$c_{\alpha}=\frac{1}{\alpha!}D_{x^{\alpha}}(f)_{|H}\in{\mathcal{O}}_{H}(U)\subset{\mathcal{O}}_{X}(\pi^{-1}(U).$

###### Proof.

Let $H=V(\overline{x})$. Observe that $f-f_{|H}\in(\overline{x})$.

Note that

$\frac{1}{\alpha!}D_{x^{\alpha}}(f-\sum_{|\alpha|<k}c_{\alpha}x^{\alpha})_{|H}=c_{\alpha}-c_{\alpha}\frac{1}{\alpha!}D_{x^{\alpha}}(x^{\alpha})=0.$

Suppose that
$g:=(f-\sum_{|\alpha|<k}c_{\alpha}x^{\alpha})\not\in\overline{x}^{k}$. Then
there is a $l<k$, such that
$g\in\overline{x}^{l}\smallsetminus\overline{x}^{l+1}$. Then
$g=\sum_{|\alpha|=l}b_{\alpha}x^{\alpha}$, where at least for one $\alpha_{0}$
we have $b_{\alpha_{0}}\not\in(\overline{x})$. Then

$\frac{1}{\alpha_{0}!}D_{x^{\alpha_{0}}}(g)_{|H}=\sum(\frac{1}{\alpha_{0}!}D_{x^{\alpha_{0}}}b_{\alpha}x^{\alpha})_{|_{|H}}\neq
0,$

which is a contradiction. ∎

######  Definition 3.3.6.

Let $\overline{x}_{1}=(x_{11},\ldots,x_{1s_{1}})$ be a partial system of
coordinates on a a regular scheme $X$ with a set of derivations
$D_{\overline{x}_{1}}:=(D_{x_{11}},\ldots,D_{x_{1s_{1}}})$ such that
$D_{x_{1j}}(x_{1i})=\delta_{ij}$. We say that a system of derivations
$D_{\overline{x}_{1}}$ is compatible with a Rees center ${\mathcal{A}}$ on $X$
if there is a certain presentation

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

such that so that $D_{\overline{x}_{1}}(\overline{x}_{i})=0$ for $i\geq 2$.

######  Lemma 3.3.7.

Let
$A={\mathcal{O}}_{X}[\overline{x}_{1}t^{1/a_{1}},\ldots,\overline{x_{k}}t^{1/a_{k}}]^{\operatorname{int}}$
be a center. If $(\overline{x},D_{\overline{x}_{1}})$ splits in
${\mathcal{O}}_{X}$ then it is compatible with the center ${\mathcal{A}}$.

###### Proof.

Set $\overline{x}^{\prime}_{i}:=\overline{x}_{i|H_{1}}$. Then
$\overline{x}^{\prime}_{i}-\overline{x}_{i}\in(\overline{x}_{1})$, and since
$a_{i}>a_{1}$, we have that $\overline{x}_{1}t^{1/a_{i}}\in{\mathcal{A}}$. So

$\overline{x}^{\prime}_{i}t^{1/a_{i}}-\overline{x}_{i}t^{1/a_{i}}\in(\overline{x}_{1}t^{1/a_{i}})\subset{\mathcal{A}}.$

Hence $\overline{x}^{\prime}_{i}t^{1/a_{i}}\in{\mathcal{A}}$ and we have
inclusion of the centers
${\mathcal{O}}_{X}[\overline{x}_{1}t^{1/a_{1}},\ldots,\overline{x_{k}}t^{1/a_{k}}]^{\operatorname{int}}\subseteq{\mathcal{A}}$.
By symmetry we have the reverse inclusion. ∎

######  Corollary 3.3.8.

If $X$ is smooth over a field $K$ then any derivation $D_{\overline{x}_{1}}$
splits in $\widehat{{\mathcal{O}}_{X,p}}$ so it is compatible with any center
of the form

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

∎

###  3.4. Derivations on the Rees centers

For an ideal ${\mathcal{I}}$ of order $a\in{\mathbb{N}}$ at a point $p\in X$
on a smooth variety $X$ we can place it in $t^{a}$-gradation of a certain
${\mathbb{Z}}$-graded Rees algebra $R={\mathcal{O}}_{X}[{\mathcal{I}}t^{a}]$
so that we have the admissibility condition

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

with respect to maximal ideal $m_{p}\subset{\mathcal{O}}_{X,p}$. Here we
naturally associate with the coordinates gradation $t$, and since the
derivatives $D_{x_{i}}$ lower the order of ideals by $1$ and we associate with
with gradation $t^{-1}$. Consequently the graded derivations $t^{-1}D_{x_{i}}$
act on the elements of $R\subset{\mathcal{O}}_{X}[t]$ with positive
gradations. Moreover they preserve ${\mathcal{O}}_{X}[m_{p}t]$.

In our approach we place the ideal ${\mathcal{I}}$ in gradation $t$. Then the
order of the Rees algebra $R={\mathcal{O}}_{X}[{\mathcal{I}}t]$ at a point $p$
is $a_{1}={\operatorname{ord}}_{p}({\mathcal{I}})$ and thus the corresponding
coordinates and derivations shall be rescaled accordingly and we consider the
action of $t^{-1/a_{1}}D_{x_{i}}$. A similar concept was used in the first
version of [ATW19] to interpret the algebra of the center.

In general, let $R=\bigoplus R_{a}\subset{\mathcal{O}}_{X}[t^{1/w}]$ be any
Rees algebra on a smooth $X$. For any local coordinate system
$\overline{x}=(x_{1},\ldots,x_{n})$ compatible with $E$ on a smooth variety
$X$, and a given $a_{1}\in{\mathbb{Q}}$ (which is usually the order of the
Rees algebra $R$) one considers graded derivations

$D_{x_{i}t^{1/a_{1}}}:=t^{-1/a_{1}}D_{x_{i}}$

acting on the elements of $R_{a}$ of the gradations $a\geq 1/a_{1}$.

Consequently, if $ft^{a}\in{\mathcal{O}}_{X}[t^{1/w}]$, where $a\geq 1/a_{1}$
then

$D_{x_{i}t^{1/a_{1}}}(ft^{a})=D_{x_{i}}(f)t^{a-1/a_{1}}\in{\mathcal{O}}_{X}[t^{1/w}].$

The idea is the following 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 when applying the differential operators
$D_{\overline{x}_{1}t^{1/a_{1}}}$ to the both sides we preserve the right side
while enlarging the left side by adding new elements. This way we step by step
enlarge $R$ which at some point of the algorithm becomes equal to the center
${\mathcal{A}}$ on the right side.

More generally, by composing $D_{\overline{x}_{1}t^{1/a_{1}}}$ we consider
differential operators:

$D_{\overline{x}t^{1/a_{1}}}^{\alpha}:=t^{-|\alpha|/a_{i}}D_{\overline{x}^{\alpha}},\quad\mbox{where}$
$D_{\overline{x}^{\alpha}}:=\frac{1}{\alpha_{1}!\ldots\alpha_{k}!}\cdot\frac{\partial^{|\alpha|}}{\partial
x_{1}^{\alpha_{1}}\ldots\partial
x_{n}^{\alpha_{n}}},\quad|\alpha|=\alpha_{1}+\ldots+\alpha_{n}.$

acting on elements $ft^{a}\in{\mathcal{O}}_{X}[t^{1/w}]$ and
$a\geq|\alpha|/a_{1}$:

$D_{\overline{x}t^{1/a_{1}}}^{\alpha}ft^{a}=D_{\overline{x}^{\alpha}}(f)t^{a-|\alpha|/a_{1}}$

######  Example 3.4.1.

If $ft=(x^{3}+x^{2}y+y^{3})t$, and $a_{1}=3$ then

$D_{xt^{1/3}}((x^{3}+x^{2}y+y^{3})t)=(3x^{2}+2xy)t^{2/3}.$

####  3.4.2. Differential operators preserving centers

A main consequence of the condition $a_{1}\leq\ldots\leq a_{n}$ is that the
center $A$ is preserved by the action of $D_{x_{i}t^{1/a_{1}}}$ for
$i=1,\ldots,k$.

######  Lemma 3.4.3.

Let
$A={\mathcal{O}}_{X}[{x}_{1}t^{1/a_{1}},\ldots,{x_{k}}t^{1/a_{k}}]^{\operatorname{int}}$,
and $\overline{x}=({x}_{1},\ldots,{x}_{n})$, where $n\geq k$, be a complete
coordinate system on $X$ extending $({x}_{1},\ldots,{x}_{k})$. If $ft^{a}\in
A_{a}t^{a}$, and $|\alpha|/a_{1}<a$ then

$D_{\overline{x}t^{1/a_{1}}}^{\alpha}(ft^{a})\in
A_{a-(|\alpha|/a_{1})}t^{a-(|\alpha|/a_{1})},$

###### Proof.

The property can be verified on monomials $(x_{1}^{b_{1}}\cdot\ldots\cdot
x_{k}^{b_{k}})t^{a}\in A_{a}t^{a}$ and on the differential operators
$D_{x_{i}t^{1/a_{1}}}$ for $i=1,\ldots,k$. If $x_{1}^{b_{1}}\cdot\ldots\cdot
x_{k}^{b_{k}}\in A_{a}$ then $b_{1}/a_{1}+\ldots b_{k}/a_{k}\geq a$. So

$b_{1}/a_{1}+\ldots+(b_{i}-1)/a_{i}+\ldots+b_{k}/a_{k}\geq a-(1/a_{1})$

and

$D_{x_{i}t^{1/a_{1}}}(x_{1}^{b_{1}}\cdot\ldots\cdot x_{k}^{b_{k}}\cdot
t^{a})\sim(x_{1}^{b_{1}}\cdot\ldots\cdot x_{i}^{b_{i}-1}\cdot\ldots\cdot
x_{k}^{b_{k}}\cdot t^{a-(1/a_{1})})\in A_{a-(1/a_{1})}$

∎

###  3.5. Coefficient ideal of Rees algebra

The origin of the coefficient ideals concept can be traced back to the work of
Abhyankhar and Hironaka, as seen in [Hir64]. Various definitions of this
notion have been explored in multiple studies, including [Vil89], [BM97],
[Wło05], and [Kol07], among others.

In the context of our methodology, the coefficient ideal is used to derive a
better approximation of a maximal admissible center. As demonstrated in
Section 3.7.1, the maximal admissible center ${\mathcal{A}}$ can be obtained
by the recursive application of coefficient ideals to a specified Rees algebra
$R={\mathcal{O}}_{X}[{\mathcal{I}}t]$. The coefficient ideal, as per our
definition, is a non-canonical concept that is relatively straightforward to
compute in the completion $\widehat{{\mathcal{O}}}_{X,p}$ in what we refer to
as a ’split form’ (as discussed in Section 3.5.4).

This version of the coefficient ideal simplifies and streamlines computations
and can be likened to the approach used by Bierstone-Milman in their
resolution algorithm, as referenced in [BM91] and [BM97], albeit in a
differing framework.

######  Remark 3.5.1.

If $D_{\overline{x}}$ splits in ${\mathcal{O}}_{X}$ then any
$f\in{\mathcal{O}}_{X}$ can be written in the coefficient form

$f=\sum c_{\alpha}\overline{x}^{\alpha},$

where $c_{\alpha}\in{\mathcal{O}}_{V({\overline{x}})}\subset{\mathcal{O}}_{X}$
so we can obtain a particularly nice description of the center on $X$. This
can be always done in the completion $\widehat{{\mathcal{O}}}_{X,p}$.

######  Definition 3.5.2.

Let $a_{1}\in{\mathbb{Q}}_{>0}$, and
$R={\mathcal{O}}_{X}[f_{j}t^{b_{j}}]_{j=1,\ldots,s}$ be a Rees algebra
generated by $f_{j}t^{b_{j}}$ for $j=1,\ldots,s$ and let $\overline{x}$ be any
partial system of local coordinates compatible with $E$ at $p$, and set
$H:=V(\overline{x})$. By the coefficient ideal with respect to
$\overline{x}t^{{1/a_{1}}}$ we mean the Rees algebra:

$C_{\overline{x}t^{1/a_{1}}}(R):={\mathcal{O}}_{X}[(\,\,\overline{x}t^{1/a_{1}})^{\operatorname{Int}},D_{\overline{x}t^{1/a_{1}}}^{\alpha}(f_{j}t^{b_{j}}),\quad|\alpha|<b_{j}a_{1},\,\,j=1,\ldots,s\,]$

Recall that 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 ${\mathcal{O}}_{X}[t^{1/w}]$ containing all the
generators in $C_{\overline{x}t^{1/a_{1}}}(R)$. Thus

${\mathcal{O}}_{X}[\,\,\overline{x}t^{1/a_{1}}]^{\operatorname{Int}}={\mathcal{O}}_{X}[\overline{x}t^{1/w},\ldots,\overline{x}t^{w_{1}/w}],$

where $w_{1}=w/a_{1}$, so $\overline{x}t^{w_{1}/w}=\overline{x}t^{1/a_{1}}$.

######  Remark 3.5.3.

The key concept of the coefficient ideal as well as the role of the graded
local parameters $\overline{x}_{1}t^{1/a_{1}}$ are encapsulated by Lemma
3.5.11. The coefficient ideal maintains admissibility property, while
containing already some generators $\overline{x}_{1}t^{1/a_{1}}$ from a to be
constructed maximal admissible center ${\mathcal{A}}$. Moreover after
splitting it is generated by
$(\overline{x}_{1}t^{1/a_{1}})^{\operatorname{Int}}$ and the part
$C_{\overline{x}_{1}t^{1/a_{1}}}(R)_{|V(\overline{x}_{1}}$ which is
independent of the coordinates $\overline{x}_{1}$.

####  3.5.4. Coefficient ideal in the split form

The process of computing the coefficient ideal becomes remarkably
straightforward when utilizing splitting derivations $D_{\overline{x}}$ in
${\mathcal{O}}_{X}$. This can be always done by passing to the completion
$\widehat{{\mathcal{O}}}_{X,p}$ or to a corresponding étale neighborhood.
Assume that $(\overline{x},D_{\overline{x}})$ splits on $X$. For any element
$f_{j}t^{b_{j}}\in R_{b_{j}}$, one can write $f_{j}$ in ${{\mathcal{O}}}_{X}$
we can write

$f_{j}\equiv\sum
c_{j\alpha}\overline{x}^{\alpha}=\sum_{|\alpha|<b_{j}a_{1}}c_{j\alpha}\overline{x}^{\alpha}+\sum_{|\alpha|\geq
b_{j}a_{1}}c_{j\alpha}\overline{x}^{\alpha}\in\widehat{{\mathcal{O}}}_{X},p$

where $c_{j\alpha}\in\widehat{{\mathcal{O}}}_{H}\subset{\mathcal{O}}_{X}$ for
$H:=V(\overline{x})$. Then, for $|\alpha|<b_{j}a_{1}$ we have

$D^{\alpha}_{\overline{x}t^{1/a_{1}}}(f_{j}t^{b_{j}})_{|H}=c_{j\alpha}t^{b_{j}-|\alpha|/a_{1}}$

Thus the restricted coefficient ideal can be represented in the format which
is simpler for computations and justifies the name ”coefficient”:

######  Lemma 3.5.5.

$C_{\overline{x}t^{1/a_{1}}}(R)_{|H}:={\mathcal{O}}_{H}[\,D_{\overline{x}t^{1/a_{1}}}^{\alpha}(f_{j}t^{b_{j}})_{|H},|\alpha|<b_{j}a_{1}]={\mathcal{O}}_{H}[\,c_{j\alpha}t^{b_{j}-|\alpha|/a_{1}},|\alpha|<b_{j}a_{1}\,]$

As a corollary from the above we obtain:

######  Lemma 3.5.6.

Suppose $D_{\overline{x}}$ splits in ${\mathcal{O}}_{X}$. Then, with the above
notations and the assumptions the coefficient ideal
$C_{\overline{x}t^{1/a_{1}}}(R)$ can be written in the split form:

$\displaystyle
C_{\overline{x}t^{1/a_{1}}}(R)={\mathcal{O}}_{X}[(\overline{x}t^{1/a_{1}})^{\operatorname{Int}},C_{\overline{x}t^{1/a_{1}}}(R)_{|H}]=$
$\displaystyle={\mathcal{O}}_{X}[(\overline{x}t^{1/a_{1}})^{\operatorname{Int}},\,c_{j\alpha}t^{b_{j}-|\alpha|/a_{1}},|\alpha|<b_{j}a_{1}\,]$

where $c_{j\alpha}\in{\mathcal{O}}_{H}\subset{\mathcal{O}}_{X}$ , for
$H=V(\overline{x})$.

###### Proof.

Let $R={\mathcal{O}}_{X}[f_{j}t^{b_{j}}]$. In the presentation

$f_{j}=\sum
c_{j\alpha}\overline{x}^{\alpha}=\sum_{|\alpha|<aa_{1}}c_{j\alpha}\overline{x}^{\alpha}+\sum_{|\alpha|\geq
aa_{1}}c_{j\alpha}\overline{x}^{\alpha}\in{\mathcal{O}}_{X}$

the term

$(\sum_{|\alpha|\geq
b_{j}a_{1}}c_{j\alpha}\overline{x}^{\alpha})t^{b_{j}}=(f_{j}-\sum_{|\alpha|<b_{j}a_{1}}c_{j\alpha}\overline{x}^{\alpha})t^{b_{j}}\in\,\,{\mathcal{O}}_{X}[\overline{x}t^{1/a_{1}}]^{\operatorname{Int}}$

Then we have

$f^{0}_{j}t^{b_{j}}:=\sum_{|\alpha|<b_{j}a_{1}}c_{j\alpha}\overline{x}^{\alpha}t^{b_{j}}\,\,\in\,\,C_{\overline{x}t^{1/a_{1}}}(R)$

Thus for any maximal $\alpha_{j}^{0}$ in the above presentation we have

$\frac{1}{\alpha_{0}!}D^{\alpha_{j}^{0}}_{\overline{x}t^{1/a}}(f^{0}_{j}t^{b_{j}})=c_{j\alpha_{j}^{0}}t^{b_{j}-|\alpha_{j}^{0}|/a_{1}}\in
C_{\overline{x}t^{1/a_{1}}}(R).$

Thus

$c_{j\alpha_{j}^{0}}x^{\alpha_{0}}t^{b_{j}}=c_{j\alpha_{j}^{0}}t^{b_{j}-|\alpha_{j}^{0}|/a_{1}}\cdot
x^{\alpha_{j}^{0}}t^{|\alpha_{j}^{0}|/a_{1}}\in
C_{\overline{x}t^{1/a_{1}}}(R).$

Next consider

$f^{1}_{j}t^{b_{j}}:=f^{0}_{j}t^{b_{j}}-c_{j\alpha_{j}^{0}}x^{\alpha_{0}}t^{b_{j}}\in
C_{\overline{x}t^{1/a_{1}}}(R),$

and repeat the process for maximal $\alpha_{j}^{1}$ in $f^{1}_{j}$, and
continue inductively for $\alpha_{j}^{i}$ in $f^{i}_{j}$ until $f^{k}_{j}=0$.
∎

In particular, we have

######  Corollary 3.5.7.

$\displaystyle\widehat{{\mathcal{O}}_{X,p}}\cdot{\mathcal{C}}_{\overline{x}t^{1/a_{1}}}(R)=\widehat{{\mathcal{O}}_{X,p}}[(\overline{x}_{1}t^{1/a_{1}})^{\operatorname{Int}},{\mathcal{C}}_{\overline{x}_{1}t^{1/a_{1}}}(R)_{|H}]=$
$\displaystyle=\widehat{{\mathcal{O}}}_{X,p}[(\overline{x}t^{1/a_{1}})^{\operatorname{Int}},\,c_{j\alpha}t^{b_{j}-|\alpha|/a_{1}},|\alpha|<b_{j}a_{1}\,].$

####  3.5.8. Split vs non-split form of the coefficient ideal

The split form of the coefficient ideal is highly efficient for computations.
The subsequent example provides a comparison between two methods used for
computing the coefficient ideal:

######  Example 3.5.9.

Let $a_{1}=3$ then $H=V(x)$ splits in $X=\operatorname{Spec}(K[x,y,z]$ and

$ft=(x^{3}+x^{2}yw+z^{n})t=(xt^{1/3})^{3}+(xt^{1/3})^{2}ywt^{1/3}+z^{n}t$

has coefficients $ywt^{1/3}$, and $z^{n}t$. Thus the split form of the
coefficient ideal :

$C_{xt^{1/3}}(x^{3}+x^{2}yw+z^{n})t={\mathcal{O}}_{X}[(xt^{1/3})^{\operatorname{Int}},ywt^{1/3},z^{n}t]$

is generated by $(xt^{1/3})^{\operatorname{Int}}$ and the coefficients
$ywt^{2/3},z^{n}t$.

Note that the standard form of the coefficient ideal is more complicated,
though still easily computable in this case:

$\displaystyle
C_{xt^{1/3}}(x^{3}+x^{2}yw+z^{n})t={\mathcal{O}}_{X}[(xt^{1/3})^{\operatorname{Int}},ft,D_{xt^{1/3}}(ft),D^{2}_{xt^{1/3}}(ft)]=$
$\displaystyle={\mathcal{O}}_{X}[(xt^{1/3})^{\operatorname{Int}},(x^{3}+x^{2}yw+z^{n})t,(3x^{2}+2xyw)t^{2/3},ywt^{1/3}],$

####  3.5.10. Coefficient ideals and admissibility

######  Lemma 3.5.11.

Let $R={\mathcal{O}}_{X}[f_{j}t^{b_{j}}]$ be the Rees algebra on $X$. The
following conditions are equivalent in a neighborhood of $p\in X$.

  1. (1)

$R\subseteq{\mathcal{A}}^{\operatorname{Int}}={\mathcal{O}}_{X}[\overline{x}_{1}t^{1/a_{1}},\ldots,\overline{x_{k}}t^{1/a_{k}}]^{{\operatorname{Int}}}$.

  2. (2)

${\mathcal{C}}_{\overline{x}_{1}t^{1/a_{1}}}(R)\subseteq{\mathcal{A}}={\mathcal{O}}_{X}[\overline{x}_{1}t^{1/a_{1}},\ldots,\overline{x_{k}}t^{1/a_{k}}]^{\operatorname{Int}}$

  3. (3)

${\mathcal{C}}_{\overline{x}_{1}t^{1/a_{1}}}((R)_{|H}\subseteq{\mathcal{A}}_{|H}={\mathcal{O}}_{H}[\overline{x}_{2|H}t^{1/a_{2}},\ldots,\overline{x}_{k|H}t^{1/a_{k}}]^{\operatorname{Int}}$.

###### Proof.

We can pass to the completion $\widehat{{\mathcal{O}}}_{X,p}$, and replace
$\overline{x}_{i}$ with $\overline{x_{i|H}}$, so that $D_{\overline{x}_{1}}$
split and are compatible with ${\mathcal{A}}$. Then
${\mathcal{C}}_{\overline{x}_{1}t^{1/a_{1}}}(R)$ is obtained from $R$ by
applying ${\mathcal{D}}_{\overline{x}_{1}t^{1/a_{1}}}$. Note that
$D_{\overline{x}_{1}}$ splits, it is compatible with $\overline{x}_{i|H}$ in
$\overline{{\mathcal{O}}}_{X,p}$.

$(1)\Rightarrow(2)$ by Lemma 3.4.3, the operator
${\mathcal{D}}_{\overline{x}_{1}t^{1/a_{1}}}$ preserves both sides of the
inclusion in (1) giving (2).

$(2)\Rightarrow(3)$ The assertion in (3) follows from (2) simply by taking the
restriction.

$(3)\Rightarrow(1)$ By Lemma 3.5.6,
${\mathcal{C}}_{\overline{x}_{1}t^{1/a_{1}}}(R)\supset R$ can be written in
the split form

${\mathcal{C}}_{\overline{x}_{1}t^{1/a_{1}}}(R)={\mathcal{O}}_{X}[(\overline{x}t^{1/a_{1}})^{\operatorname{Int}},{\mathcal{C}}_{\overline{x}_{1}t^{1/a_{1}}}((R)_{|H}]\subset{\mathcal{O}}_{X}[\overline{x}_{1}t^{1/a_{1}},\ldots,\overline{x_{k}}t^{1/a_{k}}]^{\operatorname{Int}}.$

∎

####  3.5.12. Nested Rees algebras

The following definition is motivated by the construction of the coefficient
ideal.

######  Definition 3.5.13.

Given a partial system $\overline{x}_{1},\ldots,\overline{x}_{n}$ of local
parameters compatible with $E$. Let

$R_{|H_{k}}=\bigoplus R_{a|H_{k}}t^{a}$

be the restriction of $R$ to
$H_{k}:=V(\overline{x}_{1},\ldots,\overline{x}_{k})$, for $k\leq n$, and let

$0<a_{1}<a_{2}\ldots<a_{k}$

be a sequence of rational numbers.

We say that the Rees algebra $R$ is strictly nested at $H_{k}$ if
$H_{k}\subset X$ splits and

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

We say that the Rees algebra $R$ is nested at $H_{k}$ if it contains
subalgebra

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

and such that $\widehat{{\mathcal{O}}}_{X,p}\cdot R$ is strictly nested at
$H_{k}$:

${\widehat{{\mathcal{O}}_{X,p}}}\cdot
R={\widehat{{\mathcal{O}}_{X,p}}}[(\overline{x}_{1}t^{1/a_{1}},\ldots,\overline{x}_{k}t^{1/a_{k}})^{\operatorname{Int}},R_{|H_{k}}],$

The order ${\operatorname{ord}}_{p}(R_{|H_{k}})$ will be called the nested
order of $R$ at $H_{k}$ and $p$.

Using this terminology one can rephrase the Lemma 3.5.6 we get

######  Lemma 3.5.14.

The coefficient ideal $C_{\overline{x}t^{1/a_{1}}}(R)$ is nested at
$H=V(\overline{x})$.

Moreover, if $(H,D_{\overline{x}})$ splits in ${{\mathcal{O}}}_{X}$ then
$C_{\overline{x}t^{1/a_{1}}}(R)$ is strictly nested at $H$:

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

∎

In particular we can always write

$\widehat{{\mathcal{O}}}_{X,p}\cdot
C_{\overline{x}t^{1/a_{1}}}(R)=\widehat{{\mathcal{O}}}_{X,p}[(\overline{x}t^{1/a_{1}})^{\operatorname{Int}},C_{\overline{x}t^{1/a_{1}}}(R)_{|H}]$

####  3.5.15. Nested coefficient ideals

The definition of the coefficient ideal can be extended to the setting of
nested Rees algebras.

######  Definition 3.5.16.

Let $\overline{x}_{1},\ldots,\overline{x}_{n}$ be a local system of local
coordinates. Let

$R={\mathcal{O}}_{X}[(\overline{x}_{1}t^{1/a_{1}},\ldots,\overline{x}_{k}t^{1/a_{k}})^{\operatorname{Int}},f_{j}t^{b_{j}}]_{j=1,\ldots,s}$

be the Rees algebra nested at
$H_{k}=V(\overline{x}_{1},\ldots,\overline{x}_{k})$. Assume $a_{k+1}\geq
a_{k}\geq\ldots\geq a_{1}$. Then we shall call the coefficient ideal
$C_{\overline{x}_{k+1}t^{1/a_{k+1}}}(R)$ at $\overline{x}_{k+1}t^{1/a_{k+1}}$
nested for $R$ at $H_{k}=V(\overline{x}_{1},\ldots,\overline{x}_{1})$.

######  Lemma 3.5.17.

With the above notation the nested coefficient ideal of $R$ at
$\overline{x}_{k+1}t^{1/a_{k+1}},$ with respect to $H_{k}$ can be written as

$\displaystyle C_{\overline{x}_{k+1}t^{1/a_{k+1}}}(R)$ $\displaystyle:=$
$\displaystyle:=$
$\displaystyle{\mathcal{O}}_{X}[(x_{1}t^{1/a_{1}},\ldots,x_{k+1}t^{1/a_{k+1}})^{\operatorname{Int}},D_{\overline{x}_{k+1}t^{1/a_{k+1}}}^{\alpha}(f_{j}t^{b_{j}}),|\alpha|/a_{k+1}\leq
b_{j}]$

The subsequent statement is a straightforward extension of Lemma 3.5.7:

######  Lemma 3.5.18.

Let $R$ be a Rees algebra nested at
$H_{k}=V(\overline{x}_{1},\ldots,\overline{x}_{k+1})$. The nested coefficient
ideal $C_{\overline{x}_{k+1}t^{1/a_{k+1}}}(R)$ at
$\overline{x}_{k+1}t^{1/a_{k+1}}$ is a nested Rees algebra at
$H_{k+1}=V(\overline{x}_{1},\ldots,\overline{x}_{k+1})$.

###### Proof.

By the assumption

$\widehat{{\mathcal{O}}}_{X,p}\cdot
R=\widehat{{\mathcal{O}}}_{X,p}[(\overline{x}_{1}t^{1/a_{1}},\ldots,\overline{x}_{k}t^{1/a_{k}})^{\operatorname{Int}},R_{|H_{k}}]$

Then by the above

$\widehat{{\mathcal{O}}}_{X,p}\cdot
C_{\overline{x}_{k+1}t^{1/a_{k+1}}}(R)=\widehat{{\mathcal{O}}}_{X,p}[(\overline{x}_{1}t^{1/a_{1}},\ldots,\overline{x}_{k}t^{1/a_{k}})^{\operatorname{Int}},C_{\overline{x}_{k+1}t^{1/a_{k+1}}}(R_{|H_{k}})]$

is also nested in $H_{k}$. On the other hand, by Lemma 3.5.7,
$C_{\overline{x}_{k+1}t^{1/a_{k+1}}}(R_{|H_{k}})$ is nested in $H_{k+1}$ on
$H_{k}$. Consequently $C_{\overline{x}_{k+1}t^{1/a_{k+1}}}(R)$ is nested on
$X$ at $H_{k+1}$. ∎

####  3.5.19. Nested coefficient ideals in the split form

We can adapt Lemma 3.5.6 to strictly nested Rees algebras: Let
$\overline{x}_{1},\ldots,\overline{x}_{n}$ be a system of local coordinates
compatible with $E$ at $p$. Let

$\displaystyle
R={\mathcal{O}}_{X}[(\overline{x}_{1}t^{1/a_{1}},\ldots,\overline{x}_{k}t^{1/a_{k}})^{\operatorname{Int}},f_{j}t^{b_{j}}]_{j=1,\ldots,s}$

be a Rees algebra nested at
$H_{k}=V(\overline{x}_{1},\ldots,\overline{x}_{k})$, for $k+1\leq n$. Then

$\widehat{{\mathcal{O}}}_{X,p}\cdot
R=\widehat{{\mathcal{O}}}_{X,p}[(\overline{x}_{1}t^{1/a_{1}},\ldots,\overline{x}_{k}t^{1/a_{k}})^{\operatorname{Int}},f_{j|H_{k}}t^{b_{j}}]_{j=1,\ldots,s}$

with $f_{j|H_{k}}\in{\mathcal{O}}_{H_{k}}$. Moreover
$({x}_{k+1},D_{\overline{x}_{k+1}})$ splits in
$\operatorname{Spec}(\widehat{{\mathcal{O}}}_{X,p})$ and we can write each
generator $f_{j|H_{k}}t^{b_{j}}$ as

$\displaystyle
f_{j|H_{k}}t^{b_{j}}\equiv\sum_{|\alpha|<aa_{k+1}}c_{j\alpha}\overline{x}_{k+1}^{\alpha}t^{b_{j}}\equiv\sum_{|\alpha|<aa_{k+1}}(c_{j\alpha}t^{b_{j}-|\alpha|/a_{k+1}})\overline{x}_{k+1}^{\alpha}t^{|\alpha|/a_{k+1}}$

modulo the ideal

$\widehat{{\mathcal{O}}}_{X,p}((\overline{x}_{k+1})^{\alpha}t^{b_{j}}\mid|\alpha|\geq
b_{j}a_{k+1})\subset\widehat{{\mathcal{O}}}_{X,p}[\overline{x}_{1}t^{1/a_{1}},\ldots,\overline{x}_{k}t^{1/a_{k}}]^{\operatorname{Int}},$

where
$c_{j\alpha}=c_{j\alpha}(\overline{x}_{k+2},\ldots,\overline{x}_{n})\in{\mathcal{O}}_{H_{k+1}}\subset\widehat{{\mathcal{O}}}_{X,p}$.
Thus we obtain the result:

######  Lemma 3.5.20.

With the above assumptions and notations the completion of the nested
coefficient ideal with respect to $\overline{x}_{k+1}t^{1/a_{k+1}}$ is equal
to:

$\displaystyle\widehat{{\mathcal{O}}}_{X,p}\cdot
C_{\overline{x}_{k+1}t^{1/a_{k+1}}}(R)=\widehat{{\mathcal{O}}}_{X,p}[(\overline{x}_{1}t^{1/a_{1}},\ldots,\overline{x}_{k}t^{1/a_{k}})^{\operatorname{Int}},C_{\overline{x}_{k+1}t^{1/a_{k+1}}}(R)_{|H}]=$
$\displaystyle=\widehat{{\mathcal{O}}}_{X,p}[(\overline{x}_{1}t^{1/a_{1}},\ldots,\overline{x}_{k+1}t^{1/a_{k+1}})^{\operatorname{Int}},\,c_{j\alpha}t^{b_{j}-|\alpha|/a_{k+1}},|\alpha|<b_{j}a_{k+1}\,]$

with
$c_{j\alpha}=c_{j\alpha}(\overline{x}_{k+2},\ldots,\overline{x}_{n})\in{\mathcal{O}}_{H_{k+1}}\subset\widehat{{\mathcal{O}}}_{X,p}$.

∎

####  3.5.21. Strictly nested coefficient ideals in the split form

The result can be stated for the strictly nested Rees algebras in view of
Lemma 3.3.4.

######  Lemma 3.5.22.

Let $\overline{x}_{1},\ldots,\overline{x}_{n}$ be a complete coordinate
system. Let

$\displaystyle
R={\mathcal{O}}_{X}[(\overline{x}_{1}t^{1/a_{1}},\ldots,\overline{x}_{k}t^{1/a_{k}})^{\operatorname{Int}},f_{j}t^{b_{j}}]_{j=1,\ldots,s}$

be a Rees algebra nested at
$H_{k}=V(\overline{x}_{1},\ldots,\overline{x}_{k})$, for $k\leq n-1$, where
$f_{j}\in{\mathcal{O}}_{H_{k}}\subset{\mathcal{O}}_{X}$. Assume that
$H_{k+1}\subset X$ splits. Then upon the coordinate change
$\overline{x}_{1},\ldots,\overline{x}_{k},\overline{x}_{k+1|H_{k}},\ldots,\overline{x}_{n|H_{k}}$,
$D_{\overline{x}_{k+1}}=\partial_{\overline{x}_{k+1}}$ splits in
${\mathcal{O}}_{X}$ and the coefficient ideal
$C_{\overline{x}_{k+1}t^{1/a_{k+1}}}(R)$ is strictly nested at $H_{k+1}$ and
admits the split form

$C_{\overline{x}_{k+1}t^{1/a_{k+1}}}(R)={{\mathcal{O}}}_{X}[(\overline{x}_{1}t^{1/a_{1}},\ldots,\overline{x}_{k+1}t^{1/a_{k+1}})^{\operatorname{Int}},\,c_{j\alpha}t^{b_{j}-|\alpha|/a_{k+1}},|\alpha|<b_{j}a_{k+1}\,]$

with
$c_{j\alpha}=c_{j\alpha}(\overline{x}_{k+2},\ldots,\overline{x}_{n})\in{\mathcal{O}}_{H_{k+1}}\subset\widehat{{\mathcal{O}}}_{X}$.
∎

######  Example 3.5.23.

Let

$R={\mathcal{O}}_{X}[(xt^{1/3})^{\operatorname{int}},(y_{1}^{5}+y_{1}y_{2}^{6}+y_{3}^{7})t]$

be a Rees algebra strictly nested at $H_{1}=V(x)$. The subvariety
$H_{2}=V(x,y_{1})$ splits in $X=\operatorname{Spec}(K[x,y_{1},y_{2},y_{3}])$
and $D_{y_{1}}$ splits on $H_{1}$ (so $D_{y_{1}}$ vanishes on
${\mathcal{O}}_{H_{2}}=K[y_{1},y_{3}]\subset K[x,y_{1},y_{2},y_{3}]$). Then
the generator $(y_{1}^{5}+y_{1}y_{2}^{6}+y_{3}^{7})$ can be written in the
coefficient form with respect $y_{2}t^{1/5}$:

$(y_{1}^{5}+y_{1}y_{2}^{6}+y_{3}^{7})=(y_{1}t^{1/5})^{5}+(y_{1}t^{1/5})y_{2}^{6}t^{4/5}+y_{3}^{7}t],$

with the coefficients $c_{y^{1}}=y_{2}^{6}t^{4/5},c_{0}=y_{3}^{7}t$.
Consequently the nested coefficient ideal with respect to $yt^{1/5}$ is equal
to:

$C_{yt^{1/5}}{\mathcal{O}}_{X}[(xt^{1/3})^{\operatorname{Int}},(y_{1}^{5}+y_{1}y_{2}^{6}+y_{3}^{7})t]={\mathcal{O}}_{X}[(xt^{1/3},yt^{1/5})^{\operatorname{Int}},y_{2}^{6}t^{4/5},y_{3}^{7}t]$

is generated by $(xt^{1/3},yt^{1/5})^{\operatorname{Int}},$ and
$,y_{2}^{6}t^{4/5}$ and $y_{3}^{7}t$.

As an immediate corollary from Lemma 3.5.11, by considering the restriction
$R_{|H_{k}}$ we obtain its extension:

######  Lemma 3.5.24.

Let $R$ be the Rees algebra on $X$ nested at $H_{k}$, so that

$\widehat{{\mathcal{O}}}_{X,p}R=\widehat{{\mathcal{O}}}_{X,p}[(x_{1}t^{1/a_{1}},\ldots,x_{k}t^{1/a_{k}})^{\operatorname{Int}},R_{|H_{k}}].$

The following conditions are equivalent.

  1. (1)

$R\subset{\mathcal{A}}={\mathcal{O}}_{X}[\overline{x}_{1}t^{1/a_{1}},\ldots,\overline{x_{n}}t^{1/a_{n}}]^{\operatorname{Int}}$.

  2. (2)

${\mathcal{C}}_{\overline{x}_{k+1}t^{1/a_{k+1}}}(R)\subset{\mathcal{O}}_{X}[\overline{x}_{1}t^{1/a_{1}},\ldots,\overline{x_{n}}t^{1/a_{n}}]^{\operatorname{Int}}$

  3. (3)

${\mathcal{C}}_{\overline{x}_{k+1}t^{1/a_{k+1}}}(R)_{|H_{k+1}}\subset{\mathcal{O}}_{H_{k+1}}[\overline{x}_{k+2|H_{k+1}}t^{1/a_{k+2}},\ldots,\overline{x}_{n|H_{k+1}}t^{1/a_{n}}]^{\operatorname{Int}}$.

###### Proof.

Passing to the completion $\widehat{{\mathcal{O}}}_{X,p}$ we may assume that
$R$ is strictly nested. Moreover, by replacing $\overline{x}_{k+2}$ by
$\overline{x}_{k+2|H_{k+1}}$ we can assume that $D_{\overline{x}_{k+1}}$ is
compatible with ${\mathcal{A}}$. Let $R^{\prime}:=R_{|H_{k}}$. Observe that

${\mathcal{C}}_{\overline{x}_{k+1}t^{1/a_{k+1}}}(R^{\prime})=({\mathcal{C}}_{\overline{x}_{k+1}t^{1/a_{k+1}}}(R))_{|H_{k}}.$

By Lemma 3.5.11, we have the equivalent conditions for $R^{\prime}$. On the
other hand, since $R$ is strictly nested the condition

$R\subset{\mathcal{O}}_{X}[\overline{x}_{1}t^{1/a_{1}},\ldots,\overline{x_{n}}t^{1/a_{n}}]^{\operatorname{Int}}$

is equivalent to

$R^{\prime}=R_{|H_{k}}\subset{\mathcal{O}}_{X}[\overline{x}_{k+1|H_{k}}t^{1/a_{k+1}},\ldots,\overline{x}_{n|H_{k}}t^{1/a_{n}}]^{\operatorname{Int}}.$

Thus the result from Lemma 3.5.11. ∎

###  3.6. Maximal contact

####  3.6.1. Cotangent ideal of Rees algebra

From now on until the end of Chapter 3.7 we shall assume that the
characteristic of the base field $K$ is zero.

First, recall that given an ideal ${\mathcal{I}}$ on smooth variety $X$, and a
positive integer $a$, the set of points where
${\operatorname{ord}}_{p}({\mathcal{I}})\geq a$ is described as
$V({\mathcal{D}}_{X}^{\leq a-1}({\mathcal{I}}))$, where
${\mathcal{D}}_{X}^{\leq a-1}({\mathcal{I}})$ is the ideal generated by all
the derivatives $D_{\overline{x}^{\alpha}}(f)$, where $f\in{\mathcal{I}}$ and
$0\leq|\alpha|\leq a-1$. If the order of ${\mathcal{I}}$ is $a$ at a point
then the order ${\mathcal{D}}_{X}^{\leq a-1}({\mathcal{I}})$ is equal to $1$
and thus it contains a local parameter $u$, called a maximal contact, such
that $V(u)$ contains $V({\mathcal{D}}_{X}^{\leq a-1}({\mathcal{I}}))$. In such
a case we shall call the ideal
$T^{1/a}({\mathcal{I}}):={\mathcal{D}}_{X}^{\leq a-1}({\mathcal{I}})$
cotangent. The maximal contact plays a critical role in the classical Hironaka
resolution algorithm. It allows for the inductive reduction of the process to
the hypersurface of maximal contact.

In our context, the meaning of the maximal contact is different. It is simply
the coordinate in the gradation $t^{1/a_{1}}$ occurring a some presentation of
a maximal ${\mathcal{R}}$-admissible center
${\mathcal{A}}={\mathcal{O}}_{X}[\overline{x}_{1}t^{1/a_{1}},\ldots,\overline{x}_{k}t^{1/a_{k}}]$
for any Rees algebra $R$

Consider the admissibility condition

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

The idea of the cotangent ideal $T^{1/a_{1}}(R)$ is to gather the elements
occurring in gradation $t^{1/a_{1}}$, obtained by applying differential
operators ${\mathcal{D}}_{\overline{x}t^{1/a_{1}}}^{\alpha}$ to the generators
of $R$, for $\overline{x}$ being a complete system of coordinates. The
differential operators ${\mathcal{D}}_{\overline{x}t^{1/a_{1}}}^{\alpha}$
preserve the right side of the admissibility condition (See Lemma 3.4.3). Then
the ideal $T^{1/a_{1}}(R)$ is necessarily contained in the gradation
${\mathcal{A}}^{\operatorname{Int}}_{1/a_{1}}$. Conversely by the maximal
admissibility condition $T^{1/a_{1}}(R)t^{1/a_{1}}$ contains all the
coordinates in $\overline{x}_{1}t^{1/a_{1}}$ up to the higher order terms of
the gradation ${\mathcal{A}}^{\operatorname{Int}}_{1/a_{1}}$. So, by the
Replacement Lemma 3.1.29, we can assume that the $T^{1/a_{1}}(R)$ contains
$\overline{x}_{1}t^{1/a_{1}}$. This way we will determine
$\overline{x}_{1}t^{1/a_{1}}$\- part of the center ${\mathcal{A}}$ on the
right side of the inclusion with $\overline{x}_{1}\in T^{1/a_{1}}$, which
would be called a maximal contact at a point. A similar idea was used in the
first version of [ATW19] to interpret the algebra of the center.

Consequently $T^{1/a_{1}}(R)$ generalizes ${\mathcal{D}}_{X}^{\leq
a_{1}-1}({\mathcal{I}})$ for the ideal ${\mathcal{I}}$ of order $a$ at $p$. In
particular, if
${\mathcal{I}}t\subset{\mathcal{A}}={\mathcal{O}}_{X}[\overline{x}_{1}t^{1/a_{1}},\ldots,\overline{x}_{k}t^{1/a_{k}}]^{\operatorname{int}}$,
then by Lemma 3.4.3,

$t^{-(a_{1}-1/a_{1})}{\mathcal{D}}_{X}^{\leq
a_{1}-1}({\mathcal{I}})={\mathcal{D}}_{X}^{\leq
a_{1}-1}({\mathcal{I}})t^{1/a_{1}}\subset{\mathcal{O}}_{X}[\overline{x}_{1}t^{1/a_{1}},\ldots,\overline{x}_{k}t^{1/a_{k}}]^{\operatorname{int}}.$

Thus ${\mathcal{D}}_{X}^{\leq
a_{1}-1}({\mathcal{I}})\subset{\mathcal{A}}^{\operatorname{int}}_{1/a_{1}}$.

######  Definition 3.6.2.

Given a Rees algebra
$R={\mathcal{O}}_{X}[f_{j}t^{b_{j}}]_{j=1,\ldots,s}=\bigoplus R_{a}t^{a}$ on
an open affine subset $U$ of $X$, and a rational number $a_{1}>0$.

$\displaystyle
T^{1/a_{1}}(R):=\sum_{|\alpha|=b_{j}a_{1}-1}{\mathcal{O}}_{X}{\mathcal{D}}_{\overline{x}^{\alpha}}(f_{j})\quad\subseteq\quad{{\mathcal{T}}}^{\leq
1/a_{1}}(R):=\sum_{\,\,|\alpha|<b_{j}a_{1}}{\mathcal{O}}_{X}{\mathcal{D}}_{\overline{x}^{\alpha}}(f_{j})\subseteq$
$\displaystyle\subseteq{{\mathcal{T}}}^{<1/a_{1}}(R):=\sum_{\,\,|\alpha|\leq
b_{j}a_{1}}{\mathcal{O}}_{X}{\mathcal{D}}_{\overline{x}^{\alpha}}(f_{j}).$

######  Remark 3.6.3.

As the ideal $T^{1/a_{1}}(R)$ is in the gradation $t^{1/a_{1}}$, it can be
written in the graded form

$T^{1/a_{1}}(R)t^{1/a_{1}}:=\sum_{|\alpha|=b_{j}a_{1}-1}{\mathcal{D}}_{\overline{x}t^{1/a_{1}}}^{\alpha}(f_{j}t^{b_{j}}).$

######  Remark 3.6.4.

The definition intentionally uses a specific representation of Rees algebras
for the purpose of streamlining computations. The uniqueness of the center
obtained during the resolution process in Section 3.7.1 is automatic and
unaffected by any choices made.

The set of generators for $R$ can be chosen arbitrarily, and we have the
option to select the entire set of homogeneous elements of $R$ as generators
instead.

In the following examples, when we refer to $D_{X}^{i}({\mathcal{I}})\subset
D_{X}^{\leq i}({\mathcal{I}})$, as the ideal generated by all the derivatives
$D_{\overline{x}^{\alpha}}(f_{i})$, where $f_{i}\in{\mathcal{I}}$ are
generators of ${\mathcal{I}}$, and $|\alpha|=i$. This ideal is consistent with
the definition of $T^{1/a_{1}}(R)$.

######  Example 3.6.5.

Let $a_{1}=3={\operatorname{ord}}_{0}(R)$, where
$R={\mathcal{O}}_{X}[(x^{3}+y^{4}+z^{5})t]$.

$T^{1/3}(R)t^{1/3}={\mathcal{O}}_{X}[{\mathcal{D}}_{X}^{2}(x^{3}+y^{4}+z^{5})t^{1/3}]={\mathcal{O}}_{X}[(x,y^{2},z^{3})t^{1/3}]$

######  Example 3.6.6.

$a_{1}=3={\operatorname{ord}}_{0}(R)$, where

$R={\mathcal{O}}_{X}([(x^{3}+y^{4}+z^{5})t,(w^{2}x^{4}+z^{7})t^{2},(y^{2}+xz^{2})t^{2/3},w_{1}^{3}t^{3/4}]$

Then

$\displaystyle
T^{1/3}(R)t^{1/3}=(D_{X}^{2}(x^{3}+y^{4}+z^{5}),D_{X}^{5}(w^{2}x^{4}+v^{7}),D_{X}(v^{2}+xz^{2}))t^{1/3}=$
$\displaystyle=(x,w,v,y^{2},z^{2})t^{1/3},$

with local paremeters $x,w,v$ in gradation $t^{1/3}$. The element
$w_{1}^{3}t^{3/4}$ is of higher order $4>3$. It is thus ignored as
$ba_{1}=3/4\cdot 3=9/4$ is not integral. It will not contribute to the set of
local parameters of gradation $t^{1/3}$. Corresponding local parameter is
$w_{1}t^{1/4}$ is of order $4$.

On the other hand the ideal $T^{\leq 1/3}(R)=(x,w,v,y^{2},z^{2},w_{1})$ is
generated by $T^{1/3}(R)$, and $D^{2}(w_{1}^{3})\sim w_{1}$, for the
derivation order $2<ba_{1}=9/4$. The element $w_{1}$ is however, in gradation
$t^{1/4}$. The ideal $T^{\leq 1/3}(R)$ is used only in the context of Lemma
3.6.9.

In a more general context, we can state the following:

######  Lemma 3.6.7.

Let us assume that ${\operatorname{ord}}_{p}(R)=a_{1}$. Then

$T^{1/a_{1}}(R)+m_{p}^{2}={\bf C}_{1/a_{1}}+m_{p}^{2}$

where ${\bf C}_{1/a_{1}}t^{1/a_{1}}$ is the gradation of the algebra

${\bf C}_{\overline{x}t^{1/a_{1}}}=\bigoplus{\bf C}_{a}t^{a},$

generated by all the elements $D_{\overline{x}t^{1/a_{1}}}^{\alpha}(ft^{a})$,
where $f\in R_{a}$, and $|\alpha|<aa_{1}$. In particular,
$T^{1/a_{1}}(R)+m_{p}^{2}$ is independent of the choice of generators. ∎

####  3.6.8. Singular locus, and the cotangent ideal

The following lemma extends the analogous result for the ideals.

######  Lemma 3.6.9.

  1. (1)

${\operatorname{supp}}{\operatorname{ord}}(R,\geq a_{1}):=\\{q\in
U\mid{\operatorname{ord}}_{q}(R)\geq a_{1}\\}=V(\overline{T}^{\leq
1/a_{1}}(R))$.

  2. (2)

${\operatorname{supp}}{\operatorname{ord}}(R,>a_{1}):=\\{q\in
U\mid{\operatorname{ord}}_{q}(R)>a_{1}\\}=V(\overline{T}^{<1/a_{1}}(R)).$

###### Proof.

(1)${\operatorname{ord}}_{q}(R)\geq a_{1}$ if and only if
${\operatorname{ord}}_{q}(f_{j})\geq b_{j}a_{1}$ for any $j$ iff all
${\mathcal{D}}_{\overline{x}^{\alpha}}(f_{j})$ vanish at $q$ for
$|\alpha|<b_{j}a_{1}$. (2) The reasoning is similar. ∎

####  3.6.10. Maximal contact of Rees algebra

The concept of hypersurfaces of maximal contact was initially introduced by
Hironaka, Abhyankhar, and Giraud, and further developed in the works of
Bierstone-Milman, Villamayor, and others. In the case where ${\mathcal{I}}$ is
an ideal of order $a_{1}$, and no divisors are present a maximal contact is
defined as a local parameter $u\in
T^{1/a}({\mathcal{I}})={\mathcal{D}}_{X}^{a-1}({\mathcal{I}})\subset{\mathcal{D}}_{X}^{\leq
a-1}({\mathcal{I}})$. Its zero locus $V(u)$ includes the set
$V({\mathcal{D}}_{X}^{\leq a-1}({\mathcal{I}}))$, which consists of the
equimultiple locus of all points where
${\operatorname{ord}}_{p}({\mathcal{I}})=a$.

Our definition of maximal contact is intended for Rees algebras. From our
perspective, a maximal contact refers to a partial coordinate system
$\overline{x}_{1}$ found in $t^{1/a_{1}}$-gradation of a maximal
$R$-admissible center

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

at a point $p$. It can be characterized using Corollary 3.6.20. This
definition a priori does not need the property that $V(\overline{x}_{1})$
contains the equimultiple locus and thus can be used in a nonzero
characteristic. (See for instance Example 4.5.4)

The definition is straightforward when no divisors are present. In such cases,
”maximal contact” refers to a maximal partial set of coordinates contained in
$T^{1/a_{1}}({\mathcal{R}})$. However, in general, maximal contact at a point
$p$ is a partial set of coordinates that is compatible with an SNC divisor,
consisting of free coordinates in $T^{1/a_{1}}({\mathcal{R}})$ along with the
divisorial coordinates contained in $T^{1/a_{1}}({\mathcal{R}})+m_{p}^{2}$,
thus tangential to $T^{1/a_{1}}({\mathcal{R}})$.

The concept is specifically designed to enhance the efficiency of computations
by eliminating unnecessary steps. Unlike in the standard approach, the notion
incorporates multiple maximal contact coordinates simultaneously, which
simplifies the representation of the main invariant and significantly aids in
its computation. The idea of utilizing multiple maximal contacts is well-known
among experts and is commonly employed in practical implementations for its
effectiveness.

######  Definition 3.6.11.

Given a Rees algebra $R$ on $X$ and a rational number $a_{1}>0$. By a partial
maximal contact of $(R,a_{1})$ on an open affine subset $U$ we mean a partial
system of coordinates on $U$ compatible with $E$:

$\overline{x}=(x_{1},\ldots,x_{s},x_{s+1},\ldots,x_{r}),$

such that

  1. (1)

If $i\leq s$ then $x_{i}\in T^{1/a_{1}}(R)(U)$ and $x_{i}$ is free on $U$. In
particular $\frac{\partial}{\partial x_{i}}(T^{1/a_{1}}R)={\mathcal{O}}_{U}$.

  2. (2)

If $s+1\leq i\leq r$, then $V(x_{i})\in E$, and $\frac{\partial}{\partial
x_{i}}(T^{1/a_{1}}R)={\mathcal{O}}_{U}$,

If $p\in V(\overline{x})$, and ${\operatorname{ord}}_{p}(R)=a_{1}$, and the
conditions (1) and (2) are satisfied in a neighborhood of a point $p$ then
$\overline{x}$ is a partial maximal contact of $R$ at $p$.

If additionally $\frac{\partial}{\partial
x_{j}}(T^{1/a_{1}}R)\neq{\mathcal{O}}_{X}$ at $p$ for $j>r$, where
$(x_{1},\ldots,x_{n})$ is a local system of parameters extending
$(x_{1},\ldots,x_{r})$ then we say that $\overline{x}$ is a maximal contact of
$R$ at $p$.

We shall associate partial maximal contact with respect to $(R,a)$ gradation
$t^{1/a_{1}}$ and write it in the graded form $\overline{x}t^{1/a_{1}}$.

######  Lemma 3.6.12.

Let $\overline{x}_{1}t^{1/a_{1}}$ be a maximal contact for $R$ at $p$, with
${\operatorname{ord}}_{p}(R)=a_{1}$. Then it is a partial maximal contact for
$(R,a_{1})$ in a certain neighborhood $U$ of $p$. ∎

###### Proof.

Let us consider an open neighborhood where the conditions (1) and (2) of
Definition 3.6.11 are met. ∎

######  Remark 3.6.13.

Note that a partial maximal contact on $U$ exist if $a_{1}$ is an integer such
that ${\operatorname{ord}}_{p}(R)\leq a_{1}$ for all $p\in U$. More
specifically we have:

######  Lemma 3.6.14.

If $\overline{x}$ is a partial maximal contact of $(R,a_{1})$ on $U$, then
${\operatorname{ord}}_{q}(R)\leq a_{1}$ for any $q\in U$.

###### Proof.

It follows from the definition of the partial maximal contact that
${\mathcal{D}}_{X}(T^{1/a_{1}}R)={\mathcal{O}}_{U}$. As a consequence, for
every point $q$ in $U$, there exists
${\mathcal{D}}_{\overline{x}^{\alpha}}(f_{j})$ that is invertible at $q$,
where $f_{j}t^{b_{j}}$ is a generator of $R$ and $|\alpha|=b_{j}a_{1}$.

∎

######  Example 3.6.15.

Let

$R={\mathcal{O}}_{X}[(x^{3}+y^{5}+z^{7})t,(v^{4}w^{2})t^{2}]$

where $a_{1}=3={\operatorname{ord}}_{0}(R)$, as in Example 3.6.6.

Then $T^{1/3}(R)t^{1/3}=(x,y^{3},z^{4},v,w)t^{1/3},$ and the maximal contact
at $0$ is given by $(x,v,w)t^{1/3}$ in gradation $t^{1/3}$. It is a partial
maximal contact for $(R,3)$ on $X$.

######  Lemma 3.6.16.

  1. (1)

A maximal contact of $R$ at $p\in X$ exists.

  2. (2)

Any partial maximal contact of $R$ at $p$ can be extended to a maximal contact
of $R$ at $p$.

  3. (3)

The image of a maximal contact $\overline{x}$ at $p$ in $m_{p}/m_{p}^{2}$ is
determined uniquely. It is the smallest subspace compatible with $E_{p}$
containing the image of $T^{1/a_{1}}_{p}(R)$.

  4. (4)

The divisorial coordinates in the maximal contact $\overline{x}$ at $p$ are
uniquely determined and do not depend upon the maximal contact.

###### Proof.

According to the definition, for a given $a$, we have
${\operatorname{ord}}_{p}(R_{a})=aa_{1}$. Consequently,
$D^{aa_{1}-1}(R_{a})\subset T^{1/a_{1}}R$, and
${\operatorname{ord}}_{p}(D^{aa_{1}-1}(R_{a}))={\operatorname{ord}}_{p}(T^{1/a_{1}}R)=1$.

Let ${\mathcal{I}}_{E,p}$ denote the ideal generated by the divisorial
coordinates in $E_{p}$ (i.e., in $E$ passing through $p$). Consider the image

$\overline{T}^{a_{1}}R:=(T^{1/a_{1}}R+{\mathcal{I}}_{E,p}+m_{p}^{2})/({\mathcal{I}}_{E,p}+m_{p}^{2})$

of $T^{1/a_{1}}R$ in $m_{p}/({\mathcal{I}}_{E,p}+m_{p}^{2})$. Its basis is
determined by a partial system of local free parameters $x_{1},\ldots,x_{s}\in
T^{1/a_{1}}(R)$. Then we can extend it to a partial coordinate system

$\overline{x}:=(x_{1},\ldots,x_{s},x_{s+1},\ldots,x_{r}),$

where $x_{i}$, for $i\geq s+1$ are divisorial at $p$ with the smallest image
in $m_{p}/m_{p}^{2}$ containing the image of $T^{1/a_{1}}(R)$. Consequently,
for any divisorial $x_{j}$, there exists an element $v=\sum c_{i}x_{i}\in
T^{1/a_{1}}+m_{p}^{2}$ with $c_{j}\neq 0$ at $p$, and all $x_{i}$ are
divisorial.

Furthermore, $\overline{x}$ precisely represents a maximal contact of $R$ at
$p$. Conversely, any maximal contact of $R$ at $p$ can be constructed in this
manner.

If $\overline{x}$ is any partial maximal contact, its image in
$\overline{T}^{1/a_{1}}R$ can be extended to form a basis for
$\overline{T}^{1/a_{1}}R$. This allows us to extend the free part of
$\overline{x}$ to a maximal contact $\overline{x^{\prime}}$. We need to show
that $\overline{x^{\prime}}$ contains the entirety of $\overline{x}$.

If $x_{j}\in\overline{x}$ is a divisorial coordinate, according to the
previous discussion, $D_{x_{j}}(v)\neq 0$, where $v\in T^{1/a_{1}}(R)$ and
$v=\sum a_{i}x_{i},({\rm mod}\,\,m_{p}^{2})$, with $x_{i}$ all being
divisorial coordinates that include $x_{j}$. As the image of the maximal
contact $\overline{x^{\prime}}$ in $m_{p}/m_{p}^{2}$ necessarily includes the
image of $v$, it also contains the images of all $x_{i}$ with $a_{i}\neq 0$,
including the coordinate $x_{j}$. Hence,
$\overline{x}\subseteq\overline{x^{\prime}}$.

In conclusion, the divisorial coordinates are precisely those that appear in
the presentation of some $v\in T^{1/a_{1}}(R)$, where $v=\sum a_{i}x_{i},({\rm
mod}\,m_{p}^{2})$, with all $x_{i}$ being divisorial. Therefore, the
divisorial coordinates are uniquely determined.

∎

######  Lemma 3.6.17.

Let
$R\subset{\mathcal{A}}^{\operatorname{Int}}={\mathcal{O}}_{X}[(\overline{x}_{1}t^{1/a_{1}},\ldots,\overline{x}_{k}t^{1/a_{k}}]^{\operatorname{Int}}$
at a point $p\in X$ such that $a_{1}={\operatorname{ord}}_{p}(R)$. Then upon
the change of the coordinate representation of ${\mathcal{A}}$,
$\overline{x}_{1}$ contains a maximal contact $\overline{x}$ of $R$ at $p$.

###### Proof.

According to Lemma 3.4.3, the operators
$t^{-|\alpha|/a_{1}}{\mathcal{D}}_{x^{\alpha}}$ in the definition of
$T^{1/a_{1}}(R)$ preserve ${\mathcal{A}}^{\operatorname{Int}}$. Therefore, we
have the following inclusion:

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

implies that
$T^{1/a_{1}}(R)t^{1/a_{1}}\subset{\mathcal{A}}^{\operatorname{Int}}$. Consider
the free part $(x_{1},\ldots,x_{r})$ of the maximal contact $\overline{x}$ in
$T^{1/a_{1}}(R)$. According to Lemma 3.1.29, we can assume that

$(x_{1},\ldots,x_{r})t^{1/a_{1}}\subseteq\overline{x}_{1}t^{1/a_{1}}\subseteq{\mathcal{A}}^{\operatorname{Int}}_{1/a_{1}},$

after a coordinate change. Since $a_{i}>a_{1}$ for $i\geq 2$,the image of
$\overline{x}_{1}$ in $m_{p}/m_{p}^{2}$ is the same as the image of
${\mathcal{A}}^{{\operatorname{Int}}}_{1/a_{1}}$ in $m_{p}/m_{p}^{2}$ and
includes the image of $T^{1/a_{1}}(R)$. By Lemma 3.6.16, it also includes the
image of the maximal contact $\overline{x}$ of $R$. Moreover,
$\overline{x}_{1}$ is compatible with the divisorial structure, indicating
that it must also contain the divisorial part of the maximal contact itself.
Therefore, $\overline{x}_{1}\supseteq\overline{x}$.

∎

######  Definition 3.6.18.

To a partial maximal contact $\overline{x}=(x_{1},\ldots,x_{r})$ of $R$ on
$U$, we can associate the invariant

${\operatorname{inv}}^{1}(\overline{x})=(1,\ldots,1,1_{+},\ldots 1_{+}),$

where the $1$’s correspond to the free maximal coordinates in $\overline{x}$,
and the $1_{+}$’s correspond to the coordinates in $\overline{x}$ that define
the divisors on $U$.

######  Lemma 3.6.19.

Given a Rees algebra $R$ of order $a_{1}$ at $p\in X$ with a maximal contact
$\overline{x}$ of $R$ at $p$. Then

  1. (1)

${\operatorname{inv}}_{p}^{1}(R)=a_{1}{\operatorname{inv}}(\overline{x}).$

  2. (2)

$R\subset{\mathcal{O}}_{X}[\overline{x}t^{1/a_{1}},\overline{y}t^{1/a_{2}}]^{\operatorname{Int}},$
for some $a_{2}>a_{1}$, and a coordinate system $(\overline{x},\overline{y})$
at $p$ compatible with $E$.

  3. (3)

If $\overline{x}$ is a partial maximal contact for $(R,a_{1})$ on $U$ then
${\operatorname{inv}}_{p}^{1}(R)\leq a_{1}{\operatorname{inv}}(\overline{x})$.

###### Proof.

(1) and (2). If
$R\subset{\mathcal{O}}_{X}[(\overline{x}_{1}t^{1/a_{1}},\ldots,\overline{x}_{k}t^{1/a_{k}}])$
at $p$, where $a_{1}={\operatorname{ord}}_{p}(R)$, then according to Lemma
3.6.17, $\overline{x}_{1}$ contains a maximal contact of $R$ after a
coordinate change. Therefore, based on Definition 3.6.18, we have
${\operatorname{inv}}^{1}_{p}(R)\leq a_{1}{\operatorname{inv}}(\overline{x})$.

Moreover, note that since ${\operatorname{ord}}_{p}(R)\geq a_{1}$, we have
${\operatorname{ord}}_{p}(C_{\overline{x}t^{1/a_{1}}}R)\geq a_{1}$ and
${\operatorname{ord}}_{p}(C_{\overline{x}t^{1/a_{1}}}R)_{|H}\geq a_{1}$.
Suppose the latter order is equal to $a_{1}$. Then, since
$C_{\overline{x}t^{1/a_{1}}}(R)_{|H}$ is generated by the coefficients as
shown in Lemma 3.5.5, there exists a generator

$ft^{a}=\sum c_{\alpha}\overline{x}^{\alpha}t^{a}\in R_{a}t^{a}$

of $R$, with $c_{\alpha}=c_{\alpha}(\overline{y}),$ for the corresponding
system of local parameters $(\overline{x},\overline{y})$, such that
${\operatorname{ord}}_{p}(c_{\alpha})=a_{1}a-|\alpha|$ for some $\alpha$.
Moreover, for some $\beta$ with
$|\beta|={\operatorname{ord}}_{p}(c_{\alpha})=a_{1}a-|\alpha|$, the derivative
$D_{\overline{y}^{\beta}}(c_{\alpha})=D_{y_{1}^{b_{1}},\ldots
y_{k}^{b_{k}}}(c_{\alpha})$ is invertible. Consequently,
$D_{\overline{y}^{\beta}}D_{x^{\alpha}}(f)$ is invertible, and we have
$D_{y_{1}^{b_{1}-1},\ldots y_{k}^{b_{k}}}D_{x^{\alpha}}(f)\in T^{1/a_{1}}(R)$.
Thus, $D_{y_{1}}(T^{1/a_{1}}(R))$ is invertible, contradicting the condition
of maximal contact at a point as defined in Definition 3.6.11.

This implies that
${\operatorname{ord}}_{p}(C_{\overline{x}t^{1/a_{1}}}R){|H}=a_{2}>a_{1}$. By
Lemma 3.1.24, this further implies that
$C_{\overline{x}t^{1/a_{1}}|H}R\in{\mathcal{O}}_{H}[\overline{y}t^{1/a_{2}}]^{\operatorname{Int}}$.
Hence, based on Lemma 3.5.11,

$R\subseteq{\mathcal{O}}_{X}[\overline{x}t^{1/a_{1}},\overline{y}t^{1/a_{2}}]^{\operatorname{Int}},$

which yields,by Lemma 3.1.21, that ${\operatorname{inv}}^{1}_{p}(R)\geq
a_{1}{\operatorname{inv}}(\overline{x})$, resulting in equality.

(3) If $\overline{x}$ represents a partial maximal contact on $U$, then for
any point $p\in U$, either ${\operatorname{ord}}_{p}({\mathcal{A}})<a_{1}$ or
${\operatorname{ord}}_{p}({\mathcal{A}})=a_{1}$. According to Lemma 3.6.17, in
the latter case, $\overline{x}$ can be extended to a maximal contact
$\overline{x^{\prime}}$. Thus, we obtain the following:

${\operatorname{inv}}_{p}^{1}(R)=a_{1}{\operatorname{inv}}(\overline{x^{\prime}})\leq
a_{1}{\operatorname{inv}}(\overline{x}).$

∎

######  Corollary 3.6.20.

Let $\overline{x}$ be a maximal contact of $R$ at $p$. 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 maximal $R$-admissible center at a point $p\in X$. Then upon the change
of the coordinate representation of ${\mathcal{A}}$,
$\overline{x}_{1}=\overline{x}$ is the maximal contact of $R$ at $p$.

###### Proof.

According to Lemma 3.6.17, $\overline{x}_{1}$ contains $\overline{x}$ after
the coordinate change. Furthermore, based on Definition 3.6.18 and Lemma
3.6.19(1), we have

${\operatorname{inv}}_{p}^{1}(R)=a_{1}({\operatorname{inv}}(\overline{x}_{1}))=a_{1}({\operatorname{inv}}(\overline{x})).$

This implies the equality $\overline{x}_{1}=\overline{x}$. ∎

####  3.6.21. Support of the invariant

Let

${\operatorname{supp}}{\operatorname{inv}}^{1}(R,\overline{b}_{1}):=\\{p\in
X\mid{\operatorname{inv}}^{1}_{p}(R)=\overline{b}_{1}\\}$

The following lemma highlights a fundamental property of maximal contact
within our context:

######  Lemma 3.6.22.

Let $\overline{x}$ be a partial maximal contact of $(R,a_{1})$ on $U$, and set
$b_{1}:=a_{1}{\operatorname{inv}}(\overline{x})$. Then we have

${\operatorname{supp}}{\operatorname{inv}}^{1}(R,\overline{b}_{1})\subseteq
V(\overline{x}).$

###### Proof.

Let’s denote $\overline{x}=(x_{1},\ldots,x_{r},x_{r+1},\ldots,x_{s})$, where
$x_{i}$ are free on $U$ for $1\leq i\leq r$ and divisorial for $r+1\leq i\leq
s$. Consider $q\in U\smallsetminus V(x_{i})$, where $x_{i}\in\overline{x}$ is
a free coordinate.

In this case, we have $x_{i}\in T^{1/a_{1}}(R)={\mathcal{O}}_{X}$, and
${\mathcal{D}}^{a_{1}b_{j}-1}(f_{j})$ is invertible for some generator
$f_{j}t^{b_{j}}\in R_{b_{j}}$. It follows that

${\operatorname{ord}}_{q}(f_{j}t^{b_{j}})\leq(a_{1}b_{j}-1)/b_{j}<a_{1},$

which implies ${\operatorname{ord}}_{q}(R)<a_{1}$. Therefore, we have
${\operatorname{inv}}^{1}_{q}(R)<\overline{b}_{1}$.

Consider $q\in V(x_{1},\ldots,x_{r})\cap U\smallsetminus V(x_{i})$, where
$x_{1},\ldots,x_{r}$ are free coordinates, and $x_{i}$ with $i>r$ is a
divisorial coordinate on $U$. Additionally, we have
$D_{x_{i}}(T^{1/a_{1}}(R))={\mathcal{O}}_{X}$, which implies that $x_{i}$ is a
free coordinate at $q$ since $q\not\in V(x_{i})$. Furthermore, for a certain
local parameter $u\in T^{1/a_{1}}(R)$, we have $D_{x_{i}}(u)$ invertible.
Consequently, $u$ is linearly independent from $x_{1},\ldots,x_{r}$ as well as
from the other divisorial coordinates at $q$.

As a result, in a partial maximal contact $(x_{1},\ldots,x_{r},u)$ of $R$ at
$q$, there are at least $r+1$ free coordinates. According to Lemma 3.6.19(3),
this implies that

${\operatorname{inv}}_{q}^{1}(R)\leq
a_{1}{\operatorname{inv}}_{1}(x_{1},\ldots,x_{r},u)<\overline{b}_{1}=a_{1}{\operatorname{inv}}(\overline{x}).$

In this context, it is worth noting that the $(r+1)$-th component of
$a_{1}{\operatorname{inv}}_{1}(x_{1},\ldots,x_{r},u)$ is $a_{1}$, while for
$\overline{b}_{1}=a_{1}{\operatorname{inv}}(\overline{x})$, it is $a_{1+}$.

∎

######  Example 3.6.23.

Consider the ideal ${\mathcal{I}}=(x_{1}^{2}+x_{2})\subset K[x_{1},x_{2}]$,
where $x_{1}$ is a free coordinate and $x_{2}$ is a divisorial coordinate.
Then

$T^{1}({\mathcal{I}}t)=(x_{1}^{2}+x_{2})t$

At the point $0$, $(x_{2})$ represents a maximal contact, while on the scheme
$X=\operatorname{Spec}(K[x_{1},x_{2}])$, $(x_{2})$ corresponds to a partial
maximal contact. Thus, we obtain the following:

${\operatorname{inv}}^{1}_{0}({\mathcal{I}}t)={\operatorname{inv}}^{1}(x_{2})=(1_{+})\quad\mbox{and}$
${\operatorname{supp}}{\operatorname{inv}}^{1}({\mathcal{I}}t,(1_{+}))\subseteq
V(x_{2}).$

When calculating the invariant ${\operatorname{inv}}^{1}_{p}({\mathcal{I}}t)$
away from the origin, such as at $p\in V({\mathcal{I}})\smallsetminus{0}$, the
coordinate $x_{1}^{\prime}:=x_{2}+x_{1}^{2}$ is free. Therefore,
$(x^{\prime}_{1},\in T^{2}({\mathcal{I}}t))$ represents a maximal contact at
$p$, which yields the following:

${\operatorname{inv}}^{1}_{p}({\mathcal{I}}t)={\operatorname{inv}}^{1}(x^{\prime}_{1})=(1)<(1_{+}).$

Consequently, the invariant drops in a neighborhood of $0$.

This example illustrates the concept of assigning heavier weights: $1_{+}>1$,
to the divisorial coordinates.

######  Lemma 3.6.24.

Let $\overline{x}=(x_{1},\ldots,x_{r})$ be a partial maximal contact for
$(R,a_{1})$ on an open subset $U$. Then

  1. (1)

${\operatorname{supp}}{\operatorname{inv}}^{1}(R,\overline{b}_{1})\subseteq
V({T}^{\leq 1/a_{1}}(R))\cap V(\overline{x}).$

  2. (2)

$\overline{x}$ is a partial maximal contact at all points $q\in V({T}^{\leq
1/a_{1}}(R))\cap V(\overline{x})$ for which
${\operatorname{ord}}_{q}(R)=a_{1}$.

  3. (3)

${\operatorname{supp}}{\operatorname{inv}}^{1}(R,\overline{b}_{1})=V({T}^{\leq
1/a_{1}}(R))\cap V(\overline{x})\cap
V(D_{x_{r+1},\ldots,x_{k}}(T^{1/a_{1}}R)))$ is closed on $U$.

  4. (4)

$\overline{x}$ is a maximal contact for the points in
${\operatorname{supp}}{\operatorname{inv}}^{1}(R,\overline{b}_{1})$.

###### Proof.

(1) and (2) Based on Lemmas 3.6.22 and 3.6.9, we have

${\operatorname{supp}}{\operatorname{inv}}^{1}(R,\overline{b}_{1})\subset
V({T}^{\leq 1/a_{1}}(R))\cap V(\overline{x}).$

Furthermore, according to Lemma 3.6.14, for points $q\in V({T}^{\leq
1/a_{1}}(R))\cap V(\overline{x})$, we have ${\operatorname{ord}}_{q}(R)\leq
a_{1}$, and by Definition 3.6.11, $\overline{x}$ represents a partial maximal
contact at $q$ as long as ${\operatorname{ord}}_{q}(R)=a_{1}$.

(3) and (4). Let’s assume that $D_{x_{j}}({T}^{1/a_{1}}(R))={\mathcal{O}}_{X}$
at $q\in V({T}^{\leq 1/a_{1}}(R))\cap V(\overline{x})$ for some $j>r$.

If $x_{j}$ is a divisorial coordinate, then $(\overline{x},x_{j})$ represents
a partial maximal contact at $q$, and according to Lemma 3.6.19, we have
${\operatorname{inv}}_{q}(R)\leq
a_{1}{\operatorname{inv}}(\overline{x},x_{j})<\overline{b}_{1}$.

If $x_{j}$ is a free coordinate at $q$, there exists a free local parameter
$u\in{T}^{1/a_{1}}(R)$ such that $D_{x_{j}}(u)$ is invertible. In this case,
$(x_{1},\ldots,x_{r},u)$ represents a partial maximal contact at $q$, and we
have ${\operatorname{inv}}_{q}(R)\leq
a_{1}{\operatorname{inv}}(\overline{x},x_{j})<\overline{b}_{1}$.

Conversely, if $q\in V({T}^{\leq 1/a_{1}}(R))\cap V(\overline{x})\cap
V(D_{x_{r+1},\ldots,x_{k}}(T^{1/a_{1}}R))$, then the condition (3) of
Definition 3.6.11 is satisfied, and $\overline{x}$ represents a maximal
contact at $q$. According to Lemma 3.6.19, we have
${\operatorname{inv}}_{q}^{1}(R)=a_{1}({\operatorname{inv}}(\overline{x}))$.

∎

####  3.6.25. Maximal contact for nested ideals

We can directly extend the definition of maximal contact to nested Rees
algebras. Consider a coordinate system
$\overline{x}_{1},\ldots,\overline{x}_{n}$ of local parameters that is
compatible with $E$. Let

$R={\mathcal{O}}_{X}[(x_{1}t^{1/a_{1}},\ldots,x_{k}t^{1/a_{k}})^{\operatorname{Int}},f_{j}t^{b_{j}}]_{j=1,\ldots,s}$

be a Rees algebra nested at
$H_{k}:=V(\overline{x}_{1},\ldots,\overline{x}_{k})$, with the nested order
${\operatorname{ord}}_{p}(R_{|H_{k}})=a_{k+1}$. The nested cotangent ideal
$T^{1/a_{k+1}}(R)$ of $R$ can be defined as follows:

$T_{H_{K}}^{1/a_{k+1}}(R):=\sum_{|\alpha|=b_{j}a_{k+1}-1}{\mathcal{O}}_{X}{\mathcal{D}}_{\overline{x}_{k+1},\ldots,\overline{x}_{n}}^{\alpha}(f_{j})$

in $t^{1/a_{k+1}}$\- gradation. Its restriction
$T_{H_{K}}^{1/a_{k+1}}(R)_{|H_{k}}$ defines the cotangent ideal of
$R_{|H_{k}}$ on $H_{k}$. Consequently we define a nested maximal contact for
$R$ at $H_{i}$ given by a partial system of coordinates

$\overline{x}_{k+1}=(x_{k+1,1},\ldots,x_{k+1,s},x_{k+1,s+1},\ldots,x_{k+1,r}))$

on $X$, with free coordinates $x_{k+1,1},\ldots,x_{k+1,s}\in
T_{H_{K}}^{a_{k+1}}(R)$, and divisorial
$x_{k+1,s+1},\ldots,x_{k+1,r}$ such that its restriction
$\overline{x}_{k+1|H_{k}}$ is a maximal contact for $R_{|H_{k}}$.

######  Corollary 3.6.26.

Let

$R={\mathcal{O}}_{X}[(x_{1}t^{1/a_{1}},\ldots,x_{k}t^{1/a_{k}})^{\operatorname{Int}},f_{j}t^{b_{j}}]_{j=1,\ldots,s}$

be a Rees algebra nested at
$H_{k}:=V(\overline{x}_{1},\ldots,\overline{x}_{k})$, and assume that

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

with $n\geq k$, is a maximal $R$\- admissible center at $p\in X$ so that

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

Then the nested order ${\operatorname{ord}}_{p}(R_{|H_{k}})=a_{k+1}$. Let
$\overline{x}$ be a nested maximal contact of $R$ at $p$ and $H_{k}$. Then
upon the change of the coordinate representation of ${\mathcal{A}}$,
$\overline{x}_{k+1}=\overline{x}$ is the maximal contact of $R$ at $p$.

###### Proof.

Given that $R$ is nested at $H_{k}$, it can be expressed in the split form in
$\widehat{{\mathcal{O}}}_{X,p}$ as:

$\widehat{{\mathcal{O}}}_{X,p}\cdot
R=\widehat{{\mathcal{O}}}_{X,p}[(x_{1}t^{1/a_{1}},\ldots,x_{k}t^{1/a_{k}})^{\operatorname{Int}},R_{|H_{k}}]$

Consequently,

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

serves as a maximal admissible center for $R_{|H_{k}}$. By applying Lemma
3.1.24, we can observe that ${\operatorname{ord}}_{p}(R_{|H_{k}})=a_{k}$.
According to Lemma 3.1.29, the free coordinates in $\overline{x}$ are a part
of $\overline{x}_{k+1}$ after a coordinate change. Moreover, since
$\overline{x}_{|H_{k}}$ represents a maximal contact for $R_{|H_{k}}$, we can
deduce from Lemma 3.6.16(4) that $\overline{x}_{k+1|H_{k}}$ and
$\overline{x}_{|H_{k}}$ share the same divisorial coordinates. Consequently,
we obtain the equality $\overline{x}_{k+1}=\overline{x}$.

∎

###  3.7. Effective algorithm

####  3.7.1. The algorithm

Consider an arbitrary Rees algebra $R$ on a smooth variety $X$. In particular
case, if an ideal ${\mathcal{I}}$ is given we consider the Rees algebra
$R={\mathcal{O}}_{X}[{\mathcal{I}}t]$. We shall construct a maximal
${\mathcal{A}}$-admissible center ${\mathcal{R}}$ at a given point $p\in X$.
Assume, at first that the Rees center

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

which is maximal admissible at the point $p\in X$ for ${\mathcal{R}}$ on $X$
exists. We shall use ${\mathcal{A}}$ only as the reference for our
construction. All the operations are done for $R$ without using or modifying
the algebra ${\mathcal{A}}$. We will however modify the presentation of
${\mathcal{A}}$ so it will be compatible with the presentations of
modifications of $R$.

The procedure is recursive and we put $R=R_{1}$. Assume that the center
${\mathcal{A}}$ is maximal admissible $R$, so that

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

Using Lemma 3.1.24, we see that ${\operatorname{ord}}_{p}(R_{1})=a_{1}.$ Let
$\overline{x}^{\prime}_{1}\subseteq T^{1/a_{1}}(R_{1})$ be a maximal contact
at $p\in X$, and put $H_{1}:=V(\overline{x}^{\prime}_{1})$.

By Lemma 3.6.20, one can change the presentation of the algebra
${\mathcal{A}}$ so that $\overline{x}_{1}=\overline{x}^{\prime}_{1}$ is a
maximal contact. This modification does not affect the algebra ${\mathcal{A}}$
on the right, but only changes its presentation.

Let $R_{2}:=C_{\overline{x}_{1}t^{1/{a_{1}}}}(R_{1})$ be the coefficient ideal
with respect to the maximal contact $\overline{x}_{1}$. By Lemma 3.5.7,
$R_{2}$ is nested at $H_{1}:=V(\overline{x}_{1})$, and can be written in
$\widehat{{\mathcal{O}}}_{X,p}$ in the split form

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

Furthermore, according to Lemma 3.5.24, the same center ${\mathcal{A}}$
remains maximal admissible for
$R_{2}=C_{\overline{x}_{1}t^{1/{a_{1}}}}(R_{1})$.

This recursive procedure can be summarized as follows. Let $R_{i}$, for $i\geq
2$, be the algebra nested at
$H_{i-1}=V(\overline{x}_{1},\ldots,\overline{x}_{i-1})$ such that

$\displaystyle\widehat{{\mathcal{O}}}_{X,p}R_{i}=\widehat{{\mathcal{O}}}_{X,p}[(\overline{x}_{1}t^{1/a_{1}},\ldots,\overline{x}_{i-1}t^{1/a_{i-1}})^{\operatorname{Int}},R_{i|H_{i-1}}]\subseteq$
$\displaystyle\subseteq\widehat{{\mathcal{O}}}_{X,p}{\mathcal{A}}=\widehat{{\mathcal{O}}}_{X,p}[\overline{x}_{1}t^{1/a_{1}},\ldots,\overline{x}_{k}t^{1/a_{k}}]^{{\operatorname{Int}}},$

where ${\mathcal{A}}$ is a maximal admissible center for $R_{i}$. According to
Lemma 3.6.26, we have ${\operatorname{ord}}_{p}(R_{i})=a_{i}$. We find a
nested maximal contact $\overline{x}^{\prime}_{i}$ for $R_{i}$ at $H_{i-1}$
and $p$. Using Lemma 3.6.26 we can modify the coordinates
$\overline{x}_{i},\ldots,\overline{x}_{k}$ in the presentation of
${\mathcal{A}}$ and assume that the maximal contact
$\overline{x^{\prime}}_{i}=\overline{x}_{i}$ occurs in the presentation of
${\mathcal{A}}$. Consequently, by Lemma 3.5.24,

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

where ${\mathcal{A}}$ on the right represents a maximal admissible center for
$R_{i+1}$. Furthermore, by Lemma 3.5.18, $R_{i+1}$ is nested at
$H_{i}=V(\overline{x}_{1},\ldots,\overline{x}_{i})$, and can be written in
$\widehat{{\mathcal{O}}}_{X,p}$ as

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

We continue the procedure until we reach $R_{k+1}$ which is nested at
$H_{k}=V(\overline{x}_{1},\ldots,\overline{x}_{k})$, such that
$R_{k+1|H_{k}}=0$. Finally

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

with the extended Rees algebra

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

where $w_{R,A}={\operatorname{lcm}}(a_{1},\ldots,a_{r},w_{R}).$

####  3.7.2. Uniqueness

Throughout this process, we do not alter the algebra
${\mathcal{A}}={\mathcal{O}}_{X}[\overline{x}_{1}t^{1/a_{1}},\ldots,\overline{x}_{k}t^{1}]^{\operatorname{int}}$.
However, we do modify the coordinates within ${\mathcal{A}}$, which gradually
transform into maximal contacts for the nested Rees algebra on the left.

This demonstrates that the procedure, which is independent of ${\mathcal{A}}$,
results in a unique, predetermined center ${\mathcal{A}}$ that is maximal
admissible for $R$ at $p$.

####  3.7.3. Existence

######  Proposition 3.7.4.

(see also [ATW17, Theorem 5.3.1] (in the language of ${\mathbb{Q}}$-ideals)),
). For any Rees algebra $R$ on a smooth variety $X$ over $K$ with a SNC
divisor $E$, and for any point $p\in X$ there exists a uniquely determined
Rees center

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

which is a maximal admissible for $R$ at $p$.

Moreover ${\mathcal{A}}^{\operatorname{Int}}=R_{k+1}$ is obtained by the
recursive procedure $R_{1}=R$, and
$R_{i+1}=C_{\overline{x}_{i}t^{1/a_{1}}}(R_{i})$ independent of
${\mathcal{A}}$, where

  1. (1)

$R_{i}$ is nested at $H_{i-1}:=V(\overline{x}_{1},\ldots,\overline{x}_{i-1})$,
for $i-1=1,\ldots,k$, and $H_{0}:=X$.

  2. (2)

$a_{i}={\operatorname{ord}}_{p}(R_{i|H_{i-1}})$

  3. (3)

$\overline{x}_{i}$ is a maximal contact for $R_{i|H_{i-1}}$.

  4. (4)

${\operatorname{inv}}_{p}(R)=(a_{1}{\operatorname{inv}}(\overline{x}_{1}),\ldots,a_{k}{\operatorname{inv}}(\overline{x}_{k}))$.

###### Proof.

The construction does not rely on the Rees center ${\mathcal{A}}$ on the right
side of the admissibility condition at any step. Moreover, the inductive
process leads to a Rees center ${\mathcal{A}}^{\prime}$ that is admissible for
$R$ at $p$. We obtain a sequence of nested Rees algebras $R_{i}$ at
$H_{i-1}=V(\overline{x}_{1},\ldots,\overline{x}_{i-1})$ such that
${\operatorname{ord}}_{p}(R_{i|H_{i-1}})=a_{i}$ and $\overline{x}_{i}$ is a
maximal contact for $R_{i|H_{i-1}}$. The final Rees center
${\mathcal{A}}^{\prime}=R_{k+1}^{\operatorname{int}}$ is an admissible center
for $R$.

Now, consider another admissible center

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

for $R$ at $p$. Suppose
${\operatorname{inv}}({\mathcal{A}}^{\prime\prime})\geq{\operatorname{inv}}({\mathcal{A}}^{\prime})$.
We run the recursive procedure using ${\mathcal{A}}^{\prime\prime}$ on the
right side. By Lemma 3.1.24, we conclude that $a_{1}=a^{\prime\prime}_{1}$.
Furthermore, according to Lemma 3.6.17(2), after a coordinate change,
$\overline{x}^{\prime\prime}_{1}$ contains a maximal contact
$\overline{x}_{1}$ such that
${\operatorname{inv}}^{1}(\overline{x}_{1})\geq{\operatorname{inv}}^{1}(\overline{x}^{\prime\prime}_{1})$.
Since
${\operatorname{inv}}({\mathcal{A}}^{\prime\prime})\geq{\operatorname{inv}}({\mathcal{A}}^{\prime})$,
it follows that
${\operatorname{inv}}^{1}(\overline{x}^{\prime\prime}_{1})\geq{\operatorname{inv}}^{1}(\overline{x}_{1})$.
Therefore, we have $\overline{x}^{\prime\prime}_{1}=\overline{x}_{1}$. We
continue this process step by step, demonstrating that
$a_{i}=a_{i}^{\prime\prime}$ and
$\overline{x}^{\prime\prime}_{i}=\overline{x}_{i}$ after a possible change in
the coordinate representation of ${\mathcal{A}}^{\prime}$. This shows that
${\mathcal{A}}^{\prime\prime}={\mathcal{A}}^{\prime}=R_{k+1}^{\operatorname{int}}$
is the maximal admissible center for ${\mathcal{I}}$ at $p$. ∎

####  3.7.5. The inductive principle

[ATW19],

Let $R$ be a Rees algebra of order $a_{1}$ at $p$ with a maximal contact
$\overline{x}_{1}$, and let $H_{1}:=V(\overline{x}_{1})$. By Lemma 3.5.11, a
center
${\mathcal{A}}={\mathcal{O}}_{X}[\overline{x}_{1}t^{1/a_{1}},\ldots,\overline{x}_{k}t^{1/a_{k}}]^{\operatorname{int}}$
is maximal admissible for $R$ if and only if
${\mathcal{A}}_{|H_{1}}={\mathcal{O}}_{H_{1}}[\overline{x}_{2|H_{1}}t^{1/a_{2}},\ldots,\overline{x}_{k|H_{1}}t^{1/a_{k}}]^{{\operatorname{int}}}$
is maximal admissible for $C_{\overline{x}_{1}(R)|H_{1}}$.

This leads to the following inductive formula, as stated in [ATW19]:

${\operatorname{inv}}_{p}(R)=({\operatorname{inv}}^{1}_{p}(R),{\operatorname{inv}}_{p}(C_{\overline{x}_{1}}(R)_{|H_{1}})),$

####  3.7.6. Semicontinuity of canonical invariant. Local admissibility

The inductive principle above ensures the semicontinuity of the invariant
${\operatorname{inv}}_{p}$, as in [ATW19]. We proceed with an induction on
$n=\dim(X)$ and assume that the function $p\mapsto{\operatorname{inv}}_{p}(R)$
is upper semicontinuous for any Rees algebra $R^{\prime}$ on any smooth
variety $H$ of dimension $n-1$. Let $R$ be a Rees algebra on a smooth $X$ of
dimension $n$.

The condition
${\operatorname{inv}}_{p}(R)\geq(\overline{b}_{1},\ldots,\overline{b}_{k})$
implies either ${\operatorname{inv}}^{1}_{p}(R)>\overline{b}_{1}$, or
${\operatorname{inv}}^{1}_{p}(R)=\overline{b}_{1}$ and
${\operatorname{inv}}_{p}(C_{\overline{x}_{1}|H_{1}}(R))\geq(\overline{b}_{2},\ldots,\overline{b}_{k})$.
This description corresponds to a closed subset, which follows from the
semicontinuity of ${\operatorname{inv}}^{1}_{p}$ as established in Lemma
3.6.24, and with the inductive assumption.

####  3.7.7. Duality of Rees centers

The Rees centers $\mathcal{A}$ possess a dual interpretation: they serve as
admissible centers with a dummy variable $t$, and they also represent the
extended algebras of the full cobordant blow-ups, where $t^{-1}$ serves as the
introduced coordinate on $B$. These two concepts are intimately connected and
can be described by identical formulas, up to rescaling. In order to prevent
any confusion, we will introduce a distinct variable $t_{B}$ specifically for
the algebras on $B$, where the differentiation between $t$ and $t_{B}$ becomes
necessary.

####  3.7.8. Controlled transforms of ideals

[ATW19], Let ${\mathcal{I}}$ be an ideal on regular $X$. One writes
${\mathcal{I}}$-admissibility condition as:

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

By employing the variable $t_{B}$ and applying rescaling, we can rewrite this
inclusion as follows:

${\mathcal{I}}\cdot
t_{B}^{w_{A}}\,\,\subset\,\,{\mathcal{O}}_{B}\,=\,{\mathcal{O}}_{X}[t_{B}^{-1},\overline{x}_{1}t_{B}^{w_{1}},\ldots,\overline{x}_{k}t_{B}^{w_{k}}]={\mathcal{O}}_{X}[t_{B}^{-1},\overline{x}^{\prime}_{1},\ldots,\overline{x}^{\prime}_{k}],$

where $x_{i}^{\prime}:=x_{i}t^{w_{i}}$.

According to Lemma 2.5.1, the exceptional divisor on $B_{+}$ can be expressed
as $t_{B}^{-1}$, which serves as a local parameter on $B$. Using this
information, we can deduce that the full transform $\mathcal{O}_{B}\cdot I$ is
divisible by $t_{B}^{-w_{A}}$ since
$t^{w_{A}}\cdot\mathcal{O}_{B}\cdot\mathcal{I}=\mathcal{O}_{B}\cdot
t_{B}^{w_{A}}\cdot\mathcal{I}\subset\mathcal{O}_{B}$.

Now, we introduce the concept of the controlled transform of the ideal
$\mathcal{I}$, which is defined as:

$\sigma^{c}({\mathcal{I}}):={\mathcal{O}}_{B}\cdot
t_{B}^{w_{A}}\cdot{\mathcal{I}}\subset B.$

####  3.7.9. Strict transform of ideals

Recall that

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

is trivial over $X$. By the strict transform $\sigma^{s}({\mathcal{I}})$ of an
ideal ${\mathcal{I}}$ on $X$ under a full cobordant blow-up $\sigma:B\to X$ of
${{\mathcal{A}}}$ on $X$ we mean the schematic closure of
$({\mathcal{O}}_{B}\cdot{\mathcal{I}}_{X})_{|B_{-}}={\mathcal{O}}_{B_{-}}\cdot{\mathcal{I}}_{X}$.
Thus

$\sigma^{s}({\mathcal{I}}):=\\{t^{a}f\in{\mathcal{O}}_{B}\mid\quad
f\,\in\,{\mathcal{O}}_{B}\cdot{\mathcal{I}}_{Y},\quad a\geq 0\\}.$

Consequently, the strict transform of a closed subscheme $Y$ of $X$ is the
schematic closure $Y^{s}$ of $Y\times G_{m}\subset B_{-}=X\times G_{m}$ in
$B$. It is defined by ${\mathcal{I}}_{Y^{s}}=\sigma^{s}({\mathcal{I}}_{Y})$.
This implies that
$\sigma^{c}({\mathcal{I}})\subseteq\sigma^{s}({\mathcal{I}})$, and thus
${\operatorname{inv}}(\sigma^{c}({\mathcal{I}}))\geq{\operatorname{inv}}(\sigma^{s}({\mathcal{I}}))$
for any $p\in B$.

######  Remark 3.7.10.

Observe that for the cobordant blow-up $\sigma_{+}:B_{+}\to X$ the induced
strict transform coincides with the standard definition
$\sigma_{+}^{s}({\mathcal{I}})$ as the schematic closure on $B_{+}$ of

${\mathcal{O}}_{\sigma_{+}^{-1}(X\smallsetminus
V({\mathcal{J}}))}\cdot{\mathcal{I}}_{|X\smallsetminus
V({\mathcal{J}})}=({\mathcal{O}}_{B}\cdot{\mathcal{I}})_{|{B_{+}\cap B_{-}}}.$

####  3.7.11. Controlled transforms of Rees algebras and double gradation

The relationship between admissibility and the controlled transforms of ideals
can be described using the concept of double gradation. Let $\mathcal{I}$
correspond to the gradation $\mathcal{I}t$ in the Rees algebra
$\mathcal{A}_{\mathcal{I}}=\mathcal{O}_{X}[\mathcal{I}t]$. We can assign a
double gradation $\mathcal{O}_{B}\cdot\mathcal{I}\cdot t^{w_{A}}_{B}t$ in the
Rees algebra $\mathcal{O}_{B}[\mathcal{I}\cdot t^{w_{A}}_{B}t]$ on $B$, which
can be interpreted as the controlled transform denoted by

$\sigma^{c}({\mathcal{I}}t):=({\mathcal{O}}_{B}\cdot
t_{B}^{w}\cdot{\mathcal{I}})t\subset\mathcal{O}_{B}[\mathcal{I}\cdot
t^{w_{A}}_{B}t]$

under the gradation $t$. One can simply write it as

$\sigma^{c}({\mathcal{I}}):=({\mathcal{O}}_{B}\cdot
t_{B}^{w}\cdot{\mathcal{I}})\subset{\mathcal{O}}_{B}$

In a more general setting, suppose we have a Rees algebra $R=\bigoplus
R_{a}t^{a}$, and $\mathcal{A}$ is an $R$-admissible center. Then, we have the
inclusion

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

where $w$ is a multiple of $w_{R,A}$. By rescaling $\mathcal{A}$ with
$t_{B}\mapsto t_{B}^{w}$ and defining $w_{i}:=w/a_{i}\in\mathbb{Z}_{\geq 0}$,
we obtain the cobordant blow-up of $\mathcal{A}^{\text{ext}}$:

$B=\operatorname{Spec}([t_{B}^{-1},\overline{x}_{1}t_{B}^{w_{1}},\ldots,\overline{x}_{k}t_{B}^{w_{k}}]=\operatorname{Spec}{\mathcal{O}}_{B}[t_{B}^{-1},\overline{x}^{\prime}_{1},\ldots,\overline{x}^{\prime}_{k}],$

where $x_{i}^{\prime}:=x_{i}t_{B}^{w_{i}}$. By combining the gradations on
both $t$ and $t^{w}_{B}$, we can express the admissibility of Rees algebras on
$B$ as follows:

$\displaystyle\bigoplus R_{a}t_{B}^{a\cdot
w}t^{a}\,\,\subset\,\,{\mathcal{O}}_{B}[t_{B}^{-1}t^{-1/w},\overline{x}_{1}t_{B}^{w_{1}}t^{1/a_{1}},\ldots,\overline{x}_{k}t_{B}^{w_{k}}t^{1/a_{k}}]\,\,\subset$
$\displaystyle\subset\,\,{\mathcal{O}}_{B}[t^{-1/w},\overline{x}^{\prime}_{1}t^{1/a_{1}},\ldots,\overline{x}^{\prime}_{k}t^{1/a_{k}}].$

Hence, the Rees algebra on $B$ can be denoted as:

$\sigma^{c}(R):=\bigoplus({\mathcal{O}}_{B}\cdot R_{a}\cdot t_{B}^{a\cdot
w})t^{a},$

and it will be referred to as the controlled transform of $R$.

####  3.7.12. Cobordant blow-ups and admissibility

Consequently, based on the previous discussion, we can conclude that:

######  Lemma 3.7.13.

If
${\mathcal{A}}^{\operatorname{ext}}={\mathcal{O}}_{X}[t^{-1/w},\overline{x}_{1}t^{1/a_{1}},\ldots,\overline{x}_{k}t^{1/a_{k}}]^{\operatorname{int}}$
is admissible for $R$ , and
$\sigma:B=\operatorname{Spec}_{X}({\mathcal{O}}_{B}[t_{B}^{-1},\overline{x}_{1}t_{B}^{w_{1}},\ldots,\overline{x}_{k}t_{B}^{w_{k}}])$
is the full cobordant blow-up of ${\mathcal{A}}^{\operatorname{ext}}$ then the
Rees center on $B$:

$\displaystyle{\mathcal{A}}^{\operatorname{ext}}_{B}:=\sigma^{c}({\mathcal{A}}^{\operatorname{ext}}):={\mathcal{O}}_{B}[t^{-1/w},\overline{x}_{1}t_{B}^{w_{1}}t^{1/a_{1}},\ldots,\overline{x}_{k}t_{B}^{w_{k}}t^{1/a_{k}}]^{\operatorname{int}}=$
$\displaystyle{\mathcal{O}}_{B}[t^{-1/w},\overline{x}^{\prime}_{1}t^{1/a_{1}},\ldots,\overline{x}^{\prime}_{k}t^{1/a_{k}}]$

is admissible for
$\sigma^{c}(R)=\bigoplus{\mathcal{O}}_{B}R_{a}t_{B}^{wa}t^{a}$. ∎

####  3.7.14. Derivations on cobordant blow-up

[ATW19],

Consider be the full cobordant blow-up

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

of a center
${\mathcal{A}}^{\operatorname{ext}}={\mathcal{O}}_{X}[t^{-1/w},x_{1}t^{1/a_{1}},\ldots,x_{k}t^{1/a_{k}}]$.
The sheaf $\mathcal{D}_{X}$ of derivations on $X$ is a coherent
$\mathcal{O}_{X}$-module that is locally generated by the derivations
$D_{x_{i}}$. Using chain rule we can write derivations $D_{x_{i}^{\prime}}$ on
$B$ as $D_{x_{i}^{\prime}}=t_{B}^{-w_{i}}D_{x_{i}}$, and
$D_{x_{j}^{\prime}}=D_{x_{j}}$.

This above formula can be extended using a double gradation principle: To
obtain the controlled transform, we associate the derivations
$t^{-1/a_{i}}D_{x_{i}}$ in the graded form with a double gradation with
respect to $t$ and $t_{B}^{w}$.

$\sigma^{c}(t^{-1/a_{i}}D_{x_{i}})=t^{-1/a_{i}}t_{B}^{-w_{i}}D_{x_{i}}=t^{-1/a_{i}}D_{x_{j}^{\prime}}$

In a similar manner, we define the ”controlled transform” of
$t^{-1/a_{1}}\mathcal{D}_{X}$ to be the subsheaf

$\sigma^{c}(t^{-1/a_{1}}{\mathcal{D}}_{X}):=t^{-1/a_{1}}{\mathcal{O}}_{B}t_{B}^{-w_{1}}{\mathcal{D}}_{X}\subseteq
t^{-1/a_{1}}{\mathcal{D}}_{B}$

of the sheaf ${\mathcal{D}}_{B}$ of the derivations on $B$ in gradation
$t^{-1/a_{1}}$. Note that the sheaf
${\mathcal{O}}_{B}t_{B}^{-w_{1}}\cdot{\mathcal{D}}_{X}$ is generated by
$t_{B}^{-(w_{1}-w_{i})}t_{B}^{-w_{i}}D_{x_{i}}=t_{B}^{-(w_{1}-w_{i})}D_{x^{\prime}_{i}}$
for $i=1,\ldots,k$ and
$t_{B}^{-w_{1}}D_{x_{i}}=t_{B}^{-w_{1}}D_{x^{\prime}_{i}}$ for
$i=k+1,\ldots,n$.

####  3.7.15. The order of the controlled transforms

######  Lemma 3.7.16.

[ATW19], Let $\sigma:B\to X$ be a cobordant blow-up of $R$-admissible center
${\mathcal{A}}^{\operatorname{ext}}={\mathcal{O}}_{X}[t^{1/w},\overline{x}_{1}t^{1/a_{1}},\ldots,\overline{x}_{k}t^{1/a_{k}}]$,
where ${\operatorname{ord}}_{p}(R)\leq a_{1}$ for $p\in X$. Then
${\operatorname{ord}}_{p^{\prime}}(\sigma^{c}(R))\leq a_{1}$ for
$p^{\prime}\in B$.

###### Proof.

Write $R={\mathcal{O}}_{X}[f_{j}t^{b_{j}}]_{j=1,\ldots,s}$. If
${\operatorname{ord}}_{p}(R)=a_{1}$ then
${\operatorname{ord}}_{p}(f_{j}t^{b_{j}})=b_{j}a_{1}$ for some $j$. Thus there
exists $\alpha$, with $|\alpha|=b_{j}a_{1}$, such that
${\mathcal{D}}_{x^{\alpha}}f_{j}$ is invertible. Consequently

$\displaystyle 1\sim$
$\displaystyle{\mathcal{D}}_{x^{\alpha}}f_{j}=(t_{B}^{-b_{j}w}t^{-b_{j}}{\mathcal{D}}_{x^{\alpha}})(f_{j}t_{B}^{b_{j}w}t^{b_{j}})\in\sigma^{c}({\mathcal{D}}_{X}^{b_{j}a_{1}}t^{-b_{j}})(\sigma^{c}(f_{j}t^{b_{j}}))$
$\displaystyle\subseteq{\mathcal{D}}_{B}^{b_{j}a_{1}}t^{-b_{j}}(\sigma^{c}(R_{b_{j}})t^{b_{j}})={\mathcal{D}}_{B}^{b_{j}a_{1}}(\sigma^{c}(R_{b_{j}}))={\mathcal{O}}_{B},$

which shows that ${\operatorname{ord}}_{p}\sigma^{c}(R)\leq a_{1}$. ∎

####  3.7.17. Controlled transforms of cotangent ideal

Similarly we have

######  Lemma 3.7.18.

[ATW19], Let $\sigma:B\to X$ be a cobordant blow-up of
$R={\mathcal{O}}_{X}[f_{j}t^{b_{j}}]$-admissible center
${\mathcal{A}}^{\operatorname{ext}}={\mathcal{O}}_{X}[t^{1/w},\overline{x}_{1}t^{1/a_{1}},\ldots,\overline{x}_{k}t^{1/a_{k}}]$.
If $T^{1/a_{1}}(R)$ is the cotangent ideal for $R$ then

$\sigma^{c}(T^{1/a_{1}}(R)t^{1/a_{1}})\subseteq(T^{1/a_{1}}(\sigma^{c}(R))t^{1/a_{1}}.$

###### Proof.

Let $f_{j}t^{b_{j}}\in R_{b_{j}}$, and $|\alpha|=b_{j}a_{1}-1$. Then

$\displaystyle({\mathcal{D}}_{x^{\alpha}}f_{j})t^{1/a_{1}}t_{B}^{w_{1}}=\sigma^{c}(({\mathcal{D}}_{x^{\alpha}}f_{j})t^{1/a_{1}})=\sigma^{c}({\mathcal{D}}^{\alpha}_{\overline{x}t^{1/a_{1}}}f_{j}t^{b_{j}})=$
$\displaystyle=\sigma^{c}({\mathcal{D}}^{\alpha}_{\overline{x}t^{1/a_{1}}})(\sigma^{c}(f_{j}t^{b_{j}})\in{\mathcal{D}}_{B}^{|\alpha|}t^{-|\alpha|/a_{1}}(\sigma^{c}(R_{b_{j}})t^{b_{j}})=$
$\displaystyle={\mathcal{D}}_{B}^{|\alpha|}(\sigma^{c}(R_{b_{j}}))t^{1/a_{1}}\subseteq(T^{1/a_{1}}(\sigma^{c}(R))t^{1/a_{1}}.$

∎

####  3.7.19. Controlled transforms of a partial maximal contact

######  Lemma 3.7.20.

[ATW19], Let $\sigma:B\to X$ be a cobordant blow-up of
$R={\mathcal{O}}_{X}[f_{j}t^{b_{j}}]$-admissible center
${\mathcal{A}}^{\operatorname{ext}}={\mathcal{O}}_{X}[t^{1/w},\overline{x}_{1}t^{1/a_{1}},\ldots,\overline{x}_{k}t^{1/a_{k}}]$.
Assume that $\overline{x}_{1}=(x_{1},\ldots,x_{r})$ is a partial maximal
contact for $(R,a_{1})$ on an open affine subset $U$ then
$\sigma^{c}(\overline{x}_{1})$ is a partial maximal contact for
$\sigma^{c}(R)$ on $\sigma^{-1}(U)$.

###### Proof.

$\sigma^{c}(\overline{x}_{1})=(x_{1}^{\prime},\ldots,x^{\prime}_{r})$ is a
local system of coordinates. So if $x_{i}\in
T^{1/a_{1}}(R)\cap\overline{x}_{1}$ is free then
$x_{i}^{\prime}=\sigma^{c}(x_{i})\in
T^{1/a_{1}}(\sigma^{c}(R))\cap\sigma^{c}(\overline{x}_{1})$ is free. Thus
condition (1) of Definition 3.6.11 is satisfied.

If $x_{i}\in\overline{x}_{1}$ is divisorial, then using Lemma 3.7.18, we have

$\displaystyle{\mathcal{O}}_{B}={\mathcal{O}}_{B}\cdot
D_{x_{i}}(T^{1/a_{1}}(R))=\sigma^{c}(D_{x_{i}t^{1/a_{1}}}(T^{1/a_{1}}(R)t^{1/a_{1}})=$
$\displaystyle=\sigma^{c}(D_{x_{i}t^{1/a_{1}}})\sigma^{c}(T^{1/a_{1}}(R)t^{1/a_{1}})=$
$\displaystyle=t^{-1/a_{1}}D_{x^{\prime}_{i}}\sigma^{c}((T^{1/a_{1}}(R)t^{1/a_{1}})\subset
D_{x^{\prime}_{i}}(T^{1/a_{1}}(\sigma^{c}(R))$

and thus condition (2) of Lemma 3.6.11 is also satisfied.

∎

####  3.7.21. Controlled transform of the coefficient ideal

######  Lemma 3.7.22.

(see also [ATW19]) Let $\sigma:B\to X$ be a cobordant blow-up of
$R={\mathcal{O}}_{X}[f_{j}t^{b_{j}}]$-admissible algebra
${\mathcal{A}}={\mathcal{O}}_{X}[\overline{x}_{1}t^{1/a_{1}},\ldots,\overline{x}_{k}t^{1/a_{k}}]$
then $\sigma^{c}(R)={\mathcal{O}}_{B}[\sigma^{c}(f_{j}t^{b_{j}})]$, and we
have commutativity:

$\sigma^{c}(C_{\overline{x}_{1}t^{1/a_{1}}}(R))=C_{\overline{x}^{\prime}_{1}t^{1/a_{1}}}(\sigma^{c}(R)).$

###### Proof.

The assertion follows from the of the commutativity of the controlled
transforms with derivations:

$\sigma^{c}(D^{\alpha}_{\overline{x}_{1}t^{1/a_{1}}}(f_{j}t^{b_{j}}))=\sigma^{c}(D^{\alpha}_{\overline{x}_{1}t^{1/a_{1}}})(\sigma^{c}(f_{j}t^{b_{j}}))=D^{\alpha}_{\overline{x}^{\prime}_{1}t^{1/a_{1}}}(\sigma^{c}(f_{j}t^{b_{j}}))$

in $t^{|\alpha|}$-gradation for $|\alpha|<b_{j}a_{1}$. ∎

####  3.7.23. Restriction of cobordant blow-up to a maximal contact

######  Lemma 3.7.24.

[ATW19], If ${\mathcal{A}}^{{\operatorname{ext}}}$ is admissible for $R$ then
${\mathcal{A}}^{{\operatorname{ext}}}_{|H}$ is admissible for $R_{|H}$ and
$\sigma^{c}(R)_{|H}=\sigma_{H}^{c}(R_{|H})$, where $H:=V(\overline{x}_{1})$.

The restriction of the blow-up $\sigma_{X}:B\to X$ of
${\mathcal{A}}^{{\operatorname{ext}}}$ to the strict transform
$H_{B}=V(\overline{x}^{\prime}_{1})$ of $H=V(\overline{x}_{1})$ is the
cobordant blow-up $\sigma_{H}:H_{B}\to H$ of the restriction
${\mathcal{A}}^{{\operatorname{ext}}}_{|H}$. ∎

####  3.7.25. The centers with maximal invariant

######  Lemma 3.7.26.

(see also [ATW19]) Let $R$ be a Rees algebra, and ${\mathcal{A}}$ be a maximal
admissible center for $R$ at a point $p\in X$. Then there exists an open
neighborhood $U$ of $p$ such that

${\operatorname{maxinv}}_{U}(R)={\operatorname{inv}}_{p}(R)={\operatorname{inv}}({\mathcal{A}})$

on $U$ is attained at $V({\mathcal{A}})$.

###### Proof.

Let $\overline{x}_{1}$ be a maximal contact at $p$ for $R$. Then, by Lemma
3.6.12, it is a partial maximal contact on an open subset $U$ of $p$. Thus by
Lemma 3.6.19(3),

${\operatorname{inv}}^{1}(R)\leq\overline{b}_{1}:=a_{1}{\operatorname{inv}}(\overline{x}_{1}).$

In addition, by Lemma 3.6.24, the maximum
$\max{\operatorname{inv}}^{1}(R)=\overline{b}_{1}$ of the invariant
${\operatorname{inv}}^{1}_{q}(R)$, where $q\in X$, is attained at the closed
subset
${\operatorname{supp}}{\operatorname{inv}}^{1}(R,\overline{b}_{1})\subset
H_{1}=V(\overline{x}_{1})$, and $\overline{x}_{1}$ is a maximal contact along
${\operatorname{supp}}{\operatorname{inv}}^{1}(R,\overline{b}_{1})$.
Furthermore for the points in
${\operatorname{supp}}{\operatorname{inv}}^{1}(R,\overline{b}_{1})$, we have

${\operatorname{inv}}_{p}(R)=({\operatorname{inv}}^{1}_{p}(R),{\operatorname{inv}}_{p}(C_{\overline{x}_{1}t^{1/a_{1}}}(R)_{|H_{1}}),$

Moreover ${\mathcal{A}}_{|H_{1}}$ is maximal admissible for $R$ at $p$. It
suffices to use induction on dimension to find an open neighborhood $U$ of $p$
such that
${\operatorname{inv}}_{p}(C_{\overline{x}_{1}t^{1/a_{1}}}(R)_{|H_{1}})$
attains its maximal on $V({\mathcal{A}}_{|H_{1}})=V({\mathcal{A}})\subset H$.

∎

######  Proposition 3.7.27.

, Let $R=\bigoplus R_{a}$ be a Rees algebra on a smooth variety $X$ over a
field $K$ such that $R_{a}\neq{\mathcal{O}}_{X}$ for any $a\in A$. There
exists a unique center ${\mathcal{A}}(R)={\mathcal{A}}$ such that

  1. (1)

The maximum
${\operatorname{maxinv}}(R)=(\overline{b}_{1},\ldots,\overline{b}_{k})$ of the
invariant ${\operatorname{inv}}_{p}(R)$, where $p\in X$, is attained at
$V({\mathcal{A}})$.

  2. (2)

${\mathcal{A}}={\mathcal{O}}_{X}[\overline{x}_{1}t^{1/a_{1}},\ldots,\overline{x}_{k}t^{1/a_{k}}]$
is a maximal admissible center for $R$, with
${\operatorname{inv}}({\mathcal{A}})=(\overline{b}_{1},\ldots,\overline{b}_{k})$,
where $\overline{b_{i}}=a_{i}{\operatorname{inv}}(\overline{x}_{i})$.

###### Proof.

Lemma 3.7.26 implies that ${\operatorname{inv}}_{p}(R)$ is upper
semicontinuous, and admits finitely many values. Consider the closed point set

$S:={\operatorname{supp}}{\operatorname{inv}}(R,(\overline{b}_{1},\ldots,\overline{b}_{k})):=\\{p\in
X\mid{\operatorname{inv}}(R)=(\overline{b}_{1},\ldots,\overline{b}_{k})\\},$

where ${\operatorname{inv}}_{p}(R)$, where $p\in X$, attains its maximal value
${\operatorname{maxinv}}(R)=(\overline{b}_{1},\ldots,\overline{b}_{k})$. Then
for any point $p\in S$ let ${\mathcal{A}}$ be a maximal admissible center for
$R$ at $p$. Then

$V({\mathcal{A}})={\operatorname{supp}}{\operatorname{inv}}(R,(\overline{b}_{1},\ldots,\overline{b}_{k}))$

locally around $p$. Moreover, ${\mathcal{A}}$ is a unique maximal admissible
center for $R$ at all points of $V({\mathcal{A}})$ in a neighborhood of $p$.
Thus ${\mathcal{A}}$ glues to a unique maximal admissible center for $R$ along
$V({\mathcal{A}})={\operatorname{supp}}{\operatorname{inv}}(R,(\overline{b}_{1},\ldots,\overline{b}_{k}))$
as desired.

∎

####  3.7.28. Cobordant blow-ups of the centers with maximal invariant

######  Proposition 3.7.29.

(see also [ATW19]) Let $R=\bigoplus R_{a}$ be a Rees algebra on a smooth
variety $X$ over a field $K$ such that $R_{a}\neq{\mathcal{O}}_{X}$ for any
$a\in A$. Let ${\mathcal{A}}={\mathcal{A}}(R)$ be maximal admissible center
for $R$ at $V({\mathcal{A}})$, and $B\to X$ be a full cobordant blow-up of
${\mathcal{A}}^{\operatorname{ext}}$. Then in a neighborhood of
$V({\mathcal{A}})$ such that ${\operatorname{inv}}_{p}(R)$ attains its maximum
${\operatorname{maxinv}}(\sigma^{c}(R))=(\overline{b}_{1},\ldots,\overline{b}_{k})$
at $V({\mathcal{A}})$ we have

  1. (1)

The maximum
${\operatorname{maxinv}}(\sigma^{c}(R))=(\overline{b}_{1},\ldots,\overline{b}_{k})$
is attained at $V(\sigma^{c}(A))$ with the invariant
${\operatorname{inv}}(\sigma^{c}(R))$, and such that

$\sigma^{c}(A^{{\operatorname{ext}}})={\mathcal{O}}_{B}[t^{-1/w},x^{\prime}_{1}t^{1/a_{1}},\ldots,x^{\prime}_{k}t^{1/a_{k}}]$

is maximal admissible center for $\sigma^{c}(R)$

  2. (2)

${\operatorname{maxinv}}(\sigma^{c}(R))<(\overline{b}_{1},\ldots,\overline{b}_{k})$
on $B_{+}=B\smallsetminus V(\sigma^{c}(A))$.

###### Proof.

We shall use induction on dimension of $X$. If $\dim(X)=0$ then
${\mathcal{A}}=R={\mathcal{O}}_{X}={\mathcal{O}}_{X}[0]$, $p=X$, and
${\operatorname{inv}}_{p}(R)=()$ has no entries and thus corresponds to the
infinite sequence of $\infty$. Let $p\in V({\mathcal{A}})$, and set
${\operatorname{ord}}_{p}({\mathcal{A}})=a_{1}$.

Consider a neighborhood $U$ of $p$ such that $\overline{x}_{1}$ is a maximal
contact on $U$ at $p$, and a partial maximal contact on $U$. Then, by Lemma
3.6.24, ${\operatorname{inv}}^{1}$ attains its maximum $\overline{b}_{1}$ on
${\operatorname{supp}}{\operatorname{inv}}^{1}(R,\overline{b}_{1})\subset
V(\overline{x}_{1})$. Moreover $\overline{x}_{1}$ is a maximal contact along
${\operatorname{supp}}{\operatorname{inv}}^{1}(R,\overline{b}_{1})$.

Consequently for the points
$p\in{\operatorname{supp}}{\operatorname{inv}}^{1}(R,\overline{b}_{1})$ we
have

${\operatorname{inv}}_{p}(R)=({\operatorname{inv}}^{1}_{p}(R),{\operatorname{inv}}_{p}(C_{\overline{x}_{1}t^{1/a_{1}}}(R)_{|H_{1}})).$

By Lemma 3.7.20, the controlled transform
$\sigma^{c}(\overline{x}_{1})=\overline{x}^{\prime}_{1}$ is a partial maximal
contact on $B_{U}=\sigma^{-1}(U)$ for $\sigma^{c}(R)$. Then, by Lemma 3.6.24,
${\operatorname{maxinv}}^{1}(\sigma^{c}(R))\leq\overline{b}_{1}$, and it
attains its value $\overline{b}_{1}$ on

${\operatorname{supp}}{\operatorname{inv}}^{1}(\sigma^{c}(R),\overline{b}_{1})\subseteq
H^{\prime}_{1}=V(\overline{x}^{\prime}_{1}),$

and for the points $p^{\prime}$ in
${\operatorname{supp}}{\operatorname{inv}}^{1}(R,\overline{b}_{1})$ we have
that $\overline{x}^{\prime}_{1}$ is a maximal contact. Moreover, by Lemma
3.7.22, and Section 3.7.5 we get

${\operatorname{inv}}_{p^{\prime}}(\sigma^{c}(R))=({\operatorname{inv}}^{1}_{p^{\prime}}(\sigma^{c}(R)),{\operatorname{inv}}_{p^{\prime}}(C_{\overline{x}^{\prime}_{1}t^{1/a_{1}}}(\sigma^{c}(R))_{|H^{\prime}_{1}}).$

By Lemma 3.7.24, the restriction of $B\to X$ to
$H^{\prime}_{1}=V_{B}(\overline{x}^{\prime}_{1})$ is the cobordant blow-up
$\sigma_{H^{\prime}_{1}}:H^{\prime}_{1}\to H_{1}$ of
${\mathcal{A}}^{\operatorname{ext}}_{|H_{1}}$. By the inductive assumption on
dimension the conditions (1), (2) of the Proposition are satisfied on $H_{1}$
for $C_{\overline{x}_{1}t^{1/a_{1}}}(R)_{|H_{1}}$ and for the cobordant blow-
up $\sigma_{H^{\prime}_{1}}:H^{\prime}_{1}\to H_{1}$. Moreover, by Lemma
3.5.11, and the assumption, ${\mathcal{A}}^{\operatorname{ext}}_{|H_{1}}$ is
maximal admissible for $C_{\overline{x}_{1}t^{1/a_{1}}}(R)_{|H_{1}}$ on
$H_{1}$, associated with
${\operatorname{maxinv}}(C_{\overline{x}_{1}t^{1/a_{1}}}(R)_{|H_{1}})$.

Consequently, by the inductive argument
$\sigma^{c}({\mathcal{A}}^{\operatorname{ext}})_{|H_{1}}={\mathcal{A}}^{\operatorname{ext}}_{B|H^{\prime}_{1}}$
is maximal admissible for

$\sigma_{H^{\prime}_{1}}^{c}(C_{\overline{x}_{1}t^{1/a_{1}}}(R)_{|H_{1}})=C_{\overline{x}^{\prime}_{1}t^{1/a_{1}}}(\sigma^{c}(R))_{|H^{\prime}_{1}}.$

By Lemma 3.5.11 and the fact that $\overline{x}^{\prime}_{1}$ is a partial
maximal contact associated with
${\operatorname{maxinv}}^{1}(\sigma^{c}(R))=\overline{b}_{1}$ we conclude that
${\mathcal{A}}_{B}^{\operatorname{ext}}$ is maximal admissible for
$\sigma^{c}(R)$. By the inductive assumption, the maximum

${\operatorname{maxinv}}(\sigma^{c}(C_{\overline{x^{\prime}}_{1}t^{1/a_{1}}}(R)_{|H_{1}}))=(\overline{b}_{2},\ldots,\overline{b}_{k})$

on $B$ is attained at $V(\sigma_{H_{1}}^{c}({\mathcal{A}}_{|H_{1}})$. Thus

${\operatorname{maxinv}}(\sigma^{c}(R)_{|H^{\prime}_{1}})=\max(\overline{b}_{1},{\operatorname{inv}}(C_{\overline{x^{\prime}}_{1}t^{1/a_{1}}}(R)_{|H_{1}})=(\overline{b}_{1},\ldots,\overline{b}_{k})$

is attained at

$V(\sigma_{H_{1}}^{c}({\mathcal{A}}_{|H_{1}})=V(\sigma^{c}({\mathcal{A}})_{|H^{\prime}_{1}})=V(\sigma^{c}({\mathcal{A}}))\subseteq
H^{\prime}_{1},$

and $\sigma^{c}({\mathcal{A}})$ is a maximal admissible center for
$\sigma^{c}(R)$ associated with ${\operatorname{maxinv}}\sigma^{c}(R)$. ∎

####  3.7.30. Resolution principle

(see also [ATW19] for (1), (2) and the second part of (4)) Summarizing the
above, the resolution process consists of the following:

Given a rational Rees algebra on a smooth $X$ with SNC divisor $E$ over a
field $K$ of characteristic zero.

  1. (1)

The invariant ${\operatorname{inv}}_{p}(R)$ on $X$ is semicontinuous. (Section
3.7.6)

  2. (2)

It attains its maximum ${\operatorname{maxinv}}_{X}(R)$ at a certain unique
center ${\mathcal{A}}^{{\operatorname{ext}}}(R)$ on $X$. (Proposition 3.7.27)

  3. (3)

The invariant ${\operatorname{inv}}_{p}(\sigma^{c}(R))$ on the full cobordant
blow-up $B\to X$ at ${\mathcal{A}}^{{\operatorname{ext}}}(R)$, attains its
maximum

${\operatorname{maxinv}}_{X}(R)={\operatorname{maxinv}}_{B}(\sigma^{c}(R))$

exactly at the center $\sigma^{c}({\mathcal{A}}^{{\operatorname{ext}}}(R))$
associated with the vertex
${\operatorname{Vert}}(B)=V(\sigma^{c}({\mathcal{A}}^{{\operatorname{ext}}}))$
of the full cobordant blow-up $B\to X$.(Proposition 3.7.29(1))

  4. (4)

The invariant ${\operatorname{inv}}_{p}(.)$ drops on the cobordant blow-up

$B_{+}=B\smallsetminus
V(\sigma^{c}({\mathcal{A}}^{{\operatorname{ext}}}(R)))=B\smallsetminus{\operatorname{Vert}}(B)$

after removing the vertex ${\operatorname{Vert}}(B)$, so that

${\operatorname{maxinv}}_{X}(R)>{\operatorname{maxinv}}_{B_{+}}(\sigma^{c}(R)).$

(Proposition 3.7.29(2))

###  3.8. Properties of the invariant

####  3.8.1. The invariant ${\operatorname{inv}}$ at the smooth points

Assume that $Y$ is a smooth subvariety of codimension $k$ on a smooth variety
$X$ and is described at $p\in Y$ by a partial set of free local parameters
$Y=V(u_{1},\ldots,u_{k})$ compatible with an SNC divisor $E$. Then

${\mathcal{A}}:={\mathcal{O}}_{X}[(u_{1},\ldots,u_{k})t]$

is a maximal ${\mathcal{I}}_{Y}$\- admissible center at $p$, with

${\operatorname{inv}}_{p}({\mathcal{I}})=(1,\ldots,1),$

with $k$ entries equal $1$. Conversely, if
${\operatorname{inv}}_{p}({\mathcal{I}}_{Y})=(1,\ldots,1)$ is as above then
there exists a partial system of free local parameters
$u_{1},\ldots,u_{k}\in{\mathcal{I}}$ compatible with $E$, such that

${\mathcal{O}}_{X}[{\mathcal{I}}_{Y}t]\subseteq{\mathcal{O}}_{X}[(u_{1},\ldots,u_{k})t]^{\operatorname{int}}={\mathcal{O}}_{X}[(u_{1},\ldots,u_{k})t].$

So ${\mathcal{I}}_{Y}=(u_{1},\ldots,u_{k})$ is smooth generated by free
coordinates and compatible with $E$ for $Y$ having SNC with $E$ at $p\in X$.

####  3.8.2. Torus action

If $X$ admits a torus action and $R$ is $T$-stable, then the maximal
admissible centers are canonical and unique and thus $T$-stable. One can run
the algorithm in Section 3.7.1 using seminvariant maximal contacts and
seminvariant derivations, with all intermediate Rees algebras $R_{i}$ being
$T$-stable. Thus one can choose inductively semiinvariant coordinates of the
maximal admissible centers. These are the nested maximal contacts in the
$T$-stable cotangent ideals $T^{1/a_{i}}(R_{i})$.

Moreover, if additionally $X$ admits a geometric quotient $X/T$ with all
orbits of dimension $\dim(T)$, then the number of semiinvariant coordinates in
a $T$-stable center at a given point cannot exceed
$\dim(X/T)=\dim(X)-\dim(T)$, as it describes a smooth $T$-stable subvariety of
$X$ of dimension at least $\dim(T)$. Consequently the set of values of
${\operatorname{inv}}_{p}({\mathcal{I}})$ in the resolution process

$X_{0}\leftarrow X_{1}\leftarrow X_{2}\leftarrow\ldots$

is contained in $({\mathbb{Q}}_{+})_{\geq 0}^{k}$, where

$k=\dim(X_{0})=\ldots=\dim(X_{i})-\dim({\mathbb{G}}_{m}^{i}).$

####  3.8.3. The descending chain condition

######  Lemma 3.8.4.

The set $\Gamma\subset({\mathbb{Q}}_{+})_{\geq 0}^{k}$, of the possible values
of ${\operatorname{inv}}_{p}(R)$ of the $T$-stable Rees algebras on a smooth
varieties $X$ with action of torus $T$, and having a geometric quotient $X/T$
of dimension $k$, which are generated by the gradations $R_{a^{i}}t^{a^{i}}$,
for $i=1,\ldots,s$ satisfies dcc.

###### Proof.

Donote by $a:={\operatorname{lcm}}(a^{1},\ldots,a^{s}))$. We use induction on
$k$. For $k=1$, the possible values of
${\operatorname{inv}}_{p}(R)={\operatorname{inv}}^{1}_{p}(R)$ are contained in
$(1/a)\cdot{\mathbb{N}}_{+}$.

In general, we note that ${\mathcal{A}}$ is $R$-admissible if and only if
$R_{a}t^{a}\subset{\mathcal{A}}$. Thus we have

${\operatorname{inv}}_{p}(R)=1/a\cdot({\operatorname{inv}}_{p}(R_{a}),$

and the problem reduces to the ideals ${\mathcal{I}}=R_{a}$ on $X$.

Now, for any ideal ${\mathcal{I}}$ on $X$ we have

${\operatorname{inv}}_{p}({\mathcal{I}})=({\operatorname{inv}}^{1}_{p}({\mathcal{I}}),{\operatorname{inv}}_{p}(\sigma_{H_{1}}^{c}({\mathcal{C}}_{\overline{x}_{1}t^{1/a_{1}}}({\mathcal{O}}_{X}[({\mathcal{I}}t)]_{|H_{1}})),$

where the restricted coefficient ideal

$C:={\mathcal{C}}_{\overline{x}_{1}t^{1/a_{1}}}{\mathcal{O}}_{X}[({\mathcal{I}}t)]_{|H_{1}})=\bigoplus
C_{a/a_{1}}\cdot t^{a/a_{1}}$

is generated at gradations

$C_{1/a_{1}}t^{1/a_{1}},\ldots,C_{(a_{1}-1)/a_{1}}t^{(a_{1}-1)/a_{1}}$

on $H_{1}$ with $\dim(H_{1}/T)<k$. By the inductive assumption the set of
values of ${\operatorname{inv}}_{p}(C)$ satisfies dcc. Now since the set of
values ${\operatorname{inv}}^{1}_{p}({\mathcal{I}})$ satisfies dcc condition
we conclude that the set of values ${\operatorname{inv}}_{p}({\mathcal{I}})$
on $X$ satisfies dcc.

Consequently the set of values of ${\operatorname{inv}}_{p}(R)$ in a
resolution sequence also satisfies dcc. ∎

####  3.8.5. Functoriality of the invariant

[ATW19]

######  Lemma 3.8.6.

The invariant ${\operatorname{inv}}_{p}(R)$ and the maximal admissible center
${\mathcal{A}}$ are functorial under smooth morphisms, field extensions, and
group actions.

The functoriality of ${\operatorname{inv}}_{p}(R)$ and the maximal admissible
centers ${\mathcal{A}}$ at $p$ is a consequence of the functoriality of the
resolution algorithm in Section 3.7.1 and functoriality of derivations, Rees
algebras, coefficient ideals, and the maximal contacts.

###  3.9. Final Conclusions

####  3.9.1. Functorial Principalization

[ATW19] (in a non-SNC setting),

Let ${\mathcal{I}}$ be an ideal on $X$. We initiate the SNC resolution
algorithm using cobordant blow-ups $\sigma_{i}:X_{i+1}\to X_{i}$ at the
maximal ${\mathcal{I}}_{i}$-admissible centers
${\mathcal{A}}({\mathcal{I}}_{i})$ on $X_{i}$, associated with the maximum
${\operatorname{maxinv}}_{X_{i}}({\mathcal{I}}_{i})$ of
${\operatorname{inv}}_{p}({\mathcal{I}}_{i})$ on $X_{i}$. They form a sequence

$X=X_{0}\stackrel{{\scriptstyle\sigma_{0}}}{{\leftarrow}}X_{1}\stackrel{{\scriptstyle\sigma_{1}}}{{\leftarrow}}\ldots\stackrel{{\scriptstyle\sigma_{k-1}}}{{\leftarrow}}X_{k}=X^{\prime},$

Here the ideal ${\mathcal{I}}_{0}:={\mathcal{I}}$, and
${\mathcal{I}}_{i+1}:=\sigma_{i}^{c}({\mathcal{I}}_{i})$ is the controlled
transform of ${\mathcal{I}}_{i}$. Each such blow-up reduces
${\operatorname{maxinv}}_{X_{i}}({\mathcal{I}}_{i})$ so that

${\operatorname{maxinv}}_{X_{i}}({\mathcal{I}}_{i})>{\operatorname{maxinv}}_{X_{i+1}}({\mathcal{I}}_{i+1}),$

where $X_{i+1}$ is the coborandant blow-ups $X_{i}$ at the center
${\mathcal{A}}({\mathcal{I}}_{i})$. By dcc property this is a finite process
which continues until the maximum value

${\operatorname{maxinv}}_{X_{k}}(\sigma^{c}({\mathcal{I}}_{k}))=0.$

In this situation, the controlled transform

$\sigma^{c}({\mathcal{I}}):=\sigma_{k-1}^{c}({\mathcal{I}}_{i})\circ\ldots\circ\sigma_{1}^{c}({\mathcal{I}})={\mathcal{O}}_{X}$

of ${\mathcal{I}}$ becomes ${\mathcal{O}}_{X}$, and the full transform
${\mathcal{O}}_{X^{\prime}}\cdot{\mathcal{I}}$ is locally the product of
divisorial components. Note that each cobordant blow-up $X_{i+1}\to X_{i}$
inductively creates a scheme $X_{i+1}$ with an action of $T_{i+1}=T_{i}\times
G_{m}$. Since all constructions are canonical and functorial for smooth
morphisms, the centers, the ideals ${\mathcal{I}}_{i}$ and the exceptional
divisors are automatically $T_{i}$-stable. This proves Theorem 1.11.2.

####  3.9.2. Embedded Desingularization

[ATW19] In the process of embedded resolution of an irreducible subvariety
$Y\subset X$ of codimension $k$, we consider the principalization of
${\mathcal{I}}={\mathcal{I}}_{Y}$ of $Y$ on $X$. We execute the algorithm
applying the cobordant blow-ups at maximal ${\mathcal{I}}$-admissible centers
${\mathcal{A}}={\mathcal{A}}({\mathcal{I}})$. In this process, we use the
strict transforms of the ideal ${\mathcal{I}}={\mathcal{I}}_{Y}$ instead of
the controlled transforms. We stop the algorithm when the invariant

${\operatorname{maxinv}}_{X}({\mathcal{I}})=(1,\ldots,1),$

with $k$ entries equal $1$. In such a case, the associated maximal admissible
center with the invariant ${\operatorname{inv}}({\mathcal{A}})=(1,\ldots,1)$
for the ideal ${\mathcal{I}}_{Y^{\prime}}$ of the strict transform
$Y^{\prime}$ of $Y$ coincides with
${\mathcal{O}}_{X}[{\mathcal{I}}_{Y^{\prime}}t]$, and thus, by Section 3.8.1,
the strict transform $Y^{\prime}$ of $Y$ has SNC with the exceptional divisor,
which is also SNC.

Alternatively one can use the controlled transforms of ideals instead of the
strict transforms in the process without changing the strategy. The process is
slower and some centers need not to be contained in the strict transform of
$Y$.

Both procedures show the existence of the functorial embedded
desingularization by the smooth cobordant blow-ups as in Theorem 1.11.4. As a
result, we obtain a smooth subvariety $Y^{\prime}$ having SNC with an SNC
exceptional divisor $E^{\prime}$ on $X^{\prime}$, and with the torus action.

Each consecutive cobordant blow-up $X_{i+1}\to X_{i}$ will create inductively
a variety $X_{i+1}$ with an action of $T_{i+1}=T_{i}\times G_{m}$. Since all
the constructions are canonical and functorial for smooth morphisms, the
centers and subvariety $Y_{i}\subset X_{i}$ are automatically $T_{i}$-stable.
On the level of the geometric quotients, we obtain the sequence of weighted
blow-ups

$X=X_{0}\stackrel{{\scriptstyle\sigma_{0}}}{{\leftarrow}}X_{1}/T_{1}\stackrel{{\scriptstyle\sigma_{1/T_{1}}}}{{\leftarrow}}\ldots\stackrel{{\scriptstyle\sigma_{k-1/T_{k-1}}}}{{\leftarrow}}X_{k}/T_{k}=X^{\prime}/T,$

of varieties with quotient singularities such that the subvariety
$Y^{\prime}/T$ of $X^{\prime}/T$ admits quotient singularities. When
considering stack-theoretic quotients $[X_{i}/T_{i}]$, we obtain smooth stacks
with a smooth substack $[Y^{\prime}/T]\subset[X^{\prime}/T]$ having SNC with
the exceptional divisors. A similar process without exceptional SNC divisor
was considered in [ATW19].

####  3.9.3. Nonembedded SNC Resolution

The nonembedded resolution is a consequnce of the embedded resolution using
local embeddings and the functoriality properties. It will have SNC
exceptional divisor.

We associate with a variety $Y$ over $K$, initially without any divisors, a
modified invariant $\widetilde{{\operatorname{inv}}}_{p}(Y)$ in the following
way. We locally embed $Y$ into a smooth variety $X$. Any two such embeddings
$Y\subset X_{1}$ and $Y\subset X_{2}$ into smooth varieties of the same
dimension are étale equivalent. If $\dim(X_{1})+m=\dim(X_{2})$, and $m\geq 0$
then the induced embeddings $Y\subseteq
X_{1}\subseteq{\mathbf{A}}^{m}_{X_{1}}$ and $Y\subset X_{2}$ are ’etale
equivalent. Here the embedding

$X_{1}\subseteq{\mathbf{A}}^{m}_{X_{1}}=\operatorname{Spec}({\mathcal{O}}_{X}[x_{1},\ldots,x_{m}])$

is defined by $V(x_{1},\ldots,x_{m})$.

For any local embeddings $Y\subset
X_{1}=V(x_{1},\ldots,x_{m})\subset{\mathbf{A}}^{m}_{X_{1}}$ and $Y\subset
X_{2}$ with $\dim(X_{1})+m=\dim(X_{2})$ we get that $(x_{1},\ldots,x_{m})$ is
partial maximal contact in gradation $t$. Then passing to the completion
$\widehat{{\mathcal{O}}}_{X,p}$ the maximal admissibility condition

$\widehat{{\mathcal{O}}}_{X,p}\cdot
R_{Y}=\widehat{{\mathcal{O}}}_{X,p}[(x_{1},\ldots,x_{m})t,R_{Y|X_{1}}]\subset\widehat{{\mathcal{O}}}_{X,p}\cdot{\mathcal{A}}:=\widehat{{\mathcal{O}}}_{X,p}[\overline{x}_{1}t,\ldots\overline{x}_{k}t^{1/a_{k}}],$

with $(x_{1},\ldots,x_{m})\subset\overline{x}_{1}$ is equivalent to the
maximal admissibility $R_{Y|X_{1}}\subset{\mathcal{A}}_{Y|X_{1}}$, and thus

${\operatorname{inv}}_{p}(R_{Y,{\mathbf{A}}^{m}_{X_{1}}})=(1,\ldots,1,{\operatorname{inv}}_{p}(R_{Y|X_{1}})=$
$=(1,\ldots,1,{\operatorname{inv}}_{p}({\mathcal{I}}_{Y,X_{1}}).$

Moreover, by functoriality, we have

${\operatorname{inv}}_{p}({\mathcal{I}}_{Y,{X_{2}}})={\operatorname{inv}}_{p}({\mathcal{I}}_{Y,{\mathbf{A}}^{m}_{X_{1}}})=(1,\ldots,1,{\operatorname{inv}}_{p}({\mathcal{I}}_{Y,X_{1}})),$

For a closed embedding $Y\subset X$ into a smooth $X$ of dimension $n$, and
$p\in Y$, let
${\operatorname{inv}}_{p}({\mathcal{I}}_{Y})={\operatorname{inv}}(b_{1},\ldots,b_{k})$.
We define the invariant $\widetilde{{\operatorname{inv}}}_{p}(Y)$ on $Y$ to be
the equivalence class of the sequences $(b_{1},\ldots,b_{k})_{n}$, marked by
$n=\dim(X)$, where $Y\subset X$ and
${\operatorname{inv}}_{p}({\mathcal{I}}_{Y})=(b_{1},\ldots,b_{k})$, with the
equivalence relation

$(b_{1},\ldots,b_{k})_{n}\simeq(1,\ldots,1,b_{1},\ldots,b_{k})_{n+m},$

where $m$ is the number of ones in front of $b_{1}$. By the above the
invariant is independent of embeddings and functorial for smooth morphisms.

One can compare two equivalence classes lexicographically, as before, by
fixing their representatives with the same marking $n$.

We shall use the following

######  Lemma 3.9.4.

Let $Y\subset X_{1}$, and $Y\subset X_{2}$ be two different embeddings. For a
point $p\in Y$ consider two maximal admissible centers at $p$:
${\mathcal{A}}_{1}$ for ${\mathcal{I}}_{Y,X_{1}}$ on $X_{1}$ and
${\mathcal{A}}_{2}$ for ${\mathcal{I}}_{Y,X_{1}}$. Denote by $B_{1+}\to
X_{1}$, and $B_{2+}\to X_{2}$ the cobordant blow-ups respectively at
${\mathcal{A}}_{1}$ and ${\mathcal{A}}_{2}$, Then

${\mathcal{A}}_{1|Y}={\mathcal{A}}_{2|Y},$

and consequently the restrictions of the cobordant blow-ups $B_{1+}\to X_{1}$,
and $B_{2+}\to X_{2}$ to the strict transforms of $Y$ coincide with the
cobordant blow-up $B_{Y+}\to Y$ of ${\mathcal{A}}_{1|Y}={\mathcal{A}}_{2|Y}$
on $Y$.

###### Proof.

We can embed further $X_{1}$ and $X_{2}$ into an affine space
${\mathbb{A}}^{N}$ in such a way that the induced embeddings of $Y$ to
${\mathbb{A}}^{N}$ coincide (see for intance [Wło05, Lemma 2.5.3]). This
reduces the situation to the embedding $X_{1}\subset X_{2}={\mathbb{A}}^{N}$.

Now ${\mathcal{I}}_{Y,X_{1}}={\mathcal{I}}_{Y,X_{2}|X_{1}}$, and similarly
${\mathcal{A}}_{1}={\mathcal{A}}_{2|X_{1}}$ implying
${\mathcal{A}}_{1|Y}={\mathcal{A}}_{2|Y}$ on $Y$. Consequently the morphism on
the strict transforms $\sigma^{c}(Y)\to Y$ on $X_{1}$ or $X_{2}$ coincides, by
Lemma 5.1.5, with the cobordant blow-up of
${\mathcal{A}}_{1|Y}={\mathcal{A}}_{2|Y}$.

∎

We shall run the embedded algorithm using the centers associated with
$\max\widetilde{{\operatorname{inv}}}$ on $Y$. It locally corresponds to the
center ${\mathcal{A}}^{\operatorname{ext}}$ associated with
${\operatorname{maxinv}}({\mathcal{I}}_{Y})$ for a certain embedding $Y\subset
X$.

Consider an open affine cover of $Y$ by $Y^{j}$ admitting closed embeddings
$Y^{j}\subset X^{j}$ into smooth varieties $X^{j}$. Let
${\mathcal{I}}_{Y^{j}}$ be the corresponding ideal on $X^{j}$. We can assume
that all $X^{j}$ are of the same dimension. Let

$\overline{Y}:=\coprod Y_{j}\hookrightarrow\overline{X}:=\coprod X_{j}$

be the induced embedding of disjoint unions. Consider the natural étale
projection $\overline{Y}\to Y$.

Now we shall run the embedded desingularization with the strict transforms for
$\overline{Y}\hookrightarrow\overline{X}$, that the sequence of maximal
${\mathcal{I}}_{\overline{Y}_{i}}$-admissible centers
$\overline{{\mathcal{A}}}_{i}$:

$\overline{X}=\overline{X}_{0}\stackrel{{\scriptstyle\sigma_{0}}}{{\leftarrow}}\overline{X}_{1}\stackrel{{\scriptstyle\sigma_{1}}}{{\leftarrow}}\ldots\stackrel{{\scriptstyle\sigma_{k-1}}}{{\leftarrow}}\overline{X}_{k}=\overline{X}^{\prime},$

with ideals ${\mathcal{I}}_{\overline{Y}_{i}}$ being the strict transform of
${\mathcal{I}}_{\overline{Y}_{i-1}}$, which induces the sequence of the strict
transforms

$\overline{Y}=\overline{Y}_{0\overline{Y}}\stackrel{{\scriptstyle\sigma_{1\overline{Y}}}}{{\leftarrow}}\overline{Y}_{1}\stackrel{{\scriptstyle\sigma_{{1}\overline{Y}}}}{{\leftarrow}}\ldots\stackrel{{\scriptstyle\sigma_{k-1\overline{Y}}}}{{\leftarrow}}\overline{Y}_{k}=\overline{Y}^{\prime},$

The latter is the sequence of the cobordant blow-ups with the induced centers
$\overline{{\mathcal{A}}}_{i|\overline{Y}_{1}}$. By functoriality and Lemma
3.9.4, it descends to the sequence of the induced cobordant blow-ups

${Y}={Y}_{0}\stackrel{{\scriptstyle\sigma_{0Y}}}{{\leftarrow}}{Y}_{1}\stackrel{{\scriptstyle\sigma_{1Y}}}{{\leftarrow}}\ldots\stackrel{{\scriptstyle\sigma_{k-1Y}}}{{\leftarrow}}{Y}_{k}={Y}^{\prime},$

on $Y$. Then, $\max\widetilde{{\operatorname{inv}}}$ drops to the minimal
value $(1,1,\ldots,1)_{n}$ on $\overline{Y}^{\prime}$ and on $Y^{\prime}$,
where $n$ is the dimension of the ambient variety $\overline{X}$, and the
number of ones is equal to the codimension of the smooth subvariety
$\overline{Y}\subset\overline{X}$.

The exceptional SNC divisor on a local ambient space $\overline{X}^{\prime}$
is compatible with a smooth subvariety $\overline{Y}^{\prime}$. Consequently,
its restriction to $\overline{Y}^{\prime}$ defines an SNC-exceptional divisor
on $\overline{Y}^{\prime}$ and determines a functorial nonembedded SNC
resolution of $\overline{Y}$, while its descent determines a functorial
nonembedded SNC resolution of ${Y}$. This completes the proof of Theorem
1.11.7. A similar non-SNC resolution was considered in [ATW19] ,

##  4\. Free characteristic resolution of almost homogenous singularities

###  4.1. The weighted normal bundles to the centers

####  4.1.1. Weighted normal bundle

Let $X$ be a regular scheme.

Let ${\mathcal{J}}$ be a ${\mathbb{Q}}$-ideal locally written as
${\mathcal{J}}=(u_{1}^{1/w_{1}},\ldots,u_{k}^{1/w_{k}})$, and let
${\mathcal{A}}_{\mathcal{J}}={\mathcal{O}}_{X}[t^{-1},u_{1}t^{w_{1}},\ldots,u_{k}t^{w_{k}}]^{\operatorname{int}}$
be the associated Rees algebra. Once can consider the filtration

$\\{({\mathcal{J}}^{a})_{X}\\}_{a\in{\mathbb{Z}}_{\geq
0}}=\\{{\mathcal{A}}_{{\mathcal{J}},a}\\}_{a\in{\mathbb{Z}}_{\geq 0}}$

of ${\mathcal{O}}_{X}$. Then
${\operatorname{gr}}_{{\mathcal{J}}}({\mathcal{O}}_{X})$ is a sheaf of graded
${\mathcal{O}}_{X}/{\mathcal{J}}_{X}={\mathcal{O}}_{V({\mathcal{J}})}$-modules
on $V({\mathcal{J}})$, which is locally on $X$, isomorphic to the graded ring

$\displaystyle{\operatorname{gr}}_{{\mathcal{J}}}({\mathcal{O}}_{X})={\mathcal{O}}_{X}/{\mathcal{J}}_{X}\oplus({\mathcal{J}}_{X}/{\mathcal{J}}^{2}_{X})t\oplus\ldots\oplus({\mathcal{J}}^{i}_{X}/{\mathcal{J}}^{i+1})_{X}t^{i}\oplus\ldots=$
$\displaystyle=({\mathcal{O}}_{X}\oplus({\mathcal{J}}_{X})t\oplus\ldots\oplus({\mathcal{J}}^{i}_{X}/t^{i}\oplus\ldots)\,\,/\,\,({\mathcal{J}}_{X}\oplus\ldots\oplus({\mathcal{J}}^{i}_{X})t^{i-1}\oplus\ldots=$
$\displaystyle={\mathcal{A}}_{{\mathcal{J}}}[t^{-1}]/(t^{-1}{\mathcal{A}}_{{\mathcal{J}}}[t^{-1}])={\mathcal{A}}_{\mathcal{J}}^{{\operatorname{ext}}}/(t^{-1}{\mathcal{A}}_{\mathcal{J}}^{{\operatorname{ext}}})={\mathcal{O}}_{B}/(t^{-1}\cdot{\mathcal{O}}_{B})=$
$\displaystyle={\mathcal{O}}_{X}[t^{-1},u_{1}t^{w_{1}},\ldots,u_{k}t^{w_{k}}]/(t^{-1})={\mathcal{O}}_{V({\mathcal{J}})}[u_{1}t^{w_{1}},\ldots,u_{k}t^{w_{k}}],$

where $u_{i}$ is in $w_{i}$ gradation. We shall call the corresponding scheme
the weighted normal bundle at the center ${\mathcal{J}}$, and denote it by

$N_{{\mathcal{J}}}(X):=\operatorname{Spec}({\operatorname{gr}}_{{\mathcal{J}}}({\mathcal{O}}_{X})).$

######  Remark 4.1.2.

A similar idea was considered independently in [QR22].

####  4.1.3. The ideal of the initial forms

With any function $f\in{\mathcal{O}}_{X,p}$, regular at $p\in
V({\mathcal{J}})$, such that
$f\in(({\mathcal{J}}^{a})_{X}\smallsetminus({\mathcal{J}}^{a+1})_{X})$, for a
certain $a\in{\mathbb{N}}$ one can associate the unique homogenous element-
the initial form

$\operatorname{in}(f)\in(({\mathcal{J}}^{a})_{X}/({\mathcal{J}}^{a+1})_{X})t^{a}\subset{\operatorname{gr}}_{{\mathcal{J}}}({\mathcal{O}}_{X}).$

Similarly, we associate with an ideal sheaf ${\mathcal{I}}$, the filtration
${\mathcal{I}}_{a}:={\mathcal{I}}\cap({\mathcal{J}}^{a})_{X}$, where
$a\in{\mathbb{N}}$, and its ideal of the initial forms on
$N_{{\mathcal{J}}}(X)$:

$\operatorname{in}({\mathcal{I}})=\bigoplus_{a\in{\mathbb{Z}}{\geq
0}}({\mathcal{I}}_{a}/({\mathcal{I}}_{a+1})t^{a}=\bigoplus_{a\in{\mathbb{Z}}_{\geq
0}}({\mathcal{I}}_{a}+({\mathcal{J}}^{a+1})_{X})/({\mathcal{J}}^{a+1})_{X}t^{a}\subset{\operatorname{gr}}_{{\mathcal{J}}}({\mathcal{O}}_{X}).$

####  4.1.4. The weighted normal bundle and the exceptional divisor

The computations above imply the following extension of a classical result of
Huneke-Swanson on extended Rees algebras and smooth blow-ups [HS06, Definition
5.1.5].

######  Lemma 4.1.5.

Let $X$ be a regular scheme and $\sigma:B\to X$ be the full cobordant blow-up
of the center ${\mathcal{J}}$. Then the exceptional divisor $V_{B}(t^{-1})$ is
isomorphic to $N_{{\mathcal{J}}}(X)$:

$V_{B}(t^{-1})=\operatorname{Spec}_{X}({\mathcal{O}}_{B}/(t^{-1}))\simeq
N_{{\mathcal{J}}}(X)$

∎

####  4.1.6. The strict transform and the ideal of the initial form

We will need the above identification in the context of the strict transforms
of the ideals.

######  Lemma 4.1.7.

With the previous assumptions and notation. Let
${\mathcal{I}}\subset{\mathcal{O}}_{X}$ be an ideal sheaf on $X$. Let
$\sigma^{s}({\mathcal{I}})\subset{\mathcal{O}}_{B}$ be the strict transform of
${\mathcal{I}}$. Then the natural isomorphism
${\mathcal{O}}_{B}/(t^{-1})\to{\operatorname{gr}}_{{\mathcal{J}}}({\mathcal{O}}_{X})$
takes $\sigma^{s}({\mathcal{I}})_{|V(t^{-1})}$ to
$\operatorname{in}({\mathcal{I}})$.

###### Proof.

Let $f\in{\mathcal{I}}$ such that
$f\in({\mathcal{J}}^{a})_{X}\smallsetminus({\mathcal{J}}^{{a}+1})_{X}$. Then
the strict transform
$\sigma^{s}(f)=t^{a}f\in({\mathcal{J}}^{a})_{X}t^{a}\subset{\mathcal{O}}_{B}$,
and its reduction modulo $t^{-1}$ is in
${\mathcal{O}}_{B}/(t^{-1}\cdot{\mathcal{O}}_{B})={\operatorname{gr}}_{{\mathcal{J}}}({\mathcal{O}}_{X})$
in the gradation

$({\mathcal{J}}^{a})_{X}t^{a}/(t^{-1}{\mathcal{O}}_{B}\cap{\mathcal{O}}_{X}t^{a})=(({\mathcal{J}}^{a})_{X}/({\mathcal{J}}^{{a}+1})_{X})\cdot
t^{a},$

and naturally and bijectively corresponds to
$\operatorname{in}(f)\in({\mathcal{J}}^{a})_{X}/({\mathcal{J}}^{{a}+1})_{X}t^{a}\subset{\operatorname{gr}}_{\mathcal{J}}({\mathcal{O}}_{X}).$

∎

####  4.1.8. Weighted normal cone

One can extend the construction of the weighted normal bundle of a center
${\mathcal{J}}$ on a regular scheme to the weighted normal cone at any Rees
algebra $R=\bigoplus_{i\in{\mathbb{Z}}_{\geq
0}}R_{a}t^{a}\subset{\mathcal{O}}_{X}[t]$ on any noetherian scheme $X$.
Consider the associated gradation

${\operatorname{gr}}_{R}({\mathcal{O}}_{X})=\bigoplus_{a\in{\mathbb{Z}}_{\geq
0}}((R_{a})_{X}/(R_{a+1})_{X})t^{a}=R/(R\cap t^{-1}R).$

Then by the weighted normal cone of $X$ at $R$ we shall mean

$C_{R}(X):=\operatorname{Spec}_{V(R)}({\operatorname{gr}}_{R}({\mathcal{O}}_{X}))$

######  Definition 4.1.9.

Let $X$ be a regular scheme, and $Y\subset X$ be an integral, closed subscheme
with an ideal sheaf ${\mathcal{I}}_{Y}$. Let ${\mathcal{J}}$ be a center on
$X$ with $V({\mathcal{J}})\subset Y$. Denote by

$R_{{\mathcal{J}},Y}:={\mathcal{O}}_{Y}\cdot{\mathcal{A}}_{{\mathcal{J}}}={\mathcal{O}}_{Y}({\mathcal{O}}_{X}[{\mathcal{J}}t]_{X})$

the induced Rees algebra on $Y$. Then by the weighted normal cone of $Y$ at
$V({\mathcal{J}})$ we mean

$C_{{\mathcal{J}}}(Y):=\operatorname{Spec}_{V({\mathcal{J}})}(C_{R_{{\mathcal{J}},Y}}(Y))=\operatorname{Spec}_{V(R)}({\operatorname{gr}}_{R_{{\mathcal{J}},Y}}({\mathcal{O}}_{Y})).$

######  Lemma 4.1.10.

With the above notation and assumptions

$C_{{\mathcal{J}}}(Y)=\operatorname{Spec}_{V(R)}({\operatorname{gr}}_{R_{{\mathcal{J}},Y}}({\mathcal{O}}_{Y}))$

is the closed subscheme of the weighted normal bundle
${\mathcal{N}}_{{\mathcal{J}}}(X)$ which is defined by
$\operatorname{in}({\mathcal{I}}_{Y})\subset{\operatorname{gr}}_{{\mathcal{J}}}({\mathcal{O}}_{X})$.

###### Proof.

Consider the surjective morphism of sheaves
$\phi:{\mathcal{A}}_{{\mathcal{J}}}\to R_{{\mathcal{J}},Y}$. Its kernel is
generated by
$({\mathcal{I}}_{Y}\cdot{\mathcal{O}}_{X}[t])\cap{\mathcal{A}}_{{\mathcal{J}}}$.
So

$R_{{\mathcal{J}},Y}\simeq\bigoplus\frac{({\mathcal{J}}^{a})_{X}}{{\mathcal{I}}_{Y}\cap({\mathcal{J}}^{a})_{X}}\cdot
t^{a}.$

The morphism $\phi$ defines the surjective morphism

${\operatorname{gr}}_{{\mathcal{J}}}({\mathcal{O}}_{X})=\bigoplus({\mathcal{J}}^{a})_{X}/({\mathcal{J}}^{a+1})_{X}t^{a}\to{\operatorname{gr}}_{R}({\mathcal{O}}_{Y})=\bigoplus\frac{({\mathcal{J}}^{a})_{X}}{({\mathcal{J}}^{a+1})_{X}+({\mathcal{I}}_{Y}\cap({\mathcal{J}}^{a})_{X})}\cdot
t^{a}.$

Its kernel is exactly

$\operatorname{in}({\mathcal{I}}_{Y})=\bigoplus\frac{(({\mathcal{I}}_{Y}\cap({\mathcal{J}}^{a})_{X})+({\mathcal{J}}^{a+1})_{X}}{({\mathcal{J}}^{a+1})_{X}}\cdot
t^{a}.$

∎

######  Remark 4.1.11.

Assume that $X$ is a variety over an algebraically closed field $K$ and $B$ is
the cobordant blow-up of a center ${\mathcal{J}}$. It follows from Lemmas
4.1.5, 4.1.7, that when considering the natural projection
$\pi:B\to{\mathbb{A}}^{1}=\operatorname{Spec}(K[t^{-1}]$ we obtain the
deformation of the $X=\pi^{-1}(a)$, where $a\neq 0$ to the the normal weighted
cone $N_{{\mathcal{J}}}(X)=\pi^{-1}(0)=V_{B}(t^{-1})$. Its restriction to
$\pi_{C}:C:=V(\sigma^{s}({\mathcal{I}}))\to{\mathbb{A}}^{1}$ determines the
deformation of $Y$ to the normal weighted cone $C_{{\mathcal{J}}}(Y)$, with
$Y=\pi_{C}^{-1}(a)$ for $a\neq 0$ and
$C_{{\mathcal{J}}}(Y)=\pi_{C}^{-1}(0)=V_{C}(t^{-1})$. This observation was
made by Quek and Rydh in [QR22].

###  4.2. Almost homogenous singularities and their resolution

For any ideal ${\mathcal{I}}$ on a regular scheme $X$ let

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

denote the singular locus of the scheme
$\operatorname{Spec}_{X}({\mathcal{O}}_{X}/{\mathcal{I}}_{Y})$.

######  Definition 4.2.1.

Let $X$ be a regular scheme. Let $Y\subset X$ be an integral closed subscheme
with an ideal sheaf ${\mathcal{I}}_{Y}$. We say that a regular subscheme
$Z\subset Y$ is an almost homogenous singularity of $Y$ if

  1. (1)

The singular locus of $Y$ is ${\operatorname{Sing}}(Y)=Z$.

  2. (2)

There is a center ${\mathcal{J}}$ on $X$, such that $V({\mathcal{J}})=Z$, and

${\operatorname{Sing}}(C_{{\mathcal{J}}}(Y))=V(\operatorname{in}({\mathcal{J}}))=Z\subseteq
N_{{\mathcal{J}}}(X).$

######  Theorem 4.2.2.

Let $X$ be a regular scheme. Let $Y\subset X$ be an integral closed subscheme.
Let $Z\subset Y$ be an almost homogenous singularity of $Y$ for a center
${\mathcal{J}}$. Let $N_{{\mathcal{J}}}(X)$ be the weighted normal bundle of
$X$ at ${\mathcal{J}}$, and $C_{{\mathcal{J}}}(Y)\subset N_{{\mathcal{J}}}(X)$
be the weighted normal cone of $Y$ at ${\mathcal{J}}$. Assume that either $X$
is universally catenary or the codimension of each component of
$C_{{\mathcal{J}}}(Y)\smallsetminus Z$ in $N_{{\mathcal{J}}}(X)$ is equal to
the codimension of $Y$ in $X$.

Then the cobordant blow-up $B_{+}\to X$ defines a cobordant resolution of $Y$.
That is, the strict transform $Y^{\prime}$ of $Y$ is a regular subscheme of
$B_{+}$ of the codimension equal to the codimension of $Y$ in $X$.

###### Proof.

The problem is local on $X$. Thus we can assume that
${\mathcal{J}}=(u_{1}^{1/w_{1}},\ldots,u_{k}^{1/w_{k}})$ and the full
cobordant blow-up of ${\mathcal{J}}$ is given by

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

Assume that $X$ is universally catenary, and thus $B$ is catenary. Let $d$ be
the codimension of $Y$ in $X$. Then for the morphism
$\sigma_{-}:B_{-}=X\times{\mathbb{G}}_{m}\to X$, the inverse image
$\sigma_{-}^{-1}(Y)$ is irreducible of codimension $d$. So it is its closure
$Y^{\prime}:=\overline{\sigma^{-1}(Y)}$, which is the strict transform of $Y$:

$\operatorname{codim}_{B}(Y^{\prime})=d$

Note that by the definition of the strict transform, $t^{-1}$ is not a zero
divisor of ${\mathcal{O}}_{B}/{\mathcal{I}}_{Y^{\prime}}$. Then, by the Krull
Hauptidealsatz, we have that each component of $Y^{\prime}\cap V(t^{-1})$ is
of codimension $1$ in $Y^{\prime}$, and the codimension $d+1$ in $B$. We
conclude that each component of $Y^{\prime}\cap V(t^{-1})$ is of codimension
$d$ in $V(t^{-1})$.

Note that ${\operatorname{Sing}}(Y^{\prime})\smallsetminus
V(t^{-1})={\operatorname{Sing}}(Y^{\prime})\cap B_{-}$ is contained in

$V_{B_{-}}(u_{1},\ldots,u_{k})=V_{B_{-}}(u_{1}t^{w_{1}},\ldots,u_{k}t^{w_{k}})=V_{B_{-}}(u^{\prime}_{1},\ldots,u^{\prime}_{k}),$

where $u_{i}^{\prime}=u_{i}t^{w_{i}}$. On the other hand, by the assumption,

${\operatorname{Sing}}(\operatorname{in}(C_{\mathcal{J}}(Y))\subset
V(u_{1}t^{w_{1}},\ldots,u_{n}t^{w_{n}})$

so via isomorphism from and Lemma 4.1.7 we obtain

${\operatorname{Sing}}(Y^{\prime}\cap V_{B}(t^{-1}))\subset
V_{B}(u_{1}^{\prime},\ldots,u_{n}^{\prime})\cap V_{B}(t^{-1}).$

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

∎

As a corollary from Theorem 4.2.2, we obtain the following:

######  Theorem 4.2.3.

Let $X$ be a smooth scheme over a field $K$ of any characteristic. Let
$Y\subset X$ be a closed integral subscheme of $X$ admitting an almost
homogenous singularity $Z\subset Y$. There is a resolution of $Y$ at $Z$, that
is, a projective birational morphism $\phi:Y^{res}\to Y$ from a smooth variety
$Y^{res}$ with the exceptional locus $Z\subset Y$, such that $\phi^{-1}(Z)$ is
an SNC divisor on $Y^{\prime}$.

###### Proof.

Take the cobordant resolution $B_{+}\to X$ from Theorem 4.2.2. By Corollary
5.2.9, the geometric quotient $B_{+}/{\mathbb{G}}_{m}$ is locally toric, and
thus it can be canonically resolved by the combinatorial method of [Wło20,
Theorem 7.17.1]. This produces the projective birational resolution
$Y^{\prime}\to Y$ of $Y$ such that the inverse image of the singular point is
an SNC divisor.

∎

A typical situation where the theorem can be applied is an isolated
singularity.

######  Example 4.2.4.

In Example 3.2.2 with the singularity

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

we can assume that the characteristic of the field is nonzero and the singular
locus is the origin $V(x_{1},\ldots,x_{n})$. Then the the cobordant blow-up of
${\mathcal{J}}=(x_{1}^{1/w_{1}},\ldots,x_{n}^{1/w_{n}})$ with
$w_{1}c_{1}=\ldots=w_{n}c_{n}$ resolves the singularity. The same is still
valid if

$\operatorname{in}_{{\mathcal{J}}}(f)=x_{1}^{c_{1}}+\ldots+x_{n}^{c_{n}},$

and we assume that both
${\operatorname{Sing}}(V(\operatorname{in}_{{\mathcal{J}}}(f))$ and
${\operatorname{Sing}}(V(f))$ is the origin.

######  Example 4.2.5.

Consider the mixed characteristic hypersurface $Y$ in
$X={\mathbb{A}}_{{\mathbb{Z}}}^{2}\smallsetminus V(k)$, where
$k\in{\mathbb{Z}}$, defined by

$f=x^{p}+p^{p}+y^{k}\in{\mathbb{Z}}[1/k][x,y],$

with $p\not|k$. Taking the derivative of the ideal $(f)$ over
${\mathbb{Z}}[1/k]$ we get

${\mathcal{D}}(f)=(x^{p}+p^{p}+y^{k},px^{p-1},ky^{k-1})$

we see that ${\operatorname{Sing}}(f)\subseteq
V(x^{p}+p^{p}+y^{k},px^{p-1},ky^{k-1})=V(x,y,p)$. Thus $Y:=V(f)$ has an
isolated singularity $V(x,y,p)$. After the coordinate change $x^{\prime}:=x+p$
we get the equation:

$f={x^{\prime}}^{p}-p\cdot p{x^{\prime}}^{p-1}+\ldots+p\cdot
p^{p-1}x^{\prime}+y^{k}$

Consider the center

${{\mathcal{J}}}:=((x^{\prime})^{1/w_{1}},p^{1/w_{2}},y^{1/w_{3}})$

with the integral weights satisfying

$pw_{1}=pw_{2}+w_{1}=kw_{3}.$

We obtain the initial form

$\operatorname{in}f={x^{\prime}}^{p}+p^{p}x^{\prime}+y^{k}\in{\mathbb{Z}}_{p}[x^{\prime}t^{w_{1}},pt^{w_{2}},zt^{w_{3}}].$

Here $x^{\prime},p,y$ correspond to the graded elements
$x^{\prime}t^{w_{1}},pt^{w_{2}},zt^{w_{3}}$. Taking derivatives over
${\mathbb{Z}}_{p}$ with respect to the free variables $x^{\prime},p,y$ on
$Z_{p}[x^{\prime},p,z]$ we obtain

${\mathcal{D}}(\operatorname{in}f)=({x^{\prime}}^{p}+p^{p}x^{\prime}+y^{k},p^{p},ky^{k-1}).$

This implies that

${\operatorname{Sing}}(\operatorname{in}(f))\subset
V({x^{\prime}}^{p}+p^{p}x^{\prime}+y^{k},p^{p},ky^{k-1})=V(x^{\prime},y,p)=V({{\mathcal{J}}}).$

Thus $f$ defines an almost homogenous singularity for ${\mathcal{J}}$ and can
be resolved by a single cobordant blow-up of ${{\mathcal{J}}}$.

Observe that the seemingly natural center
${{\mathcal{J}}}_{1}=(x^{1/k},p^{1/k},y^{1/p})$ does not define an almost
homogenous singularity (for ${\mathcal{J}}_{1}$), and the corresponding
cobordant blow-up does not resolve singularity. In such a case

$\operatorname{in}(f)=x^{p}+p^{p}+y^{k}=(x+p)^{p}+z^{k}\in{\mathbb{Z}}_{p}[x,y,p],$

whence ${\operatorname{Sing}}(\operatorname{in}(f))=V(x+p,z)\supsetneq
V({\mathcal{J}}_{1})$. This also motivates the coordinate change above.

###  4.3. Partial resolution by the invariants

The method can be linked to different invariants and is related to the
characteristic zero method.

For any ideal ${\mathcal{I}}$ on a regular scheme $X$, and an integer $d$ let

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

######  Theorem 4.3.1.

Let ${\mathcal{I}}$ be an ideal on a regular scheme $X$. Let $d$ be any
natural integer. Assume that there exists a center ${{\mathcal{J}}}$ such that

  1. (1)

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

  2. (2)

${\operatorname{supp}}(\operatorname{in}({\mathcal{I}}),d)=V({\mathcal{J}})\subset
C_{J}(X)$.

Then for the cobordant blow-up $\sigma_{+}:B_{+}\to X$ of ${\mathcal{J}}$, we
have:

${\operatorname{ord}}_{B_{+}}(\sigma^{s}({\mathcal{I}}))<d$

###### Proof.

Let ${\mathcal{J}}=(x_{1}^{1/w_{1}},\ldots,x_{k}^{1/w_{k}})$ be the center. If
$q\in V(t^{-1})\smallsetminus V(x^{\prime}_{1},\ldots,x^{\prime}_{k})$, then

${\operatorname{ord}}_{q}(\sigma^{s}({\mathcal{I}})\leq{\operatorname{ord}}_{q}(\sigma^{s}({\mathcal{I}})_{|V(t^{-1})})={\operatorname{ord}}_{q}(\operatorname{in}({\mathcal{I}}))<d.$

If $q\in B\smallsetminus V(t^{-1})\smallsetminus
V(x^{\prime}_{1},\ldots,x^{\prime}_{k})=B_{-}\smallsetminus
V(x^{\prime}_{1},\ldots,x^{\prime}_{k})=(X\smallsetminus
V({\mathcal{J}}))\times{\mathbb{G}}_{m}$, then since $\sigma(q)\in
X\smallsetminus V({\mathcal{J}})$ we conclude that

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

∎

###  4.4. Resolution principle in any characteristic

In practice, one can replace $"{\operatorname{ord}}"$ with a different type of
invariants satisfying the properties used in the proof of the above lemma,
like some modifications of the Hilbert-Samuel function or
${\operatorname{inv}}$ and ${\operatorname{inv}}_{E}$ for smooth schemes in
characteristic zero or certain schemes in positive or mixed characteristic.
One can link it to a more general principle:

Let $\sigma:B\to X$ be a full cobordant blow-up of a center ${\mathcal{J}}$.
Let $Y\subset X$ be a closed subscheme. Let $D=V_{B}(t^{-1})$ be the
exceptional divisor of $\pi$.

  * •

The behavior of the strict transform $\sigma^{s}(Y)$ at the part of the vertex
${\operatorname{Vert}}(B)\smallsetminus D$ of $B$ is controlled by the
behavior of $Y$ at $V({\mathcal{J}})$.

  * •

The behavior of the strict transform $\sigma^{s}(Y)$ at the part of the vertex
${\operatorname{Vert}}(B)\cap D$ is to a great extent controlled by the
weighted normal cone $C_{{\mathcal{J}}}(Y)$ to the center $V({\mathcal{J}})$.

######  Example 4.4.1.

Let $k$ be a field of characteristic $p>2$. Let $Y\subset
X=\operatorname{Spec}k[x,y,z]$ be a closed subscheme defined by

$f=x^{p}+y^{p}z+z^{k}+x^{p-1}y^{2}+x^{p+1}yz,$

of order $p>2$ with $p\not|k(k-1)$, $k\geq 2p+1$. Computing
${\mathcal{D}}^{2}(f)$ we obtain that

${\operatorname{supp}}(f,3)=V({\mathcal{D}}^{2}(f))=V(x,y,z).$

Then by solving the equations for the weights

$pw_{1}=pw_{2}+w_{3}=kw_{3}$

we obtain that the initial form with respect to