# Open loci of ideals with applications to Birational maps

S. Hamid Hassanzadeh and Maral Mostafazadehfard<EMAIL_ADDRESS><EMAIL_ADDRESS>Departamento de Matemática, Centro de Tecnologia, Cidade
Universitária da Universidade Federal do Rio de Janeiro, 21941-909 Rio de
Janeiro, RJ, Brazil  Dedicated to Aron Simis on the occasion of his eightieth
birthday.

(Date: August 27, 2024)

###### Abstract.

In this work we show that the loci of ideals in principal class, ideals of
grade at least two, and ideals of maximal analytic spread are Zariski open
sets in the parameter space. As an application, we show that the set of
birational maps of clear polynomial degree $d$ over an arbitrary projective
variety $X$, denoted by $\operatorname{Bir}(X)_{d}$, is a constructible set.
This extends a previous result by Blanc and Furter.

##  1\. Introduction

In classical commutative ring theory, an open locus of a ring $R$ refers to a
set of prime ideals $\mathfrak{p}\in\operatorname{Spec}(R)$ such that
$R_{\mathfrak{p}}$ satisfies a particular property. However, in subjects such
as Hilbert schemes, the focus is on sets of ideals (in a polynomial ring) that
satisfy specific properties, such as having the same Hilbert function. The
term ”open loci of ideals” refers to the open Zariski subsets of a parametric
space whose points correspond to ideals with such properties.

In this paper, we study three properties of homogeneous ideals in a positively
graded standard Noetherian domain over a field. The first property, described
in 2.3, is belonging to the principal class. The principal class refers to the
class of ideals whose codimension is equal to the number of their generators.
The second property, presented in 2.4, is having grade at least 2. The grade
of an ideal is the maximum length of a regular sequence inside the ideal.
Finally, we investigate the Noether Normalization locus in 2.5. This locus is
determined by ideals with maximum analytic spread, which is defined as the
dimension of the special fiber of the ideal over the base ring. We show all of
these loci are open.

Despite the interesting algebraic nature of the aforementioned open loci, we
have discovered a significant application of them in the study of the group of
birational maps over an arbitrary projective variety. In this context, it is
worth mentioning some historical background.

Let $k$ be a field and $X$ be an algebraic variety over $k$. The group of
birational transformations on $X$ is denoted by $\operatorname{Bir}_{k}(X)$ or
simply $\operatorname{Bir}(X)$. Although this group has been the subject of
research for over 160 years, many mysteries about it remain, even in the case
where $X=\mathbb{P}^{n}$.

Regarding the group theoretic structures, there are several classical results
for $\operatorname{Bir}(\mathbb{P}^{2})$, dating back to Noether and
Castelnuovo. However, it was only in 2013 that Cantat and Lamy discovered the
non-simplicity and some unusual behaviors of normal subgroups in this group
[CL13]. The case of $\operatorname{Bir}(\mathbb{P}^{n})$ is more intricate,
and less is known about the group structure of $\operatorname{Bir}(X)$ when
$X$ is not of general type.

In 2010, J.-P. Serre [Ser10] introduced a topology on $\operatorname{Bir}(X)$
by considering morphisms from algebraic varieties. In 2013, Blanc and Furter
proved that there is no structure of an algebraic variety on
$\operatorname{Bir}(\mathbb{P}^{n})$ for $n\geq 2$, but on the positive side,
they showed that for any fixed integer $d$, the set of birational
transformations of degree $d$, denoted by
$\operatorname{Bir}(\mathbb{P}^{n})_{d}$, has the structure of an algebraic
variety. This led to the classification of the irreducible components of
$\operatorname{Bir}(\mathbb{P}^{2})_{d}$ by Bisi, Calabri, and Mella in 2015
[BCM15]. In 2017, Blanc raised the question of what kind of algebraic
structure can be put on $\operatorname{Bir}_{k}(X)$ in the survey [Bla17].

Let $R=k[x_{0},\ldots,x_{n}]/\mathcal{I}(X)$ be the coordinate ring of $X$,
and let $d$ be an integer. An element of $\operatorname{Bir}_{k}(X)$ can be
presented by a set of $n+1$ homogeneous polynomials in $R$ of the same degree
$d$. The parameter space is the space of coefficients of $(n+1)$ polynomials
of degree $d$. Using the openness of loci introduced above, we prove in 3.10
that for any irreducible projective variety $X$ over any infinite field $k$
and any integer $d$, $\operatorname{Bir}(X)_{d}$ has the structure of a
constructible algebraic set. This result recovers [BF13, Lemma 2.4 (2)]. In
[PR12, Theorem 34], Pan and Rittatore mentioned the constructibility of some
families of subsets of $\operatorname{Bir}(X)$ for a rational variety $X$. We
show further in 3.10(iii) that
$\operatorname{edim}(\operatorname{Bir}(X)_{d})\leq(n+1)\operatorname{HF}_{R}(d)-1$
where $\operatorname{HF}_{R}$ is the Hilbert function of $R$.

The first subtlety in proving this result is defining the degree of a
birational map over an arbitrary variety $X$. We introduced the concept of
“clear polynomial degree” of a rational map, as defined in 3.2. We want to
generalize the notion of degree of birational maps for any variety, not just
those with factorial coordinate rings. To do this, we need a natural and non-
trivial way to measure the degree of a birational map. We took into account
the topology introduced by Serre and the construction presented in [BF13]. In
this way, a crucial step is to find a bound for the degree of the inverse of a
birational map. This is the content of 3.9.

3.11 states that when $\operatorname{HF}_{\mathcal{I}(X)}(d)=0$,
$\operatorname{Bir}(X)_{d}$ is a quasi-projective variety. For instance,
$\operatorname{Bir}(\mathbb{P}^{n})_{d}$ is a quasi-projective variety for all
$d$.

Acknowledgement. The authors would like to thank C. Araujo, I. Pan, and J.
Blanc for their helpful discussions and motivating questions on the topic.
Both authors are partially supported by the Coordenação de Aperfeiçoamento de
Pessoal de Nível Superior-Brasil (CAPES)-Finance Code 001. The first author
was partially supported by the MathAmSud project “ALGEO” and CNPq Grant Number
406377/2021-9.

##  2\. Open loci of ideals

We will be using the following notation throughout this section: $k$ is a
field, $S=k[x_{0},\cdots,x_{n}]$ is a standard polynomial ring, and
$R=S/\mathfrak{p}$ is a graded integral domain of dimension $r$. Let $d$ be an
integer and let $M_{1},\cdots,M_{N}$ be all of the monomials of degree $d$ in
$S$, explicitly $N={n+d\choose d}$. For any ${\bf a}=(a_{1},\cdots,a_{N})\in
k^{N}$, we define $f_{{\bf a}}=\sum_{i=1}^{N}a_{i}M_{i}\in S$.

Given a set of $m$ polynomials of degree $d$ in $S$, we can identify a
corresponding point in the affine space $k^{mN}$. Specifically, let
$g_{1},\ldots,g_{m}$ be a sequence of homogeneous polynomials of degree $d$ in
$S$. For any index $i$, there exists ${\bf a}_{i}=(a_{i1},\cdots,a_{iN})\in
k^{N}$ such that $g_{i}=f_{{\bf a}_{i}}$. Conversely, any point $({\bf
a}_{1},\ldots,{\bf a}_{m})\in k^{mN}$ determines a sequence of polynomials
$f_{{\bf a}_{1}},\ldots,f_{{\bf a}_{m}}\in S$ and, consequently, an ideal in
$R$.

A locus of ideals is defined as the set of points $({\bf a}_{1},\ldots,{\bf
a}_{m})\in\mathbb{P}^{mN-1}$ such that the corresponding ideal $(f_{{\bf
a}_{1}},\ldots,f_{{\bf a}_{m}})\subseteq R$ satisfies a particular property.

To begin, we state a computational algebraic property.

######  Proposition 2.1.

Let $m$ be an integer, $p(z_{1},\ldots,z_{m})\in k[z_{1},\ldots,z_{m}]$ a
homogeneous polynomial, $\mathfrak{b}$ a homogeneous ideal of $S$ and keep the
other notation as above. Then

$V_{\mathfrak{b}}^{p}:=\\{({\bf a}_{1},\ldots,{\bf
a}_{m})\in\mathbb{P}^{mN-1}_{k}:p(f_{{\bf a}_{1}},\ldots,f_{{\bf
a}_{m}})\in\mathfrak{b}\\}$

is a closed algebraic subset of $\mathbb{P}^{mN-1}_{k}$. Consequently,
$U_{\mathfrak{b}}^{p}:=\\{({\bf a}_{1},\ldots,{\bf
a}_{m})\in\mathbb{P}^{mN-1}_{k}:p(f_{{\bf a}_{1}},\ldots,f_{{\bf
a}_{m}})\not\in\mathfrak{b}\\}$ is open in $\mathbb{P}^{mN-1}_{k}$.

###### Proof.

Fix a monomial order on $x_{0},\ldots,x_{n}$ and consider a Groebner basis for
the ideal $\mathfrak{b}$. Let $g=p(f_{{\bf a}_{1}},\ldots,f_{{\bf a}_{m}})$.
If the remainder, after doing the division algorithm, is zero then
$g\in\mathfrak{b}$. If the leading term of the polynomial $g$ is not divisible
by those of the Groebner basis of $\mathfrak{b}$ then $g\not\in\mathfrak{b}$.
Hence, if the leading coefficient of $g$ is not divisible, we set the
corresponding coefficient equal to zero and continue with the rest of the
polynomial. Notice that this coefficient is a polynomial in terms of
$a_{i}$’s. We continue dividing by polynomials in the Groebner basis of
$\mathfrak{b}$. The reminder is a polynomial whose coefficients are
polynomials in terms of $a_{i}$’s. The result follows by induction on the
number of terms of $g$. ∎

######  Remark 2.2.

The special case where $m=1$ and $p(z)=z$ is of particular interest. In this
case the elements of $V_{\mathfrak{b}}^{z}$ define the vectors of the
$k$-vector space $\mathfrak{b}_{d}\setminus\\{0\\}$. Hence
$V_{\mathfrak{b}}^{z}$ is a linear subspace of $\mathbb{P}^{N-1}_{k}$ of
(Krull) dimension HF${}_{\mathfrak{b}}(d)-1$ where HFb is the Hilbert function
of $\mathfrak{b}$.

Next, we study ideals that are traditionally called ideals of the principal
class. That means an ideal whose number of generators is the same as its
codimension.

######  Theorem 2.3.

For any $1\leq j\leq\dim(R)$, the following set, $\mathcal{C}_{j}$, is a non-
empty proper open subset of $\mathbb{P}^{jN-1}$:

$\mathcal{C}_{j}=\\{({\bf a}_{1},\ldots,{\bf
a}_{j})\in\mathbb{P}^{jN-1}:\operatorname{codim}_{R}(f_{{\bf
a}_{1}},\ldots,f_{{\bf a}_{j}})=j\\}.$

###### Proof.

For any integer $t$, $S_{t}$ is the $k$-vector space generated by monomials of
degree $t$. We denote by $R_{t}$ the $k$-vector space generated by the class
of the same monomials of $S_{t}$ modulo $\mathfrak{p}$ (these classes are not
necessarily linearly independent).

Recall that $R=S/\mathfrak{p}$ is a graded integral domain of dimension $r$.
Let $\mathfrak{p}$ be generated by $p_{1},\ldots,p_{q}$ of degrees
$d_{1},\ldots,d_{q}$ respectively. We first prove that $\mathcal{C}_{r}$ is
open. For any integer $m\geq\max\\{d,d_{1},\ldots,d_{q}\\}$ and $({\bf
a}_{1},\ldots,{\bf a}_{r})\in\mathbb{P}^{rN-1}$, we define a map

$\displaystyle\tau^{m}_{({\bf a}_{1},\ldots,{\bf
a}_{r})}:\underbrace{R_{m-d}\oplus\cdots\oplus R_{m-d}}_{r}$
$\displaystyle\longrightarrow R_{m}$ $\displaystyle(g_{1},\ldots,g_{r})$
$\displaystyle\mapsto(\sum_{j=1}^{r}g_{j}f_{{\bf a}_{j}}).$

For simplicity, we use the notation $\tau^{m}$ instead of $\tau^{m}_{({\bf
a}_{1},\ldots,{\bf a}_{r})}$.

$\tau^{m}$ is surjective if and only if
$(x_{0},\ldots,x_{n})^{m}\subseteq(f_{{\bf a}_{1}},\ldots,f_{{\bf
a}_{r}})+\mathfrak{p}$. The latter is equivalent to
$\operatorname{codim}_{R}(f_{{\bf a}_{1}},\ldots,f_{{\bf a}_{r}})=r$. We show
that the surjectivity of $\tau^{m}$ is an open property.

Using the class of the basis of $S_{i}$’s to generate $R_{i}$’s, the
presentation matrix of $\tau^{m}$ is a matrix with ${m+n\choose n}$ rows and
${m-d+n\choose n}r$ columns. One can regard the columns in $r$ partitions. The
$i$th partition is described by considering the multiplication of the basis of
$R_{m-d}$ into $f_{{\bf a}_{i}}$. Hence the entries of the columns of the
$i$’th partition are either $a_{ij}$ for some $j$, or zero.

Let assume $\dim_{k}(R_{m})=N_{m}$ and let $I_{N_{m}}(\tau^{m})$ be the ideal
generated by the $N_{m}$-minors of the representation matrix of $\tau^{m}$.
Then $\tau^{m}$ is surjective if and only if $I_{N_{m}}(\tau^{m})\neq 0$.
Consider the polynomial ring $A=k[a_{ij}:1\leq i\leq r,\leavevmode\nobreak\
\leavevmode\nobreak\ 1\leq j\leq N]$ and $I_{N_{m}}(\tau^{m})$ as an ideal of
$A$. Besides the standard graded structure of $A$, we consider an
$\mathbb{N}^{r}$-graded structure for $A$ as follows: for all $1\leq j\leq N$
and $1\leq i\leq r$,

$\deg(a_{ij})=(\underbrace{0,\ldots,\underbrace{1}_{i},0\ldots,0}_{r}).$

That means the columns of the $i$’th partition are all of the same multi-
degree.

With respect to either grading, $I_{N_{m}}(\tau^{m})$ is a
($\mathbb{N}^{r}$-)homogeneous ideal of $A$.

Let

$\mathcal{I}=\sum_{m\geq\max\\{d,d_{1},\ldots,d_{r}\\}}I_{N_{m}}(\tau^{m}).$

Since $A$ is Noetherian, $\mathcal{I}$ is finitely generated. Let
$\mathcal{I}=(Q_{1},\ldots,Q_{v})$ for some ($\mathbb{N}^{r}$-)homogeneous
polynomials $Q_{i}\in A$. We then have
$\mathcal{C}_{r}=(V(Q_{1},\ldots,Q_{v}))^{c}$ that is an open Zariski set of
$\mathbb{P}^{rN-1}$.

Now, fix $j<r$. Consider the polynomial ring $A^{\prime}=k[a_{st}:1\leq s\leq
j,\leavevmode\nobreak\ \leavevmode\nobreak\ 1\leq t\leq N]$, let ${\bf a}_{i}$
to be the sequence of variables $a_{i,1},\ldots,a_{i,N}$ for all $i$, and
define

$\mathcal{P}_{j}:=(Q_{i}({\bf a}_{1},\ldots,{\bf
a}_{j},\operatorname{\boldsymbol{\alpha}}_{j+1},\ldots,\operatorname{\boldsymbol{\alpha}}_{r})\in
A^{\prime}:1\leq i\leq
v,\operatorname{\boldsymbol{\alpha}}_{t}\in\mathbb{P}^{N-1}\text{ and
}\\{f_{\operatorname{\boldsymbol{\alpha}}_{j+1}},\ldots,f_{\operatorname{\boldsymbol{\alpha}}_{r}}\\}\in\mathfrak{S})$

where
$\mathfrak{S}=\\{\\{f_{\operatorname{\boldsymbol{\alpha}}_{j+1}},\ldots,f_{\operatorname{\boldsymbol{\alpha}}_{r}}\\}:f_{\operatorname{\boldsymbol{\alpha}}_{j+1}},\ldots,f_{\operatorname{\boldsymbol{\alpha}}_{r}}\text{\leavevmode\nobreak\
\leavevmode\nobreak\ is a part of a system of parameters
of\leavevmode\nobreak\ \leavevmode\nobreak\ }R\\}$. We claim that
$\mathcal{C}_{j}=(V(\mathcal{P}_{j}))^{c}\subseteq\mathbb{P}^{jN-1}$.

We first notice that $\mathcal{P}_{j}$ is a homogeneous ideal of $A^{\prime}$.
This is due to the $\mathbb{N}^{r}$-graded structure of $A$ and homogeneity of
$I_{N_{m}}(\tau^{m})$, as explained in the previous paragraph. So that
$(V(\mathcal{P}_{j}))^{c}\subseteq\mathbb{P}^{jN-1}$.

Since $R$ is a quotient of a polynomial ring,
$(\operatorname{\boldsymbol{\alpha}}_{1},\ldots,\operatorname{\boldsymbol{\alpha}}_{j})\in\mathcal{C}_{j}$
if and only if
$(f_{\operatorname{\boldsymbol{\alpha}}_{1}},\ldots,f_{\operatorname{\boldsymbol{\alpha}}_{j}})$
is a part of a system of parameters of $R$ that is
$\dim(R/(f_{\operatorname{\boldsymbol{\alpha}}_{1}},\ldots,f_{\operatorname{\boldsymbol{\alpha}}_{j}}))=r-j$.
Hence there exists
$(\operatorname{\boldsymbol{\alpha}}_{j+1},\ldots,\operatorname{\boldsymbol{\alpha}}_{r})\in\mathfrak{S}$
such that
$\operatorname{codim}(f_{\operatorname{\boldsymbol{\alpha}}_{1}},\ldots,f_{\operatorname{\boldsymbol{\alpha}}_{j}},\ldots,f_{\operatorname{\boldsymbol{\alpha}}_{r}})=r$;
so that there exists $1\leq i\leq v$ with
$Q_{i}({\operatorname{\boldsymbol{\alpha}}_{1}},\ldots,{\operatorname{\boldsymbol{\alpha}}_{j}},\ldots,{\operatorname{\boldsymbol{\alpha}}_{r}})\neq
0$. The latter says that $\mathcal{C}_{j}\subseteq(V(\mathcal{P}_{j}))^{c}$.
The other inclusion is more visible. ∎

Now, we pass from ”codim” to ”grade”.

######  Proposition 2.4.

Keeping the same notation in this section, the set

$G_{2}:=\\{({\bf a},{\bf b})\in\mathbb{P}^{2N-1}_{k}:(f_{{\bf a}},f_{{\bf
b}})\text{\leavevmode\nobreak\ \leavevmode\nobreak\ is a regular sequence
in\leavevmode\nobreak\ \leavevmode\nobreak\ }R\\}$

is a Zariski open subset of $\mathbb{P}^{2(N-1)}_{k}$. $G_{2}$ is non-empty if
and only if $\operatorname{depth}(R)\geq 2$.

###### Proof.

It is known that for a ring such as $R$ (homomorphic image of Gorenstein
ring), the $(S_{2})$-locus of $\operatorname{Spec}(R)$ is an open subset, c.f.
[BS13, 12.3.10]. Let $V(I)$ be the non-$(S_{2})$-locus of $R$ then $I$ is a
homogeneous ideal of $R$ of codimension at least $2$.

Let
$\mathcal{A}=\\{\mathfrak{q}\in\operatorname{Spec}(R):\operatorname{codim}(\mathfrak{q})=2\text{
and}\operatorname{grade}(\mathfrak{q})=1\\}$. $\mathcal{A}$ may be empty. If
not, for any $\mathfrak{q}\in\mathcal{A}$, $R_{\mathfrak{q}}$ is not $S_{2}$,
hence $\mathfrak{q}\supset I$, and since $\operatorname{codim}(I)\geq 2$,
$\mathfrak{q}$ is a minimal prime over $I$. Thence $\mathcal{A}$ is a finite
set (or empty).

Recalling the definition of $\mathcal{C}_{2}$ from 2.3,

$G_{2}=\mathcal{C}_{2}\setminus\\{({\bf a},{\bf
b})\in\mathbb{P}^{2N-1}_{k}:\operatorname{codim}(f_{{\bf a}},f_{{\bf
b}})=2\text{\leavevmode\nobreak\ and\leavevmode\nobreak\
}\operatorname{grade}(f_{{\bf a}},f_{{\bf b}})=1\\}.$

We show that the take-out part is closed.

$\displaystyle\\{({\bf a},{\bf
b})\in\mathbb{P}^{2N-1}_{k}:\operatorname{codim}(f_{{\bf a}},f_{{\bf
b}})=2\text{\leavevmode\nobreak\ and\leavevmode\nobreak\
}\operatorname{grade}(f_{{\bf a}},f_{{\bf b}})=1\\}$ $\displaystyle=$
$\displaystyle\\{({\bf a},{\bf
b})\in\mathbb{P}^{2N-1}_{k}:\operatorname{codim}(f_{{\bf a}},f_{{\bf
b}})=2\leavevmode\nobreak\ \text{and}\leavevmode\nobreak\ f_{{\bf a}},f_{{\bf
b}}\in\mathfrak{q}\leavevmode\nobreak\ \text{\leavevmode\nobreak\ for
some\leavevmode\nobreak\ }\mathfrak{q}\in\mathcal{A}\\}.$

By 2.1, $f_{{\bf a}},f_{{\bf b}}\in\mathfrak{q}$ if and only if $({\bf a},{\bf
b})\in V^{z}_{\mathfrak{p}+\mathfrak{q}}\times
V^{z}_{\mathfrak{p}+\mathfrak{q}}$ where $\mathfrak{p}$ is the defining ideal
of $R$. Therefore

$G_{2}=\mathcal{C}_{2}\setminus\bigcup_{\mathfrak{q}\in\mathcal{A}}(V^{z}_{\mathfrak{p}+\mathfrak{q}}\times
V^{z}_{\mathfrak{p}+\mathfrak{q}}).$

This completes the proof since $\mathcal{A}$ is a finite set and
$\mathcal{C}_{2}$ is open by 2.3.

∎

Initially, it might appear enticing to establish both 2.3 and 2.4 through
“prime avoidance lemma”. However, it’s worth noting that the prime avoidance
lemma constructs the sought-after sequences in an inductive manner. On the
contrary, the above mentioned results construct these desired sequences
simultaneously.

Should we opt for an inductive approach, it would necessitate a considerable
endeavor to demonstrate the openness of the assembled sets.

The next locus we discuss is about analytic spread. We present more details
about its definition. Let $(A,\mathfrak{m})$ denote a standard graded
Noetherian algebra over the field $k$ and $I=(f_{1},\ldots,f_{s})$ stand for a
graded ideal of $A$. Set $\mathcal{R}_{A}(I):=A[It]\subset A[t]$, the Rees
algebra of $I$ over $A$. The special fiber of $I$ is the algebra
$\mathcal{F}(I)=\mathcal{R}_{A}(I)/\mathfrak{m}\mathcal{R}_{A}(I)$. The Krull
dimension of $\mathcal{F}(I)$ is called the analytic spread of $I$. When $I$
is generated in a single degree, $\mathcal{F}(I)\simeq k[f_{1},\ldots,f_{s}]$.
We say that $I$ has maximum analytic spread if
$\ell_{A}(I)=\min\\{\operatorname{dim}(A),s\\}$. For more details on this
notation see [HS06]. The maximum analytic spread is closely related to the
Noether normalization theorem.

######  Theorem 2.5.

Let $k$ be an infinite field. The following set that we call the Noether
Normalization locus is a non-empty open subset of $\mathbb{P}^{(n+1)N-1}$

$\mathcal{N}_{n+1}=\\{({\bf a}_{0},\ldots,{\bf
a}_{n})\in\mathbb{P}^{(n+1)N-1}:\ell_{R}(f_{{\bf a}_{0}},\cdots,f_{{\bf
a}_{n}})=r\\}.$

###### Proof.

We prove by induction on $i\geq r$ that $\mathcal{N}_{i}$ is a Zariski open
where

$\mathcal{N}_{i}=\\{({\bf a}_{0},\ldots,{\bf
a}_{i-1})\in\mathbb{P}^{iN-1}:\ell_{R}(f_{{\bf a}_{0}},\cdots,f_{{\bf
a}_{i-1}})=r\\}.$

First, we verify the case $i=r$. Let
$\\{\theta_{1},\ldots,\theta_{r}\\}\subset R_{d}$ such that
$\operatorname{dim}(k[\theta_{1},\ldots,\theta_{r}])=r$. Since
$k[\theta_{1},\ldots,\theta_{r}]$ is integrally closed and the extension
$k[\theta_{1},\ldots,\theta_{r}]\subset R$ is integral, the going-down theorem
holds for this extension, in particular, the ideal
$(\theta_{1},\ldots,\theta_{r})R$ has codimension $r$ i.e it is a system of
parameters. On the other hand, any system of parameters is analytically
independent [Mat86, Theorem 14.5]. Therefore

$\mathcal{N}_{r}=\\{({\bf a}_{0},\ldots,{\bf a}_{r-1})\in
k^{rN}:\operatorname{codim}_{R}(f_{{\bf a}_{0}},\cdots,f_{{\bf
a}_{r-1}})=r\\}=\mathcal{C}_{r}.$

The latter determines an open subset of $\mathbb{P}^{rN-1}$ by 2.3.

Next, we show that $\mathcal{N}_{r+1}$ is open Zariski. The proof of the
general inductive pass is similar.

Let ${\bf a}_{1},\ldots,{\bf a}_{r+1}\in k^{N}$ be such that $\ell_{R}(f_{{\bf
a}_{1}},\ldots,f_{{\bf a}_{r+1}})=r$. Then using the same technique as that of
the proof of the Noether normalization theorem [Mat86, Lemma 2, page 262], one
can show that there exists $(c_{1},\ldots,c_{r})\in k^{r}$ such that
$\ell_{R}(f_{{\bf a}_{1}}-c_{1}f_{{\bf a}_{r+1}},\ldots,f_{{\bf
a}_{r}}-c_{r}f_{{\bf a}_{r+1}})=r$–(here we need an infinite field). We repeat
this statement in other words: we denote the points in $k^{(r+1)N},$ $k^{rN}$
and $k^{r(r+1)}$ by matrices of sizes $(r+1)\times N$, $r\times N$ and
$r\times(r+1)$, moreover, $M=(m_{1},\ldots,m_{N})^{t}$ the $N\times 1$ matrix
of the monomials of degree $d$ in $R$. Then for any $(r+1)\times N$-matrix
$A=({\bf a}_{ij})$ with $\ell_{R}(AM)=r$, there exists an
$r\times{(r+1)}$-matrix $C$ such that $CA\in\mathcal{N}_{r}$.

Conversely, if for an $(r+1)\times N$-matrix $A=({\bf a}_{ij})$ there exists
an $r\times{(r+1)}$-matrix $C$ such that $CA\in\mathcal{N}_{r}$. Then
$r=\ell(CAM)\leq\ell(AM)\leq\ell(M)=r$. So that the rows of $A$ constitute
points of $k^{(r+1)N}$ that belong to $\mathcal{N}_{r+1}$. In brief

(2.1) $\mathcal{N}_{r+1}=\\{A\in k^{(r+1)N}:\text{\leavevmode\nobreak\
\leavevmode\nobreak\ there exists\leavevmode\nobreak\ \leavevmode\nobreak\
}C\in k^{r(r+1)}\text{\leavevmode\nobreak\ \leavevmode\nobreak\
with\leavevmode\nobreak\ \leavevmode\nobreak\ }CA\in\mathcal{N}_{r}\\}.$

Restating the above equality, any $r\times(r+1)$ matrix $C$ determines a
linear map $\varPhi_{C}:k^{r+1}\to k^{r}$. $N$ copies such a map define a
linear map

$\bigoplus^{N}\varPhi_{C}:k^{(r+1)N}\to k^{rN}.$

The equality (Equation 2.1) states that

$\mathcal{N}_{r+1}=\bigcup_{C}(\bigoplus^{N}\varPhi_{C})^{-1}(\mathcal{N}_{r}).$

The linearity of $\bigoplus^{N}\varPhi_{C}$ guarantees that
$\mathcal{N}_{r+1}$ is an open Zariski subset of $\mathbb{P}^{(r+1)N-1}$. The
inductive pass of the induction argument follows by showing that

$\mathcal{N}_{r+j}=\bigcup_{C}(\bigoplus^{N}\varPhi_{C})^{-1}(\mathcal{N}_{r+j-1}).$

∎

In the following remark, we provide an alternative, constructive proof for
2.5.

######  Remark 2.6.

Consider three sets of independent variables

$\begin{pmatrix}Z_{1,1}&\ldots&\ldots&Z_{1,N}\\\
\cdots&\cdots&\cdots&\cdots\\\
Z_{r,1}&\ldots&\ldots&Z_{r,N}\end{pmatrix},\quad\begin{pmatrix}X_{1,1}&\ldots&\ldots&X_{1,N}\\\
\cdots&\cdots&\cdots&\cdots\\\ X_{r,1}&\ldots&\ldots&X_{r,N}\\\
X_{r+1,1}&\ldots&\ldots&X_{r+1,N}\end{pmatrix},\quad\begin{pmatrix}Y_{1,1}&\ldots&Y_{1,r+1}\\\
\cdots&\cdots&\cdots\\\ Y_{r,1}&\ldots&Y_{r,r+1}\end{pmatrix},$

and set ${\bf Z}_{i}=(Z_{i,1},\ldots,Z_{i,N})$, ${\bf
X}^{i}=(X_{i,1},\ldots,X_{r+1,i})^{t}$ and ${\bf Y}_{i}=(Y_{i,1},\ldots,{\bf
Y}_{i,r+1})$. Assume that
$\mathbb{P}^{rN-1}\setminus\mathcal{N}_{r}=V(\mathfrak{n}_{r})$ where
$\mathfrak{n}_{r}\subset k[{\bf Z}_{1},\ldots,{\bf Z}_{r}]$ is a homogeneous
ideal. For any $F\in\mathfrak{n}_{r}$, set

$g_{F}({\bf Y}_{1},\ldots,{\bf Y}_{r},{\bf X}^{1},\ldots,{\bf X}^{N})=F(({\bf
Y}_{1}{\bf X}^{1},\ldots,{\bf Y}_{1}{\bf X}^{N}),\ldots,({\bf Y}_{r}{\bf
X}^{1},\ldots,{\bf Y}_{r}{\bf X}^{N}))).$

Let $\mathfrak{n}=(g_{F}:F\in\mathfrak{n}_{r})$ be the homogeneous ideal in
$k[{\bf Y}_{1},\ldots,{\bf Y}_{r},{\bf X}^{1},\ldots,{\bf X}^{N}]$. The
projection $\pi:\operatorname{Proj}(k[{\bf Y}_{1},\ldots,{\bf Y}_{r},{\bf
X}^{1},\ldots,{\bf X}^{N}])\to\operatorname{Proj}(k[{\bf X}^{1},\ldots,{\bf
X}^{N}])$ is a closed map. Hence, there is a homogeneous ideal
$\mathfrak{n}_{r+1}\subset k[{\bf X}^{1},\ldots,{\bf X}^{N}]$ with
$V(\mathfrak{n}_{r+1})=\pi(V(\mathfrak{n}))$.

The explanation of $\mathcal{N}_{r+1}$ in (Equation 2.1) shows that
$\mathbb{P}^{(r+1)N-1}\setminus\mathcal{N}_{r+1}=V(\mathfrak{n}_{r+1}),$ as
desired .

##  3\. Application to Birational Maps

We fix a field $k$, of any characteristic and not necessarily algebraically
closed. Let $X\subseteq\mathbb{P}^{n}_{k}$ be a reduced and irreducible
projective variety. We assume $X$ is non-degenerated i.e. $X$ is not contained
in any hyperplane, and denote the coordinate ring of $X$ by $R$. I.e.
$R:=k[x_{0},\ldots,x_{n}]/\mathcal{I}(X)$.

###  3.1. Clear Degree

A rational map $h$ from $X$ to $X$ is presented by

$h:(x_{0}:\cdots:x_{n})\mapsto(h_{0}(x_{0}:\cdots:x_{n}):\cdots:h_{n}(x_{0}:\cdots:x_{n})).$

Here $h_{0},\cdots,h_{n}$ belong to $R$ and all of them are of the same
degree. One faces preliminarily the fact that the map can be defined by
different sets of forms of fixed degree, where this degree may vary from set
to set. One way to define the degree of a birational map is to consider the
minimum possible degree among all of the representative of the map. This is
the natural way, that coincide with the usual way of definition for the case
where $X=\mathbb{P}^{n}$. However, in the case of arbitrary variety $X$, such
as $X_{2}$ in the example below, it is possible that a map admits two or more
representative of minimal degree. Non-uniqueness of the representative causes
difficulties to study the locus of birational maps.

We present two examples:

######  Example 3.1.

Consider rational maps $\sigma_{i}:X_{i}\dasharrow X_{i}$,
$(x:y:z)\mapsto(yz:xz:xy),$ in the cases where $X_{1}=V(y^{2}-xz)$, a smooth
curve in $\mathbb{P}^{2}$, and $X_{2}=V(y^{3}-x^{2}z)$ a cuspidal curve in
$\mathbb{P}^{2}$. $\sigma_{1}$ is also represented by $(z:y:x)$; while
$\sigma_{2}$ admits another representative of degree $2$ that is not obtained
by multiplication, namely $\sigma_{2}$ is the same as the map

$\displaystyle\tau:X_{2}$ $\displaystyle\dasharrow X_{2}$
$\displaystyle(x:y:z)$ $\displaystyle\mapsto(xz:y^{2}:x^{2}).$

A strategy to circumvent the complexities associated with defining degree of a
birational map is to consider the grade of the ideal of the base of the map.
Recall that for an ideal $I$ in a ring $R$, $\operatorname{grade}(I,R)$ is the
maximum length of a proper regular sequence inside $I$. For the map $h$
defined earlier, let $I=(h_{0},\ldots,h_{n})\subset R$. If
$\operatorname{grade}(I,R)\geq 2$, then the map $h$ is presented uniquely up
to multiplication by an element of $R$ (rather than an element of $Q(R)$), as
shown in [Sim04, Proposition 1.1]. Based on this, we present the following
definition:

######  Definition 3.2.

Let $h:X\dasharrow X$ be a rational map. We say the map $h$ has clear
polynomial degree $d$, if $h$ admits a representative $(h_{0}:\cdots:h_{n})$
such that the ideal in $R$ generated by $h_{0},\ldots,h_{n}$ has grade at
least $2$ and $\deg(h_{i})=d$.

One says $X$ satisfies the Serre’s $(S_{2})$-condition if for all prime ideal
$\mathfrak{p}\subset R$,
$\operatorname{depth}(R_{\mathfrak{p}})\geq\min\\{\operatorname{codim}(\mathfrak{p}),2\\}.$
In the case where $X$ is $S_{2}$ or at least $S_{2}$ in codimension $2$, then
$\operatorname{grade}(I,R)\geq 2$ if and only if $\operatorname{codim}(I)\geq
2$. In particular, if $R$ is factorial, then $\operatorname{grade}(I,R)\geq 2$
if and only if $h_{0},\ldots,h_{n}$ have no common factor.

######  Definition 3.3.

We denote by $\operatorname{Bir}(X)_{d}$ the set of birational maps from $X$
to $X$ of clear polynomial degree $d$.
$\operatorname{Bir}(X)_{\ast}:=\cup_{d=1}^{\infty}\operatorname{Bir}(X)_{d}$.
If, for example, $R$ is factorial, then
$\operatorname{Bir}(X)_{\ast}=\operatorname{Bir}(X)$.

In 3.1,
$\sigma_{1}\in\operatorname{Bir}(X_{1})_{1}\setminus\operatorname{Bir}(X_{1})_{2}$
while
$\sigma_{2}\in\operatorname{Bir}(X_{2})\setminus\operatorname{Bir}(X_{2})_{*}$.

######  Remark 3.4.

Once we study the entire group $\operatorname{Bir}(X)$, it is possible and
useful to replace $X$ with a smooth variety, using the resolution of
singularities, or in some cases even replace $\operatorname{Bir}(X)$ with
$\mathrm{Aut}(X)$, using the minimal model. However, such a replacement is not
as functional in the study of $\operatorname{Bir}(X)_{d}$. For example, the
varieties $X_{1}$ and $X_{2}$ in 3.1 are birationally equivalent. The maps
$\theta:X_{2}\dasharrow X_{1}$ that sends $(x:y:z)$ to $(yz:xz:y^{2})$ and
$\theta^{-1}:X_{1}\dasharrow X_{2}$ that sends $(x:y:z)$ to $(yz:xz:x^{2})$
define the birational equivalence. Using these birational maps, we have an
isomorphism $\Psi:\operatorname{Bir}(X_{1})\to\operatorname{Bir}(X_{2})$ given
by
$\Psi(\varphi)=\theta^{-1}\circ\varphi\circ\theta\in\operatorname{Bir}(X_{2})$.
Interestingly, in this example, $\Psi(\sigma_{1})=\sigma_{2}$. Thus the
restriction of $\Psi$ to $\operatorname{Bir}(X_{1})_{*}$ does not define a
well-defined map to $\operatorname{Bir}(X_{2})_{*}$.

A natural question is the following

######  Question 3.5.

For which variety $X$ does $\operatorname{Bir}(X)_{*}$ admit the structure of
a group?

An immediate answer to this question is the varieties $X$ whose coordinate
ring $R$ is a unique factorization domain, since
$\operatorname{Bir}(X)_{*}=\operatorname{Bir}(X)$.

A remarkable class of varieties to be studied is the class of projective
smooth complete intersection $F$: when $\operatorname{dim}(F)\geq 3$ the
coordinate ring of $F$ is a factorial ring due to the theorem of Grothendieck
c.f. [Cal94]; when the sum of degrees is at most $n$ these varieties are Fano
varieties. Recall that for a Fano variety $F$, $\operatorname{Bir}(F)$ is not
a finite set, unlike varieties of general type.

When $X$ is a curve, $\operatorname{Bir}(X)_{*}$ coincides with linear
automorphism over $X$. Hence it is a group. We prove the latter in the
following proposition.

######  Proposition 3.6.

Let $X$ be an irreducible curve. The following statements are equivalent

  * •

$R=k[x_{0},\ldots,x_{n}]/\mathcal{I}(X)$ is a Cohen-Macaulay ring (of Krull
dimension $2$).

  * •

$\operatorname{Bir}(X)_{*}\neq\emptyset$.

  * •

$\operatorname{Bir}(X)_{*}$ is the same as linear automorphisms of $X$.

###### Proof.

The identity map has clear polynomial degree if and only if the irrelevant
maximal ideal has grade at least $2$. Since $R$ is of dimension $2$, it is
equivalent to Cohen-Maculayness of $R$. Moreover, the base ideal of any
rational map is contained in the irrelevant maximal ideal, hence if
$\operatorname{Bir}(X)_{*}\neq\emptyset$ then the identity map has clear
polynomial degree. This proves the equality of the first two parts. The third
statement clearly implies the second one. Hence it remains to show that the
first statement implies the third one.

Now assume that $R$ is Cohen-Macaulay. To prove the last assertion, we use the
birationality criterion of Simis-Ulrich-Vasconcelos [SUV01, Theorem 6.6].
Since the latter criterion holds in more general cases than our setting, we
present the adapted version for our purpose in 3.7. Let $\varphi:X\dasharrow
X$ be a dominant rational map of clear degree $d$. Regarding the notation of
3.7, $r=2$, $g=\operatorname{ht}(I)=2$ and $\mathfrak{R}=\emptyset$.Therefore,
$e(R)=e(R)d$ if and only if $\varphi$ is birational. That is $d=1$ if and only
if $\varphi$ is birational.

∎

######  Theorem 3.7.

[SUV01, Theorem 6.6]. Let $X$ be an irreducible projective variety with
coordinate ring $R$ of dimension $r$. Let $\varphi:X\dasharrow X$ be a
dominant rational map of polynomial degree $d$. Let $I$ be the ideal generated
by that presentation. Set $g=\operatorname{ht}(I)$ and
$\mathfrak{R}:=\\{\mathfrak{p}\in\operatorname{Proj}(R):\mathfrak{p}\supseteq
I{\text{\leavevmode\nobreak\ }and\leavevmode\nobreak\
}\dim(R/P)=\dim(R/I)\\}.$ Then

$e(R)\leq
e(R)d^{r-1}-\sum_{\mathfrak{p}\in\mathfrak{R}}e(I_{\mathfrak{p}},R_{\mathfrak{p}})e(R/P)d^{r-g-1}.$

Equality happens if and only if $\varphi$ is birational and $r\leq g+1$.

In dimension $2$, there are some varieties that have nice properties but do
not have a group structure on their birational maps. We used Macaulay-2[GS98]
to construct the following example.

######  Example 3.8.

Let $V\subset\mathbb{P}^{5}$ be the Veronese surface realized by the Veronese
embedding. Let $V_{1}$ be the projection of $V$ into $\mathbb{P}^{4}$. $V_{1}$
is the image of the following map $\nu$

$\begin{array}[]{rcl}\nu:\mathbb{P}^{2}&\dasharrow&V_{1}\\\
(s:t:u)&\mapsto&(st:su:t^{2}:tu:u^{2})\end{array}$

$\nu^{-1}:V_{1}\dasharrow\mathbb{P}^{2}$ is given by
$(a_{0}:\cdots:a_{4})\mapsto(a_{1}:a_{3}:a_{4})$. Consider the birational map
$f:\mathbb{P}^{2}\dasharrow\mathbb{P}^{2}$, $(s:t:u)\mapsto(st:su:t^{2}).$ Let
$\varphi=\nu f\nu^{-1}:V_{1}\dasharrow V_{1}$. $\varphi$ is a birational map
represented by

$(a_{0}a_{1}:a_{0}a_{2}:a_{1}^{2}:a_{0}a_{3}:a_{2}^{2}).$

Calculations show that $\varphi$ has clear degree $2$. However, having a clear
degree does not extend either to its inverse or its successive powers. This
example shows that $\operatorname{Bir}(V_{1})_{*}$ does not admit any
structure as a group or a monoid. Furthermore, we mention that $V_{1}$ is a
quadric rational smooth surface. Its coordinate ring is Cohen-Macaulay and not
factorial.

In our experience with Veronese surface, we could not find any such map, as
the one in 3.8. However, we still could not show that
$\operatorname{Bir}(V)_{*}$ is a group.

###  3.2. The structure of $\operatorname{Bir}(X)_{d}$

The following proposition has a critical role in the proof of 3.10. It has
been already stated in [HS17, Proposition 1.2], and its proof is based on the
main result of [DHS12].

######  Proposition 3.9.

Let $X\subseteq\mathbb{P}_{k}^{n}$ denote a closed reduced and irreducible
subvariety, and $\mathfrak{F}:X\dasharrow\mathbb{P}_{k}^{m}$ a birational map
admitting a representative of polynomial degree $d\geq 1$. Let
$Y=\operatorname{Im}\mathcal{F}$. Assume that $X$ and $Y$ are non-degenerated
and $\delta=\max\\{d+1,d_{0}\\}$ where $d_{0}$ is the maximum degree among the
degrees of a minimal generating set of $\mathcal{I}(X)$.

Then the inverse map $\mathfrak{F}^{-1}$ admits a representative of polynomial
degree at most

(3.1)
$B(X,d,n)=2n\bigg{[}\frac{1}{2}\delta^{2(n+m+1-\operatorname{dim}(X))^{2}}+\delta\bigg{]}^{2^{(\operatorname{dim}(X)+2)}}.$

In the case where $X=Y$, accordingly $n=m$, we call the formula in Equation
3.1, the inverse bound and denote it by $B(X,d)$.

In terms of determining an upper bound, this result is a generalization of a
result attributed to O.Gabber. Gabber’s result states that for
$\mathfrak{F}:\mathbb{P}^{n}\dasharrow\mathbb{P}^{n}$ of degree $d$ the
inverse map is of degree at most $d^{n-1}$. It is clear that the bound
$B(\mathbb{P}^{n},d)$, above, is much higher than $d^{n-1}$.

To prove the main theorem, we adapt some of the notation of [BF13, Lemma 2.4].
Returning the notation in Section 2, we fix an integer $d$ and denote by
$M_{1},\ldots,M_{N}$ all of the monomials of degree $d$ in the polynomial ring
$S=k[x_{0},\cdots,x_{n}]$. Consider the projective space
$\mathbb{P}_{k}^{(n+1)N-1}$. Any point ${\bf a}$ of
$\mathbb{P}_{k}^{(n+1)N-1}$ is an $(n+1)$-uple $({\bf a}_{0},\ldots,{\bf
a}_{n})$ with ${\bf a}_{i}\in\mathbb{A}^{N}_{k}$. For each index ${\bf a}_{i}$
we construct a polynomial

(3.2) $f_{{\bf a}_{i}}=\sum_{j=1}^{N}({\bf a}_{i})_{j}M_{j}.$

We denote $W_{d}$ to be the set of equivalence classes of nonzero $(n+1)$-uple
$(f_{0},\ldots,f_{n})$ of homogeneous polynomials $f_{i}\in S$ of degree $d$
where $(f_{0},\ldots,f_{n})$ is equivalent to $(\lambda f_{0},\ldots,\lambda
f_{n})$ for any $\lambda\in k\setminus{\\{0\\}}$. The equivalence class of
$(f_{0},\ldots,f_{n})$ will be denoted by $(f_{0}:\ldots:f_{n})$. It is clear
that $\mathbb{P}^{(n+1)N-1}$ is isomorphic to $W_{d}$ via the correspondence:

$\displaystyle\theta:\mathbb{P}^{(n+1)N-1}$ $\displaystyle\to W_{d}$
$\displaystyle{\bf a}=({\bf a}_{0},\ldots,{\bf a}_{n})$ $\displaystyle\mapsto
f_{{\bf a}}=(f_{{\bf a}_{0}}:\cdots:f_{{\bf a}_{n}}).$

We define $H_{d}\subseteq W_{d}$ to be the set of elements
$f=(f_{0},\ldots,f_{n})\in W_{d}$ such that the rational map

(3.3) $\displaystyle\psi_{f}:X$ $\displaystyle\dasharrow X$
$\displaystyle(x_{0}:\cdots:x_{n})$
$\displaystyle\mapsto(f_{0}(x_{0},...,x_{n}):\cdots:f_{n}(x_{0},...,x_{n}))$

is a well-defined birational map of polynomial degree $d$.

$\tilde{H}_{d}\subseteq H_{d}$ is the set of elements
$f=(f_{0},\ldots,f_{n})\in H_{d}$ such that $\psi_{f}$ is birational of clear
polynomial degree $d$.

The last concept before stating the main theorem is the definition of
constructible set. A subset of a topological space is called constructible
when it is a finite union of locally closed sets. A set is locally closed if
it is an intersection of a closed set and an open set. For a constructible
subset $\mathcal{C}$ of a projective space. The embedding dimension of
$\mathcal{C}$ is defined by
$\operatorname{edim}(\mathcal{C}):=\min\\{m:\mathcal{C}\subseteq\mathbb{P}^{m}\text{\leavevmode\nobreak\
\leavevmode\nobreak\ as a closed embedding}\\}$.

######  Theorem 3.10.

Let $d$ be an integer, $k$ an infinite field and $X\subseteq\mathbb{P}^{n}$ an
irreducible projective non-degenerated variety with coordinate ring $R$ and
Hilbert function $\operatorname{HF}_{R}$. Then

  * (i)

$H_{d}$ and $\tilde{H}_{d}$ (defined above) are quasi-projective varieties;

  * (ii)

$\operatorname{Bir}(X)_{d}$ has the structure of a constructible algebraic
set;

  * (iii)

$\operatorname{edim}(\operatorname{Bir}(X)_{d})\leq(n+1)\operatorname{HF}_{R}(d)-1$.

###### Proof.

(i) The first paragraph of the proof follows the argument of [BF13, Lemma
2.4]. Let $R=k[x_{0},\cdots,x_{n}]/\mathfrak{p}$ be the coordinate ring of
$X$. Let $\mathfrak{p}=(p_{1},\ldots,p_{t})$ and
$r=\operatorname{dim}(R)=\operatorname{dim}(X)+1$. Let $d^{\prime}=B(X,d)$ be
the inverse bound introduced in 3.9. We denote by $Y^{\prime}\subseteq
W_{d^{\prime}}\times W_{d}$ the set consisting of elements $(g,f)$ such that
$h:=(g_{0}(f_{0},\ldots,f_{n}),\ldots,g_{n}(f_{0},\ldots,f_{n}))$ and
$h^{\prime}:=(f_{0}(g_{0},\ldots,g_{n}),\ldots,f_{n}(g_{0},\ldots,g_{n}))$ are
multiples of the identity modulo $\mathfrak{p}$; i.e.,
$h_{i}x_{j}-h_{j}x_{i}\in\mathfrak{p}$ and
$h^{\prime}_{i}x_{j}-h^{\prime}_{j}x_{i}\in\mathfrak{p}$ for all $i,j$.

In order to $f:X\dasharrow X$ and $g:X\dasharrow X$ be well-defined maps, the
coordinates must satisfy $p_{i}(f_{0},\ldots,f_{n})\in\mathfrak{p}$ and
$p_{i}(g_{0},\ldots,g_{n})\in\mathfrak{p}$ for all $i$.

Let

$\mathcal{F}_{i}:=\\{(f_{{\bf a}_{0}},\ldots,f_{{\bf a}_{n}})\in
W_{d}:p_{i}(f_{{\bf a}_{0}},\ldots,f_{{\bf a}_{n}})\in\mathfrak{p}\\}$

and

$\mathcal{G}_{i}:=\\{(f_{{\bf a}^{\prime}_{0}},\ldots,f_{{\bf
a}^{\prime}_{n}})\in W_{d^{\prime}}:p_{i}(f_{{\bf
a}^{\prime}_{0}},\ldots,f_{{\bf a}^{\prime}_{n}})\in\mathfrak{p}\\}.$

2.1 shows that $\mathcal{F}_{i}$ and $\mathcal{G}_{i}$ are closed varieties of
$W_{d}$ and $W_{d^{\prime}}$ respectively. Similarly, $Y^{\prime}$ is a closed
subset of $W_{d^{\prime}}\times W_{d}$. Hence

$Y:=Y^{\prime}\cap(\bigcap_{i=1}^{t}\mathcal{F}_{i}\times
W_{d^{\prime}})\cap(W_{d}\times\bigcap_{i=1}^{t}\mathcal{G}_{i})$

is a closed subset of $W_{d^{\prime}}\times W_{d}$. Now, applying the
principal theorem of the elimination theory, the image of $Y$ by the
projection $\pi_{2}:W_{d^{\prime}}\times W_{d}\to W_{d}$ is closed in $W_{d}$.

Next, we look for $f=(f_{0}:\ldots:f_{n})\in W_{d}$ such that the rational map
$f:X\dasharrow X$ is dominant. Being dominant means that the Krull dimension
of the image of $f$ is $r-1$. The coordinate ring of the closure of the image
of $f$ is $\operatorname{Proj}(\mathcal{F}_{R}(I))$ (see definitions after
3.9) wherein $I$ is the ideal generated by the classes of $f_{0},\ldots,f_{n}$
in $R$. Therefore the map $f$ is dominant if and only if $\ell_{R}(I)=r$.
According to 2.5, this defines the set $\mathcal{N}_{n+1}$. Therefore,

$H_{d}=\\{h\in W_{d}:\psi_{h}:X\dasharrow X\text{\leavevmode\nobreak\
\leavevmode\nobreak\ is birational of polynomial degree\leavevmode\nobreak\
\leavevmode\nobreak\ }d\\}=\pi_{2}(Y)\cap\theta(\mathcal{N}_{n+1}).$

That is an open subset of the algebraic set $\pi_{2}(Y)$ thence a quasi-
projective variety.

Next, we show that $\tilde{H}_{d}$ is a quasi-projective variety.
$\tilde{H}_{d}$ is the intersection of $H_{d}$ with those birational maps
whose base ideal is of grade at least $2$. The regular sequence of length $2$
may not be among the generators, however it is inside a linear combination of
them. Hence we need to work with

$U_{2}=\\{({\bf a}_{0},\ldots,{\bf a}_{n})\in\mathbb{P}^{(n+1)N-1}:\sigma({\bf
a}_{0},\ldots,{\bf a}_{n})\in
G_{2}\times\mathbb{A}^{(n-1)N}\text{\leavevmode\nobreak\ for
some\leavevmode\nobreak\ }\sigma\in{\rm GL}_{n+1}(k)\\}.$

That is

$U_{2}=\bigcup_{\sigma\in{\rm
GL}_{n+1}(k)}\sigma^{-1}(G_{2}\times\mathbb{A}^{(n-1)N}).$

Since $G_{2}$ is an open subset, by 2.4, $U_{2}\subseteq\mathbb{P}^{(n+1)N-1}$
is open, as well.

We now have $\tilde{H}_{d}=\pi_{2}(Y)\cap\theta(\mathcal{N}_{n+1}\cap U_{2})$
thence $\tilde{H}_{d}$ is a quasi-projective variety.

(ii) In the case where $X=\mathbb{P}^{n}$, the proof of this part is to show
that there is a bijection between $\tilde{H}_{d}$ and
$\operatorname{Bir}(X)_{d}$. However, when $X\neq\mathbb{P}^{n}$, even if a
map $f=(f_{0},\ldots,f_{n}):X\dasharrow X$ belongs to $\tilde{H}_{d}$, the
defining polynomials of $f$ in $S$ are not uniquely defined. In other words,
two sets of polynomials in $S$ say $f_{0},\ldots,f_{n}$ and
$f^{\prime}_{0},\ldots,f^{\prime}_{n}$, define the same map over $X$ if
$f_{i}-f^{\prime}_{i}\in\mathfrak{p}$ for all $i$.

To address this ambiguity, we revisit 2.2. Let

(3.4) $s=N-{\operatorname{HF}}_{\mathfrak{p}}(d).$

Then the ideal of definition of the linear variety
$V_{\mathfrak{p}}^{z}:=\\{(a_{1},\ldots,a_{N})\in\mathbb{P}^{N-1}_{k}:\sum_{i=1}^{N}a_{i}M_{i}\in\mathfrak{p}\\}$
is determined by $s$ linear equations $l_{1},\ldots,l_{s}$. Let
$\mathcal{L}:k^{N}\to k^{s}$ be the linear map defined by
$l_{1},\ldots,l_{s}$. By repeating $\mathcal{L}$ on the diagonal, we extend it
to a map $\mathcal{L}^{\prime}:k^{(n+1)N}\to k^{(n+1)s}$ thence to a map
$\mathcal{D}:W_{d}\dasharrow\mathbb{P}^{(n+1)s-1}$.

More precisely, let $y_{1},\ldots,y_{N}$ be the coordinates of
$\mathbb{A}_{k}^{N}$, $l_{i}=\sum_{j=1}^{N}l_{ij}y_{j}$ for $i=1,\ldots,s$ and
$(f_{{\bf a}_{0}}:\ldots:f_{{\bf a}_{n}})\in W_{d}$. Then

$\mathcal{D}(f_{{\bf a}_{0}}:\cdots:f_{{\bf a}_{n}})=(l_{1}({\bf
a}_{0}):\cdots:l_{s}({\bf a}_{0}):\cdots:l_{1}({\bf a}_{n}):\cdots:l_{s}({\bf
a}_{n})).$

Let $\mathcal{D}^{\prime}=\mathcal{D}|_{\tilde{H}_{d}}$.

Notice that $\mathcal{D}^{\prime}$ is a (regular) morphism. Since for any
$0\leq j\leq n$, the linear forms $l_{i}({\bf a}_{j})$ all vanish for $0\leq
i\leq s$ if and only if $f_{{\bf a}_{j}}\in\mathfrak{p}$. This provides a
contradiction to the choice of $(f_{{\bf a}_{0}}:\cdots:f_{{\bf a}_{n}})$ as a
rational map over $X$ of grade at least $2$ in the coordinate ring
$R=S/\mathfrak{p}$.

Recalling the definition of the clear polynomial degree of a birational map,
we have the natural map

$\displaystyle\Psi_{d}:{\tilde{H}}_{d}$
$\displaystyle\longrightarrow\operatorname{Bir}(X)_{d}$ $\displaystyle h$
$\displaystyle\mapsto\psi_{h}.$

The map $\Psi_{d}$ induces a well-defined map $\Phi$, according to the
following triangle

(3.5)
$\textstyle{\tilde{H}_{d}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Psi_{d}}$$\textstyle{\operatorname{Im}(\mathcal{D}^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{D}^{\prime}}$$\scriptstyle{\Phi}$$\textstyle{\operatorname{Bir}(X)_{d}.}$

We verify that $\Phi$ is bijective.

Well-definedness: Let
$g=\mathcal{D}^{\prime}(f)=\mathcal{D}^{\prime}(f^{\prime})$ for $f=(f_{{\bf
a}_{0}}:\cdots:f_{{\bf a}_{n}})$ and $f^{\prime}=(f_{{\bf
a}^{\prime}_{0}}:\ldots:f_{{\bf a}^{\prime}_{n}})$ in $\tilde{H}_{d}$. We have
to show that $\Phi(g):=\Psi_{d}(f)=\Psi_{d}(f^{\prime})=:\Phi(g)$. Since
$\mathcal{D}^{\prime}$ is a linear map,
$\mathcal{D}^{\prime}(f-f^{\prime})=0$. Therefore, $l_{j}({\bf a}_{i}-{\bf
a}_{i}^{\prime})=0$ for all $i$ and $j$. Hence ${\bf a}_{i}-{\bf
a}_{i}^{\prime}\in V_{\mathfrak{p}}$. Equivalently, $f_{{\bf
a}^{\prime}_{i}}-f_{{\bf a}_{i}}\in\mathfrak{p}$. Therefore the images of $f$
and $f^{\prime}$ in $R$ are identical. In particular,
$\Psi_{d}(f)=\Psi_{d}(f^{\prime})\in\operatorname{Bir}(X)_{d}$.

Injectivity: Suppose that $f=(f_{{\bf a}_{0}}:\cdots:f_{{\bf a}_{n}})$ and
$f^{\prime}=(f_{{\bf a}^{\prime}_{0}}:\ldots:f_{{\bf a}^{\prime}_{n}})$ are
two elements of $\tilde{H}_{d}$ such that
$\Phi(\mathcal{D}^{\prime}(f))=\Phi(\mathcal{D}^{\prime}(f^{\prime}))$.
Equivalently, $\Psi_{d}(f)=\Psi_{d}(f^{\prime})$, or
$\psi_{f}=\psi_{f^{\prime}}\in\operatorname{Bir}(X)_{d}$. Since the base ideal
of $\psi_{f}$ and $\psi_{f^{\prime}}$, as ideals of $R$, have grade at least
$2$ over $R$, they differ by multiplication into a constant. Thus, if the
difference depends on a constant multiplication, lifting to the
(quasi-)projective space $\tilde{H}_{d}$, we have $(f_{{\bf
a}_{0}}:\ldots:f_{{\bf a}_{n}})=(f_{{\bf a}^{\prime}_{0}}:\ldots:f_{{\bf
a}^{\prime}_{n}})$–here is the place where we need to separate maps of clear
degree from other maps. The other point is that lifting from
$\operatorname{Bir}(X)_{d}$ to ${\tilde{H}}_{d}$ depends on the natural ring
map $S\to R$. Hence, the two representatives may differ by the equations of
$\mathfrak{p}$, that is, (looking at the representatives as elements of $S$)
$f_{{\bf a}_{i}}-f_{{\bf a}_{i}^{\prime}}\in\mathfrak{p}$ for all $i$. Hence
${\bf a}_{i}-{\bf a}_{i}^{\prime}\in V_{\mathfrak{p}}$. Therefore, $l_{j}({\bf
a}_{i}-{\bf a}_{i}^{\prime})=0$ for all $i$ and $j$. It then follows that

$\mathcal{L}^{\prime}(f_{{\bf a}_{0}}-f_{{\bf a}^{\prime}_{0}},\ldots,f_{{\bf
a}_{n}}-f_{{\bf a}^{\prime}_{n}})=0.$

Since $\mathcal{D}$ is a linear map, we get

$\mathcal{D}^{\prime}(f)=\mathcal{D}^{\prime}(f^{\prime}).$

Surjectivity: Let $h=(f_{{\bf a}_{0}}:\cdots:f_{{\bf a}_{n}}):X\dasharrow X$
be a birational map of clear degree $d$. Since
$\operatorname{grade}_{R}(f_{{\bf a}_{0}},\ldots,f_{{\bf a}_{n}})\geq 2$, we
have $(f_{{\bf a}_{0}}:\cdots:f_{{\bf a}_{n}})\in\theta(U_{2})$. Since $h$ is
dominant $\operatorname{dim}(\operatorname{Proj}(k[f_{{\bf
a}_{0}},\ldots,f_{{\bf a}_{n}}]))=\operatorname{dim}(X)$ hence
$\ell_{R}(f_{{\bf a}_{0}},\ldots,f_{{\bf a}_{n}})=r$; so that $(f_{{\bf
a}_{0}}:\cdots:f_{{\bf a}_{n}})\in\theta(\mathcal{N}_{n+1})$, and finally
since the birational map $h$ has an inverse of polynomial degree at most
$d^{\prime}=B(X,d)$ by 3.9, $(f_{{\bf a}_{0}}:\cdots:f_{{\bf
a}_{n}})\in\pi_{2}(Y)$; accordingly $(f_{{\bf a}_{0}}:\cdots:f_{{\bf
a}_{n}})\in\pi_{2}(Y)\cap\theta(\mathcal{N}_{n+1}\cap U_{2})=\tilde{H}_{d}$.
Therefore $\Phi(\mathcal{D}^{\prime}((f_{{\bf a}_{0}}:\cdots:f_{{\bf
a}_{n}})))=h$.

The bijectivity between $\operatorname{Im}(\mathcal{D}^{\prime})$ and
$\operatorname{Bir}(X)_{d}$ ensues that the latter inherits the same
topological structure as any one of the
$\operatorname{Im}(\mathcal{D}^{\prime})$. Since $\tilde{H}_{d}$ is quasi-
projective, by part (i), then Chevalley’s Theorem implies that
$\operatorname{Im}(\mathcal{D}^{\prime})$ is constructible, and so is
$\operatorname{Bir}(X)_{d}$.

(iii). According to the proof of part (ii), $\operatorname{Bir}(X)_{d}$ is
embedded in the projective space where
$\operatorname{Im}(\mathcal{D}^{\prime})$ is embedded. Since
$\mathcal{D}^{\prime}:\tilde{H}_{d}\longrightarrow\mathbb{P}^{(n+1)s-1}$, we
have $\operatorname{edim}(\operatorname{Bir}(X)_{d})\leq(n+1)s-1$. According
to Equation 3.4, $s=N-{\operatorname{HF}}_{\mathfrak{p}}(d)$. The latter is
equal to $\operatorname{HF}_{R}(d)$, since $N={n+d\choose
d}=\operatorname{HF}_{S}(d)$. ∎

An immediate corollary is to decide when $\operatorname{Bir}(X)_{d}$ is indeed
quasi-projective.

######  Corollary 3.11.

With the same token as 3.10, if $\operatorname{HF}_{\mathfrak{p}}(d)=0$, then
$\operatorname{Bir}(X)_{d}$ has the structure of a quasi-projective variety.

###### Proof.

According to Equation 3.4, $\operatorname{HF}_{\mathfrak{p}}(d)=N-s$. Hence,
in the course of the proof of 3.10(ii), the map $\mathcal{D}^{\prime}$ is the
identity map. So that $\Psi_{d}$ is a bijection. ∎

We continue with another immediate corollary of 3.10(iii).

######  Corollary 3.12.

Let $d$ be an integer, $k$ an infinite field and $X\subseteq\mathbb{P}^{n}$ an
irreducible projective non-degenerated variety with coordinate ring $R$,
Hilbert function $\operatorname{HF}_{R}$ and multiplicity $e(R)$. Then

(3.6)
$\limsup\limits_{d\rightarrow\infty}\frac{\operatorname{edim}(\operatorname{Bir}(X)_{d})}{d^{\dim(X)}}\leq\frac{(n+1)e(R)}{\dim(X)!}.$

The natural question arising by the above corollary is the following

######  Question 3.13.

For a projective variety $X$ what is the big O value of function
$\operatorname{edim}(\operatorname{Bir}(X)_{d})$. In other words, what is

(3.7)
$\mathcal{O}(\operatorname{edim}(\operatorname{Bir}(X)_{d}):=\min\\{m:\limsup\limits_{d\rightarrow\infty}\frac{\operatorname{edim}(\operatorname{Bir}(X)_{d})}{d^{m}}<\infty\\}?$

When $X$ is a curve then 3.6 shows that
$\mathcal{O}(\operatorname{edim}(\operatorname{Bir}(X)_{d})=0$. The same holds
when $X$ is a varity of general type. For $X=\mathbb{P}^{2}$, Bisi et al. in
[BCM15] show that $\dim(\operatorname{Bir}(X)_{d})=4d+6$. Hence in conjunction
with Equation 3.6, we have
$\mathcal{O}(\operatorname{edim}(\operatorname{Bir}(\mathbb{P}^{2})_{d})\in\\{1,2\\}.$

To follow the positive results of [BF13] for an arbitrary variety $X$, the
next structural property, 3.15, is essential. To state this result, we recall
the Demazure-Serre definition of the topology of $\operatorname{Bir}(X)$. We
slightly modify the original definition as follows.

######  Definition 3.14.

Let $X$ be an irreducible projective variety. Let $\mathfrak{A}$ be a
projective constructible set. A Demazure-Serre morphism from $\mathfrak{A}$ to
$\operatorname{Bir}(X)$ is a map $\zeta:\mathfrak{A}\to\operatorname{Bir}(X)$,
$a\mapsto\zeta_{a},$ such that for any irreducible component $A$ of
$\mathfrak{A}$, the induced map $\mathfrak{F}_{\zeta}:A\times X\dasharrow
A\times X,(a,x)\mapsto(a,\zeta_{a}(x))$ is a birational map.

A subset $F\subset\operatorname{Bir}(X)$ is closed if and only if for any
Demazure-Serre morphism $\zeta:\mathfrak{A}\to\operatorname{Bir}(X)$, the pre-
image $\zeta^{-1}(F)$ is closed in $\mathfrak{A}$.

Next theorem is a generalization of [BF13, Lemma 2.4(c)] for any irreducible
variety $X\subseteq\mathbb{P}^{n}$. Here, due to the generality, we present a
different proof from [BF13] .

######  Proposition 3.15.

With the same token as in 3.10, the map $\psi:{H}_{d}\to\operatorname{Bir}(X)$
defined in LABEL:psif is a Demazure-Serre morphism.

###### Proof.

To prove that $\mathfrak{F}_{\psi}:{H}_{d}\times X\to{H}_{d}\times X$,
$(f,x)\mapsto(x,\psi_{f(x)})$ is birational, we take an irreducible open
affine subset $A\subseteq{H}_{d}$ and show that the map
$\mathfrak{F}_{\psi}:A\times X\to A\times X$ is birational. To do this, we
first show that this map is dominant.

Let $\\{y_{ij}:i=0,\ldots,n;j=1,\ldots,N\\}$ be a set of variables defining
the coordinate ring of $A$. Let ${\bf y}_{i}=(y_{i1},\ldots,y_{iN})$
$i=0,\ldots,n$ and $k[A]=k[{\bf y}_{0},\ldots,{\bf y}_{n}]/\mathfrak{a}$ for
some ideal $\mathfrak{a}$. For any ${\bf a}\in A$, $\psi_{{\bf a}}:X\dasharrow
X$ is presented by $f_{{\bf a}}=(f_{{\bf a}_{0}}:\cdots:f_{{\bf a}_{n}})$ as
defined in (Equation 3.2). Assume that the closure of the image of
$\mathfrak{F}_{\psi}$ is contained in a closed subset of $A\times X$ defined
by a polynomial $Q({\bf y},{\bf x})$. We have $Q({\bf y},{\bf x})$ is
homogeneous in terms of ${\bf x}$-variables and $Q({\bf a},f_{{\bf a}}(u))=0$
for any $u\in U$ for some open subset $U\subseteq X$. Since $X$ is
irreducible, $Q({\bf a},f_{{\bf a}}({\bf x}))=0$. Since $f_{{\bf a}}$ is
birational there exists a polynomial $\alpha({\bf x})$ with $f_{{\bf
a}}(f^{-1}_{{\bf a}}({\bf x}))=\alpha({\bf x}){\bf x}$. Hence for some integer
$m$, $\alpha({\bf x})^{m}Q({\bf a},{\bf x})=0$ for all ${\bf a}$. Thus $Q({\bf
a},{\bf x})=0$ for all ${\bf a}$; yielding $Q({\bf y},{\bf x})=0$. This proves
that $\mathfrak{F}_{\psi}$ is dominant.

Since $\mathfrak{F}_{\psi}$ is a dominant map of varieties of the same
dimension, it is a generically finite map. That means there exists an open
subset $T\subset A\times X$ such that

$\mathfrak{F}_{\psi}:\mathfrak{F}_{\psi}^{-1}(T)\rightarrow T$

is a finite morphism. We show that the degree of this finite morphism is $1$.

For each ${\bf a}\in A$, there exist open subsetes $U_{{\bf a}}\subseteq X$
and $V_{{\bf a}}\subseteq X$ with $f_{{\bf a}}:U_{{\bf a}}\rightarrow V_{{\bf
a}}$ is an isomorphism. We claim that for some ${\bf a}\in A$, the
intersection $({\bf a}\times V_{{\bf a}})\cap T$ is not empty. Since for any
point in $({\bf a}\times V_{{\bf a}})\cap T$ the pre-image is a single set
this proves that the degree of the above finite morphism is $1$. To prove the
claim, let $P({\bf y},{\bf x})$ be a polynomial homogeneous in ${\bf x}$
variables that defines one of the equations of the complement of $T$. If by
contrary, we suppose that for any ${\bf a}\in A$ , $({\bf a}\times V_{{\bf
a}})\cap T=\emptyset$. Then for any ${\bf a}\in A$ and $u\in U_{{\bf a}}$,
$P({\bf a},f_{{\bf a}}(u))=0$, using the birationality of $f_{{\bf a}}$, we
find a polynomial $\alpha({\bf x})$ and an integer $m$ such that $\alpha({\bf
x})^{m}P({\bf a},{\bf x})=0$ for all ${\bf a}\in A$, accordingly $P({\bf
y},{\bf x})=0$ that implies that $T=\emptyset$ which is absurd. Hence there
exists an ${\bf a}\in A$ such that $({\bf a}\times V_{{\bf a}})\cap
T\neq\emptyset$.

∎

Now, having 3.10 and 3.15, one can follow the techniques in [BF13, 2.6–2.10]
to prove the following corollary.

######  Notation 3.16.

We denote $\operatorname{Im}(\mathcal{D}^{\prime})$ in 3.10, by
$\tilde{\tilde{H}}_{d}$, and the map
$\operatorname{Im}(\mathcal{D}^{\prime})\xrightarrow{\Phi}\operatorname{Bir}(X)_{d}$
by $\Phi_{d}:\tilde{\tilde{H}}_{d}\to\operatorname{Bir}(X)_{d}$.

######  Corollary 3.17.

With the same notation introduced before, during and after 3.10, the following
hold

  1. (1)

Let $A$ and $X$ be irreducible algebraic varieties and
$\rho:A\to\operatorname{Bir}(X)$ be a Demazure-Serre morphism whose image is
inside $\operatorname{Bir}(X)_{*}$. There exists an open affine covering
$\\{A_{i}\\}$ of $A$ such that for each $i$, there exists an integer $d_{i}$
and a morphism $\rho_{i}:A_{i}\to\tilde{\tilde{H}}_{d_{i}}$ such that the
restriction of $\rho$ to $A_{i}$ is equal to $\Phi_{d_{i}}\circ\rho_{i}$.

  2. (2)

A set $F\subseteq\operatorname{Bir}(X)_{*}$ is closed if and only if
$\Phi_{d}^{-1}(F)$ is closed in $\tilde{\tilde{H}}_{d}$ for any $d$.

  3. (3)

For any $d$, the set $\operatorname{Bir}(X)_{\leq d}$ is closed in
$\operatorname{Bir}(X)_{*}$.

  4. (4)

The topology of $\operatorname{Bir}(X)_{*}$ is the inductive limit topology of
$\operatorname{Bir}(X)_{\leq d}$.

## References

  * [BCM15] Cinzia Bisi, Alberto Calabri, and Massimiliano Mella.  On plane Cremona transformations of fixed degree.  J. Geom. Anal., 25(2):1108–1131, 2015.
  * [BF13] Jérémy Blanc and Jean-Philippe Furter.  Topologies and structures of the Cremona groups.  Ann. of Math. (2), 178(3):1173–1198, 2013.
  * [Bla17] Jérémy Blanc.  Algebraic structures of groups of birational transformations.  In Algebraic groups: structure and actions, volume 94 of Proc. Sympos. Pure Math., pages 17–30. Amer. Math. Soc., Providence, RI, 2017\.
  * [BS13] M. P. Brodmann and R. Y. Sharp.  Local cohomology, volume 136 of Cambridge Studies in Advanced Mathematics.  Cambridge University Press, Cambridge, second edition, 2013.  An algebraic introduction with geometric applications.
  * [Cal94] Frederick W. Call.  A theorem of Grothendieck using Picard groups for the algebraist.  Math. Scand., 74(2):161–183, 1994.
  * [CL13] Serge Cantat and Stéphane Lamy.  Normal subgroups in the Cremona group.  Acta Math., 210(1):31–94, 2013.  With an appendix by Yves de Cornulier.
  * [DHS12] A. V. Doria, S. H. Hassanzadeh, and A. Simis.  A characteristic-free criterion of birationality.  Adv. Math., 230(1):390–413, 2012.
  * [GS98] D Grayson and M Stillman.  Macaulay 2–a system for computation in algebraic geometry and commutative algebra, 1997.  Available via anonymous ftp from math. uiuc. edu, 1998.
  * [HS06] Craig Huneke and Irena Swanson.  Integral closure of ideals, rings, and modules, volume 336 of London Mathematical Society Lecture Note Series.  Cambridge University Press, Cambridge, 2006.
  * [HS17] S. H. Hassanzadeh and A. Simis.  Bounds on degrees of birational maps with arithmetically Cohen-Macaulay graphs.  J. Algebra, 478:220–236, 2017.
  * [Mat86] Hideyuki Matsumura.  Commutative ring theory, volume 8 of Cambridge Studies in Advanced Mathematics.  Cambridge University Press, Cambridge, 1986.  Translated from the Japanese by M. Reid.
  * [PR12] Ivan Pan and Alvaro Rittatore.  Some remarks about the zariski topology of the cremona group.  arXiv preprint arXiv:1212.5698, 2012.
  * [Ser10] Jean-Pierre Serre.  Le groupe de Cremona et ses sous-groupes finis.  Number 332, pages 75–100. 2010.  Séminaire Bourbaki. Volume 2008/2009. Exposés 997–1011.
  * [Sim04] Aron Simis.  Cremona transformations and some related algebras.  J. Algebra, 280(1):162–179, 2004.
  * [SUV01] Aron Simis, Bernd Ulrich, and Wolmer V. Vasconcelos.  Codimension, multiplicity and integral extensions.  Math. Proc. Cambridge Philos. Soc., 130(2):237–257, 2001.