headcorheadthm headcor headconjheadthm headconj corollarytheorem corollary
claimtheorem claim lemmatheorem lemma conjecturetheorem conjecture
propositiontheorem proposition definitiontheorem definition notationtheorem
notation exampletheorem example examplestheorem examples remarktheorem remark
questiontheorem question questionstheorem questions problemtheorem problem
constructiontheorem construction setuptheorem setup algorithmtheorem algorithm
observationtheorem observation defproptheorem defprop

# A study of nonlinear multiview varieties

Yairon Cid-Ruiz  Department of Mathematics, KU Leuven, Celestijnenlaan 200B,
3001 Leuven, Belgium<EMAIL_ADDRESS>,  Oliver Clarke  Department
of Mathematics: Algebra and Geometry, Ghent University, S25, 9000 Gent,
Belgium<EMAIL_ADDRESS>and  Fatemeh Mohammadi  Department of
Mathematics, KU Leuven, Celestijnenlaan 200B, Leuven, Belgium and Department
of Computer Science, KU Leuven, Celestijnenlaan 200A, 3001 Leuven, Belgium
<EMAIL_ADDRESS>
###### Abstract.

We study the nonlinear generalization of the classical multiview variety,
which is a fundamental concept in computer vision. In this paper, we take the
first comprehensive step to develop the nonlinear analogue of multiview
varieties. To this end, we introduce a multigraded version of the saturated
special fiber ring. By applying this tool, we are able to compute the
multidegrees of several families of nonlinear multiview varieties.

###### Key words and phrases:

nonlinear multiview varieties, rational maps, multidegrees, mixed
multiplicities, saturated special fiber ring, syzygies, blow-up algebras.

###### 2010 Mathematics Subject Classification:

Primary: 14E05, Secondary: 13D02, 13D45, 13P99.

###### Contents

  1. 1 Introduction
  2. 2 Notations and preliminaries
  3. 3 Our main tool: the saturated special fiber ring
  4. 4 Linear rational maps
  5. 5 Certain special families of rational maps
  6. 6 Monomial rational maps
  7. 7 Application / Interpretation

##  1\. Introduction

Motivation. Let $\mathbb{k}$ be a field. The classical _multiview variety_ is
an important tool in computer vision, see [aholt2013hilbert]. It is defined as
the closure of the image $Y$ of a linear rational map
$\mathcal{G}:{\mathbb{P}}_{\mathbb{k}}^{3}\dashrightarrow{\mathbb{P}}_{\mathbb{k}}^{2}\times_{\mathbb{k}}\cdots\times_{\mathbb{k}}{\mathbb{P}}_{\mathbb{k}}^{2}$.
Conceptually, the image of $\mathcal{G}$ is a sequence of cameras that take
$2$-dimensional images of $3$-dimensional space. Naturally, properties about
$Y$ are crucial to solving the problem of reconstructing a $3$-dimensional
object from a collection of images.

Since $\mathcal{G}$ is linear, the defining ideal $J$ of $Y$ has a structure
that is amenable to study via a number of different methods. Firstly, $J$ is
an example of a Cartwright-Sturmfels ideal, see [aholt2013hilbert,
conca2018cartwright, conca2021radical]. And so, $J$ satisfies a number of
pleasant properties, see [conca2021radical]. Secondly, it is possible to give
an explicit description of the generators of $J$, see [agarwal2019ideals]. In
particular, if the focal points of the cameras are distinct then $J$ is
generated by certain determinants of degree $2$ and $3$, see
[agarwal2019ideals, Theorem 3.7]. Thirdly, one can study the structure of the
coordinate ring of $Y$ in terms of products of linear ideals, see
[conca2019resolution]. In particular, it is possible to give an explicit
formula for the Betti numbers in terms of the polymatroidal data defined by
the positions of the cameras. These results demonstrate connections between
properties of $Y$ and properties of the base locus of $\mathcal{G}$. The goal
and purpose of this paper is to extend such connections for when $\mathcal{G}$
is nonlinear. Therefore, our approach embodies the general and fruitful
philosophy of studying nonlinear algebra [NONLINEAR_BOOK,
breiding2021nonlinear].

Previous Work. Let us consider a rational map
$\mathcal{G}:{\mathbb{P}}_{\mathbb{k}}^{3}\dashrightarrow{\mathbb{P}}_{\mathbb{k}}^{2}\times_{\mathbb{k}}\cdots\times_{\mathbb{k}}{\mathbb{P}}_{\mathbb{k}}^{2}$,
as above, which is not necessarily linear. Note that, none of the methods
above can be extended to study this case. In particular, it is difficult to
determine the generators of $J$ and, in general, this ideal is not necessarily
a Cartwright-Sturmfels ideal. So, it is natural to ask how to compute its
Hilbert function. More specifically, it is important to study the
_multidegrees_ of $Y$ as a subvariety of the target
${\mathbb{P}}_{\mathbb{k}}^{2}\times_{\mathbb{k}}\cdots\times_{\mathbb{k}}{\mathbb{P}}_{\mathbb{k}}^{2}$
of $\mathcal{G}$. Our approach to determine the multidegrees of $Y$ is based
on the study of blow-up algebras and syzygies of the ideals generated by the
defining polynomials of the rational map $\mathcal{G}$.

A syzygy-based approach to study rational maps seems to have been initiated in
[HULEK_KATZ_SCHREYER_SYZ], and it has now become an active and fruitful
research area, see e.g. [AB_INITIO, Simis_cremona, KPU_blowup_fibers,
EISENBUD_ULRICH_ROW_IDEALS, Hassanzadeh_Simis_Cremona_Sat, SIMIS_RUSSO_BIRAT,
EFFECTIVE_BIGRAD, SIMIS_PAN_JONQUIERES, HASSANZADEH_SIMIS_DEGREES, MULTPROJ,
MULT_SAT_PERF_HT_2, SAT_FIB_GOR_3, MIXED_MULT, BOTBOL_ALICIA_RAT_SURF,
BOTBOL_ALICIA_MAT, KPU_BIGRAD_STRUCT, KPU_GOR3, KPU_NORMAL_SCROLL,
CARLOS_MONO, CARLOS_MONOID, CARLOS_MU2]. In the area of Computer Aided
Geometric Design (CAGD), a somewhat similar story has emerged where the
concept of syzygies is substituted by an equivalent one in $\mu$-bases, see
[COX_EQ_PARAM, cox1998moving_line]. In all these previous works, the main goal
is to study a (singly projective) rational map of the form
$\mathcal{G}:{\mathbb{P}}_{\mathbb{k}}^{r}\dashrightarrow{\mathbb{P}}_{\mathbb{k}}^{s}$.
To illustrate the general idea, we recall an instance where the degree of the
syzygies of the base ideal completely determines the degree of the image. In
[COX_EQ_PARAM], Cox considers a parametric surface $Y$ given by the image of
$\mathcal{G}:{\mathbb{P}}_{\mathbb{k}}^{2}\dashrightarrow{\mathbb{P}}_{\mathbb{k}}^{3}$.
In this case, under some technical assumptions, the degree of $Y$ can be
computed by an elementary symmetric polynomial evaluated at the degrees of the
syzygies of the base ideal, see [COX_EQ_PARAM, Proposition 5.3]. In
[MULT_SAT_PERF_HT_2, Theorem A], this result is widely generalized using the
saturated special fiber ring to the case of rational maps
$\mathcal{G}:{\mathbb{P}}_{\mathbb{k}}^{r}\dashrightarrow{\mathbb{P}}_{\mathbb{k}}^{s}$
where the base locus is given by a perfect ideal of height two. The _saturated
special fiber ring_ has been recently used to successfully study rational maps
in many different contexts, see [MULT_SAT_PERF_HT_2, SAT_FIB_GOR_3,
MIXED_MULT, DEGREE_SPECIALIZATION]. It was introduced by Busé – Cid-Ruiz –
D’Andrea [MULTPROJ] for studying the degree and birationality of rational
maps. In this paper we define the multigraded version of the saturated special
fiber ring, see §missing 3.2.

Main Results. We consider rational maps
$\mathcal{G}:{\mathbb{P}}_{\mathbb{k}}^{r}\dashrightarrow{\mathbb{P}}_{\mathbb{k}}^{m_{1}}\times_{\mathbb{k}}\cdots\times_{\mathbb{k}}{\mathbb{P}}_{\mathbb{k}}^{m_{p}}$
whose image can be thought of as the nonlinear generalization of a multiview
variety. Note that, the closure of the image $Y$ of $\mathcal{G}$ naturally
admits a ${\mathbb{Z}}^{p}$-grading, and so we will be concerned with the
multidegrees of $Y$ instead of the degree. Our first main result is Theorem
3.1 which relates the mixed multiplicities, see §missing 2.1, of the saturated
special fiber ring with the multidegrees of $Y$. In particular, if the special
fiber ring is integrally closed, we obtain a criterion for birationality of
$\mathcal{G}$ in terms of the saturated special fiber ring.

Our second main result is Theorem 3.2 which gives an upper bound for the
multidegrees of $Y$, and, if the base locus of $\mathcal{G}$ is zero-
dimensional, then it yields an exact formula for the multidegrees of $Y$ in
terms of the mixed multiplicities of the base points. This generalizes the
well-known degree formula for rational maps with finite base locus to the
multiprojective setting, see [Laurent_Jouanolou_Closed_Image, Theorem 2.5],
[Sim_Ulr_Vasc_mult, Theorem 6.6] and [MULTPROJ, Theorem 3.3]. Note that, if
$\mathcal{G}$ is a linear rational map, then $Y$ is multiplicity-free, see
Theorem 4.1 which recovers [li2013images, Theorem 1.1] and
[conca2019resolution, Theorem 3.9].

Our third main result is Theorem 5.2 which gives an explicit formula for the
multidegrees of $Y$ in the case that the base ideals $I_{1},\dots,I_{p-1}$ are
zero-dimensional and $I_{p}$ is either a height two perfect ideal or a height
three Gorenstein ideal. In particular, this result generalizes
[MULT_SAT_PERF_HT_2, Theorem A] and [SAT_FIB_GOR_3, Theorem A] to the
multigraded setting.

Lastly, for the case of monomial rational maps, Theorem 6.3 provides a closed
formula for the multidegrees of $Y$ in terms of mixed volumes of some
naturally constructed lattice polytopes.

Outline. In Section missing 2, we fix our notation and recall the definitions
of multidegrees and mixed multiplicities, see §missing 2.1. In Section missing
3, we begin by fixing Section missing 3 which details the generality for which
our main results hold and, following this, we recall some fundamental
properties of the multigraded Rees algebra. In §missing 3.2, we introduce our
main tool for studying the multidegrees of the image of rational maps, namely
the saturated special fiber ring, see §missing 3.2. Using this tool, we state
and prove our first main result: Theorem 3.1. In §missing 3.4, we define the
mixed multiplicities of the base locus of $\mathcal{G}$ and prove our second
main result: Theorem 3.2, which allows us to compute the multidegrees of $Y$
in terms of the mixed multiplicities of the base locus of $\mathcal{G}$.

We proceed to apply Theorem 3.2 to important families of examples. In Section
missing 4, we show that our results recover previous results for linear
rational maps, i.e. for classical multiview varieties. In Section missing 5,
we prove our third main result, Theorem 5.2, which computes the multidegrees
of $Y$ for a large family of rational maps extending previous work in
[MULT_SAT_PERF_HT_2, SAT_FIB_GOR_3] to the multigraded setting. In Section
missing 6, we consider monomial rational maps $\mathcal{G}$, and we show that
in this case, the multidegrees of the image can be computed in terms of the
mixed volumes of certain lattice polytopes.

In Section missing 7, we further elaborate on the previous works, along with
interpretations and applications related to our results.

Acknowledgments. Y.C.R. was partially supported by an FWO Postdoctoral
Fellowship (1220122N). F.M. was partially supported by KU Leuven iBOF/23/064,
and FWO grants (G023721N, G0F5921N).

##  2\. Notations and preliminaries

We begin by fixing some common notation and recalling some preliminary
results. Let $p\geq 1$ be a positive integer and, for each $1\leq i\leq p$,
let ${\mathbf{e}}_{i}\in\mathbb{N}^{p}$ be the $i$-th elementary vector
${\mathbf{e}}_{i}=\left(0,\ldots,1,\ldots,0\right)$. For any
${\mathbf{n}}=(n_{1},\ldots,n_{p})\in{\mathbb{Z}}^{p}$, we define its weight
as $\lvert{\mathbf{n}}\rvert:=n_{1}+\cdots+n_{p}$. For two vectors
${\mathbf{n}}=(n_{1},\ldots,n_{p})\in\mathbb{N}^{p}$ and
$\delta=(\delta_{1},\ldots,\delta_{p})\in\mathbb{N}^{p}$, we denote the dot
product by ${\mathbf{n}}\cdot\delta:=n_{1}\delta_{1}+\cdots+n_{p}\delta_{p}$.
If
${\mathbf{n}}=(n_{1},\ldots,n_{p}),\,{\mathbf{m}}=(m_{1},\ldots,m_{p})\in{\mathbb{Z}}^{p}$
are two vectors, we write ${\mathbf{n}}\geq{\mathbf{m}}$ whenever $n_{i}\geq
m_{i}$ for all $1\leq i\leq p$, and ${\mathbf{n}}>{\mathbf{m}}$ whenever
$n_{j}>m_{j}$ for all $1\leq j\leq p$. We write $\mathbf{0}\in\mathbb{N}^{p}$
for the zero vector $\mathbf{0}=(0,\ldots,0)$.

###  2.1. Multigraded schemes and multidegrees

We fix the following setup during this subsection.

######  Setup .

Let $\mathbb{k}$ be a field and $S$ a finitely generated standard
$\mathbb{N}^{p}$-graded algebra, i.e. $\left[S\right]_{\mathbf{0}}=\mathbb{k}$
and $S$ is finitely generated over $\mathbb{k}$ by elements of degree
${\mathbf{e}}_{i}$ with $1\leq i\leq p$.

Given a standard $\mathbb{N}^{p}$-graded $\mathbb{k}$-algebra $S$, we consider
the corresponding multigraded scheme.

######  Definition .

The multiprojective scheme $\text{MultiProj}(S)$ is given by
$\text{MultiProj}(S):=\big{\\{}\mathfrak{P}\in{\text{Spec}}(S)\mid\mathfrak{P}\text{
is $\mathbb{N}^{p}$-graded and
}\mathfrak{P}\not\supseteq{\mathfrak{N}}\big{\\}},$ and its scheme structure
is obtained by using multi-homogeneous localizations, see e.g.
[HYRY_MULTIGRAD, §1]. The multigraded irrelevant ideal ${\mathfrak{N}}\subset
S$ is given by
${\mathfrak{N}}:=\left([S]_{{\mathbf{e}}_{1}}\right)\cap\cdots\cap\left([S]_{{\mathbf{e}}_{p}}\right)$.

Let
$P_{S}(\mathbf{t})=P_{S}(t_{1},\ldots,t_{p})\in\mathbb{Q}[\mathbf{t}]=\mathbb{Q}[t_{1},\ldots,t_{p}]$
be the _(multigraded) Hilbert polynomial_ of $S$, see e.g. [HERMANN_MULTIGRAD,
Theorem 4.1] and [MIXED_MULT, Theorem 3.4]. Then, the degree of $P_{S}$ is
equal to
$r:=\dim(\text{MultiProj}(S))=\dim(S/(0:_{S}{\mathfrak{N}}^{\infty}))-p$ and
$P_{S}(\nu)=\dim_{\mathbb{k}}\left([S]_{\nu}\right)$ for all
$\nu\in\mathbb{N}^{p}$ such that $\nu\gg\mathbf{0}$. Furthermore, if we write

(1) $P_{S}(\mathbf{t})=\sum_{d_{1},\ldots,d_{p}\geq
0}e(d_{1},\ldots,d_{p})\binom{t_{1}+d_{1}}{d_{1}}\cdots\binom{t_{p}+d_{p}}{d_{p}},$

then $0\leq e(d_{1},\ldots,d_{p})\in{\mathbb{Z}}$ for all
$d_{1}+\cdots+d_{p}=r$. Interchangeably, $P_{S}(\mathbf{t})$ is the Hilbert
polynomial $P_{X}(\mathbf{t})$ of the multiprojective scheme
$X=\text{MultiProj}(S)$.

We embed $X=\text{MultiProj}(S)$, as a closed subscheme, into the
multiprojective space
${\mathbb{P}}:={\mathbb{P}}_{\mathbb{k}}^{m_{1}}\times_{\mathbb{k}}\cdots\times_{\mathbb{k}}{\mathbb{P}}_{\mathbb{k}}^{m_{p}}$.
Under the notation of (missing) 1, we define the following invariants.

######  Definition .

Let ${\mathbf{d}}=(d_{1},\ldots,d_{p})\in\mathbb{N}^{p}$ with
$\lvert{\mathbf{d}}\rvert=r$. Then:

  1. (i)

$e({\mathbf{d}},S):=e(d_{1},\ldots,d_{p})$ is the mixed multiplicity of $S$ of
type ${\mathbf{d}}$.

  2. (ii)

$\deg_{\mathbb{P}}^{\mathbf{d}}(X):=e(d_{1},\ldots,d_{p})$ is the multidegree
of $X=\text{MultiProj}(S)$ of type ${\mathbf{d}}$ with respect to
${\mathbb{P}}$.

Whenever the chosen embedding of $X$ into a product of projective spaces
${\mathbb{P}}:={\mathbb{P}}_{\mathbb{k}}^{m_{1}}\times_{\mathbb{k}}\cdots\times_{\mathbb{k}}{\mathbb{P}}_{\mathbb{k}}^{m_{p}}$
is clear from the context, we simply write $\deg^{\mathbf{d}}(X)$ instead of
$\deg_{\mathbb{P}}^{\mathbf{d}}(X)$. In classical geometrical terms, when
$\mathbb{k}$ is algebraically closed, $\deg_{\mathbb{P}}^{\mathbf{d}}(X)$ is
also equal to the number of points (counting multiplicity) in the intersection
of $X$ with the product
$L_{1}\times_{\mathbb{k}}\cdots\times_{\mathbb{k}}L_{p}\subset{\mathbb{P}}$,
where $L_{i}\subseteq{\mathbb{P}}_{\mathbb{k}}^{m_{i}}$ is a general linear
subspace of dimension $m_{i}-d_{i}$ for each $1\leq i\leq p$, see
[VAN_DER_WAERDEN] and [MIXED_MULT, Theorem 4.7].

###  2.2. Mixed multiplicities of ideals

Let us recall the definition of mixed multiplicities of ideals in the setting
that we are considering. For more details, we refer the reader to
[TRUNG_VERMA_SURVEY]. Let $R=\mathbb{k}[x_{0},\ldots,x_{r}]$ be a standard
graded polynomial ring, ${\mathfrak{m}}=(x_{0},\ldots,x_{r})$ the graded
irrelevant ideal, and $I_{1},\ldots,I_{p}\subset R$ a sequence of homogeneous
ideals. We consider the $\mathbb{N}^{p+1}$-graded $\mathbb{k}$-algebra

$T({\mathfrak{m}}\mid
I_{1},\ldots,I_{p})\,:=\,\bigoplus_{(n_{0},n_{1},\ldots,n_{p})\in\mathbb{N}^{p+1}}\frac{{\mathfrak{m}}^{n_{0}}I_{1}^{n_{1}}\cdots
I_{p}^{n_{p}}}{{\mathfrak{m}}^{n_{0}+1}I_{1}^{n_{1}}\cdots I_{p}^{n_{p}}}.$

The mixed multiplicities of the ideals $I_{1},\ldots,I_{p}$ with respect to
${\mathfrak{m}}$ are defined as the mixed multiplicities of the algebra
$T({\mathfrak{m}}\mid I_{1},\ldots,I_{p})$.

######  Definition .

Let $(d_{0},{\mathbf{d}})=(d_{0},d_{1},\ldots,d_{p})\in\mathbb{N}^{p+1}$ with
$d_{0}+\lvert{\mathbf{d}}\rvert=r$. The _mixed multiplicity of
$I_{1},\ldots,I_{p}$ with respect to ${\mathfrak{m}}$ of type
$(d_{0},{\mathbf{d}})$_ is $e_{(d_{0},{\mathbf{d}})}\left({\mathfrak{m}}\mid
I_{1},\ldots,I_{p}\right):=e\left((d_{0},{\mathbf{d}}),\,T({\mathfrak{m}}\mid
I_{1},\ldots,I_{p})\right)$.

##  3\. Our main tool: the saturated special fiber ring

In this section, we develop and study the multigraded saturated special fiber
ring. Originally, the saturated special fiber ring was introduced in
[MULTPROJ], and has been successfully used to study several classes of
rational maps [MULT_SAT_PERF_HT_2, GEN_FREENESS_LOC_COHOM, MIXED_MULT,
DEGREE_SPECIALIZATION, SAT_FIB_GOR_3, cid2019blow].

We begin by fixing the following setup, which is used throughout this section.

######  Setup .

Let $\mathbb{k}$ be an arbitrary field, $R=\mathbb{k}[x_{0},\ldots,x_{r}]$ a
standard graded polynomial ring and
${\mathfrak{m}}=(x_{0},\ldots,x_{r})\subset R$ the graded irrelevant ideal of
$R$. For each $1\leq i\leq p$, let
$\\{f_{i,0},f_{i,1},\ldots,f_{i,m_{i}}\\}\subset R$ be a set of forms of the
same degree $\delta_{i}\geq 1$,
$I_{i}=(f_{i,0},f_{i,1},\ldots,f_{i,m_{i}})\subset R$ the ideal generated by
these forms and
$\mathcal{G}_{i}:{\mathbb{P}}_{\mathbb{k}}^{r}\dashrightarrow{\mathbb{P}}_{\mathbb{k}}^{m_{i}}$
the rational map determined by
$(\mathbf{x})=(x_{0},\ldots,x_{r})\mapsto(f_{i,0}(\mathbf{x}),f_{i,1}(\mathbf{x}),\ldots,f_{i,m_{i}}(\mathbf{x}))$.
Let

$\mathcal{G}=(\mathcal{G}_{1},\ldots,\mathcal{G}_{p})\;:\;{\mathbb{P}}_{\mathbb{k}}^{r}\dashrightarrow{\mathbb{P}}_{\mathbb{k}}^{m_{1}}\times_{\mathbb{k}}\cdots\times_{\mathbb{k}}{\mathbb{P}}_{\mathbb{k}}^{m_{p}}$

be the rational map given by the product of
$\mathcal{G}_{1},\ldots,\mathcal{G}_{p}$. Let
$Y\subset{\mathbb{P}}_{\mathbb{k}}^{m_{1}}\times_{\mathbb{k}}\cdots\times_{\mathbb{k}}{\mathbb{P}}_{\mathbb{k}}^{m_{p}}$
be the closure of the image of $\mathcal{G}$ and
$\Gamma=\overline{\\{(\mathbf{x},\mathcal{G}(\mathbf{x}))\\}}\subset{\mathbb{P}}_{\mathbb{k}}^{r}\times_{\mathbb{k}}{\mathbb{P}}_{\mathbb{k}}^{m_{1}}\times_{\mathbb{k}}\cdots\times_{\mathbb{k}}{\mathbb{P}}_{\mathbb{k}}^{m_{p}}$
the closure of the graph of $\mathcal{G}$. Let ${\mathbb{P}}$ be the
multiprojective space
${\mathbb{P}}:={\mathbb{P}}_{\mathbb{k}}^{m_{1}}\times_{\mathbb{k}}\cdots\times_{\mathbb{k}}{\mathbb{P}}_{\mathbb{k}}^{m_{p}}$.

With respect to the above setting, we call
$Y\subset{\mathbb{P}}_{\mathbb{k}}^{m_{1}}\times_{\mathbb{k}}\cdots\times_{\mathbb{k}}{\mathbb{P}}_{\mathbb{k}}^{m_{p}}$
the _nonlinear multiview variety_ associated to $\mathcal{G}$. Note that, if
$\delta_{i}=1$ for all $i$, then we are in the classical case of _multiview
varieties_.

###  3.1. Rational maps and some related algebras

In this subsection, we recall some basic facts regarding rational maps. The
_(multigraded) Rees algebra_ of the ideals $I_{1},\ldots,I_{p}$ is given by

$\mathcal{R}(I_{1},\ldots,I_{p}):=R[I_{1}t_{1},\ldots,I_{p}t_{p}]=\bigoplus_{n_{1}\geq
0,\ldots,n_{p}\geq 0}I_{1}^{n_{1}}\dots I_{p}^{n_{p}}t_{1}^{n_{1}}\dots
t_{p}^{n_{p}}\;\subset\;R[t_{1},\ldots,t_{p}].$

Let $\mathfrak{A}$ be the standard $\mathbb{N}^{p+1}$-graded polynomial ring

$\mathfrak{A}\,:=\,R\otimes_{\mathbb{k}}\mathbb{k}[\mathbf{y}_{1}]\otimes_{\mathbb{k}}\cdots\otimes_{\mathbb{k}}\mathbb{k}[\mathbf{y}_{p}]\,=\,R\otimes_{\mathbb{k}}\mathbb{k}[y_{1,0},y_{1,1},\ldots,y_{1,m_{1}}]\otimes_{\mathbb{k}}\cdots\otimes_{\mathbb{k}}\mathbb{k}[y_{p,0},y_{p,1},\ldots,y_{p,m_{1}}],$

where $\deg(x_{i})={\mathbf{e}}_{1}\in\mathbb{N}^{p+1}$ for all $0\leq i\leq
r$, and $\deg(y_{i,j})={\mathbf{e}}_{i+1}\in\mathbb{N}^{p+1}$ for all $1\leq
i\leq p,0\leq j\leq m_{i}$. By setting
$\deg(t_{i})=-\delta_{i}{\mathbf{e}}_{1}+{\mathbf{e}}_{i+1}\in\mathbb{N}^{p+1}$
for all $1\leq i\leq p$, the Rees algebra $\mathcal{R}(I_{1},\ldots,I_{p})$
inherits a natural $\mathbb{N}^{p+1}$-graded structure. And so, we can present
$\mathcal{R}(I_{1},\ldots,I_{p})$ as quotient of $\mathfrak{A}$ via the
following $\mathbb{N}^{p+1}$-homogeneous surjective $R$-homomorphism

$\mathfrak{A}\twoheadrightarrow\mathcal{R}(I_{1},\ldots,I_{p}),\qquad
y_{i,j}\mapsto f_{i,j}t_{i}.$

Thus, the $\mathbb{N}^{p+1}$-graded structure of
$\mathcal{R}(I_{1},\ldots,I_{p})$ is given by

$\mathcal{R}(I_{1},\ldots,I_{p})\;=\;\bigoplus_{c\in\mathbb{N},\;{\mathbf{n}}\in\mathbb{N}^{p}}{\big{[}\mathcal{R}(I_{1},\ldots,I_{p})\big{]}}_{c,{\mathbf{n}}}\
,$

where each $\mathbb{N}^{p+1}$-graded part is

${\big{[}\mathcal{R}(I_{1},\ldots,I_{p})\big{]}}_{c,{\mathbf{n}}}\;=\;{\big{[}\mathbf{I}^{\mathbf{n}}\big{]}}_{c+{\mathbf{n}}\cdot\delta}\mathbf{t}^{\mathbf{n}}\;=\;{\big{[}I_{1}^{n_{1}}\cdots
I_{p}^{n_{p}}\big{]}}_{c+n_{1}\delta_{1}+\cdots+n_{p}\delta_{p}}t_{1}^{n_{1}}\cdots
t_{p}^{n_{p}}.$

Since we are primarily interested in the $R$-grading of the Rees algebra, we
fix the following notation.

######  Notation .

Let $\mathbb{M}$ be an $\mathbb{N}^{p+1}$-graded module over $\mathfrak{A}$.
We denote by ${\left[\mathbb{M}\right]}_{c}$ the “one-sided” $R$-graded part
${\left[\mathbb{M}\right]}_{c}=\bigoplus_{{\mathbf{n}}\in{\mathbb{Z}}^{p}}{\left[\mathbb{M}\right]}_{c,{\mathbf{n}}}.$
Then, ${\left[\mathbb{M}\right]}_{c}$ has a natural $\mathbb{N}^{p}$-graded
structure, and its ${\mathbf{n}}$-th graded part is given by
${\big{[}{\left[\mathbb{M}\right]}_{c}\big{]}}_{\mathbf{n}}={\left[\mathbb{M}\right]}_{c,{\mathbf{n}}}$
where ${\mathbf{n}}\in\mathbb{N}^{p}$.

The _(multigraded) special fiber ring_ of the ideals $I_{1},\ldots,I_{p}$ is
given by

$\mathfrak{F}(I_{1},\ldots,I_{p})\,:=\,\big{[}\mathcal{R}(I_{1},\ldots,I_{p})\big{]}_{0}\,\cong\,\mathbb{k}\big{[}[I_{1}]_{\delta_{1}}t_{1},\ldots,[I_{p}]_{\delta_{p}}t_{p}\big{]}\,\cong\,\bigoplus_{n_{i}\geq
0}\left[I_{1}^{n_{1}}\cdots
I_{p}^{n_{p}}\right]_{n_{1}\delta_{1}+\cdots+n_{p}\delta_{p}}.$

Note that $\mathfrak{F}(I_{1},\ldots,I_{p})$ is naturally a standard
$\mathbb{N}^{p}$-graded $\mathbb{k}$-algebra. As the ideals $I_{i}$’s are
equigenerated in degrees $\delta_{i}$’s, Nakayama’s lemma yields the natural
isomorphism
$\mathfrak{F}(I_{1},\ldots,I_{p})\cong\mathcal{R}(I_{1},\ldots,I_{p})\otimes_{R}R/\mathfrak{m}$.

In the following proposition, we recall the known relationships between the
rational map $\mathcal{G}$ and the algebras $\mathcal{R}(I_{1},\ldots,I_{p})$
and $\mathfrak{F}(I_{1},\ldots,I_{p})$.

######  Proposition .

  1. (i)

The closure of the image of $\mathcal{G}$ coincides with
$Y=\text{MultiProj}(\mathfrak{F}(I_{1},\ldots,I_{p}))$.

  2. (ii)

The closure of the graph of $\mathcal{G}$ coincides with
$\Gamma=\text{MultiProj}(\mathcal{R}(I_{1},\ldots,I_{p}))$.

###### Proof.

For details, see e.g. [DEGREE_SPECIALIZATION, §3]. ∎

We will now recall some equivalent conditions for a rational map to be
generically finite. For any homogeneous ideal $J\subset R$, we denote by
$\ell(J):=\dim\left(\mathcal{R}(J)/{\mathfrak{m}}\mathcal{R}(J)\right)$ the
_analytic spread_ of $J$. We have the following equality

$\dim(Y)=\dim\left(\mathbb{k}\Big{[}[I_{1}]_{\delta_{1}}t_{1},\ldots,[I_{p}]_{\delta_{p}}t_{p}\Big{]}\right)-p.$

The Segre embedding yields the isomorphism

$Y=\text{MultiProj}\left(\mathbb{k}\Big{[}[I_{1}]_{\delta_{1}}t_{1},\ldots,[I_{p}]_{\delta_{p}}t_{p}\Big{]}\right)\cong\text{Proj}\left(\mathbb{k}\Big{[}[I_{1}\cdots
I_{p}]_{\delta_{1}+\cdots+\delta_{p}}\Big{]}\right).$

Thus, we obtain that $\dim(Y)=\ell(I_{1}\cdots I_{p})-1$. Finally, we say that
the rational map $\mathcal{G}$ is _generically finite_ when any of the
following equivalent conditions is satisfied:

  1. (i)

$\left[K({\mathbb{P}}_{\mathbb{k}}^{r}):K(Y)\right]<\infty$ where
$K({\mathbb{P}}_{\mathbb{k}}^{r})$ and $K(Y)$ denote the fields of rational
functions of ${\mathbb{P}}_{\mathbb{k}}^{r}$ and $Y$, respectively.

  2. (ii)

$\dim(Y)=r$.

  3. (iii)

$\ell(I_{1}\cdots I_{p})-1=r$.

Whenever $\mathcal{G}$ is generically finite, we define the degree of
$\mathcal{G}:{\mathbb{P}}_{\mathbb{k}}^{r}\dashrightarrow{\mathbb{P}}_{\mathbb{k}}^{m_{1}}\times_{\mathbb{k}}\cdots\times_{\mathbb{k}}{\mathbb{P}}_{\mathbb{k}}^{m_{p}}$
to be

$\deg(\mathcal{G}):=\left[K({\mathbb{P}}_{\mathbb{k}}^{r}):K(Y)\right].$

The map $\mathcal{G}$ is said to be _birational_ if $\deg(\mathcal{G})=1$. If
the degree of the field extension $K({\mathbb{P}}_{\mathbb{k}}^{r})|K(Y)$ is
infinite, then we say that $\mathcal{G}$ has no well-defined degree and, by
convention, we write $\deg(\mathcal{G})=0$.

###  3.2. A multigraded version of the saturated special fiber ring

Our main tool to study the rational map
$\mathcal{G}:{\mathbb{P}}_{\mathbb{k}}^{r}\dashrightarrow{\mathbb{P}}_{\mathbb{k}}^{m_{1}}\times_{\mathbb{k}}\cdots\times_{\mathbb{k}}{\mathbb{P}}_{\mathbb{k}}^{m_{p}}$
is the following algebra.

######  Definition .

The _(multigraded) saturated special fiber ring_ of the ideals
$I_{1},\ldots,I_{p}$ is the $\mathbb{N}^{p}$-graded $\mathbb{k}$-algebra given
by

$\widetilde{\mathfrak{F}}(I_{1},\ldots,I_{p})\,:=\,\bigoplus_{{\mathbf{n}}\in\mathbb{N}^{p}}\Big{[}\left(\mathbf{I}^{\mathbf{n}}:_{R}{\mathfrak{m}}^{\infty}\right)\Big{]}_{{\mathbf{n}}\cdot\delta}=\bigoplus_{n_{1}\geq
0,\ldots,n_{p}\geq 0}\Big{[}\left(I_{1}^{n_{1}}\cdots
I_{p}^{n_{p}}:_{R}{\mathfrak{m}}^{\infty}\right)\Big{]}_{n_{1}\delta_{1}+\cdots+n_{p}\delta_{p}}.$

An important result regarding the saturated special fiber ring is the
following.

######  Proposition .

We have that $\widetilde{\mathfrak{F}}(I_{1},\ldots,I_{p})$ is a finitely
generated $\mathbb{N}^{p}$-graded module over the standard
$\mathbb{N}^{p}$-graded $\mathbb{k}$-algebra
$\mathfrak{F}(I_{1},\ldots,I_{p})$. In particular,
$\widetilde{\mathfrak{F}}(I_{1},\ldots,I_{p})$ has a Hilbert polynomial
$P_{\widetilde{\mathfrak{F}}(I_{1},\ldots,I_{p})}(\mathbf{t})$.

###### Proof.

The proof is analogous to the proof of [MULTPROJ, Proposition 2.7]. ∎

Our main result is the following theorem, which says that the mixed
multiplicities of the saturated special fiber ring are equal to the product of
the multidegrees of $Y$ and the degree of $\mathcal{G}$. This provides an
extension [MULTPROJ, Theorem 2.4] for nonlinear multiview varieties.

######  Theorem 3.1.

Assume Section missing 3 and suppose that
$\mathcal{G}:{\mathbb{P}}_{\mathbb{k}}^{r}\dashrightarrow{\mathbb{P}}_{\mathbb{k}}^{m_{1}}\times_{\mathbb{k}}\cdots\times_{\mathbb{k}}{\mathbb{P}}_{\mathbb{k}}^{m_{p}}$
is generically finite. Then, the following statements hold:

  1. (i)

For all ${\mathbf{d}}\in\mathbb{N}^{p}$ with $\lvert{\mathbf{d}}\rvert=r$, we
have the equality

$e\big{(}{\mathbf{d}},\widetilde{\mathfrak{F}}(I_{1},\ldots,I_{p})\big{)}\;=\;\deg(\mathcal{G})\cdot\deg_{\mathbb{P}}^{\mathbf{d}}(Y).$

  2. (ii)

Under the additional condition of $\mathfrak{F}(I_{1},\ldots,I_{p})$ being
integrally closed, $\mathcal{G}$ is birational if and only if
$\widetilde{\mathfrak{F}}(I_{1},\ldots,I_{p})=\mathfrak{F}(I_{1},\ldots,I_{p})$.

###### Proof.

(i) For ease of notation, let $S:=\mathfrak{F}(I_{1},\ldots,I_{p})$ and
$T:=\widetilde{\mathfrak{F}}(I_{1},\ldots,I_{p})$. We consider the following
functions

$F_{S}({\mathbf{n}})=\lim_{n\rightarrow\infty}\frac{\dim_{\mathbb{k}}\left(\left[S\right]_{n\cdot{\mathbf{n}}}\right)}{n^{r}}\quad\text{
and }\quad
F_{T}({\mathbf{n}})=\lim_{n\rightarrow\infty}\frac{\dim_{\mathbb{k}}\left(\left[T\right]_{n\cdot{\mathbf{n}}}\right)}{n^{r}}.$

These functions coincide with the top-degree parts of the Hilbert polynomials
of $S$ and $T$. In other words, we can write

$F_{S}({\mathbf{n}})=\sum_{{\mathbf{d}}\in\mathbb{N}^{p},\,\lvert{\mathbf{d}}\rvert=r}\frac{\deg^{\mathbf{d}}(Y)}{{\mathbf{d}}!}\,{\mathbf{n}}^{\mathbf{d}}\quad\text{
and }\quad
F_{T}({\mathbf{n}})=\sum_{{\mathbf{d}}\in\mathbb{N}^{p},\,\lvert{\mathbf{d}}\rvert=r}\frac{e\big{(}{\mathbf{d}},\widetilde{\mathfrak{F}}(I_{1},\ldots,I_{p})\big{)}}{{\mathbf{d}}!}\,{\mathbf{n}}^{\mathbf{d}}.$

Since we have the equalities

$S^{({\mathbf{n}})}=\bigoplus_{n=0}^{\infty}\left[I_{1}^{nn_{1}}\cdots
I_{p}^{nn_{p}}\right]_{nn_{1}\delta_{1}+\cdots+nn_{p}\delta_{p}}=\mathfrak{F}(I_{1}^{n_{1}}\cdots
I_{p}^{n_{p}})$

and

$T^{({\mathbf{n}})}=\bigoplus_{n=0}^{\infty}\left[\left(I_{1}^{nn_{1}}\cdots
I_{p}^{nn_{p}}:_{R}{\mathfrak{m}}^{\infty}\right)\right]_{nn_{1}\delta_{1}+\cdots+nn_{p}\delta_{p}}=\widetilde{\mathfrak{F}}(I_{1}^{n_{1}}\cdots
I_{p}^{n_{p}}),$

it follows that the functions $F_{S}$ and $F_{T}$ can be obtained by computing
the multiplicities of the special fiber ring and the saturated special fiber
ring of products of powers of the ideals $I_{i}$’s. More precisely, we have
that

$F_{S}({\mathbf{n}})=e\left(\mathfrak{F}(I_{1}^{n_{1}}\cdots
I_{p}^{n_{p}})\right)\quad\text{ and }\quad
F_{T}({\mathbf{n}})=e\left(\widetilde{\mathfrak{F}}(I_{1}^{n_{1}}\cdots
I_{p}^{n_{p}})\right).$

For each ${\mathbf{n}}\in{\mathbb{Z}}_{+}^{r}$, we compose Veronese and Segre
embeddings to obtain the following natural embedding

$\Delta:{\mathbb{P}}_{\mathbb{k}}^{m_{1}}\times_{\mathbb{k}}\cdots\times_{\mathbb{k}}{\mathbb{P}}_{\mathbb{k}}^{m_{p}}\;\hookrightarrow\;{\mathbb{P}}_{\mathbb{k}}^{\binom{n_{1}+m_{1}}{m_{1}}-1}\times_{\mathbb{k}}\cdots\times_{\mathbb{k}}{\mathbb{P}}_{\mathbb{k}}^{\binom{n_{p}+m_{p}}{m_{p}}-1}\;\hookrightarrow\;{\mathbb{P}}_{\mathbb{k}}^{m}$

where $m=\binom{n_{1}+m_{1}}{m_{1}}\cdots\binom{n_{p}+m_{p}}{m_{p}}-1$. Hence,
we obtain the following commutative diagram

${{\mathbb{P}}_{\mathbb{k}}^{r}}$${Y}$${Y_{\mathbf{n}}^{\prime},}$$\cong$$\mathcal{G}$$\mathcal{G}_{\mathbf{n}}^{\prime}$

where $Y_{\mathbf{n}}^{\prime}\subset{\mathbb{P}}_{\mathbb{k}}^{m}$ is the
image of
$Y\subset{\mathbb{P}}_{\mathbb{k}}^{m_{1}}\times_{\mathbb{k}}\cdots\times_{\mathbb{k}}{\mathbb{P}}_{\mathbb{k}}^{m_{p}}$
under the embedding $\Delta$ and
$\mathcal{G}_{\mathbf{n}}^{\prime}:{\mathbb{P}}_{\mathbb{k}}^{r}\dashrightarrow
Y_{\mathbf{n}}^{\prime}\subset{\mathbb{P}}_{\mathbb{k}}^{m}$ is the rational
map induced by the set of generators naturally obtained for the ideal
$I_{1}^{n_{1}}\cdots I_{p}^{n_{p}}$. It is clear that
$\deg(\mathcal{G})=\deg(\mathcal{G}_{\mathbf{n}}^{\prime})$. So, for each
${\mathbf{n}}\in{\mathbb{Z}}_{+}^{p}$, [MULTPROJ, Theorem 2.4] may be applied
to the rational map $\mathcal{G}_{\mathbf{n}}^{\prime}$ and gives the
following equalities

$F_{T}({\mathbf{n}})=e\left(\widetilde{\mathfrak{F}}(I_{1}^{n_{1}}\cdots
I_{p}^{n_{p}})\right)=\deg(\mathcal{G}^{\prime})\cdot\deg(Y_{\mathbf{n}}^{\prime})=\deg(\mathcal{G})\cdot
e\left(\mathfrak{F}(I_{1}^{n_{1}}\cdots
I_{p}^{n_{p}})\right)=\deg(\mathcal{G})\cdot F_{S}({\mathbf{n}}).$

Finally, by comparing the coefficients of the polynomials
$F_{S}({\mathbf{n}})$ and $F_{T}({\mathbf{n}})$, the claimed equality

$e\big{(}{\mathbf{d}},\widetilde{\mathfrak{F}}(I_{1},\ldots,I_{p})\big{)}\;=\;\deg(\mathcal{G})\cdot\deg^{\mathbf{d}}(Y)$

follows for all ${\mathbf{d}}\in\mathbb{N}^{p}$ with
$\lvert{\mathbf{d}}\rvert=r$.

(ii) The proof follows similarly to the proof of [MULTPROJ, Theorem 2.4(iv)].
∎

###  3.3. A cohomological formula relating the degree of the rational map and
the multidegrees of the image

Throughout this subsection, we continue using Section missing 3. Here, we give
a formula that relates the degree of $\mathcal{G}$ with the mixed
multiplicities of
${\left[\text{H}_{{\mathfrak{m}}}^{1}(\mathcal{R}(I_{1},\ldots,I_{p}))\right]}_{0}$
and the multidegrees of the image $Y$. Notice that
${\left[\text{H}_{{\mathfrak{m}}}^{1}(\mathcal{R}(I_{1},\ldots,I_{p}))\right]}_{0}$
is a finitely generated $\mathbb{N}^{p}$-graded module over the special fiber
ring $\mathfrak{F}(I_{1},\ldots,I_{p})$, and so we can consider its mixed
multiplicities. The following result extends [MULTPROJ, Corollary 2.12] into
the current setting.

######  Corollary .

Assume Section missing 3 and suppose that
$\mathcal{G}:{\mathbb{P}}_{\mathbb{k}}^{r}\dashrightarrow{\mathbb{P}}_{\mathbb{k}}^{m_{1}}\times_{\mathbb{k}}\cdots\times_{\mathbb{k}}{\mathbb{P}}_{\mathbb{k}}^{m_{p}}$
is generically finite. Then, for all ${\mathbf{d}}\in\mathbb{N}^{p}$ with
$\lvert{\mathbf{d}}\rvert=r$, we have the equality

$\deg^{\mathbf{d}}(Y)\cdot\big{(}\deg(\mathcal{G})-1\big{)}\,=\,e\left({\mathbf{d}},{\left[\text{H}_{{\mathfrak{m}}}^{1}(\mathcal{R}(I_{1},\ldots,I_{p}))\right]}_{0}\right).$

In particular, $\mathcal{G}$ is birational if and only if
$\dim\left(\text{Supp}\left({\left[\text{H}_{{\mathfrak{m}}}^{1}(\mathcal{R}(I_{1},\ldots,I_{p}))\right]}_{0}\right)\cap
Y\right)<r$.

###### Proof.

We have the short exact sequence

$0\,\rightarrow\,\bigoplus_{{\mathbf{n}}\in\mathbb{N}^{p}}\big{[}\mathbf{I}^{\mathbf{n}}\big{]}_{{\mathbf{n}}\cdot\delta}\,\rightarrow\,\bigoplus_{{\mathbf{n}}\in\mathbb{N}^{p}}\Big{[}\left(\mathbf{I}^{\mathbf{n}}:_{R}{\mathfrak{m}}^{\infty}\right)\Big{]}_{{\mathbf{n}}\cdot\delta}\,\rightarrow\,\bigoplus_{{\mathbf{n}}\in\mathbb{N}^{p}}\Big{[}\text{H}_{{\mathfrak{m}}}^{0}\left(R/\mathbf{I}^{\mathbf{n}}\right)\Big{]}_{{\mathbf{n}}\cdot\delta}\,\rightarrow\,0,$

which can then be written as

$0\rightarrow\mathfrak{F}(I_{1},\ldots,I_{p})\rightarrow\widetilde{\mathfrak{F}}(I_{1},\ldots,I_{p})\rightarrow{\left[\text{H}_{{\mathfrak{m}}}^{1}(\mathcal{R}(I_{1},\ldots,I_{p}))\right]}_{0}\rightarrow
0.$

The additivity of mixed multiplicities gives the equality

$e({\mathbf{d}},\widetilde{\mathfrak{F}}(I_{1},\ldots,I_{p}))=e({\mathbf{d}},\mathfrak{F}(I_{1},\ldots,I_{p}))+e({\mathbf{d}},{\left[\text{H}_{{\mathfrak{m}}}^{1}(\mathcal{R}(I_{1},\ldots,I_{p}))\right]}_{0}),$

and so the result follows from Theorem 3.1. ∎

###  3.4. A degree formula for rational maps

Here, we study the case when
$\mathcal{G}:{\mathbb{P}}_{\mathbb{k}}^{r}\dashrightarrow{\mathbb{P}}_{\mathbb{k}}^{m_{1}}\times_{\mathbb{k}}\cdots\times_{\mathbb{k}}{\mathbb{P}}_{\mathbb{k}}^{m_{p}}$
is not defined only in a finite number of points. In particular, we provide a
formula that relates the degree of $\mathcal{G}$, the multidegrees of the
image, and the mixed multiplicities of the base points. This can be seen as an
extension of a known degree formula, see e.g. [Laurent_Jouanolou_Closed_Image,
Theorem 2.5], [Sim_Ulr_Vasc_mult, Theorem 6.6] and [MULTPROJ, Theorem 3.3]).
This formula could also be derived by utilizing [FULTON_INTERSECTION_THEORY,
Proposition 4.4]. We also give a general upper bound for the degree of the
image for arbitrary rational maps that are generically finite.

Hereafter, we use the same notations and conventions of Section missing 3. We
have that the _base locus of $\mathcal{G}$_ (i.e. the points where
$\mathcal{G}$ is not well-defined) is given by

$\mathcal{B}(\mathcal{G}):=V(I_{1}\cdots
I_{p})\,\subset\,{\mathbb{P}}_{\mathbb{k}}^{r}.$

We then have that $\dim(\mathcal{B}(\mathcal{G}))=0$ (i.e. the base locus is
_finite_) if and only if $\dim(R/I_{i})\leq 1$ for all $1\leq i\leq p$.
Assuming that $\dim(\mathcal{B}(\mathcal{G}))=0$, we then have the equalities

$\displaystyle\dim_{\mathbb{k}}\Big{(}\text{H}^{0}\Big{(}{\mathbb{P}}_{\mathbb{k}}^{r},\mathcal{O}_{{\mathbb{P}}_{\mathbb{k}}^{r}}/(I_{1}^{n_{1}}\cdots
I_{p}^{n_{p}})^{\sim}\Big{)}\Big{)}$
$\displaystyle=\sum_{\mathfrak{p}\in\mathcal{B}(\mathcal{G})}\dim_{\mathbb{k}}\Big{(}{\big{(}\mathcal{O}_{{\mathbb{P}}_{\mathbb{k}}^{r}}/(I_{1}^{n_{1}}\cdots
I_{p}^{n_{p}})^{\sim}\big{)}}_{\mathfrak{p}}\Big{)}$
$\displaystyle=\sum_{\mathfrak{p}\in\mathcal{B}(\mathcal{G})}\left[\kappa(\mathfrak{p}):\mathbb{k}\right]\cdot\text{length}_{\mathcal{O}_{{\mathbb{P}}_{\mathbb{k}}^{r},\mathfrak{p}}}\Big{(}{\big{(}\mathcal{O}_{{\mathbb{P}}_{\mathbb{k}}^{r}}/(I_{1}^{n_{1}}\cdots
I_{p}^{n_{p}})^{\sim}\big{)}}_{\mathfrak{p}}\Big{)}$
$\displaystyle=\sum_{\mathfrak{p}\in\mathcal{B}(\mathcal{G})}\left[\kappa(\mathfrak{p}):\mathbb{k}\right]\cdot\text{length}_{R_{\mathfrak{p}}}\Big{(}R_{\mathfrak{p}}/(I_{1}^{n_{1}}\cdots
I_{p}^{n_{p}})R_{\mathfrak{p}}\Big{)}$

for all $n_{1},\ldots,n_{p}\in\mathbb{N}$, where $\kappa(\mathfrak{p})$
denotes the residue field of the local ring
$\mathcal{O}_{{\mathbb{P}}_{\mathbb{k}}^{r},\mathfrak{p}}$. Since we have
$\dim(R_{\mathfrak{p}}/I_{i}R_{\mathfrak{p}})=0$ by assumption, it follows
that the function

$P_{R_{\mathfrak{p}};I_{1},\ldots,I_{p}}(n_{1},\ldots,n_{p})\;:=\;\lim\limits_{n\rightarrow\infty}\frac{\text{length}_{R_{\mathfrak{p}}}\Big{(}R_{\mathfrak{p}}/(I_{1}^{nn_{1}}\cdots
I_{p}^{nn_{p}})R_{\mathfrak{p}}\Big{)}}{n^{r}}$

is a polynomial in $\mathbb{Q}[n_{1},\ldots,n_{p}]$. For further details, we
refer the reader to the survey [TRUNG_VERMA_SURVEY] and Chapter 17 of the book
[huneke2006integral]. The polynomial
$P_{R_{\mathfrak{p}};I_{1},\ldots,I_{p}}(n_{1},\ldots,n_{p})$ is of degree $r$
if there is some $i$ with
$I_{i}R_{\mathfrak{p}}\neq\mathfrak{p}R_{\mathfrak{p}}$, and the zero
polynomial otherwise. We write the polynomial
$P_{R_{\mathfrak{p}};I_{1},\ldots,I_{p}}$ as

$P_{R_{\mathfrak{p}};I_{1},\ldots,I_{p}}(n_{1},\ldots,n_{p})\,=\,\sum_{{\mathbf{d}}=(d_{1},\ldots,d_{p})\in\mathbb{N}^{p},\,|{\mathbf{d}}|=r}\frac{e_{{\mathbf{d}}}(R_{\mathfrak{p}};I_{1},\ldots,I_{p})}{d_{1}!\cdots
d_{p}!}n_{1}^{d_{1}}\cdots n_{p}^{d_{p}},$

where the numbers $e_{{\mathbf{d}}}(R_{\mathfrak{p}};I_{1},\ldots,I_{p})$ are
nonnegative integers called the mixed multiplicities of $R_{\mathfrak{p}}$
with respect to $I_{1},\ldots,I_{p}$. By summarizing the above results and
discussion, it follows that the following expression

$P_{\mathcal{B}(\mathcal{G})}(n_{1},\ldots,n_{p})\;:=\;\lim\limits_{n\rightarrow\infty}\frac{\dim_{\mathbb{k}}\Big{(}\text{H}^{0}\Big{(}{\mathbb{P}}_{\mathbb{k}}^{r},\mathcal{O}_{{\mathbb{P}}_{\mathbb{k}}^{r}}/(I_{1}^{nn_{1}}\cdots
I_{p}^{nn_{p}})^{\sim}\Big{)}\Big{)}}{n^{r}}$

is a polynomial in $\mathbb{Q}[n_{1},\ldots,n_{p}]$. We have that
$P_{\mathcal{B}(\mathcal{G})}$ is a polynomial of degree $r$ if
$\mathcal{B}(\mathcal{G})\neq\emptyset$, and the zero polynomial otherwise.
Furthermore, we can write $P_{\mathcal{B}(\mathcal{G})}$ as

$P_{\mathcal{B}(\mathcal{G})}(n_{1},\ldots,n_{p})\,=\,\sum_{{\mathbf{d}}=(d_{1},\ldots,d_{p})\in\mathbb{N}^{p},\,|{\mathbf{d}}|=r}\frac{e_{{\mathbf{d}}}(\mathcal{B}(\mathcal{G}))}{d_{1}!\cdots
d_{p}!}n_{1}^{d_{1}}\cdots n_{p}^{d_{p}},$

where the numbers $e_{{\mathbf{d}}}(\mathcal{B}(\mathcal{G}))$ are nonnegative
integers and given by

$e_{{\mathbf{d}}}(\mathcal{B}(\mathcal{G}))\;:=\;\sum_{\mathfrak{p}\in\mathcal{B}(\mathcal{G})}\left[\kappa(\mathfrak{p}):\mathbb{k}\right]\cdot
e_{{\mathbf{d}}}(R_{\mathfrak{p}};I_{1},\ldots,I_{p}).$

So, the nonnegative integers $e_{{\mathbf{d}}}(\mathcal{B}(\mathcal{G}))$
gather the mixed multiplicities of the base points, and we call them the
_mixed multiplicities of the base locus of $\mathcal{G}$_.

We are now ready to state the main result of this subsection.

######  Theorem 3.2.

Assume Section missing 3 and suppose that
$\mathcal{G}:{\mathbb{P}}_{\mathbb{k}}^{r}\dashrightarrow{\mathbb{P}}_{\mathbb{k}}^{m_{1}}\times_{\mathbb{k}}\cdots\times_{\mathbb{k}}{\mathbb{P}}_{\mathbb{k}}^{m_{p}}$
is generically finite. Then, for all
${\mathbf{d}}=(d_{1},\ldots,d_{p})\in\mathbb{N}^{p}$ with
$\lvert{\mathbf{d}}\rvert=r$, the following statements hold:

  1. (i)

$\deg_{\mathbb{P}}^{\mathbf{d}}(Y)\deg(\mathcal{G})\,\leq\,\delta_{1}^{d_{1}}\cdots\delta_{p}^{d_{p}}$.

  2. (ii)

If $\dim(\mathcal{B}(\mathcal{G}))=0$, then we have the equality

$\delta_{1}^{d_{1}}\cdots\delta_{p}^{d_{p}}\,=\,\deg_{\mathbb{P}}^{\mathbf{d}}(Y)\deg(\mathcal{G})+e_{\mathbf{d}}(\mathcal{B}(\mathcal{G})).$

###### Proof.

For any ${\mathbf{n}}=(n_{1},\ldots,n_{p})\in{\mathbb{Z}}_{+}^{p}$, we have
the exact sequence of sheaves

$0\rightarrow{(I_{1}^{n_{1}}\cdots
I_{p}^{n_{p}})}^{\sim}({\mathbf{n}}\cdot\delta)\rightarrow\mathcal{O}_{{\mathbb{P}}_{\mathbb{k}}^{r}}({\mathbf{n}}\cdot\delta)\rightarrow\frac{\mathcal{O}_{{\mathbb{P}}_{\mathbb{k}}^{r}}}{{(I_{1}^{n_{1}}\cdots
I_{p}^{n_{p}})}^{\sim}}({\mathbf{n}}\cdot\delta)\rightarrow 0,$

that yields the following exact sequence in cohomology

$0\rightarrow\text{H}^{0}\big{(}{\mathbb{P}}_{\mathbb{k}}^{r},(\mathbf{I}^{\mathbf{n}})^{\sim}({\mathbf{n}}\cdot\delta)\big{)}\rightarrow\text{H}^{0}\big{(}{\mathbb{P}}_{\mathbb{k}}^{r},\mathcal{O}_{{\mathbb{P}}_{\mathbb{k}}^{r}}({\mathbf{n}}\cdot\delta)\big{)}\rightarrow\text{H}^{0}\big{(}{\mathbb{P}}_{\mathbb{k}}^{r},\frac{\mathcal{O}_{{\mathbb{P}}_{\mathbb{k}}^{r}}}{(\mathbf{I}^{\mathbf{n}})^{\sim}}({\mathbf{n}}\cdot\delta)\big{)}\rightarrow\text{H}^{1}\big{(}{\mathbb{P}}_{\mathbb{k}}^{r},(\mathbf{I}^{\mathbf{n}})^{\sim}({\mathbf{n}}\cdot\delta)\big{)}\rightarrow
0.$

Notice that
$\dim_{\mathbb{k}}(\text{H}^{0}({\mathbb{P}}_{\mathbb{k}}^{r},\mathcal{O}_{{\mathbb{P}}_{\mathbb{k}}^{r}}({\mathbf{n}}\cdot\delta)))=\binom{{\mathbf{n}}\cdot\delta+r}{r}$
is a polynomial of degree $r$ in $\mathbb{Q}[{\mathbf{n}}]$. By
[DEGREE_SPECIALIZATION, Theorem 4.4], we have that

$\dim\left(\left[\text{H}_{{\mathfrak{m}}}^{i}(\mathcal{R}(I_{1},\ldots,I_{p}))\right]_{0}\right)\leq(\dim(R)+p)-i=r+1+p-i.$

This implies that for any $i\geq 1$ and
${\mathbf{n}}=(n_{1},\ldots,n_{p})\in{\mathbb{Z}}_{+}^{p}$ with $n_{i}\gg 0$,
the expression

$\dim_{\mathbb{k}}\big{(}\text{H}^{i}\big{(}{\mathbb{P}}_{\mathbb{k}}^{r},{(I_{1}^{n_{1}}\cdots
I_{p}^{n_{p}})}^{\sim}({\mathbf{n}}\cdot\delta)\big{)}\big{)}=\dim_{\mathbb{k}}\big{(}{\left[\text{H}_{{\mathfrak{m}}}^{i+1}(\mathcal{R}(I_{1},\ldots,I_{p}))\right]}_{0,{\mathbf{n}}}\big{)}$

is a polynomial in ${\mathbf{n}}$ of degree strictly less than $r$. So, for
any ${\mathbf{n}}=(n_{1},\ldots,n_{p})\in{\mathbb{Z}}_{+}^{p}$ we have the
equalities

$\displaystyle\lim_{n\rightarrow\infty}\frac{h^{0}\big{(}{\mathbb{P}}_{\mathbb{k}}^{r},(\mathbf{I}^{n\cdot{\mathbf{n}}})^{\sim}(n\cdot{\mathbf{n}}\cdot\delta)\big{)}-h^{1}\big{(}{\mathbb{P}}_{\mathbb{k}}^{r},(\mathbf{I}^{n\cdot{\mathbf{n}}})^{\sim}(n\cdot{\mathbf{n}}\cdot\delta)\big{)}}{n^{r}}=\lim_{n\rightarrow\infty}\frac{h^{0}\big{(}{\mathbb{P}}_{\mathbb{k}}^{r},(\mathbf{I}^{n\cdot{\mathbf{n}}})^{\sim}(n\cdot{\mathbf{n}}\cdot\delta)\big{)}}{n^{r}}=F_{T}({\mathbf{n}}),$

where
$F_{T}({\mathbf{n}})=\lim_{n\rightarrow\infty}\frac{\dim_{\mathbb{k}}\left(\left[T\right]_{n\cdot{\mathbf{n}}}\right)}{n^{r}}$
is the polynomial function that encodes the mixed multiplicities of the
saturated special fiber ring $T=\widetilde{\mathfrak{F}}(I_{1},\ldots,I_{p})$.
Since $F_{T}({\mathbf{n}})$ is a homogeneous polynomial of degree $r$ in
$\mathbb{Q}[{\mathbf{n}}]$, it follows that the following function

$Q({\mathbf{n}})\,:=\,\lim_{n\rightarrow\infty}\frac{h^{0}\big{(}{\mathbb{P}}_{\mathbb{k}}^{r},\,\mathcal{O}_{{\mathbb{P}}_{\mathbb{k}}^{r}}/(\mathbf{I}^{n\cdot{\mathbf{n}}})^{\sim}({\mathbf{n}}\cdot\delta)\big{)}}{n^{r}}\,=\,\lim_{n\rightarrow\infty}\frac{\binom{n\cdot{\mathbf{n}}\cdot\delta+r}{r}}{n^{r}}-F_{T}({\mathbf{n}})$

is also a homogeneous polynomial of degree $r$ in $\mathbb{Q}[{\mathbf{n}}]$.
The fact that $Q({\mathbf{n}})\geq 0$ for all ${\mathbf{n}}\in\mathbb{N}^{r}$
implies that the coefficients of $Q({\mathbf{n}})$ are all nonnegative.
Therefore, the equality
$F_{T}({\mathbf{n}})+Q({\mathbf{n}})\,=\,\lim_{n\rightarrow\infty}\frac{\binom{n\cdot{\mathbf{n}}\cdot\delta+r}{r}}{n^{r}}$
and Theorem 3.1 yield the general upper bounds
$\deg_{\mathbb{P}}^{\mathbf{d}}(Y)\deg(\mathcal{G})\,\leq\,\delta_{1}^{d_{1}}\cdots\delta_{p}^{d_{p}}$.
This completes the proof of part (i).

Next, we assume that the base locus of $\mathcal{G}$ is zero-dimensional.
Since $\dim(\mathcal{B}(\mathcal{G}))=0$, we obtain the equalities

$Q({\mathbf{n}})\,:=\,\lim_{n\rightarrow\infty}\frac{h^{0}\big{(}{\mathbb{P}}_{\mathbb{k}}^{r},\,\mathcal{O}_{{\mathbb{P}}_{\mathbb{k}}^{r}}/(\mathbf{I}^{n\cdot{\mathbf{n}}})^{\sim}({\mathbf{n}}\cdot\delta)\big{)}}{n^{r}}=\lim_{n\rightarrow\infty}\frac{h^{0}\big{(}{\mathbb{P}}_{\mathbb{k}}^{r},\,\mathcal{O}_{{\mathbb{P}}_{\mathbb{k}}^{r}}/(\mathbf{I}^{n\cdot{\mathbf{n}}})^{\sim}\big{)}}{n^{r}}=P_{\mathcal{B}(\mathcal{G})}({\mathbf{n}}).$

So, combining the above expressions for $Q({\mathbf{n}})$, we obtain the
following equality of polynomials

$F_{T}({\mathbf{n}})+P_{\mathcal{B}(\mathcal{G})}({\mathbf{n}})\,=\,\lim_{n\rightarrow\infty}\frac{\binom{n\cdot{\mathbf{n}}\cdot\delta+r}{r}}{n^{r}}.$

And so, the result of part (ii) follows from Theorem 3.1. This completes the
proof of the theorem. ∎

As a direct consequence of the above theorem, we obtain the following result.

######  Corollary .

Assume Section missing 3 and suppose that all the ideals $I_{i}$ are
${\mathfrak{m}}$-primary (i.e. the rational map $\mathcal{G}$ is actually a
morphism
$\mathcal{G}:{\mathbb{P}}_{\mathbb{k}}^{r}\rightarrow{\mathbb{P}}_{\mathbb{k}}^{m_{1}}\times_{\mathbb{k}}\cdots\times_{\mathbb{k}}{\mathbb{P}}_{\mathbb{k}}^{m_{p}}$).
Then, for all ${\mathbf{d}}=(d_{1},\ldots,d_{p})\in\mathbb{N}^{p}$ with
$\lvert{\mathbf{d}}\rvert=r$, we have the equality

$\deg_{\mathbb{P}}^{\mathbf{d}}(Y)\deg(\mathcal{G})\,=\,\delta_{1}^{d_{1}}\cdots\delta_{p}^{d_{p}}.$

##  4\. Linear rational maps

In this section, we revisit the case of linear multiview varieties that has
been extensively studied. Here, we use the following setup.

######  Setup .

Assume Section missing 3 and let $\delta_{i}=1$ for each $i$. That is, the
forms $f_{i,j}$ defining the rational map
$\mathcal{G}:{\mathbb{P}}_{\mathbb{k}}^{r}\dashrightarrow{\mathbb{P}}_{\mathbb{k}}^{m_{1}}\times_{\mathbb{k}}\cdots\times_{\mathbb{k}}{\mathbb{P}}_{\mathbb{k}}^{m_{p}}$
are linear. In addition, we will assume that the field is algebraically
closed.

We start by recalling that, in this setting, the mixed multiplicities of the
special fiber ring are either zero or one.

######  Theorem 4.1 ([li2013images, Theorem 1.1] and [conca2019resolution,
Theorem 3.9]).

Assume Section missing 4 and suppose that the map
$\mathcal{G}:{\mathbb{P}}_{\mathbb{k}}^{r}\dashrightarrow{\mathbb{P}}_{\mathbb{k}}^{m_{1}}\times_{\mathbb{k}}\cdots\times_{\mathbb{k}}{\mathbb{P}}_{\mathbb{k}}^{m_{p}}$
is generically finite. Then, the closure $Y$ of the image of $\mathcal{G}$ is
multiplicity-free. That is, $\deg_{\mathbb{P}}^{\mathbf{d}}(Y)\in\\{0,1\\}$
for all ${\mathbf{d}}\in\mathbb{N}^{p}$ with $\lvert{\mathbf{d}}\rvert=r$.

###### Proof.

This result follows directly from Theorem 3.2(i). ∎

Our next observation is the following interesting corollary of Theorem 3.1.

######  Corollary .

Assume Section missing 4 and suppose that
$\mathcal{G}:{\mathbb{P}}_{\mathbb{k}}^{r}\dashrightarrow{\mathbb{P}}_{\mathbb{k}}^{m_{1}}\times_{\mathbb{k}}\cdots\times_{\mathbb{k}}{\mathbb{P}}_{\mathbb{k}}^{m_{p}}$
is generically finite. Then, the saturated special fiber ring and the special
fiber ring coincide. Explicitly, for each
$(n_{1},\dots,n_{p})\in\mathbb{N}^{p}$, we have

$\left[\left(I_{1}^{n_{1}}\cdots
I_{p}^{n_{p}}:_{R}{\mathfrak{m}}^{\infty}\right)\right]_{n_{1}+n_{2}+\dots+n_{p}}=\left[\left(I_{1}^{n_{1}}\cdots
I_{p}^{n_{p}}\right)\right]_{n_{1}+n_{2}+\dots+n_{p}}.$

###### Proof.

By Theorem 4.1 we have that $\mathfrak{F}(I_{1},\dots,I_{p})$ is multiplicity-
free. Hence, by a theorem of Brion, see [conca2018cartwright, Theorem 1.11]
and [brion2002multiplicity, Theorem 2], the special fiber ring
$\mathfrak{F}(I_{1},\dots,I_{p})$ is integrally closed. We may therefore apply
Theorem 3.1(ii) to deduce that
$\mathfrak{F}(I_{1},\dots,I_{p})=\widetilde{\mathfrak{F}}(I_{1},\dots,I_{p})$
if and only if $\mathcal{G}$ is birational. By assumption, all forms $f_{i,j}$
defining $\mathcal{G}$ are linear and so the $\mathcal{G}$ is birational. ∎

We now restrict our attention to the case where the base locus of
$\mathcal{G}$ is zero-dimensional. For each $1\leq i\leq p$, the rational map
$\mathcal{G}_{i}:{\mathbb{P}}_{\mathbb{k}}^{r}\dashrightarrow{\mathbb{P}}_{\mathbb{k}}^{m_{i}}$
is then given by

$(x_{0},\dots,x_{r})\mapsto A_{i}(x_{0},\dots,x_{r})^{T}\quad\text{for some
matrix }A_{i}\in\mathbb{k}^{(m_{i}+1)\times(r+1)}.$

Under the assumption that $\dim(\mathcal{B}(\mathcal{G}))=0$, we set-
theoretically obtain $\mathcal{B}(\mathcal{G})=\\{\text{Ker}(A_{i})\mid 1\leq
i\leq p\\}$ and we have the equivalent conditions $\dim(R/I_{i})\leq 1$ and
$\text{rank}(A_{i})\geq r$. We note that each ideal $I_{i}$ is linear, hence
prime.

######  Remark .

Suppose that $r=3$ and $m_{i}=2$ for all $i$. The assumption that the base
locus is zero-dimensional is equivalent to the defining matrices $A_{i}$ of
the rational map $\mathcal{G}_{i}$ be of rank $3$. In particular, this is the
case studied in [agarwal2019ideals] which we generalize in Section missing 4.

By applying Theorem 3.2, we can give an explicit description of the degrees
${\mathbf{d}}$, where $|{\mathbf{d}}|=r$, such that
$\deg_{\mathbb{P}}^{\mathbf{d}}(Y)=1$.

######  Corollary .

Assume Section missing 4 and suppose that $\dim(\mathcal{B}(\mathcal{G}))=0$.
Let ${\mathbf{d}}=(d_{1},\dots,d_{p})\in\mathbb{N}^{p}$ with
$|{\mathbf{d}}|=r$. Then $\deg_{\mathbb{P}}^{{\mathbf{d}}}(Y)=1$ if and only
if for each point $\mathfrak{p}\in\mathcal{B}(\mathcal{G})$ there exists
$1\leq i\leq p$ such that $I_{i}\neq\mathfrak{p}$ and $d_{i}\geq 1$.

###### Proof.

Fix a point $\mathfrak{p}\in\mathcal{B}(\mathcal{G})$ in the base locus. We
write $I_{i_{1}}=I_{i_{2}}=\dots=I_{i_{t}}=\mathfrak{p}$ for all of the ideals
which correspond to the same point in the base locus. We proceed by
calculating the value of
$e_{{\mathbf{d}}}(R_{\mathfrak{p}};I_{1},\dots,I_{p})$. Note that for each
$1\leq j\leq p$ we have

$I_{j}R_{\mathfrak{p}}=\left\\{\begin{array}[]{ll}\mathfrak{p}R_{\mathfrak{p}}&\text{if
}j\in\\{i_{1},i_{2},\dots,i_{t}\\},\\\
R_{p}&\text{otherwise}.\end{array}\right.$

So, it follows that

$e_{{\mathbf{d}}}(R_{\mathfrak{p}};I_{1},\dots,I_{p})=\begin{cases}e(\mathfrak{p}R_{\mathfrak{p}})=1\quad\text{if
}d_{i_{1}}+d_{i_{2}}+\cdots+d_{i_{t}}=r\\\
0\quad\qquad\qquad\text{otherwise},\end{cases}$

for more details see [huneke2006integral, §17.4].

By Theorem 3.2, we have
$1=\deg_{\mathbb{P}}^{{\mathbf{d}}}(Y)\deg(\mathcal{G})+e_{{\mathbf{d}}}(\mathcal{B}(\mathcal{G})).$
The mixed multiplicities of the base locus are given by
$\sum[\kappa(\mathfrak{p}):\mathbb{k}]e_{{\mathbf{d}}}(R_{\mathfrak{p}};I_{1},\dots,I_{p})$
where the sum is taken over the set of primes
$\mathfrak{p}\in\mathcal{B}(\mathcal{G})$. Since, $\mathbb{k}$ is
algebraically closed, for all $\mathfrak{p}\in\mathcal{B}(\mathcal{G})$, we
have $[\kappa(\mathfrak{p}):\mathbb{k}]=1$. Finally, it follows that
$\deg_{\mathbb{P}}^{\mathbf{d}}(Y)=1$ if and only if for each point
$\mathfrak{p}\in\mathcal{B}(\mathcal{G})$ there exists some $1\leq i\leq p$
such that $I_{i}\neq\mathfrak{p}$ and $d_{i}\geq 1$. ∎

##  5\. Certain special families of rational maps

In this section, we consider some special families of rational maps for which
we give exact formulas for the product of the degree of the rational map with
the multidegrees of its image. We shall extend the computations obtained in
[MULT_SAT_PERF_HT_2, SAT_FIB_GOR_3] for perfect ideals of height two and
Gorenstein ideals of height three. Throughout this section we continue using
Section missing 3.

We begin by studying the multidegrees of the graph
$\Gamma\subset{\mathbb{P}}_{\mathbb{k}}^{r}\times_{\mathbb{k}}{\mathbb{P}}_{\mathbb{k}}^{m_{1}}\times_{\mathbb{k}}\cdots\times_{\mathbb{k}}{\mathbb{P}}_{\mathbb{k}}^{m_{p}}$
of the rational map
$\mathcal{G}:{\mathbb{P}}_{\mathbb{k}}^{r}\dashrightarrow{\mathbb{P}}_{\mathbb{k}}^{m_{1}}\times_{\mathbb{k}}\cdots\times_{\mathbb{k}}{\mathbb{P}}_{\mathbb{k}}^{m_{p}}$.
To simplify notation, let

$\mathbb{D}:={\mathbb{P}}_{\mathbb{k}}^{r}\times_{\mathbb{k}}{\mathbb{P}}_{\mathbb{k}}^{m_{1}}\times_{\mathbb{k}}\cdots\times_{\mathbb{k}}{\mathbb{P}}_{\mathbb{k}}^{m_{p}}={\mathbb{P}}_{\mathbb{k}}^{r}\times_{\mathbb{k}}{\mathbb{P}}.$

Then, for any $(d_{0},{\mathbf{d}})\in\mathbb{N}^{p+1}$ with
$d_{0}+|{\mathbf{d}}|=r$, we shall consider the multidegrees
$\deg_{\mathbb{D}}^{(d_{0},{\mathbf{d}})}(\Gamma)=e\big{(}(d_{0},{\mathbf{d}}),\,\mathcal{R}(I_{1},\ldots,I_{p})\big{)}$.
We proceed by relating the multidegrees of the graph $\Gamma$ with the mixed
multiplicities of $\widetilde{\mathfrak{F}}(I_{1},\ldots,I_{p})$. The
following result provides an extension of [MIXED_MULT, Theorem 5.4] to our
setting.

######  Theorem 5.1.

Assume Section missing 3. Then, for all
${\mathbf{d}}=(d_{1},\ldots,d_{p})\in\mathbb{N}^{p}$ with $|{\mathbf{d}}|=r$,
we have

$\deg_{\mathbb{D}}^{(0,{\mathbf{d}})}(\Gamma)\,=\,e\big{(}{\mathbf{d}},\,\widetilde{\mathfrak{F}}(I_{1},\ldots,I_{p})\big{)}.$

###### Proof.

This proof follows similarly to [MIXED_MULT, Theorem 5.4]. Let
$\mathfrak{b}=(\mathbf{y}_{1})\cap\cdots\cap(\mathbf{y}_{p})$,
${\mathfrak{N}}={\mathfrak{m}}\cap\mathfrak{b}$ and
$\mathfrak{M}={\mathfrak{m}}+{\mathfrak{N}}$. Note that ${\mathfrak{N}}$ is
the irrelevant ideal in $\mathbb{D}=\text{MultiProj}(\mathfrak{A})$. From the
Mayer-Vietoris exact sequence

$\text{H}_{\mathfrak{M}}^{i}(\mathcal{R}(I_{1},\ldots,I_{p}))\,\rightarrow\,\text{H}_{\mathfrak{m}}^{i}(\mathcal{R}(I_{1},\ldots,I_{p}))\oplus\text{H}_{\mathfrak{b}}^{i}(\mathcal{R}(I_{1},\ldots,I_{p}))\,\rightarrow\,\text{H}_{\mathfrak{N}}^{i}(\mathcal{R}(I_{1},\ldots,I_{p}))\,\rightarrow\,\text{H}_{\mathfrak{M}}^{i+1}(\mathcal{R}(I_{1},\ldots,I_{p}))$

and the fact that
$\left[\text{H}_{\mathfrak{M}}^{i}(\mathcal{R}(I_{1},\ldots,I_{p}))\right]_{0,{\mathbf{n}}}=\left[\text{H}_{\mathfrak{b}}^{i}(\mathcal{R}(I_{1},\ldots,I_{p}))\right]_{0,{\mathbf{n}}}=0$
for all ${\mathbf{n}}\gg\mathbf{0}$, we obtain the isomorphism

$\left[\text{H}_{{\mathfrak{m}}}^{i}(\mathcal{R}(I_{1},\ldots,I_{p}))\right]_{0,{\mathbf{n}}}\cong\left[\text{H}_{{\mathfrak{N}}}^{i}(\mathcal{R}(I_{1},\ldots,I_{p}))\right]_{0,{\mathbf{n}}}\quad\text{for
all }{\mathbf{n}}\gg\mathbf{0}.$

Let $X=\text{Proj}_{R\text{-gr}}\left(\mathcal{R}(I_{1},\ldots,I_{p})\right)$
be the projective scheme obtained by considering $\mathcal{R}(I)$ as a single-
graded $\mathbb{k}$-algebra with the grading of $R$. By using the relations
between sheaf and local cohomologies, see e.g. [HYRY_MULTIGRAD, Corollary 1.5]
and [EISEN_COMM, Appendix A4.1], we have that

$\left[\widetilde{\mathfrak{F}}(I_{1},\ldots,I_{p})\right]_{\mathbf{n}}\;\cong\;\left[\text{H}^{0}(X,\mathcal{O}_{X})\right]_{\mathbf{n}}\;\cong\;\text{H}^{0}(\Gamma,\mathcal{O}_{\Gamma}(0,{\mathbf{n}}))\quad\text{for
all }{\mathbf{n}}\gg\mathbf{0}.$

The multigraded Hilbert polynomial of $\mathcal{R}(I_{1},\ldots,I_{p})$ is
given by

$P_{\mathcal{R}(I_{1},\ldots,I_{p})}(c,{\mathbf{n}})\,=\,\sum_{i\geq
0}{(-1)}^{i}\dim_{\mathbb{k}}\Big{(}\text{H}^{i}\left(\Gamma,\mathcal{O}_{\Gamma}(c,{\mathbf{n}})\right)\Big{)},$

see e.g. [Kleiman_geom_mult, Lemma 4.3]. By [DEGREE_SPECIALIZATION, Theorem
4.4], we have that
$\left[\text{H}_{\mathfrak{m}}^{i}(\mathcal{R}(I_{1},\ldots,I_{p}))\right]_{0}$
is a finitely generated $\mathbb{N}^{p}$-graded module over
$\mathfrak{F}(I_{1},\ldots,I_{p})$ such that

$\dim\left(\left[\text{H}_{\mathfrak{m}}^{i}(\mathcal{R}(I_{1},\ldots,I_{p}))\right]_{0}\right)\,\leq\,(\dim(R)+p)-i=r+1+p-i.$

Hence, the fact that
$\text{H}^{i}\left(\Gamma,\mathcal{O}_{\Gamma}(0,{\mathbf{n}})\right)\cong\left[\text{H}_{{\mathfrak{N}}}^{i+1}(\mathcal{R}(I_{1},\ldots,I_{p}))\right]_{0,{\mathbf{n}}}\cong\left[\text{H}_{{\mathfrak{m}}}^{i+1}(\mathcal{R}(I_{1},\ldots,I_{p}))\right]_{0,{\mathbf{n}}}$
for all $i\geq 1$ and ${\mathbf{n}}\gg\mathbf{0}$ gives the equality

$\lim_{n\rightarrow\infty}\frac{P_{\mathcal{R}(I_{1},\ldots,I_{p})}(0,n\cdot{\mathbf{n}})}{n^{r}}\;=\;\lim_{n\rightarrow\infty}\frac{\dim_{\mathbb{k}}\left(\left[\widetilde{\mathfrak{F}}(I_{1},\ldots,I_{p})\right]_{n\cdot{\mathbf{n}}}\right)}{n^{r}}$

for all ${\mathbf{n}}\in{\mathbb{Z}}_{+}^{p}$. So, the equality
$\deg_{\mathbb{D}}^{(0,{\mathbf{d}})}(\Gamma)\,=\,e\big{(}{\mathbf{d}},\,\widetilde{\mathfrak{F}}(I_{1},\ldots,I_{p})\big{)}$
holds for all ${\mathbf{d}}\in\mathbb{N}^{p}$ with $|{\mathbf{n}}|=r$. ∎

We also relate the multidegrees of the graph $\Gamma$ with the mixed
multiplicities of the ideals $I_{1},\ldots,I_{p}$ with respect to
${\mathfrak{m}}$.

######  Lemma .

Assume Section missing 3. Then, for all
$(d_{0},{\mathbf{d}})\in\mathbb{N}^{p+1}$ with $d_{0}+|{\mathbf{d}}|=r$, we
have

$\deg_{\mathbb{D}}^{(d_{0},{\mathbf{d}})}(\Gamma)\,=\,e_{(d_{0},{\mathbf{d}})}\left({\mathfrak{m}}\mid
I_{1},\ldots,I_{p}\right).$

###### Proof.

Consider the $\mathbb{N}^{p+1}$-graded $\mathbb{k}$-algebra
$T({\mathfrak{m}}\mid I_{1},\ldots,I_{p})$. Since the ideals
$I_{1},\ldots,I_{p}$ are equally generated in degrees
$\delta_{1},\ldots,\delta_{p}$, by Nakayama’s lemma we obtain

$\displaystyle\dim_{\mathbb{k}}\left(\left[T({\mathfrak{m}}\mid
I_{1},\ldots,I_{p})\right]_{n_{0},{\mathbf{n}}}\right)$
$\displaystyle=\dim_{\mathbb{k}}\left(\frac{{\mathfrak{m}}^{n_{0}}I_{1}^{n_{1}}\cdots
I_{p}^{n_{p}}}{{\mathfrak{m}}^{n_{0}+1}I_{1}^{n_{1}}\cdots
I_{p}^{n_{p}}}\right)$
$\displaystyle=\dim_{\mathbb{k}}\left(\left[{\mathfrak{m}}^{n_{0}}I_{1}^{n_{1}}\cdots
I_{p}^{n_{p}}\right]_{n_{0}+{\mathbf{n}}\cdot\delta}\right)$
$\displaystyle=\dim_{\mathbb{k}}\left(\left[I_{1}^{n_{1}}\cdots
I_{p}^{n_{p}}\right]_{n_{0}+{\mathbf{n}}\cdot\delta}\right)$
$\displaystyle=\dim_{\mathbb{k}}\left(\left[\mathcal{R}(I_{1},\ldots,I_{p})\right]_{n_{0},{\mathbf{n}}}\right)$

for all $(n_{0},{\mathbf{n}})=(n_{0},n_{1},\ldots,n_{p})\in\mathbb{N}^{p+1}$.
This shows that $T({\mathfrak{m}}\mid I_{1},\ldots,I_{p})$ and
$\mathcal{R}(I_{1},\ldots,I_{p})$ have the same Hilbert function, and so the
result of the lemma follows. ∎

The following proposition deals with the process of cutting the graph $\Gamma$
with general hyperplanes in the target space ${\mathbb{P}}$. Note that this is
similar to [MIXED_MULT, Proposition 5.6], where the case of cutting $\Gamma$
with general hyperplanes in the source ${\mathbb{P}}_{\mathbb{k}}^{r}$ was
treated.

######  Proposition .

Assume Section missing 3 with $\mathbb{k}$ being an infinite field. Fix $1\leq
i\leq p$. Let
$h=\alpha_{0}y_{i,0}+\alpha_{1}y_{i,1}+\cdots+\alpha_{m_{i}}y_{i,m_{i}}$ with
$\alpha_{i}\in\mathbb{k}$ such that
$V(h)\subset{\mathbb{P}}_{\mathbb{k}}^{m_{i}}$ is a general hyperplane. Let
$f=\alpha_{0}f_{i,0}+\alpha_{1}f_{i,1}+\cdots+\alpha_{m_{i}}f_{i,m_{i}}\in
I_{i}$, $S=R/fR$ and $J_{j}=I_{j}S\subset S$ for all $1\leq j\leq p$. Then,
for all $(d_{0},{\mathbf{d}})=(d_{0},d_{1},\ldots,d_{p})\in\mathbb{N}^{p+1}$
with $d_{0}+|{\mathbf{d}}|=r$ and $d_{i}\geq 1$, we have

$\deg_{\mathbb{D}}^{(d_{0},{\mathbf{d}})}(\Gamma)\,=\,e\big{(}(d_{0},\ldots,d_{i}-1,\ldots,d_{p}),\mathcal{R}_{S}(J_{1},\ldots,J_{p})\big{)}.$

###### Proof.

By using [MIXED_MULT, Lemma 3.7], we choose
$(\alpha_{0},\ldots,\alpha_{m_{i}})$ in such a way that

$h=\alpha_{0}y_{i,0}+\alpha_{1}y_{i,1}+\cdots+\alpha_{m_{i}}y_{i,m_{i}}\in\mathbb{k}[\mathbf{y}_{i}]\subset\mathfrak{A}=R\otimes_{\mathbb{k}}\mathbb{k}[\mathbf{y}_{1}]\otimes_{\mathbb{k}}\cdots\otimes_{\mathbb{k}}\mathbb{k}[\mathbf{y}_{p}]$

becomes a filter-regular element on $\mathcal{R}(I_{1},\ldots,I_{p})$ and on

$\text{gr}_{I_{i}}\big{(}\mathcal{R}(I_{1},\ldots,\widehat{I_{i}},\ldots,I_{p})\big{)}=\mathcal{R}(I_{1},\ldots,I_{p})\otimes_{R}R/I_{i}=\bigoplus_{n_{1},\ldots,n_{p}\geq
0}\frac{I_{1}^{n_{1}}\cdots I_{i}^{n_{i}}\cdots
I_{p}^{n_{p}}}{I_{1}^{n_{1}}\cdots I_{i}^{n_{i}+1}\cdots I_{p}^{n_{p}}},$

where
$\mathcal{R}(I_{1},\ldots,\widehat{I_{i}},\ldots,I_{p})=\mathcal{R}(I_{1},\ldots,I_{i-1},I_{i+1},\ldots,I_{p})$.
So, by [MIXED_MULT, Lemma 3.9], we have that

$\deg_{\mathbb{D}}^{(d_{0},{\mathbf{d}})}(\Gamma)\,=\,e\big{(}(d_{0},{\mathbf{d}}-{\mathbf{e}}_{i}),\mathcal{R}(I_{1},\ldots,I_{p})/h\,\mathcal{R}(I_{1},\ldots,I_{p})\big{)}$

for all $(d_{0},{\mathbf{d}})\in\mathbb{N}^{p+1}$ with
$d_{0}+|{\mathbf{d}}|=r$ and $d_{i}\geq 1$. Notice that we have the following
natural surjection

$\mathfrak{s}\,:\;\frac{\mathcal{R}(I_{1},\ldots,I_{p})}{h\,\mathcal{R}(I_{1},\ldots,I_{p})}\cong\bigoplus_{{\mathbf{n}}\in\mathbb{N}^{p}}\frac{\mathbf{I}^{\mathbf{n}}}{f\,\mathbf{I}^{{\mathbf{n}}-{\mathbf{e}}_{i}}}\mathbf{t}^{\mathbf{n}}\;\;\twoheadrightarrow\;\;\mathcal{R}_{S}(J_{1},\ldots,J_{p})\cong\bigoplus_{{\mathbf{n}}\in\mathbb{N}^{p}}\frac{\mathbf{I}^{\mathbf{n}}+fR}{fR}\mathbf{t}^{\mathbf{n}}.$

For any $\mathfrak{p}\in{\text{Spec}}(R)\setminus V(I_{i})$, the localization
$\mathfrak{s}\otimes_{R}R_{\mathfrak{p}}$ of the surjection $\mathfrak{s}$
becomes an isomorphism. Hence, there is some $l>0$ such that
$I_{i}^{l}\cdot\text{Ker}(\mathfrak{s})=0$. Known dimension computations give
us that $\dim\left(\mathcal{R}(I_{1},\ldots,I_{p})\right)=r+1+p$ and
$\dim\left(\text{gr}_{I_{i}}\big{(}\mathcal{R}(I_{1},\ldots,\widehat{I_{i}},\ldots,I_{p})\big{)}\right)=r+p$,
see e.g. [huneke2006integral, §17.5]. For any finitely generated
$\mathbb{N}^{p+1}$-graded $\mathfrak{A}$-module $M$, we set
$\text{Supp}_{++}(M):=\text{Supp}(M)\cap\text{MultiProj}(\mathfrak{A})$. Since
$h$ is a filter-regular element on
$\text{gr}_{I_{i}}\big{(}\mathcal{R}(I_{1},\ldots,\widehat{I_{i}},\ldots,I_{p})\big{)}$
and on $\mathcal{R}(I_{1},\ldots,I_{p})$, [MIXED_MULT, Lemma 3.9] and
[HYRY_MULTIGRAD, Lemma 1.2] imply that

$\displaystyle\dim\left(\text{Supp}_{++}\left(\text{Ker}(\mathfrak{s})\right)\right)$
$\displaystyle\leq\dim\left(\text{Supp}_{++}\left(\frac{\mathcal{R}(I_{1},\ldots,I_{p})}{h\,\mathcal{R}(I_{1},\ldots,I_{p})}\otimes_{R}R/I_{i}\right)\right)$
$\displaystyle=\dim\left(\text{Supp}_{++}\left(\text{gr}_{I_{i}}\big{(}\mathcal{R}(I_{1},\ldots,\widehat{I_{i}},\ldots,I_{p})\big{)}\otimes_{\mathfrak{A}}\mathfrak{A}/h\mathfrak{A}\right)\right)$
$\displaystyle\leq(r+p)-(p+1)-1=r-2$

and

$\dim\left(\text{Supp}_{++}\left(\frac{\mathcal{R}(I_{1},\ldots,I_{p})}{h\,\mathcal{R}(I_{1},\ldots,I_{p})}\right)\right)=(r+1+p)-(p+1)-1=r-1.$

Finally, the short exact sequence
$0\rightarrow\text{Ker}(\mathfrak{s})\rightarrow\frac{\mathcal{R}(I_{1},\ldots,I_{p})}{h\,\mathcal{R}(I_{1},\ldots,I_{p})}\xrightarrow{\mathfrak{s}}\mathcal{R}_{S}(J_{1},\ldots,J_{p})\rightarrow
0$ and the additivity of mixed multiplicities yield

$\deg_{\mathbb{D}}^{(d_{0},{\mathbf{d}})}(\Gamma)\,=\,e\left((d_{0},{\mathbf{d}}-{\mathbf{e}}_{i}),\frac{\mathcal{R}(I_{1},\ldots,I_{p})}{h\,\mathcal{R}(I_{1},\ldots,I_{p})}\right)\,=\,e\left((d_{0},{\mathbf{d}}-{\mathbf{e}}_{i}),\mathcal{R}_{S}(J_{1},\ldots,J_{p})\right),$

and so the proof of the proposition is complete. ∎

The next lemma will allow us to simplify the families of rational maps we
consider in this section. It shows that the mixed multiplicities of
$\widetilde{\mathfrak{F}}(I_{1},\ldots,I_{p})$ depend only on a certain part
of the data given and is similar to [SAT_FIB_GOR_3, Proposition 2.11].

######  Lemma .

Assume Section missing 3 and suppose that the ideals $I_{1},\ldots,I_{p-1}$
are ${\mathfrak{m}}$-primary. Then, the following statements hold:

  1. (i)

We have the equality

$\widetilde{\mathfrak{F}}(I_{1},\ldots,I_{p})\,=\,\bigoplus_{n_{1},\ldots,n_{p}\geq
0}\left[\big{(}I_{p}^{n_{p}}\big{)}^{{\text{sat}}}\right]_{n_{1}\delta_{1}+\cdots+n_{p}\delta_{p}}.$

In particular, for all ${\mathbf{d}}\in\mathbb{N}^{p}$ with
$|{\mathbf{d}}|=r$, the value of
$e\big{(}{\mathbf{d}},\widetilde{\mathfrak{F}}(I_{1},\ldots,I_{p})\big{)}\,=\,\deg(\mathcal{G})\cdot\deg_{\mathbb{P}}^{\mathbf{d}}(Y)$
depends only on the degrees $\delta_{1},\ldots,\delta_{p-1}$ and on the ideal
$I_{p}$.

  2. (ii)

$e\big{(}(0,\ldots,0,r),\,\widetilde{\mathfrak{F}}(I_{1},\ldots,I_{p})\big{)}\;=\;e\big{(}(0,r),\,\mathcal{R}(I_{p})\big{)}\;=\,e\big{(}\widetilde{\mathfrak{F}}(I_{p})\big{)}.$

###### Proof.

(i) By the assumption of $I_{1},\ldots,I_{p-1}$ being ${\mathfrak{m}}$-primary
ideals, we have $I_{1}^{n_{1}}\cdots
I_{p}^{n_{p}}R_{\mathfrak{p}}=I_{p}^{n_{p}}R_{\mathfrak{p}}$ for all
$\mathfrak{p}\in{\text{Spec}}(R)\setminus\\{{\mathfrak{m}}\\}$ and
$n_{1},\ldots,n_{p}\in\mathbb{N}$. We then obtain the equality
$\big{(}I_{1}^{n_{1}}\cdots
I_{p}^{n_{p}}\big{)}^{\sim}=\big{(}I_{p}^{n_{p}}\big{)}^{\sim}$ of sheaves. It
follows that

$\widetilde{\mathfrak{F}}(I_{1},\ldots,I_{p})=\bigoplus_{{\mathbf{n}}\in\mathbb{N}^{p}}\text{H}^{0}\left({\mathbb{P}}_{\mathbb{k}}^{r},\big{(}I_{1}^{n_{1}}\cdots
I_{p}^{n_{p}}\big{)}^{\sim}({\mathbf{n}}\cdot\delta)\right)=\bigoplus_{{\mathbf{n}}\in\mathbb{N}^{p}}\text{H}^{0}\left({\mathbb{P}}_{\mathbb{k}}^{r},\big{(}I_{p}^{n_{p}}\big{)}^{\sim}({\mathbf{n}}\cdot\delta)\right)=\bigoplus_{{\mathbf{n}}\in\mathbb{N}^{p}}\left[\big{(}I_{p}^{n_{p}}\big{)}^{{\text{sat}}}\right]_{{\mathbf{n}}\cdot\delta}.$

So, the first statement of the lemma follows.

(ii) Consider the Rees algebra $\mathcal{R}(I_{p})$ with the usual bigrading
$\mathcal{R}(I_{p})=\bigoplus_{c,n\in\mathbb{N}}[\mathcal{R}(I_{p})]_{c,n}$
where $[\mathcal{R}(I_{p})]_{c,n}=\big{[}I_{p}\big{]}_{c+n\delta_{p}}$. Let
$Q(c,n):=P_{\mathcal{R}(I)}(c,n)$ be the bigraded Hilbert polynomial of
$\mathcal{R}(I_{p})$. Choose $e>0$ such that
$\left[\text{H}_{{\mathfrak{m}}}^{1}\left(\mathcal{R}(I_{p})\right)\right]_{k}=0$
for all $k\geq e$. From the grading chosen for the Rees algebra, it follows
that

$\left[\text{H}_{{\mathfrak{m}}}^{0}(R/I_{p}^{n})\right]_{n\delta_{p}+k}\cong\left[\text{H}_{{\mathfrak{m}}}^{1}(I_{p}^{n})\right]_{n\delta_{p}+k}=0$

for all $n\geq 0$ and $k\geq e$. This implies that
$\left[I_{p}^{n}\right]_{n\delta_{p}+k}=\left[(I_{p}^{n})^{\text{sat}}\right]_{n\delta_{p}+k}$
for all $n\geq 0$ and $k\geq e$.

Let
$P({\mathbf{n}}):=P_{\widetilde{\mathfrak{F}}(I_{1},\ldots,I_{p})}(n_{1},\ldots,n_{p})$
be the multigraded Hilbert polynomial of
$\widetilde{\mathfrak{F}}(I_{1},\ldots,I_{p})$. Choose positive integers
$n_{1}^{\prime},\ldots,n_{p}^{\prime}$ with
$n_{1}^{\prime}\delta_{1}+\cdots+n_{p-1}^{\prime}\delta_{p-1}\geq e$ such that
$P(n_{1},\ldots,n_{p})$ equals the Hilbert function of
$\widetilde{\mathfrak{F}}(I_{1},\ldots,I_{p})$ when $n_{i}\geq
n_{i}^{\prime}$. We can also assume that $Q(c,n)$ equals the Hilbert function
of $\mathcal{R}(I_{p})$ when $c\geq
c^{\prime}:=n_{1}^{\prime}\delta_{1}+\cdots+n_{p-1}^{\prime}\delta_{p-1}$ and
$n\geq n_{p}^{\prime}$. Therefore, with the above choices in place, we obtain
the following equalities

$\displaystyle P(n_{1}^{\prime},\ldots,n_{p-1}^{\prime},n)$
$\displaystyle=\dim_{\mathbb{k}}\left(\left[\left(I_{p}^{n}\right)^{\text{sat}}\right]_{n_{1}^{\prime}\delta_{1}+\cdots+n_{p-1}^{\prime}\delta_{p-1}+n\delta_{p}}\right)$
$\displaystyle=\dim_{\mathbb{k}}\left(\left[I_{p}^{n}\right]_{n_{1}^{\prime}\delta_{1}+\cdots+n_{p-1}^{\prime}\delta_{p-1}+n\delta_{p}}\right)$
$\displaystyle=\dim_{\mathbb{k}}\left([\mathcal{R}(I_{p})]_{c^{\prime},\,n}\right)=Q(c^{\prime},n)$

for all $n\geq n_{p}^{\prime}$. As a consequence, we obtain

$e\big{(}(0,\ldots,0,r),\,\widetilde{\mathfrak{F}}(I_{1},\ldots,I_{p})\big{)}\,=\,\lim_{n\rightarrow\infty}\,\frac{P(n_{1}^{\prime},\ldots,n_{p-1}^{\prime},n)}{n^{r}/r!}\,=\,\lim_{n\rightarrow\infty}\,\frac{Q(c^{\prime},n)}{n^{r}/r!}\,=\,e\big{(}(0,r),\,\mathcal{R}(I_{p})\big{)}.$

Finally, [MIXED_MULT, Theorem 5.4] (or Theorem 5.1 above) implies that
$e\big{(}(0,r),\,\mathcal{R}(I_{p})\big{)}=e\big{(}\widetilde{\mathfrak{F}}(I_{p})\big{)}$,
and so we are done. ∎

We are now ready to state our main result in this section. We say that an
ideal $I\subset R$ satisfies the condition $G_{r+1}$ when
$\mu(I_{\mathfrak{p}})\leq\dim(R_{\mathfrak{p}})$ for all $\mathfrak{p}\in
V(I)\subset{\text{Spec}}(R)$ with $\text{ht}(\mathfrak{p})<r+1$, where
$\mu(I_{\mathfrak{p}})$ is the minimal number of generators of
$I_{\mathfrak{p}}$. The following theorem gives exact formulas for the
multidegrees of the saturated special fiber ring
$e\big{(}{\mathbf{d}},\,\widetilde{\mathfrak{F}}(I_{1},\ldots,I_{p})\big{)}$
for several families of rational maps.

######  Theorem 5.2.

Assume Section missing 3 and suppose that the ideals $I_{1},\ldots,I_{p-1}$
are ${\mathfrak{m}}$-primary and that $m_{p}\geq r$.

  1. (I)

Suppose that the following conditions hold:

    1. (a)

$I_{p}$ is perfect of height two with Hilbert-Burch resolution of the form

$0\rightarrow\bigoplus_{i=1}^{m_{p}}R(-\delta_{p}-\mu_{i})\rightarrow{R(-\delta_{p})}^{m_{p}+1}\rightarrow
I_{p}\rightarrow 0.$

    2. (b)

$I$ satisfies the condition $G_{r+1}$.

Then, for all ${\mathbf{d}}=(d_{1},\ldots,d_{p})\in\mathbb{N}^{p}$ with
$|{\mathbf{d}}|=r$, we have

$\deg(\mathcal{G})\cdot\deg_{\mathbb{P}}^{\mathbf{d}}(Y)\,=\,\delta_{1}^{d_{1}}\cdots\delta_{p-1}^{d_{p-1}}\cdot
e_{d_{p}}(\mu_{1},\ldots,\mu_{m_{p}}),$

where $e_{d_{p}}(\mu_{1},\ldots,\mu_{m_{p}})$ denotes the elementary symmetric
polynomial

$e_{d_{p}}(\mu_{1},\ldots,\mu_{m_{p}})=\sum_{1\leq j_{1}<\cdots<j_{d_{p}}\leq
m_{p}}\mu_{j_{1}}\cdots\mu_{j_{d_{p}}}.$

  2. (II)

Suppose that the following conditions hold:

    1. (a)

$I_{p}$ is a Gorenstein ideal of height three.

    2. (b)

Every nonzero entry of an alternating minimal presentation matrix of $I_{p}$
has degree $D\geq 1$.

    3. (c)

$I_{p}$ satisfies the condition $G_{r+1}$.

Then, for all ${\mathbf{d}}=(d_{1},\ldots,d_{p})\in\mathbb{N}^{p}$ with
$|{\mathbf{d}}|=r$, we have

$\deg(\mathcal{G})\cdot\deg_{\mathbb{P}}^{\mathbf{d}}(Y)\,=\,\begin{cases}\delta_{1}^{d_{1}}\cdots\delta_{p-1}^{d_{p-1}}\cdot
D^{d_{p}}\sum_{k=0}^{\lfloor\frac{m_{p}-d_{p}}{2}\rfloor}\binom{m_{p}-1-2k}{d_{p}-1}\quad\text{
if }d_{p}\geq 3\\\
\delta_{1}^{d_{1}}\cdots\delta_{p-1}^{d_{p-1}}\delta_{p}^{d_{p}}\qquad\qquad\quad\;\qquad\qquad\qquad\text{
otherwise.}\end{cases}$

###### Proof.

Due to Theorem 3.1, it suffices to compute the mixed multiplicities
$e\big{(}{\mathbf{d}},\,\widetilde{\mathfrak{F}}(I_{1},\ldots,I_{p-1},I_{p})\big{)}$.
The assumption of $I_{1},\ldots,I_{p}$ being ${\mathfrak{m}}$-primary allows
us to reduce to the case where $I_{1}=\cdots=I_{p-1}={\mathfrak{m}}$. By
Section missing 5(i), we have the following relation of multigraded Hilbert
polynomials

$P_{\widetilde{\mathfrak{F}}(I_{1},\ldots,I_{p-1},I_{p})}(n_{1},\ldots,n_{p-1},n_{p})\;=\;P_{\widetilde{\mathfrak{F}}({\mathfrak{m}},\ldots,{\mathfrak{m}},I_{p})}(n_{1}\delta_{1},\ldots,n_{p-1}\delta_{p-1},n_{p})$

for all $n_{i}\gg 0$. This implies that

$e\big{(}{\mathbf{d}},\,\widetilde{\mathfrak{F}}(I_{1},\ldots,I_{p-1},I_{p})\big{)}\;=\;\delta_{1}^{d_{1}}\cdots\delta_{p-1}^{d_{p-1}}\cdot
e\big{(}{\mathbf{d}},\,\widetilde{\mathfrak{F}}({\mathfrak{m}},\ldots,{\mathfrak{m}},I_{p})\big{)}$

for all ${\mathbf{d}}=(d_{1},\ldots,d_{p})\in\mathbb{N}^{p}$ with
$|{\mathbf{d}}|=r$. Therefore, for the rest of the proof, we assume that
$I_{1}=\cdots=I_{p-1}={\mathfrak{m}}$.

(I) We can assume that $\mathbb{k}$ is an infinite field. By Theorem 5.1, we
have that
$e\big{(}{\mathbf{d}},\,\widetilde{\mathfrak{F}}(I_{1},\ldots,I_{p-1},I_{p})\big{)}=\deg_{\mathbb{D}}^{(0,{\mathbf{d}})}(\Gamma)$.
Since $I_{1}=\cdots=I_{p}={\mathfrak{m}}$, by applying Section missing 5
successively we have that

$\deg_{\mathbb{D}}^{(0,d_{1},\ldots,d_{p-1},d_{p})}(\Gamma)\;=\;e\big{(}(0,0,\ldots,0,d_{p}),\,\mathcal{R}_{S}(J_{1},\ldots,J_{p-1},J_{p})\big{)}.$

Here $S=R/(l_{1},\ldots,l_{s})R$, $s=d_{1}+\cdots+d_{p-1}$, each
$l_{i}=\alpha_{i,0}x_{0}+\cdots+\alpha_{i,r}x_{r}$ gives a general hyperplane
$V(l_{i})\subset{\mathbb{P}}_{\mathbb{k}}^{r}$, and
$J_{1}=\cdots=J_{p-1}={\mathfrak{m}}S$, $J_{p}=I_{p}S$. Then Section missing
5(ii) gives

$\deg_{\mathbb{D}}^{(0,d_{1},\ldots,d_{p-1},d_{p})}(\Gamma)\;=\;e\big{(}(0,0,\ldots,0,d_{p}),\,\mathcal{R}_{S}(J_{1},\ldots,J_{p-1},J_{p})\big{)}\;=\;e\big{(}(0,d_{p}),\,\mathcal{R}_{S}(J_{p})\big{)}.$

By [MIXED_MULT, Proposition 5.6(i)], we have
$e\big{(}(0,d_{p}),\,\mathcal{R}_{S}(J_{p})\big{)}=e\big{(}(s,d_{p}),\,\mathcal{R}(I_{p})\big{)}=d_{s}(\mathcal{G}_{i})$,
where $d_{s}(\mathcal{G}_{i})$ denotes the $s$-th projective degree of the
rational map
$\mathcal{G}_{i}:{\mathbb{P}}_{\mathbb{k}}^{r}\dashrightarrow{\mathbb{P}}_{\mathbb{k}}^{m_{p}}$.
Finally, by applying [MIXED_MULT, Theorem 5.7], we obtain the formula

$d_{s}(\mathcal{G}_{i})=e_{d_{p}}(\mu_{1},\ldots,\mu_{m_{p}}),$

and so the result follows.

(II) The proof follows similarly to (I), however we use [MIXED_MULT, Theorem
5.8] instead of [MIXED_MULT, Theorem 5.7]. ∎

##  6\. Monomial rational maps

In this section, we study the case of a monomial rational map. We shall
express the mixed multiplicities of the saturated special fiber ring in terms
of mixed volumes of some naturally constructed polytopes. The following setup
is used throughout this section.

######  Setup .

Assume Section missing 3 and suppose that the polynomials $f_{i,k}\in I_{i}$
are monomials. So, $I_{1},\ldots,I_{p}\subset R$ are monomial ideals.

First, we need to recall some important results and fix our notation. Let
$\mathbf{K}=(K_{1},\ldots,K_{p})$ be a sequence of convex bodies in
$\mathbb{R}^{r}$. For any sequence
$\lambda=(\lambda_{1},\ldots,\lambda_{p})\in\mathbb{N}^{p}$ of nonnegative
integers, we denote by $\lambda\mathbf{K}$ the Minkowski sum
$\lambda\mathbf{K}:=\lambda_{1}K_{1}+\cdots+\lambda_{p}K_{p}$ and by
$\mathbf{K}_{\lambda}$ the multiset
$\mathbf{K}_{\lambda}:=\bigcup_{i=1}^{p}\bigcup_{j=1}^{\lambda_{i}}\\{K_{i}\\}$
of $\lambda_{i}$ copies of $K_{i}$ for each $1\leq i\leq p$. Let us recall
Minkowski’s classic theorem, see e.g. [EWALD, Theorem 3.2, page 116].

######  Theorem 6.1 (Minkowski).

${\text{Vol}}_{r}(\lambda\mathbf{K})$ is a homogeneous polynomial of degree
$r$.

We write the polynomial ${\text{Vol}}_{r}(\lambda\mathbf{K})$ as

${\text{Vol}}_{r}(\lambda\mathbf{K})=\sum_{{{\mathbf{d}}\in\mathbb{N}^{p}\;\lvert{\mathbf{d}}\rvert=r}}\;\frac{1}{{\mathbf{d}}!}\,{\text{MV}}_{r}(\mathbf{K}_{\mathbf{d}})\,\lambda^{\mathbf{d}},$

where ${\text{MV}}_{r}(-)$ denotes the _mixed volume_.

For a monomial ideal $J\in R$ with monomial generating set $G(J)$, we define
the following lattice convex polytope

$\Gamma(J)\,:=\,\text{ConvexHull}\left(\big{\\{}(n_{0},n_{1},\ldots,n_{r})\in\mathbb{N}^{r}\,\mid\,x_{0}^{n_{0}}x_{1}^{n_{1}}\cdots
x_{r}^{n_{r}}\in G(J)\big{\\}}\right)\,\subset\,\mathbb{R}^{r+1}.$

We denote by $\pi:\mathbb{R}^{r+1}\twoheadrightarrow\mathbb{R}^{r}$ the
natural projection given by
$(x_{0},x_{1}\ldots,x_{r})\mapsto(x_{1}\ldots,x_{r})$. We have the following
important result that expresses the mixed multiplicities of
$I_{1},\ldots,I_{p}$ with respect to ${\mathfrak{m}}$ in terms of mixed
volumes.

######  Theorem 6.2 ([TRUNG_VERMA_MIXED_VOL, Theorem 2.4, Corollary 2.5]).

Assume Section missing 6. Then, for all
${\mathbf{d}}=(d_{1},\ldots,d_{p})\in\mathbb{N}^{p}$ with $|{\mathbf{d}}|=r$,
we have

$e_{(0,{\mathbf{d}})}\left({\mathfrak{m}}\mid
I_{1},\ldots,I_{p}\right)\,=\,{\text{MV}}_{r}\left({\Big{(}\pi(\Gamma(I_{1})),\,\ldots,\,\pi(\Gamma(I_{p}))\Big{)}}_{\mathbf{d}}\right).$

It should be mentioned that this result was generalized in [cid2020convex,
cid2021multigraded] for not-necessarily-Noetherian graded families of ideals.

The following theorem is the main result of this section.

######  Theorem 6.3.

Assume Section missing 6. Then, for all
${\mathbf{d}}=(d_{1},\ldots,d_{p})\in\mathbb{N}^{p}$ with $|{\mathbf{d}}|=r$,
we have

$\deg(\mathcal{G})\cdot\deg_{\mathbb{P}}^{\mathbf{d}}(Y)\,=\,{\text{MV}}_{r}\left({\Big{(}\pi(\Gamma(I_{1})),\,\ldots,\,\pi(\Gamma(I_{p}))\Big{)}}_{\mathbf{d}}\right).$

###### Proof.

The result follows directly by combining Theorem 3.1, Theorem 5.1, Section
missing 5 and Theorem 6.2. More precisely, we obtain the following sequence of
equalities

$\displaystyle\deg(\mathcal{G})\cdot\deg_{\mathbb{P}}^{\mathbf{d}}(Y)$
$\displaystyle=e\big{(}{\mathbf{d}},\widetilde{\mathfrak{F}}(I_{1},\ldots,I_{p})\big{)}$
$\displaystyle=\deg_{\mathbb{D}}^{(0,{\mathbf{d}})}(\Gamma)$
$\displaystyle=e_{(0,{\mathbf{d}})}\left({\mathfrak{m}}\mid
I_{1},\ldots,I_{p}\right)$
$\displaystyle={\text{MV}}_{r}\left({\big{(}\pi(\Gamma(I_{1})),\,\ldots,\,\pi(\Gamma(I_{p}))\big{)}}_{\mathbf{d}}\right)$

for all ${\mathbf{d}}\in\mathbb{N}^{p}$ with $|{\mathbf{d}}|=r$. ∎

We now illustrate the above result with three simple but instructive examples.

######  Example .

Consider the following classical rational maps

$\mathcal{G}_{1}:{\mathbb{P}}_{\mathbb{k}}^{2}\dashrightarrow{\mathbb{P}}_{\mathbb{k}}^{2},\quad(x,y,z)\mapsto(x^{2},y^{2},z^{2})\quad\textrm{
and
}\quad\mathcal{G}_{2}:{\mathbb{P}}_{\mathbb{k}}^{2}\dashrightarrow{\mathbb{P}}_{\mathbb{k}}^{2},\quad(x,y,z)\mapsto(xy,xz,yz)$

and denote their corresponding base ideals $I_{1}=(x^{2},y^{2},z^{2})$ and
$I_{2}=(xy,xz,yz)$. Let $\pi$ be the map that projects onto the $x$ and $y$
coordinates. The polytopes $\Gamma(I_{1})$ and $\Gamma(I_{2})$ are depicted in
Figure 1 along with their projections.

Figure 1. The polytopes associated to the ideals in Section missing 6 and
their projections.

Since the image $Y={\mathbb{P}}_{\mathbb{k}}^{2}$ of $\mathcal{G}_{1}$ has
degree $1$, by Theorem 6.3 we have that

$\deg(\mathcal{G}_{1})={\text{MV}}_{2}\left(\pi(\Gamma(I_{1}),\pi(\Gamma(I_{1}))=2!{\text{Vol}}_{2}(\pi(\Gamma(I_{1}))\right)=4.$

Similarly, for the rational map $\mathcal{G}_{2}$, by Theorem 6.3 we have

$\deg(\mathcal{G}_{2})={\text{MV}}_{2}\left(\pi(\Gamma(I_{2}),\pi(\Gamma(I_{2})\right)=2!{\text{Vol}}_{2}(\pi(\Gamma(I_{2})))=1.$

We note that the equalities $\deg(\mathcal{G}_{1})=4$ and
$\deg(\mathcal{G}_{2})=1$ are also confirmed by Theorem 3.2, since
$\mathcal{G}_{1}$ has no base points and $\mathcal{G}_{2}$ has three base
points.

######  Example .

Let us consider the following rational map

$\mathcal{G}:{\mathbb{P}}_{\mathbb{k}}^{2}\dashrightarrow{\mathbb{P}}_{\mathbb{k}}^{2}\times{\mathbb{P}}_{\mathbb{k}}^{2},\qquad(x,y,z)\mapsto((x,y,z),(x^{3},yz^{2},xy^{2}))$

which we note has degree $1$. Let $I_{1}=(x,y,z)$ and
$I_{2}=(x^{3},yz^{2},xy^{2})$ be the corresponding base ideals. Let $\pi$ be
the map that projects onto the $x$ and $y$ coordinates. In Figure 2 we depict
the polytopes $\pi(\Gamma(I_{1}))$ and $\pi(\Gamma(I_{2}))$ along with their
Minkowski sum.

Figure 2. The polytopes $\pi(\Gamma(I_{1}))$ and $\pi(\Gamma(I_{2}))$ from
Section missing 6 and their Minkowski sum. The depiction of the Minkowski sum
(right) contains disjoint copies of the polytopes $\pi(\Gamma(I_{1}))$ and
$\pi(\Gamma(I_{2}))$ indicated by the faint dotted lines.

By the definition of the mixed volume, we have

${\text{MV}}_{2}\left(\pi(\Gamma(I_{1})),\pi(\Gamma(I_{2}))\right)={\text{Vol}}_{2}\left(\pi(\Gamma(I_{1})+\pi(\Gamma(I_{2}))\right)-{\text{Vol}}_{2}\left(\pi(\Gamma(I_{1}))\right)-{\text{Vol}}_{2}\left(\pi(\Gamma(I_{2}))\right)=\frac{11}{2}-\frac{1}{2}-2=3.$

And so, by Theorem 6.3, we have that the multidegrees of the image of
$\mathcal{G}$ are given by

$\deg_{\mathbb{P}}^{(2,0)}(Y)=1,\quad\deg_{\mathbb{P}}^{(0,2)}(Y)=4\quad\text{
and }\quad\deg_{\mathbb{P}}^{(1,1)}(Y)=3.$

######  Example .

Let us consider the following rational map

$\mathcal{G}:{\mathbb{P}}_{\mathbb{k}}^{3}\dashrightarrow{\mathbb{P}}_{\mathbb{k}}^{3}\times{\mathbb{P}}_{\mathbb{k}}^{3},\qquad(x,y,z,w)\mapsto((x,y,z,w),(xy^{2},xw^{2},yzw,z^{2}w)).$

Let $I_{1}=(x,y,z,w)$ and $I_{2}=(xy^{2},xw^{2},yzw,z^{2}w)$ be the
corresponding base ideals. Let $\pi$ be the map that projects onto the $x$,
$y$ and $z$ coordinates. The polytopes $\pi(\Gamma(I_{1}))$ and
$\pi(\Gamma(I_{2}))$ and the Minkowski sums
$\pi(\Gamma(I_{1}))+\pi(\Gamma(I_{2}))$ and
$2\pi(\Gamma(I_{1}))+\pi(\Gamma(I_{1}))$ are depicted in Figure 3. For ease of
notation, let us write $A=\pi(\Gamma(I_{1}))$ and $B=\pi(\Gamma(I_{2}))$. The
volumes of the polytopes are

${\text{Vol}}_{3}(A)=\frac{1}{6},\quad{\text{Vol}}_{3}(B)=\frac{1}{3},\quad{\text{Vol}}_{3}(A+B)=\frac{9}{2}\quad\text{and}\quad{\text{Vol}}_{3}(2A+B)=\frac{56}{3}.$

Figure 3. The polytopes $\pi(\Gamma(I_{1}))$ and $\pi(\Gamma(I_{2}))$ and
their Minkowski sums from Section missing 6.

By definition of mixed volumes, we have

${\text{Vol}}_{3}(A+B)={\text{Vol}}_{3}(A)+{\text{Vol}}_{3}(B)+\frac{1}{2}{\text{MV}}_{3}(A,A,B)+\frac{1}{2}{\text{MV}}_{3}(A,B,B)$
${\text{Vol}}_{3}(2A+B)=8{\text{Vol}}_{3}(A)+{\text{Vol}}_{3}(B)+4{\text{MV}}_{3}(A,A,B)+{\text{MV}}_{3}(A,B,B)$

Solving these equations, we find that ${\text{MV}}_{3}(A,A,B)=3$ and
${\text{MV}}_{3}(A,B,B)=5$. It is straightforward to see that
$\deg(\mathcal{G})=1$. And so, by Theorem 6.3 we have

$\deg_{\mathbb{P}}^{(3,0)}(Y)=1,\quad\deg_{\mathbb{P}}^{(2,1)}(Y)=3,\quad\deg_{\mathbb{P}}^{(1,2)}(Y)=5,\quad\text{and}\quad\deg_{\mathbb{P}}^{(0,3)}(Y)=2.$

##  7\. Application / Interpretation

The problem of implicitization is fundamental in the area of Computer Aided
Geometric Design (CAGD). The main problem, in the case of curves, can be
stated as follows. Let $a_{0},a_{1},a_{2}\in\mathbb{k}[t]$ be polynomials,
then $(x,y):=(a_{1}/a_{0},a_{2}/a_{0})$ is a rational parametric curve in
$\mathbb{k}^{2}$. The implicitization of the curve is a polynomial in $x$ and
$y$ which carves out this curve. This problem is studied in [COX_EQ_PARAM,
cox1998moving_line] using a collection of generators of the syzygy module of
$I=(a_{0}x-a_{1},a_{0}y-a_{2})\subseteq\mathbb{k}[x,y,t]$ called a
$\mu$-basis. This setup can be translated into the language used throughout
this paper by homogenizing. Let
$d=\max\\{\deg(a_{0}),\deg(a_{1}),\deg(a_{2})\\}$ and, for each
$i\in\\{0,1,2\\}$, if $a_{i}=\sum_{i}\alpha_{i}t^{i}$ then define
$\overline{a_{i}}=\sum_{i}\alpha_{i}t^{i}s^{d-i}\in\mathbb{k}[s,t]$. The
parametric curve is the image of the rational map
$\mathcal{G}:{\mathbb{P}}_{\mathbb{k}}^{1}\dashrightarrow{\mathbb{P}}_{\mathbb{k}}^{2}$
given by
$\mathcal{G}(s,t)=(\overline{a_{0}},\overline{a_{1}},\overline{a_{2}})$,
restricted to a particular affine patch.

The concept of a $\mu$-basis naturally extends to higher dimensions. For
example, $\mu$-bases for surfaces
$\mathcal{G}:{\mathbb{P}}_{\mathbb{k}}^{2}\dashrightarrow{\mathbb{P}}_{\mathbb{k}}^{3}:(s,t)\mapsto(a,b,c,d)$
are studied in [COX_EQ_PARAM, Section 5]. The $\mu$-basis allows us to
determine properties about the parametric surface, i.e. the image of
$\mathcal{G}$. Under particular assumptions, Cox computes in [COX_EQ_PARAM,
Proposition 5.3] that the degree of the image of $\mathcal{G}$ can be
determined completely in terms of the degrees of a $\mu$-basis. Explicitly,
Cox requires that $\deg(\mathcal{G})=1$ and that the base locus
$V(a,b,c,d)\subseteq{\mathbb{P}}_{\mathbb{k}}^{2}$ of the rational map is a
local complete intersection. As a result, the formula for the degree is
$\deg_{{\mathbb{P}}_{\mathbb{k}}^{3}}(Y)=\mu_{1}\mu_{2}+\mu_{1}\mu_{3}+\mu_{2}\mu_{3}=e_{2}(\mu_{1},\mu_{2},\mu_{3})$,
where the $\mu$-basis of $(a,b,c,d)$ has degrees $\mu_{1},\mu_{2},\mu_{3}$. In
more recent work, [MULT_SAT_PERF_HT_2, Theorem A] gives a generalization of
this result. It is shown that, for a rational map
$\mathcal{G}:{\mathbb{P}}_{\mathbb{k}}^{r}\dashrightarrow{\mathbb{P}}_{\mathbb{k}}^{s}$
determined by a perfect base ideal of height two, the degree of the image
times the degree of the map equals an elementary symmetric polynomial
evaluated at the degrees of the $\mu$-basis. The main tool used in the proof
of this formula is the saturated special fiber ring, see Section missing 3.

The study of $\mu$-bases has played an important role in a number of
applications. As mentioned above, they appear in the implicitization problem
of parametric surfaces and curves. But, they are also an important tool for
detecting singularities and in studying the degree and birationality of
rational maps. They are nowadays an essential tool in many developments in
CAGD, see e.g. [survey_mu_basis, CHEN_SEDERB_IMP, CHEN_WANG_YANG_SING,
GOLDMAN_JIA, MU_BASIS_RULED_SURF, SEDERBERG20151, SEDERBERG20161].

More generally, in a somehow parallel story, $\mu$-bases form part of the
general syzygy-based approach for studying rational maps. This method appears
to have been originally initiated in [HULEK_KATZ_SCHREYER_SYZ]. In this paper,
Hulek, Katz and Schreyer give a characterization of when a map is a Cremona
transformation in terms of properties of the syzygies. The use of syzygies in
studying rational maps has become an active and particularly fruitful research
area, see e.g. [AB_INITIO, Simis_cremona, KPU_blowup_fibers,
EISENBUD_ULRICH_ROW_IDEALS, Hassanzadeh_Simis_Cremona_Sat, SIMIS_RUSSO_BIRAT,
EFFECTIVE_BIGRAD, SIMIS_PAN_JONQUIERES, HASSANZADEH_SIMIS_DEGREES, MULTPROJ,
MULT_SAT_PERF_HT_2, SAT_FIB_GOR_3, MIXED_MULT, BOTBOL_ALICIA_RAT_SURF,
BOTBOL_ALICIA_MAT, KPU_BIGRAD_STRUCT, KPU_GOR3, KPU_NORMAL_SCROLL,
CARLOS_MONO, CARLOS_MONOID, CARLOS_MU2].

In this paper, we further extend the syzygy-based approach (a.k.a $\mu$-bases)
by considering a family of rational maps which has, until this point, been
mostly unobserved. We are interested in rational maps from a projective
variety to a multiprojective variety, typically, a rational map of the form
$\mathcal{G}:{\mathbb{P}}_{\mathbb{k}}^{r}\dashrightarrow{\mathbb{P}}_{\mathbb{k}}^{m_{1}}\times_{\mathbb{k}}\cdots\times_{\mathbb{k}}{\mathbb{P}}_{\mathbb{k}}^{m_{p}}$.
It should be mentioned that rational maps from a multiprojective variety to a
projective variety (i.e. a rational map typically of the form
$\mathcal{H}:{\mathbb{P}}_{\mathbb{k}}^{m_{1}}\times_{\mathbb{k}}\cdots\times_{\mathbb{k}}{\mathbb{P}}_{\mathbb{k}}^{m_{p}}\dashrightarrow{\mathbb{P}}_{\mathbb{k}}^{r}$)
are an important gadget in the field of Geometric Modeling, see e.g.
[EFFECTIVE_BIGRAD, SEDERBERG20151, SEDERBERG20161]. We define and use a
_multigraded generalization of the saturated special fiber ring_ that allows
us to obtain important results regarding the _(nonlinear) multiview variety_
given as the image of $\mathcal{G}$.

Our work is motivated by the study of multiview varieties that were considered
in [aholt2013hilbert] and in [agarwal2019ideals]. The classical multiview
variety is the image of a linear rational map
$\mathcal{G}:{\mathbb{P}}_{\mathbb{k}}^{3}\dashrightarrow{\mathbb{P}}_{\mathbb{k}}^{2}\times\cdots\times{\mathbb{P}}_{\mathbb{k}}^{2}$.
Each component of this map
$\mathcal{G}_{i}:{\mathbb{P}}_{\mathbb{k}}^{3}\dashrightarrow{\mathbb{P}}_{\mathbb{k}}^{2}$
is called a (pinhole) camera and is given by a full-rank $3\times 4$ matrix
$A_{i}$. This setup naturally arises in the study of computer vision, and the
multiview variety is the closure of the image $Y$ of $\mathcal{G}$. In
general, the defining ideal $J$ of $Y$ is a Cartwright-Sturmfels ideal, and so
it has many desirable properties, see [conca2018cartwright, conca2021radical].
The kernel of $A_{i}$ is a single point in ${\mathbb{P}}_{\mathbb{k}}^{3}$ and
the base locus of $\mathcal{G}$ is the collection of these points. In the case
that these points are distinct, by [aholt2013hilbert, Theorem 3.7], we have an
explicit description for the multigraded Hilbert function of $J$. This result
follows from explicitly computing the generic initial ideal of the defining
ideal $J$ of $Y$. Our primary goal is to show that similar relationships hold
even in the nonlinear case. In particular, we focus on the highest degree
coefficients of the multigraded Hilbert polynomial of $J$, which are the
multidegrees of the multiprojective variety $Y$.

To study the ideal of the nonlinear multiview variety, see Section missing 3,
it is not possible to take the same approach as [aholt2013hilbert] since the
generic initial ideals are not radical. Instead, we are able to compute the
multidegrees of the image of $\mathcal{G}$ using the saturated special fiber
ring. We show in Theorem 3.1 that our multigraded version of the saturated
special fiber ring maintains similar important properties (cf. [MULTPROJ,
Theorem 2.4]). Similarly to [aholt2013hilbert, Theorem 3.7], we show in
Theorem 3.2 that the multidegrees of $Y$ are related to the mixed
multiplicities of the base points. In Section missing 4, we recover several
known results in the setting of the classical linear multiview varieties (see
Theorem 4.1, Section missing 4, Section missing 4). In Section missing 5, we
provide explicit formulas for several interesting families of rational maps.
The techniques used in Section missing 5 convey the depth and scalability of
the saturated special fiber ring to study rational maps. We also prove some
technical results that might be interesting in their own right, see Theorem
5.1, Section missing 5 and Section missing 5. In Section missing 6, we focus
on the case where the rational map $\mathcal{G}$ is given by monomials, i.e.
the ideals $I_{1},\dots,I_{p}$ are monomial ideals. Using convex geometry, we
relate the multidegrees of the image with the mixed volumes of the associated
Newton polytopes of the ideals $I_{i}$ after dehomogenizing.

Future directions. In this paper, we have computed for large families of
rational maps, the multidegrees of their image. Following this, it is natural
to ask how we may compute the other coefficients of the multigraded Hilbert
polynomial. In particular, we ask whether these coefficients can be determined
from the points in the base locus analogously to Theorem 3.2. Of course, it is
also quite desirable to study other families of nonlinear multiview varieties.
And so, we ask for which families of multiview varieties is possible to find
explicit formulas for their multidegrees.

## References