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

# Multidegrees, prime ideals, and non-standard gradings

Alessio Caminata  Dipartimento di Matematica, Università di Genova, via
Dodecaneso 35, 16146 Genova, Italy<EMAIL_ADDRESS>,  Yairon Cid-
Ruiz  Department of Mathematics, KU Leuven, Celestijnenlaan 200B, 3001 Leuven,
Belgium<EMAIL_ADDRESS>and  Aldo Conca  Dipartimento di
Matematica, Università di Genova, via Dodecaneso 35, 16146 Genova, Italy
<EMAIL_ADDRESS>
###### Abstract.

We study several properties of multihomogeneous prime ideals. We show that the
multigraded generic initial ideal of a prime has very special properties, for
instance, its radical is Cohen-Macaulay. We develop a comprehensive study of
multidegrees in arbitrary positive multigraded settings. In these
environments, we extend the notion of Cartwright-Sturmfels ideals by means of
a standardization technique. Furthermore, we recover or extend important
results in the literature, for instance: we provide a multidegree version of
Hartshorne’s result stating the upper semicontinuity of arithmetic degree
under flat degenerations, and we give an alternative proof of Brion’s result
regarding multiplicity-free varieties.

###### Key words and phrases:

multidegrees, prime ideals, non-standard gradings, multigraded generic initial
ideals, projections, standardization, Cartwright-Sturmfels ideals,
polymatroids, multiplicity-free varieties.

###### 2020 Mathematics Subject Classification:

13H15, 13P10, 14C17, 05E40, 52B40.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 A multidegree version of a result of Hartshorne
  4. 4 Multidegrees of projections
  5. 5 The multidegrees of prime ideals
  6. 6 Multiplicity-free varieties
  7. 7 Standardization of ideals
  8. 8 Examples of determinantal ideals with a fine grading

##  1\. Introduction

This paper is concerned with several aspects of the theory of multidegrees.
The concept of multidegree provides the right generalization of the degree of
a variety to a multiprojective setting, and its study goes back to seminal
work by van der Waerden [VAN_DER_WAERDEN] in 1929. Multidegrees have found
applications in several areas (e.g., algebraic geometry, commutative algebra,
combinatorics, convex geometry, and more recently, algebraic statistics), and
there is a long list of more recent papers where the notion of multidegree (or
mixed multiplicity) plays a fundamental role (see, e.g., [Bhattacharya,
VERMA_BIGRAD, HERMANN_MULTIGRAD, trung2001positivity, cidruiz2021mixed,
EXPONENTIAL_VARIETIES, KNUTSON_MILLER_SCHUBERT, michalek2020maximum,
POSITIVITY, CDNG_MINORS, CDNG_CS_IDEALS, Huh12, huh2019logarithmic, CS_PAPER,
EQ_MDEG_SYM_MAT, TRUNG_VERMA, manivel2020complete, KNUTSON_MILLER_YONG,
STURMFELS_UHLER]).

The goal of this paper is twofold. Firstly, we concentrate on studying several
interesting properties that prime ideals enjoy in a multigraded setting.
Secondly, we make a comprehensive study of multidegrees in non-standard
multigradings.

For organizational purposes, we divide the Introduction into two shorter
subsections.

###  1.1. Prime ideals

Let $\mathbb{k}$ be a field and $S=\mathbb{k}[x_{i,j}\mid 1\leq i\leq p,0\leq
j\leq m_{i}]$ be a standard $\mathbb{N}^{p}$-graded polynomial ring with
grading induced by setting $\deg(x_{i,j})={\mathbf{e}}_{i}\in\mathbb{N}^{p}$,
where ${\mathbf{e}}_{i}$ is the $i$-th standard basis vector. Then $S$ is
naturally seen as the multihomogeneous coordinate ring of the product of
projective spaces
${\mathbb{P}}={\mathbb{P}}_{\mathbb{k}}^{m_{1}}\times_{\mathbb{k}}\cdots\times_{\mathbb{k}}{\mathbb{P}}_{\mathbb{k}}^{m_{p}}$.

Let $P\subset S$ be an $S$-homogeneous prime ideal. A fundamental goal of this
paper is to study several properties of $P$. Although our results regarding
prime ideals do not involve in principle multidegrees, our proofs heavily
depend on the use of this concept. The fact that prime ideals are quite
special from a multigraded point of view has already been noticed. To
highlight some of these we point out: Castillo–Cid-Ruiz–Li–Montaño–Zhang’s
result that the support of the positive multidegrees of $S/P$ forms a discrete
polymatroid [POSITIVITY, Theorem A], Brändén–Huh’s result that the volume
polynomial (a different encoding of multidegrees) of $S/P$ is Lorentzian
[HUH_BRANDEN, Theorem 4.6], and Brion’s result on multiplicity-free varieties
[BRION_MULT_FREE].

Our first main result is regarding _multigraded generic initial ideals_. Given
a monomial order $>$ on $S$, one may define the multigraded generic initial
ideal in analogy with the singly-graded setting (see Section missing 5 for
details).

######  Theorem A (Theorem 5.2).

($\mathbb{k}$ infinite) We have that $\sqrt{{\text{gin}}_{>}(P)}$ is a Cohen-
Macaulay ideal.

This result can be seen as yet another manifestation of the fact that the
initial ideals of primes are “special”. For example:

  * –

In [KS_INIT_PRIMES], Kalkbrener and Sturmfels showed that the radical of the
initial ideal of a prime is equidimensional and connected in codimension one.

  * –

In [STURMFELS_GROBNER, Chapter 8], Sturmfels proved that the radical of the
initial ideal of a toric ideal is Cohen-Macaulay.

  * –

In [HOSTEN_THOMAS], Hoşten and Thomas proved that the associated primes of the
initial ideal of a toric ideal satisfy a saturated chain property.

On the other hand, it is not difficult to find a prime ideal $P$ such that
${\text{in}}_{>}(P)$ and its radical are not Cohen-Macaulay (see Section
missing 5). In this sense, Theorem A is a sharp result.

We also discovered an interesting (and perhaps unexpected) behavior of the
minimal primary components of ${\text{gin}}_{>}(P)$. Let
${\mathfrak{J}}=\\{j_{1},\ldots,j_{k}\\}\subset[p]=\\{1,\ldots,p\\}$ be a
subset. We denote by $S_{({\mathfrak{J}})}$ the polynomial subring generated
by the variables with degrees
${\mathbf{e}}_{j_{1}},\ldots,{\mathbf{e}}_{j_{k}}$, and we write
$I_{({\mathfrak{J}})}=I\cap S_{({\mathfrak{J}})}$ for the corresponding
contraction of an $S$-homogeneous ideal $I\subset S$. For an ideal $I\subset
S$, let $\text{MLength}(I)$ denote the maximal length of the minimal primary
components of $I$. The following theorem shows that MLength cannot increase
under the natural projections of ${\text{gin}}_{>}(P)$.

######  Theorem B (Theorem 5.3).

($\mathbb{k}$ infinite) We have the inequality

${\rm
MLength}\left({\text{gin}}_{>}\left(P_{({\mathfrak{J}})}\right)\right)\;\leq\;{\rm
MLength}\left({\text{gin}}_{>}\left(P\right)\right).$

Moreover, for every
${\mathfrak{p}}\in{\text{Min}}_{S_{({\mathfrak{J}})}}\left(S_{({\mathfrak{J}})}/{\text{gin}}_{>}(P_{({\mathfrak{J}})})\right)$,
we can find
$\mathfrak{q}\in{\text{Min}}_{S}\left(S/{\text{gin}}_{>}(P)\right)$ such that
the length of the ${\mathfrak{p}}$-primary component of
${\text{gin}}_{>}\left(P_{({\mathfrak{J}})}\right)$ divides the length of the
$\mathfrak{q}$-primary component of ${\text{gin}}_{>}(P)$.

To better appreciate the complexity of the process described in Theorem B, the
reader is referred to Section missing 5. Also, Theorem B may not hold for non
prime ideals (see Section missing 5).

Let
$X=\text{MultiProj}(S/P)\subset{\mathbb{P}}={\mathbb{P}}_{\mathbb{k}}^{m_{1}}\times_{\mathbb{k}}\cdots\times_{\mathbb{k}}{\mathbb{P}}_{\mathbb{k}}^{m_{p}}$
be the integral closed subscheme corresponding to $P$. We also consider the
behavior of the multidegrees of $X$ under the natural projections. It turns
out that our proof of Theorem B is a consequence of this study of multidegrees
under projections. The projection corresponding to
${\mathfrak{J}}=\\{j_{1},\ldots,j_{k}\\}$ is given by
$\Pi_{\mathfrak{J}}:{\mathbb{P}}={\mathbb{P}}_{\mathbb{k}}^{m_{1}}\times_{\mathbb{k}}\cdots\times_{\mathbb{k}}{\mathbb{P}}_{\mathbb{k}}^{m_{p}}\rightarrow{\mathbb{P}}^{\prime}={\mathbb{P}}_{\mathbb{k}}^{m_{j_{1}}}\times_{\mathbb{k}}\cdots\times_{\mathbb{k}}{\mathbb{P}}_{\mathbb{k}}^{m_{j_{k}}}$.
For each ${\mathbf{n}}=(n_{1},\ldots,n_{p})\in\mathbb{N}^{p}$ with
$|{\mathbf{n}}|=n_{1}+\cdots+n_{p}=\dim(X)$, one says that
$\deg_{\mathbb{P}}^{\mathbf{n}}(X)$ is the multidegree of $X$ of type
${\mathbf{n}}$ with respect to ${\mathbb{P}}$ (see Section missing 2 for
precise definitions). Geometrically speaking, when $\mathbb{k}$ is
algebraically closed, $\deg_{\mathbb{P}}^{\mathbf{n}}(X)$ equals 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 $L_{i}\subset{\mathbb{P}}_{\mathbb{k}}^{m_{i}}$ is a general linear
subspace of dimension $m_{i}-n_{i}$. Let $\text{MDeg}_{\mathbb{P}}(X)$ be the
maximum of the multidegrees of $X\subset{\mathbb{P}}$. The next theorem shows
that MDeg cannot increase under the natural projections.

######  Theorem C (Theorem 4.1).

We have the inequality

${\rm
MDeg}_{{\mathbb{P}}^{\prime}}\left(\Pi_{\mathfrak{J}}(X)\right)\;\leq\;{\rm
MDeg}_{\mathbb{P}}(X).$

Moreover, for any ${\mathbf{d}}\in\mathbb{N}^{k}$ with
$|{\mathbf{d}}|=\dim\left(\Pi_{\mathfrak{J}}(X)\right)$ and such that
$\deg_{{\mathbb{P}}^{\prime}}^{\mathbf{d}}\left(\Pi_{\mathfrak{J}}(X)\right)>0$,
there exists some ${\mathbf{n}}\in\mathbb{N}^{p}$ with
$|{\mathbf{n}}|=\dim(X)$ and such that $\deg_{\mathbb{P}}^{\mathbf{n}}(X)>0$
and
$\deg_{{\mathbb{P}}^{\prime}}^{\mathbf{d}}\left(\Pi_{\mathfrak{J}}(X)\right)$
divides $\deg_{\mathbb{P}}^{\mathbf{n}}(X)$.

If we drop the condition that $P$ is a prime ideal, then one can find fairly
simple instances where multidegrees do increase under projections (see Section
missing 4).

Finally, we also provide an alternative proof of the following remarkable
result of Brion [BRION_MULT_FREE] regarding multiplicity-free varieties. We
say that $X\subset{\mathbb{P}}$ is _multiplicity-free_ if
$\deg_{\mathbb{P}}^{{\mathbf{n}}}(X)\in\\{0,1\\}$ for all
${\mathbf{n}}\in\mathbb{N}^{p}$ with $|{\mathbf{n}}|=\dim(X)$. Let
$\text{MSupp}_{\mathbb{P}}(X)=\\{{\mathbf{n}}\in\mathbb{N}^{p}\mid|{\mathbf{n}}|=\dim(X)\text{
and }\deg_{{\mathbb{P}}}^{\mathbf{n}}(X)>0\\}$ be the support of positive
multidegrees of $X$.

######  Theorem D (Theorem 6.2; Brion [BRION_MULT_FREE]).

If
$X\subset{\mathbb{P}}={\mathbb{P}}_{\mathbb{k}}^{m_{1}}\times_{\mathbb{k}}\cdots\times_{\mathbb{k}}{\mathbb{P}}_{\mathbb{k}}^{m_{p}}$
is multiplicity-free, then:

  1. (i)

$X$ is arithmetically Cohen-Macaulay (i.e., $S/P$ is a Cohen-Macaulay ring).

  2. (ii)

$X$ is arithmetically normal (i.e., $S/P$ is a normal domain).

  3. (iii)

($\mathbb{k}$ infinite) There is flat degeneration of $X$ to the following
reduced union of multiprojective spaces

$H\;=\;\bigcup_{{\mathbf{n}}=(n_{1},\ldots,n_{p})\,\in\,\text{MSupp}_{\mathbb{P}}(X)}{\mathbb{P}}_{\mathbb{k}}^{n_{1}}\times_{\mathbb{k}}\cdots\times_{\mathbb{k}}{\mathbb{P}}_{\mathbb{k}}^{n_{p}}\;\,\subset\;\,{\mathbb{P}}={\mathbb{P}}_{\mathbb{k}}^{m_{1}}\times_{\mathbb{k}}\cdots\times_{\mathbb{k}}{\mathbb{P}}_{\mathbb{k}}^{m_{p}}$

where
${\mathbb{P}}_{\mathbb{k}}^{n_{i}}=\text{Proj}\left(\mathbb{k}[x_{i,m_{i}-n_{i}},\ldots,x_{i,m_{i}}]\right)\subset{\mathbb{P}}_{\mathbb{k}}^{m_{i}}=\text{Proj}\left(\mathbb{k}[x_{i,0},\ldots,x_{i,m_{i}}]\right)$
only uses the last $n_{i}+1$ coordinates.

Our proof of Theorem D uses techniques quite different from the ones used in
[BRION_MULT_FREE], and ideas related to the fiber-full scheme
([LOC_FIB_FULL_SCHEME, FIBER_FULL, FIB_FULL_SCHEME]) play an important role.
We also point out that our proof is valid for arbitrary fields.

###  1.2. Non-standard gradings

Multidegrees in non-standard multigradings are considerably more complicated
as they lack a clear multiprojective geometrical content, but they are still
important because of the flexibility they provide in terms of applications.
One prime example is Knutson–Miller’s result [KNUTSON_MILLER_SCHUBERT, Theorem
A] expressing double Schubert polynomials and double Grothendieck polynomials
as the multidegree polynomials and $\mathcal{K}$-polynomials, respectively, of
matrix Schubert varieties with a fine grading. This fine grading takes into
account both row and column position; we provide some examples that explore
the technicalities of this fine grading (up to certain sign changes) in
Section missing 8. These fine gradings play a key role in the recent proof by
Castillo–Cid-Ruiz–Mohammadi–Montaño [DOUBLE_SCHUBERT] of a conjecture of
Monical–Tokcan–Yong [monical2019newton] stating the saturated Newton polytope
property of double Schubert polynomials.

Let $R=\mathbb{k}[x_{1},\ldots,x_{n}]$ be a positively $\mathbb{N}^{p}$-graded
polynomial ring (that is,
$\deg(x_{i})\in\mathbb{N}^{p}\setminus\\{\mathbf{0}\\}$ for all $1\leq i\leq
n$ and $\deg(\alpha)=\mathbf{0}\in\mathbb{N}^{p}$ for all
$\alpha\in\mathbb{k}$). Let $I\subset R$ be an $R$-homogeneous ideal. The
_multidegree polynomial_
$\mathcal{C}(R/I;{\mathbf{t}})=\mathcal{C}(R/I;t_{1},\ldots,t_{p})$ of $R/I$
is defined in terms of the multigraded Hilbert series of $R/I$ (see Section
missing 2 for details). In the standard multigraded setting it may be seen as
an algebraic encoding of the multidegrees of the corresponding multiprojective
scheme (see Theorem 2.1).

Our approach is to associate to $I$ an ideal $J$ in a standard
$\mathbb{N}^{p}$-graded polynomial ring $S$ such that
$\mathcal{C}(R/I;{\mathbf{t}})=\mathcal{C}(S/J;{\mathbf{t}})$. We call $J$ the
_standardization_ of $I$ (see Section missing 7). This construction is
inspired by the techniques _step-by-step homogenization_ introduced by
McCullough–Peeva in [McCULLOUGH_PEEVA] for singly-graded settings and
_standardization_ introduced in [DOUBLE_SCHUBERT] for certain multigraded
settings. It turns out that, there is a tight relation between $I\subset R$
and its standardization $J\subset S$, as it may be witnessed from the
following statements:

  1. (i)

${\text{codim}}(I)={\text{codim}}(J)$.

  2. (ii)

$I\subset R$ and $J\subset S$ have the same $\mathbb{N}^{p}$-graded Betti
numbers.

  3. (iii)

$\mathcal{K}(R/I;{\mathbf{t}})=\mathcal{K}(S/J;{\mathbf{t}})$ and
$\mathcal{C}(R/I;{\mathbf{t}})=\mathcal{C}(S/J;{\mathbf{t}})$.

  4. (iv)

$R/I$ is a Cohen-Macaulay ring if and only if $S/J$ is a Cohen-Macaulay ring.

  5. (v)

Initial ideals are compatible with standardization.

  6. (vi)

If $I\subset R$ is a prime ideal and it does not contain any variable, then
$J\subset S$ is also a prime ideal.

For more details, see Theorem 7.1.

These results allow us to deduce in Theorem 7.2 that the support of
$\mathcal{C}(R/P;{\mathbf{t}})$ is a discrete polymatroid for any prime ideal
$P\subset R$, which constitutes an extension of [POSITIVITY, Theorem A].

We introduce the notion of _arithmetic multidegree polynomial_ , which can be
seen as a generalization of the notion of arithmetic degree considered by
Sturmfels–Trung–Vogel in [STV_DEGREE]. It is defined as follows

$\mathcal{A}(R/I;{\mathbf{t}})\;=\;\sum_{P\in{\text{Ass}}_{R}(R/I)}\text{length}_{R_{P}}\left(\text{H}_{P}^{0}((R/I)_{P})\right)\,\mathcal{C}(R/P;{\mathbf{t}}).$

Motivated by the work of Hartshorne on the connectivity of Hilbert schemes
[HARTSHORNE_CONNNECT], we show in Theorem 3.1 that arithmetic multidegree is
upper semicontinuous under flat degenerations. As a consequence, in the case
of Gröbner degenerations, we obtain the coefficient-wise inequality

$\mathcal{A}\left(R/I;{\mathbf{t}}\right)\,\leq_{c}\,\mathcal{A}\left(R/{\text{in}}_{>}(I);{\mathbf{t}}\right).$

This upper semicontinuity result was obtained by Hartshorne
[HARTSHORNE_CONNNECT, Chapter 2] in the singly-graded setting.

In §missing 7.1, we extend the notion of Cartwright–Sturmfels ideals to
arbitrary positive multigradings. The original definition was given over a
standard multigraded setting by Conca–De Negri–Gorla in a series of papers
[CDNG_CS_IDEALS, CDNG_GIN, CDNG_GRAPH, CDNG_MINORS, conca2022radical]. This
was obtained as an abstraction of previous work of Cartwright and Sturmfels
[CS_PAPER]; thus the chosen name.

In the current arbitrary positive multigraded setting, we say that $I\subset
R$ is _Cartwright–Sturmfels_ (CS for short) if there is a radical Borel-fixed
ideal in the standard $\mathbb{N}^{p}$-graded polynomial ring $S$ that has the
same $\mathcal{K}$-polynomial as $I$ (see §missing 7.1). It should be
mentioned that $I\subset R$ is CS if and only if its standardization $J\subset
S$ is CS (in the sense of [CDNG_CS_IDEALS]). By utilizing the close relation
between $I\subset R$ and its standardization, in Theorem 7.3, we prove the
following statements when $I$ is CS:

  1. (i)

${\text{in}}_{>}(I)$ is radical and CS for any monomial order $>$ on $R$; in
particular, $I$ is radical.

  2. (ii)

The $\mathbb{N}$-graded Castelnuovo-Mumford regularity $\text{reg}(I)$ is
bounded from above by $p$.

  3. (iii)

$I$ is a multiplicity-free ideal.

  4. (iv)

If $P\subset R$ is a minimal prime of $I$, then $P$ is CS.

  5. (v)

All reduced Gröbner bases of $I$ consist of elements of multidegree
$\leq(1,\ldots,1)\in\mathbb{N}^{p}$. In particular, $I$ has a universal
Gröbner basis of elements of multidegree $\leq(1,\ldots,1)\in\mathbb{N}^{p}$.

This shows that CS ideals in arbitrary positive multigradings enjoy similar
desirable properties as in the standard multigraded setting.

Finally, as a consequence of our work, we portray a family of multigraded
Hilbert functions whose ideals have very rigid properties. It is most
convenient to enunciate this result in terms of the multigraded Hilbert scheme
of Haiman and Sturmfels [HAIMAN_STURMFELS]. Let
$h:\mathbb{N}^{p}\rightarrow\mathbb{N}$ be a function and consider the
corresponding multigraded Hilbert scheme ${\text{HS}}_{R/\mathbb{k}}^{h}$
parametrizing $R$-homogeneous ideals with Hilbert function equal to $h$.

######  Theorem E (§missing 7.1).

If ${\text{HS}}_{R/\mathbb{k}}^{h}$ contains a $\mathbb{k}$-point that
corresponds to a CS ideal and a $\mathbb{k}$-point that corresponds to a prime
ideal, then any $\mathbb{k}$-point in ${\text{HS}}_{R/\mathbb{k}}^{h}$
corresponds to an ideal that is Cohen-Macaulay and CS.

This theorem can be seen as a generalization of a previous result by
Cartwright and Sturmfels [CS_PAPER, Theorem 2.1 and Corollary 2.6].

### Outline

The basic outline of this paper is as follows. In Section missing 2, we set
the notation and recall basic definitions. In Section missing 3, we prove the
upper semicontinuity of arithmetic multidegree under flat degenerations.
Section missing 4 is dedicated to prove Theorem C. We establish Theorem A and
Theorem B in Section missing 5. We study multiplicity-free varieties and prove
Theorem D in Section missing 6. In Section missing 7, we develop the new
notion of CS ideals and prove Theorem E. Finally, in Section missing 8, we
present several examples of determinantal ideals with a fine grading.

##  2\. Preliminaries

In this preparatory section, we briefly recall some basic concepts and we fix
the notation. In particular, we establish the initial stage regarding the main
theme of this paper, that of multidegrees. For more details on multidegrees
the reader is referred to [miller2005combinatorial, cidruiz2021mixed].

We use a multi-index notation. 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 ${\mathbf{e}}_{i}=\left(0,\ldots,1,\ldots,0\right)$. 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 $1\leq j\leq p$. For any
${\mathbf{n}}=(n_{1},\ldots,n_{p})\in\mathbb{N}^{p}$, we set
${\mathbf{n}}!=n_{1}!\cdots n_{p}!$. We write $\mathbf{0}\in\mathbb{N}^{p}$
and $\mathbf{1}\in\mathbb{N}^{p}$ for the vectors $\mathbf{0}=(0,\ldots,0)$
and $\mathbf{1}=(1,\ldots,1)$, respectively.

A discrete polymatroid $\mathcal{P}$ on $[p]:=\\{1,\ldots,p\\}$ is a
collection of points in $\mathbb{N}^{p}$ of the following form

$\mathcal{P=}\left\\{(n_{1},\ldots,n_{p})\in\mathbb{N}^{p}\;\mid\;\sum_{j\in{\mathfrak{J}}}n_{j}\leq
r({\mathfrak{J}}),\;\forall{\mathfrak{J}}\subsetneq[p],\;\sum_{i\in[p]}n_{i}=r([p])\right\\}$

with $r:2^{[p]}\rightarrow\mathbb{N}$ being a rank function on $[p]$. A _rank
function on_ $[p]$ is a function $r:2^{[p]}\rightarrow\mathbb{N}$ satisfying
the following three properties:

  1. (i)

$r(\emptyset)=0$.

  2. (ii)

$r({\mathfrak{J}}_{1})\leq r({\mathfrak{J}}_{2})$ if
${\mathfrak{J}}_{1}\subseteq{\mathfrak{J}}_{2}\subseteq[p]$.

  3. (iii)

$r({\mathfrak{J}}_{1}\cap{\mathfrak{J}}_{2})+r({\mathfrak{J}}_{1}\cup{\mathfrak{J}}_{2})\leq
r({\mathfrak{J}}_{1})+r({\mathfrak{J}}_{2})$ if
${\mathfrak{J}}_{1},{\mathfrak{J}}_{2}\subseteq[p]$.

A comprehensive discussion regarding polymatroids can be found in
[schrijver2003combinatorial].

######  Remark .

The following statements hold:

  1. (i)

Let $\mathcal{P}$ be a discrete polymatroid on $[p]$ with rank function
$r:2^{[p]}\rightarrow\mathbb{N}$. Let $m_{1},\ldots,m_{p}$ be positive
integers such that $r(\\{i\\})\leq m_{i}$. Then the function
$s:2^{[p]}\rightarrow\mathbb{N}$ given by

$s({\mathfrak{J}})\;:=\;\sum_{j\in{\mathfrak{J}}}m_{j}+r([p]\setminus{\mathfrak{J}})-r([p])$

is a rank function. The discrete polymatroid $\mathcal{P}^{*}$ on $[p]$
determined by $s$ is said to be a dual of $\mathcal{P}$.

  2. (ii)

If $\mathcal{P}_{1}$ and $\mathcal{P}_{2}$ are two discrete polymatroids on
$[p]$, then the Minkowski sum $\mathcal{P}_{1}+\mathcal{P}_{2}$ is a discrete
polymatroid on $[p]$.

###### Proof.

(i) See [schrijver2003combinatorial, §44.6f].

(ii) See [schrijver2003combinatorial, Theorem 44.6, Corollary 46.2c]. ∎

Following [monical2019newton], we say that a polynomial
$f=\sum_{{\mathbf{n}}}c_{{\mathbf{n}}}{\mathbf{t}}^{\mathbf{n}}\in\mathbb{Z}[t_{1},\ldots,t_{p}]$
has the _Saturated Newton Polytope property_ (SNP property for short) if the
support ${\text{supp}}(f)=\\{{\mathbf{n}}\in\mathbb{N}^{p}\mid
c_{\mathbf{n}}\neq 0\\}$ of $f$ is equal to
$\text{Newton}(f)\cap\mathbb{N}^{p}$, where
$\text{Newton}(f)=\text{ConvexHull}\\{{\mathbf{n}}\in\mathbb{N}^{n}\mid
c_{\mathbf{n}}\neq 0\\}$ denotes the _Newton polytope_ of $f$; in other words,
if the support of $f$ consists of the integer points of a polytope.

######  Remark .

Given a homogeneous polynomial $f\in\mathbb{Z}[t_{1},\ldots,t_{p}]$, if
${\text{supp}}(f)$ is a discrete polymatroid on $[p]$, then $f$ has the SNP
property.

###  2.1. Multidegrees in positive multigradings

Here we briefly review the notion of multidegrees in positive multigradings
that are not necessarily standard.

Let $\mathbb{k}$ be a field and $R=\mathbb{k}[x_{1},\ldots,x_{n}]$ be a
_positively $\mathbb{N}^{p}$-graded_ polynomial ring (that is,
$\deg(x_{i})\in\mathbb{N}^{p}\setminus\\{\mathbf{0}\\}$ for all $1\leq i\leq
n$ and $\deg(\alpha)=\mathbf{0}\in\mathbb{N}^{p}$ for all
$\alpha\in\mathbb{k}$). Let $M$ be a finitely generated
$\mathbb{Z}^{p}$-graded $R$-module and $F_{\bullet}$ be a
$\mathbb{Z}^{p}$-graded free $R$-resolution $F_{\bullet}:\;\cdots\rightarrow
F_{i}\rightarrow F_{i-1}\rightarrow\cdots\rightarrow F_{1}\rightarrow F_{0}$
of $M$. Let $t_{1},\ldots,t_{p}$ be variables over $\mathbb{Z}$ and consider
the polynomial ring $\mathbb{Z}[{\mathbf{t}}]=\mathbb{Z}[t_{1},\ldots,t_{p}]$,
where the variable $t_{i}$ corresponds with the $i$-th elementary vector
${\mathbf{e}}_{i}\in\mathbb{N}^{p}$. If we write
$F_{i}=\bigoplus_{j}R(-\mathbf{b}_{i,j})$ with
$\mathbf{b}_{i,j}=(\mathbf{b}_{i,j,1},\ldots,\mathbf{b}_{i,j,p})\in\mathbb{Z}^{p}$,
then we define the Laurent polynomial
$\left[F_{i}\right]_{\mathbf{t}}\,:=\,\sum_{j}{\mathbf{t}}^{\mathbf{b}_{i,j}}=\sum_{j}t_{1}^{\mathbf{b}_{i,j,1}}\cdots
t_{p}^{\mathbf{b}_{i,j,p}}.$ Then, the _$\mathcal{K}$ -polynomial_ of $M$ is
defined by

$\mathcal{K}(M;{\mathbf{t}})\,:=\,\sum_{i}{(-1)}^{i}\left[F_{i}\right]_{\mathbf{t}}.$

######  Remark .

Since $R$ is assumed to be positively graded, there is a well-defined notion
of Hilbert series

${\text{Hilb}}_{M}({\mathbf{t}})\;:=\;\sum_{{\mathbf{n}}\in\mathbb{Z}^{p}}\dim_{\mathbb{k}}\left(\left[M\right]_{\mathbf{n}}\right){\mathbf{t}}^{\mathbf{n}},$

and then we can write

${\text{Hilb}}_{M}({\mathbf{t}})\;=\;\frac{\mathcal{K}(M;{\mathbf{t}})}{\prod_{i=1}^{n}\left(1-{\mathbf{t}}^{\deg(x_{i})}\right)}.$

In particular, this shows that $\mathcal{K}(M;{\mathbf{t}})$ does not depend
on the specific free resolution $F_{\bullet}$ (also, see
[miller2005combinatorial, Theorem 8.34]).

######  Definition .

The _multidegree polynomial_ of a finitely generated $\mathbb{Z}^{p}$-graded
$R$-module $M$ is the homogeneous polynomial
$\mathcal{C}(M;{\mathbf{t}})\in\mathbb{Z}[{\mathbf{t}}]$ given as the sum of
all terms in

$\mathcal{K}(M;\mathbf{1}-{\mathbf{t}})=\mathcal{K}(M;1-t_{1},\ldots,1-t_{p})$

having total degree ${\text{codim}}(M)$, which is the lowest degree appearing.

An $R$-homogeneous ideal $I\subset R$ is said to be _multiplicity-free_ if the
coefficients of $\mathcal{C}(R/I;{\mathbf{t}})$ belong to the set $\\{0,1\\}$.

We also have the following variation of the multidegree polynomial that was
introduced in [CDNG_MINORS]. This notion was introduced with the goal of
measuring the contribution of all the minimal primes (possibly of not maximal
dimension), and in the case of Borel-fixed ideals, it does precisely that (see
[CDNG_MINORS, Proposition 3.12]).

######  Definition .

The _$\mathcal{G}$ -multidegree polynomial_ of a finitely generated
$\mathbb{Z}^{p}$-graded $R$-module $M$ is the polynomial
$\mathcal{G}(M;{\mathbf{t}})\in\mathbb{Z}[{\mathbf{t}}]$ given as the sum of
all terms in $\mathcal{K}(M;\mathbf{1}-{\mathbf{t}})$ whose corresponding
monomial is minimal in the support of
$\mathcal{K}(M;\mathbf{1}-{\mathbf{t}})$.

For a polynomial $p\in\mathbb{Z}[{\mathbf{t}}]$, we denote by
$\left[p\right]_{i}$ the sum of the terms of $p$ with total degree equal to
$i$. So, for any finitely generated $\mathbb{Z}^{p}$-graded module $M$ with
${\text{codim}}(M)\geq i$, we obtain that
$\left[\mathcal{K}(M;\mathbf{1}-{\mathbf{t}})\right]_{i}$ equals
$\mathcal{C}(M;{\mathbf{t}})$ if ${\text{codim}}(M)=i$ and
$0\in\mathbb{Z}[{\mathbf{t}}]$ otherwise. We collect the following basic
facts.

######  Remark .

Let $M$ be a finitely generated $\mathbb{Z}^{p}$-graded $R$-module. The
following statements hold:

  1. (i)

$\mathcal{C}(M;{\mathbf{t}})$ is the sum of the terms of
$\mathcal{G}(M;{\mathbf{t}})$ having total degree equal to
${\text{codim}}(M)$.

  2. (ii)

If $z\in R$ is a homogeneous non-zero-divisor on $M$ with
$\deg(z)=(a_{1},\ldots,a_{p})\in\mathbb{N}^{p}$, then

    1. (a)

$\mathcal{K}(M/zM;{\mathbf{t}})=(1-t_{1}^{a_{1}}\cdots
t_{p}^{a_{p}})\,\mathcal{K}(M;{\mathbf{t}})$,

    2. (b)

$\mathcal{C}(M/zM;{\mathbf{t}})=\langle\deg(z),{\mathbf{t}}\rangle\,\mathcal{C}(M;{\mathbf{t}})$
where $\langle\deg(z),{\mathbf{t}}\rangle=a_{1}t_{1}+\cdots+a_{p}t_{p}$.

  3. (iii) (additivity)

Let $0\rightarrow L\rightarrow M\rightarrow N\rightarrow 0$ be a short exact
sequence of finitely generated $\mathbb{Z}^{p}$-graded $R$-modules. Then we
have the equality
$\mathcal{C}(M;{\mathbf{t}})=\left[\mathcal{K}(L;\mathbf{1}-{\mathbf{t}})\right]_{i}+\left[\mathcal{K}(N;\mathbf{1}-{\mathbf{t}})\right]_{i}$
with $i={\text{codim}}(M)$.

  4. (iv) (associativity formula)

$\mathcal{C}(M;{\mathbf{t}})=\sum_{\begin{subarray}{c}P\in{\text{Ass}}_{R}(M)\\\
\dim(R/P)=\dim(M)\end{subarray}}\text{length}_{R_{P}}\left(M_{P}\right)\,\mathcal{C}(R/P;{\mathbf{t}}).$

  5. (v) (positivity)

$\mathcal{C}(M;{\mathbf{t}})$ is a nonzero polynomial with non-negative
coefficients.

###### Proof.

(i) This part follows from the fact that any term in
$\mathcal{K}(M;\mathbf{1}-{\mathbf{t}})$ has total degree at least
${\text{codim}}(M)$ (see [miller2005combinatorial, Claim 8.54]).

(ii) It can be obtained by simple algebraic manipulations with Hilbert series
(see [miller2005combinatorial, Exercise 8.12]).

(iii) It is a direct consequence of the additivity of Hilbert series and the
inequalities ${\text{codim}}(L)\geq{\text{codim}}(M)$ and
${\text{codim}}(N)\geq{\text{codim}}(M)$.

(iv) This is proved by a standard use of prime filtrations; see
[miller2005combinatorial, Theorem 8.53].

(v) Write $M\cong F/K$ with $F$ a finite rank $\mathbb{Z}^{p}$-graded free
$R$-module. By considering a Gröbner degeneration we may assume that $K$ is a
monomial submodule (see, e.g., [EISEN_COMM, Chapter 15]). Then the
associativity formula (part (iv)) reduces to the case $M=R/P$ and $P$ a
monomial prime ideal. For a monomial prime ideal
$P=(x_{i_{1}},\ldots,x_{i_{m}})$, part (ii)(b) yields
$\mathcal{C}(R/P;{\mathbf{t}})=\langle\deg(x_{i_{1}}),{\mathbf{t}}\rangle\cdots\langle\deg(x_{i_{m}}),{\mathbf{t}}\rangle$,
and so the result follows from the fact that $R$ is positively
$\mathbb{N}^{p}$-graded. Alternatively, see Section missing 7. ∎

We introduce yet another notion of multidegree polynomial. It can be seen as a
generalization of the notion of arithmetic degree from [STV_DEGREE] (also, see
[HARTSHORNE_CONNNECT]). The goal of this notion is to capture the contribution
of all the associated primes.

######  Definition .

The _arithmetic multidegree polynomial_ of a finitely generated
$\mathbb{Z}^{p}$-graded $R$-module $M$ is given by

$\mathcal{A}(M;{\mathbf{t}})\;:=\;\sum_{P\in{\text{Ass}}_{R}(M)}\text{length}_{R_{P}}\left(\text{H}_{P}^{0}(M_{P})\right)\,\mathcal{C}(R/P;{\mathbf{t}}).$

###  2.2. Multidegrees in standard multigradings

We now concentrate on standard multigradings. The study of standard
multigraded algebras is of utmost importance as they correspond with closed
subschemes of a product of projective spaces (see, e.g., [POSITIVITY] and the
references therein).

Let $S=\mathbb{k}[x_{1},\ldots,x_{n}]$ be a _standard $\mathbb{N}^{p}$-graded_
polynomial ring over a field $\mathbb{k}$. That is, the total degree of each
variable $x_{i}$ is equal to one (i.e., for each $1\leq i\leq n$, we have
$\deg(x_{i})={\mathbf{e}}_{k_{i}}\in\mathbb{N}^{p}$ with $1\leq k_{i}\leq p$)
and $\deg(\alpha)=\mathbf{0}\in\mathbb{N}^{p}$ for all $\alpha\in\mathbb{k}$.

Suppose that for each $1\leq i\leq p$ there are exactly $m_{i}+1$ variables
$x_{j}$’s with degree ${\mathbf{e}}_{i}$ (i.e., $m_{i}+1=\lvert\\{1\leq j\leq
n\mid\deg(x_{j})={\mathbf{e}}_{i}\\}\rvert$). Thus the corresponding
multiprojective scheme is the product of projective spaces
$\text{MultiProj}(S)={\mathbb{P}}_{\mathbb{k}}^{m_{1}}\times_{\mathbb{k}}\cdots\times_{\mathbb{k}}{\mathbb{P}}_{\mathbb{k}}^{m_{p}}$.
In this setting, the irrelevant ideal is given by
${\mathfrak{N}}:=\left([S]_{{\mathbf{e}}_{1}}\right)\cap\cdots\cap\left([S]_{{\mathbf{e}}_{p}}\right)\subset
S$. An $S$-homogeneous prime ideal $P\subset S$ is said to be _relevant_ if
$P\not\supseteq{\mathfrak{N}}$. Relevant prime ideals are important as they
are the ones that have a well-defined geometrical counterpart. For more
details on the MultiProj construction, the reader is referred to [POSITIVITY,
HYRY_MULTGRAD]. The next remark shows that one can always restrict to relevant
prime ideals when considering the multidegree polynomial.

######  Remark .

Let $S^{\prime}=S[x_{n+1},\ldots,x_{n+p}]$ be a standard
$\mathbb{N}^{p}$-graded polynomial ring extending the grading of $S$ and with
$\deg(x_{n+i})={\mathbf{e}}_{i}$ for all $1\leq i\leq p$; accordingly, we have
that
$\text{MultiProj}(S^{\prime})={\mathbb{P}}_{\mathbb{k}}^{m_{1}+1}\times_{\mathbb{k}}\cdots\times_{\mathbb{k}}{\mathbb{P}}_{\mathbb{k}}^{m_{p}+1}$.
Let $P\subset S$ be an $S$-homogeneous (not necessarily relevant) prime ideal,
and set $P^{\prime}=PS^{\prime}$ to be the extension of $P$ to $S^{\prime}$.
Then the following statements hold:

  1. (i)

$P^{\prime}\subset S^{\prime}$ is a relevant prime ideal.

  2. (ii)

$\mathcal{C}(S/P;{\mathbf{t}})=\mathcal{C}(S^{\prime}/P^{\prime};{\mathbf{t}})$.

###### Proof.

(i) It is clear that $P^{\prime}$ is a prime ideal, and it is relevant because
$x_{n+i}\notin P^{\prime}$ for all $1\leq i\leq p$.

(ii) Given a $\mathbb{Z}^{p}$-graded free $S$-resolution $F_{\bullet}$ of
$S/P$, then $F_{\bullet}\otimes_{S}S^{\prime}$ is a $\mathbb{Z}^{p}$-graded
free $S^{\prime}$-resolution of $S^{\prime}/P^{\prime}$ with the same Betti
numbers. Therefore,
$\mathcal{K}(S/P;{\mathbf{t}})=\mathcal{K}(S^{\prime}/P^{\prime};{\mathbf{t}})$
and consequently
$\mathcal{C}(S/P;{\mathbf{t}})=\mathcal{C}(S^{\prime}/P^{\prime};{\mathbf{t}})$.
∎

Let $M$ be a finitely generated $\mathbb{Z}^{p}$-graded $S$-module, and set
${\mathbb{P}}:=\text{MultiProj}(S)={\mathbb{P}}_{\mathbb{k}}^{m_{1}}\times_{\mathbb{k}}\cdots\times_{\mathbb{k}}{\mathbb{P}}_{\mathbb{k}}^{m_{p}}$.
The relevant support of $M$ is given by
$\text{Supp}_{++}(M):=\text{Supp}(M)\cap{\mathbb{P}}$. There is a polynomial
$P_{M}({\mathbf{t}})=P_{M}(t_{1},\ldots,t_{p})\in\mathbb{Q}[{\mathbf{t}}]=\mathbb{Q}[t_{1},\ldots,t_{p}]$,
called the _Hilbert polynomial_ of $M$ (see, e.g., [HERMANN_MULTIGRAD, Theorem
4.1], [cidruiz2021mixed, Theorem 3.4]), such that the degree of $P_{M}$ is
equal to $r=\dim\left(\text{Supp}_{++}(M)\right)$ and

$P_{M}(\nu)=\dim_{\mathbb{k}}\left([M]_{\nu}\right)$

for all $\nu\in\mathbb{N}^{p}$ such that $\nu\gg\mathbf{0}$. Furthermore, if
we write

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

then $0\leq e(n_{1},\ldots,n_{p})\in\mathbb{Z}$ for all
$n_{1}+\cdots+n_{p}=r$.

######  Definition .

Under the notation of (missing) 1, we define the following invariants:

  1. (i)

For ${\mathbf{n}}=(n_{1},\ldots,n_{p})\in\mathbb{N}^{p}$ with
$\lvert{\mathbf{n}}\rvert=\dim\left(\text{Supp}_{++}(M)\right)$,
$e({\mathbf{n}};M):=e(n_{1},\ldots,n_{p})$ is the mixed multiplicity of $M$ of
type $\mathbf{n}$.

  2. (ii)

Let $R$ be an $\mathbb{N}^{p}$-graded quotient ring of $S$ and
$X=\text{MultiProj}(R)\subset{\mathbb{P}}$ be the corresponding closed
subscheme. For each ${\mathbf{n}}\in\mathbb{N}^{p}$ with
$\lvert{\mathbf{n}}\rvert=\dim(X)$,
$\deg_{\mathbb{P}}^{\mathbf{n}}(X):=e({\mathbf{n}};R)$ is the multidegree of
$X$ of type ${\mathbf{n}}$ with respect to ${\mathbb{P}}$.

######  Definition .

For a multiprojective scheme
$X\subset{\mathbb{P}}={\mathbb{P}}_{\mathbb{k}}^{m_{1}}
\times_{\mathbb{k}}\cdots\times_{\mathbb{k}}{\mathbb{P}}_{\mathbb{k}}^{m_{p}}$,
the support of positive multidegrees of $X$ is denoted by
$\text{MSupp}_{\mathbb{P}}(X):=\\{{\mathbf{n}}\in\mathbb{N}^{p}\mid|{\mathbf{n}}|=\dim(X)\text{
and }\deg_{{\mathbb{P}}}^{\mathbf{n}}(X)>0\\}$.

######  Remark .

In a multigraded setting one needs to be careful because the notions of
multidegrees introduced in §missing 2.1 and §missing 2.2 may not agree. This
pathology stems from the fact that the support $\text{Supp}(M)$ and the
relevant support $\text{Supp}_{++}(M)$ may be quite different for a
$\mathbb{Z}^{p}$-graded $S$-module $M$. Nevertheless, we have the following
unifying result.

######  Theorem 2.1 ([cidruiz2021mixed]).

Let $M$ be a finitely generated $\mathbb{Z}^{p}$-graded $S$-module and set
$r=\dim\left(\text{Supp}_{++}(M)\right)$. If
$\text{H}_{{\mathfrak{N}}}^{0}(M)=0$, then we have equality

$\mathcal{C}(M;{\mathbf{t}})\;=\;\sum_{\begin{subarray}{c}{\mathbf{n}}\in\mathbb{N}^{p}\\\
|{\mathbf{n}}|=r\end{subarray}}e({\mathbf{n}};M)\,t_{1}^{m_{1}-n_{1}}\cdots
t_{p}^{m_{p}-n_{p}}.$

###### Proof.

This is a consequence of [cidruiz2021mixed, Theorem A]. Also, for more
details, see [POSITIVITY, Remark 2.9]. ∎

For a closed subscheme $X\subset{\mathbb{P}}={\mathbb{P}}_{\mathbb{k}}^{m_{1}}
\times_{\mathbb{k}}\cdots\times_{\mathbb{k}}{\mathbb{P}}_{\mathbb{k}}^{m_{p}}$,
we say that the _multidegree polynomial of $X$_ is defined as

$\mathcal{C}(X;{\mathbf{t}})\;:=\;\sum_{\begin{subarray}{c}{\mathbf{n}}\in\mathbb{N}^{p}\\\
|{\mathbf{n}}|=\dim(X)\end{subarray}}\deg_{\mathbb{P}}^{\mathbf{n}}(X)\,t_{1}^{m_{1}-n_{1}}\cdots
t_{p}^{m_{p}-n_{p}}.$

The above theorem implies that
$\mathcal{C}(X;{\mathbf{t}})=\mathcal{C}(R;{\mathbf{t}})$ for an
$\mathbb{N}^{p}$-graded quotient ring $R$ of $S$ such that
$X=\text{MultiProj}(R)$ and $(0:_{R}{\mathfrak{N}}^{\infty})=0$.

The following piece of notation will be useful throughout the paper.

######  Notation  (Natural projections).

Let $\mathfrak{J}=\\{j_{1},\ldots,j_{k}\\}\subseteq[p]=\\{1,\ldots,p\\}$ be a
subset.

  1. (i)

Let
$\Pi_{\mathfrak{J}}:{\mathbb{P}}_{\mathbb{k}}^{m_{1}}\times_{\mathbb{k}}\cdots\times_{\mathbb{k}}{\mathbb{P}}_{\mathbb{k}}^{m_{p}}\rightarrow{\mathbb{P}}_{\mathbb{k}}^{m_{j_{1}}}\times_{\mathbb{k}}\cdots\times_{\mathbb{k}}{\mathbb{P}}_{\mathbb{k}}^{m_{j_{k}}}$
denote the natural projection.

  2. (ii)

Denote by $S_{({\mathfrak{J}})}$ the $\mathbb{N}^{k}$-graded
$\mathbb{k}$-algebra given by

$S_{({\mathfrak{J}})}:=\bigoplus_{\begin{subarray}{c}i_{1}\geq
0,\ldots,i_{p}\geq 0\\\ i_{j}=0\text{ if
}j\not\in{\mathfrak{J}}\end{subarray}}{\left[S\right]}_{(i_{1},\ldots,i_{p})},$

and denote by $I_{({\mathfrak{J}})}$ the contraction
$I_{({\mathfrak{J}})}:=I\cap S_{({\mathfrak{J}})}$ for any $S$-homogeneous
ideal $I\subset S$.

Since the coefficients of the multidegree polynomial are non-negative in
positive gradings, it becomes natural to address the positivity of these
coefficients. The next result characterizes the positivity of multidegrees,
and it follows from [POSITIVITY].

######  Theorem 2.2 ([POSITIVITY]).

Let $P\subset S$ be an $S$-homogeneous (not necessarily relevant) prime ideal.
Then the support of $\mathcal{C}(S/P;{\mathbf{t}})$ is a discrete polymatroid.

###### Proof.

By §missing 2.2, we may adjoin new variables to $S$ and assume that $P\subset
S$ is a relevant prime ideal. Consider the closed subscheme
$X=\text{MultiProj}(S/P)\subset{\mathbb{P}}={\mathbb{P}}_{\mathbb{k}}^{m_{1}}\times_{\mathbb{k}}\cdots\times_{\mathbb{k}}{\mathbb{P}}_{\mathbb{k}}^{m_{p}}$.
Then Theorem 2.1 implies that
$\mathcal{C}(S/P;{\mathbf{t}})\,=\,\sum_{\begin{subarray}{c}{\mathbf{n}}\in\mathbb{N}^{p}\\\
|{\mathbf{n}}|={\text{codim}}(P)\end{subarray}}\deg_{\mathbb{P}}^{{\mathbf{m}}-{\mathbf{n}}}(X){\mathbf{t}}^{\mathbf{n}}\;\in\;\mathbb{N}[t_{1},\ldots,t_{p}].$

Fix ${\mathbf{n}}\in\mathbb{N}^{p}$ with $|{\mathbf{n}}|={\text{codim}}(P)$.
From [POSITIVITY, Theorem A] we obtain that
$\deg_{\mathbb{P}}^{{\mathbf{m}}-{\mathbf{n}}}(X)>0$ if and only if
$\sum_{j\in{\mathfrak{J}}}m_{j}-\sum_{j\in{\mathfrak{J}}}n_{j}\leq\dim\big{(}\Pi_{\mathfrak{J}}(X)\big{)}$
for all ${\mathfrak{J}}\subset[p]$. We have that
$r:2^{[p]}\rightarrow\mathbb{N},\,r({\mathfrak{J}}):=\dim\big{(}\Pi_{\mathfrak{J}}(X)\big{)}\big{)}$
is a rank function (see [POSITIVITY, Proposition 5.1]). Equivalently, we get
that $\deg_{\mathbb{P}}^{{\mathbf{m}}-{\mathbf{n}}}(X)>0$ if and only if

$\sum_{j\in{\mathfrak{J}}}n_{j}\;\leq\;s({\mathfrak{J}}):=\sum_{j\in{\mathfrak{J}}}m_{j}+r([p]\setminus{\mathfrak{J}})-r([p])\quad\text{
for all \quad${\mathfrak{J}}\subseteq[p]$},$

where $s:2^{[p]}\rightarrow\mathbb{N}$ is a rank function by Section missing
2(i). Therefore the support of $\mathcal{C}(S/P;{\mathbf{t}})$ is a discrete
polymatroid, as claimed. ∎

##  3\. A multidegree version of a result of Hartshorne

In this section, we extend some results of Hartshorne into the world of
multidegrees. More precisely, by extending the results of
[HARTSHORNE_CONNNECT, Chapter 2], we study the behavior of the arithmetic
multidegree under flat degenerations (§missing 2.1). We shall see that the
arithmetic multidegree is the multigraded extension of the numbers $n_{i}$’s
introduced by Hartshorne in [HARTSHORNE_CONNNECT, Chapter 2].

###  3.1. The functor $R^{i}$

Here we recall the definition of the functors $R^{i}$ and some of their basic
properties. Essentially the same construction was considered by Schenzel
[SCHENZEL_CMF], under the name of _dimension filtration_ , in his study of
Cohen-Macaulay filtered modules.

Let $R$ be a Noetherian ring and $M$ be a finitely generated $R$-module.

######  Definition .

For any $i\geq 0$, we set

$R^{i}(M)\;:=\;\big{\\{}m\in
M\mid{\text{codim}}\left(\text{Supp}_{R}(m)\right)\geq i\big{\\}}.$

We first pause to point out some of the basic properties of the functor
$R^{i}$.

######  Remark .

The following statements hold:

  1. (i)

$R^{i}(M)$ is an $R$-module.

  2. (ii)

$R^{i}$ is a left exact functor.

  3. (iii)

We have the equality $R^{i}(M)\,=\,\text{H}_{J^{i}(M)}^{0}(M)$ where

$J^{i}(M)\,:=\bigcap_{\begin{subarray}{c}P\in{\text{Ass}}_{R}(M)\\\
{\text{codim}}(P)\geq i\end{subarray}}P$

is the intersection of the associated primes of $M$ of codimension $\geq i$.
By convention, we set $J^{i}(M)=R$ if there is no $P\in{\text{Ass}}_{R}(M)$
with ${\text{codim}}(P)\geq i$.

  4. (iv)

Let $0=\bigcap_{j}M_{j}$ be an irredundant primary decomposition of the
submodule $0\subset M$, and suppose that $M_{j}$ is $P_{j}$-primary. Then
$R^{i}(M)$ is the intersection of those $M_{j}$ such that
${\text{codim}}(P_{j})<i$.

  5. (v)

${\text{Ass}}_{R}\left(R^{i}(M)\right)=\\{P\in{\text{Ass}}_{R}(M)\mid{\text{codim}}(P)\geq
i\\}$.

  6. (vi)

${\text{Ass}}_{R}\left(M/R^{i}(M)\right)=\\{P\in{\text{Ass}}_{R}(M)\mid{\text{codim}}(P)<i\\}$.

  7. (vii)

If $N$ is an $R$-submodule of $M$ supported on codimension $\geq i$, then
$N\subset R^{i}(M)$.

###### Proof.

(i) and (ii) are immediate from the definition of $R^{i}$.

(iii) It is clear that $R^{i}(M)\supseteq\text{H}_{J^{i}(M)}^{0}(M)$. Let
$m\in R^{i}(M)$. By definition,
${\text{codim}}(\text{Ann}_{R}(m))={\text{codim}}(\text{Supp}_{R}(m))\geq i$,
and so it follows that
${\text{Ass}}_{R}(R/\text{Ann}_{R}(m))\subseteq\\{P\in{\text{Ass}}_{R}(M)\mid{\text{codim}}(P)\geq
i\\}$. This implies that $J^{i}(M)\subseteq\sqrt{\text{Ann}_{R}(m)}$ and
consequently $m\in\text{H}_{J^{i}(M)}^{0}(M)$. Thus we have the claimed
equality.

(iv), (v) and (vi) follow from the already proved part (iii) and known
behavior of the zeroth local cohomology with respect to associated primes and
primary decompositions (see [EISEN_COMM, Proposition 3.13]).

(vii) follows directly from the definition of $R^{i}$. ∎

The proposition below extends [HARTSHORNE_CONNNECT, Proposition 2.2].

######  Lemma .

Let $\phi:A\rightarrow B$ be homomorphism of Noetherian rings which is either
faithfully flat or a localization map with $B=W^{-1}A$ and $W\subset A$ a
multiplicatively closed set. Let $R$ be a finitely generated $A$-algebra and
$M$ be a finitely generated $R$-module. Set $R_{B}=R\otimes_{A}B$ and
$M_{B}=M\otimes_{A}B$. Then

$R^{i}(M_{B})\,\cong\,R^{i}(M)\otimes_{A}B$

for all $i\geq 0$.

###### Proof.

From §missing 3.1(iii) and the base change property of local cohomology (in
either case $\phi$ is flat), we obtain

$R^{i}(M)\otimes_{A}B\,=\,\text{H}_{J^{i}(M)}^{0}(M)\otimes_{A}B\,\cong\,\text{H}_{J^{i}(M)R_{B}}^{0}(M_{B}).$

First, we assume that $\phi:A\rightarrow B=W^{-1}B$ is a localization map.
Then, by basic properties of localizations,
${\text{Ass}}_{R_{B}}(M_{B})=\\{PR_{B}\mid P\in{\text{Ass}}_{R}(M)\text{ and
}P\not\supset W\\}$ and ${\text{codim}}(PR_{B})={\text{codim}}(P)$ for any
$P\in\text{Spec}(R)$. As a consequence, the equality
$J^{i}(M)R_{B}=J^{i}(M_{B})$ follows, and so we get
$R^{i}(M)\otimes_{A}B\cong\text{H}_{J^{i}(M)R_{B}}^{0}(M_{B})=\text{H}_{J^{i}(M_{B})}^{0}(M_{B})=R^{i}(M_{B})$.

Next, we assume that $\phi:A\rightarrow B$ is faithfully flat. By [MATSUMURA,
Theorem 23.2] the associated primes of $M_{B}$ are given by

${\text{Ass}}_{R_{B}}(M_{B})\,=\,\bigcup_{P\in{\text{Ass}}_{R}(M)}{\text{Ass}}_{R_{B}}\left(R_{B}/PR_{B}\right).$

On the other hand, since $R\rightarrow R_{B}$ is faithfully flat, it follows
that
${\text{codim}}\left(J^{i}(M)\right)={\text{codim}}\left(J^{i}(M)R_{B}\right)$
(see [MATSUMURA_OLD, (4.D) Theorem 3, page 28] and [MATSUMURA_OLD, (13.B)
Theorem 19(3), page 79]). This yields the equality
$\sqrt{J^{i}(M)R_{B}}=\sqrt{J^{i}(M_{B})}$ (see §missing 3.1(iii)). So, we
obtain
$R^{i}(M)\otimes_{A}B\cong\text{H}_{J^{i}(M)R_{B}}^{0}(M_{B})=\text{H}_{J^{i}(M_{B})}^{0}(M_{B})=R^{i}(M_{B})$,
and the result follows. ∎

We now make the connection between the functors $R^{i}$ and the arithmetic
multidegree introduced in §missing 2.1.

######  Definition-Proposition .

Let $\mathbb{k}$ be a field and $R=\mathbb{k}[x_{1},\ldots,x_{n}]$ be a
positively $\mathbb{N}^{p}$-graded polynomial ring. For any $i\geq 0$, we
define the _$i$ -th truncation multidegree polynomial_ by

$\mathcal{C}^{i}(M;{\mathbf{t}})\,:=\,\left[\mathcal{C}(R^{i}(M);{\mathbf{t}})\right]_{i}\,=\,\begin{cases}\mathcal{C}(R^{i}(M);{\mathbf{t}})&\text{
if }{\text{codim}}(R^{i}(M))=i\\\ 0&\text{ otherwise.}\end{cases}$

Then one has the equality

$\mathcal{A}(M;{\mathbf{t}})\,=\,\sum_{i\geq
0}\mathcal{C}^{i}(M;{\mathbf{t}}).$

###### Proof.

For any $P\in{\text{Ass}}_{R}(M)$ with ${\text{codim}}(P)=i$, §missing
3.1(iii) yields the following equality

$R^{i}(M)_{P}\,=\,\text{H}_{P}^{0}(M_{P}).$

Thus by §missing 3.1(v) and the associativity formula of multidegrees (see
§missing 2.1(iv)), we obtain

$\left[\mathcal{C}(R^{i}(M);{\mathbf{t}})\right]_{i}\,=\,\sum_{\begin{subarray}{c}P\in{\text{Ass}}_{R}(M)\\\
{\text{codim}}(P)=i\end{subarray}}\text{length}_{R_{P}}\left(\text{H}_{P}^{0}(M_{P})\right)\mathcal{C}(R/P;{\mathbf{t}}).$

Hence summing over $i\geq 0$ gives the claimed equality. ∎

###  3.2. Behavior under flat degenerations

In this subsection, we study the functor $R^{i}$ under a flat degeneration.
The following setup is assumed throughout the present subsection.

######  Setup .

Let $A$ be a universally catenary Noetherian domain. Let
$R=A[x_{1},\ldots,x_{n}]$ be a positively $\mathbb{N}^{p}$-graded polynomial
ring over $A$. Let $Q=\text{Quot}(A)$ be the field of fractions of $A$. For
any ${\mathfrak{p}}\in\text{Spec}(A)$, let
$\kappa({\mathfrak{p}}):=A_{\mathfrak{p}}/{\mathfrak{p}}A_{\mathfrak{p}}$ be
the residue field at ${\mathfrak{p}}$.

Notice that, if $M$ is a finitely generated $\mathbb{Z}^{p}$-graded $R$-module
and ${\mathfrak{p}}\in\text{Spec}(A)$, then
$M\otimes_{A}\kappa({\mathfrak{p}})$ is a finitely generated
$\mathbb{Z}^{p}$-graded module over the positively $\mathbb{N}^{p}$-graded
polynomial ring
$R\otimes_{A}\kappa({\mathfrak{p}})\cong\kappa({\mathfrak{p}})[x_{1},\ldots,x_{n}]$.
The goal of this subsection is to study the behavior of
$\mathcal{A}(M\otimes_{A}\kappa({\mathfrak{p}});{\mathbf{t}})$ when varying
${\mathfrak{p}}\in\text{Spec}(A)$. We have the following generic base change
property of $R^{i}$.

######  Proposition  ([HARTSHORNE_CONNNECT, Proposition 2.3]).

Let $M$ be a finitely generated $\mathbb{Z}^{p}$-graded $R$-module. Then there
is a dense open subset $\mathcal{U}\subset\text{Spec}(A)$ such that
$R^{i}(M)\otimes_{A}\kappa({\mathfrak{p}})\cong
R^{i}(M\otimes_{A}\kappa({\mathfrak{p}}))$ for all $i\geq 0$ and
${\mathfrak{p}}\in\text{Spec}(A)$.

In the polynomial ring
$\mathbb{Z}[{\mathbf{t}}]=\mathbb{Z}[t_{1},\ldots,t_{p}]$ we consider a
coefficient-wise order. Let $\geq_{c}$ be the partial order on
$\mathbb{Z}[{\mathbf{t}}]$ such that for any two polynomials
$f=\sum_{\mathbf{n}}f_{\mathbf{n}}{\mathbf{t}}^{\mathbf{n}}$ and
$g=\sum_{\mathbf{n}}g_{\mathbf{n}}{\mathbf{t}}^{\mathbf{n}}$ in
$\mathbb{Z}[{\mathbf{t}}]$ we have $f\geq_{c}g$ if and only if
$f_{\mathbf{n}}\geq g_{\mathbf{n}}$ for all ${\mathbf{n}}\in\mathbb{N}^{p}$.
The next proposition shows that the arithmetic multidegree polynomial cannot
decrease under flat degenerations.

######  Proposition .

Let $M$ be a finitely generated $\mathbb{Z}^{p}$-graded $R$-module, and
suppose that $M$ is flat over $A$. Then we have the coefficient-wise
inequality

$\mathcal{A}\left(M\otimes_{A}Q;{\mathbf{t}}\right)\;\leq_{c}\;\mathcal{A}\left(M\otimes_{A}\kappa({\mathfrak{p}});{\mathbf{t}}\right)$

for all ${\mathfrak{p}}\in\text{Spec}(A)$.

###### Proof.

Fix ${\mathfrak{p}}\in\text{Spec}(A)$. By [HARTSHORNE, Exercise II.4.11] or
[EGAII, Proposition 7.1.7] there is a discrete valuation ring $V$ of $Q$ that
dominates $A_{\mathfrak{p}}$; that is, $A_{\mathfrak{p}}\subset V$ and
${\mathfrak{p}}A_{\mathfrak{p}}=\mathfrak{n}\cap A_{\mathfrak{p}}$ where
$\mathfrak{n}$ is the closed point of $\text{Spec}(V)$. We have a field
extension $\kappa({\mathfrak{p}})\hookrightarrow\kappa(\mathfrak{n})$ and by
utilizing §missing 3.1, we obtain that

$R^{i}\left(M_{V}\otimes_{V}\kappa(\mathfrak{n})\right)\,\cong\,R^{i}\left(M\otimes_{A}\kappa({\mathfrak{p}})\right)\otimes_{\kappa({\mathfrak{p}})}\kappa(\mathfrak{n}),$

where $M_{V}=M\otimes_{A}V$. Since multigraded Hilbert functions are preserved
under base field extensions, we get the equality of polynomials
$\mathcal{A}\left(M\otimes_{A}\kappa({\mathfrak{p}});{\mathbf{t}}\right)=\mathcal{A}\left(M_{V}\otimes_{V}\kappa(\mathfrak{n});{\mathbf{t}}\right)$.
Therefore, we can substitute $A$ by $V$, and for the rest of the proof we
assume that $A$ is a discrete valuation ring with maximal ideal
${\mathfrak{p}}$.

Over the discrete valuation ring $A$, a module is $A$-flat if and only if it
is $A$-torsion-free. Since $M$ is $A$-flat by assumption, we have $P\cap A=0$
for all $P\in{\text{Ass}}_{R}(M)$. Then §missing 3.1(v)(vi) imply that both
$R^{i}(M)$ and $M/R^{i}(M)$ are $A$-flat for all $i\geq 0$. From §missing 3.1,
we get $R^{i}(M)\otimes_{A}Q\cong R^{i}(M\otimes_{A}Q)$. The $A$-flatness of
$R^{i}(M)$ yields the equalities

$\mathcal{C}\left(R^{i}(M\otimes_{A}Q);{\mathbf{t}}\right)\,=\,\mathcal{C}\left(R^{i}(M)\otimes_{A}Q;{\mathbf{t}}\right)\,=\,\mathcal{C}\left(R^{i}(M)\otimes_{A}\kappa({\mathfrak{p}});{\mathbf{t}}\right),$

and so we obtain
${\text{codim}}(R^{i}(M)\otimes_{A}\kappa({\mathfrak{p}}))\geq i$. The
$A$-flatness of $M/R^{i}(M)$ gives the injection

$R^{i}(M)\otimes_{A}\kappa({\mathfrak{p}})\hookrightarrow
M\otimes_{A}\kappa({\mathfrak{p}}).$

Then we obtain the injection
$R^{i}(M)\otimes_{A}\kappa({\mathfrak{p}})\hookrightarrow
R^{i}(M\otimes_{A}\kappa({\mathfrak{p}}))$ (see §missing 3.1(vii)), and so the
additivity and positivity of multidegree polynomials give the inequality

$\mathcal{C}^{i}\left(R^{i}(M)\otimes_{A}\kappa({\mathfrak{p}});{\mathbf{t}}\right)\,\leq_{c}\,\mathcal{C}^{i}\left(R^{i}(M\otimes_{A}\kappa({\mathfrak{p}}));{\mathbf{t}}\right)$

(see §missing 2.1(iii)(v)). By combining everything together we get the
inequality

$\mathcal{C}^{i}\left(M\otimes_{A}Q;{\mathbf{t}}\right)\,=\,\mathcal{C}^{i}\left(R^{i}(M\otimes_{A}Q);{\mathbf{t}}\right)\,\leq_{c}\,\mathcal{C}^{i}\left(R^{i}(M\otimes_{A}\kappa({\mathfrak{p}}));{\mathbf{t}}\right)\,=\,\mathcal{C}^{i}\left(M\otimes_{A}\kappa({\mathfrak{p}});{\mathbf{t}}\right).$

This concludes the proof of the proposition. ∎

We are now ready to present the main result of this section.

######  Theorem 3.1.

Assume §missing 3.2. Let $M$ be a finitely generated $\mathbb{Z}^{p}$-graded
$R$-module, and suppose that $M$ is flat over $A$. Then the function

$\text{Spec}(A)\rightarrow\mathbb{Z}[{\mathbf{t}}],\quad{\mathfrak{p}}\mapsto\mathcal{A}\left(M\otimes_{A}\kappa({\mathfrak{p}});{\mathbf{t}}\right)$

is upper semicontinuous with respect to the component-wise order $\geq_{c}$ on
$\mathbb{Z}[{\mathbf{t}}]$.

###### Proof.

It suffices to show that

$E_{h}\,:=\,\left\\{{\mathfrak{p}}\in\text{Spec}(A)\mid\mathcal{A}\left(M\otimes_{A}\kappa({\mathfrak{p}});{\mathbf{t}}\right)\leq_{c}h\right\\}$

is an open subset for an arbitrary $h\in\mathbb{Z}[{\mathbf{t}}]$. Let
$h\in\mathbb{Z}[{\mathbf{t}}]$. The topological Nagata criterion for openness
(see, e.g., [MATSUMURA, Theorem 24.2]) says that $E_{h}$ is open if and only
if the following two conditions are satisfied:

  1. (i)

If ${\mathfrak{p}},\mathfrak{q}\in\text{Spec}(A)$, ${\mathfrak{p}}\in E_{h}$
and ${\mathfrak{p}}\supseteq\mathfrak{q}$, then $\mathfrak{q}\in E_{h}$.

  2. (ii)

If ${\mathfrak{p}}\in E_{h}$, then $E_{h}$ contains a non-empty open subset of
$V({\mathfrak{p}})\subset\text{Spec}(A)$.

Condition (i) follows from §missing 3.2. Condition (ii) is obtained by
utilizing §missing 3.2 and Grothendieck’s generic freeness lemma (see, e.g.,
[MATSUMURA, Theorem 24.1]). This establishes the result of the theorem. ∎

###  3.3. Behavior under Gröbner degenerations

For the sake of completeness, we restate our results in terms of Gröbner
degenerations.

######  Theorem 3.2.

Let $\mathbb{k}$ be a field, $R=\mathbb{k}[x_{1},\ldots,x_{n}]$ be a
positively $\mathbb{N}^{p}$-graded polynomial ring and $>$ be a monomial order
(or a weight order) on $R$. For any $R$-homogeneous ideal $I\subset R$, we
have that

$\mathcal{A}\left(R/I;{\mathbf{t}}\right)\,\leq_{c}\,\mathcal{A}\left(R/{\text{in}}_{>}(I);{\mathbf{t}}\right).$

###### Proof.

It is a direct consequence of [EISEN_COMM, Theorem 15.17] and §missing 3.2. ∎

This result portrays a different behavior of the arithmetic multidegree, since
we always have the equalities
$\mathcal{C}\left(R/I;{\mathbf{t}}\right)=\mathcal{C}\left(R/{\text{in}}_{>}(I);{\mathbf{t}}\right)$
and
$\mathcal{G}\left(R/I;{\mathbf{t}}\right)=\mathcal{G}\left(R/{\text{in}}_{>}(I);{\mathbf{t}}\right)$.
These two equalities follow from the fact that these two notions of
multidegree are determined by the Hilbert series (see §missing 2.1 and
§missing 2.1).

##  4\. Multidegrees of projections

In this section, we study the multidegrees of all the possible natural
projections of an integral multiprojective scheme
$X\subset{\mathbb{P}}_{\mathbb{k}}^{m_{1}}\times_{\mathbb{k}}\cdots\times_{\mathbb{k}}{\mathbb{P}}_{\mathbb{k}}^{m_{p}}$.
We show that the maximal multidegree of a projection is always smaller or
equal than the maximal multidegree of $X$. Even more, we prove that any
multidegree of a projection must divide some multidegree of $X$. The setup
below is used throughout this section.

######  Setup .

Let $\mathbb{k}$ be a field and $S=\mathbb{k}[x_{1},\ldots,x_{n}]$ be a
standard $\mathbb{N}^{p}$-graded polynomial ring. Suppose that
$\text{MultiProj}(S)={\mathbb{P}}={\mathbb{P}}_{\mathbb{k}}^{m_{1}}\times_{\mathbb{k}}\cdots\times_{\mathbb{k}}{\mathbb{P}}_{\mathbb{k}}^{m_{p}}$.
Let $X=\text{MultiProj}(R)\subset{\mathbb{P}}$ be an integral closed subscheme
and $R$ be an $\mathbb{N}^{p}$-graded quotient ring of $S$ that is a domain.

For a closed subscheme
$Z\subset{\mathbb{P}}^{\prime}={\mathbb{P}}_{\mathbb{k}}^{r_{1}}\times_{\mathbb{k}}\cdots\times_{\mathbb{k}}{\mathbb{P}}_{\mathbb{k}}^{r_{k}}$,
we consider the following number

$\text{MDeg}_{{\mathbb{P}}^{\prime}}(Z)\;:=\;\max\\{\deg_{{\mathbb{P}}^{\prime}}^{\mathbf{n}}(Z)\mid{\mathbf{n}}\in\mathbb{N}^{k}\text{
with }|{\mathbf{n}}|=\dim(Z)\\}.$

The main result of this section is the theorem below.

######  Theorem 4.1.

Assume Section missing 4. Let
${\mathfrak{J}}=\\{j_{1},\ldots,j_{k}\\}\subset[p]$ and set
${\mathbb{P}}^{\prime}={\mathbb{P}}_{\mathbb{k}}^{m_{j_{1}}}\times_{\mathbb{k}}\cdots\times_{\mathbb{k}}{\mathbb{P}}_{\mathbb{k}}^{m_{j_{k}}}$.
Then we have the inequality

${\rm
MDeg}_{{\mathbb{P}}^{\prime}}\left(\Pi_{\mathfrak{J}}(X)\right)\;\leq\;{\rm
MDeg}_{\mathbb{P}}(X).$

Moreover, for any ${\mathbf{d}}\in\mathbb{N}^{k}$ with
$|{\mathbf{d}}|=\dim\left(\Pi_{\mathfrak{J}}(X)\right)$ and such that
$\deg_{{\mathbb{P}}^{\prime}}^{\mathbf{d}}\left(\Pi_{\mathfrak{J}}(X)\right)>0$,
there exists some ${\mathbf{n}}\in\mathbb{N}^{p}$ with
$|{\mathbf{n}}|=\dim(X)$ and such that $\deg_{\mathbb{P}}^{\mathbf{n}}(X)>0$
and
$\deg_{{\mathbb{P}}^{\prime}}^{\mathbf{d}}\left(\Pi_{\mathfrak{J}}(X)\right)$
divides $\deg_{\mathbb{P}}^{\mathbf{n}}(X)$.

Before providing the proof of the theorem we need some preparatory results.

######  Lemma .

Consider the natural projection
$\Pi_{[p-1]}:{\mathbb{P}}\rightarrow{\mathbb{P}}^{\prime}={\mathbb{P}}_{\mathbb{k}}^{m_{1}}\times_{\mathbb{k}}\cdots{\mathbb{P}}_{\mathbb{k}}^{m_{p-1}}$
and let $Y=\Pi_{[p-1]}(X)$ be the corresponding image. Let $\Phi:X\rightarrow
Y$ be the restriction of $\Pi_{[p-1]}$ to $X$ and $Y$. If $\dim(X)=\dim(Y)$,
then

$\deg_{\mathbb{P}}^{{\mathbf{n}},0}(X)\;=\;\deg(\Phi)\deg_{{\mathbb{P}}^{\prime}}^{\mathbf{n}}(Y)$

for all ${\mathbf{n}}\in\mathbb{N}^{p-1}$ with $|{\mathbf{n}}|=\dim(X)$; here
$\deg(\Phi)=\left[K(X):K(Y)\right]$ denotes the degree of $\Phi$.

###### First proof..

Let $r=\dim(X)$. We write the Hilbert polynomial of $X$ as follows

$P_{X}(t_{1},\ldots,t_{p})\;=\;\sum_{\begin{subarray}{c}{\mathbf{n}}\in\mathbb{N}^{p}\\\
|{\mathbf{n}}|\leq
r\end{subarray}}e({\mathbf{n}})\binom{t_{1}+n_{1}}{n_{1}}\cdots\binom{t_{p}+n_{p}}{n_{p}}.$

Choose a positive integer $\omega$ such that
$P_{X}(\nu)=\dim_{\mathbb{k}}\left(\left[R\right]_{\nu}\right)$ for all
$\nu\in\mathbb{N}^{p}$ satisfying
$\nu\geq\omega\cdot\mathbf{1}\in\mathbb{N}^{p}$. Let $A\subset R$ be the
standard $\mathbb{N}^{p-1}$-graded $\mathbb{k}$-algebra given by

$A\,=\,R_{([p-1])}\,=\,\bigoplus_{\nu_{1},\ldots,\nu_{p-1}\geq
0}\left[R\right]_{\nu_{1},\ldots,\nu_{p-1},0}.$

Consider the finitely generated $\mathbb{N}^{p-1}$-graded $A$-module
$M=\bigoplus_{\nu_{1},\ldots,\nu_{p-1}\geq
0}\left[R\right]_{\nu_{1},\ldots,\nu_{p-1},\omega}$. By construction the
Hilbert polynomial of $M$ is given by

$P_{M}({\mathbf{t}})\,=\,P_{X}({\mathbf{t}},\omega)\,=\,\sum_{\begin{subarray}{c}{\mathbf{n}}\in\mathbb{N}^{p-1},\,m\in\mathbb{N}\\\
|{\mathbf{n}}|+m\leq
r\end{subarray}}e({\mathbf{n}},m)\binom{t_{1}+n_{1}}{n_{1}}\cdots\binom{t_{p-1}+n_{p-1}}{n_{p-1}}\binom{\omega+m}{m}.$

Regrouping the terms yields the following expression

$P_{M}({\mathbf{t}})\,=\,\sum_{\begin{subarray}{c}{\mathbf{n}}\in\mathbb{N}^{p-1}\\\
|{\mathbf{n}}|\leq
r\end{subarray}}\left(\sum_{m=0}^{r-|{\mathbf{n}}|}e({\mathbf{n}},m)\binom{\omega+m}{m}\right)\cdot\binom{t_{1}+n_{1}}{n_{1}}\cdots\binom{t_{p-1}+n_{p-1}}{n_{p-1}}.$

This forces the equality
$e({\mathbf{n}};M)=e({\mathbf{n}},0)=\deg_{\mathbb{P}}^{{\mathbf{n}},0}(X)$
for all ${\mathbf{n}}\in\mathbb{N}^{p-1}$ with $|{\mathbf{n}}|=r$. Since $M$
is a torsion-free module over the domain $A$, the associativity formula for
mixed multiplicities gives the equalities
$\deg_{\mathbb{P}}^{{\mathbf{n}},0}(X)=e({\mathbf{n}};M)=\text{rank}_{A}(M)e({\mathbf{n}};A)=\text{rank}_{A}(M)\deg_{{\mathbb{P}}^{\prime}}^{\mathbf{n}}(Y)$
for all ${\mathbf{n}}\in\mathbb{N}^{p-1}$ with $|{\mathbf{n}}|=r$. As
$\dim(X)=\dim(Y)$, the morphism $\Phi$ is generically finite. Let
$\phi:\text{Proj}(R)\rightarrow\text{Spec}(A)$ be the natural morphism
obtained by considering $R$ as an $\mathbb{N}$-graded ring after setting
$[R]_{\mu}=\bigoplus_{\nu_{1},\ldots,\nu_{p-1}\geq
0}[R]_{\nu_{1},\ldots,\nu_{p-1},\mu}$ for all $\mu\in\mathbb{N}$. Notice that
$\phi$ is a generically finite since $\Phi$ is, and that
$\deg(\Phi)=\deg(\phi)$. Let $\eta$ the be the generic point of
$\text{Spec}(A)$ and $Q=\text{Quot}(A)$ be the field of fractions of $A$. Then
the generic fiber $Z=\phi^{-1}(\eta)=\text{Proj}(R\otimes_{A}Q)$ is zero-
dimensional (and thus affine, see, e.g., [GORTZ_WEDHORN, Proposition 5.20])
and we have the equality

$\deg(\phi)\,=\,\dim_{Q}(\phi^{-1}(\eta))\,=\,\dim_{Q}\left(\text{H}^{0}(Z,\mathcal{O}_{Z}(\mu))\right)$

for all $\mu\in\mathbb{Z}$. In particular, this implies that
$\deg(\phi)=\dim_{Q}\left(\text{H}^{0}(Z,\mathcal{O}_{Z}(\omega))\right)=\dim_{Q}\left(\left[R\otimes_{A}Q\right]_{\omega}\right)=\text{rank}_{A}(M)$,
after possibly increasing $\omega$ and making it large enough. By combining
everything, we obtain the claimed equality
$\deg_{\mathbb{P}}^{{\mathbf{n}},0}(X)\;=\;\deg(\Phi)\deg_{{\mathbb{P}}^{\prime}}^{\mathbf{n}}(Y)$
for all ${\mathbf{n}}\in\mathbb{N}^{p-1}$ with $|{\mathbf{n}}|=r$. ∎

###### Second proof..

For each $1\leq i\leq p-1$, let
$L_{i}=\mathcal{O}_{{\mathbb{P}}^{\prime}}\left({\mathbf{e}}_{i}^{\prime}\right)$
be the line bundle on
${\mathbb{P}}^{\prime}={\mathbb{P}}_{\mathbb{k}}^{m_{1}}\times_{\mathbb{k}}\cdots\times_{\mathbb{k}}{\mathbb{P}}_{\mathbb{k}}^{m_{p-1}}$
corresponding with the $i$-th elementary vector
${\mathbf{e}}_{i}^{\prime}\in\mathbb{N}^{p-1}$. To simplify notation, set
$f=\Pi_{[p-1]}$ and let $r=\dim(X)$. From the projection formula (see
[FULTON_INTER, Proposition 2.5]) and the fact that $f_{*}[X]=\deg(\Phi)[Y]$
(see [FULTON_INTER, §1.4]), we obtain the equalities

$\displaystyle\deg_{\mathbb{P}}^{{\mathbf{n}},0}(X)$
$\displaystyle=\int_{{\mathbb{P}}}c_{1}(f^{*}(L_{1}))^{n_{1}}\cdots
c_{1}(f^{*}(L_{p-1}))^{n_{p-1}}\cap[X]$
$\displaystyle=\deg(\Phi)\int_{{\mathbb{P}}^{\prime}}c_{1}(L_{1})^{n_{1}}\cdots
c_{1}(L_{p-1})^{n_{p-1}}\cap[Y]$
$\displaystyle=\deg(\Phi)\deg_{{\mathbb{P}}^{\prime}}^{\mathbf{n}}(Y)$

for all ${\mathbf{n}}=(n_{1},\ldots,n_{p-1})\in\mathbb{N}^{p-1}$ with
$|{\mathbf{n}}|=r$. So, the result follows. ∎

We shall use a standard type of generic Bertini theorem, much in the spirit of
[JOINS_INTERS, §1.5], that is quite flexible and useful for our purposes. To
that end, we introduce the following notation.

######  Notation .

Let $\\{y_{0},\ldots,y_{\ell}\\}$ be a basis of the $\mathbb{k}$-vector space
$[S]_{{\mathbf{e}}_{p}}$. Consider a purely transcendental field extension
$\mathbb{L}:=\mathbb{k}(z_{0},\ldots,z_{\ell})$ of $\mathbb{k}$, and set
$S_{\mathbb{L}}:=S\otimes_{\mathbb{k}}\mathbb{L}$,
$R_{\mathbb{L}}:=R\otimes_{\mathbb{k}}\mathbb{L}$ and
$X_{\mathbb{L}}:=X\otimes_{\mathbb{k}}\mathbb{L}=\text{MultiProj}\left(R_{\mathbb{L}}\right)\subset{\mathbb{P}}_{\mathbb{L}}:={\mathbb{P}}\otimes_{\mathbb{k}}\mathbb{L}={\mathbb{P}}_{\mathbb{L}}^{m_{1}}\times_{\mathbb{L}}\cdots\times_{\mathbb{L}}{\mathbb{P}}_{\mathbb{L}}^{m_{p}}$.
We say that
$y:=z_{0}y_{0}+\cdots+z_{\ell}y_{\ell}\in{\left[S_{\mathbb{L}}\right]}_{{\mathbf{e}}_{p}}$
is the _generic element_ of
${\left[S_{\mathbb{L}}\right]}_{{\mathbf{e}}_{p}}$.

######  Remark .

If $\mathbb{k}(\xi)$ is a purely transcendental field extension over
$\mathbb{k}$, then $R\otimes_{\mathbb{k}}\mathbb{k}(\xi)$ is also a domain
(see, e.g., [POSITIVITY, Remark 3.9]).

######  Remark  ([JOINS_INTERS, Corollary 1.5.9, Corollary 1.5.10]).

Assume $\dim(X)\geq 1$. Then $X_{\mathbb{L}}\cap V(y)$ is also an integral
scheme.

###### Proof.

We may restrict to an affine open subscheme $U=\text{Spec}(A)$ with $A$ a
domain, and assume that the restriction to $A$ of one of the $y_{i}$’s, say
$y_{0}$, becomes a unit. Notice that $U_{\mathbb{L}}\cap
V(y)=\text{Spec}\left((A\otimes_{\mathbb{k}}\mathbb{L})/(y)\right)$ and that
$(A\otimes_{\mathbb{k}}\mathbb{L})/(y)$ is a localization of
$A[z_{0},z_{1}\ldots,z_{\ell}]/(z_{0}+\frac{y_{1}}{y_{0}}z_{1}+\cdots+\frac{y_{\ell}}{y_{0}}z_{\ell})$.
From the fact that
$(z_{0}+\frac{y_{1}}{y_{0}}z_{1}+\cdots+\frac{y_{\ell}}{y_{0}}z_{\ell})$ is a
prime ideal, it follows that $(A\otimes_{\mathbb{k}}\mathbb{L})/(y)$ is a
domain. This concludes the proof. ∎

We recall the following result from [POSITIVITY] that will be essential.

######  Theorem 4.2.

Assume Section missing 4 and Section missing 4, and suppose that
$\dim(\Pi_{p}(X))\geq 1$. Then the following statements hold:

  1. (i)

$\dim\left(\Pi_{{\mathfrak{J}}}(X_{\mathbb{L}}\cap
V(y))\right)\;=\;\min\big{\\{}\dim(\Pi_{{\mathfrak{J}}}(X)),\dim(\Pi_{{\mathfrak{J}}\cup\\{p\\}}(X))-1\big{\\}}$
for all ${\mathfrak{J}}\subseteq[p]$.

  2. (ii)

$\deg_{{\mathbb{P}}_{\mathbb{L}}}^{{\mathbf{n}}-{\mathbf{e}}_{p}}\left(X_{\mathbb{L}}\cap
V(y)\right)=\deg_{{\mathbb{P}}_{\mathbb{L}}}^{\mathbf{n}}(X_{\mathbb{L}})$ for
all ${\mathbf{n}}=(n_{1},\ldots,n_{p})\in\mathbb{N}^{p}$ with
$|{\mathbf{n}}|=\dim(X)$ and $n_{p}\geq 1$.

###### Proof.

We may assume that the field $\mathbb{k}$ is infinite; indeed, by changing
$\mathbb{k}$ by a purely transcendental field extension we can keep all the
assumptions (see Section missing 4).

Let $x\in[R]_{{\mathbf{e}}_{p}}$ be a general element. From [POSITIVITY,
Theorem 3.7], we obtain that

$\dim\left(\Pi_{{\mathfrak{J}}}(X\cap
V(x))\right)\;=\;\min\big{\\{}\dim(\Pi_{{\mathfrak{J}}}(X)),\dim(\Pi_{{\mathfrak{J}}\cup\\{p\\}}(X))-1\big{\\}}$

for all ${\mathfrak{J}}\subset[p]$. As a consequence of [cidruiz2021mixed,
Lemma 3.7, Lemma 3.9], we get that
$\deg_{{\mathbb{P}}}^{{\mathbf{n}}-{\mathbf{e}}_{p}}\left(X\cap
V(x)\right)=\deg_{{\mathbb{P}}}^{\mathbf{n}}(X)$ for all
${\mathbf{n}}=(n_{1},\ldots,n_{p})\in\mathbb{N}^{p}$ with
$|{\mathbf{n}}|=\dim(X)$ and $n_{p}\geq 1$. Therefore, the results of both
parts (i) and (ii) follow by applying [POSITIVITY, Lemma 3.10] and by
utilizing the two latter formulas for cutting with a general element
$x\in[R]_{{\mathbf{e}}_{p}}$. ∎

We are now ready for the proof of the main result of this section.

###### Proof of Theorem 4.1.

Notice that it suffices to consider the case ${\mathfrak{J}}=[p-1]\subset[p]$.
Let $Y$ be the image of $X$ under the natural projection
$\Pi_{[p-1]}:{\mathbb{P}}\rightarrow{\mathbb{P}}^{\prime}={\mathbb{P}}_{\mathbb{k}}^{m_{1}}\times_{\mathbb{k}}\cdots\times_{\mathbb{k}}{\mathbb{P}}_{\mathbb{k}}^{m_{p-1}}$.
We proceed by induction on $s=\dim(X)-\dim(Y)$. If $s=0$, then the result
follows directly from Section missing 4.

Suppose that $s>0$. This implies that $\dim(\Pi_{p}(X))\geq 1$ since the
function $r({\mathfrak{J}})=\dim\left(\Pi_{{\mathfrak{J}}}(X)\right)$ is a
rank function (see [POSITIVITY, Proposition 5.1]). We substitute $X$ and $Y$
by $X_{\mathbb{L}}$ and $Y_{\mathbb{L}}$, respectively, and we consider the
generic element $y$ in $\left[S_{\mathbb{L}}\right]_{{\mathbf{e}}_{p}}$. Let
$X^{\prime}=X_{\mathbb{L}}\cap V(y)\subset{\mathbb{P}}_{\mathbb{L}}$ and
$Y^{\prime}=\Pi_{[p-1]}(X^{\prime})\subset{\mathbb{P}}_{\mathbb{L}}^{\prime}$.
From Section missing 4 and Theorem 4.2, we obtain that $X^{\prime}$ is an
integral scheme and the equalities

$\dim\left(X^{\prime}\right)\,=\,\dim(X)-1\quad\text{ and
}\quad\dim(Y^{\prime})\,=\,\min\\{\dim(X)-1,\dim(Y)\\}=\dim(Y).$

This implies that $Y^{\prime}$ and $Y_{\mathbb{L}}$ have the same dimension,
and thus we necessarily get $Y^{\prime}=Y_{\mathbb{L}}$ because
$Y^{\prime}\subset Y_{\mathbb{L}}$ by construction. Also, for all
${\mathbf{n}}=(n_{1},\ldots,n_{p})\in\mathbb{N}^{p}$ such that
$|{\mathbf{n}}|=\dim(X)$ and $n_{p}\geq 1$, we have that
$\deg_{{\mathbb{P}}_{\mathbb{L}}}^{{\mathbf{n}}-{\mathbf{e}}_{p}}(X^{\prime})=\deg_{\mathbb{P}}^{{\mathbf{n}}}(X)$.
Therefore, since $\dim(X^{\prime})-\dim(Y^{\prime})=s-1$, we can apply the
inductive hypothesis to the scheme $X^{\prime}$ and the corresponding image
$Y^{\prime}=Y_{\mathbb{L}}$ under the projection $\Pi_{[p-1]}$. This completes
the proof of the theorem. ∎

We illustrate the result of Theorem 4.1 with the following example that
constitutes a variation of [POSITIVITY, Example 1.1].

######  Example .

Let $A=\mathbb{k}[v_{1},v_{2},v_{3}][w_{1},w_{2},w_{3}]$ be
$\mathbb{N}^{3}$-graded with $\deg(v_{i})=\mathbf{0}\in\mathbb{N}^{3}$ and
$\deg(w_{i})={\mathbf{e}}_{i}\in\mathbb{N}^{3}$. Let
$S=\mathbb{k}\left[x_{0},\ldots,x_{3}\right]\left[y_{0},\ldots,y_{3}\right]\left[z_{0},\ldots,z_{3}\right]$
be $\mathbb{N}^{3}$-graded with $\deg(x_{i})={\mathbf{e}}_{1}$,
$\deg(y_{i})={\mathbf{e}}_{2}$ and $\deg(z_{i})={\mathbf{e}}_{3}$. Consider
the $\mathbb{N}^{3}$-graded $\mathbb{k}$-algebra homomorphism

$\varphi=S\rightarrow A,\qquad\begin{array}[]{llll}x_{0}\mapsto
w_{1},&x_{1}\mapsto v_{1}^{2}w_{1},&x_{2}\mapsto v_{1}^{2}w_{1},&x_{3}\mapsto
v_{1}w_{1},\\\ y_{0}\mapsto w_{2},&y_{1}\mapsto v_{1}w_{2},&y_{2}\mapsto
v_{2}w_{2},&y_{3}\mapsto(v_{1}^{2}+v_{2}^{2})w_{2},\\\ z_{0}\mapsto
w_{3},&z_{1}\mapsto v_{1}^{2}w_{3},&z_{2}\mapsto v_{2}w_{3},&z_{3}\mapsto
v_{3}^{2}w_{3}.\end{array}$

Let $P\subset S$ be the $\mathbb{N}^{3}$-graded prime ideal
$P=\text{Ker}(\varphi)$, which is given by

$P\,=\,\left(\begin{array}[]{c}{x}_{1}-{x}_{2},\,{y}_{3}{z}_{0}-{y}_{0}{z}_{1}-{y}_{2}{z}_{2},\,{y}_{2}{z}_{0}-{y}_{0}{z}_{2},\,{x}_{2}{z}_{0}-{x}_{0}{z}_{1},\,{y}_{1}^{2}+{y}_{2}^{2}-{y}_{0}{y}_{3},\,{x}_{3}{y}_{0}-{x}_{0}{y}_{1}\\\
{x}_{2}{y}_{0}-{x}_{3}{y}_{1},\,{x}_{0}{x}_{2}-{x}_{3}^{2},\,{y}_{0}{y}_{2}{z}_{1}+{y}_{2}^{2}{z}_{2}-{y}_{0}{y}_{3}{z}_{2},\,{x}_{3}{y}_{2}{z}_{1}-{x}_{2}{y}_{1}{z}_{2},\,{x}_{0}{y}_{2}{z}_{1}-{x}_{3}{y}_{1}{z}_{2},\\\
{x}_{3}{y}_{1}{z}_{1}-{x}_{0}{y}_{3}{z}_{1}+{x}_{2}{y}_{2}{z}_{2},\,{x}_{3}{y}_{1}{z}_{0}-{x}_{0}{y}_{0}{z}_{1},\,{x}_{3}^{2}{z}_{0}-{x}_{0}^{2}{z}_{1}\end{array}\right).$

Let
$X=\text{MultiProj}(S/P)\subset{\mathbb{P}}_{\mathbb{k}}^{3}\times_{\mathbb{k}}{\mathbb{P}}_{\mathbb{k}}^{3}\times_{\mathbb{k}}{\mathbb{P}}_{\mathbb{k}}^{3}$
be the corresponding integral closed subscheme. The multidegree polynomial of
$X$ is equal to

$\mathcal{C}(X;t_{1},t_{2},t_{3})\,=\,2\,{t}_{1}^{3}{t}_{2}^{3}+4\,{t}_{1}^{3}{t}_{2}^{2}{t}_{3}+2\,{t}_{1}^{3}{t}_{2}{t}_{3}^{2}+2\,{t}_{1}^{2}{t}_{2}^{3}{t}_{3}+4\,{t}_{1}^{2}{t}_{2}^{2}{t}_{3}^{2}.$

For ease of exposition, we use the variables indexed by ${\mathfrak{J}}$ to
express the multidegree polynomial of a projection $\Pi_{{\mathfrak{J}}}(X)$.
Under this notation, we obtain that the multidegree polynomials of the
possible projections are described in the following table:

$\begin{array}[]{lll}\Pi_{1}(X)\subset{\mathbb{P}}_{\mathbb{k}}^{3},&\dim(\Pi_{1}(X))=1,&\mathcal{C}(\Pi_{1}(X);t_{1})=2\,{t}_{1}^{2}\\\
\Pi_{2}(X)\subset{\mathbb{P}}_{\mathbb{k}}^{3},&\dim(\Pi_{2}(X))=2,&\mathcal{C}(\Pi_{2}(X);t_{2})=2\,{t}_{2}\\\
\Pi_{3}(X)\subset{\mathbb{P}}_{\mathbb{k}}^{3},&\dim(\Pi_{3}(X))=3,&\mathcal{C}(\Pi_{3}(X);t_{3})=1\\\
\Pi_{1,2}(X)\subset{\mathbb{P}}_{\mathbb{k}}^{3}\times_{\mathbb{k}}{\mathbb{P}}_{\mathbb{k}}^{3},&\dim(\Pi_{1,2}(X))=2,&\mathcal{C}(\Pi_{1,2}(X);t_{1},t_{2})=2\,{t}_{1}^{3}{t}_{2}+4\,{t}_{1}^{2}{t}_{2}^{2}\\\
\Pi_{1,3}(X)\subset{\mathbb{P}}_{\mathbb{k}}^{3}\times_{\mathbb{k}}{\mathbb{P}}_{\mathbb{k}}^{3},&\dim(\Pi_{1,3}(X))=3,&\mathcal{C}(\Pi_{1,3}(X);t_{1},t_{3})=2\,{t}_{1}^{3}+2\,{t}_{1}^{2}{t}_{3}\\\
\Pi_{2,3}(X)\subset{\mathbb{P}}_{\mathbb{k}}^{3}\times_{\mathbb{k}}{\mathbb{P}}_{\mathbb{k}}^{3},&\dim(\Pi_{2,3}(X))=3,&\mathcal{C}(\Pi_{2,3}(X);t_{2},t_{3})=2\,{t}_{2}^{3}+4\,{t}_{2}^{2}{t}_{3}+2\,{t}_{2}{t}_{3}^{2}.\\\
\end{array}$

Here one may see that the multidegrees of the projections behave as predicted
by Theorem 4.1. The computations of this example can be carried out by hand or
by utilizing Macaulay2 [M2].

The next simple example shows that, if we drop the condition of Theorem 4.1
that $R$ is a domain, then the multidegrees can grow arbitrarily under
projections.

######  Example .

Suppose $S=\mathbb{k}[x_{0},x_{1},x_{2}][y_{0},y_{1},y_{2}]$ is
$\mathbb{N}^{2}$-graded with $\deg(x_{i})={\mathbf{e}}_{1}$ and
$\deg(y_{i})={\mathbf{e}}_{2}$. Let $a>0$ be a positive integer. Consider the
monomial ideal
$J=\left(x_{0}^{2},x_{0}x_{1},x_{1}y_{0},y_{0}^{a}\right)\subset S$ and its
corresponding closed subscheme
$Z=\text{MultiProj}(S/J)\subset{\mathbb{P}}_{\mathbb{k}}^{2}\times_{\mathbb{k}}{\mathbb{P}}_{\mathbb{k}}^{2}$.
One can check that

$\mathcal{C}(Z;t_{1},t_{1})=t_{1}t_{2}\quad\text{ and
}\quad\deg_{{\mathbb{P}}_{\mathbb{k}}^{2}}(\Pi_{2}(Z))=a.$

This follows at once from the fact
$J=(x_{0},y_{0})\cap(x_{0}^{2},x_{1},y_{0}^{a})$ is a primary decomposition of
$J$.

As a direct consequence of Theorem 4.1, we obtain that the projection of a
multiplicity-free variety is also multiplicity-free. A closed subscheme
$Z\subset{\mathbb{P}}^{\prime}={\mathbb{P}}_{\mathbb{k}}^{r_{1}}\times_{\mathbb{k}}\cdots\times_{\mathbb{k}}{\mathbb{P}}_{\mathbb{k}}^{r_{k}}$
is said to be _multiplicity-free_ if
$\deg_{{\mathbb{P}}^{\prime}}^{\mathbf{n}}(Z)$ is either $0$ or $1$ for all
${\mathbf{n}}\in\mathbb{N}^{k}$ with $|{\mathbf{n}}|=\dim(Z)$ (that is, ${\rm
MDeg}_{{\mathbb{P}}^{\prime}}(Z)=1$).

######  Corollary .

If $X\subset{\mathbb{P}}$ is multiplicity-free, then $\Pi_{\mathfrak{J}}(X)$
is multiplicity-free for all ${\mathfrak{J}}\subset[p]$.

##  5\. The multidegrees of prime ideals

In this section, we investigate several special properties that are enjoyed by
the multidegrees of prime ideals. We use the following setup throughout this
section.

######  Setup .

Let $\mathbb{k}$ be a field and $S=\mathbb{k}[x_{1},\ldots,x_{n}]$ be a
standard $\mathbb{N}^{p}$-graded polynomial ring.

The theorem below proves that the multidegree polynomial and the
$\mathcal{G}$-multidegree polynomial coincide for a prime ideal.

######  Theorem 5.1.

Assume Section missing 5. Let $P\subset S$ be an $S$-homogeneous prime ideal.
Then we have the equality

$\mathcal{C}(S/P;{\mathbf{t}})\,=\,\mathcal{G}(S/P;{\mathbf{t}}).$

###### Proof.

Choose a monomial order $>$ on $S$. By using [KS_INIT_PRIMES], we obtain that
$\sqrt{{\text{in}}(P)}$ is equidimensional and connected in codimension one
(see also [HUNEKE_LOC_COHOM, Appendix 1]). For any monomial prime ideal
$L=(x_{j_{1}},\ldots,x_{j_{k}})\subset S$, we have

$\mathcal{C}(S/L;{\mathbf{t}})\,=\,{\mathbf{t}}^{\mathbf{a}}\quad\text{ and
}\quad\mathcal{G}(S/L;{\mathbf{t}})\,=\,{\mathbf{t}}^{\mathbf{a}}$

where $\mathbf{a}=(a_{1},\ldots,a_{p})\in\mathbb{N}^{p}$ with
$a_{i}=|\\{l\mid\deg(x_{j_{l}})={\mathbf{e}}_{i}\\}|$ (see §missing 2.1).

Let $P_{1},\ldots,P_{h},P_{h+1},\ldots,P_{h+s}$ be the associated primes of
${\text{in}}(P)$, where $P_{1},\ldots,P_{h}$ are the minimal ones. We choose a
prime filtration

$0=M_{0}\subset M_{1}\subset\cdots\subset M_{r}=S/{\text{in}}(P)$

where $M_{j}/M_{j-1}\cong S/L_{j}$ and $L_{j}\subset S$ is a monomial prime
ideal (see, e.g., [EISEN_COMM, Proposition 3.7]). All the associated primes of
${\text{in}}(P)$ must appear among the $L_{j}$’s and all the $L_{j}$’s must
contain a minimal prime of ${\text{in}}(P)$. Therefore, by utilizing the
additivity of Hilbert functions over this prime filtration and the above
remarks, we obtain

$\mathcal{C}\left(S/{\text{in}}(P);{\mathbf{t}}\right)\,=\,\mathcal{G}\left(S/{\text{in}}(P);{\mathbf{t}}\right)\,=\,\sum_{j=1}^{h}\text{length}_{S_{P_{j}}}\left((S/{\text{in}}(P))_{P_{j}}\right)\mathcal{C}(S/P_{j};{\mathbf{t}});$

indeed: if $L_{k_{1}}\subset L_{k_{2}}$ then
$\mathcal{C}(S/L_{k_{1}};{\mathbf{t}})$ divides
$\mathcal{C}(S/L_{k_{2}};{\mathbf{t}})$; all the minimal $P_{j}$’s have the
same codimension; and by localizing we get that the number of times that a
minimal $P_{j}$ appears among the $L_{k}$’s is equal to the length of the
Artinian local ring $(S/{\text{in}}(P))_{P_{j}}$. This concludes the proof
because $S/P$ and $S/{\text{in}}(P)$ have the same multigraded Hilbert
function. ∎

The remainder of this section is dedicated to multigraded generic initial
ideals. Thus we further specify the notation by fixing the following setup.

######  Setup .

Let $\mathbb{k}$ be an infinite field,
${\mathbf{m}}=(m_{1},\ldots,m_{p})\in\mathbb{Z}_{+}^{p}$ be a vector of
positive integers, and

$S=\mathbb{k}\left[x_{i,j}\mid 1\leq i\leq p,0\leq j\leq m_{i}\right]$

be a standard $\mathbb{N}^{p}$-graded polynomial with
$\deg(x_{i,j})={\mathbf{e}}_{i}\in\mathbb{N}^{p}$. Hence we have
$\text{MultiProj}(S)={\mathbb{P}}={\mathbb{P}}_{\mathbb{k}}^{m_{1}}\times_{\mathbb{k}}\cdots\times_{\mathbb{k}}{\mathbb{P}}_{\mathbb{k}}^{m_{p}}$.
The group naturally acting on $S$ as the group of multigraded
$\mathbb{k}$-algebra isomorphisms is
$G=\text{GL}_{m_{1}+1}(\mathbb{k})\times\cdots\times\text{GL}_{m_{p}+1}(\mathbb{k})$,
where $\text{GL}_{m_{i}+1}(\mathbb{k})$ is the group of invertible
$(m_{i}+1)\times(m_{i}+1)$ matrices over $\mathbb{k}$. The _Borel subgroup of
$G$_ is given by $B=B_{m_{1}+1}(\mathbb{k})\times\cdots\times
B_{m_{p}+1}(\mathbb{k})$, where $B_{m_{i}+1}(\mathbb{k})$ is the subgroup of
$\text{GL}_{m_{i}+1}(\mathbb{k})$ consisting of upper triangular invertible
matrices. Let $>$ be a monomial order on $S$ that satisfies
$x_{i,0}>x_{i,1}>\cdots>x_{i,m_{i}}$ for all $1\leq i\leq p$.

Much as in the singly-graded case (see [EISEN_COMM, §15.9]), we have the
following definition.

######  Definition .

Let $I\subset S$ be an $S$-homogeneous ideal. The _multigraded generic initial
ideal_ ${\text{gin}}_{>}(I)$ of $I$ with respect to $>$ is the ideal
${\text{in}}_{>}(g(I))$, where $g$ belongs to a Zariski dense open subset
$\mathcal{U}\subset G$.

Whenever the used monomial order $>$ is clear from the context, we shall write
${\text{in}}(I)$ and ${\text{gin}}(I)$ instead of ${\text{in}}_{>}(I)$ and
${\text{gin}}_{>}(I)$, respectively. An $S$-homogeneous ideal $I\subset S$ is
said to be _Borel-fixed_ if $g(I)=I$ for all $g\in B$. Similarly to the
singly-graded setting, it can be shown that ${\text{gin}}(I)$ is Borel-fixed
(see [EISEN_COMM, Theorem 15.20]).

######  Remark .

One can extend the known properties of Borel-fixed ideals in the singly-graded
setting (see [EISEN_COMM, Chapter 15]) to obtain the following statements:

  1. (i)

Any Borel-fixed prime ideal in $S$ is of the form

$P_{\mathbf{a}}\,=\,\left(x_{i,j}\mid 1\leq i\leq p\text{ and }0\leq
j<a_{i}\right)$

for some $\mathbf{a}=(a_{1},\ldots,a_{p})\in\mathbb{N}^{p}$ (see [CDNG_MINORS,
Lemma 3.1]).

  2. (ii)

An $S$-homogeneous Borel-fixed ideal $I\subset S$ is a monomial ideal and all
its associated primes are also Borel-fixed (see [CDNG_MINORS, Lemma 3.2]).

The following theorem shows that the radical of the multigraded generic
initial ideal of any prime is Cohen-Macaulay. This demonstrates a very special
behavior of prime ideals in a multigraded setting.

######  Theorem 5.2.

Assume Section missing 5. Let $P\subset S$ be an $S$-homogeneous prime ideal.
Then $\sqrt{{\text{gin}}(P)}$ is a Cohen-Macaulay ideal.

###### Proof.

Let $J=\sqrt{{\text{gin}}(P)}\subset S$. Let $P_{1},\ldots,P_{h}$ be the
minimal primes of $J$. As a consequence of [KS_INIT_PRIMES], all the $P_{k}$’s
have the same codimension. Each $P_{k}$ is Borel-fixed, and so we can write

$P_{k}\,=\,\left(x_{i,j}\mid 1\leq i\leq p,0\leq j<a_{k,i}\right)$

for some $\mathbf{a}_{k}=(a_{k,1},\ldots,a_{k,p})\in\mathbb{N}^{p}$ with
$|\mathbf{a}_{k}|={\text{codim}}(P)$. With this notation we now have

$\mathcal{C}(S/P;{\mathbf{t}})\,=\,\mathcal{C}(S/{\text{gin}}(P);{\mathbf{t}})\,=\,\sum_{k=1}^{h}\text{length}_{S_{P_{k}}}\left((S/{\text{gin}}(P))_{P_{k}}\right)\,{\mathbf{t}}^{\mathbf{a}_{k}}\quad\text{
and
}\quad\mathcal{C}(S/J;{\mathbf{t}})\,=\,\sum_{k=1}^{h}\,{\mathbf{t}}^{\mathbf{a}_{k}}.$

The Alexander dual of $J$ is a monomial ideal $K\subset S$ given by
$K=\left({\mathbf{x}}_{\mathbf{a}_{1}},\ldots,{\mathbf{x}}_{\mathbf{a}_{h}}\right)$
where

${\mathbf{x}}_{\mathbf{a}_{k}}=\prod_{1\leq i\leq p,\,0\leq
j<a_{k,i}}x_{i,j};$

see [HERZOG_HIBI, Corollary 1.5.5]. The Eagon-Reiner theorem (see, e.g.,
[HERZOG_HIBI, Theorem 8.1.9]) yields that $J$ is Cohen-Macaulay if and only if
$K$ has a linear resolution. Thus we concentrate on proving that $K$ has a
linear resolution.

From Theorem 2.2, we obtain that the support of
$\mathcal{C}(S/P;{\mathbf{t}})$, which coincides with the set of lattice
points $\\{\mathbf{a}_{1},\ldots,\mathbf{a}_{h}\\}\subset\mathbb{N}^{p}$, is a
discrete polymatroid. We construct the following monomial ideal

$M=\left({\mathbf{x}}_{\mathbf{a}_{1}}^{\prime},\ldots,{\mathbf{x}}_{\mathbf{a}_{h}}^{\prime}\right)\quad\text{
with
}\quad{\mathbf{x}}_{\mathbf{a}_{k}}^{\prime}=x_{1,0}^{a_{k,1}}x_{2,0}^{a_{k,2}}\cdots
x_{p,0}^{a_{k,p}}$

in the polynomial subring $\mathbb{k}[x_{1,0},x_{2,0},\ldots,x_{p,0}]\subset
S$. By construction and following the notation of [HERZOG_HIBI, §12.6], we say
that that $M$ is a polymatroidal ideal, and so the ideal $M$ has a linear
resolution by [HERZOG_HIBI, Theorem 12.6.2, Proposition 8.2.1]. Notice that
$K$ can be seen naturally as the polarization of $M$ by sending
${\mathbf{x}}_{\mathbf{a}_{k}}^{\prime}$ to ${\mathbf{x}}_{\mathbf{a}_{k}}$.
Finally, by standard properties of polarization (see [HERZOG_HIBI, §1.6]), it
follows that $K$ also has a linear resolution. This concludes the proof of the
theorem. ∎

######  Remark .

For the more combinatorially inclined reader it should be the mentioned that
the same proof of Theorem 5.2 shows that the simplicial complex associated to
the radical monomial ideal $\sqrt{{\text{gin}}(P)}$ is shellable (see
[HERZOG_HIBI, Proposition 8.2.5]).

######  Remark .

One may find many examples of a prime ideal $P$ and a monomial order $>$ such
that $\sqrt{{\text{in}}_{>}(P)}$ is not Cohen-Macaulay. For a simple instance,
see [CV_SQRFREE, Remark 2.8(1)]. Also, for more examples of this type and
related results, see the book [BOOK_UPCOMMING, Chapter 5].

For an ideal $I\subset S$, we consider the following invariant

$\text{MLength}(I)\;:=\;\max\big{\\{}\text{length}_{S_{\mathfrak{p}}}\left((S/I)_{\mathfrak{p}}\right)\mid{\mathfrak{p}}\in{\text{Min}}_{S}(S/I)\big{\\}}$

that measures the maximal length of the minimal primary components of $I$. The
next theorem shows that, for a given $S$-homogeneous prime ideal, the
multiplicities of the minimal primary components of the multigraded generic
initial ideal have a very restrictive behavior with respect to the natural
projections.

######  Theorem 5.3.

Assume Section missing 5. Let $P\subset S$ be an $S$-homogeneous prime ideal.
Let ${\mathfrak{J}}=\\{j_{1},\ldots,j_{k}\\}\subset[p]$ be a subset. Then we
have the inequality

${\rm
MLength}\left({\text{gin}}\left(P_{({\mathfrak{J}})}\right)\right)\;\leq\;{\rm
MLength}\left({\text{gin}}\left(P\right)\right).$

Moreover, for a given minimal prime
${\mathfrak{p}}\in{\text{Min}}_{S_{({\mathfrak{J}})}}\left(S_{({\mathfrak{J}})}/{\text{gin}}(P_{({\mathfrak{J}})})\right)$
of ${\text{gin}}(P_{({\mathfrak{J}})})$, we can find a minimal prime
$\mathfrak{q}\in{\text{Min}}_{S}\left(S/{\text{gin}}(P)\right)$ of
${\text{gin}}(P)$ such that the length of the ${\mathfrak{p}}$-primary
component of ${\text{gin}}\left(P_{({\mathfrak{J}})}\right)$ divides the
length of the $\mathfrak{q}$-primary component of ${\text{gin}}(P)$.

###### Proof.

As in §missing 2.2, we can add new variables $x_{i,m_{i}+1}$ to $S$ with
$\deg(x_{i,m_{i}+1})={\mathbf{e}}_{i}$, and then assume that $P$ is relevant.
By further specifying $x_{i,m_{i}}>x_{i,m_{i}+1}$, we may see that
${\text{gin}}(P)$ will not involve the new variables $x_{i,m_{i}+1}$. So,
without any loss of generality, we assume that $P$ is a relevant prime ideal.

Let $I={\text{gin}}(P)\subset S$, and $P_{1},\ldots,P_{h}$ be the minimal
primes of $I$. For each $1\leq k\leq h$, we know that $P_{k}=\left(x_{i,j}\mid
1\leq i\leq p,0\leq j<a_{k,i}\right)$ for some
$\mathbf{a}_{k}=(a_{k,1},\ldots,a_{k,p})\in\mathbb{N}^{p}$ with
$|\mathbf{a}_{k}|={\text{codim}}(P)$. Hence by utilizing the associativity
formula for multidegrees (§missing 2.1(iv)) and §missing 2.1(ii), we obtain

(2)
$\mathcal{C}(S/P;{\mathbf{t}})\,=\,\mathcal{C}(S/I;{\mathbf{t}})\,=\,\sum_{k=1}^{h}\text{length}_{S_{P_{k}}}\left((S/I)_{P_{k}}\right)\,{\mathbf{t}}^{\mathbf{a}_{k}}.$

Let $X=\text{MultiProj}(S/P)\subset{\mathbb{P}}$ be the integral closed
subscheme corresponding with $P$. Set
$r=\dim(X)=m_{1}+\cdots+m_{p}-{\text{codim}}(P)$. From Theorem 2.1 it follows
that

$\mathcal{C}(S/P;{\mathbf{t}})\,=\,\sum_{\begin{subarray}{c}{\mathbf{n}}\in\mathbb{N}^{p}\\\
|{\mathbf{n}}|=r\end{subarray}}\deg_{{\mathbb{P}}}^{\mathbf{n}}(X){\mathbf{t}}^{{\mathbf{m}}-{\mathbf{n}}}.$

Then comparing coefficients yields the equality

$\text{length}_{S_{P_{k}}}\left((S/I)_{P_{k}}\right)\,=\,\deg_{{\mathbb{P}}}^{{\mathbf{m}}-\mathbf{a}_{k}}(X)$

for all $1\leq k\leq h$.

Set
${\mathbb{P}}^{\prime}={\mathbb{P}}_{\mathbb{k}}^{m_{j_{1}}}\times_{\mathbb{k}}\cdots\times_{\mathbb{k}}{\mathbb{P}}_{\mathbb{k}}^{m_{j_{k}}}$.
By applying the same argument to the projection corresponding to
${\mathfrak{J}}$, we obtain that for any minimal prime ${\mathfrak{p}}\subset
S_{({\mathfrak{J}})}$ of ${\text{gin}}(P_{({\mathfrak{J}})})$, there exists
some ${\mathbf{d}}\in\mathbb{N}^{k}$ with
$|{\mathbf{d}}|=\dim\left(\Pi_{\mathfrak{J}}(X)\right)$ such that

$\text{length}_{{\left(S_{({\mathfrak{J}})}\right)}_{{\mathfrak{p}}}}\left(\left(S_{({\mathfrak{J}})}/{\text{gin}}(P_{({\mathfrak{J}})})\right)_{{\mathfrak{p}}}\right)\,=\,\deg_{{\mathbb{P}}^{\prime}}^{{\mathbf{d}}}\left(\Pi_{\mathfrak{J}}(X)\right).$

Therefore, the result follows as a consequence of Theorem 4.1. ∎

We single out the following result that was obtained in (missing) 2.

######  Remark .

Let $P\subset S$ be an $S$-homogeneous prime ideal. Let
$I={\text{gin}}(P)\subset S$. Let $P_{1},\ldots,P_{h}$ be the minimal primes
of $I$ and $Q_{1},\ldots,Q_{h}$ be the corresponding minimal primary
components. The non-zero coefficients of $\mathcal{C}(S/P;{\mathbf{t}})$ are
in one-to-one correspondence with the multiplicities
$\text{length}_{{S_{P_{k}}}}\left((S/Q_{k})_{P_{k}}\right)$.

The next example shows that the minimal primary components of
${\text{gin}}(P_{({\mathfrak{J}})})$ may be determined by embedded primary
components of ${\text{gin}}(P)$. However, Theorem 5.3 still gives a tight
relation between the minimal primary components of
${\text{gin}}(P_{({\mathfrak{J}})})$ and ${\text{gin}}(P)$.

######  Example .

Here we continue using the prime ideal $P\subset S$ from Section missing 4,
where $S$ is the $\mathbb{N}^{3}$-graded polynomial ring
$S=\mathbb{k}\left[x_{0},\ldots,x_{3}\right]\left[y_{0},\ldots,y_{3}\right]\left[z_{0},\ldots,z_{3}\right]$
with $\deg(x_{i})={\mathbf{e}}_{1}$, $\deg(y_{i})={\mathbf{e}}_{2}$ and
$\deg(z_{i})={\mathbf{e}}_{3}$. The corresponding multidegree polynomial is
given by

$\mathcal{C}(R/P;t_{1},t_{2},t_{3})\,=\,2\,{t}_{1}^{3}{t}_{2}^{3}+4\,{t}_{1}^{3}{t}_{2}^{2}{t}_{3}+2\,{t}_{1}^{3}{t}_{2}{t}_{3}^{2}+2\,{t}_{1}^{2}{t}_{2}^{3}{t}_{3}+4\,{t}_{1}^{2}{t}_{2}^{2}{t}_{3}^{2}.$

As we have pointed out before in this section, since ${\text{gin}}(P)$ is
Borel-fixed and $\sqrt{{\text{gin}}(P)}$ is equidimensional (we proved that it
is even Cohen-Macaulay in Theorem 5.2), it follows that the multidegree
polynomial determines the minimal primes of ${\text{gin}}(P)$. More precisely,
we have the following correspondence between terms of
$\mathcal{C}(R/P;{\mathbf{t}})$ and minimal primary components of
${\text{gin}}(P)$:

$\begin{array}[]{lll}2\,t_{1}^{3}t_{2}^{3}&\longleftrightarrow&\left({x}_{0},\,{x}_{1},\,{x}_{2}^{2},\,{y}_{0},\,{y}_{1},\,{y}_{2}\right)\,=:\,M_{1}\\\
4\,t_{1}^{3}t_{2}^{2}t_{3}&\longleftrightarrow&\left({x}_{0},\,{x}_{1}^{2},\,{x}_{1}{x}_{2},\,{x}_{2}^{3},\,{y}_{0},\,{y}_{1},\,{z}_{0}\right)\,=:\,M_{2}\\\
2\,t_{1}^{3}t_{2}t_{3}^{2}&\longleftrightarrow&\left({x}_{0},\,{x}_{1},\,{x}_{2}^{2},\,{y}_{0},\,{z}_{0},\,{z}_{1}\right)\,=:\,M_{3}\\\
2\,t_{1}^{2}t_{2}^{3}t_{3}&\longleftrightarrow&\left({x}_{0},\,{x}_{1}^{2},\,{y}_{0},\,{y}_{1},\,{y}_{2},\,{z}_{0}\right)\,=:\,M_{4}\\\
4\,t_{1}^{2}t_{2}^{2}t_{3}^{2}&\longleftrightarrow&\left({x}_{0},\,{x}_{1}^{2},\,{y}_{0},\,{y}_{1}^{2},\,{z}_{0},\,{z}_{1}\right)\,=:\,M_{5}.\\\
\end{array}$

One can compute the whole expression of ${\text{gin}}(P)$ by utilizing
Macaulay2 [M2], and one may check that ${\text{gin}}(P)$ has a total of $9$
primary components and that all the embedded ones have codimension $7$. We
consider the projection ${\mathfrak{J}}=\\{2,3\\}$. The corresponding
projected ideal is given by

$Q=P_{({2,3})}=\left({y}_{3}{z}_{0}-{y}_{0}{z}_{1}-{y}_{2}{z}_{2},\,{y}_{2}{z}_{0}-{y}_{0}{z}_{2},\,{y}_{1}^{2}+{y}_{2}^{2}-{y}_{0}{y}_{3}\right)\,\subset
T=S_{(2,3)}.$

We already saw that the multidegree polynomial of this projection is equal to

$\mathcal{C}(T/Q;t_{2},t_{3})=2\,{t}_{2}^{3}+4\,{t}_{2}^{2}{t}_{3}+2\,{t}_{2}{t}_{3}^{2}.$

The correspondence between terms of $\mathcal{C}(T/Q;t_{2},t_{3})$ and minimal
primary components of ${\text{gin}}(Q)$ is depicted below:

$\begin{array}[]{lll}2\,t_{2}^{3}&\longleftrightarrow&\left({y}_{0},\,{y}_{1},\,{y}_{2}^{2}\right)\\\
4\,t_{2}^{2}t_{3}&\longleftrightarrow&\left({y}_{0}^{2},\,{y}_{0}{y}_{1},\,{y}_{1}^{3},\,{z}_{0}\right)\\\
2\,t_{2}t_{3}^{2}&\longleftrightarrow&\left({y}_{0}^{2},\,{z}_{0},\,{z}_{1}\right).\\\
\end{array}$

Again, ${\text{gin}}(Q)$ can be computed by using Macaulay2 [M2], or just by
realizing that ${\text{gin}}(Q)={\text{gin}}(P)_{(2,3)}$. It turns out that
${\text{gin}}(Q)$ has only three primary components and that all of them are
minimal. Notice that $M_{1}\cap T=(y_{0},y_{1},y_{2})$, $M_{2}\cap
T=(y_{0},y_{1},z_{0})$, $M_{3}\cap T=(y_{0},z_{0},z_{1})$, $M_{4}\cap
T=(y_{0},y_{1},y_{2},z_{0})$ and $M_{5}\cap T=(y_{0},y_{1}^{2},z_{0},z_{1})$.
So, it follows that all the primary components of ${\text{gin}}(Q)$ come from
embedded components of ${\text{gin}}(P)$, and yet the length of any (minimal)
component of ${\text{gin}}(Q)$ divides the length of some minimal component of
${\text{gin}}(P)$; as predicted by Theorem 5.3.

######  Remark .

If we drop the prime condition, then the result of Theorem 5.3 may not hold.
Indeed, let $J=\left(x_{0}^{2},x_{0}x_{1},x_{1}y_{0},y_{0}^{a}\right)\subset
S$ be the $\mathbb{N}^{2}$-graded ideal of Section missing 4, then we have
that $\text{MLength}\left({\text{gin}}(J)\right)=1$ and
$\text{MLength}\left({\text{gin}}(J_{(2)})\right)=a$.

##  6\. Multiplicity-free varieties

The main goal of this section is to obtain an alternative proof of the result
of Brion [BRION_MULT_FREE] regarding multiplicity-free varieties. For
organizational purposes we divide the section into two subsections.

###  6.1. Cohomology and associated primes under flat degenerations

This subsection contains some technical results that will be needed in our
treatment of multiplicity-free varieties. Here we study the behavior of
cohomology and associated primes under flat degenerations. Our approach is
inspired by ideas related to the fiber-full scheme ([LOC_FIB_FULL_SCHEME,
FIBER_FULL, FIB_FULL_SCHEME]).

######  Proposition  (cf. [LOC_FIB_FULL_SCHEME, Theorem A]).

Let $(A,{\mathfrak{n}},\mathbb{k})$ be a Noetherian local ring with residue
field $\mathbb{k}$. Let $\mathbb{Y}\subset{\mathbb{P}}_{A}^{r}$ be a closed
subscheme and $\mathfrak{F}$ be a coherent sheaf on $\mathbb{Y}$. Suppose that
$\mathfrak{F}$ is flat over $A$. Let
$Y=\mathbb{Y}\times_{\text{Spec}(A)}\text{Spec}(\mathbb{k})\subset{\mathbb{P}}_{\mathbb{k}}^{r}$
be the closed subscheme given as the special fiber and
$F=\mathfrak{F}\otimes_{A}\mathbb{k}$ be the corresponding coherent sheaf on
$Y$.

If
$\text{Hom}_{\mathbb{k}}\left(\text{H}^{i}(Y,F),\,\text{H}^{i+1}(Y,F)\right)=0$
for all $i\geq 0$, then the following statements hold:

  1. (i)

$\text{H}^{i}(\mathbb{Y},\mathfrak{F})$ is $A$-flat for all $i\geq 0$.

  2. (ii)

The natural map
$\text{H}^{i}(\mathbb{Y},\mathfrak{F})\otimes_{A}B\rightarrow\text{H}^{i}(\mathbb{Y}\times_{\text{Spec}(A)}\text{Spec}(B),\mathfrak{F}\otimes_{A}B)$
is an isomorphism for all $i\geq 0$ and any $A$-algebra $B$.

###### Proof.

For any $A$-algebra $B$, set
$\mathbb{Y}_{B}:=\mathbb{Y}\times_{\text{Spec}(A)}\text{Spec}(B)$ and
$\mathfrak{F}_{B}:=\mathfrak{F}\otimes_{A}B$.

First, we prove that for any Artinian quotient ring $C$ of $A$ both statements
of the proposition hold. We proceed by induction on $\text{length}(C)$. The
base case $\text{length}(C)=1$ is clear because we necessarily have
$C=\mathbb{k}$. Thus, we assume that $\text{length}(C)>1$. We may choose an
ideal $\mathfrak{a}\subset C$ generated by a socle element of $C$ to obtain a
short exact sequence

$0\rightarrow\mathfrak{a}\rightarrow C\rightarrow C^{\prime}\rightarrow 0$

with $\mathfrak{a}\cong\mathbb{k}$ and
$\text{length}(C^{\prime})=\text{length}(C)-1$. By tensoring with
$-\otimes_{A}\mathfrak{F}$, since $\mathfrak{F}$ is flat over $A$, we obtain
the short exact sequence

$0\rightarrow
F\rightarrow\mathfrak{F}_{C}\rightarrow\mathfrak{F}_{C^{\prime}}\rightarrow
0.$

Consequently, we get an induced long exact sequence in cohomology

$\text{H}^{i-1}(\mathbb{Y}_{C^{\prime}},\mathfrak{F}_{C^{\prime}})\,\xrightarrow{\delta_{i-1}}\,\text{H}^{i}(Y,F)\,\rightarrow\,\text{H}^{i}(\mathbb{Y}_{C},\mathfrak{F}_{C})\,\rightarrow\,\text{H}^{i}(\mathbb{Y}_{C^{\prime}},\mathfrak{F}_{C^{\prime}})\,\xrightarrow{\delta_{i}}\,\text{H}^{i+1}(Y,F).$

The inductive hypothesis yields the isomorphism
$\text{H}^{i}(\mathbb{Y}_{C^{\prime}},\mathfrak{F}_{C^{\prime}})\otimes_{C^{\prime}}\mathbb{k}\xrightarrow{\cong}\text{H}^{i}(Y,F)$.
Then the map $\delta_{i}$ gets the following factorization

$\text{H}^{i}(\mathbb{Y}_{C^{\prime}},\mathfrak{F}_{C^{\prime}})\;\twoheadrightarrow\;\text{H}^{i}(\mathbb{Y}_{C^{\prime}},\mathfrak{F}_{C^{\prime}})\otimes_{C^{\prime}}\mathbb{k}\cong\text{H}^{i}(Y,F)\;\xrightarrow{\beta_{i}}\;\text{H}^{i+1}(Y,F)$

where $\beta_{i}$ is a $\mathbb{k}$-linear map. The condition
$\text{Hom}_{\mathbb{k}}\left(\text{H}^{i}(Y,F),\,\text{H}^{i+1}(Y,F)\right)=0$
implies that $\beta_{i}=0$, hence $\delta_{i}=0$. For all $i\geq 0$, we have
proved that the natural map
$\text{H}^{i}(\mathbb{Y}_{C},\mathfrak{F}_{C})\rightarrow\text{H}^{i}(\mathbb{Y}_{C^{\prime}},\mathfrak{F}_{C^{\prime}})$
is surjective, and the natural map
$\text{H}^{i}(\mathbb{Y}_{C^{\prime}},\mathfrak{F}_{C^{\prime}})\rightarrow\text{H}^{i}(Y,F)$
is surjective by induction. Therefore, it follows that the natural map

$\text{H}^{i}(\mathbb{Y}_{C},\mathfrak{F}_{C})\,\rightarrow\,\text{H}^{i}(Y,F)$

is also surjective for all $i\geq 0$. Finally, by applying [HARTSHORNE,
Theorem 12.11], we obtain both statements for $C$. This establishes the
proposition for any Artinian quotient ring of $A$.

For each $q\geq 1$, let
$\mathbb{Y}_{q}=\mathbb{Y}\times_{\text{Spec}(A)}\text{Spec}(A/{\mathfrak{n}}^{q})$
and $\mathfrak{F}_{q}=\mathfrak{F}\otimes_{A}A/{\mathfrak{n}}^{q}$. By the
above step we have that $\text{H}^{i}(\mathbb{Y}_{q},\mathfrak{F}_{q})$ is
flat over $A/{\mathfrak{n}}^{q}$. The Theorem on Formal Functions (see
[HARTSHORNE, Theorem III.11.1], [stacks-project, Tag 02OC]) yields the
following isomorphism

$\text{H}^{i}(\mathbb{Y},\mathfrak{F})\otimes_{A}\widehat{A}\;\;\xrightarrow{\;\cong\;}\;\;\underset{\leftarrow}{\lim}\;\text{H}^{i}(\mathbb{Y}_{q},\mathfrak{F}_{q}).$

So by applying [stacks-project, Tag 0912] to the inverse system
$\left(\text{H}^{i}(\mathbb{Y}_{q},\mathfrak{F}_{q})\right)_{q\geq 1}$, we
obtain that $\text{H}^{i}(\mathbb{Y},\mathfrak{F})\otimes_{A}\widehat{A}$ is a
flat $A$-module. Hence $\text{H}^{i}(\mathbb{Y},\mathfrak{F})$ is $A$-flat for
all $i\geq 0$. This settles part (i) of the proposition. Then part (ii)
follows from (i) (see, e.g., [LOC_FIB_FULL_SCHEME, Lemma 2.6]). ∎

######  Notation .

Given projective scheme $Y\subset{\mathbb{P}}_{\mathbb{k}}^{r}=\text{Proj}(B)$
over a field $\mathbb{k}$, we denote by $I_{Y}\subset B$ the corresponding
saturated homogeneous ideal.

######  Lemma .

Let $Y\subset{\mathbb{P}}_{\mathbb{k}}^{r}=\text{Proj}(B)$ be a projective
scheme over a field $\mathbb{k}$. The following statements hold:

  1. (i)

For $1\leq i\leq r$, the ideal $I_{Y}$ has an associated prime of codimension
$i$ if and only if the following limit

$\underset{\nu\to\infty}{\lim}\,\frac{\dim_{\mathbb{k}}\left(\text{H}^{r-i}\left(Y,\mathcal{O}_{Y}(-\nu)\right)\right)}{\nu^{r-i}/(r-i)!}\;\in\;\mathbb{Z}_{+}$

is a positive integer.

  2. (ii)

$Y$ is Cohen-Macaulay and equidimensional if and only if
$\text{H}^{i}(Y,\mathcal{O}_{Y}(-\nu))=0$ for all $i<\dim(Y)$ and $\nu\gg 0$.

###### Proof.

Let ${\mathfrak{m}}=[B]_{+}$ be the graded irrelevant ideal of $B$. Set
$C=B/I_{Y}$.

(i) We recall the following known properties of Ext modules:

  * -

${\text{codim}}\left(\text{Ext}_{B}^{i}(C,B)\right)\geq i$;

  * -

a prime ${\mathfrak{p}}\in\text{Spec}(B)$ of codimension $i$ is an associated
prime of $C$ if and only if it is a minimal prime of
$\text{Ext}_{B}^{i}(C,B)$;

see, e.g., [EHV, Theorem 1.1]. Since
$\delta_{i}=\dim\left(\text{Ext}_{B}^{i}(C,B)\right)\leq r+1-i$, we consider
the usual truncated notion of multiplicity

$e_{r+1-i}\left(\text{Ext}_{B}^{i}(C,B)\right)\,=\,\underset{\nu\to\infty}{\lim}\,\frac{\dim_{\mathbb{k}}\left(\left[\text{Ext}_{B}^{i}(C,B)\right]_{\nu}\right)}{\nu^{r-i}/(r-i)!}\,=\,\begin{cases}e\left(\text{Ext}_{B}^{i}(C,B)\right)&\text{if
}\delta_{i}=r+1-i\\\ 0&\text{otherwise}\end{cases}$

(see, e.g., [SERRE_LOC]). Then the graded local duality theorem (see
[BRUNS_HERZOG, Theorem 3.6.19]) implies that

$e_{r+1-i}\left(\text{Ext}_{B}^{i}(C,B)\right)\;=\;\underset{\nu\to\infty}{\lim}\,\frac{\dim_{\mathbb{k}}\left(\left[\text{H}_{{\mathfrak{m}}}^{r+1-i}(C)\right]_{-\nu}\right)}{\nu^{r-i}/(r-i)!}.$

There is a short exact sequence $0\rightarrow
C\rightarrow\bigoplus_{\nu\in\mathbb{Z}}\text{H}^{0}(Y,\mathcal{O}_{Y}(\nu))\rightarrow\text{H}_{{\mathfrak{m}}}^{1}(C)\rightarrow
0$ and an isomorphism

$\text{H}_{{\mathfrak{m}}}^{j+1}(C)\;\cong\;\bigoplus_{\nu\in\mathbb{Z}}\text{H}^{j}(Y,\mathcal{O}_{Y}(\nu))\quad\text{
for all $j\geq 1$;}$

see [EISEN_COMM, Theorem A4.1]. As a consequence, we have the equality

$e_{r+1-i}\left(\text{Ext}_{B}^{i}(C,B)\right)\;=\;\underset{\nu\to\infty}{\lim}\,\frac{\dim_{\mathbb{k}}\left(\text{H}^{r-i}(Y,\mathcal{O}_{Y}(-\nu))\right)}{\nu^{r-i}/(r-i)!}.$

Finally, the result follows because
$e_{r+1-i}\left(\text{Ext}_{B}^{i}(C,B)\right)$ is a positive integer if and
only if $\text{Ext}_{B}^{i}(C,B)$ has dimension $r+1-i$ (i.e., when $C$ has an
associated prime of codimension $i$).

(ii) This is the content of the proof of [HARTSHORNE, Theorem III.7.6(b)].
Alternatively, we can argue as follows. Let $y\in Y$ be a point and
${\mathfrak{p}}\in\text{Spec}(B)$ the corresponding prime in $B$. By
definition we have that $\mathcal{O}_{Y,y}=\big{\\{}\frac{f}{g}\mid f,g\in
C\text{ homogeneous
elements},\,g\not\in{\mathfrak{p}}C,\,\deg(f)=\deg(g)\big{\\}}$. Let
$C_{({\mathfrak{p}})}$ be the localization of $C$ at all the homogeneous
elements not in ${\mathfrak{p}}$ (we follow the notation of [BRUNS_HERZOG,
§1.5]); this is an ∗local ring ring with ∗maximal ideal
${\mathfrak{p}}C_{({\mathfrak{p}})}$. By [BRUNS_HERZOG, Exercise 2.1.27],
$C_{({\mathfrak{p}})}$ is Cohen-Macaulay if and only if $C_{\mathfrak{p}}$ is.
Let $w\in[C]_{1}$ be a linear form not in ${\mathfrak{p}}C$. Notice that we
have the equality $C_{({\mathfrak{p}})}=\mathcal{O}_{Y,y}[w,w^{-1}]$ and that
$w$ can be seen as an indeterminate over $\mathcal{O}_{Y,y}$. Therefore,
$\mathcal{O}_{Y,y}$ is Cohen-Macaulay if and only if $C_{\mathfrak{p}}$ is
Cohen-Macaulay.

The same arguments in the proof of part (i) imply that
$\text{H}^{i}(Y,\mathcal{O}_{Y}(-\nu))=0$ for all $i<\dim(Y),\nu\gg 0$ if and
only if
$\text{Supp}_{B}\left(\text{Ext}_{B}^{r-i}(C,B)\right)\subseteq\\{{\mathfrak{m}}\\}$
for all $i<\dim(Y)$. The latter condition is equivalent to the following two
conditions:

  1. (a)

all minimal primes of $I_{Y}$ have codimension equal to $r-\dim(Y)$;

  2. (b)

$C_{\mathfrak{p}}$ is Cohen-Macaulay for all ${\mathfrak{p}}\in
V(I_{Y})\setminus\\{{\mathfrak{m}}\\}\subset\text{Spec}(B)$;

indeed, this follows from the local duality theorem and the depth sensitivity
of local cohomology (see [BRUNS_HERZOG, Theorem 3.5.7]). So, the proof of part
(ii) is complete. ∎

By combining the above results we obtain the following theorem.

######  Theorem 6.1.

Let $(A,{\mathfrak{n}},\mathbb{k})$ be a Noetherian local ring with residue
field $\mathbb{k}$. Let $\mathbb{Y}\subset{\mathbb{P}}_{A}^{r}$ be a closed
subscheme and suppose that $\mathbb{Y}$ is flat over $A$. Let
$Y=\mathbb{Y}\times_{\text{Spec}(A)}\text{Spec}(\mathbb{k})\subset{\mathbb{P}}_{\mathbb{k}}^{r}$
be the special fiber. Let $\mathfrak{q}\in\text{Spec}(A)$ and
$Z=\mathbb{Y}\times_{\text{Spec}(A)}\text{Spec}(\kappa(\mathfrak{q}))\subset{\mathbb{P}}_{\kappa(\mathfrak{q})}^{r}$.

If
$\text{Hom}_{\mathbb{k}}\left(\text{H}^{i}(Y,\mathcal{O}_{Y}(-\nu)),\,\text{H}^{i+1}(Y,\mathcal{O}_{Y}(-\nu))\right)=0$
for all $i\geq 0$ and $\nu\gg 0$, then the following statements hold:

  1. (i)

$Y$ is Cohen-Macaulay and equidimensional if and only if $Z$ is both.

  2. (ii)

For $1\leq i\leq r$, $I_{Y}$ has an associated prime of codimension $i$ if and
only if $I_{Z}$ has one.

###### Proof.

By §missing 6.1, we obtain the following equality

$\dim_{\mathbb{k}}\left(\text{H}^{i}(Y,\mathcal{O}_{Y}(-\nu))\right)\;=\;\dim_{\kappa(\mathfrak{q})}\left(\text{H}^{i}(Z,\mathcal{O}_{Z}(-\nu))\right)$

for all $i\geq 0$ and $\nu\gg 0$. Then the statements of the theorem follow
from §missing 6.1. ∎

###  6.2. Properties of multiplicity-free varieties

Throughout this subsection we shall use the following setup.

######  Setup .

Let $\mathbb{k}$ be a field and $S=\mathbb{k}\left[x_{i,j}\mid 1\leq i\leq
p,0\leq j\leq m_{i}\right]$ be a standard $\mathbb{N}^{p}$-graded polynomial
ring over $\mathbb{k}$, such that
$\text{MultiProj}(S)={\mathbb{P}}={\mathbb{P}}_{\mathbb{k}}^{m_{1}}\times_{\mathbb{k}}\cdots\times_{\mathbb{k}}{\mathbb{P}}_{\mathbb{k}}^{m_{p}}$.
Let $X=\text{MultiProj}(R)\subset{\mathbb{P}}$ be an integral closed subscheme
and $R$ be an $\mathbb{N}^{p}$-graded quotient ring of $S$ that is a domain.

Our next goal is to obtain an alternative proof of the following beautiful
result.

######  Theorem 6.2 (Brion [BRION_MULT_FREE]).

Assume §missing 6.2. If $X\subset{\mathbb{P}}$ is multiplicity-free, then:

  1. (i)

$X$ is arithmetically Cohen-Macaulay.

  2. (ii)

$X$ is arithmetically normal.

  3. (iii)

($\mathbb{k}$ infinite) There is a flat degeneration of $X$ to the following
reduced union of multiprojective spaces

$H\;=\;\bigcup_{{\mathbf{n}}=(n_{1},\ldots,n_{p})\,\in\,\text{MSupp}_{\mathbb{P}}(X)}{\mathbb{P}}_{\mathbb{k}}^{n_{1}}\times_{\mathbb{k}}\cdots\times_{\mathbb{k}}{\mathbb{P}}_{\mathbb{k}}^{n_{p}}\;\,\subset\;\,{\mathbb{P}}={\mathbb{P}}_{\mathbb{k}}^{m_{1}}\times_{\mathbb{k}}\cdots\times_{\mathbb{k}}{\mathbb{P}}_{\mathbb{k}}^{m_{p}}$

where
${\mathbb{P}}_{\mathbb{k}}^{n_{i}}=\text{Proj}\left(\mathbb{k}[x_{i,m_{i}-n_{i}},\ldots,x_{i,m_{i}}]\right)\subset{\mathbb{P}}_{\mathbb{k}}^{m_{i}}=\text{Proj}\left(\mathbb{k}[x_{i,0},\ldots,x_{i,m_{i}}]\right)$
only uses the last $n_{i}+1$ coordinates.

The proof of this theorem is divided in several steps. We first consider the
case of one-dimensional multiplicity-free varieties.

######  Lemma .

Suppose that $\dim(X)=1$ and that $X\subset{\mathbb{P}}$ is multiplicity-free.
Then $X$ is isomorphic to the image of a diagonal morphism
${\mathbb{P}}_{\mathbb{k}}^{1}\rightarrow\left({\mathbb{P}}_{\mathbb{k}}^{1}\right)^{e}\subset{\mathbb{P}}$
for some $0<e\leq p$.

###### Proof.

The basic idea is to express $X$ as a blow-up (see [HARTSHORNE, Theorem
II.7.17]). We have that $\Pi_{i}(X)\cong\text{Proj}\left(R_{(i)}\right)$ with
$R_{(i)}=\bigoplus_{k\geq 0}{\left[R\right]}_{k\cdot{\mathbf{e}}_{i}}$ (see
§missing 2.2) and the natural surjection
$R_{(1)}\otimes_{\mathbb{k}}\cdots\otimes_{\mathbb{k}}R_{(p)}\twoheadrightarrow
R$ of standard $\mathbb{N}^{p}$-graded $\mathbb{k}$-algebras, and so as a
consequence we obtain that the natural morphism
$X\rightarrow\Pi_{1}(X)\times_{\mathbb{k}}\cdots\times_{\mathbb{k}}\Pi_{p}(X)$
is a closed immersion. By Section missing 4, for each $1\leq i\leq p$, we have
that $\Pi_{i}(X)$ equals either ${\mathbb{P}}_{\mathbb{k}}^{1}$ or
$\text{Spec}(\mathbb{k})$. Since
$X\hookrightarrow\Pi_{1}(X)\times_{\mathbb{k}}\cdots\times_{\mathbb{k}}\Pi_{p}(X)$
is a closed immersion, we may assume that
$X\subset\left({\mathbb{P}}_{\mathbb{k}}^{1}\right)^{e}\subset{\mathbb{P}}$
for some $0<e\leq p$. Let
$\alpha:X\subset\left({\mathbb{P}}_{\mathbb{k}}^{1}\right)^{e}\rightarrow{\mathbb{P}}_{\mathbb{k}}^{1}$
be the projection into the last component and
$\beta:X\subset\left({\mathbb{P}}_{\mathbb{k}}^{1}\right)^{e}\rightarrow\left({\mathbb{P}}_{\mathbb{k}}^{1}\right)^{e-1}$
into the first $e-1$ components. Due to Section missing 4, $\alpha$ is a
birational morphism, and so there exists a dense open subset
$\mathcal{U}\subset{\mathbb{P}}_{\mathbb{k}}^{1}$ such that
$\alpha_{|\mathcal{V}}:\mathcal{V}=\alpha^{-1}(\mathcal{U})\rightarrow\mathcal{U}$
is an isomorphism. Then the composition $\beta\circ\alpha_{|\mathcal{V}}^{-1}$
gives a morphism
$\mathcal{U}\rightarrow\left({\mathbb{P}}_{\mathbb{k}}^{1}\right)^{e-1}$ and
$X$ can be realized as its graph. That is, we have the following commutative
diagram:

${X\subset\left({\mathbb{P}}_{\mathbb{k}}^{1}\right)^{e}}$${{\mathbb{P}}_{\mathbb{k}}^{1}}$${\left({\mathbb{P}}_{\mathbb{k}}^{1}\right)^{e-1}.}$$\alpha$$\beta$$\mathcal{F}$

Set ${\mathbb{P}}_{\mathbb{k}}^{1}=\text{Proj}(B)$ with
$B=\mathbb{k}[z_{0},z_{1}]$. The morphism $\mathcal{F}$ is given by $e-1$
pairs of homogeneous polynomials $\\{f_{i,0},f_{i,1}\\}\subset B$ with
$\delta_{i}=\deg(f_{i,0})=\deg(f_{i,1})$ and $\gcd(f_{i,0},f_{i,1})=1$. Let
$Y\subset{\mathbb{P}}^{\prime}=\left({\mathbb{P}}_{\mathbb{k}}^{1}\right)^{e-1}$
be the image of $\mathcal{F}$. As a consequence of Section missing 4, $Y$ is
multiplicity-free. The degree formula of [cid2021study, Corollary 3.9] yields
the equalities
$\delta_{i}=\deg_{{\mathbb{P}}^{\prime}}^{{\mathbf{e}}_{i}^{\prime}}(Y)\deg(\mathcal{F})=1$
for all $1\leq i\leq e-1$, where
${\mathbf{e}}_{i}^{\prime}\in\mathbb{N}^{e-1}$ denotes the standard basis
vector. Therefore, up to a $\mathbb{k}$-linear change of coordinates we may
assume that $f_{i,0}=z_{0}$ and $f_{i,1}=z_{1}$. This shows that $X$ is
isomorphic to the image of the diagonal morphism
${\mathbb{P}}_{\mathbb{k}}^{1}\rightarrow\left({\mathbb{P}}_{\mathbb{k}}^{1}\right)^{e}\subset{\mathbb{P}}$.
∎

The next proposition deals with the Hilbert polynomial of $X$. This is the
main step in the proof of Theorem 6.2, and it is the place where we utilize
the tools developed in §missing 6.1.

######  Proposition .

Under the notation and assumptions of Theorem 6.2, we have
$P_{X}({\mathbf{t}})=P_{H}({\mathbf{t}})$.

###### Proof.

We proceed by induction on $\dim(X)$.

If $\dim(X)=0$, then the result is clear since we have
$P_{X}({\mathbf{t}})=1=P_{H}({\mathbf{t}})$.

If $\dim(X)=1$, then $X$ is isomorphic to the image of a diagonal morphism
${\mathbb{P}}_{\mathbb{k}}^{1}\rightarrow\left({\mathbb{P}}_{\mathbb{k}}^{1}\right)^{e}\subset{\mathbb{P}}$
by §missing 6.2. Thus, we get
$P_{X}({\mathbf{t}})=1+\sum_{i\,\mid\,{\mathbf{e}}_{i}\in\text{MSupp}_{{\mathbb{P}}}(X)}t_{i}=P_{H}({\mathbf{t}})$.

Suppose that $d=\dim(X)\geq 2$. Here we use the conventions of Section missing
4. Write $R=S/P$ and notice that
$P_{\mathbb{L}}:=P\otimes_{\mathbb{k}}\mathbb{L}\subset S_{\mathbb{L}}$ is
again a prime ideal (see Section missing 4). One way of relating $X$ and $H$
is by using the multigraded generic initial ideal (see Section missing 5). Let
$>$ be a monomial order on $S$ that satisfies $y_{0}>\cdots>y_{\ell}$, where
$y_{j}=x_{p,j}$. Let $I={\text{gin}}(P_{\mathbb{L}})\subset S_{\mathbb{L}}$.
Let $P_{1},\ldots,P_{h}$ be the minimal primes of $I$ and $Q_{1},\ldots,Q_{h}$
be the corresponding minimal primary components. Since $X$ is assumed to be
multiplicity-free, Section missing 5 yields that $Q_{k}=P_{k}$ for all $1\leq
k\leq h$. Set $J=Q_{1}\cap\cdots\cap Q_{h}=\sqrt{I}\subset S_{\mathbb{L}}$. By
construction we have
$H_{\mathbb{L}}:=H\otimes_{\mathbb{k}}\mathbb{L}\cong\text{MultiProj}(S_{\mathbb{L}}/J)$.
Let $G:=\text{MultiProj}(S_{\mathbb{L}}/I)$. There exists a one-parameter flat
family
$\mathfrak{X}\subset{\mathbb{P}}_{\mathbb{L}}\times_{\mathbb{L}}\mathbb{A}_{\mathbb{L}}^{1}$
such that $X_{\mathbb{L}}$ is the general fiber and $G$ is the special fiber
(see [EISEN_COMM, §15.8]). Then we write

$P_{X}({\mathbf{t}})\,=\,P_{G}({\mathbf{t}})\,=\,\sum_{\begin{subarray}{c}{\mathbf{n}}\in\mathbb{N}^{p}\\\
|{\mathbf{n}}|\leq
d\end{subarray}}e({\mathbf{n}})\binom{t_{1}+n_{1}}{n_{1}}\cdots\binom{t_{p}+n_{p}}{n_{p}}$

and

$P_{H}({\mathbf{t}})\,=\,\sum_{\begin{subarray}{c}{\mathbf{n}}\in\mathbb{N}^{p}\\\
|{\mathbf{n}}|\leq
d\end{subarray}}h({\mathbf{n}})\binom{t_{1}+n_{1}}{n_{1}}\cdots\binom{t_{p}+n_{p}}{n_{p}}.$

Since $G_{\text{red}}=H_{\mathbb{L}}$, it follows that
$\dim\left(\Pi_{\mathfrak{J}}(G)\right)=\dim\left(\Pi_{\mathfrak{J}}(H)\right)$
for all ${\mathfrak{J}}\subset[p]$. From [POSITIVITY, Proposition 3.1], we
have that $\dim\left(\Pi_{i}(X)\right)$ equals the degree of
$P_{X}({\mathbf{t}})$ in the variable $t_{i}$; and the same holds for $G$ and
$H$. Suppose that $\dim\left(\Pi_{p}(X)\right)\geq 1$. We have the equalities

$P_{X_{\mathbb{L}}\cap V(y)}({\mathbf{t}})\,=\,P_{G\cap
V(y)}({\mathbf{t}})\,=\,\sum_{\begin{subarray}{c}{\mathbf{n}}\in\mathbb{N}^{p}\\\
|{\mathbf{n}}|\leq d,n_{p}\geq
1\end{subarray}}e({\mathbf{n}})\binom{t_{1}+n_{1}}{n_{1}}\cdots\binom{t_{p-1}+n_{p-1}}{n_{p-1}}\binom{t_{p}+n_{p}-1}{n_{p}-1}$

and

$P_{H_{\mathbb{L}}\cap
V(y)}({\mathbf{t}})\,=\,\sum_{\begin{subarray}{c}{\mathbf{n}}\in\mathbb{N}^{p}\\\
|{\mathbf{n}}|\leq d,n_{p}\geq
1\end{subarray}}h({\mathbf{n}})\binom{t_{1}+n_{1}}{n_{1}}\cdots\binom{t_{p-1}+n_{p-1}}{n_{p-1}}\binom{t_{p}+n_{p}-1}{n_{p}-1}.$

As no monomial of $J$ involves the variable $y_{\ell}$, we obtain
$J+(y)=\bigcap_{k=1}^{h}\left(Q_{k}+(y)\right)$. This implies that

$H_{\mathbb{L}}\cap
V(y)\;=\;\bigcup_{\begin{subarray}{c}{\mathbf{n}}=(n_{1},\ldots,n_{p})\,\in\,\text{MSupp}_{\mathbb{P}}(X)\\\
n_{p}\geq
1\end{subarray}}{\mathbb{P}}_{\mathbb{L}}^{n_{1}}\times_{\mathbb{L}}\cdots\times_{\mathbb{L}}{\mathbb{P}}_{\mathbb{L}}^{n_{p-1}}\times_{\mathbb{L}}{\mathbb{P}}_{\mathbb{L}}^{n_{p}-1}\;\subset\;{\mathbb{P}}_{\mathbb{L}}.$

On the other hand, $X_{\mathbb{L}}\cap V(y)$ is an integral scheme, and
Theorem 4.2 and [POSITIVITY, Theorem A] yield that
$\text{MSupp}_{{\mathbb{P}}_{\mathbb{L}}}\left(X_{\mathbb{L}}\cap
V(y)\right)=\\{{\mathbf{n}}-{\mathbf{e}}_{p}\mid{\mathbf{n}}\in\text{MSupp}_{\mathbb{P}}(X)\text{
and }n_{p}\geq 1\\}$. Therefore, the inductive hypothesis gives the equality
$P_{X_{\mathbb{L}}\cap V(y)}({\mathbf{t}})=P_{H_{\mathbb{L}}\cap
V(y)}({\mathbf{t}})$, and so we obtain that $e({\mathbf{n}})=h({\mathbf{n}})$
for all $n_{p}\geq 1$.

For $1\leq i<p$, we can repeat the above arguments and consider the $i$-th
component of
${\mathbb{P}}={\mathbb{P}}_{\mathbb{k}}^{m_{1}}\times_{\mathbb{k}}\cdots\times_{\mathbb{k}}{\mathbb{P}}_{\mathbb{k}}^{m_{p}}$.
So, we conclude that $e({\mathbf{n}})=h({\mathbf{n}})$ for all
${\mathbf{n}}\neq\mathbf{0}$. Thus $c=P_{X}({\mathbf{t}})-P_{H}({\mathbf{t}})$
is a constant polynomial.

We have a short exact sequence $0\rightarrow J/I\rightarrow
S_{\mathbb{L}}/I\rightarrow S_{\mathbb{L}}/J\rightarrow 0$. Hence
$P_{J/I}({\mathbf{t}})=c$ and the corresponding coherent sheaf
$\mathfrak{K}=(J/I)^{\sim}$ on $G$ is supported on dimension zero. By Theorem
5.2, $S_{\mathbb{L}}/J$ is Cohen-Macaulay, and thus $H_{\mathbb{L}}$ is Cohen-
Macaulay.

Let
$\mathfrak{s}:{\mathbb{P}}_{\mathbb{L}}={\mathbb{P}}_{\mathbb{L}}^{m_{1}}\times_{\mathbb{L}}\cdots\times_{\mathbb{L}}{\mathbb{P}}_{\mathbb{L}}^{m_{p}}\rightarrow{\mathbb{P}}_{\mathbb{L}}^{r}$
be the Segre embedding, where $r=(m_{1}+1)\cdots(m_{p}+1)-1$. Let
$Z=\mathfrak{s}(X_{\mathbb{L}})$, $Y=\mathfrak{s}(G)$ and
$Y^{\prime}=\mathfrak{s}(H_{\mathbb{L}})$. Let
$\mathfrak{S}=\mathfrak{s}\times_{\mathbb{L}}\mathbb{A}_{\mathbb{L}}^{1}:{\mathbb{P}}_{\mathbb{L}}\times_{\mathbb{L}}\mathbb{A}_{\mathbb{L}}^{1}\rightarrow{\mathbb{P}}_{\mathbb{L}}^{r}\times_{\mathbb{L}}\mathbb{A}_{\mathbb{L}}^{1}$,
and notice that
$\mathbb{Y}=\mathfrak{S}(\mathfrak{X})\subset{\mathbb{P}}_{\mathbb{L}}^{r}\times_{\mathbb{L}}\mathbb{A}_{\mathbb{L}}^{1}$
is a one-parameter flat family with general fiber
$Z\subset{\mathbb{P}}_{\mathbb{L}}^{r}$ and special fiber
$Y\subset{\mathbb{P}}_{\mathbb{L}}^{r}$. We get a short exact sequence
$0\rightarrow\mathfrak{s}_{*}\mathfrak{K}\rightarrow\mathcal{O}_{Y}\rightarrow\mathcal{O}_{Y^{\prime}}\rightarrow
0$. As $\mathfrak{s}_{*}\mathfrak{K}$ is supported on dimension zero, we have
$\text{H}^{i}(Y,(\mathfrak{s}_{*}\mathfrak{K})(\nu))=0$ for all $i\geq
1,\nu\in\mathbb{Z}$. On the other hand, §missing 6.1(ii) yields
$\text{H}^{i}(Y^{\prime},\mathcal{O}_{Y^{\prime}}(-\nu))=0$ for all
$i<d,\nu\gg 0$. From the induced long exact sequence in cohomology

$\cdots\,\rightarrow\,\text{H}^{i}(Y,(\mathfrak{s}_{*}\mathfrak{K})(-\nu))\,\rightarrow\,\text{H}^{i}(Y,\mathcal{O}_{Y}(-\nu))\,\rightarrow\,\text{H}^{i}(Y^{\prime},\mathcal{O}_{Y^{\prime}}(-\nu))\,\rightarrow\,\cdots$

we obtain that $\text{H}^{i}(Y,\mathcal{O}_{Y}(-\nu))=0$ for all $1\leq
i<d,\nu\gg 0$. Since $\dim(Y)=d\geq 2$, it follows that the condition of
Theorem 6.1 holds for our current $Y$. The ideal $I_{Z}$ does not have an
associated prime of codimension $r$ because $Z$ is an integral scheme, and so
Theorem 6.1(ii) implies that $I_{Y}$ does not have an associated prime of
codimension $r$. Therefore $\mathfrak{s}_{*}\mathfrak{K}=0$ and
$Y=Y^{\prime}$.

Finally, we have shown that $c=P_{X}({\mathbf{t}})-P_{H}({\mathbf{t}})=0$, and
this concludes the proof of the proposition. ∎

We can now complete our proof of Brion’s result.

###### Proof of Theorem 6.2.

Let $P\subset S$ be a prime ideal with $X=\text{MultiProj}(S/P)$. Let $>$ be a
monomial order on $S$ and assume that $\mathbb{k}$ is infinite (see Section
missing 4). As in §missing 2.2 and the beginning of the proof of Theorem 5.3,
we may adjoin new variables to $S$ in such a way that all the associated
primes of ${\text{gin}}(P)$ are relevant. Let $I={\text{gin}}(P)$ and
$J=\sqrt{{\text{gin}}(P)}$. Since $X$ is multiplicity-free, Section missing 5
implies that $J$ is the intersection of the minimal primary components of $I$.
By §missing 6.2, we have the equalities
$P_{S/I}({\mathbf{t}})=P_{S/J}({\mathbf{t}})=P_{S/P}({\mathbf{t}})$, and since
all the associated primes of $I$ are assumed to be relevant, we obtain $I=J$.

The result of part (iii) now follows at once (see [EISEN_COMM, §15.8]).
Moreover, we have proved that the multigraded Hilbert function of $S/P$ equals
the one of $S/J$. From Theorem 5.2 we have that $S/J$ is Cohen-Macaulay and as
a consequence $S/P$ also is. This settles part (i).

Let $\overline{R}$ be the integral closure of $R=S/P$ in its field of
fractions $\text{Quot}(R)$. We have that $\overline{R}$ is a finitely
generated $\mathbb{N}^{p}$-graded $R$-module (see [SwHu, §2.3]). Write
$\overline{R}=R[w_{1},\ldots,w_{b}]\subset\text{Quot}(R)$ where
$w_{1},\ldots,w_{b}$ is a minimal set of homogeneous generators of
$\overline{R}$ as an $R$-algebra. Suppose by contradiction that each
$w_{l}\not\in R$. Let $T=S[z_{1},\ldots,z_{b}]$ be an $\mathbb{N}^{p}$-graded
polynomial ring over $S$ with a surjective $S$-algebra homomorphism
$\psi:T\twoheadrightarrow\overline{R},\,z_{l}\rightarrow w_{l}$. Since
$\text{rank}_{R}\left(\overline{R}\right)=1$, the associativity formula for
mixed multiplicities yields the equality
$e({\mathbf{n}};\overline{R})=e({\mathbf{n}};R)$ for all
${\mathbf{n}}\in\mathbb{N}^{p}$ with $|{\mathbf{n}}|=\dim(X)$. Hence we
consider the multiplicity-free prime ideal
$Q=\text{Ker}\left(\psi\right)\subset T$. Choose a lexicographical monomial
order $>^{\prime}$ on $T$ such that $z_{l}>^{\prime}x_{i,j}$ for all $1\leq
l\leq b,1\leq i\leq p,0\leq j\leq m_{i}$. The assumption $w_{l}\not\in R$
would imply that a monomial of the form $z_{l}^{g_{l}}$ with $g_{l}\geq 2$ is
among the minimal generators of ${\text{in}}_{>^{\prime}}(Q)$. However, this
would contradict the fact that ${\text{in}}_{>^{\prime}}(Q)$ is radical by
Theorem 7.3(i) and §missing 7.1; notice that the proof of §missing 7.1 only
depends on the fact that ${\text{gin}}(P)$ is radical, which we already
proved. So, $R$ is a normal domain and the proof of the remaining part (ii) is
complete. ∎

Finally, we restate Brion’s theorem in a more algebraic language.

######  Theorem 6.3.

Assume §missing 6.2. If $P\subset S$ is an $S$-homogeneous multiplicity-free
prime ideal, then:

  1. (i)

$S/P$ is Cohen-Macaulay.

  2. (ii)

$S/P$ is a normal domain.

  3. (iii)

($\mathbb{k}$ infinite) ${\text{gin}}_{>}(P)$ is a radical monomial ideal for
any monomial order $>$ on $S$.

###### Proof.

This follows from the proof of Theorem 6.2 as we can always make $P$ a
relevant ideal. ∎

##  7\. Standardization of ideals

In this section, we develop a process of standardization of ideals in an
_arbitrary positive (possibly non-standard) multigrading_. Here we use a
mixture of the techniques _step-by-step homogenization_ introduced in
[McCULLOUGH_PEEVA] for singly-graded settings and _standardization_ introduced
in [DOUBLE_SCHUBERT] for certain multigraded settings. The following setup and
construction is used throughout this section.

######  Setup .

Let $\mathbb{k}$ be a field and $R=\mathbb{k}[x_{1},\ldots,x_{n}]$ be a
positively $\mathbb{N}^{p}$-graded polynomial ring. For $1\leq i\leq n$, let
$\ell_{i}=|\deg(x_{i})|$ be the total degree of the variable $x_{i}$. Let

$S\;=\;\mathbb{k}\left[y_{i,j}\mid 1\leq i\leq n\text{ and }1\leq
j\leq\ell_{i}\right]$

be a standard $\mathbb{N}^{p}$-graded polynomial ring such that

$\deg(x_{i})\;=\;\sum_{j=1}^{\ell_{i}}\deg(y_{i,j})\quad\text{for all
\;\;$1\leq i\leq n$.}$

We define the $\mathbb{N}^{p}$-graded $\mathbb{k}$-algebra homomorphism

$\phi:R=\mathbb{k}[{\mathbf{x}}]\longrightarrow
S=\mathbb{k}[\mathbf{y}],\quad\phi(x_{i})=y_{i,1}y_{i,2}\cdots
y_{i,\ell_{i}}.$

For an $R$-homogeneous ideal $I\subset R$, we say that the extension
$J=\phi(I)S$ is the _standardization_ of $I$, as $J\subset S$ is an
$S$-homogeneous ideal in the standard $\mathbb{N}^{p}$-graded polynomial ring
$S$. Let ${\mathbf{t}}=\\{t_{1},\ldots,t_{p}\\}$ be variables indexing the
$\mathbb{Z}^{p}$-grading, where $t_{i}$ corresponds with
${\mathbf{e}}_{i}\in\mathbb{Z}^{p}$. Given a finitely generated
$\mathbb{Z}^{p}$-graded $R$-module $M$ and a finitely generated
$\mathbb{Z}^{p}$-graded $S$-module $N$, by a slight abuse of notation, we
consider both multidegree polynomials $\mathcal{C}(M;{\mathbf{t}})$ and
$\mathcal{C}(N;{\mathbf{t}})$ as elements of the same polynomial ring
$\mathbb{Z}[{\mathbf{t}}]=\mathbb{Z}[t_{1},\ldots,t_{p}]$.

The following theorem contains some of the basic and desirable properties that
the standardization process satisfies.

######  Theorem 7.1.

Assume Section missing 7. Let $I\subset R$ be an $R$-homogeneous ideal and
$J=\phi(I)S$ be its standardization. Then the following statements hold:

  1. (i)

${\text{codim}}(I)={\text{codim}}(J)$.

  2. (ii)

$I\subset R$ and $J\subset S$ have the same $\mathbb{N}^{p}$-graded Betti
numbers.

  3. (iii)

$\mathcal{K}(R/I;{\mathbf{t}})=\mathcal{K}(S/J;{\mathbf{t}})$ and
$\mathcal{C}(R/I;{\mathbf{t}})=\mathcal{C}(S/J;{\mathbf{t}})$.

  4. (iv)

$R/I$ is a Cohen-Macaulay ring if and only if $S/J$ is a Cohen-Macaulay ring.

  5. (v)

Let $>$ be a monomial order on $R$ and $>^{\prime}$ be a monomial order on $S$
which is compatible with $\phi$ (i.e., if $f,g\in R$ with $f>g$, then
$\phi(f)>^{\prime}\phi(g)$). Then
${\text{in}}_{>^{\prime}}(J)=\phi({\text{in}}_{>}(I))S$.

  6. (vi)

If $I\subset R$ is a prime ideal and it does not contain any variable, then
$J\subset S$ is also a prime ideal.

###### Proof.

Let $T$ be the polynomial ring $T=\mathbb{k}[{\mathbf{x}},\mathbf{y}]\cong
R\otimes_{\mathbb{k}}S$ with its natural $\mathbb{N}^{p}$-grading induced from
the ones of $R$ and $S$. We see $R$ and $S$ as subrings of $T$. We consider
the quotient ring $T/IT$ and notice that
$\\{x_{i}-\prod_{j=1}^{\ell_{i}}y_{i,j}\\}_{1\leq i\leq n}$ is a regular
sequence of homogeneous elements on $T/IT$. We also have the following natural
isomorphism

$\frac{T}{IT+\left(\\{x_{i}-\prod_{j=1}^{\ell_{i}}y_{i,j}\\}_{1\leq i\leq
n}\right)}\;\cong\;S/J.$

As the natural inclusion $R\hookrightarrow T$ is a polynomial extension, we
have that $\dim(T/IT)=\dim(R/I)+\dim(S)=\dim(R)+\dim(S)-{\text{codim}}(I)$ and
that $T/IT$ is Cohen-Macaulay if and only if $R/I$ is Cohen-Macaulay. So, by
cutting out with the regular sequence described above, we obtain that
$\dim(S/J)=\dim(T/IT)-\dim(R)=\dim(S)-{\text{codim}}(I)$ and that $S/J$ is
Cohen-Macaulay if and only if $T/IT$ is Cohen-Macaulay. This completes the
proofs of parts (i) and (iv).

Let $F_{\bullet}:\cdots\xrightarrow{f_{2}}F_{1}\xrightarrow{f_{1}}F_{0}$ be a
$\mathbb{Z}^{p}$-graded free $R$-resolution of $R/I$. Since
$\\{x_{i}-\prod_{j=1}^{\ell_{i}}y_{i,j}\\}_{1\leq i\leq n}$ is a regular
sequence on both $T$ and $T/IT$, it follows that

${\text{Tor}}_{k}^{T}\Big{(}T/IT,T/\Big{(}\\{x_{i}-\prod_{j=1}^{\ell_{i}}y_{i,j}\\}_{1\leq
i\leq n}\Big{)}\Big{)}=0\quad\text{ for all $k>0$,}$

and so
$G_{\bullet}=F_{\bullet}\otimes_{R}T/(\\{x_{i}-\prod_{j=1}^{\ell_{i}}y_{i,j}\\}_{1\leq
i\leq n})$ provides (up to isomorphism) a $\mathbb{Z}^{p}$-graded free
$S$-resolution of $S/J$. The identification of $G_{\bullet}$ as a resolution
of $S$-modules is the same as $\phi(F_{\bullet})$ (more precisely,
$G_{\bullet}$ has the same Betti numbers as $F_{\bullet}$ and the $i$-th
differential matrix of $G_{\bullet}$ is given by the substitution
$\phi(f_{i})$ of the differential matrix $f_{i}$). We obtained the result of
part (ii). Therefore, by definition, we have the equalities
$\mathcal{K}(R/I;{\mathbf{t}})=\mathcal{K}(S/J;{\mathbf{t}})$ and
$\mathcal{C}(R/I;{\mathbf{t}})=\mathcal{C}(S/J;{\mathbf{t}})$ that settle part
(iii).

To show part (v) we can use Buchberger’s algorithm (see, e.g., [EISEN_COMM,
Chapter 15]). Indeed, we can perform essentially the same steps of the
algorithm in a set of generators of $I$ and the corresponding set of
generators for $J$; for instance, for any two polynomials $f,g\in R$ we have
the following relation of $S$-polynomials $S(\phi(f),\phi(g))=\phi(S(f,g))$.

Lastly, we concentrate on the proof of part (vi). Suppose that $I\subset R$ is
a prime ideal not containing any variable. It then follows that $T/IT$ is a
domain and that $\\{y_{1,2}\cdots y_{1,\ell_{1}},\,x_{1}\\}$ is a regular
sequence on $T/IT$, and so [EISEN_COMM, Exercise 10.4] implies that
$T/(IT,x_{1}-\prod_{j=1}^{\ell_{1}}y_{1,j})$ is a also a domain. Finally, by
repeating iteratively this argument we obtain that
$T/\big{(}IT+\big{(}\\{x_{i}-\prod_{j=1}^{\ell_{i}}y_{i,j}\\}_{1\leq i\leq
n}\big{)}\big{)}\cong S/J$ is a domain. This concludes the proof of part (vi).
∎

######  Remark .

Without too many changes the above arguments could be used to define the
_standardization of a module_. In that case, one could proceed by
standardizing a presentation matrix of a module.

A direct known consequence of the above theorem is the following remark.

######  Remark .

For any $R$-homogeneous ideal $I\subset R$ the multidegree polynomial
$\mathcal{C}(R/I;{\mathbf{t}})$ has non-negative coefficients. Indeed, it
follows from Theorem 7.1(iii) and the (well-known) fact that multidegrees are
non-negative in a standard multigraded setting.

The following theorem shows that the support of the multidegree polynomial is
a discrete polymatroid for prime ideals in a polynomial ring with arbitrary
positive multigrading. It is a consequence of [POSITIVITY, Theorem A].

######  Theorem 7.2.

Assume Section missing 7. Let $P\subset R$ be an $R$-homogeneous prime ideal.
Then the support of the multidegree polynomial $\mathcal{C}(R/P;{\mathbf{t}})$
is a discrete polymatroid.

###### Proof.

Let $\mathcal{L}=\\{i\mid x_{i}\in P\\}$ be the set of indices such that the
corresponding variables belong to $P$. We consider the polynomial ring
$R^{\prime}=\mathbb{k}[x_{i}\mid i\notin\mathcal{L}]\subset R.$ Let
$P^{\prime}\subset R^{\prime}$ be the (unique) ideal that satisfies the
condition $P=P^{\prime}R+\left(x_{i}\mid i\in\mathcal{L}\right)$. By
construction $P^{\prime}\subset R^{\prime}$ does contain any variable. Since
$R/P\cong R^{\prime}/P^{\prime}$, it follows that $P^{\prime}$ is also a prime
ideal. As a consequence of Theorem 7.1(vi) the standardization
$Q=\phi(P^{\prime}R)\subset S$ is a prime ideal.

Since the variables $x_{i}$ with indices in $\mathcal{L}$ form a regular
sequence on $R/P^{\prime}R$, we obtain the equation

$\mathcal{C}(R/P;{\mathbf{t}})\;=\;\prod_{i\in\mathcal{L}}\langle\deg(x_{i}),{\mathbf{t}}\rangle\,\cdot\,\mathcal{C}(R/P^{\prime}R;{\mathbf{t}})$

where
$\langle\deg(x_{i}),{\mathbf{t}}\rangle=a_{i,1}t_{1}+\cdots+a_{i,p}t_{p}\in\mathbb{N}[{\mathbf{t}}]$
after writing $(a_{i,1},\ldots,a_{i,p})=\deg(x_{i})\in\mathbb{N}^{p}$ (see
§missing 2.1(ii)). From Theorem 2.2 we have that the support of
$\mathcal{C}(S/Q;{\mathbf{t}})$ is a discrete polymatroid, and then Theorem
7.1(iii) implies that the support of $\mathcal{C}(R/P^{\prime}R;{\mathbf{t}})$
is also a discrete polymatroid. On the other hand, it is clear that the
support of each linear polynomial $\langle\deg(x_{i}),{\mathbf{t}}\rangle$ is
a discrete polymatroid. Finally, Section missing 2(ii) yields that the support
of $\mathcal{C}(R/P;{\mathbf{t}})$ is a discrete polymatroid. ∎

######  Corollary .

Assume Section missing 7 with $\mathbb{k}$ an infinite field. Let $P\subset R$
be an $R$-homogeneous prime ideal, and $Q=\phi(P)S$ be its standardization.
Then $\sqrt{{\text{gin}}(Q)}$ is a Cohen-Macaulay ideal.

###### Proof.

By Theorem 7.2 and Theorem 7.1(iii), the support of
$\mathcal{C}(S/Q;{\mathbf{t}})$ is a discrete polymatroid. One may notice that
the proof of Theorem 5.2 only depends on the fact that the support of the
multidegree polynomial of the ideal is a discrete polymatroid. Therefore, by
using the same arguments applied to $Q$ (which may not be prime), we conclude
that $\sqrt{{\text{gin}}(Q)}$ is a Cohen-Macaulay ideal. ∎

###  7.1. Cartwright-Sturmfels ideals in positive multigradings

The purpose of this subsection is to extend the notion of Cartwright-Sturmfels
ideals to arbitrary positive multigradings. This family of ideals was defined
and studied in a series of papers [CDNG_CS_IDEALS, CDNG_GIN, CDNG_GRAPH,
CDNG_MINORS, conca2022radical] over a standard multigraded setting.

######  Definition .

An $R$-homogeneous ideal $I\subset R$ is said to be _Cartwright-Sturmfels_ (CS
for short) if there exists an $S$-homogeneous radical Borel-fixed ideal
$K\subset S$ such that
$\mathcal{K}(R/I;{\mathbf{t}})=\mathcal{K}(S/K;{\mathbf{t}})$.

The first basic (yet important) observation that one can make is the following
lemma.

######  Lemma .

Let $I\subset R$ be an $R$-homogeneous ideal and $J=\phi(I)S$ be its
standardization. Then, $I$ is CS (in the sense of §missing 7.1) if and only if
$J$ is CS (in the sense of [CDNG_CS_IDEALS]).

###### Proof.

By Theorem 7.1(iii), we have
$\mathcal{K}(R/I;{\mathbf{t}})=\mathcal{K}(S/J;{\mathbf{t}})$. Thus any
$S$-homogeneous ideal $K\subset S$ has the same $\mathcal{K}$-polynomial as
$I\subset R$ if and only if it has the same multigraded Hilbert function as
$J\subset S$. So, the equivalence is clear because $J$ is CS when there is a
radical Borel-fixed ideal with the same multigraded Hilbert function as $J$. ∎

######  Proposition .

Let $P\subset R$ be a multiplicity-free prime ideal, then $P$ is ${\rm CS}$.

###### Proof.

As in the proof of Theorem 7.2, we may get rid of the variables that belong to
$P$. Let $\mathcal{L}=\\{i\mid x_{i}\in P\\}$ be the set that indexes the
variables belonging to $P$. Write $P=P^{\prime}+\left(x_{i}\mid
i\in\mathcal{L}\right)$ with $P^{\prime}\subset R$ only involving the
variables not in $\mathcal{L}$. Let $T=R\left[z_{i}\mid
i\in\mathcal{L}\right]$ be a positively $\mathbb{N}^{p}$-graded polynomial
ring extending the grading of $R$ and with $\deg(z_{i})=\deg(x_{i})$. Let
$\mathfrak{P}=P^{\prime}T+\left(x_{i}-z_{i}\mid i\in\mathcal{L}\right)\subset
T$. Notice that $\mathfrak{P}$ is a prime ideal containing no variable and
that $\mathcal{K}(T/\mathfrak{P};{\mathbf{t}})=\mathcal{K}(R/P;{\mathbf{t}})$.
Let $W=S\left[w_{i,j}\mid i\in\mathcal{L},1\leq j\leq\ell_{i}\right]$ be a
standard $\mathbb{N}^{p}$-graded polynomial ring where we consider the
corresponding standardization $\mathfrak{Q}\subset W$ of $\mathfrak{P}\subset
T$. From Theorem 7.1(iii)(vi) the standardization $\mathfrak{Q}\subset W$ is a
multiplicity-free prime ideal. Then Theorem 6.3(iii) implies that