# Trends of neutron skins and radii of mirror nuclei from first principles

S. J. Novario  Theoretical Division, Los Alamos National Laboratory, Los
Alamos, NM 87545, USA  D. Lonardoni  Theoretical Division, Los Alamos National
Laboratory, Los Alamos, NM 87545, USA  S. Gandolfi  Theoretical Division, Los
Alamos National Laboratory, Los Alamos, NM 87545, USA  G. Hagen  Physics
Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA  Department
of Physics and Astronomy, University of Tennessee, Knoxville, TN 37996, USA

###### Abstract

The neutron skin of atomic nuclei impacts the structure of neutron-rich
nuclei, the equation of state of nucleonic matter, and the size of neutron
stars. Here we predict the neutron skin of selected light and medium-mass
nuclei using coupled-cluster theory and the auxiliary field diffusion Monte
Carlo method with two- and three-nucleon forces from chiral effective field
theory. We find a linear correlation between the neutron skin and the isospin
asymmetry and compare with data. We also find a linear relationship between
proton and neutron radii of mirror nuclei and quantify the effect of charge
symmetry breaking. Our results for the mirror-difference charge radii and
binding energies per nucleon agree with existing data.

††preprint: LA-UR-21-31426

Introduction.—  The size of neutron stars and the distributions of excess
neutrons in medium-mass and heavy nuclei can be linked to the microscopic
forces between the constituent nucleons that build up the atomic nucleus and
the equation-of-state (EoS) of nucleonic matter [1, 2, 3, 4]. With recent
advancements in both observational astrophysics and experimental nuclear
physics, investigating this link and constraining nuclear models is now
becoming possible. The advent of multimessenger astronomy, established with
the simultaneous observation of a binary neutron star merger by the LIGO
gravitational-wave observatory [5, 6] and gamma-ray astronomy, has opened up
the possibility of examining the structure of neutron stars in new detail [7].
Furthermore, state-of-the-art experiments designed to precisely measure the
neutron distributions in nuclei, including the recent neutron skin measurement
of [208]Pb by PREX [8, 9] and [48]Ca by CREX [10, 11], are now being
confronted by theory.

The neutron skin thickness, $\Delta R_{\mathrm{np}}$, is defined as the
difference between the root-mean-squared (rms) point radii of the neutron and
proton density distributions. Collectively, the neutron skin results from a
balance between the inward pressure of the surface tension on excess neutrons
on the edge of the nucleus and outward degeneracy pressure from excess
neutrons within the core of the nucleus. The same balance is reached in
neutron stars with the inward pressure of gravity. The relationship between
this pressure and the neutron density is quantified in the slope of the
symmetry energy at saturation density, $L$. Therefore, this bulk property of
neutron stars should be related to the neutron skin of nuclei [1, 2, 12, 13,
14, 15, 16].

Figure 1: Neutron skin thickness plotted against the isospin asymmetry from ab
initio calculations. Coupled-cluster results of nuclei with $14\leq A\leq 56$
using the $\Delta$N2LOGO(394), $\Delta$N2LOGO(450), and N2LOsat(450)
interactions are shown as solid blue circles, empty orange squares, and right-
filled red triangles, respectively. Left-filled green triangles represent
auxiliary field diffusion Monte Carlo results for $6\leq A\leq 18$ using the
N2LOE1 local chiral interaction. The linear regression is printed in the
bottom right with $1\sigma$ uncertainty, and the $1\sigma$, $2\sigma$, and
$3\sigma$ confidence levels are shown as overlapping gray bands. Select nuclei
are labeled.

Recent high-precision measurements of charge radii in isotope chains of
potassium [17], calcium [18, 19], nickel [20], copper [21, 22], and silver
[23], have revealed that they carry information about changes in shell-
structure, deformation, and pairing effects. Charge radii can be easily probed
with the well-known electromagnetic interaction [24, 25]. Conversely,
measuring the neutron (weak-charge) radius is more difficult, but can be
accomplished using either strong or electroweak probes, each with their own
shortcomings. Hadronic probes suffer from model-dependent uncertainties [26],
and electroweak probes, like those used in PREX [8, 9] and CREX [10, 11], are
less model-dependent but are hindered by one-part-per-million partiy-violating
asymmetry [27]. This hurdle can be overcome in some cases [8, 9], but
precision measurements of the neutron radius for short-lived isotopes remain
elusive.

In this Letter, we study trends in the neutron skin thickness of light and
medium-mass mirror nuclei with mass number $6\leq A\leq 56$ (shown in Fig. 1)
by computing the proton and neutron radii using the ab initio coupled-cluster
(CC) [28, 29, 30, 31, 32, 33, 34] and auxiliary field diffusion Monte Carlo
(AFDMC) methods [35, 36, 37], each with different nuclear interactions derived
from chiral effective field theory (described below). We then establish a
simple and robust relation between these data and the isospin asymmetry of the
various nuclei and use this relation to estimate the neutron skin of relevant
heavy nuclei not directly calculated in this work. Additionally, we use the
isospin symmetry of mirror nuclei to develop a relation between the proton and
neutron radii of mirror partners and compare our computed mirror-difference
charge radii and binding energies per nucleon with available data. Finally, we
estimate the contribution of the isospin-symmetry-breaking terms by turning
off the Coulomb interaction.

Methods.—  We describe nuclei as a collection of $A$ point-like nucleons of
mass $m$ interacting according to the non-relativistic intrinsic Hamiltonian

$\displaystyle
H=-\frac{\hbar^{2}}{2mA}\sum_{i<j}(\nabla_{i}-\nabla_{j})^{2}+\sum_{i<j}V_{ij}+\sum_{i<j<k}V_{ijk}.$
(1)

Here, $V_{ij}$ and $V_{ijk}$ are the nucleon-nucleon (NN) and three-nucleon
(NNN) potentials, respectively, the former of which also includes the Coulomb
force. Interactions used in this work are derived from chiral effective field
theory (EFT) [38, 39, 40, 41, 42, 43, 44, 45]. The CC calculations use three
different chiral interactions. The first, N2LOsat [46], was optimized to few-
body observables as well as radii and ground-state energies of selected carbon
and oxygen isotopes. The second and third, $\Delta$N2LO${}_{\rm GO}(394)$ and
$\Delta$N2LO${}_{\rm GO}(450)$ [47, 48, 49], include explicit $\Delta$-isobars
and were optimized to few-body observables and properties of nucleonic matter,
and use momentum cutoff of $\Lambda=394$ MeV/c and $\Lambda=450$ MeV/c,
respectively.

After establishing the nuclear interactions, we perform a Hartree-Fock
diagonalization from a spherical harmonic-oscillator basis and then construct
a prolate, axially-deformed product state built from natural orbitals [50,
51]. We then normal-order the Hamiltonian with respect to the reference state,
denoted by $|\Phi_{0}\rangle$, and retain one- and two-body terms [52, 53].
Next, we use the coupled-cluster method in the singles-and-doubles (CCSD)
approximation to construct the similarity-transformed Hamiltonian,
$\overline{H}_{N}=e^{-\hat{T}}{H_{N}}e^{\hat{T}}$, which decouples the
reference state from excitations around it. In the CCSD approximation the
cluster operator, $\hat{T}$, is truncated at the $2p$–$2h$ excitation level.
Because $\hat{T}$ is asymmetric, $\overline{H}_{N}$ is non-Hermitian, and the
left ground state must be decoupled separately using
$\langle\Phi_{0}|(1+\hat{\Lambda})$, where $\hat{\Lambda}$ is a de-excitation
analogue to $\hat{T}$ [33, 34] and is also truncated at the $2p$–$2h$ level.
In this work we are interested in computing the ground-state expectation value
of the squared intrinsic point-nucleon radius operator, $\hat{r}^{2}$. In CC
theory this operator is similarity transformed according to,
$\overline{r}^{2}\equiv e^{-\hat{T}}\hat{r}^{2}e^{\hat{T}}$, and applied
between the left and right CC ground states, i.e.,
$\langle\hat{r}^{2}\rangle\equiv\langle\Phi_{0}|(1+\hat{\Lambda})\overline{r}^{2}|\Phi_{0}\rangle.$

We also perform AFDMC calculations of the point-proton and neutron radii (see
Ref. [37] for more details) using a local chiral nucleon-nucleon and three-
nucleon interaction at next-to-next-to-leading order, N2LOE1 [54, 55, 56, 37].
This interaction has been used to study the ground-state properties of nuclei
up to [16]O [57, 56, 58, 59, 37, 60, 61, 62, 63, 64], few neutron systems [65,
66, 67], and neutron star matter [55, 68, 56, 69, 70, 71, 72, 73, 74, 75, 76,
77].

Results.—  Our results for the neutron skin thickness of selected nuclei with
$6\leq A\leq 56$ are shown in Fig. 1. These data are plotted against the
isospin asymmetry, $\delta=(N-Z)/A$, with $N$ and $Z$ being the number of
neutrons and protons, respectively. As can be seen, there is a linear relation
between the neutron skin thickness and the isospin asymmetry which is robust
across different ab initio methods and chiral interactions. This correlation
was first derived in Ref. [78] from the liquid-drop model and demonstrated
with mean-field methods. We fit the data using linear regression which is
shown in Fig. 1 as shaded gray bands at $1\sigma$, $2\sigma$, and $3\sigma$
confidence levels.

There are several distinct sources of uncertainties in the CC calculations.
First, the error associated with the finite size of the employed model space
is estimated by taking the difference between the point-nucleon radius
calculated in the model spaces of $N_{\rm max}=12$ and $N_{\rm max}=10$,
respectively, where $N_{\rm max}=(2n+l)_{\rm max}$ indicates the maximum
harmonic oscillator energy level. We also note that in this work we utilize a
basis with an oscillator energy of $\hbar\omega=16\ \rm MeV$, which might not
be optimal for all calculated nuclei. The additional uncertainty associated
with this choice is estimated to be 1% [79, 51]. Second, the uncertainty
associated with truncating the CC excitation operator at the $2p$–$2h$ level
amounts to another 1% [51, 80, 79]. Finally, the error from breaking
rotational symmetry is estimated to be less than 1% on computed radii (see
Ref. [81] for details). Uncertainties on AFDMC results are statistical, and
are reported in this work at the $1\sigma$ confidence level. Errors on radii
also include an extrapolation uncertainty from mix estimates, and those on
binding energies from unconstrained evolution (see Ref. [37] for more
details).

To test the validity of our linear relationship between neutron skin thickness
and the isospin asymmetry, we plot the linear regression of our results at
$3\sigma$ confidence level with experimental data of neutron-rich nuclei in
Fig. 2. Most of this data is extracted from $X$-ray spectroscopy following
antiprotonic annihilation [82]. These data are used to fit a linear
relationship which is shown as a green band [83]. Also shown are the results
from PREX-I [8] and PREX-II [9], which measure the neutron radius of [208]Pb
using parity-violating electron scattering.

Figure 2: Neutron skin thickness plotted against the isospin asymmetry from
available experimental data. Solid black diamonds are the results extracted
from antiprotonic $X$-ray data [82, 83]. The right-filled purple star and the
left-filled brown star are the values for [208]Pb from PREX-I [8] and PREX-II
[9], respectively. The green dashed line with band is the empirical evaluation
of the relation, printed in the bottom right, from Ref. [83]. These
experimental data are compared with the theoretical relation derived in Fig.
1, which is shown by the gray band.

While our linear fit has a slightly larger slope than the antiprotonic X-ray
data [83], our results overlap significantly with the experimental data
and—using [208]Pb as an indicator—lie in between the hadronic and electroweak
data. Using our extracted linear relationship to predict the neutron-skin of
[208]Pb with $\delta=44/208$ we obtain $\Delta
R_{\mathrm{np}}\left({}^{208}\mathrm{Pb}\right)=0.241\pm 0.028\ \mathrm{fm}$,
consistent with the PREX-II result, $\Delta
R_{\mathrm{np}}\left({}^{208}\mathrm{Pb}\right)=0.283\pm 0.071\ \mathrm{fm}$
[9].

While this comparison relies on a precarious extrapolation from medium-mass to
heavy nuclei, it could indicate the possibility of both a relatively large
neutron skin in [208]Pb with an interaction that corresponds to a realistic
EoS of nucleonic matter. Specifically, the interactions used in this work give
symmetry energy slopes at saturation of $L=58.4,65.2\ \mathrm{MeV}$ for the
$\Delta$N2LOGO interactions [49], $L=40.8\ \mathrm{MeV}$ for N2LOsat [13], and
$L=59(9)\ \mathrm{MeV}$ for N2LOE1 [76]. This is in contrast to results from
energy density functional theory that suggest that the relatively large
neutron skin measurement from PREX-II corresponds to an unrealistic symmetry
energy slope of $L=107\pm 37\ \mathrm{MeV}$ [84].

Figure 3: Top panel: Point-neutron radius plotted against the point-proton
radius of the corresponding mirror nucleus. The data markers are the same as
those in Fig. 1. The linear regression is printed in the bottom right with
$1\sigma$ uncertainty, and the $1\sigma$, $2\sigma$, and $3\sigma$ confidence
levels are shown as overlapping gray bands. Selected neutron-deficient nuclei
are labeled, and the $R_{\mathrm{p}}=R^{\mathrm{mirror}}_{\mathrm{n}}$ line is
shown as a visual reference. Bottom panel: Residuals between data and linear
model of the top panel, plotted against the isospin asymmetry.

In addition to establishing a general equation for the neutron skin, we can
also use our results to establish a relationship between the point-proton
radius of a nucleus, $R_{\mathrm{p}}$ and the point-neutron radius of its
mirror partner, $R^{\mathrm{mirror}}_{\mathrm{n}}$. In the limit of conserved
isospin symmetry, these two quantities should be exactly equal. We show the
resulting linear relationship in Fig. 3, the slope of which is just less than
unity. This highlights the deviation from isospin symmetry, mostly due to the
Coulomb interaction pushing the protons out to an extended distribution. The
residuals between the data and the predictions from the linear model are also
shown in Fig. 3, plotted against the isospin asymmetry. This linear
relationship again highlights the underlying dependence on the isospin
asymmetry for the neutron skin thickness. Combining the two linear fits in
Fig. 3 results in the overall linear model connecting mirror radii with
isospin-symmetry dependence, $R^{\mathrm{mirror}}_{\mathrm{n}}=0.952\cdot
R_{\mathrm{p}}+0.146\cdot\delta+0.108\pm 0.011\ \mathrm{fm}$. This analysis is
useful because neutron distributions are very difficult to measure
experimentally, so the proton radius, much easier to measure, can be used as a
surrogate.

Figure 4: Top panel: Mirror-difference charge radii plotted against the
isospin asymmetry compared with experiment. The data markers are the same as
those in Fig. 1. The linear regression of our data is printed in the bottom
left with $1\sigma$ uncertainty, and the $1\sigma$, $2\sigma$, and $3\sigma$
confidence levels are shown as overlapping gray bands. Selected neutron-
deficient nuclei are labeled. Experimental data are shown as open white
diamonds, taken from Ref. [25] except for those of [36]Ca [19] and [54]Ni
[85]. Bottom panel: Mirror-difference binding energy per nucleon
$\Delta(\mathrm{BE}/A)$ as a function of the Coulomb asymmetry. Experimental
data are taken from Ref. [86].

The near-equivalence between $R^{\mathrm{mirror}}_{\mathrm{n}}$ and
$R_{\mathrm{p}}$, shown in Fig. 3, indicates a direct relation between the
neutron skin and the mirror-difference charge radii, $\Delta
R_{\mathrm{ch}}^{\mathrm{mirror}}=\left[R_{\mathrm{ch}}({}^{A}_{N}\mathrm{Y}_{Z})-R_{\mathrm{ch}}({}^{A}_{Z}\mathrm{X}_{N})\right]$.
Recently, calculations based on energy density functional theories [14, 87,
85] have shown a correlation between $\Delta
R_{\mathrm{ch}}^{\mathrm{mirror}}$ and the symmetry energy slope at
saturation, $L$. This finding has enabled recent high-precision measurements
of charge radii of the neutron-deficient isotopes [36]Ca [19] and [54]Ni [85]
to constrain $L$ in the range $20\leq L\leq 70\ \mathrm{MeV}$. We note that
this range is consistent with the range for $L$ from the different
interactions used in this work, $40.8\leq L\leq 65.2\ \mathrm{MeV}$. To
compare with data and the conclusions drawn from the mean-field calculations
of [85], we compute the mirror-difference charge radii for selected nuclei
shown in Fig. 4. The charge radius is computed using the formula
$R_{\mathrm{ch}}^{2}=R_{\mathrm{p}}^{2}+\langle
r_{\mathrm{p}}^{2}\rangle+{N\over Z}\langle r_{\mathrm{n}}^{2}\rangle+\langle
r_{\mathrm{DF}}^{2}\rangle+\langle r_{\mathrm{SO}}^{2}\rangle$ where the spin-
orbit correction, $\langle r_{\mathrm{SO}}^{2}\rangle$, is calculated with the
CC method [13], and $\langle r_{\mathrm{p}}^{2}\rangle=0.709$ fm2, $\langle
r_{\mathrm{n}}^{2}\rangle=-0.106$ fm2, and $\langle
r_{\mathrm{DF}}^{2}\rangle=3/(4m^{2})=0.033$ fm2 are the charge radius squared
of the proton [88, 89], the neutron [90], and the Darwin-Foldy term,
respectively, with $m$ being the nucleon mass. Our results agree well with the
sparse data, most of which were taken from Ref. [25] except for those of
[36]Ca [19] and [54]Ni [85]. Because $\Delta
R_{\mathrm{ch}}^{\mathrm{mirror}}$ is an analogue of the neutron skin, it also
has a linear relation to the isospin asymmetry. Also shown in Fig. 4 is the
comparison of our results with data from Ref. [86] of mirror-difference
binding energies per nucleon
($\Delta(\mathrm{BE}/A)=\left[\mathrm{BE}({}^{A}_{Z}\mathrm{X}_{N})-\mathrm{BE}({}^{A}_{N}\mathrm{Y}_{Z})\right]/A$).
This quantity exhibits a linear relationship with the Coulomb asymmetry,
$\delta_{\mathrm{C}}=[N(N-1)-Z(Z-1)]/A^{4/3}$, which stems from the only
mirror-asymmetric term in the liquid-drop model.

Finally, we also study the extent of the isospin-symmetry-breaking terms in
the nuclear interaction by computing ground-state energies and radii of select
mirror nuclei using CC theory with and without the Coulomb term. We show these
results for $A=42-48$ using $\Delta$N2LOGO(394) in Table 1, as the neutron
skin thickness, the mirror-difference charge radius, and the mirror-difference
binding energy per nucleon. Like their corresponding quantities calculated
with the Coulomb term, $\Delta R_{\mathrm{np}}$ and $\Delta
R_{\mathrm{ch}}^{\mathrm{mirror}}$ both exhibit a linear relationship with the
isospin asymmetry. However, unlike the relation shown in Fig. 4,
$\Delta(\mathrm{BE}/A)$ also exhibits a linear relationship with the isospin
asymmetry without the Coulomb term. By removing the Coulomb term to
approximate isospin symmetry, the near equivalence between $\Delta
R_{\mathrm{np}}$ and $\Delta R_{\mathrm{ch}}^{\mathrm{mirror}}$ shows the
validity of this approximation. Additionally, the $\Delta(\mathrm{BE}/A)$
results without the Coulomb interaction entirely reflect the effects of the
isospin-symmetry-breaking terms in the chiral interaction, accounting for only
$\sim 1\%$ of $\Delta(\mathrm{BE}/A)$. These results can be used to constrain
the charge-symmetry-breaking term in Skyrme density functional theory
according to Ref. [91].

Table 1: Neutron skin thickness ($\Delta R_{\mathrm{np}}$), mirror-difference charge radius ($\Delta R_{\mathrm{ch}}^{\mathrm{mirror}}$), and mirror-difference binding energy per nucleon ($\Delta({\rm BE}/A)$) of select nuclei using $\Delta$N2LO${}_{\rm GO}(394)$ with and without the Coulomb term. $\Delta R_{\mathrm{np}}\ \rm(fm)$ | Full | w/o Coulomb
---|---|---
${}^{42}\rm Ca$ | 0.019(35) | 0.060(34)
${}^{46}\rm Ca$ | 0.113(35) | 0.151(35)
${}^{48}\rm Cr$ | -0.048(38) | -0.002(36)
${}^{48}\rm Ti$ | 0.049(37) | 0.090(36)
${}^{48}\rm Ca$ | 0.150(36) | 0.186(35)
$\Delta R_{\mathrm{ch}}^{\mathrm{mirror}}\ \rm(fm)$ |  |
${}^{42}\rm Ca$ | 0.075(36) | 0.082(33)
${}^{46}\rm Ca$ | 0.191(38) | 0.209(35)
${}^{48}\rm Ti$ | 0.119(38) | 0.128(36)
${}^{48}\rm Ca$ | 0.238(38) | 0.261(35)
$\Delta({\rm BE}/A)\ ({\rm MeV})$ |  |
${}^{42}\rm Ca$ | 0.349(04) | 0.0038(5)
${}^{46}\rm Ca$ | 1.039(11) | 0.0113(1)
${}^{48}\rm Ti$ | 0.689(75) | 0.0075(1)
${}^{48}\rm Ca$ | 1.381(15) | 0.0150(2)

Conclusions.— We performed CC and AFDMC calculations of neutron skins of a
wide range of light- and medium-mass nuclei using different interactions from
chiral EFT, and found a robust linear relationship with the isospin asymmetry.
When extrapolated to heavy nuclei, this relationship is consistent with the
available data. Additionally, we found a linear relationship between the
neutron and proton radii of mirror nuclei pairs which can be used to estimate
neutron radii of rare isotopes not yet accessible by experiment. Our
calculations are in good agreement with available data for the difference in
charge radii and binding energies of mirror nuclei. Finally, we estimated the
contributions from charge-symmetry-breaking terms in the nuclear interaction
which can be useful for constraining those terms in Skyrme density
functionals.

###### Acknowledgements.

We would like to thank J. Carlson, I. Tews, and R. F. Garcia Ruiz for helpful
discussions. The work of S. G., S. N., and D. L. was supported by the DOE
NUCLEI SciDAC Program, and by the DOE Early Career Research Program. The work
of S. G. was also supported by U.S. Department of Energy, Office of Science,
Office of Nuclear Physics, under Contract No. DE-AC52-06NA25396. G. H. was
supported by the Office of Nuclear Physics, U.S. Department of Energy, under
grants DE-SC0018223 (NUCLEI SciDAC-4 collaboration) and by the Field Work
Proposal ERKBP72 at Oak Ridge National Laboratory (ORNL). Computer time was
provided by the Innovative and Novel Computational Impact on Theory and
Experiment (INCITE) program. This research used resources of the Oak Ridge
Leadership Computing Facility located at ORNL, which is supported by the
Office of Science of the Department of Energy under Contract No. DE-
AC05-00OR22725. This research also used resources provided by the Los Alamos
National Laboratory Institutional Computing Program, which is supported by the
U.S. Department of Energy National Nuclear Security Administration under
Contract No. 89233218CNA000001.

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# Equinormalizability and topologically triviality of deformations of isolated
curve singularities over smooth base spaces

Lê Công-Trình  Department of Mathematics, Quy Nhon University, Vietnam
<EMAIL_ADDRESS>Dedicated to Professor Gert-Martin Greuel on the
occasion of his 70th birthday

###### Abstract.

We give a $\delta$-constant criterion for equinormalizability of deformations
of isolated (not necessarily reduced) curve singularities over smooth base
spaces of dimension $\geq 1$. For one-parametric families of isolated curve
singularities, we show that their topologically triviality is equivalent to
the admission of weak simultaneous resolutions.

###### Key words and phrases:

Isolated curve singularities; generically reduced; weak simultaneous
resolution; equinormalizable deformation; $\mu$-constant; $\delta$-constant;
topologically trivial

###### 2010 Mathematics Subject Classification:

14B07, 14B12, 14B25

##  1\. Introduction

The theory of equinormalizable deformations has been initiated by B. Teissier
([14]) in the late 1970’s for deformations of reduced curve singularities over
$(\mathbb{C},0)$. It is generalized to higher dimensional base spaces by M.
Raynaud and Teissier himself ([15]; some insight into the background of
Raynaud’s argument might be gleaned from the introduction to [10]). Recently,
it is developed by Chiang-Hsieh and Lipman ([5], 2006) for projective
deformations of reduced complex spaces over normal base spaces, and it is
studied by Kollár ([11], 2011) for projective deformations of generically
reduced algebraic schemes over semi-normal base spaces.

Each reduced curve singularity is associated with a $\delta$ number (see
Definition 3.3), which is a finite number and it is a topological invariant of
reduced curve singularities. Teissier-Raynaud-Chiang-Hsieh-Lipman ([14], [15],
[5]) showed that a deformation of a reduced curve singularity over a normal
base space is equinormalizable (see Definition 3.1) if and only if it is
$\delta$-constant, that is the $\delta$ number of all of its fibers are the
same. This is so-called the $\delta$-constant criterion for
equinormalizability of deformations of reduced curve singularities.

For isolated curve singularities with embedded components, Brücker and Greuel
([3], 1990) gave a similar $\delta$-constant criterion (with a new definition
of the $\delta$ number, see Definition 3.3) for equinormalizability of
deformations of isolated (not necessarily reduced) curve singularities over
$(\mathbb{C},0)$. The author considered in [12] (2012) deformations of plane
curve singularities with embedded components over smooth base spaces of
dimension $\geq 1$, and gave a similar $\delta$-constant criterion for
equinormalizability of these deformations, using special techniques (e.g. a
corollary of Hilbert-Burch theorem), which are effective only for plane curve
singularities.

The first purpose of this paper is to generalize the $\delta$-constant
criterion given in [3] and [12] to deformations of isolated (not necessarily
reduced) curve singularities over normal or smooth base spaces of dimension
$\geq 1$. In Proposition 3.4 we show that equinormalizability of deformations
of isolated curve singularities over normal base spaces implies the constancy
of the $\delta$ number of fibers of these deformations. Moreover, in Theorem
3.6 we show that if the normalization of the total space of a deformation of
an isolated curve singularity over $(\mathbb{C}^{k},0)$, $k\geq 1$, is Cohen-
Macaulay then the converse holds. The assumption on Cohen-Macaulayness of the
normalization of the total space ensures for flatness of the composition map.
Moreover, Cohen-Macaulayness of the normalization of the total space is always
satified for deformations over $(\mathbb{C},0)$, because in this case, the
total space is a normal surface singularity, which is Cohen-Macaulay.

In all of known results for the $\delta$-constant criterion for
equinormalizability of deformations of isolated curve singularities, the total
spaces of these deformations are always assumed to be reduced and pure
dimensional. It is necessary to weaken the hypothesis on reducedness or purity
of the dimension of total spaces. In section 2 we study the relationship
between reducedness of the total space and that of the generic fibers of a
flat morphism, and show in Theorem 2.5 that if the generic fibers of a flat
morphism over a reduced Cohen-Macaulay space are reduced then the total space
is reduced. In particular, if there exists a representative of a deformation
of an isolated singularity over a reduced Cohen-Macaulay base space such that
the total space is generically reduced over the base space then the total
space is reduced (see Corollary 2.6). This gives a way to check reducedness of
the total space of a deformation, and to weaken the hypothesis on reducedness
of the total space of a deformation.

For families of isolated curve singularities, one of the most important things
is the admission of weak simultaneous resolutions ([15]) of these families.
Buchweitz and Greuel ([2], 1980) gave a list of criteria for the admission of
weak simultaneous resolutions of one-parametric families of reduced curve
singularities, namely, the constancy of the Milnor number, the constancy of
the $\delta$ number as well as the number of branches of all fibers, and the
topologically triviality of these families (see Theorem 4.3). In the last
section, we use a very new result of Bobadilla, Snoussi and Spivakovsky (2014)
to show that these criteria are also true for one-parametric families of
isolated (not necessarily reduced) curve singularities (see Theorem 4.5).

Notation: Let $f:(X,x)\rightarrow(S,0)$ be a morphism of complex germs. Denote
by $(X^{red},x)$ the reduction of $(X,x)$ and
$i:(X^{red},x)\hookrightarrow(X,x)$ the inclusion. Let
$\nu^{red}:(\overline{X},\overline{x})\rightarrow(X^{red},x)$ be the
normalization of $(X^{red},x)$, where $\overline{x}:=(\nu^{red})^{-1}(x)$.
Then the composition
$\nu:(\overline{X},\overline{x})\overset{\nu^{red}}{\rightarrow}(X^{red},x)\overset{i}{\hookrightarrow}(X,x)$
is called the normalization of $(X,x)$. Denote
$\bar{f}:=f\circ\nu:(\overline{X},\overline{x})\rightarrow(S,0).$ For each
$s\in S$, we denote

$X_{s}:=f^{-1}(s),\quad\overline{X}_{s}:=\bar{f}^{-1}(s).$

##  2\. Generic reducedness

Let $f:(X,x)\rightarrow(S,0)$ be a flat morphism of complex germs. In this
section we study the relationship between reducedness of the total space
$(X,x)$ and that of the generic fibers of $f$. This gives a way to check
reducedness of the total space of a flat morphism.

######  Definition 2.1.

Let $f:X\rightarrow S$ be a morphism of complex spaces. Denote by
$\mbox{Red}(X)$ the set of all reduced points of $X$ and

$\mbox{Red}(f)=\\{x\in X|f\mbox{ is flat at }x\mbox{ and }f^{-1}(f(x))\mbox{
is reduced at }x\\}$

the _reduced locus_ of $f$. We say

  * (1)

$X$ is _generically reduced_ if $\mbox{Red}(X)$ is open and dense in $X$;

  * (2)

$X$ is _generically reduced over_ $S$ if there is an analytically open dense
set $V$ in $S$ such that $f^{-1}(V)$ is contained in $\mbox{Red}(X)$;

  * (3)

the _generic fibers of $f$ are reduced_ if there is an analytically open dense
set $V$ in $S$ such that $X_{s}:=f^{-1}(s)$ is reduced for all $s$ in $V$.

We show in the following that under properness of the restriction of a flat
morphism $f:(X,x)\rightarrow(S,0)$ to its non-reduced locus, the generically
reducedness of $X$ over $S$ implies reducedness of the generic fibers of $f$.

######  Proposition 2.2.

Let $f:(X,x)\rightarrow(S,0)$ be flat with $(S,0)$ reduced. Assume that there
is a representative $f:X\rightarrow S$ such that its restriction on the non-
reduced locus $\mbox{NRed}(f):=X\setminus\mbox{Red}(f)$ is proper and $X$ is
generically reduced over $S$. Then the generic fibers of $f$ are reduced.

###### Proof.

$\mbox{NRed}(f)$ is analytically closed in $X$ (cf. [8, Corollary I.1.116]).
Moreover, since $X$ is generically reduced over $S$, there exists an
analytically open dense set $U$ in $S$ such that
$f^{-1}(U)\subseteq\mbox{Red}(X)$. Then, by properness of the restriction
$\mbox{NRed}(f)\rightarrow S$, $f(\mbox{NRed}(f))$ is analytically closed and
nowhere dense in $S$ by [1, Theorem 2.1(3), p.56]. This implies that
$V:=S\setminus f(\mbox{NRed}(f))$ is analytically open dense in $S$, and for
all $s\in V$, $X_{s}:=f^{-1}(s)$ is reduced. Therefore the generic fibers of
$f$ are reduced. ∎

######  Corollary 2.3.

Let $f:(X,x)\rightarrow(S,0)$ be flat with $(S,0)$ reduced. Assume that
$X_{0}\setminus\\{x\\}$ is reduced and there exists a representative
$f:X\rightarrow S$ such that $X$ is generically reduced over $S$. Then the
generic fibers of $f$ are reduced.
In particular, if $X_{0}\setminus\\{x\\}$ and $(X,x)$ are reduced, then the
generic fibers of $f$ are reduced.

###### Proof.

Since $f$ is flat, we have

$\mbox{NRed}(f)\cap X_{0}=\mbox{NRed}(X_{0})\subseteq\\{x\\},$

where $\mbox{NRed}(X_{0})$ denotes the set of non-reduced points of $X_{0}$.
This implies that the restriction $f:\mbox{NRed}(f)\rightarrow S$ is finite,
hence proper. Then the first assertion follows from Proposition 2.2. Moreover,
if $(X,x)$ is reduced then there exists a representative $X$ of $(X,x)$ which
is reduced. Then $X$ is obviously generically reduced over some representative
$S$ of $(S,s)$. Hence we have the latter assertion. ∎

######  Remark 2.4.

The assumption on reducedness of $X_{0}\setminus\\{x\\}$ in Corollary 2.3 is
necessary for reducedness of generic fibers, even for the case $S=\mathbb{C}$.
In fact, let $(X_{0},0)\subseteq(\mathbb{C}^{3},0)$ be defined by the ideal

$I_{0}=\left<x^{2},y\right>\cap\left<y^{2},z\right>\cap\left<z^{2},x\right>\subseteq\mathbb{C}\\{x,y,z\\}$

and $(X,0)\subseteq(\mathbb{C}^{4},0)$ defined by the ideal

$I=\left<x^{2}-t^{2},y\right>\cap\left<y^{2}-t^{2},z\right>\cap\left<z^{2},x\right>\subseteq\mathbb{C}\\{x,y,z,t\\}.$

Let $f:(X,0)\rightarrow(\mathbb{C},0)$ be the restriction on $(X,0)$ of the
projection on the fourth component
$\pi:(\mathbb{C}^{4},0)\rightarrow(\mathbb{C},0),~{}(x,y,z,t)\mapsto t$. Then
$f$ is flat, $X\setminus X_{0}$ is reduced, hence $X$ is generically reduced
over some representative $T$ of $(\mathbb{C},0)$. However the fiber
$(X_{t},0)$ is not reduced for any $t\not=0$. Note that in this case
$X_{0}\setminus\\{0\\}$ is not reduced.

As we have seen from Corollary 2.3, if the total space of a flat morphism over
a reduced base space is reduced, then the generic fibers of that morphism are
reduced. In the following we shows that over a reduced Cohen-Macaulay base
space, the converse is also true. This generalizes [3, Proposition 3.1.1 (3)]
to deformations over higher dimensional base spaces.

######  Theorem 2.5.

Let $f:(X,x)\rightarrow(S,0)$ be flat with $(S,0)$ reduced Cohen-Macaulay of
dimension $k\geq 1$. If there exists a representative $f:X\rightarrow S$ whose
generic fibers are reduced, then $(X,x)$ is reduced.

###### Proof.

We divide the proof of this part into two steps.
Step 1: $\mathbb{S=\mathbb{C}^{k}}.$ Then
$f=(f_{1},\cdots,f_{k}):(X,x)\rightarrow(\mathbb{C}^{k},0)$ is flat.
For $k=1$, assume that there exists a representative $f:X\rightarrow T$ such
that $X_{t}:=f^{-1}(t)$ is reduced for every $t\not=0$. Then for any $y\in
X\setminus X_{0}$ we have $(X_{f(y)},y)$ is reduced. It follows that $(X,y)$
is reduced (cf. [8, Theorem I. 1.101]). Thus $X\setminus X_{0}$ is reduced. To
show that $(X,x)$ is reduced, let $g$ be a nilpotent element of
$\mathcal{O}_{X,x}$. Then we have

$supp(g)=V(\mbox{Ann}(g))\subseteq X_{0}=V(f).$

It follows from Hilbert-Rückert’s Nullstellensatz (cf. [8, Theorem I.1.72])
that $f^{n}\in\mbox{Ann}(g)$ for some $n\in\mathbb{Z}_{+}$. Hence $f^{n}g=0$
in $\mathcal{O}_{X,x}$. Since $f$ is flat, it is a non-zerodivisor in
$\mathcal{O}_{X,x}$. Then $f^{n}$ is also a non-zerodivizor in
$\mathcal{O}_{X,x}$. It follows that $g=0$. Thus $(X,x)$ is reduced, and the
statement is true for $k=1$.
For $k\geq 2$, suppose there is a representative $f:X\rightarrow S$ and an
analytically open dense set $V$ in $S$ such that $X_{s}$ is reduced for all
$s\in V$. Let us denote by $H$ the line

$H:=\\{(t_{1},\cdots,t_{k})\in\mathbb{C}^{k}|t_{1}=\cdots=t_{k-1}=0\\}.$

Denote by $A$ the complement of $V$ in $S$. Then $A$ is analytically closed
and nowhere dense in $S$. We can choose coordinates $t_{1},\cdots,t_{k}$ and a
representative of $(\mathbb{C}^{k},0)$ such that $A\cap H=\\{0\\}$.
Denote $f^{\prime}:=(f_{1},\cdots,f_{k-1})$. Since $f$ is flat,
$f_{1},\cdots,f_{k-1}$ is an $\mathcal{O}_{X,x}$-regular sequence, hence
$f^{\prime}:(X,x)\rightarrow(\mathbb{C}^{k-1},0)$ is flat with the special
fiber $(X^{\prime},x):=(f^{\prime-1}(0),x)=(f^{-1}(H),x)$. Since $f$ is flat,
$f_{k}$ is a non-zerodivisor in
$\mathcal{O}_{X,x}/f^{\prime}\mathcal{O}_{X,x}=\mathcal{O}_{X^{\prime},x}$,
hence the morphism $f_{k}:(X^{\prime},x)\rightarrow(\mathbb{C},0)$ is flat.
For any $t\in\mathbb{C}\setminus\\{0\\}$ close to $0$, we have
$(0,\cdots,0,t)\not\in A$, hence $f_{k}^{-1}(t)=f^{-1}(0,\cdots,0,t)$ is
reduced. It follows from the case $k=1$ that the total space $(X^{\prime},x)$
of $f_{k}$ is reduced. Since $f^{\prime}:(X,x)\rightarrow(\mathbb{C}^{k-1},0)$
is flat whose special fiber is reduced, $(X,x)$ is reduced (cf. [8, Theorem
I.1.101]), and we have the proof for this step.
Step 2: $\mathbb{(S,0)}$ is Cohen-Macaulay of dimension  $\mathbb{k\geq 1}.$
Since $(S,0)$ is Cohen-Macaulay, there exists an $\mathcal{O}_{S,0}$-regular
sequence $g_{1},\cdots,g_{k}$, where $g_{i}\in\mathcal{O}_{S,0}$ for every
$i=1,\cdots,k$. Then the morphism

$g=(g_{1},\cdots,g_{k}):(S,0)\longrightarrow(\mathbb{C}^{k},0),t\longmapsto\big{(}g_{1}(t),\cdots,g_{k}(t)\big{)}$

is flat. We have

$\dim(g^{-1}(0),0)=\dim\mathcal{O}_{S,0}/(g_{1},\cdots,g_{k})\mathcal{O}_{S,0}=0$

(cf. [8, Prop. I.1.85]). This implies that $g$ is finite. Let $g:S\rightarrow
T$ be a representative which is flat and finite, where $T$ is an open
neighborhood of $0\in\mathbb{C}^{k}$. Then the composition $h=g\circ
f:X\longrightarrow T$ (for some representative) is flat. To apply Step 1 for
$h$, we need to show the existence of an analytically open dense set $U$ in
$T$ such that all fibers over $U$ are reduced. In fact, since $S$ is reduced,
its singular locus $\mbox{Sing}(S)$ is closed and nowhere dense in $S$ (cf.
[8, Corollary I.1.111]). It follows that $A\cup\mbox{Sing}(S),$ $A$ as in Step
1, is closed and nowhere dense in $S$. Then the set $U:=T\setminus
g(A\cup\mbox{Sing}(S))$ is open and dense in $T$ by the finiteness of $g$.
Furthermore, for any $t\in U$, $g^{-1}(t)=\\{t_{1},\cdots,t_{r}\\}$, $t_{i}\in
V\cap(S\setminus\mbox{Sing}(S))$. It follows that
$h^{-1}(t)=f^{-1}(t_{1})\cup\cdots\cup f^{-1}(t_{r})$ is reduced.
Now applying Step 1 for the flat map $h:X\rightarrow T$, we have reducedness
of $(X,x)$. The proof is complete. ∎

The following result is a direct consequence of Corollary 2.3 and Theorem 2.5.

######  Corollary 2.6.

Let $f:(X,x)\rightarrow(S,0)$ be flat with $(S,0)$ reduced Cohen-Macaulay of
dimension $k\geq 1$. Suppose $X_{0}\setminus\\{x\\}$ is reduced and there
exists a representative $f:X\rightarrow S$ such that $X$ is generically
reduced over $S$. Then $(X,x)$ is reduced.

Since normal surface singularities are reduced and Cohen-Macaulay, we have

######  Corollary 2.7.

Let $f:(X,x)\rightarrow(S,0)$ be flat with $(S,0)$ a normal surface
singularity. If there exists a representative $f:X\rightarrow S$ whose generic
fibers are reduced, then $(X,x)$ is reduced.

##  3\. Equinormalizable deformations of isolated curve singularities over
smooth base spaces

In this section we focus on equinormalizability of deformations of isolated
(not necessarily reduced) curve singularities over smooth base spaces of
dimension $\geq 1$. Because of isolatedness of singularities in the special
fibers of these deformations, by Corollary 2.6, instead of assuming
reducedness of the total spaces, we need only assume the generically
reducedness of the total spaces over the base spaces.

First we recall a definition of equinormalizable deformations which follows
Chiang-Hsieh-Lipman ([5]) and Kollár ([11]).

######  Definition 3.1.

Let $f:X\longrightarrow S$ be a morphism of complex spaces. A _simultaneous
normalization of $f$ _ is a morphism $n:\widetilde{X}\longrightarrow X$ such
that

  * (1)

$n$ is finite,

  * (2)

$\tilde{f}:=f\circ n:\widetilde{X}\rightarrow S$ is _normal_ , i.e., for each
$z\in\widetilde{X}$, $\tilde{f}$ is flat at $z$ and the fiber
$\widetilde{X}_{\tilde{f}(z)}:=\tilde{f}^{-1}(\tilde{f}(z))$ is normal,

  * (3)

the induced map $n_{s}:\widetilde{X}_{s}:=\tilde{f}^{-1}(s)\longrightarrow
X_{s}$ is bimeromorphic for each $s\in f(X)$.

The morphism $f$ is called _equinormalizable_ if the normalization
$\nu:\overline{X}\rightarrow X$ is a simultaneous normalization of $f$. It is
called _equinormalizable at $x\in X$_ if the restriction of $f$ to some
neighborhood of $x$ is equinormalizable.
If $f:(X,x)\longrightarrow(S,s)$ is a morphism of germs, then a _simultaneous
normalization of $f$_ is a morphism $n$ from a multi-germ
$(\widetilde{X},n^{-1}(x))$ to $(X,x)$ such that some representative of $n$ is
a simultaneous normalization of a representative of $f$. The germ $f$ is
_equinormalizable_ if some representative of $f$ is equinormalizable.

The following lemma allows us to do base change, reducing deformations over
higher dimensional base spaces to those over smooth 1-dimensional base spaces
with similar properties.

######  Lemma 3.2.

Let $f:(X,x)\rightarrow(S,0)$ be a deformation of an isolated singularity
$(X_{0},x)$ with $(S,0)$ normal. Suppose that there exists some representative
$f:X\rightarrow S$ such that $X$ is generically reduced over $S$. Then there
exists an open and dense set $U$ in $S$ such that $X_{s}:=f^{-1}(s)$ is
reduced, $\overline{X}_{s}:=\bar{f}^{-1}(s)$ is normal for all $s\in U$.
Moreover, for each $s\in U$, the induced morphism on the fibers
$\nu_{s}:\overline{X}_{s}\rightarrow X_{s}$ is the normalization of $X_{s}$.

Here, we recall that $\nu:(\overline{X},\overline{x})\rightarrow(X,x)$ is the
normalization of $(X,x)$ and
$\bar{f}:=f\circ\nu:(\overline{X},\overline{x})\rightarrow(S,0)$.

###### Proof.

Since $X_{0}\setminus\\{x\\}$ is reduced, it follows from the proof of
Corollary 2.3 that the set $f(\mbox{NRed}(f))$ is closed and nowhere dense in
$S$. Denote by $\mbox{NNor}(f)$ (resp. $\mbox{NNor}(\bar{f})$) the non-normal
locus of $f$ (resp. $\bar{f}$), the set of points $z$ in $X$ (resp.
$\overline{X}$) at which either $f$ (resp. $\bar{f}$) is not flat or
$X_{f(z)}$ (resp. $\overline{X}_{\bar{f}(z)}$) is not normal. Since $f$ is
flat and $S$ is normal, we have
$\nu(\mbox{NNor}(\bar{f})\cap\overline{X}_{0})\subseteq\mbox{NNor}(f)\cap
X_{0}=\mbox{NNor}(X_{0})$. Equivalently,
$\mbox{NNor}(\bar{f})\cap\overline{X}_{0}\subseteq\nu^{-1}(\mbox{NNor}(X_{0}))$
which is finite since $\nu$ is finite and $X_{0}$ has an isolated singularity
at $x$. It follows that the restriction of $\bar{f}$ on $\mbox{NNor}(\bar{f})$
is finite. Then $\bar{f}(\mbox{NNor}(\bar{f}))$ is closed and nowhere dense in
$S$ by [1, Theorem 2.1(3), p.56]. The set
$U:=S\setminus\big{(}f(\mbox{NRed}(f))\cup\bar{f}(\mbox{NNor}(\bar{f}))\big{)}$
satisfies all the required properties. ∎

For deformations of isolated curve singularities we have the following
necessary condition for their equinormalizability, in terms of the constancy
of the $\delta$-invariant of fibers. For the reader’s convenience we recall
the definition of the $\delta$-invariant of isolated (not necessarily reduced)
curve singularities, which is defined by Brücker and Greuel in [3].

######  Definition 3.3.

Let $X$ be a complex curve and $x\in X$ an isolated singular point. Denote by
$X^{red}$ its reduction and let $\nu^{red}:\overline{X}\rightarrow X^{red}$ be
the normalization of the reduced curve $X^{red}$. The number

$\delta(X^{red},x):=\dim_{\mathbb{C}}(\nu^{red}_{*}\mathcal{O}_{\overline{X}})_{x}/\mathcal{O}_{X^{red},x}$

is called the _delta-invariant of $X^{red}$ at $x$_,

$\epsilon(X,x):=\dim_{\mathbb{C}}H_{\\{x\\}}^{0}(\mathcal{O}_{X})$

is called the _epsilon-invariant of $X$ at $x$_, where
$H_{\\{x\\}}^{0}(\mathcal{O}_{X})$ denotes local cohomology, and

$\delta(X,x):=\delta(X^{red},x)-\epsilon(X,x)$

is called the _delta-invariant of $X$ at $x$_.
If $X$ has only finitely many singular points then the number

$\delta(X):=\sum_{x\in\mbox{Sing}(X)}\delta(X,x)$

is called the _delta-invariant_ of $X$.

It is easy to see that $\delta(X^{red},x)\geq 0$, and $\delta(X^{red},x)=0$ if
and only if $x$ is an isolated point of $X$ or the germ $(X^{red},x)$ is
smooth. Hence, if $x\in X$ is an isolated point of $X$ then
$\delta(X,x)=-\dim_{\mathbb{C}}\mathcal{O}_{X,x}=-\epsilon(X,x)$. In
particular, $\delta(X,x)=-1$ for $x$ an isolated and reduced (hence normal)
point of $X$.

######  Proposition 3.4.

Let $f:(X,x)\rightarrow(S,0)$ be a deformation of an isolated curve
singularity $(X_{0},x)$ with $(X,x)$ pure dimensional, $(S,0)$ normal. Suppose
that there exists some representative $f:X\rightarrow S$ such that $X$ is
generically reduced over $S$. If $f$ is equinormalizable, then it is
$\delta$-constant, that is, $\delta(X_{s})=\delta(X_{0})$ for every $s\in S$
close to $0$.

###### Proof.

(Compare to the proof of [12, Theorem 4.1 (2)])
It follows from Lemma 3.2 that there exists an open and dense set $U$ in $S$
such that $X_{s}$ is reduced and $\overline{X}_{s}$ is normal for all $s\in
U$.
We first show that $f$ is $\delta$-constant on $U$, i.e.
$\delta(X_{s})=\delta(X_{0})$ for any $s\in U$. In fact, for any $s\in U$,
$s\not=0$, there exist an irreducible reduced curve singularity $C\subseteq S$
passing through $0$ and $s$. Let $\alpha:T\longrightarrow C\subseteq S$ be the
normalization of this curve singularity such that
$\alpha(T\setminus\\{0\\})\subseteq U$, where $T\subseteq\mathbb{C}$ is a
small disc with center at $0$. Denote

$X_{T}:=X\times_{S}T,~{}\overline{X}_{T}:=\overline{X}\times_{S}T.$

Then we have the following Cartesian diagram:

$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\overline{X}_{T}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Box}$$\scriptstyle{\nu_{T}}$$\scriptstyle{\bar{f}_{T}}$$\textstyle{\overline{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\nu}$$\scriptstyle{\bar{f}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces
X_{T}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Box}$$\scriptstyle{f_{T}}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{T\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S}$

For any $t\in T,s=\alpha(t)\in S$, we have

$\mathcal{O}_{(X_{T})_{t}}:=\mathcal{O}_{f_{T}^{-1}(t)}\cong\mathcal{O}_{X_{s}},~{}\mathcal{O}_{(\overline{X}_{T})_{t}}:=\mathcal{O}_{\bar{f}_{T}^{-1}(t)}\cong\mathcal{O}_{\overline{X}_{s}}.$
(3.1)

Since $f$ is flat by hypothesis and $\bar{f}$ is flat by equinormalizability,
it follows from the preservation of flatness under base change (cf. [8, Prop.
I. 1.87]) that the induced morphisms $f_{T}$ and $\bar{f}_{T}$ are flat over
$T$. Hence, it follows from equinormalizability of $f$ and (3.1) that
$f_{T}:X_{T}\rightarrow T$ is equinormalizable.
For any $t\in T\setminus\\{0\\}$, $s=\alpha(t)\in U$, hence $(X_{T})_{t}\cong
X_{s}$ is reduced by the existence of $U$. It follows from Theorem 2.5 that
$X_{T}$ is reduced. On the other hand, since $X$ and $S$ are pure dimensional,
all fibers of $f$, hence of $f_{T}$, are pure dimensional by the dimension
formula ([7, Lemma, p.156]). Then $X_{T}$ is also pure dimensional because $T$
is pure 1-dimensional. Therefore it follows from [3, Korollar 2.3.5] that
$f_{T}:X_{T}\longrightarrow T$ is $\delta$-constant, hence $f:X\longrightarrow
S$ is $\delta$-constant on $U$.
Let us now take $s_{0}\in S\setminus U$. Since $U$ is dense in $S$, $s_{0}\in
S$, there exists always a point $s_{1}\in U$ which is close to $s_{0}$. It
follows from the semi-continuity of the $\delta$-function (cf. [12, Lemma
4.2]) that

$\delta(X_{0})\geq\delta(X_{s_{0}})\geq\delta(X_{s_{1}}).$

Moreover, $\delta(X_{0})=\delta(X_{s_{1}})$ as shown above. It implies that
$\delta(X_{s_{0}})=\delta(X_{0})$. Hence $f:X\longrightarrow S$ is
$\delta$-constant. ∎

######  Remark 3.5.

The complex spaces $X_{T}$ and $\overline{X}_{T}$ appearing in the proof of
Proposition 3.4 have the following properties:

  * (1)

$X_{T}$ is reduced; $\overline{X}_{T}$ is reduced if $\bar{f}_{T}$ is flat;

  * (2)

they have the same normalization $\widetilde{X_{T}}$;

  * (3)

fibers of the compositions
$\widetilde{X_{T}}\overset{\mu_{T}}{\rightarrow}\overline{X}_{T}\overset{\bar{f}_{T}}{\rightarrow}T$
and
$\widetilde{X_{T}}\overset{\theta_{T}}{\rightarrow}X_{T}\overset{f_{T}}{\rightarrow}T$
coincide.

In fact, as we have seen in the proof of Proposition 3.4, $X_{T}$ is reduced.
Moreover, if $\bar{f}_{T}$ is flat, since its generic fibers are reduced
(actually normal), $\overline{X}_{T}$ is reduced by Theorem 2.5. Therefore we
have (1).
Now we show (2). Since finiteness and surjectivity are preserved under base
change, $\nu_{T}$ is finite and surjective. Let us denote by
$\mu_{T}:\widetilde{X_{T}}\rightarrow\overline{X}_{T}$ the normalization of
$\overline{X}_{T}$. Then the composition $\theta_{T}:=\mu_{T}\circ\nu_{T}$ is
finite and surjective.
Denote $A:=\mbox{NNor}(f_{T})$. Since $X_{T}$ is reduced, $A$ is nowhere dense
in $X_{T}$. Moreover, since $\nu_{T}$ is finite and surjective, it follows
from Ritt’s lemma (cf. [9, Chapter 5, §3, p.102]) that the preimage
$A^{\prime}:=\nu_{T}^{-1}(A)$ is nowhere dense in $\overline{X}_{T}$.
Furthermore, for any $z\not\in A^{\prime}$, $y=\nu_{T}(z)\not\in A$, hence the
fiber $(X_{T})_{t}$ resp. $X_{s}$ is normal at $y$ resp. $\alpha_{T}(y)$,
where $t=f_{T}(y),s=\alpha(t)$. Thus
$(X,\alpha_{T}(y))\cong(\overline{X},\bar{\alpha}_{T}(z))$. It follows that
$(X_{T},y)\cong(\overline{X}_{T},z)$. Therefore $\overline{X}_{T}\setminus
A^{\prime}\cong X_{T}\setminus A$. Then $(\mu_{T}\circ\nu_{T})^{-1}(A)$ is
nowhere dense in $\widetilde{X_{T}}$ and we have the isomorphism

$\widetilde{X_{T}}\setminus(\mu_{T}\circ\nu_{T})^{-1}(A)=\widetilde{X_{T}}\setminus\mu_{T}^{-1}(A^{\prime})\cong\overline{X}_{T}\setminus
A^{\prime}\cong X_{T}\setminus A.$

Therefore $\theta_{T}$ is bimeromorphic, whence it is the normalization of
$X_{T}$. (3) is obvious.

The following theorem is the main result of this section, which asserts that
under certain conditions, the $\delta$-criterion is sufficient for
equinormalizability of deformations of isolated curve singularities over
smooth base spaces of dimension $\geq 1$. This gives a generalization of [3,
Korollar 2.3.5].

######  Theorem 3.6.

Let $f:(X,x)\rightarrow(\mathbb{C}^{k},0)$, $k\geq 1$, be a deformation of an
isolated curve singularity $(X_{0},x)$ with $(X,x)$ pure dimensional. Suppose
that there exists a representative $f:X\rightarrow S$ such that $X$ is
generically reduced over $S$. If the normalization $\overline{X}$ of $X$ is
Cohen-Macaulay 111This holds always for $k=1$, since normal surfaces are
Cohen-Macaulay. and $f$ is $\delta$-constant, then $f$ is equinormalizable.

###### Proof.

First we show that Cohen-Macaulayness of $\overline{X}$ implies flatness of
the composition $\bar{f}$. Since $\overline{X}$ is Cohen-Macaulay and $S$ is
smooth, it is sufficient to check that the dimension formula holds for
$\bar{f}$ (cf. [7, Proposition, p.158]). But it is always the case, since for
any $z\in\nu^{-1}(x)$, we have

$\displaystyle\dim(\overline{X},z)$
$\displaystyle=\dim(X,x)=\dim(X_{0},x)+k\quad\quad\mbox{by flatness of $f$}$
$\displaystyle=\dim(\overline{X}_{0},z)+k.$

The latter equality follows from finiteness and surjectivity of
$\nu_{0}:(\overline{X}_{0},z)\rightarrow(X_{0},x)$.

Let $U\subseteq S$ be the open dense set with properties described as in Lemma
3.2. For any $s\in U$, let $C\subseteq S$ be an irreducible reduced curve
singularity passing through $s$ and $0$ such that $C\cap(S\setminus
U)=\\{0\\}$. Let $\alpha:T\longrightarrow C\subseteq S$ be the normalization
of this curve singularity such that $\alpha(T\setminus\\{0\\})\subseteq U$,
where $T\subseteq\mathbb{C}$ is a small disc with center at $0$. Denote
$X_{T}$ and $\overline{X}_{T}$ as in the proof of Proposition 3.4. Then, since
$\bar{f}$ is flat, it follows from Remark 3.5 that $X_{T}$ and
$\overline{X}_{T}$ are reduced and they have the same normalization
$\widetilde{X}_{T}$. Consider the following Cartesian diagram:

$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\widetilde{X_{T}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{T}}$$\scriptstyle{\theta_{T}}$$\scriptstyle{\widetilde{f}_{T}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\overline{X}_{T}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Box}$$\scriptstyle{\bar{\alpha}_{T}}$$\scriptstyle{\nu_{T}}$$\scriptstyle{\bar{f}_{T}}$$\textstyle{\overline{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\nu}$$\scriptstyle{\bar{f}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces
X_{T}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Box}$$\scriptstyle{f_{T}}$$\scriptstyle{\alpha_{T}}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{T\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{S}$

Since fibers of $f$ and $f_{T}$ are isomorphic, $f_{T}$ is $\delta$-constant
and $X_{T}$ is pure dimensional. Then it follows from [3, Korollar 2.3.5] that
$f_{T}$ is equinormalizable. Therefore, by definition, for each $t\in T$,
$(\widetilde{X})_{t}:=(\widetilde{f}_{T})^{-1}(t)$ is normal, and it is the
normalization of $(X_{T})_{t}$.

Let us consider the flat map $\bar{f}_{T}:\overline{X}_{T}\rightarrow T$ and
consider the normalization
$\mu_{T}:\widetilde{X_{T}}\rightarrow\overline{X}_{T}$ of $\overline{X}_{T}$.
It follows from [3, Proposition 1.2.2] that the composition
$\bar{f}_{T}\circ\mu_{T}:\widetilde{X_{T}}\rightarrow T$ is flat. Moreover, by
the same argument as given in Remark 3.5, we can show that $(X_{T})_{t}$ and
$(\overline{X}_{T})_{t}$ have the same normalization for each $t\in T$. Hence
the restriction on the fibers
$(\widetilde{X})_{t}\rightarrow(\overline{X}_{T})_{t}$ is the normalization.
Thus by definition, $\bar{f}_{T}$ is equinormalizable. Then $\bar{f}_{T}$ is
$\delta$-constant by Proposition 3.4 (or by [3, Korollar 2.3.5]). This implies
that for any $t\in T\setminus\\{0\\}$, we have

$\delta(\overline{X}_{0})=\delta((\overline{X}_{T})_{0})=\delta((\overline{X}_{T})_{t})=0\mbox{
(since }(\overline{X}_{T})_{t}\mbox{ is normal).}$

Now we show that $\overline{X}_{0}$ is reduced. First we show that
$\nu(\mbox{NNor}(\overline{X}_{0}))\subseteq\mbox{NNor}(X_{0})$. In fact, if
$y\not\in\mbox{NNor}(X_{0})$ then $X_{0}$ is normal at $y$. Since $f$ is flat
and $S$ is normal at $0$, $X$ is normal at $y$ (cf. [8, Theorem I.1.101]).
Therefore we have the isomorphism
$(\overline{X},z)\overset{\cong}{\longrightarrow}(X,y)$ for every
$z\in\nu^{-1}(y)$. It induces an isomorphism on the fibers
$(\overline{X}_{0},z)\overset{\cong}{\longrightarrow}(X_{0},y)$, hence
$\overline{X}_{0}$ is normal at every point $z\in\nu^{-1}(y)$. It follows that
$y\not\in\nu(\mbox{NNor}(\overline{X}_{0}))$.
Then, for any $z\in\mbox{NNor}(\overline{X}_{0})$, since $\mbox{NNor}(X_{0})$
is nowhere dense in $X_{0}$, by Ritt’s lemma (cf. [9, Chapter 5, §3, 2,
p.103]) and by the dimension formula (when $f$ is flat) we have

$\displaystyle\dim(\nu(\mbox{NNor}(\overline{X}_{0})),\nu(z))\leq\dim(\mbox{NNor}(X_{0}),\nu(z))<\dim(X_{0},\nu(z))$
$\displaystyle=\dim(X,\nu(z))-\dim(S,0)=\dim(\overline{X},z)-\dim(S,0)\leq\dim(\overline{X}_{0},z).$

Furthermore, the restriction $\nu_{0}:\overline{X}_{0}\longrightarrow X_{0}$
is finite. Hence

$\dim(\nu(\mbox{NNor}(\overline{X}_{0})),\nu(z))=\dim(\mbox{NNor}(\overline{X}_{0}),z)~{}\mbox{(cf.
\cite[cite]{[\@@bibref{}{Fi}{}{}, Corollary, p.141]})}.$

It follows that for any $z\in\mbox{NNor}(\overline{X}_{0})$ we have
$\dim(\mbox{NNor}(\overline{X}_{0}),z)<\dim(\overline{X}_{0},z)$, i.e.,
$\mbox{NNor}(\overline{X}_{0})$ is nowhere dense in $\overline{X}_{0}$ by
Ritt’s lemma. This implies that $\overline{X}_{0}$ is generically normal,
whence generically reduced.
Moreover, for each $z\in\nu^{-1}(x)$, since $\bar{f}$ is flat and
$\dim(\overline{X},z)=\dim(X,x)=k+1$, we have

$\mbox{depth}(\mathcal{O}_{\overline{X}_{0},z})=\mbox{depth}(\mathcal{O}_{\overline{X},z})-k\geq(k+1)-k=1.$

On the other hand, we have

$\dim(\overline{X}_{0},z)=\dim(\overline{X},z)-k=1.$

Hence $\mbox{depth}(\mathcal{O}_{\overline{X}_{0},z})\geq
1=\min\\{1,\dim(\overline{X}_{0},z)\\}$, i.e. $\overline{X}_{0}$ satisfies
$(S_{1})$ at every point $z\in\nu^{-1}(x)$. This implies that
$\overline{X}_{0}$ is reduced at every point of $\nu^{-1}(x)$. Then
$\overline{X}_{0}$ is normal, and it is the normalization of $X_{0}$. It
follows that $f$ is equinormalizable. The proof is complete. ∎

The following example illustrates our main theorem.

######  Example 3.7 ([13], cf. [12, Example 4.2]).

Let us consider the curve singularity $(X_{0},0)\subseteq(\mathbb{C}^{4},0)$
defined by the ideal

$I_{0}:=\left<x^{2}-y^{3},z,w\right>\cap\left<x,y,w\right>\cap\left<x,y,z,w^{2}\right>\subseteq\mathbb{C}\\{x,y,z,w\\}.$

The curve singularity $(X_{0},0)$ is a union of a cusp $C$ in the plane
$z=w=0,$ a straight line $L=\\{x=y=w=0\\}$ and an embedded non-reduced point
$O=(0,0,0,0)$. Now we consider the restriction
$f:(X,0)\rightarrow(\mathbb{C}^{2},0)$ of the projection
$\pi:(\mathbb{C}^{6},0)\rightarrow(\mathbb{C}^{2},0),~{}(x,y,z,w,u,v)\mapsto(u,v),$
to the complex germ $(X,0)$ defined by the ideal

$I=\left<x^{2}-y^{3}+uy^{2},z,w\right>\cap\left<x,y,w-v\right>\subseteq\mathbb{C}\\{x,y,z,w,u,v\\}.$

It is easy to check that $f$ is flat, $f^{-1}(0,0)=(X_{0},0)$, the total space
$(X,0)$ is reduced and pure $3$-dimensional, with two 3-dimensional
irreducible components.

We have $\delta((X_{0})^{red})=2$, $\epsilon(X_{0})=1$, hence
$\delta(X_{0})=1$. Moreover, for each $u,v\in\mathbb{C}\setminus\\{0\\}$, we
have

$\delta(X_{(u,v)})=\delta((X_{(u,v)})^{red})-\epsilon(X_{(u,v)})=1-0=1;$
$\delta(X_{(u,0)})=2-1=1;\quad\delta(X_{(0,v)})=1-0=1.$

Hence $f$ is $\delta$-constant.

Moreover, the normalizations of the first component $(X_{1},0)$ and the second
component $(X_{2},0)$ of $(X,0)$ are given respectively by

$\nu_{1}:(\mathbb{C}^{3},0)\rightarrow(X_{1},0),\quad(T_{1},T_{2},T_{3})\mapsto(0,0,T_{1},T_{3},T_{2},T_{3})$

and

$\nu_{2}:(\mathbb{C}^{3},0)\rightarrow(X_{2},0),\quad(T_{1},T_{2},T_{3})\mapsto(T_{3}^{3}+T_{1}T_{3},T_{3}^{2}+T_{1},0,0,T_{1},T_{2}).$

Hence the composition maps are given respectively by

$\bar{f}_{1}:(\mathbb{C}^{3},0)\rightarrow(\mathbb{C},0),\quad(T_{1},T_{2},T_{3})\mapsto(T_{2},T_{3})$

and

$\bar{f}_{2}:(\mathbb{C}^{3},0)\rightarrow(\mathbb{C},0),\quad(T_{1},T_{2},T_{3})\mapsto(T_{1},T_{2}).$

On both components, $\bar{f}$ is flat with normal fibers, hence $f$ is
equinormalizable. Note that, in this example, the normalization of $(X,0)$ is
smooth. All the computation given above can be easily done by SINGULAR ([6]).

##  4\. Topologically triviality of one-parametric families of isolated curve
singularities

In this section we consider one-parametric families of isolated (not
necessarily reduced) curve singularities and show that the topologically
triviality of these families is equivalent to the admission of weak
simultaneous resolutions ([15]).

Let $f:(X,x)\rightarrow(\mathbb{C},0)$ be a deformation of an isolated curve
singularity $(X_{0},x)$ with $(X,x)$ pure dimensional. Let $f:X\rightarrow T$
be a good representative (in the sense of [2, §2.1, p.248]) such that $X$ is
generically reduced over $T$. Then $X$ is reduced by Corollary 2.6. Let
$\nu:\overline{X}\rightarrow X$ be the normalization of $X$. Denote
$\bar{f}:=f\circ\nu:\overline{X}\rightarrow T$.

######  Definition 4.1 (cf. [3]).

  * (1)

$f$ is said to be topologically trivial  if there is a homeomorphism
$h:X\overset{\approx}{\rightarrow}X_{0}\times T$ such that $f=\pi\circ h$,
where $\pi:X_{0}\times T\rightarrow T$ is the projection.

  * (2)

Assume that $f$ admits a section $\sigma:T\rightarrow X$ such that
$X_{t}\setminus\sigma(t)$ is smooth for all $t\in T$. Then $f$ admits a weak
simultaneous resolution if $f$ is equinormalizable and

$\big{(}\nu^{-1}(\sigma(T))\big{)}^{red}\cong\big{(}\nu^{-1}(\sigma(0))\big{)}^{red}\times
T\quad(\mbox{over $T$}).$

######  Remark 4.2 (cf. [15]).

$f$ admits a weak simultaneous resolution if and only if $f$ is
equinormalizable and the number of branches $r(X_{t},\sigma(t))$ of
$(X_{t},\sigma(t))$ is constant for all $t\in T$.

Buchweitz and Greuel (1980) proved the following result for families of
reduced curve singularities.

######  Theorem 4.3 ([2, Theorem 5.2.2]).

Let $f:X\rightarrow T$ be a good representative of a flat family of reduced
curves with section $\sigma:T\rightarrow X$ such that
$X_{t}\setminus\sigma(t)$ is smooth for each $t\in T$. Then the following
conditions are equivalent:

  * (1)

$f$ admits a weak simultaneous resolution;

  * (2)

the delta number $\delta(X_{t},\sigma(t))$ and the number of branches
$r(X_{t},\sigma(t))$ are constant for $t\in T$;

  * (3)

the Milnor number $\mu(X_{t},\sigma(t))$ is constant for $t\in T$;

  * (4)

$f$ is topologically trivial.

We shall show that this result is also true for families of isolated (not
necessarily reduced) curve singularities. Due to Brücker and Greuel ([3]), we
give a new definition for the Milnor number of a curve singulariy $C$ at an
isolated singular point $c\in C$, namely,

$\mu(C,c):=2\delta(C,c)-r(C,c)+1.$

The Milnor number of $C$ is defined to be

$\mu(C):=\sum_{c\in\mbox{Sing}(C)}\mu(C,c).$

To state and prove a similar result to Theorem 4.3 we need the following
result of Bobadilla, Snoussi and Spivakovsky (2014).

######  Lemma 4.4 ([4, Theorem 4.4]).

Let $f:(X,x)\rightarrow(\mathbb{C},0)$ be a deformation of an isolated curve
singularity $(X_{0},x)$ with $(X,x)$ reduced. Assume that the singular locus
$\mbox{Sing}(X,x)$ of $(X,x)$ is smooth of dimension 1. If $f$ is
topologically trivial, then for any $z\in\nu^{-1}(x)$,
$\bar{f}:(\overline{X},z)\rightarrow(\mathbb{C},0)$ is topologically trivial,
and the normalization $(\overline{X},\nu^{-1}(x))$ of $(X,x)$ is smooth.

The following theorem is the main result of this section.

######  Theorem 4.5.

Let $f:(X,x)\rightarrow(\mathbb{C},0)$ be a deformation of an isolated curve
singularity $(X_{0},x)$ with $(X,x)$ pure dimensional. Let $f:X\rightarrow T$
be a good representative with section $\sigma:T\rightarrow X$ such that
$X_{t}\setminus\sigma(t)$ is smooth for each $t\in T$ and $X$ is generically
reduced over $T$. Assume that $\mbox{Sing}(X,x)$ is smooth of dimension 1.
Then the following conditions are equivalent:

  * (1)

$f$ admits a weak simultaneous resolution;

  * (2)

the delta number $\delta(X_{t},\sigma(t))$ and the number of branches
$r(X_{t},\sigma(t))$ are constant for $t\in T$;

  * (3)

the Milnor number $\mu(X_{t},\sigma(t))$ is constant for $t\in T$;

  * (4)

$f$ is topologically trivial.

###### Proof.

The equivalence of (1) and (2) follows from Theorem 3.6 (for $k=1$) and Remark
4.2. (2) $\Longleftrightarrow(3)$ because of the definition of the Milnor
number. The implication $(1)\Longrightarrow(4)$ is proved by the same way for
families of reduced curve singularities as given in the proof of the
implication $(4)\Longrightarrow(6)$ of [2, Theorem 5.2.2]. Now we prove that
$(4)\Longrightarrow(1)$.

For convenience, let us assume that $\nu^{-1}(x)=\\{z_{1},\cdots,z_{r}\\}$.
Note that $\overline{X}_{0}:=\bar{f}^{-1}(0)$ is reduced,
$\overline{X}_{t}:=\bar{f}^{-1}(t)$ is smooth for every $t\not=0$ by [3, Lemma
2.1.1]. Therefore for every $i=1,\cdots,r$,
$\bar{f}:(\overline{X},z_{i})\rightarrow(\mathbb{C},0)$ is a family of reduced
curve singularities with smooth general fibers, and there exist sections
$\bar{\sigma}_{1},\cdots,\bar{\sigma}_{r}:T\rightarrow\overline{X}$ such that
$\bar{\sigma}_{i}(0)=z_{i}$,
$\nu^{-1}(\sigma(t))=\\{\bar{\sigma}_{1}(t),\cdots,\bar{\sigma}_{r}(t)\\}$,
and $\overline{X}_{t}\setminus\bar{\sigma}_{i}(t)$ is smooth for every $t\in
T$ and for every $i=1,\cdots,r$.
Assume that $f$ is topologically trivial. Then it follows from Lemma 4.4 that
the deformation $\bar{f}:(\overline{X},z_{i})\rightarrow(\mathbb{C},0)$ of
$(\overline{X}_{0},z_{i})$ is also topologically trivial for every
$i=1,\cdots,r$. Hence it follows from Theorem 4.3, applying for the flat
family of reduced curve singularities
$\bar{f}:(\overline{X},z_{i})\rightarrow(\mathbb{C},0)$ with section
$\bar{\sigma}_{i}:(\mathbb{C},0)\rightarrow(\overline{X},z_{i})$, that the
delta number $\delta(\overline{X}_{t},\bar{\sigma}_{i}(t))$ and the number of
branches $r(\overline{X}_{t},\bar{\sigma}_{i}(t))$ are constant for $t\in T$.
Then for $t\not=0$ we have

$\delta(\overline{X}_{0})=\delta(\overline{X}_{t})=0.$

Hence $\overline{X}_{0}$ is normal. It follows that $f$ is equinormalizable.
On the other hand, the equinormalizability of $f$ over the smooth base space
$(\mathbb{C},0)$ implies that for every $t\in T$ and for each $i=1,\cdots,r$,
the induced map of $\nu$ on the fibers
$\nu_{t}:(\overline{X}_{t},\bar{\sigma}_{i}(t))\rightarrow(X_{t},\sigma(t))$
is the normalization of the corresponding irreducible component of
$(X_{t},\sigma(t))$. It follows that the number of irreducible components of
$(X_{t},\sigma(t))$ is equal to the cardinality of $\nu^{-1}(\sigma(t))$,
which is equal to $r$ for every $t\in T$. Hence $r(X_{t},\sigma(t))$ is
constant for every $t\in T$. It follows that $f$ admits a weak simultaneous
resolution, and we have (1). ∎

######  Example 4.6.

Let us consider again the curve singularity
$(X_{0},0)\subseteq(\mathbb{C}^{4},0)$ considered in Example 3.7 which is
defined by the ideal

$I_{0}:=\left<x^{2}-y^{3},z,w\right>\cap\left<x,y,w\right>\cap\left<x,y,z,w^{2}\right>\subseteq\mathbb{C}\\{x,y,z,w\\}.$

Now we consider the restriction $f:(X,0)\rightarrow(\mathbb{C},0)$ of the
projection
$\pi:(\mathbb{C}^{5},0)\rightarrow(\mathbb{C},0),~{}(x,y,z,w,t)\mapsto t,$ to
the complex germ $(X,0)$ defined by the ideal

$I=\left<x^{2}-y^{3}+ty^{2},z,w\right>\cap\left<x,y,w-t\right>\subseteq\mathbb{C}\\{x,y,z,w,t\\}.$

We can check the following (all of them can be checked easily by SINGULAR):

  * (1)

$f$ is flat;

  * (2)

$(X,0)$ is reduced and pure $2$-dimensional, with two 2-dimensional
irreducible components;

  * (3)

$f$ is $\delta$-constant with $\delta(X_{t})=1$ for all $t\in\mathbb{C}$ close
to $0$;

  * (4)

$r(X_{t})=2$ for all $t\in\mathbb{C}$ close to $0$;

  * (5)

$f$ is equinormalizable;

  * (6)

the normalization of each component of $(X,0)$ is $(\mathbb{C}^{2},0)$, which
is smooth.

By Theorem 4.5, $f$ is topologically trivial.

Acknowledgements. The author would like to express his gratitude to Professor
Gert-Martin Greuel for his valuable discussions, careful proof-reading and a
lot of precise comments. He would also like to thank the anonymous referees
for their careful proof-reading and suggestions. This research is funded by
Vietnam National Foundation for Science and Technology Development (NAFOSTED)
under the grant number 101.99-2013.24. This work is finished during the
author’s postdoctoral fellowship at the Vietnam Institute for Advanced Study
in Mathematics (VIASM). He thanks VIASM for financial support and hospitality.

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