# Local volumes, equisingularity and generalized smoothability

Antoni Rangachev  Department of Mathematics
University of Chicago
Chicago, IL 60637
Institute of Mathematics
and Informatics, Bulgarian Academy of Sciences
Akad. G. Bonchev, Sofia 1113, Bulgaria

###### Abstract.

We introduce the restricted local volume of a relatively very ample invertible
sheaf as an invariant of equisingularity by determining its change across
families. We apply this result to give numerical control of Whitney–Thom
(differential) equisingularity for families of isolated complex analytic
singularities. The characterization of the vanishing of the local volume gives
rise to the class of deficient conormal (dc) singularities. We introduce a
notion of generalized smoothability by considering the class of singularities
that deform to dc singularities. Using Whitney stratifications and the
functoriality properties of conormal spaces we show that fibers of conormal
spaces are well-behaved under transverse maps. Then by Thom’s transversality,
the structure theorems of Hilbert–Burch and Buchsbaum–Eisenbud, we show that
all smoothable singularities of dimension at least $2$, Cohen–Macaulay
codimension $2$, Gorenstein codimension $3$, and more generally determinantal
and Pfaffian singularities deform to dc singularities.

###### Key words and phrases:

Equisingularity, Excess–Degree Formula, local volume of an invertible sheaf,
Local Volume Formula, deficient conormal (dc) singularities, generalized
smoothability, Thom’s transversality

###### 2010 Mathematics Subject Classification:

32S15, 32S30, 32S60, 14C17, 14B15 13A30, 13B22, 13H15

###### Contents

  1. 1 Introduction
  2. 2 The Excess–Degree Formula and applications
  3. 3 Computing and vanishing of local volumes
  4. 4 The Local Volume Formula
  5. 5 Volume Stability and the Principle of Specialization of Integral Dependence
  6. 6 Whitney–Thom conditions, Jacobian modules, conormal spaces and integral dependence
  7. 7 Numerical control of Whitney–Thom equisingularity
  8. 8 Deficient conormal singularities and generalized smoothability

##  1\. Introduction

A principal goal of equisingularity theory is to decide if two germs of sets
or maps look alike in some sense. In general, this is a hard problem, but if
the germs are part of a family, then it is somewhat easier to predict when the
members of the family are the “same” [Z71]. The conditions that will guarantee
this “similiarity” depend on the total space of the family. Nevertheless, we
would like to control these conditions by fiberwise dependent numerical
invariants.

The main notion of equsingularity theory that will be addressed in this work
is that of Whitney–Thom equisingularity, also known as differential
equisingularity. We will use its algebro-geometric formulation and interpret
the problem of finding numerical control for it as a problem of intersection
theory. Variations of this problem appear in resolutions of singularities,
equiresolutions, numerical control of flatness and existence of simultaneous
canonical models.

For the most part we will work in a fairly general setup: the schemes
considered are of finite type over an arbitrary field. The applications of our
results though will be in the complex analytic category via a classical
translation.

Let’s fix some notation. Let $X$ and $Y$ be affine reduced schemes of finite
type over a field $\Bbbk$. Assume $X$ is equidimensional and $Y$ is regular
and integral of dimension one. Suppose $h\colon X\rightarrow Y$ is a morphism
with equidimensional fibers of positive dimension. All points considered are
assumed to be closed unless stated otherwise. Let $S$ be a subscheme of $X$
that is proper over $Y$ such that $S_{y}$ is nowhere dense in $X_{y}$ for each
point $y\in Y$. Let $C$ be an equidimensional reduced scheme, projective over
$X$ such that the structure morphism $c\colon C\rightarrow X$ maps each
irreducible component of $C$ to an irreducible component of $X$. Set
$D:=c^{-1}S$ and $\dim C:=r+1$.

Fix a point $y_{0}$ in $Y$. Denote by $D_{\mathrm{vert}}$ the union of
components of $D$ of top dimension $r$ that map to $y_{0}$ under $h\circ c$.
Our goal is to control numerically the presence of $D_{\mathrm{vert}}$.

In the applications to equisingularity, $S$ is a subscheme supported over the
singular locus of a complex analytic variety $X$, $Y$ is a smooth subspace of
$X$, and $X$ is viewed as the total space of a family obtained by a transverse
retraction from $X$ to $Y$. The scheme $C$ is a conormal modification of $X$.
Then the condition for $X$ to be equisingular along $Y$ in a neighborhood of
$y_{0}$ is expressed by asking $D_{\mathrm{vert}}$ to be empty.

The first case where one can use basic intersection theory to control
numerically the presence of $D_{\mathrm{vert}}$ is when $C:=\mathrm{Bl}_{S}X$.
In this case $D$ is the exceptional divisor of the blowup. For each $y$ denote
by $C(y)$ the blowup of $X_{y}$ by $S_{y}$ and by $D(y)$ the exceptional
divisor. Set $l:=c_{1}\mathcal{O}_{C}(1)$ and
$l_{y}:=c_{1}\mathcal{O}_{C(y)}(1)$. Let $U$ be a small enough affine
neighborhood of $y_{0}$. Then we have the following Excess–Degree Formula
(EDF):

$\int_{S_{y_{0}}}l_{y_{0}}^{r-1}[D(y_{0})]-\int_{S_{y}}l_{y}^{r-1}[D(y)]=\int_{S_{y_{0}}}l^{r}[D_{\mathrm{vert}}]$

for $y\in U-\\{y_{0}\\}$. Suppose $\mathcal{O}_{C}(1)$ is ample on
$D_{\mathrm{vert}}$. Then $D_{\mathrm{vert}}$ is empty if and only if the
intersection number $\int_{S_{y}}l_{y}^{r-1}[D(y)]$ is constant for $y\in Y$.
A typical situation when $\mathcal{O}_{C}(1)$ is ample on $D_{\mathrm{vert}}$
is when $S$ is finite over $Y$, for example.

What happens if $D$ is more generally a Weil divisor in $C$, and if $C$ is not
birational to $X$? If $\dim X=2$ we show that the EDF still applies. In
general, to obtain a formula similar to the EDF, one needs to introduce a
volume-type invariant that generalizes the top self-intersection number of a
Cartier divisor. The volume of a line bundle is an invariant studied
extensively by birational geometers (cf. [Laz04, Chp. 2] for definitions,
basic properties and constructions). Its local counterpart the local volume
was introduced by Fulger [Ful13]. The local volume’s algebraic analogue was
studied by [UV11].

From now on assume $S$ is finite over $Y$ and for simplicity of the exposition
assume that $\Bbbk$ is the residue field of each closed point $y$ and the
points in $S_{y}$. Let $\mathcal{L}$ be an invertible very ample sheaf on $C$
relative to $X$. Let $\mathcal{A}:=\oplus_{n\geq
0}\Gamma(C,\mathcal{L}^{\otimes n})$ be the ring of sections of $\mathcal{L}$.
Denote by $\mathcal{A}_{n}$ the $n$th graded piece of $\mathcal{A}$. Inspired
by [Ful13], [UV11] and [ELMNP], for each point $y\in Y$ define the restricted
local volume of $\mathcal{L}$ at $S_{y}$ as

$\mathrm{vol}_{C_{y}}(\mathcal{L}):=\limsup_{n\to\infty}\frac{r!}{n^{r}}\dim_{\Bbbk}H_{S}^{1}(\mathcal{A}_{n})\otimes_{\mathcal{O}_{Y}}k(y).$

The existence of the volumes as a limit has been a topic of extensive research
(cf. [Cut15] and Kaveh and Khovanskii [KK12]). The local volume might be an
irrational number as shown by Cutkosky, Herzog and Srinivasan [CHS10].

One of the main results of this paper is the following relation which we refer
simply as the Local Volume Formula (LVF):

$\mathrm{vol}_{C_{y_{0}}}(\mathcal{L})-\mathrm{vol}_{C_{y}}(\mathcal{L})=\int_{S_{y_{0}}}l^{r}[D_{\mathrm{vert}}].$

Particular instances of the LVF and the EDF were known before as results of
Hironaka, Schickhoff (see Rmk. 2.6 in [Lip82] and [Hir70] for a related
result) and Teissier [T81] for the Hilbert–Samuel multiplicity, Kleiman and
Gaffney [GK99], and Gaffney ([Gaff04] and [Gaf08]) for the Buchsbaum–Rim
multiplicity. For related work in the projective setting see [Kol18] and
[HMX18, Sct. 4].

A direct consequence of the proof of the LVF is the following result: $D$ is
flat over $Y$ if and only if
$\dim_{\Bbbk}H_{S}^{1}(\mathcal{A}_{n})\otimes_{\mathcal{O}_{Y}}k(y)$ is
constant for $n\gg 0$. This is a local counterpart to the classical result of
Hironaka that says that a family of projective schemes over an integral base
is flat if and only if the Hilbert polynomials of the fibers remain the same
(see [Hir58] or [Har77, Thm. 9.9, Chp. III ]). An important local analogue of
Hironaka’s flatness result appears in [Hir64] in regard to what Hironaka calls
normal flatness, which is the flatness of the normal cone of a smooth
subvariety in a smooth ambient variety. Hironaka controls normal flatness
using the Hilbert–Samuel function. His result can be derived from the proof of
the LVF.

How do we control numerically the presence of $D_{\mathrm{vert}}$ when $Y$ is
integral of arbitrary dimension? In the case of the EDF assuming $Y$ is
regular, we reduce to the case $\dim Y=1$. For the LVF the major obstacle is
to show that the restricted local volume is constant over Zariski open dense
subset of $Y$. From now on assume $\Bbbk=\mathbb{C}$. View $X\rightarrow Y$ as
a part of a larger family $\mathcal{X}\rightarrow W$, where $W$ is integral
and $Y\subset W$. Assume that $S$ and $C$ are defined in the same way for
$\mathcal{X}$. We say that $\mathrm{vol}_{C_{w}}(\mathcal{L})$ is stable if it
is constant for $w$ in a Zariski open dense subset of $W$. In this case we say
that $W$ is a good base space.

Following [GR16] we show that we can control the presence of vertical
components of $D$ for $X\rightarrow Y$ by computing the restricted local
volumes from generic one-parameter families connecting $X_{y_{0}}$ and $X_{y}$
to fibers $\mathcal{X}_{w}$ for which the local volume stabilizes. This gives
an extension of the LVF to good base spaces of arbitrary dimension.

The extension of the LVF to the complex analytic setting is obtained in a
standard fashion using compact Stein neighborhoods. The main application of
the LVF in this work is obtaining numerical control for Whitney–Thom
(differential) equisingularity. Let’s review briefly its definition and some
of its applications. By a classical result of Whitney every complex analytic
variety $V$ can be partitioned into a locally finite family of submanifolds,
called strata, so that any pair of incident strata gives rise to a family of
singularities obtained as the fibers of a retraction to the lower dimensional
stratum and such that the total space of the family satisfies certain
geometric compatibility conditions.

More precisely, let $X\subset V$ be a complex analytic variety and $Y$ a
smooth subvariety of $X$ of dimension $k$ such that $X-Y$ is smooth and
$(X-Y,Y)$ is a pair of strata. Embed $X$ in $\mathbb{C}^{n+k}$ in such a way
that so that $Y$ contains $0$ and $(Y,0)$ is linear subspace of dimension $k$.
We say $H$ is a tangent hyperplane at $x\in X-Y$ if $H$ is a hyperplane in
$\mathbb{C}^{n+k}$ that contains the tangent space $T_{x}X$. Let $(x_{i})$ be
a sequence of points from $X-Y$ and $(y_{i})$ be a sequence of points from $Y$
both converging to $0$. Suppose that the sequence of secants
$(\overline{x_{i}y_{i}})$ has limit $l$ and the sequence of tangent
hyperplanes $\\{T_{x_{i}}X\\}$ has limit $T$. We say that $X$ is Whitney
equisingular along $Y$ at $0$, or that the pair of strata $(X-Y,Y)$ satisfies
Whitney conditions at $0$, if $l\subset T$.

The Whitney stratifications play an important role in the classification of
differentiable maps ([M73], [M76], cf. [Gaf93]), in $D$-module theory and the
solution of the Riemann–Hilbert problem [LM83], and in the Goresky–McPherson
theory of intersection homology [GM80], to mention few.

Let $f$ be a function on $X$ of constant rank off $Y$. The relative form of
Whitney conditions, called the $W_{f}$ condition, is defined in the same way
as the Whitney condition for the pair $(X-Y,Y)$ by replacing the tangent
hyperplanes that contain $T_{x_{i}}X$ by tangent hyperplanes that contain the
tangent space $T_{x_{i}}f^{-1}fx_{i}$ to the level surface $f^{-1}fx_{i}$,
where each $x_{i}$ is a smooth point of $f^{-1}fx_{i}$.

Denote by $X_{y}$ the fiber of a transverse projection to $Y$ and set
$f_{y}:=f|X_{y}$. Then the Thom-Mather second isotopy lemma yields the
following result: If $W_{f}$ holds, then there exists a homeomorphism $q$ from
$X_{0}\times Y$ onto $X$ such that $fq=f_{0}\times 1_{Y}$. Hence, for all
$y\in Y$ close enough to $0$ the pairs $X_{y},f_{y}$ are topologically the
same.

Our goal is to find numerical invariants depending on the fibers $X_{y},f_{y}$
and use the LVF to show that their constancy across $Y$ is equivalent to
$W_{f}$. As $W_{f}$ involves limits of tangent hyperplanes and secant lines,
it’s natural to expect that the invariants should be defined in terms of
conormal spaces. Define the conormal variety $C(X,f)$ (see Sct. 6) relative to
$f$ as the closure in $X\times\mathbb{P}^{n+k-1}$ of the set of pairs $(x,H)$
such that $x$ is a smooth point of the level set $f^{-1}fx$ and $H$ is a
tangent hyperplane to $f^{-1}fx$. Let $c_{X,f}\colon C(X,f)\rightarrow X$ be
structure morphism. Consider the blowup of $C(X,f)$ with center
$c_{X,f}^{-1}(Y)$. Teissier shows that $W_{f}$ holds at $0$ if and only if the
exceptional divisor of the blowup does not have vertical components of top
dimension. To control the presence of vertical components with invariants
depending on the fibers $X_{y},f_{y}$ one needs to replace $C(X,f)$ with its
relative version. Define the relative conormal variety $C_{\mathrm{rel}}(X,f)$
of $X$ relative to $h:X\rightarrow Y$ in the same way but this time requiring
that $H$ contains a parallel to $Y$. This replacement is achieved by
Teissier’s Principle of Specialization of Integral Dependence.

Define the conormal variety $C_{\mathrm{rel}}(X)$ of $X$ relative to
$h:X\rightarrow Y$ to be the closure in $X\times\mathbb{P}^{n+k-1}$ of the set
of pairs $(x,H)$ where $x$ is a smooth point of $X$ and $H$ is a tangent
hyperplane at $x$ containing a parallel to $Y$. Denote by
$J_{\mathrm{rel}}(X)$ the Jacobian module, and by $J_{\mathrm{rel}}(X,f)$ the
augmented Jacobian module of $X$ and $f$ with respect to the fiber coordinates
(see Sct. 6). Both modules are contained in a free module $\mathcal{F}$. Then
$C_{\mathrm{rel}}(X,f)=\mathrm{Projan}(\mathcal{R}(J_{\mathrm{rel}}(X,f)))$
where $\mathcal{R}(J_{\mathrm{rel}}(X,f))$ is the Rees algebra of
$J_{\mathrm{rel}}(X,f)$. Denote by $C$ the blowup of $C_{\mathrm{rel}}(X,f)$
with center the inverse image of $Y$. Then
$C:=\mathrm{Projan}(\mathcal{R}(m_{Y}J_{\mathrm{rel}}(X,f)))$ where $m_{Y}$ is
the ideal of $Y$ in $\mathcal{O}_{X,0}$.

Assume $X_{y}$ and $f_{y}$ have isolated singularities at $y$. Let $m_{y}$ be
the ideal of $y$ in $\mathcal{O}_{X_{y},y}$. For $C$ denote by
$\varepsilon(m_{y}J_{\mathrm{rel}}(X,f))(y)$ the restricted local volume of
$\mathcal{O}_{C}(1)$ at $y$. For $C_{\mathrm{rel}}(X)$ denote by
$\varepsilon(J_{\mathrm{rel}}(X))(y)$ the restricted local volume of
$\mathcal{O}_{C_{\mathrm{rel}}(X)}(1)$ at $y$. This notation aligns with the
notation of $\varepsilon$-multiplicity of modules (see [UV11]).

Assume $X\rightarrow Y$ can be read from the family $\mathcal{X}\rightarrow
W$, where $W$ is a component of the miniversal base space of $X_{0}$ and
$\mathcal{X}$ is the total deformation space of $X_{0}$ over $W$. Assume
$\varepsilon(J_{\mathrm{rel}}(\mathcal{X},\tilde{f}))(w)$ is stable for
generic $w\in W$ and generic deformation $\tilde{f}$ of $f$. By generic
specialization, the stability condition means that the local volume
$\varepsilon(J(\mathcal{X}_{w},\tilde{f}_{w}))$ (or the
$\varepsilon$-multiplicity of the $J(\mathcal{X}_{w},\tilde{f}_{w})$) is
constant for generic $w$ where $J(\mathcal{X}_{w},\tilde{f}_{w})$ is the
augmented Jacobian module of $\mathcal{X}_{w},\tilde{f}_{w}$. As an
application of the LVF we show that $(X-Y,Y)$ satisfies $W_{f}$ if and only if

$y\rightarrow\varepsilon(m_{y}J_{\mathrm{rel}}(X,f))(y)$

is contanst along $Y$. Thus owing to Grauert’s theorem [Gra72], in the case of
volume stability, to each isolated singularity $X_{0}$ we can associated
finitely many invariants, each associated with an irreducible component of the
base space of miniversal deformations of $X_{0}$, which allow to recognize all
equisingular deformations of $X_{0}$.

We prove a similar statement for Thom’s $A_{f}$ condition, which is a relative
stratification condition for the study of functions and mappings on stratified
sets. It plays an important role in Thom’s second isotopy theorem, and
provides a transversality condition in the development of the Milnor
fibration.

Our result for $W_{f}$ was first proved for isolated hypersurface
singularities by Teissier [T73] using the Hilbert-Samuel multiplicity. Gaffney
([Gaf92] and [Gaf96] based on ideas conceived in [Gaf93]) and Gaffney and
Kleiman ([GK99]) treated the case of isolated complete-intersection
singularities using the Buchsbaum–Rim (BR) multiplicity. More recently,
Gaffney and Rangachev [GR16] adressed the case of families of isolated
determinantal singularities using Gaffney’s Multiplicity-Polar Theorem for the
relative BR multiplicity. In all this cases the base space $W$ of miniversal
deformations of $X_{0}$ is smooth, $\mathcal{X}_{w}$ is smooth for generic
$w$, and $\varepsilon(J_{\mathrm{rel}}(\mathcal{X},\tilde{f})(w)=0$ which as
we show in Rmk. 7.2 is equivalent to $\varepsilon(J(\mathcal{X}_{w}))=0$ where
$J(\mathcal{X}_{w})$ is the Jacobian module of $\mathcal{X}_{w}$.

Thus it is natural to consider the class of isolated singularities $X_{0}$ for
which $\varepsilon(J(\mathcal{X}_{w}))=0$ for generic $w$, which is the
simplest instance of volume stability. Assume
$\mathcal{X}_{w}\subset\mathbb{C}^{n}$. Let $c_{w}\colon
C(\mathcal{X}_{w})\rightarrow\mathcal{X}_{w}$ be the structure morphism for
the conormal space of $\mathcal{X}_{w}$ in $\mathbb{C}^{n}$ and let $S_{w}$ be
the singular locus of $\mathcal{X}_{w}$. We show that

$\varepsilon(J(\mathcal{X}_{w}))=0\ \ \text{if and only if}\ \
\operatorname{codim}(c_{w}^{-1}S_{w},C(\mathcal{X}_{w}))\geq 2.$

In other words the volume vanishes if and only if the fibers of
$C(\mathcal{X}_{w})$ over the singular points of $\mathcal{X}_{w}$ are of
dimension less than expected. We call such $\mathcal{X}_{w}$ deficient
conormal (dc) singularity. We show that the dc property is independent of the
embedding of $\mathcal{X}_{w}$ in affine space and stable under infinitesimal
deformations. Furthermore, if $\mathcal{X}_{w}$ is smooth of dimension at
least $2$, then $\mathcal{X}_{w}$ is a dc singularity. Thus the class of
singularities that admit deformations to dc singularities is a natural
generalization of the class of smoothable singularities of dimension at least
$2$. Finally, we show that Cohen–Macaulay codimension $2$, Gorenstein
codimension $3$, and determinantal singularities belong to this class. This
allows us to propose a notion of generalized smoothability by considering all
singularities that deform to dc singularities.

The paper is organized as follows. In Sct. 2 we prove the EDF. Based on
Ramanujam’s interpretation of the Hilbert–Samuel multiplicity we recover and
generalize formulas due to Teissier for the Hilbert–Samuel multiplicity and
Gaffney’s Multiplicity-Polar Theorem. Our approach relies on basic
intersection theory as developed by Fulton in [Ful84], Bertini type theorems,
and the geometric theory of the multiplicity of pairs of standard graded
algebras developed by Kleiman and Thorup in [KT94]. We show how to derive the
EDF in the complex analytic setting and discuss some of its applications to
the deformation theory of singularities.

In Sct. 3 we compute the local volume using a Noether normalization-type
result of the author ([Ran19, Prp. 2.6 ]) that shows that every reduced
standard graded algebra $\mathcal{A}$ admits a homogeneous embedding in a
standard graded algebra $\mathcal{B}$ that behaves like a polynomial ring. We
show that in the computation for the local volume we can replace
$H_{S}^{1}(\mathcal{A}_{n})$ by $H_{S}^{0}(\mathcal{B}_{n}/\mathcal{A}_{n})$
which is more manageable. The section culminates with a result characterizing
the vanishing of the local volume. The main technical ingredient here are
results of the author ([Ran19] and [Ran18]) about the structure of
$\mathrm{Ass}_{\mathcal{A}}(\mathcal{B}_{n}/\overline{\mathcal{A}_{n}})$. We
pay special attention to the case when $\mathcal{A}$ is the Rees algebra of a
module, in which case the local volume was introduced by [UV11] under the name
of epsilon multiplicity. Its relevance to equisingularity was discovered by
Kleiman, Ulrich and Validashti [KUV]. Their work served as an inspiration for
this work.

In Sct. 4 we prove the LVF. We show that the restricted local volume
specializes with passage to the generic fibers using a result of the author
about the finiteness of
$\mathrm{Ass}_{\mathcal{A}}(\mathcal{B}_{n}/\overline{\mathcal{A}_{n}})$ (see
[Ran19, Thm. 1.1 (ii)]). Also, we show that the restricted local volume exists
as a limit.

In Sct.5 we show how to extend the LVF to the case when $\dim Y>1$ under the
assumption of volume stability using a covering argument due to Gaffney and
the author [GR16] and by computing the local volumes through one-parameter
generic deformations. We prove a general version of Teissier’s Principle of
Specialization of Integral Dependence.

In Sct. 6 we review the algebraic and analytic formulations of Whitney–Thom
equisingularity (the $W_{f}$ and Thom’s $A_{f}$ conditions) and we state the
necessary results about integral closure of modules and conormal geometry
needed for their numerical characterization. In Sct. 7 we show how to
characterize numerically $W_{f}$ and $A_{f}$ using the LVF and following an
approach pioneered by Teissier, which was subsequently simplified and
generalized by Gaffney and Kleiman.

In Sct. 8 we introduce the notion of deficient conormal (dc) singularities. We
give examples coming from affine cones over projective varieties having duals
with positive defect. We show that the dc property is intrinsic, using a
result of Teissier about polar varieities, and that it is stable under
infinitesimal deformations using the LVF. We show that Cohen–Macaulay
codimension $2$, Gorensten codimension $3$, and determinantal singularities
admit deformations to dc singularities. The proof is based on several results.

First we show that the fibers of conormal spaces pullback set-theoretically
under morphisms between affine spaces satisfying certain transversality
conditions. In particular, in our setup a dc singularity pulls back to a dc
singularity. We prove all this using stratification theory, the Lagrangian and
functoriality nature of conormal spaces and a recent result of Gaffney and the
author (see [GR19, Thm. 3.1]).

Then using the structure theorems of Hilbert-Burch and Buchsbaum-Eisenbud, we
show that the classes of singularities under consideration can be obtained as
pullbacks of generic determinantal and Pfaffian singularities by holomorphic
maps between affine spaces. Unfoldings of these maps gives rise deformations
of the singularities under consideration by a result of Buchweitz. By a
complex analytic version of Thom’s transversality due to Trivedi, the
unfoldings can be done in such a way so that the resulting maps are
generically transverse to the strata of a Whitney stratification of the
generic singularities. Finally, we are left with showing that the generic
determinantal and Pfaffian singularities are dc. This is done by computations
of Gaffney and the author, and Lakshmibai and Singh.

Inspired by a result of Kollár and Kovács [KK18] we show that affine cones
over normally embedded abelian varieties of dimension at least $2$ cannot be
deformed to dc singularities. We finish the section with showing how to
compute the restricted local volume associated with the conormal space of an
isolated nonsmoothable Cohen–Macaulay codimension $2$ singularity.

Acknowledgements. It’s a pleasure to thank Terence Gaffney, Steven Kleiman and
Bernard Teissier for countless helpful and stimulating discussions and for
their support and encouragement. Special thanks go to David Massey and Madhav
Nori for many fruitful and stimulating conversations. I would like to thank
Lawrence Ein, Mihai Fulger, Sándor Kovács, Rahul Singh, Bernd Ulrich, Javid
Validashti and Mathias Zach for helpful conversations. I was partially
supported by the University of Chicago FACCTS grant “Conormal and Arc Spaces
in the Deformation Theory of Singularities” and NEU summer and travel grants.
I would like to acknowledge the hospitality of IMPA and IMJ-PRG where part of
this work was completed.

##  2\. The Excess–Degree Formula and applications

In this section we prove the Excess–Degree Formula (EDF) and generalize and
recover multiplicity-polar results by Gaffney for the Buchsbaum–Rim
multiplicity and by Teissier for the Hilbert–Samuel multiplicity.

The following problem is at the heart of equisingularity theory. Let $h\colon
X\rightarrow Y$ be a morphism of schemes of finite type over a field $\Bbbk$
with equidimensional fibers of positive dimension $r$. Assume $X$ is
equidimensional and $Y$ is regular. Let $S$ be a closed subscheme of $X$
proper over $Y$ such that for each point $y\in Y$ the fiber $S_{y}$ is nowhere
dense in $X_{y}$. Denote by $C$ the blowup of $X$ with center $S$, by $D$ the
exceptional divisor, and by $c$ the blowup map $c\colon C\rightarrow X$. For
each $y\in Y$ denote by $C(y)$ the blowup of $X_{y}$ with center $S_{y}$ and
by $D(y)$ the exceptional divisor. Let $y_{0}$ be a closed point of $Y$. We
can ask: when do we have an equality of fundamental cycles

(1) $[C_{y_{0}}]=[C(y_{0})]?$

It’s not hard to see that (1) holds if and only if the fiber of the
exceptional divisor $D_{y_{0}}$ is equidimensional. More precisely, we would
like to find a numerical invariant depending solely on $D(y)$ whose constancy
across $Y$ guarantees (1) for each closed point $y_{0}\in Y$.

Set $l:=c_{1}\mathcal{O}_{C}(1)$ and $l_{y}:=c_{1}\mathcal{O}_{C(y)}(1)$ for
each $y\in Y$. Denote by $D_{\mathrm{vert}}$ the union of the components of
$D$ that surject onto $y_{0}$. For a closed point $y_{0}\in Y$ and affine
neighborhood $U$ of $y_{0}$ denote by $U^{\prime}$ the punctured neighborhood
$U-\\{y_{0}\\}$. A partial answer to our question is given by the following
result.

######  Theorem 2.1 (Excess–Degree Formula).

The following holds.

  * (i)

Assume $Y$ is integral and regular of dimension one and $y_{0}$ is a closed
point in $Y$. Let $U$ be a small enough affine neighborhood of $y_{0}$ such
that $C_{y_{0}}$ is principal Cartier divisor in $C$ after base change
$U\rightarrow Y$ and $C$ is flat over $U^{\prime}$. Then we have the following
Excess–Degree Formula (EDF):

$\int_{S_{y_{0}}}l_{y_{0}}^{r-1}[D(y_{0})]-\int_{S_{y}}l_{y}^{r-1}[D(y)]=\int_{S_{y_{0}}}l^{r}[D_{\mathrm{vert}}]$

for $y\in U^{\prime}$.

  * (ii)

Assume $Y$ is regular and $S$ is finite over $Y$. Then (1) holds if and only
if $\int_{S_{y}}l_{y}^{r-1}[D(y)]$ is constant for each closed $y$ is a
sufficiently small affine neighborhood $U$ of $y_{0}$.

###### Proof.

Preserve the setup from the introduction. For each $k$ denote by $Z_{k}(C)$
the group of $k$-cycles on $C$ and by $A_{k}(C)$ the group of $k$-cycles
modulo rational equivalence. For each $y\in Y$ consider the refined Gysin
homomorphism $i_{y}^{!}\colon A_{k}(C)\rightarrow A_{k-1}(C_{y})$ defined from
the fiber square

$\begin{CD}C_{y}@>{}>{}>C\\\ @V{}V{}V@V{}V{}V\\\ \\{y\\}@>{}>{}>Y\end{CD}$

(see [Ful84, Sct. 6.2]). For a $k$-cycle class $Z$ on $C$ set
$Z_{y}:=i_{y}^{!}(Z).$ Finally, denote by $D_{\mathrm{hor}}$ the union of the
irreducible components of $D$ that surject onto $Y$.

First, note that $C(y_{0})$ and $C_{y_{0}}$ are isomorphic over points $x\in
X_{y_{0}}$ with $x\notin S_{y_{0}}$. Also, the irreducible components of
$C(y_{0})$ surject onto those of $X_{y_{0}}$. But $S_{y_{0}}$ is nowhere dense
in $X_{y_{0}}$ by hypothesis. Hence the irreducible components of $C_{y_{0}}$
are either vertical components of $D$ or components that surject onto the
irreducible components of $X_{y_{0}}$. Thus we have

(2) $[C(y_{0})]-[C_{y_{0}}]=-[D_{\mathrm{vert}}]\ \text{in}\ Z_{r}(C).$

Second, $D\cdot[C_{y_{0}}]=C_{y_{0}}\cdot[D]$ because $C_{y_{0}}$ is Cartier.
Write $[D]=[D_{\mathrm{hor}}]+[D_{\mathrm{vert}}]$. Then
$C_{y_{0}}\cdot[D]=C_{y_{0}}\cdot[D_{\mathrm{hor}}]$ in $Z_{r-1}(C)$ because
$C_{y_{0}}$ is principal in $C$, so $C_{y_{0}}\cdot[D_{\mathrm{vert}}]=0$ (see
Rmk. 2.3 in [Ful84]). Thus, $C_{y_{0}}\cdot[D]=[D_{\mathrm{hor}}]_{y_{0}}$.
Hence $D\cdot[C_{y_{0}}]=[D_{\mathrm{hor}}]_{y_{0}}.$

Additionally, observe that $D\cdot[C(y_{0})]=[D(y_{0})]$. Hence by
intersecting each term in (2) with $D$ we get

$[D(y_{0})]-[D_{\mathrm{hor}}]_{y_{0}}=-D\cdot[D_{\mathrm{vert}}]\ \text{in}\
A_{r-1}(C)$

or equivalently

(3) $[D(y_{0})]-[D_{\mathrm{hor}}]_{y_{0}}=l[D_{\mathrm{vert}}].$

as $D$ is dual to $\mathcal{O}_{C}(1)$. Because $\mathcal{O}_{C}(1)$ restricts
to $O_{C(y_{0})}(1)$ on $C(y_{0})$ then
$l^{r-1}[D(y_{0})]=l_{y_{0}}^{r-1}[D(y_{0})].$ Apply $l^{r-1}$ to both sides
of (3) to get

(4)
$l_{y_{0}}^{r-1}[D(y_{0})]-l^{r-1}[D_{\mathrm{hor}}]_{y_{0}}=l^{r}[D_{\mathrm{vert}}]\
\text{in}\ A_{0}(C).$

Next apply [Ful84, Prp.10.2] to the $1$-cycle $l^{r-1}[D_{\mathrm{hor}}]$ in
$D$ and the map $D\rightarrow Y$ which is proper because it is composition of
two proper maps: $D\rightarrow S$ and $S\rightarrow Y$. We have

(5)
$\int_{S_{y_{0}}}(l^{r-1}[D_{\mathrm{hor}}])_{y_{0}}=\int_{S_{y}}(l^{r-1}[D_{\mathrm{hor}}])_{y}.$

Note that $[D]_{y}=[D(y)]$ because by assumption $C$ is flat over $U^{\prime}$
so $C_{y}=C(y)$ as schemes. Then by [Ful84, Prop. 10.1 (d)] about
specialization of Chern classes, we have

(6) $l^{r-1}[D_{\mathrm{hor}}]_{y}=l_{y}^{r-1}[D(y)].$

Combining (4), (6) and (5) we get

$\int_{S_{y_{0}}}l_{y_{0}}^{r-1}[D(y_{0})]-\int_{S_{y}}l_{y}^{r-1}[D(y)]=\int_{S_{y_{0}}}l^{r}[D_{\mathrm{vert}}].$

The proof of $\rm{(i)}$ is complete.

Consider $\rm{(ii)}$. Suppose that for each affine neighborhood $U$ of $y_{0}$
there exists $y\in U$ such that

$\int_{S_{y_{0}}}l_{y_{0}}^{r-1}[D(y_{0})]\neq\int_{S_{y}}l_{y}^{r-1}[D(y)].$

Suppose $\dim D_{y_{0}}<r$. Because $D\rightarrow Y$ is proper, by upper semi-
continuity $\dim D_{y}<r$ for each closed $y$ in a sufficiently small
neighborhood $U$ of $y_{0}$. Then $[D_{y}]=[D(y)]$ for $y\in U$. Because
$i_{y}\colon y\hookrightarrow Y$ is a regular embedding, the operation of
pullback along $i_{y}$ commutes with that of pushforth along the proper map
$D\rightarrow Y$. Thus by [Ful84, Prp.10.2] $\int_{S_{y}}l_{y}^{r-1}[D(y)]$ is
constant along $U$ which is a contradiction. Therefore, $\dim D_{y_{0}}\geq r$
which implies $[C_{y_{0}}]\neq[C(y_{0})]$.

Next, suppose $[C_{y_{0}}]\neq[C(y_{0})]$. Then $\dim D_{y_{0}}\geq r$.
Replace $Y$ with an affine open neighborhood of $y_{0}$. Then $h_{|S}^{-1}Y$
is affine because $S\rightarrow Y$ is finite. Replace $X$ with an affine open
subset containing $h_{|S}^{-1}Y$. As $X$ is affine we can fix an embedding
$C\hookrightarrow X\times\mathbb{P}^{m}$. Denote by
$A(C):=\mathcal{O}_{X}[v_{0},\ldots,v_{m}]/\mathcal{I}_{C}$ the homogeneous
coordinate ring of $C$ in $X\times\mathbb{P}^{m}$. Select $r$ general (to be
specified below) hypersurfaces $V_{1},\ldots,V_{r}$ in $X\times\mathbb{P}^{m}$
such that each hypersurface is defined by a general $\Bbbk$-linear combination
of the generators of a sufficiently high power of the irrelevant ideal
$(v_{0},\ldots,v_{m})$.

Let’s specify the genericity conditions on the $V_{i}$s. We rely on the fact
that irrelevant ideal $(v_{0},\ldots,v_{m})$ is maximal in $A(C)$. By prime
avoidance we can select the defining equations for the $V_{i}$s in $A(C)$ in
such a way that the ideal of $V:=V_{1}\cap\ldots\cap V_{r}$ is of height $r$
modulo each minimal prime. Then by the dimension formula [Mat70, Ch. 5, §14]
$V\cap C$ is equidimensional of dimension equal to that of $Y$. Furthermore,
by prime avoidance we can ensure that $\dim(V\cap D)=\dim Y-1$. Thus, the
generic point of each irreducible component of $V\cap C$ is not contained in
$D$. Because $C$ is birational with $X$, we get that $c(V\cap C)$ is
equidimensional and $\dim c(V\cap C)=\dim Y$. Because $\dim D(y_{0})=r-1$ by
prime avoidance we can select the $V_{i}$s so that $V\cap D(y_{0})=\emptyset$.
Thus $c(V\cap C(y_{0}))$ is either empty or consists of finitely many points
which do not lie in $S_{y_{0}}$. By shrinking $X$ a little bit we get $V\cap
C(y_{0})=\emptyset$. But $[C_{y_{0}}]-[C(y_{0})]\subset D_{y_{0}}$. Because
$\dim D_{y_{0}}\geq r$ we have $V\cap C_{y_{0}}\neq\emptyset$. Thus, $V\cap
C_{y_{0}}\subset D_{y_{0}}$. Hence $c(V\cap C)_{y_{0}}\subset S_{y_{0}}$.
Therefore, $(h\circ c)(V\cap C)$ is dense in $Y$.

We have the following local version of Bertini’s theorem. Denote by
$\mathfrak{m}_{y_{0}}$ the maximal ideal of $\mathcal{O}_{Y,y_{0}}$. Let
$P_{1},\ldots,P_{n}$ be prime ideals in $\mathcal{O}_{Y,y_{0}}$ different from
$\mathfrak{m}_{y_{0}}$. Let
$h_{1}\in\mathfrak{m}_{y_{0}}-\mathfrak{m}_{y_{0}}^{2}$. Note that the ideal
$(h_{1})+\mathfrak{m}_{y_{0}}^{2}$ avoids each $P_{i}$. Then by E. Davis’
version of prime avoidance [Mat86, Ex. 16.8], there exists
$h_{2}\in\mathfrak{m}_{y_{0}}^{2}$ such that $h:=h_{1}+h_{2}$ avoids each
$P_{i}$. Furthermore, because $h_{1}\not\in\mathfrak{m}_{y_{0}}^{2}$ the ring
$\mathcal{O}_{Y,y_{0}}/h\mathcal{O}_{Y,y_{0}}$ is regular. We will apply this
observation to reduce the proof of part $\rm{(ii)}$ to the case $\dim Y=1$.

Say that a component of $D$ is vertical if it maps to a proper closed subset
of $Y$. Set $\dim Y:=k$. Using repeatedly the local version of Bertini’s
theorem we select a nested sequence $Y:=Y_{0}\supset Y_{1}\ldots\supset
Y_{k-1}$ where $Y_{i}$ is regular hypersurface in $Y_{i-1}$ through $y_{0}$
for $i=1,\ldots,k-2$ subject to the following genericity conditions:

  * (1)

$Y_{1}$ is chosen so that it intersects the images under $g\circ c$ of all
vertical components of $D$.

  * (2)

For each $i=1,\ldots,r-1$ set $C_{Y_{i}}:=C\times_{Y}Y_{i}$. Denote by
$C(Y_{i})$ the blowup of $X\times_{Y}Y_{i}$ with center $S\times_{Y}Y_{i}$ and
by $D(Y_{i})$ the corresponding exceptional divisor. We choose $Y_{i}$ so that
it intersects properly the closures of the images under $h\circ c$ of the
vertical components of $D(Y_{i-1})$.

  * (3)

$Y_{k-1}$ is not in the closure of $Y\setminus(h\circ c)(V\cap C)$.

Then by applying repeatedly Bertini’s theorem for extreme morphisms [Ran18,
Thm. 4.3] for the semi-local scheme $(X\times_{Y}Y_{i},S\times_{Y}Y_{i})$ we
get $C_{Y_{i}}=D_{y_{0}}\cup C(Y_{i})$. In particular,
$C_{Y_{k-1}}=D_{y_{0}}\cup C(Y_{k-1})$ where $Y_{k-1}$ is a regular curve.

We can assume that $(C\cap V)\times_{Y}Y_{k-1}$ surjects onto $Y_{r-1}$ after
possibly replacing the latter with a smaller affine neighborhood containing
$y_{0}$. But $(C\cap V)\times_{Y}Y_{k-1}=C_{Y_{k-1}}\cap V$. Thus $V\cap
C(Y_{k-1})$ is a curve that surjects onto $Y_{k-1}$. In particular, $V\cap
C(Y_{k-1})_{y_{0}}\neq\emptyset$. But $V\cap C(y_{0})=\emptyset$ and $V$ is
generic. Thus $D(Y_{k-1})$ has a vertical component $D_{\mathrm{vert}}$. But
$D_{\mathrm{vert}}$ is supported over a subscheme of
$S_{y_{0}}\times\mathbb{P}^{m}$, so $\mathcal{O}_{C(Y_{k-1})}(1)$ is ample on
$D_{\mathrm{vert}}$. Set $l:=c_{1}\mathcal{O}_{C(Y_{k-1})}(1)$. Then
$l^{r}[D_{\mathrm{vert}}]>0$. By part $\rm{(i)}$ we obtain

$\int_{S_{y_{0}}}l_{y_{0}}^{r-1}[D(y_{0})]\neq\int_{S_{y}}l_{y}^{r-1}[D(y)]$

for generic $y\in Y_{k-1}$. ∎

Below we prove strengthening of Thm. 2.7 (i). We show that the assumption that
$S_{y_{0}}$ is nowhere dense in $X_{y_{0}}$ in part $\rm{(i)}$ can be
substantially relaxed which is what’s needed for applications to
equisingularity theory.

######  Corollary 2.2.

Let $X_{y_{0}}^{\prime}$ be the union of those components of $X_{y_{0}}$ that
are not irreducible components of $S$. Set
$S_{y_{0}}^{\prime}=S_{y_{0}}\times_{X_{y_{0}}}X_{y_{0}}^{\prime}$. Denote by
$C^{\prime}(y_{0})$ the blowup of $X_{y_{0}}^{\prime}$ with center
$S_{y_{0}}^{\prime}$ and denote by $D^{\prime}(y_{0})$ its exceptional
divisor. Then the EDF remains valid after replacing $D(y_{0})$ with
$D^{\prime}(y_{0})$.

###### Proof.

Note that $C_{y_{0}}$ and $C^{\prime}(y_{0})$ are isomorphic over points $x\in
X_{y_{0}}$ with $x\not\in S_{y_{0}}$. Therefore the cycle
$[C(y_{0})]-[C_{y_{0}}]$ is supported over the irreducible components of $S$
that are also irreducible components of $X_{y_{0}}$. Hence, once again
$[C(y_{0})]-[C_{y_{0}}]=-[D_{\mathrm{vert}}].$ Additionally, observe that
$D\cdot[C^{\prime}(y_{0})]=[D^{\prime}(y_{0})]$ because
$D\times_{C}C^{\prime}(y_{0})=D^{\prime}(y_{0})$. The rest of the proof goes
through unchanged. ∎

In equisingularity theory we apply Cor. 2.2 with $C$ the blowup of the
relative conormal space of a family $(\mathfrak{X},x_{0})\rightarrow(Y,y_{0})$
with center the singular locus of $\mathfrak{X}$. In this setting
$X_{y_{0}}^{\prime}$ is the conormal space of $\mathfrak{X}_{y_{0}}$. So the
intersection numbers appearing on the left-hand side of the EDF depend only on
the fibers.

For related results to Thm. 2.1 see [Kol18, Chp. 5]. As a first application of
Theorem 2.7 (i) we recover a result by Teissier for the Hilbert– Samuel
multiplicity. Preserve the setup of Thm. 2.1. Assume $S$ is finite over $Y$.
Following Fulton (see [Ram73], and §4.3 and Ex. 4.3.4 in [Ful84]) define the
Hilbert–Samuel multiplicity

$e(S_{y},X_{y}):=\int_{S_{y}}l_{y}^{r-1}[D(y)].$

######  Theorem 2.3 (Teissier, Prp. 3.1 in [T73] and Rmk. 5.1.1 in [T81]).

We have

$e(S_{y_{0}},X_{y_{0}})-e(S_{y},X_{y})=\mathrm{deg}(D_{\mathrm{vert}})$

where $\mathrm{deg}(D_{\mathrm{vert}}):=\int l^{r}[D_{\mathrm{vert}}].$

###### Proof.

Follows immediately from Thm. 2.1 (i). ∎

######  Remark 2.4.

In complex analytic singularity theory one works with a complex analytic germ
$(X,0)\subset(\mathbb{C}^{n+k},0)$ and a smooth subspace $(Y,0)\subset(X,0)$
which can be taken to be $(\mathbb{C}^{k},0)$. In this situation $h\colon
X\rightarrow Y$ is obtained from a transverse projection to $Y$ (see the
beginning of Sct. 6). To obtain the EDF in this setting one needs to work with
the right analogue of an affine subsets of $X$ and $Y$. This is done via
distinguished compact Stein sets: one takes an open neighborhood $U$ of $0$ in
$\mathbb{C}^{n+k}$ and considers $Q\cap X$ where $Q$ is a compact stone in $U$
(see Chp. II, Sct. 1 and Chp. III in Moonen’s appendix in [HIO88]). We do the
same for $(Y,0)$. Next one considers the corresponding analytic spectrums and
the projective analytic spectrum on the blowup. The basic intersection theory
we used in the proof of the EDF extends to this new setting (cf. [Mas03, App.
A]).

######  Example 2.5.

Thm. 2.3 provides a useful tool in the deformation theory of singularities.
Consider an analytic function
$f\colon(\mathbb{C}^{n},0)\rightarrow(\mathbb{C},0)$ with an isolated critical
point at the origin. Let ${\bf x}:=(x_{1},\ldots,x_{n})$ be a linear choice of
coordinates. Denote by $J(f):=(\partial f/\partial x_{1},\ldots,\partial
f/\partial x_{n})$ the Jacobian ideal of $f$ in
$\mathcal{O}_{\mathbb{C}^{n},0}$. Because the generators for $J(f)$ form a
regular sequence in the regular ring $\mathcal{O}_{\mathbb{C}^{n},0}$, the
Hilbert–Samuel multiplicity $e(J(f),\mathcal{O}_{\mathbb{C}^{n},0})$ is the
Milnor number
$\mu(f):=\dim_{\mathbb{C}}\mathbb{C}\\{x_{1},\ldots,x_{n}\\}/J(f)$. Consider
$f_{t}\colon(\mathbb{C}^{n}\times\mathbb{C},(0,0))\rightarrow(\mathbb{C},0)$
where $f_{0}=f$ and $f_{t}$ for $t\neq 0$ is a generic perturbation of $f$
such that its critical points are nondegenerate, or Morse. Let $J_{{\bf
x}}(f_{t})$ be the relative Jacobian ideal, i.e. the ideal generated by the
partials of $f_{t}$ with respect to the ${\bf x}$ coordinates. Let $S$ be the
subscheme of $\mathbb{C}^{n}\times\mathbb{C}$ defined by $J_{{\bf x}}(f_{t})$.
There are no vertical components of the exceptional divisor $D$ of
$\mathrm{Bl}_{S}(\mathbb{C}^{n}\times\mathbb{C})$ because $D_{0}$ is set-
theoretically in $\\{0\\}\times\mathbb{P}^{n-1}$. Applying Thm. 2.3 we get
that $\mu(f)=\mu(f_{t})$ for generic $t$. But $f_{t}$ has only Morse critical
points and their number is $\mu(f_{t})$. Thus $\mu(f)$ equals the number of
critical points of a generic perturbation of $f$.

As another example, let $(X,0)\rightarrow(Y,0)$ be a smoothing of an isolated
hypersurface $(X_{0},0)$ singularity, defined as the zero locus in
$(\mathbb{C}^{N},0)$ of a complex analytic function $F$, and $S$ be the
subscheme defined by the partials of $F$ with respect to the fiber
coordinates. Applying the EDF to this setting we get that the first left-hand
side term is equal to the Hilbert–Samuel multiplicity of the Jacobian ideal in
$\mathcal{O}_{X_{0},0}$ associated with $X_{0}$, whereas the second term is
zero because $X_{y}$ is smooth. By conservation of number argument, the right-
hand side of the EDF is equal to the intersection multiplicity of the relative
polar curve associated with the total space of the family with a generic
fiber. In turn, by [T73, Chp. 2, Prp. 2.1] this intersection multiplicity is
the sum of the Milnor number of $(X_{0},0)$ and the Milnor number of a generic
hyperplane slice of $(X_{0},0)$.

In what follows we extend Thm. 2.1 to the more general case when we work with
a projective scheme over $X$. Let $X\rightarrow Y$ be a morphism of schemes of
finite type over a field $\Bbbk$ with equidimensional fibers. Assume $X$ is
equidimensional and reduced and $Y$ is integral and regular of dimension one.
Consider two graded sheaves of $\mathcal{O}_{X}$-algebras
$\mathcal{A}:=\bigoplus\mathcal{A}_{n}$ and
$\mathcal{A}^{\prime}:=\bigoplus\mathcal{A}_{n}^{\prime}$ with
$\mathcal{A}_{0}=\mathcal{A}_{0}^{\prime}=\mathcal{O}_{X}$ such that
$\mathcal{A}$ and $\mathcal{A}^{\prime}$ are locally generated by
$\mathcal{A}_{1}$ and $\mathcal{A}_{1}^{\prime}$ as
$\mathcal{O}_{X}$-algebras. Assume $\mathcal{A}_{1}$ and
$\mathcal{A}_{1}^{\prime}$ are coherent $\mathcal{O}_{X}$-modules and
$\mathcal{A}\subset\mathcal{A}^{\prime}$ is a homogeneous and birational
inclusion. Set $P:=\mathrm{Proj}(\mathcal{A})$ and
$Q:=\mathrm{Proj}(\mathcal{A}^{\prime})$. Assume that the irreducible
components of $P$ and $Q$ surject onto those of $X$. Denote by
$Z:=\mathbb{V}(\mathcal{A}_{1})$ the variety of the ideal sheaf in
$\mathcal{A}^{\prime}$ generated by $\mathcal{A}_{1}$. Assume $Z$ is proper
over $Y$. Let $C$ be the blowup of $Q$ with center $Z$ and let $D$ be its
exceptional divisor. Denote by $c_{p}$ and $c_{q}$ the projections of $C$ to
$P$ and $Q$ respectively. Let $p\colon P\rightarrow X$ and $q\colon
Q\rightarrow X$ be structure morphisms. Set $\mathcal{L}:=\mathcal{O}_{P}(1)$
and $l:=c_{1}(\mathcal{L})$, and $\mathcal{L}^{\prime}:=\mathcal{O}_{Q}(1)$
and $l^{\prime}:=c_{1}(\mathcal{L}^{\prime})$.

Further, we assume that $\mathcal{A}$ and $\mathcal{A}^{\prime}$ are contained
in a graded $\mathcal{O}_{X}$-algebra sheaf
$\mathcal{B}:=\oplus\mathcal{B}_{n}$ satisfying locally the properties
described in Prp. 3.2. If $\mathcal{A}^{\prime}$ is reduced, such an embedding
exists by Prp. 3.2. For each $y\in Y$ denote by $\mathcal{B}_{y}$ the induced
sheaf on $X_{y}$ and by $\mathcal{A}(y)$ and $\mathcal{A}^{\prime}(y)$ the
images of $\mathcal{A}$ and $\mathcal{A}^{\prime}$ in $\mathcal{B}_{y}$. Set
$P(y):=\mathrm{Proj}(\mathcal{A}(y))$ and
$Q(y):=\mathrm{Proj}(\mathcal{A}^{\prime}(y))$. Consider the following diagram

$\begin{CD}C(y)@>{c_{q}}>{}>Q(y)\\\ @V{}V{c_{p}}V@V{}V{q}V\\\
P(y)@>{p}>{}>X_{y}\end{CD}$

where $C(y)$ is the blowup of $Q(y)$ with center $Z\times_{Q}Q(y)$ and $D(y)$
is the corresponding exceptional divisor. Set
$\mathcal{L}_{y}:=\mathcal{O}_{P(y)}(1)$ and
$\mathcal{L}_{y}^{\prime}:=\mathcal{O}_{Q(y)}(1)$ and
$l_{y}:=c_{1}(\mathcal{L}_{y}^{\prime})$ and
$l_{y}^{\prime}:=c_{1}(\mathcal{L}_{y}^{\prime})$. Assume $P(y)$ and $Q(y)$
are birational and equidimensional of dimension $r$. Define the generalized
Buchsbaum–Rim multiplicity (see [KT94, Sct. 5])

(7)
$e(\mathcal{A}(y),\mathcal{A}^{\prime}(y)):=\sum_{i=0}^{r-1}\int_{Z_{y}}(c_{p}^{*}l_{y})^{r-i-1}(c_{q}^{*}l_{y}^{\prime})^{i}[D(y)].$

where $c_{p}^{*}l_{y}=c_{1}(c_{p}^{*}\mathcal{L}_{y})$ and
$c_{q}^{*}l_{y}^{\prime}=c_{1}(c_{q}^{*}\mathcal{L}_{y}^{\prime})$. Finally,
denote by $D_{\mathrm{vert}}^{P}$ and $D_{\mathrm{vert}}^{Q}$ the projections
of $D_{\mathrm{vert}}$ to $P$ and $Q$.

######  Theorem 2.6.

There exists an affine neighborhood $U$ of $y_{0}$ in $Y$ such that

(8)
$e(\mathcal{A}(y_{0}),\mathcal{A}^{\prime}(y_{0}))-e(\mathcal{A}(y),\mathcal{A}^{\prime}(y))=l^{r}[D_{\mathrm{vert}}^{P}]-l^{\prime
r}[D_{\mathrm{vert}}^{Q}].$

for each closed point $y\in U-\\{y_{0}\\}$.

###### Proof.

By Prp. 3.2 the irreducible components of $Q(y_{0})$ surject onto those of
$X_{y_{0}}$. Because $P(y_{0})$ and $Q(y_{0})$ are birational $q(Z_{y_{0}})$
and $X_{y_{0}}$ do not share irreducible components. By Cor. 2.2 applied to
the family $Q\rightarrow Y$ with $S:=Z$ and by (3) we get

(9)
$(c_{p}^{*}l_{y_{0}})^{r-i-1}(c_{q}^{*}l_{y_{0}}^{\prime})^{i}[D(y_{0})]-(c_{p}^{*}l_{y})^{r-i-1}(c_{q}^{*}l_{y}^{\prime})^{i}[D_{\mathrm{hor}}]_{y_{0}}=-(c_{p}^{*}l_{y})^{r-i-1}(c_{q}^{*}l_{y}^{\prime})^{i}D[D_{\mathrm{vert}}].$

By (5) and (6) we get

(10)
$\int_{Z_{y_{0}}}(c_{p}^{*}l_{y_{0}})^{r-i-1}(c_{q}^{*}l_{y_{0}}^{\prime})^{i}[D_{\mathrm{hor}}]_{y_{0}}=\int_{Z_{y}}(c_{p}^{*}l_{y})^{r-i-1}(c_{q}^{*}l_{y}^{\prime})^{i}[D(y)].$

By [KT94, Prp. 2.2 ] we have

(11) $\mathcal{O}_{C}(D)=c_{p}^{*}\mathcal{L}\otimes
c_{q}^{*}\mathcal{L}^{\prime-1}.$

Then (11) yields

(12)
$\sum_{i=1}^{r-1}-(c_{p}^{*}l_{y})^{r-i-1}(c_{q}^{*}(l_{y}^{\prime})^{i}D[D_{\mathrm{vert}}]=(c_{p}^{*}l)^{r}[D_{\mathrm{vert}}]-(c_{q}^{*}l^{\prime})^{r}[D_{\mathrm{vert}}].$

Summing over $i$ in (9) and plugging (12) and (10) in (9) we get

$e(\mathcal{A}(y_{0}),\mathcal{A}^{\prime}(y_{0}))-e(\mathcal{A}(y),\mathcal{A}^{\prime}(y))=(b_{p}^{*}l)^{r}[D_{\mathrm{vert}}]-(b_{c}^{*}l^{\prime})^{r}[D_{\mathrm{vert}}].$

Applying the projection formula (see [Ful84, Prp. 2.5 (c)]) to each term of
the right-hand side of the last equality we get (8). ∎

Suppose $X$ and $Y$ are local with closed points $x_{0}$ and $y_{0}$
respectively and suppose the fibers of $X\rightarrow Y$ are equidimensional of
positive dimension $d$. Let $\mathcal{M}\subset\mathcal{N}\subset\mathcal{F}$
be coherent $\mathcal{O}_{X}$-modules such that $\mathcal{F}$ is free and
$\mathcal{M}$ and $\mathcal{N}$ are free of constant rank $e$ at the generic
point of each irreducible component of $X$. Let $\mathcal{R}(\mathcal{M})$ and
$\mathcal{R}(\mathcal{N})$ be the Rees algebras of $\mathcal{M}$ and
$\mathcal{N}$ respectively. These are defined as the subalgebras of
$\mathrm{Sym}(\mathcal{F})$ generated in degree one by the generators of each
of the two modules.

Now work in the setup of Thm. 2.6. Set
$C:=\mathrm{Proj}(\mathcal{R}(\mathcal{M}))$ and
$P:=\mathrm{Proj}(\mathcal{R}(\mathcal{N}))$. Assume $q(Z)$ is finite over
$Y$. One can check that this is equivalent to assuming that
$\mathrm{Supp}_{X}(\overline{\mathcal{N}}/\overline{\mathcal{M}})$ is finite
over $Y$, where $\overline{\mathcal{N}}$ and $\overline{\mathcal{M}}$ are the
integral closure of the two modules in $\mathcal{F}$ (see Sct. 6 for the
definition of integral closure of a module). For each $y\in Y$ denote by
$\mathcal{F}_{y}$ the restriction of $\mathcal{F}$ to $X_{y}$ and by
$\mathcal{M}(y)$ and $\mathcal{N}(y)$ the images of $\mathcal{M}$ and
$\mathcal{N}$ in $\mathcal{F}_{y}$. Define the Buchsbaum–Rim multiplicity
$e(\mathcal{M}(y),\mathcal{N}(y))$ as in (7) with
$\mathcal{A}(y)=\mathcal{R}(\mathcal{M}(y))$ and
$\mathcal{A}^{\prime}(y)=\mathcal{R}(\mathcal{N}(y))$. Here we turn results of
Kleiman and Thorup (see [KT94, Sct. 5]) into a definition. In the original
treatment of Buchsbaum and Rim [BR64], the multiplicity
$e(\mathcal{M}(y),\mathcal{N}(y))$ is defined as the normalized leading
coefficient of the length $\lambda(\mathcal{N}^{i}(y)/\mathcal{M}^{i}(y))$
which is a polynomial of degree $r:=d+e-1$ for $i$ large enough, where
$\mathcal{N}^{i}(y)$ and $\mathcal{M}^{i}(y)$ are the $i$th graded components
of the respective Rees algebras.

Suppose $\Bbbk$ is algebraically closed field of characteristic zero. Assume
$X$ is generically reduced and $Y$ is smooth of arbitrary dimension. Let
$NF(\mathcal{M})$ be the nonfree locus of $\mathcal{M}$ in $X$. Consider the
composition of maps

$p^{-1}(NF(\mathcal{M}))\hookrightarrow
X\times\mathbb{P}^{g(M)-1}\xrightarrow{pr_{2}}\mathbb{P}^{g(M)-1}$

where $g(M)$ is the number of a generating set for $\mathcal{M}$ as an
$\mathcal{O}_{X}$-module. As $\mathcal{M}$ is of generic rank $e$, by the
Kleiman Transversality Theorem [Kle74], the intersection of
$p^{-1}(NF(\mathcal{M}))$ with a general plane $H_{r}$ from
$\mathbb{P}^{g(M)-1}$ of codimension $r$, is of dimension at most $\dim Y-1$.
Denote by $\Gamma_{d}(\mathcal{M})$ the projection of
$\mathrm{Proj}(\mathcal{R}(\mathcal{M}))\cap H_{r}$ to $X$. This is what
Gaffney [Gaff04] calls the polar curve of $\mathcal{M}$. For a generic $y\in
Y$ the fiber of $\Gamma_{d}(\mathcal{M})$ over $y$ consists of the same number
of points, each of them appearing with multiplicity one because $X$ is
generically reduced, and because at each one of them $\mathcal{M}$ is free.
Denote this number by $\mathrm{mult}_{Y}\Gamma_{d}(\mathcal{M})$. Similarly,
define $\mathrm{mult}_{Y}\Gamma_{d}(\mathcal{N}).$ Thm. 2.6 yields Gaffney’s
Multiplicity-Polar Theorem.

######  Corollary 2.7.

(Gaffney, [Gaf04]) Suppose $\Bbbk$ is algebraically closed of characteristic
zero. Suppose $X$ is generically reduced, $Y$ is smooth, and
$\mathrm{Supp}_{X}(\overline{\mathcal{N}}/\overline{\mathcal{M}})$ is finite
over $Y$. Then

$e(\mathcal{M}(y_{0}),\mathcal{N}(y_{0}))-e(\mathcal{M}(y),\mathcal{N}(y))=\mathrm{mult}_{Y}\Gamma_{d}(\mathcal{M})-\mathrm{mult}_{Y}\Gamma_{d}(\mathcal{N}).$

###### Proof.

Let $P_{\mathrm{vert}}$ and $Q_{\mathrm{vert}}$ be the union of irreducible
components of $p^{-1}X_{y_{0}}$ and $q^{-1}X_{y_{0}}$ that are supported over
$x_{0}$. Then $D_{\mathrm{vert}}^{P}=P_{\mathrm{vert}}$ and
$D_{\mathrm{vert}}^{Q}=Q_{\mathrm{vert}}$. By a conservation of number, for
generic smooth curve $Y^{\prime}$ passing through $y_{0}$, the degrees on the
right-hand side of (8) are equal to $\mathrm{mult}_{Y}\Gamma_{d}(\mathcal{M})$
and $\mathrm{mult}_{Y}\Gamma_{d}(\mathcal{N})$, respectively. Then the
statement follows from (8). ∎

If $\mathcal{N}$ is free, then $\dim q^{-1}x_{0}=e-1<r$, so
$\mathrm{mult}_{Y}\Gamma_{d}(\mathcal{N})=0$. Additionally, if $e=1$ one
recovers Thm. 2.3. In these situations the change of multiplicities detects
precisely the presence of $P_{\mathrm{vert}}$. In general, the multiplicities
may fail to detect $P_{\mathrm{vert}}$ because of the contribution of
$\mathrm{mult}_{Y}\Gamma_{d}(\mathcal{N})$. To get rid of
$\mathrm{mult}_{Y}\Gamma_{d}(\mathcal{N})$, instead of the Buchsbaum–Rim
multiplicity we use the local volume which absorbs
$\mathrm{mult}_{Y}\Gamma_{d}(\mathcal{N})$. This is done by the LVF in Thm.
4.1. Another way to do this is indicated in Rmk. 4.11. There an asymptotic
version of (8) is indicated in which the contribution coming from the various
$D_{\mathrm{vert}}^{Q}$ “vanishes asymptotically”.

##  3\. Computing and vanishing of local volumes

Let $(R,\mathfrak{m})$ be a reduced Noetherian local equidimensional ring of
dimension at least $2$. Let
$\mathcal{A}:=\oplus_{i=0}^{\infty}\mathcal{A}_{i}$ be a reduced
equidimensional standard graded $R$-algebra of dimension $r$. Denote by
$\lambda_{R}(-)$ the length function. In this section we will be interested in
computing

(13)
$\varepsilon(\mathcal{A}):=\limsup_{n\to\infty}\frac{r!}{n^{r}}\lambda_{R}(H_{\mathfrak{m}}^{1}(\mathcal{A}_{n})).$

In our applications to geometry, $R$ is be assumed to be essentially of finite
type over a field and $\mathcal{A}$ is the homogeneous coordinate ring of
$\mathrm{Proj}(\mathcal{A})$ associated with an invertible very ample sheaf
$\mathcal{L}$ on $\mathrm{Proj}(\mathcal{A})$ relative to $\mathrm{Spec}(R)$.
Then $(\ref{epsilon})$ is what Fulger[Ful13] calls the local volume of
$\mathcal{L}$.

When $\mathcal{A}$ is the Rees algebra of a torsion-free finitely generated
$R$-module $\mathcal{M}$ (see [EHU03] for a definition), then (13) is called
the epsilon multiplicity of $\mathcal{M}$. It is denoted by
$\varepsilon(\mathcal{M})$ (see [UV11] and the remark below).

Often it’s preferable to work with an $H_{\mathfrak{m}}^{0}$ instead of
$H_{\mathfrak{m}}^{1}$. For example, we can do this if we assume that the
minimal primes of $\mathcal{A}$ contract to minimal primes of $R$. Set
$\mathfrak{X}:=\mathrm{Spec}(R)$. Let $x_{0}$ be the closed point of
$\mathfrak{X}$. Then $\mathcal{A}$ viewed as
$\mathcal{O}_{\mathfrak{X},x_{0}}$-module is torsion free. Set
$U:=\mathfrak{X}-x_{0}$. For each $n$ consider the standard exact sequence

$H_{x_{0}}^{0}(\mathfrak{X},\widetilde{\mathcal{A}_{n}})\rightarrow
H^{0}(\mathfrak{X},\widetilde{\mathcal{A}_{n}})\rightarrow
H^{0}(U,\widetilde{\mathcal{A}_{n})}\rightarrow
H_{x_{0}}^{1}(\mathfrak{X},\widetilde{\mathcal{A}_{n}})\rightarrow
H^{1}(\mathfrak{X},\widetilde{\mathcal{A}_{n}})$

where $\widetilde{A_{n}}$ is the associated sheaf of $\mathcal{A}_{n}$. The
two extreme terms vanish because $\mathcal{A}$ is torsion free and
$\mathfrak{X}$ is affine. Thus we can compute $H_{x_{0}}^{1}(\mathcal{A}_{n})$
as $H^{0}(U,\widetilde{\mathcal{A}_{n}})/\widetilde{\mathcal{A}_{n}}$ (see Ex.
3.3 (b) Chp. III in [Har77]).

The existence of (13) as a limit has been established in some cases by Das
[Das21] and Cutkosky [Cut15] based on ideas of Kaveh and Khovanskii [KK12],
and Okounkov [Oko03], and by Fulger [Ful13].

Assume $\mathcal{B}$ is a graded $R$-algebra, not necessarily finitely
generated over $R$, such that $\mathcal{A}\subset\mathcal{B}$ is a homogeneous
inclusion and

(14)
$\lim_{n\to\infty}\frac{r!}{n^{r}}\lambda_{R}(H_{\mathfrak{m}}^{i}(\mathcal{B}_{n}))=0$

for $i=1$ and $2$. Then an exact sequence of local cohomology yields

(15)
$\limsup_{n\to\infty}\frac{r!}{n^{r}}\lambda_{R}(H_{\mathfrak{m}}^{1}(\mathcal{A}_{n}))=\limsup_{n\to\infty}\frac{r!}{n^{r}}\lambda_{R}(H_{\mathfrak{m}}^{0}(\mathcal{B}_{n}/\mathcal{A}_{n})).$

The reason we will make use of (15) is that
$H_{\mathfrak{m}}^{0}(\mathcal{B}_{n}/\mathcal{A}_{n})$ is more manageable
than $H_{\mathfrak{m}}^{1}(\mathcal{A}_{n})$ when studying the behavior of
$\varepsilon(\mathcal{A})$ in families. Below we list several instances when
the representation (15) is possible. Denote by $c_{\mathcal{A}}$ and by
$c_{\mathcal{B}}$ the structure morphisms from $\mathrm{Proj}(\mathcal{A})$
and $\mathrm{Proj}(\mathcal{B})$ to $\mathrm{Spec}(R)$, respectively.

######  Proposition 3.1.

Assume that $(R,\mathfrak{m})$ is a local Noetherian equidimensional ring with
$\dim R\geq 2$. The following holds:

  1. (i)

Assume $\mathrm{depth}(R)\geq 2$. Suppose the minimal primes of $\mathcal{A}$
contract to minimal primes of $R$. Then (15) holds with
$\mathcal{B}_{n}:=\mathcal{A}_{n}^{**}$, where $\mathcal{A}_{n}^{**}$ is the
double dual of the $R$-module $\mathcal{A}_{n}$.

  2. (ii)

Assume that $R$ is Nagata and $\mathcal{B}$ is reduced, equidimensional of
dimension $r$. Suppose that
$\operatorname{codim}c_{\mathcal{B}}^{-1}(\mathfrak{m})\geq 2$ in
$\mathrm{Proj}(\mathcal{B})$ and the minimal primes of $\mathcal{B}$ contract
to minimal primes of $R$. Then (15) holds.

  3. (iii)

Assume $R$ is essentially of finite type over a field. Let $\mathcal{M}$ be an
$R$-module free of rank $e$ at locally at each minimal prime of $R$. Then
$\mathcal{M}$ admits an embedding into an $R$-free module $\mathcal{F}$ of
rank $e$ such that (15) holds with $\mathcal{A}:=\mathcal{R}(\mathcal{M})$ and
$\mathcal{B}:=\mathrm{Sym}(\mathcal{F})$. Moreover, the right-hand side of
(15) is in fact a limit.

###### Proof.

Consider $\rm{(i)}$. Because $\mathcal{A}$ is reduced and its minimal primes
contract to minimal primes of $R$, then $\mathcal{B}$ and $\mathcal{A}$ are
torsion free. Because $X$ is reduced, the natural map
$\mathcal{A}_{n}\rightarrow\mathcal{A}_{n}^{**}$ is injection by a
straightforward generalization of [Stks, Tag 0AV0]. But $\mathcal{A}_{n}^{**}$
is reflexive, so any $2$-regular sequence from $\mathfrak{m}$ lifts to a
$2$-regular sequence of $\mathcal{A}_{n}^{**}$ (see [Stks, Tag 0AV5]). Thus,
$H_{x_{0}}^{i}(\mathcal{B}_{n})=0$ for $i=0,1$ by [Gr67, Thm. 3.8].

Consider $\rm{(ii)}$. Observe that the total ring of fractions
$Q(\mathcal{B})$ of $\mathcal{B}$ is a graded ring. Denote by
$\overline{\mathcal{B}}$ the integral closure of $\mathcal{B}$ in
$Q(\mathcal{B})$. Then $\overline{\mathcal{B}}$ is graded by Prp. 2.3.5 in
[SH06]. Because $\mathcal{B}$ is reduced and is of finite type over $R$, which
is Nagata, then $\overline{\mathcal{B}}$ is module-finite over $\mathcal{B}$
by [Stks, Tag 03GH]. Thus
$H_{\mathfrak{m}}^{0}(\overline{\mathcal{B}}/\mathcal{B})$ is a finite
$\mathcal{B}/\mathfrak{m}^{l}\mathcal{B}$-module for some positive $l$. This
implies that
$\lambda_{R}(H_{\mathfrak{m}}^{0}(\overline{\mathcal{B}_{n}}/\mathcal{B}_{n}))$
is a polynomial of degree at most $\dim c_{\mathcal{B}}^{-1}(\mathfrak{m})$.
Because $\mathcal{B}$ is equidimensional of dimension $r$, and the minimal
primes of $\mathcal{B}$ contract to minimal primes of $R$, then $\dim
c_{\mathcal{B}}^{-1}(\mathfrak{m})\leq r-1$. Hence

(16)
$\limsup_{n\to\infty}\frac{r!}{n^{r}}\lambda_{R}(H_{\mathfrak{m}}^{0}(\overline{\mathcal{B}_{n}}/\mathcal{B}_{n}))=0.$

Consider the sequence

(17)
$H_{\mathfrak{m}}^{0}(\overline{\mathcal{B}_{n}}/\mathcal{B}_{n})\rightarrow
H_{\mathfrak{m}}^{1}(\mathcal{B}_{n})\rightarrow
H_{\mathfrak{m}}^{1}(\overline{\mathcal{B}_{n}}).$

Because $\overline{\mathcal{B}}$ and $\mathcal{B}$ are generically equal, each
minimal prime of the former ring contracts to a minimal prime of the latter.
Select $x_{1}\in\mathfrak{m}$ that avoids the minimal primes of $R$. Hence
$x_{1}$ avoids the minimal primes of $\overline{\mathcal{B}}$. Because
$\overline{\mathcal{B}}$ is normal, it follows that the associated primes of
the ideal $(x_{1})$ generated by $x_{1}$ in $\overline{\mathcal{B}}$ are of
height one. But $\operatorname{codim}c_{\mathcal{B}}^{-1}(\mathfrak{m})\geq 2$
in $\mathrm{Proj}(\mathcal{B})$, so
$\operatorname{codim}c_{\overline{\mathcal{B}}}^{-1}(\mathfrak{m})\geq 2$ in
$\mathrm{Proj}(\overline{\mathcal{B}})$. Therefore, none of the associated
primes of $(x_{1})$ in $\overline{\mathcal{B}}$ contracts to $\mathfrak{m}$.
Hence by prime avoidance we can select $x_{2}\in\mathfrak{m}$ such that
$x_{2}$ is a nonzero divisor of
$\overline{\mathcal{B}}/x_{1}\overline{\mathcal{B}}$. In this way we have
constructed a $2$-regular sequence of $\overline{\mathcal{B}_{n}}$ for each
$n$. Thus $H_{\mathfrak{m}}^{i}(\overline{\mathcal{B}_{n}})=0$ for $i=0,1$ by
[Gr67, Thm. 3.8]. Finally, (17) and (16) yield (15).

Consider $\rm{(iii)}$. Note that the Rees algebra of
$\mathcal{R}(\mathcal{M})$ is reduced because $X$ is reduced. Thus
$\mathcal{R}(\mathcal{M})\hookrightarrow\mathcal{R}(\mathcal{M})\otimes Q(R)$
where $Q(R)$ is the total ring of fractions of $R$. Let
$\mathfrak{p}_{1},\ldots,\mathfrak{p}_{q}$ be the minimal primes of $R$. By
hypothesis $\mathcal{M}_{\mathfrak{p}_{i}}=e$ for each for each minimal prime
$\mathfrak{p}_{i}$. Then there exists $e$ generic combinations of the
generators of $\mathcal{M}$ that generate $\mathcal{M}_{\eta_{i}}$ for each
$i$. Consider the module $\mathcal{F}^{\prime}$ generated by these elements in
$\mathcal{M}\otimes Q(R)$. Scale the generators of $\mathcal{F}^{\prime}$ with
elements from $Q(R)$ in such a way that all of the remaining generators of
$\mathcal{M}$ can be expressed as a linear combination of the scaled
generators $\mathcal{F}^{\prime}$ with coefficients in $R$. Denote by
$\mathcal{F}$ the $R$-module generated by the scaled generators of
$\mathcal{F}^{\prime}$. Then $\mathcal{F}$ is free of rank $e$. At this point
one can apply part (ii) with $\mathcal{A}:=\mathcal{R}(\mathcal{M})$ and
$\mathcal{B}:=\mathrm{Sym}(\mathcal{F})$ to derive (15), or proceed directly
as follows. Let $\mathcal{F}^{n}$ be the $n$th symmetric power of
$\mathcal{F}$ and $\mathcal{M}^{n}$ be the $n$th homogeneous component of the
subalgebra of $\mathrm{Sym}(\mathcal{F})$ generated by $\mathcal{M}$. Consider
the exact sequence

(18) $H_{\mathfrak{m}}^{0}(\mathcal{F}^{n})\rightarrow
H_{\mathfrak{m}}^{0}(\mathcal{F}^{n}/\mathcal{M}^{n})\rightarrow
H_{\mathfrak{m}}^{1}(\mathcal{M}^{n})\rightarrow
H_{\mathfrak{m}}^{1}(\mathcal{F}^{n}).$

Because $X$ is reduced of positive dimension and $\mathcal{F}^{n}$ is free,
then $H_{\mathfrak{m}}^{0}(\mathcal{F}^{n})=0$. Because $\mathcal{F}^{n}$ is
free of rank $\binom{e+n-1}{n}$, then $H_{\mathfrak{m}}^{1}(\mathcal{F}^{n})$
is equal to the direct sum of $\binom{e+n-1}{n}$ copies of
$H_{\mathfrak{m}}^{1}(R)$. Because $R$ is esentially of finite type over a
field, then $R$ is a homomorphic image of a regular ring. Because $\dim R\geq
2$, Grothendieck’s finiteness theorem (see Exposé VIII, Corollaire 2.3 in
[Gr68]) implies that $H_{\mathfrak{m}}^{1}(R)$ is finitely generated. Hence
$\lambda_{R}(H_{\mathfrak{m}}^{1}(\mathcal{F}^{n}))=O(n^{e-1})$. Recall that
by [Ran18, Prp. 5.1 $\rm{(2)}$] we have $\dim\mathcal{R}(\mathcal{M})=\dim
X+e-1$. Because $R$ is of positive dimension, then $r>e-1$, so
$\lim_{n\to\infty}\frac{r!}{n^{r}}\lambda_{R}(H_{\mathfrak{m}}^{1}(\mathcal{F}^{n}))=0$
which proves (15) for $\mathcal{A}:=\mathcal{R}(\mathcal{M})$ and
$\mathcal{B}:=\mathrm{Sym}(\mathcal{F})$.

Note that $R$ is analytically unramified because $R$ is reduced and of finite
type over a field. So by Theorem 3.2 in [Cut15]
$\limsup_{n\to\infty}\frac{r!}{n^{r}}\lambda_{R}(H_{\mathfrak{m}}^{0}(\mathcal{F}^{n}/\mathcal{M}^{n}))$
exists as a limit. ∎

The next proposition guarantees the existence of a $\mathcal{B}$ satisfying
the hypothesis of Prp. 3.1 (ii). The proof is based on Noether normalization.

######  Proposition 3.2 (Prp. 2.6 in [Ran19]).

Suppose $R$ is a reduced equidimensional universally catenary Noetherian ring
of positive dimension or an infinite field. Assume
$\mathcal{A}=\oplus_{i=0}^{\infty}\mathcal{A}_{i}$ is a reduced
equidimensional standard graded algebra over $R$. Assume that the minimal
primes of $\mathcal{A}$ contract to minimal primes of $R$. Then there exists a
standard graded $R$-algebra $\mathcal{B}=\oplus_{i=0}^{\infty}\mathcal{B}_{i}$
such that

  1. (1)

$\mathcal{B}$ is a birational extension of $\mathcal{A}$, and the inclusion
$\mathcal{A}\subset\mathcal{B}$ is homogeneous;

  2. (2)

For each prime $\mathfrak{p}$ in $R$ the minimal primes of
$\mathfrak{p}\mathcal{B}$ are of height at least
$\mathrm{ht}(\mathfrak{p}/\mathfrak{p}_{\mathrm{min}})$ where
$\mathfrak{p}_{\mathrm{min}}$ is a minimal prime of $R$ contained in
$\mathfrak{p}$.

######  Corollary 3.3.

Suppose $(R,\mathfrak{m})$ is excellent and equidimensional with $\dim R\geq
2$. Suppose $\mathcal{A}$ is reduced. Then $\varepsilon(\mathcal{A})$ exists
as a finite limit.

###### Proof.

Because $R$ is excellent, then $R$ is Nagata and universally catenary. By Prp.
3.2 applied with $\mathfrak{p}=\mathfrak{m}$ there exists a standard graded
$R$-algebra $\mathcal{B}$ such that $\mathcal{A}\subset\mathcal{B}$ is a
homogeneous inclusion, and $\mathcal{B}$ satisfies the hypothesis of Prp. 3.1
(ii) because $R$ is local and $\dim R\geq 2$. Then by Prp. 3.1 (ii) (15)
holds. By [Das21, Thm. 5] $\varepsilon(\mathcal{A})$ exists as a finite limit.
∎

The following proposition is key to proving our vanishing result for
$\varepsilon(\mathcal{A})$. It’s essentially the content of the two main
results of [Ran19]. Let $\mathcal{A}\subset\mathcal{B}$ be commutative rings
with identity. Denote by $\overline{\mathcal{A}}$ the integral closure of
$\mathcal{A}$ in $\mathcal{B}$.

######  Proposition 3.4.

Let $(R,\mathfrak{m})$ be a local Noetherian equidimensional ring. Let
$\mathcal{A}\subset\mathcal{B}$ be finitely generated $R$-algebras.

  1. (i)

Suppose $R$ is universally catenary and the minimal primes of $\mathcal{B}$
contract to minimal primes of $\mathcal{A}$. If
$\mathfrak{m}\in\mathrm{Ass}_{R}(\mathcal{B}/\overline{\mathcal{A}})$, then
$\operatorname{codim}c_{\mathcal{A}}^{-1}(\mathfrak{m})\leq 1$.

  2. (ii)

Suppose $\operatorname{codim}c_{\mathcal{B}}^{-1}(\mathfrak{m})\geq 2$. If
$\operatorname{codim}c_{\mathcal{A}}^{-1}(\mathfrak{m})\leq 1$, then
$\mathfrak{m}\in\mathrm{Ass}_{R}(\mathcal{B}/\overline{\mathcal{A}}).$

###### Proof.

By the second part of [Ran19, Prp. 2.1] or [Stks, Tag 05DZ] it’s enough to
consider primes in
$\mathrm{Ass}_{\mathcal{A}}(\mathcal{B}/\overline{\mathcal{A}})$ contracting
to $\mathfrak{m}$. Then part (i) follows from [Ran19, Thm. 1.1 (iii)]. Part
(ii) follows from [Ran19, Thm. 1.2]. ∎

The following theorem characterizes the vanishing of
$\varepsilon(\mathcal{A})$. The result is the main inspiration for introducing
the class of deficient conormal singularities in Sct. 8.

######  Theorem 3.5.

Let $(R,\mathfrak{m})$ be a reduced Noetherian local equidimensional ring with
$\dim R\geq 2$. Assume $R$ is Nagata and universally catenary. Assume
$\mathcal{A}$ is a reduced standard graded $R$-algebra such that its minimal
primes contract to minimal primes of $R$. Then

(19) $\varepsilon(\mathcal{A})=0$

if and only if $\operatorname{codim}c_{\mathcal{A}}^{-1}(\mathfrak{m})\geq 2$.

###### Proof.

As in the proof of Cor. 3.3 there exists a $\mathcal{B}$ with the properties
prescribed in Prp. 3.2 such that (15) holds. Assume
$\operatorname{codim}c_{\mathcal{A}}^{-1}(\mathfrak{m})\geq 2$. Denote by
$\overline{\mathcal{A}_{n}}$ the integral closure of $\mathcal{A}_{n}$ in
$\mathcal{B}_{n}$. Note that
$\overline{\mathcal{A}}=\oplus_{n=0}^{\infty}\overline{\mathcal{A}_{n}}$ with
$\overline{\mathcal{A}_{0}}=R$ by [SH06, Prp. 2.3.5].

By Prp. 3.4 (i)
$\mathfrak{m}\not\in\mathrm{Ass}_{R}(\mathcal{B}_{n}/\overline{\mathcal{A}_{n}})$.
Thus
$H_{\mathfrak{m}}^{0}(\mathcal{B}_{n}/\mathcal{A}_{n})\subset\overline{\mathcal{A}_{n}}/\mathcal{A}_{n}$
for each $n$. But $R$ is Nagata. Then so is $\mathcal{B}$. Furthermore,
$\mathcal{B}$ is reduced, because $\mathcal{A}$ is. So,
$\overline{\mathcal{A}}$ is module-finite over $\mathcal{A}$. Therefore, by
repeating the argument from the proof of Prp. 3.1 $\rm{(ii)}$ we obtain (19).

Conversely, assume that (19) holds. Suppose $b$ is an element from
$\mathcal{B}_{k}$ for some $k$ such that there exists positive $l$ with
$\mathfrak{m}^{l}b_{k}\in\overline{\mathcal{A}_{k}}$ and
$b_{k}\not\in\overline{\mathcal{A}_{k}}$. Set
$G^{\prime}:=\bigoplus_{i=1}^{\infty}\mathcal{A}_{ik}$ and let $G$ be the
algebra generated by $G$ and $b_{k}$. By shift in degrees assume $G^{\prime}$
and $G$ are standard graded. Then by [KT94, Cor. 5.10 ]

$\lambda_{R}(G_{n}/G^{\prime}_{n}):=e(G^{\prime},G)n^{r}/r!+\cdots$

where the Buchsbaum–Rim multiplicity $e(G^{\prime},G)$ is a positive integer.
Thus

$\limsup_{n\to\infty}\frac{r!}{(nk)^{r}}\lambda_{R}(H_{\mathfrak{m}}^{0}(\mathcal{B}_{nk}/\mathcal{A}_{nk}))\geq\lim_{n\to\infty}\frac{r!}{n^{r}}\lambda_{R}(G_{n}/G^{\prime}_{n})=e(G^{\prime},G)>0$

which contradicts our assumption. Hence
$\mathfrak{m}\not\in\bigcup_{n=1}^{\infty}\mathrm{Ass}_{R}(\mathcal{B}_{n}/\overline{\mathcal{A}_{n}})$
which by reasons of grading is equivalent to
$\mathfrak{m}\not\in\bigcup_{n=1}^{\infty}\mathrm{Ass}_{R}(\mathcal{B}/\overline{\mathcal{A}})$.
So by Prp. 3.4 (ii)
$\operatorname{codim}c_{\mathcal{A}}^{-1}(\mathfrak{m})\geq 2$ in
$\mathrm{Proj}(\mathcal{A})$. ∎

The forward direction of Thm. 3.5 was proved for the epsilon multiplicity
$\varepsilon(\mathcal{A}|\mathcal{B})$ with the additional hypothesis that
$\mathcal{B}$ is $R$-flat by Ulrich and Validashti (see [UV11, Thm. 4.2]). We
record two observations about primary decomposition and local cohomology that
will be used in the proof of the Local Volume Formula in the next section.
Their statements and proofs can be found in [AK12, Chp. 18] for example.

######  Proposition 3.6.

Let $X$ be an affine Noetherian scheme and let $S$ be a subscheme of $X$ with
an ideal sheaf $\mathcal{I}_{S}$ in $\mathcal{O}_{X}$. Assume $\mathcal{M}$
and $\mathcal{F}$ are coherent $\mathcal{O}_{X}$-modules with
$\mathcal{M}\subset\mathcal{F}$. Let $\mathcal{M}=\cap\mathcal{M}_{i}$ be a
primary decomposition of $\mathcal{M}$ in $\mathcal{F}$ with
$x_{i}=\mathrm{Ass}_{X}(\mathcal{F}/\mathcal{M}_{i})$. The following holds

$H_{S}^{0}(\mathcal{F}/\mathcal{M})=(\cap_{\\{j|x_{j}\not\subset
S\\}}\mathcal{M}_{j})/\mathcal{M}.$

######  Proposition 3.7.

Let $X$ be an affine Noetherian scheme, and let $x$ be a point in $X$.
Consider the nested chain of coherent $\mathcal{O}_{X}$-modules
$\mathcal{M}\subset\mathcal{N}\subset\mathcal{F}$. Assume that
$\mathcal{M}_{z}=N_{z}$ for every point $z\notin\overline{\\{x\\}}$. Then

$\mathcal{N}/\mathcal{M}\subset H_{x}^{0}(\mathcal{F}/\mathcal{M}).$

Furthermore, $H_{x}^{0}(\mathcal{F}/\mathcal{M})$ is the largest submodule of
$\mathcal{F}/\mathcal{M}$ equal to $\mathcal{M}$ locally off
$\overline{\\{x\\}}$.

Sometimes it will be more convenient to write the local cohomology in terms of
saturations. Recall that
$H_{S}^{0}(\mathcal{F}/\mathcal{M})=\cup_{n=0}^{\infty}(\mathcal{M}:_{\mathcal{F}}\mathcal{I}_{S}^{n})/\mathcal{M}$
and
$H_{x}^{0}(\mathcal{F}/\mathcal{M})=\cup_{n=0}^{\infty}(\mathcal{M}:_{\mathcal{F}}\mathcal{I}_{x}^{n})/\mathcal{M}.$

######  Remark 3.8.

We conclude this section with some historical background for the $\varepsilon$
invariant discussed above. Assume that $\mathcal{M}$ is an $R$-module embedded
in a free module $\mathcal{F}:=R^{p}$ such that $\mathcal{M}$ is free of rank
$e$ at each minimal prime of $R$. Set $r:=d+e-1$. Inspired by work of Cutkosky
and collaborators (cf. [CHS10]), Ulrich and Validashti defined the
$\varepsilon$-multiplicity of $\mathcal{M}$ as

$\varepsilon(\mathcal{M}):=\limsup_{n\to\infty}\frac{r!}{n^{r}}\lambda_{R}(H_{\mathfrak{m}}^{0}(\mathcal{F}^{n}/\mathcal{M}^{n})).$

They proved that $\varepsilon(\mathcal{M})$ satisfies the Rees criterion for
integral dependence [UV11] and together with Kleiman established in [KUV] that
it is upper semi-continuous. These two properties allowed them to prove a
generalization of Teissier’s Principle of Specialization of Integral
Dependence. Thus, it seemed that $\varepsilon(\mathcal{M})$ was the right
generalization of the Buchsbaum–Rim multiplicity for applications to
equisingularity. However, Kleiman found the following counterexample (see Ex.
4.5 on p. 46 of [BGG80]): a family $(X,0)\rightarrow(\mathbb{C},0)$ of germs
of curves in $4$-space with parametrization given by

$x=t^{4},y=t^{6}+2ut^{7},z=t^{11},w=t^{13}+ut^{14}$

where $u$ is the parameter of the family. Eliminating $t$ gives a one-
parameter family of curves defined by $6$ equations. The $\delta$ invariant
equals $12$ and the multiplicity of the fibers is $4$ for any value of $u$. So
the family is Whitney equisingular by Thm. III. 3 on p. 23 in [BGG80].

Denote by $\mathcal{M}$ the relative Jacobian module of
$(X,0)\rightarrow(\mathbb{C},0)$ (see Sct. 6). It’s contained in a free module
$\mathcal{F}$ of rank $6$. Denote by $\mathcal{M}(u)$ the image of
$\mathcal{M}$ in $\mathcal{F}(u):=\mathcal{F}\otimes\mathcal{O}_{X_{u}}$ One
can compute $\varepsilon(\mathcal{M}(u))$ with $R$ being the local ring of the
singular point of $X_{u}$. Using Singular, Kleiman found that
$\varepsilon(\mathcal{M}(u))=21$ if $u=0$, but
$\varepsilon(\mathcal{M}(u))=19$ for generic $u\neq 0$. Kleiman’s computation
is based on an observation of Ulrich who showed that
$\varepsilon(\mathcal{M}(u))$ can be computed as the multiplicity of the ideal
of maximal nonvanishing minors of the presentation matrix of $\mathcal{M}(u)$.

However, if we instead embed $\mathcal{M}$ in a free module $\mathcal{F}$ of
rank $3$ as we did in Prp. 3.1 (iii), then $\varepsilon(\mathcal{M}(u))$ is
the Buchsbaum–Rim multiplicity $e(\mathcal{M}(u),\mathcal{F}(u))$, because
$\mathcal{M}(u)$ and $\mathcal{F}(u)$ are generically equal, so
$\mathcal{F}(u)/\mathcal{M}(u)$ is supported over the origin. An easy
computations shows that $e(\mathcal{M}(u),\mathcal{F}(u))=19$ for each $u$ as
predicted by Cor. 2.7.

This example shows that one should use in general the homogeneous embedding of
$\mathcal{A}$ constructed in Prp. 3.2. Also, the example shows that the right
invariant for equisingularity is not the local volume of the restrictions of
the inverible sheaf $\mathcal{L}$ but the restricted local volume of
$\mathcal{L}$.

##  4\. The Local Volume Formula

In this section we prove the Local Volume Formula (LVF). We show that the
restricted local volume exists as a limit.

Let $X$ and $Y$ be affine reduced schemes of finite type over a field $\Bbbk$.
Assume $X$ is equidimensional and $Y$ is regular and integral of dimension
one. Suppose $h\colon X\rightarrow Y$ is a morphism with equidimensional
fibers of positive dimension. Let $S$ be a subscheme of $X$ that is quasi-
finite over $Y$. Assume that $\Bbbk$ is the residue field of each closed point
$y$ and the points in $S_{y}$. Let $C$ be an equidimensional reduced scheme
projective over $X$ such that the structure morphism $c\colon C\rightarrow X$
maps each irreducible component of $C$ to an irreducible component of $X$. Set
$D:=c^{-1}S$ and $\dim C=r+1$. Fix a closed point $y_{0}$ in $Y$ such that
$S_{y_{0}}$ is nonempty. For simplicity we will assume that $y_{0}$ and the
points in $S_{y_{0}}$ are rational. For an affine neighborhood $U$ of $y_{0}$
set $U^{\prime}:=U-\\{y_{0}\\}$. Denote by $D_{\mathrm{vert}}$ the union of
components of $D$ that maps to $y_{0}$ under $h\circ c$.

Let $\mathcal{L}$ be an invertible very ample sheaf on $C$ relative to $X$.
Let $\mathcal{A}:=\oplus_{n\geq 0}\Gamma(C,\mathcal{L}^{\otimes n})$ be the
ring of sections of $\mathcal{L}$. Denote by $\mathcal{A}_{n}$ the $n$th
graded piece of $\mathcal{A}$. Recall that restricted local volume of
$\mathcal{L}$ at $S_{y}$ is defined as

$\mathrm{vol}_{C_{y}}(\mathcal{L}):=\limsup_{n\to\infty}\frac{r!}{n^{r}}\dim_{\Bbbk}H_{S}^{1}(\mathcal{A}_{n})\otimes_{\mathcal{O}_{Y}}k(y).$

In the definition of the restricted local volume we can assume that
$\mathcal{A}$ is the coordinate ring of $C$ as the $n$th graded component of
the coordinate ring and $\Gamma(C,\mathcal{L}^{\otimes n})$ coincide for $n\gg
0$ (cf. Chp. II Ex. 5.9 in [Har77]).

Let $\mathcal{B}$ be a standard graded algebra containing $\mathcal{A}$
satisfying the properties listed in Prp. 3.2. Denote by
$\overline{\mathcal{A}}$ the integral closure of $\mathcal{A}$ in
$\mathcal{B}$. Set $\mathcal{B}(y):=\mathcal{B}\otimes_{\mathcal{O}_{Y}}k(y)$
and $\mathcal{B}_{n}(y):=\mathcal{B}_{n}\otimes_{\mathcal{O}_{Y}}k(y)$. Denote
the images of $\mathcal{A}$ in $\mathcal{B}(y)$ and $\mathcal{A}_{n}$ in
$\mathcal{B}_{n}(y)$ by $\mathcal{A}(y)$ and $\mathcal{A}_{n}(y)$,
respectively. We say that the restricted volume specializes with passage to
the fiber $X_{y}$ if

$\mathrm{vol}_{C_{y}}(\mathcal{L}):=\limsup_{n\to\infty}\frac{r!}{n^{r}}\dim_{\Bbbk}H_{S_{y}}^{1}(\mathcal{A}_{n}(y)).$

######  Theorem 4.1 (Local Volume Formula).

The following holds.

  1. (i)

Suppose $\dim X=2$ and
$S:=\mathrm{Supp}_{\mathcal{O}_{X}}(\mathcal{B}/\overline{\mathcal{A}})$. Then
there exists a an affine neighborhood $U$ of $y_{0}$ in $Y$ such that

$e(\mathcal{A}(y_{0}),\mathcal{B}(y_{0}))-e(\mathcal{A}(y),\mathcal{B}(y))=\int_{S_{y_{0}}}l^{r}[D_{\mathrm{vert}}]$

for each $y\in U^{\prime}$.

  2. (ii)

Suppose $\dim X\geq 3$. Then there exists an affine neighborhood $U$ of
$y_{0}$ in $Y$ such that

$\mathrm{vol}_{C_{y_{0}}}(\mathcal{L})-\mathrm{vol}_{C_{y}}(\mathcal{L})=\int_{S_{y_{0}}}l^{r}[D_{\mathrm{vert}}]$

for each $y\in U^{\prime}$ and the restricted local volume specializes with
passage to $X_{y}$. Furthermore, $\mathrm{vol}_{C_{y}}(\mathcal{L})$ is finite
and exists as a limit for each $y\in U$.

Proof

Consider $\rm{(i)}$. Denote by
$c_{\mathcal{B}}\colon\mathrm{Proj}(\mathcal{B})\rightarrow X$ the structure
morphism. Because $\mathcal{B}$ is a birational extension of $\mathcal{A}$, it
follows that the irreducible components of $\mathrm{Proj}(\mathcal{B})$
surject onto those of $X$. By Prp. 3.2 $\rm{(ii)}$ and the fact that $X$ and
$X_{y}$ are equidimensional, and $Y$ is regular, it follows that the
irreducible components of $\mathrm{Proj}(\mathcal{B}(y_{0}))$ surject onto
those of $X_{y_{0}}$. So no irreducible component of
$\mathrm{Proj}(\mathcal{B}(y_{0}))$ is supported over $S_{y_{0}}$. As
$\mathcal{B}(y_{0})$ is integral over $\mathcal{A}(y_{0})$ locally off
$S_{y_{0}}$ it follows that
$c^{-1}X_{y_{0}}=\mathrm{Proj}(\mathcal{A}(y_{0}))\cup D_{\mathrm{vert}}$ (cf.
Prp. 4.4). Applying Thm. 2.6 we get the desired result.

Consider $\rm{(ii)}$. We break the proof of the LVF into several parts.
Because $\dim X\geq 3$ using the analysis in Prp. 3.1, we can replace in the
LVF $H_{S}^{1}(\mathcal{A}_{n})$ by
$H_{S}^{0}(\mathcal{B}_{n}/\mathcal{A}_{n}).$ In Prp. 4.2 for each $n$ we
relate the change of the dimension of
$H_{S}^{0}(\mathcal{B}_{n}/\mathcal{A}_{n})\otimes_{\mathcal{O}_{Y}}k(y_{0})$
as $y$ “moves” from $y_{0}$ to a generic $y\in U^{\prime}$, to the dimension
of
$H_{S_{y_{0}}}^{0}(\mathcal{B}_{n}/\mathcal{A}_{n})\otimes_{\mathcal{O}_{Y}}k(y_{0})$.
Here the main algebraic tool we use is the well-known fact that a torsion-free
module over a principal ideal domain is free. Then we relate the limit of
normalized vector space dimensions of
$H_{S_{y_{0}}}^{0}(\mathcal{B}_{n}/\mathcal{A}_{n})\otimes_{\mathcal{O}_{Y}}k(y_{0})$
to the degree of the cycle $[c^{-1}(S_{y_{0}})]_{r}$ using two reductions. The
first relates the dimensions of
$H_{S_{y_{0}}}^{0}(\mathcal{B}_{n}/\mathcal{A}_{n})\otimes_{\mathcal{O}_{Y}}k(y_{0})$
to the dimension of the $n$th graded piece of the ideal defining the residual
scheme in $c^{-1}X_{y_{0}}$ to the union of components that surject onto
$S_{y_{0}}$. The second one relates the limit of normalized vector space
dimensions of the graded pieces of the ideal of the residual scheme to
$\int_{S_{y_{0}}}l^{r}[D_{\mathrm{vert}}]$. Finally, we prove that the
formation of the “generic” limit term in the LVF specializes with passage to
generic fibers using in [Ran19, Prp. 2.1], which along with the LVF implies
that $\mathrm{vol}_{C_{y}}(\mathcal{L})$ is finite and exists as a limit for
each $y\in U$.

Let $\mathcal{P}_{n}$ and $\mathcal{N}_{n}$ be submodules of $\mathcal{B}_{n}$
such that

$H_{S}^{0}(\mathcal{B}_{n}/\mathcal{A}_{n})=\mathcal{N}_{n}/\mathcal{A}_{n}\
\text{and}\
H_{S_{y_{0}}}^{0}(\mathcal{B}_{n}/\mathcal{A}_{n})=\mathcal{P}_{n}/\mathcal{A}_{n}.$

Let $V$ be an affine neighborhood in $X$ containing $S_{y_{0}}$. Denote by
$A[V]$ and $A[U]$ the homogeneous coordinate rings of $V$ and $U$. Because
$S\rightarrow Y$ is quasi-finite and $S_{y_{0}}$ is nonempty, then by
shrinking $U$ if necessary, we can assume that that $S\rightarrow Y$ is
finite. Note that the support of each of the quotients
$\mathcal{N}_{n}/\mathcal{P}_{n}$, $\mathcal{N}_{n}/\mathcal{A}_{n}$ and
$\mathcal{P}_{n}/\mathcal{A}_{n}$ is in $S$. Because $S$ is finite over $Y$,
the direct image of each of these quotient by $h$ is a coherent
$\mathcal{O}_{Y}$-module.

Flatness

######  Proposition 4.2.

Assume $U$ is small enough so that for each $n$, the associated points of
$\mathcal{B}_{n}/\mathcal{A}_{n}$ viewed as $A[V]$-module map to $y_{0}$ or
the generic point of $U$. Then

(20)
$\dim_{\Bbbk}(\mathcal{N}_{n}/\mathcal{A}_{n})\otimes_{\mathcal{O}_{Y}}k(y_{0})-\dim_{\Bbbk}(\mathcal{N}_{n}/\mathcal{A}_{n})\otimes_{\mathcal{O}_{Y}}k(y)=\dim_{\Bbbk}(\mathcal{P}_{n}/\mathcal{A}_{n})\otimes_{\mathcal{O}_{Y}}k(y_{0}).$

###### Proof.

By [Ran19, Prp. 2.1] there are finitely many points in
$\bigcup_{i=0}^{\infty}\mathrm{Ass}_{X}(\mathcal{B}_{i}/\mathcal{A}_{i})$.
Each of them maps to a closed point in $Y$, or the generic point of $Y$. Thus
we can select $U$ so that the only associated points of
$\mathcal{B}_{n}/\mathcal{A}_{n}$ are those that map to $y_{0}$ or to the
generic point of $U$. Therefore, the only associated points of
$\mathcal{N}_{n}/\mathcal{A}_{n}$ viewed now as $A[U]$-module are $y_{0}$ or
the generic point of $U$.

Consider the nested chain of modules

$\mathcal{A}_{n}\subset\mathcal{P}_{n}\subset\mathcal{N}_{n}$

Form the exact sequence

(21)
$\setcounter{MaxMatrixCols}{11}\begin{CD}0@>{}>{}>\mathcal{P}_{n}/\mathcal{A}_{n}@>{}>{}>\mathcal{N}_{n}/\mathcal{A}_{n}@>{}>{}>\mathcal{N}_{n}/\mathcal{P}_{n}@>{}>{}>0\\\
@V{}V{\mu}V@V{}V{\mu}V@V{}V{\mu}V\\\
0@>{}>{}>\mathcal{P}_{n}/\mathcal{A}_{n}@>{}>{}>\mathcal{N}_{n}/\mathcal{A}_{n}@>{}>{}>\mathcal{N}_{n}/\mathcal{P}_{n}@>{}>{}>0\end{CD}$

where the vertical maps $\mu$ are multiplication by the ideal
$\mathfrak{m}_{y_{0}}$ of $y_{0}$. Since $\mu$ is injective on
$\mathcal{N}_{n}/\mathcal{P}_{n}$, then the Snake Lemma yields

(22)
$\begin{CD}0@>{}>{}>\mathcal{P}_{n}/\mathcal{A}_{n}\otimes_{\mathcal{O}_{Y}}k(y_{0})@>{}>{}>\mathcal{N}_{n}/\mathcal{A}_{n}\otimes_{\mathcal{O}_{Y}}k(y_{0}<EMAIL_ADDRESS>
Therefore,

(23)
$\dim_{\Bbbk}\mathcal{N}_{n}/\mathcal{A}_{n}\otimes_{\mathcal{O}_{Y}}k(y_{0})=\dim_{\Bbbk}\mathcal{P}_{n}/\mathcal{A}_{n}\otimes_{\mathcal{O}_{Y}}k(y_{0})+\dim_{\Bbbk}\mathcal{N}_{n}/\mathcal{P}_{n}\otimes_{\mathcal{O}_{Y}}k(y_{0}).$

Now let $y\in Y$ be a point from $U^{\prime}$. Then

(24)
$\begin{CD}\mathcal{P}_{n}/\mathcal{A}_{n}\otimes_{\mathcal{O}_{Y}}k(y)@>{}>{}>\mathcal{N}_{n}/\mathcal{A}_{n}\otimes_{\mathcal{O}_{Y}}k(y<EMAIL_ADDRESS>
However, $\mathcal{P}_{n}/\mathcal{A}_{n}\otimes_{\mathcal{O}_{Y}}k(y)=0$ as
the support of $\mathcal{P}_{n}/\mathcal{A}_{n}$ is $S_{y_{0}}$. Thus,

$\mathcal{N}_{n}/\mathcal{A}_{n}\otimes_{\mathcal{O}_{Y}}k(y)\cong\mathcal{N}_{n}/\mathcal{P}_{n}\otimes_{\mathcal{O}_{Y}}k(y).$

But the only associated point of the $A[U]$-module
$\mathcal{N}_{n}/\mathcal{P}_{n}$ is the generic point of $U$, hence
$\mathcal{N}_{n}/\mathcal{P}_{n}$ is torsion-free $\mathcal{O}_{Y,y}$ module
for each closed point $y\in U$. Because $\mathcal{O}_{Y,y}$ is a DVR, and $Y$
is reduced, then $\mathcal{N}_{n}/\mathcal{P}_{n}$ is locally free. Hence
$\dim_{\Bbbk}\mathcal{N}_{n}/\mathcal{P}_{n}\otimes_{\mathcal{O}_{Y}}k(y)=\dim_{\Bbbk}\mathcal{N}_{n}/\mathcal{P}_{n}\otimes_{\mathcal{O}_{Y}}k(y_{0})$
(cf. Ex. 5.8 in Chp. II in [Har77]). So,

(25)
$\dim_{\Bbbk}\mathcal{N}_{n}/\mathcal{A}_{n}\otimes_{\mathcal{O}_{Y}}k(y)=\dim_{\Bbbk}\mathcal{N}_{n}/\mathcal{P}_{n}\otimes_{\mathcal{O}_{Y}}k(y_{0})$

Subtracting (25) from (23) we get the desired result. ∎

Let $x\in S_{y_{0}}$. Define $\mathcal{P}_{n}(x)$ so that
$H_{x}^{0}(\mathcal{B}_{n}/\mathcal{A}_{n})=\mathcal{P}_{n}(x)/\mathcal{A}_{n}$.
Clearly,

$\dim_{\Bbbk}(\mathcal{P}_{n}/\mathcal{A}_{n})\otimes_{\mathcal{O}_{Y}}k(y_{0})=\sum_{x\in
S_{y_{0}}}\dim_{\Bbbk}(\mathcal{P}_{n}(x)/\mathcal{A}_{n})\otimes_{\mathcal{O}_{Y}}k(y_{0}).$

Let $D_{\mathrm{vert}}(x)$ be the union of components of $D_{\mathrm{vert}}$
that map to $x$. We will show that

$\lim_{n\to\infty}\frac{r!}{n^{r}}\dim_{\Bbbk}(\mathcal{P}_{n}(x)/\mathcal{A}_{n})\otimes_{\mathcal{O}_{Y}}k(y_{0})=\int_{x}l^{r}[D_{\mathrm{vert}}(x)].$

Thus we can assume that $S_{y_{0}}$ consists of a single point $x_{0}$.

Key Isomorphism

The next proposition provides a key isomorphism that allows to connect
$\mathcal{P}_{n}/\mathcal{A}_{n}$ to the ideal of the residual scheme to the
union of components of dimension $r$ in $c^{-1}(x_{0})$. Let
$t\in\mathfrak{m}_{y_{0}}-\mathfrak{m}_{y_{0}}^{2}$ be a uniformizing
parameter of $\mathcal{O}_{Y,y_{0}}$. Identify $t$ with its image $h^{\\#}t$
in $\mathfrak{m}_{x_{0}}$. Replace $\mathcal{P}_{n}$ and $\mathcal{A}_{n}$ by
their localizations at $x_{0}$. This does not affect the length of
$\mathcal{P}_{n}/\mathcal{A}_{n}\otimes_{\mathcal{O}_{Y}}k(y_{0})$.

######  Proposition 4.3.

We have

(26)
$\mathcal{P}_{n}/\mathcal{A}_{n}\otimes_{\mathcal{O}_{Y}}k(y_{0})\simeq(t\mathcal{P}_{n}\cap\mathcal{A}_{n})/t\mathcal{A}_{n}$

as $\Bbbk$-vector spaces.

###### Proof.

To begin, observe that both quotients
$\mathcal{P}_{n}/\mathcal{A}_{n}/t(\mathcal{P}_{n}/\mathcal{A}_{n})$ and
$(t\mathcal{P}_{n}\cap\mathcal{A}_{n})/t\mathcal{A}_{n}$ are supported over
$x_{0}$. Because $k(y_{0})=\Bbbk$ is contained in $\mathcal{O}_{X,x_{0}}$ and
equals $k(x_{0})$, then two the quotients are $\Bbbk$-vector spaces of finite
dimension. Throughout the proof we will use repeatedly the fact that $t$ is a
nonzero divisor of $\mathcal{B}$. Indeed, $X$ and the fibers of $X\rightarrow
Y$ are equidimensional. So there is no component of $X$ supported over
$X_{y_{0}}$. Also, $\mathrm{Proj}(\mathcal{B})$ is reduced and its components
surject onto those of $X$. Thus $t$ is a nonzero divisor of $\mathcal{B}$.

The isomorphism is built from the obvious correspondence: for
$b\in\mathcal{P}_{n}$, not divisible by $t$ in $\mathcal{B}_{n}$, associate
$t^{s}b\in t\mathcal{P}_{n}\cap\mathcal{A}_{n}$ where $s$ is the smallest
integer such that $t^{s}b\in\mathcal{A}_{n}$ but $t^{s}b\not\in
t\mathcal{A}_{n}$. Below we make this precise.

For each $b$ in $\mathcal{P}_{n}$ define $\mathrm{ord}_{t}(b)$ to be the
smallest integer such that $t^{\mathrm{ord}_{t}(b)}b\in\mathcal{A}_{n}$. Set
$s_{n}:=\max\\{\mathrm{ord}_{t}(b)|b\in\mathcal{P}_{n}\\}$ and let
$\mathcal{P}_{n}^{(i)}$ be the image of
$(\mathcal{A}_{n}:_{\mathcal{P}^{n}}t^{i})$ in
$\mathcal{P}_{n}/\mathcal{A}_{n}\otimes_{\mathcal{O}_{Y}}k(y_{0})$. Let
$L_{n}^{(i)}$ be submodule of
$(t\mathcal{P}_{n}\cap\mathcal{A}_{n})/t\mathcal{A}_{n}$ generated by the
images of all elements $a\in\mathcal{A}_{n}$ such that
$a/t^{i}\in\mathcal{P}_{n}$. Consider the following filtrations

$\mathcal{P}_{n}/\mathcal{A}_{n}\otimes_{\mathcal{O}_{Y}}k(y_{0})=:\mathcal{P}_{n}^{(s_{n})}\supset\mathcal{P}_{n}^{(s_{n}-1)}\supset\cdots\supset\mathcal{P}_{n}^{(1)}\supset
0$

and

$0\subset L_{n}^{(s_{n})}\subset\cdots\subset
L_{n}^{(1)}:=(t\mathcal{P}_{n}\cap\mathcal{A}_{n})/t\mathcal{A}_{n}.$

To prove the isomorphism in question, it’s enough to show

$\mathcal{P}_{n}^{(i)}/\mathcal{P}_{n}^{(i-1)}\simeq
L_{n}^{(i)}/L_{n}^{(i+1)}$

as $\Bbbk$-vector spaces for each $i=1,\ldots s_{n}$, where we assume that
$\mathcal{P}_{n}^{(0)}=L_{n}^{(s_{n}+1)}=0$. Let $\hat{b_{i}}$ be an element
in $\mathcal{P}_{n}^{(i)}/\mathcal{P}_{n}^{(i-1)}$ and $b_{i}$ be an element
$(\mathcal{A}_{n}:_{\mathcal{P}^{n}}t^{i})$ that maps to it. Denote the image
of $t^{i}b_{i}$ in $L_{n}^{(i)}/L_{n}^{(i+1)}$ by $\widetilde{t^{i}b_{i}}$.
Define the $\Bbbk$-linear map $\psi_{i}(c\hat{b_{i}})=c\widetilde{t^{i}b_{i}}$
for $c\in\Bbbk$. To see that the map is well-defined suppose there exists
$b_{i}^{\prime}\in(\mathcal{A}_{n}:_{\mathcal{P}^{n}}t^{i})$ such that
$\hat{b_{i}^{\prime}}=\hat{b_{i}}$. Then
$b_{i}-b_{i}^{\prime}\in(\mathcal{A}_{n}:_{\mathcal{P}^{n}}t^{i-1})$ and so
$t^{i}(b_{i}-b_{i}^{\prime})\in t\mathcal{A}_{n}$ which implies that
$\widetilde{t^{i}b_{i}}=\widetilde{t^{i}b_{i}^{\prime}}$. The map $\psi_{i}$
is a $\Bbbk$-linear because the composition
$(\mathcal{A}_{n}:_{\mathcal{P}^{n}}t^{i})\rightarrow\mathcal{A}_{n}\rightarrow(t\mathcal{P}_{n}\cap\mathcal{A}_{n})/t\mathcal{A}_{n}$
is a $\Bbbk$-homomorphism. Also, $\psi_{i}$ is obviously surjective. Suppose
$\psi_{i}(\hat{b_{i}})=0$. Then $t^{i}b_{i}\in t\mathcal{A}_{n}$. As $t$ is
nonzero divisor in $\mathcal{B}_{n}$ it follows that
$t^{i-1}b_{i}\in\mathcal{A}_{n}$, or
$b_{i}\in(\mathcal{A}_{n}:_{\mathcal{P}^{n}}t^{i-1})$ which implies that
$\hat{b_{i}}=0$. Thus $\psi_{i}$ is an isomorphism. ∎

The Residual Scheme

The next proposition identifies the ideal of the residual scheme to the union
of components of $c^{-1}(X_{y_{0}})$ that surject onto $x_{0}$. In what
follows we will replace $\mathcal{B}$ with its integral closure in its total
ring of fractions and redefine $\mathcal{P}_{n}$ and $\mathcal{N}_{n}$
accordingly. By the analysis in Prp. 3.1 $\rm{(ii)}$ applied to the family
setting, the terms participating in the LVF will be unaffected.

Let $V$ be the union of the components of $c^{-1}(X_{y_{0}})$ whose projection
to $X_{y_{0}}$ is $x_{0}$. Let $W:=(c^{-1}(X_{y_{0}})-V)^{-}$ be the residual
scheme of $V$ in $c^{-1}(X_{y_{0}})$. As the problem is local at $x_{0}$,
assume $X_{y_{0}}$ is local with closed point $x_{0}$. Denote by
$A[C_{y_{0}}]$ the homogeneous coordinate ring of $c^{-1}(X_{y_{0}})$. Set
$\mathcal{P}:=\mathcal{O}_{X,x_{0}}\oplus\mathcal{P}_{1}\oplus\mathcal{P}_{2}\oplus\cdots$

######  Proposition 4.4.

The ideal $\mathcal{I}_{W}$ of $W$ in $A[C_{y_{0}}]$ is
$(t\mathcal{P}\cap\mathcal{A})/t\mathcal{A}.$

###### Proof.

Since $A[C_{y_{0}}]=\mathcal{A}/t\mathcal{A}$, then by Prp. 3.6
$\mathcal{I}_{W}=\bigcup_{i=0}^{\infty}(t\mathcal{A}:_{\mathcal{A}}\mathfrak{m}_{x_{0}}^{i})/t\mathcal{A}$
where $\mathfrak{m}_{x_{0}}$ is the ideal of $x_{0}$ in
$\mathcal{O}_{X,x_{0}}$. We claim that

$\bigcup_{i=0}^{\infty}(t\mathcal{A}:_{\mathcal{A}}\mathfrak{m}_{x_{0}}^{i})=t\mathcal{P}\cap\mathcal{A}.$

Indeed, let $tp\in t\mathcal{P}\cap\mathcal{A}$ where $p\in\mathcal{P}$. Let
$j\gg 0$ such that $\mathfrak{m}_{x_{0}}^{j}p\in\mathcal{A}$. Then
$\mathfrak{m}_{x_{0}}^{j}tp\in t\mathcal{A}$ and hence
$tp\in\bigcup_{i=0}^{\infty}(t\mathcal{A}:_{\mathcal{A}}\mathfrak{m}_{x_{0}}^{i})$.

As in the proof of Prp. 3.1 $\rm{(ii)}$ we can find $\mathfrak{z}$ such that
$t,\mathfrak{z}$ is a $2$-regular sequence for $\mathcal{B}$. Let $q$ be an
element from
$\bigcup_{i=0}^{\infty}(t\mathcal{A}:_{\mathcal{A}}\mathfrak{m}_{x_{0}}^{i})$.
For $j\gg 0$ we have $\mathfrak{m}_{x_{0}}^{j}q\in t\mathcal{A}$ so we can
write $\mathfrak{z}^{j}q=ta$ for $a\in\mathcal{A}$. Consider the last identity
as an identity in $\mathcal{B}$. Because $\mathfrak{z}$ is a nonzero divisor
modulo $t$, then $q/t\in\mathcal{B}$. We claim that $q/t\in\mathcal{P}$.
Indeed, because $\mathfrak{m}_{x_{0}}^{j}q\in t\mathcal{A}$ we have
$\mathfrak{m}_{x_{0}}^{j}q/t\in\mathcal{A}$ which yields $q/t\in\mathcal{P}$.
Hence $q\in t\mathcal{P}\cap\mathcal{A}$. ∎

Let $Z:=\mathrm{Spec}(R)$ be an affine Noetherian scheme of finite type over a
field $K$. Let $C$ be a subscheme of $Z\times\mathbb{P}_{K}^{u}$, where $u$ is
a positive integer. Denote the homogeneous coordinate ring of $C$ by $A[C]$.
It is a graded ring with respect to the coordinates of $\mathbb{P}_{K}^{u}$.
Denote its $n$th graded piece by $A[C]_{n}$. Let $W$ be a closed subscheme of
$C$ and denote its ideal in $A[C]$ by $\mathcal{I}_{W}$. Set
$(\mathcal{I}_{W})_{n}=A[C]_{n}\cap\mathcal{I}_{W}$.

Let $\mathrm{pr}_{1}$ be the projection from $Z\times\mathbb{P}_{K}^{u}$ onto
the first factor. Suppose $z$ is a rational point of $Z$. Set
$V:=\mathrm{pr}_{1}^{-1}(z)\cap C$ and $r:=\dim V$. Let
$l:=c_{1}\mathcal{O}_{C}(1)$ and let $[V]_{r}$ be the dimension $r$ part of
the fundamental cycle of $V$. Finally, define $\deg[V]_{r}:=\int l^{r}V$.

######  Proposition 4.5.

Assume $C=V\cup W$ where $W$ is a closed subscheme of
$Z\times\mathbb{P}_{K}^{u}$ with $\dim(V\cap W)<r$. Then
$(\mathcal{I}_{W})_{n}$ is a finite-dimensional $K$-vector space for each $n$,
and

$\lim_{n\to\infty}\frac{r!}{n^{r}}\dim_{K}(\mathcal{I}_{W})_{n}=\deg[V]_{r}.$

###### Proof.

Because $\mathcal{I}_{V}\cap\mathcal{I}_{W}=0$ in $A[C]$, then as $R$-modules

(27)
$(\mathcal{I}_{W})_{n}\simeq((\mathcal{I}_{W})_{n}+(\mathcal{I}_{V})_{n})/(\mathcal{I}_{V})_{n}$

for each $n$. Set $A[V]=A[C]/\mathcal{I}_{V}$. Since the residue field of $x$
is $K$, it follows that $A[V]_{n}$ is finite-dimensional $K$-vector space. The
inclusion
$((\mathcal{I}_{W})_{n}+(\mathcal{I}_{V})_{n})/(\mathcal{I}_{V})_{n}\subset
A[V]_{n}$ shows that
$(\mathcal{I}_{W})_{n}+(\mathcal{I}_{V})_{n})/(\mathcal{I}_{V})_{n}$ is finite
dimensional $K$-vector space as well. But $R$ is a $K$-algebra. Thus, by (27)
the $R$-module $(\mathcal{I}_{W})_{n}$ is a finite-dimensional $K$-vector
space and

(28)
$\dim_{K}(\mathcal{I}_{W})_{n}=\dim_{K}((\mathcal{I}_{W})_{n}+(\mathcal{I}_{V})_{n})/(\mathcal{I}_{V})_{n}.$

Next, consider the exact sequence
$0\rightarrow(\mathcal{I}_{W}+\mathcal{I}_{V})/\mathcal{I}_{V}\rightarrow
A[V]\rightarrow A[V\cap W]\rightarrow 0$ where $A[V\cap
W]=A[C]/\mathcal{I}_{V\cap W}$. As $\dim(V\cap W)<r$, we get
$\lim_{n\to\infty}\frac{r!}{n^{r}}\dim_{K}A[V\cap W]_{n}=0.$ Thus

(29)
$\lim_{n\to\infty}\frac{r!}{n^{r}}\dim_{K}((\mathcal{I}_{W})_{n}+(\mathcal{I}_{V})_{n})/(\mathcal{I}_{V})_{n}=\lim_{n\to\infty}\frac{r!}{n^{r}}\dim_{K}A[V]_{n}.$

But

(30) $\lim_{n\to\infty}\frac{r!}{n^{r}}\dim_{K}A[V]_{n}=\deg[V]_{r}.$

Combining (28), (29) and (30) we get the desired result. ∎

######  Proposition 4.6.

We have

$\lim_{n\to\infty}\frac{r!}{n^{r}}\dim_{\Bbbk}\mathcal{P}_{n}/\mathcal{A}_{n}\otimes_{\mathcal{O}_{Y}}k(y_{0})=\int_{S_{y_{0}}}l^{r}[D_{\mathrm{vert}}].$

###### Proof.

By Prp. 4.3
$\dim_{\Bbbk}\mathcal{P}_{n}/\mathcal{A}_{n}\otimes_{\mathcal{O}_{Y}}k(y_{0})=\dim_{\Bbbk}(t\mathcal{P}_{n}\cap\mathcal{A}_{n})/t\mathcal{A}_{n}.$
By Prp. 4.4 we have
$(\mathcal{I}_{W})_{n}=(t\mathcal{P}_{n}\cap\mathcal{A}_{n})/t\mathcal{A}_{n}$.
Therefore,
$\dim_{\Bbbk}\mathcal{P}_{n}/\mathcal{A}_{n}\otimes_{\mathcal{O}_{Y}}k(y_{0})=\dim_{\Bbbk}(\mathcal{I}_{W})_{n}$.
Finally, Prp. 4.5 applied with $Z=X_{y_{0}}$ and $z=x_{0}$, $K=\Bbbk$ and
$C=c^{-1}(X_{y_{0}})$ give the desired equality. ∎

Specialization to the generic fiber

######  Proposition 4.7.

Let $U$ be a sufficiently small neighborhood of $y_{0}$. Then

$H_{S}^{0}(\mathcal{B}_{n}/\mathcal{A}_{n})\otimes_{\mathcal{O}_{Y}}k(y)=H_{S_{y}}^{0}((\mathcal{B}_{n}/\mathcal{A}_{n})\otimes_{\mathcal{O}_{Y}}k(y))$

for $y\in U^{\prime}$ and each $n$.

###### Proof.

As usual identify $H_{S}^{0}(\mathcal{B}_{n}/\mathcal{A}_{n})$ with
$\mathcal{N}_{n}/\mathcal{A}_{n}$. We want to show that

$\mathcal{N}_{n}/\mathcal{A}_{n}\otimes_{\mathcal{O}_{Y}}k(y)=H_{S_{y}}^{0}((\mathcal{B}_{n}/\mathcal{A}_{n})\otimes_{\mathcal{O}_{Y}}k(y))$

for each $y\in U$. First, because the support of
$\mathcal{N}_{n}/\mathcal{A}_{n}\otimes_{\mathcal{O}_{Y}}k(y)$ is in $S_{y}$,
then by Prp. 3.7 it follows that
$\mathcal{N}_{n}/\mathcal{A}_{n}\otimes_{\mathcal{O}_{Y}}k(y)\hookrightarrow
H_{S_{y}}^{0}((\mathcal{B}_{n}/\mathcal{A}_{n})\otimes_{\mathcal{O}_{Y}}k(y))$.
Suppose $\mathcal{N}_{n}^{\prime}$ is such that

$\mathcal{N}_{n}^{\prime}/\mathcal{A}_{n}\otimes_{\mathcal{O}_{Y}}k(y)=H_{S_{y}}^{0}((\mathcal{B}_{n}/\mathcal{A}_{n})\otimes_{\mathcal{O}_{Y}}k(y)).$

By [Ran19, Prp. 2.1]
$\bigcup_{n=1}^{\infty}\mathrm{Ass}_{X}(\mathcal{B}_{n}/\mathcal{A}_{n})$ is
finite (we have better qualitative result if instead we appeal to [Ran19, Thm.
1.1 (ii)] which describes the associated primes
$\mathrm{Ass}_{X}(\mathcal{B}_{n}/\overline{\mathcal{A}_{n}})$ as the generic
points of the irreducible components where the fiber dimension of $c$ jumps
(cf. [Ran18, Thm. 7.8]). Shrink $V$ and $U$ so that for each associated point
$z\in\bigcup_{n=1}^{\infty}\mathrm{Ass}_{X}(\mathcal{B}_{n}/\mathcal{A}_{n})$,
the image $h(\overline{\\{z\\}})$ surjects onto $U$ or onto $y_{0}$. Then
further shrink $U$ if necessary, so that each $\overline{\\{z\\}}$ of
dimension one that is not a component of $S$ intersects $S$ at $x_{0}$ only.
Set $S^{\prime}:=\mathrm{Supp}_{X}(\mathcal{N}_{n}^{\prime}/\mathcal{A}_{n})$.
The support of
$\mathcal{N}_{n}^{\prime}/\mathcal{A}_{n}\otimes_{\mathcal{O}_{Y}}k(y)$ is
$S_{y}^{\prime}$. Also, the support of
$H_{S_{y}}^{0}((\mathcal{B}_{n}/\mathcal{A}_{n})\otimes_{\mathcal{O}_{Y}}k(y))$
is $S_{y}$. This forces $S_{y}^{\prime}=S_{y}$. Therefore, the union of
components of $S^{\prime}$ that do not surject onto $x_{0}$ is of dimension
one and hence equal to $S$. By Prp. 3.7 $\mathcal{N}_{n}$ is the maximal
submodule of $\mathcal{B}_{n}$ that contains $\mathcal{A}_{n}$ with
$\mathrm{Supp}(\mathcal{N}_{n}/\mathcal{A}_{n})=S$. Thus
$(\mathcal{N}_{n}^{\prime}/\mathcal{N}_{n})\otimes_{\mathcal{O}_{Y}}k(y)=0$. ∎

Completing the proof of the LVF

Now we are in position to finish the proof of Thm. 4.1. By Prp. 4.7 and by our
hypothesis

$\limsup_{n\to\infty}\frac{r!}{n^{r}}\dim_{\Bbbk}H_{S}^{0}(\mathcal{B}_{n}/\mathcal{A}_{n})\otimes_{\mathcal{O}_{Y}}k(y)$

is finite for $y\in U^{\prime}$. Therefore, by Prp. 4.2 and Prp. 4.6

$\limsup_{n\to\infty}\frac{r!}{n^{r}}\dim_{\Bbbk}H_{S}^{0}(\mathcal{B}_{n}/\mathcal{A}_{n})\otimes_{\mathcal{O}_{Y}}k(y_{0})-\limsup_{n\to\infty}\frac{r!}{n^{r}}\dim_{\Bbbk}H_{S}^{0}(\mathcal{B}_{n}/\mathcal{A}_{n})\otimes_{\mathcal{O}_{Y}}k(y)=\int_{S_{y_{0}}}l^{r}[D_{\mathrm{vert}}].$

This completes the proof of the LVF. By Prp. 4.7 and Cor. 3.3
$\mathrm{vol}_{C_{y}}(\mathcal{L})$ exists as a finite limit for $y\in
U^{\prime}$. By the LVF $\mathrm{vol}_{C_{y_{0}}}(\mathcal{L})$ exists as a
finite limit, too. ∎

We conclude this section with two immediate applications of Thm. 4.1.

######  Corollary 4.8.

The restricted local volume $\mathrm{vol}_{C_{y}}(\mathcal{L})$ is constant
for $y\in U^{\prime}$.

###### Proof.

Indeed, both $\mathrm{vol}_{C_{y_{0}}}(\mathcal{L})$ and
$\int_{S_{y_{0}}}l^{r}[D_{\mathrm{vert}}]$ remain constant as $y$ moves around
in $U^{\prime}$. ∎

The following is an immediate corollary from the proof of the LVF.

######  Corollary 4.9.

In the setup of Thm. 4.1 the scheme $c^{-1}S$ is flat over $Y$ if and only if
$\dim_{\Bbbk}H_{S}^{0}(\mathcal{B}_{n}/\mathcal{A}_{n})\otimes_{\mathcal{O}_{Y}}k(y)$
is constant over $Y$ for $n\gg 0$.

###### Proof.

We have
$\dim_{\Bbbk}H_{S}^{0}(\mathcal{B}_{n}/\mathcal{A}_{n})\otimes_{\mathcal{O}_{Y}}k(y)$
is constant over $Y$ for $n\gg 0$ iff $(\mathcal{I}_{W})_{n}=0$ for $n\gg 0$
iff $V$ is empty iff $c^{-1}S$ is flat (see [Har77, Prp. 9.7]). ∎

######  Remark 4.10.

The proof of the LVF translates almost verbatim to the complex analytic
setting of Rmk. 2.4. As before one should work with sufficiently small
distinguished compact Stein subsets of $X$ and $Y$ containing $S_{y_{0}}$ and
$y_{0}$. The theory of local cohomology and in particular of gap-sheaves
(saturations) of the corresponding analytic sheaves is developed in [ST71].
The rest of the proof remains the same.

######  Remark 4.11.

In [Ran21a] we give a Fujita-type version of the LVF by replacing local
cohomology with intersection numbers. For each closed point $y$ denote by
$\mathcal{A}_{n}(y)$ and $\mathcal{N}_{n}(y)$ the images of $\mathcal{A}_{n}$
and $\mathcal{N}_{n}$ in $\mathcal{B}_{n}\otimes_{\mathcal{O}_{Y}}k(y)$. For
each $n$ denote by $\langle\mathcal{A}_{n}\rangle$ and
$\langle\mathcal{N}_{n}\rangle$ the subalgebras of $\mathcal{B}$ generated by
$\mathcal{A}_{n}$ and $\mathcal{N}_{n}$. Denote by
$c_{\mathcal{A}_{n}}\colon\mathrm{Proj}(\langle\mathcal{A}_{n}\rangle)\rightarrow
X$ and
$c_{\mathcal{N}_{n}}\colon\mathrm{Proj}(\langle\mathcal{N}_{n}\rangle)\rightarrow
X$ the corresponding structure morphisms. Denote the union of components of
$c_{\mathcal{A}_{n}}^{-1}S$ and $c_{\mathcal{N}_{n}}^{-1}S$ supported over
$S_{y_{0}}$ by $D_{\mathrm{vert}}^{\mathcal{A}_{n}}$ and
$D_{\mathrm{vert}}^{\mathcal{N}_{n}}$, respectively. Applying Thm. 2.6 we get

(31)
$e(\mathcal{A}_{n}(y_{0}),\mathcal{N}_{n}(y_{0}))-e(\mathcal{A}_{n}(y),\mathcal{N}_{n}(y))=\int_{S_{y_{0}}}l^{r}[D_{\mathrm{vert}}^{\mathcal{A}_{n}}]-\int_{S_{y_{0}}}l^{r}[D_{\mathrm{vert}}^{\mathcal{N}_{n}}].$

Because $\langle\mathcal{A}_{n}\rangle$ is the $n$th Veronese subalgebra of
$\mathcal{A}$ we have

(32)
$\int_{S_{y_{0}}}l^{r}[D_{\mathrm{vert}}^{\mathcal{A}_{n}}]=n^{r}\int_{S_{y_{0}}}l^{r}[D_{\mathrm{vert}}].$

In [Ran21a] we prove that

(33)
$\limsup_{n\to\infty}\frac{r!}{n^{r}}\int_{S_{y_{0}}}l^{r}[D_{\mathrm{vert}}^{\mathcal{N}_{n}}]=0.$

This is easy to see when $\bigoplus_{i=0}^{\infty}\mathcal{N}_{i}$ is finitely
generated, for example. Let $\mathcal{I}_{S_{y_{0}}}$ be the ideal of
$S_{y_{0}}$ in $\mathcal{O}_{X}$. By Prp. 3.6 and Prp. 2.1 in [Ran19] one can
find an element $x\in\mathcal{I}_{S_{y_{0}}}$ such that $x$ is not a zero
divisor of $\mathcal{B}_{i}/\mathcal{N}_{i}$ for each $i$. Then
$\bigoplus_{i=0}^{\infty}\mathcal{N}_{i}/x\bigoplus_{i=0}^{\infty}\mathcal{N}_{i}$
injects into $\mathcal{B}/x\mathcal{B}$. By Prp. 3.2 and the assumption $\dim
X\geq 2$ we can find $y\in\mathcal{I}_{S_{y_{0}}}$ which avoids the minimal
primes of $\mathcal{B}/x\mathcal{B}$ and hence those of
$\bigoplus_{i=0}^{\infty}\mathcal{N}_{i}/x\bigoplus_{i=0}^{\infty}\mathcal{N}_{i}$.
Denote by
$c_{\mathcal{N}}\colon\mathrm{Proj}(\bigoplus_{i=0}^{\infty}\mathcal{N}_{i})\rightarrow
X$ the structure morphism. Then
$\operatorname{codim}(c_{\mathcal{N}}^{-1}S_{y_{0}},\mathrm{Proj}(\bigoplus_{i=0}^{\infty}\mathcal{N}_{i}))\geq
2$. But $\bigoplus_{i=0}^{\infty}\mathcal{N}_{i}$ is birational extension of
$\mathcal{A}$ because $\mathcal{B}$ is. So
$\bigoplus_{i=0}^{\infty}\mathcal{N}_{i}$ is of pure dimension $r$. Also,
$D_{\mathrm{vert}}^{\mathcal{N}_{n}}$ is a subscheme of
$c_{\mathcal{N}}^{-1}S_{y_{0}}$. Thus $\dim
D_{\mathrm{vert}}^{\mathcal{N}_{n}}<r$, which implies that all terms in (33)
vanish for $n$ sufficiently large.

Divide both sides of (31) by $n^{r}$. Then by (32) and (33), and by taking
limit superiors we obtain

$\limsup_{n\to\infty}\frac{e(\mathcal{A}_{n}(y_{0}),\mathcal{N}_{n}(y_{0}))}{n^{r}}-\limsup_{n\to\infty}\frac{e(\mathcal{A}_{n}(y),\mathcal{N}_{n}(y))}{n^{r}}=\int_{S_{y_{0}}}l^{r}[D_{\mathrm{vert}}].$

##  5\. Volume Stability and the Principle of Specialization of Integral
Dependence

In this section we show how to extend the LVF to the case $\dim Y>1$ under the
assumption of volume stability. As an application we prove a general version
of Teissier’s Principle of Specialization of Integral Dependence.

Let $X\rightarrow Y$ be a family of reduced complex analytic spaces such that
$X$ is equidimensional, $Y$ is smooth of diomension one, and the fibers
$X_{y}$ are equidimensional of positive dimension. Fix a point $y_{0}\in Y$.
Assume $Y$ is contained in a smooth complex analytic space $W$ and

(34) $\mathcal{X}\longrightarrow W$

is an equidimensional reduced family with equidimensional fibers of positive
dimension such that $\mathcal{X}_{y_{0}}=X_{y_{0}}$. Assume $S(W)$ is a
subspace of $\mathcal{X}$ finite over $W$. Let $C(W)$ be reduced,
equidimensional and projective space over $\mathcal{X}$ such that its
irreducible components surject onto irreducible components of $\mathcal{X}$.
Let $\mathcal{L}$ be an invertible very ample sheaf on $C(W)$ relative to $X$.
Set $\mathcal{A}:=\oplus_{n\geq 0}\Gamma(C(W),\mathcal{L}^{\otimes n})$. For
each closed point $w$ define the restricted local volume of $\mathcal{L}$ at
$S(W)_{w}$ as

$\mathrm{vol}_{C(W)_{w}}(\mathcal{L}):=\limsup_{n\to\infty}\frac{r!}{n^{r}}\dim_{\Bbbk}H_{S(W)}^{1}(\mathcal{A}_{n})\otimes_{\mathcal{O}_{W}}k(w).$

Set $\mathcal{B}(w):=\mathcal{B}\otimes_{\mathcal{O}_{W}}k(w)$. For each $n$
denote the image of $\mathcal{A}_{n}$ in $\mathcal{B}_{n}(w)$ by
$\mathcal{A}_{n}(w)$ We say that the restricted local volume specializes with
passage to the fiber $\mathcal{X}_{w}$ if

$\mathrm{vol}_{C(W)_{w}}(\mathcal{L}):=\limsup_{n\to\infty}\frac{r!}{n^{r}}\dim_{\Bbbk}H_{S(W)_{w}}^{1}(\mathcal{A}_{n}(w)).$

Replacing $Y$ by $W$ in Prp. 4.7 we get that there exists a Zariski open
subset $U$ of $W$ such that the restricted local volume specializes with
passage to $\mathcal{X}_{w}$ for each $w\in U$.

######  Definition 5.1.

We say $W$ is a good base space for $\mathrm{vol}_{C(W)_{w}}(\mathcal{L})$ if
there exists a Zariski open dense subset $U$ of $W$ such that
$\mathrm{vol}_{C(W)_{w}}(\mathcal{L})$ is constant for all $w\in U$. In this
case we say $\mathrm{vol}_{C(W)_{w}}(\mathcal{L})$ is stable.

Let $\mathcal{B}$ be a birational extension of $\mathcal{A}$ satisfying the
properties listed in Prp. 3.2. Set
$\mathcal{B}(Y):=\mathcal{B}\otimes_{\mathcal{O}_{\mathcal{X}}}\mathcal{O}_{X}$.
Let $\mathcal{A}(Y)$ be the image of $\mathcal{A}$ in $\mathcal{B}(Y)$. Set
$C:=\mathrm{Proj}(\mathcal{A}(Y))$ and $S:=S(W)\cap X$. Denote by
$D_{\mathrm{vert}}$ the union of components in the inverse image of $S$ in $C$
supported over $y_{0}$.

Let $W_{1}$ and $W_{2}$ be generic curves in $W$ passing through $y_{0}$ and a
generic point $y$ in $Y$ close enough to $y_{0}$. Consider the families
$\mathcal{X}(W_{i}):=\mathcal{X}\times_{W}W_{i}$. Let $\mathcal{A}(W_{i})$ be
the image of $\mathcal{A}$ in
$\mathcal{B}\otimes_{\mathcal{O}_{\mathcal{X}}}\mathcal{O}_{\mathcal{X}_{i}(W_{i})}$.
Set $C(W_{i}):=\mathrm{Proj}(\mathcal{A}(W_{i}))$ and $S(W_{i}):=S\cap W_{i}$.
Denote by $D(W_{1})_{\mathrm{vert}}$ the union of components of the inverse
image of $S(W_{1})$ supported in $C(W_{1})$ over $S(W)_{y_{0}}$, and by
$D(W_{2})_{\mathrm{vert}}$ the union of components of the inverse image of
$S(W_{2})$ supported in $C(W_{2})$ over $S(W)_{y}$.

######  Theorem 5.2.

Suppose $W$ is a good base space for $\mathrm{vol}_{C(W)_{w}}(\mathcal{L})$.
Assume that the set of points $x$ in $\mathcal{X}$ with $\dim c^{-1}x\geq r$
is contained in $S(W)$. Then we can replace
$\mathrm{vol}_{C_{y_{0}}}(\mathcal{L})$ and
$\mathrm{vol}_{C_{y}}(\mathcal{L})$ in the LVF applied to $X\rightarrow Y$,
without violating its validity, by the restricted local volumes
$\mathrm{vol}_{C(W_{1})_{y_{0}}}(\mathcal{L})$ and
$\mathrm{vol}_{C(W_{2})_{y}}(\mathcal{L})$, i.e.

$\mathrm{vol}_{C(W_{1})_{y_{0}}}(\mathcal{L})-\mathrm{vol}_{C(W_{2})_{y}}(\mathcal{L})=\int_{S_{y_{0}}}l^{r}[D_{\mathrm{vert}}].$

###### Proof.

Select $W_{1}$ and $W_{2}$ generic enough so that $W_{1}-\\{y_{0}\\}$ and
$W_{2}-\\{y\\}$ lie in the Zariski open subset $U$ over which the volume
$\mathrm{vol}_{C(W)_{w}}(\mathcal{L})$ is stable and specializes with passage
to $\mathcal{X}_{w}$. Replace $W_{i}$ by $W_{i}\cap U$. Let $w_{1}$ and
$w_{2}$ be points from $W_{1}$ and $W_{2}$ close enough to $y_{0}$ and $y$,
respectively. Apply the LVF to the families $\mathcal{X}(W_{i})\rightarrow
W_{i}$ to get

$\mathrm{vol}_{C(W_{1})_{y_{0}}}(\mathcal{L})-\mathrm{vol}_{C(W_{1})_{w_{1}}}(\mathcal{L})=\int_{S_{y_{0}}}l^{r}[D(W_{1})_{\mathrm{vert}}]$

and

$\mathrm{vol}_{C(W_{2})_{y}}(\mathcal{L})-\mathrm{vol}_{C(W_{2})_{w_{2}}}(\mathcal{L})=\int_{S_{y}}l^{r}[D(W_{2})_{\mathrm{vert}}].$

Because of volume stability and specialization
$\mathrm{vol}_{C(W_{1})_{w_{1}}}(\mathcal{L})=\mathrm{vol}_{C(W_{2})_{w_{2}}}(\mathcal{L})$.
Subtracting the second identity from the first we get

(35)
$\mathrm{vol}_{C(W_{1})_{y_{0}}}(\mathcal{L})-\mathrm{vol}_{C(W_{2})_{y}}(\mathcal{L})=\int_{S_{y_{0}}}l^{r}[D(W_{1})_{\mathrm{vert}}]-\int_{S_{y}}l^{r}[D(W_{2})_{\mathrm{vert}}].$

Our goal is to show that the right-hand side (35) equals
$\int_{S_{y_{0}}}l^{r}[D_{\mathrm{vert}}].$ To do this we interpret each of
the three intersection numbers as the local degrees of a covering of $W$.

Let $W(y_{0})$ be a small enough neighborhood of $W$ around $y_{0}$. Let $y\in
W(y_{0})\cap Y$ be a generic point of $Y$ to be specified later, and let
$W(y)$ be a small enough neighborhood of $y$ such that $W(y)\subset W(y_{0})$.
Let $c\colon C(W)\rightarrow\mathcal{X}$ be the structure morphism. Let
$C(W)\hookrightarrow\mathcal{X}\times\mathbb{P}^{k}$ be the embedding of $C$
induced by $\mathcal{L}$.

Let $H_{r}$ be a general plane in $\mathbb{P}^{k}$ of codimension $r$, with
finitely many genericity conditions to be specified later. Set
$\Gamma(C(W)):=C(W)\cap H_{r}$. By Kleiman’s transversality theorem (see Thm.
2 (ii) and Rmk. 7 in [Kle74]), $\Gamma(C(W))$ is rreduced of pure dimension
equal to $\dim W$. Furthermore, the intersection of $c^{-1}(S(W))$ with a
general plane $H_{r}$ is of dimension at most $\dim W-1$. Therefore, for
generic $w\in W(y_{0})$ the fiber $\Gamma(C)_{w}$ consists of the same number
of points, each of them appearing with multiplicity one, and none of them
lying in $c^{-1}S(W)$. Set
$\mathrm{deg}_{W(y_{0})}\Gamma(C(W)):=\mathrm{Card}(\Gamma(C)_{w})$. Define
$\Gamma(C(W_{1})):=C(W_{1})\cap H_{r}$ and set
$\mathrm{deg}_{W_{1}(y_{0})}\Gamma(C(W_{1})):=\mathrm{Card}(\Gamma(C(W_{1}))_{w_{1}})$
for generic $w_{1}\in W_{1}$, where $W_{1}(y_{0})$ is a small enogh
neighborhood of $y_{0}$ in $W_{1}$. Then for a generic $W_{1}$ we have
$\Gamma(C(W))_{w_{1}}=\Gamma(C(W_{1}))_{w_{1}}$ for generic $w_{1}\in W_{1}$.
Because by assumption the only points in $X_{y_{0}}$ over which the fiber of
$c$ is of dimension at least $r$ are contained in $S_{y_{0}}$ and by
conservation of number we obtain

$\mathrm{deg}_{W(y_{0})}\Gamma(C(W))=\mathrm{deg}_{W_{1}(y_{0})}\Gamma(C(W_{1}))=\int_{S_{y_{0}}}l^{r}[D(W_{1})_{\mathrm{vert}}].$

‘

In the same way define $\Gamma(C)=C\cap H_{r}$ and $\mathrm{deg}_{Y}\Gamma(C)$
where $H_{r}$ is generic enough so that the cover $\Gamma(C)\rightarrow Y$ is
unramified for generic $y\in U:=W(w_{0})\cap Y$ and $\Gamma(C)\cap
S=\emptyset$. By conservation of number

$\mathrm{deg}_{Y}\Gamma(C)=\int_{S_{y_{0}}}l^{r}[D_{\mathrm{vert}}].$

Set $\Gamma(C(W_{2})):=C(W_{2})\cap H_{r}$ and
$\mathrm{deg}_{W_{2}(y)}\Gamma(C(W_{2})):=\mathrm{Card}(\Gamma(C(W_{2}))_{w})$
where $W_{2}(y)$ is small enough neighborhood of $y$ in $W_{2}$ and $w\in
W_{2}$ is generic.

We claim that

$\mathrm{deg}_{W_{2}(y)}\Gamma(C(W_{2}))=\int_{S_{y}}l^{r}[D(W_{2})_{\mathrm{vert}}].$

Note $\int_{S_{y}}(H_{r}\cap
c^{-1}S_{y_{0}})=\int_{S_{y}}l^{r}[D(W_{2})_{\mathrm{vert}}]$ because the
codimension of $H_{r}$ is right. Denote by $s_{1},s_{2},\ldots s_{q}$ to be
the points in $S_{y}$ such that $\dim c^{-1}s_{i}\geq r$. Because $W_{2}$ is
generic and because the cover the cover $\Gamma(C(W))\rightarrow(W,w_{0})$ is
unramified for generic $w\in W(y)$, we get that the branches passing through
$s_{i}$ form $\Gamma(C(W_{2}))$.

The following relation (see [GR16, Thm. 2.8]) shows that the presence of
$\mathrm{deg}_{Y}\Gamma(C)$ is controlled by
$\mathrm{deg}_{W(y_{0})}\Gamma(C(W))$ and
$\mathrm{deg}_{W_{2}(y)}\Gamma(C(W_{2}))$. We claim

(36)
$\mathrm{deg}_{W(y_{0})}\Gamma(C(W))-\mathrm{deg}_{W_{2}(y)}\Gamma(C(W_{2}))=\mathrm{deg}_{Y}\Gamma(C).$

Recall that $H_{r}$ was chosen so that it produces the covers
$\Gamma(C(W))\rightarrow(W,y_{0})$ and $\Gamma(C)\rightarrow(Y,y_{0})$.
Observe that for generic $w\in W(y)$ close enough to $y$ the degree of the
cover $\Gamma(C(W))\rightarrow(W,w_{0})$ is
$\mathrm{deg}_{W(y_{0})}\Gamma(C(W))$. Over $y$ some of these branches merge
at the $s_{i}$s and their number is $\mathrm{deg}_{W_{2}(y)}\Gamma(C(W_{2}))$
as shown above. The rest of the branches intersect $\mathcal{X}_{y}$ at points
away from $S(W)$. Their number is $\mathrm{deg}_{Y}\Gamma(C)$ by our choice of
$H_{r}$. The proof of the theorem is now complete. ∎

The figure summarizes the proof of (36) where for convenience it’s assumed
that there exists a section $\sigma\colon W\rightarrow\mathcal{X}$ and $W$ is
identified with $\sigma(W)$.

In complex analytic singularity theory we apply Thm. 5.2 in the following
setting: $X,x_{0}\rightarrow Y,y_{0}$ is a family of isolated singularties,
$W$ is a larger deformation base space containing $Y$, usually taken to be a
component of the miniversal base space of $X_{y_{0}}$, $C(W)$ is the relative
conormal space of $\mathcal{X}\rightarrow W$, and $S(W)$ is the singular locus
of $\mathcal{X}$. Assume that $W$ is a good deformation space. Thus for
$X_{y_{0}}$ and $X_{y}$ for $y\in Y$ close enough to $y_{0}$, we can associate
unique nonnegative real numbers depending solely on $W$ such that
$\int_{S_{y_{0}}}[D_{\mathrm{vert}}]$ vanishes if and only if the
corresponding restricted volumes associated with one-parameter generic
deformations of $X_{y_{0}}$ and $X_{y}$ in $W$ are the same.

We finish this section by obtaining a numerical control of Teissier’s
Principle of Specialization of Integral Dependence. Preserve the setup of Thm.
5.2. When we write $\mathrm{vol}_{C_{y}}(\mathcal{L})$ we really mean the
restricted local volume computed through the family
$\mathcal{X}\times_{W}W^{\prime}$ with special fiber $X_{y}$ where
$W^{\prime}$ is a generic smooth curve in $W$ passing through $y$. As usual
denote by $\overline{\mathcal{A}(Y)}$ the integral closure of $\mathcal{A}(Y)$
in $\mathcal{B}(Y)$.

######  Theorem 5.3 (Principle of Specialization of Integral Dependence).

Assume $W$ is a good base space for $\mathrm{vol}_{C(W)_{w}}(\mathcal{L})$.
Let $g\in\mathcal{B}(Y)$ is such that
$\mathrm{Supp}_{\mathcal{O}_{\mathcal{X}}}(\overline{(g,\mathcal{A}(Y))}/\overline{\mathcal{A}(Y)})\subset
S(W)$. Assume that there exists a Zariski open set $U$ in $Y$ such that for
each $y\in U$ the image of $g$ in $\mathcal{B}\otimes_{\mathcal{O}_{Y}}k(y)$
is integral over $\mathcal{A}(y)$. If $\mathrm{vol}_{C_{y}}(\mathcal{L})$ is
constant, then $g$ is integral over $\mathcal{A}(Y)$.

###### Proof.

By Thm. 5.2, constancy of $\mathrm{vol}_{C(W)_{y}}(\mathcal{L})$ implies that
$\dim c^{-1}S_{y}<r$ for each $y$. Set
$Z:=\mathrm{Supp}_{\mathcal{O}_{\mathcal{X}}}(\overline{(g,\mathcal{A}(Y))}/\overline{\mathcal{A}(Y)})$.
By [GK99, Lem. 1.2] there exists a smaller Zariski subset $U^{\prime}$ of $Y$
such that $(g,\mathcal{A})$ is integral over $\mathcal{A}$. Thus $Z$ is proper
closed subset of $S$. In particular, $\dim Z<\dim S=\dim Y$. Let $y_{0}\in Y$
be such that $\dim c^{-1}S_{y_{0}}$ is maximal. Because $c^{-1}Z\subset
c^{-1}S$, by upper semi-continuity of fiber dimension we have

$\dim c^{-1}Z\leq\dim c^{-1}S_{y_{0}}+\dim Z\leq r-1+\dim Y-1=\dim C-2.$

Then by [KT94, Cor. 10.7] (cf. [SUV01, Thm. 4.1] or [Ran19, Thm. 1.1 (iii)])
$Z$ is empty, and thus $g$ is integral over $\mathcal{A}(Y)$. ∎

##  6\. Whitney–Thom conditions, Jacobian modules, conormal spaces and
integral dependence

First we review the analytic and algebro-geometric formulations of the Whitney
conditions and their connection to integral closure of modules. For a thorough
overview of the history of differential equisingularity theory we refer the
reader to the masterful treatments of Kleiman [Kle99], Gaffney and Massey
[GaM99] and Teissier [T75]. For a more recent account see [TF18].

Let $X\subset\mathbb{C}^{n+k}$ be a equidimensional complex analytic space,
and let $Y$ be a smooth subspace of $X$ of dimension $k$, so that $X-Y$ is
smooth. Choose an embedding of $(X,0)$ in
$\mathbb{C}^{n}\times\mathbb{C}^{k}$, so that $(Y,0)$ is represented by
$0\times V$, where $V$ is an open neighborhood of $0$ in $\mathbb{C}^{k}$. Let
$\mathrm{pr}:\mathbb{C}^{n}\times\mathbb{C}^{k}\rightarrow\mathbb{C}^{k}$ be
the projection. Set $h:=\mathrm{pr}_{|X}$. View $X$ as the total space of the
family $h:(X,0)\rightarrow(Y,0)$. For each closed point $y\in Y$ set
$X_{y}:=X\cap\mathrm{pr}^{-1}(y)$. Whitney [Wh64] introduced the following
conditions on the regularity of the triple $(X-Y,Y,0)$.

Whitney Condition A. Let $(x_{i})$ be a sequence of points from $X-Y$ that
converges to $0$, and suppose that the sequence $\\{T_{x_{i}}X\\}$ of tangent
hyperplanes has a limit $T$ in the corresponding Grassmannian. Then
$T_{0}Y\subset T$.

Whitney Condition B. Let $(x_{i})$ be a sequence of points from $X-Y$ and let
$(y_{i})$ be a sequence of points from $Y$ both converging to $0$. Suppose
that the sequence of secants $({\overline{x_{i}y_{i}}})$ has limit $l$ and the
sequence of tangents $\\{T_{x_{i}}X\\}$ has limit $T$. Then $l\subseteq T$.

Whitney conjectured that these conditions would ensure the existence of a
homeomorphism $h$ from $X_{0}\times Y$ onto $X$. In other words the family
$X_{y}$ is topologically equisingular. The conjecture is nowadays known as the
Thom–Mather [Tm69] first isotopy theorem. Thom and Mather proved it by
considering constant tangent vector field to $Y$ and lifting it carefully to
$X$ such that the lift is integrable. The integral provides a continuous flow
on $X$ which is $\mathbb{C}^{\infty}$ away from $Y$.

Hironaka [Hir70] introduced an intrinsic modified verson of the Whitney
conditions. We say that $X$ satisfies the strict Whitney condition A if the
distance from $Y$ to the tangent space $T_{x}X$ approaches $0$ as $x$
approaches $0$ along any analytic path
$\phi:(\mathbb{C},0)\longrightarrow(X,0)$ such that $\phi(u)\subset X-Y$ for
any $u\neq 0$, i.e.

(37) $\mathrm{dist}(Y,T_{x}X)\leq c\ \mathrm{dist}(x,Y)^{e},$

holds where the exponent $e$ and the constant $c$ depend on the path $\phi$.
Hironaka proved that if $(X-Y,Y,0)$ satisfies both Whitney condition A and B,
then it satisfies the strict Whitney condition A with exponent $e$ independent
of the path $\phi$. The strict Whitney condition with exponent $e=1$ and $c$
independent of the path $\phi$ is called Verdier’s W condition. Verdier [V76]
showed that his condition implies Whitney A and B. Teissier (Thm. 1.2 in Chap.
V of [T81]) showed that in the complex-analytic case $W$ is in fact equivalent
to Whitney $A$ and $B$.

Let $f$ be a germ of complex analytic function
$f\colon\mathbb{C}^{n+k}\rightarrow(\mathbb{C},0)$. Identify $f$ with its
restriction to $(X,0)$. Assume $f$ is of constant rank off $Y$. In (37)
replace $T_{x}X$ by $T_{x}f^{-1}fx$. The “relative” form of the Whitney
condition A, called Thom’s $A_{f}$ condition, holds at $0$ if (37) is
satisfied. The “relative” form of the Whitney condition B, called the $W_{f}$
condition, holds at $0$ if (37) is satisfied with $e=1$ and $c$ independent of
the path $\phi$.

Let $U$ be an open set of $\mathbb{C}^{n+k}$ containing a representative of
$(X,0)$. The differential $df$ of $f$ defines an embedding of $X$ in
$\mathbb{C}^{n+k}\times{\mathbb{C}^{n+k}}^{*}$ by the graph map. Let
$x_{1},\ldots,x_{n+k}$ be coordinates on $U$ and $w_{1},\ldots,w_{n+k}$ be the
cotangent coordinates. Consider the blowup of $T_{X}^{*}U$ along the image of
the graph map. It is the blowup of $T_{X}^{*}U$ by the ideal
$(w_{1}-\frac{\partial f}{\partial x_{1}},\ldots,w_{n+k}-\frac{\partial
f}{\partial x_{n+k}})$ in $T_{X}^{*}U$. We denote this blowup by
$\mathrm{Bl}_{\mathrm{im}\ df}T_{X}^{*}U$. Thus, the blowup is contained in
$X\times{\mathbb{C}^{n+k}}^{*}\times\mathbb{P}^{n+k-1}$. Denote the
exceptional divisor of this blowup by $E_{f}$. Denote the projection on $X$ of
this exceptional divisor to $X$ by $\Sigma(f)$ and call it the critical locus
of $f$.

Define the the absolute conormal space $C(X,f)$ as the closure in
$X\times\mathbb{P}^{n+k-1}$ of the set of pairs $(x,H)$ such that $x$ is a
point in $X-\Sigma(f)$ and $H$ is a hyperplane tangent at $x$ to the level
hypersurface $f^{-1}fx$. Teissier inspired by some work of Hironaka,
considered the following normal-conormal diagram

$\begin{CD}\mathrm{Bl}_{c^{-1}(Y)}C(X,f)@>{a^{\prime}}>{}>C(X,f)\\\
@V{}V{c^{\prime}}V@V{}V{c}V\\\ \mathrm{Bl}_{Y}X@>{a}>{}>X\end{CD}$

Set $\xi:=c\circ a^{\prime}$. Teissier [T81] in the case of Whitney B, and
Henry, Merle and Sabbah [HMS84] in the case of $W_{f}$ obtained the following
equivalence for the Whitney conditions.

######  Theorem 6.1 (Teissier).

Preserving the setup from above, the following conditions are equivalent:

  1. (i)

The triple $(X-Y,Y,0)$ satisfies $W_{f}$.

  2. (ii)

Let $\xi_{Y}:D_{Y}\rightarrow Y$ be the restriction of $\xi$ to the
exceptional divisor $D_{Y}$ of $\mathrm{Bl}_{c^{-1}(Y)}C(X,f)$. Then $\xi_{Y}$
is equidimensional and $\dim\xi_{Y}^{-1}(0)=n-2$.

Here is a way to see the relation between $\rm{(i)}$ and $\rm{(ii)}$ in the
Whitney B case. Observe that set-theoretically $D_{Y}$ and
$\mathbb{P}(C_{Y}X)\times_{Y}c^{-1}(Y)$ have the same irreducible components.
Each point of $\mathbb{P}(C_{Y}X)\times_{Y}c^{-1}(Y)$ has the form $(y,l,H)$
where $y$ is point of $Y$, $l$ belongs to $(\mathbb{P}(C_{Y}X))_{y}$ and $H\in
c^{-1}(y)$. Then the condition is equivalent to $l\in H$. The triple
$(X-Y,Y,0)$ satisfies $W_{f}$ if $(D_{Y})_{\mathrm{red}}$ is contained in the
incidence correspondence of
$\mathbb{P}(m_{Y}/(m_{Y})^{2})\times_{Y}\mathbb{P}((m_{Y}/(m_{Y})^{2})^{*})$.
Then a simple dimension count yields $\rm{(ii)}$.

Next we provide a concrete algebraic description of the conormal variety
$C(X)$ using the Jacobian module of $X$. Suppose $X$ is reduced. Let $X$ be
defined by the vanishing of some analytic functions $f_{1},\ldots,f_{p}$ on a
Euclidean neighborhood of $0$ in $\mathbb{C}^{n+k}$. Consider the following
exact sequence

(38)
$\begin{CD}I/I^{2}@>{\delta}>{}>\Omega_{\mathbb{C}^{n+k}}^{1}|X\longrightarrow\Omega_{X}^{1}\longrightarrow
0\end{CD}$

where $I$ is the ideal of $X$ in $\mathcal{O}_{\mathbb{C}^{n+k},0}$ and the
map $\delta$ sends a function $f$ vanishing on $X$ to its differential $df$.
Dualizing we obtain the following nested sequence of torsion-free sheaves:

$\mathrm{Image}(\delta^{*})\subset(\mathrm{Image}\
\delta)^{*}\subset(I/I^{2})^{*}.$

Observe that locally the sheaf $\mathrm{Image}(\delta^{*})$ can be viewed as
the column space of the (absolute) Jacobian module of $X$ which we denote by
$J(X)$. It’s a module contained in the free module $\mathcal{O}_{X,0}^{p}$.
Define the Rees algebra $\mathcal{R}(J(X))$ to be the subalgebra of
$\mathrm{Sym}(\mathcal{O}_{X,0}^{p})$ generated by the generators of $J(X)$
(for a general definition see [EHU03]). Define the conormal space $C(X)$ of
$X$ to be the closure in $X\times\mathbb{P}^{n+k-1}$ of the set of pairs
$(x,H)$ where $x$ is a smooth point of $X$ and $H$ is a tangent hyperplane at
$x$. Algebraically,

$C(X)=\mathrm{Projan}(\mathcal{R}(J(X))).$

To see that equality holds observe that both sides are equal over $X-Y$ to the
set of pairs $(x,H)$ where $H$ is a tangent hyperplane to the simple point
$x$. The left side is the closure of this set, and so is the right side simply
because the Rees algebra is by construction a subalgbebra of the symmetric
algebra $\mathrm{Sym}(\mathcal{O}_{X,0}^{p})$. Similarly, one sees that
$C(X,f)=\mathrm{Projan}(\mathcal{R}(J(X,f)))$ where $J(X,f)$ is the augmented
Jacobian module with the partials of $f$.

Consider the family setup $h:(X,0)\rightarrow(Y,0)$ from the beginning of the
section. Assume each $f_{i}$ is analytic function on two sets of variables:
the fiber variables $(x_{1},\ldots,x_{n})$ and the parameter variables
$y_{1},\ldots,y_{k}$. Form the augmented Jacobian matrix of $X$ and $f$ with
respect to the fiber variables:

$\begin{pmatrix}\partial f_{1}/\partial x_{1}&\cdots&\partial f_{1}/\partial
x_{n}\\\ \vdots&\ddots&\vdots\\\ \partial f_{p}/\partial x_{1}&\cdots&\partial
f_{p}/\partial x_{n}\\\ \partial f/\partial x_{1}&\cdots&\partial f/\partial
x_{n}\end{pmatrix}.$

The column space of this matrix generates a module over the local ring
$\mathcal{O}_{X,0}$, which we denote by

$J_{\mathrm{rel}}(X,f):=JM_{x}(f_{1},\ldots,f_{p};f)\subset\mathcal{O}_{X,0}^{p+1}$

and call the relative Jacobian module of $X$ and $f$. Given $y\in Y$ form the
image of the module $JM_{x}(f_{1},\ldots,f_{p};f)$ in the free module
$\mathcal{O}_{X_{y}}^{p+1}$:

$J(X_{y},f_{y}):=JM_{x}(f_{1},\ldots,f_{p};f)|X_{y}\subset{O}_{X_{y}}^{p}$

Obviously, $J(X_{y},f_{y})$ is the augmented Jacobian module of
$(X_{y},f_{y})$. Analogously to our discussion above we have
$C(X_{y},f_{y})=\mathrm{Projan}(\mathcal{R}(J(X_{y},f_{y})))$. Define the
relative conormal space
$C_{\mathrm{rel}}(X,f):=\mathrm{Projan}(\mathcal{R}(J_{\mathrm{rel}}(X,f)))$.

Dropping the row with partials of $f$ in $J_{\mathrm{rel}}(X,f)$ we obtain the
relative Jacobian module $J_{\mathrm{rel}}(X)$. Then the relative conormal
space $C_{\mathrm{rel}}(X)$ of $X$ is defined as the closure of the set of
pairs $(x,H)$ where $x$ is a smooth of $X$ and $H$ is tangent hyperplane
containing a parallel to $Y$. Algebraically,
$C_{\mathrm{rel}}(X)=\mathrm{Projan}(\mathcal{R}(J_{\mathrm{rel}}(X)))$.

The next ingredient we need in the proof of Thm. 7.1 is the relation between
$W_{f}$ and integral dependence of modules. Keep the setup from above. Let
$\mathcal{F}$ be a free module, $\mathcal{M}$ be a submodule of $\mathcal{F}$,
and let $g$ be an element from $\mathcal{F}$. Assume $X$ is reduced. Form the
Rees algebra $\mathcal{R}(\mathcal{M})$ of $\mathcal{M}$. It is a subalgebra
of the symmetric algebra $\mathrm{Sym}(\mathcal{F})$ (see [EHU03, Prp. 1.8 ]).
View $g$ as a degree one element in $\mathrm{Sym}(\mathcal{F})$. We say that
$g$ is integrally dependent on $\mathcal{M}$ if it satisfies an equation of
integral dependence,

$g^{u}+m_{1}g^{u-1}+\cdots+m_{u}=0$

where $u\geq 1$ and each $m_{i}$ belongs to the $i$th homogeneous component of
$\mathcal{R}(\mathcal{M})$. The following well-known criteria tell us that we
can check integral dependence on curves or by considering speeds of vanishing.

(Valuative criterion) An element $g\in\mathcal{F}$ is integrally dependent on
$\mathcal{M}$ if for any map germ $\phi:(\mathbb{C},0)\rightarrow(X,0)$ the
following holds

$\phi^{*}g\in\phi^{*}\mathcal{M}\subset\phi^{*}\mathcal{F}.$

(Analytic criterion) A necessary condition for $g\in\mathcal{F}$ to be
integrally dependent on $\mathcal{M}$ is that for any finite set of generators
$m_{i}$ of $\mathcal{M}$, there exists an Euclidean neighborhood $U$ of $0$ in
$X$ and a constant $c$ such that

$|g(x)|\leq c\max|m_{i}(x)|$

for each $x\in U$. Conversely, it suffices that this inequality holds for a
finite generating set of $\mathcal{M}$.

Now for each $j=1,\ldots,k$ let $g_{j}$ be the column vector

(39) $g_{j}:=\begin{pmatrix}\partial f_{1}/\partial y_{j}\\\ \vdots\\\
\partial f_{p}/\partial y_{j}\\\ \partial f/\partial y_{j}\end{pmatrix}.$

Denote the ideal of $Y$ in $\mathcal{O}_{X,0}$ by $m_{Y}$. The following
result (see [Gaf92, Thm. 2.5] and [GK99, Prp. 2.3]) characterizes the Whitney
conditions by the ingtegral dependence of $g$ on the module
$m_{Y}\mathcal{M}$. The main idea of its proof is to connect (37) with
integral dependence of modules through the analytic criterion mentioned above.
This was done by Teissier in the case when $X$ is a of codimension one in
$\mathbb{C}^{n+k}$ using integral closure of ideals, and generalized by
Gaffney and Kleiman [GK99, Prp. 6.1] to arbitrary codimension using integral
closure of modules.

######  Theorem 6.2 (Gaffney–Kleiman).

The triple $(X-Y,Y,0)$ satisfies $W_{f}$ if and only if $g_{j}$ is integrally
dependent on $m_{Y}J_{x}(X,f)$ for each $j=1,\ldots,p$.

Hironaka [Hir76] observed that the Whitney conditions hold generically, i.e.
there exists a Zariski dense open subset $U$ of $Y$ such that the Whitney
conditions hold at every $y\in U$. The same result for $W_{f}$ was obtained by
Henry, Merle and Sabbah ([HMS84, Thm. 5.1]). Therefore, the proof of Thm. 6.2
is reduced to show that the closed set in $Y$ where the integral dependence of
$g_{j}$ on $m_{Y}J_{x}(X,f)$ fails is empty. The following fundamental result
is a geometric criterion for the existence of this closed set.

Let $\mathcal{M}$ and $\mathcal{N}$ be two $\mathcal{O}_{X,0}$ modules with
$\mathcal{M}\neq\mathcal{N}$ such that
$\mathcal{M}\subset\mathcal{N}\subset\mathcal{O}_{X,0}^{p}$. Assume that $X$
is equidimensional and $\mathcal{M}$ and $\mathcal{N}$ are generically equal
and free of constant rank. Denote by
$c_{\mathcal{M}}\colon\mathrm{Projan}(\mathcal{R}(\mathcal{M}))\rightarrow X$
the structure morphism. Since $\mathcal{M}$ is generically free of constant
rank on a dense open subset of $X$, then
$\mathrm{Projan}(\mathcal{R}(\mathcal{M}))$ is equidimensional. Let $Z$ be the
closed set where $\mathcal{N}$ is not integral over $\mathcal{M}$ and let
$E:=c_{\mathcal{M}}^{-1}Z$. The following is a result of Kleiman and Thorup
([KT94] and [KT01]). For a generalization of this result see [Ran18, Thm.
6.1].

######  Theorem 6.3 (Kleiman–Thorup).

If $\mathcal{N}$ is not integral over $\mathcal{M}$, then $E$ has codimension
$1$.

##  7\. Numerical control of Whitney–Thom equisingularity

Preserve the setup from the previous section. In this section we apply the LVF