main .3pt [0][tri-linear birational maps in dimension three][] laurent busé,
pablo gonzález-mazón, josef schicho 0

# Tri-linear birational maps in dimension three

Laurent Busé  ,  Pablo González-Mazón  Université Côte d’Azur, Inria, 2004
route des Lucioles, 06902 Sophia Antipolis, France<EMAIL_ADDRESS><EMAIL_ADDRESS>and  Josef Schicho  Johannes Kepler University,
RISC, Austria<EMAIL_ADDRESS>
###### Abstract.

A tri-linear rational map in dimension three is a rational map
$\phi:(\mathbb{P}_{\mathbb{C}}^{1})^{3}\dashrightarrow\mathbb{P}_{\mathbb{C}}^{3}$
defined by four tri-linear polynomials without a common factor. If $\phi$
admits an inverse rational map $\phi^{-1}$, it is a tri-linear birational map.
In this paper, we address computational and geometric aspects about these
transformations. We give a characterization of birationality based on the
first syzygies of the entries. More generally, we describe all the possible
minimal graded free resolutions of the ideal generated by these entries. With
respect to geometry, we show that the set $\mathfrak{Bir}_{(1,1,1)}$ of tri-
linear birational maps, up to composition with an automorphism of
$\mathbb{P}_{\mathbb{C}}^{3}$, is a locally closed algebraic subset of the
Grassmannian of $4$-dimensional subspaces in the vector space of tri-linear
polynomials, and has eight irreducible components. Additionally, the group
action on $\mathfrak{Bir}_{(1,1,1)}$ given by composition with automorphisms
of $(\mathbb{P}_{\mathbb{C}}^{1})^{3}$ defines 19 orbits, and each of these
orbits determines an isomorphism class of the base loci of these
transformations.

##  1\. Introduction

### Motivation

Rational maps between multi-projective and projective spaces have been studied
recently, with a special emphasis on the computation of the degrees and
birationality (e.g. [6, 8, 5, 21, 13]). The introduction of the multi-
projective setting is motivated by applications of these rational maps to
Computer-Aided Geometric Design (CAGD), computer vision, shape optimization,
animation, and other related fields ([21] and references therein). In
particular, maps defined over products of projective lines are ubiquitous in
geometric modeling. For instance, tensor-product rational Bézier surfaces and
volumes are used intensively for the definition of geometric objects. More
specifically, dominant rational maps
$(\mathbb{P}_{\mathbb{C}}^{1})^{2}\dashrightarrow\mathbb{P}_{\mathbb{C}}^{2}$
and
$(\mathbb{P}_{\mathbb{C}}^{1})^{3}\dashrightarrow\mathbb{P}_{\mathbb{C}}^{3}$
provide simple and meaningful control points for the deformation of 2D and 3D
shapes, in a technique known as Free-Form Deformation (FFD) (e.g. [22, 21]).
One of the requirements for these applications is the injectivity of the
rational map. More interestingly, birational maps are globally injective, and
have an inverse rational map. Moreover, the computation of preimages is
necessary in some applications ([21] and references), and the inverse map
allows the computation of these preimages avoiding the numerical resolution of
equations.

### Previous works and context

Approximately thirty years ago, modern techniques from commutative algebra
were incorporated to the study of birational maps, primarily motivated by
computational purposes. Syzygy-based birationality criteria date back to [20],
and have been further developed ever since (e.g. [19, 23]). In this context,
the Jacobian Dual Criterion (JDC) is a general method to decide if a rational
map between projective varieties, or from a multi-projective variety to a
projective variety, is birational [10, 6]. The JDC relies on the computation
of the defining equations of the Rees algebra associated to the rational map.
Unfortunately, the derivation of these equations entails a high computational
cost, and for some applications specialized methods are preferred. In
dimension two, there are efficient strategies for the manipulation of bi-
linear birational maps
$\phi:(\mathbb{P}_{\mathbb{C}}^{1})^{2}\dashrightarrow\mathbb{P}_{\mathbb{C}}^{2}$,
whose entries are polynomials of bi-degree $(1,1)$ without a common factor,
where a syzygy-based characterization of birationality is known [22, 5].
Larger bi-degrees have also been considered, but only partially. Namely,
sufficient conditions under which birationality holds are known for maps with
entries of bi-degree $(2,2)$ [5] and $(1,n)$ [21]. In dimension three, no
syzygy-based criteria for the birationality of a rational map
$\phi:(\mathbb{P}_{\mathbb{C}}^{1})^{3}\dashrightarrow\mathbb{P}_{\mathbb{C}}^{3}$
are known yet, probably because the situation is much more involved. On the
other hand, the study of groups and spaces of birational maps has a long
history in algebraic geometry, and there is an extensive literature about the
topic (e.g. [17, 1, 7, 3, 4]). The $n$-dimensional Cremona group over
$\mathbb{C}$, denoted by either $\text{Cr}_{n}(\mathbb{C})$ or
$\mathrm{Bir}(\mathbb{P}_{\mathbb{C}}^{n})$, is the group of birational
endomorphisms of $\mathbb{P}_{\mathbb{C}}^{n}$. Given $d\geq 0$, the subset
$\mathfrak{Bir}_{d}(\mathbb{P}_{\mathbb{C}}^{n})\subset\text{Cr}_{n}(\mathbb{C})$
consists of all the birational endomorphisms that are determined by $n+1$
homogeneous polynomials of degree $d$, without a common factor. Remarkably, we
have a good understanding of $\mathfrak{Bir}_{d}(\mathbb{P}_{\mathbb{C}}^{2})$
[3] and a complete classification of the base loci is known for the degrees
$d=2,3$ [7]. However, for $n\geq 3$ the situation quickly becomes more
complicated. For $n=3$, the algebraic structure of
$\mathfrak{Bir}_{2}(\mathbb{P}_{\mathbb{C}}^{3})$ has been described together
with a complete classification of the base loci of birational endomorphisms
with quadratic entries [18], but this is no longer the case if $d\geq 3$. For
$d=3$, some of the irreducible components of
$\mathfrak{Bir}_{3}(\mathbb{P}_{\mathbb{C}}^{3})$ have also been described
[9], but only a partial classification of the base loci is known [17, 9]. With
respect to birational maps from a multi-projective space to a projective
target, only the structure and classification of the base loci of bi-linear
birational maps
$(\mathbb{P}_{\mathbb{C}}^{1})^{2}\dasharrow\mathbb{P}_{\mathbb{C}}^{2}$ have
been addressed [22, 5]. Namely, the set of bi-linear birational maps is an
irreducible hypersurface in $\mathbb{P}_{\mathbb{C}}^{11}$, and the base locus
is always a closed point. In this paper, we study the algebraic set of tri-
linear birational maps
$\phi:(\mathbb{P}_{\mathbb{C}}^{1})^{3}\dasharrow\mathbb{P}_{\mathbb{C}}^{3}$
and we give the complete classification of the base loci of such
transformations, up to isomorphism of schemes. This geometric analysis targets
the derivation of computational methods for the manipulation of these
birational maps. Namely, we provide a syzygy-based characterization for
birationality, whose proof strongly relies on the classification of the base
loci.

### Organization of the paper

In §2 we introduce tri-linear birational maps in dimension three and the
concepts of base ideal, base locus, and type of a tri-linear birational map.
Additionally, we briefly review the Jacobian Dual Criterion. In §3 we explain
our strategy for the factorization of tri-linear birational maps, which is
crucial for the later analysis. The algebraic structure of the set
$\mathfrak{Bir}_{(1,1,1)}$, consisting of tri-linear birational maps up to
composition with automorphisms of $\mathbb{P}_{\mathbb{C}}^{3}$, is studied in
§4. Here, we also describe the group action on $\mathfrak{Bir}_{(1,1,1)}$
given by the automorphisms of $(\mathbb{P}_{\mathbb{C}}^{1})^{3}$, that we
call right-action. Our first main result is Theorem 4.1, which states that
$\mathfrak{Bir}_{(1,1,1)}$ has the structure of a locally closed algebraic
subset of a certain Grassmannian and describes its irreducible components.
Later, §5 is devoted to the classification of the base loci. In Theorems 5.1,
5.4, 5.6, and 5.8 we derive the complete list of the orbits that the right-
action defines on $\mathfrak{Bir}_{(1,1,1)}$, which are in correspondence with
the isomorphism classes of the base loci. On the computational side, in §6 we
study the syzygies of the entries of tri-linear birational maps. Namely,
Theorem 6.1 is a characterization of birationality based on the first
syzygies, whose proof relies strongly on the classification of §5. More
generally, in Propositions 6.2, 6.4, 6.5, and 6.8 we also classify all the
possible minimal tri-graded free resolutions of the base ideal of a tri-linear
birational map.

### Acknowledgements

The authors are grateful to the anonymous reviewers for their comments which
significantly improved the content and presentation of this paper. The three
authors have received funding from the European Union’s “Horizon 2020 research
and innovation programme”, under the Marie Skłodowska-Curie grant agreement n∘
860843\. All the examples, and multiple computations not explicitly included,
were performed using the computer algebra software Macaulay2 [14].

##  2\. Preliminars

###  2.1. Tri-linear birational maps in dimension three

A tri-linear rational map in dimension three $\phi$ is a rational map

(2.1)
$\displaystyle\phi:\mathbb{P}_{\mathbb{C}}^{1}\times\mathbb{P}_{\mathbb{C}}^{1}\times\mathbb{P}_{\mathbb{C}}^{1}$
$\displaystyle\dasharrow$ $\displaystyle\mathbb{P}_{\mathbb{C}}^{3}$
$\displaystyle(x_{0}:x_{1})\times(y_{0}:y_{1})\times(z_{0}:z_{1})$
$\displaystyle\mapsto$ $\displaystyle(f_{0}:f_{1}:f_{2}:f_{3})\ ,$

where $f_{i}=f_{i}(x_{0},x_{1},y_{0},y_{1},z_{0},z_{1})$ is a tri-linear
polynomial, i.e. is linear and homogeneous with respect to the three pairs of
variables $\\{x_{0},x_{1}\\}$, $\\{y_{0},y_{1}\\}$, and $\\{z_{0},z_{1}\\}$,
and the $f_{i}$’s do not have a common factor. For simplicity in the notation,
we will refer to them simply as tri-linear rational maps. If $\phi$ admits an
inverse rational map, then $\phi$ is a tri-linear birational map. Let
$A_{1}=\mathbb{C}[x_{0},x_{1}]$, $A_{2}=\mathbb{C}[y_{0},y_{1}]$, and
$A_{3}=\mathbb{C}[z_{0},z_{1}]$ be the graded homogeneous coordinate rings of
each $\mathbb{P}_{\mathbb{C}}^{1}$, with the standard grading in each ring
given by the corresponding pair of variables. The tri-homogeneous coordinate
ring of the product $X\coloneqq(\mathbb{P}_{\mathbb{C}}^{1})^{3}$ is the
tensor product ring $R\coloneqq
A_{1}\otimes_{\mathbb{C}}A_{2}\otimes_{\mathbb{C}}A_{3}$, which inherits a
tri-graded structure

$R=\bigoplus_{(i,j,k)\in\mathbb{Z}^{3}}\,R_{(i,j,k)}=\bigoplus_{(i,j,k)\in\mathbb{Z}^{3}}\,(A_{1})_{i}\otimes_{\mathbb{C}}(A_{2})_{j}\otimes_{\mathbb{C}}(A_{3})_{k}\
,$

where subindices stand for graded components. The irrelevant ideal of $R$ is

$\mathfrak{N}=\bigoplus_{i>0,\,j>0,\,k>0}\,(A_{1})_{i}\otimes_{\mathbb{C}}(A_{2})_{j}\otimes_{\mathbb{C}}(A_{3})_{k}=(x_{0},x_{1})\cap(y_{0},y_{1})\cap(z_{0},z_{1})\
.$

The tri-homogeneous ideal $B_{\phi}=(f_{0},f_{1},f_{2},f_{3})\subset R$ is the
base ideal of $\phi$. The subscheme $Z_{\phi}$ of $X$ defined by the base
ideal $B_{\phi}$ is the base locus or base locus scheme of $\phi$. The
codimension of $Z_{\phi}$ is always at least two, as otherwise we find a
common factor to the $f_{i}$’s. On the other hand, a surface $S\subset X$ is
the vanishing locus of a tri-homogeneous polynomial in $R$. The tri-degree of
$S$, denoted by tri-deg$(S)$, is the tri-degree of its defining polynomial.

###  2.2. Curves in $(\mathbb{P}_{\mathbb{C}}^{1})^{3}$

Let $Y$ be a one-dimensional subscheme of $X$. The tri-degree of $Y$, denoted
by $\text{tri-deg}(Y)$, is the triple $(d_{1},d_{2},d_{3})$ where $d_{i}$ is
the number of points, counted with multiplicity, in the intersection of $Y$
with the subspace of $X$ determined by a general linear form in $A_{i}$.
Equivalently, $d_{i}$ is the degree of the projection of $Y$ onto the $i$-th
$\mathbb{P}_{\mathbb{C}}^{1}$ factor of $X$. For zero-dimensional subschemes
of $X$, we define the tri-degree as $(0,0,0)$. If all the irreducible
components of $Y$ have dimensione one, we say that $Y$ is a curve. In this
situation, we write $I_{Y}\subset R$ for its defining ideal. On the other
hand, recall that $Y$ is Cohen-Macaulay if for every point $P\in Y$ the local
ring of $Y$ at $P$ is Cohen-Macaulay. In particular, $Y$ is a Cohen-Macaulay
curve if and only if it does not have embedded nor isolated points [24, Lemma
31.4.4]. We define $C_{Y}\subset X$ as the subscheme determined by the
intersection of all the associated primes to $I_{Y}$ in $R$ of codimension
exactly two. Clearly, $C_{Y}\subset Y$ is a Cohen-Macaulay curve.
Additionally, when $Y=Z_{\phi}$ we denote $C_{Y}$ simply by $C_{\phi}$, and
$I_{C_{Y}}$ simply by $I_{\phi}$.

###  2.3. The Jacobian Dual Criterion

Let $J$ be the kernel of the ring homomorphism

$\displaystyle R\,[t_{0},t_{1},t_{2},t_{3}]$ $\displaystyle\rightarrow$
$\displaystyle R\,[u]$ $\displaystyle t_{i}$ $\displaystyle\mapsto$
$\displaystyle f_{i}\,u\ \ \ \ \ \ \ .$

The ideal $J$ is the Rees ideal associated to $\phi$. It is a prime ideal
hence saturated with respect to $\mathfrak{N}$. Moreover, it defines the
scheme-theoretic graph of $\phi$ in the product
$X\times\mathbb{P}_{\mathbb{C}}^{3}$. The Rees ideal $J$ is equipped with a
$\mathbb{Z}^{4}$-graded structure, as it inherits the $\mathbb{Z}^{3}$-grading
from $R$ and the standard $\mathbb{Z}$-grading from
$\mathbb{C}[t_{0},t_{1},t_{2},t_{3}]$. We use the notation
$(d_{1},d_{2},d_{3};d_{4})$ for this multi-degree in $\mathbb{Z}^{4}$. More
explicitly, we have

$g=g(x_{0},x_{1},y_{0},y_{1},z_{0},z_{1},t_{0},t_{1},t_{2},t_{3})\in
J_{(d_{1},d_{2},d_{3};d_{4})}$

if $g$ is homogeneous of degree $d_{1}$ in $x_{0},x_{1}$, of degree $d_{2}$ in
$y_{0},y_{1}$, of degree $d_{3}$ in $z_{0},z_{1}$, and of degree $d_{4}$ in
$t_{0},t_{1},t_{2},t_{3}$, and additionally

$g(x_{0},x_{1},y_{0},y_{1},z_{0},z_{1},f_{0},f_{1},f_{2},f_{3})=0\ .$

The elements in $J_{(d_{1},d_{2},d_{3};1)}$ are in correspondence with the
first syzygies of the $f_{i}$’s with coefficients of tri-degree
$(d_{1},d_{2},d_{3})$ in $R$. We refer to these as syzygies of tri-degree
$(d_{1},d_{2},d_{3})$. The multi-graded Jacobian Dual Criterion (JDC) [6,
Theorem 4.4] characterizes birationality in terms of the ideal $J$. Namely,
given $(d_{1},d_{2},d_{3})\in\mathbb{Z}^{3}$ consider the
$\mathbb{C}[t_{0},t_{1},t_{2},t_{3}]$-module

$J_{(d_{1},d_{2},d_{3};*)}\coloneqq\bigoplus_{d_{4}\in\,\mathbb{Z}}J_{(d_{1},d_{2},d_{3};d_{4})}\
.$

Assuming that $\phi$ is dominant, the JDC establishes that $\phi$ is
birational if and only if the modules

$J_{(1,0,0;*)}\ ,\ J_{(0,1,0;*)}\ ,\ J_{(0,0,1;*)}$

all have rank one when regarded as vector spaces over the field
$\mathbb{C}(t_{0},t_{1},t_{2},t_{3})$. In particular, we find minimal integers
$r_{1},r_{2},r_{3}\in\mathbb{Z}$ such that

(2.2) $\begin{vmatrix}x_{0}&x_{1}\\\ a_{0}&a_{1}\end{vmatrix}\in
J_{(1,0,0;\,r_{1})}\ ,\ \begin{vmatrix}y_{0}&y_{1}\\\
b_{0}&b_{1}\end{vmatrix}\in J_{(0,1,0;\,r_{2})}\ ,\
\begin{vmatrix}z_{0}&z_{1}\\\ c_{0}&c_{1}\end{vmatrix}\in J_{(0,0,1;\,r_{3})}\
,$

for some polynomials
$a_{i},b_{j},c_{k}\in\mathbb{C}[t_{0},t_{1},t_{2},t_{3}]$. Thus, we find
$s,t,u\in R$ satisfying

$a_{i}\xmapsto{t_{l}\mapsto f_{l}}x_{i}\,s\ ,\ b_{j}\xmapsto{t_{l}\mapsto
f_{l}}y_{j}\,t\ ,\ c_{k}\xmapsto{t_{l}\mapsto f_{l}}z_{k}\,u\ ,$

and the rational map

(2.3) $\displaystyle\phi^{-1}:\mathbb{P}_{\mathbb{C}}^{3}$
$\displaystyle\dasharrow$ $\displaystyle
X=\mathbb{P}_{\mathbb{C}}^{1}\times\mathbb{P}_{\mathbb{C}}^{1}\times\mathbb{P}_{\mathbb{C}}^{1}$
$\displaystyle(t_{0}:t_{1}:t_{2}:t_{3})$ $\displaystyle\mapsto$
$\displaystyle(a_{0}:a_{1})\times(b_{0}:b_{1})\times(c_{0}:c_{1})$

must be the inverse of $\phi$, as the composition $\phi^{-1}\circ\phi$ yields
the identity in any open subset of $X$ where $s,t$, and $u$ are all non-zero.

###  2.4. The type of $\phi$

Assume that $\phi$ is birational, and maintain the notation of §2.3. We define
the type of $\phi$ to be the triple $(r_{1},r_{2},r_{3})=(\deg a_{i},\deg
b_{j},\deg c_{k})\in\mathbb{Z}^{3}$. Notice that this is well defined, as we
are assuming $a_{0},a_{1}$ (resp. $b_{0},b_{1}$ and $c_{0},c_{1}$) of minimal
degree. Given a general line $\ell\subset\mathbb{P}_{\mathbb{C}}^{3}$ the
image $\phi^{-1}(\ell)$ is a closed rational curve in $X$. Additionally,
writing $\pi_{i}:X\xrightarrow{}\mathbb{P}_{\mathbb{C}}^{1}$ for the
projection of $X$ onto the $i$-th factor, the composition
$\pi_{1}\circ\phi^{-1}|_{\ell}$

$\displaystyle\mathbb{P}_{\mathbb{C}}^{1}\cong\ell\subset\mathbb{P}_{\mathbb{C}}^{3}$
$\displaystyle\xrightarrow{\phi^{-1}|_{\ell}}$
$\displaystyle\phi^{-1}(\ell)\subset
X=\mathbb{P}_{\mathbb{C}}^{1}\times\mathbb{P}_{\mathbb{C}}^{1}\times\mathbb{P}_{\mathbb{C}}^{1}$
$\displaystyle\xrightarrow{\pi_{1}}$ $\displaystyle\ \ \
\mathbb{P}_{\mathbb{C}}^{1}$ $\displaystyle(t_{0}:t_{1}:t_{2}:t_{3})$
$\displaystyle\mapsto$
$\displaystyle(a_{0}:a_{1})\times(b_{0}:b_{1})\times(c_{0}:c_{1})$
$\displaystyle\mapsto$ $\displaystyle(a_{0}:a_{1})\ ,$

yields an endomorphism of $\mathbb{P}_{\mathbb{C}}^{1}$ of degree $r_{1}$.
Similarly, the projections onto the other factors yield endomorphisms of
degree $r_{2}$ and $r_{3}$, implying that the tri-degree of $\phi^{-1}(\ell)$
is $(r_{1},r_{2},r_{3})$. On the other hand, let
$(\sum_{i=0}^{3}\alpha_{i}t_{i},\sum_{i=0}^{3}\beta_{i}t_{i})$ be the defining
ideal of $\ell$ in $\mathbb{C}[t_{0},t_{1},t_{2},t_{3}]$. Then, the tri-
homogeneous ideal

$\left(\alpha_{0}f_{0}+\alpha_{1}f_{1}+\alpha_{2}f_{2}+\alpha_{3}f_{3}\,,\,\beta_{0}f_{0}+\beta_{1}f_{1}+\beta_{2}f_{2}+\beta_{3}f_{3}\right)$

defines a scheme in $X$ of tri-degree $(2,2,2)$, which is the scheme-theoretic
union of $\phi^{-1}(\ell)$ and the base locus $Z_{\phi}$. Therefore, it
follows the identity

(2.4) $\text{tri-deg}(\phi^{-1}(\ell))+\text{tri-
deg}(Z_{\phi})=(r_{1},r_{2},r_{3})+\text{tri-deg}(Z_{\phi})=(2,2,2)$

(a projective version of (2.4) can be found in [9, §$1$]). In particular, we
obtain the following pairing between the possible types of $\phi$ and the tri-
degree of the base locus $Z_{\phi}$.

Type of $\phi$ | Tri-degree of $Z_{\phi}$
---|---
(1,1,1) | (1,1,1)
(1,1,2) (and permutations) | (1,1,0) (and permutations)
(1,2,2) (and permutations) | (1,0,0) (and permutations)
(2,2,2) | (0,0,0)

Table 1. All the possible types of $\phi$, and the tri-degrees of $Z_{\phi}$.

From Table 1 it follows that if $\phi$ has type $(2,2,2)$ the base locus
$Z_{\phi}$ is zero-dimensional.

###  2.5. Automorphisms of $(\mathbb{P}_{\mathbb{C}}^{1})^{3}$

Let $\mathrm{Aut}(X)$ be the group of automorphisms of $X$. Consider the
normal subgroup $H\trianglelefteq\mathrm{Aut}(X)$ of all the automorphisms
that preserve the tri-degrees in $X$, i.e. those of the form

$(x_{0}:x_{1})\times(y_{0}:y_{1})\times(z_{0}:z_{1})\mapsto\eta_{1}(x_{0}:x_{1})\times\eta_{2}(y_{0}:y_{1})\times\eta_{3}(z_{0}:z_{1})\
,$

for some $\eta_{i}$’s in the projective general linear group
$\mathrm{PGL}(2,\mathbb{C})$. Clearly, we have an isomorphism
$H\cong\mathrm{PGL}(2,\mathbb{C})^{3}$. On the other hand, the automorphism
given by

$\displaystyle X$ $\displaystyle\xrightarrow{}$ $\displaystyle X$
$\displaystyle(x_{0}:x_{1})\times(y_{0}:y_{1})\times(z_{0}:z_{1})$
$\displaystyle\mapsto$
$\displaystyle(y_{0}:y_{1})\times(z_{0}:z_{1})\times(x_{0}:x_{1})$

sends curves of tri-degree $(d_{1},d_{2},d_{3})$ to curves of tri-degree
$(d_{2},d_{3},d_{1})$, so it does not preserve the tri-degrees in $X$. In
general, every automorphism of $X$ sends curves of tri-degree
$(d_{1},d_{2},d_{3})$ to curves of tri-degree $(d_{i},d_{j},d_{k})$ for some
indices $i,j,k$ such that $\\{i,j,k\\}=\\{1,2,3\\}$, i.e. it might permute the
tri-degrees. Let $K\leq\mathrm{Aut}(X)$ be the subgroup consisting of the
automorphisms that simply permute the factors of $X$, which is isomorphic to
the symmetric group $\mathfrak{S}_{3}$ acting on three elements. In
particular, the intersection $H\cap K$ is trivial. Moreover, the association

(2.5) $\displaystyle K\times H$ $\displaystyle\xrightarrow{}$
$\displaystyle\mathrm{Aut}(X)$ $\displaystyle(\sigma,\eta)$
$\displaystyle\mapsto$ $\displaystyle\sigma\circ\eta$

is a bijection of sets. Therefore, the semidirect product of groups $K\rtimes
H\cong\mathfrak{S}_{3}\rtimes\mathrm{PGL}(2,\mathbb{C})^{3}$ is isomorphic to
$\mathrm{Aut}(X)$.

##  3\. Factorization of tri-linear birational maps

In this section we explain the strategy followed in our geometric analysis. It
is inspired by [18], where the classification of birational endomorphisms of
$\mathbb{P}_{\mathbb{C}}^{3}$ with quadratic entries is addressed. The idea is
the factorization of tri-linear rational maps through a linear system and a
projection.

###  3.1. Factorization of tri-linear rational maps

Let $W$ be a $\mathbb{C}$-vector space satisfying

(3.1) $(B_{\phi})_{(1,1,1)}\subset W\subset R_{(1,1,1)}\ ,$

and let $N+1=\dim_{\mathbb{C}}W$. Given a basis of the dual vector space
$W^{\vee}$, let $\zeta_{W}:X\dashrightarrow\mathbb{P}(W^{\vee})$ be the
rational map whose entries are the dual vectors in this basis, and write
$Y_{W}$ for the Zariski closure in $\mathbb{P}(W^{\vee})$ of the image of
$\zeta_{W}$. In particular, we have the commutative diagram

(3.2)
${Y_{W}}$${\mathbb{P}\left(W^{\vee}\right)\cong\mathbb{P}_{\mathbb{C}}^{N}}$${\text{Bl}_{W}X\subset
X\times\mathbb{P}_{\mathbb{C}}^{N}}$${X}$${\mathbb{P}_{\mathbb{C}}^{3}}$$\scriptstyle{\pi_{L}}$$\scriptstyle{\pi_{W}}$$\scriptstyle{\Pi_{W}}$$\scriptstyle{\phi}$$\scriptstyle{\zeta_{W}}$

where $\pi_{W}:\text{Bl}_{W}X\xrightarrow{}X$ is the blow-up of $X$ along the
subscheme defined by the entries of $\zeta_{W}$,
$\Pi_{W}:\text{Bl}_{W}X\xrightarrow{}Y_{W}$ is the projection onto the second
factor, and
$\pi_{L}:\mathbb{P}(W^{\vee})\dashrightarrow\mathbb{P}_{\mathbb{C}}^{3}$ is
the linear projection from some suitable subspace
$L\subset\mathbb{P}(W^{\vee})$. If $N=3$, then
$\mathbb{P}(W^{\vee})\cong\mathbb{P}_{\mathbb{C}}^{3}$ and $\pi_{L}$ is just
an automorphism of $\mathbb{P}_{\mathbb{C}}^{3}$. It follows from Diagram
(3.2) that $\phi$ is birational if and only if both $\zeta_{W}$ and the
restriction of $\pi_{L}$ to $Y_{W}$ are birational. On the other hand, given a
point

(3.3)
$Q=(\alpha_{0}:\alpha_{1})\times(\beta_{0}:\beta_{1})\times(\gamma_{0}:\gamma_{1})\in
X$

we define the polynomials in $R$

$\Delta_{1}^{Q}\coloneqq\begin{vmatrix}x_{0}&x_{1}\\\
\alpha_{0}&\alpha_{1}\end{vmatrix}\ ,\
\Delta_{2}^{Q}\coloneqq\begin{vmatrix}y_{0}&y_{1}\\\
\beta_{0}&\beta_{1}\end{vmatrix}\ ,\
\Delta_{3}^{Q}\coloneqq\begin{vmatrix}z_{0}&z_{1}\\\
\gamma_{0}&\gamma_{1}\end{vmatrix}\ .$

Before making specific choices for the vector space $W$, we first prove the
following result.

######  Lemma 3.1.

Let $\phi$ be birational of type $(2,2,2)$. Then, there exists a unique point
$Q$, as in (3.3), such that for every $0\leq l\leq 3$ the entries of $\phi$
satisfy

(3.4) $f_{l}\in\left(\lambda\ \Delta_{1}^{Q}\,y_{j}\,z_{k}+\mu\
x_{i}\,\Delta_{2}^{Q}\,z_{k}+\nu\
x_{i}\,y_{j}\,\Delta_{3}^{Q}\right)+\left(\Delta_{1}^{Q},\Delta_{2}^{Q},\Delta_{3}^{Q}\right)^{2}$

for some $\lambda,\mu,\nu\in\mathbb{C}^{*}$ and indices $0\leq i,j,k\leq 1$
where $\alpha_{i},\beta_{j},\gamma_{k}$ are all non-zero.

###### Proof.

Consider the Segre embedding
$\sigma:X\hookrightarrow\Sigma\subset\mathbb{P}_{\mathbb{C}}^{7}$ of $X$,
where the Segre variety $\Sigma$ is smooth of degree 6. From this isomorphism,
it follows that the Hilbert series of $\Sigma$ is

$\mathrm{HS}_{\Sigma}(t)=\sum_{d\geq
0}\dim_{\mathbb{C}}\Gamma(X,\mathcal{O}_{X}(d,d,d))\ t^{d}=\sum_{d\geq
0}(d+1)^{3}\,t^{d}=\frac{1+4t+t^{2}}{(1-t)^{4}}\ ,$

where $(1-t)^{4}\,\mathrm{HS}_{\Sigma}(t)$ is a polynomial because
$\dim\Sigma=3$. On the other hand, given a general line
$\ell\subset\mathbb{P}_{\mathbb{C}}^{3}$ the image $\phi^{-1}(\ell)$ is a
closed, irreducible curve in $X$. Moreover, if
$(\sum_{i=0}^{3}\alpha_{i}\,t_{i},\sum_{i=0}^{3}\beta_{i}\,t_{i})$ is the
defining ideal of $\ell$, then the ideal

$I^{\prime}=\left(\sum_{i=0}^{3}\alpha_{i}\,f_{i},\sum_{i=0}^{3}\beta_{i}\,f_{i}\right)$

defines $\phi^{-1}(\ell)$. More specifically, $I^{\prime}$ is the ideal of the
scheme-theoretic union of $\phi^{-1}(\ell)$ and the base locus $Z_{\phi}$. As
$\text{codim}\,I^{\prime}=2$ and $R$ is a Cohen-Macaulay ring, all the
associated primes to $I^{\prime}$ have codimension two. However we have $\dim
Z_{\phi}=0$, so $\text{codim}\,B_{\phi}=3$. Therefore, $I^{\prime}$ must be
the defining ideal of $\phi^{-1}(\ell)$. In particular, the curve
$\sigma(\phi^{-1}(\ell))$ is determined by two hyperplane sections in
$\mathbb{P}_{\mathbb{C}}^{7}$, and its Hilbert series [15, Chapter 5] is then

$\mathrm{HS}_{\sigma(\phi^{-1}(\ell))}(t)=\mathrm{HS}_{\phi^{-1}(\ell)}(t)=(1-t)^{2}\,\mathrm{HS}_{\Sigma}(t)=\frac{1+4t+t^{2}}{(1-t)^{2}}\
,$

implying that the Hilbert polynomial of $\phi^{-1}(\ell)$ is
$\mathrm{HP}_{\phi^{-1}(\ell)}(t)=6\,t$. In particular, the arithmetic genus
of $\phi^{-1}(\ell)$ is one. By definition $\phi^{-1}(\ell)$ is a rational
curve and its normalization is $\mathbb{P}_{\mathbb{C}}^{1}$. Then, from [16,
Chapter 4, Exercise 1.8] $\phi^{-1}(\ell)$ has a unique singular point $Q$. As
$\Sigma$ is smooth, by the strong Bertini’s theorem [12, Theorem 0.5] the
curve $\phi^{-1}(\ell)$ is smooth outside $Z_{\phi}$. Therefore, $Q$ is
supported in $Z_{\phi}$. Without loss of generality, we can assume that $Q$
belongs to the affine subset $U\subset X$ given by $x_{0}\not=0,$
$y_{0}\not=0,$ and $z_{0}\not=0$, i.e. we can assume indices $(i,j,k)=(0,0,0)$
in the statement. From the Jacobian criterion [16, §$1$ Exercise $5.8$] in
$U$, as $Q$ is a singular point of $\phi^{-1}(\ell)$ the rank of the Jacobian
matrix

$\displaystyle\text{Jac}_{\phi^{-1}(\ell)}=\begin{pmatrix}\sum_{l=0}^{3}\alpha_{l}\,\partial
f_{l}/\partial x_{1}&\sum_{l=0}^{3}\alpha_{l}\,\partial f_{l}/\partial
y_{1}&\sum_{l=0}^{3}\alpha_{l}\,\partial f_{l}/\partial z_{1}\\\
\sum_{l=0}^{3}\beta_{l}\,\partial f_{l}/\partial
x_{1}&\sum_{l=0}^{3}\beta_{l}\,\partial f_{l}/\partial
y_{1}&\sum_{l=0}^{3}\beta_{l}\,\partial f_{l}/\partial z_{1}\end{pmatrix}$

evaluated at $Q$ is one. Equivalently, writing $\nabla f_{l}=(\partial
f_{l}/\partial x_{1}\,,\,\partial f_{l}/\partial y_{1}\,,\,\partial
f_{l}/\partial z_{1})$ then

$\sum_{l=0}^{3}\alpha_{l}\,\nabla f_{l}(Q)\ ,\ \sum_{l=0}^{3}\beta_{l}\,\nabla
f_{l}(Q)$

must be proportional vectors. This is true for any choice of a general
$\ell\subset\mathbb{P}_{\mathbb{C}}^{3}$, i.e. for general values of the
$\alpha_{l}$’s and $\beta_{l}$’s. Therefore, it can only occur if the $\nabla
f_{l}(Q)$’s are all proportional to a vector
$(\lambda,\mu,\nu)\in\mathbb{C}^{3}$. In particular, from the Taylor expansion
of $f_{l}$ centered at $Q$ we can write

$f_{l}=\delta_{l}\,(\lambda\,\Delta_{1}^{Q}\,y_{0}\,z_{0}+\mu\,x_{0}\,\Delta_{2}^{Q}\,z_{0}+\nu\,x_{0}\,y_{0}\,\Delta_{3}^{Q})+h_{l}\
,$

for some $\delta_{l}\in\mathbb{C}$ and
$h_{l}\in(\Delta_{1}^{Q},\Delta_{2}^{Q},\Delta_{3}^{Q})^{2}$. Moreover, if
$\lambda=0$ (resp. $\mu=0$ or $\nu=0$) we find a line in the base locus
$Z_{\phi}$, against $\dim Z_{\phi}=0$. Hence, $\lambda,\mu,\nu$ are all non-
zero. ∎

######  Remark 3.2.

Recall that a point $Q$ in the base locus $Z_{\phi}$ is a point of contact to
a surface $S\subset X$ smooth at $Q$ if $B_{\phi}\subset I(S)+I(Q)^{2}$.
Geometrically, if $\phi$ is birational of type $(2,2,2)$, Lemma 3.1
establishes that every surface in the linear system spanned by the $f_{l}$’s
has $Q$ as a point of contact to the surface given by

$\lambda\ \Delta_{1}^{Q}\,y_{j}\,z_{k}+\mu\ x_{i}\,\Delta_{2}^{Q}\,z_{k}+\nu\
x_{i}\,y_{j}\,\Delta_{3}^{Q}=0\ .$

######  Setup 3.3.

Given a point $Q$ as in (3.3) and $\lambda,\mu,\nu\in\mathbb{C}^{*}$, we
define the ideal in $R$

$I_{Q(\lambda,\mu,\nu)}\coloneqq\left(\lambda\
\Delta_{1}^{Q}\,y_{j}\,z_{k}+\mu\ x_{i}\,\Delta_{2}^{Q}\,z_{k}+\nu\
x_{i}\,y_{j}\,\Delta_{3}^{Q}\right)+\left(\Delta_{1}^{Q}\,,\,\Delta_{2}^{Q}\,,\,\Delta_{3}^{Q}\right)^{2}$

for the smallest indices $0\leq i,j,k\leq 1$ such that
$\alpha_{i},\beta_{j},\gamma_{k}$ are all non-zero. The graded component of
$I_{Q(\lambda,\mu,\nu)}$ in tri-degree $(1,1,1)$ is generated by the five
independent polynomials

$\displaystyle\lambda\ \Delta_{1}^{Q}\,y_{j}\,z_{k}+\mu\
x_{i}\,\Delta_{2}^{Q}\,z_{k}+\nu\ x_{i}\,y_{j}\,\Delta_{3}^{Q}\ ,$
$\displaystyle\Delta_{1}^{Q}\,\Delta_{2}^{Q}\,z_{k}\ ,\
\Delta_{1}^{Q}\,y_{j}\,\Delta_{3}^{Q}\ ,\
x_{i}\,\Delta_{2}^{Q}\,\Delta_{3}^{Q}\ ,\
\Delta_{1}^{Q}\,\Delta_{2}^{Q}\,\Delta_{3}^{Q}\ .$

In particular, there is always an automorphism of $R$ preserving the tri-
graded structure (equivalently, an automorphism of $X$) that sends
$I_{Q(\lambda,\mu,\nu)}$ into the ideal $I_{Q^{\prime}(1,1,1)}$, where
$Q^{\prime}=(1:0)^{3}$.

The following is a useful description of the Cohen-Macaulay curves in $X$ of
the tri-degrees of interest for the later analysis.

######  Lemma 3.4.

Let $C$ be a Cohen-Macaulay curve in $X$. Then, we have the following:

  * •

If $C$ has tri-degree $(1,1,1)$ there is an automorphism of $X$ sending it to
the curve defined by one of the following ideals:  (1)
$(x_{0}y_{1}-x_{1}y_{0},x_{0}z_{1}-x_{1}z_{0},y_{0}z_{1}-y_{1}z_{0})$ (2)
$(x_{0}y_{1}-x_{1}y_{0},z_{1})\cap(x_{1},y_{0})$ (3)
$(x_{0}y_{1}-x_{1}y_{0},z_{1})\cap(x_{1},y_{1})$ (4)
$(x_{1},y_{1})\cap(x_{0},z_{1})\cap(y_{0},z_{0})$ (5)
$(x_{1},y_{1})\cap(x_{1},z_{1})\cap(y_{0},z_{0})$ (6)
$(x_{1},y_{1})\cap(x_{1},z_{1})\cap(y_{1},z_{0})$ (7)
$(x_{1},y_{1})\cap(x_{1},z_{1})\cap(y_{1},z_{1})$

  * •

If $C$ has tri-degree either $(1,1,0)$, $(1,0,1)$, or $(0,1,1)$ there is an
automorphism of $X$ sending it to the curve defined by one of the following
ideals:  (8) $(x_{0}y_{1}-x_{1}y_{0},z_{1})$ (9)
$(x_{1},z_{0})\cap(y_{1},z_{1})$ (10) $(x_{1},z_{1})\cap(y_{1},z_{1})$

  * •

If $C$ has tri-degree either $(1,0,0)$, $(0,1,0)$, or $(0,0,1)$ it is a
projective line, and it can be transformed by an automorphism of $X$ into the
line of ideal $(y_{1},z_{1})$.

###### Proof.

Up to an automorphism of $X$ that simply permutes the factors (see §2.5), we
can assume that $C$ has tri-degree either $(1,1,1)$, $(1,1,0)$, or $(1,0,0)$.
In the latter case, the projection
$\pi_{1}:C\xrightarrow{}\mathbb{P}_{\mathbb{C}}^{1}$ onto the first factor of
$X$ is an isomorphism, so $C$ is a projective line. In particular, there is an
automorphism of $X$ sending $C$ to the line $(y_{1},z_{1})$. Now, we assume
that $C$ is irreducible of tri-degree either $(1,1,1)$ or $(1,1,0)$. The
projections $\pi_{i}:X\xrightarrow{}\mathbb{P}_{\mathbb{C}}^{1}$ onto the
first two factors of $X$ have degree one, so $C$ is rational. Hence, $C$ is
the image of a regular map
$\varphi:\mathbb{P}_{\mathbb{C}}^{1}\xrightarrow{}C$ given by

(3.5)
$(t_{0}:t_{1})\mapsto(u_{0}(t_{0},t_{1}):u_{1}(t_{0},t_{1}))\times(v_{0}(t_{0},t_{1}):v_{1}(t_{0},t_{1}))\times(w_{0}(t_{0},t_{1}):w_{1}(t_{0},t_{1}))\
,$

for some linear forms $u_{i},v_{j},w_{k}\in\mathbb{C}[t_{0},t_{1}]$ (if $C$
has tri-degree $(1,1,0)$, the $w_{k}$’s are proportional and we have
$(w_{0}(t_{0},t_{1}):w_{1}(t_{0},t_{1}))=(\gamma_{0}:\gamma_{1})$ for some
$(\gamma_{0}:\gamma_{1})\in\mathbb{P}_{\mathbb{C}}^{1}$). Therefore, any two
parametrizations of curves of the same tri-degree coincide up to the action of
$\mathrm{PGL}(2,\mathbb{C})^{3}\leq\mathrm{Aut}(X)$. In particular, if $C$ has
tri-degree $(1,1,1)$ (resp$.\,(1,1,0)$) there is an automorphism of $X$
sending it to the curve defined by the ideal $(1)$ (resp$.\,(8)$) in the
statement. Now, assume that $C$ has exactly two irreducible components of tri-
degrees $(1,1,0)$ and $(0,0,1)$. By the argument above, the component of tri-
degree $(1,1,0)$ can be transformed into the curve $D\subset X$ defined by the
ideal $(x_{0}y_{1}-x_{1}y_{0},z_{1})$. Moreover, the component of tri-degree
$(0,0,1)$ can be parametrized as

$(t_{0}:t_{1})\mapsto(\alpha_{0}:\alpha_{1})\times(\beta_{0}:\beta_{1})\times(w_{0}^{\prime}(t_{0},t_{1}):w_{1}^{\prime}(t_{0},t_{1}))\
,$

for some
$(\alpha_{0}:\alpha_{1})\times(\beta_{0}:\beta_{1})\in\mathbb{P}_{\mathbb{C}}^{1}\times\mathbb{P}_{\mathbb{C}}^{1}$
and linear forms $w_{0}^{\prime},w_{1}^{\prime}$ in $\mathbb{C}[t_{0},t_{1}]$.
In particular, if
$(\alpha_{0}:\alpha_{1})\times(\beta_{0}:\beta_{1})\in\mathbb{P}_{\mathbb{C}}^{1}\times\mathbb{P}_{\mathbb{C}}^{1}$
satisfies $\alpha_{0}\beta_{1}-\alpha_{1}\beta_{0}=0$ we find an automorphism
of $X$ that stabilizes $D$ and sends

$(\alpha_{0}:\alpha_{1})\times(\beta_{0}:\beta_{1})\times(w_{0}^{\prime}(t_{0},t_{1}):w_{1}^{\prime}(t_{0},t_{1}))\mapsto(1:0)\times(1:0)\times(w_{0}^{\prime}(t_{0},t_{1}):w_{1}^{\prime}(t_{0},t_{1}))\
.$

On the other hand, if $\alpha_{0}\beta_{1}-\alpha_{1}\beta_{0}\not=0$ then we
find an automorphism sending

$(\alpha_{0}:\alpha_{1})\times(\beta_{0}:\beta_{1})\times(w_{0}^{\prime}(t_{0},t_{1}):w_{1}^{\prime}(t_{0},t_{1}))\mapsto(1:0)\times(0:1)\times(w_{0}^{\prime}(t_{0},t_{1}):w_{1}^{\prime}(t_{0},t_{1}))$

that stabilizes $D$. The images of the two last parametrizations are the
projective lines in $X$ respectively defined by $(x_{1},y_{1})$ and
$(x_{1},y_{0})$. Therefore, $C$ can be transformed into the curve defined by
either the ideal $(3)$ or $(2)$ in the statement by means of an automorphism
of $X$. Proceeding similarly with curves of tri-degree $(1,1,1)$ with three
irreducible components, one derives the curves from $(4)$ to $(7)$. On the
other hand, if $C$ is reducible of tri-degree $(1,1,0)$ it must have exactly
two irreducible components of tri-degrees $(1,0,0)$ and $(0,1,0)$. Depending
on whether these components intersect, $C$ can be transformed into the curve
defined by either $(9)$ or $(10)$. ∎

Recall from §2.2 that if $\dim Z_{\phi}=1$ then $I_{\phi}$ stands for the
intersection of all the associated primes to $B_{\phi}$ in $R$ of codimension
exactly two. On the other hand, if $\dim Z_{\phi}=0$ we define
$I_{\phi}\coloneqq I_{Q(\lambda,\mu,\nu)}$ where $Q\in X$ and
$\lambda,\mu,\nu\in\mathbb{C}^{*}$ are as in the statement of Lemma 3.1.
Additionally, in all the cases we define
$W_{\phi}\coloneqq(I_{\phi})_{(1,1,1)}$. In particular, if $W=W_{\phi}$ then
(3.1) is satisfied. Moreover, vectors in $W_{\phi}$ are identified to surfaces
in the linear system that the $f_{i}$’s define in $X$.

The following result provides useful information about Diagram (3.2),
according to the type of $\phi$. Recall that a subvariety
$V\subset\mathbb{P}_{\mathbb{C}}^{N}$ has minimal degree if it is non-
degenerate and $\deg V=1+\text{codim }V$.

######  Proposition 3.5.

Assume that $\phi$ is birational, and set $W=W_{\phi}$. We have the following:

  * •

If $\phi$ has type $(1,1,1)$, then $N=3$ and
$Y_{W}\cong\mathbb{P}_{\mathbb{C}}^{3}$.

  * •

If $\phi$ has type $(1,1,2)$, then $N=4$ and $\deg Y_{W}=2$.

  * •

If $\phi$ has type $(1,2,2)$, then $N=5$ and $\deg Y_{W}=3$.

  * •

If $\phi$ has type $(2,2,2)$, then $N=4$ and $\deg Y_{W}=2$.

In all the cases, $Y_{W}$ has minimal degree and $\zeta_{W}$ is birational for
any choice of basis in $W^{\vee}$. Moreover, if $\dim Z_{\phi}=1$ the curve
$C_{\phi}$ is connected.

###### Proof.

We prove each statement separately:

  * •

If $\phi$ has type $(1,1,1)$, the base locus $Z_{\phi}$ has tri-degree
$(1,1,1)$. Therefore, up to an automorphism of $X$ the curve $C_{\phi}$ is one
of the seven listed in Lemma 3.4. The ideals $(1),(3),(6)$, and $(7)$
determine connected curves, and satisfy $\dim_{\mathbb{C}}W=4$. On the other
hand, the ideals $(2),(4),(5)$ yield $\dim_{\mathbb{C}}W=3$. As the entries
$f_{l}$’s of $\phi$ are independent and they belong to $W$, we must have
$\dim_{\mathbb{C}}W\geq 4$. Hence, $\dim_{\mathbb{C}}W=4$ and $C_{\phi}$ is a
connected curve.

  * •

If $\phi$ has type $(1,1,2)$, the base locus $Z_{\phi}$ has tri-degree
$(1,1,0)$. Therefore, up to an automorphism of $X$ the curve $C_{\phi}$ is one
of the three listed in Lemma 3.4. The ideals $(8)$ and $(10)$ determine
connected curves and satisfy $\dim_{\mathbb{C}}W=5$. On the other hand, the
ideal $(9)$ defines two skew lines. Moreover, the ideal generated by the
polynomials of tri-degree $(1,1,1)$ in $(9)$ defines a curve of tri-degree
$(1,1,1)$ in $X$. From (3.1), it follows that if $C_{\phi}$ is equivalent to
$(9)$ then $Z_{\phi}$ has tri-degree $(1,1,1)$. Therefore, we must have
$\dim_{\mathbb{C}}W=5$ and $C_{\phi}$ is again a connected curve. The degree
of $Y_{W}$ follows from a direct computation (performed with the help of
Macaulay2) with the ideals $(8)$ and $(10)$.

  * •

If $\phi$ has type $(1,2,2)$, the base locus $Z_{\phi}$ has tri-degree
$(1,0,0)$. By Lemma 3.4 the curve $C_{\phi}$ is a projective line, so it is
connected and $\dim_{\mathbb{C}}W=6$. Similarly, the degree of $Y_{W}$ follows
from a direct computation on the ideal $(y_{0},z_{0})$.

  * •

If $\phi$ has type $(2,2,2)$, it is immediate from Setup 3.3 that
$\dim_{\mathbb{C}}W=5$. Similarly, it follows from a direct computation that
$\deg Y_{W}=2$.

On the other hand, two distinct basis in $W^{\vee}$ yield rational maps
$\zeta_{W}$ that coincide up to an automorphism of
$\mathbb{P}_{\mathbb{C}}^{N}$. The birationality of $\zeta_{W}$ then follows
from the explicit construction of this rational map when $W$ is the component
in tri-degree $(1,1,1)$ of the ideals listed above. ∎

##  4\. The algebraic set of tri-linear birational maps

We now describe the algebraic set $\mathfrak{Bir}_{(1,1,1)}$ of tri-linear
birational maps after quotient by the equivalence relation given by
composition with an automorphism of $\mathbb{P}_{\mathbb{C}}^{3}$.
Additionally, we introduce the group action of $\mathrm{Aut}(X)$ on
$\mathfrak{Bir}_{(1,1,1)}$.

###  4.1. $\mathfrak{Bir}_{(1,1,1)}$ and $\mathrm{Bir}_{(r_{1},r_{2},r_{3})}$

We write $\mathfrak{Rat}_{(1,1,1)}$ for the set of ordered tuples of four tri-
linear polynomials in the variables of $X$ without a common factor, up to non-
zero scalar. The elements in $\mathfrak{Rat}_{(1,1,1)}$ are identified with
tri-linear rational maps. In this set, we introduce the equivalence relation
$\sim$ given by setting $\phi\sim\phi^{\prime}$ if
$W_{\phi}=W_{\phi^{\prime}}$. Equivalently, $\phi\sim\phi^{\prime}$ if there
is $\rho\in\mathrm{Aut}(\mathbb{P}_{\mathbb{C}}^{3})$ such that
$\phi^{\prime}=\rho\circ\phi$. We define $\mathfrak{Bir}_{(1,1,1)}$ as the
quotient by $\sim$ of the subset of $\mathfrak{Rat}_{(1,1,1)}$ given by the
tuples associated to birational maps. In particular, if $V_{(1,1,1)}$ is the
$\mathbb{C}$-vector space of tri-linear polynomials in the variables of $X$,
there is an injective map of sets from $\mathfrak{Bir}_{(1,1,1)}$ to the
Grassmannian $\mathbb{G}(4,V_{(1,1,1)})$ given by $[\phi]\mapsto W_{\phi}$.
Therefore, we can write $W_{\phi}$ instead of $[\phi]$. Clearly, two
birational maps $\phi$ and $\phi^{\prime}$ in the same class of
$\mathfrak{Bir}_{(1,1,1)}$ have the same type. Thus, given $1\leq
r_{1},r_{2},r_{3}\leq 2$ it makes sense to define the sets

$\mathrm{Bir}_{(r_{1},r_{2},r_{3})}\coloneqq\\{W_{\phi}\in\mathfrak{Bir}_{(1,1,1)}:\,\phi\text{
has type }(r_{1},r_{2},r_{3})\\}\ .$

From Table 1, these subsets form a partition of $\mathfrak{Bir}_{(1,1,1)}$.

###  4.2. The group action of $\mathrm{Aut}(X)$

The map

(4.1) $\displaystyle\mathrm{Aut}(X)\times\mathfrak{Bir}_{(1,1,1)}$
$\displaystyle\xrightarrow{}$ $\displaystyle\mathfrak{Bir}_{(1,1,1)}$
$\displaystyle\xi\times W_{\phi}$ $\displaystyle\mapsto$ $\displaystyle
W_{\phi\circ\xi}$

is a well-defined group action of $\mathrm{Aut}(X)$ on
$\mathfrak{Bir}_{(1,1,1)}$, that we call right-action. Namely, given
birational maps $\phi$ and $\phi^{\prime}$, the vector spaces $W_{\phi}$ and
$W_{\phi^{\prime}}$ belong to the same orbit of the right-action if and only
if there are $\rho\in\mathrm{Aut}(\mathbb{P}_{\mathbb{C}}^{3})$ and
$\xi\in\mathrm{Aut}(X)$ such that $\phi^{\prime}=\rho\circ\phi\circ\xi$. With
a small abuse in the notation, we also say that $\phi$ and $\phi^{\prime}$
belong to the same orbit. In particular, the base loci of two birational maps
in the same orbit coincide up to automorphism of $X$.

###  4.3. The algebraic set $\mathfrak{Bir}_{(1,1,1)}$

We write $\mathscr{S}(Y_{W})$ for the variety of $(N-4)$-dimensional subspaces
secant to $Y_{W}$. Now, we prove the main result of this section.

######  Theorem 4.1.

Let $V_{(1,1,1)}$ be the $\mathbb{C}$-vector space of tri-linear polynomials
in the variables of $X$. Then, the set $\mathfrak{Bir}_{(1,1,1)}$ has the
structure of a locally closed algebraic subset of the Grassmannian
$\mathbb{G}(4,V_{(1,1,1)})$. Moreover, its irreducible components are the
following:

  * •

$\mathrm{Bir}_{(1,1,1)}$, of dimension 6.

  * •

$\mathrm{Bir}_{(1,1,2)}$, $\mathrm{Bir}_{(1,2,1)}$, and
$\mathrm{Bir}_{(2,1,1)}$, all of dimension $7$.

  * •

$\mathrm{Bir}_{(1,2,2)}$, $\mathrm{Bir}_{(2,1,2)}$, and
$\mathrm{Bir}_{(2,2,1)}$, all of dimension $8$.

  * •

$\mathrm{Bir}_{(2,2,2)}$, of dimension $8$.

###### Proof.

The birational map

(4.2) $\displaystyle\Phi:X$ $\displaystyle\dasharrow$
$\displaystyle\mathbb{P}_{\mathbb{C}}^{3}$
$\displaystyle(x_{0}:x_{1})\times(y_{0}:y_{1})\times(z_{0}:z_{1})$
$\displaystyle\mapsto$
$\displaystyle(x_{0}y_{0}z_{0}:x_{1}y_{0}z_{0}:x_{0}y_{1}z_{0}:x_{0}y_{0}z_{1})$

is an isomorphism between the affine subset $U\subset X$ determined by
$x_{0}\not=0,y_{0}\not=0,z_{0}\not=0$ and the affine subset of
$\mathbb{P}_{\mathbb{C}}^{3}=\text{Proj
}(\mathbb{C}[t_{0},t_{1},t_{2},t_{3}])$ given by $t_{0}\not=0$. Write $V_{3}$
for the $\mathbb{C}$-vector space of cubic forms in $t_{0}$,$t_{1}$,$t_{2}$,
and $t_{3}$. We have the injective linear map
$\chi:V_{(1,1,1)}\xrightarrow{}V_{3}$ given by

$\displaystyle x_{0}y_{0}z_{0}\mapsto t_{0}^{3}\ ,\ x_{1}y_{0}z_{0}\mapsto
t_{0}^{2}t_{1}\ ,\ x_{0}y_{1}z_{0}\mapsto t_{0}^{2}t_{2}\ ,\
x_{0}y_{0}z_{1}\mapsto t_{0}^{2}t_{3}\ ,$ $\displaystyle
x_{1}y_{1}z_{0}\mapsto t_{0}t_{1}t_{2}\ ,\ x_{1}y_{0}z_{1}\mapsto
t_{0}t_{1}t_{3}\ ,\ x_{0}y_{1}z_{1}\mapsto t_{0}t_{2}t_{3}\ ,\
x_{1}y_{1}z_{1}\mapsto t_{1}t_{2}t_{3}\ .$

In particular, the rational endomorphism
$\psi:\mathbb{P}_{\mathbb{C}}^{3}\dasharrow\mathbb{P}_{\mathbb{C}}^{3}$ that
sends

$(t_{0}:t_{1}:t_{2}:t_{3})\mapsto(\chi(f_{0}):\chi(f_{1}):\chi(f_{2}):\chi(f_{3}))$

coincides with $\phi\circ\Phi^{-1}$ on the open set $\Phi(U)$. Therefore, the
association $\phi\mapsto\phi\circ\Phi^{-1}$ is an identification between tri-
linear rational maps and rational endomorphisms of
$\mathbb{P}_{\mathbb{C}}^{3}$ whose entries belong to the vector subspace

$V^{\prime}=\langle
t_{0}^{3}\,,\,t_{0}^{2}t_{1}\,,\,t_{0}^{2}t_{2}\,,\,t_{0}^{2}t_{3}\,,\,t_{0}t_{1}t_{2}\,,\,t_{0}t_{1}t_{3}\,,\,t_{0}t_{2}t_{3}\,,\,t_{1}t_{2}t_{3}\rangle\subset
V_{3}\ .$

By [18, Proposition B], the classes (up to composition with an automorphism of
$\mathbb{P}_{\mathbb{C}}^{3}$) of birational automorphisms of
$\mathbb{P}_{\mathbb{C}}^{3}$ with cubic entries form a locally closed
algebraic set $Y^{\prime}\subset\mathbb{G}(4,V_{3})$. On the other hand, the
Grassmannian $\mathbb{G}(4,V^{\prime})$ is a subvariety of
$\mathbb{G}(4,V_{3})$. Hence, the intersection

$\mathfrak{Bir}_{(1,1,1)}=Y^{\prime}\cap\mathbb{G}(4,V^{\prime})$

is also a locally closed algebraic subset of the Grassmannian
$\mathbb{G}(4,V^{\prime})\cong\mathbb{G}(4,V_{(1,1,1)})$.

Now, we prove the statements about the irreducible components of
$\mathfrak{Bir}_{(1,1,1)}$. Recall from Proposition 3.5 that if $\dim
Z_{\phi}=1$ the Cohen-Macaulay curve $C_{\phi}$ is connected. Let
$\mathscr{C}_{(1,1,1)}$ be the variety of connected Cohen-Macaulay curves of
tri-degree $(1,1,1)$, which is irreducible of dimension 6 [2, Proposition
4.9]. Additionally, consider the algebraic set

$\mathscr{V}_{(1,1,1)}=\\{W_{\phi}\times
C\in\mathfrak{Bir}_{(1,1,1)}\times\mathscr{C}_{(1,1,1)}:W_{\phi}\subset{(I_{C})}_{(1,1,1)}\\}\
,$

together with the canonical projections
$\Pi_{1}:\mathscr{V}_{(1,1,1)}\xrightarrow{}\mathfrak{Bir}_{(1,1,1)}$ and
$\Pi_{2}:\mathscr{V}_{(1,1,1)}\xrightarrow{}\mathscr{C}_{(1,1,1)}$. Given a
curve $C\in\mathscr{C}_{(1,1,1)}$ we have $\Pi_{2}^{-1}(C)=W\times C$, where
$W={(I_{C})}_{(1,1,1)}$ as $\dim_{\mathbb{C}}W=4$ by Proposition 3.5. Hence,
$\mathscr{V}_{(1,1,1)}$ is irreducible of dimension $6$. On the other hand,
given $W_{\phi}\in\mathfrak{Bir}_{(1,1,1)}$ we have
$\Pi_{1}^{-1}(W_{\phi})=W_{\phi}\times C_{\phi}$ if $\phi$ has type $(1,1,1)$
and $\Pi_{1}^{-1}(W_{\phi})=\varnothing$ otherwise. Therefore, it follows that
$\mathrm{Bir}_{(1,1,1)}\cong\mathscr{C}_{(1,1,1)}$. The study of the
irreducible components $\mathrm{Bir}_{(1,1,2)}$, $\mathrm{Bir}_{(1,2,1)}$, and
$\mathrm{Bir}_{(2,1,1)}$ is identical, as the associated birational maps
differ by a permutation of the factors of $X$. Therefore, we can focus on
$\mathrm{Bir}_{(1,1,2)}$. By Lemma 3.4, the ideal of a connected Cohen-
Macaulay curve of tri-degree $(1,1,0)$ is of the form

$(\alpha_{0}\,x_{0}y_{0}+\alpha_{1}\,x_{1}y_{0}+\alpha_{2}\,x_{0}y_{1}+\alpha_{3}\,x_{1}y_{1},\gamma_{0}z_{1}-\gamma_{1}z_{0})\
,$

for some
$(\alpha_{0}:\alpha_{1}:\alpha_{2}:\alpha_{3})\times(\gamma_{0}:\gamma_{1})\in\mathbb{P}_{\mathbb{C}}^{3}\times\mathbb{P}_{\mathbb{C}}^{1}$,
where the first generator might be reducible. Thus, we can identify the
variety $\mathscr{C}_{(1,1,0)}$ of such curves with
$\mathbb{P}_{\mathbb{C}}^{3}\times\mathbb{P}_{\mathbb{C}}^{1}$. Consider now
the algebraic set

$\mathscr{V}_{(1,1,2)}=\\{W_{\phi}\times
C\in\mathfrak{Bir}_{(1,1,1)}\times\mathscr{C}_{(1,1,0)}:W_{\phi}\subset{(I_{C})}_{(1,1,1)}\\}\
,$

together with the projections as before. Given a curve
$C\in\mathscr{C}_{(1,1,0)}$, by [18, Proposition A] the fiber
$\Pi_{2}^{-1}(C)$ is isomorphic to an open subset of $\mathscr{S}(Y_{W})$,
where $W={(I_{C})}_{(1,1,1)}$. Hence, $\mathscr{V}_{(1,1,2)}$ is irreducible
of dimension $\dim\mathscr{C}_{(1,1,0)}+\dim\mathscr{S}(Y_{W})=4+3=7$. On the
other hand, we have the identity of sets

$\mathrm{Bir}_{(1,1,2)}=\Pi_{1}(\mathscr{V}_{(1,1,2)})\,\backslash\,\mathrm{Bir}_{(1,1,1)}\
,$

so given a general $W_{\phi}\in\Pi_{1}(\mathscr{V}_{(1,1,2)})$ the base locus
$Z_{\phi}$ has tri-degree $(1,1,0)$. Thus, we have
$\Pi_{1}^{-1}(W_{\phi})=(W_{\phi},C_{\phi})$ and $\Pi_{1}$ is birational,
implying that $\mathrm{Bir}_{(1,1,2)}$ is irreducible of dimension 7.
Similarly, one proves that $\mathrm{Bir}_{(1,2,2)}$, $\mathrm{Bir}_{(2,1,2)}$,
and $\mathrm{Bir}_{(2,2,1)}$ are all irreducible of dimension 8. For the
irreducible component $\mathrm{Bir}_{(2,2,2)}$, we consider the algebraic set

$\mathscr{V}_{(2,2,2)}=\\{W_{\phi}\times
Q\times(\lambda,\mu)\in\mathfrak{Bir}_{(1,1,1)}\times
X\times(\mathbb{C}^{*})^{2}:W_{\phi}\subset{(I_{Q(\lambda,\mu,1)})}_{(1,1,1)}\\}\
,$

with the projections
$\Pi_{1}:\mathscr{V}_{(2,2,2)}\xrightarrow{}\mathfrak{Bir}_{(1,1,1)}$ and
$\Pi_{2}:\mathscr{V}_{(2,2,2)}\xrightarrow{}X\times(\mathbb{C}^{*})^{2}$. By
[18, Proposition A], given $Q\times(\lambda,\mu)\in
X\times(\mathbb{C}^{*})^{2}$ the fiber $\Pi_{2}^{-1}(Q\times(\lambda,\mu))$ is
isomorphic to an open subset of $\mathscr{S}(Y_{W})$, where
$W={(I_{Q(\mu,\nu,1)})}_{(1,1,1)}$. Hence, $\mathscr{V}_{(2,2,2)}$ is
irreducible of dimension

$\dim X+\dim(\mathbb{C}^{*})^{2}+\dim\mathscr{S}(Y_{W})=3+2+3=8\ .$

On the other hand, given a general $W_{\phi}\in\Pi_{1}(\mathscr{V}_{(2,2,2)})$
we have $\Pi_{1}^{-1}(W_{\phi})=W_{\phi}\times Q\times(\lambda,\mu)$ for the
unique singular point $Q\in Z_{\phi}$ and some
$(\lambda,\mu)\in(\mathbb{C}^{*})^{2}$. In particular, $\Pi_{1}$ is
birational. Moreover, as a general point in $\Pi_{1}(\mathscr{V}_{(2,2,2)})$
belongs to $\mathrm{Bir}_{(2,2,2)}$ the latter is irreducible of dimension 8.
∎

##  5\. Geometric classification of tri-linear birational maps

In this section, we study the right-action on $\mathfrak{Bir}_{(1,1,1)}$ and
give the complete list of its orbits. More specifically we find exactly 19
orbits, and each of these determines an isomorphism class of the possible base
loci of a tri-linear birational map. Recall that a point $Q$ in the base locus
$Z_{\phi}$ is a point of contact to a surface $S\subset X$ smooth at $Q$ if
$B_{\phi}\subset I(S)+I(Q)^{2}$. Similarly, $Q$ is a point of tangency to a
curve $C\subset X$ smooth at $Q$ if $B_{\phi}\subset I(C)+I(Q)^{2}$. On the
other hand, when we refer to “a surface in $W_{\phi}$” we mean a surface in
$X$ whose equation belongs to the linear system $W_{\phi}$ spanned by the
entries of $\phi$.

###  5.1. Orbits of birational maps of type $(1,1,1)$

The classification of the orbits in $\mathrm{Bir}_{(1,1,1)}$ is the most
straightforward, as from Proposition 6.2 (that we postpone until §6) the base
locus is a Cohen-Macaulay curve. Namely, we have the following.

######  Theorem 5.1.

Let $\phi$ be birational of type $(1,1,1)$. Then, $\phi$ belongs to the orbit
of one of the following birational maps:

  * •

$\rho_{1}^{(1,1,1)}\equiv(x_{1}y_{0}z_{1}-x_{0}y_{1}z_{1}:x_{1}y_{1}z_{0}-x_{0}y_{1}z_{1}:x_{0}y_{1}z_{0}-x_{0}y_{0}z_{1}:x_{1}y_{0}z_{0}-x_{0}y_{0}z_{1})$
The entries of $\rho_{1}^{(1,1,1)}$ are given by the $3\times 3$ minors of the
matrix

$\begin{pmatrix}0&-y_{1}&-z_{1}\\\ -x_{1}&0&z_{1}\\\ x_{0}&y_{0}&0\\\
-x_{0}&0&z_{0}\end{pmatrix}\ .$

A surface in $W_{\rho_{1}}$ contains an irreducible curve of tri-degree
$(1,1,1)$. The base ideal is

$(x_{0}y_{1}-x_{1}y_{0},x_{0}z_{1}-x_{1}z_{0},y_{0}z_{1}-y_{1}z_{0})\ .$

  * •

$\rho_{2}^{(1,1,1)}\equiv(x_{1}y_{1}z_{1}:x_{0}y_{1}z_{1}:x_{1}y_{0}z_{1}:x_{1}y_{0}z_{0}-x_{0}y_{1}z_{0})$
The entries of $\rho_{2}^{(1,1,1)}$ are given by the $3\times 3$ minors of the
matrix

$\begin{pmatrix}0&0&-z_{1}\\\ 0&-y_{1}&z_{0}\\\ -x_{1}&0&-z_{0}\\\
x_{0}&y_{0}&0\end{pmatrix}\ ,$

A surface in $W_{\rho_{2}}$ contains an irreducible curve of tri-degree
$(1,1,0)$ and a line of tri-degree $(0,0,1)$, these two intersecting. The base
ideal is

$(x_{1}y_{0}-x_{0}y_{1},z_{1})\cap(x_{1},y_{1})\ .$

  * •

$\rho_{3}^{(1,1,1)}\equiv(x_{1}y_{1}z_{1}:x_{0}y_{1}z_{1}:x_{1}y_{1}z_{0}:x_{1}y_{0}z_{0})$
The entries of $\rho_{3}^{(1,1,1)}$ are given by the $3\times 3$ minors of the
matrix

$\begin{pmatrix}0&-y_{1}&0\\\ 0&y_{0}&-z_{1}\\\ -x_{1}&0&0\\\
x_{0}&0&z_{0}\end{pmatrix}\ .$

A surface in $W_{\rho_{3}}$ contains three lines of tri-degrees
$(1,0,0),(0,1,0),$ and $(0,0,1)$, one of them intersecting the other two at
distinct points. The base ideal is

$(x_{1},y_{1})\cap(x_{1},z_{1})\cap(y_{1},z_{0})\ .$

  * •

$\rho_{4}^{(1,1,1)}\equiv(x_{1}y_{1}z_{1}:x_{0}y_{1}z_{1}:x_{1}y_{0}z_{1}:x_{1}y_{1}z_{0})$
The entries of $\rho_{4}^{(1,1,1)}$ are given by the $3\times 3$ minors of the
matrix

$\begin{pmatrix}0&0&-z_{1}\\\ 0&-y_{1}&0\\\ -x_{1}&0&0\\\
x_{0}&y_{0}&z_{0}\end{pmatrix}\ .$

A surface in $W_{\rho_{4}}$ contains three lines of tri-degrees
$(1,0,0),(0,1,0),$ and $(0,0,1)$ that intersect at a common point. The base
ideal is

$(x_{1},y_{1})\cap(x_{1},z_{1})\cap(y_{1},z_{1})\ .$

###### Proof.

From Proposition 6.1, the base locus of $\phi$ is a connected Cohen-Macaulay
curve of tri-degree $(1,1,1)$, i.e. we have $Z_{\phi}=C_{\phi}$. Additionally,
from the proof of Proposition 3.5 given a connected Cohen-Macaulay curve $C$
of tri-degree $(1,1,1)$ we have $\dim_{\mathbb{C}}(I_{C})_{(1,1,1)}=4$. Hence,
any such curve determines a unique $W_{\phi}$ in $\mathfrak{Bir}_{(1,1,1)}$.
On the other hand, by Lemma 3.4 any connected Cohen-Macaulay curve of tri-
degree $(1,1,1)$ is equivalent, by means of an automorphism of $X$, to the
base loci of one of the birational maps in the statement. ∎

###  5.2. Orbits of birational maps of type $(1,1,2)$, $(1,2,1)$ and $(2,1,1)$

Let $C\subset X$ be a connected Cohen-Macaulay curve of tri-degree either
$(1,1,0)$, $(1,0,1)$, or $(0,1,1)$. From Lemma 3.4 and Proposition 3.5, by
means of an automorphism of $X$ the curve $C$ can be transformed into either
$C_{o}$ or $C_{\times}$, respectively defined by the ideals
$I_{C_{o}}=(x_{0}y_{1}-x_{1}y_{0},z_{1})$ and
$I_{C_{\times}}=(x_{1}y_{1},z_{1})$. In particular, any birational map of type
$(1,1,2)$, $(1,2,1)$, or $(2,1,1)$ lies in the orbit of a birational map with
either $C_{o}$ or $C_{\times}$ in its base locus. According to Diagram (3.2),
in order to study the orbits of the right-action in $\mathrm{Bir}_{(1,1,2)}$,
$\mathrm{Bir}_{(1,2,1)}$ and $\mathrm{Bir}_{(2,1,1)}$ it is sufficient to
compute the orbits of the group action given by the stabilizer
$\text{Stab}(C_{o})\leq\mathrm{Aut}(X)$ of the curve $C_{o}$ acting on
$Y_{W_{o}}$, where $W_{o}\coloneqq{(I_{C_{o}})}_{(1,1,1)}$, and the orbits of
the group action given by the stabilizer
$\text{Stab}(C_{\times})\leq\mathrm{Aut}(X)$ of the curve $C_{\times}$ acting
on $Y_{W_{\times}}$, where $W_{\times}\coloneqq{(I_{C_{\times}})}_{(1,1,1)}$.

####  5.2.1. Orbits in $Y_{W_{o}}$

In this subsection, we set $W=W_{o}$. The birational map $\zeta_{W}$ appearing
in Diagram (3.2) is

$\zeta_{W}:(x_{0}:x_{1})\times(y_{0}:y_{1})\times(z_{0}:z_{1})\mapsto(x_{0}y_{0}z_{1}:x_{1}y_{0}z_{1}:x_{0}y_{1}z_{1}:x_{1}y_{1}z_{1}:(x_{0}y_{1}-x_{1}y_{0})z_{0})\
,$

and $Y_{W}\subset\mathbb{P}_{\mathbb{C}}^{4}$ is the cone determined by
$t_{0}t_{3}-t_{1}t_{2}=0$. Equivalently, $Y_{W}$ is the image of the
projection $\Pi_{W}:\text{Bl}_{W}X\subset
X\times\mathbb{P}_{\mathbb{C}}^{4}\xrightarrow{}\mathbb{P}_{\mathbb{C}}^{4}$,
where $\pi_{W}:\text{Bl}_{W}X\xrightarrow{}X$ is the blow-up of $X$ along
$C_{o}$. The exceptional divisor $E_{W}$ of $\text{Bl}_{W}X$ projects onto a
two-dimensional cone in $Y_{W}$. Namely, the defining equation of
$\Pi_{W}(E_{W})$ in $Y_{W}$ is $t_{1}-t_{2}=0$, and its vertex is the point
$P=$ ($0\,$:$\,0\,$:$\,0\,$:$\,0\,$:$\,1$). We write $H_{1-2}$ and $H_{4}$ for
the divisors in $Y_{W}$ determined by $t_{1}-t_{2}=0$ and $t_{4}=0$,
respectively.

######  Lemma 5.2.

The group action of $\text{Stab}(C_{o})$ on $Y_{W}$ determines the four
orbits:

$Y_{W}\backslash H_{1-2}\ ,\ H_{1-2}\backslash(H_{4}\cup P)\ ,\ H_{1-2}\cap
H_{4}\ ,\ P$

###### Proof.

Write $D_{xy}$ and $D_{z}$ for the divisors in $X$ defined by
$x_{0}y_{1}-x_{1}y_{0}=0$ and $z_{1}=0$, respectively. The group action of
$\text{Stab}(C_{o})$ on $X$ determines the four orbits

(5.1) $C_{o}=D_{xy}\cap D_{z}\ ,\ D_{z}\backslash C_{o}\ ,\ D_{xy}\backslash
C_{o}\ ,\ X\backslash(D_{z}\cup D_{xy})\ .$

Namely, the action of an automorphism in $\text{Stab}(C_{o})$ on the point
$(x_{0}:x_{1})\times(y_{0}:y_{1})\times(z_{0}:z_{1})$ is explicitly given by
either

$\begin{pmatrix}\lambda_{00}&\lambda_{01}\\\
\lambda_{10}&\lambda_{11}\end{pmatrix}\begin{pmatrix}x_{0}\\\
x_{1}\end{pmatrix}\times\begin{pmatrix}\lambda_{00}&\lambda_{01}\\\
\lambda_{10}&\lambda_{11}\end{pmatrix}\begin{pmatrix}y_{0}\\\
y_{1}\end{pmatrix}\times\begin{pmatrix}\mu_{00}&\mu_{01}\\\
0&\mu_{11}\end{pmatrix}\begin{pmatrix}z_{0}\\\ z_{1}\end{pmatrix}$

or

$\begin{pmatrix}\lambda_{00}&\lambda_{01}\\\
\lambda_{10}&\lambda_{11}\end{pmatrix}\begin{pmatrix}y_{0}\\\
y_{1}\end{pmatrix}\times\begin{pmatrix}\lambda_{00}&\lambda_{01}\\\
\lambda_{10}&\lambda_{11}\end{pmatrix}\begin{pmatrix}x_{0}\\\
x_{1}\end{pmatrix}\times\begin{pmatrix}\mu_{00}&\mu_{01}\\\
0&\mu_{11}\end{pmatrix}\begin{pmatrix}z_{0}\\\ z_{1}\end{pmatrix}\ ,$

for some matrices $(\lambda_{ij})_{0\leq i,j\leq 1},\,(\mu_{ij})_{0\leq
i,j\leq 1}$ in $\mathrm{PGL}(2,\mathbb{C})$, depending on whether the tri-
degrees in $X$ are preserved or permuted. In particular, it follows that
$\text{Stab}(C_{o})$ acts transitively on each of the orbits in (5.1). On the
other hand, the last three orbits are respectively transformed by $\zeta_{W}$
into $P$, $H_{1-2}\cap H_{4}$, and $Y_{W}\backslash H_{1-2}$. Thus, we find
three of the orbits in the statement. The orbit $H_{1-2}\backslash(H_{4}\cup
P)$ corresponds to the projection by $\Pi_{W}$ of general orbit in $E_{W}$,
when we let $\text{Stab}(C_{o})$ act on $\text{Bl}_{W}X$. ∎

####  5.2.2. Orbits in $Y_{C_{\times}}$

In this subsection, we set $W=W_{\times}$. The birational map $\zeta_{W}$ in
Diagram (3.2) is

$\zeta_{W}:(x_{0}:x_{1})\times(y_{0}:y_{1})\times(z_{0}:z_{1})\mapsto(x_{0}y_{0}z_{1}:x_{1}y_{0}z_{1}:x_{0}y_{1}z_{1}:x_{1}y_{1}z_{1}:x_{1}y_{1}z_{0})\
.$

Once more, $Y_{W}$ is determined by $t_{0}t_{3}-t_{1}t_{2}=0$. However,
$\Pi_{W}(E_{W})$ is now the union of the loci $t_{1}=t_{3}=0$ and
$t_{2}=t_{3}=0$ in $\mathbb{P}_{\mathbb{C}}^{4}$. We maintain the notation of
§5.2.1, and write $H_{i}$ for the divisor in $Y_{W}$ defined by $t_{i}=0$.
Additionally, we set $O=$ ($1\,$:$\,0\,$:$\,0\,$:$\,0\,$:$\,0$) and write $r$
for the projective line $\overline{OP}=H_{1}\cap H_{2}\cap H_{3}$.

######  Lemma 5.3.

The group action of $\text{Stab}(C_{\times})$ on $Y_{W}$ determines the six
orbits:

$\displaystyle Y_{W}\backslash((H_{1}\cup H_{2})\cap H_{3})\,,\,((H_{1}\cup
H_{2})\cap H_{3})\backslash(H_{4}\cup r)\ ,$ $\displaystyle((H_{1}\cup
H_{2})\cap H_{3}\cap H_{4})\backslash O\ ,\ r\,\backslash(O\cup
P)\,,\,O\,,\,P$

###### Proof.

Write $D_{x}$, $D_{y}$, and $D_{z}$ for the divisors in $X$ respectively
determined by $x_{1}=0$, $y_{1}=0$, and $z_{1}=0$, and moreover let
$Q^{\prime}=(1:0)^{3}\in X$. The group action of $\text{Stab}(C_{\times})$ on
$X$ determines the five orbits

$Q^{\prime}\ ,\ C_{\times}\backslash Q^{\prime}\ ,\ D_{z}\backslash
C_{\times}\ ,\ (D_{x}\cup D_{y})\backslash C_{\times}\ ,\
X\backslash(D_{x}\cup D_{y}\cup D_{z})\ .$

The last three orbits are respectively transformed by $\zeta_{W}$ into $P$,
$((H_{1}\cup H_{2})\cap H_{3}\cap H_{4})\backslash O$, and
$Y_{W}\backslash((H_{1}\cup H_{2})\cap H_{3})$. Therefore, we find three of
the orbits in the statement. The orbits $((H_{1}\cup H_{2})\cap
H_{3})\backslash(H_{4}\cup r)$, $r\backslash(O\cup P)$, and $O$ correspond to
the projection by $\Pi_{W}$ of the orbits in $E_{W}$, when we let
$\text{Stab}(C_{\times})$ act on $\text{Bl}_{W}X$. ∎

The following is a classification theorem for the tri-linear birational maps
of type $(1,1,2)$, $(1,2,1)$, and $(2,1,1)$. Equivalently, it provides the
orbits of the right-action in $\mathrm{Bir}_{(1,1,2)}$,
$\mathrm{Bir}_{(1,2,1)}$, and $\mathrm{Bir}_{(2,1,1)}$.

######  Theorem 5.4.

Let $\phi$ be birational of type either $(1,1,2)$, $(1,2,1)$ or $(2,1,1)$.
Then, $\phi$ belongs to the orbit of one of the following birational maps:

  * •

$\rho_{1}^{(1,1,2)}\equiv(x_{1}y_{1}z_{1}:x_{0}y_{1}z_{1}:x_{0}y_{0}z_{1}:x_{1}y_{0}z_{0}-x_{0}y_{1}z_{0})$
A surface in $W_{\rho_{1}}$ contains an irreducible curve of tri-degree
$(1,1,0)$ and an isolated point. The base ideal is

$(x_{0}y_{1}-x_{1}y_{0},z_{1})\cap(x_{0},y_{1},z_{0})\ .$

  * •

$\rho_{2}^{(1,1,2)}\equiv(x_{0}y_{1}z_{1}:x_{1}y_{0}z_{1}:x_{0}y_{0}z_{1}:x_{1}y_{0}z_{0}-x_{0}y_{1}z_{0}-x_{1}y_{1}z_{1})$
A surface in $W_{\rho_{2}}$ contains an irreducible curve of tri-degree
$(1,1,0)$ and has contact to a surface of tri-degree $(1,1,1)$ at a point of
the curve. The base ideal is

$(x_{0}y_{1}-x_{1}y_{0},z_{1})\,\cap(z_{1}^{2},y_{0}z_{1},x_{0}z_{1},y_{0}^{2},x_{0}y_{0},x_{0}^{2},x_{1}y_{0}z_{0}-x_{0}y_{1}z_{0}-x_{1}y_{1}z_{1})\
.$

  * •

$\rho_{3}^{(1,1,2)}\equiv(x_{0}y_{1}z_{1}:x_{1}y_{0}z_{1}:x_{0}y_{0}z_{1}:x_{1}y_{1}z_{0})$
A surface in $W_{\rho_{3}}$ contains a pair of intersecting lines, of tri-
degrees $(1,0,0)$ and $(0,1,0)$, and an isolated point. The base ideal is

$(x_{1},z_{1})\cap(y_{1},z_{1})\cap(x_{0},y_{0},z_{0})\ .$

  * •

$\rho_{4}^{(1,1,2)}\equiv(x_{1}y_{1}z_{1}:x_{0}y_{1}z_{1}:x_{0}y_{0}z_{1}:x_{1}y_{1}z_{0}-x_{1}y_{0}z_{1})$
A surface in $W_{\rho_{4}}$ contains a pair of intersecting lines, of tri-
degrees $(1,0,0)$ and $(0,1,0)$, and is tangent to a curve of tri-degree
$(0,1,1)$ at one of the points of the line of tri-degree $(1,0,0)$. The base
ideal is

$(x_{1},z_{1})\cap(y_{1},z_{1})\cap(x_{0},y_{0}z_{1}-y_{1}z_{0},y_{1}^{2},y_{1}z_{1},y_{1}^{2})\
.$

  * •

$\rho_{5}^{(1,1,2)}\equiv(x_{1}y_{1}z_{1}:x_{0}y_{1}z_{1}:x_{1}y_{0}z_{1}:x_{1}y_{1}z_{0}+x_{0}y_{0}z_{1})$
A surface in $W_{\rho_{5}}$ contains a pair of intersecting lines, of tri-
degrees $(1,0,0)$ and $(0,1,0)$, and has contact to a surface of tri-degree
$(1,1,1)$ at the point of intersection of the lines. The base ideal is

$(x_{1},z_{1})\cap(y_{1},z_{1})\,\cap(x_{1}y_{1}z_{0}+x_{0}y_{0}z_{1},x_{1}^{2},x_{1}z_{1},y_{1}^{2},y_{1}z_{1},z_{1}^{2})\
.$

###### Proof.

From Diagram (3.2), $\phi$ lies in the orbit of a birational map that factors
as $\zeta_{W}\circ\pi_{L}$ for either $W=W_{o}$ or $W=W_{\times}$ and some
point $L\in Y_{W}$. In particular, the orbit of $\phi$ is determined by one of
the orbits listed in Lemmas 5.2 and 5.3. In the case that $L=P$, we find a
common factor to the entries of $\phi$. Excluding this case, the possibilities
when $W=W_{o}$ are the following:

  * •

If $L\in Y_{W}\backslash H_{1-2}$, $\phi$ lies in the orbit of
$\rho_{1}^{(1,1,2)}$

  * •

If $L\in H_{1-2}\backslash(H_{4}\cup P)$, $\phi$ lies in the orbit of
$\rho_{2}^{(1,1,2)}$

  * •

If $L\in H_{1-2}\cap H_{4}$, $\phi$ lies in the orbit of $\rho_{2}^{(1,1,1)}$

Secondly, if $W=W_{\times}$ the possibilities are the following:

  * •

If $L\in Y_{W}\backslash((H_{1}\cup H_{2})\cap H_{3})$, $\phi$ lies in the
orbit of $\rho_{3}^{(1,1,2)}$

  * •

If $L\in((H_{1}\cup H_{2})\cap H_{3})\backslash(H_{4}\cup r)$, $\phi$ lies in
the orbit of $\rho_{4}^{(1,1,2)}$

  * •

If $L\in((H_{1}\cup H_{2})\cap H_{3}\cap H_{4})\backslash O$, $\phi$ lies in
the orbit of $\rho_{3}^{(1,1,1)}$

  * •

If $L\in r\backslash(O\cup P)$, $\phi$ lies in the orbit of
$\rho_{5}^{(1,1,2)}$

  * •

If $L=O$, $\phi$ lies in the orbit of $\rho_{4}^{(1,1,1)}$

∎

###  5.3. Orbits of birational maps of type $(1,2,2)$, $(2,1,2)$ and $(2,2,1)$

By Lemma 3.4, any Cohen-Macaulay curve in $X$ of tri-degree either $(1,0,0)$,
$(0,1,0)$, or $(0,0,1)$ is a projective line, and by means of an automorphism
of $X$ it can be transformed into the line $\ell$ of ideal
$I_{\ell}=(y_{1},z_{1})$. In particular, any birational map of type $(1,2,2)$,
$(2,1,2)$, or $(2,2,1)$ lies in the orbit of a birational map with $\ell$ in
its base locus. According to Diagram (3.2), in order to study the orbits of
the right-action in $\mathrm{Bir}_{(1,2,2)}$, $\mathrm{Bir}_{(2,1,2)}$, and
$\mathrm{Bir}_{(2,2,1)}$ we can compute the orbits of the group action given
by the stabilizer $\text{Stab}(\ell)\leq\mathrm{Aut}(X)$ of $\ell$ acting on
$Y_{W}$, where $W={(I_{\ell})}_{(1,1,1)}$. Namely, the birational map in the
diagram $\zeta_{W}:X\dashrightarrow Y_{W}\subset\mathbb{P}_{\mathbb{C}}^{5}$
is

$\zeta_{W}:(x_{0}:x_{1})\times(y_{0}:y_{1})\times(z_{0}:z_{1})\mapsto(x_{0}y_{0}z_{1}:x_{1}y_{0}z_{1}:x_{0}y_{1}z_{1}:x_{1}y_{1}z_{1}:x_{0}y_{1}z_{0}:x_{1}y_{1}z_{0})\
,$

and $Y_{W}$ is determined by the ideal
$(t_{0}t_{5}-t_{1}t_{4},t_{2}t_{5}-t_{3}t_{4},t_{1}t_{2}-t_{0}t_{3})$. The
defining equations of $\Pi_{W}(E_{W})$ in $Y_{W}$ are $t_{2}=t_{3}=0$. We
write $H_{01}$, $H_{23}$, and $H_{45}$ for the subvarieties of $Y_{W}$ given
respectively by $t_{0}=t_{1}=0$, $t_{2}=t_{3}=0$, and $t_{4}=t_{5}=0$.

######  Lemma 5.5.

The group action of $\text{Stab}(\ell)$ on $Y_{W}$ determines the three
orbits:

$Y_{W}\backslash H_{23}\ ,\ H_{23}\backslash(H_{01}\cup H_{45})\ ,\
H_{23}\cap(H_{01}\cup H_{45})$

###### Proof.

Write $D_{y}$ and $D_{z}$ for the divisors in $X$ respectively determined by
$y_{1}=0$ and $z_{1}=0$. The group action of $\text{Stab}(\ell)$ on $X$
determines the three orbits

$\ell\ ,\ (D_{y}\cup D_{z})\backslash\ell\ ,\ X\backslash(D_{y}\cup D_{z})\ .$

The last two orbits are respectively transformed by $\zeta_{W}$ into
$H_{23}\cap(H_{01}\cup H_{45})$ and $Y_{W}\backslash H_{23}$. Thus, we find
two of the orbits in the statement. The orbit $H_{23}\backslash(H_{01}\cup
H_{45})$ corresponds to the projection by $\Pi_{W}$ of the general orbit in
$E_{W}$, when we let $\text{Stab}(\ell)$ act on $\text{Bl}_{W}X$. ∎

The following is a classification theorem for the tri-linear birational maps
of type $(1,2,2)$, $(2,1,2)$, and $(2,2,1)$. Equivalently, it provides the
orbits of the right-action in $\mathrm{Bir}_{(1,2,2)}$,
$\mathrm{Bir}_{(2,1,2)}$, and $\mathrm{Bir}_{(2,2,1)}$.

######  Theorem 5.6.

Let $\phi$ be birational of type either $(1,2,2)$, $(2,1,2)$ or $(2,2,1)$.
Then, $\phi$ belongs to the orbit of one of the following birational maps:

  * •

$\rho_{1}^{(1,2,2)}\equiv(x_{1}y_{0}z_{1}-x_{0}y_{1}z_{1}:x_{0}y_{0}z_{1}-x_{0}y_{1}z_{1}:x_{1}y_{1}z_{0}-x_{0}y_{1}z_{1}:x_{0}y_{1}z_{0}-x_{0}y_{1}z_{1})$
A surface in $W_{\rho_{1}}$ contains a line and two isolated points with all
the coordinates different, i.e. the projections onto the factors of $X$ are
all different. The base ideal is

$(y_{1},z_{1})\cap(x_{0},y_{0},z_{0})\cap(x_{0}-x_{1},y_{0}-y_{1},z_{0}-z_{1})\
.$

  * •

$\rho_{2}^{(1,2,2)}\equiv(x_{1}y_{0}z_{1}:x_{0}y_{0}z_{1}:x_{1}y_{1}z_{0}-x_{0}y_{1}z_{1}:x_{0}y_{1}z_{0}+x_{0}y_{1}z_{1})$
A surface in $W_{\rho_{2}}$ contains a line and two isolated points with the
same coordinate in exactly one of the factors of $X$. The base ideal is

$(y_{1},z_{1})\cap(x_{0},y_{0},z_{0})\cap(x_{0}+x_{1},y_{0},z_{0}+z_{1})\ .$

  * •

$\rho_{3}^{(1,2,2)}\equiv(x_{1}y_{0}z_{1}-x_{0}y_{1}z_{1}:x_{0}y_{0}z_{1}:x_{1}y_{1}z_{0}-x_{0}y_{1}z_{1}:x_{0}y_{1}z_{0})$
A surface in $W_{\rho_{3}}$ contains a line, and is tangent to a curve of tri-
degree $(1,1,1)$ at an isolated point. The base ideal is

$(y_{1},z_{1})\cap(x_{0}y_{1}-x_{1}y_{0},x_{0}z_{1}-x_{1}z_{0},y_{0}z_{1}-y_{1}z_{0},x_{0}^{2},x_{0}y_{0},x_{0}z_{0},y_{0}^{2},y_{0}z_{0},z_{0}^{2})\
.$

  * •

$\rho_{4}^{(1,2,2)}\equiv(x_{1}y_{0}z_{1}-x_{0}y_{1}z_{1}:x_{0}y_{0}z_{1}:x_{1}y_{1}z_{0}:x_{0}y_{1}z_{0})$
A surface in $W_{\rho_{4}}$ contains a line of tri-degree $(1,0,0)$ and is
tangent to a curve of tri-degree $(1,1,0)$ at an isolated point. The base
ideal is

$(y_{1},z_{1})\cap(z_{0},x_{0}y_{1}-x_{1}y_{0},x_{0}^{2},x_{0}y_{0},y_{0}^{2})\
.$

  * •

$\rho_{5}^{(1,2,2)}\equiv(x_{0}y_{1}z_{1}:x_{1}y_{0}z_{1}:x_{1}y_{1}z_{0}:x_{0}y_{1}z_{0}-x_{0}y_{0}z_{1})$
A surface in $W_{\rho_{5}}$ contains a line of tri-degree $(1,0,0)$ and an
isolated point, and is tangent to a curve of tri-degree $(0,1,1)$ at a point
of the line. The base ideal is

$(y_{1},z_{1})\cap(x_{0},y_{0},z_{0})\cap(x_{1},y_{0}z_{1}-y_{1}z_{0},y_{1}^{2},y_{1}z_{1},z_{1}^{2})\
.$

  * •

$\rho_{6}^{(1,2,2)}\equiv(x_{1}y_{1}z_{1}:x_{0}y_{1}z_{1}:x_{1}y_{1}z_{0}+x_{1}y_{0}z_{1}:x_{0}y_{1}z_{0}-x_{0}y_{0}z_{1}-2x_{1}y_{0}z_{1})$
A surface in $W_{\rho_{6}}$ contains a line of tri-degree $(1,0,0)$ and is
tangent to two curves of tri-degree $(0,1,1)$ at two distinct points of the
line. The base ideal is

$(y_{1},z_{1})\cap(x_{1},y_{0}z_{1}-y_{1}z_{0},y_{1}^{2},y_{1}z_{1},z_{1}^{2})\cap(x_{0}+x_{1},y_{0}z_{1}+y_{1}z_{0},y_{1}^{2},y_{1}z_{1},z_{1}^{2})\
.$

  * •

$\rho_{7}^{(1,2,2)}\equiv(x_{1}y_{1}z_{1}:x_{0}y_{1}z_{1}:x_{1}y_{1}z_{0}-x_{1}y_{0}z_{1}:x_{0}y_{1}z_{0}-x_{0}y_{0}z_{1}+x_{1}y_{0}z_{1})$
A surface in $W_{\rho_{7}}$ contains a line and has contact to a surface of
tri-degree $(0,1,1)$ at a point of the line. The base ideal is

$(y_{1},z_{1})\cap(x_{1}y_{1}z_{0}-x_{1}y_{0}z_{1},x_{0}y_{1}z_{0}-x_{0}y_{0}z_{1}+x_{1}y_{0}z_{1},x_{1}^{2},y_{1}^{2},y_{1}z_{1},z_{1}^{2})\
.$

  * •

$\rho_{8}^{(1,2,2)}\equiv(x_{1}y_{1}z_{1}:x_{1}y_{0}z_{1}-x_{0}y_{1}z_{1}:x_{1}y_{1}z_{0}-x_{0}y_{1}z_{1}:x_{0}y_{1}z_{0}-x_{0}y_{0}z_{1}-x_{0}y_{1}z_{1})$
A surface in $W_{\rho_{8}}$ contains a line and has contact to a surface of
tri-degree $(0,1,1)$ at a point of the line. The base ideal is

$\displaystyle(y_{1},z_{1})\,\cap(y_{1}z_{0}-y_{0}z_{1}-y_{1}z_{1},x_{1}^{2},z_{1}^{3},y_{1}z_{1}^{2},x_{1}z_{1}^{2},x_{1}z_{0}z_{1}-x_{0}z_{1}^{2},$
$\displaystyle
y_{1}^{2}z_{1},x_{1}y_{1}z_{1},x_{1}y_{0}z_{1}-x_{0}y_{1}z_{1},y_{1}^{3},x_{1}y_{1}^{2},x_{1}y_{0}y_{1}-x_{0}y_{1}^{2})\
.$

###### Proof.

From Diagram (3.2), $\phi$ lies in the orbit of a birational map that factors
as $\zeta_{W}\circ\pi_{L}$ for some line $L\subset\mathbb{P}_{\mathbb{C}}^{5}$
such that the restriction $\pi_{L}|_{Y_{W}}$ has degree one, i.e.
$\pi_{L}|_{Y_{W}}$ is birational. From Proposition 3.5, we have $\deg
Y_{W}=3$. If $L\not\subset Y_{W}$, then $\deg(L\cap Y_{W})\geq 2$ as otherwise
$\deg(\pi_{L}|_{Y_{W}})>1$. On the other hand, consider a linear
parametrization $\mathbb{P}_{\mathbb{C}}^{1}\xrightarrow{}L$. As the ideal of
$Y_{W}$ is generated by three quadratic forms, the pull-back of $L\cap Y_{W}$
yields three quadratic forms in $\mathbb{P}_{\mathbb{C}}^{1}$, implying that
$\deg(L\cap Y_{W})\leq 2$. Then, we must have either $L\subset Y_{W}$ or
$L\not\subset Y_{W}$ and $\deg(L\cap Y_{W})=2$. Given a point $P\in
Y_{W}\backslash H_{23}$, we find
$Q=(\alpha_{0}:\alpha_{1})\times(\beta:1)\times(\gamma:1)\in X$ such that
$\zeta_{W}(Q)=P$. Write $Y_{y}(P)\subset Y_{W}$ for the closure of the image
of the restriction of $\zeta_{W}$ to
$\mathbb{P}_{\mathbb{C}}^{1}\times(\beta:1)\times\mathbb{P}_{\mathbb{C}}^{1}$.
Analogously, define $Y_{z}(P)\subset Y_{W}$ by restricting $\zeta_{W}$ to
$\mathbb{P}_{\mathbb{C}}^{1}\times\mathbb{P}_{\mathbb{C}}^{1}\times(\gamma:1)$.
Notice that the restriction of $\zeta_{W}$ to
$(\alpha_{0}:\alpha_{1})\times\mathbb{P}_{\mathbb{C}}^{1}\times\mathbb{P}_{\mathbb{C}}^{1}$
is a projective plane. Both $Y_{y}(P)$ and $Y_{z}(P)$ are smooth subvarieties
of $Y_{W}$. Given a subvariety $V\subset Y_{W}$ and a smooth point $P\in V$,
we write $T_{P}(V)$ for the tangent space of $V$ at $P$. Now, we discuss all
the possible intersections between $L$ and $Y_{W}$. In the first place, we
assume that $L\not\subset Y_{W}$:

  * •

If $L$ meets $Y_{W}$ transversally at $P_{1},P_{2}\in Y_{W}\backslash H_{23}$,
and moreover we have $P_{2}\not\in Y_{y}(P_{1})$ and $P_{2}\not\in
Y_{z}(P_{1})$, $\phi$ lies in the orbit of $\rho_{1}^{(1,2,2)}$

  * •

If $L$ meets $Y_{W}$ transversally at $P_{1},P_{2}\in Y_{W}\backslash H_{23}$,
and either $P_{2}\in Y_{y}(P_{1})$ or $P_{2}\in Y_{z}(P_{1})$, $\phi$ lies in
the orbit of $\rho_{2}^{(1,2,2)}$

  * •

If $L\subset T_{P}(Y_{W})$ at a point $P\in Y_{W}\backslash H_{23}$, and
moreover we have $L\not\subset T_{P}(Y_{y}(P))$ and $L\not\subset
T_{P}(Y_{z}(P))$, $\phi$ lies in the orbit of $\rho_{3}^{(1,2,2)}$

  * •

If $L\subset T_{P}(Y_{W})$ at a point $P\in Y_{W}\backslash H_{23}$, and
either $L\subset T_{P}(Y_{y}(P))$ or $L\subset T_{P}(Y_{z}(P))$,

$\phi$ lies in the orbit of $\rho_{4}^{(1,2,2)}$

  * •

If $L$ meets $Y_{W}$ transversally at $P_{1}\in Y_{W}\backslash H_{23}$,
$P_{2}\in H_{23}\backslash(H_{01}\cup H_{45})$, $\phi$ lies in the orbit of
$\rho_{5}^{(1,2,2)}$

  * •

If $L$ meets $Y_{W}$ transversally at $P_{1},P_{2}\in
H_{23}\backslash(H_{01}\cup H_{45})$, $\phi$ lies in the orbit of
$\rho_{6}^{(1,2,2)}$

  * •

If $L\subset T_{P}(Y_{W})$ at a point $P\in H_{23}\backslash(H_{01}\cup
H_{45})$ and $L\subset T_{P}(H_{23})$, $\phi$ lies in the orbit of
$\rho_{7}^{(1,2,2)}$

  * •

If $L\subset T_{P}(Y_{W})$ at a point $P\in H_{23}\backslash(H_{01}\cup
H_{45})$ and $L\not\subset T_{P}(H_{23})$, $\phi$ lies in the orbit of
$\rho_{8}^{(1,2,2)}$

  * •

If $L$ meets $Y_{W}$ transversally at $P_{1}\in Y_{W}\backslash H_{23}$,
$P_{2}\in H_{23}\cap(H_{01}\cup H_{45})$, $\phi$ lies in the orbit of
$\rho_{3}^{(1,1,2)}$

  * •

If $L$ meets $Y_{W}$ transversally at $P_{1}\in H_{23}\backslash(H_{01}\cup
H_{45})$, $P_{2}\in H_{23}\cap(H_{01}\cup H_{45})$, $\phi$ lies in the orbit
of $\rho_{4}^{(1,1,2)}$

  * •

If $L\subset T_{P}(H_{23})$ at a point $P\in H_{23}\cap(H_{01}\cup H_{45})$,
$\phi$ lies in the orbit of $\rho_{5}^{(1,1,2)}$

Secondly, we study the cases where $L\subset Y_{W}$:

  * •

If $L\subset Y_{W}\backslash H_{23}$, $\phi$ is not dominant

  * •

If $L\subset Y_{W}$ and intersects transversally $H_{23}\backslash(H_{01}\cup
H_{45})$, then $\phi$ lies in the orbit of $\rho_{2}^{(1,1,1)}$

  * •

If $L\subset Y_{W}$ and intersects transversally $H_{23}\cap(H_{01}\cup
H_{45})$, $\phi$ lies in the orbit of $\rho_{3}^{(1,1,1)}$

  * •

If $L\subset H_{23}\backslash(H_{01}\cup H_{45})$, and $L\not\subset
H_{23}\cap(H_{01}\cup H_{45})$, $\phi$ lies in the orbit of
$\rho_{4}^{(1,1,1)}$

∎

###  5.4. Orbits of birational maps of type $(2,2,2)$

Given a point $Q\in X$ and $\lambda,\mu,\nu\in\mathbb{C}^{*}$, by means of an
automorphism of $X$ the ideal $I_{Q(\lambda,\mu,\nu)}$ can be transformed into

$I^{\prime}=I_{Q^{\prime}(1,1,1)}=(x_{1}y_{0}z_{0}+x_{0}y_{1}z_{0}+x_{0}y_{0}z_{1}\,,\,x_{0}y_{1}z_{1}\,,\,x_{1}y_{0}z_{1}\,,\,x_{1}y_{1}z_{0}\,,\,x_{1}y_{1}z_{1})\
,$

where $Q^{\prime}=(1:0)^{3}\in X$. In particular, any birational map of type
$(2,2,2)$ lies in the orbit of a birational map with the non-reduced point
defined by $I_{Q^{\prime}(1,1,1)}$ in its base locus. Let
$\text{Stab}(Q^{\prime}(1,1,1))\leq\mathrm{Aut}(X)$ be the stabilizer of the
non-reduced point defined by $I^{\prime}$. In order to study the orbits of the
right-action in $\mathrm{Bir}_{(2,2,2)}$, we can compute the orbits of the
group action given by the stabilizer $\text{Stab}(Q^{\prime}(1,1,1))$ acting
on $Y_{W}$, where we set $W={I^{\prime}}_{(1,1,1)}$. The birational map
$\zeta_{W}:X\dashrightarrow Y_{W}\subset\mathbb{P}_{\mathbb{C}}^{5}$ in
Diagram (3.2) is now

$(x_{0}:x_{1})\times(y_{0}:y_{1})\times(z_{0}:z_{1})\mapsto(x_{1}y_{0}z_{0}+x_{0}y_{1}z_{0}+x_{0}y_{0}z_{1}:x_{0}y_{1}z_{1}:x_{1}y_{0}z_{1}:x_{1}y_{1}z_{0}:x_{1}y_{1}z_{1})\
,$

and the defining equation of $Y_{W}$ in $\mathbb{P}_{\mathbb{C}}^{4}$ is
$t_{1}t_{2}+t_{1}t_{3}+t_{2}t_{3}-t_{0}t_{4}=0$. On the other hand,
$\Pi_{W}(E_{W})$ is the hyperplane section in $Y_{W}$ given by $t_{4}=0$, and
we write $H_{4}=\Pi_{W}(E_{W})$. The divisor $H_{4}$ is again a two-
dimensional cone, of vertex $O=$ ($1\,$:$\,0\,$:$\,0\,$:$\,0\,$:$\,0$).
Similarly, we define $r_{12}$, $r_{13}$ and $r_{23}$ to be the projective
lines in $Y_{W}$ defined by $t_{1}=t_{2}=t_{4}=0$, $t_{1}=t_{3}=t_{4}=0$, and
$t_{2}=t_{3}=t_{4}=0$, respectively.

######  Lemma 5.7.

The group action of $\text{Stab}(Q^{\prime}(1,1,1))$ on $Y_{W}$ determines the
four orbits:

$Y_{W}\backslash H_{4}\ ,\ H_{4}\backslash(r_{12}\cup r_{13}\cup r_{23})\ ,\
(r_{12}\cup r_{13}\cup r_{23})\backslash O\ ,\ O$

###### Proof.

Write $D_{x}$, $D_{y}$, and $D_{z}$ for the divisors in $X$ defined
respectively by $x_{1}=0$, $y_{1}=0$, and $z_{1}=0$. Similarly, write
$D_{xy}$, $D_{xz}$ and $D_{yz}$ for their intersections, with the obvious
notation. The group action of $\text{Stab}(Q^{\prime}(1,1,1))$ on $X$
determines the four orbits

$Q^{\prime}\ ,\ (D_{xy}\cup D_{xz}\cup D_{yz})\backslash Q^{\prime}\ ,\
(D_{x}\cup D_{y}\cup D_{z})\backslash(D_{xy}\cup D_{xz}\cup D_{yz})\ ,\
X\backslash(D_{x}\cup D_{y}\cup D_{z})\ .$

The last three orbits are respectively transformed by $\zeta_{W}$ into $O$,
$(r_{12}\cup r_{13}\cup r_{23})\backslash O$, and $Y_{W}\backslash H_{4}$.
Therefore, we find three of the orbits in the statement. The orbit
$H_{4}\backslash(r_{12}\cup r_{13}\cup r_{23})$ corresponds to the projection
by $\Pi_{W}$ of the general orbit in $E_{W}$, when we let
$\text{Stab}(Q(1,1,1))$ act on $\text{Bl}_{W}X$. ∎

The following is a classification theorem for the tri-linear birational maps
of type $(2,2,2)$. Equivalently, it provides the orbits of the right-action in
$\mathrm{Bir}_{(2,2,2)}$.

######  Theorem 5.8.

Let $\phi$ be birational of type $(2,2,2)$. Then, $\phi$ belongs to the orbit
of one of the following birational maps:

  * •

$\rho_{1}^{(2,2,2)}\equiv(x_{0}y_{1}z_{1}:x_{1}y_{0}z_{1}:x_{1}y_{1}z_{0}:x_{1}y_{0}z_{0}+x_{0}y_{1}z_{0}+x_{0}y_{0}z_{1})$
A surface in $W_{\rho_{1}}$ contains a point and has contact to a surface of
tri-degree $(1,1,1)$ at a distinct point. The base ideal is

$(x_{0},y_{0},z_{0})\cap(x_{1}y_{0}z_{0}+x_{0}y_{1}z_{0}+x_{0}y_{0}z_{1}\,,\,x_{1}^{2}\,,\,x_{1}y_{1}\,,\,x_{1}z_{1}\,,\,y_{1}^{2}\,,\,y_{1}z_{1}\,,\,z_{1}^{2})\
.$

  * •

$\rho_{2}^{(2,2,2)}\equiv(x_{1}y_{1}z_{1}:x_{1}y_{0}z_{1}-x_{0}y_{1}z_{1}:2x_{1}y_{1}z_{0}+x_{0}y_{1}z_{1}:x_{1}y_{0}z_{0}+x_{0}y_{1}z_{0}+x_{0}y_{0}z_{1})$
A surface in $W_{\rho_{2}}$ has contact to a surface of tri-degree $(1,1,1)$
at a point. The base ideal is

$\displaystyle(z_{1}^{3}\,,\,y_{1}z_{1}^{2}\,,\,x_{1}z_{1}^{2}\,,\,2y_{1}z_{0}z_{1}+y_{0}z_{1}^{2}\,,\,2x_{1}z_{0}z_{1}+x_{0}z_{1}^{2}\,,\,y_{1}^{2}z_{1}\,,\,x_{1}y_{1}z_{1}\,,\,x_{1}y_{0}z_{1}-x_{0}y_{1}z_{1}\,,\,$
$\displaystyle
x_{1}^{2}z_{1}\,,\,2y_{1}^{2}z_{0}+y_{0}y_{1}z_{1}\,,\,2x_{1}y_{1}z_{0}+x_{0}y_{1}z_{1}\,,\,x_{1}y_{0}z_{0}+x_{0}y_{1}z_{0}+x_{0}y_{0}z_{1}\,,\,$
$\displaystyle
2x_{1}^{2}z_{0}+x_{0}x_{1}z_{1}\,,\,y_{1}^{3}\,,\,x_{1}y_{1}^{2}\,,\,x_{1}y_{0}y_{1}-x_{0}y_{1}^{2}\,,\,x_{1}^{2}y_{1}\,,\,x_{1}^{2}y_{0}-x_{0}x_{1}y_{1}\,,\,x_{1}^{3})\
.$

###### Proof.

From Diagram (3.2), $\phi$ lies in the orbit of a birational map that factors
as $\zeta_{W}\circ\pi_{L}$ for some point $L\in Y_{W}$. We have the following
four possibilities:

  * •

If $L\in Y_{W}\backslash H_{4}$, $\phi$ lies in the orbit of
$\rho_{1}^{(2,2,2)}$

  * •

If $L\in H_{4}\backslash(r_{12}\cup r_{13}\cup r_{23})$, $\phi$ lies in the
orbit of $\rho_{2}^{(2,2,2)}$

  * •

If $L\in(r_{12}\cup r_{13}\cup r_{23})\backslash O$, $\phi$ lies in the orbit
of $\rho_{2}^{(1,1,2)}$

  * •

If $L=O$, $\phi$ lies in the orbit of $\rho_{4}^{(1,1,1)}$

∎

Altogether, Theorems 5.1, 5.4, 5.6, and 5.8 provide the complete list of the
orbits of the right-action in $\mathfrak{Bir}_{(1,1,1)}$. Interestingly, we
find the following corollary from this classification.

######  Corollary 5.9.

All the irreducible components of $\mathfrak{Bir}_{(1,1,1)}$ are unirational.

###### Proof.

Given an irreducible component $V$ of $\mathfrak{Bir}_{(1,1,1)}$, the action
of the subgroup $H\trianglelefteq\mathrm{Aut}(X)$ consisting of the
automorphisms that preserve the tri-degrees in $X$ on any representative of
the general orbit of $V$, as listed in either Theorem 5.1, 5.4, 5.6, or 5.8,
yields a dominant rational map from $H$ to $V$. As we find a dense affine set
in $H\cong\mathrm{PGL}(2,\mathbb{C})^{3}$, the result follows. ∎

###  5.5. Degenerations of the base loci

Let $O_{1},O_{2}\subset\mathfrak{Bir}_{(1,1,1)}$ be two orbits of the right-
action. We say that the orbit $O_{1}$ degenerates to $O_{2}$ if the Zariski
closure of $O_{1}$ in $\mathfrak{Bir}_{(1,1,1)}$ contains $O_{2}$. From the
proofs of Theorems 5.1, 5.4, 5.6, and 5.8 we deduce most of the degenerations
of the orbits. Diagram 1 depicts geometric illustrations of all the
degenerations of the possible base loci of a tri-linear birational map. The
symbols
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\bullet}$
and ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}-}$
respectively represent non-singular points and curves contained in any surface
of the linear system, and
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\circ}$
represents a point of contact to a surface. Similarly, an arrow
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\rightarrow}$
represents a tangency at a point to a curve of tri-degree $(1,1,1)$, and a
dashed arrow
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\dasharrow}$
a tangency to a curve of tri-degree $(1,1,0)$ or permutation.

Diagram 1\. Degenerations of the base loci of tri-linear birational maps. The
dimension of the associated orbit of the right-action in
$\mathfrak{Bir}_{(1,1,1)}$ is written on the left.

The degenerations of the zero-dimensional base loci require further analysis.
Namely, we can derive the degenerations to base loci of tri-degree $(1,0,0)$
(resp. $(0,1,0)$, $(0,0,1)$) by studying the orbits in the closure
$Y^{\prime}$ of the image of

$\displaystyle\zeta^{\prime}:X$ $\displaystyle\dasharrow$ $\displaystyle
Y^{\prime}\subset\mathbb{P}_{\mathbb{C}}^{4}$
$\displaystyle(x_{0}:x_{1})\times(y_{0}:y_{1})\times(z_{0}:z_{1})$
$\displaystyle\mapsto$
$\displaystyle(x_{1}y_{1}z_{1}:x_{0}y_{1}z_{1}:x_{1}y_{0}z_{1}:x_{1}y_{1}z_{0}:x_{0}y_{1}z_{0}+x_{0}y_{0}z_{1})\
.$

In this case, the entries of $\zeta^{\prime}$ determine a basis for the graded
component of the ideal

$I^{\prime}=(x_{0}y_{1}z_{0}+x_{0}y_{0}z_{1})+(x_{1},y_{1},z_{1})^{2}$

in tri-degree $(1,1,1)$. Similarly, the degenerations to base loci of tri-
degree $(1,1,0)$ (resp. $(1,0,1)$, $(0,1,1)$) can be deduced from the orbits
in the closure $Y^{\prime\prime}$ of the image of

$\displaystyle\zeta^{\prime\prime}:X$ $\displaystyle\dasharrow$ $\displaystyle
Y^{\prime\prime}\subset\mathbb{P}_{\mathbb{C}}^{4}$
$\displaystyle(x_{0}:x_{1})\times(y_{0}:y_{1})\times(z_{0}:z_{1})$
$\displaystyle\mapsto$
$\displaystyle(x_{1}y_{1}z_{1}:x_{0}y_{1}z_{1}:x_{1}y_{0}z_{1}:x_{1}y_{1}z_{0}:x_{0}y_{0}z_{1})\
,$

where the entries of $\zeta^{\prime\prime}$ now determine a basis of the
graded component of the ideal

$I^{\prime\prime}=(x_{0}y_{0}z_{1})+(x_{1},y_{1},z_{1})^{2}$

in tri-degree $(1,1,1)$. As an example, the orbits on $Y^{\prime}$ of the
action given by the stabilizer of the subscheme defined by $I^{\prime}$ are

(5.2) $Y^{\prime}\backslash H_{4}\ ,\ (H_{1}\cap H_{4})\backslash H_{2-3}\ ,\
(H_{2-3}\cap H_{4})\backslash(H_{1}\cup H_{2})\ ,\ r\backslash H_{1}\ ,\
H_{1}\cap H_{2-3}\cap H_{4}\ ,\ O\ ,$

where $H_{1},H_{2-3},H_{4}$ are the divisors in $Y^{\prime}$ respectively
given by $t_{1}=0$, $t_{2}-t_{3}=0$, and $t_{4}=0$, and $r\subset Y^{\prime}$
is the line determined by $t_{2}=t_{3}=t_{4}=0$. Representatives associated to
each orbit in (5.2), in the same order, are

$\rho_{5}^{(1,2,2)}\ ,\ \rho_{6}^{(1,2,2)}\ ,\ \rho_{8}^{(1,2,2)}\ ,\
\rho_{2}^{(1,1,1)}\ ,\ \rho_{7}^{(1,2,2)}\ ,\ \rho_{4}^{(1,1,1)}.$

As $r\subset H_{2-3}\cap H_{4}$ but $r\not\subset H_{1}\cap H_{4}$, it follows
that $\rho_{8}^{(1,2,2)}$ degenerates to $\rho_{2}^{(1,1,1)}$, but
$\rho_{6}^{(1,2,2)}$ does not.

##  6\. Syzygies of tri-linear birational maps

Finally, we study the syzygies of the entries of tri-linear birational maps.
Interestingly, the following theorem yields a birationality criterion for tri-
linear rational maps, based on the computation of the first syzygies in some
tri-degrees. Furthermore, novel methods for the manipulation of birational
volumes might be devised from this theorem, relying on the imposition of
specific syzygies. In order to simplify the statement, we set

$e_{1}=(1,0,0)\ ,\ e_{2}=(0,1,0)\ ,\ e_{3}=(0,0,1)\ .$

######  Theorem 6.1.

Assume that $\phi$ is dominant, and let $i,j,k$ be indices such that
$\\{i,j,k\\}=\\{1,2,3\\}$. Then, $\phi$ is birational if and only if one of
the following conditions holds:

  1. (1)

The $f_{i}$’s have syzygies of tri-degrees $e_{1}$, $e_{2}$, and $e_{3}$. In
this case, the type of $\phi$ is $(1,1,1)$.

  2. (2)

The $f_{i}$’s have syzygies of tri-degree $e_{i}$ and $e_{j}$, but not
$e_{k}$. In this case, the type of $\phi$ is $(1,1,1)+e_{k}$.

  3. (3)

The $f_{i}$’s have a syzygy of tri-degree $e_{i}$, but neither $e_{j}$ nor
$e_{k}$. Moreover, the $f_{i}$’s also have a syzygy of tri-degree either
$e_{i}+e_{j}$ or $e_{i}+e_{k}$, independent from the first one. In this case,
the type of $\phi$ is $(1,1,1)+e_{j}+e_{k}$.

  4. (4)

The $f_{i}$’s do not have syzygies of tri-degree $e_{1}$, $e_{2}$, or $e_{3}$.
Moreover, the $f_{i}$’s have two independent syzygies in each of the tri-
degrees $e_{1}+e_{2}$, $e_{1}+e_{3}$, and $e_{2}+e_{3}$ satisfying the
splitting property in Condition 6.7 (see §6.4). In this case, the type of
$\phi$ is $(2,2,2)$.

The proof of Theorem 6.1 follows immediately from Propositions 6.2, 6.4, 6.5,
and 6.8, and it relies strongly on the geometric classification of §5. More
explicitly, the composition of a rational map with automorphisms of
$(\mathbb{P}_{\mathbb{C}}^{1})^{3}$ and $\mathbb{P}_{\mathbb{C}}^{3}$
preserves the structure of the module of first syzygies, with maybe a
permutation in the tri-degrees (if this is not preserved by the automorphism
of $(\mathbb{P}_{\mathbb{C}}^{1})^{3}$). Therefore, the syzygies of the
entries can be computed for the representatives of the finitely many orbits of
the right-action in $\mathfrak{Bir}_{(1,1,1)}$, listed in Theorems 5.1, 5.4,
5.6, and 5.8. On the other hand, although Theorem 6.1 is adequate for
computational purposes, we provide further cohomological information about the
base loci of tri-linear birational maps. Namely, we will prove that there is a
one-to-one correspondence between the type of $\phi$ and the shape of the
minimal tri-graded free resolution of the base ideal $B_{\phi}$. For the sake
of comparison, the minimal free resolution of a tri-linear rational map $\phi$
with general entries is

$0\xrightarrow[]{}R\xrightarrow[]{}R^{21}\xrightarrow[]{}R^{62}\xrightarrow[]{}R^{69}\xrightarrow[]{}R^{30}\xrightarrow[]{}R^{4}\xrightarrow[]{}B_{\phi}\xrightarrow{}0\
.$

As we shall see, the resolution is remarkably simpler when $\phi$ is
birational.

###  6.1. Syzygies of birational maps of type $(1,1,1)$

For birational maps of type $(1,1,1)$, the minimal tri-graded free resolution
of the base ideal $B_{\phi}$ is Hilbert-Burch.

######  Proposition 6.2.

Assume that $\phi$ is dominant. The following are equivalent:

  1. (1)

$\phi$ is birational of type $(1,1,1)$.

  2. (2)

The minimal tri-graded free resolution of the base ideal $B_{\phi}$ is
Hilbert-Burch, i.e.

(6.1) $0\xrightarrow{}\begin{matrix}R(-2,-1,-1)\\\ \oplus\\\ R(-1,-2,-1)\\\
\oplus\\\ R(-1,-1,-2)\\\
\end{matrix}\xrightarrow{}R(-1,-1,-1)^{4}\xrightarrow{(f_{0}\ f_{1}\ f_{2}\
f_{3})}B_{\phi}\xrightarrow{}0\ .$

In particular, the base locus $Z_{\phi}$ is a Cohen-Macaulay curve.

  3. (3)

The $f_{i}$’s have syzygies of tri-degrees $e_{1}$, $e_{2}$, and $e_{3}$.

###### Proof.

We first prove that $(1)$ implies $(2)$, so we assume that $\phi$ is
birational of type $(1,1,1)$. Maintaining the notation of (2.3) for
$\phi^{-1}$, in the Rees ideal $J$ we find the polynomials

(6.2) $g_{1}=\begin{vmatrix}x_{0}&x_{1}\\\ a_{0}&a_{1}\end{vmatrix}\ ,\
g_{2}=\begin{vmatrix}y_{0}&y_{1}\\\ b_{0}&b_{1}\end{vmatrix}\ ,\
g_{3}=\begin{vmatrix}z_{0}&z_{1}\\\ c_{0}&c_{1}\end{vmatrix}\ .$

Consider the $4\times 3$ matrix $M=M(x_{i},y_{j},z_{k})$, whose $(i,j)$-th
entry is the coefficient of $g_{j}$ in $t_{i}$. By definition, the tuple
$(f_{0},f_{1},f_{2},f_{3})$ lies in the cokernel of $M$. Now, write
$\Delta_{i}$ for the $3\times 3$ signed minor obtained by deleting the $i$-th
row of $M$. Then, the tuple $(\Delta_{0},\Delta_{1},\Delta_{2},\Delta_{3})$
also lies in the cokernel of $M$. Moreover, as $\phi$ is birational the
specialization of the $g_{i}$’s at a general point of $X$ yields three
independent linear forms, so the matrix $M$ has generically rank $3$. In
particular $M$ has no kernel in $R^{3}$, and its cokernel is a free $R$-module
of rank one generated by $(\Delta_{0},\Delta_{1},\Delta_{2},\Delta_{3})$. As
the tuples $(f_{0},f_{1},f_{2},f_{3})$ and
$(\Delta_{0},\Delta_{1},\Delta_{2},\Delta_{3})$ have tri-linear entries, they
must differ by a non-zero constant. Therefore, as $\text{codim}\,B_{\phi}\geq
2$ the complex

(6.3)
$0\xrightarrow{}R^{3}\xrightarrow{M}R^{4}\xrightarrow{(\Delta_{0}\,\Delta_{1}\,\Delta_{2}\,\Delta_{3})}B_{\phi}\xrightarrow{}0$

is a minimal free resolution of $B_{\phi}$. On the other hand, from the
Auslander-Buchbaum formula the ring $R/B_{\phi}$ is Cohen-Macaulay of
codimension two (see e.g. [11, §20.4]), which implies that $Z_{\phi}$ is a
Cohen-Macaulay curve. Clearly $(2)$ implies $(3)$, so we now assume $(3)$ and
prove $(1)$. The syzygies of tri-degree $e_{1}$, $e_{2}$, and $e_{3}$ can be
written as

$x_{0}\,a_{1}-x_{1}\,a_{0}=\begin{vmatrix}x_{0}&x_{1}\\\
a_{0}&a_{1}\end{vmatrix}\ ,\
y_{0}\,b_{1}-y_{1}\,b_{0}=\begin{vmatrix}y_{0}&y_{1}\\\
b_{0}&b_{1}\end{vmatrix}\ ,\
z_{0}\,c_{1}-z_{1}\,c_{0}=\begin{vmatrix}z_{0}&z_{1}\\\
c_{0}&c_{1}\end{vmatrix}\ ,$

for some linear forms $a_{i},b_{j},c_{k}$ in
$\mathbb{C}[t_{0},t_{1},t_{2},t_{3}]$. Then, the rational map from
$\mathbb{P}_{\mathbb{C}}^{3}$ to $X$ given by

$(t_{0}:t_{1}:t_{2}:t_{3})\mapsto(a_{0}:a_{1})\times(b_{0}:b_{1})\times(c_{0}:c_{1})$

must be $\phi^{-1}$, so $\phi$ is birational of type $(1,1,1)$. ∎

######  Remark 6.3.

Notice that the proof of Proposition 6.2 is independent from the
classification of §5. In particular, it is valid to use Proposition 6.2 in the
proof of Theorem 5.1. On the other hand, it follows from the proof that the
base locus $Z_{\phi}$ of a birational map $\phi$ of type $(1,1,1)$ is an
arithmetically Cohen-Macaulay curve, i.e. the tri-homogeneous coordinate ring
$R/B_{\phi}$ is Cohen-Macaulay, which is stronger than requiring that
$Z_{\phi}$ is Cohen-Macaulay.

###  6.2. Syzygies of birational maps of type $(1,1,2)$, $(1,2,1)$, and
$(2,1,1)$

The study of the syzygies of birational maps of type either $(1,1,2)$,
$(1,2,1)$, or $(2,1,1)$ is equivalent, by means of an automorphism of $X$. In
particular, the shape of the minimal tri-graded free resolution of the base
ideal $B_{\phi}$ is the same, up to a permutation of the Betti numbers. Thus,
we restrict to birational maps of type $(1,1,2)$.

######  Proposition 6.4.

Assume that $\phi$ is dominant. The following are equivalent:

  1. (1)

$\phi$ is birational of type $(1,1,2)$.

  2. (2)

The minimal tri-graded free resolution of the base ideal $B_{\phi}$ is

(6.4) $0\rightarrow R(-2,-2,-2)\xrightarrow{}\begin{matrix}R(-2,-1,-1)\\\
\oplus\\\ R(-1,-2,-1)\\\ \oplus\\\ R(-2,-1,-2)\\\ \oplus\\\
R(-1,-2,-2)\end{matrix}\xrightarrow{}R(-1,-1,-1)^{4}\xrightarrow{(f_{0}\
f_{1}\ f_{2}\ f_{3})}B_{\phi}\xrightarrow{}0\ .$

  3. (3)

The $f_{i}$’s have syzygies of tri-degree $e_{1}$ and $e_{2}$, but not
$e_{3}$.

###### Proof.

The implication from $(1)$ to $(2)$ follows from the computation of the
minimal tri-graded free resolutions of the base ideal of all the
representatives listed in Theorem 5.4 (performed with the help of Macaulay2).
Additionally, it is immediate that $(2)$ implies $(3)$. Now, we assume $(3)$.
The syzygies of tri-degree $e_{1}$ and $e_{2}$ can be respectively written as

(6.5) $x_{0}\,a_{1}-x_{1}\,a_{0}=\begin{vmatrix}x_{0}&x_{1}\\\
a_{0}&a_{1}\end{vmatrix}\ ,\
y_{0}\,b_{1}-y_{1}\,b_{0}=\begin{vmatrix}y_{0}&y_{1}\\\
b_{0}&b_{1}\end{vmatrix}\ ,$

for some linear forms $a_{i},b_{j}$ in $\mathbb{C}[t_{0},t_{1},t_{2},t_{3}]$.
Let $P$ be a general point in $\mathbb{P}_{\mathbb{C}}^{3}$, and
$(\lambda_{0}:\lambda_{1})$, $(\mu_{0}:\mu_{1})\in\mathbb{P}_{\mathbb{C}}^{1}$
be the projective points such that

$\begin{vmatrix}\lambda_{0}&\lambda_{1}\\\ a_{0}(P)&a_{1}(P)\end{vmatrix}=0\
,\ \begin{vmatrix}\mu_{0}&\mu_{1}\\\ b_{0}(P)&b_{1}(P)\end{vmatrix}=0\ .$

Thus, any point $Q$ in the pullback $\phi^{-1}(P)$ is of the form
$Q=(\lambda_{0}:\lambda_{1})\times(\mu_{0}:\mu_{1})\times(z_{0}:z_{1})$ for
some $(z_{0}:z_{1})\in\mathbb{P}_{\mathbb{C}}^{1}$, as the point $Q\times P$
in $X\times\mathbb{P}_{\mathbb{C}}^{3}$ must vanish the polynomials in (6.5).
Moreover, as $\phi$ is dominant the restriction given by

(6.6)
$\displaystyle\phi^{\prime}:(\lambda_{0}:\lambda_{1})\times(\mu_{0}:\mu_{1})\times\mathbb{P}_{\mathbb{C}}^{1}$
$\displaystyle\dasharrow$ $\displaystyle\mathbb{P}_{\mathbb{C}}^{3}$
$\displaystyle(z_{0}:z_{1})$ $\displaystyle\mapsto$
$\displaystyle(f^{\prime}_{0}:f^{\prime}_{1}:f^{\prime}_{2}:f^{\prime}_{3})\
,$

where
$f^{\prime}_{i}(z_{0},z_{1})=f_{i}(\lambda_{0},\lambda_{1},\mu_{0},\mu_{1},z_{0},z_{1})$,
is an isomorphism of projective lines. In particular, there is a unique point
in the pullback $\phi^{-1}(P)$, implying that $\phi$ is birational. The
syzygies of tri-degree $e_{1}$ and $e_{2}$ determine $\phi^{-1}$ on the first
two factors of $X$. Moreover there is no syzygy of tri-degree $e_{3}$, so the
type of $\phi$ must be $(1,1,2)$. ∎

###  6.3. Syzygies of birational maps of type $(1,2,2)$, $(2,1,2)$, and
$(2,2,1)$

Similarly to §6.2, the analysis of the syzygies of birational maps of type
either $(1,2,2)$, $(2,1,2)$, or $(2,2,1)$ is equivalent, and the minimal tri-
graded free resolution of the base ideal $B_{\phi}$ is the same up to
permutation of the Betti numbers. Thus, we restrict to birational maps of type
$(1,2,2)$.

######  Proposition 6.5.

Assume that $\phi$ is dominant. The following are equivalent:

  1. (1)

$\phi$ is birational of type $(1,2,2)$.

  2. (2)

The minimal tri-graded free resolution of the base ideal $B_{\phi}$ is

(6.7) $0\rightarrow R(-2,-2,-2)^{2}\xrightarrow{}\begin{matrix}R(-2,-1,-1)\\\
\oplus\\\ R(-2,-2,-1)\\\ \oplus\\\ R(-2,-1,-2)\\\ \oplus\\\
R(-1,-2,-2)^{2}\end{matrix}\xrightarrow{}R(-1,-1,-1)^{4}\xrightarrow{(f_{0}\
f_{1}\ f_{2}\ f_{3})}B_{\phi}\xrightarrow{}0\ .$

  3. (3)

The $f_{i}$’s have a syzygy of tri-degree $e_{1}$, but neither $e_{2}$ nor
$e_{3}$. Moreover, the $f_{i}$’s also have a syzygy of tri-degree either
$e_{1}+e_{2}$ or $e_{1}+e_{3}$, independent from the first one.

###### Proof.

The implication from $(1)$ to $(2)$ follows from the computation of the
minimal tri-graded free resolutions of the base ideal of all the
representatives listed in Theorem 5.6 (performed with the help of Macaulay2).
Additionally, it is immediate that $(2)$ implies $(3)$. Now, we assume $(3)$.
Without loss of generality, we suppose that the $f_{i}$’s have a syzygy of
tri-degree $e_{1}+e_{2}$. Thus, the syzygies of tri-degree $e_{1}$ and
$e_{1}+e_{2}$ can be respectively written as

(6.8) $\displaystyle x_{0}\,a_{1}-x_{1}\,a_{0}=\begin{vmatrix}x_{0}&x_{1}\\\
a_{0}&a_{1}\end{vmatrix}\ ,$ (6.9) $\displaystyle
y_{0}\,(x_{0}\,h_{01}+x_{1}h_{11})-y_{1}\,(x_{0}\,h_{00}+x_{1}h_{10})=\begin{vmatrix}y_{0}&y_{1}\\\
x_{0}\,h_{00}+x_{1}h_{10}&x_{0}\,h_{01}+x_{1}h_{11}\end{vmatrix}\ ,$

for some linear forms $a_{i},h_{ij}$ in $\mathbb{C}[t_{0},t_{1},t_{2},t_{3}]$.
Let $P$ be a general point in $\mathbb{P}_{\mathbb{C}}^{3}$, and
$(\lambda_{0}:\lambda_{1})\in\mathbb{P}_{\mathbb{C}}^{1}$ be the projective
point such that

$\begin{vmatrix}\lambda_{0}&\lambda_{1}\\\ a_{0}(P)&a_{1}(P)\end{vmatrix}=0\
.$

Similarly, write $(\mu_{0}:\mu_{1})\in\mathbb{P}_{\mathbb{C}}^{1}$ for the
projective point such that

$\begin{vmatrix}\mu_{0}&\mu_{1}\\\
\lambda_{0}\,h_{00}(P)+\lambda_{1}\,h_{10}(P)&\lambda_{0}\,h_{01}(P)+\lambda_{1}\,h_{11}(P)\end{vmatrix}=0\
.$

Thus, any point $Q$ in the pullback $\phi^{-1}(P)$ is of the form
$Q=(\lambda_{0}:\lambda_{1})\times(\mu_{0}:\mu_{1})\times(z_{0}:z_{1})$ for
some $(z_{0}:z_{1})\in\mathbb{P}_{\mathbb{C}}^{1}$, as the point $Q\times P$
in $X\times\mathbb{P}_{\mathbb{C}}^{3}$ must vanish the polynomials (6.8) and
(6.9). Repeating the argument in the proof of Proposition 6.4 with the
restriction map $\phi^{\prime}$, it follows that there is a unique point in
the pullback $\phi^{-1}(P)$, so $\phi$ is birational. The syzygy of tri-degree
$e_{1}$ determines $\phi^{-1}$ on the first factor of $X$. On the other hand,
there are no syzygies of tri-degree $e_{2}$ nor $e_{3}$, so the type of $\phi$
must be $(1,2,2)$. ∎

###  6.4. Syzygies of birational maps of type $(2,2,2)$

Finally, we study the syzygies of birational maps of type $(2,2,2)$. In this
case, the entries of $\phi$ do not have syzygies of tri-degree $e_{1},e_{2}$,
or $e_{3}$, and the analysis is more complicated. Moreover, the tri-degree of
the syzygies alone is not enough to decide birationality, as it is shown by
Example 6.9. However, from Proposition 6.8 it follows that birationality can
still be decided from the first syzygies, by checking a splitting condition on
three related polynomials. Before stating this characterization, we first
establish some hypotheses on the first syzygies.

######  Setup 6.6.

Assume that the $f_{i}$’s do not have syzygies of tri-degree $e_{1}$, $e_{2}$,
or $e_{3}$ but have a pair of independent syzygies in each of the tri-degrees
$e_{1}+e_{2}$, $e_{1}+e_{3}$, and $e_{2}+e_{3}$. In particular, the syzygies
of tri-degree $e_{1}+e_{2}$ can be written as

(6.10)
$x_{0}y_{0}\,g_{00}+x_{1}y_{0}\,g_{10}-x_{0}y_{1}\,g_{01}-x_{1}y_{1}\,g_{11}\
,\
x_{0}y_{0}\,h_{00}+x_{1}y_{0}\,h_{10}-x_{0}y_{1}\,h_{01}-x_{1}y_{1}\,h_{11}\
,$

for some linear forms $g_{ij},h_{ij}$ in
$\mathbb{C}[t_{0},t_{1},t_{2},t_{3}]$. We define $u_{xy}=u_{xy}(x_{i},t_{j})$
as

$u_{xy}\coloneqq\begin{vmatrix}x_{0}\,g_{00}+x_{1}g_{10}&x_{0}\,g_{01}+x_{1}g_{11}\\\
x_{0}\,h_{00}+x_{1}h_{10}&x_{0}\,h_{01}+x_{1}h_{11}\end{vmatrix}\ .$

The polynomial $u_{xy}$ is non-zero, as the syzygies in (6.10) are independent
by assumption.
Moreover, we have

$\displaystyle\begin{pmatrix}x_{0}\,g_{00}+x_{1}g_{10}&x_{0}\,g_{01}+x_{1}g_{11}\\\
x_{0}\,h_{00}+x_{1}h_{10}&x_{0}\,h_{01}+x_{1}h_{11}\end{pmatrix}\begin{pmatrix}y_{0}\\\
y_{1}\end{pmatrix}\xmapsto{t_{j}\mapsto f_{j}}\begin{pmatrix}0\\\
0\end{pmatrix}$

implying that $u_{xy}$ is a relation in the Rees ideal $J$ of multi-degree
$(2,0,0;2)$. Similarly, we define the relations $u_{yz},u_{zx}$ from the pair
of syzygies of tri-degree $e_{2}+e_{3}$ and $e_{1}+e_{3}$, respectively. In
particular, the multi-degree in $J$ of $u_{yz}$ (resp. $u_{zx}$) is
$(0,2,0;2)$ (resp. $(0,0,2;2)$).

######  Condition 6.7.

Assuming Setup 6.6, at least two of the $u_{xy},u_{yz},$ and $u_{zx}$ have a
linear factor in $\\{x_{0},x_{1}\\}$, $\\{y_{0},y_{1}\\}$, and
$\\{z_{0},z_{1}\\}$, respectively.

######  Proposition 6.8.

Assume that $\phi$ is dominant. The following are equivalent:

  1. (1)

$\phi$ is birational of type $(2,2,2)$.

  2. (2)

The $f_{i}$’s satisfy Condition 6.7. In this case, the minimal tri-graded free
resolution of the base ideal $B_{\phi}$ is

(6.11) $0\rightarrow
R(-2,-2,-2)^{3}\xrightarrow{}\begin{matrix}R(-2,-2,-1)^{2}\\\ \oplus\\\
R(-2,-1,-2)^{2}\\\ \oplus\\\
R(-1,-2,-2)^{2}\end{matrix}\xrightarrow{}R(-1,-1,-1)^{4}\xrightarrow{(f_{0}\
f_{1}\ f_{2}\ f_{3})}B_{\phi}\xrightarrow{}0\ .$

###### Proof.

The implication from $(1)$ to $(2)$ follows from the computation of the
minimal tri-graded free resolutions of the base ideal of all the
representatives listed in Theorem 5.8, and the polynomials $u_{xy}$, $u_{yz}$,
and $u_{zx}$ always have a linear factor as in Condition 6.7 (these
computations have been performed with the help of Macaulay2). Now, we assume
Condition 6.7. Without loss of generality, the polynomials $u_{xy}$ and
$u_{yz}$ have linear factors $l_{1}=l_{1}(x_{0},x_{1})$ and
$l_{2}=l_{2}(y_{0},y_{1})$, i.e. we can write

(6.12) $u_{xy}=l_{1}(x_{0},x_{1})\,\begin{vmatrix}x_{0}&x_{1}\\\
a_{0}&a_{1}\end{vmatrix}\ ,\
u_{yz}=l_{2}(y_{0},y_{1})\,\begin{vmatrix}y_{0}&y_{1}\\\
b_{0}&b_{1}\end{vmatrix}\ ,$

for some quadratic forms $a_{i},b_{j}$ in
$\mathbb{C}[t_{0},t_{1},t_{2},t_{3}]$. Let $P$ given a general point in
$\mathbb{P}_{\mathbb{C}}^{3}$. We now prove that $\phi^{-1}(P)$ has a unique
preimage in the open set $U\subset X$ where $l_{1}$ and $l_{2}$ are both non-
zero. Given $Q\in U\cap\phi^{-1}(P)$, the point $Q\times P\in
U\times\mathbb{P}_{\mathbb{C}}^{3}$ must vanish the two polynomials in (6.12).
By assumption we have $l_{1}(Q)\not=0$ and $l_{2}(Q)\not=0$, implying that
$Q\times P$ must vanish the second factor in each polynomial. In particular,
$Q$ must be of the form
$Q=(\lambda_{0}:\lambda_{1})\times(\mu_{0}:\mu_{1})\times(z_{0}:z_{1})$ for
some $(z_{0}:z_{1})\in\mathbb{P}_{\mathbb{C}}^{1}$, where
$(\lambda_{0}:\lambda_{1}),(\mu_{0}:\mu_{1})\in\mathbb{P}_{\mathbb{C}}^{1}$
are the projective points satisfying

$\begin{vmatrix}\lambda_{0}&\lambda_{1}\\\ a_{0}(P)&a_{1}(P)\end{vmatrix}=0\
,\ \begin{vmatrix}\mu_{0}&\mu_{1}\\\ b_{0}(P)&b_{1}(P)\end{vmatrix}=0\ .$

Repeating the argument in the proof of Proposition 6.4 with the restriction
map $\phi^{\prime}$, it follows that the pullback $\phi^{-1}(P)$ defines a
unique point in $U$, so $\phi$ is birational. On the other hand there are no
syzygies of tri-degree $e_{1},e_{2}$ or $e_{3}$, so the type of $\phi$ must be
$(2,2,2)$. ∎

######  Example 6.9.

From Proposition 6.8, the minimal tri-graded free resolution of the base ideal
of a tri-linear birational map of type $(2,2,2)$ is always (6.11). However,
given the rational map

$\displaystyle\phi:\mathbb{P}_{\mathbb{C}}^{1}\times\mathbb{P}_{\mathbb{C}}^{1}\times\mathbb{P}_{\mathbb{C}}^{1}\dashrightarrow\mathbb{P}_{\mathbb{C}}^{3}$
$\displaystyle(x_{0}:x_{1})\times(y_{0}:y_{1})\times(z_{0}:z_{1})\mapsto(x_{0}y_{0}z_{1}:x_{1}y_{1}z_{0}+x_{1}y_{0}z_{1}+x_{0}y_{1}z_{1}+x_{1}y_{1}z_{1}:x_{0}y_{1}z_{0}:x_{1}y_{0}z_{0})$

the minimal tri-graded free resolution of $B_{\phi}$ is also as (6.11), but
$\phi$ is not birational as Condition 6.7 is not satisfied.

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