NB

# On the dependence of the affine Springer fibers on their root valuation
datum

Zongbin Chen

###### Abstract

For the group $\mathrm{GL}_{d}$, we confirm a conjecture of Goresky, Kottwitz
and MacPherson, which states that the cohomology of the affine Springer fibers
depend only on the root valuation datum of their defining elements. The proof
relies on a microlocal analysis of the intermediate extensions appearing in
Ngô’s support theorem, and a Whitney regularity property of the union of the
strict $\delta$-strata over the equisingular strata for the miniversal
deformation of plane curve singularities.

###### Contents

  1. 1 Introduction
  2. 2 The affine Springer fibers and the compactified Jacobians
    1. 2.1 Laumon’s work
    2. 2.2 Ngô’s support theorem
  3. 3 Microlocal analysis of the intermediate extensions in Ngô’s support theorem
    1. 3.1 Calculation of some vanishing cycles
    2. 3.2 Reduction to the nearby fiber
    3. 3.3 Iterated fiberations of the nearby fiber
  4. 4 Whitney regularity along the equisingular strata
    1. 4.1 The geometry of the discriminant
    2. 4.2 Whitney regularity
  5. 5 Appendix: Microlocal analysis of the intermediate extension

##  1 Introduction

Let $F=\mathbf{C}(\\!(\varepsilon)\\!)$ be the field of Laurent series over
$\mathbf{C}$, $\mathcal{O}=\mathbf{C}[\\![\varepsilon]\\!]$ its ring of
integers. We fix an algebraic closure
$\overline{F}=\bigcup_{n\in\mathbf{N}}\mathbf{C}(\\!(\varepsilon^{1/n})\\!)$
of $F$, let $\tau\in{\rm Gal}(\overline{F}/F)$ be the unique element that
sends $\varepsilon^{1/n}$ to $\zeta_{n}\varepsilon^{1/n}$ for any
$n\in\mathbf{N}$, where $\\{\zeta_{n}\\}_{n\in\mathbf{N}}$ is a compatible
family of primitive $n$-th root of unity. Let ${\rm
val}:\overline{F}\to\mathbf{Q}$ be the valuation normalized by the condition
${\rm val}(\varepsilon)=1$.

Let $G$ be the general linear group $\mathrm{GL}_{d}$ over $\mathbf{C}$, and
let $\mathfrak{g}$ be its Lie algebra. Let
$\gamma\in\mathfrak{g}[\\![\varepsilon]\\!]$ be a regular semisimple element.
The _affine Springer fiber_ at $\gamma$ is the closed sub
ind-$\mathbf{C}$-scheme of the _affine grassmannian_
$\mathcal{X}=G(\\!(\varepsilon)\\!)/G[\\![\varepsilon]\\!]$ defined by

$\mathcal{X}_{\gamma}=\Big{\\{}g\in\mathcal{X}\,\big{|}\,\mathrm{Ad}(g^{-1})\gamma\in\mathfrak{g}[\\![\varepsilon]\\!]\Big{\\}}.$

It is a scheme locally of finite type over $\mathbf{C}$. Their analogues over
finite fields have been used to study the orbital integrals [GKM1]. Let $A$ be
the maximal torus of $G$ consisting of the diagonal matrices, let $\Phi$ be
the root system of $G$ with respect to $A$, let $W$ be the associated Weyl
group. Goresky, Kottwitz and MacPherson [GKM2] have defined the _root
valuation datum_ of $\gamma$: Choose an element
$\gamma^{\prime}\in{\mathfrak{a}}(\overline{F})$ conjugate to $\gamma$, we can
attach to $\gamma^{\prime}$ a pair $(w,r)$, where $w\in W$ is the unique
element such that $w\tau(\gamma^{\prime})=\gamma^{\prime}$, and
$r:\Phi\to\mathbf{Q}$ is the function sending $\alpha\in\Phi$ to ${\rm
val}(\alpha(\gamma^{\prime}))$. The element $\gamma^{\prime}$ is not unique,
we can replace it with any element of the form $w^{\prime}\gamma^{\prime}$,
$w^{\prime}\in W$, for which the pair $(w,r)$ is replaced by
$(w^{\prime}w{w^{\prime}}^{-1},w^{\prime}r)$, where
$w^{\prime}r(\alpha)=r({w^{\prime}}^{-1}(\alpha))$. The equivalence class of
the pair $(w,r)$ under the above action is called the _root valuation datum_
of $\gamma$.

######  Theorem 1 (Conjecture of Goresky-Kottwitz-MacPherson, [GKM2]).

The cohomology of the affine Springer fiber $\mathcal{X}_{\gamma}$ depends
only on the root valuation datum of the element $\gamma$.

The theorem is proven via deformation of the compactified Jacobians. Recall
that $\mathcal{X}_{\gamma}\cong\mathcal{X}_{\gamma}^{0}\times\mathbf{Z}$ with
$\mathcal{X}_{\gamma}^{0}$ being the central connected component of
$\mathcal{X}_{\gamma}$, and $\mathcal{X}_{\gamma}^{0}$ admits the action of a
discrete free abelian group $\Lambda^{0}$. According to Laumon [L], the
quotient $\Lambda^{0}\backslash\mathcal{X}_{\gamma}^{0}$ is homeomorphic to
the compactified Jacobian $\overline{P}_{C_{\gamma}}$ of a spectral curve
$C_{\gamma}$ over $\mathbf{C}$, and $\mathcal{X}_{\gamma}^{0}$ is homeomorphic
to a $\Lambda^{0}$-torsor $\overline{P}{}_{C_{\gamma}}^{\natural}$ over
$\overline{P}_{C_{\gamma}}$, which can be seen as a universal abelian covering
of $\overline{P}_{C_{\gamma}}$. Moreover, the construction of
$\overline{P}{}_{C_{\gamma}}^{\natural}$ can be put in family, and theorem 1
can be reduced to:

######  Theorem 2.

The cohomology of $\overline{P}_{C_{\gamma}}$ depends only on the root
valuation datum of the element $\gamma$.

In terms of plane curve singularities, the root valuation datum of $\gamma$
determines the datum consisting of the Puiseux characteristics of the
irreducible components (called _branches_) of the singularity
$\mathrm{Spf}(\mathcal{O}[\gamma])$ and the intersection numbers between pairs
of the branches. According to Zariski [Z1], plane curve singularities with the
same such datum will be _equisingular_. Hence it is enough to show that the
cohomology of $\overline{P}_{C_{\gamma}}$ is locally constant under
equisingular deformation of $C_{\gamma}$.

Let $\pi:\mathcal{C}\to{\mathcal{B}}$ be an algebraization of a miniversal
deformation of $C_{\gamma}$, with $C_{\gamma}$ lying over $0\in{\mathcal{B}}$.
It is known that ${\mathcal{B}}$ is smooth over $\mathbf{C}$ and we can even
take ${\mathcal{B}}=\mathbf{A}^{\tau_{\gamma}}$, where $\tau_{\gamma}$ is the
Tjurina number of the singularity $\mathrm{Spf}(\mathcal{O}[\gamma])$. Then
$\overline{P}_{C_{\gamma}}$ fits in the projective flat family of relative
compactified Jacobians

$f:\overline{\mathcal{P}}=\overline{\mathrm{Pic}}{}^{\,0}_{\mathcal{C}/{\mathcal{B}}}\to{\mathcal{B}}.$

Let $j:{\mathcal{B}}^{\rm sm}\to{\mathcal{B}}$ be the open subscheme over
which $f$ is smooth, and let $f^{\rm sm}$ be the restriction of $f$ to the
inverse image of ${\mathcal{B}}^{\rm sm}$. Ngô’s support theorem [N] states
that

$Rf_{*}\mathbf{Q}_{\ell}=\bigoplus_{i=0}^{2\delta_{\gamma}}j_{!*}(R^{\,i}f^{\mathrm{sm}}_{*}\mathbf{Q}_{\ell})[-i],$

where $\delta_{\gamma}$ is the $\delta$-invariant of $C_{\gamma}$. In
particular,

$H^{i}\big{(}\overline{P}_{C_{\gamma}},\mathbf{Q}_{\ell}\big{)}=\bigoplus_{i^{\prime}=0}^{i}\mathcal{H}^{i-i^{\prime}}\big{(}j_{!*}R^{i^{\prime}}f^{\mathrm{sm}}_{*}\mathbf{Q}_{\ell}\big{)}_{0},\quad
i=0,\cdots,2\delta_{\gamma}.$

Consider the partition of the base ${\mathcal{B}}$ as union of locally closed
subschemes such that fibers of the family $\pi:\mathcal{C}\to{\mathcal{B}}$
are equisingular over each subscheme, we call it the _equisingular
stratification_ , and call the resulting strata _equisingular strata_ 111Here
the word _stratification_ is used in an abusive manner, it seems unknown
whether the equisingular stratification satisfies the frontier condition for a
stratification.. To prove theorem 2, we need to show that the sheaves
$\mathcal{H}^{*}(j_{!*}R^{\,i}f^{\mathrm{sm}}_{*}\mathbf{Q}_{\ell})$ are
locally constant over the equisingular strata for all $i$. If the equisingular
stratification was a Whitney stratification, theorem 2 will be an immediate
consequence of Ngô’s support theorem, as the intermediate extension is locally
constant along the Whitney strata. But this is too much to ask, there is a
counter-example of Frédérique Pham (cf. [Ber]), which shows that the
equisingular stratification is not fine enough to be a Whitney stratification.

Here some miracle happens. The local regularity criteria of Fantechi-Göttsche-
van Straten [FGS] about the family $f:\overline{\mathcal{P}}\to{\mathcal{B}}$
implies a local acyclicity property of the complex
$j_{!*}R^{\,i}f^{\mathrm{sm}}_{*}\mathbf{Q}_{\ell}$ (theorem 3.3), to the
effect that it is enough to show that the union of the _strict
$\delta$-strata_ are Whitney regular over the equisingular strata which lie at
their frontier. Here the _strict $\delta$-strata_ refer to the equisingular
strata over which the curves have $\delta$ ordinary double points as
singularities. With Teissier’s criteria [T3], the Whitney regularity property
in question follows from the smoothness of the equisingular strata and a
wonderful property of the discriminant $\Delta$ of the family $\pi$, that its
Nash blow-up is smooth [T1].

### Notations, conventions and useful facts

#### (1)

Let $X$ be a locally noetherian scheme, we denote by $X^{\mathrm{reg}}$ its
regular locus and $X^{\rm sing}=X-X^{\mathrm{reg}}$ the singular locus.

#### (2)

Let $f:X\to S$ be a morphism of schemes, we will denote by $\Delta_{f}$ the
discriminant locus of $f$, and by $S^{\mathrm{sm}}$ the open subscheme
$S-\Delta_{f}$ over which $f$ is smooth, and $f^{\mathrm{sm}}:X^{\rm sm}\to
S^{\mathrm{sm}}$ the restriction of $f$ to the inverse image of
$S^{\mathrm{sm}}$. For any point $s\in S$, we denote by $X_{s}$ the fiber of
$f$ at $s$.

#### (3)

Let $C^{\circ}=\mathrm{Spf}(\mathbf{C}[\\![x,y]\\!]/(f(x,y)))$ be a germ of
plane curve singularity. The _Milnor number_ and _Tjurina number_ of
$C^{\circ}$ are defined respectively by

$\mu=\dim(\mathbf{C}[\\![x,y]\\!]/(\partial_{x}f,\partial_{y}f)),\quad\tau=\dim(\mathbf{C}[\\![x,y]\\!]/(f,\partial_{x}f,\partial_{y}f)).$

Let $A=\mathbf{C}[\\![x,y]\\!]/(f(x,y))$ and $\widetilde{A}$ the normalization
of $A$, the _conductor_ of $A$ is the ideal

${\mathfrak{a}}=\big{\\{}a\in A\mid a\widetilde{A}\subset A\big{\\}}.$

It is known that $\dim(\widetilde{A}/A)=\dim(A/{\mathfrak{a}})$, we call it
the _$\delta$ -invariant_ of $C^{\circ}$, denoted $\delta(C^{\circ})$ or
$\delta(A)$.

#### (4)

Let $C^{\circ}$ be a germ of plane curve singularity over $\mathbf{C}$. Its
irreducible components are called the _branches_. It is known that every
branch admits a _Puiseux parametrization_

$\begin{cases}X=t^{n},&\\\ Y=\sum_{i=m}^{\infty}a_{i}t_{i},&\text{for some
}m>n.\end{cases}$

#### (5)

Let $C^{\circ}=\mathrm{Spf}(A)$ be a germ of plane curve singularity over
$\mathbf{C}$. We denote by $\mathrm{Def}^{\rm top}_{C^{\circ}}$ or
$\mathrm{Def}^{\rm top}_{A}$ the deformation functor of $C^{\circ}$. The
functor $\mathrm{Def}^{\rm top}_{C^{\circ}}$ is known to be representable and
we have a miniversal deformation of $C^{\circ}$.

#### (6)

Let $C_{0}$ be a projective algebraic curve over $\mathbf{C}$ with planar
singularities, its _$\delta$ -invariant_ $\delta(C_{0})$ is defined as the sum
of the $\delta$-invariants of its singularities. Let $\psi:C\to S$ be a
deformation of $C_{0}$ with smooth base $S$, then the function $s\in
S\mapsto\delta(C_{s})$ is upper semi-continuous according to Teissier [T2].
The subscheme $S_{\delta}$ parametrizing curves with $\delta$-invariant
$\delta$ is then locally closed, and we obtain a stratification
$S=\bigsqcup_{\delta=0}^{\delta(C_{0})}S_{\delta}$, called the _$\delta$
-stratification_ of $S$.

#### (7)

We denote $\mathbf{N}$ for the set of natural numbers and
$\mathbf{N}_{0}=\mathbf{N}\cup\\{0\\}$.

#### Acknowledgement

We want to thank Prof. Gérard Laumon for his constant support and
encouragements during the years. Part of the work is done during the author’s
visit to the Research Centre for Mathematics and Interdisciplinary Sciences at
Shandong University, we want to thank the institute for the wonderful
hospitality.

##  2 The affine Springer fibers and the compactified Jacobians

Let $\gamma\in\mathfrak{g}[\\![\varepsilon]\\!]$ be a regular semisimple
element. Let $\mathrm{T}$ be the centralizer of $\gamma$ in $G$, then
$\mathrm{T}$ acts on $\mathcal{X}_{\gamma}$ by left translation. Let
$\mathrm{S}$ be the maximal $F$-split subtorus of $\mathrm{T}$. Let
$\Lambda\subset\mathrm{S}(F)$ be the subgroup generated by
$\chi(\varepsilon),\,\chi\in X_{*}(\mathrm{S})$, then $\Lambda$ acts simply
transitively on the irreducible components of $\mathcal{X}_{\gamma}$. The
connected components of $\mathcal{X}_{\gamma}$ is naturally parametrized by
$\mathbf{Z}$ with the morphism

$\mathcal{X}_{\gamma}\to\mathbf{Z},\quad[g]\mapsto{\rm val}(\det(g)).$

Let $\mathcal{X}_{\gamma}^{0}$ be the central connected component of
$\mathcal{X}_{\gamma}$, then
$\mathcal{X}_{\gamma}\cong\mathcal{X}_{\gamma}^{0}\times\mathbf{Z}$. Let

$\Lambda^{0}=\\{\lambda\in\Lambda\mid{\rm val}(\det(\lambda))=0\\},$

then $\Lambda^{0}$ acts naturally on $\mathcal{X}_{\gamma}^{0}$. The quotient
$\Lambda^{0}\backslash\mathcal{X}_{\gamma}^{0}$ is a projective algebraic
variety over $\mathbf{C}$ and
$\mathcal{X}_{\gamma}^{0}\to\Lambda^{0}\backslash\mathcal{X}_{\gamma}^{0}$ is
an étale Galois covering of Galois group $\Lambda^{0}$.

###  2.1 Laumon’s work

In an attempt to deform the affine Springer fibers, Laumon [L] discovered a
remarkable relationship between the affine Springer fibers and the
compactified Jacobian of an algebraic curve. By definition, the affine
Springer fiber $\mathcal{X}_{\gamma}$ parametrizes the lattices $L$ in $F^{n}$
such that $\gamma L\subset L$. Such lattices can be naturally identified with
sub-$\mathcal{O}[\gamma]$-modules of finite type in $E:=F[\gamma]\cong F^{d}$.
Moreover, such $\mathcal{O}[\gamma]$-modules can be globalized. Let
$C_{\gamma}$ be an irreducible projective algebraic curve over $\mathbf{C}$,
with two points $c$ and $\infty$ such that

  1. (i)

$C_{\gamma}$ has unique singularity at $c$ and
$\widehat{\mathcal{O}}_{C_{\gamma},\,c}\cong\mathcal{O}[\gamma]$,

  2. (ii)

the normalization of $C_{\gamma}$ is isomorphic to $\mathbf{P}^{1}$ by an
isomorphism sending $\infty\in\mathbf{P}^{1}$ to $\infty\in C_{\gamma}$.

The curve $C_{\gamma}$ will be called the _spectral curve_ of $\gamma$. Let
$M$ be a sub-$\mathcal{O}[\gamma]$-modules of finite type in $F[\gamma]$.
Similar to the obvious fact that $\mathcal{O}_{C_{\gamma}}$ can be obtained by
gluing $\widehat{\mathcal{O}}_{C_{\gamma},c}=\mathcal{O}[\gamma]$ and
${\mathcal{O}}_{C_{\gamma}\backslash\\{c\\}}$, we can glue $M$ with the sheaf
$\mathcal{O}_{C_{\gamma}\backslash\\{c\\}}$ along
$(C_{\gamma}\backslash\\{c\\})\times_{C_{\gamma}}\mathrm{Spec}(\widehat{\mathcal{O}}_{C_{\gamma},c})=\mathrm{Spec}(E)$,
to get a torsion-free coherent $\mathcal{O}_{C_{\gamma}}$-module
${\mathcal{M}}$ of generic rank $1$. Recall that the compactified Jacobian
$\overline{P}_{C_{\gamma}}:=\overline{\mathrm{Pic}}{}^{0}_{C_{\gamma}/\mathbf{C}}$
parametrizes the isomorphism class of the torsion-free coherent
$\mathcal{O}_{C_{\gamma}}$-modules of generic rank $1$ and degree $0$.
Consider the functor $\overline{P}{}_{C_{\gamma}}^{\natural}$ which associates
to an affine $\mathbf{C}$-scheme $S$ the groupoïd of couples $(M,\iota)$,
where $M$ is a torsion free coherent $\mathcal{O}_{C_{\gamma}\times S}$-module
of generic rank $1$ and degree $0$, and $\iota:M|_{(C_{\gamma}\backslash
c)\times S}\cong\mathcal{O}_{(C_{\gamma}\backslash c)\times S}$ is a
trivialization of $M$ over $(C_{\gamma}\backslash c)\times S$. One verifies
that $\overline{P}{}_{C_{\gamma}}^{\natural}$ is representable by a
$\mathbf{C}$-scheme. With the glueing construction, we get a natural morphism

$\mathcal{X}_{\gamma}^{0}\to\overline{P}_{C_{\gamma}}^{\natural},$

which sends a lattice $L\subset E$ satisfying $\gamma L\subset L$ to the
torsion-free $\mathcal{O}_{C_{\gamma}}$-module obtained by gluing $L$ and
$\mathcal{O}_{C_{\gamma}\backslash\\{c\\}}$ along $\mathrm{Spec}(E)$, together
with the obvious trivialization. Forgetting the trivialization $\iota$, we get
a morphism $\overline{P}_{C_{\gamma}}^{\natural}\to\overline{P}_{C_{\gamma}}$.
For the affine Springer fiber, this corresponds to the quotient
$\mathcal{X}_{\gamma}^{0}\to\Lambda^{0}\backslash\mathcal{X}_{\gamma}^{0}$.

######  Theorem 2.1 (Laumon [L]).

The morphism $\mathcal{X}_{\gamma}^{0}\to\overline{P}_{C_{\gamma}}^{\natural}$
is finite radical and surjective, and so is the induced morphism
$\Lambda^{0}\backslash\mathcal{X}_{\gamma}^{0}\to\overline{P}_{C_{\gamma}}$.
In particular, both morphisms are universal homeomorphisms.

Unlike the affine Springer fiber or its quotients, the compactified Jacobian
can be deformed naturally. Let
$\pi:(\mathcal{C},C_{\gamma})\to({\mathcal{B}},0)$ be an algebraization of a
miniversal deformation of $C_{\gamma}$. Then $\overline{P}_{C_{\gamma}}$ fits
in the projective flat family of relative compactified Jacobians

$f:\overline{\mathcal{P}}=\overline{\mathrm{Pic}}{}^{\,0}_{\mathcal{C}/{\mathcal{B}}}\to{\mathcal{B}}.$

The compactified Jacobians of curves with planar singularities have very nice
properties, which we can exploit to understand the affine Springer fibers.

On the other hand, it is more difficult to deform
$\overline{P}{}_{C_{\gamma}}^{\natural}$. According to Laumon [L], it doesn’t
admit deformation over the whole base ${\mathcal{B}}$ but only over the
normalization $\widetilde{{\mathcal{B}}}_{\delta_{\gamma}}$ of the
$\delta$-stratum ${\mathcal{B}}_{\delta_{\gamma}}$. We will not need the full
deformation of $\overline{P}{}_{C_{\gamma}}^{\natural}$ over
$\widetilde{{\mathcal{B}}}_{\delta_{\gamma}}$, but only its restriction to
$\Delta_{\mu}$, where $\Delta_{\mu}$ is the closed subscheme of
${\mathcal{B}}$ parametrizing equisingular deformations of $C_{\gamma}$ (see
§4.2 for the notion of equisingularity). According to Teissier [Z2],
$\Delta_{\mu}$ is smooth over $\mathbf{C}$. Let
$\overline{\mathcal{P}}_{\mu}=\overline{\mathcal{P}}\times_{{\mathcal{B}}}\Delta_{\mu}$,
and let $f_{\mu}:\overline{\mathcal{P}}_{\mu}\to\Delta_{\mu}$ be the
restriction of the family $f:\overline{\mathcal{P}}\to{\mathcal{B}}$ to
$\Delta_{\mu}$.

######  Theorem 2.2 (Laumon [L]).

There exists a $\Lambda^{0}$-torsor
$\overline{\mathcal{P}}{}^{\natural}_{\mu}\to\overline{\mathcal{P}}_{\mu}$
such that its fiber over any geometric point $t\in\Delta_{\mu}$ coincides with
the $\Lambda^{0}$-torsor
$\overline{P}{}^{\natural}_{\mathcal{C}_{t}}\to\overline{P}_{\mathcal{C}_{t}}$.

######  Remark 2.3.

As we will see in §4.2, for
$\gamma,\gamma^{\prime}\in\mathfrak{g}[\\![\varepsilon]\\!]$ with the same
root valuation datum, the spectral curves $C_{\gamma}$ and
$C_{\gamma^{\prime}}$ are equisingular. As a corollary of theorem 2.1 and 2.2,
theorem 2 implies theorem 1.

The glueing construction of Laumon works for curves with multiple planar
singularities. We will not go into details here, but refer the reader to the
original work of Laumon.

###  2.2 Ngô’s support theorem

Consider the family $\pi:\mathcal{C}\to{\mathcal{B}}$ and its relative
compactified Jacobians $f:\overline{\mathcal{P}}\to{\mathcal{B}}.$ According
to Fantechi-Göttsche-van Straten [FGS], the total space
$\overline{\mathcal{P}}$ is smooth over $\mathbf{C}$. As the morphism
$f:\overline{\mathcal{P}}\to{\mathcal{B}}$ is projective, the complex
$Rf_{*}\mathbf{Q}_{\ell}$ is pure by Deligne’s Weil-II [D]. The decomposition
theorem of Beilinson, Bernstein, Deligne and Gabber [BBDG] then implies that

$Rf_{*}\mathbf{Q}_{\ell}=\bigoplus_{n=0}^{2\delta_{\gamma}}{}^{p}\mathcal{H}^{n}(Rf_{*}\mathbf{Q}_{\ell})[-n],$

where $\delta_{\gamma}$ is the $\delta$-invariant of $C_{\gamma}$, which is
also the relative dimension of the flat morphism $f$.

######  Theorem 2.4 (Ngô’s support theorem [N]).

We have decomposition

$Rf_{*}\mathbf{Q}_{\ell}=\bigoplus_{n=0}^{2\delta_{\gamma}}j_{!*}(R^{n}f^{\mathrm{sm}}_{*}\mathbf{Q}_{\ell})[-n],$

where $j:{\mathcal{B}}^{\rm sm}\to{\mathcal{B}}$ is the natural inclusion. In
particular,

$H^{n}\big{(}\overline{P}_{C_{\gamma}},\mathbf{Q}_{\ell}\big{)}=\bigoplus_{i=0}^{n}\mathcal{H}^{n-i}\big{(}j_{!*}{\textstyle{\bigwedge^{i}}}R^{1}f^{\mathrm{sm}}_{*}\mathbf{Q}_{\ell}\big{)}_{0},\quad
n=0,\cdots,2\delta_{\gamma}.$

######  Remark 2.5.

We refer the reader to our work [C] for a sketch of the proof. Actually, the
theorem extends to the following situation: Let $B\to{\mathcal{B}}$ be an
embedding of a smooth algebraic variety over $\mathbf{C}$, suppose that $B$ is
transversal to all the $\delta$-strata ${\mathcal{B}}_{\delta}$, i.e. for any
point $x\in B\cap{\mathcal{B}}_{\delta}$ the tangent space $T_{x}B$ is
transversal to the tangent cone of ${\mathcal{B}}_{\delta}$ at $x$. Then for
the sub-family $f_{B}:\overline{\mathcal{P}}\times_{{\mathcal{B}}}B\to B$,
Ngô’s support theorem continues to hold. Indeed, by the local regularity
criteria of Fantechi-Göttsche-van Straten [FGS] (to be recalled below as
theorem 3.1), the total space $\overline{\mathcal{P}}\times_{{\mathcal{B}}}B$
is smooth over $\mathbf{C}$. Moreover, Severi’s inequality
$\mathrm{Codim}(B_{\delta})\geqslant\delta$ continues to hold by the
transversality condition. We get then all the main ingredients in the proof of
Ngô’s support theorem.

##  3 Microlocal analysis of the intermediate extensions in Ngô’s support
theorem

We will exploit Ngô’s support theorem to understand the cohomology of
$\overline{P}_{C_{\gamma}}$. For this, we make a microlocal analysis of the
fiber $(j_{!*}R^{i}f_{*}^{\mathrm{sm}}\mathbf{Q}_{\ell})_{0}$. For simplicity
we denote $\mathcal{F}^{i}=R^{i}f_{*}^{\mathrm{sm}}\mathbf{Q}_{\ell}$.

The main ingredient for the analysis is the local regularity criteria of
Fantechi-Göttsche-van Straten [FGS], we make a brief recall here. Let $C_{0}$
be an irreducible projective algebraic curve over $\mathbf{C}$ with _planar
singularities_. For every $c\in C_{0}^{\rm sing}$, let
$R_{c}:=\widehat{\mathcal{O}}_{C_{0},c}\cong\mathbf{C}[\\![x,y]\\!]/(f_{c}(x,y))$
for some $f_{c}(x,y)\in\mathbf{C}[\\![x,y]\\!]$, then the germ of singularity
$\mathrm{Spf}(R_{c})$ admits a miniversal deformation
$\widehat{\phi}_{c}:\widehat{C}_{c}\to\widehat{B}_{c}$, and the tangent space
of $\widehat{B}_{c}$ at its unique close point $0$ can be identified naturally
with $\mathbf{C}[\\![x,y]\\!]/(f_{c},\partial_{x}f_{c},\partial_{y}f_{c})$.
Let $EG_{c}$ be the closed sub formal scheme of $\widehat{B}_{c}$ which
parametrizes the deformation of $\mathrm{Spf}(R_{c})$ with constant geometric
genus. According to Diaz-Harris [DH], the tangent cone of $EG_{c}$ at $0$ can
be naturally identified with the sub vector space

$V(R_{c}):={\mathfrak{a}}_{c}/(f_{c},\partial_{x}f_{c},\partial_{y}f_{c})\subset
T_{0}(\widehat{B}_{c})=\mathbf{C}[\\![x,y]\\!]/(f_{c},\partial_{x}f_{c},\partial_{y}f_{c}),$

where ${\mathfrak{a}}_{c}$ is the _conductor_ of $R_{c}$. Note that $V(R_{c})$
is then of codimension $\delta(R_{c})$ in $T_{0}(\widehat{B}_{c})$. Let
$\phi:(C,C_{0})\to(B,0)$ be a deformation of $C_{0}$ such that the restriction
of $\phi$ to the formal neighbourhood $\widehat{B}_{0}$ of $B$ at $0$ is
isomorphic to the deformation $\prod_{c\in C_{0}^{\rm
sing}}\widehat{\phi}_{c}.$ Let $V(C_{0})$ be the union of the linear subspaces
of $T_{0}(B)$ which is the preimage of $V(R_{c})$ under the natural surjection
$T_{0}(B)\cong\bigoplus_{c\in C_{0}^{\rm
sing}}T_{0}(\widehat{B}_{c})\twoheadrightarrow T_{0}(\widehat{B}_{c})$ for all
$c\in C_{0}^{\rm sing}$.

######  Theorem 3.1 (Fantechi-Göttsche-van Straten [FGS]).

Let $C_{T}\to T$ be a relative curve with base
$T=\mathrm{Spec}(\mathbf{C}[\\![t_{1},\cdots,t_{m}]\\!])$ which comes from the
family $\phi:C\to B$ by the base change with a morphism $T\to B$, then
$\overline{\mathrm{Pic}}{}^{0}_{C_{T}/T}$ is smooth over $\mathbf{C}$ if and
only if the image of the tangent space of $T$ is transversal to the subspace
$V(C_{0})$ of $T_{0}(B)$.

###  3.1 Calculation of some vanishing cycles

By the general theory of stratification (cf. [M3], proposition 10.1), for the
family $\pi:\mathcal{C}\to{\mathcal{B}}$, let $\Sigma$ be the critical locus
of $\pi$, then there exists a unique pair of Whitney stratifications
$\big{(}\\{\mathcal{C}_{\alpha}\\}_{\alpha\in{\mathfrak{A}}},\\{{\mathcal{B}}_{\beta}\\}_{\beta\in{\mathfrak{B}}}\big{)}$
of $\mathcal{C}$ and ${\mathcal{B}}$ respectively, such that $\Sigma$ is a
union of strata in $\\{\mathcal{C}_{\alpha}\\}_{\alpha\in{\mathfrak{A}}}$, and
that

  1. $(1)$

if $\mathcal{C}_{\alpha}\subset\Sigma$, then $\pi|_{\mathcal{C}_{\alpha}}$ is
an immersion and $\pi(\mathcal{C}_{\alpha})$ is a stratum in
$\\{{\mathcal{B}}_{\beta}\\}_{\beta\in{\mathfrak{B}}}$,

  2. $(2)$

if $\mathcal{C}_{\alpha}\cap\Sigma=\emptyset$, then
$\pi(\mathcal{C}_{\alpha})$ is a stratum in
$\\{{\mathcal{B}}_{\beta}\\}_{\beta\in{\mathfrak{B}}}$ and
$\mathcal{C}_{\alpha}=\pi^{-1}(\pi(\mathcal{C}_{\alpha}))\cap(\mathcal{C}-\Sigma)$,

  3. $(3)$

all the other pairs of Whitney stratifications of $\mathcal{C}$ and
${\mathcal{B}}$ satisfying the above conditions are refinements of the above
one.

This unique pair of Whitney stratifications will be called the _canonical_
Whitney stratifications of the family $\pi:\mathcal{C}\to{\mathcal{B}}$. By
construction, ${\mathcal{B}}^{\mathrm{sm}}$ is one of the strata in
$\\{{\mathcal{B}}_{\beta}\\}_{\beta\in{\mathfrak{B}}}$, and for any stratum
$\mathcal{C}_{\alpha}$ with image ${\mathcal{B}}_{\beta}$ the subspace
$\mathcal{C}\times_{{\mathcal{B}}}{\mathcal{B}}_{\beta}$ is Whitney regular
along $\mathcal{C}_{\alpha}$. By theorem 4.15, the family
$\pi:\mathcal{C}\times_{{\mathcal{B}}}{\mathcal{B}}_{\beta}\to{\mathcal{B}}_{\beta}$
is equisingular along $\mathcal{C}_{\alpha}$. As
$\mathcal{F}^{i}=\bigwedge^{i}\mathcal{F}^{1}$ and
$\mathcal{F}^{1}=R^{1}\pi^{\mathrm{sm}}_{*}\mathbf{Q}_{\ell}$ is lisse over
${\mathcal{B}}^{\mathrm{sm}}$, the intermediate extension
$j_{!*}\mathcal{F}^{i}$ is constructible with respect to the Whitney
stratification $\\{{\mathcal{B}}_{\beta}\\}_{\beta\in{\mathfrak{B}}}$.

Fix a stratum ${\mathcal{B}}_{\beta}$ of dimension $d_{\beta}$, let $x$ be a
point on it, we take a generic affine subspace of
${\mathcal{B}}=\mathbf{A}^{\tau_{\gamma}}$ of codimension $d_{\beta}$ passing
through $x$ such that it is transversal at $x$ to all the strata
${\mathcal{B}}_{\beta},\beta\in{\mathfrak{B}}$. This is possible because the
stratification is locally finite. We then cut off from this affine subspace
the closed subscheme where its intersection with the strata
$\\{{\mathcal{B}}_{\beta}\\}_{\beta\in{\mathfrak{B}}}$ is not transversal, to
get a dense open subscheme $V$ of this affine subspace. By construction, $V$
intersects transversally with all the $\delta$-strata and it contains $0$. Let
$f_{V}:\overline{\mathcal{P}}_{V}\to V$ be the restriction of the family
$f:\overline{\mathcal{P}}\to{\mathcal{B}}$ to $V$.

######  Proposition 3.2.

The subspace $\overline{\mathcal{P}}_{V}$ is nonsingular over $\mathbf{C}$,
and Ngô’s support theorem holds for the family $f_{V}$, i.e.

$R(f_{V})_{*}\mathbf{Q}_{\ell}=\bigoplus_{i=0}^{2\delta_{\gamma}}(j_{V})_{!*}R^{i}(f^{\mathrm{sm}}_{V})_{*}\mathbf{Q}_{\ell}[-i],$

where $j_{V}:V^{\mathrm{sm}}\to V$ is the natural inclusion.

###### Proof.

By construction, for any point $z\in V\cap\Delta$, $V$ is transversal to the
subspace $V(\mathcal{C}_{z})$ of $T_{z}({\mathcal{B}})$. By theorem 3.1,
$\overline{\mathcal{P}}_{V}$ is nonsingular over $\mathbf{C}$. As a
consequence, the complex $R(f_{V})_{*}\mathbf{Q}_{\ell}$ is pure. Moreover,
Severi’s inequality
$\mathrm{Codim}(V\cap{\mathcal{B}}_{\delta})\geqslant\delta$ continues to hold
for all the $\delta$-strata as $V$ intersects transversally with them. Hence
the reasoning of theorem 2.4 continues to work and we get the support theorem.

∎

Let $p:V\to\mathbf{A}^{1}$ be a generic linear projection sending $x$ to $0$,
such that its geometric fibers over a sufficiently small open neighborhood of
$0\in\mathbf{A}^{1}$ intersect transversally with the strata
${\mathcal{B}}_{\beta}$ which contains $0$ in their closure. Let
$\bar{\eta}_{0}$ be a geometric generic point of the strict Henselization
$\mathbf{A}^{1}_{\\{0\\}}$. Let $R\Psi_{\bar{\eta}_{0}}$ (resp.
$R\Phi_{\bar{\eta}_{0}}$) be the nearby cycle (resp. vanishing cycle) functor
with respect to the projection $p$. We denote
$\mathcal{F}_{V}^{i}=R^{i}(f^{\mathrm{sm}}_{V})_{*}\mathbf{Q}_{\ell}$, it
equals the restriction of $\mathcal{F}^{i}$ to $V^{\mathrm{sm}}$.

######  Theorem 3.3.

Under the above settings, we have

  1. $(1)$

If ${\mathcal{B}}_{\beta}$ is not one of the strict $\delta$-strata, then

$(R\Phi_{\bar{\eta}_{0}}R(f_{V})_{*}\mathbf{Q}_{\ell})_{x}=0.$

Consequently, for $i=1,\cdots,2\delta_{\gamma}$, we have

$(R\Phi_{\bar{\eta}_{0}}(j_{V})_{!*}\mathcal{F}_{V}^{i})_{x}=0.$

  2. $(2)$

If ${\mathcal{B}}_{\beta}$ is one of the strict $\delta$-strata
${\mathcal{B}}_{\delta}^{\circ}$, then

$(R\Phi_{\bar{\eta}_{0}}R(f_{V})_{*}\mathbf{Q}_{\ell})_{x}=\boldsymbol{\Lambda}_{x}^{\bullet}\otimes\mathbf{Q}_{\ell}(-\delta)[-2\delta+1],$

where $\boldsymbol{\Lambda}_{x}^{\bullet}$ is the exterior algebra $\bigoplus
H^{*}\big{(}\mathrm{Jac}_{\widetilde{\mathcal{C}}_{x}},\mathbf{Q}_{\ell}\big{)}$
with $\widetilde{\mathcal{C}}_{x}$ being the normalization of the curve
$\mathcal{C}_{x}$. Consequently, we have

$(R^{\delta-1}\Phi_{\bar{\eta}_{0}}(j_{V})_{!*}\mathcal{F}_{V}^{i})_{x}=\boldsymbol{\Lambda}_{x}^{{i-\delta}}\otimes\mathbf{Q}_{\ell}(-\delta),$

and all the other vanishing cycles vanish.

###### Proof.

Let $x^{\prime}$ be a geometric point of
$V_{\\{x\\}}\times_{\mathbf{A}^{1}}\mathbf{A}_{\\{0\\}}^{1}$, and
$\mathcal{C}_{x^{\prime}}$ the fiber at $x^{\prime}$ of the family
$\pi:\mathcal{C}\to{\mathcal{B}}$. Let $\\{c_{i}\\}_{i\in I_{x^{\prime}}}$ be
the set of singular points of $\mathcal{C}_{x^{\prime}}$, let $R_{i}$ be the
completed local ring of $\mathcal{C}_{x^{\prime}}$ at $c_{i}$. By the
deformation theory of algebraic curves, we have a natural surjection of
tangent spaces

$T_{x^{\prime}}({\mathcal{B}})\twoheadrightarrow\bigoplus_{i\in
I_{x^{\prime}}}\mathrm{Def}_{R_{i}}^{\rm top}(k[\epsilon]/\epsilon^{2}).$
(3.1)

Let $\tau_{i}$ be the Tjurina number of the local singularity at $c_{i}$, and
$\delta_{i}$ the $\delta$-invariant. It is known that the tangent space
$\mathrm{Def}_{R_{i}}^{\rm top}(k[\epsilon]/\epsilon^{2})$ is of dimension
$\tau_{i}$ and that $\tau_{i}\geqslant\delta_{i}$.

Let $H_{x^{\prime}}$ be the fiber of $p$ passing through $x^{\prime}$, it is
of codimension $d_{\beta}+1$. Note that ${\mathcal{B}}_{\beta}$ is contained
in the $\delta$-stratum with $\delta=\delta(\mathcal{C}_{x})$, we get then the
estimate

$d_{\beta}\leqslant\tau_{\gamma}-\delta(\mathcal{C}_{x})\leqslant\tau_{\gamma}-\delta(\mathcal{C}_{x^{\prime}})=\tau_{\gamma}-\sum_{i\in
I_{x^{\prime}}}\delta_{i},$ (3.2)

where the second inequality is due to the semicontinuity of the
$\delta$-invariant ([T2], I-1.3.2). In particular, we have

$\dim(H_{x^{\prime}})=\tau_{\gamma}-(d_{\beta}+1)\geqslant\sum_{i\in
I_{x^{\prime}}}\delta_{i}-1.$ (3.3)

$(1)$ In case that ${\mathcal{B}}_{\beta}$ is not one of the strict
$\delta$-strata, the first inequality in the equation (3.2) is strict, hence
also the inequality of (3.3), and so

$\dim(H_{x^{\prime}})\geqslant\sum_{i\in I_{x^{\prime}}}\delta_{i}.$

We claim that the image of $H_{x^{\prime}}$ in $\mathrm{Def}_{R_{i}}^{\rm
top}(\mathbf{C}[\epsilon]/\epsilon^{2})$ under the projection (3.1) is
transversal to the subspace $V(R_{i})$ for all $i\in I_{x^{\prime}}$. For
this, we need to verify that the morphism

$H_{x^{\prime}}\to\bigoplus_{i\in I_{x^{\prime}}}\mathrm{Def}_{R_{i}}^{\rm
top}(k[\epsilon]/\epsilon^{2})/V(R_{i})$ (3.4)

is surjective. But this is clear because the vector space at the right hand
side is a quotient of $T_{x^{\prime}}{\mathcal{B}}$ of dimension $\sum_{i\in
I_{x^{\prime}}}\delta_{i}$, and $H_{x^{\prime}}$ is a generic subspace of
$T_{x^{\prime}}{\mathcal{B}}$ of dimension at least $\sum_{i\in
I_{x^{\prime}}}\delta_{i}$. With this transversality result, by theorem 3.1,
the subspace $(f_{V})^{-1}(H_{x^{\prime}})$ is nonsingular at the points of
the fiber $(f_{V})^{-1}(x^{\prime})$. This implies that the composite $p\circ
f_{V}$ is smooth at the points of $(f_{V})^{-1}(x)$ as it is clearly flat at
these points. The local acyclicity of the smooth morphism then implies that

$R\Phi^{\prime}_{\bar{\eta}_{0}}\mathbf{Q}_{\ell}\big{|}_{(f_{V})^{-1}(x)}=0,$

where $R\Phi^{\prime}_{\bar{\eta}_{0}}$ is the vanishing cycle functor for the
morphism $p\circ f_{V}$. As $f_{V}$ is proper, by proper base change theorem,
we get

$(R\Phi_{\bar{\eta}_{0}}R(f_{V})_{*}\mathbf{Q}_{\ell})_{x}=0.$

With the decomposition of proposition 3.2, we obtain

$\big{(}R\Phi_{\bar{\eta}_{0}}R(f_{V})_{*}\mathbf{Q}_{\ell}\big{)}_{x}=\bigoplus_{i=0}^{2\delta_{\gamma}}\big{(}R\Phi_{\bar{\eta}_{0}}(j_{V})_{!*}\mathcal{F}_{V}^{i}\big{)}_{x}[-i].$
(3.5)

By theorem 5.1, we have

$R^{q}\Phi_{\bar{\eta}_{0}}(j_{V})_{!*}\mathcal{F}_{V}^{i}[\dim(V)]=\begin{cases}{\rm
Im}\big{\\{}T-{\rm
Id};R^{-1}\Psi_{\bar{\eta}_{0}}(j_{V})_{!*}\mathcal{F}_{V}^{i}[\dim(V)]\big{\\}},&\text{
for }q=-1,\\\ 0,&\text{ otherwise,}\end{cases}$

where $T$ is the monodromy action on the nearby cycle. Hence

$\big{(}R\Phi_{\bar{\eta}_{0}}R(f_{V})_{*}\mathbf{Q}_{\ell}\big{)}_{x}=\bigoplus_{i=0}^{2\delta_{\gamma}}\big{(}R^{\dim(V)-1}\Phi_{\bar{\eta}_{0}}(j_{V})_{!*}\mathcal{F}_{V}^{i}\big{)}_{x}[-i].$
(3.6)

As the left hand side vanishes, so does each term on the right.

$(2)$ In case that ${\mathcal{B}}_{\beta}$ is one of the strict
$\delta$-strata ${\mathcal{B}}_{\delta}^{\circ}$, we get $\dim(V)=\delta$ and
that the curve $\mathcal{C}_{x}$ is irreducible with $\delta$ ordinary double
points as its singularity. We have then

$R_{i}\cong\mathbf{C}[\\![x,y]\\!]/(xy),\quad\forall i\in I_{x},$

and so

$\mathrm{Def}_{R_{i}}^{\rm
top}\cong\mathrm{Spf}(\mathbf{C}[\\![t]\\!]),\quad\forall i\in I_{x},$

and we calculate directly that $V(R_{i})=0$. Moreover, the transversal slice
$V$ can be identified with the base of a miniversal deformation of the
singularities of $\mathcal{C}_{x}$. Then the fiber $H_{x}$, being of dimension
$\delta-1$, can not make the morphism (3.4) surjective. But it does intersect
transversally with each subspace $\mathrm{Def}_{R_{i}}^{\rm
top}(\mathbf{C}[\epsilon]/\epsilon^{2})$. Hence by theorem 3.1 all the
singularities of the fiber $(p\circ f_{V})^{-1}(0)$ are smoothed when we
deform it to the fiber $(p\circ f_{V})^{-1}(\bar{\eta}_{0})$. This deformation
process can be understood more concretely as follows: Let $C_{xy}$ be the
irreducible projective rational curve over $\mathbf{C}$ with a unique ordinary
double point as singularity, let $\overline{P}_{xy}$ be its compactified
Jacobian, then $\overline{P}_{xy}\cong C_{xy}$. With the gluing construction
of Laumon [L], we have the dévissage

$1\to\underbrace{\overline{P}_{xy}\times\cdots\times\overline{P}_{xy}}_{\delta}\to\overline{\mathrm{Jac}}_{\mathcal{C}_{x}}\to\mathrm{Jac}_{\widetilde{\mathcal{C}}_{x}}\to
1,$ (3.7)

and the deformation is only for the first factor. Let
$\hat{f}_{xy}:\widehat{\mathcal{P}}_{xy}\to\widehat{\mathbf{A}}_{0}^{1}$ be
the miniversal deformation of $\overline{P}_{xy}$. The deformation in question
is then equivalent to

$\underbrace{\widehat{\mathcal{P}}_{xy}\times\cdots\times\widehat{\mathcal{P}}_{xy}}_{\delta}\xrightarrow{\hat{f}_{xy}\times\cdots\times\hat{f}_{xy}}\underbrace{\widehat{\mathbf{A}}_{0}^{1}\times\cdots\times\widehat{\mathbf{A}}_{0}^{1}}_{\delta}\xrightarrow{\sum_{i=1}^{\delta}a_{i}}\widehat{\mathbf{A}}_{0}^{1}.$
(3.8)

As $\overline{P}_{xy}\cong C_{xy}$, the deformation
$\hat{f}_{xy}:\widehat{\mathcal{P}}_{xy}\to\widehat{\mathbf{A}}_{0}^{1}$ can
be described explicitly as $\mathbf{C}^{2}\to\mathbf{C}:(x,y)\mapsto xy$, and
the above composite is nothing but

$\mathbf{C}^{2\delta}\to\mathbf{C},\quad(x_{1},y_{1},\cdots,x_{\delta},y_{\delta})\mapsto\sum_{i=1}^{\delta}x_{i}y_{i},$

which describes the miniversal deformation of an ordinary quadratic
singularity. Its vanishing cycle and the monodromy action is well known. Let
$R\overline{\Phi}_{\bar{\eta}_{0}}$ be the vanishing cycle functor for the
composite (3.8), then $R^{m}\bar{\Phi}_{\bar{\eta}_{0}}\mathbf{Q}_{\ell}$ is
supported at the point $\bar{c}:=(c,\cdots,c)$ with $c$ being the unique
double point of $\overline{P}_{xy}$, and that

$(R^{m}\bar{\Phi}_{\bar{\eta}_{0}}\mathbf{Q}_{\ell})_{\bar{c}}=\begin{cases}\mathbf{Q}_{\ell}(-\delta),&\text{if
}m=2\delta-1,\\\ 0,&\text{otherwise}.\end{cases}$

With the dévissage (3.7) and the freeness property in [N], proposition 7.4.10,
we obtain that

$(R\Phi_{\bar{\eta}_{0}}R(f_{V})_{*}\mathbf{Q}_{\ell})_{x}=\boldsymbol{\Lambda}_{x}^{\bullet}\otimes\mathbf{Q}_{\ell}(-\delta)[-2\delta+1].$

With the same argument as in the first case, we have the decomposition (3.6),
hence

$\bigoplus_{i=0}^{2\delta_{\gamma}}\big{(}R^{\delta-1}\Phi_{\bar{\eta}_{0}}j_{V})_{!*}\mathcal{F}_{V}^{i}\big{)}_{x}[-i]=\boldsymbol{\Lambda}_{x}^{\bullet}\otimes\mathbf{Q}_{\ell}(-\delta)[-2\delta+1],$

where we have used the equality $\dim(V)=\delta$. Compare the terms of the
same degree on both sides, we get the second assertion.

∎

######  Remark 3.4.

The results above are intrinsic to the singularities of $\mathcal{C}_{x}$.
Indeed, by the openness of versality, the restriction of the family
$\pi:\mathcal{C}\to{\mathcal{B}}$ to the formal neighbourhood
$\widehat{{\mathcal{B}}}_{x}$ of $x$ in ${\mathcal{B}}$ defines a versal
deformation for the singularities of the fiber $\mathcal{C}_{x}$. More
precisely, let $\\{c_{i}\\}_{i\in I_{x}}$ be the set of singularities of
$\mathcal{C}_{x}$, let $R_{i}=\widehat{\mathcal{O}}_{\mathcal{C}_{x},c_{i}}$
and $\widehat{\phi}_{i}:\widehat{C}_{i}\to\widehat{B}_{i}$ be the miniversal
deformation of the singularity $\mathrm{Spf}(R_{i})$. Then there exists a
smooth morphism $h:\widehat{{\mathcal{B}}}_{x}\to\prod_{i\in
I_{x}}\widehat{B}_{i}$ such that the family
$\mathcal{C}\times_{{\mathcal{B}}}\widehat{{\mathcal{B}}}_{x}\to\widehat{{\mathcal{B}}}_{x}$
is the pull-back of the family $\prod_{i\in I_{x}}\widehat{\phi}_{i}$ along
$h$. For each family $\widehat{\phi}_{i}$, we have the canonical Whitney
stratification and we can make transversal slice $V_{i}$ to the stratum
containing the origin as in the theorem. Then the restriction of $\pi$ to the
transversal slice $V$ is essentially the same as the product of the
restriction of $\widehat{\phi}_{i}$ to $V_{i}$, similar factorization holds
for $\mathcal{F}^{1}\cong R^{1}\pi^{\mathrm{sm}}_{*}\mathbf{Q}_{\ell}$ and
also the complex $j_{!*}\mathcal{F}^{i}$ as
$\mathcal{F}^{i}=\bigwedge^{i}\mathcal{F}^{1}$.

###  3.2 Reduction to the nearby fiber

Following the general procedure described in the appendix §5, we make a
microlocal analysis of the fiber
$(j_{!*}R^{i}f_{*}^{\mathrm{sm}}\mathbf{Q}_{\ell})_{0}$. Note that
$\dim({\mathcal{B}})=\tau_{\gamma}\geqslant\delta_{\gamma}$, and the equality
holds if and only if the singularity $\mathrm{Spf}(\mathcal{O}[\gamma])$ is a
node, i.e. $\gamma$ is of the form
$\gamma=\mathrm{diag}(\gamma_{1},\gamma_{2})\in\mathfrak{gl}_{2}(\mathcal{O})$
with ${\rm val}(\gamma_{1}-\gamma_{2})=1$. Indeed, the $\delta$-stratum
containing $0$ is of codimension $\delta_{\gamma}$ in ${\mathcal{B}}$
according to Diaz-Harris [DH], hence in case $\tau_{\gamma}=\delta_{\gamma}$
the singularity $\mathrm{Spf}(\mathcal{O}[\gamma])$ will be a node. The
geometry of $\mathcal{X}_{\gamma}$ is clear for such elements, and we will
therefore skip this case and assume that $\tau_{\gamma}>\delta_{\gamma}$. As
there are only finitely many $\delta$-strata, the linear subspaces of
${\mathcal{B}}=\mathbf{A}^{\tau_{\gamma}}$ of dimension $\delta_{\gamma}+1$
which intersects transversally with the tangent cone at $0$ of all the
$\delta$-strata form a dense open subvariety of the Grassmannian
$\mathrm{Gr}_{\delta_{\gamma}+1,\tau_{\gamma}}$. Take such a linear subspace
$H_{0}$ and cut off the locus where its intersection with the $\delta$-strata
is not transversal, we get a dense open subscheme ${\mathcal{B}}^{\prime}$ of
it. By construction, ${\mathcal{B}}^{\prime}$ intersects transversally with
all the $\delta$-strata and it contains $0$. Let
$\mathcal{C}^{\prime}=\mathcal{C}\times_{{\mathcal{B}}}{\mathcal{B}}^{\prime}$,
consider the restriction

$f^{\prime}:\overline{\mathcal{P}}{}^{\prime}=\overline{\mathrm{Pic}}{}^{\,0}_{\mathcal{C}^{\prime}/{\mathcal{B}}^{\prime}}\to{\mathcal{B}}^{\prime}.$

######  Proposition 3.5.

The subspace $\overline{\mathcal{P}}{}^{\prime}$ is nonsingular over
$\mathbf{C}$, and Ngô’s support theorem holds for the family $f^{\prime}$,
i.e.

$Rf^{\prime}_{*}\mathbf{Q}_{\ell}=\bigoplus_{i=0}^{2\delta_{\gamma}}j^{\prime}_{!*}(R^{\,i}{f^{\prime}}^{\mathrm{sm}}_{*}\mathbf{Q}_{\ell})[-i],$

where
$j^{\prime}:({\mathcal{B}}^{\prime})^{\mathrm{sm}}\to{\mathcal{B}}^{\prime}$
is the natural inclusion.

###### Proof.

By construction, for any point $z\in{\mathcal{B}}^{\prime}\cap\Delta$,
$T_{z}({\mathcal{B}}^{\prime})$ is transversal to the subspace
$V(\mathcal{C}_{x})$ of $T_{x}({\mathcal{B}})$. By theorem 3.1,
$\overline{\mathcal{P}}^{\prime}$ is nonsingular over $\mathbf{C}$. As a
consequence, the complex $Rf^{\prime}_{*}\mathbf{Q}_{\ell}$ is pure. To get
the support theorem, note that Severi’s inequality continues to hold as
${\mathcal{B}}^{\prime}$ intersects transversally with the $\delta$-strata,
hence the reasoning of theorem 2.4 works in this case as well.

∎

Let $p:{\mathcal{B}}^{\prime}\to\mathbf{A}^{1}$ be a generic projection. Let
$\eta_{0}$ be the generic point of the strict Henselization
$\mathbf{A}^{1}_{\\{0\\}}$ of $\mathbf{A}^{1}$ at $0$, and let
$\bar{\eta}_{0}$ be a geometric point over it. Let $R\Psi_{\bar{\eta}_{0}}$
(resp. $R\Phi_{\bar{\eta}_{0}}$) be the nearby cycle (resp. vanishing cycle)
functor with respect to the projection $p$. We keep the notation
$\mathcal{F}^{i}$ for its restriction to ${\mathcal{B}}^{\prime}$.

######  Proposition 3.6.

For $i=1,\cdots,2\delta_{\gamma}$, we have
$(R\Phi_{\bar{\eta}_{0}}j^{\prime}_{!*}\mathcal{F}^{i})_{0}=0,$ and so

$(j^{\prime}_{!*}\mathcal{F}^{i})_{0}=(R\Psi_{\bar{\eta}_{0}}j^{\prime}_{!*}\mathcal{F}^{i})_{0},\quad
i=0,\cdots,2\delta_{\gamma}.$

Consequently, let $B_{0}(\epsilon)$ be a sufficiently small open ball around
$0\in{\mathcal{B}}=\mathbf{A}^{\tau_{\gamma}}$ and $D_{0}(\epsilon^{\prime})$
a sufficiently small open interval around $0\in\mathbf{A}^{1}$, then

$\mathcal{H}^{*}(j^{\prime}_{!*}\mathcal{F}^{i})_{0}=H^{*}\big{(}{\mathcal{B}}^{\prime}\cap
B_{0}(\epsilon)\cap p^{-1}(t),j^{\prime}_{!*}\mathcal{F}^{i}\big{)},\quad t\in
D_{0}(\epsilon^{\prime}).$

###### Proof.

By the choice of $p:{\mathcal{B}}^{\prime}\to\mathbf{A}^{1}$, the tangent
space at $0$ of its fiber at $0$ is generic of dimension $\delta_{\gamma}$ in
$T_{0}({\mathcal{B}})$, hence transversal to the codimension $\delta_{\gamma}$
subspace $V(C_{\gamma})$ of $T_{0}({\mathcal{B}})$. By theorem 3.1, the
composite $p\circ f^{\prime}:\overline{\mathcal{P}}^{\prime}\to\mathbf{A}^{1}$
is smooth at an étale neighborhood of $(f^{\prime})^{-1}(0)$. By the local
acyclicity of the smooth morphism, we have

$R\Phi^{\prime}_{\bar{\eta}_{0}}\mathbf{Q}_{\ell}\big{|}_{(f^{\prime})^{-1}(0)}=0,$

where $R\Phi^{\prime}_{\bar{\eta}_{0}}$ is the vanishing cycle functor for the
morphism $p\circ f^{\prime}$. As $f^{\prime}$ is proper, by proper base change
theorem, we get

$(R\Phi_{\bar{\eta}_{0}}Rf^{\prime}_{*}\mathbf{Q}_{\ell})_{0}=0.$

With the decomposition in proposition 3.5, we deduce that
$(R\Phi_{\bar{\eta}_{0}}j^{\prime}_{!*}\mathcal{F}^{i})_{0}=0$ for all $i$ as
in theorem 3.3. All the other assertions then follow.

∎

######  Corollary 3.7.

With the above notations, for $i=0,\cdots,2\delta_{\gamma}$, we have the
decomposition

$H^{i}(\overline{P}_{C_{\gamma}},\mathbf{Q}_{\ell})=\bigoplus_{i^{\prime}=0}^{i}H^{i-i^{\prime}}\big{(}{\mathcal{B}}^{\prime}\cap
B_{0}(\epsilon)\cap p^{-1}(t),j^{\prime}_{!*}\mathcal{F}^{i^{\prime}}\big{)}.$

###### Proof.

This is a corollary of proposition 3.5 and 3.6, and the equality
$H^{*}(\overline{P}_{C_{\gamma}},\mathbf{Q}_{\ell})=(R^{*}f^{\prime}_{*}\mathbf{Q}_{\ell})_{0}.$

∎

###  3.3 Iterated fiberations of the nearby fiber

We make further analysis of the cohomologies in corollary 3.7 by iterated
fiberations of the nearby fiber as explained in the appendix §5. We take a
generic flag of linear subspaces

$\\{0\\}=H_{\delta_{\gamma}+1}\subsetneq
H_{\delta_{\gamma}}\subsetneq\cdots\subsetneq H_{2}\subsetneq H_{1}\subsetneq
H_{0},\quad\dim(H_{k})=\delta_{\gamma}+1-k\text{ for all }1\leqslant
k\leqslant\delta_{\gamma}+1,$

for which the conclusions of proposition 4.11 and 4.14 hold for the germs of
${\mathcal{B}}^{\prime}\cap{\mathcal{B}}_{\beta}$ at $0$,
$\beta\in{\mathfrak{B}}$. This is possible as there are only finitely many
such germs. Let $\pi_{k}:H_{0}\to\mathbf{C}^{k}$ be the projection with kernel
$H_{k}$. Let $t_{k}\in\mathbf{C}^{k}$ be a point sufficiently close to $0$,
the projection $\pi_{k+1}$ induces an affine map

$p_{k+1}:{\mathcal{B}}^{\prime}\cap
B_{0}(\epsilon)\cap\pi_{k}^{-1}(t_{k})\to\mathbf{C}^{1}.$

Its critical locus equals the intersection of $\pi_{k}^{-1}(t_{k})$ with the
local polar variety $P_{0}({\mathcal{B}}^{\prime}\cap
B_{0}(\epsilon),H_{k+1})$ (see §4.2 for a brief review of local polar
varieties), which is finite over $\mathbf{C}$. The points
$\\{t_{k}\\}_{k=1}^{\delta_{\gamma}+1}$ have been taken such that
${\mathcal{B}}^{\prime}\cap B_{0}(\epsilon)\cap\pi_{k+1}^{-1}(t_{k+1})$
coincides with the fiber of $p_{k+1}$ at a generic point in ${\rm
Im}(p_{k+1})$. We use the iterated fibrations
$p_{1},\cdots,p_{\delta_{\gamma}+1}$ to calculate the cohomology

$\mathcal{H}^{*}(j^{\prime}_{!*}\mathcal{F}^{i})_{0}=H^{*}\big{(}{\mathcal{B}}^{\prime}\cap
B_{0}(\epsilon),j^{\prime}_{!*}\mathcal{F}^{i}\big{)}.$ (3.9)

Note that the first fibration $p_{1}$ is just the projection $p$ in the
previous section. By proposition 3.6, we have

$H^{*}\big{(}{\mathcal{B}}^{\prime}\cap
B_{0}(\epsilon),j^{\prime}_{!*}\mathcal{F}^{i}\big{)}=H^{*}\big{(}{\mathcal{B}}^{\prime}\cap
B_{0}(\epsilon)\cap p_{1}^{-1}(t_{1}),j^{\prime}_{!*}\mathcal{F}^{i}\big{)}.$

With the fibration $p_{2}:{\mathcal{B}}^{\prime}\cap B_{0}(\epsilon)\cap
p_{1}^{-1}(t_{1})\to\mathbf{C}^{1}$, the above cohomological group can be
built up from

$H^{*}\big{(}{\mathcal{B}}^{\prime}\cap B_{0}(\epsilon)\cap
p_{2}^{-1}(t_{2}),j^{\prime}_{!*}\mathcal{F}^{i}\big{)}$

and the vanishing cycles of the complex $j^{\prime}_{!*}\mathcal{F}^{i}$ with
respect to $p_{2}$. By construction, the vanishing cycles are supported at the
finite subscheme $\pi_{1}^{-1}(t_{1})\cap P_{0}({\mathcal{B}}^{\prime}\cap
B_{0}(\epsilon),H_{2})$. This process can be iterated with the fibrations
$p_{3}$, $p_{4}$ and so on until $p_{\delta_{\gamma}+1}$.

We can simplify the calculations with stratified Morse theory. Indeed, the
ramification of $p_{k+1}:{\mathcal{B}}^{\prime}\cap
B_{0}(\epsilon)\cap\pi_{k}^{-1}(t_{k})\to\mathbf{C}^{1}$ can be very
complicated. We replace the projection $p_{k+1}$ by a stratified Morse
function

$\varphi_{k+1}:{\mathcal{B}}^{\prime}\cap
B_{0}(\epsilon)\cap\pi_{k}^{-1}(t_{k})\to\mathbf{R}^{1}$

with respect to the stratification of ${\mathcal{B}}^{\prime}\cap
B_{0}(\epsilon)\cap\pi_{k}^{-1}(t_{k})$ induced from the canonical Whitney
stratification $\\{{\mathcal{B}}_{\beta}\\}_{\beta\in{\mathfrak{B}}}$, which
approximates sufficiently well to the function ${\rm Re}(p_{k+1})$. Let
$v_{k}$ be the smallest critical value of $\varphi_{k+1}$, then

$H^{*}\big{(}{\mathcal{B}}^{\prime}\cap
B_{0}(\epsilon)\cap\pi_{k}^{-1}(t_{k})\cap\varphi_{k+1}^{-1}(-\infty,v_{k}),j^{\prime}_{!*}\mathcal{F}^{i}\big{)}=H^{*}\big{(}{\mathcal{B}}^{\prime}\cap
B_{0}(\epsilon)\cap
p_{k+1}^{-1}(t_{k+1}),j^{\prime}_{!*}\mathcal{F}^{i}\big{)},$

and the cohomology group $H^{*}\big{(}{\mathcal{B}}^{\prime}\cap
B_{0}(\epsilon)\cap\pi_{k}^{-1}(t_{k}),j^{\prime}_{!*}\mathcal{F}^{i}\big{)}$
can be built up from it and the Morse groups at the critical points of
$\varphi_{k+1}$. For simplicity, we denote $F^{k}={\mathcal{B}}^{\prime}\cap
B_{0}(\epsilon)\cap\pi_{k}^{-1}(t_{k})$.

######  Proposition 3.8.

Under the above setting, let $x\in F^{k}\cap{\mathcal{B}}_{\beta}$ be a
critical point of $\varphi_{k+1}$ with critical value $v$, let $\lambda$ be
the Morse index of the restriction of $\varphi_{k+1}$ to
$F^{k}\cap{\mathcal{B}}_{\beta}$. Let $a$ be a sufficiently small positive
real number.

  1. $(1)$

If ${\mathcal{B}}_{\beta}$ is not one of the strict $\delta$-strata, then

$H^{*}\big{(}F^{k}_{>v+a},F^{k}_{<v-a};j^{\prime}_{!*}\mathcal{F}^{i}\big{)}=0,\quad\text{for
all }i.$

  2. $(2)$

If ${\mathcal{B}}_{\beta}$ is one of the strict $\delta$-strata
${\mathcal{B}}_{\delta}^{\circ}$, then for all $i$ we have

$H^{q}\big{(}F^{k}_{>v+a},F^{k}_{<v-a};j^{\prime}_{!*}\mathcal{F}^{i}\big{)}=\begin{cases}\boldsymbol{\Lambda}_{x}^{{i-\delta}},&\text{
if }q=\lambda+\delta,\\\ 0,&\text{otherwise}.\end{cases}$

###### Proof.

According to Goresky and MacPherson [GM], part II, 6.4 and the appendix 6.A,
to calculate the Morse group, we take a transversal slice $N$ to
$F^{k}\cap{\mathcal{B}}_{\beta}$ in $F_{k}$ and a generic projection
$p:N\to\mathbf{C}^{1}$ sending $x$ to $0$, then the Morse group in question is
equal to the vanishing cycle

$\big{(}R^{d_{N}-1}\Phi_{\bar{\eta}_{0}}j^{\prime}_{!*}\mathcal{F}^{i}\big{)}_{x},$
(3.10)

where $R\Phi_{\bar{\eta}_{0}}$ is the vanishing cycle for the projection $p$
at the point $0$ and $d_{N}$ is the dimension of $N$. Note that the projection
$p_{k}$ and the point $t_{k}$ have been chosen generic, hence the fiber
$F^{k}$ intersects transversally with the discriminant $\Delta$. Consequently,
for any point $x^{\prime}\in F^{k}\cap\Delta$, the restriction of
$\pi:\mathcal{C}\to{\mathcal{B}}$ to the formal neighbourhood of $x^{\prime}$
in $F^{k}$ defines a versal deformation of the singularities of
$\mathcal{C}_{x^{\prime}}$ by the openness of versality. As explained in
remark 3.4, this implies that the restriction of the complex
$j^{\prime}_{!*}\mathcal{F}^{i}$ to the transversal slice $N$ depends only on
the singularities of $\mathcal{C}_{x}$, and it is essentially the same as the
restriction of $j_{!*}\mathcal{F}^{i}$ to the transversal slice $V$ in theorem
3.3, as both can be reduced to transversal slices in the miniversal
deformations of the singularities of $\mathcal{C}_{x}$. In particular, the
vanishing cycle (3.10) equals that of theorem 3.3 and the assertion follows
from the theorem.

∎

######  Remark 3.9.

In conclusion, the cohomology groups

$H^{*}(F^{k},j^{\prime}_{!*}\mathcal{F}^{i})=H^{*}\big{(}{\mathcal{B}}^{\prime}\cap
B_{0}(\epsilon)\cap\pi_{k}^{-1}(t_{k}),j^{\prime}_{!*}\mathcal{F}^{i}\big{)},\quad
k=1,\cdots,\delta_{\gamma}+1,$

can be calculated inductively, starting from
$H^{*}(F^{\delta_{\gamma}+1},j^{\prime}_{!*}\mathcal{F}^{i})$ and arriving at

$H^{*}(F^{1},j^{\prime}_{!*}\mathcal{F}^{i})=H^{*}\big{(}{\mathcal{B}}^{\prime}\cap
B_{0}(\epsilon)\cap\pi_{1}^{-1}(t_{1}),j^{\prime}_{!*}\mathcal{F}^{i}\big{)}=\mathcal{H}^{*}(j^{\prime}_{!*}\mathcal{F}^{i})_{0}.$

With stratified Morse theory, the cohomology
$H^{*}(F^{k},j^{\prime}_{!*}\mathcal{F}^{i})$ can be built up from

$H^{*}\big{(}F^{k}\cap\varphi_{k+1}^{-1}(-\infty,v_{k}),j^{\prime}_{!*}\mathcal{F}^{i}\big{)}=H^{*}(F^{k+1},j^{\prime}_{!*}\mathcal{F}^{i})$

and the Morse groups
$H^{*}\big{(}F^{k}_{>v+a},F^{k}_{<v-a};j^{\prime}_{!*}\mathcal{F}^{i}\big{)}$
at the critical values $v$ via the long exact sequence

$\cdots\to H^{*}\big{(}F^{k}_{<v-a},j^{\prime}_{!*}\mathcal{F}^{i}\big{)}\to
H^{*}\big{(}F^{k}_{>v+a},j^{\prime}_{!*}\mathcal{F}^{i}\big{)}\to
H^{*}\big{(}F^{k}_{>v+a},F^{k}_{<v-a};j^{\prime}_{!*}\mathcal{F}^{i}\big{)}\to\cdots.$

To actually carry out this process, we still need to know how to attach

$H^{q}\big{(}F^{k}_{>v+a},F^{k}_{<v-a};j^{\prime}_{!*}\mathcal{F}^{i}\big{)}\to
H^{q+1}\big{(}F^{k}_{<v-a},j^{\prime}_{!*}\mathcal{F}^{i}\big{)},\quad\text{for
all }q,$

which seems to be out of reach now. Nonetheless, the above proposition tells
us that the Morse groups are non-trivial only over the strict $\delta$-strata,
which is enough for our purpose.

##  4 Whitney regularity along the equisingular strata

Based on a study of the geometry of the discriminant $\Delta$, we will show
that the union of the strict $\delta$-strata is Whitney regular over the
equisingular strata. In conjunction with the microlocal analysis of the
complexes $j_{!*}\mathcal{F}^{i}$, this implies that the sheaves
$\mathcal{H}^{*}(j_{!*}\mathcal{F}^{i})$ are locally constant over the
equisingular strata.

###  4.1 The geometry of the discriminant

Teissier [T1] has discovered a very remarkable property of the discriminant of
the miniversal deformation of isolated hypersurface singularities, which is
further extended to the isolated complete intersection singularities by
Looijenga [Loo]. We make a recount of their result.

Let $(X_{0},0)\subset(\mathbf{C}^{n},0)$ be a germ of isolated complete
intersection singularity. Let
$\mathrm{Def}_{(X_{0},0)\hookrightarrow(\mathbf{C}^{n},0)}$ be the deformation
functor for the embedding $(X_{0},0)\hookrightarrow(\mathbf{C}^{n},0)$, and
let $\mathrm{Def}_{(X_{0},0)}$ be the deformation functor for the germ
$(X_{0},0)$. Let $T^{1}_{(X_{0},0)\hookrightarrow(\mathbf{C}^{n},0)}$ and
$T_{(X_{0},0)}^{1}$ be their tangent spaces, i.e.

$T^{1}_{(X_{0},0)\hookrightarrow(\mathbf{C}^{n},0)}=\mathrm{Def}_{(X_{0},0)\hookrightarrow(\mathbf{C}^{n},0)}(\mathbf{C}[\varepsilon]/\varepsilon^{2}),\quad
T_{(X_{0},0)}^{1}=\mathrm{Def}_{(X_{0},0)}(\mathbf{C}[\varepsilon]/\varepsilon^{2}).$

Let $I_{0}\subset\mathcal{O}_{\mathbf{C}^{n},0}$ be the defining ideal of
$X_{0}$, then its conormal sheaf $I_{0}/I_{0}^{2}$ is locally free, and we get
an exact sequence

$0\to
I_{0}/I_{0}^{2}\to\Omega_{\mathbf{C}^{n}}^{1}\otimes_{\mathcal{O}_{\mathbf{C}^{n}}}\mathcal{O}_{X_{0}}\to\Omega_{X_{0}}^{1}\to
0.$ (4.1)

Let
$\mathcal{N}_{X_{0}/\mathbf{C}^{n}}:=\mathcal{H}om_{\mathcal{O}_{X_{0}}}(I_{0}/I_{0}^{2},\mathcal{O}_{X_{0}})$,
we call it the normal sheaf of $X_{0}$. According to the general deformation
theory (cf. [GLS]), we can identify the tangent space

$T^{1}_{(X_{0},0)\hookrightarrow(\mathbf{C}^{n},0)}=\mathcal{N}_{X_{0}/\mathbf{C}^{n},0},$

and the tangent space $T^{1}_{(X_{0},0)}$ fits in the exact sequence

$0\to\Theta_{X_{0},0}\to\Theta_{\mathbf{C}^{n},0}\otimes_{\mathcal{O}_{\mathbf{C}^{n},0}}\mathcal{O}_{X_{0},0}\to\mathcal{N}_{X_{0}/\mathbf{C}^{n},0}\to
T^{1}_{(X_{0},0)}\to 0,$ (4.2)

which is obtained from the exact sequence (4.1) by taking dual. Note that this
leads to the isomorphism

$T^{1}_{(X_{0},0)}\cong\mathcal{E}xt_{\mathcal{O}_{X_{0},0}}^{1}(\Omega_{X_{0},0}^{1},\mathcal{O}_{X_{0},0}).$

Let $f=(f_{1},\cdots,f_{k})$ be a minimal set of generators for the defining
ideal $I_{0}$ of $X_{0}$, let $Df$ be the Jacobian matrix

$Df=\begin{bmatrix}\frac{\partial f_{1}}{\partial x_{1}}&\cdots&\frac{\partial
f_{1}}{\partial x_{n}}\\\ \vdots&\ddots&\vdots\\\ \frac{\partial
f_{k}}{\partial x_{1}}&\cdots&\frac{\partial f_{k}}{\partial
x_{n}}\end{bmatrix}.$

With the exact sequence (4.2), we calculate

$T_{(X_{0},0)}^{1}=\mathcal{O}_{\mathbf{C}^{n},0}^{k}/(Df\cdot\mathcal{O}_{\mathbf{C}^{n},0}^{n}+I_{0}\cdot\mathcal{O}_{\mathbf{C}^{n},0}^{k}).$

The $\mathcal{O}_{X_{0},0}$-module at the right hand side is traditionally
called _Tjurina module_. Let
$g_{1},\cdots,g_{\tau}\in\mathcal{O}_{\mathbf{C}^{n},0}^{k}$,
$g_{i}=(g_{i,1},\cdots,g_{i,k})^{t}$, be a set of representatives for a basis
of the above quotient as a vector space over $\mathbf{C}$. Let

$F_{i}(\mathbf{x},\mathbf{t})=f_{i}(x_{1},\cdots,x_{n})+\sum_{j=1}^{\tau}t_{j}g_{i,j}(x_{1},\cdots,x_{n}),\quad
i=1,\cdots,k.$

Let $I\subset\mathcal{O}_{\mathbf{C}^{n}\times\mathbf{C}^{\tau}}$ be the ideal
generated by $F_{1},\cdots,F_{k}$, and let $X$ be the closed subscheme of
$\mathbf{C}^{n}\times\mathbf{C}^{\tau}$ defined by $I$. With the second
projection $\mathbf{C}^{n}\times\mathbf{C}^{\tau}\to\mathbf{C}^{\tau}$, we get
a family

$\psi:(X,0)\to(S=\mathbf{C}^{\tau},0),$

which is a miniversal deformation of $X_{0}$. One verifies that $X$ is
nonsingular. Let $\Sigma\subset X$ be the _critical locus_ of the morphism
$\psi$, i.e. it is the closed subscheme defined by the Fitting ideal ${\rm
F}_{n-k}(\Omega^{1}_{X/S})$. Then all the geometric fibers of $\psi$ have
isolated complete intersection singularities at its intersection with
$\Sigma$. In concrete terms, $\Sigma$ is the closed subscheme of $X$ defined
by the condition that the Jacobian matrix

$\begin{bmatrix}\frac{\partial F_{1}}{\partial x_{1}}&\cdots&\frac{\partial
F_{1}}{\partial x_{n}}\\\ \vdots&\ddots&\vdots\\\ \frac{\partial
F_{k}}{\partial x_{1}}&\cdots&\frac{\partial F_{k}}{\partial
x_{n}}\end{bmatrix}$

is of rank $<k$, i.e. all the minors of the Jacobian matrix of size $k\times
k$ vanish.

At each geometric point $x\in\Sigma$, let $s=\psi(x)$, we get the Tjurina
module $T^{1}_{(X_{s},x)}$. They can be put in family as $x$ varies. We start
from the exact sequence

$0\to\psi^{*}\Omega_{S}^{1}\to\Omega_{X}^{1}\to\Omega_{X/S}^{1}\to 0.$

Take its dual, we get

$\cdots\to\Theta_{X}\to\psi^{*}\Theta_{S}\to\mathcal{E}xt_{\mathcal{O}_{X}}^{1}(\Omega_{X/S}^{1},\mathcal{O}_{X})\to
0.$ (4.3)

Let
$\mathcal{T}_{X/S}^{1}=\mathcal{E}xt_{\mathcal{O}_{X}}^{1}(\Omega_{X/S}^{1},\mathcal{O}_{X})$,
then

$\mathcal{T}_{X/S}^{1}\otimes_{\mathcal{O}_{X}}\mathcal{O}_{X_{s},x}=\mathcal{E}xt_{\mathcal{O}_{X_{s},x}}^{1}(\Omega_{\mathcal{O}_{X_{s},x}}^{1},\mathcal{O}_{\mathcal{O}_{X_{s},x}})=T^{1}_{(X_{s},x)}.$
(4.4)

Let
$F=(F_{1},\cdots,F_{k})^{t}\in\mathcal{O}^{k}_{\mathbf{C}^{n}\times\mathbf{C}^{\tau},0}$,
and let $DF$ be its Jacobian matrix with respect to the variables
$x_{1},\cdots,x_{n}$. The exact sequence (4.2) can be put in families as well,
and from it we calculates

$\mathcal{T}_{X/S}^{1}=\mathcal{O}_{\mathbf{C}^{n}\times\mathbf{C}^{\tau},0}^{k}/(DF\cdot\mathcal{O}_{\mathbf{C}^{n}\times\mathbf{C}^{\tau},0}^{n}+I\cdot\mathcal{O}_{\mathbf{C}^{n}\times\mathbf{C}^{\tau},0}^{k}).$

The $\mathcal{O}_{X}$-module at the right hand side is called the _relative
Tjurina module_ of the family $\psi:X\to S$.

Actually the above construction of $\mathcal{T}_{X/S}^{1}$ works for any
deformation $\psi^{\prime}:X^{\prime}\to S^{\prime}$ of $X_{0}$. From the
analogue of the exact sequence (4.3), for any geometric point $x\in
X^{\prime}$ with image $s\in S^{\prime}$, we deduce a morphism

$\Theta_{S^{\prime},s}\xrightarrow{\psi^{-1}}(\psi^{*}\Theta_{S^{\prime}})_{x}\to\mathcal{T}^{1}_{X^{\prime}/S^{\prime},x}.$

Modulo the maximal ideal ${\mathfrak{m}}_{S^{\prime},s}$ of
$\mathcal{O}_{S^{\prime},s}$, we get the _Kodaira-Spencer_ map

$\kappa_{x}:T_{s}S^{\prime}\to T_{(X^{\prime}_{s},x)}^{1}$

It is known that $\kappa_{x}$ is surjective if $\psi^{\prime}$ is versal and
isomorphic if $\psi^{\prime}$ is miniversal (cf. [GLS]).

Let $\Delta\subset S$ be the image of $\Sigma$, called the _discriminant_ of
the morphism $\psi$, it is the closed subscheme defined by the Fitting ideal
${\rm F}_{0}(\psi_{*}\mathcal{O}_{\Sigma})$.

######  Theorem 4.1 (Purity of the discriminant, Teissier [T1]).

Let $\psi:(X,0)\to(S,0)$ be a miniversal deformation of an isolated complete
intersection singularity as above. Then $\Delta\subset(S,0)$ is a hypersurface
which is locally analytically irreducible at $0$, and the induced morphism
$\Sigma\to\Delta$ is the normalization map.

###### Proof.

We give a sketch of the proof. From the above concrete description of $\psi$,
one deduce that $X$ is regular and it is of dimension $n-k+\tau$, and that the
restriction $\psi:\Sigma\to\Delta$ is a finite morphism. In particular,
$\dim(\Sigma)=\dim(\Delta)$. As the generic fiber of $\psi$ is smooth, we
obtain that $\dim(\Delta)\leqslant\tau-1$. On the other hand, for any
geometric point $x\in X$, take a local coordinate of $X$ at $x$, the tangent
map induces a morphism ${\rm d}\psi_{x}:(X,x)\to{\rm
Hom}(\mathbf{C}^{n-k+\tau},\mathbf{C}^{\tau})$. For $m,d\in\mathbf{N}$ and
$r=0,\cdots,\min(m,d)$, let

$\mathcal{O}_{r}(m,d)=\big{\\{}A\in{\rm
Hom}(\mathbf{C}^{m},\mathbf{C}^{d})\mid\mathrm{rk}(A)=r\big{\\}}.$

Consider the action of $\mathrm{GL}_{m}\times\mathrm{GL}_{d}$ on ${\rm
Hom}(\mathbf{C}^{m},\mathbf{C}^{d})$, one verifies that $\mathcal{O}_{r}(m,d)$
is a smooth irreducible affine algebraic variety of codimension $(m-r)(d-r)$
in ${\rm Hom}(\mathbf{C}^{m},\mathbf{C}^{d})$, and that the Zariski closure
satisfies
$\overline{\mathcal{O}}_{r}(m,d)=\bigsqcup_{r^{\prime}=0}^{r}\mathcal{O}_{r^{\prime}}(m,d)$.
Then the germ $(\Sigma,x)$ is the inverse image of
$\overline{\mathcal{O}}_{\tau-1}(n-k+\tau,\tau)$ under the morphism ${\rm
d}\psi_{x}$. Hence the codimension of $(\Sigma,x)$ in $(X,x)$ is at most
$n-k+1$, i.e. $\dim(\Sigma)\geqslant\tau-1$. Combining the two inequalities,
we obtain that both $\Sigma$ and $\Delta$ are of pure dimension $\tau-1$. In
particular, $\Delta$ is a hypersurface in $(\mathbf{C}^{\tau},0)$.

On verifies then that $\Sigma$ is Cohen-MaCaulay. With the concrete
description of $\psi$, one verifies that ${\rm F}_{n-k}(\Omega_{X/S}^{1})={\rm
F}_{0}(\mathcal{T}_{X/S}^{1})$. As $X$ is regular and $\Sigma$ is of pure
codimension $n-k+1$, we have

${\rm depth}\big{(}{\rm
F}_{0}(\mathcal{T}_{X/S,x}^{1}),\mathcal{O}_{X,x}\big{)}=n-k+1.$ (4.5)

By definition, $\mathcal{T}_{X/S}^{1}$ has presentation

$\mathcal{O}_{X}^{n}\xrightarrow{DF}\mathcal{O}_{X}^{k}\to\mathcal{T}_{X/S}^{1}\to
0.$

According to Buchsbaum-Rim [BR], Corollary 2.7, the equality (4.5) implies
that the homological dimension ${\rm
hd}_{\mathcal{O}_{X,x}}(\mathcal{O}_{\Sigma,x})$ is $n-k+1$. As $X$ is
regular, we have the equality of Auslander-Buchsbaum

${\rm hd}_{\mathcal{O}_{X,x}}(\mathcal{O}_{\Sigma,x})+{\rm
depth}_{\mathcal{O}_{X,x}}(\mathcal{O}_{\Sigma,x})=\dim(X),$

whence ${\rm
depth}_{\mathcal{O}_{X,x}}(\mathcal{O}_{\Sigma,x})=\tau-1=\dim(\Sigma)$ and so
$\Sigma$ is Cohen-MaCaulay.

By Serre’s criteria of normality, we need to verify that the singular locus of
$\Sigma$ is of codimension at least $2$. For this we introduce the _Thom
strata_ : Let $\Theta_{X}$ and $\Theta_{S}$ be the sheaves of holomorphic
vector fields on $X$ and $S$ respectively, the pull-back and the tangent map
induces the morphisms of sheaves

$\psi^{-1}:\Theta_{S}\to\psi^{*}(\Theta_{S})\quad\text{and}\quad
t\psi:\Theta_{X}\to\psi^{*}\Theta_{S}.$

######  Lemma 4.2.

The miniversal deformation $\psi$ is _infinitesimally stable_ , i.e. for any
point $x\in X$ with image $s$, we have

$\psi^{*}(\Theta_{S,s})=\psi^{-1}(\Theta_{S,s})+t\psi(\Theta_{X,x}).$

###### Proof.

Let ${\mathfrak{m}}_{x}\subset\mathcal{O}_{X,x}$ be the maximal ideal at $x$
and ${\mathfrak{m}}_{s}\subset\mathcal{O}_{S,s}$ the maximal ideal at $s$. We
identify ${\mathfrak{m}}_{s}$ with its image in $\mathcal{O}_{X,x}$ via the
morphism $\mathcal{O}_{S,s}\to\mathcal{O}_{X,x}$. We claim that the assertion
is equivalent to

$\psi^{*}(\Theta_{S,s})=\psi^{-1}(\Theta_{S,s})+t\psi(\Theta_{X,x})+{\mathfrak{m}}_{s}\psi^{*}(\Theta_{S,s}).$
(4.6)

Indeed, the above equation is equivalent to

$\frac{\psi^{*}(\Theta_{S,s})}{t\psi(\Theta_{X,x})}=\frac{\psi^{-1}(\Theta_{S,s})+t\psi(\Theta_{X,x})}{t\psi(\Theta_{X,x})}+\frac{{\mathfrak{m}}_{s}\psi^{*}(\Theta_{S,s})+t\psi(\Theta_{X,x})}{t\psi(\Theta_{X,x})}.$

Let $M=\frac{\psi^{*}(\Theta_{S,s})}{t\psi(\Theta_{X,x})}$, the equation can
be rewritten as

$M=\frac{\psi^{-1}(\Theta_{S,s})+t\psi(\Theta_{X,x})}{t\psi(\Theta_{X,x})}+{\mathfrak{m}}_{s}M.$
(4.7)

One verifies that $M$ is an $\mathcal{O}_{S,s}$-module of finite type. By
Nakayama’s lemma, the above equation is equivalent to

$M=\frac{\psi^{-1}(\Theta_{S,s})+t\psi(\Theta_{X,x})}{t\psi(\Theta_{X,x})},$

which is exactly the equation in the lemma.

For the equation (4.6) or equivalently (4.7), it is enough to verify the
surjectivity of

$T_{s}S\xrightarrow{\psi^{-1}}M/{\mathfrak{m}}_{s}M.$ (4.8)

Note that

$M/{\mathfrak{m}}_{s}M=\frac{\psi^{*}(\Theta_{S,s})}{{\mathfrak{m}}_{y}\psi^{*}(\Theta_{S,s})+t\psi(\Theta_{X,x})}=T_{(X_{s},x)}^{1},$

and that the morphism (4.8) is just the Kodaira-Spencer map, here we have used
the exact sequence (4.3) and the isomorphism (4.4). Now the surjectivity of
(4.8) follows from the versality of $\psi$.

∎

According to Mather [M1], the morphism $\psi$ is _stable_ , i.e. it is
analytically equivalent to any sufficiently small perturbation of it. This
implies that the tangent map ${\rm d}\psi_{x}:(X,x)\to{\rm
Hom}(\mathbf{C}^{n-k+\tau},\mathbf{C}^{\tau})$ is transversal to the
stratification ${\rm
Hom}(\mathbf{C}^{n-k+\tau},\mathbf{C}^{\tau})=\bigsqcup_{r=0}^{\tau}\mathcal{O}_{r}(n-k+\tau,\tau)$.
Consequently, for $r=0,\cdots,\tau-1$, the inverse image

$\Sigma_{r,x}=({\rm
d}\psi_{x})^{-1}\big{(}\mathcal{O}_{r}(n-k+\tau,\tau)\big{)}$

is smooth. $\Sigma_{r,x}$ is clearly independent of the local coordinate of
$X$ at $x$, hence when $x$ runs through points on $X$ they glue together to a
smooth locally closed subscheme $\Sigma_{r}$ of $X$, called _Thom strata_. It
is clear that $\Sigma$ equals the Zariski closure of $\Sigma_{\tau-1}$. As
$\Sigma_{\tau-1}$ is itself smooth, the singularity of $\Sigma$ is contained
in the closure of $\Sigma_{\tau-2}$, which is of codimension
$2(n-k+2)-(n-k+1)=n-k+3\geqslant 3$ in $\Sigma$. This finishes the proof that
$\Sigma$ is normal.

As the fibers of $\psi$ have at worst isolated complete intersection
singularities and the singularities can be deformed independently, the fibers
of $\psi$ over the generic point of each local irreducible component of
$(\Delta,0)$ can have only a unique singularity. Moreover, for any point $x$
of $\Sigma_{\tau-1}$, one verifies by the openness of versality that the
singularity $(X_{\psi(x)},x)$ must be ordinary quadratic. With the deformation
theory of ordinary quadratic singularity, the restriction of $\psi$ to
$\Sigma_{\tau-1}$ must be analytically locally an isomorphism. Hence $\psi$ is
generically an isomorphism over each local irreducible component of
$(\Delta,0)$. The normality of $\Sigma$ implies that it is locally
analytically irreducible, this implies that $\Delta$ is locally analytically
irreducible at $0$ as well. As $\psi$ is finite generically an isomorphism and
$\Sigma$ is normal, $\psi$ must be the normalization.

∎

The Thom stratification inspires the following construction: For
$m,d\in\mathbf{N}$ with $m>d$, let

$\mathcal{N}_{m,d}=\big{\\{}A\in{\rm
Hom}(\mathbf{C}^{m},\mathbf{C}^{d})\mid\mathrm{rk}(A)<d\big{\\}},$

which is also the Zariski closure of ${\mathcal{O}_{d-1}(m,d)}$. It admits a
resolution called _Springer resolution_ : Let
$\widetilde{\mathcal{N}}_{m,d}=\big{\\{}(M,H)\in{\rm
Hom}(\mathbf{C}^{m},\mathbf{C}^{d})\times\check{\mathbf{P}}^{d-1}\mid{\rm
Im}(M)\subset H\big{\\}}$, where $\check{\mathbf{P}}^{d-1}$ parametrizes the
hyperplanes in $\mathbf{C}^{d}$. With the projection
$\widetilde{\mathcal{N}}_{m,d}\to\check{\mathbf{P}}^{d-1}$, one verifies that
$\widetilde{\mathcal{N}}_{m,d}$ is smooth irreducible. Hence the projection
$\widetilde{\mathcal{N}}_{m,d}\to\mathcal{N}_{m,d}$ is a resolution of
singularity as it is proper and generically isomorphic. We define
$\widetilde{\Sigma}$ with the Cartesian diagram

${\widetilde{\Sigma}}$${\widetilde{\mathcal{N}}_{n-k+\tau,\tau}}$${X}$${{\rm
Hom}(\mathbf{C}^{n-k+\tau},\mathbf{C}^{\tau}).}$${\square}$$\scriptstyle{{\rm
d}\psi}$

As $X$ is transversal to the stratification of ${\rm
Hom}(\mathbf{C}^{n-k+\tau},\mathbf{C}^{\tau})$ by the rank, the fiber product
$\widetilde{\Sigma}$ is smooth and the resulting morphism
$\widetilde{\Sigma}\to\Sigma$ is a resolution of singularity. More
intrinsically,

$\widetilde{\Sigma}=\big{\\{}(x,H)\in\Sigma\times\check{\mathbf{P}}^{\tau-1}\mid{\rm
Im}{({\rm d}\psi_{x})}\subset H\big{\\}},$

where $\check{\mathbf{P}}^{\tau-1}$ parametrizes the hyperplanes in
$\mathbf{C}^{\tau}$ and we have moved ${\rm Im}{({\rm d}\psi_{x})}$ to the
plane parallel to it passing through $0$.

On the other hand, let $\widetilde{\Delta}$ be the Nash blow-up of $\Delta$,
i.e. it is the Zariski closure of the graph
$\widetilde{\Delta}^{\mathrm{reg}}$ of the embedding

$\Delta^{\mathrm{reg}}\hookrightarrow\Delta\times\check{\mathbf{P}}^{\tau-1},\quad
y\mapsto(y,T_{y}\Delta),$

where $\Delta^{\mathrm{reg}}$ is the regular locus of $\Delta$. Note that
there exists a natural morphism
$\widetilde{\psi}:\widetilde{\Sigma}\to\widetilde{\Delta}$. Indeed, the
composite
$\widetilde{\Sigma}\hookrightarrow\Sigma\times\check{\mathbf{P}}^{\tau-1}\to\Delta\times\check{\mathbf{P}}^{\tau-1}$
is clearly proper, hence its image in
$\Delta\times\check{\mathbf{P}}^{\tau-1}$ is closed. It is even irreducible as
$\widetilde{\Sigma}$ is irreducible. Let $\widetilde{\Sigma}_{\tau-1}$ be the
inverse image of ${\Sigma}_{\tau-1}$ for the projection
$\widetilde{\Sigma}\to\Sigma$. For any point $x\in\Sigma_{\tau-1}$ such that
$\psi(x)\in\Delta^{\mathrm{reg}}$, the singularity of the fiber $X_{\psi(x)}$
at $x$ is ordinary quadratic, hence the rank of ${\rm d}\psi_{x}$ is $\tau-1$
and ${\rm Im}({\rm d}\psi_{x})=T_{\psi(x)}\Delta^{\mathrm{reg}}$. This implies
that the image of $\widetilde{\Sigma}_{\tau-1}$ under the morphism
$\widetilde{\Sigma}\to\Delta\times\check{\mathbf{P}}^{\tau-1}$ contains
$\widetilde{\Delta}^{\mathrm{reg}}$. As the image of
$\widetilde{\Sigma}\to\Delta\times\check{\mathbf{P}}^{\tau-1}$ is closed
irreducible and $\widetilde{\Delta}^{\mathrm{reg}}$ is dense in
$\widetilde{\Delta}$, we obtain that $\widetilde{\Sigma}$ does map onto
$\widetilde{\Delta}$ and this defines $\widetilde{\psi}$.

######  Theorem 4.3 (Teissier [T1] for the hypersurface, Looijenga [Loo]).

The morphism $\widetilde{\psi}:\widetilde{\Sigma}\to\widetilde{\Delta}$ is an
isomorphism. In particular, the Nash blow-up of $\Delta$ is smooth.

###### Proof.

As $\widetilde{\psi}$ is already known to be surjective, it is enough to show
that it is an immersion at any point $\tilde{x}\in\widetilde{\Sigma}$. Let
$\tilde{s}=\widetilde{\psi}(\tilde{x})\in\widetilde{\Delta}$, we need to show
that
$\widetilde{\psi}^{*}{\mathfrak{m}}_{\widetilde{\Delta},\tilde{s}}={\mathfrak{m}}_{\widetilde{\Sigma},\tilde{x}}$.
The inclusion
$\widetilde{\psi}^{*}{\mathfrak{m}}_{\widetilde{\Delta},\tilde{s}}\subset{\mathfrak{m}}_{\widetilde{\Sigma},\tilde{x}}$
is clear and it is enough to show the inclusion in the other direction. Let
$\tilde{x}=(x,H)\in\Sigma\times\check{\mathbf{P}}^{\tau-1}$ and let
$s=\psi(x)$, then $\tilde{s}=(s,H)$. Let $\delta\in\mathcal{O}_{S}$ be a
generator of the defining ideal of $\Delta\subset S$. Let
$t_{1},\cdots,t_{\tau}$ be a coordinate at $0$ of $S$, and let
$[\xi_{1}:\cdots:\xi_{\tau}]$ be the associated coordinate for
$\check{\mathbf{P}}^{\tau-1}$, then $\widetilde{\Delta}$ can be described as

$\widetilde{\Delta}=\Big{\\{}\big{(}s,[\xi_{1}:\cdots:\xi_{\tau}]\big{)}\in\mathbf{C}^{\tau}\times\check{\mathbf{P}}^{\tau-1}\,\big{|}\,\delta(s)=0,\,\xi_{i}\textstyle{\frac{\partial\delta}{\partial{t_{j}}}}=\xi_{j}\textstyle{\frac{\partial\delta}{\partial{t_{i}}}},i,j=1,\cdots,\tau\Big{\\}}.$

Without loss of generality, we can take the coordinate such that the
hyperplane $H$ is defined by $t_{\tau}=0$. Let
$\eta_{i}=\xi_{i}/\xi_{\tau},i=1,\cdots,\tau-1$. Then

$\mathcal{O}_{\widetilde{\Delta},\tilde{s}}=\mathbf{C}[t_{1},\cdots,t_{\tau},\eta_{1},\cdots,\eta_{\tau-1}]_{(s,0)}\Big{/}\Big{(}\delta,\textstyle{\frac{\partial\delta}{\partial{t_{i}}}}-\eta_{i}\textstyle{\frac{\partial\delta}{\partial{t_{\tau}}}},i=1,\cdots,\tau-1\Big{)}.$

On the other hand, with the inclusion $\widetilde{\Sigma}\subset
X\times\check{\mathbf{P}}^{\tau-1}$, we obtain that
${\mathfrak{m}}_{\widetilde{\Sigma},\tilde{x}}$ is generated by the
restriction of ${\mathfrak{m}}_{X,x}$ to $\Sigma$ and the functions
$\eta_{1},\cdots,\eta_{\tau-1}$.

By lemma 4.2, we have
$\psi^{*}(\Theta_{S,s})=\psi^{-1}(\Theta_{S,s})+t\psi(\Theta_{X,x})$. Applying
the vector fields on both sides at any element
$t\in{\mathfrak{m}}_{S,s}-{\mathfrak{m}}_{S,s}^{2}$, we obtain that

$\mathcal{O}_{X,x}=\psi^{-1}(\mathcal{O}_{S,s})+\Theta_{X,x}(\psi^{*}(t)),$

from which we deduce that

${\mathfrak{m}}_{X,x}={\mathfrak{m}}_{X,x}\cdot\big{[}\psi^{-1}(\mathcal{O}_{S,s})+\Theta_{X,x}(\psi^{*}(t))\big{]}\subset\psi^{*}{\mathfrak{m}}_{S,s}+\Theta_{X,x}(\psi^{*}(t)).$
(4.9)

We look at the ideal
$\Theta_{X,x}(\psi^{*}(t))\cdot\mathcal{O}_{\widetilde{\Sigma},\tilde{x}}$ for
$t=t_{i},i=1,\cdots,\tau$. For any vector field $v\in\Theta_{X,x}$, we have

$v(\psi^{*}(\delta))=\sum_{i=1}^{\tau}v\big{(}\psi^{*}(t_{i})\big{)}\psi^{*}\big{(}\textstyle{\frac{\partial\delta}{\partial
t_{i}}}\big{)}.$

As an element in $\mathcal{O}_{\widetilde{\Sigma},\tilde{x}}$, it is equal to

$\displaystyle v(\psi^{*}(\delta))$
$\displaystyle=v\big{(}\psi^{*}(t_{\tau})\big{)}\psi^{*}\big{(}{\textstyle{\frac{\partial\delta}{\partial
t_{\tau}}}}\big{)}+\sum_{i=1}^{\tau-1}v\big{(}\psi^{*}(t_{i})\big{)}\cdot\eta_{i}\psi^{*}\big{(}\textstyle{\frac{\partial\delta}{\partial
t_{\tau}}}\big{)}$
$\displaystyle=\Big{[}v\big{(}\psi^{*}(t_{\tau})\big{)}+\sum_{i=1}^{\tau-1}v\big{(}\psi^{*}(t_{i})\big{)}\cdot\eta_{i}\Big{]}\psi^{*}\big{(}{\textstyle{\frac{\partial\delta}{\partial
t_{\tau}}}}\big{)},$ (4.10)

due to the relation
$\textstyle{\frac{\partial\delta}{\partial{t_{i}}}}=\eta_{i}\textstyle{\frac{\partial\delta}{\partial{t_{\tau}}}}$
in $\mathcal{O}_{\widetilde{\Delta},\tilde{s}}$. One verifies that
$v(\psi^{*}(\delta))|_{\Sigma}=0$, i.e.
$\Theta_{X}(\psi^{*}(\delta))|_{\Sigma}=0$222In fact, one can show that ${\rm
Im}\\{{\rm
d}\psi:\Theta_{X}\to\psi^{*}\Theta_{S}\\}=\psi^{*}(\Theta_{S}(-\log\Delta))$,
where $\Theta_{S}(-\log\Delta)$ is the sheaf of holomorphic vector fields
preserving $\Delta$ (cf. [Loo], Lem. 6.18).. Indeed, as $\Sigma$ is normal, it
is enough to verify it on the dense open subscheme $\Sigma_{\tau-1}$. It is
known that the singularity of the geometric fibers of $\psi$ at points in
$\Sigma_{\tau-1}$ is ordinary quadratic. By the openness of versality, it is
then enough to verify the assertion for the miniversal deformation of ordinary
quadratic singularity, which is clear. Consequently, as
$\psi^{*}\big{(}{\textstyle{\frac{\partial\delta}{\partial
t_{\tau}}}}\big{)}\neq 0$ in $\mathcal{O}_{\widetilde{\Sigma},\tilde{x}}$, we
deduce from the equation (4.10) the equality in
$\mathcal{O}_{\widetilde{\Sigma},\tilde{x}}$:

$v\big{(}\psi^{*}(t_{\tau})\big{)}=-\sum_{i=1}^{\tau-1}v\big{(}\psi^{*}(t_{i})\big{)}\cdot\eta_{i},$

which implies that

$\Theta_{X,x}(\psi^{*}(t_{\tau}))\subset(\eta_{1},\cdots,\eta_{\tau-1})\quad\text{in
}\mathcal{O}_{\widetilde{\Sigma},\tilde{x}}.$

With the equation (4.9), we obtain

${\mathfrak{m}}_{X,x}\cdot\mathcal{O}_{\widetilde{\Sigma},\tilde{x}}\subset\psi^{*}{\mathfrak{m}}_{S,s}\cdot\mathcal{O}_{\widetilde{\Sigma},\tilde{x}}+(\eta_{1},\cdots,\eta_{\tau-1})\subset\widetilde{\psi}^{*}({\mathfrak{m}}_{\widetilde{\Delta},\tilde{s}}).$

As ${\mathfrak{m}}_{\widetilde{\Sigma},\tilde{x}}$ is generated by
${\mathfrak{m}}_{X,x}\cdot\mathcal{O}_{\widetilde{\Sigma},\tilde{x}}$ and
$\eta_{1},\cdots,\eta_{\tau-1}$, this implies that
${\mathfrak{m}}_{\widetilde{\Sigma},\tilde{x}}\subset\widetilde{\psi}^{*}({\mathfrak{m}}_{\widetilde{\Delta},\tilde{s}})$
as required.

∎

######  Remark 4.4.

In the hypersurface case, one can show that ${\rm d}\psi$ is of rank
$\geqslant\tau-1$. Indeed, let $g_{1},\cdots,g_{\tau}$ be a set of
representative for the Tjurina module
$\mathcal{O}_{X_{0}}/Df\cdot\mathcal{O}_{X,0}^{n}$. Without loss of
generality, we can take $g_{\tau}=1$. Let

$F(x_{1},\cdots,x_{n},t_{1},\cdots,t_{\tau})=f(x_{1},\cdots,x_{n})+\sum_{i=1}^{\tau-1}t_{i}g_{i}+t_{\tau}.$

Let $X\subset\mathbf{C}^{n}\times\mathbf{C}^{\tau}$ be the closed subscheme
defined by $F=0$. The composite
$\psi:X\hookrightarrow\mathbf{C}^{n}\times\mathbf{C}^{\tau}\to\mathbf{C}^{\tau}$
is then a miniversal deformation of $X_{0}$. Note that $\partial F/\partial
t_{\tau}=1$, the germ $(X,0)$ then admits a coordinate
$\mathbf{C}^{n+\tau-1}\to(X,0)$ defined by

$(x_{1},\cdots,x_{n},t_{1},\cdots,t_{\tau-1})\mapsto\Big{(}x_{1},\cdots,x_{n},t_{1},\cdots,t_{\tau-1},-f(x_{1},\cdots,x_{n})-\sum_{i=1}^{\tau-1}t_{i}g_{i}\Big{)}.$

With respect to this coordinate, the morphism $\psi:X\to\mathbf{C}^{\tau}$ is
nothing but the morphism $\mathbf{C}^{n+\tau-1}\to\mathbf{C}^{\tau}$ defined
by

$(x_{1},\cdots,x_{n},t_{1},\cdots,t_{\tau-1})\mapsto\Big{(}t_{1},\cdots,t_{\tau-1},-f(x_{1},\cdots,x_{n})-\sum_{i=1}^{\tau-1}t_{i}g_{i}\Big{)}$

Its Jacobian matrix can be calculated to be

$D\psi=\begin{bmatrix}0&\cdots&0&1&&\\\ 0&\cdots&0&&\ddots&\\\
0&\cdots&0&&&1\\\ -\frac{\partial F}{\partial x_{1}}&\cdots&-\frac{\partial
F}{\partial
x_{n}}&-g_{1}&\cdots&-g_{\tau-1}\end{bmatrix}_{\tau\times(n+\tau-1)},$

which is clearly of rank $\geqslant\tau-1$. This implies that
$\widetilde{\Sigma}=\Sigma$, hence the morphism
$\psi|_{\Sigma}:\Sigma\to\Delta$ can be identified with the Nash blow-up of
$\Delta$.

######  Example 4.5.

Let $X_{0}$ be the germ of singularity defined by $y^{2}-x^{n}=0,n\geqslant
2$. Let $X\subset\mathbf{C}^{2}\times\mathbf{C}^{n-1}$ be the hypersurface
defined by the equation

$F(x,y;a_{2},\cdots,a_{n}):=y^{2}-(x^{n}+a_{2}x^{n-2}+\cdots+a_{n-1}x+a_{n})=0.$

Then the composite
$\psi:X\hookrightarrow\mathbf{C}^{2}\times\mathbf{C}^{n-1}\to\mathbf{C}^{n-1}$
defines a miniversal deformation of $X_{0}$. Notice that $\partial F/\partial
a_{n}=1$, whence the germ of $X$ at $0$ admits a coordinate
$\mathbf{C}^{2}\times\mathbf{C}^{n-2}\mapsto(X,0)$ defined by

$(x,y;a_{2},\cdots,a_{n-1})\mapsto\big{(}x,y,a_{2},\cdots,a_{n-1},y^{2}-(x^{n}+a_{2}x^{n-2}+\cdots+a_{n-1}x)\big{)}.$

With this coordinate, the map $\psi$ becomes
$\psi:\mathbf{C}^{2}\times\mathbf{C}^{n-2}\mapsto\mathbf{C}^{n-1}$, which can
be described as

$(x,y;a_{2},\cdots,a_{n-1})\mapsto\big{(}a_{2},\cdots,a_{n-1},y^{2}-(x^{n}+a_{2}x^{n-2}+\cdots+a_{n-1}x)\big{)}.$

The induced tangent map is then described by the Jacobian matrix

$D\psi=\begin{bmatrix}0&0&1&&\\\ 0&0&&\ddots&\\\ 0&0&&&1\\\ \frac{\partial
F}{\partial x}&2y&-x^{n-2}&\cdots&-x\end{bmatrix}_{(n-1)\times n},$ (4.11)

Note that $D\psi$ is of rank at least $n-2$, and that it is singular if and
only if

$\begin{cases}\frac{\partial F}{\partial y}=2y=0,&\\\ \frac{\partial
F}{\partial
x}=-\big{(}nx^{n-1}+(n-2)a_{2}x^{n-3}+\cdots+a_{n-1}\big{)}=0.\end{cases}$

These equations then define $\Sigma$ as a closed subscheme of
$\mathbf{C}^{2}\times\mathbf{C}^{n-2}$. Note that $\frac{\partial}{\partial
a_{n-1}}\big{(}\frac{\partial F}{\partial x}\big{)}=-1$, we obtain that
$\Sigma$ admits a coordinate $\mathbf{C}^{n-2}\to\Sigma$ defined by

$(x,a_{2},\cdots,a_{n-2})\mapsto\big{(}x,0;a_{2},\cdots,a_{n-2},-(nx^{n-1}+(n-2)a_{2}x^{n-3}+\cdots+2a_{n-2}x^{1})\big{)}.$

Compose with the map
$\psi:\mathbf{C}^{2}\times\mathbf{C}^{n-2}\to\mathbf{C}^{n-1}$, we find that
the discriminant $\Delta$ can be parametrized by the map
$\varphi:\mathbf{C}^{n-2}\to\Delta$ defined by

$(x,a_{2},\cdots,a_{n-2})\mapsto\big{(}a_{2},\cdots,a_{n-2},\varphi_{n-1}(x,a_{2},\cdots,a_{n-2}),\varphi_{n}(x,a_{2},\cdots,a_{n-2})\big{)},$

where

$\displaystyle\varphi_{n-1}(x,a_{2},\cdots,a_{n-2})$
$\displaystyle=-(nx^{n-1}+(n-2)a_{2}x^{n-3}+\cdots+2a_{n-2}x^{1}),$
$\displaystyle\varphi_{n}(x,a_{2},\cdots,a_{n-2})$
$\displaystyle=-\big{(}x^{n}+a_{2}x^{n-2}+\cdots+a_{n-2}x^{2}+\varphi_{n-1}(x,a_{2},\cdots,a_{n-2})x\big{)}$
$\displaystyle=(n-1)x^{n}+(n-3)a_{2}x^{n-2}+\cdots+a_{n-2}x^{2}.$

Observe with the concrete description (4.11) of $D\psi$ that ${\rm Im}(D\psi)$
at $\varphi(x,a_{2},\cdots,a_{n-2})$, which is also the tangent space of
$\Delta$ at that point in case that the point lies in $\Delta^{\mathrm{reg}}$,
equals the hyperplane of $\mathbf{C}^{n-1}$ generated by the vectors

$\begin{bmatrix}1\\\ 0\\\ \vdots\\\ 0\\\
-x^{n-2}\end{bmatrix},\begin{bmatrix}0\\\ 1\\\ \vdots\\\ 0\\\
-x^{n-1}\end{bmatrix},\cdots\cdots,\begin{bmatrix}0\\\ \vdots\\\ 0\\\ 1\\\
-x\end{bmatrix},$

which clearly depends only on $x$. In particular, the coordinate $x$ can be
recovered from the tangent space. This observation explains why the Nash blow-
up of $\Delta$ equals exactly $\Sigma$ as asserted in theorem 4.3.

Consider the parametrization
$\varphi:\mathbf{C}^{n-2}\to\Delta\subset\mathbf{C}^{n-1}$, it has Jacobian
matrix

$D\varphi=\begin{bmatrix}0&1&0&\cdots&0\\\ 0&0&1&\cdots&0\\\
\vdots&\vdots&\vdots&\ddots&\vdots\\\ 0&0&0&\cdots&1\\\
\frac{\partial\varphi_{n-1}}{\partial
x}&-(n-2)x^{n-3}&-(n-3)x^{n-4}&\cdots&2x\\\
\frac{\partial\varphi_{n}}{\partial
x}&(n-3)x^{n-2}&(n-4)x^{n-3}&\cdots&x^{2}\end{bmatrix}_{(n-1)\times(n-2)}.$
(4.12)

Note that $D\varphi$ is of rank at least $n-3$, and that it is singular if and
only if

$\begin{cases}\frac{\partial\varphi_{n-1}}{\partial
x}=-\big{(}n(n-1)x^{n-2}+(n-2)(n-3)a_{2}x^{n-4}+\cdots+2\cdot
1a_{n-2}\big{)}=0,&\\\ \frac{\partial\varphi_{n}}{\partial
x}=n(n-1)x^{n-1}+(n-2)(n-3)a_{2}x^{n-3}+\cdots+2\cdot
1a_{n-2}x=-x\frac{\partial\varphi_{n-1}}{\partial x}=0.\end{cases}$

Let $\Sigma^{1,1}\subset\Sigma$ be the closed subscheme defined by these two
equations, or equivalently by the equation
$\frac{\partial\varphi_{n-1}}{\partial x}=0$, it is the critical locus of the
morphism $\varphi:\mathbf{C}^{n-2}\to\Delta$. Note that
$\frac{\partial}{\partial a_{n-2}}\big{(}\frac{\partial\varphi_{n-1}}{\partial
x}\big{)}=-2$, we obtain that $\Sigma^{1,1}$ admits a coordinate
$\phi^{1,1}:\mathbf{C}^{n-3}\to\Sigma^{1,1}\subset\mathbf{C}^{n-2}$ defined by

$(x,a_{2},\cdots,a_{n-3})\mapsto\Big{(}x,a_{2},\cdots,a_{n-3},-\frac{1}{2}\big{(}n(n-1)x^{n-2}+(n-2)(n-3)a_{2}x^{n-4}+\cdots+3\cdot
2a_{n-3}x\big{)}\Big{)}.$

In particular, $\Sigma^{1,1}$ is smooth. It is an example of the _Thom-
Boardman strata_. Let $\Delta^{1,1}$ be the image of $\Sigma^{1,1}$ under the
morphism $\psi$, let $\varphi^{1,1}$ be the composite
$\mathbf{C}^{n-3}\xrightarrow{\phi^{1,1}}\Sigma^{1,1}\to\Delta^{1,1}$, it is
the restriction of $\psi$ to $\Sigma^{1,1}$ written with the coordinate
$\phi^{1,1}$. As before, with the concrete description (4.12) of $D\varphi$,
we obtain that ${\rm Im}(D\varphi)$ at $\phi(x,a_{2},\cdots,a_{n-3})$ is the
linear space of $\mathbf{C}^{n-1}$ generated by the vectors

$\begin{bmatrix}1\\\ 0\\\ \vdots\\\ 0\\\ -(n-2)x^{n-3}\\\
(n-3)x^{n-2}\end{bmatrix},\begin{bmatrix}0\\\ 1\\\ \vdots\\\ 0\\\
-(n-3)x^{n-4}\\\ (n-4)x^{n-3}\end{bmatrix},\cdots\cdots,\begin{bmatrix}0\\\
\vdots\\\ 0\\\ 1\\\ 2x\\\ x^{2}\end{bmatrix},$

which depends only on $x$. Moreover, in case that
$\phi(x,a_{2},\cdots,a_{n-3})$ is a smooth point of $\Delta^{1,1}$, ${\rm
Im}(D\varphi)$ coincides with the tangent space of $\Delta^{1,1}$ at this
point. Again, this implies that the Nash blow-up of $\Delta^{1,1}$ is exactly
$\Sigma^{1,1}$ as the coordinate $x$ can be recovered from the tangent space.

We can proceed to consider the critical locus $\Sigma^{1,1,1}$ of the morphism
$\varphi^{1,1}:\Sigma^{1,1}\to\Delta^{1,1}$ and this process can be iterated,
we get a sequence of smooth manifolds
$\Sigma\supsetneq\Sigma^{1,1}\supsetneq\Sigma^{1,1,1}\supsetneq\cdots$, called
_Thom-Boardman strata_. Let $\Delta^{1,\cdots,1}$ be the image of
$\Sigma^{1,\cdots,1}$ under the morphism $\psi$. With the same reasoning as
before, we can show that $\Sigma^{1,\cdots,1}$ is the Nash blow-up of
$\Delta^{1,\cdots,1}$.

Despite this beautiful property of the Thom-Boardman strata, they are not
sufficient to understand the geometry of the discriminant $\Delta$. For
example, they miss completely the strict $\delta$-strata, none of them appears
as $\Delta^{1,\cdots,1}$.

$\Box$

###  4.2 Whitney regularity

Whitney stratification is the most widely accepted notion of
“equisingularity”. In this section, we make a brief recall of Thom-Mather’s
theory of Whitney stratification and Teissier’s numerical criteria for the
Whitney regularity condition. They are used to show the Whitney regularity of
the union of the strict $\delta$-strata over the equisingular strata. We then
use this result to show that the sheaves
$\mathcal{H}^{*}(j_{!*}\mathcal{F}^{i})$ are locally constant over the
equisingular strata.

#### (1) Whitney stratification

Let $X$ be a reduced complex analytic variety, a _stratification_ of $X$ is a
decomposition into disjoint union $X=\bigsqcup_{\alpha\in A}X_{\alpha}$ of
locally closed smooth complex analytic subvarieties, called _strata_ ,
satisfying the _frontier condition_ :

$X_{\alpha}\cap\overline{X_{\beta}}\neq\emptyset\iff
X_{\alpha}\subset\overline{X_{\beta}},\quad\forall\,\alpha,\beta\in A.$

The stratification is said to be _locally finite_ if the number of strata is
locally finite.

######  Definition 4.6.

Let $X$ be a reduced locally closed subvariety of a smooth complex analytic
variety $M$, let $\\{X_{\alpha}\\}_{\alpha\in A}$ be a stratification of $X$.
A pair of strata $(X_{\alpha},X_{\beta})$ with
$X_{\alpha}\subset\overline{X_{\beta}}$ is said to satisfy Whitney’s condition
at $y\in X_{\alpha}$ if

  1. $(a)$

For any sequence $\\{x_{i}\\}_{i\in\mathbf{N}}\subset X_{\beta}$ tending to
$y$, we have

$T_{y}X_{\alpha}\subset\lim_{i\to\infty}T_{x_{i}}X_{\beta}.$

  2. $(b)$

For any sequences $\\{y_{i}\\}_{i\in\mathbf{N}}\subset
X_{\alpha},\\{x_{i}\\}_{i\in\mathbf{N}}\subset X_{\beta}$ tending to $y$, we
have

$\lim_{i\to\infty}\overline{x_{i}y_{i}}\subset\lim_{i\to\infty}T_{x_{i}}X_{\beta},$

where $\overline{x_{i}y_{i}}$ is the secant line passing through $x_{i}$ and
$y_{i}$ for some local coordinate system of $M$.

In case that the conditions are satisfied for all points of $X_{\alpha}$, we
say that $X_{\beta}$ is _Whitney regular_ over $X_{\alpha}$. If the Whitney
conditions are satisfied for any pair of strata $(X_{\alpha},X_{\beta})$ with
$X_{\alpha}\subset\overline{X_{\beta}}$, we call the stratification
$\\{X_{\alpha}\\}_{\alpha\in A}$ a _Whitney stratification_.

The above conditions will be called Whitney’s condition $(a)$ and $(b)$. It is
known that the condition $(b)$ implies condition $(a)$.

######  Theorem 4.7 (Whitney [W]).

Let $X$ be a reduced locally closed subvariety of a smooth complex analytic
variety $M$, then there exists a locally finite Whitney stratification of $M$
such that $X$ is a union of strata.

The Whitney stratified spaces enjoy a nice locally topologically trivial
property along the strata:

######  Theorem 4.8 (Thom-Mather [M2], Consequence of Thom’s first isotopy
lemma).

Let $X$ be a reduced locally closed subvariety of a smooth complex analytic
variety $M$. Let $\\{X_{\alpha}\\}_{\alpha\in A}$ be a Whitney stratification
of $X$. For any point $x$, let $X_{\alpha}$ be the stratum containing $x$.
Then for any local embedding $(X,x)\subset(\mathbf{C}^{n},0)$ induced from a
local coordinate of $M$, and any local retraction
$\rho:(\mathbf{C}^{n},0)\to(X_{\alpha},0)$, there exists $\epsilon_{0}>0$ such
that for any $0<\epsilon<\epsilon_{0}$ there exists $\eta_{\epsilon}>0$ such
that for any $0<\eta<\eta_{\epsilon}$ there exists a homeomorphism $h$ making
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which induces for any stratum $X_{\beta}$ containing $X_{\alpha}$ at its
frontier a homeomorphism

$\overline{X_{\beta}}\cap B_{0}(\epsilon)\cap\rho^{-1}(X_{\alpha}\cap
B_{0}(\eta))\xrightarrow{h}\big{(}\overline{X_{\beta}}\cap
B_{0}(\epsilon)\cap\rho^{-1}(x)\big{)}\times(X_{\alpha}\cap B_{0}(\eta)),$

where $B_{0}(r)$ is the open ball in $\mathbf{C}^{n}$ with center $0$ and
radius $r$.

Conversely, the Whitney stratification can be characterized topologically as
follows: Let $\\{X_{\alpha}\\}_{\alpha\in A}$ be a stratification of a reduced
complex analytic variety $X$, let $(X_{\alpha},X_{\beta})$ be any pair
satisfying $X_{\alpha}\subset\overline{X_{\beta}}$. The pair is said to
satisfy the condition $(TT)$ (local topological triviality) if it satisfies
the conclusion of Thom-Mather’s theorem. It is said to satisfy the condition
$(TT^{*})$ if for any $x\in X_{\alpha}$, any local embedding
$(X,x)\subset(\mathbf{C}^{n},0)$ and any $i=\dim(X_{\alpha}),\cdots,n$, there
exists a dense open subscheme $U_{\alpha}$ of the Grassmannian of linear
subspaces of $\mathbf{C}^{n}$ of dimension $i$ containing $T_{x}(X_{\alpha})$,
such that for any submanifold $H\subset\mathbf{C}^{n}$ containing
$(X_{\alpha},x)$ with $T_{x}(H)\in U_{\alpha}$, the slice $(X_{\alpha}\cap
H,X_{\beta}\cap H)$ satisfies the condition $(TT)$.

######  Theorem 4.9 (Lê-Teissier [LT]).

Let $\\{X_{\alpha}\\}_{\alpha\in A}$ be a stratification of a complex analytic
variety $X$, then it is a Whitney stratification if and only if any pair
$X_{\alpha}\subset\overline{X_{\beta}}$ of it satisfies the condition
$(TT^{*})$.

#### (2) Teissier’s criteria

Teissier [T3] has found a beautiful characterization of the Whitney regularity
condition, we make a brief recall of his result.

Let $X$ be a reduced complex analytic variety of pure dimension $d$, let $Y$
be a smooth irreducible closed subvariety of $X^{\rm sing}$ of dimension
$d^{\prime}$. For $y\in Y$, without loss of generality, we can assume the
existence of a local embedding $(X,y)\subset(\mathbf{C}^{n},0)$ such that $Y$
is identified with the subspace
$\mathbf{C}^{d^{\prime}}\subset\mathbf{C}^{n}$. Henceforth we denote by
$(X,y)$ the germ at $y$ of $X$ equipped with this local embedding. The Whitney
condition involves taking limit of secant lines and tangent spaces, this led
to the consideration of the blow-up ${\rm Bl}_{Y}(X)$ of $X$ with center $Y$
and the Nash blow-up $N(X)$ of $X$, which is the Zariski closure in
$X\times\mathrm{Gr}_{d,n}$ of the graph of the Gauss map

$X^{\mathrm{reg}}\to\mathrm{Gr}_{d,n},\quad x\mapsto T_{x}(X^{\mathrm{reg}}),$

where $\mathrm{Gr}_{d,n}$ is the Grassmannian of $d$-dimensional linear
subspace of $\mathbf{C}^{n}$, and the tangent space $T_{x}(X^{\mathrm{reg}})$
is identified with the $d$-dimensional linear subspace of $\mathbf{C}^{n}$
parallel to it. For technical reasons, we consider the _projective conormal
space_ $C(X)$ of $X$ instead of the Nash blow-up. Recall that the _conormal
bundle_ $T_{X^{\mathrm{reg}}}^{*}\mathbf{C}^{n}$ of $X^{\mathrm{reg}}$ in the
ambient space $\mathbf{C}^{n}$ is defined as

$T_{X^{\mathrm{reg}}}^{*}\mathbf{C}^{n}=\big{\\{}(x,\xi)\in
T^{*}\mathbf{C}^{n}\mid x\in X,\xi\in T_{x}^{*}(\mathbf{C}^{n})\text{ such
that }\langle v,\xi\rangle=0\text{ for all }v\in
T_{x}(X^{\mathrm{reg}})\big{\\}},$

i.e. at each point $x\in X^{\mathrm{reg}}$, it parametrizes the hyperplanes in
$\mathbf{C}^{n}$ containing the tangent space $T_{x}(X^{\mathrm{reg}})$. Let
$T_{X}^{*}\mathbf{C}^{n}$ be the Zariski closure of
$T^{*}_{X^{\mathrm{reg}}}\mathbf{C}^{n}$ in $T^{*}\mathbf{C}^{n}$, and let
$C(X)$ be the projectivization of $T^{*}_{X}(\mathbf{C}^{n})$. It is clear
that $C(X)$ is of dimension $n-1$. By construction, we get the diagram

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Let $e_{Y}:{\rm Bl}_{Y}(X)\to X$ be the blow-up of $X$ with center $Y$, let
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{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
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}\hbox{$\scriptstyle{\hat{e}_{Y}}$} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{ {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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}\hbox{$\scriptstyle{\kappa^{\prime}}$} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{ {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
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}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-52.82678pt}{19.05005pt}\pgfsys@lineto{-5.30939pt}{-32.3317pt}\pgfsys@stroke\pgfsys@invoke{
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}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{}}{} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$\scriptstyle{\xi}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{ {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
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}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{2.5331pt}{19.05005pt}\pgfsys@lineto{2.5331pt}{-32.22537pt}\pgfsys@stroke\pgfsys@invoke{
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}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{ {}{}{}}{}{ {}{}{}}
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}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{}}{} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{$\scriptstyle{\lambda}$} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{ {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@transformcm{0.0}{-1.0}{1.0}{0.0}{63.5477pt}{1.64998pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{65.90047pt}{10.25697pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{$\scriptstyle{\mathrm{pr}_{2}}$} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{ {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-40.9071pt}{-40.81836pt}\pgfsys@lineto{-6.90712pt}{-40.81836pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-6.70714pt}{-40.81836pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-26.94238pt}{-37.09892pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{$\scriptstyle{{e}_{Y}}$} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}},$ (4.13)

where the existence of the morphism $\kappa^{\prime}$ is due to the universal
property of the blow-up. Note that $E_{Y}C(X)$ coincides with the Zariski
closure of the inverse image of ${\rm Bl}_{Y}(X)-e_{Y}^{-1}(Y)$ in ${\rm
Bl}_{Y}(X)\times_{X}C(X)$. By construction, the pre-image $\xi^{-1}(Y)$ and
the morphisms

${\xi^{-1}(Y)}$${{\rm
Bl}_{Y}(X)}$${C(X)}$$\scriptstyle{\kappa^{\prime}}$$\scriptstyle{\hat{e}_{Y}}$

encodes the information about the limit secant lines and limit tangent spaces.
For any $y\in Y$, the above diagram induces morphisms

$\leavevmode\hbox to145.37pt{\vbox
to53.11pt{\pgfpicture\makeatletter\hbox{\hskip
72.68636pt\lower-26.55498pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{\offinterlineskip{}{}{{{}}{{}}{{}}}{{{}}}{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-72.68636pt}{-26.55498pt}\pgfsys@invoke{
}\hbox{\vbox{\halign{\pgf@matrix@init@row\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding&&\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding\cr\hfil\hskip
0.0pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
0.0pt\hfil&\hfil\hskip 60.00009pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-11.04053pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${\xi^{-1}(y)}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
0.0pt\hfil&\hfil\hskip 60.00009pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
0.0pt\hfil\cr\vskip 18.00005pt\cr\hfil\hskip
14.55301pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{${\mathbf{P}^{n-d^{\prime}-1}}$} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
0.0pt\hfil&\hfil\hskip 60.00009pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
0.0pt\hfil&\hfil\hskip 60.00009pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{${\check{\mathbf{P}}^{n-1}}$} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
10.81953pt\hfil\cr}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}{{{{}}}{{}}{{}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} {}{ {}{}{}}{}{ {}{}{}} {{{{{}}{
{}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-12.47728pt}{4.12698pt}\pgfsys@lineto{-43.04001pt}{-14.7393pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-0.85094}{-0.52528}{0.52528}{-0.85094}{-43.21017pt}{-14.84435pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ }}{ } {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-35.65761pt}{-1.69734pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$\scriptstyle{q_{1}}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{ {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
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}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{16.21564pt}{4.12698pt}\pgfsys@lineto{50.50699pt}{-17.0436pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.8509}{-0.52531}{0.52531}{0.8509}{50.67714pt}{-17.14864pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{}}{} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{35.88423pt}{-2.8495pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$\scriptstyle{q_{2}}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys<EMAIL_ADDRESS>(4.14)

To analyse the fiber $\xi^{-1}(y)$ via the above diagram, we are led to the
local polar varieties:

######  Definition 4.10.

Let $H_{d-k+1}$ be a linear subspace of $\mathbf{C}^{n}$ of codimension
$d-k+1$, let $\pi:\mathbf{C}^{n}\to\mathbf{C}^{d-k+1}$ be the linear
projection with kernel $H_{d-k+1}$. The Zariski closure in $(X,y)$ of the
critical locus for the restriction
$\pi:(X^{\mathrm{reg}},y)\subset\mathbf{C}^{n}\to\mathbf{C}^{d-k+1}$ is called
the _local polar variety_ of $X$ at $y$ with respect to $H_{d-k+1}$, denoted
$P_{y}(X,H_{d-k+1})$.

Let $\check{H}^{d-k}\subset\check{\mathbf{P}}^{n-1}$ be the dual of
$H_{d-k+1}\subset\mathbf{C}^{n}$, it consists of the hyperplanes in
$\mathbf{C}^{n}$ containing $H_{d-k+1}$, hence is a linear projective subspace
of dimension $d-k$.

######  Proposition 4.11 (Teissier [T3], Chap. IV, corollaire 1.3.2).

For a generic choice of $H_{d-k+1}$, the local polar variety
$P_{y}(X,H_{d-k+1})$ coincides with the image
$\kappa(\lambda^{-1}(\check{H}^{d-k}))$ in $(X,y)$. It is a reduced closed