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# Multidegrees, families, and integral dependence

Yairon Cid-Ruiz  Department of Mathematics, North Carolina State University,
Raleigh, NC 27695, USA<EMAIL_ADDRESS>,  Claudia Polini  Department of
Mathematics, University of Notre Dame, Notre Dame, IN 46556, USA
<EMAIL_ADDRESS>and  Bernd Ulrich  Department of Mathematics, Purdue
University, West Lafayette, IN 47907, USA<EMAIL_ADDRESS>
###### Abstract.

We study the behavior of multidegrees in families and the existence of
numerical criteria to detect integral dependence. We show that mixed
multiplicities of modules are upper semicontinuous functions when taking
fibers and that projective degrees of rational maps are lower semicontinuous
under specialization. We investigate various aspects of the polar
multiplicities and Segre numbers of an ideal and introduce a new invariant
that we call polar-Segre multiplicities. In terms of polar multiplicities and
our new invariants, we provide a new integral dependence criterion for certain
families of ideals. By giving specific examples, we show that the Segre
numbers are the only invariants among the ones we consider that can detect
integral dependence. Finally, we generalize the result of Gaffney and Gassler
regarding the lexicographic upper semicontinuity of Segre numbers.

###### Key words and phrases:

multidegrees, families, integral dependence, mixed multiplicities, polar
multiplicities, Segre numbers, rational maps, projective degrees,
specialization

###### 2020 Mathematics Subject Classification:

13H15, 14C17, 13B22, 13D40, 13A30

##  1\. Introduction

This paper is concerned with _the behavior of multidegrees in families_ and
with _the search for criteria to detect integral dependence_. Although these
two themes are not typically studied together, the backbone of our work is the
delicate interplay between them. Multidegrees provide the natural
generalization of the degree of a projective variety to a multiprojective
setting, and their study goes back to classical work of van der Waerden
[VAN_DER_WAERDEN]. The notion of multidegrees (or mixed multiplicities) has
become of importance in several areas of mathematics (e.g., algebraic
geometry, commutative algebra and combinatorics; see [Bhattacharya,
VERMA_BIGRAD, HERMANN_MULTIGRAD, Trung2001, MS, EXPONENTIAL_VARIETIES,
KNUTSON_MILLER_SCHUBERT, POSITIVITY, CDNG_MINORS, CDNG_CS_IDEALS, Huh12,
LORENTZIAN, CS_PAPER, TRUNG_VERMA, KNUTSON_MILLER_YONG, STURMFELS_UHLER]). On
the other hand, the idea of detecting integral dependence with numerical
invariants was initiated with seminal work of Rees [REES]. Considerable effort
has been made to extend Rees’ theorem to the case of arbitrary ideals,
modules, and, more generally, algebras (see [KLEIMAN_THORUP_GEOM,
KLEIMAN_THORUP_MIXED, UV_NUM_CRIT, SUV_MULT, UV_CRIT_MOD, BOEGER,
FLENNER_MANARESI, GAFFNEY, GG, PTUV, cidruiz2024polar]).

Teissier’s Principle of Specialization of Integral Dependence (PSID) can be
seen as the first indication of a fruitful connection between the behavior of
multiplicities in families and the detection of integral dependence (see
[TEISSIER_CYC], [TEISSIER_RES2, Appendice I]). Indeed, in an analytic setting,
the original PSID states that for a family of zero-dimensional ideals with
constant Hilbert-Samuel multiplicity, a section is integrally dependent on the
total family if and only if it is integrally dependent on the fibers
corresponding to a Zariski-open dense subset of the base. For families of not
necessarily zero-dimensional ideals (also in an analytic setting), the PSID
was extended by Gaffney and Gassler [GG] using Segre numbers. This paper
continues the line of research traced by the aforementioned works of Teissier,
Gaffney and Gassler.

We now describe the results of this paper more precisely.

1.1. The behavior of multidegrees and projective degrees of rational maps in
families.

Let $\mathbb{k}$ be a field and $T$ be a finitely generated standard
${\mathbb{Z}}_{\geq 0}^{p}$-graded $\mathbb{k}$-algebra. Let
$X=\text{MultiProj}(T)$ be the corresponding multiprojective scheme embedded
in a product of projective spaces
${\mathbb{P}}={\mathbb{P}}_{\mathbb{k}}^{m_{1}}\times_{\mathbb{k}}\cdots\times_{\mathbb{k}}{\mathbb{P}}_{\mathbb{k}}^{m_{p}}$.
For each ${\mathbf{n}}=(n_{1},\ldots,n_{p})\in{\mathbb{Z}}_{\geq 0}^{p}$ with
$|{\mathbf{n}}|=n_{1}+\cdots+n_{p}=\dim(X)$, one denotes by
$\deg_{\mathbb{P}}^{\mathbf{n}}(X)$ the _multidegree of $X$ of type
${\mathbf{n}}$ with respect to ${\mathbb{P}}$_. In classical geometric terms,
if $\mathbb{k}$ is algebraically closed, then
$\deg_{\mathbb{P}}^{\mathbf{n}}(X)$ is equal to the number of points (counted
with multiplicities) in the intersection of $X$ with the product
$L_{1}\times_{\mathbb{k}}\cdots\times_{\mathbb{k}}L_{p}\subset{\mathbb{P}}$,
where $L_{i}\subseteq{\mathbb{P}}_{\mathbb{k}}^{m_{i}}$ is a general linear
subspace of dimension $m_{i}-n_{i}$ for each $1\leq i\leq p$. More generally,
given a finitely generated ${\mathbb{Z}}^{p}$-graded $T$-module $M$, one
denotes by $e({\mathbf{n}};M)$ the _mixed multiplicity of $M$ of type
${\mathbf{n}}$_, for each ${\mathbf{n}}\in{\mathbb{Z}}_{\geq 0}^{p}$ with
$|{\mathbf{n}}|=\dim(\text{Supp}_{++}(M))$. Let
$\Psi:{\mathbb{P}}_{\mathbb{k}}^{r}\dashrightarrow{\mathbb{P}}_{\mathbb{k}}^{s}$
be a rational map and
$\Gamma\subset{\mathbb{P}}_{\mathbb{k}}^{r}\times_{\mathbb{k}}{\mathbb{P}}_{\mathbb{k}}^{s}$
be the closure of the graph of $\mathcal{F}$. For each $0\leq i\leq r$, the
$i$-th projective degree of
$\Psi:{\mathbb{P}}_{\mathbb{k}}^{r}\dashrightarrow{\mathbb{P}}_{\mathbb{k}}^{s}$
is given by the following multidegree

$d_{i}(\Psi)\,=\,\deg_{{\mathbb{P}}_{\mathbb{k}}^{r}\times_{\mathbb{k}}{\mathbb{P}}_{\mathbb{k}}^{s}}^{i,r-i}\left(\Gamma\right).$

For more details on these notions, see Section missing 2.

In Section missing 3, we study the behavior of mixed multiplicities under the
process of taking fibers with respect to a base ring. The idea of studying the
multiplicities of families is now classical (see, e.g., [LIPMAN_EQUIMULT],
[COX_KPU, Chapter 5]), but the case of mixed multiplicities does not seem to
have been considered before. Let $A$ be a Noetherian ring and $\mathscr{T}$ be
a finitely generated standard ${\mathbb{Z}}_{\geq 0}^{p}$-graded $A$-algebra.
Denote by $\mathscr{X}=\text{MultiProj}(\mathscr{T})$ the corresponding
multiprojective scheme. Let $\mathscr{M}$ be a finitely generated
${\mathbb{Z}}^{p}$-graded $\mathscr{T}$-module. For each
$\mathfrak{p}\in{\text{Spec}}(A)$, let
$\kappa(\mathfrak{p})=A_{\mathfrak{p}}/\mathfrak{p}A_{\mathfrak{p}}$ be the
residue field of $\mathfrak{p}$, and consider the finitely generated
${\mathbb{Z}}^{p}$-graded module $\mathscr{M}\otimes_{A}\kappa(\mathfrak{p})$
over the finitely generated standard ${\mathbb{Z}}_{\geq 0}^{p}$-graded
$\kappa(\mathfrak{p})$-algebra
$\mathscr{T}(\mathfrak{p})=\mathscr{T}\otimes_{A}\kappa(\mathfrak{p})$. Then,
for all ${\mathbf{n}}\in{\mathbb{Z}}_{\geq 0}^{p}$, we introduce a function

$e_{\mathbf{n}}^{\mathscr{M}}\,:\,{\text{Spec}}(A)\rightarrow{\mathbb{Z}}\cup\\{\infty\\}$

that naturally measures the mixed multiplicities of the fibers
$\mathscr{M}\otimes_{A}\kappa(\mathfrak{p})$ (see Section missing 3). Our
first main result is the following.

######  Theorem A (Theorem 3.1).

For all ${\mathbf{n}}\in{\mathbb{Z}}_{\geq 0}^{p}$, the function
$e_{\mathbf{n}}^{\mathscr{M}}:{\text{Spec}}(A)\rightarrow{\mathbb{Z}}\cup\\{\infty\\}$
is upper semicontinuous.

A direct consequence of the above theorem is the upper semicontinuity of the
respective functions

$\deg_{\mathscr{X},{\mathbb{P}}_{A}}^{\mathbf{n}}:{\text{Spec}}(A)\rightarrow{\mathbb{Z}}\cup\\{\infty\\}$

measuring the multidegrees of the fibers
$\mathscr{X}\times_{{\text{Spec}}(A)}{\text{Spec}}(\kappa(\mathfrak{p}))$ (see
Section missing 3 and Section missing 3). On the other hand, under certain
conditions, in Section missing 3 we show that mixed multiplicities are lower
semicontinuous under the process of taking specializations.

We study rational maps and their specializations in Section missing 4. In
particular, we are interested in the behavior of projective degrees with
respect to specializations. Since projective degrees are the mixed
multiplicities of the corresponding Rees algebra (i.e., the multidegrees of
the graph), this type of question can be traced back to the problem of
specializing Rees algebras (see [EH_SPECIALIZ, KSU]). More recently,
specializations of rational maps were studied in [SPECIALIZATION_ARON,
SPECIALIZATION_MARC_ARON]. Let $A$ be a Noetherian domain,
$S=A[x_{0},\ldots,x_{r}]$ be a standard graded polynomial ring and
${\mathbb{P}}_{A}^{r}=\text{Proj}(S)$. Let
$\mathcal{F}:{\mathbb{P}}_{A}^{r}\dashrightarrow{\mathbb{P}}_{A}^{s}$ be a
rational map with representative $\mathbf{f}=(f_{0}:\cdots:f_{s})$ such that
$\\{f_{0},\ldots,f_{s}\\}\subset S$ are homogeneous elements of the same
degree. Denote by $I=(f_{0},\ldots,f_{s})\subset S$ the base ideal of
$\mathcal{F}$. For any $\mathfrak{p}\in{\text{Spec}}(A)$, we get the rational
map
$\mathcal{F}(\mathfrak{p}):{\mathbb{P}}_{\kappa(\mathfrak{p})}^{r}\dashrightarrow{\mathbb{P}}_{\kappa(\mathfrak{p})}^{s}$
with representative
$\pi_{\mathfrak{p}}(\mathbf{f})=\left(\pi_{\mathfrak{p}}(f_{0}):\cdots:\pi_{\mathfrak{p}}(f_{s})\right)$
where $\pi_{\mathfrak{p}}(f_{i})$ is the image of $f_{i}$ under the natural
map $\pi_{\mathfrak{p}}:S\rightarrow
S(\mathfrak{p})=S\otimes_{A}\kappa(\mathfrak{p})$. Then, we introduce the
functions

$\displaystyle{\rm
degIm}^{\mathcal{F}}:{\text{Spec}}(A)\rightarrow{\mathbb{Z}}$ (measures the
degree of the image of $\mathcal{F}(\mathfrak{p})$) $\displaystyle
d_{i}^{\mathcal{F}}:{\text{Spec}}(A)\rightarrow{\mathbb{Z}}$ (measures the
projective degrees of $\mathcal{F}(\mathfrak{p})$) $\displaystyle
j^{I}:{\text{Spec}}(A)\rightarrow{\mathbb{Z}}$
$\displaystyle\qquad\qquad\text{(measures the $j$-multiplicity of
$I(\mathfrak{p})\subset S(\mathfrak{p})$)};$

see Section missing 4. Our second main result deals with the behavior of the
last three functions.

######  Theorem B (Theorem 4.1).

The following statements hold:

  1. (i)

${\text{degIm}}^{\mathcal{F}}:{\text{Spec}}(A)\rightarrow{\mathbb{Z}}$ is a
lower semicontinuous function.

  2. (ii)

$d_{i}^{\mathcal{F}}:{\text{Spec}}(A)\rightarrow{\mathbb{Z}}$ is a lower
semicontinuous function for all $0\leq i\leq r$.

  3. (iii)

$j^{I}:{\text{Spec}}(A)\rightarrow{\mathbb{Z}}$ is a lower semicontinuous
function.

In Section missing 4, we use Theorem B to give sharp upper bounds for the
projective degrees of several families of rational maps (the list includes
perfect ideals of height two and Gorenstein ideals of height three). The basic
idea is that for several families of rational maps under generic conditions we
can compute projective degrees, and then Theorem B yields an upper bound for
any specialization.

1.2. Polar multiplicities, Segre numbers and integral dependence.

Our next objective is to study various aspects of the _polar multiplicities_
and _Segre numbers_ of an ideal and introduce a new invariant that plays an
important role in our work. One technical goal of our work is to extend
several of the results of Gaffney and Gassler [GG] from their analytic setting
to an algebraic one over a Noetherian local ring. Let
$(R,{\mathfrak{m}},\kappa)$ be a Noetherian local ring with maximal ideal
${\mathfrak{m}}$ and residue field $\kappa$. Let $d=\dim(R)$,
$X={\text{Spec}}(R)$ and $I\subset R$ be a proper ideal. Consider the blow-up
$\pi:P=\text{Proj}(\mathscr{R}(I))\rightarrow X$ and the exceptional divisor
$E=\text{Proj}(\text{gr}_{I}(R))$ of $X$ along $I$. Following the general
notion of polar multiplicities due to Kleiman and Thorup [KLEIMAN_THORUP_GEOM,
KLEIMAN_THORUP_MIXED], one defines

$\text{(\emph{polar multiplicity})
\;}m_{i}(I,R)\;=\;m_{d}^{d-i}(\mathscr{R}(I))\quad\text{ and
}\qquad\text{(\emph{Segre number})
\;}c_{i}(I,R)\;=\;m_{d-1}^{d-i}(\text{gr}_{I}(R))$

as polar multiplicities of $\mathscr{R}(I)$ and $\text{gr}_{I}(R)$,
respectively. It should be mentioned that the polar multiplicities of a
standard graded $R$-algebra can be seen as the multidegrees of a biprojective
scheme over the residue field $\kappa$. We introduce the new invariant

$\nu_{i}(I,R)\;=\;m_{i}(I,R)+c_{i}(I,R)$

that we call _polar-Segre multiplicity_.

Section missing 5 is dedicated to establish several properties of the
invariants $m_{i}(I,R)$, $c_{i}(I,R)$ and $\nu_{i}(I,R)$. Let $\delta=o(I)$ be
the order of $I$ (see Section missing 5). In Section missing 5, we show the
inequality

$\delta\cdot m_{i-1}(I,R)\;\leq\;m_{i}(I,R)+c_{i}(I,R)\;=\;\nu_{i}(I,R),$

and that equality holds for all $1\leq i\leq d$ if and only if $I$ satisfies
the _$\mathscr{G}$ -parameter condition generically_ (see Section missing 5).
Next, we express all these numbers as the multiplicities of the push-forward
via $\pi$ of certain cycles obtained by making general cuttings; thus
following general tradition when studying local invariants (see, e.g., [GG],
[KLEIMAN_THORUP_GEOM, §8]). Assume that $\kappa$ is an infinite field,
$\underline{H}=H_{1},\ldots,H_{d}$ is a sequence of general hyperplanes, and
denote by $\underline{g}=g_{1},\ldots,g_{d}$ the associated sequence of
elements in $I$ (see Section missing 5). We introduce the following objects:

(polar scheme) $\displaystyle\qquad
P_{i}(I,X)\;=\;P_{i}^{\underline{H}}(I,X)\;=\;\pi\left(H_{1}\cap\cdots\cap
H_{i}\right)$ (Segre cycle)
$\displaystyle\qquad\Lambda_{i}(I,X)\;=\;\Lambda_{i}^{\underline{H}}(I,X)\;=\;\pi_{*}\left(\big{[}E\cap
H_{1}\cap\cdots\cap H_{i-1}\big{]}_{d-i}\right)\;\in\;Z_{d-i}(X)$ (polar-Segre
cycle) $\displaystyle\qquad
V_{i}(I,X)\;=\;V_{i}^{\underline{H}}(I,X)\;=\;\pi_{*}\left(\big{[}H_{1}\cap\cdots\cap
H_{i-1}\cap\pi^{*}D_{i}\big{]}_{d-i}\right)\;\in\;Z_{d-i}(X);$

for more details, see Section missing 5. We have the following unifying
result.

######  Theorem C (Theorem 5.1).

($\kappa$ infinite). The following statements hold:

  1. (i)

$m_{i}(I,R)=e_{d-i}\left(P_{i}(I,X)\right)$ and
$P_{i}(I,X)\;=\;{\text{Spec}}\left(R/(g_{1},\ldots,g_{i}):_{R}I^{\infty}\right).$

  2. (ii)

$c_{i}(I,R)=e_{d-i}\left(\Lambda_{i}(I,X)\right)$ and

$\Lambda_{i}(I,X)\;=\;\sum_{\begin{subarray}{c}\mathfrak{p}\in
V\left((g_{1},\ldots,g_{i-1}):_{R}I^{\infty}\right)\\\ \mathfrak{p}\in
V(I),\;\dim(R/\mathfrak{p})=d-i\end{subarray}}e\big{(}I,\,R_{\mathfrak{p}}/(g_{1},\ldots,g_{i-1})R_{\mathfrak{p}}:_{R_{\mathfrak{p}}}I^{\infty}R_{\mathfrak{p}}\big{)}\cdot\left[R/\mathfrak{p}\right]\;\in\;Z_{d-i}(X).$

  3. (iii)

$\nu_{i}(I,R)=e_{d-i}(V_{i}(I,X))$ and

$V_{i}(I,X)\;=\;\left[P_{i}(I,X)\right]_{d-i}+\Lambda_{i}(I,X)\;=\;\big{[}{\text{Spec}}\left(R/(g_{1},\ldots,g_{i-1}):_{R}I^{\infty}+g_{i}R\right)\big{]}_{d-i}\;\in\,Z_{d-i}(X).$

A particular consequence of the above theorem is that it recovers known
formulas for the polar multiplicities and Segre numbers of an ideal (see
Section missing 5).

In Section missing 6, we introduce new numerical criteria for integral
dependence. From the main result of [PTUV], we know that Segre numbers detect
integral dependence (also, in an analytic setting, see [GG]). In Theorem 6.2,
we prove a PSID that is similar to the ones in [GG, Theorem 4.7] and [PTUV,
Theorem 4.4]. Then a driving question for our work is: _can we detect integral
dependence with the invariants $m_{i}(I,R)$ and $\nu_{i}(I,R)$?_ In the
particular case of the polar multiplicities $m_{i}(I,R)$ this has been a
folklore question for many years. Here we give a definitive and perhaps
unfortunate answer:

  * •

_“only Segre numbers detect integral dependence”_.

Indeed, in Section missing 6 and Section missing 6, we provide examples where
the invariants $m_{i}(I,R)$ and $\nu_{i}(I,R)$ do not detect integral
dependence. On the other hand, we have the following criterion where these two
invariants can be used.

######  Theorem D (Theorem 6.3).

Assume that $R$ is equidimensional and universally catenary, and let $I\subset
J$ be two $R$-ideals. Suppose the following two conditions:

  1. (a)

$o(I)=o(J)$.

  2. (b)

$I$ satisfies the $\mathscr{G}$-parameter condition generically (see Section
missing 5).

Then the following are equivalent:

  1. (i)

$J$ is integral over $I$.

  2. (ii)

$m_{i}(I,R)=m_{i}(J,R)$ for all $0\leq i\leq d-1$.

  3. (iii)

$\nu_{i}(I,R)=\nu_{i}(J,R)$ for all $0\leq i\leq d$.

An interesting family of ideals where the above theorem applies is that of
equigenerated ideals (see §missing 6.2).

Finally, we study the behavior of Segre numbers in families. More precisely,
we generalize the result of Gaffney and Gassler [GG] regarding the
lexicographic upper semicontinuity of Segre numbers. We now introduce a
suitable algebraic notation to extend their original result expressed in an
analytic setting. Let $\iota:A\hookrightarrow R$ be a flat injective
homomorphism of finite type of Noetherian rings and assume that
$\pi:R\twoheadrightarrow A$ is a section of $\iota$. Let $Q=\text{Ker}(\pi)$.
For each $\mathfrak{p}\in{\text{Spec}}(A)$, consider the Noetherian local ring
$S(\mathfrak{p})=R(\mathfrak{p})_{QR(\mathfrak{p})}$ that we call the
_distinguished fiber_ of $\mathfrak{p}$ (see §missing 6.1). Our last main
result is the following.

######  Theorem E (Theorem 6.4).

Assume that for all $\mathfrak{p}\in{\text{Spec}}(A)$, the fibers
$R(\mathfrak{p})$ are equidimensional of the same dimension $d$ and
$\text{ht}(I(\mathfrak{p}))>0$. Then the function

$\mathfrak{p}\in{\text{Spec}}(A)\;\mapsto\;\big{(}c_{1}\left(I,S(\mathfrak{p})\right),c_{2}\left(I,S(\mathfrak{p})\right),\ldots,c_{d}\left(I,S(\mathfrak{p})\right)\big{)}\in{\mathbb{Z}}_{\geq
0}^{d}$

is upper semicontinuous with respect to the lexicographic order.

We give related lexicographic upper semicontinuity results for Segre numbers
in Section missing 6 and Section missing 6.

##  2\. Preliminaries and notation

Here we recall the concepts of mixed multiplicities, multidegrees and
projective degrees. We also set up the notation that is used throughout the
paper. Let $p\geq 1$ be a positive integer and, for each $1\leq i\leq p$, let
${\mathbf{e}}_{i}\in{\mathbb{Z}}_{\geq 0}^{p}$ 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 $1\leq j\leq p$. We write
$\mathbf{0}\in{\mathbb{Z}}_{\geq 0}^{p}$ for the zero vector
$\mathbf{0}=(0,\ldots,0)$.

Let $\mathbb{k}$ be a field and $T$ be a finitely generated standard
${\mathbb{Z}}_{\geq 0}^{p}$-graded algebra over $\mathbb{k}$, that is,
$\left[T\right]_{\mathbf{0}}=\mathbb{k}$ and $T$ is finitely generated over
$\mathbb{k}$ by elements of degree ${\mathbf{e}}_{i}$ with $1\leq i\leq p$.
The multiprojective scheme $X=\text{MultiProj}(T)$ corresponding to $T$ is
given by the set of all multihomogeneous prime ideals in $T$ not containing
the irrelevant ideal
${\mathfrak{N}}:=\left([T]_{{\mathbf{e}}_{1}}\right)\cap\cdots\cap\left([T]_{{\mathbf{e}}_{p}}\right)$,
that is,

$X=\text{MultiProj}(T):=\big{\\{}\mathfrak{P}\in{\text{Spec}}(T)\mid\mathfrak{P}\text{
is multihomogeneous and }\mathfrak{P}\not\supseteq{\mathfrak{N}}\big{\\}},$

and its scheme structure is obtained by using multihomogeneous localizations
(see, e.g., [HYRY_MULTIGRAD, §1]). We embed $X$ as a closed subscheme of a
multiprojective space
${\mathbb{P}}:={\mathbb{P}}_{\mathbb{k}}^{m_{1}}\times_{\mathbb{k}}\cdots\times_{\mathbb{k}}{\mathbb{P}}_{\mathbb{k}}^{m_{p}}$.

Let $M$ be a finitely generated ${\mathbb{Z}}^{p}$-graded $T$-module. A
homogeneous element is said to be filter-regular on $M$ (with respect to
${\mathfrak{N}}$; see [STUCKRAD_VOGEL_BUCHSBAUM_RINGS, Appendix]) if
$z\not\in\mathfrak{P}$ for all associated primes
$\mathfrak{P}\in\text{Ass}_{T}(M)$ of $M$ such that
$\mathfrak{P}\not\supseteq{\mathfrak{N}}$. In terms of the multiprojective
scheme $X$, this means that $zT_{\mathfrak{P}}$ is a nonzerodivisor on
$M_{\mathfrak{P}}$ for all $\mathfrak{P}\in X$. A sequence of homogeneous
elements $z_{1},\ldots,z_{m}$ in $T$ is said to be filter-regular on $M$ (with
respect to ${\mathfrak{N}}$) if $z_{j}$ is a filter-regular element on
$M/\left(z_{1},\ldots,z_{j-1}\right)M$ for all $1\leq j\leq m$. The relevant
support of $M$ is given by $\text{Supp}_{++}(M):=\text{Supp}(M)\cap X$. There
is a polynomial
$P_{M}({\textbf{t}})=P_{M}(t_{1},\ldots,t_{p})\in\mathbb{Q}[{\textbf{t}}]=\mathbb{Q}[t_{1},\ldots,t_{p}]$,
called the _Hilbert polynomial_ of $M$ (see, e.g., [HERMANN_MULTIGRAD, Theorem
4.1], [KLEIMAN_THORUP_GEOM, §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 $\nu\in{\mathbb{Z}}^{p}$ such that $\nu\gg\mathbf{0}$. Furthermore, if
we write

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

then $e(n_{1},\ldots,n_{p})\in{\mathbb{Z}}_{\geq 0}$ for all
$n_{1}+\cdots+n_{p}=r$. From this, we obtain the following invariants:

######  Definition .

  1. (i)

For ${\mathbf{n}}=(n_{1},\ldots,n_{p})\in{\mathbb{Z}}_{\geq 0}^{p}$ 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)

For ${\mathbf{n}}\in{\mathbb{Z}}_{\geq 0}^{p}$ with
$\lvert{\mathbf{n}}\rvert=\dim(X)$,
$\deg_{\mathbb{P}}^{\mathbf{n}}(X):=e({\mathbf{n}};T)$ is the multidegree of
$X=\text{MultiProj}(T)\subset{\mathbb{P}}$ of type ${\mathbf{n}}$ with respect
to ${\mathbb{P}}$.

We recall the following basic concepts related to rational maps.

######  Definition .

Let
$\Psi:{\mathbb{P}}_{\mathbb{k}}^{r}\dashrightarrow{\mathbb{P}}_{\mathbb{k}}^{s}$
be a rational map, $Y\subset{\mathbb{P}}_{\mathbb{k}}^{s}$ be the closure of
the image of $\Psi$, and
$\Gamma\subset{\mathbb{P}}_{\mathbb{k}}^{r}\times_{\mathbb{k}}{\mathbb{P}}_{\mathbb{k}}^{s}$
be the closure of the graph of $\Psi$. The rational map $\Psi$ is generically
finite if one of the following equivalent conditions is satisfied:

  1. (i)

The field extension $K(Y)\hookrightarrow K({\mathbb{P}}_{\mathbb{k}}^{r})$ is
finite, where $K({\mathbb{P}}_{\mathbb{k}}^{r})$ and $K(Y)$ denote the fields
of rational functions of ${\mathbb{P}}_{\mathbb{k}}^{r}$ and $Y$,
respectively.

  2. (ii)

$\dim(Y)=\dim({\mathbb{P}}_{\mathbb{k}}^{r})=r$.

The degree of $\Psi$ is defined as
$\deg(\Psi):=\left[K({\mathbb{P}}_{\mathbb{k}}^{r}):K(Y)\right]$ when $\Psi$
is generically finite. Otherwise, by convention, we set $\deg(\Psi):=0$. For
each $0\leq i\leq r$, the $i$-th projective degree of
$\Psi:{\mathbb{P}}_{\mathbb{k}}^{r}\dashrightarrow{\mathbb{P}}_{\mathbb{k}}^{s}$
is given by

$d_{i}(\Psi)\;:=\;\deg_{{\mathbb{P}}_{\mathbb{k}}^{r}\times_{\mathbb{k}}{\mathbb{P}}_{\mathbb{k}}^{s}}^{i,r-i}\left(\Gamma\right).$

For more details on projective degrees, the reader is referred to [HARRIS,
Example 19.4] and [DOLGACHEV, §7.1.3]. Of particular interest is the $0$-th
projective degree $d_{0}(\Psi)$ as it is equal to the product of the degree of
the map times the degree of the image

(1) $d_{0}(\Psi)\;=\;\deg(\Psi)\cdot\deg_{{\mathbb{P}}_{\mathbb{k}}^{s}}(Y)$

(e.g., this follows from [MIXED_MULT, Theorem 5.4] and [MULTPROJ, Theorem
2.4]).

When we work with families of ideals, we shall use the following notation.

######  Notation .

Let $A$ be a ring and $R$ be an $A$-algebra. For any prime
$\mathfrak{p}\in{\text{Spec}}(A)$, let
$\kappa(\mathfrak{p}):=A_{\mathfrak{p}}/\mathfrak{p}A_{\mathfrak{p}}$ be the
residue field of $\mathfrak{p}$, and set
$R_{\mathfrak{p}}:=R\otimes_{A}A_{\mathfrak{p}}$,
$R(\mathfrak{p}):=R\otimes_{A}\kappa(\mathfrak{p})$ and
$I(\mathfrak{p}):=IR(\mathfrak{p})\subset R(\mathfrak{p})$ for any ideal
$I\subset R$.

The notion general element will be quite useful in our treatment, thus we
recall the following definition.

######  Definition .

Let $R$ be a Noetherian local ring infinite residue field $\kappa$. Let
$I\subset R$ be a proper ideal generated by elements $f_{1},\ldots,f_{m}\in
R$.

  * –

We say that a property $\mathscr{P}$ holds for a _general element_ $g\in I$ if
there exists a dense Zariski-open subset $U\subset\kappa^{e}$ such that
whenever $g=a_{1}f_{1}+\cdots+a_{m}f_{m}$ and the image of $(a_{1},...,a_{m})$
belongs to $U$, then the property $\mathscr{P}$ holds for $g$.

  * –

We say that $\underline{g}=g_{1},\ldots,g_{k}$ is a _sequence of general
elements_ in $I$ if the image of $g_{i}$ in the ideal
$I/(g_{1},\ldots,g_{i-1})\subset R/(g_{1},\ldots,g_{i-1})$ is a general
element for all $1\leq i\leq k$. We also say that
$\underline{g}=g_{1},\ldots,g_{k}$ are _sequentially general elements_ in $I$.

Given a multihomogeneous ideal $\mathcal{J}\subset T$, we say that an element
$z\in T$ is general in $\mathcal{J}$ if its image is general in the
localization $\mathcal{J}T_{\mathfrak{M}}$ where
$\mathfrak{M}:=\left([T]_{{\mathbf{e}}_{1}}\right)+\cdots+\left([T]_{{\mathbf{e}}_{p}}\right)$.

##  3\. The behavior of mixed multiplicities

In this section, we study the behavior of mixed multiplicities under the
processes of taking fibers and performing specializations, both with respect
to a base ring. This section revisits and generalizes some results from
[COX_KPU, Chapter 5] and [SPECIALIZATION_MARC_ARON, SPECIALIZATION_ARON]. We
fix the following setup throughout this section.

######  Setup .

Let $A$ be a Noetherian ring and $\mathscr{T}$ be a finitely generated
standard ${\mathbb{Z}}_{\geq 0}^{p}$-graded algebra over $A$. Let
$\mathscr{X}=\text{MultiProj}(\mathscr{T})$ be the corresponding
multiprojective scheme.

Given a topological space $Z$ and a totally ordered set $(\mathfrak{S},<)$, we
say that a function $f:Z\rightarrow\mathfrak{S}$ is _upper semicontinuous_ if
$\\{z\in Z\mid f(z)\geq s\\}$ is a closed subset of $Z$ for each
$s\in\mathfrak{S}$; on the other hand, a function $f:Z\rightarrow\mathfrak{S}$
is said to be _lower semicontinuous_ if $\\{z\in Z\mid f(z)\leq s\\}$ is a
closed subset of $Z$ for each $s\in\mathfrak{S}$.

An important basic tool in this paper is the topological Nagata criterion
(see, e.g., [MATSUMURA, Theorem 24.2]).

######  Remark  (topological Nagata criterion for openness).

A subset $U\subset{\text{Spec}}(A)$ is open if and only if the following two
conditions are satisfied:

  1. (i)

If $\mathfrak{q}\in U$, then $U$ contains a nonempty open subset of
$V(\mathfrak{q})\subset{\text{Spec}}(A)$.

  2. (ii)

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

Given a finitely generated ${\mathbb{Z}}^{p}$-graded $\mathscr{T}$-module
$\mathscr{M}$, we seek to study the behavior of mixed multiplicities
$e({\mathbf{n}};\bullet)$ when considering the family of modules
$\mathscr{M}\otimes_{A}\kappa(\mathfrak{p})$ with
$\mathfrak{p}\in{\text{Spec}}(A)$. Notice that
$\mathscr{M}\otimes_{A}\kappa(\mathfrak{p})$ is a finitely generated
${\mathbb{Z}}^{p}$-graded $\mathscr{T}(\mathfrak{p})$-module and that
$\mathscr{T}(\mathfrak{p})$ is a finitely generated standard
${\mathbb{Z}}_{\geq 0}^{p}$-graded algebra over the field
$\kappa(\mathfrak{p})$.

######  Definition .

We consider the functions

$d_{++}^{\mathscr{M}}:{\text{Spec}}(A)\rightarrow{\mathbb{Z}},\quad\mathfrak{p}\mapsto\dim\left(\text{Supp}_{++}\left(\mathscr{M}\otimes_{A}\kappa(\mathfrak{p})\right)\right)$

and

$e_{\mathbf{n}}^{\mathscr{M}}:{\text{Spec}}(A)\rightarrow{\mathbb{Z}}\cup\\{\infty\\},\qquad\mathfrak{p}\mapsto\begin{cases}e\left({\mathbf{n}};\mathscr{M}\otimes_{A}\kappa(\mathfrak{p})\right)&\text{
if }|{\mathbf{n}}|=d_{++}^{\mathscr{M}}(\mathfrak{p})\\\ 0&\text{ if
}|{\mathbf{n}}|>d_{++}^{\mathscr{M}}(\mathfrak{p})\\\ \infty&\text{ if
}|{\mathbf{n}}|<d_{++}^{\mathscr{M}}(\mathfrak{p})\end{cases}$

for every ${\mathbf{n}}\in{\mathbb{Z}}_{\geq 0}^{p}$. We use the natural
ordering on the set ${\mathbb{Z}}\cup\\{\infty\\}$.

The following result extends [COX_KPU, Theorem 5.13] to a multigraded setting.

######  Theorem 3.1.

Assume Section missing 3. Let $\mathscr{M}$ be a finitely generated
${\mathbb{Z}}^{p}$-graded $\mathscr{T}$-module. Then the following statements
hold:

  1. (i)

$d_{++}^{\mathscr{M}}:{\text{Spec}}(A)\rightarrow{\mathbb{Z}}$ is an upper
semicontinuous function.

  2. (ii)

$e_{\mathbf{n}}^{\mathscr{M}}:{\text{Spec}}(A)\rightarrow{\mathbb{Z}}\cup\\{\infty\\}$
is an upper semicontinuous function for every
${\mathbf{n}}\in{\mathbb{Z}}_{\geq 0}^{p}$.

###### Proof.

We prove both statements by utilizing the topological Nagata criterion (see
Section missing 3) and Grothendieck’s Generic Freeness Lemma (see, e.g.,
[MATSUMURA, Theorem 24.1], [EISEN_COMM, Theorem 14.4]).

Fix elements $d\in{\mathbb{Z}}$,
${\mathbf{n}}=(n_{1},\ldots,n_{p})\in{\mathbb{Z}}_{\geq 0}^{p}$ and
$e\in{\mathbb{Z}}\cup\\{\infty\\}$. We need to show that

$U_{d}:=\big{\\{}\mathfrak{p}\in{\text{Spec}}(A)\mid
d_{++}^{\mathscr{M}}(\mathfrak{p})\leq d\big{\\}} \quad\text{ and }\quad
V_{{\mathbf{n}},e}:=\big{\\{}\mathfrak{p}\in{\text{Spec}}(A)\mid
e_{{\mathbf{n}}}^{\mathscr{M}}(\mathfrak{p})\leq e\big{\\}}$

are open subsets of ${\text{Spec}}(A)$.

First, we verify condition (i) of Section missing 3 for both subsets $U_{d}$
and $V_{{\mathbf{n}},e}$. Let $\mathfrak{q}\in{\text{Spec}}(A)$ and
$\overline{A}:=A/\mathfrak{q}$. The Generic Freeness Lemma applied to the
module $\overline{\mathscr{M}}:=\mathscr{M}/\mathfrak{q}\mathscr{M}$ gives a
nonzero element $0\neq a\in\overline{A}$ such that each graded component of
$\overline{\mathscr{M}}_{a}$ is a finitely generated free
$\overline{A}_{a}$-module. It follows that
$P_{\mathscr{M}\otimes_{A}\kappa(\mathfrak{p})}=P_{\mathscr{M}\otimes_{A}\kappa(\mathfrak{q})}$
for every $\mathfrak{p}\in D(a)\subset
V(\mathfrak{q})\subset{\text{Spec}}(A)$, which verifies the validity of
condition (i) of Section missing 3 for both $U_{d}$ and $V_{{\mathbf{n}},e}$.

Next, we show that condition (ii) of Section missing 3 also holds. Due to
Nakayama’s lemma, for any two primes
$\mathfrak{p},\mathfrak{q}\in{\text{Spec}}(A)$ with
$\mathfrak{p}\supseteq\mathfrak{q}$, we have that

$\dim_{\kappa(\mathfrak{p})}\left(\left[\mathscr{M}\otimes_{A}\kappa(\mathfrak{p})\right]_{\nu}\right)=\mu_{A_{\mathfrak{p}}}\left(\left[\mathscr{M}\otimes_{A}A_{\mathfrak{p}}\right]_{\nu}\right)\geq\mu_{A_{\mathfrak{q}}}\left(\left[\mathscr{M}\otimes_{A}A_{\mathfrak{q}}\right]_{\nu}\right)=\dim_{\kappa(\mathfrak{q})}\left(\left[\mathscr{M}\otimes_{A}\kappa(\mathfrak{q})\right]_{\nu}\right)$

for all $\nu\in{\mathbb{Z}}^{p}$. The dimension of the relevant support equals
the degree of the Hilbert polynomial, and the latter can be read-off from the
Hilbert function. For any $\mathfrak{p},\mathfrak{q}\in{\text{Spec}}(A)$ with
$\mathfrak{p}\supseteq\mathfrak{q}$, it follows that
$P_{\mathscr{M}\otimes_{A}\kappa(\mathfrak{p})}(\nu)\geq
P_{\mathscr{M}\otimes_{A}\kappa(\mathfrak{q})}(\nu)$ for all
$\nu\gg\mathbf{0}$, and so $d_{++}^{\mathscr{M}}(\mathfrak{p})\geq
d_{++}^{\mathscr{M}}(\mathfrak{q})$. This shows that condition (ii) of Section
missing 3 is satisfied for the subset $U_{d}$.

Given two primes $\mathfrak{p},\mathfrak{q}\in{\text{Spec}}(A)$ with
$\mathfrak{p}\supseteq\mathfrak{q}$ and
$d_{++}^{\mathscr{M}}(\mathfrak{p})>d_{++}^{\mathscr{M}}(\mathfrak{q})$, we
easily check from the definition of the function
$e_{\mathbf{n}}^{\mathscr{M}}$ that
$e_{\mathbf{n}}^{\mathscr{M}}(\mathfrak{p})\geq
e_{\mathbf{n}}^{\mathscr{M}}(\mathfrak{q})$.

Next, consider the case $\mathfrak{p},\mathfrak{q}\in{\text{Spec}}(A)$ with
$\mathfrak{p}\supseteq\mathfrak{q}$ and
$d_{++}^{\mathscr{M}}(\mathfrak{p})=d_{++}^{\mathscr{M}}(\mathfrak{q})$. (That
$P_{\mathscr{M}\otimes_{A}\kappa(\mathfrak{p})}(\nu)-P_{\mathscr{M}\otimes_{A}\kappa(\mathfrak{q})}(\nu)\geq
0$ for all $\nu\gg\mathbf{0}$ does not necessarily imply that the coefficients
of the monomials of highest degree of
$P_{\mathscr{M}\otimes_{A}\kappa(\mathfrak{p})}$ are bigger or equal than the
ones of $P_{\mathscr{M}\otimes_{A}\kappa(\mathfrak{q})}$; for instance,
$f(x,y)=(x-y)^{2}=x^{2}-2xy+y^{2}\in\mathbb{Q}[x,y]$.)

Let $r:=|{\mathbf{n}}|$. We only need to consider the case where
$r=d_{++}^{\mathscr{M}}(\mathfrak{p})=d_{++}^{\mathscr{M}}(\mathfrak{q})$.
Notice that we can reduce modulo $\mathfrak{q}$ and localize at
$\mathfrak{p}$. Hence we assume $A$ is a local domain with maximal ideal
$\mathfrak{p}$ and $\mathfrak{q}=0$. By utilizing the faithfully flat
extension $A\rightarrow A[y]_{\mathfrak{p}A[y]}$, we may assume that the
residue field of $A$ is not an algebraic extension of a finite field. From
[PTUV, Lemma 2.6], for any multihomogeneous ideal
$\mathfrak{J}\subset\mathscr{T}$, we can choose an element $z\in\mathfrak{J}$
whose image is general in both $\mathfrak{J}\mathscr{T}(\mathfrak{p})$ and
$\mathfrak{J}\mathscr{T}(\mathfrak{q})$. Therefore, by prime avoidance, there
exists a sequence of homogeneous elements $z_{1},\ldots,z_{r}$ in
$\mathscr{T}$ such that the following three conditions are satisfied:

  1. (1)

each $z_{j}\in\mathscr{T}$ has degree
$\deg(z_{j})={\mathbf{e}}_{l_{j}}\in{\mathbb{Z}}_{\geq 0}^{p}$ where $1\leq
l_{j}\leq p$;

  2. (2)

$n_{i}$ equals the number $|\\{j\mid 1\leq j\leq r\text{ and }l_{j}=i\\}|$;

  3. (3)

$\\{z_{1},\ldots,z_{r}\\}\mathscr{T}(\mathfrak{p})$ and
$\\{z_{1},\ldots,z_{r}\\}\mathscr{T}(\mathfrak{q})$ are filter-regular
sequences on the modules $\mathscr{M}\otimes_{A}\kappa(\mathfrak{p})$ and
$\mathscr{M}\otimes_{A}\kappa(\mathfrak{q})$, respectively.

To simplify notation, let
$\mathscr{M}_{j,\mathfrak{p}}:=\mathscr{M}/(z_{1},\ldots,z_{j})\mathscr{M}\otimes_{A}\kappa(\mathfrak{p})$.
By successively applying [MIXED_MULT, Lemma 3.9], we obtain

$e\left({\mathbf{n}};\mathscr{M}\otimes_{A}\kappa(\mathfrak{p})\right)=e\left(\mathbf{0};\mathscr{M}_{r,\mathfrak{p}}\right)\quad\text{
and }\quad
e\left({\mathbf{n}};\mathscr{M}\otimes_{A}\kappa(\mathfrak{q})\right)=e\left(\mathbf{0};\mathscr{M}_{r,\mathfrak{q}}\right).$

We choose $\mathbf{0}\ll\nu\in{\mathbb{Z}}_{\geq 0}^{p}$ with the property
that
$e\left(\mathbf{0};\mathscr{M}_{r,\mathfrak{p}}\right)=\dim_{\kappa(\mathfrak{p})}\left(\left[\mathscr{M}_{r,\mathfrak{p}}\right]_{\nu}\right)$
and
$e\left(\mathbf{0};\mathscr{M}_{r,\mathfrak{q}}\right)=\dim_{\kappa(\mathfrak{q})}\left(\left[\mathscr{M}_{r,\mathfrak{q}}\right]_{\nu}\right)$.
Finally, Nakayama’s lemma yields that

$\displaystyle
e\left({\mathbf{n}};\mathscr{M}\otimes_{A}\kappa(\mathfrak{p})\right)=e\left(\mathbf{0};\mathscr{M}_{r,\mathfrak{p}}\right)=\dim_{\kappa(\mathfrak{p})}\left(\left[\mathscr{M}_{r,\mathfrak{p}}\right]_{\nu}\right)=\mu_{A_{\mathfrak{p}}}\left(\left[\mathscr{M}/(z_{1},\ldots,z_{r})\mathscr{M}\otimes_{A}A_{\mathfrak{p}}\right]_{\nu}\right)$
$\displaystyle\quad\geq\mu_{A_{\mathfrak{q}}}\left(\left[\mathscr{M}/(z_{1},\ldots,z_{r})\mathscr{M}\otimes_{A}A_{\mathfrak{q}}\right]_{\nu}\right)=\dim_{\kappa(\mathfrak{q})}\left(\left[\mathscr{M}_{r,\mathfrak{q}}\right]_{\nu}\right)=e\left(\mathbf{0};\mathscr{M}_{r,\mathfrak{q}}\right)=e\left({\mathbf{n}};\mathscr{M}\otimes_{A}\kappa(\mathfrak{q})\right).$

Therefore, the subset $V_{{\mathbf{n}},e}$ satisfies condition (ii) of Section
missing 3. This completes the proof of the theorem. ∎

We restate the above theorem for the case of multidegrees. As before, we embed
$\mathscr{X}$ as a closed subscheme of a multiprojective space
${\mathbb{P}}_{A}:={\mathbb{P}}_{A}^{m_{1}}\times_{A}\cdots\times_{A}{\mathbb{P}}_{A}^{m_{p}}$.
We seek to study the multidegrees of the fibers
$\mathscr{X}_{\mathfrak{p}}:=\mathscr{X}\times_{{\text{Spec}}(A)}{\text{Spec}}(\kappa(\mathfrak{p}))=\text{MultiProj}(\mathscr{T}(\mathfrak{p}))\subset{\mathbb{P}}_{\mathfrak{p}}:={\mathbb{P}}_{A}\times_{{\text{Spec}}(A)}{\text{Spec}}(\kappa(\mathfrak{p}))$.

######  Definition .

We define the functions

$d^{\mathscr{X}}:{\text{Spec}}(A)\rightarrow{\mathbb{Z}},\quad\mathfrak{p}\mapsto\dim\left(\mathscr{X}_{\mathfrak{p}}\right)$

and

$\deg_{\mathscr{X},{\mathbb{P}}_{A}}^{\mathbf{n}}:{\text{Spec}}(A)\rightarrow{\mathbb{Z}}\cup\\{\infty\\},\qquad\mathfrak{p}\mapsto\begin{cases}\deg_{{\mathbb{P}}_{\mathfrak{p}}}^{\mathbf{n}}(\mathscr{X}_{\mathfrak{p}})&\text{
if }|{\mathbf{n}}|=\dim\left(\mathscr{X}_{\mathfrak{p}}\right)\\\ 0&\text{ if
}|{\mathbf{n}}|>\dim\left(\mathscr{X}_{\mathfrak{p}}\right)\\\ \infty&\text{
if }|{\mathbf{n}}|<\dim\left(\mathscr{X}_{\mathfrak{p}}\right)\end{cases}$

for every ${\mathbf{n}}\in{\mathbb{Z}}_{\geq 0}^{p}$.

We have the following direct consequence of Theorem 3.1.

######  Corollary .

Assume Section missing 3. Then the following statements hold:

  1. (i)

$d^{\mathscr{X}}:{\text{Spec}}(A)\rightarrow{\mathbb{Z}}$ is an upper
semicontinuous function.

  2. (ii)

$\deg_{\mathscr{X},{\mathbb{P}}_{A}}^{\mathbf{n}}:{\text{Spec}}(A)\rightarrow{\mathbb{Z}}\cup\\{\infty\\}$
is an upper semicontinuous function for every
${\mathbf{n}}\in{\mathbb{Z}}_{\geq 0}^{p}$.

We quickly revisit the notion of specialization of a module. Here we follow
the same setting and notations as in [SPECIALIZATION_MARC_ARON]. Let
$\mathscr{M}$ be a finitely generated torsionless ${\mathbb{Z}}^{p}$-graded
$\mathscr{T}$-module, with a fixed injection
$\iota:\mathscr{M}\hookrightarrow\mathscr{F}$ into a free
${\mathbb{Z}}^{p}$-graded $\mathscr{T}$-module of finite rank. For any
$\mathfrak{p}\in{\text{Spec}}(A)$, the _specialization of $\mathscr{M}$ with
respect to $\mathfrak{p}$_ is defined as

$\mathbb{S}_{\mathfrak{p}}(\mathscr{M})\,:=\,\text{Im}\Big{(}\iota\otimes_{A}\kappa(\mathfrak{p}):\mathscr{M}\otimes_{A}\kappa(\mathfrak{p})\rightarrow\mathscr{F}\otimes_{A}\kappa(\mathfrak{p})\Big{)}.$

We now consider the following function

$\mathbb{S}e_{\mathbf{n}}^{\mathscr{M}}:{\text{Spec}}(A)\rightarrow{\mathbb{Z}},\qquad\mathfrak{p}\mapsto
e\left({\mathbf{n}};\mathbb{S}_{\mathfrak{p}}(\mathscr{M})\right)$

for every ${\mathbf{n}}\in{\mathbb{Z}}_{\geq 0}^{p}$ with
$|{\mathbf{n}}|\geq\dim\left(\text{Supp}_{++}\left(\mathbb{S}_{\mathfrak{p}}(\mathscr{M})\right)\right)$.

The following result deals with the behavior of mixed multiplicities with
respect to specializations.

######  Corollary .

Assume Section missing 3 and that each graded component of $\mathscr{T}$ is a
free $A$-module. Let $\mathscr{M}$ be a finitely generated torsionless
${\mathbb{Z}}^{p}$-graded $\mathscr{T}$-module, with a fixed injection
$\iota:\mathscr{M}\hookrightarrow\mathscr{F}$ into a free
${\mathbb{Z}}^{p}$-graded $\mathscr{T}$-module of finite rank. Let $r$ be the
common dimension
$\dim(\text{Supp}_{++}(\mathscr{F}\otimes_{A}\kappa(\mathfrak{p})))$ for all
$\mathfrak{p}\in{\text{Spec}}(A)$. Then the function

$\mathbb{S}e_{\mathbf{n}}^{\mathscr{M}}:{\text{Spec}}(A)\rightarrow{\mathbb{Z}}$

is lower semicontinuous for every ${\mathbf{n}}\in{\mathbb{Z}}_{\geq 0}^{p}$
with $|{\mathbf{n}}|\geq r$.

###### Proof.

For any $\mathfrak{p}\in{\text{Spec}}(A)$, we have the short exact sequence

$0\rightarrow\mathbb{S}_{\mathfrak{p}}(\mathscr{M})\rightarrow\mathscr{F}\otimes_{A}\kappa(\mathfrak{p})\rightarrow\mathscr{F}/\mathscr{M}\otimes_{A}\kappa(\mathfrak{p})\rightarrow
0.$

From the additivity of mixed multiplicities, we get
$e_{\mathbf{n}}^{\mathscr{F}}(\mathfrak{p})=e_{\mathbf{n}}^{\mathscr{F}/\mathscr{M}}(\mathfrak{p})+\mathbb{S}e_{\mathbf{n}}^{\mathscr{M}}(\mathfrak{p})$.
Since $e_{\mathbf{n}}^{\mathscr{F}}$ is a constant function by assumption and
$e_{{\mathbf{n}}}^{\mathscr{F}/\mathscr{M}}$ is upper semicontinuous by
Theorem 3.1, the result of the corollary follows. ∎

##  4\. Rational maps and their specializations

Here we concentrate on a specialization process of rational maps. The next
setup is now in place.

######  Setup .

Let $r<s$ be two positive integers. Let $A$ be a Noetherian domain,
$S=A[x_{0},\ldots,x_{r}]$ be a standard graded polynomial ring,
${\mathbb{P}}_{A}^{r}=\text{Proj}(S)$, and
${\mathfrak{m}}=(x_{0},\ldots,x_{r})\subset S$ be the graded irrelevant ideal.
Let $\mathcal{F}:{\mathbb{P}}_{A}^{r}\dashrightarrow{\mathbb{P}}_{A}^{s}$ be a
rational map with representative $\mathbf{f}=(f_{0}:\cdots:f_{s})$ such that
$\\{f_{0},\ldots,f_{s}\\}\subset S$ are homogeneous elements of degree
$\delta>0$.

We specialize this rational map as follows. For any
$\mathfrak{p}\in{\text{Spec}}(A)$, we get the rational map

$\mathcal{F}(\mathfrak{p})\,:\,{\mathbb{P}}_{\kappa(\mathfrak{p})}^{r}\dashrightarrow{\mathbb{P}}_{\kappa(\mathfrak{p})}^{s}$

with representative
$\pi_{\mathfrak{p}}(\mathbf{f})=\left(\pi_{\mathfrak{p}}(f_{0}):\cdots:\pi_{\mathfrak{p}}(f_{s})\right)$
where $\pi_{\mathfrak{p}}(f_{i})$ is the image of $f_{i}$ under the natural
map $\pi_{\mathfrak{p}}:S\rightarrow S(\mathfrak{p})$.

Let $I=(f_{0},\ldots,f_{s})\subset S$ be the base ideal of the rational map
$\mathcal{F}:{\mathbb{P}}_{A}^{r}\dashrightarrow{\mathbb{P}}_{A}^{s}$. The
closure of the graph of $\mathcal{F}$ is given as
$\Gamma={\text{BiProj}}(\mathscr{R}(I))\subset{\mathbb{P}}_{A}^{r}\times_{A}{\mathbb{P}}_{A}^{s}$
where $\mathscr{R}(I):=\bigoplus_{n=0}^{\infty}I^{n}T^{n}\subset S[T]$ is the
Rees algebra of $I$. As customary, $\mathscr{R}(I)$ is presented as a quotient
of a standard bigraded polynomial ring
$\mathscr{T}:=S\otimes_{A}A[y_{0},\ldots,y_{s}]$ by using the $A$-algebra
homomorphism

$\mathscr{T}\twoheadrightarrow\mathscr{R}(I),\quad x_{i}\mapsto
x_{i},\;y_{j}\mapsto f_{j}t.$

We have the equalities
$I(\mathfrak{p})=\left(\pi_{\mathfrak{p}}(f_{0}),\ldots,\pi_{\mathfrak{p}}(f_{s})\right)\subset
S(\mathfrak{p})$ and $(I^{k})(\mathfrak{p})=I(\mathfrak{p})^{k}\subset
S(\mathfrak{p})$ for all $\mathfrak{p}\in{\text{Spec}}(A)$ and $k\geq 0$. For
any $\mathfrak{p}\in{\text{Spec}}(A)$, let
$\Gamma(\mathfrak{p})\subset{\mathbb{P}}_{\kappa(\mathfrak{p})}^{r}\times_{\kappa(\mathfrak{p})}{\mathbb{P}}_{\kappa(\mathfrak{p})}^{s}$
and $Y(\mathfrak{p})\subset{\mathbb{P}}_{\kappa(\mathfrak{p})}^{s}$ be the
closures of the graph and the image of the rational map
$\mathcal{F}(\mathfrak{p})$. We have that $\dim(\Gamma(\mathfrak{p}))\leq r$
and $\dim(Y(\mathfrak{p}))\leq r$ for all $\mathfrak{p}\in{\text{Spec}}(A)$.
We also consider the _j-multiplicity_ of ideal $I(\mathfrak{p})\subset
S(\mathfrak{p})$ for all $\mathfrak{p}\in{\text{Spec}}(A)$ (see
[ACHILLES_MANARESI_J_MULT], [FLENNER_O_CARROLL_VOGEL, §6.1]).

######  Definition .

We have the following three functions:
$j^{I}:{\text{Spec}}(A)\rightarrow{\mathbb{Z}},\;\mathfrak{p}\mapsto
j\left(I(\mathfrak{p})\right),$

${\text{degIm}}^{\mathcal{F}}:{\text{Spec}}(A)\rightarrow{\mathbb{Z}},\qquad\mathfrak{p}\mapsto\begin{cases}\deg_{{\mathbb{P}}_{\kappa(\mathfrak{p})}^{s}}\left(Y(\mathfrak{p})\right)&\text{
if }\dim(Y(\mathfrak{p}))=r\\\ 0&\text{ if
}\dim(Y(\mathfrak{p}))<r\end{cases}$

and
$d_{i}^{\mathcal{F}}:{\text{Spec}}(A)\rightarrow{\mathbb{Z}},\;d_{i}\left(\mathcal{F}(\mathfrak{p})\right)$
for all $0\leq i\leq r$.

Our main result regarding these functions is the following theorem.

######  Theorem 4.1.

Assume Section missing 4. Then the following statements hold:

  1. (i)

${\text{degIm}}^{\mathcal{F}}:{\text{Spec}}(A)\rightarrow{\mathbb{Z}}$ is a
lower semicontinuous function.

  2. (ii)

$d_{i}^{\mathcal{F}}:{\text{Spec}}(A)\rightarrow{\mathbb{Z}}$ is a lower
semicontinuous function for all $0\leq i\leq r$.

  3. (iii)

$j^{I}:{\text{Spec}}(A)\rightarrow{\mathbb{Z}}$ is a lower semicontinuous
function.

###### Proof.

By [KPU_blowup_fibers, Theorem 5.3], we have that
$j^{I}(\mathfrak{p})=\delta\cdot d_{0}^{\mathcal{F}}(\mathfrak{p})$ for all
$\mathfrak{p}\in{\text{Spec}}(A)$. Thus, we only need to prove parts (i) and
(ii) of the theorem.

Fix $0\leq i\leq r$, $e\in{\mathbb{Z}}$ and $h\in{\mathbb{Z}}$. It remains to
show that

$D_{e}:=\big{\\{}\mathfrak{p}\in{\text{Spec}}(A)\mid{\text{degIm}}^{\mathcal{F}}(\mathfrak{p})\geq
e\big{\\}} \quad\text{ and }\quad
E_{i,h}:=\big{\\{}\mathfrak{p}\in{\text{Spec}}(A)\mid
d_{i}^{\mathcal{F}}(\mathfrak{p})\geq h\big{\\}}$

are open subsets of ${\text{Spec}}(A)$. Again, to prove this we utilize a
combination of the topological Nagata criterion and the Generic Freeness
Lemma.

First, we verify condition (i) of Section missing 3 for both $D_{e}$ and
$E_{i,h}$. Let $\mathfrak{q}\in{\text{Spec}}(A)$, and set
$\overline{A}=A/\mathfrak{q}$, $\overline{S}=S/\mathfrak{q}S$ and
$\overline{I}=I\overline{S}\subset\overline{S}$. We use the version of the
Generic Freeness Lemma given in [HOCHSTER_ROBERTS_INVARIANTS, Lemma 8.1]
applied to the inclusion of algebras
$\mathscr{R}_{\overline{S}}\left(\overline{I}\right)\hookrightarrow\overline{S}[t]$,
and we find a nonzero element $0\neq a\in\overline{A}$ such that each graded
component of
$\overline{S}[t]/\mathscr{R}_{\overline{S}}\left(\overline{I}\right)\otimes_{\overline{A}}\overline{A}_{a}$
is a finitely generated free $\overline{A}_{a}$-module (also, see
[SPECIALIZATION_MARC_ARON, Theorem 3.5]). For every $\mathfrak{p}\in
D(a)\subset V(\mathfrak{q})\subset{\text{Spec}}(A)$, we obtain that
$\dim_{\kappa(\mathfrak{p})}\left(\left[I(\mathfrak{p})^{n}\right]_{\nu}\right)=\dim_{\kappa(\mathfrak{q})}\left(\left[I(\mathfrak{q})^{n}\right]_{\nu}\right)$
for all $n\geq 0$ and $\nu\in{\mathbb{Z}}$. Accordingly, condition (i) of
Section missing 3 holds for both $D_{e}$ and $E_{i,h}$.

Next, we show that condition (ii) of Section missing 3 also holds for $D_{e}$
and $E_{i,h}$.

For any $\mathfrak{p}\in{\text{Spec}}(A)$, we have that
${\text{degIm}}^{\mathcal{F}}(\mathfrak{p})=e_{r+1}\left(L_{\mathfrak{p}}\right)$
and
$d_{i}^{\mathcal{F}}(\mathfrak{p})=e\left(i,r-i;\mathscr{R}_{S(\mathfrak{p})}(I(\mathfrak{p}))\right)$,
where
$L_{\mathfrak{p}}:=\kappa(\mathfrak{p})\left[\pi_{\mathfrak{p}}(f_{0}),\ldots,\pi_{\mathfrak{p}}(f_{s})\right]=\bigoplus_{n=0}^{\infty}\left[I(\mathfrak{p})^{n}\right]_{n\delta}$.
For each $n\geq 0$, we have a short exact sequence

$0\rightarrow I(\mathfrak{p})^{n}\rightarrow S(\mathfrak{p})\rightarrow
S/I^{n}\otimes_{A}\kappa(\mathfrak{p})\rightarrow 0.$

Since we know that
$\dim_{\kappa(\mathfrak{p})}\left(\left[S/I^{n}\otimes_{A}\kappa(\mathfrak{p})\right]_{\nu}\right)\geq\dim_{\kappa(\mathfrak{q})}\left(\left[S/I^{n}\otimes_{A}\kappa(\mathfrak{q})\right]_{\nu}\right)$
for all $n\geq 0$, $\nu\in{\mathbb{Z}}$ and
$\mathfrak{p},\mathfrak{q}\in{\text{Spec}}(A)$ with
$\mathfrak{p}\supseteq\mathfrak{q}$, it follows that $D_{e}$ satisfies
condition (ii) of Section missing 3.

Fix two primes $\mathfrak{p},\mathfrak{q}\in{\text{Spec}}(A)$ with
$\mathfrak{p}\supseteq\mathfrak{q}$. As in the proof of Theorem 3.1, we may
assume that $\mathfrak{q}=0$ and that $A$ is a local domain with maximal ideal
$\mathfrak{p}$. Moreover, by making a purely transcendental field extension,
we may assume that for any multihomogeneous ideal
$\mathfrak{J}\subset\mathscr{T}$, we can choose an element $z\in\mathfrak{J}$
whose image is general in both $\mathfrak{J}\mathscr{T}(\mathfrak{p})$ and
$\mathfrak{J}\mathscr{T}(\mathfrak{q})$ (see [PTUV, Lemma 2.6]).

By applying [MIXED_MULT, Proposition 5.6], we find a sequence
$\\{z_{1},\ldots,z_{i}\\}\subset{\mathfrak{m}}\subset S\subset\mathscr{T}$ of
homogeneous elements of degree one such that

$d_{i}^{\mathcal{F}}(\mathfrak{p})=e\left(0,r-i;\,\mathscr{R}_{R_{\mathfrak{p}}}(J_{\mathfrak{p}})\right)\quad\text{
and }\quad
d_{i}^{\mathcal{F}}(\mathfrak{q})=e\left(0,r-i;\,\mathscr{R}_{R_{\mathfrak{q}}}(J_{\mathfrak{q}})\right)$

where $R_{\mathfrak{p}}:=S(\mathfrak{p})/(z_{1},\ldots,z_{i})S(\mathfrak{p})$,
$R_{\mathfrak{q}}:=S(\mathfrak{q})/(z_{1},\ldots,z_{i})S(\mathfrak{q})$,
$J_{\mathfrak{p}}:=IR_{\mathfrak{p}}\subset R_{\mathfrak{p}}$ and
$J_{\mathfrak{q}}:=IR_{\mathfrak{q}}\subset R_{\mathfrak{q}}$. Since $i\leq
r$, we may further assume that $\\{z_{1},\ldots,z_{i}\\}S(\mathfrak{p})$ and
$\\{z_{1},\ldots,z_{i}\\}S(\mathfrak{q})$ are regular sequences on the
polynomial rings $S(\mathfrak{p})$ and $S(\mathfrak{q})$, respectively.

There exists a positive integer $m>0$ such that

$e\left(0,r-i;\,\mathscr{R}_{R_{\mathfrak{p}}}(J_{\mathfrak{p}})\right)=\lim_{n\to\infty}\frac{\dim_{\kappa(\mathfrak{p})}\left(\left[\mathscr{R}_{R_{\mathfrak{p}}}(J_{\mathfrak{p}})\right]_{(m,n)}\right)}{n^{r-i}/(r-i)!}=\lim_{n\to\infty}\frac{\dim_{\kappa(\mathfrak{p})}\left(\left[J_{\mathfrak{p}}^{n}\right]_{m+n\delta}\right)}{n^{r-i}/(r-i)!}$

and

$e\left(0,r-i;\,\mathscr{R}_{R_{\mathfrak{q}}}(J_{\mathfrak{q}})\right)=\lim_{n\to\infty}\frac{\dim_{\kappa(\mathfrak{q})}\left(\left[\mathscr{R}_{R_{\mathfrak{q}}}(J_{\mathfrak{q}})\right]_{(m,n)}\right)}{n^{r-i}/(r-i)!}=\lim_{n\to\infty}\frac{\dim_{\kappa(\mathfrak{q})}\left(\left[J_{\mathfrak{q}}^{n}\right]_{m+n\delta}\right)}{n^{r-i}/(r-i)!}.$

So, to show that $d_{i}^{\mathcal{F}}(\mathfrak{p})\leq
d_{i}^{\mathcal{F}}(\mathfrak{q})$, it suffices to verify that
$\dim_{\kappa(\mathfrak{p})}\left(\left[J_{\mathfrak{p}}^{n}\right]_{\nu}\right)\leq\dim_{\kappa(\mathfrak{q})}\left(\left[J_{\mathfrak{q}}^{n}\right]_{\nu}\right)$
for all $n\geq 0$ and $\nu\in{\mathbb{Z}}$. Notice that the Hilbert functions
of $R_{\mathfrak{p}}$ and $R_{\mathfrak{q}}$ are equal. Then, from Nakayama’s
lemma we get

$\displaystyle\dim_{\kappa(\mathfrak{p})}\left(\left[R_{\mathfrak{p}}/J_{\mathfrak{p}}^{n}\right]_{\nu}\right)$
$\displaystyle=\mu_{A_{\mathfrak{p}}}\left(\left[\frac{S}{\left(z_{1},\ldots,z_{i},I^{n}\right)S}\otimes_{A}A_{\mathfrak{p}}\right]_{\nu}\right)$
$\displaystyle\geq\mu_{A_{\mathfrak{q}}}\left(\left[\frac{S}{\left(z_{1},\ldots,z_{i},I^{n}\right)S}\otimes_{A}A_{\mathfrak{q}}\right]_{\nu}\right)=\dim_{\kappa(\mathfrak{q})}\left(\left[R_{\mathfrak{q}}/J_{\mathfrak{q}}^{n}\right]_{\nu}\right)$

for all $n\geq 0$ and $\nu\in{\mathbb{Z}}$. Finally, this implies that
$d_{i}^{\mathcal{F}}(\mathfrak{p})\leq d_{i}^{\mathcal{F}}(\mathfrak{q})$, and
so it follows that condition (ii) of Section missing 3 holds for $E_{i,h}$.
So, we are done with the proof of the theorem. ∎

We single out an important corollary of Theorem 4.1. When $r=s$, and we
consider a rational map of the form
$\mathcal{F}:{\mathbb{P}}_{A}^{r}\dashrightarrow{\mathbb{P}}_{A}^{r}$, we have
some control over the degree of the specialized rational maps
$\mathcal{F}(\mathfrak{p}):{\mathbb{P}}_{\kappa(\mathfrak{p})}^{r}\dashrightarrow{\mathbb{P}}_{\kappa(\mathfrak{p})}^{r}$.

######  Corollary .

Assume Section missing 4. Let
$\mathcal{F}:{\mathbb{P}}_{A}^{r}\dashrightarrow{\mathbb{P}}_{A}^{r}$ be a
rational map. Then the function

$\deg^{\mathcal{F}}:{\text{Spec}}(A)\rightarrow{\mathbb{Z}},\qquad\mathfrak{p}\mapsto\deg\left(\mathcal{F}(\mathfrak{p})\right)$

is lower semicontinuous.

###### Proof.

We have that
$\deg^{\mathcal{F}}(\mathfrak{p})=d_{0}^{\mathcal{F}}(\mathfrak{p})$ for all
$\mathfrak{p}\in{\text{Spec}}(A)$. Indeed, if $\dim(Y(\mathfrak{p}))<r$, both
$\deg^{\mathcal{F}}(\mathfrak{p})$ and $d_{0}^{\mathcal{F}}(\mathfrak{p})$ are
equal to zero; and if $\dim(Y(\mathfrak{p}))=r$,
$\deg_{{\mathbb{P}}_{\kappa(\mathfrak{p})}^{r}}(Y(\mathfrak{p}))=1$ and so
$d_{0}(\mathcal{F}(\mathfrak{p}))=\deg(\mathcal{F}(\mathfrak{p}))$ according
to (missing) 1. Thus, the result follows from Theorem 4.1. ∎

We now apply the above results to different families of rational maps. We
obtain generalizations of [SPECIALIZATION_ARON, Theorems 6.3, 6.8], and we
eliminate the conditions assumed there. The following corollary yields
significant upper bounds for the projective degrees of certain families of
rational maps. It should be mentioned that these inequalities are sharp for
the general members of the considered families (see [MIXED_MULT, Theorems 5.7,
5.8]).

######  Corollary .

Let $\mathbb{k}$ be a field, $R=\mathbb{k}[x_{0},\ldots,x_{r}]$ be a standard
graded polynomial ring, ${\mathbb{P}}_{\mathbb{k}}^{r}=\text{Proj}(R)$,
$\Psi:{\mathbb{P}}_{\mathbb{k}}^{r}\dashrightarrow{\mathbb{P}}_{\mathbb{k}}^{s}$
be a rational map with representative $\mathbf{g}=(g_{0}:\cdots:g_{s})$ and
base ideal $J=(g_{0},\ldots,g_{s})\subset R$, and suppose that
$\delta=\deg(g_{j})>0$. Then the following statements hold:

  1. (i)

$d_{i}(\Psi)\leq\delta^{r-i}$ for all $0\leq i\leq r$.

  2. (ii)

Suppose that $J$ is a perfect ideal of height two with Hilbert-Burch
resolution of the form

$0\rightarrow\bigoplus_{i=1}^{s}R(-\delta-\mu_{i})\rightarrow{R(-\delta)}^{s+1}\rightarrow
J\rightarrow 0.$

Then, for all $0\leq i\leq r$, we have

$d_{i}(\Psi)\leq e_{r-i}(\mu_{1},\ldots,\mu_{s})$

where $e_{r-i}(\mu_{1},\ldots,\mu_{s})$ denotes the elementary symmetric
polynomial

$e_{r-i}(\mu_{1},\ldots,\mu_{s})=\sum_{1\leq j_{1}<\cdots<j_{r-i}\leq
s}\mu_{j_{1}}\cdots\mu_{j_{r-i}}.$

In particular, if $r=s$, then $\deg(\Psi)\leq\mu_{1}\cdots\mu_{r}$.

  3. (iii)

Suppose that $J$ is a Gorenstein ideal of height three. Let $D\geq 1$ be the
degree of every nonzero entry of an alternating minimal presentation matrix of
$J$. Then, for all $0\leq i\leq r$, we have

$d_{i}(\Psi)\leq\begin{cases}D^{r-i}\sum_{k=0}^{\lfloor\frac{s-r+i}{2}\rfloor}\binom{s-1-2k}{r-i-1}&\text{
if }0\leq i\leq r-3\\\ \delta^{r-i}&\text{ if }r-2\leq i\leq r.\end{cases}$

In particular, if $r=s$, then $\deg(\Psi)\leq D^{r}$.

###### Proof.

(i) For $0\leq j\leq s$, consider a set of variables
$\mathbf{z}_{j}=\\{z_{j,1},\ldots,z_{j,m}\\}$ over $\mathbb{k}$ with
$m=\binom{\delta+r}{r}$, and set
$\mathbf{z}=\mathbf{z}_{0}\cup\cdots\cup\mathbf{z}_{s}$. Let
$A=\mathbb{k}[\mathbf{z}]$, $S=A[x_{0},\ldots,x_{r}]$ and consider the generic
polynomials

$G_{j}\,:=\,z_{j,1}x_{0}^{\delta}+z_{j,2}x_{0}^{\delta-1}x_{1}+\cdots+z_{j,m}x_{r}^{\delta}\in
S.$

Let $\mathcal{F}:{\mathbb{P}}_{A}^{r}\dashrightarrow{\mathbb{P}}_{A}^{s}$ be a
rational map with base ideal $I=(G_{0},\ldots,G_{s})\subset S$. Let
$\xi=(0)\subset{\text{Spec}}(A)$ be the generic point and
${\mathfrak{m}}_{\alpha}=\left(\\{z_{j,k}-\alpha_{j,k}\\}_{j,k}\right)\in{\text{Spec}}(A)$
be a rational maximal ideal such that $J=I({\mathfrak{m}}_{\alpha})\subset R$.
It is known that the projective degrees of the morphism
$\mathcal{F}(\xi):{\mathbb{P}}_{\mathbb{k}(\mathbf{z})}^{r}\rightarrow{\mathbb{P}}_{\mathbb{k}(\mathbf{z})}^{s}$
(it is base point free as $I\otimes_{A}\mathbb{k}(\mathbf{z})\subset
S(\xi)=\mathbb{k}(\mathbf{z})[x_{0},\ldots,x_{r}]$ is a zero-dimensional
ideal) are equal to $d_{i}\left(\mathcal{F}(\xi)\right)=\delta^{r-i}$ (see,
e.g., [KPU_blowup_fibers, Observation 3.2]). By using Theorem 4.1, we obtain
$d_{i}(\Psi)=d_{i}^{\mathcal{F}}({\mathfrak{m}}_{\alpha})\leq
d_{i}^{\mathcal{F}}(\xi)=\delta^{r-i}$ for all $0\leq i\leq r$.

(ii) For $1\leq j\leq s+1$ and $1\leq k\leq s$, let
$\mathbf{z}_{j,k}=\\{z_{j,k,1},z_{j,k,2},\ldots,z_{j,k,m_{k}}\\}$ denote a set
of variables over $\mathbb{k}$ of cardinality $m_{k}=\binom{\mu_{k}+r}{r}$,
and set $\mathbf{z}=\bigcup_{j,k}\mathbf{z}_{j,k}$. Let
$A=\mathbb{k}[\mathbf{z}]$, $S=A[x_{0},\ldots,x_{r}]$ and consider the generic
$(s+1)\times s$ Hilbert-Burch matrix

$\mathcal{M}=\left(\begin{array}[]{cccc}p_{1,1}&p_{1,2}&\cdots&p_{1,s}\\\
p_{2,1}&p_{2,2}&\cdots&p_{2,s}\\\ \vdots&\vdots&&\vdots\\\
p_{s+1,1}&p_{s+1,2}&\cdots&p_{s+1,s}\\\ \end{array}\right)$

where each polynomial $p_{j,k}\in S$ is given by

$p_{j,k}=z_{j,k,1}x_{0}^{\mu_{k}}+z_{j,k,2}x_{0}^{\mu_{k}-1}x_{1}+\cdots+z_{j,k,m_{k}}x_{r}^{\mu_{k}}.$

Let $\mathcal{F}:{\mathbb{P}}_{A}^{r}\dashrightarrow{\mathbb{P}}_{A}^{s}$ be a
rational map with base ideal $I=I_{s}(\mathcal{M})\subset S$. Let $\varphi\in
R^{(s+1)\times s}$ be the Hilbert-Burch presentation of $J$. Let
$\xi=(0)\subset{\text{Spec}}(A)$ be the generic point and
${\mathfrak{m}}_{\alpha}=\left(\\{z_{j,k,l}-\alpha_{j,k,l}\\}_{j,k,l}\right)\in{\text{Spec}}(A)$
be a rational maximal ideal such that $\varphi\in R^{(s+1)\times s}$ is
obtained by specializing $\mathcal{M}\in S^{(s+1)\times s}$ via the map
$S\twoheadrightarrow S/{\mathfrak{m}}_{\alpha}\cong R$. From
[SPECIALIZATION_ARON, Lemma 6.2] the generic ideal
$I\otimes_{A}\mathbb{k}(\mathbf{z})\subset
S(\xi)=\mathbb{k}(\mathbf{z})[x_{0},\ldots,x_{r}]$ is perfect of height two
and satisfies the condition $G_{r+1}$, and so [MIXED_MULT, Theorem 5.7]
implies that the projective degrees of
$\mathcal{F}(\xi):{\mathbb{P}}_{\mathbb{k}(\mathbf{z})}^{r}\dashrightarrow{\mathbb{P}}_{\mathbb{k}(\mathbf{z})}^{s}$
are equal to $d_{i}(\mathcal{F}(\xi))=e_{r-i}(\mu_{1},\ldots,\mu_{s})$.
Finally, Theorem 4.1 yields that
$d_{i}(\Psi)=d_{i}^{\mathcal{F}}({\mathfrak{m}}_{\alpha})\leq
d_{i}^{\mathcal{F}}(\xi)=e_{r-i}(\mu_{1},\ldots,\mu_{s})$ for all $0\leq i\leq
r$.

(iii) This part follows verbatim as part (ii), except that we need to use
[MULT_SAT_FIB_GOR_3, Lemma 2.12] and [MIXED_MULT, Theorem 5.8]. ∎

From Theorem 4.1, we have that the projective degrees and the degree of the
image of a rational map
$\mathcal{F}:{\mathbb{P}}_{A}^{r}\dashrightarrow{\mathbb{P}}_{A}^{s}$ behave
as lower semicontinuous functions under specialization; accordingly, these
invariants cannot increase under specialization. However, it turns out that
the degree of a rational map is a much more erratic invariant, and it seems
that Section missing 4 is the most general result one can hope for. Indeed,
the following two examples show that the degree of a rational map can either
increase or decrease under specialization.

######  Example  (The degree of a rational map can increase).

Let $\mathbb{Q}$ be the field of rational numbers, and $A=\mathbb{Q}[a]$ and
$S=A[x_{0},x_{1},x_{2}]$ be polynomial rings. Consider the following matrix

$\mathcal{M}=\begin{pmatrix}{x}_{0}&{x}_{1}&{x}_{0}^{2}\\\
{x}_{1}&{x}_{0}&{x}_{1}^{2}\\\ {x}_{0}+a\,{x}_{2}&{x}_{1}&{x}_{2}^{2}\\\
0&{x}_{0}&0\end{pmatrix},$

and let $\mathcal{F}:{\mathbb{P}}_{A}^{2}\dashrightarrow{\mathbb{P}}_{A}^{3}$
be a rational map with base ideal $I=(f_{0},f_{1},f_{2},f_{3})\subset S$ given
by

$I=I_{3}(\mathcal{M})={\small\begin{pmatrix}{x}_{0}^{2}{x}_{1}^{2}+a\,{x}_{0}{x}_{1}^{2}{x}_{2}-{x}_{0}{x}_{1}{x}_{2}^{2},\\\
-{x}_{0}^{4}-a\,{x}_{0}^{3}{x}_{2}+{x}_{0}^{2}{x}_{2}^{2},\\\
{x}_{0}^{3}{x}_{1}-{x}_{0}^{2}{x}_{1}^{2},\\\
{x}_{0}^{4}-{x}_{0}^{2}{x}_{1}^{2}+a\,{x}_{0}^{3}{x}_{2}-a\,{x}_{1}^{3}{x}_{2}-{x}_{0}^{2}{x}_{2}^{2}+{x}_{1}^{2}{x}_{2}^{2}\end{pmatrix}}.$

Let $\mathscr{T}=S\otimes_{A}A[y_{0},y_{1},y_{2},y_{3}]$ be a standard
bigraded polynomial ring over $A$ and write the Rees algebra $\mathscr{R}(I)$
as $\mathscr{R}(I)\cong\mathscr{T}/\mathcal{J}$. The ideal
$\mathcal{J}\subset\mathscr{T}$ is equal to

$\small\begin{pmatrix}\mathtt{{x}_{1}{y}_{0}+{x}_{0}{y}_{1}+{x}_{1}{y}_{2}+{x}_{0}{y}_{3}},\\\
\mathtt{{x}_{0}{y}_{0}+{x}_{1}{y}_{1}+\left({x}_{0}+a\,{x}_{2}\right){y}_{2}},\\\
\left({x}_{0}{x}_{1}-{x}_{1}^{2}\right){y}_{1}+\left({x}_{0}^{2}+a\,{x}_{0}{x}_{2}-{x}_{2}^{2}\right){y}_{2},\\\
\mathtt{{x}_{2}{y}_{0}^{2}+\left(-a\,{x}_{0}+a\,{x}_{1}-{x}_{2}\right){y}_{1}^{2}+\left(a^{2}+2\right){x}_{2}{y}_{0}{y}_{2}-a\,{x}_{1}{y}_{1}{y}_{2}+{x}_{2}{y}_{2}^{2}+\left(-a\,{x}_{0}+a\,{x}_{1}-{x}_{2}\right){y}_{1}{y}_{3}},\\\
{y}_{0}^{4}-2\,{y}_{0}^{2}{y}_{1}^{2}+{y}_{1}^{4}+\left(a^{2}+4\right){y}_{0}^{3}{y}_{2}-4\,{y}_{0}{y}_{1}^{2}{y}_{2}+a^{2}{y}_{1}^{3}{y}_{2}+\left(2\,a^{2}+6\right){y}_{0}^{2}{y}_{2}^{2}-2\,{y}_{1}^{2}{y}_{2}^{2}+\\\
\left(a^{2}+4\right){y}_{0}{y}_{2}^{3}+{y}_{2}^{4}-2\,{y}_{0}^{2}{y}_{1}{y}_{3}+2\,{y}_{1}^{3}{y}_{3}-4\,{y}_{0}{y}_{1}{y}_{2}{y}_{3}+2\,a^{2}{y}_{1}^{2}{y}_{2}{y}_{3}-2\,{y}_{1}{y}_{2}^{2}{y}_{3}+{y}_{1}^{2}{y}_{3}^{2}+a^{2}{y}_{1}{y}_{2}{y}_{3}^{2}\end{pmatrix}.$

This can be computed in a computer algebra system like Macaulay2 [MACAULAY2].
The ideal $\mathcal{J}\subset\mathscr{T}$ is generated by $5$ bihomogeneous
polynomials in $\mathscr{T}$. The first, second and fourth generators of
$\mathcal{J}$ are linear in the variables $x_{i}$. Let
$\mathbb{L}=\mathbb{Q}(a)={\text{Quot}}(A)$ and
$\mathbb{G}:{\mathbb{P}}_{\mathbb{L}}^{2}\dashrightarrow{\mathbb{P}}_{\mathbb{L}}^{3}$
be the generic rational map with base ideal
$I\otimes_{A}\mathbb{L}\subset\mathbb{L}[x_{0},x_{1},x_{2}]$. From [AB_INITIO,
Theorem 2.18], we can check that $\mathbb{G}$ is birational, i.e.,
$\deg(\mathbb{G})=1$. Alternatively, we give a short direct argument. We may
assume that $\mathbb{L}$ is algebraically closed. By considering the generator
$f_{2}=x_{0}^{3}x_{1}-x_{0}^{2}x_{1}^{2}=x_{0}^{2}x_{1}(x_{0}-x_{1})$ of $I$,
we obtain the morphism $h:D(f_{2})\rightarrow{\mathbb{P}}_{\mathbb{L}}^{3}$.
Notice that $D(f_{2})$ lies inside the affine patch
$\mathbb{A}_{\mathbb{L}}^{2}\subset{\mathbb{P}}_{\mathbb{L}}^{2}$ with
$x_{0}=1$. For any point
$p=(p_{0}:p_{1}:p_{2}:p_{3})\in{\mathbb{P}}_{\mathbb{L}}^{3}$ in the image of
$h$, if $(1,\alpha_{1},\alpha_{2})\in h^{-1}(p)$, then we get the following
linear system

$\begin{array}[]{cccc}p_{1}\alpha_{1}&+&p_{2}a\alpha_{2}&=-(p_{0}+p_{2})\\\
(p_{0}+p_{2})\alpha_{1}&&&=-(p_{1}+p_{3})\end{array}$

that is derived from the two linear syzygies in $\mathcal{M}$; this system has
a unique solution $(1,\alpha_{1},\alpha_{2})$ if the determinant
$-ap_{2}(p_{0}+p_{2})$ is non-zero. It follows that for any
$\alpha=(1,\alpha_{1},\alpha_{2})\in D(f_{2})\cap
D(af_{2}(f_{0}+f_{2}))=D(f_{2}(f_{0}+f_{2}))$, the fiber $h^{-1}(h(\alpha))$
has one element. Thus $\mathbb{G}$ is birational.

On the other hand, we make the specialization $a=0$, which gives the matrix

$M=\begin{pmatrix}{x}_{0}&{x}_{1}&{x}_{0}^{2}\\\
{x}_{1}&{x}_{0}&{x}_{1}^{2}\\\ {x}_{0}&{x}_{1}&{x}_{2}^{2}\\\
0&{x}_{0}&0\end{pmatrix}.$

Let
$\mathbb{g}:{\mathbb{P}}_{\mathbb{k}}^{2}\dashrightarrow{\mathbb{P}}_{\mathbb{k}}^{3}$
be a rational map with base ideal

$J=I_{3}(M)=\left(-{x}_{0}^{4}+{x}_{0}^{2}{x}_{1}^{2}+{x}_{0}^{2}{x}_{2}^{2}-{x}_{1}^{2}{x}_{2}^{2},\,{x}_{0}^{3}{x}_{1}-{x}_{0}^{2}{x}_{1}^{2},\,{x}_{0}^{4}-{x}_{0}^{2}{x}_{2}^{2},\,{x}_{0}^{2}{x}_{1}^{2}-{x}_{0}{x}_{1}{x}_{2}^{2}\right).$

In this case $\mathbb{g}$ is not birational, indeed $\deg(\mathbb{g})=2$.

Therefore, under the above specialization, we obtain
$\deg(\mathbb{g})=2>1=\deg(\mathbb{G})$.

######  Example  (The degree of a rational map usually decreases).

We recall an example [SPECIALIZATION_ARON, Example 6.5] where the degree of a
rational map can decrease arbitrarily under specialization. Let $m\geq 1$ be
an integer. Let $\mathbb{k}$ be a field, and $A=\mathbb{k}[a]$ and
$S=A[x_{0},x_{1},x_{2}]$ be polynomial rings. Consider the matrix

$\mathcal{M}=\begin{pmatrix}x&zy^{m-1}\\\ -y&zx^{m-1}+y^{m}\\\
az&zx^{m-1}\end{pmatrix}$

with entries in $S$. Let
$\mathcal{F}:{\mathbb{P}}_{A}^{2}\dashrightarrow{\mathbb{P}}_{A}^{2}$ be a
rational map with base ideal $I=I_{2}(\mathcal{M})\subset S$. Let
$\mathbb{L}=\mathbb{k}(a)={\text{Quot}}(A)$ and
$\mathbb{G}:{\mathbb{P}}_{\mathbb{L}}^{2}\dashrightarrow{\mathbb{P}}_{\mathbb{L}}^{2}$
be the generic rational map with base ideal
$I\otimes_{A}\mathbb{L}\subset\mathbb{L}[x_{0},x_{1},x_{2}]$. For any
$\beta\in\mathbb{k}$, let $\mathfrak{n}_{\beta}=(a-\beta)\in A$ and
$\mathbb{g}_{\beta}:{\mathbb{P}}_{\mathbb{k}}^{2}\dashrightarrow{\mathbb{P}}_{\mathbb{k}}^{2}$
be a rational map with base ideal
$I(\mathfrak{n}_{\beta})\subset\mathbb{k}[x_{0},x_{1},x_{2}]$. Then, we have
that $\deg(\mathbb{G})=m$ and

$\deg(\mathbb{g}_{\beta})=\begin{cases}1&\text{ if }\beta=0\\\ m&\text{ if
}\beta\neq 0.\end{cases}$

So, the specialization $a=0$ gives an arbitrary decrease in degree
$\deg(\mathbb{g}_{0})=1<m=\deg(\mathbb{G})$.

##  5\. Polar multiplicities, Segre numbers and new set of invariants

In this section, we prove several results regarding polar multiplicities and
Segre numbers of an ideal, and we introduce a new related invariant. These
invariants are defined as a special case of the general notion of polar
multiplicities due to Kleiman and Thorup [KLEIMAN_THORUP_GEOM,
KLEIMAN_THORUP_MIXED]. Here, an important goal for us is to extend several of
the results of Gaffney and Gassler [GG] from their analytic setting to an
algebraic one over a Noetherian local ring. The following setup is used
throughout this section.

######  Setup .

Let $(R,{\mathfrak{m}},\kappa)$ be a Noetherian local ring with maximal ideal
${\mathfrak{m}}$ and residue field $\kappa$. Let $d:=\dim(R)$ and
$X:={\text{Spec}}(R)$. Let $I\subset R$ be a proper ideal generated by
elements $f_{1},\ldots,f_{m}\in R$. We consider the Rees algebra
$B:=\mathscr{R}(I):=R[IT]=\bigoplus_{v\geq 0}I^{v}T^{v}\subset R[T]$ of the
ideal $I$. We have a natural homogeneous presentation
$W:=R[y_{1},\ldots,y_{m}]\twoheadrightarrow\mathscr{R}(I)$, $y_{i}\mapsto
f_{i}T$, where $W$ is a standard graded polynomial ring over $R$. Let $P:={\rm
Bl}_{I}(X)=\text{Proj}(B)\subset\text{Proj}(W)={\mathbb{P}}_{R}^{m-1}$ be the
blowup of $X$ along $I$ and consider the natural projection

$\pi:P\subset{\mathbb{P}}_{R}^{m-1}\rightarrow X.$

Let $E:=\pi^{-1}(V(I))\cong\text{Proj}(G)\subset P$ be the exceptional divisor
and $G:=\text{gr}_{I}(R):=\bigoplus_{v\geq 0}I^{v}/I^{v+1}$ be the
corresponding associated graded ring. The blowup $P$ has a natural affine open
cover $P=\bigcup_{i=1}^{m}U_{i}$ where
$U_{i}={\text{Spec}}\left(\big{[}B_{y_{i}}\big{]}_{0}\right)$. More precisely,
we can write

$U_{i}={\text{Spec}}\left(R[I/f_{i}]\right),$

where $R[I/f_{i}]$ denotes the $R$-subalgebra of $R_{f_{i}}$ generated by all
$f/f_{i}$ with $f\in I$. Notice that the local equation of $E$ on $U_{i}$ is
given by $f_{i}\in R[I/f_{i}]$.

We now briefly recall the general notion of polar multiplicities due to
Kleiman and Thorup [KLEIMAN_THORUP_GEOM, KLEIMAN_THORUP_MIXED]. Here we shall
freely use the results from the references [KLEIMAN_THORUP_GEOM,
KLEIMAN_THORUP_MIXED, cidruiz2024polar] regarding polar multiplicities. Let
$M$ be a finitely generated graded $B$-module and $\mathcal{F}=\widetilde{M}$
the corresponding coherent $\mathcal{O}_{P}$-module. The function
$(v,n)\mapsto{\rm length}_{R}\left(M_{v}/{\mathfrak{m}}^{n+1}M_{v}\right)$
eventually coincides with a bivariate polynomial $P_{M}(v,n)$ of degree equal
to $\dim(\text{Supp}(\mathcal{F}))$. Then, for all
$r\geq\dim(\text{Supp}(\mathcal{F}))$, we can write

$P_{M}(v,n)\;=\;\sum_{i=0}^{r}\;\frac{m_{r}^{i}(M)}{i!\,(r-i)!}\;v^{r-i}n^{i}\;+\;\text{(lower
degree terms)}.$

We say that the invariants $m_{r}^{i}(M)$ are the _polar multiplicities_ of
$M$. Recall that $\dim(P)\leq d$ and $\dim(E)\leq d-1$. Our main interest is
on the following invariants:

######  Definition .

  1. (i)

For all $0\leq i\leq d$, we say that $m_{i}(I,R):=m_{d}^{d-i}(B)$ is the _$i$
-th polar multiplicity_ of the ideal $I\subset R$.

  2. (ii)

For all $1\leq i\leq d$, we say that $c_{i}(I,R):=m_{d-1}^{d-i}(G)$ is the
_$i$ -th Segre number_ of the ideal $I\subset R$.

  3. (iii)

For all $1\leq i\leq d$, we say that $\nu_{i}(I,R):=m_{i}(I,R)+c_{i}(I,R)$ is
the _$i$ -th polar-Segre multiplicity_ of the ideal $I\subset R$. By
convention, we also set $\nu_{0}(I,R)=m_{0}(I,R)$.

By [cidruiz2024polar, Proposition 2.10], we get
$m_{d}(I,R)=m_{d}^{0}(B)=j_{d+1}(B)$, and since $\dim(B/{\mathfrak{m}}B)\leq
d$, it follows that $m_{d}(I,R)=j_{d+1}(B)=0$ (see [FLENNER_O_CARROLL_VOGEL,
§6.1]). For all $0\leq i\leq d-1$, the polar multiplicity $m_{i}(I,R)$ is also
referred to as the _mixed multiplicity_ $e_{i}({\mathfrak{m}}\mid I)$ (see,
e.g., [Trung2001]).

We shall need some very basic rudiments from intersection theory. Since we are
working over our Noetherian local ring $R$ (and not over a field), the usual
developments from Fulton’s book [FULTON_INTERSECTION_THEORY] do not suffice.
In terms of a suitable _dimension function_ , we could use available
extensions of intersection theory (see, e.g., [FULTON_INTERSECTION_THEORY,
Chapter 20], [stacks-project, Chapter 02P3], [THORUP]). However, as we shall
not require a notion of rational equivalence, we present our results in terms
of cycles and quickly develop the necessary concepts.

######  Notation .

  1. (i)

Let $Y$ be a Noetherian scheme. We denote by $Z_{k}(Y)$ the free group of
$k$-dimensional cycles. For a coherent sheaf $\mathcal{F}$ on $Y$ and an
integer $k\geq\dim(\text{Supp}(\mathcal{F}))$, we denote by
$\big{[}\mathcal{F}\big{]}_{k}\in Z_{k}(Y)$ the associated $k$-cycle. For a
closed subscheme $Z\subset Y$ and an integer $k\geq\dim(Z)$, we denote by
$\big{[}Z\big{]}_{k}=\big{[}\mathcal{O}_{Z}\big{]}_{k}\in Z_{k}(Y)$ the
associated $k$-cycle.

  2. (ii)

Given $k$-cycle $\xi=\sum_{i}l_{i}[R/\mathfrak{p}_{i}]\in Z_{k}(X)$, its
multiplicity is given by $e_{k}(\xi):=\sum_{i}l_{i}e_{k}(R/\mathfrak{p}_{i})$.

Below we give a short self-contained result regarding the push-forward of
cycles along a projection (cf., [stacks-project, Lemma 02R6], [THORUP,
Proposition 4.3]).

######  Definition-Proposition .

Let $S$ be a standard graded $R$-algebra and consider the projective morphism
$\eta:Y=\text{Proj}(S)\rightarrow X={\text{Spec}}(R)$. Given an integral
closed subscheme $Z\subset Y$, the push-forward is defined as

$\eta_{*}\left([Z]\right)\;:=\;\begin{cases}\big{[}K(Z):K(Z^{\prime})\big{]}\cdot[Z^{\prime}]&\quad\text{
if }\dim(Z)=\dim(Z^{\prime})\\\ 0&\quad\text{ otherwise},\end{cases}$

where $Z^{\prime}=\eta(Z)$, and $K(Z)$ and $K(Z^{\prime})$ denote the function
fields of $Z$ and $Z^{\prime}$, respectively. The push-forward map
$\eta_{*}:Z_{k}(Y)\rightarrow Z_{k}(X)$ is then determined by linearity. Let
$\mathcal{F}$ be a coherent $\mathcal{O}_{Y}$-module and
$k\geq\dim(\text{Supp}(\mathcal{F}))$. Then
$\eta_{*}\left(\big{[}\mathcal{F}\big{]}_{k}\right)=\big{[}\eta_{*}(\mathcal{F})\big{]}_{k}\in
Z_{k}(X).$

###### Proof.

Let $Z=\text{Proj}(S/\mathfrak{P})$ where $\mathfrak{P}\subset S$ is a
relevant prime ideal, and set
$Z^{\prime}=\eta(Z)={\text{Spec}}(R/\mathfrak{p})$ where
$\mathfrak{p}=\mathfrak{P}\cap R$. From [vasconcelos1994arithmetic, Lemma
1.2.2], we get $\dim(S/\mathfrak{P})=\dim(R/\mathfrak{p})+{\rm
trdeg}_{R/\mathfrak{p}}(S/\mathfrak{P})$. Since we have ${\rm
trdeg}_{R/\mathfrak{p}}(S/\mathfrak{P})\geq 1$, it follows that
$\dim(Z)\geq\dim(Z^{\prime})$. Let $M$ be a finitely generated graded
$S$-module with $\mathcal{F}\cong\widetilde{M}$ and
$\text{H}_{S_{+}}^{0}(M)=0$. Notice that we may substitute $S$ by
$S/\text{Ann}_{S}(M)$ and $R$ by $R/(R\cap\text{Ann}_{S}(M))$. Therefore we
assume that $k\geq\dim(Y)$ and $k\geq\dim(R)$, and that any minimal prime of
$S$ is relevant.

Let $\mathfrak{p}\subset R$ be a minimal prime of
$N=\text{H}^{0}(X,\eta_{*}(\mathcal{F}))$ of dimension $k$. Notice that any
relevant prime of $S$ contracting to $\mathfrak{p}$ should be minimal. Thus
the fiber $\eta^{-1}(\mathfrak{p})$ is finite, and so we can find an affine
open neighborhood $V\subset X$ of $\mathfrak{p}$ such that
$\eta^{-1}(V)\rightarrow V$ is finite (see [stacks-project, Lemma 02NW],
[HARTSHORNE, Exercise II.3.7]). We set $\eta^{-1}(V)={\text{Spec}}(A)$ and
choose a finitely generated $A$-module $L$ such that
$\widetilde{L}\cong{\mathcal{F}\mid}_{\eta^{-1}(V)}$. Since $R_{\mathfrak{p}}$
is an Artinian local ring and $A_{\mathfrak{p}}=A\otimes_{R}R_{\mathfrak{p}}$
is module-finite over $R_{\mathfrak{p}}$, it follows that $A_{\mathfrak{p}}$
is Artinian. We have the equality

$\sum_{\mathfrak{q}}[\kappa(\mathfrak{q}):\kappa(\mathfrak{p})]\cdot{\rm
length}_{A_{\mathfrak{q}}}(L_{\mathfrak{q}})\;=\;{\rm
length}_{R_{\mathfrak{p}}}(N_{\mathfrak{p}}),$

where the sum runs through the minimal primes of $A$ contracting to
$\mathfrak{p}$. Finally, notice that the right-hand side of the above equality
is the coefficient in $[\eta_{*}(\mathcal{F})]_{k}$ corresponding to
$\mathfrak{p}$ and that the left-hand side is the coefficient in
$\eta_{*}([\mathcal{F}]_{k})$ corresponding to $\mathfrak{p}$. ∎

######  Notation .

Given elements $a_{1},\ldots,a_{m}$ in $R$, say that
$g=a_{1}f_{1}+\cdots+a_{m}f_{m}$ is the associated element in $I$, that
$\ell=a_{1}y_{1}+\cdots+a_{m}y_{m}\in B_{1}$ is the associated linear form,
that $D=V(g)\subset X$ is the associated hypersurface in $X$, and that
$H=V_{+}(\ell)\subset P$ is the associated hyperplane in $P$. We denote by
$\pi^{*}D:=V_{+}(gB)=\text{Proj}(B/gB)$ the pullback of the hypersurface $D$.
When the residue field $\kappa$ is infinite, we use the following conventions:

  * –

We say that $D\subset X$ is general (equivalently $H\subset P$ is general) if
$g$ is a general element in $I$ (equivalently $\ell$ is a general element in
$B_{+}$).

  * –

We say that a sequence of hyperplanes $\underline{H}=H_{1},\ldots,H_{k}$ is _a
sequence of general hyperplanes_ in $P$ if the associated sequence
$\underline{g}=g_{1},\ldots,g_{k}$ is a sequence of general elements in $I$.

When $\kappa$ is infinite and $g\in I$ is a general element, the following
remark shows that $\pi^{*}D$ is an effective Cartier divisor on $P$ even if
$D=V(g)$ is not a divisor on $X$.

######  Remark .

($\kappa$ infinite). Let $\overline{R}=R/(0:_{R}I^{\infty})$ and
$\overline{X}={\text{Spec}}(\overline{R})$. Since ${\rm Bl}_{I}(X)\cong{\rm
Bl}_{I}(\overline{X})$, we may assume that $(0:_{R}I^{\infty})=0$, and so by
prime avoidance we get that $g$ is a nonzerodivisor when $g\in I$ is general.
This shows that $\pi^{*}D$ is an effective Cartier divisor when $g\in I$ is
general.

The next notation includes an inequality that will be useful in our approach.
This inequality is related to when a general element of $I$ is a
$\mathscr{G}$-parameter on the Rees algebra $B=\mathscr{R}(I)$ (in the sense
of [cidruiz2024polar, Definition 2.7]).

######  Notation .

We say that the _order_ of the ideal $I$ is given by
$o(I):=\sup\\{\beta\in{\mathbb{Z}}_{\geq 0}\mid
I\subseteq{\mathfrak{m}}^{\beta}\\}$. Let $\delta=o(I)$. Consider the standard
bigraded algebra $\mathscr{G}:=\text{gr}_{\mathfrak{m}}(B)$ with bigraded
parts
$[\mathscr{G}]_{(v,n)}={\mathfrak{m}}^{n}B_{v}/{\mathfrak{m}}^{n+1}B_{v}$.
Take the $\kappa$-vector subspace
$\mathfrak{b}=I/{\mathfrak{m}}^{\delta+1}\subset{\mathfrak{m}}^{\delta}/{\mathfrak{m}}^{\delta+1}=[\mathscr{G}]_{(0,\delta)}$.
Let

$\text{in}^{\delta}(I)\;:=\;\mathfrak{b}\cdot\mathscr{G}\;=\;\left(\text{in}(f)\mid
f\in I\text{ and }o(f)=\delta\right)\subset\mathscr{G}$

be the ideal generated by the initial forms of elements in $I$ of order
$\delta$. If the following strict inequality

$\dim\left({\text{BiProj}}\left(\mathscr{G}/\text{in}^{\delta}(I)\right)\right)\;<\;d-1$

holds, we say that $I$ satisfies the _$\mathscr{G}$ -parameter condition
generically_.

The following proposition is inspired by one of the technical steps in
Fulton’s proof of the commutativity of intersecting with Cartier divisors:
indeed, similarly to [FULTON_INTERSECTION_THEORY, Lemma 2.4], we express the
pullback $\pi^{*}D$ as the sum of the exceptional divisor $E$ and a hyperplane
$H$ in the blowup $P={\rm Bl}_{I}(X)$. As a consequence, we get inequalities
relating the polar multiplicities and the Segre numbers of $I$.

######  Proposition .

($\kappa$ infinite). Let $H\subset P$ be a general hyperplane, $D=V(g)$ be the
associated hypersurface and $\ell\in B_{+}$ be the associated linear form.
Then the following statements hold:

  1. (i)

$\pi^{*}D$ and $H$ are effective Cartier divisors.

  2. (ii)

$\pi^{*}D=E+H$.

  3. (iii)

Let $\delta=o(I)$. Then we have the inequality

$\delta\cdot m_{i-1}(I,R)\;\leq\;m_{i}(I,R)+c_{i}(I,R)\;=\;\nu_{i}(I,R),$

and equality holds for all $1\leq i\leq d$ if and only if $I$ satisfies the
$\mathscr{G}$-parameter condition generically (see Section missing 5).

###### Proof.

(i) From Section missing 5, we get that $\pi^{*}D$ is an effective Cartier
divisor. We have that $H$ is also an effective Cartier divisor by prime
avoidance.

(ii) We write $g=a_{1}f_{1}+\cdots+a_{m}f_{m}$ and
$\ell=a_{1}y_{1}+\cdots+a_{m}y_{m}$. Consider the affine open subscheme
$U_{i}={\text{Spec}}(R[I/f_{i}])$. As we mentioned before, the local equation
of $E$ on $U_{i}$ is given by $f_{i}\in R[I/f_{i}]$. On the other hand, the
local equations of $\pi^{*}D$ and $H$ on $U_{i}$ are given by

$a_{1}f_{1}+\cdots+a_{m}f_{m}\;\in\;R[I/f_{i}]\qquad\text{ and
}\qquad\frac{a_{1}f_{1}+\cdots+a_{m}f_{m}}{f_{i}}\;\in\;R[I/f_{i}],$

respectively. This shows the equality $\pi^{*}D=E+H$ on each $U_{i}$, and so
the equality holds globally on the whole blowup $P={\rm Bl}_{I}(X)$.

(iii) First, we check that in the vacuous case $\dim(P)<d$, we have that
$m_{i}(I,R)=0$, $\nu_{i}(I,R)=0$, $\dim(B)\leq d$ and
$\dim\left({\text{BiProj}}(\mathscr{G})\right)\leq d-2$ (hence the equivalence
statement holds trivially). Therefore, we assume $\dim(P)=d$.

As $g\in I$ is general, we may assume that
$g\in{\mathfrak{m}}^{\delta}\setminus{\mathfrak{m}}^{\delta+1}$. Notice that
$\big{[}\pi^{*}D\big{]}_{d-1}=\big{[}\widetilde{B/gB}\big{]}_{d-1}$ and
$\big{[}H\big{]}_{d-1}=\big{[}\widetilde{B/\ell B}\big{]}_{d-1}$, and thus
part (ii) yields the equality of cycles

$\big{[}\widetilde{B/gB}\big{]}_{d-1}\;=\;\big{[}\widetilde{B/\ell
B}\big{]}_{d-1}+\big{[}\widetilde{G}\big{]}_{d-1}\;\in\;Z_{d-1}(P);$

see, e.g., [FULTON_INTERSECTION_THEORY, Lemma A.2.5]. Hence the additivity of
polar multiplicities (see [cidruiz2024polar, Corollary 2.6]) implies that

$m_{d-1}^{i}(B/gB)\;=\;m_{d-1}^{i}(B/\ell B)+m_{d-1}^{i}(G).$

Due to [cidruiz2024polar, Theorem 2.8], we have
$m_{d-1}^{i}(B/gB)\geq\delta\cdot m_{d}^{i+1}(B)$ and an equality holds for
all $0\leq i\leq d-1$ if and only if $g$ is a $\mathscr{G}$-parameter on $B$
(in the sense of [cidruiz2024polar, Definition 2.7]). Since $\ell\in B_{+}$ is
a general element, by utilizing [cidruiz2024polar, Proposition 2.10], we
obtain $m_{d-1}^{i}(B/\ell B)=m_{d}^{i}(B)$. Finally, by combining everything
we get the inequality

$\delta\cdot m_{i-1}(I,R)\;=\;\delta\cdot
m_{d}^{d-i+1}(B)\;\leq\;m_{d}^{d-i}(B)+m_{d-1}^{d-i}(G)\;=\;m_{i}(I,R)+c_{i}(I,R),$

and an equality holds for all $1\leq i\leq d$ if and only if $g$ is a
$\mathscr{G}$-parameter on $B$. Since $g$ is a general element of $I$, it
follows that $g$ is a $\mathscr{G}$-parameter on $B$ if and only if $I$
satisfies the $\mathscr{G}$-parameter condition generically. ∎

######  Remark .

The equality of Section missing 5(ii) can also be derived as follows. Consider
the extended Rees algebra $\mathscr{R}^{+}(I):=R[IT,T^{-1}]$. Notice that the
equality $g=\ell\cdot T^{-1}$ holds in $\mathscr{R}^{+}(I)$ and recall the
isomorphism
$\text{gr}_{I}(R)\cong\mathscr{R}^{+}(I)/T^{-1}\mathscr{R}^{+}(I)$.

Next, we introduce the notion of polar schemes and Segre cycles. We also
introduce a new type of cycle that will be fundamental in our treatment. The
following additional data is fixed for the rest of the section.

######  Setup .

Assume Section missing 5 and that the residue field $\kappa$ is infinite. Let
$\underline{H}=H_{1},\ldots,H_{d}$ be a sequence of general hyperplanes, and
denote by $\underline{g}=g_{1},\ldots,g_{d}$ the associated sequence of
elements in $I$ and by $\underline{\ell}=\ell_{1},\ldots,\ell_{d}$ the
associated sequence of linear forms in $B_{+}$. We also set
$D_{i}=V(g_{i})\subset X$ and recall that the pullback $\pi^{*}D_{i}$ is an
effective Cartier divisor on $P$ (see Section missing 5). We introduce the
following objects:

  1. (i)

For $1\leq i\leq d$, we say that the _$i$ -th polar scheme_ (with respect to
$\underline{H}$) is given by the following schematic-image

$P_{i}(I,X)\;=\;P_{i}^{\underline{H}}(I,X)\;:=\;\pi\left(H_{1}\cap\cdots\cap
H_{i}\right).$

  2. (ii)

For $1\leq i\leq d$, we say that the _$i$ -th Segre cycle_ (with respect to
$\underline{H}$) is given by

$\Lambda_{i}(I,X)\;=\;\Lambda_{i}^{\underline{H}}(I,X)\;:=\;\pi_{*}\left(\big{[}E\cap
H_{1}\cap\cdots\cap H_{i-1}\big{]}_{d-i}\right)\;\in\;Z_{d-i}(X).$

  3. (iii)

For $1\leq i\leq d$, we say that the _$i$ -th polar-Segre cycle_ (with respect
to $H$) is given by

$V_{i}(I,X)\;=\;V_{i}^{\underline{H}}(I,X)\;:=\;\pi_{*}\left(\big{[}H_{1}\cap\cdots\cap
H_{i-1}\cap\pi^{*}D_{i}\big{]}_{d-i}\right)\;\in\;Z_{d-i}(X).$

By convention, we set $P_{0}(I,X):=\pi(P)$ and
$V_{0}(I,X):=\left[P_{0}(I,X)\right]_{d}\in Z_{d}(X)$.

######  Remark .

By prime avoidance, we can assume that $g_{1},\ldots,g_{i}$ is a regular
sequence on $R_{\mathfrak{p}}$ for each $\mathfrak{p}\in
V(g_{1},\ldots,g_{i})\setminus V(I)$ (we say that $g_{1},\ldots,g_{i}$ are a
filter-regular sequence with respect to $I$; see
[STUCKRAD_VOGEL_BUCHSBAUM_RINGS, Appendix]). Therefore, we have that
$(g_{1},\ldots,g_{i}):_{R}I^{\infty}$ either equals $R$ or it has height $i$,
and so we obtain $\dim(R/(g_{1},\ldots,g_{i}):_{R}I^{\infty})\leq d-i$.
Similarly, we get $\dim(R/(g_{1},\ldots,g_{i-1}):_{R}I^{\infty}+g_{i}R)\leq
d-i$.

The theorem below gives an important description of the invariants
$m_{i}(I,R)$, $c_{i}(I,R)$ and $\nu_{i}(I,R)$. It shows that these invariants
are naturally the multiplicities of the cycles introduced in Section missing
5.

######  Theorem 5.1.

Assume Section missing 5. Then the following statements hold:

  1. (i)

For all $0\leq i\leq d$, we have the equalities
$m_{i}(I,R)=e_{d-i}\left(P_{i}(I,X)\right)$ and

$P_{i}(I,X)\;=\;{\text{Spec}}\left(R/(g_{1},\ldots,g_{i}):_{R}I^{\infty}\right).$

  2. (ii)

For all $1\leq i\leq d$, we have the equalities
$c_{i}(I,R)=e_{d-i}\left(\Lambda_{i}(I,X)\right)$ and

$\Lambda_{i}(I,X)\;=\;\sum_{\begin{subarray}{c}\mathfrak{p}\in
V\left((g_{1},\ldots,g_{i-1}):_{R}I^{\infty}\right)\\\ \mathfrak{p}\in
V(I),\;\dim(R/\mathfrak{p})=d-i\end{subarray}}e\big{(}I,\,R_{\mathfrak{p}}/(g_{1},\ldots,g_{i-1})R_{\mathfrak{p}}:_{R_{\mathfrak{p}}}I^{\infty}R_{\mathfrak{p}}\big{)}\cdot\left[R/\mathfrak{p}\right]\;\in\;Z_{d-i}(X).$

  3. (iii)

For all $1\leq i\leq d$, we have the equalities
$\nu_{i}(I,R)=e_{d-i}(V_{i}(I,X))$ and

$V_{i}(I,X)\;=\;\left[P_{i}(I,X)\right]_{d-i}+\Lambda_{i}(I,X)\;=\;\big{[}{\text{Spec}}\left(R/(g_{1},\ldots,g_{i-1}):_{R}I^{\infty}+g_{i}R\right)\big{]}_{d-i}\;\in\,Z_{d-i}(X).$

###### Proof.

Let $\mathfrak{a}_{i}:=(g_{1},\ldots,g_{i})\subset R$,
$R_{i}:=R/\mathfrak{a}_{i}$, $X_{i}:={\text{Spec}}(R_{i})$ and
$\overline{X}_{i}:={\text{Spec}}(R/\mathfrak{a}_{i}:_{R}I^{\infty})$.

(i) First, set $i=0$. By [cidruiz2024polar, Proposition 2.10], we have
$m_{0}(I,R)=m_{d}^{d}(B)=e_{d}\left(\text{H}^{0}(P,\mathcal{O}_{P})\right)$.
We have that $P_{0}(I,X)=\pi(P)={\text{Spec}}(R/\mathfrak{a})$ where
$\mathfrak{a}\subset R$ is the kernel of the natural map $R\xrightarrow{\;\rm
nat\;}\text{H}^{0}(P,\mathcal{O}_{P})$ (see, e.g., [GORTZ_WEDHORN, Proposition
10.30], [HARTSHORNE, Exercise II.3.11]). On the other hand, we have a four-
term exact sequence

$0\;\longrightarrow\;\big{[}\text{H}_{B_{+}}^{0}(B)\big{]}_{0}\;\longrightarrow\;R=B_{0}\;\xrightarrow{\;\rm
nat\;}\;\text{H}^{0}(P,\mathcal{O}_{P})\;\longrightarrow\;\big{[}\text{H}_{B_{+}}^{1}(B)\big{]}_{0}\;\longrightarrow\;0.$

Since $B=R[IT]$ is the Rees algebra of $I$, it follows that
$\big{[}\text{H}_{B_{+}}^{0}(B)\big{]}_{0}=0:_{R}I^{\infty}$. Hence we have
$\mathfrak{a}=0:_{R}I^{\infty}$ and
$P_{0}(I,X)=\pi(P)={\text{Spec}}(R/0:_{R}I^{\infty})$. Moreover, we get the
short exact sequence

$0\rightarrow
R/0:_{R}I^{\infty}\rightarrow\text{H}^{0}(P,\mathcal{O}_{P})\rightarrow\big{[}\text{H}_{B_{+}}^{1}(B)\big{]}_{0}\rightarrow
0,$

and so to prove $m_{0}(I,R)=e_{d}(P_{0}(I,X))$, it suffices to show that
$\dim\big{(}\big{[}\text{H}_{B_{+}}^{1}(B)\big{]}_{0}\big{)}<d$. Let
$\overline{B}:=B/\text{H}_{B_{+}}^{0}(B)$. As
$\text{H}_{B_{+}}^{1}(B)\cong\text{H}_{B_{+}}^{1}(\overline{B})$ and
$\left[\overline{B}\right]_{0}=R/0:_{R}I^{\infty}$, we have that
$\text{H}_{B_{+}}^{1}(B)\otimes_{R}R_{\mathfrak{p}}=0$ for any minimal prime
$\mathfrak{p}\in{\rm Min}(R)$ that contains $I$. However, if a prime
$\mathfrak{p}\in{\text{Spec}}(R)$ does not contain $I$, we obtain

$\big{[}\text{H}_{B_{+}}^{1}(B)\big{]}_{0}\otimes_{R}R_{\mathfrak{p}}\;\cong\;\big{[}\text{H}_{B_{+}}^{1}(B\otimes_{R}R_{\mathfrak{p}})\big{]}_{0}\;\cong\;\big{[}\text{H}_{T}^{1}\left(R_{\mathfrak{p}}[T]\right)\big{]}_{0}=0.$

This settles the claim that
$\dim\big{(}\big{[}\text{H}_{B_{+}}^{1}(B)\big{]}_{0}\big{)}<d$, and so the
proof is complete for the case $i=0$.

Since $\ell_{1},\ldots,\ell_{i}$ is a sequence of general elements in $B_{+}$,
[cidruiz2024polar, Proposition 2.10] yields the equalities
$m_{i}(I,R)=m_{d}^{d-i}(B)=m_{d-i}^{d-i}(B/(\ell_{1},\ldots,\ell_{i})B)=e_{d-i}\left(\text{H}^{0}(P,\mathcal{O}_{H_{1}\cap\cdots\cap
H_{i}})\right)$. Consider the natural specialization map

$\mathfrak{s}\;:\;B/(\ell_{1},\ldots,\ell_{i})B\;\cong\;\bigoplus_{v\geq
0}I^{v}/\mathfrak{a}_{i}I^{v-1}\;\;\;\twoheadrightarrow\;\;\;\mathscr{R}_{R_{i}}(IR_{i})\;\cong\;\bigoplus_{v\geq
0}I^{v}/(I^{v}\cap\mathfrak{a}_{i}).$

Since $g_{1},\ldots,g_{i}$ is a sequence of general elements in $I$, we may
assume that they form a superficial sequence for $I$ (see [huneke2006integral,
Proposition 8.5.7]), and so [huneke2006integral, Lemma 8.5.11] yields the
equality $\mathfrak{a}_{i}I^{v-1}=I^{v}\cap\mathfrak{a}_{i}$ for $v\gg 0$.
Therefore, we get the following equality (as schemes)

(2) $H_{1}\cap\cdots\cap
H_{i}\;=\;\text{Proj}(B/(\ell_{1},\ldots,\ell_{i})B)\;=\;\text{Proj}(\mathscr{R}_{R_{i}}(IR_{i}))\;=\;{\rm
Bl}_{I}(X_{i})\;=:\;P_{i}.$

Since we already dealt with the initial case, by substituting $X$ by $X_{i}$
and setting $i=0$, we obtain

$P_{i}(I,X)\;=\;\pi(H_{1}\cap\cdots\cap
H_{i})\;=\;\pi(P_{i})\;=\;P_{0}(I,X_{i})\;=\;{\text{Spec}}\left(R/(g_{1},\ldots,g_{i}):_{R}I^{\infty}\right)$

and

$m_{i}(I,R)\;=\;e_{d-i}\left(\text{H}^{0}(P,\mathcal{O}_{H_{1}\cap\cdots\cap
H_{i}})\right)\;=\;e_{d-i}\left(\text{H}^{0}\left(P_{i},\mathcal{O}_{P_{i}}\right)\right)\;=\;e_{d-i}\left(P_{0}(I,X_{i})\right).$

This completes the proof of part (i).

(ii) First, set $i=1$. By [cidruiz2024polar, Proposition 2.10], we get
$c_{1}(I,R)=m_{d-1}^{d-1}(G)=e_{d-1}\left(\text{H}^{0}(E,\mathcal{O}_{E})\right)$.
From Section missing 5, we have the equality of cycles

$\Lambda_{1}(I,X)\;=\;\big{[}\pi_{*}(\mathcal{O}_{E})\big{]}_{d-1}\;=\;\big{[}\text{H}^{0}(X,\pi_{*}(\mathcal{O}_{E}))\big{]}_{d-1}\;=\;\big{[}\text{H}^{0}(E,\mathcal{O}_{E})\big{]}_{d-1}\;\in\;Z_{d-1}(X),$

and so it follows that $c_{1}(I,R)=e_{d-1}(\Lambda_{1}(I,X))$. Since we proved
$\pi(P)={\text{Spec}}(R/0:_{R}I^{\infty})$ in part (i), it follows that
$\text{Supp}\left(\text{H}^{0}(E,\mathcal{O}_{E})\right)\subset
V(I,0:_{R}I^{\infty})$. Then we get the following equality

$\left[\text{H}^{0}(E,\mathcal{O}_{E})\right]_{d-1}\;=\;\sum_{\begin{subarray}{c}\mathfrak{p}\in
V(I,0:_{R}I^{\infty})\\\ \dim(R/\mathfrak{p})=d-1\end{subarray}}{\rm
length}_{R_{\mathfrak{p}}}\left(\text{H}^{0}(E,\mathcal{O}_{E})\otimes_{R}R_{\mathfrak{p}}\right)\cdot[R/\mathfrak{p}]\;\in\;Z_{d-1}(X).$

Fix a prime $\mathfrak{p}$ in the above summation. Let
$E_{\mathfrak{p}}:=\text{Proj}\big{(}\text{gr}_{IR_{\mathfrak{p}}}(R_{\mathfrak{p}})\big{)}$
be the exceptional divisor of ${\text{Spec}}(R_{\mathfrak{p}})$ along
$IR_{\mathfrak{p}}$. Notice that $\dim(R_{\mathfrak{p}})=1$ and
$\text{H}^{0}(E,\mathcal{O}_{E})\otimes_{R}R_{\mathfrak{p}}\cong\text{H}^{0}(E_{\mathfrak{p}},\mathcal{O}_{E_{\mathfrak{p}}})$.
As a consequence, for $v\gg 0$, we obtain

${\rm
length}_{R_{\mathfrak{p}}}\left(\text{H}^{0}(E_{\mathfrak{p}},\mathcal{O}_{E_{\mathfrak{p}}})\right)\;=\;{\rm
length}_{R_{\mathfrak{p}}}\left(\text{H}^{0}(E_{\mathfrak{p}},\mathcal{O}_{E_{\mathfrak{p}}}(v))\right)\;=\;{\rm
length}_{R_{\mathfrak{p}}}\left(I^{v}R_{\mathfrak{p}}/I^{v+1}R_{\mathfrak{p}}\right);$

the first equality holds for any $v\in{\mathbb{Z}}$ because
$\dim(E_{\mathfrak{p}})\leq 0$. On the other hand, we have

$e\left(I,R_{\mathfrak{p}}/0:_{R_{\mathfrak{p}}}I^{\infty}R_{\mathfrak{p}}\right)\;=\;{\rm
length}_{R_{\mathfrak{p}}}\left(I^{v}R_{\mathfrak{p}}/I^{v+1}R_{\mathfrak{p}}\right)\quad\text{
for }\quad v\gg 0.$

This completes the proof for the case $i=1$.

Since $\ell_{1},\ldots,\ell_{i-1}$ is a sequence of general elements in
$B_{+}$, [cidruiz2024polar, Proposition 2.10] gives
$c_{i}(I,R)=m_{d-1}^{d-i}(G)=m_{d-i}^{d-i}(G/(\ell_{1},\ldots,\ell_{i-1})G)=e_{d-i}\left(\text{H}^{0}(P,\mathcal{O}_{E\cap
H_{1}\cap\cdots\cap H_{i-1}})\right)$. From (missing) 2, we obtain the
following equality (as schemes)

(3) $E\cap H_{1}\cap\cdots\cap H_{i-1}\;=\;(H_{1}\cap\cdots\cap
H_{i-1})\times_{X}{\text{Spec}}(R/I)\;=\;P_{i-1}\times_{X}{\text{Spec}}(R/I)\;=\;E_{i-1}$

where $E_{i-1}=\text{Proj}(\text{gr}_{IR_{i-1}}(R_{i-1}))$ is the exceptional
divisor of $X_{i-1}$ along $IR_{i-1}$. Then the result of part (ii) follows.

(iii) We have the following equalities

$\displaystyle V_{i}(I,X)$
$\displaystyle\;=\;\pi_{*}\left(\left[H_{1}\cap\cdots\cap
H_{i-1}\cap\pi^{*}D_{i}\right]_{d-i}\right)$ by definition of $V_{i}(I,X)$
$\displaystyle\;=\;\pi_{*}\left(\left[P_{i-1}\cap\pi^{*}D_{i}\right]_{d-i}\right)$
by (missing) 2
$\displaystyle\;=\;\left[V_{1}(I,\overline{X}_{i-1})\right]_{d-i}$ by
definition of $V_{1}(I,\overline{X}_{i-1})$
$\displaystyle\;=\;\left[P_{1}(I,\overline{X}_{i-1})\right]_{d-i}+\left[\Lambda_{1}(I,\overline{X}_{i-1})\right]_{d-i}$
by applying Section missing 5 to $\overline{X}_{i-1}$
$\displaystyle\;=\;\left[P_{i}(I,X)\right]_{d-i}+\Lambda_{i}(I,X)$ by the
formulas of part (i) and (ii).

Therefore, to complete the proof of part (iii), we substitute $X$ by
$\overline{X}_{i-1}$ and assume that $i=1$. In particular, we may assume that
$(0:_{R}I^{\infty})=0$ and that $g_{1}$ is a nonzerodivisor. Set $g:=g_{1}$
and $D:=D_{1}$. By Section missing 5, we obtain

$V_{1}(I,X)=\big{[}\pi_{*}\left(\mathcal{O}_{\pi^{*}D}\right)\big{]}_{d-1}=\sum_{\begin{subarray}{c}\mathfrak{p}\in
V(g)\\\ \dim(R/\mathfrak{p})=d-1\end{subarray}}{\rm
length}_{R_{\mathfrak{p}}}\left(\text{H}^{0}(P,\mathcal{O}_{P}/g\mathcal{O}_{P})\otimes_{R}R_{\mathfrak{p}}\right)\cdot[R/\mathfrak{p}]\;\in\;Z_{d-1}(X).$

Let $\mathfrak{p}$ be a prime in the above summation, and let
$P_{\mathfrak{p}}:=\text{Proj}(B\otimes_{R}R_{\mathfrak{p}})$ the blowup of
${\text{Spec}}(R_{\mathfrak{p}})$ along $IR_{\mathfrak{p}}$. Then the
coefficient corresponding to $\mathfrak{p}$ is given by

${\rm
length}_{R_{\mathfrak{p}}}\left(\text{H}^{0}(P_{\mathfrak{p}},\mathcal{O}_{P_{\mathfrak{p}}}/g\mathcal{O}_{P_{\mathfrak{p}}})\right)\;=\;{\rm
length}_{R_{\mathfrak{p}}}\left(\text{H}^{0}(P_{\mathfrak{p}},\mathcal{O}_{P_{\mathfrak{p}}}/g\mathcal{O}_{P_{\mathfrak{p}}}(v))\right)\;=\;{\rm
length}_{R_{\mathfrak{p}}}\left(I^{v}R_{\mathfrak{p}}/gI^{v}R_{\mathfrak{p}}\right)$

for $v\gg 0$. Expressing length as a multiplicity (see
[FLENNER_O_CARROLL_VOGEL, Corollary 1.2.14]) and utilizing the additivity of
multiplicities give the following

${\rm
length}_{R_{\mathfrak{p}}}\left(I^{v}R_{\mathfrak{p}}/gI^{v}R_{\mathfrak{p}}\right)\;=\;e\left((g),I^{v}R_{\mathfrak{p}}\right)\;=\;e\left((g),R_{\mathfrak{p}}\right)\;=\;{\rm
length}_{R_{\mathfrak{p}}}\left(R_{\mathfrak{p}}/gR_{\mathfrak{p}}\right).$

Hence the required equality
$V_{1}(I,X)=\left[{\text{Spec}}(R/gR)\right]_{d-1}$ follows. This completes
the proof of the last part of the theorem. ∎

######  Remark .

Due to Theorem 5.1, we obtain the following consequences:

  1. (i)

We have the equations

    1. (a)

$\big{[}P_{i}(I,X)\big{]}_{d-i}\;=\;\big{[}P_{1}\left(I,P_{i-1}(I,X)\right)\big{]}_{d-i}$.

    2. (b)

$\Lambda_{i}(I,X)\;=\;\big{[}\Lambda_{1}\left(I,P_{i-1}(I,X)\right)\big{]}_{d-i}$.

    3. (c)

$V_{i}(I,X)\;=\;\big{[}V_{1}\left(I,P_{i-1}(I,X)\right)\big{]}_{d-i}$.

  2. (ii)

We recover the formula of [Trung2001, Theorem 3.4] for the mixed
multiplicities of an ideal

$m_{i}(I,R)\;=\;e_{d-i}\left(R/(g_{1},\ldots,g_{i}):_{R}I^{\infty}\right).$

  3. (iii)

The new invariants (i.e., polar-Segre multiplicities) have the following
formula

$\nu_{i}(I,R)\;=\;e_{d-i}\left(R/(g_{1},\ldots,g_{i-1}):_{R}I^{\infty}+g_{i}R\right).$

  4. (iv)

We obtain the following formula for Segre numbers

$c_{i}(I,R)\;=\;\sum_{\begin{subarray}{c}\mathfrak{p}\in
V\left((g_{1},\ldots,g_{i-1}):_{R}I^{\infty}\right)\\\ \mathfrak{p}\in
V(I),\;\dim(R/\mathfrak{p})=d-i\end{subarray}}e\big{(}I,\,R_{\mathfrak{p}}/(g_{1},\ldots,g_{i-1})R_{\mathfrak{p}}:_{R_{\mathfrak{p}}}I^{\infty}R_{\mathfrak{p}}\big{)}\cdot
e(R/\mathfrak{p}).$

On the other hand, the formulas of part (i) and (iii) yield

$\displaystyle c_{i}(I,R)$
$\displaystyle\;=\;e_{d-i}\left(R/(g_{1},\ldots,g_{i-1}):_{R}I^{\infty}+g_{i}R\right)\;-\;e_{d-i}\left(R/(g_{1},\ldots,g_{i}):_{R}I^{\infty}\right)$
$\displaystyle\;=\;e_{d-i}\left(\text{H}_{I}^{0}\big{(}R/(g_{1},\ldots,g_{i-1}):_{R}I^{\infty}+g_{i}R\big{)}\right)$
$\displaystyle\;=\;\sum_{\begin{subarray}{c}\mathfrak{p}\in
V\left((g_{1},\ldots,g_{i-1}):_{R}I^{\infty}\right)\\\ \mathfrak{p}\in
V(I),\;\dim(R/\mathfrak{p})=d-i\end{subarray}}{\rm
length}_{R_{\mathfrak{p}}}\left(\frac{R_{\mathfrak{p}}}{(g_{1},\ldots,g_{i-1})R_{\mathfrak{p}}:_{R_{\mathfrak{p}}}I^{\infty}R_{\mathfrak{p}}+g_{i}R_{\mathfrak{p}}}\right)\cdot
e(R/\mathfrak{p}).$

This recovers the length formula of [PTUV, Proposition 2.1] for the Segre
numbers of an ideal. Therefore, for any prime $\mathfrak{p}$ in the summation
above, it follows that

$e(I,\,R_{\mathfrak{p}}/(g_{1},\ldots,g_{i-1})R_{\mathfrak{p}}:_{R_{\mathfrak{p}}}I^{\infty}R_{\mathfrak{p}})\;=\;{\rm
length}_{R_{\mathfrak{p}}}\left(R_{\mathfrak{p}}/\left((g_{1},\ldots,g_{i-1})R_{\mathfrak{p}}:_{R_{\mathfrak{p}}}I^{\infty}R_{\mathfrak{p}}+g_{i}R_{\mathfrak{p}}\right)\right).$

In any case, this is expected: if the residue field $\kappa$ is not an
algebraic extension of a finite field, we may assume that $g_{i}\in I$ is
general in $IR_{\mathfrak{p}}$ for each prime $\mathfrak{p}$ above (see [PTUV,
Lemma 2.6]).

##  6\. Integral dependence and specialization

In this section, we discuss how integral dependence can be detected by
utilizing the invariants that we study in Section missing 5. We also discuss
when integral closure and polar schemes specialize. Moreover, we generalize
the result of Gaffney and Gassler [GG] regarding the lexicographic upper
semicontinuity of Segre numbers (see Theorem 6.4). Throughout this section, we
continue using Section missing 5.

To deal with the case where $\text{ht}(I)=0$, we also need to consider an
additional number that is denoted as $c_{0}(I,R)$. Let
$\mathscr{H}:=\text{gr}_{\mathfrak{m}}(G)=\text{gr}_{\mathfrak{m}}\left(\text{gr}_{I}(R)\right)$
with standard bigrading
$[\mathscr{H}]_{(v,n)}={\mathfrak{m}}^{n}G_{v}/{\mathfrak{m}}^{n+1}G_{v}$. The
first sum transform of the Hilbert function of $\mathscr{H}$ with respect to
$n$ is equal to

$H_{\mathscr{H}}^{1}(v,n):=\sum_{k=0}^{n}\dim_{\kappa}\left([\mathscr{H}]_{(v,n)}\right)={\rm
length}_{R}\left(G_{v}/{\mathfrak{m}}^{n+1}G_{v}\right),$

and it encodes the polar multiplicities of the standard graded algebra
$G=\text{gr}_{I}(R)$ (i.e., the Segre numbers $c_{1}(I,R),\ldots,c_{d}(I,R)$;
see Section missing 5). If we further consider the second sum transform
$H_{\mathscr{H}}^{2}(v,n):=\sum_{k=0}^{v}H_{\mathscr{H}}^{1}(k,n)$, for $v\gg
0$ and $n\gg 0$, we get a polynomial

$H_{\mathscr{H}}^{2}(v,n)\;=\;\sum_{i=0}^{d}\;\frac{c_{i}(I,R)}{i!\,(d-i!)}\;v^{i}n^{d-i}\;+\;\text{(lower
degree terms)},$

that encodes the Segre numbers $c_{1}(I,R),\ldots,c_{d}(I,R)$ defined in
Section missing 5, but also the new number $c_{0}(I,R)$. This leads to the
following definition.

######  Definition  (Achilles–Manaresi [AM_MULT_SEQ]).

The _multiplicity sequence_ of the ideal $I\subset R$ is given by

$\big{(}c_{0}(I,R),c_{1}(I,R),\ldots,c_{d}(I,R)\big{)}\;\in\;{\mathbb{Z}}_{\geq
0}^{d+1}.$

In Section missing 5, we restricted ourselves to considering only the first
sum transform $H_{\mathscr{H}}^{1}$ (i.e., the polar multiplicities of $G$)
because then we could use many desirable properties of polar multiplicities
(see [cidruiz2024polar], [KLEIMAN_THORUP_GEOM, §8]).

We now discuss when the polar schemes specialize module an element
$t\in{\mathfrak{m}}$ that is part of a system of parameters of $R$ (i.e.,
$\dim(R/tR)=d-1$). We say that the $i$-th polar scheme $P_{i}(I,X)$
_specializes_ modulo the element $t\in{\mathfrak{m}}$ if
$\dim\left(P_{i}(I,X)\cap V(t)\right)\leq d-i-1$ and we have the equality of
cycles

$\big{[}P_{i}(I,X)\cap V(t)\big{]}_{d-i-1}\;=\;\big{[}P_{i}\left(I,X\cap
V(t)\right)\big{]}_{d-i-1}\;\in\;Z_{d-i-1}(X).$

We shall see that the question of whether polar schemes specialize is governed
by the Segre numbers $c_{i}(I,R)$ and $c_{i}(I,R/tR)$. First, we have the
following basic but useful observation.

######  Lemma .

Assume Section missing 5. Let $t\in{\mathfrak{m}}$ be such that
$\dim(R/tR)=d-1$. If
$\text{ht}\left(I,(g_{1},\ldots,g_{i}):_{R}I^{\infty},t\right)\geq i+2$, then
$P_{i}(I,X)$ specializes modulo $t$.

###### Proof.

Let $\overline{R}:=R/\left((g_{1},\ldots,g_{i}):_{R}I^{\infty},t\right)$ and
notice that
$I^{k}\overline{R}\cdot\left((t,g_{1},\ldots,g_{i}):_{R}I^{\infty}\right)\overline{R}=0$
for some $k>0$. Thus the result follows as a direct consequence of Theorem
5.1(i). ∎

A very important technical result is the following theorem. Our arguments
below follow closely the ones in the proof of [PTUV, Theorem 3.3].

######  Theorem 6.1.

Assume Section missing 5 and that $R$ is equidimensional and catenary. Let
$1\leq i\leq d-1$. Let $t\in{\mathfrak{m}}$ be such that
$\text{ht}\left((g_{1},\ldots,g_{i-1}):_{R}I^{\infty},t\right)\geq i$ and
$\text{ht}\left(I,(g_{1},\ldots,g_{i-1}):_{R}I^{\infty},t\right)\geq i+1$.
Then the following statements hold:

  1. (i)

$\text{ht}\left((g_{1},\ldots,g_{i}):_{R}I^{\infty},t\right)\geq i+1$.

  2. (ii)

$c_{i}(I,R/tR)\geq c_{i}(I,R)$.

  3. (iii)

If $c_{i}(I,R/tR)=c_{i}(I,R)$, then
$\text{ht}\left(I,(g_{1},\ldots,g_{i}):_{R}I^{\infty},t\right)\geq i+2$.

###### Proof.

We may assume that $\dim(R/(g_{1},\ldots,g_{i-1}):_{R}I^{\infty})=d-i+1$
because $R$ is equidimensional and catenary (see Section missing 5). By
utilizing Theorem 5.1(ii), we may assume $i=1$ and substitute $R$ by
$R/(g_{1},\ldots,g_{i-1}):_{R}I^{\infty}$. So, we are free to assume the
conditions $i=1$, $\dim(R/tR)=d-1$, $0:_{R}I^{\infty}=0$ and
$\text{ht}(I,t)\geq 2$. Hence $g_{1}$ is a nonzerodivisor since it is a
general element of $I$. Set $g:=g_{1}$. Since $g$ is a general element of $I$,
it follows that $\text{ht}((g):_{R}I^{\infty},t)\geq\text{ht}(g,t)\geq 2$;
thus settling part (i). We need to verify $c_{1}(I,R/tR)\geq c_{1}(I,R)$ to
prove part (ii). Part (iii) follows by showing that if
$c_{1}(I,R/tR)=c_{1}(I,R)$ then $\text{ht}(I,(g):_{R}I^{\infty},t)\geq 3$.

The rest of the proof follows verbatim the arguments in [PTUV, page 962, proof
of Theorem 3.3]. Consider the finite sets of primes
$\Lambda:=\\{\mathfrak{p}\in V(I,t)\mid\text{ht}(\mathfrak{p})=2\\}$ and
$\Gamma:=\\{\mathfrak{q}\in V(I)\mid\text{ht}(\mathfrak{q})=1\\}$. Thus, to
prove the inequality $\text{ht}(I,(g):_{R}I^{\infty},t)\geq 3$, it suffices to
show that for each $\mathfrak{p}\in\Lambda$ we have
$\mathfrak{p}\not\supset(g):_{R}I^{\infty}$ or, equivalently,
$IR_{\mathfrak{p}}\subset\sqrt{gR_{\mathfrak{p}}}$. For each
$\mathfrak{p}\in\Lambda$, we define the finite sets

$\Sigma_{\mathfrak{p}}:=\\{\mathfrak{q}\in{\rm
Min}(g)\mid\mathfrak{q}\subset\mathfrak{p}\\}\quad\text{ and
}\quad\Gamma_{\mathfrak{p}}:=\\{\mathfrak{q}\in
V(I)\mid\mathfrak{q}\subset\mathfrak{p}\text{ and
}\text{ht}(\mathfrak{q})=1\\}.$

Notice that $\Gamma_{\mathfrak{p}}\subseteq\Sigma_{\mathfrak{p}}$ and that an
equality holds if and only if
$IR_{\mathfrak{p}}\subset\sqrt{gR_{\mathfrak{p}}}$. By utilizing the
faithfully flat extension $R\rightarrow R[y]_{{\mathfrak{m}}R[y]}$, we may
assume that the residue field of $R$ is not an algebraic extension of a finite
field, and thus we may assume that $g$ is a general element of
$IR_{\mathfrak{p}}$ for each $\mathfrak{p}\in\Lambda$ (see [PTUV, Lemma 2.6]).

We have the following

$\displaystyle c_{1}(I,R/tR)$
$\displaystyle=\sum_{\mathfrak{p}\in\Lambda}e\left(I,R_{\mathfrak{p}}/tR_{\mathfrak{p}}\right)\cdot
e(R/\mathfrak{p})$ by [PTUV, Proposition 2.4(a)] or Theorem 5.1(ii)
$\displaystyle=\sum_{\mathfrak{p}\in\Lambda}e\left((g),R_{\mathfrak{p}}/tR_{\mathfrak{p}}\right)\cdot
e(R/\mathfrak{p})$ since $g$ is general in $IR_{\mathfrak{p}}$
$\displaystyle\geq\sum_{\mathfrak{p}\in\Lambda}e\left((t),R_{\mathfrak{p}}/gR_{\mathfrak{p}}\right)\cdot
e(R/\mathfrak{p})$ by [PTUV, Lemma 3.1] since $g$ is a nonzerodivisor
$\displaystyle=\sum_{\mathfrak{p}\in\Lambda}\sum_{\mathfrak{q}\in\Sigma_{\mathfrak{p}}}{\rm
length}_{R_{\mathfrak{q}}}(R_{\mathfrak{q}}/gR_{\mathfrak{q}})\cdot
e\left((t),R_{\mathfrak{p}}/\mathfrak{q}R_{\mathfrak{p}}\right)\cdot
e(R/\mathfrak{p})$ by the associativity formula
$\displaystyle\geq\sum_{\mathfrak{p}\in\Lambda}\sum_{\mathfrak{q}\in\Gamma_{\mathfrak{p}}}{\rm
length}_{R_{\mathfrak{q}}}(R_{\mathfrak{q}}/gR_{\mathfrak{q}})\cdot
e\left((t),R_{\mathfrak{p}}/\mathfrak{q}R_{\mathfrak{p}}\right)\cdot
e(R/\mathfrak{p})$ since $\Gamma_{\mathfrak{p}}\subseteq\Sigma_{\mathfrak{p}}$
$\displaystyle=\sum_{\mathfrak{q}\in\Gamma}{\rm
length}_{R_{\mathfrak{q}}}(R_{\mathfrak{q}}/gR_{\mathfrak{q}})\sum_{\mathfrak{p}\in\Lambda,\mathfrak{p}\supset\mathfrak{q}}e\left((t),R_{\mathfrak{p}}/\mathfrak{q}R_{\mathfrak{p}}\right)\cdot
e(R/\mathfrak{p})$ by switching the summation
$\displaystyle\geq\sum_{\mathfrak{q}\in\Gamma}{\rm
length}_{R_{\mathfrak{q}}}(R_{\mathfrak{q}}/gR_{\mathfrak{q}})\cdot
e(R/\mathfrak{q})$ by [PTUV, Lemma 3.2]
$\displaystyle\geq\sum_{\mathfrak{q}\in\Gamma}e((g),R_{\mathfrak{q}})\cdot
e(R/\mathfrak{q})$ by [FLENNER_O_CARROLL_VOGEL, Corollary 1.2.12]
$\displaystyle\geq\sum_{\mathfrak{q}\in\Gamma}e(IR_{\mathfrak{q}},R_{\mathfrak{q}})\cdot
e(R/\mathfrak{q})$ since $g\in I$ $\displaystyle=c_{1}(I,R)$ by [PTUV,
Proposition 2.4(a)] or Theorem 5.1(ii).

Therefore, we obtain the inequality $c_{1}(I,R/tR)\geq c_{1}(I,R)$ and that
$\Gamma_{\mathfrak{p}}=\Sigma_{\mathfrak{p}}$ if an equality holds. This
concludes the proof of the theorem. ∎

Important consequences of Theorem 5.1 and Theorem 6.1 are the following
lexicographical upper-semicontinuity results.

######  Corollary .

Assume that $R$ is equidimensional and catenary, and let $t\in{\mathfrak{m}}$
be such that $\dim(R/tR)=d-1$. Suppose that
$\text{ht}(I,0:_{R}I^{\infty},t)\geq 2$. Then we get the inequality

$\big{(}c_{1}(I,R/tR),c_{2}(I,R/tR),\ldots,c_{d-1}(I,R/tR)\big{)}\;\;\geq_{\rm
lex}\;\;\big{(}c_{1}(I,R),c_{2}(I,R),\ldots,c_{d-1}(I,R)\big{)}$

under the lexicographical order.

######  Corollary .

Assume that $R$ is equidimensional and universally catenary. Let $I\subset J$
be two $R$-ideals. Then we get the inequality

$\big{(}c_{0}(I,R),c_{1}(I,R),\ldots,c_{d}(I,R)\big{)}\;\;\geq_{\rm
lex}\;\;\big{(}c_{0}(J,R),c_{1}(J,R),\ldots,c_{d}(J,R)\big{)}$

under the lexicographical order.

###### Proof.

By replacing $R$ by $R[y]_{({\mathfrak{m}},y)}$ (where $y$ is a new variable)
and the ideals $I$ and $J$ by $(I,y)$ and $(J,y)$, we may assume that
$\text{ht}(I)\geq 1$ and $\text{ht}(J)\geq 1$ and that $c_{0}(I,R)=0$ and
$c_{0}(J,R)=0$; see [PTUV, Corollary 2.2(d)].

Consider the local ring $S=R[t]_{({\mathfrak{m}},t)}$ (where $t$ is a new
variable). Let $H=IS+tJS\subset S$. Notice that
$\text{ht}(H,0:_{S}H^{\infty},t)\geq\text{ht}(IS,t)\geq 2$, $H\cdot S/tS=I$
and $HS_{{\mathfrak{m}}S}=JR[t]_{{\mathfrak{m}}R[t]}$. Then we obtain the
following inequalities

$\displaystyle\big{(}c_{1}(I,R),c_{2}(I,R),\ldots,c_{d}(I,R)\big{)}$
$\displaystyle\;=\;\big{(}c_{1}(H,S/tS),c_{2}(H,S/tS),\ldots,c_{d}(H,S/tS)\big{)}$
$\displaystyle\;\geq_{\rm
lex}\;\big{(}c_{1}(H,S),c_{2}(H,S/tS),\ldots,c_{d}(H,S)\big{)}$ by Section
missing 6
$\displaystyle\;\geq\;\big{(}c_{1}(H,S_{{\mathfrak{m}}S}),c_{2}(H,S_{{\mathfrak{m}}S}),\ldots,c_{d}(H,S_{{\mathfrak{m}}S})\big{)}$
by [PTUV, Proposition 2.7]
$\displaystyle\;=\;\big{(}c_{1}(J,R),c_{2}(J,R),\ldots,c_{d}(J,R)\big{)}$

that settle the claim of the corollary. ∎

We now present a principle of specialization of integral dependence that we
enunciate similarly to the ones in [GG, Theorem 4.7] and [PTUV, Theorem 4.4].

######  Theorem 6.2 (Principle of Specialization of Integral Dependence –
PSID).

Assume Section missing 5 and that $R$ is equidimensional and universally
catenary. Let $t\in{\mathfrak{m}}$ be such that $\dim(R/tR)=d-1$. Suppose that
$\text{ht}(I,t)\geq 2$ and that $c_{i}(I,R/tR)=c_{i}(I,R)$ for all $1\leq
i\leq d-1$. Then, for any element $h\in R$, the following two conditions are
equivalent:

  1. (i)

$h\in\overline{I}$.

  2. (ii)

$hR_{\mathfrak{p}}\in\overline{IR_{\mathfrak{p}}}$ for all primes
$\mathfrak{p}\in{\text{Spec}}(R)$ such that $t\not\in\mathfrak{p}$.

###### Proof.

Assume momentarily that $\dim(G/tG)<d$. As $G$ is equidimensional of dimension
$d$ (see [RATLIFF_QUNMIXED_II, Theorem 3.8], [SUV_MULT, Lemma 2.2]), it
follows that the contraction of a minimal prime of $G$ to $R$ does not contain
$t$. This implies that no prime in the set $L(I):=\\{\mathfrak{p}\in
V(I)\mid\dim(R_{\mathfrak{p}})=\ell(IR_{\mathfrak{p}})\\}$ contains $t$. Since
every associated prime of $\overline{I}$ belongs to $L(I)$ (see [MCADAM, 3.9
and 4.1]), the result of the theorem follows. Hence it remains to show
$\dim(G/tG)<d$.

Since $\text{ht}(I)\geq 1$, it follows that $\dim(G/G_{+})=\dim(R/I)\leq d-1$
and so no minimal prime of $G$ contains $G_{+}$. Set
$E^{\prime}:=\text{Proj}(G/tG)$. Thus $\dim(E^{\prime})<d-1$ if and only if
$\dim(G/tG)<d$. From [cidruiz2024polar, Proposition 2.10], we obtain
$m_{d-1}^{d-1}(G/tG)=e_{d-1}(\text{H}^{0}(E^{\prime},\mathcal{O}_{E^{\prime}}))$
and so it follows that $m_{d-1}^{d-1}(G/tG)=0$ because
$\text{Supp}(\text{H}^{0}(E^{\prime},\mathcal{O}_{E^{\prime}}))\subset V(I,t)$
and by assumption we have the inequality $\text{ht}(I,t)\geq 2$. We now assume
the notation and assumptions of Section missing 5. From [cidruiz2024polar,
Proposition 2.10] and (missing) 3, we get

$m_{d-1}^{d-1-i}(G/tG)\;=\;m_{d-1-i}^{d-1-i}\left(G/(\ell_{1},\ldots,\ell_{i},t)G\right)\;=\;m_{d-1-i}^{d-1-i}(G_{i}/tG_{i}),$

where $G_{i}=\text{gr}_{IR_{i}}(R_{i})$ and $R_{i}=R/(g_{1},\ldots,g_{i})R$.
Therefore, by utilizing Theorem 5.1(ii) and Theorem 6.1, we can prove
inductively that

$m_{d-1}^{d-1-i}(G/tG)=0\quad\text{ for all }\quad 0\leq i\leq d-1.$

Due to [cidruiz2024polar, Lemma 2.3], we obtain $\dim(E^{\prime})<d-1$, and so
it follows that $\dim(G/tG)<d$. ∎

The following theorem contains our new criteria for integral dependence in
terms of the invariants $m_{i}(I,R)$ and $\nu_{i}(I,R)$.

######  Theorem 6.3.

Assume Section missing 5 and that $R$ is equidimensional and universally
catenary. Let $I\subset J$ be two $R$-ideals. Suppose the following two
conditions:

  1. (a)

$o(I)=o(J)$.

  2. (b)

$I$ satisfies the $\mathscr{G}$-parameter condition generically (see Section
missing 5).

Then the following are equivalent:

  1. (i)

$J$ is integral over $I$.

  2. (ii)

$m_{i}(I,R)=m_{i}(J,R)$ for all $0\leq i\leq d-1$.

  3. (iii)

$\nu_{i}(I,R)=\nu_{i}(J,R)$ for all $0\leq i\leq d$.

###### Proof.

Let $\delta:=o(I)=o(J)$ be the order of $I$ and $J$.

(i) $\Rightarrow$ (ii) & (iii): If $J$ is integral over $I$, then it is known
that $m_{i}(I,R)=m_{i}(J,R)$ and $c_{i}(I,R)=c_{i}(J,R)$ (see [Trung2001,
Corollary 3.8] and [PTUV, Theorem 4.2]). This settles both implications
(without assuming the conditions (a) and (b)).

(ii) $\Rightarrow$ (i): Suppose that $m_{i}(I,R)=m_{i}(J,R)$ for all $0\leq
i\leq d-1$. Then Section missing 5 yields

$\nu_{i}(I,R)\;=\;\delta\cdot m_{i-1}(I,R)\;=\;\delta\cdot
m_{i-1}(J,R)\;\leq\;\nu_{i}(J,R)$

for all $1\leq i\leq d$. This implies that
$c_{i}(I,R)=\nu_{i}(I,R)-m_{i}(I,R)\leq\nu_{i}(J,R)-m_{i}(J,R)=c_{i}(J,R)$ for
all $1\leq i\leq d$. On the other hand, we have
$c_{0}(I,R)=e(R)-m_{0}(I,R)=e(R)-m_{0}(J,R)=c_{0}(J,R)$. Therefore, $J$ is
integral over $I$ due to [PTUV, Theorem 4.2].

(iii) $\Rightarrow$ (i): Suppose that $\nu_{i}(I,R)=\nu_{i}(J,R)$ for all
$0\leq i\leq d$. From Section missing 5, we obtain

$\delta\cdot m_{i-1}(I,R)\;=\;\nu_{i}(I,R)\;=\;\nu_{i}(J,R)\;\geq\;\delta\cdot
m_{i-1}(J,R)$

for all $1\leq i\leq d$. It follows that
$c_{i}(I,R)=\nu_{i}(I,R)-m_{i}(I,R)\leq\nu_{i}(J,R)-m_{i}(J,R)=c_{i}(J,R)$ for
all $0\leq i\leq d-1$. We also have
$c_{d}(I,R)=\nu_{d}(I,R)=\nu_{d}(J,R)=c_{d}(J,R)$. Finally, $J$ is integral
over $I$ by invoking [PTUV, Theorem 4.2] once again. ∎

The next remark complements [Trung2001]*Corollary 4.3.

######  Remark .

Let $R$ be a regular local ring of dimension at least two and let $I$ be an
ideal of height one. Write $I=aH$ where $H$ is an ideal with $\text{ht}(H)\geq
2$. Then $m_{1}(I,R)=o(H)$.

###### Proof.

We may assume that the residue field of $R$ is infinite. Let $g$ be a general
element of $H$. According to Theorem 5.1(i), we have
$m_{1}(I,R)=e(R/agR:_{R}I^{\infty}).$ On the other hand,

$agR:I^{\infty}\;=\;(agR:_{R}aH):_{R}(aH)^{\infty}\;=\;(gR:H):_{R}(aH)^{\infty}\;=\;(gR:_{R}H^{\infty}):_{R}(aR)^{\infty}\;=\;gR.$

Finally, $e(R/gR)=o(g)=o(H)$. ∎

The remark above shows that the polar multiplicities of an ideal of height one
in a two-dimensional regular local ring only depend on the order of the
saturation of the ideal with respect to its unmixed part, and hence cannot
detect integral dependence.

######  Example  (Polar multiplicities do not detect integral dependence).

Let $R=\mathbb{k}[x_{0},x_{1}]$ be a polynomial ring over a field. Consider
the ideals
$I=(x_{0}^{2},\,x_{0}x_{1}^{2})\,\subsetneq\,J=(x_{0}^{2},\,x_{0}x_{1})$. We
have the following table of values:

$\begin{array}[]{l|r}\begin{array}[]{lll}c_{0}(I,R)=0&m_{0}(I,R)=1&\nu_{0}(I,R)=1\\\
c_{1}(I,R)=1&m_{1}(I,R)=1&\nu_{1}(I,R)=2\\\
c_{2}(I,R)=4&m_{2}(I,R)=0&\nu_{2}(I,R)=4\\\
\end{array}&\begin{array}[]{lll}c_{0}(J,R)=0&m_{0}(J,R)=1&\nu_{0}(J,R)=1\\\
c_{1}(J,R)=1&m_{1}(J,R)=1&\nu_{1}(I,R)=2\\\
c_{2}(J,R)=2&m_{2}(J,R)=0&\nu_{2}(I,R)=2.\\\ \end{array}\end{array}$

Here the invariants $m_{i}$ fail to detect the fact that
$\overline{I}\subsetneq\overline{J}$. However, both invariants $c_{i}$ and
$\nu_{i}$ do detect this.

From the previous example, we see that the invariants $\nu_{i}$ seem to carry
more information related to the integral closure of an ideal. Nevertheless, we
have the following more delicate example where the invariants $\nu_{i}$ fail
to detect integral dependence.

######  Example  (Polar-Segre multiplicities do not detect integral
dependence).

Let
${\mathbb{P}}_{\mathbb{k}}^{1}=\text{Proj}(\mathbb{k}[s,t])\rightarrow{\mathbb{P}}_{\mathbb{k}}^{3}=\text{Proj}(\mathbb{k}[x_{0},\ldots,x_{3}])$,
$(s:t)\mapsto(s^{4}:s^{3}t:st^{3}:t^{4})$ be the parametrization of the
rational quartic $\mathcal{C}\subset{\mathbb{P}}_{\mathbb{k}}^{3}$. Let
$R=\mathbb{k}[x_{0},\ldots,x_{3}]/\mathfrak{P}$ be the homogeneous coordinate
ring of $\mathcal{C}$. Consider the ideals
$I=(x_{0}^{2},\,x_{0}x_{2}x_{3},\,x_{1}^{2}x_{3}^{2})R\,\subsetneq\,J=(x_{0}^{2},\,x_{0}x_{2},\,x_{0}x_{3},\,x_{1}x_{3}^{2})R$.
We get the following table of values:

$\begin{array}[]{l|r}\begin{array}[]{lll}c_{0}(I,R)=0&m_{0}(I,R)=4&\nu_{0}(I,R)=4\\\
c_{1}(I,R)=5&m_{1}(I,R)=3&\nu_{1}(I,R)=8\\\
c_{2}(I,R)=14&m_{2}(I,R)=0&\nu_{2}(I,R)=14\\\
\end{array}&\begin{array}[]{lll}c_{0}(J,R)=0&m_{0}(J,R)=4&\nu_{0}(J,R)=4\\\
c_{1}(J,R)=3&m_{1}(J,R)=5&\nu_{1}(I,R)=8\\\
c_{2}(J,R)=14&m_{2}(J,R)=0&\nu_{2}(I,R)=14.\\\ \end{array}\end{array}$

Therefore the invariants $\nu_{i}$ fail to detect the fact that
$\overline{I}\subsetneq\overline{J}$.

###  6.1. Lexicographic upper semicontinuity of Segre numbers

In this subsection, we show that Segre numbers have a lexicographic upper
semicontinuous behavior in families. This generalizes a result due to Gaffney
and Gassler [GG]. We now use the following setup.

######  Setup .

Let $\iota:A\hookrightarrow R$ be a flat injective homomorphism of finite type
of Noetherian rings. Suppose that the inclusion $\iota$ has a section; i.e.,
there is a homomorphism $\pi:R\twoheadrightarrow A$ such that
$\pi\circ\iota={\rm id}_{A}$. Consider the ideal $Q:=\text{Ker}(\pi)\subset
R$.

Notice that given prime $\mathfrak{p}\in{\text{Spec}}(A)$, we always have that
$QR(\mathfrak{p})$ is a prime ideal in the fiber $R(\mathfrak{p})$. Indeed, we
get the inclusion $\iota_{\mathfrak{p}}:A/\mathfrak{p}\hookrightarrow
R/\mathfrak{p}R$ with section
$\pi_{\mathfrak{p}}:R/\mathfrak{p}R\twoheadrightarrow A/\mathfrak{p}$, and
since $\text{Ker}(\pi_{\mathfrak{p}})={Q\cdot R/\mathfrak{p}R}$, it follows
that ${Q\cdot R/\mathfrak{p}R}$ is a prime ideal satisfying
$\left(R/\mathfrak{p}R\right)_{Q\cdot
R/\mathfrak{p}R}=R(\mathfrak{p})_{QR(\mathfrak{p})}$. Moreover
${QR(\mathfrak{p})}$ is a maximal ideal in $R(\mathfrak{p})$. Thus, although
the fiber $R(\mathfrak{p})$ might not be a local ring, we do get a
distinguished local ring $R(\mathfrak{p})_{QR(\mathfrak{p})}$. To simplify
notation, we say that $S(\mathfrak{p}):=R(\mathfrak{p})_{QR(\mathfrak{p})}$ is
the _distinguished fiber_ of $\mathfrak{p}$.

Our main result regrading the behavior of Segre numbers in families is the
following.

######  Theorem 6.4.

Assume §missing 6.1 and let $I\subset R$ be an ideal. Assume that for all
$\mathfrak{p}\in{\text{Spec}}(A)$, the fibers $R(\mathfrak{p})$ are
equidimensional of the same dimension $d$ and $\text{ht}(I(\mathfrak{p}))>0$.
Then the function

$\mathfrak{p}\in{\text{Spec}}(A)\;\mapsto\;\big{(}c_{1}\left(I,S(\mathfrak{p})\right),c_{2}\left(I,S(\mathfrak{p})\right),\ldots,c_{d}\left(I,S(\mathfrak{p})\right)\big{)}\in{\mathbb{Z}}_{\geq
0}^{d}$

is upper semicontinuous with respect to the lexicographic order.

###### Proof.

By the topological Nagata criterion (see Section missing 3), it suffices to
show the following two conditions:

  1. (i)

Under the assumption that $A$ is a domain, there is a dense open subset
$U\subset{\text{Spec}}(A)$ where the above function is constant.

  2. (ii)

If $\mathfrak{p}\supset\mathfrak{q}$, then
$\big{(}c_{1}(I,S(\mathfrak{p})),\ldots,S(\mathfrak{p})\big{)}\geq_{\rm
lex}\big{(}c_{1}(I,S(\mathfrak{q})),\ldots,c_{d}(I,S(\mathfrak{q}))\big{)}$.

First, we prove the claim (i). We assume $A$ is a domain, thus by
Grothendieck’s Generic Freeness Lemma (see, e.g., [MATSUMURA, Theorem 24.1],
[EISEN_COMM, Theorem 14.4]) there is an element $0\neq a\in A$ such that the
bigraded components of $\text{gr}_{Q}(\text{gr}_{I}(R))\otimes_{A}A_{a}$ and
the graded components of $\text{gr}_{I}(R)\otimes_{A}A_{a}$ are free
$A_{a}$-modules. Set $U:=D(a)\subset{\text{Spec}}(A)$. Then, for any
$\mathfrak{p}\in U$, we have the isomorphisms

$\text{gr}_{Q}(\text{gr}_{I}(R))\otimes_{A}\kappa(\mathfrak{p})\;\cong\;\text{gr}_{Q}(\text{gr}_{I}(R)\otimes_{A}\kappa(\mathfrak{p}))\;\cong\;\text{gr}_{Q}(\text{gr}_{I}(R(\mathfrak{p}))).$

This shows shows that the above function is constant on $U$.

Next, we prove the claim (ii). Notice that we can reduce modulo $\mathfrak{q}$
and localize at $\mathfrak{p}$. Thus we assume $A$ is a local domain with
maximal ideal $\mathfrak{p}$ and $\mathfrak{q}=0$. Let $K:={\text{Quot}}(A)$.
By [HARTSHORNE, Exercise II.4.11] or [EGAII, Proposition 7.1.7], there is a
discrete valuation ring $V$ of $K$ that dominates $A$; that is, $A\subset V$
and $\mathfrak{p}=\mathfrak{n}\cap A$ where $\mathfrak{n}=(t)\subset V$ is the
closed point of ${\text{Spec}}(V)$. Let $R_{V}:=R\otimes_{A}V$. Notice that
$\iota^{\prime}:V\hookrightarrow R_{V}$ is flat and has a section
$\pi^{\prime}:R_{V}\twoheadrightarrow V$ with
$\text{Ker}(\pi^{\prime})=QR_{V}$. We get a field extension
$\kappa(\mathfrak{p})\hookrightarrow\kappa(\mathfrak{n})$ and the following
isomorphisms

$R_{V}(\mathfrak{n})\;\cong\;(R\otimes_{A}V)\otimes_{V}\kappa(\mathfrak{n})\;\cong\;R\otimes_{A}\kappa(\mathfrak{n})\;\cong\;(R\otimes_{A}\kappa(\mathfrak{p}))\otimes_{\kappa(\mathfrak{p})}\kappa(\mathfrak{n})\;\cong\;R(\mathfrak{p})\otimes_{\kappa(\mathfrak{p})}\kappa(\mathfrak{n}).$

This gives the isomorphisms $I^{v}R_{V}(\mathfrak{n})\cong
I^{v}R(\mathfrak{p})\otimes_{\kappa(\mathfrak{p})}\kappa(\mathfrak{n})$ and so
it follows

(4)
$\text{gr}_{Q}\left(\text{gr}_{I}\left(R_{V}(\mathfrak{n})\right)\right)\;\cong\;\text{gr}_{Q}\left(\text{gr}_{I}\left(R(\mathfrak{p})\right)\right)\otimes_{\kappa(\mathfrak{p})}\kappa(\mathfrak{n}).$

Let $T:={(R_{V})}_{(t,Q)R_{V}}$. From (missing) 4, we have

(5) $c_{i}(I,S(\mathfrak{p}))\;=\;c_{i}(I,T/tT).$

Since $T_{QT}=R(\mathfrak{q})_{QR(\mathfrak{q})}=S(\mathfrak{q})$, we also
obtain

(6) $c_{i}(I,S(\mathfrak{q}))\;=\;c_{i}(I,T_{QT}).$

To use Section missing 6, we now verify that the ideal $IT\subset T$ satisfies
the required assumptions. By applying §missing 6.1(i) to the ring
$R(\mathfrak{p})$, we conclude that the fibers of $R_{V}$ over $V$ are
equidimensional of dimension $d$. Hence due to §missing 6.1(ii), $T$ is
equidimensional of dimension $d+1$ and universally catenary and
$\dim(T/tT)=\dim(T)-1$. Since $R(\mathfrak{p})\hookrightarrow T/tT$ is a flat
homomorphism, $\text{ht}((I,t)T/tT)\geq\text{ht}(I(\mathfrak{p}))\geq 1$,
hence $\text{ht}(t,IT)\geq 2$ because $t$ is regular on $T$. Now (missing) 5
and Section missing 6 yield

$\big{(}c_{1}(I,S(\mathfrak{p})),\ldots,c_{d}(I,S(\mathfrak{p}))\big{)}\;=\;\big{(}c_{1}(I,T/tT),\ldots,c_{d}(I,T/tT)\big{)}\;\geq_{\rm
lex}\;\big{(}c_{1}(I,T),\ldots,c_{d}(I,T)\big{)}.$

Finally, from [PTUV, Proposition 2.7] and (missing) 6, we get

$\big{(}c_{1}(I,T),\ldots,c_{d}(I,T)\big{)}\;\geq\;\big{(}c_{1}(I,T_{QT}),\ldots,c_{d}(I,T_{QT})\big{)}\;=\;\big{(}c_{1}(I,S(\mathfrak{q})),\ldots,c_{d}(I,S(\mathfrak{q}))\big{)}.$

This concludes the proof of the theorem. ∎

######  Remark .

A basic situation where the assumptions of §missing 6.1 hold is the following.
Assume $R=A[x_{1},\ldots,x_{d}]$ is a positively graded polynomial ring over
$A$, and let ${\mathfrak{m}}=(x_{1},\ldots,x_{d})$ be the graded irrelevant
ideal. In this case, the natural projection $\pi:R\twoheadrightarrow A\cong
R/{\mathfrak{m}}$ is a section of the inclusion $\iota:A\rightarrow R$.
Therefore, given a prime $\mathfrak{p}\in{\text{Spec}}(A)$, the distinguished
fiber of $\mathfrak{p}$ is the localization
$R(\mathfrak{p})_{{\mathfrak{m}}R(\mathfrak{p})}$ of the fiber
$R(\mathfrak{p})\cong\kappa(\mathfrak{p})[x_{1},\ldots,x_{d}]$ at the graded
irrelevant ideal ${\mathfrak{m}}R(\mathfrak{p})$.

The following technical lemma was used in the proof of Theorem 6.4.

######  Lemma .

The following statements hold:

  1. (i)

Let $B$ be a finitely generated algebra over a field $\mathbb{k}$. If $B$ is
equidimensional, then $B\otimes_{\mathbb{k}}L$ is equidimensional of the same
dimension for any field extension $L$ of $\mathbb{k}$.

  2. (ii)

Let $V$ be a discrete valuation ring with uniformizing parameter $t$. Let
$j:V\hookrightarrow B$ be a flat injective homomorphism of finite type that
has a section $\tau:B\twoheadrightarrow V$, and let
$\mathfrak{Q}=\text{Ker}(\tau)$. Let $T=B_{(t,\mathfrak{Q})}$. Suppose that
the fibers of $j$ are equidimensional of dimension $e$. Then $T$ is
equidimensional of dimension $e+1$ and universally catenary, and
$\dim(T/tT)=e$.

###### Proof.

(i) By [MATSUMURA, Theorem 23.2], we have
$\text{Ass}\left(B\otimes_{\mathbb{k}}L\right)=\bigcup_{\mathfrak{p}\in\text{Ass}(B)}\text{Ass}\left(B/\mathfrak{p}\otimes_{\mathbb{k}}L\right)$.
The result follows because each minimal prime of
$B/\mathfrak{p}\otimes_{\mathbb{k}}\mathbb{L}$ has the same dimension as
$B/\mathfrak{p}$ (see [HARTSHORNE, Exercise II.3.20(f)]).

(ii) Let $K={\text{Quot}}(V)$. Since $B$ is $V$-flat, it follows that $t$ is a
nonzerodivisor on $B$. Let $\mathfrak{P}\in{\text{Spec}}(B)$ be a minimal
prime contained in $(t,\mathfrak{Q})$. Such $\mathfrak{P}$ does not contain
$t$, hence it corresponds to a minimal in the fiber $B\otimes_{V}K$. By
applying the dimension formula [MATSUMURA, Theorems 15.5, 15.6] to the
inclusion $V\hookrightarrow B/\mathfrak{P}$, we obtain

$\dim\left((B/\mathfrak{P})_{(t,\mathfrak{Q})}\right)\;=\;\dim(V)+{\rm
trdeg}_{V}(B/\mathfrak{P})=1+e.$

This implies that $T$ is equidimensional of dimension $e+1$. As $T$ is
finitely generated over a discrete valuation ring, it is universally catenary.
Observe that $\dim(T/tT)=e$ since $t$ is regular on $T$. ∎

###  6.2. The case of equigenerated ideals

This brief subsection is dedicated to the family of equigenerated ideals,
where our results are particularly satisfying.

######  Corollary .

Let $\mathbb{k}$ be a field, $R$ be a finitely generated standard graded
$\mathbb{k}$-algebra of dimension $d$ and ${\mathfrak{m}}=R_{+}$ be the graded
irrelevant ideal. Let $I\subset R$ be a homogeneous ideal generated by
elements of the same degree $\delta\geq 1$. Then the following statements
hold:

  1. (1)

$\delta\cdot m_{i-1}(I,R)=\nu_{i}(I,R)$ for all $1\leq i\leq d$.

  2. (2)

For any homogeneous ideal $J\subset R$ containing $I$ such that $o(J)=\delta$,
the following are equivalent:

    1. (i)

$J$ is integral over $I$.

    2. (ii)

$m_{i}(I,R)=m_{i}(J,R)$ for all $0\leq i\leq d-1$.

    3. (iii)

$\nu_{i}(I,R)=\nu_{i}(J,R)$ for all $0\leq i\leq d$.

###### First proof..

We may assume that $\mathbb{k}$ is infinite. Notice that part (1) implies part
(2) (indeed, by Section missing 5(iii), the statement (1) is equivalent to the
condition (b) of Theorem 6.3). Since $I$ is equigenerated in degree $\delta$,
we see the Rees algebra $\mathscr{R}(I)$ as a standard bigraded algebra with
$\big{[}\mathscr{R}(I)\big{]}_{(v,n)}=\big{[}I^{v}\big{]}_{n+v\delta}$. Thus
Nakayama’s lemma yields the isomorphisms

$\big{[}\text{gr}_{\mathfrak{m}}\left(\mathscr{R}(I)\right)\big{]}_{(v,n)}\;=\;\frac{{\mathfrak{m}}^{n}I^{v}}{{\mathfrak{m}}^{n+1}I^{v}}\;\cong\;\big{[}I^{v}\big{]}_{n+v\delta}\;=\;\big{[}\mathscr{R}(I)\big{]}_{(v,n)}.$

Let $g$ be a general element of $I$. Since
$\left[\left(0:_{\mathscr{R}(I)}g\right)\right]_{(v,*)}=0$ for $v\gg 0$ (see
Section missing 5), it follows that
$\left[\left(0:_{\text{gr}_{\mathfrak{m}}(\mathscr{R}(I))}\text{in}(g)\right)\right]_{(v,*)}=0$
for $v\gg 0$. The result follows from Section missing 5(iii). ∎

###### Second proof..

Again, we see $\mathscr{R}(I)$ as a standard bigraded algebra with
$\big{[}\mathscr{R}(I)\big{]}_{(v,n)}=\big{[}I^{v}\big{]}_{n+v\delta}$.
Therefore $\text{gr}_{I}(I)\cong\mathscr{R}(I)/I\mathscr{R}(I)$ is also
naturally a standard bigraded algebra. Notice that we have a short exact
sequence

$0\rightarrow\mathscr{R}(I)(1,-\delta)\rightarrow\mathscr{R}(I)\rightarrow\text{gr}_{I}(R)\rightarrow
0.$

Then a basic computation with the polynomial functions
$P_{\mathscr{R}(I)}(v,n)=\sum_{k=0}^{n}\dim_{\mathbb{k}}\left([\mathscr{R}(I)]_{(v,k)}\right)$
and
$P_{\text{gr}_{I}(R)}(v,n)=\sum_{k=0}^{n}\dim_{\mathbb{k}}\left([\text{gr}_{I}(R)]_{(v,k)}\right)$
gives the equality

$\delta\cdot m_{i-1}(I,R)\;=\;m_{i}(I,R)+c_{i}(I,R)\;=\;\nu_{i}(I,R)$

for all $1\leq i\leq d$ (see [cidruiz2024polar, Remark 2.9]). ∎

As a consequence, in the equigenerated case, we get an alternative proof of
the lexicographic upper semicontinuity of Segre numbers (see Theorem 6.4).

######  Remark .

Let $A$ be a Noetherian domain and $R=A[x_{0},\ldots,x_{r}]$ be a standard
graded polynomial ring. Let
$\mathcal{F}:{\mathbb{P}}_{A}^{r}\dashrightarrow{\mathbb{P}}_{A}^{s}$ be a
rational map with representative $\mathbf{f}=(f_{0}:\cdots:f_{s})$ such that
$f_{0},\ldots,f_{s}$ are homogeneous elements of degree $\delta>0$. Let
$I=(f_{0},\ldots,f_{s})\subset R$ and assume that $I(\mathfrak{p})\neq 0$ for
all $\mathfrak{p}\in{\text{Spec}}(A)$. Notice that
$d_{i}(\mathcal{F}(\mathfrak{p}))=m_{r-i}(I,R(\mathfrak{p}))$ for all $0\leq
i\leq r$ and $\mathfrak{p}\in{\text{Spec}}(A)$. By Theorem 4.1, each function
$\mathfrak{p}\mapsto m_{i}(I,R(\mathfrak{p}))$ is lower semicontinuous. Notice
that we always have $m_{0}(I,R(\mathfrak{p}))=1$. Therefore, the equality
$c_{i}(I,R(\mathfrak{p}))=\delta\cdot
m_{i-1}(I,R(\mathfrak{p}))-m_{i}(I,R(\mathfrak{p}))$ implies that the function

$\mathfrak{p}\in{\text{Spec}}(A)\;\mapsto\;\big{(}c_{1}\left(I,R(\mathfrak{p})\right),c_{2}\left(I,R(\mathfrak{p})\right),\ldots,c_{r+1}\left(I,R(\mathfrak{p})\right)\big{)}\in{\mathbb{Z}}_{\geq
0}^{r+1}$

is upper semicontinuous with respect to the lexicographic order.

## Acknowledgments

The authors were partially supported by NSF grant DMS-1928930 and Alfred P.
Sloan Foundation G-2021-16778 while in residence at SLMath.

## References