# Singularities of Frontal Surfaces

C. Muñoz-Cabello, J.J. Nuño-Ballesteros, R. Oset Sinha  Departament de
Matemàtiques, Universitat de València, Campus de Burjassot, 46100 Burjassot,
Spain<EMAIL_ADDRESS><EMAIL_ADDRESS>Departament de Matemàtiques,
Universitat de València, Campus de Burjassot, 46100 Burjassot SPAIN.
Departamento de Matemática, Universidade Federal da Paraíba CEP 58051-900,
João Pessoa - PB, BRAZIL<EMAIL_ADDRESS>
###### Abstract.

We consider singularities of frontal surfaces of corank one and finite frontal
codimension. We look at the classification under $\mathscr{A}$-equivalence and
introduce the notion of frontalisation for singularities of fold type. We
define the cuspidal and the transverse double point curves and prove that the
frontal has finite codimension if and only if both curves are reduced.
Finally, we also discuss about the frontal versions of the Marar-Mond formulas
and the Mond’s conjecture.

###### Key words and phrases:

frontals, invariants of mappings, frontal Milnor number, double point curve

###### 2000 Mathematics Subject Classification:

Primary 32S30; Secondary 32S25, 58S25

Work of Juan J. Nuño-Ballesteros and R. Oset Sinha partially supported by
Grant PGC2018-094889-B-100 funded by MCIN/AEI/ 10.13039/501100011033 and by
“ERDF A way of making Europe"

##  1\. Introduction

In this paper we are interested in the local behaviour of complex analytic
mappings $f\colon N\to Z$ of frontal type, where $N$ and $Z$ are complex
analytic manifolds of dimensions 2 and 3 respectively. In general, a mapping
$f\colon N\to Z$ is called a frontal if it admits a Legendrian lifting
$\tilde{f}:N\to PT^{*}Z$, where $PT^{*}Z$ is the projectivised cotangent
bundle of $Z$. Roughly speaking, this means that the image $f(N)$ has a well
defined tangent hyperplane at each point $f(x)$, with $x\in N$.

Singularities of frontals were considered for the first time by Zakalyukin and
Kurbatskii in [34] and they are a natural generalisation of wave fronts, which
occur in the particular case that the Legendrian lifting $\tilde{f}$ is an
immersion. There has been a great interest for frontals in the last decades,
specially in the $C^{\infty}$ real category and looking at differential
geometric properties. The fact that you have a well defined tangent plane
everywhere provides a nice starting point if you want to extend things like
first or second fundamental forms, curvatures, etc to submanifolds with
singularities (see for instance [5, 18, 27, 29, 30]).

In a forthcoming paper [26], we will develop the general Thom-Mather theory of
frontals of any dimension, but restricted to corank one singularities. Our
approach is based on Ishikawa’s work [11] about stability and infinitesimal
deformations of integral mappings $\tilde{f}\colon N\to PT^{*}Z$ under
Legendrian equivalence, although we want to understand the singularity
downstairs, at the level of the frontal, rather than upstairs, at the level of
the Legendrian lifting. In particular, we consider infinitesimal deformations
which come from frontal unfoldings of $f$ itself. The corank one assumption is
a technical but necessary condition in order to be able to apply Ishikawa’s
theory.

In our case, by taking local charts in $N$ and $Z$ we can restate our problem
in terms of classification of frontal map germs
$f\colon(\mathbb{C}^{2},S)\to(\mathbb{C}^{3},0)$ of corank one under
$\mathscr{A}$-equivalence, that is, under holomorphic coordinate changes in
the source and the target. It is well known that the only frontal stable
singularities of surfaces are cuspidal edges, swallowtails, folded Whitney
umbrellas or their transverse self-intersections (see Figure 3). By the
frontal version of the Mather-Gaffney criterion (see Theorem 3.6), if $f$ has
finite frontal codimension, then it has isolated instability and hence, the
only singularities outside the origin are cuspidal edges or transverse double
points. Our main goal is to study the geometry and the invariants of this type
of frontal surface singularities.

In Section 3, we look at the classification of simple frontal singularities.
We introduce the notion of frontalisation of a germ of fold type and we deduce
that the classification of frontals of fold type is closely related to the
classification of $\mathscr{A}$-simple singularities obtained by Mond in [21].
In particular, all simple germs of fold type in Mond’s classification
$S_{k},B_{k},C_{k}$ and $F_{4}$ have a frontal version with the same frontal
codimension.

In Section 4, we define the cuspidal curve $C(f)$ and the transverse double
point curve $D_{+}(f)$. These two curves are considered with a certain
analytic structure and they have the property that $f$ has finite frontal
codimension if and only if both curves are reduced. We recall that $f$ has
finite $\mathscr{A}_{e}$-codimension if and only if the double point curve
$D(f)$ is reduced (see [15]). If $f$ has corank one, then we can assume that
is given by $f(x,y)=(x,p(x,y),q(x,y))$, for some function germs $p$ and $q$.
The space $D^{2}(f)$ is defined in $\mathbb{C}^{3}$ by the divided
differences:

$\frac{p(x,y^{\prime})-p(x,y)}{y^{\prime}-y}=\frac{q(x,y^{\prime})-q(x,y)}{y^{\prime}-y}=0,$

and the projection to $\mathbb{C}^{2}$ given by $(x,y,y^{\prime})\mapsto(x,y)$
is precisely the double point curve $D(f)$. As set germs, $D(f)=C(f)\cup
D_{+}(f)$. However when $f$ is a frontal, $D(f)$ has a non reduced equation
$p_{y}^{2}\tau=0$, where $p_{y}=0$ is the equation of $D(f)$ and $\tau=0$ is
the equation of $D_{+}(f)$.

Finally, in Section 5 we consider a frontal stable perturbation $f_{s}$ of
$f$. This always exists and is well defined when $f$ has finite frontal
codimension. We get invariants which count the number of 0-stable
singularities of $f_{s}$ which we call $S,W,K$ and $T$ and correspond to the
number of swallowtails, folded Whitney umbrellas, cuspidal double points and
triple points, respectively. We give algebraic formulas to compute these
invariants in terms of some algebras related to $f$ and prove the frontal
version of the Marar-Mond formulas (see [17]). Moreover, as it happens with
germs of finite $\mathscr{A}_{e}$-codimension, the image of $f_{s}$ has the
homotopy type of a wedge of $2$-spheres and the number of such spheres is
called the frontal Milnor number, denoted by $\mu_{\mathscr{F}}(f)$. This is
analogous of the image Milnor number $\mu_{I}(f)$ defined by Mond in [22] in
the context of finite $\mathscr{A}_{e}$-codimension. We discuss some basic
properties of this frontal Milnor number, and propose a frontal version of
Mond’s conjecture [23], which states that the image Milnor number of
$f\colon(\mathbb{C}^{n},0)\to(\mathbb{C}^{n+1},0)$ is greater than or equal to
its $\mathscr{A}_{e}$-codimension for all values of $n$ such that $(n,n+1)$
are in Mather’s nice dimensions, with equality if and only if $f$ is
quasihomogeneous.

##  2\. Preliminaries

We refer to [24] for further definitions and proofs of the results shown here.
Throughout this article, we shall use the following notation: we set
$\mathscr{O}_{n}$ as the ring of germs of functions on $(\mathbb{C}^{n},S)$,
$\theta_{n}$, as the $\mathscr{O}_{n}$-module of germs of vector fields $\xi$
on $(\mathbb{C}^{n},S)$, and $\mathfrak{m}_{n}$, as the ideal of
$f\in\mathscr{O}_{n}$ such that $f(S)=0$. Furthermore, for
$f\colon(\mathbb{C}^{n},S)\to(\mathbb{C}^{n+1},0)$, we set $\theta(f)$ as the
$\mathscr{O}_{n}$-module of germs of vector fields $\xi$ along $f$, and
$\Sigma(f)$ as the set-germ of non-immersive points of $f$. Unless otherwise
stated, all maps are assumed holomorphic.

We say $f,g\colon(\mathbb{C}^{n},S)\to(\mathbb{C}^{n+1},0)$ are
$\mathscr{A}$-equivalent if there exist germs of diffeomorphisms $\psi$,
$\phi$ such that $g=\psi\circ f\circ\phi^{-1}$. A ($d$-parameter) unfolding of
$f\colon(\mathbb{C}^{n},S)\to(\mathbb{C}^{n+1},0)$ is a holomorphic

${F\colon(\mathbb{C}^{n}\times\mathbb{C}^{d},S\times\\{0\\})}$${(\mathbb{C}^{n+1}\times\mathbb{C}^{d},0)}$${(x,u)}$${(f_{u}(x),u)}$

such that $f_{0}=f$. We say $f$ is $\mathscr{A}$-stable if every $d$-parameter
unfolding $F$ of $f$ is $\mathscr{A}$-equivalent to
$f\times\operatorname{id}_{(\mathbb{C}^{d},0)}$.

######  Definition 2.1.

The $\mathscr{A}$-tangent space of
$f\colon(\mathbb{C}^{n},S)\to(\mathbb{C}^{n+1},0)$ is the $\mathbb{C}$-vector
space

$T\mathscr{A}_{e}f=\left\\{\left.\frac{df_{s}}{ds}\right|_{s=0}:f_{s}=\psi_{s}\circ
f\circ\phi_{s}\right\\}\subseteq\theta(f)$

where $\psi_{s}$ and $\phi_{s}$ are smooth families of diffeomorphisms. We
define the $\mathscr{A}$-codimension of $f$ as the $\mathbb{C}$-codimension of
$T\mathscr{A}_{e}f$ in $\theta(f)$. We shall say that $f$ is
$\mathscr{A}$-finite if $\operatorname{codim}_{\mathscr{A}_{e}}(f)<\infty$.

Using the chain rule, it is then easy to see that

$T\mathscr{A}_{e}f=tf(\theta_{n})+\omega f(\theta_{n+1})$

where $tf\colon\theta_{n}\to\theta(f)$ and $\omega
f\colon\theta_{n+1}\to\theta(f)$ are respectively given by
$tf(\xi)=df\circ\xi$ and $\omega f(\eta)=\omega\circ f$.

######  Theorem 2.1.

A holomorphic $f\colon(\mathbb{C}^{n},S)\to(\mathbb{C}^{n+1},0)$ is
$\mathscr{A}$-stable if and only if
$\operatorname{codim}_{\mathscr{A}_{e}}(f)=0$.

A map-germ $f\colon(\mathbb{C}^{n},S)\to(\mathbb{C}^{n+1},0)$ is finite if
there exists a closed representative $f\colon N\to Z$ with finite fibres.
Finite map-germs are an important tool in analytic geometry, since they
preserve coherent sheaves (see e.g. [9]).

######  Theorem 2.2.

Given a holomorphic $f\colon(\mathbb{C}^{n},S)\to(\mathbb{C}^{n+1},0)$, the
following statements are equivalent:

  1. (1)

$f$ is finite;

  2. (2)

$\mathscr{O}_{n}$ is finitely generated over $(f_{1},\dots,f_{n+1})$;

  3. (3)

$\dim_{\mathbb{C}}\mathscr{O}_{n}/(f_{1},\dots,f_{n+1})<\infty$;

  4. (4)

as germs at $S$, $f^{-1}(\\{0\\})=S$.

We then have the following geometric criterion for $\mathscr{A}$-finiteness:
$f\colon(\mathbb{C}^{2},S)\to(\mathbb{C}^{3},0)$ is $\mathscr{A}$-finite if
and only if it has a finite representative $f\colon N\to Z$ such that $f(N)$
only contains transversal double points outside the origin.

Another important tool which we shall use in §4 are the Fitting ideals of a
map-germ. Let $M$ be a finitely generated module on a Noetherian ring $R$. A
finite presentation on $M$ is an exact sequence in the form

${R^{b}}$${R^{a}}$${M}$${0}$$\scriptstyle{\lambda}$

The matrix $\lambda$ is known as the presentation matrix of $M$. We then
define the $k$-th Fitting ideal of $M$ as the ideal
$\operatorname{Fitt}_{k}(M)$ in $R$ generated by the $(c-k)\times(c-k)$ minors
of $\lambda$, $c=\min\\{a,b\\}$. For $k>c$, we set
$\operatorname{Fitt}_{k}(M)=0$, and for $k<0$, $\operatorname{Fitt}_{k}(M)=R$.
In particular, $\operatorname{Fitt}_{k}(M)$ is independent of the choice of
$\lambda$ ([1]), making this notation unambiguous.

Let $f\colon(X,S)\to(Y,y)$ be a finite holomorphic map-germ between complex
manifolds, $\mathscr{O}_{X}$ be the sheaf of holomorphic functions on $X$, and
$f_{*}\mathscr{O}_{X}$ be the pushforward sheaf on $Y$,

$f_{*}\mathscr{O}_{X}(U)=\mathscr{O}_{X}(f^{-1}(U))$

By Theorem 2.2,
$(f_{*}\mathscr{O}_{X})_{y}=\oplus\\{f_{*}(\mathscr{O}_{X},x)\colon x\in S\\}$
is finitely generated over $\mathscr{O}_{Y,y}$. Since $\mathscr{O}_{Y,y}$ is a
Noetherian ring, we can consider a finite resolution

${\mathscr{O}_{Y,y}^{b}}$${\mathscr{O}_{Y,y}^{a}}$${(f_{*}\mathscr{O}_{X})_{y}}$${0}$$\scriptstyle{\lambda}$

An algorithm to compute the presentation matrix for polynomial maps can be
found at [10].

######  Definition 2.2.

We define the $k$-th Fitting ideal of $f$ at $y$ as the
$\mathscr{O}_{Y,y}$-module

$\mathcal{F}_{k}(f)=\operatorname{Fitt}_{k}((f_{*}\mathscr{O}_{X})_{y})$

and the $k$-th target multiple point space of $f$ as the zero locus $M_{k}(f)$
of $\mathcal{F}_{k-1}(f)$.

Given $p,q\in\mathscr{O}_{n}[z]$, we set the $\mathscr{O}_{n}$-module
$M_{z}(p,q)=\mathscr{O}_{n}[z]/(p,q)$. The $0$th Fitting ideal of this algebra
is generated by the algebraic resultant $\operatorname{Res}_{z}(p,q)$ (see
[31]), and is thus known as the resultant ideal of $p,q$.

##  3\. Frontal map-germs

Roughly speaking, a frontal hypersurface is an analytic hypersurface
$X\subset\mathbb{C}^{n+1}$ that has a well-defined tangent space at each
point. If $X$ has a singularity at a point $x$, we consider a sequence of
regular points $(x_{m})\subseteq X$ converging to $x$.

Let $PT^{*}\mathbb{C}^{n+1}$ be the projectivized cotangent bundle of
$\mathbb{C}^{n+1}$. If $(z,[\omega])\in PT^{*}\mathbb{C}^{n+1}$, we equip
$PT^{*}\mathbb{C}^{n+1}$ with the contact structure given by the differential
form

$\alpha=\omega_{1}\,dz^{1}+\dots+\omega_{n+1}\,dz^{n+1}$

A holomorphic $F\colon N\subset\mathbb{C}^{n}\to PT^{*}\mathbb{C}^{n+1}$ is
integral if $F^{*}\alpha=0$. We also say a projection $\pi\colon
PT^{*}\mathbb{C}^{n+1}\to\mathbb{C}^{n+1}$ is a Legendrian fibration for
$\alpha$ if $\ker d\pi_{(z,[\omega])}\subseteq\ker\alpha_{(z,[\omega])}$ for
all $(z,[\omega])\in PT^{*}\mathbb{C}^{n+1}$.

######  Definition 3.1.

A holomorphic $f\colon N\subset\mathbb{C}^{n}\to\mathbb{C}^{n+1}$ is frontal
if there exists an integral $F\colon N\to PT^{*}\mathbb{C}^{n+1}$ and a
Legendrian projection $\pi\colon PT^{*}\mathbb{C}^{n+1}\to\mathbb{C}^{n+1}$
such that

$f=\pi\circ F$

If $F$ is an immersion, we say $f$ is a wavefront. Similarly, a hypersurface
$X\subset\mathbb{C}^{n+1}$ is frontal (resp. a front) if there exists a
frontal (resp. wavefront) $f\colon N\to\mathbb{C}^{n+1}$ such that $X=f(N)$.

Let $(z,,p)$ be coordinates for $PT^{*}\mathbb{C}^{n+1}$, and $\pi\colon
PT^{*}\mathbb{C}^{n+1}\to\mathbb{C}^{n+1}$ be the canonical projection,
$\pi(z,p)=z$. It is easy to see that $\pi$ is a Legendrian fibration for
$\alpha$. If $F\colon N\to PT^{*}\mathbb{C}^{n+1}$ is an integral map and
$f=\pi\circ F$,

$0=F^{*}\alpha=\sum^{n+1}_{i=1}\nu_{i}d(Z_{i}\circ
F)=\sum^{n+1}_{i=1}\sum^{n}_{j=1}\nu_{i}\frac{\partial f_{i}}{\partial
x_{j}}\,dx^{j}$

for some $\nu_{1},\dots,\nu_{n+1}\in\mathscr{O}_{n}$, not all of them zero
(since they come from a projective space). This is the same as claiming that
there exists a nowhere-vanishing $\nu\colon N\to T^{*}\mathbb{C}^{n+1}$ such
that, for all vector fields $\xi$ on $N$,

$\nu(df\circ\xi)=0$

Since $PT^{*}\mathbb{C}^{n+1}$ is a fibre bundle, we can find for each pair
$(z,[\omega])\in PT^{*}\mathbb{C}^{n+1}$ an open neighbourhood
$Z\subset\mathbb{C}^{n+1}$ of $z$ and an open $U\subseteq\mathbb{C}P^{n+1}$
such that $\pi^{-1}(Z)\cong Z\times U$. Therefore, the integral map $F$ is
$\mathscr{A}$-equivalent to $\tilde{f}(x)=(f(x),[\nu_{x}])$, known as the Nash
lift of $f$.

We shall denote the family of differential $1$-forms $\nu$ along $f$ that
vanish nowhere as $\Omega^{1}_{f}(U)$, and the family of germs of those as
$\Omega^{1}_{f}(\mathbb{C}^{n},0)$.

######  Example 3.1.

  1. (1)

Every analytic plane curve is frontal: given an analytic
$\gamma(x)=(p(x),q(x))$, let
$k=\min\\{\operatorname{ord}p,\operatorname{ord}q\\}$. The vector field
$\gamma^{\prime}$ is orthogonal to the nowhere-vanishing differential form

$\nu_{x}=\frac{1}{x^{k}}(q^{\prime}(x)\,dX-p^{\prime}(x)\,dY)$

  2. (2)

The folded Whitney umbrella can be parametrized as

${f\colon(\mathbb{C}^{2},0)}$${(\mathbb{C}^{3},0)}$${(x,y)}$${(x,y^{2},xy^{3})}$

If $\star\colon\Omega^{2}(f)\to\Omega^{1}(f)$ is the Hodge dual over germs of
differential forms along $f$ and $(X,Y,Z)$ are coordinates for
$\mathbb{C}^{3}$,

$\star\left(\frac{\partial f}{\partial x}\wedge\frac{\partial f}{\partial
y}\right)=-2y^{4}\,dX-3xy^{2}\,dY+2y\,dZ=-y\nu_{(x,y)}$

and thus $\nu_{(x,y)}$ is a germ of $1$-form along $f$ that vanishes nowhere.

######  Example 3.2.

The $F_{4}$ singularity is described by Mond [21] as the map-germ
$f\colon(\mathbb{C}^{2},0)\to(\mathbb{C}^{3},0)$ given by

$f(x,y)=(x,y^{2},y^{5}+x^{3}y)$

We wish to know whether $f$ is a frontal germ.

Let $\nu\colon(\mathbb{C}^{2},0)\to T^{*}\mathbb{C}^{3}$ be a germ of $1$-form
along $f$ such that $\nu(df\circ\xi)=0$ for all $\xi\in\theta_{2}$. Since
$\operatorname{Im}df$ has dimension $2$ almost everywhere, there exists a
$\mu\in\mathscr{O}_{2}$ such that

$\mu(x,y)\nu_{(x,y)}=\star\left(\frac{\partial f}{\partial
x}\wedge\frac{\partial f}{\partial
y}\right)(x,y)=2y^{2}\,dX+(5y^{4}+x^{3})\,dY-2y\,dZ$

Assume there is a $\nu\in\Omega^{1}_{f}(\mathbb{C}^{2})$ such that
$\nu(df\circ\xi)=0$ for all vector fields $\xi$ on $U$. If
$\nu=\nu_{1}\,dX+\nu_{2}\,dY+\nu_{3}\,dZ$,

$\displaystyle\mu\nu_{1}=2y^{2};$ $\displaystyle\mu\nu_{2}=5y^{4}+x^{3};$
$\displaystyle\mu\nu_{3}$ $\displaystyle=-2y$

for some function $\mu$. However, it is clear that $\mu$ is a unit, so $\nu$
vanishes at $0$ and $f$ is not a frontal.

During the rest of this article, we shall assume all frontals are generically
immersive.

######  Definition 3.2.

A smooth multigerm $f\colon(\mathbb{C}^{n},S)\to\mathbb{C}^{n+1}$ is frontal
if it has a frontal representative $f\colon N\to Z$. Given a hypersurface
$X\subset\mathbb{C}^{n+1}$, $(X,0)$ is a frontal if there exists a frontal
$f\colon(\mathbb{C}^{n},S)\to(\mathbb{C}^{n+1},0)$ such that
$(X,0)=f(\mathbb{C}^{n},S)$.

We now give a characterization for corank $1$ frontal map-germs:

######  Proposition 3.1 ([28]).

Let $f\colon(\mathbb{K}^{n},0)\to(\mathbb{K}^{n+1},0)$ be a corank $1$ mono-
germ. Choose local coordinates $(x_{1},\dots,x_{n-1},y)$ in the source and
$(Y_{1},\dots,Y_{n+1})$ in the target such that

(1) $f(x,y)=(x,p(x,y),q(x,y))$

for some $p,q\in\mathscr{O}_{n}$. Then $f$ is frontal if and only if
$p_{y}|q_{y}$ or $q_{y}|p_{y}$.

We shall say that $f$ is in prenormal form if it is given as in Equation 1
with $p_{y}|q_{y}$, in which case we set $\mu=q_{y}/p_{y}$. We now state a
series of definitions and results that we shall use throughout this paper.
Proofs for these statements can be found in [26].

######  Proposition 3.2.

Two frontal multi-germs $f,g\colon(\mathbb{C}^{n},S)\to(\mathbb{C}^{n+1},0)$
are $\mathscr{A}$-equivalent if and only if their Nash lifts
$\tilde{f},\tilde{g}$ are Legendrian equivalent.

######  Corollary 3.3.

Let $f,g\colon(\mathbb{C}^{n},S)\to\mathbb{C}^{n+1}$ be smooth germs. If $f$
is $\mathscr{A}$-equivalent to $g$ and $f$ is frontal, $g$ is frontal.

Given a frontal $f\colon(\mathbb{C}^{n},S)\to(\mathbb{C}^{n+1},0)$, we define
the space of infinitesimal frontal deformations of $f$ as

$\mathscr{F}(f):=\left\\{\left.\frac{df_{t}}{dt}\right|_{t=0}:f_{0}=f,(f_{t},t)\text{
frontal}\right\\}$

Using Corollary 3.3, we see that the $\mathscr{A}$-orbit of $f$ is contained
within the space of frontal mappings
$(\mathbb{C}^{n},S)\to(\mathbb{C}^{n+1},0)$. It is then clear that
$T\mathscr{A}_{e}f\subseteq\mathscr{F}(f)$, and thus we can assign a frontal
codimension to $f$ by computing the codimension of $T\mathscr{A}_{e}f$ in
$\mathscr{F}(f)$:

######  Definition 3.3.

Let $f\colon(\mathbb{C}^{n},S)\to(\mathbb{C}^{n+1},0)$ be a frontal germ. We
define the frontal codimension or $\mathscr{F}$-codimension of $f$ as

$\operatorname{codim}_{\mathscr{F}e}(f):=\dim_{\mathbb{C}}\frac{\mathscr{F}(f)}{T\mathscr{A}_{e}f}$

We say $f$ is $\mathscr{F}$-finite if
$\operatorname{codim}_{\mathscr{F}_{e}}(f)<\infty$.

Given a smooth $f\colon(\mathbb{C}^{n},S)\to(\mathbb{C}^{n+1},0)$ (not
necessarily frontal), we shall say $f$ has corank $1$ if there exists a
representative $f\colon N\to Z$ of $f$ such that the dimension of $\ker
df_{x}$ is less than or equal to $1$ for all $x\in S$. Note that this
definition includes germs with immersive branches, such as the transverse
intersection of $n$ hyperplanes.

######  Definition 3.4.

Let $f\colon(\mathbb{C}^{n},S)\to(\mathbb{C}^{n+1},0)$ be a frontal multi-
germ. We say $f$ is stable as a frontal or $\mathscr{F}$-stable if every
$d$-parameter frontal unfolding $F$ of $f$ is $\mathscr{A}$-equivalent to
$f\times\operatorname{id}_{(\mathbb{C}^{d},0)}$.

######  Theorem 3.4.

Let $f\colon(\mathbb{C}^{n},0)\to(\mathbb{C}^{n+1},0)$ be a generically
immersive frontal germ of corank $1$. Then $f$ is stable as a frontal if and
only if $\mathscr{F}(f)=T\mathscr{A}_{e}f$.

Given a frontal $f\colon(\mathbb{C}^{n},S)\to(\mathbb{C}^{n+1},0)$, we define
the vector subspace $\hat{\tau}(f)\subseteq T_{0}\mathbb{C}^{n+1}$ as follows:

$\hat{\tau}(f)=\mathrm{ev}\left\\{\omega
f^{-1}[tf(\theta_{n,S})+(f^{*}\mathfrak{m}_{n+1,0})\mathscr{F}(f)]\right\\}$

where $\mathrm{ev}\colon\theta_{n+1}\to T_{0}\mathbb{C}^{n+1}$ denotes
evaluation at $0$.

######  Proposition 3.5.

A frontal multi-germ $f\colon(\mathbb{C}^{n},S)\to(\mathbb{C}^{n+1},0)$ with
branches $f_{1},\dots,f_{r}$ is stable as a frontal if and only if
$f_{1},\dots,f_{r}$ are stable as frontals and the vector subspaces
$\hat{\tau}(f_{1}),\dots,\hat{\tau}(f_{r})\subseteq T_{0}\mathbb{K}^{n+1}$
meet in general position.

We now state an adaptation of the Mather-Gaffney criterion for frontal germs:

######  Theorem 3.6.

Let $f\colon(\mathbb{C}^{n},S)\to(\mathbb{C}^{n+1},0)$ be a corank $1$ frontal
map-germ. If $n=1$ or $V(p_{y},\mu_{y})=S$ as set-germs, $f$ is
$\mathscr{F}$-finite if and only if there exists a representative $f\colon
N^{\prime}\to Z^{\prime}$ of $f$ such that

  1. (1)

$f^{-1}(0)\subseteq S$;

  2. (2)

the restriction $f\colon N^{\prime}\backslash\\{0\\}\to
Z^{\prime}\backslash\\{0\\}$ is locally stable as a frontal.

In particular, if $f$ is a surface, we will later show (Proposition 4.7) that
$f$ can only contain cuspidal edges and transversal double points outside $S$.

###  3.1. Frontalisation of a fold surface

Mond classified the $\mathscr{A}$-simple monogerms
$(\mathbb{R}^{2},0)\to(\mathbb{R}^{3},0)$ in [21]. The resulting
classification shows that all but one of the families (labelled $H_{k}$) is
fold-type. A surface $f\colon N^{2}\to Z^{3}$ is fold-type if we can find
local coordinates $(x,y)$ for $N$ and $(X,Y,Z)$ for $Z$ such that

$f(x,y)=(x,y^{2},yp(x,y^{2}))$

for some $p\colon N\to\mathbb{R}$.

Using the characterisation of corank $1$ frontal maps given in Proposition
3.1, it follows that $f$ is frontal if and only if $p(x,y)=yq(x,y)$, in which
case $f(x,y)=(x,y^{2},y^{3}q(x,y^{2}))$. With this observation in mind, we can
_frontalise_ the fold families from Mond’s classification, leading to the
results shown in Table 3.5 and Figure 1.

Mond’s classification | Frontalised surface | Codimension | Notes
---|---|---|---
$\rule{0.0pt}{11.19443pt}S_{k}$ | $y^{3}+x^{k+1}y$ | $\rule[-5.16663pt]{0.0pt}{0.0pt}\check{S}_{k}$ | $y^{5}+x^{k+1}y^{3}$ | $k$ |
$\rule{0.0pt}{11.19443pt}B_{k}$ | $x^{2}y+y^{2k+1}$ | $\rule[-5.16663pt]{0.0pt}{0.0pt}\check{B}_{k}$ | $x^{2}y^{3}+y^{2k+3}$ | $k$ | $k\geq 2$
$\rule{0.0pt}{11.19443pt}C_{k}$ | $xy^{3}+x^{k}y$ | $\rule[-5.16663pt]{0.0pt}{0.0pt}\check{C}_{k}$ | $xy^{5}+x^{k}y^{3}$ | $k$ | $k\geq 3$
$\rule{0.0pt}{11.19443pt}F_{4}$ | $y^{5}+x^{3}y$ | $\rule[-5.16663pt]{0.0pt}{0.0pt}\check{F}_{4}$ | $y^{7}+x^{3}y^{3}$ | $4$ |
Table 1. Simple fold surfaces from [21] along with their frontal counterparts.
The $\mathscr{A}$-codimension of the former coincides with the
$\mathscr{F}$-codimension of the latter.

(a) $S_{k}$, $B_{k}$, $C_{k}$, $F_{4}$.

(b) $\check{S}_{k}$, $\check{B}_{k}$, $\check{C}_{k}$, $\check{F}_{4}$.

Figure 1. Fold surfaces from Mond’s classification (top), along with their
frontalised counterpart (bottom).

We now define a precise definition of frontalisation.

######  Definition 3.5.

Let $f\colon(\mathbb{C}^{2},S)\to(\mathbb{C}^{3},0)$ be given by
$f(x,y)=(x,y^{2},yp(x,y^{2}))$. We define the frontalisation of $f$ as the
fold surface $\check{f}\colon(\mathbb{C}^{2},S)\to(\mathbb{C}^{3},0)$ given by

$\check{f}(x,y)=(x,y^{2},y^{3}p(x,y^{2}))$

Note that none of the frontal surfaces in Table is a wavefront. More
generally, if $\check{f}$ is a wavefront, the function

$\frac{\partial}{\partial
y}\left(y^{3}p(x,y^{2})\right)=3y^{2}p(x,y^{2})+2y^{4}p_{y}(x,y^{2})$

has to be in the form $\alpha(x)+\beta y^{2}$ for some
$\alpha\in\mathscr{O}_{1}$, $\beta\in\mathbb{C}$ (so that its Nash lift is an
immersion). However, this can only happen if $p_{y}(x,y)=0$, in which case
$\check{f}$ is either an immersion or a cuspidal edge.

The goal of this section is to prove the following

######  Theorem 3.7.

Given a fold surface $f\colon(\mathbb{C}^{2},0)\to(\mathbb{C}^{3},0)$ with
frontalisation $\check{f}$,

$\frac{\theta(f)}{T\mathscr{A}_{e}f}\cong\frac{\mathscr{F}(\check{f})}{T\mathscr{A}_{e}\check{f}}$

In particular, $f$ is $\mathscr{A}$-finite if and only if $\check{f}$ is
$\mathscr{F}$-finite and
$\operatorname{codim}_{\mathscr{A}}(f)=\operatorname{codim}_{\mathscr{F}}(\check{f})$.

In order to prove this result, we consider a $\mathbb{C}$-linear mapping
$K\colon\theta(f)\to\mathscr{F}(\check{f})$ that frontalises the infinitesimal
deformations of $f$. This map will be an epimorphism by construction and will
send $T\mathscr{A}_{e}f$ onto $T\mathscr{A}_{e}\check{f}$, thus inducing the
desired isomorphism between the quotient spaces.

Let $i\colon(\mathbb{C},0)\to(\mathbb{C}^{2},0)$ be the germ of immersion
given by $i(x)=(x,0)$. If $\xi=(\xi_{1},\xi_{2},\xi_{3})\in\theta(f)$, we can
find a unique $\hat{\xi}_{3}$ such that
$\xi=(\xi_{1},\xi_{2},i^{*}\xi_{3})+(0,0,y\hat{\xi}_{3})$, thus yielding the
following decomposition of $\theta(f)$:

$\theta(f)=(\mathscr{O}_{2}^{2}\times
i^{*}\mathscr{O}_{2})\oplus\mathscr{O}_{2}\\{(0,0,y)\\}$

We then define the linear map $K$ as follows:

  1. (1)

if $\xi\in\mathscr{O}_{2}^{2}\times i^{*}\mathscr{O}_{2}$, $K(\xi)=\xi$;

  2. (2)

if $\xi\in\mathscr{O}_{2}\\{(0,0,y)\\}$, $K(\xi)=y^{2}\xi$.

As shown in [26], $\xi\in\mathscr{F}(\check{f})$ if and only if there exist
$\alpha,\beta\in\mathscr{O}_{2}$ such that $d\xi_{3}=\alpha\,dx+y\beta\,dy$.
It then follows that

$\mathscr{F}(\check{f})\cong\frac{\theta(\check{f})}{\mathscr{O}_{2}\\{(0,0,y)\\}}$

and $K$ is a $\mathbb{C}$-linear epimorphism.

######  Lemma 3.8.

Given a fold surface $f$, $K(T\mathscr{A}_{e}f)=T\mathscr{A}_{e}\check{f}$.

Throughout the proof of this lemma, we shall use the following notation,
borrowed from [21]:

  1. (1)

$\mathscr{O}_{2}^{T}:=\\{g(x,y^{2}):g\in\mathscr{O}_{2}\\}$;

  2. (2)

$\mathfrak{m}_{2}^{T}:=\\{g(x,y^{2}):g\in\mathfrak{m}_{2}\\}=\mathscr{O}_{2}^{T}\cap\mathfrak{m}_{2}$;

  3. (3)

if $h(x,y)=(x,y^{2})$, $T_{e}\mathscr{K}^{T}(p\circ
h):=h^{*}(p_{x},yp_{y},p)$.

###### Proof of Lemma 3.8.

Let $f(x,y)=(x,y^{2},yp(x,y^{2}))$, and $q(x,y)=yp(x,y)$. By definition of
$T_{e}\mathscr{K}^{T}$, we have

$\displaystyle T_{e}\mathscr{K}^{T}(q\circ h)$
$\displaystyle=h^{*}(q_{x},yq_{y},q)=h^{*}(yp_{x},y(p+yp_{y}),yp)=$
$\displaystyle=y^{2}h^{*}(p_{x},yp_{y},p)=y^{2}T_{e}\mathscr{K}^{T}(p\circ h)$

Using [21], we then have

$\displaystyle T\mathscr{A}_{e}f=$
$\displaystyle\mathscr{O}_{2}\frac{\partial}{\partial
X}\oplus\mathscr{O}_{2}\frac{\partial}{\partial
Y}\oplus(\mathscr{O}_{2}^{T}+yT_{e}\mathscr{K}^{T}(p\circ
h))\frac{\partial}{\partial Z};$ $\displaystyle T\mathscr{A}_{e}\check{f}=$
$\displaystyle\mathscr{O}_{2}\frac{\partial}{\partial
X}\oplus\mathscr{O}_{2}\frac{\partial}{\partial
Y}\oplus(\mathscr{O}_{2}^{T}+yT_{e}\mathscr{K}^{T}(q\circ
h))\frac{\partial}{\partial Z}$

Given $\xi\in T\mathscr{A}_{e}f$, there exist
$\xi_{1},\xi_{2}\in\mathscr{O}_{2}$, $\xi_{3}\in\mathscr{O}_{2}^{T}$ and
$\xi_{4}\in T_{e}\mathscr{K}^{T}(p\circ h)$ such that

$\xi=\xi_{1}\frac{\partial}{\partial X}+\xi_{2}\frac{\partial}{\partial
Y}+(\xi_{3}+y\xi_{4})\frac{\partial}{\partial Z}$

In addition, there exists a unique $\hat{\xi_{3}}\in\mathscr{O}_{2}$ such that
$\xi_{3}=i^{*}\xi_{3}+y\hat{\xi_{3}}$. We then have

$K(\xi)=\xi_{1}\frac{\partial}{\partial X}+\xi_{2}\frac{\partial}{\partial
Y}+(i^{*}\xi_{3}+y^{3}\hat{\xi_{3}}+y^{3}\xi_{4})\frac{\partial}{\partial
Z}\in T\mathscr{A}_{e}\check{f}$

and thus $K(T\mathscr{A}_{e}f)\subseteq T\mathscr{A}_{e}\check{f}$.

Conversely, let $\eta\in T\mathscr{A}_{e}\check{f}$: there exist
$\eta_{1},\eta_{2}\in\mathscr{O}_{2}$, $\eta_{3}\in\mathscr{O}_{2}^{T}$ and
$\eta_{4}\in T_{e}\mathscr{K}^{T}(p\circ h)$ such that

$\eta=\eta_{1}\frac{\partial}{\partial X}+\eta_{2}\frac{\partial}{\partial
Y}+(\eta_{3}+y^{3}\eta_{4})\frac{\partial}{\partial Z}$

In addition, there exists a unique $\hat{\eta_{3}}\in\mathscr{O}_{2}$ such
that $\eta_{3}=i^{*}\eta_{3}+y^{2}\hat{\eta_{3}}$. We then have

$\eta=K\left(\eta_{1}\frac{\partial}{\partial
X}+\eta_{2}\frac{\partial}{\partial
Y}+(i^{*}\eta_{3}+\hat{\xi_{3}}+y\xi_{4})\frac{\partial}{\partial Z}\right)\in
K(T\mathscr{A}_{e}f)$

proving the opposite inclusion. ∎

Since $K\colon\theta(f)\to\mathscr{F}(\check{f})$ is surjective and
$K(T\mathscr{A}_{e}f)=T\mathscr{A}_{e}\check{f}$, the induced map

$\hat{K}\colon\frac{\theta(f)}{T\mathscr{A}_{e}f}\to\frac{\mathscr{F}(\check{f})}{T\mathscr{A}_{e}\check{f}}$

is an isomorphism, thus proving Theorem 3.7.

##  4\. Double point curve of a frontal surface

Marar and Tari [16] studied the geometric invariants of corank $1$ smooth maps
$F\colon(\mathbb{R}^{3},0)\to(\mathbb{R}^{3},0)$. One of their findings was
that the source double point space (see §5.2 below) of the discriminant of $F$
is given by an equation in the form $dc^{2}=0$, where
$c,d\colon(\mathbb{R}^{2},0)\to\mathbb{R}$ are the equations for the cuspidal
edge and transverse double point curves in the discriminant of $F$. In
particular, the discriminant of $F$ is a front if and only if $\Sigma(F)$ is
smooth in $(\mathbb{R}^{3},0)$ (see [2]). Therefore, this formula can be also
applied to $f=F|_{\Sigma(F)}$, seen as a wavefront.

In this section, we show that this formula holds for any frontal, not just
fronts.

###  4.1. Pre-normal form of a corank $1$ injective map

Let $f\colon(\mathbb{C}^{2},0)\to(\mathbb{C}^{3},0)$ be a corank $1$ frontal
map-germ in prenormal form. We define $D^{2}(f)$ as the space in
$(\mathbb{C}^{3},0)$ given by the equations

$\frac{p(x,y)-p(x,y^{\prime})}{y-y^{\prime}}=\frac{q(x,y)-q(x,y^{\prime})}{y-y^{\prime}}=0$

We then have $D^{2}(f)=C^{2}(f)\cup D_{+}^{2}(f)$, where

$\displaystyle C^{2}(f)=\\{(x,y,y):(x,y)\in\Sigma(f)\\};$ $\displaystyle
D_{+}^{2}(f)=\overline{\\{(x,y,y^{\prime})\in D^{2}(f):y\neq y^{\prime}\\}}$

Note that $D_{+}^{2}(f)$ coincides with the double point curve for
$\mathscr{A}$-finite surfaces.

Let us first assume $f$ is injective (hence finite by the Nullstellensatz) and
let $\pi\colon D^{2}(f)\to(\mathbb{C}^{2},0)$ be given by
$(x,y,y^{\prime})\mapsto(x,y)$. Since $f$ is finite, $\pi$ is also finite and
$D(f)=\pi(D^{2}(f))$ is an analytic set by Remmert’s Proper Mapping Theorem
([13]).

######  Definition 4.1.

We define the double point space of $f$ as $D(f)$.

Since $\pi\colon D^{2}(f)\to\mathbb{C}^{2}$ is a finite germ,
$\mathscr{O}_{D^{2}(f)}$ admits a finite presentation in the form

${\mathscr{O}_{2}^{r}}$${\mathscr{O}_{2}^{s}}$${\mathscr{O}_{D^{2}(f)}}$${0}$$\scriptstyle{\lambda}$

Since $f$ is assumed to be injective, $D(f)=\Sigma(f)$. Let
$\Delta(f)=f(D(f))$. Since $f$ is complex analytic, $\Delta(f)$ has
codimension $1$, so its dimension must be exactly $1$. We can then choose a
generic plane $H_{0}\subset\mathbb{C}^{3}$ passing through $0$ such that
$H_{0}\cap\Sigma=\\{0\\}$ and $f$ is transverse to $H_{0}$ at $0$.

If $(X,Y,Z)$ are coordinates for $(\mathbb{C}^{3},0)$, we can assume $H_{0}$
is the plane of equation $X=0$, so a suitable change of coordinates transforms
$f$ into a mapping of the form

$f(x,y)=(x,\tilde{p}(x,y),\tilde{q}(x,y))$

where $\tilde{p},\tilde{q}$ have order greater than $1$. However, since
$D^{2}(f)$ is preserved by diffeomorphisms, we can assume $\tilde{p}=p$ and
$\tilde{q}=q$. We then define $L_{0}=f^{-1}(H_{0})$, which is the line of
equation $x=0$.

######  Lemma 4.1.

Suppose that we have a curve $(X,0)$ and a hypersurface $(Y,0)$ in
$(\mathbb{C}^{n},0)$ such that $X\cap Y=\\{0\\}$. Then, $i_{0}(X,Y)=1$ if and
only if $(X,0)$, $(Y,0)$ are smooth submanifold-germs of $(\mathbb{C}^{n},0)$
and $X\pitchfork Y$.

###### Proof.

Let $(X_{1},0),\dots,(X_{r},0)$ be the irreducible components of $(X,0)$.
Since the local intersection number is additive,

$i_{0}(X,Y)=\sum^{r}_{i=1}i_{0}(X_{i},Y)$

For each $i=1,\dots,r$, we can consider a holomorphic
$\gamma_{i}\colon(\mathbb{C},0)\to(\mathbb{C}^{n},0)$ which is finite,
generically $1$-to-$1$ and whose image is $(X_{i},0)$. If
$g\colon(\mathbb{C}^{n},0)\to(\mathbb{C},0)$ is the reduced equation for
$(Y,0)$, $g\circ\gamma_{i}$ is not constant, so we can write

$(g\circ\gamma_{i})(u)=a_{k}u^{k}+a_{k+1}u^{k+1}+\dots$

with $k=\operatorname{ord}_{0}(g\circ\gamma_{i})$. For $t\neq 0$ and close to
the origin, $t$ is a regular value of $g\circ\gamma_{i}$ and has exactly $k$
preimages in a neighbourhood of $0$. Therefore,
$i_{0}(X_{i},Y)=\operatorname{ord}_{0}(g\circ\gamma_{i})$.

Suppose now that $i_{0}(X,Y)=1$. Necessarily $r=1$ and
$\operatorname{ord}(g\circ\gamma_{1})=1$, so $g\circ\gamma_{1}$ is a
diffeomorphism. This means $\gamma_{1}$ is an immersion and $g$ is a
submersion. Thus, $(X,0)$ and $(Y,0)$ are both smooth and $X\pitchfork Y$. ∎

######  Lemma 4.2.

With the above notation we have:

  1. (1)

$\Delta(f)$ is smooth and transverse to $H_{0}$ at the origin.

  2. (2)

$D(f)$ is smooth and transverse to $L_{0}$ at the origin.

###### Proof.

Let $H_{t}$ be the plane in $\mathbb{C}^{3}$ given by $x=t$. We consider a
small enough representative of the form $f\colon U\to T\times V$, where
$U,T\subset\mathbb{C}$ and $V\subset\mathbb{C}^{2}$ are open neigbourhoods of
the origin, such that

  1. (1)

$f$ is transverse to $H_{t}$ for all $t\in T$,

  2. (2)

$H_{0}\cap\Delta(f)=\\{0\\}$,

  3. (3)

$H_{t}$ is transverse to $\Sigma$ for all $t\neq 0$.

For each $t\in T$, we define $Y_{t}$ as the analytic subset of $V$ such that
$X\cap H_{t}=\\{t\\}\times Y_{t}$. Then $Y_{t}$ can be parametrised as the
curve $\gamma_{t}\colon U_{t}\to V$ given by $\gamma_{t}(y)=(p(t,y),q(t,y))$,
where $U_{t}=\\{y\in\mathbb{C}\colon(t,y)\in U\\}$.

Let $t\in T\backslash\\{0\\}$ and $x_{1},\dots,x_{m}$ be the singular points
of $Y_{t}$. By the conservation of the delta invariant, we have

$\delta(Y_{0},0)=\sum^{m}_{i=1}\delta(Y_{t},x_{i})$

Since $f$ is injective, $\gamma_{0}$ and $\gamma_{t}$ are also injective.
Thus, $Y_{0}$ is irreducible at $0$ and $Y_{t}$ is also irreducible at each
$x_{i}$. By Milnor’s formula, we get

$\mu(Y_{0},0)=\sum^{m}_{i=1}\mu(Y_{i},x_{i})$

A theorem due independently to Gabrièlov, Lazzeri and Lê[7, 12, 32] states
that, in these condition, $m=1$.

On the other hand, the number $m$ is equal to the local intersection number
$i_{0}(\Sigma;H_{0})$. Since $H_{0}$ meets $\Delta(f)$ transversally, it
follows from Lemma 4.1 that $\Delta(f)$ is smooth and transverse to $H_{0}$ at
$0$.

The second item is a consequence of the first one. ∎

######  Theorem 4.3 (Pre-normal form of a corank $1$ injective map).

Let $f\colon(\mathbb{C}^{2},0)\to(\mathbb{C}^{3},0)$ be an injective
holomorphic map-germ of corank $1$. Then $f$ is a frontal and is
$\mathscr{A}$-equivalent to a map-germ in the form

$(x,y)\mapsto(x,y^{m},y^{m}h(x,y))$

for some $h\in\mathscr{O}_{2}$ and $m\geq 2$.

###### Proof.

Let $\mu\in\mathscr{O}_{2}$ such that $\mu=0$ is the reduced equation of
$D(f)$. By Lemma 4.2, $D(f)$ is smooth and transverse to $L_{0}$, so $\mu_{y}$
does not vanish at the origin. We consider the diffeomorphism
$\phi(x,y)=(x,\mu(x,y))$. We have that the set of non-immersive points of
$f\circ\phi^{-1}$ is the line $y=0$.

Assume now that $f(x,y)=(x,p(x,y),q(x,y))$ for some
$p,q\in\mathfrak{m}^{2}_{2}$. If

$\displaystyle p(x,y)=p_{0}(x)+yp_{1}(x,y);$ $\displaystyle
q(x,y)=q_{0}(x)+yq_{1}(x,y)$

we define the diffeomorphism $\psi(X,Y,Z)=(X,Y-p_{0}(X),Z-q_{0}(X))$. Then,

$(\psi\circ f)(x,y)=(x,yp_{1}(x,y),yq_{1}(x,y))$

Using the notation from Lemma 4.2, $Y_{t}$ is parametrised as the curve
$\gamma_{t}(y)=(p_{t}(y),q_{t}(y))$, where $p_{t}(y)=p(t,y)$ and
$q_{t}(y)=q(t,y)$. Then $Y_{t}$ has a unique singular point at
$\gamma_{t}(0)=0$ and $\mu(Y_{t},0)=\mu(Y_{0},0)$ for all $t\in T$. By a
result of Zariski [35], $\\{Y_{t}\\}$ also has constant multiplicity at the
origin.

For $t=0$, $\gamma_{0}$ is injective, so $p_{0}$ and $q_{0}$ are not
identically $0$. Assume then that $m=\operatorname{ord}_{0}(p_{0})$ and
$k=\operatorname{ord}_{0}(q_{0})$. This implies that $m=m(Y_{t},0)$ for all
$t\in T$.

On the other hand, by the Weierstrass Preparation Theorem, we can write
$p(x,y)=r(x,y)y^{m}$ and $q(x,y)=y^{m}h(x,y)$ as for some unit
$r\in\mathscr{O}_{2}$ and $s\in\mathscr{O}_{2}$. In conclusion,

$f(x,y)\sim_{\mathscr{A}}(x,y^{m},y^{m}h(x,y))$

as claimed. ∎

We now move onto the generically injective case.

######  Proposition 4.4.

Let $f\colon(\mathbb{C}^{2},0)\to(\mathbb{C}^{3},0)$ be a holomorphic map-germ
of corank 1. If $f$ is generically injective, there exists a representative
$f\colon N\to Z$ of $f$ such that $f$ is locally injective outside $0$.

###### Proof.

Since $f$ is generically injective, $D^{2}(f)$ is an analytic subset of
dimension $1$. Therefore, $D^{2}(f)$ admits a decomposition into irreducible
components in the form

$\displaystyle C^{2}(f)=C_{1}\cup\dots\cup C_{r};$ $\displaystyle
D_{+}^{2}(f)=D_{1}\cup\dots\cup D_{s}$

Note that every two sets in the family

$\mathscr{I}=\\{C_{1},\dots,C_{r},D_{1},\dots,D_{s},D^{2}(f)\cap\Delta\\}$

are different, so we can use the Curve Selection Lemma (see [4]) to choose
closed representatives which only meet at the origin (see Figure 2).

$D^{2}(f)$$\Delta$$C_{i}$$D_{j}$

Figure 2.  The set of points where two given curves in $\mathscr{I}$ meet
cannot accumulate into the origin. Therefore, we can choose a small enough
neighbourhood such that no two sets meet outside the origin.

Let $(x,y,y^{\prime})\in D^{2}(f)\backslash\\{(0,0,0)\\}$.

  1. (1)

If $y\neq y^{\prime}$, $(x,y,y^{\prime})\in D_{j}$ for some $1\leq j\leq s$.
We can find an open neighbourhood $V\subset\mathbb{C}^{3}$ of
$(x,y,y^{\prime})$ such that $V\cap D^{2}(f)=V\cap D_{j}$. Since $V$ does not
meet $C_{1}\cup\dots\cup C_{r}$, $f$ is immersive on $\pi(V)$ and thus locally
injective.

  2. (2)

If $y=y^{\prime}$, $(x,y,y)\in C_{i}$ for some $1\leq i\leq r$. We can find an
open neighbourhood $U\subset\mathbb{C}^{3}$ such that $U\cap D^{2}(f)=U\cap
C_{i}$.

Let $U_{1},U_{2}\subset\mathbb{C}$ be respective open neighbourhoods of $x$
and $y$ such that $U_{1}\times U_{2}\times U_{2}\subset U$, and
$U_{0}=U_{1}\times U_{2}$. If there exist
$(x^{\prime},y^{\prime}),(x^{\prime},z^{\prime})\in U_{0}$ such that
$f(x^{\prime},y^{\prime})=f(x^{\prime},z^{\prime})$, the point
$(x^{\prime},y^{\prime},z^{\prime})$ is in $U\cap D^{2}(f)$. However, we have
by construction that

$U\cap D^{2}(f)=U\cap C_{i}$

so $(x^{\prime},y^{\prime},z^{\prime})\in C_{i}$ and $y^{\prime}=z^{\prime}$.
Therefore, $f|_{U_{0}}$ is injective.

∎

###  4.2. Branches of the double point space

Our goal is to prove that, if $f$ is frontal and $D(f)$ is generated by
$\lambda\in\mathscr{O}_{2}$, then $p_{y}^{2}$ divides $\lambda$. In order to
do so, we shall consider the slices of $f$, $\gamma_{t}(y)=(p(t,y),q(t,y))$.

Let $\gamma\colon(\mathbb{C},0)\to(\mathbb{C}^{2},0)$ be a plane curve with
isolated singularity at $0$. We define

$D^{2}(\gamma)=\left\\{(t,s)\in\mathbb{C}^{2}\colon\frac{\gamma(t)-\gamma(s)}{t-s}=0\right\\}$

Once again, $D^{2}(\gamma)$ is an analytic set. Since $\gamma$ has an isolated
singularity, it is finite, so the projection-germ $\pi^{\prime}\colon
D^{2}(\gamma)\to(\mathbb{C},0)$ given by $\pi^{\prime}(t,s)=t$ is also finite
and $D(\gamma)=\pi^{\prime}(D^{2}(\gamma))$ is an analytic set by Remmert’s
Proper Mapping Theorem. We consider $D(\gamma)$ with the analytic structure
given by the Fitting ideals.

######  Lemma 4.5.

Let $\gamma\colon(\mathbb{C},0)\to(\mathbb{C}^{2},0)$ be a plane curve in the
form $\gamma(s)=(s^{m},q(s))$ with $q\in\mathfrak{m}_{1}$. If $\gamma$ has an
isolated singularity at $0$ and $\mu$ is the Milnor number of $\gamma$,
$D(\gamma)$ is the zero locus of the function $d(s)=s^{\mu}$.

###### Proof.

Since $\gamma$ has an isolated singularity, it is $\mathscr{A}$-finite and we
can find a stabilisation $\gamma_{t}$. Since $\gamma_{t}$ is stable, it only
contains $\delta$ transversal double points. Each double point along
$\gamma_{t}$ has two preimages, so $|D(\gamma_{t})|=2\delta$ and the
generating function $d_{t}$ of $D(\gamma_{t})$ has degree $2\delta$.
Therefore, we can write

$d_{t}(s)=a_{2\delta}(t)s^{2\delta}+a_{2\delta+1}(t)s^{2\delta+1}+\dots=a_{2\delta}(t)s^{2\delta}(1+R(s,t))$

where $a_{2\delta}(t)\neq 0$. In particular, for $t=0$, we have

$d_{0}(s)=a_{2\delta}(0)s^{2\delta}(1+R(s,0))$

The statement then follows from the fact that $\gamma(\mathbb{C},0)$ is
irreducible, hence $\mu=2\delta$ by Milnor’s formula (see for instance [31]).
∎

Consider the projections
${\pi_{1}\colon
D^{2}(f)}$${\mathbb{C}^{2}}$${(x,y,y^{\prime})}$${(y,y^{\prime})}$
${\pi_{1}^{\prime}\colon D(f)}$${\mathbb{C}}$${(x,y)}$${y}$
These mappings verify that $\pi_{1}(D^{2}(f))=D^{2}(\gamma_{t})$ and
$\pi_{1}^{\prime}(D(f))=D(\gamma_{t})$, so we can write
$D^{2}(f)=D^{2}(\gamma_{t})\times_{D(\gamma_{t})}D(f)$. We can now apply [24]
Proposition 11.6 to deduce that

$D(f)=M_{1}(\pi)=(\pi^{\prime}_{1})^{-1}(M_{1}(\pi^{\prime}))=(\pi^{\prime}_{1})^{-1}(D(\gamma_{t}))$

as complex space-germs. Since $\pi^{\prime}_{1}$ is a projection map,
$D(\gamma_{t})=\pi^{\prime}_{1}(D(f))$ as complex space-germs. If $D(f)$ is
generated by $\lambda$ and $D(\gamma_{t})$ is generated by $\kappa_{t}$,

(2) $\lambda=(\pi^{\prime}_{1})^{*}\kappa_{t}$

######  Theorem 4.6.

Let $f\colon(\mathbb{C}^{2},0)\to(\mathbb{C}^{3},0)$ be a generically
injective, corank $1$ frontal map-germ. If $f$ is given in prenormal form, the
generating function of $D(f)$ is given by $\lambda=\tau p_{y}^{2}$ for some
$\tau\in\mathscr{O}_{2}$.

######  Definition 4.2.

We define the cuspidal edge and transverse double point sets of $f$ as

$\displaystyle C(f)=V(p_{y});$ $\displaystyle D_{+}(f)=V(\tau)$

To prove this result, we need to give a lower bound for the Milnor number of a
complex, irreducible plane curve $(Y,0)$. By [6], we can find an $\alpha>0$
and a $h\in\mathscr{O}_{1}$ with $\operatorname{ord}_{0}(h)>\alpha$ such that
$(Y,0)$ is the image of the curve-germ
$\gamma\colon(\mathbb{C},0)\to(\mathbb{C}^{2},0)$ given by
$\gamma(t)=(t^{\alpha},h(t))$.

######  Remark 4.1.

With the notation above, let $D_{1}=\alpha$ and $D_{j+1}$ be the greatest
common divisor of $D_{j}$ and $\beta_{j}$. We have the following identity for
$\mu(Y,0)$ due to Milnor ([20] Remark 10.10):

$\mu(Y,0)=\sum_{j\geq 1}(\beta_{j}-1)(D_{j}-D_{j+1})$

It is easy to see that, for $\alpha>1$, $\mu(Y,0)\geq 2(\alpha-1)$.

###### Proof of Theorem 4.6.

Let $f\colon N\to Z$ be a representative of $f$ and $f^{\prime}$ the germ of
$f$ at $(x_{0},y_{0})\in N\backslash\\{(0,0)\\}$. By Theorem 4.3, $f^{\prime}$
is $\mathscr{A}$-equivalent to a germ in the form

$g(x,y)=(x,y^{m},y^{m}h(x,y))$

where $m\geq 2$ and $h\in\mathcal{O}_{2}$. The slices of $g$ are then given by
the curves $\gamma_{t}(y)=(y^{m},y^{m}h(t,y))$. If $\sigma$ and $\kappa_{t}$
are the generating functions for $D(g)$ and $D(\gamma_{t})$, we can apply
Equation (2) to obtain $\sigma=(\pi^{\prime}_{1})^{*}\kappa_{t}$. By Lemma
4.5, $\kappa_{t}(y)=y^{\mu}$, where $\mu$ is the Milnor number of
$\gamma_{0}$. Using Remark 4.1,

$\sigma(t,y)=y^{\mu}\implies y^{2(m-1)}|\sigma(t,y)$

Throughout the proof of Theorem 4.3, we see that
$p(x,y)-p_{0}(x)=y^{m}r(x,y)$, where $p_{0}(x)=p(x,y_{0})$ and
$r\in\mathcal{O}_{2}$ is a unit. Taking derivatives,

$p_{y}(x,y)=y^{m-1}(yr_{y}(x,y)+mr(x,y))=S(x,y)y^{m-1}$

where $S\in\mathscr{O}_{2}$ is again a unit. Since $g$ is
$\mathscr{A}$-equivalent to $f^{\prime}$, the germ of $p(x,y)$ at
$(x_{0},y_{0})$ is $\mathscr{A}$-equivalent to $y^{m}$. If $\eta$ is the
generating function for $D(f^{\prime})$,

$\eta(x,y)=r(x,y)y^{2(m-1)}=r(x,y)S^{2}(x,y)p_{y}^{2}(x,y)$

If $\lambda$ is the generating function of $D(f)$, $\tau=\lambda/p_{y}^{2}$ is
a holomorphic function with at most an isolated singularity in $N$.
Nonetheless, Hartogs’ Kugelsatz guarantees that $\tau$ can be uniquely
extended onto $(\mathbb{C}^{2},0)$.

We conclude that $\lambda=\tau p_{y}^{2}$. ∎

Singularity | Parametrization | $C$ | $D_{+}$
---|---|---|---
Cuspidal edge | $(x,y^{2},y^{3})$ | $y$ | —
Transverse double point | $(x,y,0)$ | — | $y$
| $(0,x^{\prime},y^{\prime})$ | — | $x^{\prime}$
Folded Whitney  umbrella | $(x,y^{2},xy^{3})$ | $y$ | $x$
Swallowtail  | $(x,y^{3}+3xy,y^{4}+2xy^{2})$ | $x+y^{2}$ | $3x+y^{2}$
Cuspidal double point | $(0,x,y)$ | — | $x^{3}-y^{2}$
| $(x^{\prime},(y^{\prime})^{2},(y^{\prime})^{3})$ | $y^{\prime}$ | $x^{\prime}$
Transverse triple point | $(x,y,0)$ | — | $xy$
$(x^{\prime},0,y^{\prime})$ | — | $x^{\prime}y^{\prime}$
| $(0,x^{\prime\prime},y^{\prime\prime})$ | — | $x^{\prime\prime}y^{\prime\prime}$
Table 2. Double point space of the stable frontal surface singularities.

######  Example 4.1 (Frontalised fold surface).

Let $f\colon(\mathbb{C}^{2},0)\to(\mathbb{C}^{3},0)$ be the fold surface

$f(x,y)=(x,y^{2},y^{3}h(x,y^{2}))$

Using Theorem 4.6, we have $D_{+}(f)=V\left(h(x,y^{2})\right)$, which
coincides with the double point curve of an $\mathscr{A}$-finite fold surface.

We now finish this section with a characterisation of $\mathscr{F}$-finite
surfaces in terms of $C(f)$ and $D_{+}(f)$:

######  Proposition 4.7.

Let $f\colon(\mathbb{C}^{2},S)\to(\mathbb{C}^{3},0)$ be a corank $1$ frontal
surface in prenormal form.

  1. (1)

If $D(f)$ is generated by $\lambda\in\mathscr{O}_{2}$ and $\lambda/p_{y}$ is
regular, $f$ is either a cuspidal edge or a curve of transverse double points.

  2. (2)

If $V(p_{y},\mu_{y})=\\{0\\}$, $f$ is $\mathscr{F}$-finite if and only if the
critical set of $\lambda/p_{y}$ is an isolated subset of $(\mathbb{C}^{2},0)$.

A consequence of this statement is that a frontal surface is
$\mathscr{F}$-finite if and only if it only contains cuspidal edges and
transversal double points outside the origin. The six stable frontal surfaces
(excluding immersions) are listed on Table 2, and can be seen on Figure 3.

###### Proof of Proposition 4.7.

For the first item, let us assume $\lambda/p_{y}$ is a regular function. If
$f$ has a singularity at $0$, $D(f)$ has at least a branch of non-immersive
points. Regularity of $\lambda/p_{y}$ implies that this is the only branch of
$D(f)$, and therefore $f$ does not have any self-intersections. Therefore, $f$
is $1$-to-$1$.

By Theorem 4.3, a suitable change of coordinates in the source and target
allows us to claim that $p(x,y)=y^{m}$ and $q(x,y)=y^{m}h(x,y)$ for some
$m\geq 2$ and $h\in\mathscr{O}_{2}$. In these conditions, Lemma 4.5 tells us
that $\lambda(x,y)=y^{\mu}$, $\mu$ being the Milnor number of the slice
$\gamma_{0}$. Regularity of $\lambda/p_{y}$ then implies that $\delta=1$ and
hence $\mu=2$. However, since $\mu\geq 2(m-1)$ (Lemma 4.1), we then have that
$m=2$. Therefore, $\gamma_{0}$ is a cusp and $f$ can be seen as a
$1$-parameter frontal unfolding of $\gamma_{0}$.

Since the cusp is stable as a frontal, $f$ is $\mathscr{A}$-equivalent to the
trivial unfolding of the cusp, which is a cuspidal edge.

If $f$ is not $1$-to-$1$, there is at least one transverse double point in the
image, so $D_{+}(f)\neq\emptyset$. Once again, regularity of $\lambda/p_{y}$
implies that there are no branches of non-immersive points.

For the second item, let us first assume $f$ is $\mathscr{F}$-finite. By
Theorem 3.6, there exists a representative $f\colon N\to Z$ of $f$ such that
$f^{-1}(\\{0\\})=S$ and the restriction $\tilde{f}\colon N\backslash S\to
Z\backslash\\{0\\}$ is locally stable as a frontal. In particular, we can
choose $N$ so that $\tilde{f}(N\backslash S)$ only contains transverse double
points and cuspidal edges at most.

Since $f$ is a corank $1$ frontal surface, we can take coordinates $(x,y)$ on
$N$ and $(X,Y,Z)$ on $Z$ such that $f(x,y)=(x,p(x,y),q(x,y))$ with
$p_{y}|q_{y}$. If $\lambda=0$ is the reduced equation for $D(f)$,
$\lambda/p_{y}\colon N\backslash S\to\mathbb{C}$ is a regular function, from
which it follows that the critical set of $\lambda/p_{y}$ is contained within
$S$. Nonetheless, as $S$ is an isolated subset of $N$, so is the critical set
of $\lambda/p_{y}$.

Conversely, let us assume that the critical set $C$ of $\lambda/p_{y}$ is an
isolated subset of $N$. This means that $\lambda/p_{y}\colon N\backslash
C\to\mathbb{C}$ is a regular function. By the previous item, this implies that
$f\colon N\backslash C\to Z$ is either a cuspidal edge or a curve of
transverse double points, both of which are $\mathscr{F}$-stable
singularities. Using Theorem 3.6 once again, we conclude that $f$ is
$\mathscr{F}$-finite. ∎

Figure 3. Stable frontal surface singularities.

##  5\. Vanishing homology of a frontal surface

The formulas from Proposition 5.2 and Theorems 5.4 and 5.7 have been
implemented in a Singular library, which can be found at [25]. Let
$f\colon(\mathbb{C}^{2},S)\to(\mathbb{C}^{3},0)$ be an $\mathscr{A}$-finite
multi-germ, and $f\colon N\to Z$ be a representative of $f$. A stabilisation
of $f$ is a holomorphic family $f_{t}\colon N_{t}\to Z$ with $f_{0}=f$ and
$t\in D\subset\mathbb{C}$ such that the unfolding $F=(f_{t},t)$ is stable if
$t\neq 0$. The topological space $\Delta(f)=f_{t}(N_{t})$ is known as the
disentanglement of $f$.

Mond [22] proved that, for a small enough $t\neq 0$, the space $\Delta(f)$ has
the homology type of a bouquet of $2$-spheres, the number of which is known as
the image Milnor number of $f$ (denoted as $\mu_{I}(f)$). As a consequence,
the reduced homology of $\Delta(f)$ is known as the _vanishing homology_ of
$f$.

###  5.1. Frontal disentanglement

######  Definition 5.1.

Let $f\colon(\mathbb{C}^{n},S)\to(\mathbb{C}^{n+1},0)$ be a frontal map-germ.
A smooth family of frontal germs $(f_{t})$ is an $\mathscr{F}$-stabilisation
of $f$ if the $1$-parameter unfolding $F=(f_{t},t)$ is frontal and $f_{t}$ is
frontal stable for $t\neq 0$. Given $t\neq 0$, the frontal disentanglement of
$f$ is defined as

$\Delta_{\mathscr{F}}(f)=f_{t}(\mathbb{C}^{n},S)$

Let $f\colon(\mathbb{C}^{2},0)\to(\mathbb{C}^{3},0)$ be a corank $1$ frontal
surface with isolated frontal instability (i.e., such that
$f(N\backslash\\{0\\})$ contains at most the singularities described in Table
2). By shrinking $N$ if necessary, we can assume that $f(N\backslash\\{0\\})$
only contains cuspidal edges and transverse double points.

Let $N_{0}=\\{y\in\mathbb{C}:(x,y)\in N\\}$. We consider the slice
$\gamma_{0}\colon N_{0}\to\gamma(N_{0})$ of $f$ at $x=0$, given by
$\gamma_{0}(t)=(p(0,t),q(0,t))$. The assumption that $f$ only has at most
cuspidal edges and transverse double points outside the origin implies that
the origin is an isolated point in the singular set of $\gamma$. Using the
Mather-Gaffney criterion for $\mathscr{A}$-equivalence, it follows that
$\gamma$ is $\mathscr{A}$-finite. However, since
$\mathscr{F}(\gamma)\subseteq\theta(\gamma)$,

$\frac{\mathscr{F}(\gamma)}{T\mathscr{A}_{e}\gamma}\subseteq\frac{\theta(\gamma)}{T\mathscr{A}_{e}\gamma}\implies\operatorname{codim}_{\mathscr{F}_{e}}(\gamma)\leq\operatorname{codim}_{\mathscr{A}_{e}}(\gamma)<\infty$

and thus $\gamma$ is $\mathscr{F}$-finite. Therefore, it admits a versal
$d$-parameter frontal unfolding, $\Gamma\colon U\to V$.

Since $\Gamma$ is a versal unfolding, it is in particular a stable frontal
unfolding of $\gamma$. Since $f$ is a $1$-parameter unfolding of $\gamma$,
$f\times\operatorname{id}$ is a $(d-1)$-parameter unfolding of $\gamma$. By
stability of $\Gamma$, we then have that $f\times\operatorname{id}$ is
$\mathscr{A}$-equivalent to $\Gamma$, from which follows that $\Gamma$ is a
stable frontal unfolding of $f$. We then have the following

######  Proposition 5.1.

Every corank $1$ frontal $f\colon(\mathbb{C}^{2},S)\to(\mathbb{C}^{3},0)$ with
isolated $\mathscr{F}$-instability admits an $\mathscr{F}$-stabilisation
$(f_{t})$.

Let $f\colon(\mathbb{C}^{2},S)\to(\mathbb{C}^{3},0)$ be a corank $1$ frontal
with isolated frontal instability, and let $(f_{t})$ be an
$\mathscr{F}$-stabilisation of $f$. By Theorem 3.6, if
$V(p_{y},\mu_{y})=\\{0\\}$ for all $t$, we can find a representative of
$\Delta_{\mathscr{F}}(f)$ that only contains stable frontal singularities (see
Table 2). We then set the following notation:

  * •

$S$: number of swallowtails;

  * •

$K$: number of cuspidal double points;

  * •

$T$: number of transversal triple points;

  * •

$W$: number of folded Whitney umbrellas.

Given $h\in\mathscr{O}_{2}$, $h_{y}(x,y)$ is the limit when $y^{\prime}$ goes
to $y$ of the divided difference $h[x,y,y^{\prime}]\in\mathscr{O}_{3}$ (see
§4), so there exist functions $\alpha,\beta\in\mathscr{O}_{3}$ such that
$h[x,y,y^{\prime}]\equiv\alpha_{y}(x,y,y^{\prime})(y-y^{\prime})\mod h_{y}$.
If we now consider $p,q\in\mathscr{O}_{2}$ with $p_{y}|q_{y}$, we have

$\displaystyle p[x,y,y^{\prime}]\equiv\alpha(x,y,y^{\prime})(y-y^{\prime})\mod
p_{y};$ $\displaystyle
q[x,y,y^{\prime}]\equiv\alpha^{\prime}(x,y,y^{\prime})(y-y^{\prime})\mod
p_{y},$

and thus we consider the ideal
$(p_{y},\alpha,\alpha^{\prime})\subseteq\mathscr{O}_{3}$.

######  Proposition 5.2.

Let $f\colon(\mathbb{C}^{2},0)\to(\mathbb{C}^{3},0)$ be a frontal mono-germ
with isolated instability in the form $f(x,y)=(x,p(x,y),q(x,y))$. If the
transversal point curve of $f$ is $\tau\in\mathscr{O}_{2}$,

$\displaystyle P_{3}=\dim_{\mathbb{C}}\frac{\mathscr{O}_{2}}{(p_{y},p_{yy})}$
$\displaystyle=S;$ $\displaystyle
PT=\dim_{\mathbb{C}}\frac{\mathscr{O}_{2}}{(p_{y},\tau)}$
$\displaystyle=2S+K+W;$ $\displaystyle
PAA^{\prime}=\dim_{\mathbb{C}}\frac{\mathscr{O}_{3}}{(p_{y},\alpha,\alpha^{\prime})}$
$\displaystyle=2S+K;$ $\displaystyle
F_{3}=\dim_{\mathbb{C}}\frac{\mathscr{O}_{3}}{\mathscr{F}_{2}(f)}$
$\displaystyle=T+S+K.$

###### Proof.

The ideals $(p_{y},p_{yy})$, $(p_{y},\tau)$ and
$(p_{y},\alpha,\alpha^{\prime})$ are complete intersection in their respective
algebras, and thus Cohen-Macaulay. Since $f$ has an isolated
$\mathscr{F}$-instability, it follows from Theorem 3.6 and [24] Corollary
11.11 that $\mathscr{F}_{2}(f)$ is Cohen-Macaulay. By conservation of
multiplicity (see [24] Appendix E), we then have that the integers $P_{3}$,
$PT$, $PAA^{\prime}$ and $F_{2}$ can be written as a linear combination of
$S$, $K$, $T$ and $W$, whose coefficients can be found using the data from
Table 3. ∎

######  Remark 5.1.

The identity $P_{3}=S$ can be easily derived for wavefronts using the results
from [14].

Symbol | Parametrization | $S$ | $K$ | $T$ | $W$
---|---|---|---|---|---
$\check{S}_{k}\rule{0.0pt}{11.19443pt}\rule[-5.16663pt]{0.0pt}{0.0pt}$ | $(x,y^{2},y^{5}+x^{k}y^{3})$ | $0$ | $0$ | $0$ | $k$
$4^{k}_{1}\rule{0.0pt}{11.19443pt}\rule[-5.16663pt]{0.0pt}{0.0pt}$ | $(x,2y^{3}+x^{k}y,3y^{4}+x^{k}y^{2})$ | $k$ | $0$ | $0$ | $0$
$5_{3}\rule{0.0pt}{11.19443pt}\rule[-5.16663pt]{0.0pt}{0.0pt}$ | $(x,5y^{4}+3xy^{2},4y^{5}+2xy^{3})$ | $3$ | $3$ | $0$ | $0$
$6_{1}\rule{0.0pt}{11.19443pt}\rule[-5.16663pt]{0.0pt}{0.0pt}$ | $(x,3y^{5}+xy,5y^{6}+xy^{2})$ | $3$ | $6$ | $1$ | $0$
Table 3. Some corank $1$ frontal surfaces and their dimension 0 singularities
in the $\mathscr{F}$-stabilisation.

######  Example 5.1 (Frontal fold surfaces).

Let $f\colon(\mathbb{C}^{2},0)\to(\mathbb{C}^{3},0)$ be a frontal fold surface
with prenormal form

$f(x,y)=(x,y^{2},y^{3}h(x,y^{2}))$

and let $\pi\colon(\mathbb{C}^{3},0)\to(\mathbb{C}^{2},0)$ be the projection
$\pi(X,Y,Z)=(X,Y)$. The map $\pi\circ f$ has multiplicity $2$, so the
presentation matrix for $f$ is $2\times 2$ and
$\mathscr{F}_{2}(f)=\mathscr{O}_{3}$ by definition. It follows that $S+K+T=0$
and thus $S=K=T=0$. If $i\colon(\mathbb{C},0)\to(\mathbb{C}^{2},0)$ is the
germ of immersion given by $x\mapsto(x,0)$, we conclude using Example 4.1 that
$W=\operatorname{mult}(i^{*}h)$.

###  5.2. Frontal Marar-Mond formulas

Marar and Mond gave a formula in [17] that relates the Milnor numbers of the
curves $D(f)$ and $f(D(f))$ for an $\mathscr{A}$-finite
$f\colon(\mathbb{C}^{2},0)\to(\mathbb{C}^{3},0)$. However, it is easy to
derive from Theorem 4.6 that the Milnor number of $D(f)$ is only finite when
$f$ does not feature cuspidal edges on its frontal disentanglement. Therefore,
we need to consider the branches of cuspidal edges and transverse double
points separately.

Let $f\colon(\mathbb{C}^{2},0)\to(\mathbb{C}^{3},0)$ be a corank $1$ frontal
surface with an isolated frontal instability at $0$ and $(f_{t})$ be an
$\mathscr{F}$-stabilisation of $f$. The double point space $D(f_{t})$ then
splits into $(D_{+})_{t}$ and $C_{t}$. If $f_{t}\colon N_{t}\to Z_{t}$ is a
representative of $f_{t}$ with $t\neq 0$, we set the following partition for
$N_{t}$:

  * •

$N_{t}^{0}$ is the set of $x\in D(f_{t})$ where $f_{t}$ has an isolated
singularity;

  * •

$N_{t}^{1}=D(f_{t})\backslash N_{t}^{0}$;

  * •

$N_{t}^{2}$ is the set of $x\in N_{t}$ such that $f_{t}$ is immersive at $x$.

Since $D(f_{t})=C(f_{t})\cup D_{+}(f_{t})$, $N_{t}^{1}$ splits into the
disjoint union of $C(f_{t})^{1}$ and $D_{+}(f_{t})^{1}$, which we shall simply
denote as $C_{t}^{1}$ and $D_{t}^{1}$.

######  Lemma 5.3.

Let $f\colon(\mathbb{C}^{2},0)\to(\mathbb{C}^{3},0)$ be a corank $1$ frontal
map-germ with isolated instability. If $(f_{t})$ is an
$\mathscr{F}$-stabilisation of $f$, the projection
$\pi_{X}:X(F)\to(\mathbb{C},0)$ is flat for $X\in\\{C,D\\}$.

###### Proof.

Since $f$ is in prenormal form, we can write

$F(x,y,t)=(x,p_{t}(x,y),q_{t}(x,y),t)$

and we can compute $D(F)$ using the same procedure as described in §4. We then
have that $C(F)=V((p_{t})_{y})$ and $D_{+}(F)=V(\tau_{t})$. Both of these
spaces are analytic surfaces contained in $\mathbb{C}^{2}\times\mathbb{C}$
with codimension $1$, so they are complete intersections and thus Cohen-
Macaulay.

Let $X=C,D$ and consider the projection $\pi_{X}\colon X(F)\to\mathbb{C}$
given by $\pi_{X}(x,y,t)=t$. We have $\pi^{-1}(\\{0\\})=X(f)$, which is a
subspace of codimension $1$ in $X(F)$, which matches the dimension of
$(\mathbb{C},0)$. It then follows from [19] §23 that $\pi_{X}$ is flat. ∎

######  Theorem 5.4.

Let $f\colon(\mathbb{C}^{2},0)\to(\mathbb{C}^{3},0)$ be a corank $1$ frontal
surface with isolated $\mathscr{F}$-instability. If
$V(p_{y},\mu_{y})=\\{0\\}$,

$\displaystyle\mu(f(C(f)),0)$ $\displaystyle=2S+\mu(C(f),0);$ $\displaystyle
2\mu(f(D(f)),0)$ $\displaystyle=2K+2T+\mu(D(f),0)-W-S+1$

To prove this result, we shall make use of the following

######  Theorem 5.5 ([3]).

Let $\pi\colon X\to D\subset\mathbb{C}$ be a good representative of a flat
family $\pi\colon(X,x)\to(D,0)$ of reduced curves. Given $t\in D$,

$\mu(X_{0},x)-\mu(X_{t})=1-\chi(X_{t})$

where $X_{t}=\pi^{-1}(t)$ and $\mu(X_{t})$ is the sum of the Milnor numbers
over the singular points of $X_{t}$.

###### Proof of Theorem 5.4.

On the one hand, cuspidal edges are $1$-to-$1$, hence $C_{t}^{1}\cong
f(C_{t}^{1})$. Since
$\chi(C_{t})-\chi(C_{t}^{1})=\chi(f(C_{t}))-\chi(f(C_{t}^{1}))$, it follows
that $\chi(C_{t})=\chi(f(C_{t}))$. On the other hand, transverse double points
are 2-to-1, so $\chi(D_{t}^{1})=2\chi(f(D_{t}^{1}))$. Using the identities

$\displaystyle\chi(D_{t})-\chi(D_{t}^{1})$ $\displaystyle=S+W+2K+3T;$
$\displaystyle\chi(f_{t}(D_{t}))-\chi(f(D_{t}^{1}))$ $\displaystyle=S+W+K+T$

it then follows that $\chi(D_{t})-2\chi(f_{t}(D_{t}))=T-S-W$.

Using Theorem 5.5 and the information provided in Table 2, we have

$\displaystyle\mu(C_{t})=0$ $\displaystyle\implies\chi(C_{t})=1-\mu(C,0);$
$\displaystyle\mu(f_{t}(C_{t}))=2S$
$\displaystyle\implies\chi(f_{t}(C_{t}))=2S-\mu(f(C),0)+1;$
$\displaystyle\mu(D_{t})=2K+3T$
$\displaystyle\implies\chi(D_{t})=2K+3T-\mu(D,0)+1;$
$\displaystyle\mu(f_{t}(D_{t}))=2K+2T$ $\displaystyle\implies
2\chi(f_{t}(D_{t}))=4K+4T-2\mu(f(D),0)+2$

The remaining parts are clear. ∎

###  5.3. Frontal Milnor number

Let $f\colon(\mathbb{C}^{n},S)\to(\mathbb{C}^{n+1},0)$ be a smooth multi-germ
with isolated instability and let $f\colon N\to Z$ be a representative of $f$.
Mond [22] showed that the disentanglement of $f$ has the homology type of a
wedge of $n$-spheres for a small enough $t\neq 0$. These groups are known as
the vanishing homology of $f$.

Lê [33] proved that the vanishing homology does not depend on the choice of
stabilisation, thus being an invariant for $f$. The number of spheres is known
as the image Milnor number of $f$, $\mu_{I}(f)$. Moreover, Lê’s result can
also be applied in the case when $f$ is frontal and has an isolated frontal
instability, allowing us to define the notion of a frontal Milnor number.

######  Definition 5.2.

Let $f\colon(\mathbb{C}^{n},S)\to(\mathbb{C}^{n+1},0)$ be a frontal multi-germ
with isolated $\mathscr{F}$-instability. We define the frontal Milnor number
$\mu_{\mathscr{F}}(f)$ of $f$ as the number of $n$-spheres in the image of an
$\mathscr{F}$-stabilisation of $f$.

######  Proposition 5.6.

Given a non-constant, holomorphic plane curve
$\gamma\colon(\mathbb{C},S)\to(\mathbb{C}^{2},0)$,

$\mu_{I}(\gamma)=\mu_{\mathscr{F}}(\gamma)+\kappa$

where $\kappa$ is the number of cusps in a frontal stabilisation of $\gamma$.

###### Proof.

Let $\gamma\colon N\to Z$ be a representative of $\gamma$. Since $\gamma$ is
non-constant, the Curve Selection Lemma implies that $S$ is an isolated subset
of $\Sigma(\gamma)$. Shrinking $N$ if necessary, we can further assume that
$N\cap\Sigma(\gamma)=S$, so $\gamma$ is immersive outside $S$. Since immersive
maps are both $\mathscr{A}$-stable and $\mathscr{F}$-stable, the Mather-
Gaffney criterion then gives us Item 1, and Theorem 3.6 gives us Item 2.

Let $\gamma_{t}\colon N_{t}\to Z$ be a frontal stabilisation of $\gamma$. The
curve $\gamma_{t}(N_{t})$ contains at most normal crossings and plane cusps.
By the conservation property of the image Milnor number [8], we have

$\mu_{I}(\gamma)=\mu_{\mathscr{F}}(\gamma)+\sum_{\mathclap{x\in\Sigma(\gamma_{t})}}\mu_{I}(\gamma_{t};\gamma_{t}(x))$

Let $x\in\Sigma(\gamma_{t})$. Since $\gamma_{t}$ is $\mathscr{F}$-stable, the
germ of $\gamma_{t}$ at $x$ is a cusp, so
$\mu_{I}(\gamma_{t};\gamma_{t}(x))=1$ and the statement follows. ∎

Figure 4. Frontal stabilisation of the $E_{6}$ singularity (dashed, grey)
along with a stabilisation (solid, blue).

We now move onto frontal surfaces. Let
$f\colon(\mathbb{C}^{2},0)\to(\mathbb{C}^{3},0)$ be a corank $1$ frontal map
with isolated instability, $(f_{t})$ be an $\mathscr{F}$-stabilisation for $f$
and $F=(f_{t},t)$. We consider the function $\pi\colon
F(\mathbb{C}^{2}\times\mathbb{C},0)\to(\mathbb{C},0)$ given by
$\pi(X,Y,Z,t)=t$. A result by Lê [33] states that the fibres of $\pi$ have the
homotopy type of a bouquet of $2$-spheres; however, one of such fibres is
$\Delta_{\mathscr{F}}(f)$. Therefore,

$\mu_{\mathscr{F}}(f)=\chi(f_{t}(N_{t}))-1=\chi(f_{t}(N_{t}^{2}))+\chi(f_{t}(N_{t}^{1})+\chi(f_{t}(N_{t}^{0})-1$

######  Theorem 5.7.

Given a corank $1$ frontal $f\colon(\mathbb{C}^{2},0)\to(\mathbb{C}^{3},0)$
with isolated $\mathscr{F}$-instability,

$\displaystyle\mu_{\mathscr{F}}(f)$ $\displaystyle=\mu(f(D(f)),0)-S-W+T+1=$
$\displaystyle=\frac{1}{2}\Big{(}\mu(D(f),0)+3(1-S-W)\Big{)}+K+2T$

###### Proof.

We can assume without any loss of generality that $N_{t}$ is an open, convex
neighbourhood of $0$, in which case
$\chi(N_{t}^{2})=1-\chi(N_{t}^{0})-\chi(N_{t}^{1})$. Since
$f_{t}(N_{t}^{2})\cong N_{t}^{2}$ by construction,

$\mu_{\mathscr{F}}(f)=\chi(f_{t}(N_{t}^{1}))-\chi(N_{t}^{1})+\chi(f_{t}(N_{t}^{0}))-\chi(N_{t}^{0})$

Recall that $N_{t}^{1}=C(f_{t})^{1}\sqcup D(f_{t})^{1}$. Since $C_{t}^{1}\cong
f_{t}(C_{t}^{1})$ and $D_{t}^{1}$ is a double cover for $f_{t}(D_{t}^{1})$,

$\displaystyle\chi(f_{t}(N_{t})^{0})-\chi(N_{t}^{0})$ $\displaystyle=-K-2T;$
$\displaystyle\chi(f_{t}(N_{t})^{1})-\chi(N_{t}^{1})$
$\displaystyle=\chi(f_{t}(D_{t}))-S-W-K-T;$

from which follows that $\mu_{\mathscr{F}}(f)=\chi(f_{t}(D_{t}))-S-W-2K-3T$.
Using Theorem 5.5, we have $\chi(f_{t}(D_{t}))=2K+2T-\mu(f(D),0)+1$, from
which follows that

$\mu_{\mathscr{F}}(f)=\mu(f(D),0)-S-W+T+1$

∎

We finish this section by proposing a frontal version of Mond’s conjecture
[23]:

######  Conjecture 5.1.

Let $f\colon(\mathbb{C}^{n},S)\to(\mathbb{C}^{n+1},0)$ be an
$\mathscr{F}$-finite frontal map. Then
$\mu_{\mathscr{F}}(f)\geq\operatorname{codim}_{\mathscr{F}_{E}}(f)$, with
equality if and only if $f$ is quasihomogeneous.

## References

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