5.6in

# The Delta Invariant and Simultaneous Normalization for Families of isolated
Non-Normal Singularities

Gert-Martin Greuel and Gerhard Pfister<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We consider families of schemes over arbitrary fields resp. analytic varieties
with finitely many (not necessarily reduced) isolated non-normal
singularities, in particular families of generically reduced curves. We define
a modified delta invariant for isolated non-normal singularities of any
dimension that takes care of embedded points and prove that it behaves upper
semicontinuous in flat families parametrized by an arbitrary principal ideal
domain. Moreover, if the fibers contain no isolated points, then the familly
admits a fiberwise normalization iff the delta invariant is locally constant.
The results generalize results by Teissier and Chiang-Hsieh–Lipman for
families of reduced curve singularities and provide possible improvements for
algorithms to compute the genus of a curve.

Mathematics Subject Classification – MSC2020. 13B22, 13B40, 14B05, 14B07.
Keywords. Isolated non-normal singularity, delta invariant, semicontinuity,
generically reduced curves, simultaneous normalization.

###### Contents

  1. 1 Delta for an Isolated Non-Normal Singularity
  2. 2 Semicontinuity of the Delta Invariant
  3. 3 Fiberwise and Simultaneous Normalization
  4. 4 Charcteristic 0 and the Analytic Case

## Introduction

The delta invariant, also called genus defect, is the most basic numerical
invariant of a reduced curve singularity. It is used to control the topolgy in
a family of algebraic curves over the complex numbers, but it is also used in
coding theory for curves over finite fields. Non-reduced curves appear
naturally as fibers of the projection of a surface to a curve and can in
general not be avoided. In [BG90] a delta invariant was defined for
generically reduced complex analytic curves and it was shown that it controls
the topology of the fibers in a family, e.g. the Betti numbers, in particular
the number of connected components. In [Gr17] the delta invariant was further
extended to complex ananlytic isolated non-normal singularities (e.g. isolated
singularities) of any dimension and its behavior was studied in connection
with simultaneous normalization.

The study of simultaneous normalization of deformations of a reduced curve
singularity has been initiated by Teissier in the 1970’s in the complex
analytic setting. The main result was, that a flat family of reduced curve
singularities over a normal base space admits a simultaneous normalization if
and only if the delta invariant of the curve singularities is locally
constant. This was further carried on by Chiang-Hsieh and Lipman [CL06] in the
algebraic setting for families of reduced curves defined over a perfect field,
clarifying some points in the proof given in [Te78]. In [CL06] the authors
obtain also intermediate results for families of higher dimensional reduced
and pure dimensional varieties, but the $\delta$-constant criterion for
simultaneous normalization is only proved for families of reduced curves (and
for projective morphisms with equidimensional reduced fibers of arbitrary
dimension, replacing the $\delta$-invariant by the Hilbert polynomial, see
also [Ko11]).

The results by Chiang-Hsieh and Lipman motivated us to reconsider the
$\delta$-constant criterion for families of schemes with finitely many
isolated not necessarily reduced non-normal singularities, including the case
of generically reduced curves, defined over an arbitrary field. We define and
use in the algebraic setting a modified delta invariant for an isolated non-
normal singularity (INNS) of any dimension analogous to the complex analytic
case. One of our main results are semicontinuity theorems for this new
$\delta$-invariant for families of schemes parametrized by the spectrum of a
principal ideal domain (PID), see Theorem 20 and its corollaries in arbitrary
characteristic and Theorem 33 in characteristic 0. By the same theorems the
semicontinuity holds also for what we call the $\varepsilon$-invariant, i.e.,
the length of the embedded components of the non-normal points.

We apply the semicontinuity to prove a $\delta$-constant criterion for
simultaneous normalization of a family of INNSs. This means that a family of
affine Noetherian schemes over the spectrum of an arbitrary PID with fibers
having only finitely many isolated non-normal singularities (but no isolated
points) and with singular locus finite over the base admits a fiberwise
normalization 111 The notion of “fiberwise normalization”, introduced in this
paper, is useful for schemes over an arbitray base field and coincides with
“simultaneous normalization” if the base field is perfect. if and only if the
total $\delta$-invariant of the fibers is constant (for a precise formulation
see Theorem 31 and Corollary 32).

Although we use ideas from [CL06], the proofs of our main results are quite
different. We make substantial use of the fact that our base ring is a (not
necessarily local) PID in contrast to [CL06], where the base ring may be a
local normal ring of any dimension. Thus, in our setting, the semicontinuity
of $\delta$ holds for fibers over closed and non-closed points in a
neighbourhood of a given point (while in [CL06, Proposition 3.3] the
semicontinuity holds only for generalizations).

Moreover, to prove the $\delta$-constant criterion for simultaneous
normalization of reduced curves, it is assumed in [CL06, Theorem 4.1] that the
fibers are pure dimensional 222 Pure dimensional is also assumed for
projectice families of any dimension in [CL06] and [Ko11]. and that the base
scheme is the spectrum of a complete, or Henselian, or analytic normal local
ring. The pure-dimensionality assumption is however not necessary, even in the
more general situation of families of not necessarily reduced INNSs, but we
have to take special care of isolated points in the fibers (cf. Theorem 20 and
Theorem 31).

Since our base rings include ${\mathbb{Z}}$ and $\mathbbm{k}[t]$,
$\mathbbm{k}$ any field, we just mention in passing that our results have
interesting computational applications. E.g., if an isolated non-normal
singularity is defined over ${\mathbb{Z}}$ resp. over $\mathbbm{k}[t]$, the
computation of the $\delta$-invariant over ${\mathbb{Q}}$ resp.
$\mathbbm{k}(t)$ can be estimated and speeded up by the (much cheaper)
computation modulo any (not only lucky) prime $p\in{\mathbb{Z}}$ resp. modulo
$\langle t-a\rangle$, $a$ any element in $\mathbbm{k}$, provided that $\delta$
of the special fiber is finite. This applies of course also to $\delta$ for
reduced curves (if the special fiber is reduced, the nearby fibers are reduced
too) and hence can be used to improve algorithms to compute the genus of a
curve. We refer to [GP21, Remark 24], where we considered $\delta$ for
families of parametrized curve singularities, and where this is discussed in
more detail.

The case of families of parametrized curves shows that allowing non-reduced
singularities in the fibers is not an artificial assumption but occurs
naturally. Consider e.g. a morphism $\phi:T\times S\to Y\times S$ over $S$
(where $T$ can be $\operatorname{Spec}\mathbbm{k}[t]$ in the algebraic case or
$\operatorname{Spec}\mathbbm{k}[[t]]$ in the formal case or $({\mathbb{C}},0)$
in the analytic case, with $S$ in the corresponding category) such
$C_{s}=\phi({\mathbb{C}}\times\\{s\\})$ is a reduced curve in $Y$
($Y=\operatorname{Spec}\mathbbm{k}[y_{1},...,y_{n}]$ resp.
$\operatorname{Spec}\mathbbm{k}[[y_{1},...,y_{n}]]$ resp.
$({\mathbb{C}}^{n},0)$) for $s\in S$. Let $X=\phi(T\times S)$ be closed in
$Y\times S$ and flat over $S$. Then the fibers $X_{s}$ of $X\to S$ have in
general non reduced singularities (the reduction of $X_{s}$ coincides with
$C_{s}$) and our results apply to $\delta(X_{s})$.

All rings in this paper are associative, commutative and with 1, ring maps map
1 to 1, and ring maps of local rings map the maximal ideal to the maximal
ideal. Moreover, we assume all rings, modules and schemes to be Noetherian.
$A$, $R$ denote rings, $\mathbbm{k}$ an arbitrary field, $\dim$ the Krull
dimension and $\dim_{\mathbbm{k}}$ the $\mathbbm{k}$-vector space dimension.
If $R$ is an $A$-algebra, $\mathfrak{p}$ a prime ideal of $A$ and
$k(\mathfrak{p})=A_{\mathfrak{p}}/\mathfrak{p}A_{\mathfrak{p}}=Q(A/\mathfrak{p})$
the residue field of $A$ at $\mathfrak{p}$, we set

$M(\mathfrak{p}):=M_{\mathfrak{p}}\otimes_{A_{\mathfrak{p}}}k(\mathfrak{p})=M\otimes_{A}k(\mathfrak{p})$

and call it the fiber of an $R$-module $M$ over $\mathfrak{p}$.

##  1\. Delta for an Isolated Non-Normal Singularity

Let $R$ be a reduced ring. Then $Q(R)$, the total quotient ring of $R$, is a
direct product of fields. If $P_{1},...,P_{r}$ are the minimal primes of $R$
then $Q(R)$ is the direct product of the fields $Q(R/P_{i})$. ${\overline{R}}$
denotes the integral closure of $R$ in $Q(R)$. ${\overline{R}}$ or, more
precisely, the natural inclusion $R\hookrightarrow\overline{R}$ is called the
normalization of $R$. ${\overline{R}}$ is the direct product of the integral
closures $\overline{R_{j}}$ of $R_{j}:=R/P_{j}$ in $Q(R_{j})$ (cf. [Stack, tag
035P, Lemma 28.52.3]).

If $R$ is not reduced, let $R^{red}$ be the reduction of $R$,
$\pi:R\twoheadrightarrow R^{red}$ the natural projection and
$\operatorname{nil}(R):=\ker(\pi)$ the ideal of nilpotent elements of $R$. We
denote by $\nu^{red}:R^{red}\hookrightarrow{\overline{R}}$ the normalization
of $R^{red}$ and call ${\overline{R}}$ or the composition

$\nu:=\nu^{red}\circ\pi:R\twoheadrightarrow
R^{red}\hookrightarrow\overline{R}$

the normalization of $R$. $R$ is called normal if $\nu:R\to\overline{R}$ is an
isomorphism. This is equivalent to $R_{\mathfrak{p}}$ being a normal domain
for every prime ideal $\mathfrak{p}\subset R.$ We often write
${\overline{R}}/R^{red}$ in place of ${\overline{R}}/\nu(R)$.

For an arbitrary $R$-module let $\operatorname{Ann}_{R}(M)=\\{g\in
R\,|\,gM=0\\}$ be the annihilator ideal of $M$ in $R$.

######  Definition 1.

Let $R$ be a ring. We define

  1. (1)

${\mathcal{C}}_{R}:=\operatorname{Ann}_{R}({\overline{R}}/R^{red})\subset
R,\text{ the {\em conductor ideal} of }R,\\\
\widetilde{{\mathcal{C}}}_{R}:={\mathcal{C}}_{R}\cap\operatorname{Ann}_{R}(\operatorname{nil}(R))\subset
R,\text{ the {\em extended conductor ideal} of }R,$
$C_{R}:=\operatorname{Spec}R/{\mathcal{C}}_{R}$ the conductor scheme of $R$
and
$\widetilde{C}_{R}:=\operatorname{Spec}R/\widetilde{{\mathcal{C}}}_{R}$ the
extended conductor scheme of $R$.

  2. (2)

The non-normal locus of $R$ is denoted as
$NNor(R):=\\{\mathfrak{p}\in\operatorname{Spec}R\,|\,R_{\mathfrak{p}}\text{ is
not normal}\\}.$

######  Remark 2.

(1)
${\mathcal{C}}_{R^{red}}=\operatorname{Ann}_{R^{red}}({\overline{R}}/R^{red})=\pi({\mathcal{C}}_{R})$
is the conductor ideal of $R^{red}$. We have
$\operatorname{nil}(R)\subset{\mathcal{C}}_{R}$ since
$\operatorname{nil}(R)=\ker(\nu)$, and $\pi$ induces an isomorphism
$R/{\mathcal{C}}_{R}\xrightarrow{\cong}R^{red}/{\mathcal{C}}_{R^{red}}$.

(2) The non-normal locus of $R$ coincides with the union of the non-reduced
locus of $R$ and the non-normal locus of $R^{red}$, i.e.,
$NNor(R)=\operatorname{Supp}_{R}(\operatorname{nil}(R))\cup\operatorname{Supp}_{R}({\overline{R}}/R^{red})$.

(3) We have always $NNor(R)\subset V(\widetilde{{\mathcal{C}}}_{R})$ but
equality may not hold and $NNor(R)$ may not be closed in
$\operatorname{Spec}R$. However, if ${\overline{R}}/R^{red}$ is (module-)
finite over $R$ (equivalently, ${\overline{R}}$ is finite over $R$), then
$\operatorname{Supp}_{R}({\overline{R}}/R^{red})$ coincides with
$V({\mathcal{C}}_{R})$ and is therefore closed in $\operatorname{Spec}R$. We
get:

If ${\overline{R}}$ is finite over $R$ (for examples see Remark 15), then the
non-normal locus of $R$

$\displaystyle
NNor(R)=V(\widetilde{{\mathcal{C}}}_{R})=V(\operatorname{Ann}_{R}(\operatorname{nil}(R)))\cup
V({\mathcal{C}}_{R}).$

is closed in $\operatorname{Spec}R$ and is the zero-locus of the extended
conductor ideal $\widetilde{{\mathcal{C}}}_{R}$.

######  Definition 3.

We say that $\mathfrak{p}\in\operatorname{Spec}R$ is an isolated non-normal
point of $R$, or that $R$ has an isolated non-normal singularity at
$\mathfrak{p}$, if there is an open neighbourhood $U$ of $\mathfrak{p}$ in
$\operatorname{Spec}R$ such that $R_{\mathfrak{q}}$ is normal for all
$\mathfrak{q}\in U\smallsetminus\\{\mathfrak{p}\\}$. We also say that
$\mathfrak{p}$ is an INNS (of $R$ or of $\operatorname{Spec}R$).

Let ${\overline{R}}$ be finite over $R$ and $\mathfrak{p}$ an INNS of $R$.
Then $\mathfrak{p}$ is either a normal point or $\mathfrak{p}$ is an isolated
point of $V(\widetilde{{\mathcal{C}}}_{R})$. Hence, if $\mathfrak{p}$ is non-
normal then it is a maximal ideal, i.e., a closed point of
$\operatorname{Spec}R$.

######  Lemma 4.

Let ${\overline{R}}$ be finite over $R$. Then $R$ has finitely many non-normal
points if and only if $R/\widetilde{{\mathcal{C}}}_{R}$ is Artinian.

###### Proof.

By Remark 2 we have
$NNor(R)=V(\widetilde{{\mathcal{C}}}_{R})=\operatorname{Supp}_{R}(R/\widetilde{{\mathcal{C}}}_{R})$.
The result follows follows since $R/\widetilde{{\mathcal{C}}}_{R}$ is Artinian
$\Leftrightarrow$ $\operatorname{Spec}R/\widetilde{{\mathcal{C}}}_{R}$ is a
finite set ([AM69, Prop. 8.3]). ∎

We are now going to define the delta invariant of an INNS. For this we
introduce the following

######  Notation 5.

(1) Let $R$ be a ring and $\mathfrak{p}_{1},...,\mathfrak{p}_{r}$ the minimal
prime ideals of $R$. We denote by $\mathfrak{p}^{i}$ resp. $\mathfrak{p}^{>i}$
the intersection333 The empty intersection is the whole ring $R$. E.g., if no
minimal primes $\mathfrak{p}_{j}$ with $\dim R/\mathfrak{p}_{j}=i$ exist then
$R^{i}=0$ and $X^{i}=\emptyset$. of the $\mathfrak{p}_{j}$ with $\dim
R/\mathfrak{p}_{j}=i$ resp. $\dim R/\mathfrak{p}_{j}>i$. With
$X=\operatorname{Spec}R$ and $X^{red}=\operatorname{Spec}R^{red}$ we define
for $i\geq 0$:

$\begin{array}[]{cllllll}R^{i}&:=&R/\mathfrak{p}^{i},&X^{i}&:=&\operatorname{Spec}R^{i},\\\
R^{>i}&:=&R/\mathfrak{p}^{>i},&X^{>i}&:=&\operatorname{Spec}R^{>i}.\end{array}$

Note that $R^{i}$ and $R^{>i}$ are reduced and thus $X^{i}$ and $X^{>i}$ are
reduced subschemes of $X$. In particular, $X^{0}$ is a finite set of reduced,
isolated points of $X^{red}$ and $X^{>0}=X^{red}$ iff $X$ has no isolated
points.

We set

$r_{i}(X):=\sharp\\{\text{irreducible componentes of }X^{i}\\},$

which is the number of $i$-dimensional irreducible components of $X$ 444 By
definition, the irreducible components of $X$ are the schemes
$\operatorname{Spec}R/\mathfrak{p}_{j},j=1,...,r.$.

(2) Let $\mathfrak{m}\in\operatorname{Spec}R$ be a maximal ideal of $R$ and
$M$ an $R$-module. The $0$-th local cohomology group of $M$ is the submodule

$H^{0}_{\mathfrak{m}}(M)=\\{x\in M\ |\ \mathfrak{m}^{k}x=0\text{ for some
}k\geq 0\\}.$

Since $M$ is Noetherian, $H^{0}_{\mathfrak{m}}(M)$ is Noetherian too and is
annihilated by some power of $\mathfrak{m}$; hence $H^{0}_{\mathfrak{m}}(M)$
has finite length, i.e. is Artinian.

######  Definition 6.

Let $(R,\mathfrak{m})$ be a local ring with normalization ${\overline{R}}$,
$\mathbbm{k}$ a field, and $\mathbbm{k}\to R$ a ring map. We define:

  * (i)

the epsilon invariant of $R$ (w.r.t. $\mathbbm{k}$),

$\varepsilon_{\mathbbm{k}}(R):=\dim_{\mathbbm{k}}H^{0}_{\mathfrak{m}}(R),$

  * (ii)

the delta invariant of $R$ (w.r.t. $\mathbbm{k}$),

$\delta_{\mathbbm{k}}(R):=\dim_{\mathbbm{k}}\overline{R}/R^{red}-\varepsilon_{\mathbbm{k}}(R),$

  * (iii)

the (multiplicity of the) conductor of $R$ (w.r.t. $\mathbbm{k}$)

$c_{\mathbbm{k}}(R):=\dim_{\mathbbm{k}}{\overline{R}}/{\mathcal{C}}_{R^{red}}-\varepsilon_{\mathbbm{k}}(R).$

Hence, if $R$ is reduced and $\dim R>0$ then $\varepsilon_{\mathbbm{k}}(R)=0$
and $\delta_{\mathbbm{k}}(R)=\dim_{\mathbbm{k}}\overline{R}/R$, the usual
definition of $\delta_{\mathbbm{k}}$.

######  Remark 7.

Let $K=R/\mathfrak{m}$ denote the residue field of the local ring
$(R,\mathfrak{m})$ and assume that $\dim_{\mathbbm{k}}K<\infty$.

(1) $\varepsilon_{\mathbbm{k}}(R)$ is finite while $\delta_{\mathbbm{k}}(R)$
and $c_{\mathbbm{k}}(R)$ may be infinite. If $R$ is an INNS with
${\overline{R}}$ finite over $R$, then $\delta_{\mathbbm{k}}(R)$ and
$c_{\mathbbm{k}}(R)$ are also finite (Lemma 10).

(2) If $\dim R=0$, then ${\overline{R}}=R^{red}=K$,
$\operatorname{nil}(R)=\mathfrak{m}$ and $H^{0}_{\mathfrak{m}}(R)=R$. We get
$\delta_{\mathbbm{k}}(R)=c_{\mathbbm{k}}(R)=-\varepsilon_{\mathbbm{k}}(R)=-\dim_{\mathbbm{k}}R=-\dim_{\mathbbm{k}}\operatorname{nil}(R)-\dim_{\mathbbm{k}}K<0$.

(3) Let $\dim R>0$ and let $R$ be an INNS with ${\overline{R}}$ finite over
$R$. Since $R_{\mathfrak{p}}$ is reduced for $\mathfrak{p}\in
U\smallsetminus\mathfrak{m}$, $U$ some open neighbourhood of $\mathfrak{m}$ in
$\operatorname{Spec}R$, we have
$\operatorname{nil}(R)=H^{0}_{\mathfrak{m}}(R)$ and
$\varepsilon_{\mathbbm{k}}(R)=\dim_{\mathbbm{k}}\operatorname{nil}(R)$.

If $R$ is normal and $\dim R>0$ then
$\delta_{\mathbbm{k}}(R)=c_{\mathbbm{k}}(R)=\varepsilon_{\mathbbm{k}}(R)=0$.
If $R$ is reduced, then $\delta_{\mathbbm{k}}(R)=0$ if and only if $R$ is
normal. But if $R$ is not reduced, then $\delta_{\mathbbm{k}}(R)=0$ may happen
for non-normal $R$ (see Example 8 (3)).

######  Example 8.

(1) The ideal $I=\langle x\rangle\cap\langle x^{2},y^{2},xy\rangle=\langle
x^{2},xy\rangle\subset\mathbbm{k}[[x,y]]$ defines a line with embedded
component. With $R=\mathbbm{k}[[x,y]]/I$ we get
$\delta_{\mathbbm{k}}(R^{red})=0$ and $\varepsilon_{\mathbbm{k}}(R)=1$, hence
$\delta_{\mathbbm{k}}(R)=-1$ and $c_{\mathbbm{k}}(R)=-1$.

(2) The ideal

$I=\langle x^{3}y+x^{2}y^{2},x^{2}y^{2}+xy^{3}\rangle=\langle
x+y\rangle\cap\langle x\rangle\cap\langle y\rangle\cap\langle
x^{2},y^{3}\rangle\subset\mathbbm{k}[[x,y]],$

defines 3 lines with an embedded component at $0$. For
$R=\mathbbm{k}[[x,y]]/I$ we have $\delta_{\mathbbm{k}}(R^{red})=3$ and
$\varepsilon_{\mathbbm{k}}(R)=\dim_{\mathbbm{k}}\sqrt{I}/I=1$ 555 We compute
$\varepsilon$ and $\delta$ with Singular [DGPS]: codim computes
$\dim_{\mathbbm{k}}\operatorname{nil}(R)$ = $\dim_{\mathbbm{k}}\sqrt{I}/I$ and
the procedure normal(..,"wd") computes $\delta_{\mathbbm{k}}(R^{red})$; the
number of isolated points of $\operatorname{Spec}R$ can be determined with a
primary decomposition of $I$. and hence $\delta_{\mathbbm{k}}(R)=2$. Since
$R^{red}$ is a reduced plane curve singularity, we get
$c_{\mathbbm{k}}(R^{red})=2\delta_{\mathbbm{k}}(R^{red})=6$ and
$c_{\mathbbm{k}}(R)=c_{\mathbbm{k}}(R^{red})-\varepsilon_{\mathbbm{k}}(R)=5.$

(3) $I=\langle z,x^{2}-y^{3}\rangle\cap\langle x,y,z^{2}\rangle$,
$R=\mathbbm{k}[[x,y]]/I$, defines a cusp in the $(x,y)$-plane and an embedded
point in the $z$-direction. Then $\delta_{\mathbbm{k}}(R^{red})=1$ and
$\varepsilon_{\mathbbm{k}}(R)=1$ and hence $\delta_{\mathbbm{k}}(R)=0$.

######  Lemma 9.

Let $(R,\mathfrak{m})$ be a local ring, $K=R/\mathfrak{m}$, $\mathbbm{k}\to R$
a ring map, and $M\neq 0$ a finitely generated $R$-module. Then
$\dim_{\mathbbm{k}}M<\infty$ $\Leftrightarrow$ $M$ is Artinian and
$\dim_{\mathbbm{k}}K<\infty$.

###### Proof.

If $\dim_{\mathbbm{k}}M<\infty$ then $M$ is Artinian since it satisfies
obviously the descending chain condition. By Nakayama’s lemma,
$M/\mathfrak{m}M$ is a finite dimensional $K$-vector space $\neq 0$. We have
$\dim_{\mathbbm{k}}K\leq\dim_{\mathbbm{k}}M/\mathfrak{m}M\leq\dim_{\mathbbm{k}}M<\infty$.
Conversely, if $M$ Artinian then $\mathfrak{m}^{n}M=0$ for some $n$. The
$K$-vector space $\mathfrak{m}^{k}M/\mathfrak{m}^{k+1}M$ has finite
$K$-dimension, hence finite $\mathbbm{k}$-dimension since
$\dim_{\mathbbm{k}}K<\infty$. Thus $\dim_{\mathbbm{k}}M<\infty$. ∎

######  Lemma 10.

Let $(R,\mathfrak{m})$ be a local ring with normalization ${\overline{R}}$
finite over $R$.

  1. (1)

The following are equivalent:

    1. (i)

$\mathfrak{m}$ is an isolated non-normal point of $R$;

    2. (ii)

$R/\widetilde{{\mathcal{C}}}_{R}$ is an Artinian ring;

    3. (iii)

${\overline{R}}/{\mathcal{C}}_{R^{red}}$ and $\operatorname{nil}(R)$ are
Artinian $R$-modules;

    4. (iv)

${\overline{R}}/R^{red}$ and $\operatorname{nil}(R)$ are Artinan $R$-modules.

  2. (2)

Let $\mathbbm{k}$ be a field and $\mathbbm{k}\to R$ a ring map. Then the
following are equivalent:

    1. (i)

$\mathfrak{m}$ is an isolated non-normal point of $R$ and
$\dim_{\mathbbm{k}}R/\mathfrak{m}<\infty$;

    2. (ii)

$\dim_{\mathbbm{k}}(R/\widetilde{{\mathcal{C}}}_{R})$ is finite;

    3. (iii)

$c_{\mathbbm{k}}(R^{red})$ and $\varepsilon_{\mathbbm{k}}(R)$ are finite.

    4. (iv)

$\delta_{\mathbbm{k}}(R^{red})$ and $\varepsilon_{\mathbbm{k}}(R)$ are finite;

If any of these conditions hold, $c_{\mathbbm{k}}(R)$ and
$\delta_{\mathbbm{k}}(R)$ are finite and satisfy
$c_{\mathbbm{k}}(R)=\delta_{\mathbbm{k}}(R)+\dim_{\mathbbm{k}}(R/{\mathcal{C}}_{R}).$

###### Proof.

We may assume that $(R,\mathfrak{m})$ is not normal. By Remark 2 we have
$NNor(R)=V(\widetilde{{\mathcal{C}}}_{R})=V(\operatorname{Ann}(\operatorname{nil}(R)))\cup
V({\mathcal{C}}_{R})$.

(1) It is well known that a finitely generated $R$-module $M$ is Artinian
$\Leftrightarrow$ $\mathfrak{m}^{k}M=0$ for some $k>0$ $\Leftrightarrow$ $\dim
M=0$ $\Leftrightarrow$ $\operatorname{Supp}_{R}(M)=\\{\mathfrak{m}\\}$. Now
(1) follows from
$NNor(R)=\operatorname{Supp}_{R}(R/\widetilde{{\mathcal{C}}}_{R})=\operatorname{Supp}_{R}(\operatorname{nil}(R))\cup\operatorname{Supp}_{R}({\overline{R}}/R^{red})$.

(2) The equivalence of (i) - (iv) follows from (1) for
$\mathbbm{k}=K:=R/\mathfrak{m}$, noting that
$\dim_{\mathbbm{k}}\operatorname{nil}(R)=\varepsilon_{\mathbbm{k}}(R)$ if
$\dim(R)>0$ and
$\dim_{\mathbbm{k}}\operatorname{nil}(R)=\varepsilon_{\mathbbm{k}}(R)-1$ if
$\dim(R)=0$. Together with Lemma 9 the equivalence follows for $\mathbbm{k}$
with $\dim_{\mathbbm{k}}K<\infty$. The exact sequence

$0\to
R^{red}/{\mathcal{C}}_{R^{red}}\to{\overline{R}}/{\mathcal{C}}_{R^{red}}\to{\overline{R}}/R^{red}\to
0$

implies
$c_{\mathbbm{k}}(R^{red})=\delta_{\mathbbm{k}}(R^{red})+\dim_{\mathbbm{k}}(R^{red}/{\mathcal{C}}_{R^{red}})$
and hence
$c_{\mathbbm{k}}(R)=\delta_{\mathbbm{k}}(R)+\dim_{\mathbbm{k}}(R/{\mathcal{C}}_{R})$
by definition of $c_{\mathbbm{k}}$ and $\delta_{\mathbbm{k}}$. ∎

Now let $R$ be any (not necessarily local) ring and let ${\overline{R}}$ be
finite over $R$. It follows from Lemma 4 and 10 that $R$ has only finitely
many non-normal points $\iff$ $R/\widetilde{{\mathcal{C}}}_{R}$ is Artinian.
If $R$ is a $\mathbbm{k}$-algebra then:
$\dim_{\mathbbm{k}}(R/\widetilde{{\mathcal{C}}}_{R})<\infty$ $\iff$ $NNor(R)$
is finite and $\dim_{\mathbbm{k}}k(\mathfrak{p})<\infty$ for all
$\mathfrak{p}\in NNor(R)$. The following definition generalizes
$\delta,\varepsilon$ and $c$ to non-local rings.

######  Definition 11.

Let $R$ be a $\mathbbm{k}$-algebra with normalization ${\overline{R}}$ finite
over $R$. Assume that $R$ has only finitely many isolated non-normal points
$\\{\mathfrak{p}_{1},...,\mathfrak{p}_{r}\\}$ and that
$\delta_{\mathbbm{k}}(R_{\mathfrak{p}_{i}})<\infty$ for $i=1,...,r$. We define

$\delta_{\mathbbm{k}}(R):=\sum_{1}^{r}\delta_{\mathbbm{k}}(R_{\mathfrak{p}_{i}}),$

$\varepsilon_{\mathbbm{k}}(R):=\sum_{1}^{r}\varepsilon_{\mathbbm{k}}(R_{\mathfrak{p}_{i}})$
and $c_{\mathbbm{k}}(R):=\sum_{1}^{r}c_{\mathbbm{k}}(R_{\mathfrak{p}_{i}})$,
which are all finite.

######  Example 12.

Let $R_{\mathbb{C}}={\mathbb{C}}[x,y]/I$, $I=\langle
y^{2}-2x^{2}\rangle\cap\langle y-x^{2}\rangle$. $V(I)$ consists of two
straight lines and a parabola meeting in $(0,0)$ and in $(\pm\sqrt{2},2)$. The
three INNS correspond to the maximal ideals $\mathfrak{p},\pm\mathfrak{q}$.
$\mathfrak{p}$ is a triple point with
$\delta_{\mathbb{C}}(R_{{\mathbb{C}},\mathfrak{p}})=3$, while
$\pm\mathfrak{q}$ are ordinary nodes with
$\delta_{\mathbb{C}}(R_{{\mathbb{C}},\pm\mathfrak{q}})=1$ each, hence
$\delta_{\mathbb{C}}(R_{\mathbb{C}})=5.$

Let $R_{\mathbb{Q}}={\mathbb{Q}}[x,y]/I$, with $I$ as above. Then has
$R_{\mathbb{Q}}$ has (in $\operatorname{Spec}R_{\mathbb{Q}}$) two INNS, at the
maximal ideals $\mathfrak{p}=\langle x,y\rangle$ and $\mathfrak{q}=\langle
x^{2}-2,y-2\rangle$, with $k(\mathfrak{p})={\mathbb{Q}}$ and
$k(\mathfrak{q})={\mathbb{Q}}(\sqrt{2}).$ We get
$\delta_{\mathbb{Q}}(R_{{\mathbb{Q}},\mathfrak{p}})=3$ and
$\delta_{\mathbb{Q}}(R_{{\mathbb{Q}},\mathfrak{q}})=2$, hence
$\delta_{\mathbb{Q}}(R_{\mathbb{Q}})=5.$ The equality
$\delta_{\mathbb{Q}}(R_{\mathbb{Q}})=\delta_{\mathbb{C}}(R_{\mathbb{C}})$ is a
general fact, since 666 Let $B$ be a $\mathbbm{k}$-algebra and $K$ a separable
field extension of $\mathbbm{k}$ then
$\overline{B\otimes_{\mathbbm{k}}K}=\overline{B}\otimes_{\mathbbm{k}}K$
([Stack], Lemma 32.27.4) and hence
$\delta_{K}(B\otimes_{\mathbbm{k}}K)=\delta_{\mathbbm{k}}(B).$
$R_{\mathbb{C}}=R_{\mathbb{Q}}\otimes_{\mathbb{Q}}\mathbb{C}$.

Using that $R^{red}=R^{0}\oplus R^{>0}$,
${\overline{R}}={\overline{R}}^{0}\oplus{\overline{R}}^{>0}$ and
$R^{0}={\overline{R}}^{0}$ we get

$\displaystyle\varepsilon_{\mathbbm{k}}(R)$
$\displaystyle=\dim_{\mathbbm{k}}Ker(R\to{\overline{R}})+\\#\\{\text{isolated
points of }\operatorname{Spec}R\\},$
$\displaystyle=\dim_{\mathbbm{k}}Ker(R\to{\overline{R}}^{>0})$
$\displaystyle\delta_{\mathbbm{k}}(R)$
$\displaystyle=\dim_{\mathbbm{k}}\overline{R}^{>0}/R^{>0}-\varepsilon_{\mathbbm{k}}(R)$
$\displaystyle=\dim_{\mathbbm{k}}Coker(R\to{\overline{R}}^{>0})-\dim_{\mathbbm{k}}Ker(R\to{\overline{R}}^{>0}).$

It follows that $\delta$ can be expressed as an Euler characteristic.

######  Lemma 13.

With the assumptions of Definition 11 consider the $2$–term complex with $R$
in degree 0, $R^{\bullet}:0\to R\to{\overline{R}}^{>0}\to 0$. Then

$\displaystyle\delta_{\mathbbm{k}}(R)=-\chi_{{}_{\mathbbm{k}}}(R^{\bullet}),$

where
$\chi_{{}_{\mathbbm{k}}}(L^{\bullet}):=\sum_{i}(-1)^{i}\dim_{\mathbbm{k}}H^{i}(L^{\bullet})$
for a complex $L^{\bullet}$ of $\mathbbm{k}$-modules with finite dimensional
cohomology.

The following technical lemma compares $\delta$ and $\varepsilon$ of $R$ with
that of a finite modification of $R$ whose positive dimensional part is a
partial normalization of $R^{>0}$. It is a key lemma for the semicontinuity of
$\delta$.

######  Lemma 14.

Let $R$ be a $\mathbbm{k}$-algebra with ${\overline{R}}$ finite over $R$,
having only finitely many isolated non-normal singularities, with residue
fields finite over $\mathbbm{k}$. Consider a finite morphism of
$\mathbbm{k}$-algebras $\mu:R\to{\widetilde{R}}$. Let
$N\subset\operatorname{Spec}R$ be a finite set of closed points with residue
fields finite over $\mathbbm{k}$, such that $\operatorname{Spec}\mu$ is an
isomorphism over $\operatorname{Spec}R\smallsetminus N$.

Then the positive dimensional parts $R^{>0}$ and ${\widetilde{R}}^{>0}$ have
the same normalization and $\mu$ satisfies

$\begin{array}[]{cllllll}\dim_{\mathbbm{k}}Coker(\mu)-\dim_{\mathbbm{k}}Ker(\mu)&=&\delta_{\mathbbm{k}}(R)-\delta_{\mathbbm{k}}({\widetilde{R}})\\\
&=&\varepsilon_{\mathbbm{k}}({\widetilde{R}})-\varepsilon_{\mathbbm{k}}(R)+\dim_{\mathbbm{k}}Coker(\mu^{>0})\end{array}$

with $\mu^{>0}:R^{>0}\to{\widetilde{R}}^{>0}$ the induced map. $\mu^{>0}$ is
finite and injective and a partial normalization777 Let
$\nu:R\to{\overline{R}}$ be the normalization of $R$. A partial normalization
of $R$ is a birational morphism $\mu:R\to{\widetilde{R}}$ such that
$\nu=\tilde{\nu}\circ\mu:R\to{\widetilde{R}}\to{\overline{R}}$, with
$\tilde{\nu}$ the normalization of ${\widetilde{R}}$. of the reduced positive
dimensional part of $R$ 888 In the case $R^{>0}=0$, i.e. $X^{>0}=\emptyset$,
the statements here and further on have to be interpreted approriately, e.g.
with $\delta_{\mathbbm{k}}(R^{>0})=0$ and
$\varepsilon_{\mathbbm{k}}(R^{>0})=0$. and all numbers are finite.

###### Proof.

By assumption $R$ has only finitely many non-normal singularities with
$\delta_{\mathbbm{k}}(R)$ and $\varepsilon_{\mathbbm{k}}(R)$ finite (Lemma
10). Since $\mu$ is an isomorphism outside finitely many closed points,
$Ker(\mu)$ and $Coker(\mu)$ are Artinian. Then the kernel and cokernel of
$\mu$ and of $\mu^{>0}$ are finite over $\mathbbm{k}$ (Lemma 9).

We have
$\operatorname{Spec}\widetilde{R}^{red}=\operatorname{Spec}\widetilde{R}^{>0}\,\cup$
{finitely many isolated points} and the restriction of $\mu$ induces a
birational morphism $R^{>0}\to{\widetilde{R}}^{>0}$, since it is an
isomorphism outside finitely many closed points. Let
$\nu^{>0}:R^{>0}\to{\overline{R}}^{>0}$ be the normalization of $R^{>0}$. By
[Stack, Lemma 28.52.5 (3), 035Q] $\nu^{>0}$ factors as
$\nu^{>0}=\tilde{\nu}\circ\mu^{>0}:R^{>0}\to{\widetilde{R}}^{>0}\to{\overline{R}}^{>0}$
with $\tilde{\nu}:{\widetilde{R}}^{>0}\to{\overline{R}}^{>0}$ the
normalization of ${\widetilde{R}}^{>0}$ and $\tilde{\nu}$ finite. Hence
$\mu^{>0}$ is a partial normalization. It is finite since $\mu$ is finite and
injective since $R^{>0}$ is reduced.

Now consider the $2$–term complexes (with $R$ resp. ${\widetilde{R}}$ in
degree 0)

$\begin{array}[]{l}R^{\bullet}:0\to R\to{\overline{R}}^{>0}\to 0\,,\\\
{\widetilde{R}}^{\bullet}:0\to{\widetilde{R}}\to{\overline{R}}^{>0}\to
0\end{array}$

and the morphism of complexes
$\mu^{\bullet}:R^{\bullet}\to{\widetilde{R}}^{\bullet}$ with $\mu^{0}=\mu$ and
the identity in degree 1. Let $K^{\bullet}$ resp. $C^{\bullet}$ be the
$1$–term complexes $Ker(\mu)$ resp. $Coker(\mu)$, concentrated in degree $0$.
Then we have the exact sequence of complexes

$0\to K^{\bullet}\to R^{\bullet}\to{\widetilde{R}}^{\bullet}\to C^{\bullet}\to
0\,.$

Taking Euler characteristics we get (by Lemma 13)

$\dim_{\mathbbm{k}}Coker(\mu)-\dim_{\mathbbm{k}}Ker(\mu)=\chi_{{}_{\mathbbm{k}}}({\widetilde{R}}^{\bullet})-\chi_{{}_{\mathbbm{k}}}(R^{\bullet})=\delta_{\mathbbm{k}}(R)-\delta_{\mathbbm{k}}({\widetilde{R}})$

showing the first equality. Since
$\delta_{\mathbbm{k}}(R)=\delta_{\mathbbm{k}}(R^{>0})-\varepsilon_{\mathbbm{k}}(R)$
we get

$\delta_{\mathbbm{k}}(R)-\delta_{\mathbbm{k}}({\widetilde{R}})=\varepsilon_{\mathbbm{k}}({\widetilde{R}})-\varepsilon_{\mathbbm{k}}(R)+\delta_{\mathbbm{k}}(R^{>0})-\delta_{\mathbbm{k}}({\widetilde{R}}^{>0}).$

From this and from the inclusions
$R^{>0}\hookrightarrow{\widetilde{R}}^{>0}\hookrightarrow{\overline{R}}^{>0}$
the second equality follows. ∎

######  Remark 15.

(1) For all results of this paper we have to assume that ${\overline{R}}$ is
(module-) finite over $R$. Integral domains that satisfy this conditions are
called N-1 rings. An N-2 ring (or Japanese ring) is an integral domain $R$
such for every finite field extension $L$ of $Q(R)$ the integral closure of
$R$ in $L$ is finite over $R$. $R$ is a Nagata ring if $R$ is Noetherian and
for every prime ideal $\mathfrak{p}$ the ring $R/\mathfrak{p}$ is N-2 (see
[Stack, Lemma 10.157.2, tag 03GH]).

(2) $R$ is Nagata iff (cf. [CL06, 1.4.3])

  1. (a)

for every maximal ideal $\mathfrak{n}$ of $R$ the canonical map
$R_{\mathfrak{n}}\to\widehat{R}_{\mathfrak{n}}$ from the local ring
$R_{\mathfrak{n}}$ to its completion is reduced (flat with reduced fibers) and

  2. (b)

for every reduced finitely generated $R$-algebra $R^{\prime}$ the set of
normal points is open and dense in $\operatorname{Spec}R^{\prime}$.
Condition (b) is implied by (a) if $R$ is semi-local. For further properties
of Nagata rings we refer to [Stack, Section 10.157, tag 032E].

(3) Examples of Nagata rings are:

  1. (1)

fields, ${\mathbb{Z}}$, Noetherian complete local rings,

  2. (2)

Dedekind domains with perfect999 A field $\mathbbm{k}$ is perfect if
$\mathbbm{k}$ is of characteristic 0 or of characteristic $p>0$ and every
element has a $p$-th root (e.g. if $\mathbbm{k}$ is finite). fraction
field101010 This statement is formulated in [Stack] only for Dedekind domains
with fraction fields of characteristic 0, but the proof works for perfect
fields of positive characteristic as well.,

  3. (3)

finite type ring extensions of any of the above,
(for (1) (2) (3) see [Stack, Proposition 10.157.16, tag 0335]),

  4. (4)

quasi-excellent, in particular excellent rings (e.g. analytic local rings),
([Stack, Lemma 15.51.5, tag 07QV]),

  5. (5)

localizations of a Nagata ring ([Stack, Lemma 10.157.6, tag 032U]),

  6. (6)

$A$-algebras (essentially) of finite type over a Nagata ring $A$ ([Stack,
Proposition 10.157.16, tag 0335]),

  7. (7)

$A[[x_{1},...,x_{n}]]$ is Nagata if $A$ is Nagata, ([KS19, Appendix A,
Property PSEP]).

A scheme $X$ is called Nagata if for every $x\in X$ there exists an affine
open neighbourhood $U\subset X$ of $x$ such that the ring
${\mathcal{O}}_{X}(U)$ is Nagata. Note that there are discrete valuation rings
that are not Nagata ([Stack, Example 10.157.17, tag 09E1]).

##  2\. Semicontinuity of the Delta Invariant

We consider now families of isolated non-normal singularities. The following
Proposition 16 is fundamental for the semicontinuity results of this paper.

######  Proposition 16.

Let $\varphi:A\to R$ be a flat morphism of rings with $A$ a principal ideal
domain and $\mu:R\to{\widetilde{R}}$ a finite morphism of $A$-algebras. Assume
that

  1. (1)

the composition $\widetilde{\varphi}:=\mu\circ\varphi:A\to{\widetilde{R}}$ is
flat,

  2. (2)

$Ker(\mu)$ and $Coker(\mu)$ are finite over $A$,

  3. (3)

the normalization $\overline{R(\mathfrak{q})}$ is finite over
$R(\mathfrak{q})$ and the residue fields at the non-normal points of
$R(\mathfrak{q})$ are finite over $k(\mathfrak{q})$ for $\mathfrak{q}\in
Im(\operatorname{Spec}\varphi)$.

Then, for $\mathfrak{p}\in Im(\operatorname{Spec}\varphi)$ there exists an
open neighborhood $U\subset\operatorname{Spec}A$ of $\mathfrak{p}$ such that
for $\mathfrak{q}\in U\cap Im(\operatorname{Spec}\varphi)$ the following
holds:

$\begin{array}[]{clll}(i)&\delta_{k(\mathfrak{p})}(R(\mathfrak{p}))-\delta_{k(\mathfrak{q})}(R(\mathfrak{q}))=\delta_{k(\mathfrak{p})}({\widetilde{R}}(\mathfrak{p}))-\delta_{k(\mathfrak{q})}({\widetilde{R}}(\mathfrak{q})).\\\
(ii)&\varepsilon_{k(\mathfrak{p})}(R(\mathfrak{p}))-\varepsilon_{k(\mathfrak{q})}(R(\mathfrak{q}))=\varepsilon_{k(\mathfrak{p})}({\widetilde{R}}(\mathfrak{p}))-\varepsilon_{k(\mathfrak{q})}({\widetilde{R}}(\mathfrak{q}))\\\
&+\dim_{k(\mathfrak{p})}Coker(\mu(\mathfrak{p})^{>0})-\dim_{k(\mathfrak{q})}Coker(\mu(\mathfrak{q})^{>0}).\\\
(iii)&\text{If
}\operatorname{Spec}{\widetilde{R}}^{1}\cap\operatorname{Spec}{\widetilde{R}}^{>1}=\emptyset\text{
(e.g. if ${\widetilde{R}}={\overline{R}}$ or ${\widetilde{R}}^{1}=0$),
then}\\\
&\varepsilon_{k(\mathfrak{p})}(R(\mathfrak{p}))-\varepsilon_{k(\mathfrak{q})}(R(\mathfrak{q}))=\varepsilon_{k(\mathfrak{p})}({\widetilde{R}}(\mathfrak{p}))-\varepsilon_{k(\mathfrak{q})}({\widetilde{R}}(\mathfrak{q}))\\\
&+\dim_{k(\mathfrak{p})}({\widetilde{R}}^{>1}/\mu(R^{>1}))\otimes_{A}k(\mathfrak{p})-\dim_{k(\mathfrak{q})}({\widetilde{R}}^{>1}/\mu(R^{>1}))\otimes_{A}k(\mathfrak{q})\\\
&\geq\varepsilon_{k(\mathfrak{p})}({\widetilde{R}}(\mathfrak{p}))-\varepsilon_{k(\mathfrak{q})}({\widetilde{R}}(\mathfrak{q})).\\\
(iv)&\text{If }Ker(\mu)=0\text{ then
}Ker(\mu(\mathfrak{q}):R(\mathfrak{q})\to{\widetilde{R}}(\mathfrak{q}))=0\text{
for }\mathfrak{q}\neq\mathfrak{p}.\\\ \end{array}\\\ $

Here ${\widetilde{R}}(\mathfrak{q})={\widetilde{R}}\otimes_{A}k(\mathfrak{q})$
and
$\mu(\mathfrak{q})^{>0}:R(\mathfrak{q})^{>0}\to{\widetilde{R}}(\mathfrak{p})^{>0}$
is the induced map of positive dimensional parts, which is a partial
normalization of $R(\mathfrak{q})^{>0}$.

###### Proof.

We set

$\begin{array}[]{lcl}{\mathcal{N}}:&=&Ker(\mu:R\to{\widetilde{R}}),\\\
{\mathcal{M}}:&=&Coker(\mu:R\to{\widetilde{R}}).\end{array}$

Both $R$-modules are finitely generated $A$-modules by assumption and hence
${\mathcal{N}}(\mathfrak{q})={\mathcal{N}}\otimes_{A}k(\mathfrak{q})$ and
${\mathcal{M}}(\mathfrak{q})={\mathcal{M}}\otimes_{A}k(\mathfrak{q})$ are
finite dimensional vector spaces over $k(\mathfrak{q})$ for
$\mathfrak{q}\in\operatorname{Spec}A$. Then they are Artinian
$R(\mathfrak{q})$-modules with

$N(\mathfrak{q}):=\operatorname{Supp}_{R(\mathfrak{q})}{\mathcal{N}}(\mathfrak{q})\cup\operatorname{Supp}_{R(\mathfrak{q})}{\mathcal{M}}(\mathfrak{q})$

a finite set of closed points of $R(\mathfrak{q})$. The set
$N:=\operatorname{Supp}_{R}{\mathcal{N}}\cup\operatorname{Supp}_{R}{\mathcal{M}}$
is closed in $\operatorname{Spec}R$ with $N(\mathfrak{q})=\\{\mathfrak{n}\in
N|\mathfrak{n}\cap A=\mathfrak{q}\\}=N\cap\operatorname{Spec}R(\mathfrak{q})$.
Since
$\operatorname{Spec}\mu:\operatorname{Spec}{\widetilde{R}}\smallsetminus\mu^{-1}(N)\to\operatorname{Spec}R\smallsetminus
N$ is an isomorphism, the fiber map
$\operatorname{Spec}\mu(\mathfrak{q}):\operatorname{Spec}{\widetilde{R}}(\mathfrak{q})\to\operatorname{Spec}R(\mathfrak{q})$
is an isomorphism over $\operatorname{Spec}R(\mathfrak{q})\smallsetminus
N(\mathfrak{q})$ with $\mu(\mathfrak{q})^{-1}(N(\mathfrak{q}))$ a finite set
of closed points [Stack, tag 02LS, Lemma 36.39.1.] (since $\mu$ and hence
$\mu(\mathfrak{q})$ is finite). It follows that the assumptions of Lemma 14
are satisfied for
$\mu(\mathfrak{q}):R(\mathfrak{q})\to{\widetilde{R}}(\mathfrak{q})$ and
$\mathbbm{k}=k(\mathfrak{q})$. In particular,
$\mu(\mathfrak{q}):R(\mathfrak{q})^{>0}\to{\widetilde{R}}(\mathfrak{q})^{>0}$
is a partial normalization of $R(\mathfrak{q})^{>0}$.

Let $\mathfrak{q}\in Im(\operatorname{Spec}\varphi)$ be non-zero. Since $A$ is
principal, $\mathfrak{q}$ is a maximal ideal, generated by one element
$t_{\mathfrak{q}}\in A$. We denote the image of $t_{\mathfrak{q}}$ in $R$
resp. ${\widetilde{R}}$ by $f_{\mathfrak{q}}$ resp.
$\tilde{f}_{\mathfrak{q}}$, which are non-zero divisors since $R$ and
${\widetilde{R}}$ are flat over $A$. Consider the commutative diagram

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{R\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu}$$\scriptstyle{f_{\mathfrak{q}}}$$\textstyle{R\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu}$$\textstyle{R(\mathfrak{q})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu(\mathfrak{q})}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\widetilde{R}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{f}_{\mathfrak{q}}}$$\textstyle{{\widetilde{R}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\widetilde{R}}(\mathfrak{q})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0,}$

with exact rows. Since $A$ is a principal ideal domain we have a decomposition

${\mathcal{M}}={\mathcal{F}}\oplus{\mathcal{T}}$

with ${\mathcal{F}}$ a free $A$–module and ${\mathcal{T}}$ an $A$–torsion
submodule concentrated on finitely many maximal ideals in $A$. ${\mathcal{N}}$
is a free $A$-module (since it is torsion free as a submodule of the flat,
hence torsion free $A$-module $R$). Since ${\mathcal{F}}$ and ${\mathcal{N}}$
are free, they are of constant rank $m$ and $n$ respectively and we get for
every $\mathfrak{q}\in Im(\operatorname{Spec}\varphi)$,

$m:=\dim_{k(q)}{\mathcal{F}}(\mathfrak{q}),\
n:=\dim_{k(q)}{\mathcal{N}}(\mathfrak{q}).$

Now fix a $\langle 0\rangle\neq\mathfrak{p}\in
Im(\operatorname{Spec}\varphi)$. There exists an open neighbourhood $U$ of
$\mathfrak{p}$ in $\operatorname{Spec}A$ such that
${\mathcal{T}}_{\mathfrak{q}}=0$ for $\mathfrak{q}\in
U\smallsetminus\\{\mathfrak{p}\\}$ and hence
$\dim_{k(\mathfrak{p})}{\mathcal{T}}_{\mathfrak{p}}<\infty$. The snake lemma,
applied to the diagram above, gives the exact sequence

$0\to{\mathcal{N}}\xrightarrow{f_{\mathfrak{q}}}{\mathcal{N}}\to
Ker(\mu(\mathfrak{q}))\to{\mathcal{M}}\xrightarrow{\tilde{f_{\mathfrak{q}}}}{\mathcal{M}}\to
Coker(\mu(\mathfrak{q}))\to 0,\\\ $

and from this we get

$\begin{array}[]{c}0\to{\mathcal{N}}(\mathfrak{q})\to
Ker(\mu(\mathfrak{q}))\to Ker(\tilde{f}_{\mathfrak{q}})\to 0,\\\\[2.15277pt]
0\to
Ker(\tilde{f}_{\mathfrak{q}})\to{\mathcal{F}}\oplus{\mathcal{T}}\xrightarrow{\tilde{f}_{\mathfrak{q}}}{\mathcal{F}}\oplus{\mathcal{T}}\to
Coker(\mu(\mathfrak{q}))\to 0\ .\end{array}$

$\tilde{f}_{\mathfrak{q}}$ respects the decomposition into free and torsion
part, with $Ker(\tilde{f}_{\mathfrak{q}}|{\mathcal{F}})=0$ and
$Coker(\tilde{f}_{\mathfrak{q}}|{\mathcal{F}})={\mathcal{F}}(\mathfrak{q})$.
Since ${\mathcal{T}}$ is finite dimensional, kernel and cokernel of
$\tilde{f}_{\mathfrak{q}}:{\mathcal{T}}\to{\mathcal{T}}$ have the same
dimension for each $\mathfrak{q}\in U$ (being 0 for
$\mathfrak{q}\neq\mathfrak{p}$).

If ${\mathcal{N}}=0$ then
$Ker(\mu(\mathfrak{q}))=Ker(\tilde{f}_{\mathfrak{q}})$ and $=0$ for
$\mathfrak{q}\neq\mathfrak{p}$ since ${\mathcal{T}}_{\mathfrak{q}}=0$ and
statement (iv) follows.

By Lemma 14 $Coker(\mu(\mathfrak{q}))$ and $Ker(\mu(\mathfrak{q}))$ are finite
dimensional over $k(\mathfrak{q})$ and we get

$\begin{array}[]{ll}m&=\dim_{k(\mathfrak{q})}Coker(\mu(\mathfrak{q}))-\dim_{k(\mathfrak{q})}Ker(\tilde{f}_{\mathfrak{q}})\\\
&=\dim_{k(\mathfrak{q})}Coker(\mu(\mathfrak{q}))-\dim_{k(\mathfrak{q})}Ker(\mu(\mathfrak{q}))+n.\end{array}$

It follows that
$\dim_{k(\mathfrak{q})}Coker(\mu(\mathfrak{q}))-\dim_{k(\mathfrak{q})}Ker(\mu(\mathfrak{q}))=m-n$
is independent $\mathfrak{q}\in U\smallsetminus\langle 0\rangle$. The same
holds for $\mathfrak{q}=\langle 0\rangle$ since
${\mathcal{T}}(\mathfrak{q})=0$ and hence
$Coker(\mu(\mathfrak{q}))={\mathcal{F}}(\mathfrak{q})$ and
$Ker(\mu(\mathfrak{q}))={\mathcal{N}}(\mathfrak{q})$. Lemma 14 implies now
statement (i) and (ii).

To prove (iii) assume that
$\operatorname{Spec}{\widetilde{R}}^{1}\cap\operatorname{Spec}{\widetilde{R}}^{>1}=\emptyset$.
Then ${\widetilde{R}}(\mathfrak{p})^{0}=({\widetilde{R}}^{1})(\mathfrak{p})$
and ${\widetilde{R}}(\mathfrak{p})^{>0}=({\widetilde{R}}^{>1})(\mathfrak{p})$
for $\mathfrak{p}\in Im(\operatorname{Spec}\varphi)$ ($\varphi$ is flat) and
$Coker(\mu(\mathfrak{p})^{>0})=({\widetilde{R}}^{>1}/\mu(R^{>1}))\otimes_{A}k(\mathfrak{p})$.
By assumption ${\widetilde{R}}/\mu(R)$ is a finite $A$-module and hence also
${\widetilde{R}}^{>1}/\mu(R^{>1})$. Thus
$\dim_{k(\mathfrak{p})}({\widetilde{R}}^{>1}/\mu(R^{>1}))\otimes
k(\mathfrak{p})$ is semicontinuous on $\operatorname{Spec}A$ ([GP20, Lemma
1]), which proves
$\dim_{k(\mathfrak{p})}Coker(\mu(\mathfrak{p})^{>0})\geq\dim_{k(\mathfrak{q})}Coker(\mu(\mathfrak{q})^{>0})$
and hence (iii). We notice, that if ${\widetilde{R}}^{>1}=0$ the proof gives
$\varepsilon_{k(\mathfrak{p})}(R(\mathfrak{p}))-\varepsilon_{k(\mathfrak{q})}(R(\mathfrak{q}))=\varepsilon_{k(\mathfrak{p})}({\widetilde{R}}(\mathfrak{p}))-\varepsilon_{k(\mathfrak{q})}({\widetilde{R}}(\mathfrak{q}))$.
∎

######  Lemma 17.

Let $\varphi:A\to R$ be a morphism of rings with $A$ a principal ideal domain.
Let the normalization $\nu:R\to{\overline{R}}$ be finite and
$\mu:R\to{\widetilde{R}}$ be a finite morphism, which is a partial
normalization of $R$.

  1. (1)

Let $\varphi$ be flat.

    1. (i)

Let $Q$ be the (non-empty) intersection of some associated primes of $R$ and
set $R^{\prime}:=R/Q$. Then the induced map $\varphi^{\prime}:A\to R^{\prime}$
is flat. In particular $\varphi^{red}:A\to R^{red}$ is flat.

    2. (ii)

The map $\tilde{\varphi}=\mu\circ\varphi:A\to{\widetilde{R}}$ is flat if
${\widetilde{R}}$ is reduced. In particular,
$\overline{\varphi}=\nu\circ\varphi:A\to{\overline{R}}$ is flat.

  2. (2)

Let $R$ and ${\widetilde{R}}$ be reduced. Then $\varphi$ is flat $\iff$
$\tilde{\varphi}$ is flat.

  3. (3)

Let $\mathfrak{n}\in\operatorname{Spec}R,\,\mathfrak{p}=\mathfrak{n}\cap A$
and $\varphi:A_{\mathfrak{p}}\to R_{\mathfrak{n}}$ flat.

    1. (i)

If
$R_{\mathfrak{n}}(\mathfrak{p})=R_{\mathfrak{n}}\otimes_{A_{\mathfrak{p}}}k(\mathfrak{p})$
is reduced, then $R_{\mathfrak{n}}$ is reduced.

    2. (ii)

If $\dim R_{\mathfrak{n}}\geq 2$ and $R_{\mathfrak{n}}(\mathfrak{p})$ reduced,
then depth$(R_{\mathfrak{n}})\geq 2$ and $r_{1}(R_{\mathfrak{n}})=0$ (i.e., no
1-dimensional irreducible component of $\operatorname{Spec}R$ passes through
$\mathfrak{n}$).

    3. (iii)

If $\dim R_{\mathfrak{n}}\geq 2$ and $R_{\mathfrak{n}}(\mathfrak{p})$ an INNS
then: $R_{\mathfrak{n}}(\mathfrak{p})$ is reduced $\iff$
depth$(R_{\mathfrak{n}})\geq 2$.

  4. (4)

Let $\varphi$ be flat and $\dim R/\mathfrak{n}\geq 2$ for every minimal prime
$\mathfrak{n}$ of $R$. Assume that $R(\mathfrak{p})$ has only isolated non-
normal singularities for $\mathfrak{p}\in Im(\operatorname{Spec}\varphi)$.
Then the following are equivalent:

    1. (i)

$R$ is reduced,

    2. (ii)

for each $\mathfrak{p}\in Im(\operatorname{Spec}\varphi)$, $R(\mathfrak{p})$
is reduced at all normal closed points of $R$.

###### Proof.

Since $A$ is a PID, $\varphi$ is flat $\iff$ $\varphi(a)$ is a non-zero
divisor (n.z.d.) in $R$ for each $a\neq 0$ in $A$ $\iff$ $\varphi(a)$ is not
contained in any associated prime of $R$.

(1) The above characterization implies (i). To see (ii), let
$\nu=\tilde{\nu}\circ\mu:R\to{\widetilde{R}}\to{\overline{R}}$, with
$\tilde{\nu}$ the normalization of ${\widetilde{R}}$. Consider first the case
that $\mu=\nu:R\to{\overline{R}}$ is the normalization of $R$. Let
$\mathfrak{p}_{1},...,\mathfrak{p}_{r}$ be the minimal primes of $R$. Then
${\overline{R}}=\oplus_{i}\overline{R/\mathfrak{p}_{i}}$ and
$\bar{\varphi}(a)=(b_{1},...,b_{r})$, with $b_{i}=\bar{\varphi}(a\ mod\
\mathfrak{p}_{i})$. Since $\varphi^{red}$ is flat by (i),
$\varphi(a)\notin\mathfrak{p}_{i}$ for all $i$ and hence $b_{i}\neq 0$ for all
$i$, showing that $\bar{\varphi}(a)$ is a n.z.d. in ${\overline{R}}$, i.e.,
$\bar{\varphi}$ is flat. If ${\widetilde{R}}$ is a partial normalization of
$R$ and reduced, then ${\widetilde{R}}\subset{\overline{R}}$ and hence
$\tilde{\varphi}$ is flat.

(2) By (1) (ii) the flatness of $\varphi$ implies that of $\tilde{\varphi}$.
The flatness of $\tilde{\varphi}$ implies that of $\varphi$ since $R$ is
reduced and $R\subset{\widetilde{R}}$.

(3) (i) By [Mat86, Corollary to Theorem 23.9] $R_{\mathfrak{n}}$ is reduced
(normal), if $R_{\mathfrak{n}}\otimes_{A_{\mathfrak{p}}}k(\mathfrak{q})$ is
reduced (normal) for all $\mathfrak{q}\in\operatorname{Spec}A_{\mathfrak{p}}$.
Since $A_{\mathfrak{p}}$ has only two prime ideals $R_{\mathfrak{n}}$ is
reduced by the following Lemma 19.

(ii) $R_{\mathfrak{n}}(\mathfrak{p})$ is reduced iff it satisfies Serre’s
condition $(R_{0})$ and $(S_{1})$.
If $\dim R_{\mathfrak{n}}\geq 2$ and $R_{\mathfrak{n}}(\mathfrak{p})$ reduced,
then $\dim R_{\mathfrak{n}}(\mathfrak{p})\geq 1$ and depth
$R_{\mathfrak{n}}(\mathfrak{p})\geq 1$ (from $(S_{1})$). Hence
depth$(R_{\mathfrak{n}})\geq 2$ since $\varphi$ is flat ([Mat86, Corollary to
Theorem 23.3]) and then $r_{1}(R_{\mathfrak{n}})=0$ ([BH98, Proposition
1.2.13]).

(iii) Implication $\Rightarrow$ follows direclty from (ii). For the converse
direction we note that $\operatorname{nil}(R)=H^{0}_{\mathfrak{m}}(R)$ for
$(R,\mathfrak{m})$ a local INNS of dimension $\geq 1$ (Remark 7) and that
$H^{0}_{\mathfrak{m}}(R)=0$ iff depth$(R)\geq 1$ ([BH98, Proposition 3.5.4.]).
Hence $R_{\mathfrak{n}}(\mathfrak{p})$ is reduced iff
$\text{depth}(R_{\mathfrak{n}}(\mathfrak{p}))\geq 1$, which holds since
$\mathfrak{p}$ is generated by a non-zerodivisor and therefore
$\text{depth}(R_{\mathfrak{n}}(\mathfrak{p}))=\text{depth}(R_{\mathfrak{n}})-1$
([Stack] Lemma 10.71.7).

(4) Let $R$ be reduced and $\mathfrak{n}\in\operatorname{Spec}R$ a normal
closed point of $R$. By assumption $\dim R_{\mathfrak{n}}\geq 2$ and by
Serre’s condition (S2) depth$(R_{\mathfrak{n}})\geq 2$. Hence
$R(\mathfrak{p})_{\mathfrak{n}}=R_{\mathfrak{n}}(\mathfrak{p})$ is reduced by
(3) (iii), and this proves (i) $\Rightarrow$ (ii). Conversely, as $R$ is
reduced at all normal points we consider a non-normal point $\mathfrak{n}$ of
$R$ ($\mathfrak{n}$ is then a closed point). Let
$\mathfrak{p}=\mathfrak{n}\cap A$, $t$ a generator of $\mathfrak{p}$ and $f$
the image of $t$ in $R$. Since $R_{\mathfrak{n}}$ is an INNS,
$\operatorname{nil}(R_{\mathfrak{n}})$ is concentrated on $\mathfrak{n}$ and
hence killed by a power of $f$. Since each power of $f$ is a non-zero divisor
of $R_{\mathfrak{n}}$, $\operatorname{nil}(R_{\mathfrak{n}})=0$ and (ii)
$\Rightarrow$ (i) follows. ∎

######  Example 18.

The condition in Lemma 17 (iii) that $R_{\mathfrak{n}}(\mathfrak{p})$ is an
$INNS$ is necessary:
Let $A=\mathbbm{k}[z],\mathbbm{k}$ algebraically closed and
char$(\mathbbm{k})=p>0$ and $R=A[x,y]/\langle f\rangle$, $f=y^{p}-x^{p}-z$.
Then $R$ is regular of depth 2 at every closed point, and the canonical map
$\varphi:A\to R$ is flat ($z$ is a non-zero divisor in $R$). For every closed
point $s\in\mathbbm{k}$ the fiber $R(s)=\mathbbm{k}[x,y]/y^{p}-x^{p}-s$ is not
an INNS and not reduced ($s^{1/p}\in\mathbbm{k})$ (the generic fiber is
however regular).

Such an example is not possible in characteristic 0. E.g. if
$f:({\mathbb{C}}^{n},0)\to({\mathbb{C}},0)$ is flat then $f^{-1}(t)$ is smooth
for $t\neq 0$ close to $0$.

######  Lemma 19.

Let $\varphi:(A,\mathfrak{m})\to(R,\mathfrak{n})$ be a flat morphism of local
rings.

  1. (1)

${\overline{R}}\otimes_{A}Q(A)=\overline{R\otimes_{A}Q(A)}$; in particular, if
$R$ is normal, then $R\otimes_{A}Q(A)$ is normal.

  2. (2)

If $(A,\mathfrak{m})$ is a discrete valuation ring and
$R\otimes_{A}A/\mathfrak{m}$ reduced, then $R\otimes_{A}Q(A)$ is reduced.

###### Proof.

$A\subset R$ since $R$ is flat, hence faithfully flat over $A$.

(1) We have $R\otimes_{A}Q(A)=\\{\frac{r}{a}|\,r\in R,a\in A\text{ a non-
zerodivisor}\\}$, hence $Q(R\otimes_{A}Q(A))=Q(R)$ and
$Q({\overline{R}}\otimes_{A}Q(A))=Q({\overline{R}})=Q(R)$. Thus
${\overline{R}}\otimes_{A}Q(A))$ is normal and
$\overline{R\otimes_{A}Q(A)}\subset{\overline{R}}\otimes_{A}Q(A))$. Since the
last inclusion is birational it is an equality.

(2) Since $A$ is a DVR, $\mathfrak{m}=\langle t\rangle$ for some $t$. Then
$R\otimes_{A}Q(A)=R_{t}=\\{\frac{r}{t^{\nu}}|r\in R,\nu\geq 0\\}$. Assume
$(\frac{r}{t^{\nu}})^{n}=0$ for some $n$. Then $r^{n}=0$ since $t$ is a non-
zero divisor of $R$ and $\bar{r}^{n}=0$, $\bar{r}$ the image of $r$ in
$R/\mathfrak{m}R=R/tR$. Since $R/tR$ is reduced, $r=tr^{\prime}$ for some
$r^{\prime}\in R$. By induction $r\in\cap t^{\nu}R$, and $\cap t^{\nu}R=0$ by
Krull’s intersection theorem. Hence $r=0$. ∎

We can now prove the semicontinuity of $\delta$ and $\varepsilon$ in families
over the spectrum of principal ideal domain.

######  Theorem 20.

Let $\varphi:A\to R$ be a flat morphism of rings, $A$ a principal ideal
domain, and let the normalization $\nu:R\to{\overline{R}}$ be finite. Let
$X=\operatorname{Spec}R$, ${\overline{X}}=\operatorname{Spec}{\overline{R}}$,
$n=\operatorname{Spec}\nu:{\overline{X}}\to X$ and
$f=\operatorname{Spec}\varphi:X\to S=\operatorname{Spec}A$. Assume that
$Coker(\nu)$ and $Ker(\nu^{>1}:R\to\overline{R^{>1}})$ are finite over $A$
and, moreover, that for each $s\in f(X)$ the normalization of
$X_{s}=f^{-1}(s)$ is finite over $X_{s}$ and that $X_{s}$ is normal outside
finitely many isolated non-normal singularities at which the residue fields
are finite over $k(s)$.

Then $\overline{f}:=f\circ n:{\overline{X}}\to S$ is flat,
$\delta_{k(s)}(X_{s})<\infty,\,\varepsilon_{k(s)}(X_{s})<\infty$, and for each
$s\in f(X)$ there exists an open neighbourhood $V\subset S$ of $s$ such that
the following holds for $U=V\cap f(X)$:

  1. (1)

$\delta_{k(s)}(X_{s})-\delta_{k(t)}(X_{t})\\\ \text{
}=\delta_{k(s)}((X^{red})_{s})-\delta_{k(t)}((X^{red})_{t})\\\ \text{
}=\delta_{k(s)}((X^{>1})_{s})-\delta_{k(t)}((X^{>1})_{t})\\\ \text{
}=\delta_{k(s)}(({\overline{X}})_{s})-\delta_{k(t)}(({\overline{X}})_{t})\\\
\text{
}=\delta_{k(s)}(({\overline{X}}^{>1})_{s})-\delta_{k(t)}(({\overline{X}}^{>1})_{t})$
for $t\in U$.

  2. (2)

If $(\overline{X^{>1}})_{t}$ is normal for $t\in U\smallsetminus\\{s\\}$, then
$\delta_{k(s)}(X_{s})-\delta_{k(t)}(X_{t})=\delta_{k(s)}((\overline{X^{>1}})_{s})\geq
0.$

  3. (3)

$\delta_{k(s)}(X_{s})-\delta_{k(\eta)}(X_{\eta})=\delta_{k(s)}((\overline{X^{>1}})_{s})\geq
0$, $\eta$ the generic point of $S$.

  4. (4)

$\varepsilon_{k(s)}(X_{s})-\varepsilon_{k(t)}(X_{t})=\varepsilon_{k(s)}(({X}^{>1})_{s})\geq
0$ for $t\in U\smallsetminus\\{s\\}$.

  5. (5)

$(\overline{X^{>1}})_{t}=({\overline{X}}^{>1})_{t}:=(\overline{f}|{\overline{X}}^{>1})^{-1}(t)$
is reduced for every $t\in U$. If $X$ is reduced then $X_{t}$ is reduced for
$t\in U\smallsetminus\\{s\\}$.

Before giving the proof, we’d like to comment on the result. A semicontinuity
theorem in the algebraic setting was proved by Chiang-Hsieh and Lipman in
[CL06, Proposition 3.3] with $(A,\mathfrak{m})$ a normal local ring of any
dimension (satisfying $A/\mathfrak{m}$ perfect and $A\to\hat{A}$ normal) under
the assumption that (in our notation) the fibers $X_{t}$ and
$({\overline{X}})_{t}$, $t\in\operatorname{Spec}A$, are equidimensional and
reduced. The authors prove that
$\delta_{k(t)}(X_{t})-\delta_{k(\eta)}(X_{\eta})=\delta_{k(t)}((\overline{X})_{t})$
for $t\in\operatorname{Spec}A$ (since $\operatorname{Spec}A$ has only one
closed point, the semicontinuity holds only for generalizations to the generic
point $\eta$). Apart from the fact that there is no restriction for dim A in
[CL06], our result is stronger in two ways. (a): we do not assume that the
fibers are reduced or equidimensional and (b): $A$ doesn’t have to be local,
it can be e.g. ${\mathbb{Z}}$ or $\mathbbm{k}[t]$, $\mathbbm{k}$ an arbitrary
field. Thus our semicontinuity holds also for closed points in a neighbourhood
of the given point $s$ in $\operatorname{Spec}A$.

For complex analytic map germs $f:(X,x)\to(S,0)$, with
$(S,0)=({\mathbb{C}},0)$ and $(X_{0},x)$ a reduced curve singularity, the
result is classical and due to Teissier [Te78].

######  Remark 21.

(1) We have $\operatorname{nil}(R)=Ker(R\to R^{red})=Ker(\nu)\subset
Ker(\nu^{>1})$ = $\mathfrak{p}^{1}$ since
${\overline{R}}\to{\overline{R}}^{>1}=\overline{R^{>1}}=\overline{R/\mathfrak{p}^{1}}$
is surjective. Hence, if $Ker(\nu^{>1})$ is finite over $A$ then $Ker(\nu)$ is
also finite over $A$. On the other hand, since $Coker(\nu)$ surjects onto
$Coker(\nu^{>1})$, if $Coker(\nu)$ is finite over $A$, then $Coker(\nu^{>1})$
is finite over $A$.

(2) We remark that every irreducible component of ${\overline{X}}$ has
dimension $\geq 1$ ($\overline{f}$ is flat) and that
${\overline{X}}={\overline{X}}^{>1}\sqcup{\overline{X}}^{1}$ and
$({\overline{X}})_{t}=({\overline{X}}^{>1})_{t}\sqcup({\overline{X}}^{1})_{t}$
with $({\overline{X}}^{>1})_{t}=({\overline{X}})_{t}^{>0}$ and
$({\overline{X}}^{1})_{t}=({\overline{X}})_{t}^{0}$. In particular,
$(\overline{X^{>1}})_{t}=(\overline{X})_{t}$ $\iff$ $(\overline{X})_{t}$ has
no isolated points $\iff$ $X$ has no 1-dimensional components that meet
$X_{t}$.

(3) Since $\varphi$ is injective, the generic point $\eta=\langle 0\rangle$ is
contained in $f(X)$ ([Stack, tag 00FJ, Lemma 29.4.]). It may however happen
that $f(X)$ does not contain any open subset of $S$. E.g. $f(X)=\\{\eta\\}$
for $A={\mathbb{Z}}\to R={\mathbb{Q}}[x]$ since $A\cap\mathfrak{p}=\langle
0\rangle$ for every prime ideal $\mathfrak{p}\in R$ (see also Example 23).

(4) The statement of the theorem is especially interesting if $f$ is
surjective or if $f$ is open (which holds if $X$ is of finite presentation
over $S$ by [Stack, tag 00I1, Proposition 10.40.8.], or for analytic maps).

###### Proof.

(of Theorem 20) Let $\mu:R\to{\widetilde{R}}$ be one of the maps
$\nu^{red}:R\to R^{red}$, $\nu{{}^{\prime}}:R\to R^{>1}$,
$\nu:R\to{\overline{R}}$, and $\nu^{>1}:R\to{\overline{R}}^{>1}.$ Since
$Ker(\nu^{red})=Ker(\nu)\subset Ker(\nu^{>1})=Ker(\nu^{\prime})$, it follows
that $Ker(\mu)$ and $Coker(\mu)$ are finite over $A$, since $Ker(\nu^{>1})$
and $Coker(\nu)$ are finite over $A$ by assumption.

If $R^{>1}=0$, i.e. $X$ is of pure dimension 1, then $R=Ker(\nu^{>1})$ and
hence $R$ is finite over $A$. Since $R$ is $A$-flat, it is torsion free, hence
free and
$\dim_{k(\mathfrak{p})}(R(\mathfrak{p}))=\varepsilon_{k(\mathfrak{p})}(R(\mathfrak{p}))=-\delta_{k(\mathfrak{p})}(R(\mathfrak{p}))$
is constant on $\operatorname{Spec}A$. Thus the theorem holds in this case
trivially and we will assume in the following the $R^{>1}\neq 0$.

By Lemma 17 the maps
$\tilde{f}=\operatorname{Spec}(\mu\circ\varphi):{\widetilde{X}}=\operatorname{Spec}{\widetilde{R}}\to
S$ are flat. We use the notation $({\widetilde{X}})_{t}=\tilde{f}^{-1}(t)$.

(1) We can apply Proposition 16 (i) to each map $\mu$ and we get the
equalities in statement (1).

Now consider
$\overline{f}^{>1}:=\overline{f}|{\overline{X}}^{>1}:{\overline{X}}^{>1}\to
S$. From Proposition 16 (i) and (iii) we get an open neighbourhood $U$ of $s$
such that for $t\in U\cap f(X)$

  1. (a)

$\delta_{k(s)}({X}_{s})-\delta_{k(t)}(X_{t})=\delta_{k(s)}(({\overline{X}}^{>1})_{s})-\delta_{k(t)}((\overline{X}^{>1})_{t})$.

Altogether this proves (1).

(2) $({\overline{X}}^{>1})_{t}$ normal implies
$\delta_{k(t)}(({\overline{X}}^{>1})_{t})=0$, $t\neq s$. Since
$\overline{X}^{>1}=\overline{X^{>1}}$ is normal and of dimension $\geq 2$ at
each closed point $x$, depth$(\overline{X}^{>1})\geq 2$ at $x$. By Lemma 17
(3)(iii), $({\overline{X}}^{>1})_{t},t\in f(X),$ is reduced at every closed
point, hence at every point. Thus
$\delta_{k(t)}(({\overline{X}}^{>1})_{s})\geq 0$ and (a) proves (2).

(3) By Lemma 19 the ring ${\overline{R}}^{>1}(\eta)$ and thus the generic
fiber ${\overline{X}}^{>1}_{\eta}$ is normal, and (3) follows from (2).

To get estimates for $\varepsilon$ we apply Proposition 16 (iii) to
$f^{\prime}=\operatorname{Spec}(\nu{{}^{\prime}}\circ\varphi):X^{>1}\to S$ and
get

  1. (b)

$\varepsilon_{k(s)}(X_{s})-\varepsilon_{k(t)}(X_{t})=\varepsilon_{k(s)}(({X}^{>1})_{s})-\varepsilon_{k(t)}(({X}^{>1})_{t})\\\
+\dim_{k(s)}Coker(\nu^{\prime>1})\otimes_{A}k(s)-\dim_{k(t)}Coker(\nu^{\prime>1})\otimes_{A}k(t)\\\
=\varepsilon_{k(s)}(({X}^{>1})_{s})-\varepsilon_{k(t)}(({X}^{>1})_{t}),$
since $Coker(\nu^{\prime>1})=R^{>1}/\nu^{\prime}(R^{>1})=0.$

(4) Since ${X}^{>1}$ is reduced, $(X^{>1})_{t}$ is reduced for $t\in
U\smallsetminus\\{s\\}$ by (5). Hence $\varepsilon_{k(t)}(({X}^{>1})_{t})=0$
and (4) follows from (b).

(5) If $X$ is reduced, i.e. if $R$ is reduced, then $\nu:R\to{\overline{R}}$
is injective. By Proposition 16 (iv)
$R(\mathfrak{q})\to{\overline{R}}(\mathfrak{q})$ is injective for
$\mathfrak{q}\in U\smallsetminus\\{\mathfrak{p}\\}$, $U$ some neighbourhood of
$\mathfrak{p}$. Hence $X_{t}$ is reduced for $t\in U\smallsetminus\\{s\\}$. By
the proof of (2) $({\overline{X}}^{>1})_{t}$ is reduced for all $t\in U$. ∎

We illustrate Theorem 20 with some examples.

######  Example 22.

Let $\varphi:A=\mathbbm{k}[z]\to R=A[x,y]/I$, $I=\langle
x^{2}+y^{2}-z^{2}\rangle\cap\langle x-y,y^{2}-z\rangle$. Then
$X=V(I)=X^{2}\cup X^{1}$, with $X^{2}$ the normal surface singularity defined
by $x^{2}+y^{2}-z^{2}=0$ and $X^{1}$ the smooth curve defined by
$x-y=y^{2}-z=0$, meeting $X^{2}$ at $(0,0,0)$ and $(x,y,z)=(x,y,2)$ with
$x=y=\pm\sqrt{2}$ (if $\sqrt{2}\in\mathbbm{k}$).

For $z=0$ the fiber $X_{0}$ is the nodal curve $x^{2}+y^{2}=0$ with an
embedded point, and we compute $\varepsilon_{\mathbbm{k}}(X_{0})=2$ and thus
$\delta_{\mathbbm{k}}(X_{0})=-1$. For $z=t\in\mathbbm{k},t\neq 0,2$ the fiber
$X_{t}$ is a smooth curve and two extra reduced points not on the curve. Hence
$X_{t}$ is normal with $\varepsilon_{\mathbbm{k}}(X_{t})=2$. We get
$\delta_{\mathbbm{k}}(X_{t})=-2$ and
$\delta_{\mathbbm{k}}(X_{0})-\delta_{\mathbbm{k}}(X_{t})=1$.

The normalization ${\overline{X}}$ is the disjoint union of $X^{>1}=X^{2}$ and
$X^{1}$. $({\overline{X}})_{0}$ is the disjoint union of a nodal curve and a
double point. We get $\delta_{\mathbbm{k}}(({\overline{X}}^{>1})_{0})=1$,
confirming statement (1) of Theorem 20.

######  Example 23.

Consider $\varphi:A=\mathbbm{k}[z]\to R=\mathbbm{k}[z]_{\langle
z\rangle}[x,y,z]/I$, with $I$ as in Example 22. Let $\mathfrak{m}$ a maximal
ideal in $R$. If $z\in\mathfrak{m}$ then $\mathfrak{m}\cap A=\langle
z\rangle$. If $z\not\in\mathfrak{m}$ then $\mathfrak{m}\cap A=\langle
0\rangle$ (otherwise $\mathfrak{m}\cap A=\langle p\rangle$ for some
irreducible polynomial $p(z)\not\in\langle z\rangle$ and since $p$ is a unit
in $R$, $p\not\in\mathfrak{m}$). It follows that
$f=\operatorname{Spec}\varphi:X=\operatorname{Spec}R\to
S=\operatorname{Spec}A$ is flat and $f(X)$ consists of two points $\langle
z\rangle$ and $\eta:=\langle 0\rangle$ (every point of $f(X)$ is the image of
a closed point). As in Example 22 we get $\delta_{\mathbbm{k}}(X_{0})=-1$ and
$\delta_{\mathbbm{k}}(X_{\eta})=-2.$

######  Example 24.

We provide two examples showing the necessity of the assumptions in Theorem
20.

(1) The condition that $({\overline{X}}^{>1})_{t},t\neq s$, is normal in
Theorem 20 (1) is necessary in positive characteristic: Let
$A=\mathbbm{k}[z],\mathbbm{k}$ algebraically closed, char$(\mathbbm{k})=p>2$,
and $R=A[x,y]/\langle f\rangle,f=y^{2}-x^{p}-z$. Then $R$ is regular, hence
normal, of dimension 2 and the canonical map $\varphi:A\to R$ is flat ($z$ is
a non-zero divisor in $R$). For every closed point $s\in\mathbbm{k}$ the fiber
$R(s)=\mathbbm{k}[x,y]/y^{2}-x^{p}-s$ has an isolated non-normal point, which
is not normal since $y^{2}=(x+s^{1/p})^{p}$.

We have $\delta_{\mathbbm{k}}(R(s))=(p-1)/2$ is the same for every closed
point $s\in\mathbbm{k}$. Since ${\overline{R}}^{>1}(s)=R(s)$ the equality in
Theorem 20 (1) does not hold, while the equality in (2) holds for the generic
fiber. Note that the generic fiber $\mathbbm{k}(z)[x,y]/y^{2}-x^{p}-z$ is
regular (hence normal) but not geometrically normal: it is not-normal in
$\mathbbm{k}(z^{1/p})[x,y]/y^{2}-x^{p}-z$.

We remark that such an example is not possible in characteristic 0 (cf.
Section 4).

(2) The assumption that $Ker(\nu^{>1}:R\to\overline{R^{>1}})$ is finite over
$A$ is necessary for the upper semicontinuity of $\varepsilon$ (the finiteness
over $A$ of $Ker(\nu)$ is not sufficient). For $A=\mathbbm{k}[x]$,
$R=A[y]/\langle y(xy-1)\rangle$, we have $\varepsilon_{\mathbbm{k}}(R(0))=1$
but $\varepsilon_{\mathbbm{k}}(R(s))=2$ for $s\in\mathbbm{k}-\\{0\\}$. In this
case $R^{>1}=0$ and $X^{>1}=\emptyset$. Hence $Ker(\nu^{>1})=R$, which quasi-
finite but not finite over $A$ (the class of $x$ in $R$ is not integral over
$A$), while $Ker(\nu)=0$.

For Nagata rings $R$ the assumptions simplify and we get with the notations of
Theorem 20:

######  Corollary 25.

Let $\varphi:A\to R$ be a flat morphism of rings with $A$ a PID and $R$
Nagata. Let $\nu:R\to{\overline{R}}$ be the normalization and assume that
$Coker(\nu)$ and $Ker(\nu^{>1})$ are finite over $A$ and for each $s\in f(X)$
the fiber $X_{s}$ has finitely many isolated non-normal singularities with
residue fields finite over $k(s)$. Then the conclusions of Theorem 20 hold.

###### Proof.

Since $R$ is Nagata, ${\overline{R}}$ is finite over $R$ ([Stack, Lemma
10.157.2 (03GH)]) and since
$R/\mathfrak{p}R,\mathfrak{p}\in\operatorname{Spec}A$, is Nagata ([Stack,
Lemma 10.160.16]), the normalization of $X_{s}$ is finite over $X_{s}$. Now
apply Theorem 20. ∎

Let $R$ have no 1-dimensional components. Then the formulation of Theorem 20
becomes less cumbersum and the assumption that $Coker(\nu)$ and
$Ker(\nu^{>1})=Ker(\nu)$ are finite over $A$ is implied by
$R/\widetilde{{\mathcal{C}}}_{R}$ being finite over $A$, where
$\widetilde{{\mathcal{C}}}_{R}$ is the extended conductor ideal from
Definition 1, with $V(\widetilde{{\mathcal{C}}}_{R})$ the non-normal locus of
$\operatorname{Spec}R$. In fact, any finitely generated $R$-module $M$ is
finite over $R/\operatorname{Ann}_{R}(M)$ and if $R/\operatorname{Ann}_{R}(M)$
if finite over $A$ then $M$ is finite over $A$. Hence, we get

######  Corollary 26.

Let $X$ be a Noetherian Nagata scheme with all irreducible components of
dimension $\geq 2$ and $n:{\overline{X}}\to X$ the normalization. Let $S$ be
the spectrum of a PID and $f:X\to S$ a flat morphism such that for each $s\in
f(X)$ the fiber $X_{s}$ has only finitely many isolated non-normal
singularities with residue fields finite over $k(s)$. Assume that the extended
conductor scheme $\widetilde{C}_{X}$111111 If $X$ is covered by open affine
sets $X_{i}=\operatorname{Spec}R_{i}$, then
$\widetilde{C}_{X}|X_{i}=\widetilde{C}_{R_{i}}$ as defined in Definition 1. of
$X$ is finite over $S$. Then the conclusions of Theorem 20 hold with
$(X^{>1})_{t}=X_{t}$ and $(\overline{X^{>1}})_{t}=(\overline{X})_{t}$ for
$t\in U$.

Theorem 20 and its corollaries say that over the spectrum of a PID the the
delta invariant of the generic fiber $X_{\eta}$ is minimal among all fibers
$X_{s},s\in f(X)$. It does not say, however, that the delta invariant of a
special fiber $X_{s}$ is bigger or equal than the delta invariant of the
fibers $X_{t}$ over closed points $t$ in a neighbourhood of $s$, except
$(\overline{X^{>1}})_{t}$ is normal for $t\neq s$. By Example 24 the equality
in Theorem 20 (2) does not hold in positive characteristic (nevertheless,
semicontinuity may hold in general but we do not know this). For
characterisitc 0 see section 4.

##  3\. Fiberwise and Simultaneous Normalization

While the notion of simultaneous normalization is well known, the following
(weaker) definition of fiberwise normalization is new. It is useful if the
residue fields of the base scheme are not perfect.

######  Definition 27.

Let $m:{\widetilde{X}}\to X$ and $f:X\to S$ be morphisms of schemes.

  1. (1)

We call $m:{\widetilde{X}}\to X$ a fiberwise normalization of $f$ if

    1. (a)

$m$ is finite,

    2. (b)

the composition $\widetilde{f}:=f\circ m:{\widetilde{X}}\to S$ is flat,

    3. (c)

the non-empty fibers of $\widetilde{f}$ are normal, and

    4. (d)

the induced map $m_{t}:\widetilde{f}^{-1}(t)\to f^{-1}(t)$ is birational
121212 A morphism of schemes is birational if it is a bijection between the
generic points and an isomorphism of the corresponding local rings. for every
$t\in f(X)$

  2. (2)

A fiberwise normalization of $f$ if is called a simultaneous normalization the
non-empty fibers of $\widetilde{f}$ are geometrically normal.

Recall that a $\mathbbm{k}$-algebra $R$ is called geometrically normal (resp.
geometrically reduced) if $R\otimes_{\mathbbm{k}}\mathbbm{k}^{\prime}$ is
normal (resp. reduced) for every field extension
$\mathbbm{k}\subset\mathbbm{k}^{\prime}$ (equivalently, for every finite field
extension). If $\mathbbm{k}$ is a perfect field, then a $\mathbbm{k}$-algebra
is normal (resp. reduced) iff it is geometrically normal (resp. reduced), see
[Stack, tag 030V, Lemma 10.43.3. and tag 037Z, Lemma 10.160.1.]. A morphism of
rings $\varphi:A\to R$ is called normal (resp. reduced) if it is flat and if
the non-empty fibers $R(\mathfrak{p}):=R\otimes_{A}k(\mathfrak{p})$, are
geometrically normal (resp. geometrically reduced) as
$k(\mathfrak{p})$-algebras, where
$k(\mathfrak{p})=A_{\mathfrak{p}}/\mathfrak{p}A_{\mathfrak{p}}=\operatorname{Quot}(A/\mathfrak{p})$
is the residue field of $A_{\mathfrak{p}}$,
$\mathfrak{p}\in\operatorname{Spec}A$.

A morphism of schemes $f:X\to S$ is geometrically normal (resp. geometrically
reduced), if this holds for the induced morphisms of local rings. Hence, if
the residue fields of all local rings of $S$ are perfect (e.g. of
characteristic 0), then the notions of fiberwise normalization and
simultaneous normalization coincide.

Note that simultaneous normalization is preserved under base change, while
this is in general not the case for fiberwise normalization of schemes over
non-perfect fields. On the other hand, the following results show that the
weaker assumption of a fiberwise normalization is often sufficient and useful.

######  Lemma 28.

Let $f:X\to S$ be flat and assume that $f$ admits a fiberwise normalization
$m:{\widetilde{X}}\to X$. Let $\widetilde{f}=f\circ m:{\widetilde{X}}\to S$.
Then the following holds:

  1. (1)

$m$ is birational.

  2. (2)

${\widetilde{X}}$ is reduced (resp. normal) at $\tilde{x}$ if and only if $S$
is reduced (resp. normal) at $\widetilde{f}(\tilde{x})$. If ${\widetilde{X}}$
is normal, then ${\widetilde{X}}\cong{\overline{X}}$ and $m$ is the
normalization map.

  3. (3)

The induced fiber map $m_{t}:{\widetilde{X}}_{t}=\widetilde{f}^{-1}(t)\to
f^{-1}(t)=X_{t}$ is the normalization of $X_{t}$ for every $t\in f(X)$.

  4. (4)

Let $S$ be normal, $t\in S$ and $x\in X_{t}$. Denote by $X^{i}_{t},i=1,...,r,$
the irreducible components of $X_{t}$ passing through $x$ and by
$X^{j},j=1,...,s,$ the irreducible components of $X$ passing through $x$. Then
$r=s$ and for each $j$ there exists a unique $i=i(j)$ such that
$X^{i}_{t}\subset X^{j}$. The corresponding components satisfy the dimension
formula

$\dim(X^{j},x)=\dim(X^{i}_{t},x)+\dim(S,t).$

In particular, if $\dim(X_{t},x)>0$ then each irreducible component of $X$
passing through $x$ has dimension $>\dim(S,t).$

The dimension formula in (4) is not a direct consequence of the flatness of
$f$, since the restriction of a flat map to an irreducible component need not
be flat; it is a consequence of the existence of a fiberwise normalization.

###### Proof.

(1) A flat morphism $f$ maps a generic point $x\in X$ to a generic point
$f(x)\in S$, since flat mappings have the going down property ([Stack] Lemma
10.38.19, tag 00HS). Since $m_{t}:\widetilde{f}^{-1}(t)\to f^{-1}(t),t\in
f(X)$, induces a bijection of generic points of the fibers and an isomorphism
of their local rings, this holds also for $m$ by the following Lemma 29 (see
also [CL06, Theorem 2.3]), since the minimal primes of a ring correspond
uniquely to the generic points of its spectrum.

(2) The first statement follows from [Mat86, Corollary of Theorem 23.9]. The
second statement follows since $m$ is finite by definition and birational by
(1).

(3) Since $m_{t}$ is finite and birational it is the normalization map.

(4) Since $S$ is normal, $m$ is the normalization map by (2) and thus the
number $r$ of irreducible components $X$ passing through $x$ equals
$\\#m^{-1}(x)$. In the same way $s=\\#m_{t}^{-1}(x)$ holds. Since
$\\#m^{-1}(x)=\\#m_{t}^{-1}(x)$ we get the first statement of (2). For each
$X^{j}$ there exists a unique point $\tilde{x}^{j}\in m^{-1}(x)\cap
m^{-1}(X^{j})$ and we get for $i=i(j)$

$\dim(X^{j},x)=\dim({\widetilde{X}},\tilde{x}^{j})=\dim({\widetilde{X}}_{t},\tilde{x}^{j})+\dim(S,t)=\dim(X^{i}_{t},x)+\dim(S,t),$

since $\widetilde{f}$ is flat. Finally, $\dim(X_{t},x)>0$ means of course that
all irreducible components $X^{i}_{t}$ have dimension $>0$ at $x$ and the last
statement follows. ∎

######  Lemma 29.

Let $A$ be a ring and $R$ a flat $A$-algebra. Then the total ring of fractions
satisfies $Q(R)=Q(R\otimes_{A}Q(A))$. Let ${\widetilde{R}}$ be a flat
$A$-algebra and $\mu:R\to{\widetilde{R}}$ an $A$-algebra homomorphism inducing
an isomorphism $R\otimes_{A}Q(A)\cong{\widetilde{R}}\otimes_{A}Q(A)$. Then
$\mu$ induces an isomorphism $Q(R)\cong Q({\widetilde{R}})$ and
$\mu:R\to{\widetilde{R}}$ is birational.

###### Proof.

Let $R$ be an $A$-algebra via $\varphi:A\to R$.
$R\otimes_{A}Q(A)=\\{\frac{r}{\varphi(a)}|r\in R\text{ and }a\in A\text{ a
non-zerodivisor}\\}$. Since $\varphi$ is flat, it maps non-zero divisors to
non-zero divisors. Hence $R\otimes_{A}Q(A)\subset Q(R)$ and
$Q(R)=Q(R\otimes_{A}Q(A))$. Similarly we obtain
$Q({\widetilde{R}})=Q({\widetilde{R}}\otimes_{A}Q(A))$.

By assumption $\mu$ induces an isomorphism
$R\otimes_{A}Q(A)\cong{\widetilde{R}}\otimes_{A}Q(A)$ and it follows that
$\mu$ induces an isomorphism $Q(R)\cong Q({\widetilde{R}})$. Let
$\mathfrak{p}_{1},\ldots,\mathfrak{p}_{r}$ be the associated prime ideals of
$R$. Then $\mathfrak{p}_{1}\cup...\cup\mathfrak{p}_{r}$ is the set of zero
divisors of $R$ and $\mathfrak{p}_{1}Q(R),\ldots,\mathfrak{p}_{r}Q(R)$ are the
prime ideals of $Q(R)$ ([Mat86, Theorem 4.1]). Here the minimal associated
primes of $R$ correspond to the minimal primes of $Q(R)$ (also the embedded
associated primes of $R$ correspond to the embedded primes of $Q(R)$, but this
is not relevant for us), and similarly for $Q({\widetilde{R}})$. Since
$Q(R)\cong Q({\widetilde{R}})$ the minimal prime ideals of $Q(R)$ and
$Q({\widetilde{R}})$ are in 1-1 correspondence. Since for a minimal prime
$\mathfrak{p}$ of $R$ we have $R_{\mathfrak{p}}=Q(R)_{\mathfrak{p}Q(R)}$ we
obtain all together a bijection between the minimal primes of $R$ and
${\widetilde{R}}$ and an isomorphism of the corresponding local rings. This
implies that $\mu:R\longrightarrow\tilde{R}$ is birational. ∎

We want to characterize fiberwise resp. simultaneous normalization of a family
of INNS numerically by the $\delta$-invariant of the fibers. The following
example shows that we have to be careful with families of affine fibers.

######  Example 30.

Consider Example 24 with $R=A[y]/\langle y(xy-1)\rangle$ and
$A=\mathbbm{k}[x]$. The canonical map $A\to R$ is flat, with normal fibers,
and hence the identity $R\to R$ is a fiberwise normalization (resp.
simultaneous normalization if char$(\mathbbm{k})=0$). But the
$\delta$-invariant is not constant ($\delta_{\mathbbm{k}}(R(0))=-1$ and
$\delta_{\mathbbm{k}}(R(s))=-2$ for $s\in\mathbbm{k}-\\{0\\}$). The same holds
for $A=\mathbbm{k}[x]_{\langle x\rangle}$ and $s$ the generic point of
$\operatorname{Spec}A$. This does not contradict the following Theorem 31
since $Ker(\nu^{>1})$ is not finite over $A$.

The following theorem is a numerical characterization for the existence of a
simultaneous normalization of a family of isolated normal singularities over
the spectrum of a PID.

######  Theorem 31.

Let $\varphi:A\to R$ be a flat morphism of rings with $A$ a principal ideal
domain and $R$ Nagata. Let $\nu:R\to{\overline{R}}$ be the normalization. Set
$S=\operatorname{Spec}A$, $X=\operatorname{Spec}R$,
${\overline{X}}=\operatorname{Spec}{\overline{R}}$,
$f=\operatorname{Spec}\varphi:X\to S$,
$n=\operatorname{Spec}\nu:{\overline{X}}\to X$. Assume that $Coker(\nu)$ and
$Ker(\nu^{>1}:R\to\overline{R^{>1}})$ are finite over $A$ and that for each
$t\in f(X)$ the fiber $X_{t}$ has finitely many isolated non-normal
singularities with residue fields finite over $k(t)$.

  1. (1)

Assume that $f$ admits a fiberwise normalization. Then $\delta_{k(t)}(X_{t})$
is constant on $S$.

  2. (2)

$\delta_{k(t)}(X_{t})$ is constant on $S$ if and only if $f^{>1}:X^{>1}\to S$
admits a fiberwise normalization.

  3. (3)

If $X$ has no 1-dimensional irreducible components then $f$ admits a fiberwise
normalization if and only if $\delta_{k(t)}(X_{t})$ is constant on $S$.

###### Proof.

(1) Since $A$ is regular, hence normal, any simultaneous normalization is the
normalization by Lemma 28. Since the fibers of $\overline{f}=f\circ n$ are
normal by assumption, $(\overline{X}^{>1})_{t}$ is normal and has positive
dimension. Hence $\delta_{k(t)}(({\overline{X}}^{>1})_{t})=0$ for $t\in S$.
The constancy of $\delta_{k(t)}(X_{t})$ follows from Theorem 20 (2) (resp.
Corollary 25).

(2) If $\delta_{k(t)}(X_{t})$ is constant then
$\delta_{k(t)}(X_{t})=\delta_{k(\eta)}(X_{\eta})$ for
$t\in\operatorname{Spec}A$ and $\eta$ the generic point of
$\operatorname{Spec}A$. By Theorem 20 (3) (applied to any $t\in U$)
$\delta_{k(t)}((\overline{X^{>1}})_{t})=0$ and $(\overline{X^{>1}})_{t}$ is
reduced. Hence $(\overline{X^{>1}})_{t}$ is normal and equal to the
normalization of $(X^{>1})_{t}$ (since
$(\overline{X^{>1}})_{t}\to(X^{>1})_{t}$ is birational and finite). Since
$\overline{f}$ is flat by Lemma 17, $\overline{f}^{>1}=f|\overline{X}^{>1}$ is
also flat and $n^{>1}:{\overline{X}}^{>1}\to X^{>1}$ is a fiberwise
normalization of $f^{>1}$.

By Theorem 20 (1) $\delta_{k(t)}(X_{t})$ is constant $\iff$
$\delta_{k(t)}((X^{red})_{t})$ is constant $\iff$
$\delta_{k(t)}((X^{>1})_{t})$ is constant. If $f^{>1}$ admits a fiberwise
normalization, then $\delta_{k(t)}((X^{>1})_{t})$ is constant by (1) and thus
$\delta_{k(t)}(X_{t})$ is constant.

(3) The ”only if ” direction follows from (1). $X^{1}=\emptyset$ means
$r_{1}(X)=0$ and $f^{>1}=f^{red}:X^{red}\to S$. By (2) $\delta_{k(t)}(X_{t})$
= constant implies that ${\overline{X}}\to X^{red}$ is a fiberwise
normalization of $f^{red}$. But then ${\overline{X}}\to X$ is a fiberwise
normalization of $f$ (the flatness of $f$ implies the flatness of $f^{red}$
and $\overline{f}:{\overline{X}}\to X\to S$ by Lemma 17). ∎

######  Corollary 32.

In addition to the assumptions of Theorem 31 assume that all residue fields
$k(\mathfrak{p})$, $\mathfrak{p}\in\operatorname{Spec}A$, are perfect (e.g.,
$A={\mathbb{Z}}$ or $A$ a $\mathbbm{k}$-algebra with $char(\mathbbm{k})=0$).
Then the statements (2) and (3) of Theorem 31 hold with ’fiberwise
normalization’ replaced by ’simultaneous normalization’.

##  4\. Charcteristic 0 and the Analytic Case

If the characteristic is zero then the assumptions of the semicontinuity
theorem (Theorem 20) can be weakened and the statement is stronger for
morphisms of finite type and for analytic morphisms (using theorems of Bertini
and Sard type). The main property is that reduced spaces are regular (hence
normal) on an open dense subset.

######  Theorem 33.

Let $\mathbbm{k}$ be a field of characteristic zero and let $\varphi:A\to R$
be a flat morphism of $\mathbbm{k}$-algebras. Assume that $A$ is a PID and
that $R$ is of finite type over $\mathbbm{k}$. Let $\nu:R\to{\overline{R}}$ be
the normalization and assume that $Ker(\nu^{>1})$ and $Coker(\nu)$ are finite
over $A$. Let $X=\operatorname{Spec}R$,
${\overline{X}}=\operatorname{Spec}{\overline{R}}$,
$n=\operatorname{Spec}\nu:{\overline{X}}\to X$,
$f=\operatorname{Spec}\varphi:X\to S=\operatorname{Spec}A$ and
$\overline{f}=f\circ n:{\overline{X}}\to S$. Let $X_{s}=f^{-1}(s)$, $s\in
f(X),$ have only finitely many non-normal points.

Then $\varepsilon_{k(s)}(X_{s})$ and $\delta_{k(s)}(X_{s})$ are finite for
$s\in S$ and each $s\in S$ has an open neighbourhood $U$ such that for $t\in
U\smallsetminus\\{s\\}$ the following holds:

  1. (1)

$({\overline{X}}^{>1})_{s}$ is reduced and $({\overline{X}})_{t}$ is regular,
hence normal.

  2. (2)

$\delta_{k(s)}(X_{s})-\delta_{k(t)}(X_{t})=\delta_{k(s)}((\overline{X^{>1}})_{s})\geq
0$,

  3. (3)

$\varepsilon_{k(s)}(X_{s})-\varepsilon_{k(t)}(X_{t})=\varepsilon_{k(s)}(({X}^{>1})_{s})\geq
0.$

###### Proof.

Since $X$ is of finite type over $\mathbbm{k}$, any fiber $X_{s}$, $s\in
f(X)$, is of finite type over $k(s)$. Since the non-normal points $x$ of
$X_{s}$ are closed in $X_{s}$, the field extension $k(s)\subset k(x)$ is
finite (by Hilbert’s Nullstellensatz). Moreover, X is Nagata by Remark 15 and
hence ${\overline{X}}$ is finite over $X$, as well as the normalization
$\overline{(X_{s})}$ over $X_{s}$. By Theorem 20 statements (2) and (3) hold
and $({\overline{X}}^{>1})_{s}$ is reduced for $s\in f(X)$.

It remains to show that $({\overline{X}})_{t}$ is regular for $t\neq s$ in
some neighbourhood of $s$. The map $\overline{f}:{\overline{X}}\to S$ and its
restriction to any irreducible component of ${\overline{X}}$ is flat by Lemma
17 and dominant ($A\to{\overline{R}}$ is injective). Since ${\overline{X}}$ is
reduced, there exists an open dense subset $V\subset{\overline{X}}$ such that
$V$ is smooth ([Va17, Theorem 21.3.5]) and, since $\mathbbm{k}$ is perfect, a
point $x$ is smooth iff its local ring is regular ([Stack, tag 0B8X, Lemma
32.25.8.]). Since $\overline{f}$ is flat and of finite presentation, it
follows, that $\overline{f}(V)$ is an open subset of $S$ ([Stack, tag 00I1,
Proposition 10.40.8.]). Since it is not empty, it is dense.

By ([Va17, Theorem 25.3.3]) there exists an open dense subset $W\subset S$
such that the restriction of $\overline{f}$ to the non-empty open set
$V\cap\overline{f}^{-1}(W)$ is a smooth morphism. Then $X_{t}$ is smooth,
hence regular for $t\in W$. Since $S$ is 1-dimensional, $W$ is the complement
of finitely many points, this implies (1). ∎

Let us consider the real or complex analytic case. ${\mathbb{K}}$ denotes
${\mathbb{R}}$ or ${\mathbb{C}}$ and $\delta$ resp. $\varepsilon$ denotes
$\delta_{\mathbb{K}}$ resp. $\varepsilon_{\mathbb{K}}$. The following theorems
were proved in [Gr17] for ${\mathbb{K}}={\mathbb{C}}$. The proofs follow
similar arguments as in Section 2 and 3. The proofs for
${\mathbb{K}}={\mathbb{R}}$ are analogous.

The analytic case is in some sense technically easier than the general
algebraic case: complex spaces are Nagata, the characteristic is 0, and all
points are closed. Moreover, the non-normal locus $NNor(f)$ of an analytic
morphism $f:X\to S$ is a closed analytic subset of $X$ and $f(NNor(f))$ is
neglectible 141414 i.e. contained in a countable union of nowhere dense
locally closed analytic subsets of S. in $S$, provided there is an open dense
subset $V\subset S$ consisting of smooth points of $S$ such that $f^{-1}(V)$
consists of normal points of $X$. If the restriction of $f$ to $NNor(f)$ is
proper, then ”neglectible” can be replaced by ”a nowhere dense closed analytic
subset” (see [BF93, Theorem 2.1(3)] for a proof of these statements).

######  Theorem 34.

([Gr17, Theorem 7.14]) Let $f:(X,x)\to({\mathbb{K}},0)$ be flat morphism of
${\mathbb{K}}$-analytic germs with fibre $(X_{0},x)$ an isolated non–normal
singularity. For a good representative $f:X\to T$ the following holds for
$t\neq 0$.

  1. (1)

$({\overline{X}}^{>1})_{0}$ is reduced and $({\overline{X}})_{t}$ is smooth.

  2. (2)

$\delta(X_{0})-\delta(X_{t})$ is equal to

    1. (i)

$\delta((X^{red})_{0})-\delta((X^{red})_{t})=\delta((X^{red})_{0})-\delta((X_{t})^{red})$,

    2. (ii)

$\delta((\overline{X})_{0})-\delta((\overline{X})_{t})=\delta((\overline{X})_{0})+\dim_{\mathbb{K}}{\mathcal{O}}_{(X^{1})_{t}}$,

    3. (iii)

$\delta((X^{>1})_{0})-\delta((X^{>1})_{t})=\delta((\overline{X}^{>1})_{0})\geq
0$,

  3. (3)

$\varepsilon(X_{0})-\varepsilon(X_{t})=\varepsilon((X^{>1})_{0})\geq 0$.

We use in the analytic case that a quasi-finite morphism has a finite
representative (in the Euclidean topology). For ${\mathbb{K}}={\mathbb{C}}$ we
have $\dim_{\mathbb{K}}{\mathcal{O}}_{(X^{1})_{t}}=\\#\\{$isolated points of
$X_{t}\\}$. Statement (ii) is in [Gr17, Theorem 7.14] formulated with
$r_{1}(X)$ instead of $\dim_{\mathbb{K}}{\mathcal{O}}_{(X^{1})_{t}}$, which is
wrong in general. We have
$r_{1}(X)\leq\dim_{\mathbb{K}}{\mathcal{O}}_{(X^{1})_{t}}$ and $r_{1}(X)=0$
iff $\dim_{\mathbb{K}}{\mathcal{O}}_{(X^{1})_{t}}=0$.

######  Theorem 35.

([Gr17, Theorem 7.17]) Let $f:(X,x)\to({\mathbb{K}},0)$ be flat with
$(X_{0},x)$ an INNS of dimension $\geq 1$. Then there exists a small
representative $f:X\to S$ of the germ $f$, such that $f:X\to S$ admits a
simultaneous normalization if and only if $\delta(X_{t})$ is constant and
$r_{1}(X,x)=0$.

The assumption that $(X_{0},x)$ an INNS implies that the non-normal locus of
$f$ (which is analytic in $X$) is finite over $S$ (for $S$ sufficiently
small), and hence $X_{t}$ has only isolated non-normal singularities for $t\in
S$. We recall that an excellent local ring $(R,\mathfrak{m})$ is normal if
$R/f$ is normal for $f\in\mathfrak{m}$ a non-zero divisor of $R$ ([BF93, Lemma
4.4]). Hence, for $f:X\to S$ a flat analytic morphism, with $S$ a smooth
1-dimensional manifold, the non-normal locus of $X$ is contained in the non-
normal locus of $f$. Let $n:{\overline{X}}\to X$ be the normalization of $X$.
Then the non-normal locus of $X$ is the union of the supports of the sheaves
$Coker(n_{*}:{\mathcal{O}}_{X}\to n_{*}{\mathcal{O}}_{\overline{X}})$ and
$Ker(n_{*}:{\mathcal{O}}_{X}\to n_{*}{\mathcal{O}}_{{\overline{X}}})$ and the
assumption that $(X_{0},x)$ an INNS implies that these sheaves are finite over
$S$. The necessity of $r_{1}(X,x)=0$ follows from the dimension formula of the
analytic analog of Lemma 28 (4). We thus see that analogous assumptions as in
Theorem 33 hold also in the analytic situation of Theorem 34.

It is not difficult to see that $Coker(n_{*}:{\mathcal{O}}_{X}\to
n_{*}{\mathcal{O}}_{\overline{X}})$ and $Ker(n^{>1}_{*}:{\mathcal{O}}_{X}\to
n_{*}{\mathcal{O}}_{{\overline{X}}^{>1}})$ are finite over $S$ iff the non-
normal loci of $X$ and $X^{>1}$, defined by the extended conductor schemes,
are finite over $S$ (see Remark 21 (1)). We use this in the following Theorem
36 for (global) morphisms of analytic spaces.

######  Theorem 36.

([Gr17, Theorem 7.19S]) Let $f:X\to S$ be a flat morphism of
${\mathbb{K}}$-analytic spaces with $S$ a $1$–dimensional analytic manifold.
Assume that the non-normal loci of $f$ and $f^{>1}:X^{>1}\to S$ are finite
over $S$. Then the following are equivalent:

  1. (i)

$f$ admits a simultaneous normalization.

  2. (ii)

$\delta(X_{t})$ is locally constant on $S$ and the $1$–dimensional part
$X^{1}$ of $X$ is smooth and does not meet the higher dimensional part
$X^{>1}$.

In particular, if $X$ has no 1-dimensional part, then

$f\text{ admits a simultaneous normalization}\iff\delta(X_{t})\text{ is
locally constant on }S.$

The finiteness of $NNor(f^{>1})$ over $S$ was mistakenly omitted in [Gr17,
Theorem 7.19S].

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