by the author in [Wło05], and then used in the papers of Kollar
[Kol07], Bierstone-Milman [BM08], and others. This approach is also used in
[ATW17] and [ATW19]. It relies on the existence of local automorphisms of the
homogenized ideals and functoriality for étale morphisms. Both approaches
hinge on the induction on the dimension of ambient variety. The inductively
built centers living on different hypersurfaces of maximal contact are
compared via equivalence relations or étale maps.

With our approach the centers are unique despite being constructed using
noncanonical itermediate steps. Moreover what is important no comparison
between the intermediate steps is needed. Instead all the operations are
related directly to the constructed canonical resolution center.

Recall that the integrally graded Rees algebras have been used for some time
in the resolution algorithms([EV07],[Vil08],[BV11],[CP09],[KM16]). They were
also used recently in the approach by Quek for the weighted logarithmic
resolution [Que20]. They are also considered in the paper by Quek and Rydh
[QR22]. There are however several advantages of the Rees algebras with
rational gradations. The very first one is that the resolution center and the
resolution invariant can be conveniently defined in this language as the
natural generalization of the order.

####  1.2.1. Definition of the invariant

Let us start with the definition of the order of an ideal ${\mathcal{I}}$ at a
point $p\in X$:

${\operatorname{ord}}_{p}({\mathcal{I}})=\max\\{a\in{\mathbb{Z}}_{>0}\mid{\mathcal{I}}\subset
m_{p}^{a}\\},$

where $m_{p}\subset{\mathcal{O}}_{X,p}$ is the maximal ideal of the point $p$
in the local ring ${\mathcal{O}}_{X,p}$. This can be written as

${\operatorname{ord}}_{p}({\mathcal{I}})=\max\\{a\in{\mathbb{Q}}_{>0}\mid{\mathcal{I}}t^{a}\subset{\mathcal{O}}_{X}[m_{p}t]\\},$

where $R={\mathcal{O}}_{X}[m_{p}t]$ is the Rees algebra generated by the first
gradation $m_{p}t$, where $t$ is the dummy variable. Equivalently one writes
the latter as

${\operatorname{ord}}_{p}({\mathcal{I}})=\max\\{a\in{\mathbb{Q}}_{>0}\mid{\mathcal{I}}t\subset{\mathcal{O}}_{X}[m_{p}t^{1/a}]\\},$

If the divisors are not present that the resolution invariant of
${\mathcal{I}}$ at $p$ can be written simply as

(1)
$\displaystyle{\operatorname{inv}}_{p}({\mathcal{I}}):=\max\\{(a_{1},\ldots,a_{k})\mid{\mathcal{I}}t\subset{\mathcal{O}}_{X}[x_{1}t^{1/a_{1}},\ldots,x_{k}t^{1/a_{k}}]^{{\operatorname{int}}}\\},$

where $a_{1}\leq\ldots\leq a_{k}$ are rational numbers ordered
lexicographically and the maximum is considered over all the partial
coordinate systems $x_{1},\ldots,x_{k}$. Here ”${\operatorname{int}}$” stands
for the corresponding integral closure. The present definition is equivalent
to the one from [ATW19], which was considered in the different language. Note
that the powers of $t$ indicate the weights of the elements. In particular the
weight of $x_{i}$ is $1/a_{i}$ with the weight $1$ assigned to
${\mathcal{I}}$.

####  1.2.2. Rational Rees algebras

One of the benefits of Rees algebras is that their generators lie in the
different rational gradations which often leads to a simpler presentation.
Additional advantage of the rational Rees algebras is that no rescaling is
needed and we can always get the simplest possible set of generators by
extending the gradation. This makes the computations of the centers very
straightforward.

This could be illustrated by the Veronese embedding. When passing from the
graded algebra $R=k[x_{1},\ldots,x_{k}]=\bigoplus_{d\in{\mathbb{Z}}_{\geq
0}}R_{d}$ to the subgradation $R^{(n)}=\bigoplus_{d\in{\mathbb{Z}}_{\geq
0}}R_{nd},$ as in the $n$-tuple Veronese embedding we obtain the graded
algebra $R^{(n)}$ with a very cumbersome set of generators given by all
monomials of degree $n$ and describing the same
$\operatorname{Proj}(R^{(n)})=\operatorname{Proj}(R)$. Now when we start from
${\mathbb{Z}}$-graded algebra $R^{(n)}$ written as
$R^{(n)}=\bigoplus_{d\in{\mathbb{Z}}_{\geq 0}}R_{nd}t^{d}$ for the dummy
variable $t$ then extending the gradation and passing to the integral closure
$R^{(n)}[t^{1/n}]^{{\operatorname{int}}}$ one obtains back the algebra
$R=k[x_{1}t^{1/n},\ldots,x_{k}t^{1/k}]$ in the graded form with respect to
$t^{1/k}$ and with the nice set of generators, and giving a nicer presentation
of the same object $\operatorname{Proj}(R)$.

The process avoids various multiple rescalings which often leads to the
increased computational complexity. Since no rescaling is needed the
construction of the center can be obtained in a minimalistic way by a series
of transformations called milling.

####  1.2.3. Milling

The concept behind milling is to systematically decompose the elements of the
ideal $\mathcal{I}$, as represented by the Rees algebra
$R=\mathcal{O}_{X}[\mathcal{I}t]$, step by step, into elements of smaller
gradations. This process is done using graded differential operators and is
carried out with the objective of obtaining a smooth weighted center. A
similar idea was considered in the first version of [ATW19] to illustrate a
different construction method. Note that in our process only the final stage
is canonical, and all the intermediate steps rely on the choices made for the
convenient minimalistic computations.

Consider the admissibility inclusion

(2)
$\displaystyle{\mathcal{I}}t\subset{\mathcal{O}}_{X}[x_{1}t^{1/a_{1}},\ldots,x_{k}t^{1/a_{k}}]^{{\operatorname{int}}}$

as in the above definition (1) of the resolution invariant
${\operatorname{inv}}_{p}({\mathcal{I}})$. The Rees algebra of the form

${\mathcal{A}}:={\mathcal{O}}_{X}[x_{1}t^{1/a_{1}},\ldots,x_{k}t^{1/a_{k}}]^{{\operatorname{int}}}$

is called the Rees center. If the inclusion (2) holds then we say that the
center ${\mathcal{A}}$ is admissible for ${\mathcal{I}}$ at $p$. If
additionally ${\operatorname{inv}}_{p}({\mathcal{I}})=(a_{1},\ldots,a_{k})$ as
in (1) then the Rees center ${\mathcal{A}}$ on the right will be called a
maximal admissible center. This terminology is to a great extent equivalent to
the one introduced in [ATW19] in the language of ${\mathbb{Q}}$-ideals.

The algorithm uses the basic concepts of the standard Hironaka algorithm like
the admissibility, coefficient ideal, maximal contact, from [Hir64],[Vil89],
[BM97], [Wło05], [EH02], [EV03], [Kol07], which are redefined from the
perspective of rational Rees algebras and cobordant blow-ups.

####  1.2.4. The graded differential operators

The idea of milling is to construct the maximal admissible center through a
step-by-step process that involves enlarging the initial Rees algebra
$R_{1}:={\mathcal{O}}_{X}[{\mathcal{I}}t]\subset{\mathcal{A}}$, while keeping
${\mathcal{A}}$ unchanged. To achieve this, we perform actions on both sides
of the admissibility inclusion using graded differential operators. These
operators preserve the right side ${\mathcal{A}}$ of the inclusion while
expanding the left side. They eventually lead to a situation where both sides
become equal. As a result, the maximal admissible center is attained through a
recursive process. Importantly, this process is dependent solely on
${\mathcal{I}}$, establishing the uniqueness of the maximal admissible center
and the canonical invariant.

To illustrate the action of graded operators consider the inclusion

(3) $\displaystyle{\mathcal{I}}t^{a}\subset{\mathcal{O}}_{X}[m_{p}t],$

where $a={\operatorname{ord}}_{p}({\mathcal{I}})$. Here we can act on both
sides of the inclusion by the differential operators $t^{-1}D_{X}$ for
${\mathcal{D}}_{X}$ being the sheaf of the differential operators on smooth
variety $X$ over a field $k$, and $a\in{\mathbb{N}}$. The right side is
preserved by this action as $t^{-1}D_{X}(m_{p}t)^{a}\subset(m_{p}t)^{a-1}$
while we obtain new elements on the left side. In particular, we derive the
inclusion of $t$-gradations ${\mathcal{D}}_{X}^{a-1}({\mathcal{I}})t\subset
m_{p}t$.

When rescaling (3) we obtain the inclusion

${\mathcal{I}}t\subset{\mathcal{O}}_{X}[m_{p}t^{1/a}],$

with the action of the differential operators $t^{-1/a}D_{X}$ to get

${\mathcal{D}}_{X}^{a-1}({\mathcal{I}})t^{1/a}\subset m_{p}t^{1/a}.$

Similarly if
${\mathcal{A}}={\mathcal{O}}_{X}[x_{1}t^{1/a_{1}},\ldots,x_{k}t^{1/a_{k}}]^{{\operatorname{int}}}$
is maximal admissible for ${\mathcal{I}}$ then $a_{1}$ is necessarily the
order of ${\mathcal{I}}$ at $p$. We can act by the operators $t^{-1/a}D_{X}$
on both sides of the admissibility relation

${\mathcal{I}}t\subset{\mathcal{A}}={\mathcal{O}}_{X}[x_{1}t^{1/a_{1}},\ldots,x_{k}t^{1/a_{k}}]^{{\operatorname{int}}}$

and obtain the inclusion of $t^{1/a_{1}}$ gradations

(4) $\displaystyle
D_{X}^{a_{1}-1}({\mathcal{I}})t^{1/a_{1}}\subset{\mathcal{A}}_{1/a_{1}}t^{1/a_{1}}$

where ${\mathcal{A}}=\bigoplus_{a}{\mathcal{A}}_{a}t^{a}$.

####  1.2.5. Maximal contacts

The _maximal contacts_ of ${\mathcal{I}}$ at $p$ in our approach should be
thought of as simply the graded coordinates
$x_{1}t^{1/a_{1}},\ldots,x_{k}t^{1/a_{k}}$ in a certain presentation of a
maximal admissible center ${\mathcal{A}}\subset{\mathcal{I}}t$ at $p$. Unlike
the usual definition this approach makes sense in a nonzero characteristic.
The maximal contacts are defined in terms of the recursively constructed
algebras on the left side of the admissibility inclusion. In particular, in
view of (1) the first maximal contact is defined by a coordinate

$x_{1}\in
T^{1/a_{1}}(R_{1}):=D_{X}^{a_{1}-1}({\mathcal{I}})\subset{\mathcal{A}}_{1/a_{1}}$

in gradation $t^{1/a_{1}}$.

Observe that the notions of the admissibility, the order, the invariant and
the maximal contact can be considered in the general context of Rees algebra
$R$ replacing ${\mathcal{I}}t$ with $R$ in the relevant definitions.

Consequently, the notion of $i$-th maximal contact $x_{i}$ for any Rees
algebra shall be understood as the coordinate in $t^{1/a_{i}}$-gradation of
its maximal admissible center
${\mathcal{A}}={\mathcal{O}}_{X}[x_{1}t^{1/a_{1}},\ldots,x_{k}t^{1/a_{k}}]^{{\operatorname{int}}}\supset{\mathcal{I}}t$
such that $x_{i}$ is independent of the previous maximal contacts
$x_{1},\ldots,x_{i-1}$.

####  1.2.6. Coefficient ideals

The coefficient ideal of an algebra $R$ is a way to produce larger Rees
algebras containing the relevant maximal contacts.

Given any Rees algebra $R=\bigoplus R_{a}t^{a}$, and any graded coordinate
${x}t^{1/a}$ the coefficient ideal $C_{xt^{1/a_{1}}}(R)$ of $R$ with respect
to $xt^{1/a}$ is generated by $xt^{1/a}$ and all the derivations obtained by
the action of the graded derivations $D_{xt^{1/a}}:=t^{-1/a_{1}}\partial_{x}$.
Consequently, by the above if $R\subset{\mathcal{A}}$ and $x_{1}t^{1/a_{1}}$
is the first maximal contact of $R$ then
$C_{x_{1}t^{1/a_{1}}}(R)\subset{\mathcal{A}}$.

The computation of the coefficient ideal is even simpler when we pass to the
completion of the local ring with respect to $(x)$ or maximal ideal $m_{p}$ so
that the splitting gives the inclusion
$\widehat{{\mathcal{O}}}_{X,p}/(x)\subset\widehat{{\mathcal{O}}}_{X,p}$. In
such a case $\widehat{{\mathcal{O}}}_{X,p}\cdot C_{xt^{1/a_{1}}}(R)$ is
generated by the maximal contact and the coefficients in the graded form of
the generators of $R$ with respect to the powers $(xt^{1/a})^{i}$. These
coefficients by splitting live on $\widehat{{\mathcal{O}}}_{X,p}\cdot
R_{|V(x)}\subset\widehat{{\mathcal{O}}}_{X,p}\cdot R$. (See Lemma 3.5.7,
Examples 3.2.2, 3.2.4, 3.2.6)

###  1.3. Construction of the center

Going back to the milling process we put

$\displaystyle R_{1}:={\mathcal{O}}_{X}[{\mathcal{I}}t]\subset{\mathcal{A}},$
$\displaystyle
R_{2}:=C_{x_{1}t^{1/a_{1}}}(R_{1})\subset{\mathcal{A}}=C_{x_{1}t^{1/a_{1}}}({\mathcal{A}})$
$\displaystyle\ldots$ $\displaystyle
R_{i+1}:=C_{x_{i}t^{i/a_{i}}}(R_{i})\subset{\mathcal{A}}=C_{x_{i}t^{1/a_{i}}}({\mathcal{A}})$
$\displaystyle\ldots$ $\displaystyle R_{k+1}={\mathcal{A}}.$

The process will continue until $R_{k+1|V(x_{1},\ldots,x_{k})}=0$ and thus
$R_{k+1}={\mathcal{A}}$ which completes the procedure. In this recursion,
$a_{i}={\operatorname{ord}}_{p}(R_{i|V(x_{1},\ldots,x_{i-1})})$ and the
$i$-the maximal contact $x_{i}t^{1/a_{i}}$ is defined so that the restriction
$x_{i|V(x_{1},\ldots,x_{i-1})}$ is a local parameter in the
$t^{1/a_{i}}$-gradation of the algebra obtained by the action of
$t^{-1/a_{i}}D_{x_{i}},\ldots,t^{-1/a_{i}}D_{x_{n}}$ on
$R_{i|V(x_{1},\ldots,x_{i-1})}$. In the process we change the coordinate
system in such a way that gradually step by step all the maximal contacts are
included. These changes however do not affect ${\mathcal{A}}$ just its
representation. (See Examples 3.2.2, 3.2.4, 3.2.6). Again the above process in
a non-SNC setting shall be compared to the construction of the algebra
associated with the center in the first version of [ATW19].

The whole algorithm is independent of the a priori chosen maximal admissible
center ${\mathcal{A}}$. At the same time it is done inside of ${\mathcal{A}}$
which is then uniquely determined as the terminal algebra of the recursive
process.

###  1.4. Rees centers and cobordant blow-up

The computed maximal admissible Rees center
${\mathcal{A}}={\mathcal{O}}_{X}[x_{1}t^{1/a_{1}},\ldots,x_{k}t^{1/a_{k}}]^{{\operatorname{int}}}$
determines the extended Rees algebra

${\mathcal{A}}^{\operatorname{ext}}={\mathcal{O}}_{X}[t^{-1/w_{A}},x_{1}t^{1/a_{1}},\ldots,x_{k}t^{1/a_{k}}],$

where $w_{A}={\operatorname{lcm}}(a_{1},\ldots,a_{k})$ is the least common
multiple of the rational numers $a_{1},\ldots,a_{k}$.

Now the rescaled integral extended Rees algebra algebra

${\mathcal{O}}_{B}={\mathcal{O}}_{X}[t^{-1},x_{1}t^{w_{1}},\ldots,x_{k}t^{w_{k}}],$

with $w_{i}=w_{A}/a_{i}$ defines the desired transformation of the full
cobordant blow-up

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

Here the new coordinates are $t^{-1},x_{1}t^{w_{1}},\ldots,x_{k}t^{w_{k}}$,
where $t^{-1}$ describes the exceptional divisor $D:=V_{B}(t^{-1})$, and the
vertex of the cobordant blow-up is defined as
$V:=V_{B}(x_{1}t^{w_{1}},\ldots,x_{k}t^{w_{k}})$.

###  1.5. Controlled transforms

Rescaling the admissibility inclusion
${\mathcal{I}}t\subset{\mathcal{A}}^{{\operatorname{ext}}}$ leads to
${\mathcal{I}}t^{w_{A}}\subset{\mathcal{O}}_{B}$. Thus the full transform
${\mathcal{O}}_{B}\cdot{\mathcal{I}}$ of ${\mathcal{I}}$ is divisible by the
power $(t^{-1})^{w_{A}}$ of the exceptional divisor $D=V_{B}(t^{-1})$. The
result of this factorization
$\sigma^{c}({\mathcal{I}}):={\mathcal{O}}_{B}{\mathcal{I}}t^{w_{A}}$ is called
the controlled transform of ${\mathcal{I}}$. Thus the admissibility condition
turns into the condition for the controlled transform.

Also the new coordinates
$x_{1}^{\prime}:=x_{1}t^{w_{1}},\ldots,x^{\prime}_{k}:=x_{k}t^{w_{k}}$ are
simply the controlled transforms of $x_{1},\ldots,x_{k}$. Consequently the
controlled transform $\sigma^{c}({\mathcal{A}})$ of the center ${\mathcal{A}}$
can be written at the vertex $V$ as
$\sigma^{c}({\mathcal{A}})={\mathcal{O}}_{B}[x^{\prime}_{1}t^{1/a_{1}},\ldots,x^{\prime}_{k}t^{1/a_{k}}]$.
Similarly the derivations
$D_{x_{i}^{\prime}}=t^{-w_{i}}D_{x_{i}}=\sigma^{c}(D_{x_{i}})$ on $B$ are the
controlled transforms of $D_{x_{i}}$.

As a result the coefficient ideals $C_{x_{i}t^{1/a_{i}}}(R_{i})$ commute with
the operation of the controlled transform:
$\sigma^{c}(C_{x_{i}t^{1/a_{i}}}(R_{i}))=C_{x^{\prime}_{i}t^{1/a_{i}}}(\sigma^{c}(R_{i})),$
and the entire milling algorithm commutes at the points of the vertex $V$
converging from
$\sigma^{c}(R_{1})={\mathcal{O}}_{X}[\sigma^{c}({\mathcal{I}})t]$ to the
maximal admissible center $\sigma^{c}(R_{k+1})=\sigma^{c}({\mathcal{A}})$ and
giving the same value of the invariant

${\operatorname{inv}}(\sigma^{c}(R))=(a_{1},\ldots,a_{k})={\operatorname{inv}}(R)$

for $\sigma^{c}(R)$ at the vertex $V$ as for ${\mathcal{I}}$ at the center
$V({\mathcal{A}})$.

###  1.6. Resolution principle for the resolution in characteristic zero

The resolution algorithm in characteristic zero can be stated as follows:

For any ideal ${\mathcal{I}}$ and a point $p$ consider a unique maximal
admissible center ${\mathcal{A}}$. Then

  1. (1)

The vanishing locus $V({\mathcal{A}})$ describes the locus of the points $q$
where
${\operatorname{inv}}_{q}({\mathcal{I}})={\operatorname{inv}}_{p}({\mathcal{I}})$
is constant and attains its maximal value in a certain neighborhood $U$ of
$p$. Thus ${\operatorname{inv}}_{p}({\mathcal{I}})$ is upper semicontinuous
and attains finitely many values along the maximal admissible centers.

  2. (2)

The full cobordant of $B$ at ${\mathcal{A}}$ corresponds to the rescaled
algebra ${\mathcal{O}}_{B}$ of
${\mathcal{A}}^{\operatorname{ext}}={\mathcal{O}}_{X}$.

  3. (3)

Consider the Rees center ${\mathcal{A}}$ associated with the maximum
${\operatorname{maxinv}}_{X}({\mathcal{I}})$ of the invariant
${\operatorname{inv}}$ on $X$. After the full cobordant blow-up $B$ of $X$ at
${\mathcal{A}}$ the invariant ${\operatorname{inv}}$ of the controlled
transform
$\sigma^{c}({\mathcal{I}}):={\mathcal{O}}_{B}\cdot{\mathcal{I}}t^{w_{A}}$
attains its maximum at the vertex $V$ from $B$. This maximum is equal to to

${\operatorname{maxinv}}_{B}(\sigma^{c}({\mathcal{I}}))={\operatorname{maxinv}}_{X}({\mathcal{I}})={\operatorname{inv}}_{p^{\prime}}(\sigma^{c}({\mathcal{I}}))$

for any $p^{\prime}\in V$.

  4. (4)

The invariant $``{\operatorname{inv}}"$ drops for the cobordant blow-up
$B_{+}:=B\smallsetminus V$ after removing the vertex $V$, that is

${\operatorname{maxinv}}_{B}(\sigma^{c}({\mathcal{I}}))<{\operatorname{maxinv}}_{X}({\mathcal{I}})$

leading to the resolution of singularities.

  5. (5)

The maximal admissible center for the ideal $\sigma^{c}({\mathcal{I}})$ at $V$
is given by the controlled transform $\sigma^{c}(A)$ of the center
${\mathcal{A}}$ so we have maximal admissibility inclusion
$\sigma^{c}({\mathcal{I}})\subset\sigma^{c}({\mathcal{A}})$.

###  1.7. Resolution principle in general

When considering maximal admissible centers in characteristic zero for an
ideal ${\mathcal{I}}$ we see a strong correlation between the singularities at
the vertex and the center. The singularities along the maximal admissible
center $V({\mathcal{A}})$ are the same as the singularities at the vertex $V$
in the sense they have the same invariant.

This principle is a part of a more general picture and can be pursued in a
nonzero characteristic which leads to various types of resolution and
reduction theorems. It relies on the following observations:

  1. (1)

Let $D=V_{B}(t^{-1})$ be the exceptional divisor. The vertex $V$ consists of
two parts:

$V\smallsetminus D\subset B_{-}:=B\smallsetminus D=X\times{\mathbb{A}}^{1}$

is isomorphic to $V\smallsetminus D\simeq
V({\mathcal{A}})\times{\mathbb{A}}^{1}$, showing that the singularities of
$V({\mathcal{I}})$ at $V({\mathcal{A}})$ are equivalent up to smooth
projection to the singularities of the strict transform
$V(\sigma^{s}({\mathcal{I}}))$ at $V\smallsetminus D$. Here the strict
transform $\sigma^{s}({\mathcal{I}})$ is obtained by the factoring out the
maximal powers of the exceptional divisors from the functions in the full
transform ${\mathcal{O}}_{B}\cdot{\mathcal{I}}$.

  2. (2)

The exceptional divisor $D=V_{B}(t^{-1})$ is isomorphic to the normal weighted
bundle $N_{X/V({\mathcal{A}})}$ of $X$ at $V({\mathcal{A}})$. (Lemma 4.1.5)
The singularities of the intersection
$V(\sigma^{s}({\mathcal{I}}))\cap D$ at $V_{D}:=V\cap D\subset D$ are
isomorphic to the singularities of the normal weighted cone
$C_{V({\mathcal{A}})}(V({\mathcal{I}}))\subset N_{X/V({\mathcal{A}})}$ of the
$V({\mathcal{I}})$ at $V({\mathcal{A}})$. (Lemma 4.1.7)

This allows to resolve or improve singularities if they are controlled along
the center and when the singularities of its normal weighted cone are
controlled at the center as well. The method works for some classes of
singularities. In general, in a nonzero characteristic we may have some
automorphisms of the normal cone which prevent an apparent improvement of
singularities.

###  1.8. SNC divisors

Another novelty of this paper is the direct treatment of the exceptional SNC
divisors without any additional structures. In [ATW19], or in [McQ19], this
problem is omitted and the simple resolution is obtained without any
conditions on the exceptional divisors. In the classical Hironaka approach the
algorithm requires the exceptional divisor must have SNC crossings, and the
process is controlled by a certain inductively defined filtration on the set
of the exceptionals which is arguably the most cumbersome part of the proof.
In this paper no structure is needed and a geometric invariant is a simple
function of the subvariety or the ideal and the set of the exceptional
divisors. Consequently, the embedded and nonembedded resolution of varieties
and principalization of ideals is obtained in the stronger form where the
centers have SNC with the strict transforms, and the inverse image of the
ideal is given by an effective SNC divisor.

The definition of the invariant ${\operatorname{inv}}_{p}({\mathcal{I}})$ and
the maximal admissible centers are adapted to the regular schemes with SNC
divisor. In such a case we only consider the coordinate systems where each
divisorial component at a given point is represented by one of the
coordinates. We shall also think of the divisorial coordiantes as
infinitesimally heavier than the other ones.

Consequently in the admissibility relation

${\mathcal{I}}t\subset{\mathcal{A}}={\mathcal{O}}_{X}={\mathcal{O}}_{X}[x_{1}t^{1/a_{1}},\ldots,x_{k}t^{1/a_{k}}]^{{\operatorname{int}}}$

we associate with the any free, that is non-divisorial coordinate $x_{i}$ the
element $b_{i}=a_{i}$ and with divisorial coordinate $x_{j}$ the slightly
”heavier” symbol $b_{j}=a_{j+}$. We assume here that $a_{+}>a$ for any
$a\in{\mathbb{R}}$ and if $b>a$ for $a,b\in{\mathbb{R}}$ then $b>a_{+}$ This
gives a vector $(b_{1},\ldots,b_{k})$ and we assume that $b_{1}\leq\ldots\leq
b_{k}$. The invariant in such a case can be written as

${\operatorname{inv}}_{p}({\mathcal{I}})=\max\\{(b_{1},\ldots,b_{k})\mid{\mathcal{I}}t\subset{\mathcal{O}}_{X}[x_{1}t^{1/a_{1}},\ldots,x_{k}t^{1/a_{k}}]^{{\operatorname{int}}}\\}.$

The main point of this construction is that we obtain still upper
semicontinuous invariant, which drops after a cobordant blow-up at the
associated maximal admissible center. Moreover, the whole algorithm is only
marginally affected by this modification of the invariant. The only essential
change lies in the construction of maximal contact allowing additional
divisorial coordinates tangential to $T^{1/a_{i}}(R_{i})$.

###  1.9. Comparing to existing strategies and ideas

The ideas of this paper stem mostly from [ATW19] and the joint project with
Abramovich and Temkin. The main difference here is the use of cobordant blow-
ups, rational Rees algebras for the construction and uniqueness of the
centers, and the use of the SNC divisors giving stronger results on resolution
and principalization without changes in the complexity of the algorithm.
Moreover, the paper extends some ideas to a more general context of schemes in
any characteristic. In the case where the exceptional divisors are ignored we
obtain the same centers as in [ATW19].

The cobordant blow-ups were considered independently by Quek and Rydh in
[QR22] from a slightly different angle. They provide a convenient presentation
of the stack theoretic weighted blow-ups. Again the ideas were born from the
method in [ATW19]. One shall stress that using stack-theoretic quotients does
not give here any particular benefit. Instead, some of the information is lost
in the process. It is mostly used in the historic context of the established
notion.

In [ATW19] the construction of the center was achieved by a different
inductive argument in a different language and its uniqueness was proven by
using the idea of homogenization ideals from [Wło05]. The centers were
represented by the ${\mathbb{Q}}$-ideals
${\mathcal{J}}=(x_{1}^{1/w_{1}},\ldots,x_{k}^{1/w_{k}})$. In the very first
version of [ATW19] this construction was related to the rescaled algebra of
the center
${\mathcal{A}}_{{\mathcal{J}}}={\mathcal{O}}_{X}[x_{1}t^{1/a_{1}},\ldots,x_{k}t^{1/a_{k}}]^{{\operatorname{int}}}$
for an ideal ${\mathcal{I}}$ obtained by applying the saturated weighted
derivatives ${\mathcal{D}}_{x_{i}t^{1/a_{1}}}$ to
${\mathcal{O}}_{X}[{\mathcal{I}}t]$, and the center
${\mathcal{J}}=(x_{1}^{1/w_{1}},\ldots,x_{k}^{1/w_{k}})$ in the
${\mathbb{Q}}$-ideal form with $w_{1}a_{1}=\ldots=w_{k}a_{k}$. Such an
interpretation is valid when there are no SNC divisors. This was done as an
illustration of our method, was not used in the context of the resolution
strategy and thus was omitted in the next versions of [ATW19].

The method in this paper is designed to derive directly in an efficient way
the invariant and the unique maximal center ${\mathcal{A}}$ from
${\mathcal{I}}$ using the minimalistic transformations determined solely by
${\mathcal{I}}$ without any references to prior constructions. The
transformations are done in SNC setting and applied to generators only. As a
consequence, all the intermediate steps are easy to compute but non-canonical
and only the final center is automatically unique, since the process is
independent of the maximal admissible center containing all the transitional
algebras (see Examples 3.2.2, 3.2.4, 3.2.6 ).

The compact formula for the coefficient ideal which is generated by the
coefficients and maximal contact can be linked to the Bierstone-Milman
presentation with the important distinction that it involves no factorization
of monomials and the coefficient presentation is done on the completion of the
local ring of the original variety in the language of Rees algebras. The
Bierstone-Milman coefficient ideal lives on a hypersurface of maximal contact
and is generated solely by the adjusted coefficients ([BM91],[BM97]).

This type of Bierstone-Milman approach to the coefficient ideal is much more
efficient for computations and practical implementations. Recall that most of
the resolution strategies rely on the Villamayor approach to the coefficient
ideal with rescaling of the ideals (see [Vil89], [Wło05], [EH02], [EV03],
[Kol07],[BM08]. This strategy was also used in [ATW19]. Each coefficient ideal
typically increases the grading from $n$ to $n!$ which severely restricts the
implementability of the algorithm. This is due to the simple fact that already
$(5!)!$ exceeds $10^{97}$ which is according to the Standard Model of
particles the assumed maximal estimated number of the subatomic particles in
the universe. In practice, computing even the first coefficient ideal could be
tedious for the relatively small gradings say $n\geq 7$. Here the grading $n$
is typically given by the order of the ideal ${\mathcal{I}}$ at a point and is
estimated by the degrees of its generators.

In their paper [QR22], Rydh and Quek observed that the canonical projection
$B\to{\mathbb{A}}^{1}$ determines the deformation to the normal weighted cone.
(see Remark 4.1.11). They also noticed that weighted blow-up can be
interpreted in terms of the toric combinatorial construction of Cox rings.

The approach was pursued in the joint work [AQ21] of Abramovich-Quek based on
the Satriano combinatorial method from [Sat13]. Finally in the subsequent
paper [Wło23], we give a general formula for the realization of any proper
birational morphism by the torus action using the Cox rings. In the paper
[Wło23] we also extend the methods and results of the weighted normal cone
from Section 4 (Theorems 4.2.2, 4.2.3, 4.3.1) to the context of the blow-ups
of any locally monomial ideals.

###  1.10. The output of the algorithm

The resolution algorithm outputs a smooth scheme with a torus action and
admitting a geometric quotient having abelian quotient singularities and
birational to the original scheme. The quotient can be directly resolved by
canonical combinatorial methods in any characteristic as in [Wło20, Theorem
7.17.1], or in [BR19] by the destackification method of Bergh and Rydh.

####  1.10.1. Aknowledgements

The author would like to thank Dan Abramovich, Kenji Matsuki, Mircea Mustaţǎ,
Michael Temkin, and Willem Veys and many others for helpful discussions and
suggestions.

###  1.11. Main results in characteristic zero

The following theorems extend the results and the resolution method by stack-
theoretic weighted blow-ups from
[ATW19, Section 1.2] in the language of torus actions and SNC divisors.

####  1.11.1. Functorial principalization

######  Theorem 1.11.2.

Let $X$ be a smooth variety over a field $k$ of characteristic zero, and $E$
be a simple normal crossing (SNC) divisor and ${\mathcal{I}}$ be an ideal
sheaf on $X$. There exists a canonical principalization of ${\mathcal{I}}$,
that is a sequence of cobordant blow-ups (Definition 2.4.7) at smooth weighted
centers

$X=X_{0}\leftarrow X_{1}\leftarrow\ldots\leftarrow
X_{k}=X^{\prime},\quad\hfill{(*)}$

such that:

  1. (1)

The torus $T_{i}=G_{m}^{i}$ acts on $X_{i}$, with finite stabilizers, where
$T_{0}=1$, $T_{i+1}=T_{i}\times G_{m}$, ans $T:=T_{k}$ so that the geometric
quotient (space of orbits) $X_{i}/T_{i}$ exists.

  2. (2)

Set ${\mathcal{I}}_{0}:={\mathcal{I}}$. For $i\geq 1$, the ideals
${\mathcal{I}}_{i}\subset{\mathcal{O}}_{X_{i}}$, defined as
${\mathcal{I}}_{i}:={\mathcal{O}}_{X_{i}}\cdot{\mathcal{I}}_{i-1}$ are
$T_{i}$-stable.

  3. (3)

Set $E_{0}:=E$, and let $E_{i}$, for $i\geq 1$, be the total transform of
$E_{i-1}$. The divisors $E_{i}$ on $X_{i}$ are $T_{i}$-stable and have SNC.

  4. (4)

The smooth weighted centers ${\mathcal{J}}_{i}$ of the cobordant blow-ups
$X_{i}\leftarrow X_{i+1}$ are compatible with $E_{i}$ (Definition 3.1.2), and
are $T_{i}$-stable. Moreover $V({\mathcal{J}}_{i})\subseteq
V({\mathcal{I}}_{i})$.

  5. (5)

${\mathcal{O}}_{X^{\prime}}\cdot{\mathcal{I}}={\mathcal{I}}_{D^{\prime}}:={\mathcal{O}}_{X^{\prime}}(-D^{\prime})$
is the ideal of an SNC divisor $D^{\prime}$ whose components are the strict
transforms of the components of the total transform $E^{\prime}$ of $E$.
Moreover there is a $T$-equivariant isomorphism over $X\smallsetminus
V({\mathcal{I}})$:

$X^{\prime}\smallsetminus D^{\prime}\simeq(X\smallsetminus
V({\mathcal{I}}))\times T.$

  6. (6)

The sequence (*) defines the sequence of weighted blow-ups on the induced
geometric quotients $X=X/T_{0}\leftarrow X_{1}/T_{1}\leftarrow\ldots\leftarrow
X_{k}/T_{k}=X^{\prime}/T,$ such that
${\mathcal{O}}_{X^{\prime}/T^{\prime}}\cdot{\mathcal{I}}={\mathcal{I}}_{D^{\prime\prime}}$
is the ideal of a locally principal divisor
$D^{\prime\prime}:=D^{\prime}/T\subset X^{\prime}/T^{\prime}$, where
$D^{\prime}$ is a $T$-stable SNC divisor on $X^{\prime}$, and
${\mathcal{O}}_{X}\cdot{\mathcal{I}}_{D^{\prime\prime}}={\mathcal{I}}_{D^{\prime}}$.

  7. (7)

The sequence (*) defines the sequence of stack-theoretic weighted blow-ups on
the induced smooth stack-theoretic quotients

$X=X_{0}=[X/T_{0}]\leftarrow[X_{1}/T_{1}]\leftarrow\ldots\leftarrow[X_{k}/T_{k}]=[X^{\prime}/T^{\prime}],$

such that ${\mathcal{O}}_{[X^{\prime}/T^{\prime}]}\cdot{\mathcal{I}}$ is the
ideal of an SNC divisor on $[X^{\prime}/T^{\prime}]$.

  8. (8)

The sequence (*) is functorial for smooth morphisms, field extensions, and
group actions preserving $({\mathcal{I}},E)$.

####  1.11.3. Embedded desingularization

######  Theorem 1.11.4.

Let $Y$ be a reduced closed subscheme of a smooth scheme $X$ over a field $k$
of characteristic zero. Let $E$ be an SNC divisor on $E$. There exists a
sequence of cobordant blow-ups at smooth weighted centers

$X=X_{0}\leftarrow X_{1}\leftarrow\ldots\leftarrow
X_{k}=X^{\prime}\quad\hfill{(*)}$

and the induced sequence of the strict transforms

$Y=Y_{0}\leftarrow Y_{1}\leftarrow\ldots\leftarrow Y_{k}=Y^{\prime},$

such that

  1. (1)

The torus $T_{i}=G_{m}^{i}$ acts on $X_{i}$, with finite stabilizers, where
$T_{0}=1$, $T_{i+1}=T_{i}\times G_{m}$ and $T:=T_{k}$ so that the geometric
quotient $X_{i}/T_{i}$ exists.

  2. (2)

The closed subschemes $Y_{i}\subset X_{i}$ are $T_{i}$-stable. In particular,
they admit geometric quotients $Y_{i}/T_{i}$.

  3. (3)

Set $E_{0}:=E$, and let $E_{i}$, for $i\geq 1$, be either the total or the
strict transform of $E_{i-1}$. The divisors $E_{i}$ on $X_{i}$ are
$T_{i}$-stable and have SNC.

  4. (4)

The smooth weighted centers of the cobordant blow-ups $X_{i}\leftarrow
X_{i+1}$ are compatible with $E_{i}$. Moreover, they are $T_{i}$-stable and
contained in the locus ${\operatorname{Sing}}_{E_{i}}(Y_{i})$ of the points
where either $Y_{i}$ is singular, or $Y_{i}$ is smooth but not transversal to
$E_{i}$.

  5. (5)

$Y^{\prime}$ is a smooth subvariety of $X$ and have SNC with
$E^{\prime}:=E_{k}$. If $E^{\prime}$ is the total transform of $E$ then the
exceptional divisor of the induced morphism $Y^{\prime}\to Y$ is also SNC.

  6. (6)

The sequence (*) defines the sequence of weighted blow-ups on the induced
geometric quotients $X=X/T_{0}\leftarrow X_{1}/T_{1}\leftarrow\ldots\leftarrow
X_{k}/T_{k}=X^{\prime}/T,$ such that $Y^{\prime}/T\subset X^{\prime}/T$ admit
abelian quotient singularities,

  7. (7)

The sequence (*) defines the sequence of stack- theoretic weighted blow-ups on
the induced smooth stack-theoretic quotients

$X=X_{0}=[X/T_{0}]\leftarrow[X_{1}/T_{1}]\leftarrow\ldots\leftarrow[X_{k}/T_{k}]=[X^{\prime}/T^{\prime}],$

such that $[Y^{\prime}/T^{\prime}]\subset[X^{\prime}/T^{\prime}]$ is a smooth
substack.

  8. (8)

The sequence (*) is functorial for smooth morphisms, field extensions, and
group actions preserving $Y.$

######  Remark 1.11.5.

The total transform of the divisor $E_{i}$ is the union of the strict
transforms and the exceptional divisors. Using the total transforms gives
better control over the induced exceptional divisor of the resolution morphism
$X^{\prime}\to X$ which is SNC and has simple normal crossings with
$Y^{\prime}\subset X^{\prime}$. The strict transforms of $E_{i}$ give a
slightly faster and simpler algorithm without control of the exceptional
divisors.

In particular, if $E_{0}=0$ and the strict transforms are considered we get
$E_{i}=0$, and no divisors are present in the algorithm. In this case we
obtain the resolution procedure similar to the one in [ATW19]. Moreover, when
applying the stack-theoretic quotients as in (5), we reprove the main result
of [ATW19, Corollary 1.2.3].

####  1.11.6. Nonembedded desingularization

######  Theorem 1.11.7.

Let $Y$ be a reduced scheme of finite type over a field $k$ of characteristic
zero. There exists a sequence of cobordant blow-ups of regular weighted
centers (as in Definition 2.4.7):

$Y=Y_{0}\leftarrow Y_{1}\leftarrow\ldots\leftarrow
Y_{k}=Y^{\prime},\quad\hfill{(**)}$

such that

  1. (1)

$Y^{\prime}$ is a smooth variety.

  2. (2)

The torus $T_{i}=G_{m}^{i}$ acts on $Y_{i}$, with finite stabilizers, where
$T_{0}=1$, $T_{i+1}=T_{i}\times G_{m}$ and $T:=T_{k}$, so that the geometric
quotient $Y_{i}/T_{i}$ exists.

  3. (3)

The inverse image of the exceptional locus is a $T^{\prime}$-stable SNC
divisor $E^{\prime}$ on $Y^{\prime}$

  4. (4)

The inverse image of the nonsingular locus $Y^{ns}\subseteq Y$ is equal to
$Y^{ns}\times T$, with the natural action of $T$ on the second component.

  5. (5)

The sequence (**) defines the sequence of weighted blow-ups on the induced
geometric quotients $Y=Y/T_{0}\leftarrow Y_{1}/T_{1}\leftarrow\ldots\leftarrow
Y_{k}/T_{k}=Y^{\prime}/T,$ such that $Y^{\prime}/T$ has an abelian quotient
singularities.

  6. (6)

The sequence (**) defines the sequence of stack-theoretic weighted blow-ups on
the induced stack-theoretic quotients

$Y=Y_{0}=[Y/T_{0}]\leftarrow[Y_{1}/T_{1}]\leftarrow\ldots\leftarrow[Y_{k}/T_{k}]=[Y^{\prime}/T],$

such that $[Y^{\prime}/T]$ is a smooth stack.

  7. (7)

The sequence (**) is functorial for smooth morphisms, group actions, and field
extensions.

##  2\. Cobordant blow-ups

###  2.1. ${\mathbb{Q}}$-ideals and rational powers of ideals

In [ATW19] we introduced the notion of the valuative ${\mathbb{Q}}$-ideals
${\mathcal{J}}$ or simply ${\mathbb{Q}}$-ideals ${\mathcal{J}}$. In the
simplest case they could be directly related to the notion of rational powers
of ideals considered by Huneke-Swanson ([HS06, Section 10.5]). They naturally
generalize the ideals and can be used for the compact description of the
centers. In this paper, for the most part and for computations in
characteristic zero, we predominantly focus on the rational Rees algebras.

######  Definition 2.1.1.

([ATW19]) By a $Q$-ideal ${\mathcal{J}}$ on a irreducible noetherian scheme
$X$ we mean the equivalence classes of the formal expressions
${\mathcal{J}}^{1/n}$, where ${\mathcal{J}}$ is the ideal on $X$ and
$n\in{\mathbb{N}}$ is a natural number. We say that ${\mathcal{J}}^{1/n}$ and
${\mathcal{I}}^{1/m}$ are equivalent if the integral closures of
${\mathcal{J}}^{m}$ and ${\mathcal{I}}^{n}$ are equal:

$({\mathcal{J}}^{m})^{\operatorname{int}}=({\mathcal{I}}^{m})^{\operatorname{int}}.$

######  Remark 2.1.2.

Note that if ${\mathcal{J}}^{1/n}$ and ${\mathcal{I}}^{1/m}$ represent the
same ${\mathbb{Q}}$-ideal for some ideals ${\mathcal{I}}$, and ${\mathcal{J}}$
and $m,n\in{\mathbb{N}}$ then the ideals ${\mathcal{I}}$ and ${\mathcal{J}}$
are called projectively equivalent wth respect to the coefficient $m/n$.
([Rush07])

One can associate with a ${\mathbb{Q}}$-ideal
${\mathcal{J}}={\mathcal{I}}^{1/n}$ the unique graded Rees algebra

${\mathcal{A}}_{{\mathcal{J}}}:=({\mathcal{O}}_{X}[{\mathcal{I}}t^{n}])^{{\operatorname{int}}}\subset{\mathcal{O}}_{X}[t]$

which is the integral closure of

${\mathcal{O}}_{X}[{\mathcal{I}}t^{n}]={\mathcal{O}}_{X}\oplus{\mathcal{I}}t^{n}\oplus{\mathcal{I}}^{2}t^{2n}\oplus\ldots$

in ${\mathcal{O}}_{X}[t]$. Conversely, it is worth noting that any integrally
closed Rees algebra of the form

${\mathcal{O}}_{X}\oplus{\mathcal{I}}_{1}t\oplus\ldots\oplus{\mathcal{I}}_{n}t^{n}\oplus\ldots$

can be expressed as the integral closure of
${\mathcal{O}}_{X}[{\mathcal{I}}t^{n}]^{{\operatorname{int}}}$ for a
sufficiently large $n$, where ${\mathcal{I}}={\mathcal{I}}_{n}$. This
observation was made, in particular, by Quek:

######  Proposition 2.1.3.

[Que20, Theorem 2.2.5] There is a bijective correspondence between the
${\mathbb{Q}}$-ideals ${\mathcal{J}}={\mathcal{I}}^{1/n}$ on a normal scheme
$X$ and the integrally closed algebras ${\mathbb{Z}}$-graded Rees algebras on
$X$, given by

${\mathcal{J}}\mapsto{\mathcal{A}}_{{\mathcal{J}}}:={\mathcal{O}}_{X}[{\mathcal{I}}t^{n}]^{{\operatorname{int}}}.$

Under this correspondence the ordinary ideal ${\mathcal{I}}$ corresponds to
${\mathcal{O}}_{X}[{\mathcal{I}}t]^{{\operatorname{int}}}$.∎

With any ${\mathbb{Q}}$-ideal ${\mathcal{J}}={\mathcal{I}}^{1/n}$ on $X$ one
can associate the ideal of sections

$({\mathcal{I}}^{1/n})_{X}={\mathcal{J}}_{X}:=\\{f\in{\mathcal{O}}_{X}\mid
f^{mn}\in{\mathcal{I}}^{m}\mid
m\in{\mathbb{N}}\\}=\\{f\in{\mathcal{O}}_{X}\mid
f^{n}\in{\mathcal{I}}^{\operatorname{int}}\mid m\in{\mathbb{N}}\\}.$

. It is exactly the $t$-gradation of the algebra
${\mathcal{A}}_{{\mathcal{J}}}={\mathcal{O}}_{X}[{\mathcal{I}}t^{n}]^{{\operatorname{int}}}$.
In particular, if ${\mathcal{J}}={\mathcal{I}}$ is an ideal then
${\mathcal{J}}_{X}={\mathcal{I}}^{{\operatorname{int}}}$ is it is integral
closure.

This terminology is the strictly related to the notion of rational powers of
ideals which was coined by Huneke-Swanson [HS06, Section 10.5].

######  Definition 2.1.4.

([HS06]) For any ideal ${\mathcal{I}}$ and a ${\mathbb{Q}}$-ideal
${\mathcal{I}}^{m/n}$ for $m,n\in{\mathbb{N}}$ the ideal of sections
$({\mathcal{I}}^{m/n})_{X}$ is called the $m/n$-th rational power of ideal
${\mathcal{I}}$.

Using the correspondence between the ${\mathbb{Q}}$-ideals and Rees algebras
one can associate with the formal sum
$\sum^{k}_{i=1}{\mathcal{I}}_{i}^{a_{i}/n_{i}}$, of ${\mathbb{Q}}$-ideals
${\mathcal{I}}_{i}^{a_{i}/n_{i}}$ on $X$ the Rees algebra

${\mathcal{O}}_{X}[{\mathcal{I}}_{1}^{a_{1}}t^{n_{1}},\ldots,{\mathcal{I}}_{k}^{a_{n_{k}}}t^{n_{k}}]^{{\operatorname{int}}},$

corresponding to a certain ${\mathbb{Q}}$-ideal on $X$.

This leads to the equality of ${\mathbb{Q}}$-ideals:

$\sum{\mathcal{I}}_{i}^{1/n}=(\sum{\mathcal{I}}_{i})^{1/n}\quad\mbox{and}\quad{\mathcal{I}}^{1/n}=({\mathcal{I}}^{1/mn})^{m}$

and determines the operations of addition and multiplication on the
${\mathbb{Q}}$-ideals on $X$. For any ${\mathbb{Q}}$-ideals
${\mathcal{J}}_{1}$ and ${\mathcal{J}}_{2}$ on $X$ written in a general form
as ${\mathcal{J}}_{1}:=\sum{\mathcal{I}}_{i}^{1/n_{i}}$ and
${\mathcal{J}}_{2}:=\sum{\mathcal{J}}_{j}^{1/n_{j}}$ their sum is given by

${\mathcal{J}}_{1}+{\mathcal{J}}_{2}=\sum{\mathcal{I}}_{i}^{1/n_{i}}+\sum{\mathcal{J}}_{j}^{1/n_{j}}$

and their product by

$\displaystyle{\mathcal{J}}_{1}\cdot{\mathcal{J}}_{2}=\sum{\mathcal{I}}_{i}^{1/n_{i}}\cdot\sum{\mathcal{J}}_{j}^{1/n_{j}}=\sum_{i,j}{\mathcal{I}}_{i}^{1/n_{i}}\cdot{\mathcal{J}}_{j}^{1/n_{j}}=$
$\displaystyle=\sum_{i,j}({\mathcal{I}}_{i}^{n_{j}}\cdot{\mathcal{J}}_{j}^{n_{i}})^{\frac{1}{n_{i}n_{j}}}=(\sum_{i,j}{\mathcal{I}}_{i}^{n_{j}}\cdot{\mathcal{J}}_{j}^{n_{i}})^{\frac{1}{n_{i}n_{j}}}$

and they both define ${\mathbb{Q}}$-ideals on $X$.

Note that these operations extend the standard notion of the sum and the
product of ideals. Similarly, we can extend the inclusion relation. Let
${\mathcal{J}}_{1}={\mathcal{I}}_{1}^{1/n_{1}}$, and
${\mathcal{J}}_{2}={\mathcal{I}}_{2}^{1/n_{2}}$ be two ${\mathbb{Q}}$-ideals
on $X$. We say that ${\mathcal{J}}_{1}\subseteq{\mathcal{J}}_{2}$ if we have
the inclusion of ideals
$({\mathcal{I}}_{1}^{N/n_{1}})^{\operatorname{int}}\subseteq({\mathcal{I}}_{2}^{N/n_{2}})^{\operatorname{int}}$
for sufficiently divisible $N$.

####  2.1.5. Graded algebras of ${\mathbb{Q}}$-ideals

One can associate with the ${\mathbb{Q}}$-ideal
${\mathcal{J}}={\mathcal{I}}^{1/n}$, the corresponding graded algebra of
${\mathbb{Q}}$-ideals

${\mathcal{O}}_{X}[{\mathcal{J}}t]:=\bigoplus({\mathcal{J}}^{i})t^{i},$

where $t$ is a dummy variable. This, in turn, determines the graded Rees
algebra of ideals

${\mathcal{A}}_{J}=({\mathcal{O}}_{X}[{\mathcal{J}}t])_{X}:=\bigoplus({\mathcal{J}}^{i})_{X}t^{i}$

on $X$.

####  2.1.6. Functoriality of ${\mathbb{Q}}$-ideals

If $X^{\prime}\to X$ is any morphism of integral schemes, and
${\mathcal{I}}^{1/n}$ is a ${\mathbb{Q}}$-ideal on $X$ then
$({\mathcal{O}}_{X^{\prime}}\cdot{\mathcal{I}})^{1/n}$ is the preimage of
${\mathcal{I}}^{1/n}$, which is a ${\mathbb{Q}}$-ideal on $X^{\prime}$.

####  2.1.7. Monomial valuation

######  Definition 2.1.8.

Let $u_{1},\ldots,u_{k}$ be a partial local system of parameters on a regular
irreducible scheme $X$. We say that a valuation $\nu$ such that
$\nu(u_{i})=w_{i}\in{\mathbb{Z}}_{\geq 0}$ of $K(X)$ is monomial at a point
$p\in X$, if

${\mathcal{I}}_{\nu,a,p}:=\\{f\in{\mathcal{O}}_{X,p}\mid\nu(f)\geq
a\\}={\mathcal{O}}_{X,p}\cdot(u_{1}^{a_{1}}\cdot\ldots\cdot u_{k}^{a_{k}}\mid
a_{1}w_{1}+\ldots+a_{k}w_{k}\geq a).$

Then $\nu$ is a monomial valuation on $X$ if it is monomial at all $p\in
V(u_{1},\ldots,u_{k})$.

######  Lemma 2.1.9.

Let $X$ be a any regular scheme, and $p\in X$ be a point.

  1. (1)

If $p\in V(u_{1},\ldots,u_{k})$ then assigning weights
$w(u_{i}):=w_{i}\in{\mathbb{Z}}_{\geq 0}$ determines a monomial valuation
$\nu$ on $\operatorname{Spec}({\mathcal{O}}_{X,p})$ with the ideals
${\mathcal{I}}_{\nu,a,p}$ as above.

  2. (2)

If $V(u_{1},\ldots,u_{k})$ is irreducible on $X$ then assigning weights
$w(u_{i}):=w_{i}\in{\mathbb{Z}}_{\geq 0}$ determines a unique monomial
valuation $\nu$ on $X$.

###### Proof.

(1) We need to show that if
$f\in{\mathcal{I}}_{\nu,a,p}\smallsetminus{\mathcal{I}}_{\nu,a+1,p}$ and
$g\in{\mathcal{I}}_{\nu,b,p}\smallsetminus{\mathcal{I}}_{\nu,b+1,p}$ then
$fg\in{\mathcal{I}}_{\nu,a+b,p}\smallsetminus{\mathcal{I}}_{\nu,a+b+1,p}$.
Suppose otherwise $fg\in{\mathcal{I}}_{\nu,a+b+1,p}$. By removing the extra
terms we can assume that $f=\sum_{\nu(u^{\alpha})=a}c_{\alpha}u^{\alpha}$ and
$g=\sum_{\nu(u^{\alpha})=b}d_{\alpha}u^{\alpha}$, where all
$c_{\alpha},d_{\alpha}$ are invertible. Since ${\mathcal{O}}_{X,p}$ is regular
then for the ideal ${\mathcal{I}}=(u_{1},\ldots,u_{k})$ we have that
${\operatorname{gr}}_{I}({\mathcal{O}}_{X_{p}})={\mathcal{O}}_{X_{p}}/({\mathcal{I}})[u_{1},\ldots,u_{k}]$
is a domain, and assigning weights $\nu(u_{i})=w_{i}$ determines a unique
monomial valuation $\nu_{0}$ on
${\operatorname{gr}}_{I}({\mathcal{O}}_{X_{p}})={\mathcal{O}}_{X_{p}}/({\mathcal{I}})[u_{1},\ldots,u_{k}]$
with the ideals
${\mathcal{I}}_{\nu_{0},a}=\operatorname{in}({\mathcal{I}}_{\nu,a,p})$. Then
considering the initial terms we get
$\nu_{0}(\operatorname{in}(fg))=\nu_{0}(\operatorname{in}(f))+\nu_{0}(\operatorname{in}(g))$,
which implies that
$\operatorname{in}(fg)\not\in{\mathcal{I}}_{\nu_{0},a+b+1}$. But by the
assumption
$\operatorname{in}(fg))\in\operatorname{in}({\mathcal{I}}_{\nu,a+b+1,p})={\mathcal{I}}_{\nu_{0},a+b+1}$.

(2) The valuation $\nu$ is centered at $V(u_{1},\ldots,u_{k})$ and uniquely
determined for every point. Moreover it is the same when passing to the
generic point of $V(u_{1},\ldots,u_{k})$. ∎

####  2.1.10. Regular weighted centers

For the purpose of the resolution one considers ${\mathbb{Q}}$-ideals on a
regular scheme $X$, called centers, which can be locally presented in the form
$(u_{1}^{a_{1}},\ldots,u_{k}^{a_{k}})$ where $u_{1},\ldots,u_{k}$ is a partial
family of local parameters and $a_{i}\in{\mathbb{Q}}_{>0}$. We shall always
assume that $a_{1}\leq\ldots\leq a_{k}$. Note that one can write it in an
equivalent form
$(u_{1}^{a_{1}},\ldots,u^{a_{k}})=(u_{1}^{na_{1}},\ldots,u^{na_{k}})^{1/n}$,
where $(u_{1}^{na_{1}},\ldots,u^{na_{k}})$ is an ordinary ideal for a
sufficiently divisible $n$. Then $(u_{1}^{a_{1}},\ldots,u_{k}^{a_{k}})_{X}$
will denote the ideal of sections.

One can describe these ideals in a few different equivalent ways.

######  Lemma 2.1.11.

(see also [ATW19],[Que20]) Let $X$ be a regular $X$ scheme. Let $w_{1}\geq
w_{2}\geq\ldots\geq w_{k}$ be positive integers, and $u_{1},\ldots,u_{k}$ are
corresponding local parameters on $X$ such that $V(u_{1},\ldots,u_{k})$ is
irreducible. There is a natural bijective correspondence between

  1. (1)

${\mathbb{Q}}$-ideals
${\mathcal{J}}=(u_{1}^{1/w_{1}},\ldots,u_{k}^{1/w_{k}})$.

  2. (2)

Graded Rees algebras

${\mathcal{A}}_{J}={\mathcal{O}}_{X}[u_{1}t^{w_{1}},\ldots,u_{k}t^{w_{k}}]^{\operatorname{int}}={\mathcal{O}}_{X}[u_{1}t^{c_{1}},\ldots,u_{k}t^{a_{k}}\mid
c_{1}\leq w_{1}].$

  3. (3)

Monomial valuations $\nu_{J}$, such that $\nu_{J}(u_{i})=w_{i}$, and

${\mathcal{I}}_{\nu_{J},a}:=\\{f\in{\mathcal{O}}_{X}\mid\nu_{J}(f)\geq
a\\}=(u^{\alpha}\mid\nu_{J}(u^{\alpha})\geq a).$

Moreover we get

${\mathcal{A}}_{J}=\bigoplus_{a\in{\mathbb{Z}}_{\geq
0}}{\mathcal{I}}_{\nu_{J},a}t^{a}=({\mathcal{O}}_{X}[{\mathcal{J}}\cdot
t])_{X}.$

First we prove a more explicit form of this correspondence for the centers:

######  Lemma 2.1.12.

(see also [ATW19],[Que20]) Let
${\mathcal{J}}=(u_{1}^{1/w_{1}},\ldots,u_{k}^{1/w_{k}})$ be a regular center
on a regular $X$, where $w_{i}\in{\mathbb{N}}$. Let $\nu_{J}$ be a monomial
valuation such that $\nu_{J}(u_{i})=w_{i}$, and for any $a>0$,
${\mathcal{I}}_{\nu_{J},a}:=\\{f\in{\mathcal{O}}_{X}\mid\nu_{J}(f)\geq a\\}$
is equal to $(u^{\alpha}\mid\nu_{J}(u^{\alpha})\geq a)$. Then for any
$a\in{\mathbb{Q}}_{>0}$:

$((u_{1}^{1/w_{1}},\ldots,u_{k}^{1/w_{k}})^{a})_{X}={\mathcal{I}}_{\nu_{J},a}=\\{f\in{\mathcal{O}}_{X}\mid\nu_{J}(f)\geq
a\\}=$ $(u_{1}^{c_{1}}\cdot\ldots\cdot u_{k}^{c_{k}}\mid
c_{i}\in{\mathbb{N}},\,\,c_{1}w_{1}+\ldots c_{k}w_{k}\geq a).$

In particular

$((u_{1}^{a_{1}},\ldots,u_{k}^{a_{k}}))_{X}=(u_{1}^{c_{1}}\cdot\ldots\cdot
u_{k}^{c_{k}}\mid\,\,c_{i}\in{\mathbb{N}},\,c_{1}/a_{1}+\ldots+c_{k}/a_{k}\geq
1).$

###### Proof.

Note that for any monomial $u^{c}=u_{1}^{c_{1}}\cdot\ldots\cdot u_{1}^{c_{k}}$
with integral $c_{i}$ and for $a\in{\mathbb{Q}}_{>0}$ we have :

$u^{c}\in{\mathcal{I}}_{\nu_{J},a}\,\,\Leftrightarrow\,\,c_{1}w_{1}+\ldots+c_{k}w_{k}\geq
a.$

Let $n$ be now any positive integer which is divisible by ${\rm
lcm}(w_{1},\ldots,w_{k})$, and write $a_{i}w_{i}=n$, for the relevant integers
$a_{i}$. Then, the power
$(u^{c})^{n}=(u_{1}^{a_{1}})^{c_{1}w_{1}}\cdot\ldots\cdot(u_{k}^{a_{k}})^{c_{k}w_{k}}$
is an element of an (ordinary) ideal
$(u_{1}^{a_{1}},\ldots,u_{k}^{a_{k}})^{n}$ if and only if
$c_{1}w_{1}+\ldots+c_{k}w_{1}\geq n$.

Thus if $u^{c}\in{\mathcal{I}}_{\nu_{J},n}$ then
$c_{1}w_{1}+\ldots+c_{k}w_{k}\geq n$ and
$(u^{c})^{n}\in(u_{1}^{a_{1}},\ldots,u_{k}^{a_{k}})^{n}$, whence
$u^{c}\in((u_{1}^{a_{1}},\ldots,u_{k}^{a_{k}})^{int}=(u_{1}^{a_{1}},\ldots,u_{k}^{a_{k}})_{X}$.
This shows that
${\mathcal{I}}_{\nu_{J},n}\subseteq((u_{1}^{a_{1}},\ldots,u_{k}^{a_{k}})_{X}$.
On the other hand
$(u_{1}^{a_{1}},\ldots,u_{k}^{a_{k}})\subseteq{\mathcal{I}}_{\nu_{J},n}$, and
since ${\mathcal{I}}_{\nu_{J},n}$ is integrally closed
$(u_{1}^{a_{1}},\ldots,u_{k}^{a_{k}})^{int}\subseteq{\mathcal{I}}_{\nu_{J},n}$.
This gives
${\mathcal{I}}_{\nu_{J},n}=(u_{1}^{a_{1}},\ldots,u_{k}^{a_{k}})_{X}$, for any
integral $n$, and $a_{i}$ such that $n=a_{i}w_{i}$. In general, let
$a=\frac{l}{m}\in{\mathbb{Q}}_{>0}$, with $l,m$ integral, and and let $n$ be
as above with $n=a_{i}w_{i}$. Then, by the definition, and the above
$f\in((u_{1}^{1/w_{1}},\ldots,u_{k}^{1/w_{k}})^{a})_{X}$ if and only if
$f^{mn}\in((u_{1}^{a_{1}},\ldots,u_{k}^{a_{k}})^{l})_{X}={\mathcal{I}}_{\nu_{J},l}$,
which is equivalent to $\nu_{J}(f^{mn})\geq ln\Leftrightarrow\nu_{J}(f)\geq
a.$

For the second part, consider a sufficiently divisible integer $a>0$ such that
$w_{i}:=a/a_{i}$ are integral. Then we can write

$(u_{1}^{a_{1}},\ldots,u_{k}^{a_{k}})=(u_{1}^{1/w_{1}},\ldots,u_{k}^{1/w_{k}})^{a}$

Thus, by the previous part $u^{c}\in(u_{1}^{a_{1}},\ldots,u_{k}^{a_{k}})_{X}$
is equivalent to $c_{1}w_{1}+\ldots c_{k}w_{k}\geq a$, which translates into
$c_{1}/a_{1}+\ldots+c_{k}/a_{k}\geq 1$.

∎

###### Proof.

$(1)\Leftrightarrow(3)$ By Lemma 2.1.12, we can write the Rees algebra
$({\mathcal{O}}_{X}[{\mathcal{J}}t])_{X}$ as
$({\mathcal{O}}_{X}[{\mathcal{J}}t])_{X}=\bigoplus_{a\in{\mathbb{Z}}_{\geq
0}}{\mathcal{I}}_{\nu_{J},a}t^{a}$. Thus the valuation ${\nu_{J}}$ is
determined uniquely. Conversely given a monomial valuation $\nu$ with the
weights $w_{1},\ldots,w_{k}$ one can recover a ${\mathbb{Q}}$-ideal
${\mathcal{J}}$ from $N$-th gradation
$({\mathcal{J}}^{N})_{X}={{\mathcal{I}}_{\nu_{J},N}}$ of
$({\mathcal{O}}_{X}[{\mathcal{J}}t])_{X}$ for $N=w_{1}\cdot\ldots\cdot w_{k}$
as ${\mathcal{J}}=({\mathcal{I}}_{\nu_{J},N})^{1/N}$.

$(2)\Leftrightarrow(3)$ Given a monomial valuation $\nu_{J}$, one can form the
integrally closed algebra

${\mathcal{A}}_{J}:=\bigoplus_{a\in{\mathbb{Z}}_{\geq
0}}{\mathcal{I}}_{\nu_{J},a}t^{a}=\\{f\cdot t^{a}\mid
f\in{\mathcal{O}}_{X},\mid
a\leq{\nu_{J}(f)}\\}={\mathcal{O}}_{X}[u_{1}t^{c_{1}},\ldots,u_{k}t^{c_{k}}\mid
c_{1}\leq w_{1}].$

Conversely, any algebra
${\mathcal{O}}_{X}[u_{1}t^{c_{1}},\ldots,u_{k}t^{c_{k}}\mid c_{1}\leq w_{1}]$
determines a monomial valuation defined as follows: For any
$f\in{\mathcal{O}}_{X}$, we put $\nu_{J}(f)=\max\\{a:t^{a}\cdot
f\in{\mathcal{A}}_{J}\\}$.

∎

####  2.1.13. Blow-ups of ${\mathbb{Q}}$-ideals on $X$

Any ${\mathbb{Q}}$-ideal of the form ${\mathcal{J}}={\mathcal{I}}^{1/n}$ on a
normal scheme $X$ determines a unique blow-up

$Y={{\mathcal{P}}roj}({\mathcal{A}}_{\mathcal{J}})\to X$

of ${\mathcal{J}}$ on $X$ which can be understood as the normalized blow-up of
${\mathcal{I}}$ on $X$. It transforms ${\mathcal{J}}={\mathcal{I}}^{1/n}$ into
a $({\mathcal{O}}_{X}\cdot{\mathcal{I}})^{1/n}=({\mathcal{O}}_{X}(-E))^{1/n}$
for a Cartier exceptional divisor $E$.

####  2.1.14. Weighted blow-ups of centers

In particular, the center
${\mathcal{J}}=(x_{1}^{1/w_{1}},\ldots,x_{k}^{1/w_{k}})$ on a regular scheme
$X$ determines the weighted blow-up

$Y={{\mathcal{P}}roj}({\mathcal{A}}_{\mathcal{J}})={{\mathcal{P}}roj}({{\mathcal{O}}_{X}}[t^{w_{1}}{x_{1}},\ldots,t^{w_{k}}{x_{k}}]^{\rm
int}).$

####  2.1.15. Stack-theoretic weighted blow-ups

The above formulas of the standard blow-up can be natural extended to the
stack-theoretic setting. In [ATW19, Section 3.1] the stack-theoretic blow-up
of any ${\mathbb{Q}}$\- ideal ${\mathcal{J}}$ is defined to be the stack-
theoretic quotient of the scheme (using our notation):

$[(\operatorname{Spec}_{X}({{\mathcal{O}}_{X}[{\mathcal{J}}t])_{X}\smallsetminus
V(({\mathcal{J}}t)_{X}))/G_{m}]},$

with the natural action of the multiplicative group
$G_{m}:=\operatorname{Spec}{k[t,t^{-1}]}$.

In particular, the stack-theoretic weighted blow-up of
${\mathcal{J}}=(u_{1}^{1/w_{1}},\ldots,u_{k}^{1/w_{k}})$ is defined to be the
stack-theoretic quotient

$\displaystyle=[(\operatorname{Spec}_{X}({{\mathcal{O}}_{X}}[t^{w_{1}}{x_{1}},\ldots,t^{w_{k}}{x_{k}}]^{\rm
int})\smallsetminus V(t^{w_{1}}{x_{1}},\ldots,t^{w_{k}}{x_{k}}))/G_{m}]=$
$\displaystyle=$
$\displaystyle[(\operatorname{Spec}_{X}({{\mathcal{O}}_{X}}[t^{a_{1}}{x_{1}},\ldots,t^{a_{k}}{x_{k}}\mid
c_{i}\leq w_{i}])\smallsetminus
V(t^{w_{1}}{x_{1}},\ldots,t^{w_{k}}{x_{k}})/G_{m}].$

Observe that the space
$\operatorname{Spec}_{X}({{\mathcal{O}}_{X}}[t^{a_{1}}{x_{1}},\ldots,t^{a_{k}}{x_{k}}\mid
c_{i}\leq w_{i}])$ is usually non regular even for an ordinary blow-up of a
regular center.

###  2.2. Rational Rees algebra

In this paper we are going to consider Rees algebras on a scheme $X$ with
gradations given by a finitely generated additive subsemigoups $\Gamma$ of
${\mathbb{Q}}_{\geq 0}$.

######  Definition 2.2.1.

By a rational Rees algebra or simply Rees algebra we mean a finitely generated
${\mathcal{O}}_{X}$-algebra which can be written of the form:

$R=\bigoplus_{a\in\Gamma}R_{a}t^{a}\subset{\mathcal{O}}_{X}[t^{1/w_{R}}],$

where $w_{R}\in{\mathbb{Q}}_{>0}$ is the smallest rational number such that
$\Gamma\subseteq(1/w_{R})\cdot{\mathbb{Z}}_{\geq 0}$, and the ideals
$R_{a}\subseteq{\mathcal{O}}_{X}$ satisfy

  1. (1)

$R_{0}={\mathcal{O}}_{X}$

  2. (2)

$R_{a}\cdot R_{b}\subseteq R_{a+b}$

If $R$ is a rational Rees algebra on $X$, and $w$ is a multiple of $w_{R}$
then $R^{{\operatorname{ext}}}:=R[t^{-1/w}]$ will be called an extended Rees
algebra.

By the integral closure $R^{\operatorname{int}}$ of a rational Rees algebra
$R=\bigoplus_{a\in\Gamma}R_{a}t^{a}$ we shall mean its integral closure in
${\mathcal{O}}_{X}[t^{1/w_{R}}]$. The integral closure in
${\mathcal{O}}_{X}[t^{1/w}]$, where $w=nw_{R}$ is a relevant integral multiple
of $w_{R}$ will be denoted by $R^{\operatorname{Int}}$.

By the vertex $V(R)$ of $R$ (or $V(R^{{\operatorname{ext}}})$ of
$R^{{\operatorname{ext}}}$) we mean the vanishing locus

$V(R)=V(\sum_{a>0}R_{a})$

(respectively
$V(R^{{\operatorname{ext}}})=V(\sum_{a>0}R^{\operatorname{ext}}_{a})).$

######  Remark 2.2.2.

We do not assume here that $R_{a}\subseteq R_{b}$ if $a\geq b$ for
$a,b\in\Gamma$. However this condition is satisfied if
$R=R^{\operatorname{int}}$ is integrally closed in
${\mathcal{O}}_{X}[t^{1/w_{R}}]$.

####  2.2.3. Rational Rees algebras and ideals

With any ideal ${\mathcal{I}}$ one associates the ${\mathbb{Z}}$-graded Rees
algebra
${\mathcal{A}}_{\mathcal{I}}:={\mathcal{O}}_{X}[{\mathcal{I}}t]=\bigoplus{\mathcal{I}}^{n}t^{n}$,
and the extended Rees algebra

${\mathcal{A}}_{\mathcal{I}}^{\operatorname{ext}}={\mathcal{O}}_{X}[t^{-1},{\mathcal{I}}t].$

As we mentioned before the main idea of the rational Rees algebras is to
enlarge the gradation so we obtain a simpler presentation of the graded
algebra, and thus a nicer presentation of the blow-up of ${\mathcal{I}}$.

####  2.2.4. Rees centers and ${\mathbb{Q}}$-ideals

By a Rees center we mean a Rees algebra ${\mathcal{A}}$ locally of the form

${\mathcal{A}}={\mathcal{O}}_{X}[x_{1}t^{1/a_{1}},\ldots,x_{k}t^{1/a_{k}}]^{\operatorname{int}},$

for some a local partial system of coordinates( local sytem of parameters)
$x_{1},\ldots,x_{k}$ and some positive rational numbers $a_{1},\ldots,a_{k}$,
where the integral closure is considered in ${\mathcal{O}}_{X}[t^{1/w_{A}}]$,
where $w_{A}:={\operatorname{lcm}}(a_{1},\ldots,a_{k})$ is the smallest
positive rational such that $w_{A}/a_{1},\ldots,{w_{A}/a_{k}}$ are all
integers.

By the extended center which is also called center we shall mean the extended
Rees algebra

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

where $w$ is a multiple of $w_{A}$, so $w/a_{i}$ are all positive integers.

####  2.2.5. Rescaling

######  Definition 2.2.6.

Given $w_{0}\in{\mathbb{Q}}_{>0}$, by the $t\mapsto t^{w_{0}}$ rescaling of
the rational Rees algebra
$R=\bigoplus_{a\in\Gamma}R_{a}t^{a}\subset{\mathcal{O}}_{X}[t^{1/w_{R}}]$ we
mean the Rees algebra

$R^{w_{0}}=\bigoplus_{a\in\Gamma}R_{a}t^{w_{0}a}\subseteq{\mathcal{O}}_{X}[t^{w_{0}/w_{R}}],$

######  Lemma 2.2.7.

There is a natural isomorphism $R\to R^{w_{0}}$ $ft^{a}\mapsto ft^{w_{0}a}$ .
In particular, $R$ is integrally closed in ${\mathcal{O}}_{X}[t^{1/w_{R}}]$
iff $R^{w}$ is integrally closed in ${\mathcal{O}}_{X}[t^{w_{0}/w_{R}}]$. ∎

######  Lemma 2.2.8.

Let
${\mathcal{A}}:={\mathcal{O}}_{X}[\overline{x}_{1}t^{1/a_{1}},\ldots,\overline{x}_{k}t^{1/a_{k}}]^{\operatorname{int}}$
and

${\mathcal{A}}^{\operatorname{ext}}:={\mathcal{O}}_{X}[t^{-1/w},\overline{x}_{1}t^{1/a_{1}},\ldots,\overline{x}_{k}t^{1/a_{k}}],$

where $w$ is a multiple of $w_{A}$. Then the integral closure
${\mathcal{A}}^{\operatorname{Int}}$ of ${\mathcal{A}}$ in
${\mathcal{O}}_{X}[t^{1/w}]$ is equal to
${\mathcal{A}}^{\operatorname{Int}}:={\mathcal{A}}^{\operatorname{ext}}_{\geq
0}$.

Moreover if $V({\mathcal{A}})$ is irreducible then there is a unique monomial
associated valuation $\nu_{A}$ such that $\nu_{A}(x_{i})=1/a_{i}$, and

${\mathcal{A}}_{a}=\\{f\in{\mathcal{O}}_{X}\mid\nu_{A}(f)\geq a\\}.$

###### Proof.

It is a direct consequence of Lemma 2.1.11, obtained by rescaling $t\mapsto
t^{1/w}$.

∎

###  2.3. Regular centers vs Rees centers

One can identify a regular ${\mathbb{Q}}$-ideal center locally described by
${\mathcal{J}}=(u_{1}^{1/w_{1}},\ldots,u_{k}^{1/w_{k}})$ with
$w_{i}\in{\mathbb{N}}$, with the Rees center
${\mathcal{A}}_{\mathcal{J}}={\mathcal{O}}_{X}[u_{1}t^{w_{1}},\ldots,u_{k}t^{w_{k}}]^{\operatorname{int}}$,
or the extended Rees center
${\mathcal{A}}_{\mathcal{J}}^{\operatorname{ext}}={\mathcal{O}}_{X}[t^{-1},u_{1}t^{w_{1}},\ldots,u_{k}t^{w_{k}}].$

Per analogy with the correspondence
${\mathcal{J}}\mapsto{\mathcal{A}}_{\mathcal{J}}$ we associate with arbitrary
${\mathbb{Q}}$-ideal ${\mathcal{J}}=(x_{1}^{a_{1}},\ldots,x_{k}^{a_{k}})$,
with positive rational $a_{i}$ the rational Rees algebra

${\mathcal{A}}={\mathcal{O}}_{X}[x_{1}t^{1/a_{1}},\ldots,x_{k}t^{1/a_{k}}]^{\operatorname{int}},$

and the extended center

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

where $w={\operatorname{lcm}}(a_{1},\ldots,a_{k})$.

Conversely any extended center
${\mathcal{A}}^{\operatorname{ext}}={\mathcal{O}}_{X}[t^{-1/w},x_{1}t^{1/a_{1}},\ldots,x_{k}t^{1/a_{k}}],$
with ${\operatorname{lcm}}(a_{1},\ldots,a_{k})|w$ determines a
${\mathbb{Q}}$-ideal ${\mathcal{J}}=(u_{1}^{a_{1}/w},\ldots,u_{k}^{a_{k}/w})$,
with

${\mathcal{A}}_{\mathcal{J}}=({\mathcal{A}}^{\operatorname{ext}})^{w}={\mathcal{O}}_{X}[t^{-1},x_{1}t^{w/a_{1}},\ldots,x_{k}t^{w/a_{k}}].$

Thus there is a bijective correspondence between

  * •

${\mathbb{Q}}$-ideals centers
${\mathcal{J}}=(u_{1}^{1/w_{1}},\ldots,u_{k}^{1/w_{k}})$ with
$w_{i}\in{\mathbb{N}}$

  * •

the extended integral Rees algebras
${\mathcal{A}}_{{\mathcal{J}}}={\mathcal{O}}_{X}[t^{-1},x_{1}t^{w_{1}},\ldots,x_{k}t^{w_{k}}],$
and

  * •

the extended rational Rees algebras
${\mathcal{A}}^{\operatorname{ext}}={\mathcal{O}}_{X}[t^{-1/w},x_{1}t^{1/a_{1}},\ldots,x_{k}t^{1/a_{k}}],$
with $w_{i}=w/a_{i}\in{\mathbb{N}}$, defined up to rescaling.

###  2.4. Cobordant blow-ups

####  2.4.1. Good and geometric quotient

We consider here a relatively affine action of
$T=\operatorname{Spec}({\mathbb{Z}}[t_{1},t_{1}^{-1},\ldots,t_{k},t_{k}^{-1}])$
on a scheme $X$ over ${\mathbb{Z}}$. By the good quotient (or GIT quotient) we
mean an affine $T$-invariant morphism $\pi:X\to Y=X\sslash T$ such that the
induced morphism of the sheaves
${\mathcal{O}}_{Y}\to\pi_{*}({\mathcal{O}}_{X})$ defines the isomorphism onto
the subsheaf of invariants
${\mathcal{O}}_{Y}\simeq\pi_{*}({\mathcal{O}}_{X})^{T}$.

Then $\pi:X\to Y=X/T$ will be called a geometric quotient if additionally
every fiber $X_{\overline{y}}$ of $\pi$ over a geometric point
$\overline{y}:\operatorname{Spec}(\overline{k})\to Y$ defines a single orbit
of the action of
$T_{\overline{k}}=T\times_{k}\operatorname{Spec}(\overline{k})=\operatorname{Spec}(\overline{k}[t_{1},t_{1}^{-1},\ldots,t_{k},t_{k}^{-1}])$
on $X_{\overline{y}}$.

####  2.4.2. Birational cobordisms

Since the space
$\operatorname{Spec}_{X}({{\mathcal{O}}_{X}}[t^{w_{1}}{x_{1}},\ldots,t^{w_{k}}{x_{k}}]^{\rm
int})=\\\ \operatorname{Spec}({\mathcal{A}}_{J})$ in the definition of the
stack-theoretic quotient is singular even for the standard smooth blow-up with
weights $w_{i}=1$ we consider here an alternative approach which uses the
concept of birational cobordisms. The notion was developed over a field, but
it can be extended to the schemes with the action of the group scheme
$G_{m}:=\operatorname{Spec}({\mathbb{Z}}[t,t^{-1}]$. One shall mention that
the theory of cobordisms was also considered in the language of schemes in
[AT19, Definition 4.3], but their definition of cobordism is far more
restrictive.

The following definition is motivated for the analogous definition for the
varieties with $G_{m}$-action and is adapted to the schemes.

######  Definition 2.4.3.

Let $X$ be an integral scheme with the action of $G_{m}$. Let $p\in X$ be a
point. We say that $\lim_{t\to 0}(tp)$ exists, (respectively
$\lim_{t\to\infty}(tp)$ exists) if there is an open neighborhood $U$ of $p$
such that the morphism $G_{m}\times U\to X$ extends to the
$\operatorname{Spec}({\mathbb{Z}}[t])\times U\to X$ (respectively to
$\operatorname{Spec}({\mathbb{Z}}[t^{-1}])\times U\to X$).

The following definition extends the original definition over a field.

######  Definition 2.4.4.

[Wło00, Definition 2] By a birational cobordism representing a birational map
$\phi:X_{1}\dashrightarrow X_{2}$ of schemes we mean a scheme with an action
of $T=G_{m}=\operatorname{Spec}({\mathbb{Z}}[t,t^{-1}])$ such that the sets

$\begin{array}[]{rcrl}B_{-}&:=&\\{p\in B:&\lim_{t\to 0\ }(tp)\mbox{ does not
exist in }B\\}\\\ \mbox{and }B_{+}&:=&\\{x\in B:&\lim_{t\to\infty}(tp)\mbox{
does not exist in }B\\}\end{array}$

are nonempty Zariski open subsets of $B$ , and the geometric quotients
$\alpha_{1}:B_{+}/T\simeq X_{1}$, and $\alpha_{2}:B_{-}/T\simeq X_{2}$ exists,
with the natural birational map $\psi:B_{+}/T\to B_{-}/T$ defined by open
inclusions maps $(B_{-}\cap B_{+})/T\to B_{\pm}/T$, such that

$\phi\circ\alpha_{1}=\alpha_{2}\circ\psi.$

######  Example 2.4.5.

[Wło00, Example 2] Let $T$ act on
$B={\mathbf{A}}_{k}^{n+1}=\operatorname{Spec}(k[x_{0},\ldots,x_{n}])$ by
$t(x_{0},x_{1},\ldots,x_{n})=(t^{-1}x_{0},t^{w_{1}}x_{1},\ldots,t^{w_{k}}x_{k})$,
where $w_{1}\geq w_{2}\geq\ldots\geq w_{k}>0$. Then $B_{-}=B\smallsetminus
V(x_{0})$, $B_{+}=B\smallsetminus V(x_{1},\ldots,x_{n})$.

By using toric geometry, we will see that the induced morphism $B_{+}/T\to
B_{-}/T=B\sslash T$ is the weighted blow-up at
${\mathcal{J}}=(u_{1}^{1/w_{1}},\ldots,u_{k}^{1/w_{k}})$.

First, the cobordism $B$ corresponds to the regular cone

$\sigma=\langle e_{0},\ldots,e_{n}\rangle:=\sum_{i=0}^{n}{\mathbb{Q}}e_{i}$

in ${\mathbb{Q}}^{n+1}$ with the basis $\\{e_{0},\ldots,e_{n}\\}$. The action
of $T$ is determined by the vector $v:=-e_{0}+\sum w_{i}e_{i}$, representing
the weights. Then the good quotient

$B\to B\sslash T=\operatorname{Spec}({\mathcal{O}}(B)^{T})$

corresponds to the map $\sigma\to\pi(\sigma)=\langle
e_{1},\ldots,e_{n}\rangle$, defined by the projection

$\pi:{\mathbb{Q}}^{n+1}\to{\mathbb{Q}}^{n+1}/{\rm
span}(v)\simeq{\mathbb{Q}}^{n}$

where ${\mathbb{Q}}^{n}$ is spanned by $\\{e_{1},\ldots,e_{n}\\}$. It can be
described as $\pi(x_{0},x_{1},\ldots,x_{n})=(x_{1}+w_{1},\ldots,x_{n}+w_{n})$.
Note that face $\sigma_{0}:=\langle e_{1},\ldots,e_{n}\rangle$ of $\sigma$ can
be identified with its image $\pi(\sigma_{0})=\pi(\sigma)$.

The cone $\sigma$ with a vector $v$ and the projection $\pi$ is an example of
the combinatorial cobordism in the sense of Morelli [Mor96]. The ”lower
boundary” $B_{-}$ of $B$ corresponds to the ”lower boundary” $\sigma_{-}$ of
$\sigma$, which is the complex consisting of the faces of $\sigma$ visible
from below with respect to $v$. It contains $\sigma_{0}$ and all of its faces.
The geometric quotient $B_{-}\to B_{-}/T$ corresponds to the restriction
$\sigma_{0}\to\pi(\sigma_{0})=\pi(\sigma)$ of
$\pi:{\mathbb{Q}}^{n+1}\to{\mathbb{Q}}^{n}$ mapping isomorphically
$\sigma_{0}$ to $\pi(\sigma_{0})$. So both quotients: the geometric quotient
$B_{-}/T$ and the good quotient $B\sslash
T=\operatorname{Spec}({\mathcal{O}}(B))^{T}$ can be identified as they
correspond to $\pi(\sigma_{0})=\pi(\sigma)=\sigma_{0}\subset{\mathbb{Q}}^{n}$.

The ”upper boundary” $B_{+}$ corresponds to the subcomplex $\sigma_{+}$ with
the maximal faces $\langle
e_{0},e_{1},\ldots,\check{e_{i}},\ldots,e_{n}\rangle$, for $i=1,\ldots,n$,
”visible from above”. Their projection $\pi(\sigma_{+})$ corresponds to the
star subdivision of $\sigma_{0}$ at the vector
$w:=\pi(e_{0})=(w_{1},\ldots,w_{n})$ with maximal faces $\langle
w,e_{1},\ldots,\check{e_{i}},\ldots,e_{n}\rangle$.

Consequently, the morphism

$B_{+}/T\to B_{-}/T=B\sslash
T=\operatorname{Spec}(k[x_{0},\ldots,x_{k}])^{T}=\operatorname{Spec}(k[u_{1},\ldots,u_{k}]),$

where $u_{i}:=x_{i}x_{0}^{w_{i}}$, is the weighted blow-up corresponding to
the star subdivision $\pi(\sigma_{+})$ of $\pi(\sigma)=\sigma_{0}$. The
${\mathbb{Q}}$-ideal ${\mathcal{J}}=(u_{1}^{1/w_{1}},\ldots,u_{k}^{1/w_{k}})$
defines a piecewise linear convex function

$F_{\mathcal{I}}:=\min\\{(1/w_{i})\cdot x_{i}\mid i=1,\ldots,n\\}$

on $\sigma_{0}$ such that $F_{\mathcal{I}}(e_{i})=0$, and
$F_{\mathcal{I}}(w)=1$, and which is linear exactly on the faces $\langle
w,e_{1},\ldots,\check{e_{i}},\ldots,e_{n}\rangle$. Thus we see that the blow-
up of ${\mathcal{J}}=(u_{1}^{1/w_{1}},\ldots,u_{k}^{1/w_{k}})$ corresponds to
the star subdivision of $\sigma$ at $v$, and can be identified with
$B_{+}/T\to B_{-}/T=B\sslash T$.

One can describe the morphism $B_{+}/T\to B_{-}/T=B\sslash T$ explicitly. Set
$t:=x_{0}^{-1}$. Then we can write

${\mathcal{B}}_{J}:={\mathcal{O}}(B)=k[x_{0},x_{1},\ldots,x_{k}]=k[t^{-1},t^{w_{1}}u_{1},\ldots,t^{w_{k}}u_{k}]\quad\mbox{and}$
$\displaystyle
B_{+}/T=(\operatorname{Spec}({\mathcal{B}}_{\mathcal{J}})\smallsetminus
V(t^{w_{1}}u_{1},\ldots,t^{w_{k}}u_{k}))/T\to$ $\displaystyle\to\quad
B/T=B_{-}/T=\operatorname{Spec}(k[u_{1},\ldots,u_{k}]).$

Moreover the algebra

${\mathcal{A}}_{J}=(k[u_{1},\ldots,u_{k}])[T^{1/a_{1}}u_{1},\ldots,t^{a_{k}}u_{k}\mid
a_{i}\leq w_{i}]=(k[t^{-1},t^{w_{1}}u_{1},\ldots,t^{w_{k}}u_{k}])_{\geq 0}$

is exactly the nonnegative part of the algebra $B_{J}$. Thus the natural
homomorphism $A_{J}\subset B_{J}$ defines the isomorphisms of the
localizations $(A_{J})_{t^{w_{i}}u_{i}}\to(B_{J})_{t^{w_{i}}u_{i}}$ which
gives the isomorphism of quasi-affine schemes

$\operatorname{Spec}(B_{J})\smallsetminus
V(t^{w_{1}}u_{1},\ldots,t^{w_{k}}u_{k})\to\operatorname{Spec}(A_{J})\smallsetminus
V(t^{w_{1}}u_{1},\ldots,t^{w_{k}}u_{k}),$

and their quotients $B_{+}/T\to Proj_{X}({\mathcal{A}}_{\mathcal{J}})$ over
$B/T$.

####  2.4.6. Cobordant blow-ups

The idea of cobordant blow-up is to represent a weighted blow-up by a smooth
cobordism.

Let $T:={\mathbb{G}}_{m}=Spec({\mathbb{Z}}[t,t^{-1}])$.

######  Definition 2.4.7.

Let $X$ be a scheme. A trivial cobordant blow-up of $X$ is simply the natural
projection $X\times T\to X$ from the product $X\times
T=\operatorname{Spec}{\mathcal{O}}_{X}[t,t^{-1}]$.

By the full cobordant blow-up of $X$ at an extended Rees center

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

we mean the $T$-invariant morphism $\sigma:B\to X$ defined by the birational
cobordism

$B:=\operatorname{Spec}_{X}(({\mathcal{A}}^{\operatorname{ext}})^{w})=\operatorname{Spec}_{X}({{\mathcal{O}}_{X}}[t^{-1},t^{w_{1}}{x_{1}},\ldots,t^{w_{k}}{x_{k}}]),$

where $w={\operatorname{lcm}}(a_{1},\ldots,a_{k})$, and $w_{i}=w/a_{i}$.

The upper and lower boundaries are equal to

$B_{+}=B\smallsetminus V(({\mathcal{A}}^{\operatorname{ext}}))=B\smallsetminus
V(t^{w_{1}}{x_{1}},\ldots,t^{w_{k}}{x_{k}})=B\smallsetminus{\operatorname{Vert}}(B)$
$B_{-}=B\smallsetminus
V(t^{-1})=\operatorname{Spec}_{X}({\mathcal{O}}_{X}[t,t^{-1}]).$

By the cobordant blow-up we mean the $T$-invariant morphism
$\sigma_{+}:B_{+}\to X$. Here $\sigma_{-}:B_{-}\to X$ is the trivial cobordant
blow-up.

Equivalently $B$, $B_{+}$ will be called full cobordant blow-up (resp.
cobordant blow-up) of a ${\mathbb{Q}}$-ideal center
${\mathcal{J}}=(x_{1}^{1/w_{1}},\ldots,x_{k}^{1/w_{k}})$.

We shall call the closed subscheme

$V={\operatorname{Vert}}(B):=V(t^{w_{1}}{x_{1}},\ldots,t^{w_{k}}{x_{k}})=V(({\mathcal{A}}^{\operatorname{ext}})^{w})=V({\mathcal{O}}_{B}\cdot{\mathcal{J}})=B\smallsetminus
B_{+}$

the vertex of $B$ (per analogy to the vertex of the affine cone over a
projective scheme.)

Thus, as we shall see $B$ is in fact the birational cobordism representing the
weighted blow-up of ${\mathcal{J}}$, with $B_{+}=B\smallsetminus V$, and
$B_{-}=B\smallsetminus D=B\times G_{m}$, where $D:=V_{B}(t^{-1})$ is the
exceptional divisor and such that

$B_{+}/G_{m}=B\sslash
G_{m}={{\mathcal{P}}roj}({\mathcal{O}}_{X}[t^{w_{1}}{x_{1}},\ldots,t^{w_{k}}{x_{k}}]^{\operatorname{int}}),$

and $B_{-}/G_{m}=X$.

######  Remark 2.4.8.

The author learned recently from Dan Abramovich that the algebras
${{\mathcal{O}}_{X}}[t^{-1},t^{w_{1}}{x_{1}},\ldots,t^{w_{k}}{x_{k}}]$ were
also considered in the context of stack-theoretic blow-ups by Quek and Rydh in
their upcoming paper [QR22]. In fact, Rydh noticed independently that such
algebras gives a nice and natural description of stack-theoretic weighted
blow-ups, which was part of his original goal for [QR22]. On the other hand,
cobordant blow-ups are a different point of view on the same algebra, and they
are motivated by the interpretation of the weighted blow-ups in terms of torus
action and smooth birational cobordisms to avoid the use of the Artin stacks
in positive and mixed characteristic. The cobordant blow-ups and the full
cobordant blow-ups carry additional useful information and are often simpler
for computations. One shall mention that all these ideas stem primarily from
the paper [ATW19].

On the other hand, one can link this definition to the extended Rees algebras
${\mathcal{O}}_{X}[t^{-1},{\mathcal{I}}t]$ introduced by Rees and considered
by Swanson and Huneke in [HS06, Definition 5.1], in particular, in the context
of blow-ups of smooth centers.

######  Remark 2.4.9.

  1. (1)

By Lemma 5.2.6, the geometric quotient $B_{+}/{\mathbb{G}}_{m}\to X$ exists.
Moreover, by recalling the considerations in the second part of Example 2.4.5,
it can be naturally identified with the usual weighted blow-up:

$\displaystyle
B_{+}/{\mathbb{G}}_{m}\simeq\operatorname{Spec}({\mathcal{O}}_{X}[t^{-1},t^{w_{1}}{x_{1}},\ldots,t^{w_{k}}{x_{k}}]\smallsetminus
V(t^{w_{1}}{x_{1}},\ldots,t^{w_{k}}{x_{k}})))/T\simeq$
$\displaystyle{{\mathcal{P}}roj}({\mathcal{A}}_{{\mathcal{J}}})\simeq({{\mathcal{P}}roj}({\mathcal{O}}_{X}[t^{w_{1}}{x_{1}},\ldots,t^{w_{k}}{x_{k}}]))^{nor}\to
X=B\sslash{\mathbb{G}}_{m}=B_{-}/{\mathbb{G}}_{m}.$

  2. (2)

The stack-theoretic quotient of smooth varieties over a field $k$ of
characteristic zero $[B_{+}/{\mathbb{G}}_{m}]\to X$ is equivalent to the
definition considered in [ATW19, Section 3.1] .

  3. (3)

Similarly to the standard blow-up, the cobordant blow-up is trivial over the
complement of the geometric locus of the center
$V({\mathcal{J}})=V(x_{1}\ldots,x_{k})$:

$\sigma_{\pm}:B_{+}\cap B_{-}=\sigma^{-1}(X\smallsetminus
V({\mathcal{J}}))\simeq(X\smallsetminus V({\mathcal{J}}))\times T\to
X\smallsetminus V({\mathcal{J}}).$

  4. (4)

The vertex $V={\operatorname{Vert}}(B)$ plays a crucial role in the resolution
process. It typically represents the locus of points on $B$ that exhibit the
most severe singularities. The singularities at ${\operatorname{Vert}}(B)$
usually coincide with the singularities along the center $\mathcal{J}$ and are
eliminated when transitioning to $B_{+}$.

###  2.5. Exceptional divisor

The following result explains the definition of the exceptional divisor
$D:=V_{B}(t^{-1})$ in the language of ${\mathbb{Q}}$-ideals:

######  Lemma 2.5.1.

The cobordant blow-up $B_{+}$ transforms the center
${{\mathcal{J}}}=(x_{1}^{\frac{1}{w_{1}}},\ldots,x_{k}^{\frac{1}{w_{k}}})$
into the (ordinary) ideal of the exceptional divisor $D$ on $B_{+}$ generated
by a global invariant parameter $t^{-1}$:

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

###### Proof.

We have

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

Note however that

$({\mathcal{J}}\cdot
t)=((x_{1}t^{w_{1}})^{1/w_{1}},\ldots,(x_{k}t^{w_{k}})^{1/w_{k}})={\mathcal{O}}_{B_{+}}$

is a trivial ${\mathbb{Q}}$-ideal on $B_{+}$. ∎

####  2.5.2. Cobordant blow-up: local equations

Let $x_{1},\ldots,x_{n}$ be a system of local parameters (or coordinates) at a
point $p$ on a regular $X$, and let
${\mathcal{J}}=(x_{1}^{1/w_{1}},\ldots,x_{k}^{1/w_{k}})$ be a center. Then the
full cobordant blow-up $B\to X$ of ${\mathcal{J}}$, can be represented as

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

Thus $B\subset X\times{\mathbb{A}}^{n+1}$ is locally a closed regular
subscheme of $X\times{\mathbb{A}}^{n+1}$. Moreover it has system of local
$T$-semiinvariant parameters
$t^{-1},x^{\prime}_{1},\ldots,x^{\prime}_{k},x_{k+1},\ldots,x_{n}$ at the
relevant $p^{\prime}\in
V(t^{-1},x^{\prime}_{1},\ldots,x^{\prime}_{k},x_{k+1}\ldots,x_{n})$ mapping to
$p$. Consequently, the full cobordant blow-up $B\to X$ at
$(x_{1}^{1/w_{1}},\ldots,x_{k}^{1/w_{k}})$ can be described by a single chart
with the following coordinates:

  * •

$t^{-1}$ is the inverse of the coordinate $t$ representing the action of torus
${\mathbb{G}}_{m}=\operatorname{Spec}({\mathbb{Z}}[t,t^{-1}])$.

  * •

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

  * •

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

######  Remark 2.5.3.

For some computations and considerations it is more natural to replace
$t^{-1}$ with $s$ so that $x_{i}^{\prime}=x_{i}/s^{w_{i}}$ in the above
formulas. On the other hand the construction of the center and the majority of
the computations are consistent with the graded Rees algebras and thus with
our notation.

The cobordant blow-up $B_{+}=B\smallsetminus
V(x^{\prime}_{1},\ldots,x^{\prime}_{k})$ can be covered by the open subsets
$(B_{+})_{x^{\prime}_{i}}=B\smallsetminus V(x^{\prime}_{i})$, associated with
$x^{\prime}_{i}$ producing several more specific ”charts” similarly to the
standard blow-up.

####  2.5.4. Cobordant blow-ups of toric varieties

By definition, the full cobordant blow-up of a smooth toric variety at a
weighted toric center is again a smooth toric variety. Let
$X_{\sigma}=\operatorname{Spec}k[x_{1},\ldots,x_{n}]$ be the affine toric
variety associated with $\sigma\subset N_{R}:={\mathbb{R}}_{\geq 0}^{n}$. The
full cobordant blow-up $B=X_{\tau}\to X_{\sigma}$ of the ${\mathbb{Q}}$ ideal
$(x_{1}^{1/w_{1}},\ldots,x_{n}^{1/w_{n}})$ is the morphism of toric varieties
corresponding to the inclusion of algebras:

$k[x_{1},\ldots,x_{n}]\subseteq
k[x_{1}t^{w_{1}},\ldots,x_{n}t^{w_{n}},t^{-1}]$

The standard weighted blow-up at ${\mathcal{J}}$ corresponds to the star
subdivision of $\sigma$ with the center $v:=w_{1}e_{1}+\ldots+w_{n}e_{n}$. The
full cobordant blow-up $B$ is described by the regular cone $\tau:=\langle
e_{1},\ldots,e_{n},v+e_{n+1}\rangle$ with the natural projection onto
$\sigma=\langle e_{1},\ldots,e_{n}\rangle$ along the vector $e_{n+1}$. The
cone $\tau$ is a cobordism in the sense of Morelli [Mor96]. The upper boundary
$\tau_{+}$ of $\tau$ has maximal faces $\langle
e_{1},\ldots,e_{i}^{\vee},\ldots,e_{n},v+e_{n+1}\rangle$ which are ”visible
from above” and which project exactly to the star subdivision of $\sigma$ at
at $v$. The lower boundary $\tau_{-}$ is isomorphic to $\sigma$.

The above construction extends to any smooth toric variety $X=X_{\Sigma}$, and
$v\in{\rm int}(\sigma)$, $\sigma\in\Sigma$. The fan $\Sigma_{B}$ associated
with $B$ is the union of cones in $\Sigma$ and those spanned by the vector
$v+e_{n+1}$ and the cones in the closed star
$\overline{\operatorname{Star}(\tau,\Sigma)}$.

##  3\. Resolution of singularities in characteristic zero by weighted centers

###  3.1. Resolution invariant on regular schemes with SNC divisors

####  3.1.1. Compatibility with SNC divisors

######  Definition 3.1.2.

A coordinate system (or a partial system of local parameters)
$x_{1},\ldots,x_{n}$ on an open subset $U$ of a regular scheme $X$ is
compatible with $E$ if the restriction $E_{i|U}$ of any component $E_{i}$ in
$E$ intersecting $U$ is of the form $V(x_{i})$. If $p\in V(x_{i})$, where
$V(x_{i})$ determines a component in $E_{|U}$ then we say that $x_{i}$ is
divisorial at $p$. Otherwise, we say that $x_{i}$ is free at $p$. We say that
a Rees center ${\mathcal{A}}$ is is compatible with $E$ at $p$ or on $U$ if it
can be written as

${\mathcal{A}}={\mathcal{O}}_{X}[x_{1}t^{1/a_{1}},\ldots,x_{k}t^{1/a_{k}}]^{\operatorname{int}}\,\,\mbox{
or}\quad{\mathcal{A}}^{\operatorname{ext}}={\mathcal{O}}_{X}[t^{-1/w},x_{1}t^{1/a_{1}},\ldots,x_{k}t^{1/a_{k}}]$

where $x_{1},\ldots,x_{k}$ is compatible with $E$ at $p$ or respectively on
$U$.

######  Remark 3.1.3.

This definition can be linked to the logarithmic language. The coordinate
system defines at every point $p\in U$ a regular map

$U\to\operatorname{Spec}({\mathbb{Z}}[x_{1},\ldots,x_{n}])=\operatorname{Spec}({\mathbb{Z}}[P]),$

where the divisorial coordinates $x_{i}$ at $p$ generate a monoid
$P=\overline{{\mathcal{M}}}_{p}$, and induce the chart
$U\to\operatorname{Spec}({\mathbb{Z}}[P])$. The free coordinates $x_{j}$
determine a coordinate system on the stratum $s:=V(P\smallsetminus 0)$ through
$p$ defined by the divisorial parameters.

####  3.1.4. The centers compatible with an SNC divisor

Consider the total order on the set of symbols

${\mathbb{Q}}_{+}:={\mathbb{Q}}\,\amalg\,\\{a_{+}\mid a\in{\mathbb{Q}}\\},$

by putting $a_{+}>a$, and $a_{+}<b$ if $a<b$. Similarly we define
${\mathbb{Z}}_{+}:={\mathbb{Z}}\cup\\{a_{+}\mid a\in{\mathbb{Z}}\\}$. The
following operations are defined on ${\mathbb{Q}}_{+}$:

  * •

addition

  * •

subtraction of the elements in ${\mathbb{Q}}$ from an element in
${\mathbb{Q}}_{+}$.

  * •

multiplication of elements in ${\mathbb{Q}}_{+}$ by the positive rational
elements in ${\mathbb{Q}}_{>0}$.

Given a center ${\mathcal{A}}$ compatible with $E$ and its local presentation

${\mathcal{A}}={\mathcal{O}}_{X}[x_{1}t^{1/a_{1}},\ldots,x_{k}t^{1/a_{k}}]^{\operatorname{int}}$

we associate with a divisorial coordinate $x_{i}\in E_{p}$ the symbol
$b_{i}:=a_{i+}$ and with a free coordinate $x_{i}$ simply $b_{i}:=a_{i}$. We
shall always assume that in the presentation:

$b_{1}\leq\ldots\leq b_{k}.$

For any such a center with the presentation ${\mathcal{A}}$ compatible with
$E$ we can define

${\operatorname{inv}}({\mathcal{A}}):=(b_{1},\ldots,b_{k})$

with $b_{1}\leq\ldots\leq b_{k}$.

######  Remark 3.1.5.

Recall that the logarithmic order introduced in [ATW17, Section 3.6]
associates with divisorial unknowns the infinity weight, and with free
unknowns the weight $1$. This language is compatible with the logarithmic
derivations and leads to the logarithmic resolution. Here we assign the weight
$1_{+}$ to the divisorial unknowns. This is the ”minimal weight” greater than
the free unknowns weight, which is still equal to $1$. The notion is
compatible with the standard derivations and the standard transformation rules
and gives an SNC resolution. It prioritizes the divisorial unknowns without
essentially changing their weights.

####  3.1.6. Canonical invariant

In [ATW19] we introduced the resolution invariant used for the
desingularization. The invariant is defined by the inductive procedure, and
can be described in terms of ${\mathbb{Q}}$-ideals. We shall modify this
invariant below so it can be used in the SNC setting in the context of Rees
algebra.

Consider the set

$(({\mathbb{Q}}_{+})_{\geq 0})^{\leq n}\ \ :=\ \ \bigsqcup_{k\leq
n}(({\mathbb{Q}}_{+})_{\geq 0})^{k}.$

with lexicographic order. We compare the sequences of different lengths
lexicographically by placing a sequence of $\infty$ at the end.

######  Definition 3.1.7.

Let $X$ be a regular scheme with an SNC divisors $E$, By the canonical
invariant ${\operatorname{inv}}_{p}({\mathcal{I}})$ of ${\mathcal{I}}$ at a
point $p$ we shall mean

${\operatorname{inv}}_{p}({\mathcal{I}}):=\max\\{(b_{1},\ldots,b_{k})\mid{\mathcal{I}}t\subset{\mathcal{O}}_{X}[{x}_{1}t^{1/a_{1}},\ldots,{x}_{k}t^{1/a_{k}}]^{{\operatorname{int}}}\mid(b_{1}\leq\ldots\leq
b_{k})\\}.$

where $(x_{1},\ldots,x_{k})$ are the compatible with $E$ at $p$. Then
${\operatorname{inv}}_{p}({\mathcal{I}})\in({\mathbb{Q}}^{+}_{\geq 0})^{\leq
n}\ $ if it exists. The Rees center

${\mathcal{A}}={\mathcal{O}}_{X}[{x}_{1}t^{1/a_{1}},\ldots,{x}_{k}t^{1/a_{k}}]^{{\operatorname{int}}}$

such that
${\mathcal{I}}_{p}\subseteq{\mathcal{O}}_{X}[{x}_{1}t^{1/a_{1}},\ldots,{x}_{k}t^{1/a_{k}}]^{{\operatorname{int}}}$
and If
${\operatorname{inv}}_{p}({\mathcal{I}})={\operatorname{inv}}({\mathcal{A}})=(b_{1},\ldots,b_{k})$
will be called a maximal admissible center at $p$ compatible with $E$.

######  Remark 3.1.8.

It is not clear a priori, that ${\operatorname{inv}}_{p}({\mathcal{I}})$ is
well defined since the maximum may not be attained. It will be proven in
Section 3.7.1 that there exists unique maximal admissible algebra
${\mathcal{O}}_{X}[\overline{x}_{1}t^{1/a_{1}},\ldots,\overline{x}_{k}t^{1/a_{k}}]^{{\operatorname{int}}}$
for which the maximum $(\overline{b}_{1},\ldots,\overline{b}_{k})$ is
attained.

The invariant ${\operatorname{inv}}_{p}({\mathcal{I}})$ was introduced in
[ATW19] in a different language in the case where no divisors are present
giving the embedded desingularization without the divisorial SNC structure.

On the other hand ${\operatorname{inv}}_{p}({\mathcal{I}})$ can be also
associated with the part of the Hironaka resolution invariant developed in
earlier papers on the resolution of singularities, in particular, Bierstone
and Milman’s [BM97], Villamayor’s [Vil89], Encinas-Hauser [EH02] and
Włodarczyk’s [Wło05], in the particular situation when no exceptional divisors
are present, so at the beginning of the process.

####  3.1.9. Presentation of centers

One can write centers in a more compact form:

${\mathcal{O}}_{X}[\overline{x}_{1}t^{1/a_{1}},\ldots,\overline{x}_{k}t^{1/a_{k}}]^{{\operatorname{int}}},$

where $0<a_{1}<a_{2}\ldots<a_{k}$, and
$\overline{x}_{1},\ldots,\overline{x}_{k}$ is a partial system of local
parameters compatible with $E$ on open $U$ intersecting
$V(\overline{x}_{1},\ldots,\overline{x}_{k})$ and each
$\overline{x}_{i}:=(x_{i1},\ldots,x_{ik_{i}})$ is a subsystem of coordinates.

Then we associate with each block $\overline{x}_{i}t^{1/a_{i}}$ the invariant
$\overline{b_{i}}=(b_{i1},\ldots,b_{ik_{i}})$, where $b_{i1}\leq\ldots\leq
b_{ik_{i}}$, such that $b_{ij}=a_{i}$ if $x_{ij}$ is free on $U$ or
$b_{ij}=a_{i+}$ if $x_{ij}$ is divisorial on $U$. Consequently we can rewrite
the definition of the resolution invariant in the form which is more
convenient for computations and presentation.

${\operatorname{inv}}_{p}({\mathcal{I}})=\max\\{(\overline{b}_{1},\ldots,\overline{b}_{k})\mid{\mathcal{I}}t\subset{\mathcal{O}}_{X}[\overline{x}_{1}t^{1/a_{1}},\ldots,\overline{x}_{k}t^{1/a_{k}}]^{{\operatorname{int}}}\\}$

####  3.1.10. Order of ideal revisited

One can see that the number $a_{1}$ in the maximal admissible center is simply
the order of ideal $a_{1}={\operatorname{ord}}_{p}({\mathcal{I}})$. In
particular we have:

######  Lemma 3.1.11.

${\operatorname{ord}}_{p}({\mathcal{I}})=\max\\{a_{1}\in{\mathbb{Q}}_{>0}\mid\quad{\mathcal{I}}t\subset{\mathcal{O}}_{X}[m_{p}t^{1/a_{1}}]\\}$

###### Proof.

We can write:

${\operatorname{ord}}_{p}({\mathcal{I}})=\max\\{a_{1}\in{\mathbb{N}}\mid{\mathcal{I}}_{p}\subset
m_{p}^{a_{1}}\\}=\max\\{a_{1}\in{\mathbb{Q}}_{>0}\mid{\mathcal{I}}_{p}\subset
m_{p}^{\lceil a_{1}\rceil}\\}$

On the other hand the condition by Lemma 2.2.8,
${\mathcal{I}}_{p}t\subset{\mathcal{O}}_{X}[m_{p}t^{1/a_{1}}]$ means that
${\mathcal{I}}_{p}\subset m_{p}^{\lceil a_{1}\rceil}$. ∎

####  3.1.12. Admissibility condition for ideals

######  Lemma 3.1.13.

The following conditions are equivalent in a neighborhood of $p\in X$ for an
ideal ${\mathcal{I}}$:

  1. (1)

${\mathcal{I}}t\subset{\mathcal{A}}:={\mathcal{O}}_{X}[x_{1}t^{1/a_{1}},\ldots,x_{k}t^{1/a_{k}}]^{\operatorname{int}}$

  2. (2)

${\mathcal{I}}t\subset{\mathcal{A}}^{{\operatorname{ext}}}:={\mathcal{O}}_{X}[t^{-1/w_{A}},x_{1}t^{1/a_{1}},\ldots,x_{k}t^{1/a_{k}}]$

  3. (3)

${\mathcal{I}}t^{w_{A}}\subset{\mathcal{A}}^{w_{A}}={\mathcal{O}}_{X}[x_{1}t^{w_{1}},\ldots,x_{k}t^{w_{k}}]^{\operatorname{int}}$,
where $w_{i}=w_{A}/a_{i}$

  4. (4)

${\mathcal{I}}t^{w_{A}}\subset({\mathcal{A}}^{\operatorname{ext}})^{w_{A}}={\mathcal{O}}_{X}[t^{-1},x_{1}t^{w_{1}},\ldots,x_{k}t^{w_{k}}]$

###### Proof.

The condition (3) and (4) are obtained by rescaling the conditions in (1) and
(2). On the other hand, by Lemma 2.1.11,

$({\mathcal{A}}^{{\operatorname{ext}}})_{\geq
0}={\mathcal{O}}_{X}[t^{-1},x_{1}t^{w_{1}},\ldots,x_{k}t^{w_{k}}]_{\geq
0}={\mathcal{O}}_{X}[x_{1}t^{w_{1}},\ldots,x_{k}t^{w_{k}}]^{\operatorname{int}}={\mathcal{A}}.$

This implies that condition (1) and (2) are equivalent. ∎

######  Definition 3.1.14.

We say that the center
${\mathcal{A}}:={\mathcal{O}}_{X}[x_{1}t^{1/a_{1}},\ldots,x_{k}t^{1/a_{k}}]^{\operatorname{int}}$
or ${\mathcal{A}}^{{\operatorname{ext}}}$ is admissible for ${\mathcal{I}}$ if
one of the four equivalent conditions above holds.

####  3.1.15. Admissibility of Rees algebras

We extend the admissibility definition to Rees algebras:

######  Definition 3.1.16.

We say that the Rees center
${\mathcal{A}}={\mathcal{O}}_{X}[x_{1}t^{w_{1}},\ldots,x_{k}t^{w_{k}}]^{\operatorname{int}}$
is admissible for a Rees algebra $R$ or simply $R$-admissible if one of the
equivalent condition holds:

  1. (1)

$R\subseteq{\mathcal{O}}_{X}[x_{1}t^{1/a_{1}},\ldots,x_{k}t^{1/a_{k}}]^{\operatorname{Int}},$
where the inclusion is considered with the integral closure ”Int” taken in the
smallest ring ${\mathcal{O}}_{X}[t^{1/w_{R,A}}]$ containing both algebras, or,
equivalently in ${\mathcal{O}}_{X}[t^{1/w}]$ for any multiple $w=n\cdot
w_{R,A}$ or for a sufficiently divisible $w$.

  2. (2)

$R\subset{\mathcal{O}}_{X}[t^{-1/w_{R,A}},x_{1}t^{1/a_{1}},\ldots,x_{k}t^{1/a_{k}}].$

  3. (3)

$R\subset{\mathcal{O}}_{X}[t^{-1/w},x_{1}t^{1/a_{1}},\ldots,x_{k}t^{1/a_{k}}],$
where $w=n\cdot w_{R,A}$.

######  Remark 3.1.17.

Note that the inclusion is independent of choice of sufficiently divissible
$w$.

####  3.1.18. Resolution invariant of Rees algebras

We generalize Definition 3.1.7 to the Rees algebras:

######  Definition 3.1.19.

By the resolution invariant of a Rees algebra $R$ at a point $p$ on a regular
scheme $X$ with an SNC divisor $E$ we mean

${\operatorname{inv}}_{p}(R):=\max\\{(\overline{b}_{1},\ldots,\overline{b}_{k})\mid
R\subset{\mathcal{O}}_{X}[\overline{x}_{1}t^{1/a_{1}},\ldots,\overline{x}_{k}t^{1/a_{k}}]^{{\operatorname{Int}}}\\}$

We show in Section 3.7.1 that there exists unique center
${\mathcal{O}}_{X}[\overline{x}_{1}t^{1/a_{1}},\ldots,\overline{x}_{k}t^{1/a_{k}}]^{{\operatorname{int}}}$
for which the maximum $(\overline{b}_{1},\ldots,\overline{b}_{k})$ is
attained. We shall call it the maximal admissible center for $R$ at $p$.

######  Remark 3.1.20.

In the actual resolution process of the ideals all the occurring Rees algebras
$R$ satisfy $w_{R,A}=w_{A}$ and thus the integral closure ”Int” and ”int”
coincide.

One can also define a version of the resolution invariant used for the concept
of maximal contact in Section 3.6.10.

${\operatorname{inv}}_{p}^{1}(R):=\max\\{\overline{b}_{1}\mid
R\subset{\mathcal{O}}_{X}[\overline{x}_{1}t^{1/a_{1}},\ldots,\overline{x}_{k}t^{1/a_{k}}]^{{\operatorname{Int}}}\\}$

In particular

######  Lemma 3.1.21.

If
$R\subset{\mathcal{O}}_{X}[\overline{x}_{1}t^{1/a_{1}},\ldots,\overline{x}_{k}t^{1/a_{k}}]^{{\operatorname{Int}}}$
(with $a_{1}<a_{2}\ldots<a_{k}$) then
${\operatorname{inv}}_{p}^{1}(R)\geq\overline{b}_{1}$.

####  3.1.22. Order of Rees algebras

Consequently one generalizes the notion of the order of ideals to Rees
algebras.

######  Definition 3.1.23.

Let $X$ be a regular scheme with and SNC divisor $E$. By the order of the Rees
algebra $R=\bigoplus_{a\in\Gamma}R_{a}t^{a}$ at the point $p\in X$, we mean

${\operatorname{ord}}_{p}(R):=\min_{a\in\Gamma\smallsetminus
0}\\{{\operatorname{ord}}_{p}(R_{a})/a\\}$

Since $R$ is finitely generated ${\mathcal{O}}_{X}$-algebra the order is
attained for a certain homogenous element $ft^{p}\in R_{a}t^{a}$, so that

${\operatorname{ord}}_{p}(R)={\operatorname{ord}}_{p}(ft^{a})={\operatorname{ord}}_{p}(f)/a.$

This definition is a generalization of the order of an ideal:

${\operatorname{ord}}_{p}({\mathcal{I}})={\operatorname{ord}}_{p}({\mathcal{O}}_{X}[{\mathcal{I}}t]).$

Moreover, if $R={\mathcal{O}}_{X}[{\mathcal{I}}t]$ then
${\operatorname{ord}}_{p}({\mathcal{I}})={\operatorname{ord}}_{p}(R)$.

######  Lemma 3.1.24.

If
$R\subseteq{\mathcal{A}}={\mathcal{O}}_{X}[\overline{x}_{1}t^{1/a_{1}},\ldots,\overline{x}_{k}t^{1/a_{k}}]^{\operatorname{Int}}$
then $a_{1}\leq{\operatorname{ord}}_{p}(R)$. In particular,

${\operatorname{ord}}_{p}(R)=\max\\{a_{1}\in{\mathbb{Q}}\mid
R\subset{\mathcal{O}}_{X}[m_{p}t^{1/a_{1}}]^{\operatorname{Int}}\\}.$

Moreover, if ${\mathcal{A}}$ is maximal admissible for $R$ at $p$ then
${\operatorname{ord}}_{p}(R)=a_{1}$.

We shall use the following

######  Lemma 3.1.25.

${\operatorname{ord}}_{p}(R^{w})=w\cdot{\operatorname{ord}}_{p}(R)$

∎

###### Proof.

(of Lemma 3.1.24) By the assumption,

$R=\bigoplus
R_{a}t^{a}\subseteq{\mathcal{A}}={\mathcal{O}}_{X}[\overline{x}_{1}t^{1/a_{1}},\ldots,\overline{x}_{k}t^{1/a_{k}}]^{\operatorname{int}}\subseteq{\mathcal{O}}_{X}[m_{p}t^{1/a_{1}}]^{\operatorname{Int}}.$

Rescaling $t\mapsto t^{a_{1}}$ gives $R^{a_{1}}=\bigoplus
R_{a}t^{aa_{1}}\subseteq{\mathcal{O}}_{X}[m_{p}t]^{\operatorname{Int}}$. Thus
$R_{a}\subseteq m_{p}^{\lceil aa_{1}\rceil}$ whence, by definition, for any
$a\in\Gamma_{R}$,

$(1/a_{1}){\operatorname{ord}}_{p}(R_{a}t^{a})={\operatorname{ord}}_{p}(R_{a}t^{aa_{1}})\geq{\operatorname{ord}}_{p}(m_{p}^{\lceil
aa_{1}\rceil}t^{aa_{1}})\geq 1,$

and ${\operatorname{ord}}_{p}(R)\geq a_{1}$.

On the other hand, if ${\operatorname{ord}}_{p}(R)=a_{1}$ then for any
$a\in\Gamma\smallsetminus\\{0\\}$ we have ${\operatorname{ord}}_{p}(R_{a})\geq
aa_{1}$, so $R_{a}\subseteq m_{p}^{\lceil aa_{1}\rceil}$ or

$R_{a}t^{a}\subseteq m^{\lceil
aa_{1}\rceil}_{p}t^{a}\in{\mathcal{O}}_{X}[m_{p}t^{1/a_{1}}]^{\operatorname{Int}}.$

∎

####  3.1.26. The invariants associated with Rees centers

Per analogy to the resolution invariant one can associate with the centers

${\mathcal{A}}={\mathcal{O}}_{X}[\overline{x}_{1}t^{1/a_{1}},\ldots,\overline{x}_{k}t^{1/a_{k}}]^{{\operatorname{int}}}$

the invariant

${\operatorname{inv}}({\mathcal{A}})=(\overline{b}_{1},\ldots,\overline{b}_{k}),\quad{\operatorname{inv}}^{1}({\mathcal{A}}):=(\overline{b}_{1})$

####  3.1.27. Uniqueness of presentation of the invariant of centers

The following Lemma shows that ${\operatorname{inv}}({\mathcal{A}})$ is well-
defined and independent upon presentation. We shall need the following result:

####  3.1.28. Replacement Lemma

######  Lemma 3.1.29.

[ATW19]( non-divisorial case, $i=1$).

Let
${\mathcal{A}}={\mathcal{O}}_{X}[\overline{x}_{1}t^{1/a_{1}},\ldots\overline{x}_{k}t^{1/a_{k}}]^{{\operatorname{int}}}$
be a center and $p\in V({\mathcal{A}})$ be a point. Let
$\overline{x}_{1},\ldots,\overline{x}_{i-1},\overline{x}^{\prime}$ be a system
of local parameters compatible with $E$ at a point $p$, for some $i\leq k$,
such that

$\overline{x}^{\prime}t^{a_{i}}\in{\mathcal{A}}={\mathcal{O}}_{X}[\overline{x}_{1}t^{1/a_{1}},\ldots\overline{x}_{k}t^{1/a_{k}}]^{\operatorname{int}}$

in a neighborhood of $p$ then one can find the coordinates
$\overline{x}_{1},\ldots,\overline{x}_{i-1},\overline{x}^{\prime}_{i},\overline{x}_{i+1}\ldots,\overline{x}_{k}$
such that $\overline{x}\subset\overline{x}_{i}^{\prime}$ and

${\mathcal{A}}={\mathcal{O}}_{X}[\overline{x_{1}}t^{1/a_{1}},\ldots,\overline{x}_{i-1}t^{1/a_{i-1}},\overline{x}_{i}^{\prime}t^{1/a_{i}},\overline{x}_{i+1}t^{1/a_{i+1}}\ldots,\overline{x}_{k}t^{1/a_{k}}]^{\operatorname{int}}$

in a neighborhood of $p\in X$.

###### Proof.

By Lemma 2.2.8,
${\mathcal{A}}_{a_{i}}\subset(\overline{x}_{1},\ldots,\overline{x}_{i})+m_{p}^{2}$.
Thus upon the coordinate change of $\overline{x}_{i}$ compatible with $E$ in
image in $m_{p}/(m_{p}^{2}+(\overline{x}_{1},\ldots,\overline{x}_{i-1}))$, of
the set of coordinates $\overline{x}$ is a subset of the image of coordinates
in $\overline{x}_{i}$. So one can extend $\overline{x}$, and assume that
$\overline{x}$ and $\overline{x}_{i}$ define the same images in
$m_{p}/(m_{p}^{2}+(\overline{x}_{1},\ldots,\overline{x}_{i-1}))$. Then in the
completion $\widehat{{\mathcal{O}}_{X,p}}$ we can write equality of the
vectors of the coordinates

$\overline{x}^{\prime}_{i}=\overline{x}_{i}+\overline{g},\quad\overline{x}^{\prime}_{j}=\overline{x}_{j},\quad
j\neq i,$

where the coordinates of vector $\overline{g}$ are in
${\mathcal{A}}_{a_{1}}\cap(m_{p}^{2}+(\overline{x}_{1},\ldots,\overline{x}_{i-1}))$.
This determines an automorphism of $\widehat{{\mathcal{O}}_{X,p}}$ which takes
$\widehat{{\mathcal{O}}_{X,p}}\cdot{\mathcal{A}}$ onto
$\widehat{{\mathcal{O}}_{X,p}}\cdot{\mathcal{A}}$, and determines the desired
coordinate change. ∎

######  Corollary 3.1.30.

[ATW19] Assume that a center ${\mathcal{A}}$ has two different presentations:

${\mathcal{A}}={\mathcal{O}}_{X}[\overline{x}_{1}t^{1/a_{1}},\ldots\overline{x}_{k}t^{1/a_{k}}]^{{\operatorname{int}}}={\mathcal{O}}_{X}[\overline{x}^{\prime}_{1}t^{1/a^{\prime}_{1}},\ldots\overline{x}^{\prime}_{k^{\prime}}t^{1/a_{k^{\prime}}}]^{{\operatorname{int}}}$

then the associated invariants
$(\overline{b}_{1},\ldots,\overline{b}_{k})=(\overline{b}^{\prime}_{1},\ldots,\overline{b}^{\prime}_{k^{\prime}})$
are the same.

###### Proof.

By Lemma 3.1.24,
$a_{1}=a^{\prime}_{1}={\operatorname{ord}}_{p}({\mathcal{A}})$, which can be
verifed for generators. By Lemma 3.1.29, applied to both presentations, we can
assume that $\overline{x_{1}}=\overline{x^{\prime}_{1}}$. Restricting both
algebras to $V(\overline{x}_{1})=V(\overline{x^{\prime}}_{1})$ we get the
equality by the inductive assumption. ∎

###  3.2. Motivating examples

####  3.2.1. Simple isolated singularities in characteristic zero

######  Example 3.2.2.

This is the main motivating example as it illustrates the invariant in simple
terms. The example is generalized in Sections 4.2, Example 4.2.4 and in
Section 4.5, Example 4.5.3 in any characteristic in two different ways.
Consider the isolated singularity defined at the origin $p$ by

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

on $X={\mathbb{A}}^{n}_{k}=\operatorname{Spec}k[x_{1},\ldots,x_{k}]$, where
$c_{1}\leq\ldots\leq c_{n}$. Find the partition of $(c_{1},\ldots,c_{n})$:

$c_{1}=\ldots=c_{i_{1}}<c_{i_{1}+1}=\ldots=c_{i_{2}}<\ldots<c_{i_{k-1}+1}=\ldots=c_{i_{k}}=c_{n},$

and put $\overline{x}_{1}=(x_{1},\ldots,x_{i_{1}})$,
$\overline{x}_{2}:=(x_{i_{1}+1},\ldots,x_{i_{2}})$,…,
$\overline{x}_{k}=(x_{i_{k-1}+1},\ldots,x_{i_{k}})$,
${a}_{j}:=c_{i_{j-1}+1}=\ldots=c_{i_{j}}$. By a slight abuse of notations we
shall write:

$f=\overline{x}_{1}^{{a}_{1}}+\ldots+\overline{x}_{k}^{{a}_{k}},$

where

$\overline{x}_{j}^{{a}_{j}}:=x_{i_{j-1}+1}^{{a}_{j}}+\ldots+x_{i_{j}}^{{{a}_{j}}}.$

The goal is to find the maximal admissible Rees center
${\mathcal{A}}:={\mathcal{O}}_{X}[\overline{x}^{\prime}_{1}t^{1/a^{\prime}_{1}},\ldots,\overline{x}^{\prime}_{k}t^{1/a^{\prime}_{k}}]^{{\operatorname{int}}}$
such that:

$R:={\mathcal{O}}_{X}[(ft)]\subseteq{\mathcal{A}}:={\mathcal{O}}_{X}[\overline{x}^{\prime}_{1}t^{1/a^{\prime}_{1}},\ldots,\overline{x}^{\prime}_{k}t^{1/a^{\prime}_{k}}]^{{\operatorname{Int}}},$

for a certain $k$ and $a^{\prime}_{1}<a^{\prime}_{2}<\ldots<a^{\prime}_{k}$.
The center ${\mathcal{A}}$ will be constructed in the process of
transformation of the algebra $R$ and recursive adjoining the generators
$\overline{x}^{\prime}_{1}t^{1/a^{\prime}_{1}}$.

We put $R_{1}=R$. By Lemma 3.1.24,

${\operatorname{ord}}_{p}(R_{1})={\operatorname{ord}}_{p}(ft)=a_{1}=a^{\prime}_{1}.$

Applying iteratively the differential operators
$t^{-1/a_{1}}{\mathcal{D}}_{X}$ to the homogenous elements of positive degree
of both sides of the admissibility inclusion
${\mathcal{R}}\subset{\mathcal{A}}$ we see that left side ${\mathcal{A}}$ will
be preserved, while the right side $R_{1}$ will be enlarged so that it
contains so called cotangent ideal $T^{1/a_{1}}({\mathcal{R}}_{1})\cdot
t^{1/a_{1}}$ that is the ideal $T^{1/a_{1}}({\mathcal{R}}_{1})$ in gradation
$t^{1/a_{1}}$ of obtained from $R$ by applying to ${\mathcal{I}}t$ the
iterated differential operators $t^{-1/a_{1}}{\mathcal{D}}_{X}$,
$(a-1)$-times. More specifically:

$T^{1/a_{1}}({\mathcal{R}}_{1})\cdot
t^{1/a_{1}}:=(t^{-1}{\mathcal{D}}_{X})^{a_{1}-1}({\mathcal{I}}t)={\mathcal{D}}_{X}^{a_{1}-1}({\mathcal{I}})t^{1/a_{1}}\subseteq{\mathcal{A}}_{1/a_{1}}t^{1/a_{1}}\supseteq\overline{x}^{\prime}_{1}t^{1/a_{1}}.$

Since ${\operatorname{ord}}_{p}({\mathcal{I}})=a_{1}$, we can see that
${\operatorname{ord}}_{p}(T^{1/a_{1}})={\operatorname{ord}}_{p}({\mathcal{D}}_{X}^{a_{1}-1}({\mathcal{I}}))=1$,
and thus it contains a partial set of local parameters called maximal contact.
In our case

$T^{1/a_{1}}(R_{1})={\mathcal{D}}_{X}^{a_{1}-1}(f)=(x_{1},\ldots,x_{i_{1}},x_{i_{1}+1}^{a_{2}-a_{1}},\ldots,x_{k}^{a_{k}-a_{1}})$

contains a maximal contact $\overline{x}_{1}=(x_{1},\ldots,x_{i_{1}})$. In
general, by Lemma 3.6.20, we can assume that using a simple transformation one
can change the presentation of the algebra ${\mathcal{A}}$ so that from now on
$\overline{x}^{\prime}_{1}=\overline{x}_{1}$ is a maximal contact of $R_{1}$.
This will not affect the algebra ${\mathcal{A}}$ on the right just will change
its presentation.

Next we construct $R_{2}$ as the coefficient ideal

$R_{2}:=C_{\overline{x}_{1}t^{1/{a_{1}}}}(R_{1})\subset{\mathcal{A}}$

In our case we get

$R_{2}={\mathcal{O}}_{X}[(\overline{x}_{1}t^{1/{a_{1}}})^{\operatorname{Int}},ft]={\mathcal{O}}_{X}[\overline{x}_{1}t^{1/{a_{1}}},ft].$

Here the part
${\mathcal{O}}_{X}[\,\,\overline{x}t^{1/a_{1}}]^{\operatorname{Int}}$, means
the integral closure of ${\mathcal{O}}_{X}[\,\,\overline{x}t^{1/a_{1}}]$ in
the smallest subalgebra
${\mathcal{O}}_{X}[t^{1/w}]={\mathcal{O}}_{X}[t^{1/a_{1}}]$ containing all the
generators in $C_{\overline{x}t^{1/a_{1}}}(R_{1})$.

In this example the splitting condition hold in the sense that we have
inclusion

${\mathcal{O}}_{H_{1}}:={\mathcal{O}}_{X}/(\overline{x}_{1})=K[\overline{x}_{2},\ldots,\overline{x}_{k}]\subset{\mathcal{O}}_{X},$

with $D_{\overline{x}_{1}}({\mathcal{O}}_{H_{1}})=0$. In general such a
splitting is possible in the formal (or étale) coordinate system at a point
$p\in X$. In such a case we can write

$R_{2}=C_{\overline{x}_{1}t^{1/{a_{1}}}}(R_{1})={\mathcal{O}}_{X}[(\overline{x}_{1}t^{1/{a_{1}}})^{\operatorname{Int}},C_{\overline{x}_{1}t^{1/{a_{1}}}}(R_{1})_{|H_{1}}].$

In fact the part $C_{\overline{x}_{1}t^{1/{a_{1}}}}(R_{1})_{|H_{1}}$ is
generated by the graded coefficients as in Lemma 3.5.6.

To illustrate the construction in our case we can write

$ft=(\overline{x}_{1}t^{1/a_{1}})^{a_{1}}+(\overline{x}_{2}^{a_{2}}+\ldots+x_{k}^{a_{k}})t.$

in the graded coefficient form with respect to the graded coordinate
$\overline{x}_{1}t^{1/a_{1}}$. So the only graded coefficient is
$f_{|V(x_{1})}\cdot t=(\overline{x}_{2}^{a_{2}}+\ldots+x_{n}^{a_{n}})t$.
Consequently using presentation in Lemma 3.5.6 we simply write

$R_{2}={\mathcal{O}}_{X}[(\overline{x}_{1}t^{1/a_{1}})^{\operatorname{Int}},(\overline{x}_{2}^{a_{2}}+\ldots+x_{n}^{a_{n}})t].$

Moreover the maximal admissibility condition is satisfied with
$R_{2}\subseteq{\mathcal{A}}$.

Moreover we say that $R_{2}$ is strictly nested  at
$H_{1}=V(\overline{x}_{1})$ , if it has the presentation

$R_{2}={\mathcal{O}}_{X}[(\overline{x}_{1}t^{1/a_{1}})^{\operatorname{Int}},R_{2|H_{1}}],$

where, in our case
$R_{2|H_{1}}=(\overline{x}_{2}^{a_{2}}+\ldots+\overline{x}_{n}^{a_{n}})t$.
(Definition 3.5.13)

Then, by Lemma 3.5.11,

${\mathcal{A}}_{|H_{1}}={\mathcal{O}}_{H_{1}}[\overline{x}^{\prime}_{2|H_{1}}t^{1/a^{\prime}_{2}},\ldots,\overline{x}^{\prime}_{k|H_{1}}t^{1/a^{\prime}_{k}}]^{{\operatorname{int}}}$

is a maximal admissible center, for $R_{2|H_{1}}$ and thus the restricted Rees
algebra

$R_{2|H_{1}}=(\overline{x}_{2}^{a_{2}}+\ldots+x_{n}^{a_{n}})t$

is of order $a_{2}$. This implies that

$a_{2}=a^{\prime}_{2}={\operatorname{ord}}_{p}(R_{2|H_{1}}).$

We compute the maximal contact of
$R_{2|H_{1}}=(\overline{x}_{2}^{a_{2}}+\ldots+\overline{x}_{n}^{a_{n}})t$ to
be the maximal subsystem of local parameters

$\overline{x}_{2}\subset
T^{1/a_{2}}(R_{2|H_{1}})={\mathcal{D}}_{\overline{x}_{2},\ldots,\overline{x}_{k}}^{a_{2}-1}(f)={\mathcal{D}}_{\overline{x}_{2},\ldots,\overline{x}_{k}}^{a_{2}-1}(\overline{x}_{2}^{a_{2}}+\ldots+\overline{x}_{k}^{a_{k}})\supseteq(\overline{x}_{2}).$

Here $T^{1/a_{2}}(R_{2})$ is obtained from $R_{2}$ by applying the
differential operators
$t^{-(a_{2}-1)/a_{2}}{\mathcal{D}}_{\overline{x}_{2},\ldots,\overline{x}_{k}}^{a_{2}-1}$
to the generator $ft$ of $R_{2}$ in the gradation $t$, or in the split form to
$ft_{|H_{1}}$.

Since $T^{1/a_{2}}(R_{2})\subset{\mathcal{A}}_{1/a}$ we conclude that
$\overline{x}^{\prime}_{2}=\overline{x}_{2}$, after a change of the coordinate
presentation of ${\mathcal{A}}$.

Then, as before the coordinate $\overline{x}_{2}$ splits and computing the
coefficients of the only generator

$f_{|H_{1}}=(\overline{x}_{2}^{a_{2}}+\ldots+\overline{x}_{n}^{a_{n}})t=(\overline{x}_{2}t^{1/a_{2}})^{a_{2}}+(\overline{x}_{3}^{a_{3}}+\ldots+\overline{x}_{n}^{a_{n}})t$

of $R_{2|H_{1}}$ gives the split form of the coefficient ideal as in Lemma
3.5.22:

$\displaystyle
R_{3}:=C_{\overline{x}_{2}t^{1/{a_{1}}}}(R_{2})={\mathcal{O}}_{X}[(\overline{x}_{1}t^{1/a_{1}},\overline{x}_{2}t^{1/a_{2}})^{\operatorname{Int}},f_{|H_{2}}]=$
$\displaystyle={\mathcal{O}}_{X}[(\overline{x}_{1}t^{1/a_{1}},\overline{x}_{2}t^{1/a_{2}})^{\operatorname{Int}},(\overline{x}_{3}^{a_{3}}+\ldots+x_{n}^{a_{n}})t]\subset{\mathcal{A}},$

where $H_{2}:=V(\overline{x}_{1},\overline{x}_{2})$. We maintain the maximal
admissibility condition $R_{3}\subset{\mathcal{A}}$, while enlarging $R_{2}$.
At the same time $R_{3}$ is in the strictly nested form for
$\overline{x}_{1}t^{1/a_{1}},\overline{x}_{2}t^{1/a_{2}}$ (Definition 3.5.13),
which means exactly

$R_{3}={\mathcal{O}}_{X}([\overline{x}_{1}t^{1/a_{1}},\overline{x}_{2}t^{1/a_{2}})^{\operatorname{Int}},R_{3|H_{2}}]={\mathcal{O}}_{X}([\overline{x}_{1}t^{1/a_{1}},\overline{x}_{2}t^{1/a_{2}})^{\operatorname{Int}},ft_{|H_{2}}]\subset{\mathcal{A}}$

We continue the recursive process so that

$R_{i+1}=C_{\overline{x}_{i}t^{1/{a_{i}}}}(R_{i})={\mathcal{O}}_{X}[(\overline{x}_{1}t^{1/a_{1}},\ldots,\overline{x}_{i}t^{1/a_{1}})^{\operatorname{Int}},ft_{|H_{i}}]\subset{\mathcal{A}},$

where $H_{i}=V(\overline{x}_{1},\ldots,\overline{x}_{i})$ until
$R_{k+1|H_{k}}=0$. The latter implies, after reverting to our original
notation that

$R_{k+1}={\mathcal{A}}={\mathcal{O}}_{X}[\overline{x}_{1}t^{1/{a}_{1}},\ldots,\overline{x}_{k}t^{1/a_{k}}]^{\operatorname{Int}}={\mathcal{O}}_{X}[{x}_{1}t^{1/c_{1}},\ldots,{x}_{n}t^{1/c_{n}}]^{\operatorname{int}}.$

Consequently, ${\operatorname{inv}}_{p}({\mathcal{I}})=(c_{1},\ldots,c_{n})$.
Since ${\mathcal{A}}$ is an a priori chosen maximal ${\mathcal{I}}$-admissible
center, which remains unchanged, we conclude that the process is independent
of choices and thus canonical.

The full cobordant blow-up $B$ of $X$ at

${{\mathcal{A}}}^{{\operatorname{ext}}}={\mathcal{O}}_{X}[t^{-1/w},\overline{x}_{1}t^{1/a_{1}},\ldots,\overline{x}_{i}t^{1/a_{k}}],$

where $w={\operatorname{lcm}}(a_{1},\ldots,a_{k})$ is defined by the rescaled
algebra

${\mathcal{O}}_{B}=({{\mathcal{A}}}^{{\operatorname{ext}}})^{w}={\mathcal{O}}_{X}[t^{-1},\overline{x}_{1}t^{w_{1}},\ldots,\overline{x}_{k}t^{w_{k}}],$

where $w_{i}:=w/a_{i}$.

Equivalently one can write the center in the ${\mathbb{Q}}$-ideal form
$(\overline{x}_{1}^{1/w_{1}},\ldots,\overline{x}_{k}^{1/w_{k}})$ representing
formally $t$-gradation of ${\mathcal{O}}_{B}$.

Observe that

$B=\operatorname{Spec}({\mathcal{O}}_{X}[t^{-1},\overline{x}_{1}t^{w_{1}},\ldots,\overline{x}_{i}t^{w_{k}}])=\operatorname{Spec}((K[t^{-1},\overline{x}_{1}t^{w_{1}},\ldots,\overline{x}_{i}t^{w_{k}}])$

is an affine space with coordinates $t^{-1}$ and
$x^{\prime}_{i}=x_{i}t^{w_{i}}$.

Now, the admissibility condition
$ft\subset{{\mathcal{A}}}^{{\operatorname{ext}}}$ implies that
$ft^{w}\subset{\mathcal{O}}_{B}$. Here $ft^{w}\in{\mathcal{O}}_{B}$ is so
called controlled transform $\sigma^{c}(f)$ of $f$ is obtained by factoring
the inverse image by the power $(t^{-1})^{w}$, of the exceptional divisor
$t^{-1}$.

Solving for $x_{i}$ and substituting into $f$ gives

$f=t^{-w}((x^{\prime}_{1})^{a_{1}}+\ldots+(x^{\prime}_{n})^{a_{n}}).$

After clearing the exceptional divisor we obtain the formula for

$\sigma^{c}(f)=t^{a_{1}w_{1}}f=(x^{\prime}_{1})^{a_{1}}+\ldots+(x^{\prime}_{n})^{a_{n}}$

in the new unknowns $x_{1}^{\prime},\ldots,x_{n}^{\prime}$ on $B$ which is
identical to the original formula for $f$ on $X$. Thus the singularity locus
on $B$ at the vertex $V=V(x^{\prime}_{1},\ldots,x^{\prime}_{n})$ is described
by the same formula, and the cobordant blow-up $B_{+}=B\smallsetminus
V(x^{\prime}_{1},\ldots,x^{\prime}_{n})$ is smooth so the resolution is
obtained by the single cobordant blow-up.

####  3.2.3. Generalizations

The previous example can be easily generalized without essential changes in
computations.

######  Example 3.2.4.

For the coordinate system $\overline{x}_{1},\ldots,\overline{x}_{k}$, consider
forms $F_{i}(\overline{x}_{i})$ of degree $a_{i}$ such that
$a_{1}<\ldots<a_{k}$, and for which
$(\overline{x}_{i})={\mathcal{D}}_{\overline{x}_{i}}^{a_{i}-1}(F_{i})$. Then
for $f=F_{1}+\ldots+F_{k}$ and ${\mathcal{I}}=(f)$ and
$R=R_{1}={\mathcal{O}}_{X}[ft]$, As before we obtain the recursive formula

$R_{i+1}=C_{\overline{x}_{i}t^{1/a_{i}}}(R_{i})={\mathcal{O}}_{X}[(\overline{x}_{1}t^{1/a_{1}},\ldots,\overline{x}_{i}t^{1/a_{1}})^{\operatorname{Int}},f_{|H_{i}}\cdot
t]\subset{\mathcal{A}},$

where $H_{i}=V(\overline{x}_{1},\ldots,\overline{x}_{i})$,
$f_{|H_{i}}=F_{i+1}+\ldots+F_{k}$ and $\overline{x}_{i+1}t^{1/a_{i+1}}$ is a
maximal contact for $R_{i+1|H_{i}}={\mathcal{O}}_{H_{i}}[f_{|H_{i}}t]$. In the
inductive step we write

$f_{|H_{i}}\cdot
t=F_{i+1}(\overline{x}_{i+1}t^{1/a_{i+1}})+(F_{i+2}+\ldots+F_{k})t=F_{i+1}(\overline{x}_{i+1}t^{1/a_{i+1}})+f_{|H_{i+1}}\cdot
t,$

leading to the split formula for the
$R_{i+2}=C_{\overline{x}_{i+1}t^{1/a_{i}}}(R_{i+1})$. Consequently

$R_{k+1}={\mathcal{A}}={\mathcal{O}}_{X}[(\overline{x}_{1}t^{1/\overline{a}_{1}},\ldots,\overline{x}_{k}t^{1/a_{k}})]^{\operatorname{Int}}={\mathcal{O}}_{X}[{x}_{1}t^{1/a_{1}},\ldots,{x}_{n}t^{1/a_{k}}]^{\operatorname{int}}$

is a maximal admissible center.

The full cobordant blow-up $B$ of $X$ at

${{\mathcal{A}}}^{{\operatorname{ext}}}={\mathcal{O}}_{X}[t^{-1/w},\overline{x}_{1}t^{1/a_{1}},\ldots,\overline{x}_{i}t^{1/a_{k}}],$

where $w={\operatorname{lcm}}(a_{1},\ldots,a_{k})$ is defined by the rescaled
algebra

${\mathcal{O}}_{B}=({{\mathcal{A}}}^{{\operatorname{ext}}})^{w}={\mathcal{O}}_{X}[t^{-1},\overline{x}_{1}t^{w_{1}},\ldots,\overline{x}_{k}t^{w_{k}}],$

where $w_{i}=w/a_{i}$.

It transforms $f$ into its controlled transform

$\sigma^{c}(f):=t^{w}f=F_{1}(\overline{x}^{\prime}_{1})+\ldots+F_{k}(\overline{x}^{\prime}_{k}),$

For the new coordinates
$\overline{x}^{\prime}_{i}=\sigma^{c}(\overline{x}^{\prime}_{i})=t^{w_{i}}\overline{x}_{i}$.
Removing the vertex of the transfromation we obtain $B_{+}=B\smallsetminus
V(\overline{x}^{\prime}_{1},\ldots,\overline{x}^{\prime}_{k})$, where the
order of one of the forms $F_{i}(\overline{x}^{\prime}_{i})$ drops, which
causes the invariant ${\operatorname{inv}}_{p}({\mathcal{I}})$ drop.

In particular if

$f=x_{1}x_{2}x_{3}+x_{4}^{4}+x_{5}^{2}x_{6}^{2}$

then $R_{1}={\mathcal{O}}_{X}[ft]$. The first (multiple) maximal contact is
$(x_{1},x_{2},x_{3})$ for $a_{1}={\operatorname{ord}}_{p}(f)=3$.

Since $H_{1}=V((x_{1},x_{2},x_{3})$ splits we can write

$R_{2}=C_{(x_{1},x_{2},x_{3})t^{1/3}}(R_{1})={\mathcal{O}}_{X}[(x_{1},x_{2},x_{3})t^{1/3},(x_{4}^{4}+x_{5}^{2}x_{6}^{2})t]$

for the graded coefficient decomposition for $(x_{1},x_{2},x_{3})t^{1/3}$:

$ft=x_{1}t^{1/3}x_{2}t^{1/3}x_{3}t^{1/3}+(x_{4}^{4}+x_{5}^{2}x_{6}^{2})t$

Then

$a_{2}={\operatorname{ord}}_{p}(f_{|H_{1}})\cdot
t={\operatorname{ord}}_{p}((x_{4}^{4}+x_{5}^{2}x_{6}^{2})t)=4/1=4$

with maximal contact $(x_{4},x_{5},x_{6})t^{1/4}$, giving the maximal
admissible center

$R_{3}=C_{(x_{4},x_{5},x_{6})t^{1/4}}(R_{2})={\mathcal{A}}={\mathcal{O}}_{X}[(x_{1},x_{2},x_{3})t^{1/3},(x_{4},x_{5},x_{6})t^{1/4}]^{\operatorname{int}},$

and the invariant ${\operatorname{inv}}_{p}(f)=(3,3,3,4,4,4)$. The extended
center

${\mathcal{A}}^{\operatorname{ext}}={\mathcal{O}}_{X}[t^{-1/12},(x_{1},x_{2},x_{3})t^{1/3},(x_{4},x_{5},x_{6})t^{1/4}])$

determines the full cobordant blow-up

$B=\operatorname{Spec}({\mathcal{O}}_{X}[t^{-1},(x_{1},x_{2},x_{3})t^{4},(x_{4},x_{5},x_{6})t^{3}].$

Alternatively one can write the center as the ${\mathbb{Q}}$-ideal

$((x_{1},x_{2},x_{3})^{1/4},(x_{4},x_{5},x_{6})^{1/3})$

formally representing $t$-gradation of ${\mathcal{O}}_{B}$.

####  3.2.5. Varieties with a divisorial SNC- structure

######  Example 3.2.6.

Let $f=(x_{1}+x_{2}^{2})^{2}+x_{3}^{7}$, where $x_{1}$, $x_{2}$ are divisorial
at the origin at $x_{3}$ is free. In this case the process is similar except
finding maximal contact is slightly different as it depends on the divisorial
structure. Set $R=R_{1}={\mathcal{O}}_{X}[ft]$. Then the order of $f$ and of
$R_{1}$ at the origin is equal to $2$. The computation of the cotangent ideal
is the same $T^{1/2}(R)=D(f)=((x_{1}+x_{2}^{2}),x_{3}^{6})$.

The maximal contact for the Rees algebra of order $a_{1}$ is given by the free
coordinates in $T^{1/a_{1}}(R)$ and the divisorial unknowns occurring in the
linear parts of $T^{1/a_{1}}(R)$. In our case the maximal contact
$\overline{u}:=(x_{1})$ for $R_{1}={\mathcal{O}}_{X}[ft]$ is determined by the
only divisorial coordinate $x_{1}$ tangential to
$T^{1/2}(R)=((x_{1}+x_{2}^{2}),x_{3}^{6})$ at $p$ such that
$\overline{u}+m_{p}^{2}\supset T^{1/2}(R)$. (Definitions 3.6.2, 3.6.11)

Write $ft=((x_{1}+x_{2}^{2})^{2}+x_{3}^{7})t$ in the coefficient form

$ft=((x_{1}+x_{2}^{2})^{2}+x_{3}^{7})t=(x_{1}t^{1/2})^{2}+(x_{1}t^{1/2})2x_{2}^{2}t^{1/2}+(x_{2}^{4}+x_{3}^{7})t.$

with respect to the graded maximal contact $x_{1}t^{1/2}$. We form the
coefficient ideal

$R_{2}:=C_{x_{1}t^{1/2}}(R_{1})={\mathcal{O}}_{X}[(x_{1}t^{1/2})^{{\operatorname{Int}}},x_{2}^{2}t^{1/2},(x_{2}^{4}+x_{3}^{7})t].$

Then $R_{2}$ is in a strictly nested form:
$R_{2}={\mathcal{O}}_{X}[(x_{1}t^{1/2})^{{\operatorname{Int}}},R_{2|H_{1}}]$,
where $H_{1}=V(x_{1})$.

The order ${\operatorname{ord}}_{p}(R_{2|H_{1}})=4$, with

$T^{1/4}(R_{2})={\mathcal{D}}_{(x_{2},x_{3})}(x_{2}^{2})+{\mathcal{D}}^{3}_{(x_{2},x_{3})}((x_{2}^{4}+x_{3}^{7})=(x_{2},x_{3}^{4})$

consisting of all elements in gradation $t^{1/4}$ of the algebra obtained by
applying iterated operators ${\mathcal{D}}_{(x_{2},x_{3})t^{1/4}}$ to $R_{2}$.
Thus the maximal contact is equal to $x_{2}$ in gradation $t^{1/4}$
(Definition 3.6.2). Write $(x_{2}^{4}+x_{3}^{7})t\in R_{2}$ in the coefficient
form with respect to $x_{2}t^{1/4}$:

$(x_{2}^{4}+x_{3}^{7})t=(x_{2}t^{1/4})^{2}+x_{3}^{7}t.$

Thus

$R_{3}:=C_{x_{2}t^{1/4}}(R_{3})={\mathcal{O}}_{X}[(x_{1}t^{1/2},x_{2}t^{1/4})^{\operatorname{Int}},x_{3}^{7}t].$

The order of $R_{3|H_{2}}={\operatorname{ord}}_{0}(x_{3}^{7}t)=7$. The maximal
contact is $x_{3}\in T^{7}(R_{3})={\mathcal{D}}^{6}_{x_{3}}(x_{3}^{7})$ in
gradation $t^{1/7}$. Then

$R_{4}:=C_{x_{3}t^{1/7}}(R_{3})={\mathcal{O}}_{X}[x_{1}t^{1/2},x_{2}t^{1/4},x_{3}t^{1/7}]^{{\operatorname{int}}}.$

Since $R_{4|H_{3}}=0$, for $H_{3}=V(x_{1},x_{2},x_{3})$,we conclude that
${\mathcal{A}}=R_{4}$ is a maximal admissible center for
$(x_{1}+x_{2}^{2})^{2}+x_{3}^{7})$ t . Since
$w={\operatorname{lcm}}(2,4,7)=28$ we get the formula for the extended Rees
center

${\mathcal{A}}^{\operatorname{ext}}={\mathcal{O}}_{X}[t^{-1/28},x_{1}t^{1/2},x_{2}t^{1/4},x_{3}t^{1/7}]$

Rescaling defines the cobordant blow-up
$B={\mathcal{O}}_{X}[t^{-1},x_{1}t^{14},x_{2}t^{7},x_{3}t^{4}]$ at
${\mathcal{A}}^{\operatorname{ext}}$. Since $x_{1},x_{2}$ are divisorial and
$x_{3}$ is free, and
${\mathcal{A}}={\mathcal{O}}_{X}[x_{1}t^{1/2},x_{2}t^{1/4},x_{3}t^{1/7}]^{{\operatorname{int}}}$
is a maximal admissible center one obtains

${\operatorname{inv}}_{0}(f)=(2_{+},4_{+},7).$

The effect of the full cobordant blow-up is exactly

$f=t^{-28}((x_{1}^{\prime}+(x_{2}^{\prime})^{2})^{2}+(x_{3}^{\prime})^{7}).$

Let us examine this effect on the cobordant blow-up $B_{+}=B\smallsetminus
V=B\smallsetminus V(x^{\prime}_{1},x^{\prime}_{2},x^{\prime}_{3})$ after
removing the vertex $V=V(x^{\prime}_{1},x^{\prime}_{2},x^{\prime}_{3})$.

On $B\smallsetminus V(x^{\prime}_{1},x^{\prime}_{2})$ we write the equation
$f^{\prime}:=t^{28}f$ as $f^{\prime}=u^{2}+(x_{3}^{\prime})^{7}$, where

$u:=x_{1}^{\prime}+(x_{2}^{\prime})^{2}$

is a free coordinate. Thus
${\mathcal{A}}^{\prime}={\mathcal{O}}_{X^{\prime}}[u^{1/2},(x^{\prime}_{3})^{1/7}]^{\operatorname{int}}$
is maximal admissible center for $(f^{\prime})$ at $V(u,x^{\prime}_{3})$ with
the invariant

${\operatorname{inv}}_{p^{\prime}}(f^{\prime})=(2,7)<{\operatorname{inv}}_{0}(f)=(2_{+},4_{+},7)$

for $p^{\prime}\in V(u,x^{\prime}_{3})$.

The next cobordant blow-up at
${\mathcal{A}}^{\prime}={\mathcal{O}}_{X^{\prime}}[ut^{1/2},(x^{\prime}_{1/3})t^{1/7}]^{\operatorname{int}}$
resolves the singularity.

On $B\smallsetminus V(x_{3}^{\prime})$, the derivative

${\mathcal{D}}_{x^{\prime}_{3}}(f^{\prime})={\mathcal{D}}_{x^{\prime}_{3}}(x_{1}^{\prime}+(x_{2}^{\prime})^{2})^{2}+(x^{\prime}_{3})^{7})=(7x^{\prime}_{3})^{6}$

is invertible and $f^{\prime}$ is nonsingular.

In case there are no divisors on $X$, the process is faster as we can write
immediately $f=u^{2}+x_{3}^{7}$, where $u=x_{1}+x_{2}^{2}$. So
${\operatorname{maxinv}}(f)=(2,7)$, and by the previous example,
${\mathcal{A}}^{\prime}={\mathcal{O}}_{X^{\prime}}[u^{t}{1/2},(x_{3}^{\prime})^{t}{1/7}]^{\operatorname{int}}$
is the maximal admissible center. The cobordant blow-up of
${\mathcal{J}}=(u^{1/7},x_{3}^{1/2})$ resolves the singularity.

###  3.3. Splitting of derivations and compatibility

Splitting of derivations on regular schemes allow for the coefficient
representations of functions and simplifies the computations of the
coefficient ideals. A similar concept in a different language was also used in
[ATW19].

######  Definition 3.3.1.

We say that a closed subscheme $H$ of a scheme $X$ splits if the closed
embedding $i:H\hookrightarrow X$ splits, so there is an affine morphism
$\pi:X\to H$ which is a left inverse of $i$ with $\pi i=\text{id}_{H}$.

In other words for any open $U$, and any function
$f\in{\mathcal{O}}_{X}(\pi^{-1}(U))$ its restriction

$f_{|H}\in{\mathcal{O}}_{H}(U)\subset{\mathcal{O}}_{X}(\pi^{-1}(U)).$

The splitting condition implies that there is an injective morphism
$\pi^{-1}({\mathcal{O}}_{H})\hookrightarrow{\mathcal{O}}_{X}$ of sheaves on
$X$ with the left inverse ${\mathcal{O}}_{X}\hookrightarrow
i_{*}({\mathcal{O}}_{H})$ given by the restriction.

The notion can be extended to derivations corresponding to a system of local
parameters.

######  Definition 3.3.2.

Consider a set of derivations $D_{\overline{x}}=(D_{x_{1}},\ldots,D_{x_{k}})$
of ${\mathcal{D}}_{X}$ on a regular scheme $X$ for a certain partial system of
coordinates $\overline{x}=(x_{1},\ldots,x_{k})$ such that
$D_{x_{i}}(x_{j})=\delta_{ij}$, we say that $D_{\overline{x}}$ splits in
${\mathcal{O}}_{X}$, if $H=V(\overline{x})\subset X$ splits and the
derivations in $D_{\overline{x}}$ vanish on
$\pi^{-1}({\mathcal{O}}_{H})\subset{\mathcal{O}}_{X}$.

######  Example 3.3.3.

Let $X$ be a variety smooth over a field $K$. Let $x_{1},\ldots,x_{n}\in
m_{p}$ be a complete coordinate system at a closed point $p$. Any regular
subscheme

$H=V(x_{1},\ldots,x_{k})\subset\operatorname{Spec}(\widehat{{\mathcal{O}}}_{X,p})=\operatorname{Spec}(K_{p}[[x_{1},\ldots,x_{n}]])$

for any natural $k\leq n$, splits. Here $K_{p}={\mathcal{O}}_{X,x}/m_{p,X}$ is
the residue field at $p\in X$. Indeed, we can always write

${\mathcal{O}}_{H,p}\simeq K_{p}[[x_{k+1},\ldots,x_{n}]]\subset
K_{p}[[x_{1},\ldots,x_{n}]].$

Moreover the system of derivations $D_{x_{1}},\ldots,D_{x_{k}}$ on
$\operatorname{Spec}(\widehat{{\mathcal{O}}}_{X,p})$ split as they vanish
exactly on $K_{p}[[x_{k+1},\ldots,x_{n}]]$.

Both notions are strictly related:

######  Lemma 3.3.4.

Let $H=V(\overline{x})$ be a smooth subvariety on a smooth variety $X$. If $H$
splits on variety $X$, and $\pi:X\to H$ is a splitting morphism, then there is
a unique system of derivatives $D_{\overline{x}}$ which splits in
${\mathcal{O}}_{X}$, so that $D_{x_{i}}(x_{j})=\delta_{ij}$ and
$D_{\overline{x}}$ vanishes on $\pi^{-1}({\mathcal{O}}_{H})$. Conversely, if
$(H,D_{\overline{x}})$ splits then it defines a unique splitting morphism
$\pi:X\to H$ for the closed embedding $H$. Moreover
$D_{\overline{x}}=\partial_{\overline{x}}$ is computed by any coordinate
coordinate system $(\overline{x},\overline{y})$, where
$\overline{y}=\overline{y}_{|H}$.

###### Proof.

If $H$ splits in $X$ we extend $\overline{x}$ to a complete system of
coordinates $(\overline{x},\overline{y})$ by adjoining a system of coordinates
$\overline{y}\subset\pi^{-1}({\mathcal{O}}_{H})\subset{\mathcal{O}}_{X}$. Then
$D_{\overline{x}}=\partial_{\overline{x}}$ is defined with respect to the
coordinate system $(\overline{x},\overline{y})$. Conversely if
$(H,D_{\overline{x}})$ splits then $\pi^{-1}({\mathcal{O}}_{H})$ is determined
exactly by vanishing $D_{\overline{x}}$. ∎

Recall a well known fact in our setting:

######  Lemma 3.3.5.

If $(\overline{x},D_{\overline{x}})$ splits on a smooth variety $X$ then for
any open affine subset $U$, and $f\in{\mathcal{O}}_{X}(\pi^{-1}(U)$ there is a
decomposition of $f$ up to $(x^{k})$ for any $k\in{\mathbb{N}}$.

$f\equiv\sum_{|\alpha|<k}c_{\alpha}x^{\alpha}(\mbox{mod}\,\,(x^{k})),$

where