to obtain numerical control for Whitney–Thom equisingularity (the $W_{f}$ and
$A_{f}$ conditions) for $(X,0)\rightarrow(Y,0)$ and a function $f\colon
X\rightarrow\mathbb{C}$. By this we mean finding invariants that depend only
on the fibers $X_{y},f_{y}$ whose constancy across $Y$ is equivalent to
Whitney–Thom equisingularity.

The goal is to come up with finitely many invariants associated with each
$X_{y}$ which control equisingularity for all families having $X_{y}$ as a
member. When $X_{y}$ are isolated singularities this can be done thanks to
Grauert’s theorem [Gra72] as the base space of miniversal deformations of each
$X_{y}$ is finite. Then we can view $(X,0)\rightarrow(Y,0)$ as a subfamily of
$(\mathcal{X},0)\rightarrow(W,0)$ such that $X_{0}=\mathcal{X}_{0}$, $W$ is a
component of the miniversal base space of $X_{0}$ and $\mathcal{X}$ is the
total deformation space of $X_{0}$ over $W$. In the case the volume is stable
over $W$, there is a unique invariant associated with each $X_{y}$ that
controls equisingularity for all families with irreducible bases that are
subspaces of $W$. All deformations considered below are embedded deformations
of $X_{0}\subset\mathbb{C}^{n}$.

In the setup of Thm. 6.1 set $C:=\mathrm{Bl}_{c^{-1}(Y)}C(X,f)$ Algebraically,

$C:=\mathrm{Projan}(\mathcal{R}(m_{Y}J_{\mathrm{rel}}(X,f)))$

where $m_{Y}$ is the ideal of $Y$ in $\mathcal{O}_{X}$. Let $\Sigma(f)$ be the
critical set of $f$ \- the union of singular sets of the various level
hypersurfaces. Set $Q:=X\cap f^{-1}0$ and $Q_{y}:=X_{y}\cap f^{-1}0$.

Assume $X$ is reduced and equidimensional. Embed $J_{\mathrm{rel}}(X,f))$ in a
free module $\mathcal{F}$ of the same rank (see Prp. 3.1 (iii)). To align with
previously used notation we denote the restricted local volume corresponding
to $C_{y}$ by $\varepsilon(m_{y}J_{\mathrm{rel}}(X,f))(y)$ where $m_{y}$ is
the ideal of $y$ in $\mathcal{O}_{X_{y}}$ and $J_{\mathrm{rel}}(X,f)(y)$ is
the image of the relative Jacobian module in $\mathcal{F}(y)$ because of its
connection to the $\varepsilon$-multiplicity (see [KUV] and [UV11]). Denote
the restricted local volume corresponding to $C_{\mathrm{rel}}(X,f)$ by
$\varepsilon(J_{\mathrm{rel}}(X,f))(y)$ and the one corresponding to
$C_{\mathrm{rel}}(X)_{y}$ by $\varepsilon(J_{\mathrm{rel}}(X))(y)$. We use
Thm. 5.2 and Thm. 5.3 to obtain a numerical characterization of $W_{f}$.

######  Theorem 7.1.

Assume $X$ is reduced and equidimensional and $X_{y}$ is equidimensional for
each $y$. Suppose $X_{y}$ and $Q_{y}$ are isolated singularities. Suppose
$h\colon X\rightarrow Y$ is a subfamily of $\mathcal{X}\rightarrow W$, where
$W$ is a component of the miniversal base space of $X_{0}$ and $\mathcal{X}$
is the total deformation space of $X_{0}$ over $W$. Let $\tilde{f}$ be a
generic deformation of $f$ over $\mathcal{X}$. Suppose that
$\varepsilon(J_{\mathrm{rel}}(\mathcal{X},\tilde{f}))(w)$ is stable. The
following holds:

  1. (i)

Suppose $\Sigma(f)=Y$. Assume $\varepsilon(m_{y}J_{\mathrm{rel}}(X,f))(y)$ is
independent of $y$. Then the union of the singular points of $f_{y}$ is $Y$
and the pair $(X-Y,Y)$ satisfies $W_{f}$.

  2. (ii)

Suppose $\Sigma(f)$ is equal to $Y$ or is empty and the pair $(X-Y,Y)$
satisfies $W_{f}$. Then $\varepsilon(m_{y}J_{\mathrm{rel}}(X,f))(y)$ is
independent of $y$.

###### Proof.

Note that by additivity of the length function

$\varepsilon(m_{w}J_{\mathrm{rel}}(X,f))(w)=e(m_{w}J_{\mathrm{rel}}(X,f)(w),J_{\mathrm{rel}}(X,f)(w))+\varepsilon(J_{\mathrm{rel}}(X,f))(w)$

where $e(m_{w}J_{\mathrm{rel}}(X,f)(w),J_{\mathrm{rel}}(X,f)(w))$ is the
relative Buchsbaum–Rim multiplicity. Thus the stability of
$\varepsilon(J_{\mathrm{rel}}(\mathcal{X},\tilde{f}))(w)$ implies that
$\varepsilon(m_{w}J_{\mathrm{rel}}(X,f))(w)$ is stable. By the result of
Henry, Merle and Sabbah there exists a Zariski open subset $U$ of $Y$ such
that the image of each $g_{j}$ in $\mathcal{F}(y)$ is integral over
$m_{y}J_{\mathrm{rel}}(X,f)(y)$. By replacing $\mathcal{X}$ with a smaller
neighborhood of the origin if necessary we can assume that the singular locus
$S(W)$ of $\mathcal{X}$ is finite. Because
$\varepsilon(m_{y}J_{\mathrm{rel}}(X,f))(y)$ is constant on $y$ by Thm. 5.2
applied with $C(W):=\mathrm{Bl}_{c^{-1}W}C(\mathcal{X},\tilde{f})$ and $S(W)$
the singular locus of $\mathcal{X}$, and $g:=g_{j}$ for $1\leq j\leq k$, it
follows that each $g_{j}$ is integral over $m_{Y}J_{\mathrm{rel}}(X,f)$

Set $\mathcal{M}:=J_{x}(X,f)$ and
$\mathcal{N}:=\langle\mathcal{M},g_{1},\ldots,g_{k}\rangle$. Because
$\mathcal{N}$ is integral over $m_{Y}\mathcal{M}$ it follows that the integral
closure of $\mathcal{N}$ in $\mathcal{O}_{X}^{p}$ is in the integral closure
$\overline{\mathcal{M}}$ of $J_{x}(X,f)$ in $\mathcal{O}_{X}^{p}$. Hence the
nonfree locus of the former module, which is the union of singular points of
$f_{y}$ and $X_{y}$, is equal to the nonfree locus of the latter module, which
by assumption is equal to $Y$. This completes the proof of $\rm{(i)}$.

Consider $\rm{(ii)}$. By Thm. 6.2 $\mathcal{N}$ is integral over
$\mathcal{M}$. Thus $m_{Y}\mathcal{N}$ and $m_{Y}\mathcal{M}$ are the same up
to integral closure. However, $\mathrm{Proj}(\mathcal{R}(m_{Y}\mathcal{N}))$
is equal to $\mathrm{Bl}_{m_{Y}}C(X,f)$ and
$\mathrm{Proj}(\mathcal{R}(m_{Y}\mathcal{M}))$ is equal to
$\mathrm{Bl}_{m_{Y}}C_{\mathrm{rel}}(X,f)$. By Thm. 6.1 the excetional divisor
of $\mathrm{Bl}_{m_{Y}}C(X,f)$ is equidimensional. Then so is the exceptional
divisor of $\mathrm{Bl}_{m_{Y}}C_{\mathrm{rel}}(X,f)$. Finally, by the LVF we
get that $\varepsilon(m_{y}J_{\mathrm{rel}}(X,f)(y))$ is constant. The proof
of Thm. 7.1 is now complete. ∎

######  Remark 7.2.

In the case when $\varepsilon(J_{\mathrm{rel}}(\mathcal{X}))(w)=0$, that is
the case when $\mathcal{X}_{w}$ is a deficient conormal singularity (see the
next section for a definition and Prp. 8.3), then we claim that
$\varepsilon(J_{\mathrm{rel}}(\mathcal{X},\tilde{f}))(w)=0$. In particular,
$\varepsilon(J_{\mathrm{rel}}(\mathcal{X},\tilde{f}))(w)$ is stable. To see
this choose a deformation $\tilde{f}:=f+\sum_{i=1}^{\dim W}\alpha_{i}w_{i}$,
where $w_{i}$ are coordinates for $(W,0)$ and $\alpha_{i}$ are generic
constansts so that the hyperplane determined by $\mathrm{grad}(\tilde{f})$ at
$s$ is not part of $C(\mathcal{X})_{s}$ for generic $w$ and $s\in S(W)_{w}$.
Then by [GR19, Thm. 2.2] $C(\mathcal{X},\tilde{f})_{s}$ is the join of $df(s)$
and $C(\mathcal{X})_{s}$. But $C(\mathcal{X})_{s}$ is not of maximal dimension
because $\mathcal{X}_{w}$ is deficient conormal. Because $\dim
C(\mathcal{X},\tilde{f})=\dim C(\mathcal{X})+1$, it follows that
$C(\mathcal{X},\tilde{f})_{s}$ is not of maximal dimension either. Thus,
$\varepsilon(J_{\mathrm{rel}}(\mathcal{X},\tilde{f}))(w)=0$. The existence of
Zariski open subset of $W$ where the restricted local volume vanishes follows
from upper semi-continuity of fiber dimension.

To get numerical control for the Whitney condition B replace in Thm. 7.1
$\varepsilon(m_{y}J_{\mathrm{rel}}(X,f))(y)$ by
$\varepsilon(m_{y}J_{\mathrm{rel}}(X))(y)$. A fiberwise numerical
characterization of Whitney A is not possible (see [GK99, Ex. 4.3].

Denote by $X^{i}$ the section of $X$ by a general linear space of codimension
$i$ in $\mathbb{C}^{n+k}$ containing $Y$. Set $f^{i}:=f|X^{i}$. A fundamental
result of Lê and Teissier [LT88] states that $W_{f}$ holds if and only if
$X^{i}_{y},f^{i}_{y}$ are topologically equisingular. It readily yields the
following corollary to Thm. 7.1.

######  Corollary 7.3.

The pairs $X^{i}_{y},f^{i}_{y}$ are topologically equisingular if and only if
$\varepsilon(m_{y}J_{\mathrm{rel}}(X,f))(y)$ is constant.

Next we turn our attention to providing a numerical characterization for
Thom’s $A_{f}$ condtion.

######  Definition 7.4.

We say that $A_{f}$ condition holds for the pair $X_{\mathrm{sm}},Y$ at $0$ if
$f(Y)=0$ and $Y$ lies in every hyperplane obtained as a limit of tangent
hyperplanes to a level hypersurface at a point $x\in X_{\mathrm{sm}}$ as $x$
approaches $0$.

The $A_{f}$ condition is known to hold generically along $Y$ by a result of
Hironaka [Hir76, Thm. 2, pg. 247]. We review briefly the connection between
the theory of integral closure of modules and Thom’s $A_{f}$ condition. Recall
that given a submodule $\mathcal{M}$ of a free $\mathcal{O}_{X,0}$ module
$\mathcal{F}$, we say that $u\in\mathcal{F}$ is strictly dependent on
$\mathcal{M}$ and we write $u\in\mathcal{M}^{\dagger}$, if for all analytic
path germs $\phi:(\mathbb{C},0)\to(X,0)$, $\phi^{*}u$ is contained in
$\phi^{*}(\mathcal{M})m_{1}$, where $m_{1}$ is the maximal ideal of
$\mathcal{O}_{\mathbb{C},0}.$

Denote by $c_{(X,f)}\colon\mathrm{Proj}(C(X,f))\rightarrow X$ the structure
morphism and by $C(Y)$ the conormal space of $Y$. The following result
expresses the $A_{f}$ condition in terms of strict dependence.

######  Proposition 7.5.

Assume $f(Y)=0$. The following are equivalent

  * (i)

The $A_{f}$ condition holds for the pair $X_{\mathrm{sm}},Y$ at $0$.

  * (ii)

$c_{(X,f)}^{-1}(Y)\subset C(Y)$.

  * (iii)

$g_{j}\in J(X,f)^{\dagger}$ for all $j=1,\ldots,k.$

###### Proof.

The equivalence of i) and ii) is obvious; the equivalence of i) and iii) is
Lemma 5.1 of [GK99]. ∎

A similar result holds for the Whitney A condition. The condition we need for
our main result is a much weaker version of Whitney A.

######  Definition 7.6.

We say that $(X,0)\rightarrow(Y,0)$ satisfies the infinitesimal Whitney A
fiber condition at $0$ if the image of each $g_{j}$ in $\mathcal{F}(y)$ is in
$J(X_{0})^{\dagger}$ for $j=1,\ldots,k$, where $J(X_{0})$ is the Jacobian
module of $X_{0}$.

######  Theorem 7.7.

Assume $X$ is reduced and equidimensional and $X_{y}$ is equidimensional for
each $y$. Suppose $X_{y}$ and $Q_{y}$ are isolated singularities. Suppose
$h\colon X\rightarrow Y$ is a subfamily of $\mathcal{X}\rightarrow W$, where
$W$ is a component of the miniversal base space of $X_{0}$ and $\mathcal{X}$
is the total deformation space of $X_{0}$ over $W$. Let $\tilde{f}$ be a
generic deformation of $f$ over $\mathcal{X}$. Suppose that
$\varepsilon(J_{\mathrm{rel}}(\mathcal{X},\tilde{f}))(w)$ is stable and that
the infinitesimal fiber condition hodls in case $f\notin m_{Y}^{2}$. The
following holds:

  1. (i)

Suppose $\Sigma(f)=Y$. Suppose $\varepsilon(J_{\mathrm{rel}}(X,f))(y)$ is
independent of $y$. Then the union of the singular points of $f_{y}$ is $Y$
and the pair $(X-Y,Y)$ satisfies $A_{f}$.

  2. (ii)

Suppose $\Sigma(f)$ is equal to $Y$ or is empty and the pair $(X-Y,Y)$
satisfies $A_{f}$. Then $\varepsilon(J_{\mathrm{rel}}(X,f))(y)$ is independent
of $y$.

###### Proof.

Let $K$ be the submodule of $\mathcal{F}$ generated by $g_{1},\ldots,g_{k}$.
Then $K+J_{\mathrm{rel}}(X,f)=J(X,f)$. By Prp. 7.5 if $A_{f}$ holds, then

$J(X,f)\circ\phi=K\circ\phi+J_{\mathrm{rel}}(X,f)\circ\phi\subset
m_{1}(J(X,f)\circ\phi)+J_{\mathrm{rel}}(X,f)\circ\phi$

for any analytic path $\phi\colon(\mathbb{C},0)\rightarrow(X,0)$. By
Nakayama’s lemma we get $J(X,f)\circ\phi=J_{\mathrm{rel}}(X,f)\circ\phi$. By
the valuative criterion for integral dependence we get that
$J(X,f)\subset\overline{J_{\mathrm{rel}}(X,f)}$, where
$\overline{J_{\mathrm{rel}}(X,f)}$ is the integral closure of
$J_{\mathrm{rel}}(X,f)$ in $\mathcal{F}$, if $A_{f}$ holds. In other words, if
$A_{f}$ holds then exists a finite map $C(X,f)\rightarrow
C_{\mathrm{rel}}(X,f)$.

Consider $\rm{(i)}$. By Hironaka’s genericity result [Hir76] there exists a
Zariski open subset $U$ of $Y$ such that $A_{f}$ condition holds at all points
in $U$. Then by selecting analytic paths in $U$ and the discussion above we
obtain that $J(X,f)$ and $J_{\mathrm{rel}}(X,f)$ have the same integral
closure over $U$. Because $\varepsilon(J_{\mathrm{rel}}(X,f))(y)$ is constant
on $y$ by Thm. 5.2 applied with $C(W):=C(\mathcal{X},\tilde{f})$ and $S(W)$
the singular locus of $\mathcal{X}$, and $g:=g_{j}$ for $1\leq j\leq k$, it
follows that each $g_{j}$ is integral over $J_{\mathrm{rel}}(X,f)$. Thus,
$J(X,f)\subset\overline{J_{\mathrm{rel}}(X,f)}$ and so there exists a finite
map $C(X,f)\rightarrow C_{\mathrm{rel}}(X,f)$. Because $J_{\mathrm{rel}}(X,f)$
is generated by $n$ elements $\dim C_{\mathrm{rel}}(X,f)_{0}<n$. Thus $\dim
C(X,f)_{0}<n$. Then by [GR19, Thm. 4.4 ] it follows that the pair $(X-Y,Y)$
satisfies $A_{f}$. As in the proof of Thm. 7.1 we obtain that union of
singular points of $f_{y}$ is $Y$.

Consider $\rm{(ii)}$. By Prp. 7.5 $c_{(X,f)}^{-1}(Y)\subset C(Y)$. But $\dim
C(Y)_{0}=n-1$. Also, by the discussion above $C(X,f)\rightarrow
C_{\mathrm{rel}}(X,f)$ is finite. Thus $\dim C_{\mathrm{rel}}(X,f)_{0}<n$, so
the LVF yields the constancy of $\varepsilon(J_{\mathrm{rel}}(X,f))(y)$. ∎

##  8\. Deficient conormal singularities and generalized smoothability

In this section we define the class of deficient conormal (dc) singularities.
We show that the dc propery is intrinsic and stable under infinitesimal
deformations. We show that the fibers of conormal spaces behave well under
pullbacks of transverse holomorphic maps between affine spaces. Then using
Thom’s transversality we show that determinantal and Pfaffian singularities
deform to dc singularities. As a corollary we obtain that the the generic
deformations of Cohen–Macaulay codimension $2$ and Gorenstein codimension $3$
singularities are dc. Inspired by a recent result of Kollár and Kovács and
following work of Schlessinger, we show that the affine cones over normally
embedded abelian varieties of dimension at least $2$ do not admit
infinitesimal deformations to dc singularities. We finish the section with
computing the restricted local volume associated with the conormal space of a
nonsmoothable Cohen–Macaulay codimension $2$ singularity.

Let $Z\subset\mathbb{C}^{N}$ be a reduced complex analytic variety. As before
define the conormal space $C(Z)$ of $Z$ as the closure in
$Z\times\mathbb{P}^{N-1}$ of the set of pairs $(z,H)$ where $z$ is a smooth
point of $Z$ and $H$ is a tangent hyperplane at $z$, i.e. a hyperplane
containing $T_{Z,z}$. Denote by $c$ the structure map $c:C(Z)\rightarrow Z$.

######  Definition 8.1.

We say $Z$ is deficient conormal (dc) at $z_{0}\in Z$ if

$\operatorname{codim}(c^{-1}z_{0},C(Z))\geq 2.$

We say $Z$ is dc if $Z$ is dc at each of its points.

######  Example 8.2.

If $z_{0}$ is a smooth point of $Z$ with $Z$ equidimensional and $\dim Z\geq
2$, then $Z$ is dc at $z_{0}$. If $Z$ is local complete intersection and dc at
$z_{0}$, then by Lem. 5.7 in [GaM99] $Z$ is smooth at $z_{0}$.

If $Z$ is the affine cone over a smooth projective variety
$V\subset\mathbb{P}^{N-1}$ with positive deffect (or degenerate dual) (see
[Ein86] and [Ein85] for examples and classifications), then $Z$ is dc at the
vertex of the cone. To see this it’s enough to show that the fiber of the
conormal of $Z$ over the vertex is equal set-theoretically to the dual of the
base (not requiring $V$ to be smooth). An easy computation shows that $H$ is a
tangent hyperplane at a point $v$ in $V_{\mathrm{sm}}$ if and only if $H$ is
tangent to each point of $Z$ lying on the line $0v$ and different from the
origin. If $z_{1},z_{2},\ldots$ are points from $Z$ converging to the origin
in $\mathbb{C}^{N+1}$, and $H_{1},H_{2},\ldots$ are the corresponding tangent
hyperplanes, then $H_{i}$ are tangent hyperplanes at points of $V$, and thus
their limit belongs to the dual of $V$. Conversely, if $H$ is in the dual of
$V$, say $H$ is a limit of tangent hyperplanes $H_{1},H_{2},\ldots$ at smooth
points $v_{1},v_{2},\ldots$ then $H$ is a limit of tangent hyperplanes at any
sequence of points from the lines $0v_{1},0v_{2},\ldots$, and it belongs to
the fiber $C(Z)$ over the origin.

It can be shown that many of the examples of rigid singularities considered by
Schlessinger [Sch73] like fans (eg. two planes meeting at a point in
$\mathbb{C}^{4}$), quotient singularities of dimension at least $2$, etc. are
dc.

Set $\mathcal{L}:=\mathcal{O}_{C}(1)$. The next result shows that the dc
property is characterized by the vanishing of the local volume of
$\mathcal{L}$. Recall that the local volume $\mathrm{vol}_{C}(\mathcal{L})$ at
$z$ is given by the epsilon multiplicity of the Jacobian module $J_{z}(Z)$ of
$Z$ at $z$:

$\varepsilon(J_{z}(Z)):=\limsup_{n\to\infty}\frac{r!}{n^{r}}\dim_{\mathbb{C}}H_{\mathfrak{m}_{z}}^{0}(\mathcal{F}^{n}/J(Z)^{n})$

where $\mathcal{F}$ is a free module containing $J(Z)$ having the same generic
rank, $\mathcal{F}^{n}=\mathrm{Sym}^{n}(\mathcal{F})$ and
$J(Z)^{n}:=\mathrm{Sym}^{n}(J(Z))/(\mathcal{O}_{Z}\text{-torsion})$.

######  Proposition 8.3.

Suppose $Z$ is equidimensional with $\dim Z\geq 2$. Then
$\varepsilon(J_{z}(Z))=0$ if and only if $Z$ is dc at $z$.

###### Proof.

Follows immediately from Thm. 3.5. ∎

Our next result shows that the dc property is intrinsic for $X$. Suppose we
have two reduced complex analytic equidimensional varieties
$Z_{1}\subset\mathbb{C}^{n_{1}}$ and $Z_{2}\subset\mathbb{C}^{n_{2}}$. Denote
the two conormal spaces corresponding to each of the these germs by $C(Z_{1})$
and $C(Z_{2})$. Denote by $c_{i}\colon C(Z_{i})\rightarrow Z_{i}$ for $i=1,2$
the corresponding structure morphisms.

######  Proposition 8.4.

Suppose that there exists an analytic isomorphism $\phi\colon Z_{1}\rightarrow
Z_{2}$ such that $\phi(z_{1})=z_{2}$. Then
$\operatorname{codim}c_{1}^{-1}(z_{1})=l$ in $C(Z_{1})$ for some positive
integer $l$ if and only if $\operatorname{codim}c_{2}^{-1}(z_{2})=l$ in
$C(Z_{2})$. In particular, $Z_{1}$ is dc at $z_{1}$ if and only if $Z_{2}$ is
dc at $z_{2}$.

###### Proof.

Let $Z\subset\mathbb{C}^{N}$ be a reduced equidimensional complex analytic
varieity. Let $z\in Z$. Consider the germ $(Z,z)$. Let’s recall the
construction of the local polar varieties of $(Z,z)$ as developed by Teissier
in [T81, Chp. IV]. Consider the following diagram

${C(Z)}$${Z\times\check{\mathbb{P}}^{N-1}}$${Z}$${\check{\mathbb{P}}^{N-1}}$$\scriptstyle{i}$$\scriptstyle{\lambda}$$\scriptstyle{c}$$\scriptstyle{\mathrm{pr}_{2}}$

Assume $(Z,z)$ is a germ of pure dimension $d$ and of codimension $e$ in
$\mathbb{C}^{N}$. Let $H_{e+l-1}$ be a general plane in
$\check{\mathbb{P}}^{N-1}$ of codimension $e+l-1$. Define the local polar
variety $\Gamma_{l}(Z,H_{e+l-1})$ with respect to $H_{e+l-1}$ as
$c(\lambda^{-1}(H_{e+l-1}))$. Then $\Gamma_{l}(Z,H_{e+l-1})$ is either empty
or of pure codimension $l$. Teissier [T81, Chp. IV, Thm. 3.1] showed that for
sufficiently general $H_{e+l-1}$ the multiplicity of $\Gamma_{l}(Z,H_{e+l-1})$
at $z$ depends only on the analytic type of $(Z,z)$. Often we suppres
$H_{e+l-1}$ from the notation and we simply write $\Gamma_{l}$ because in the
applications we will be interested mostly in the existence of the polar
varieties of appropriate codimension. Finally, observe that
$\Gamma_{d-l+1}(Z)$ is empty if and only if $\operatorname{codim}c^{-1}(z)\geq
l$ in $C(Z)$.

Apply Teissier’s result to $Z_{1}$ and $Z_{2}$. Because
$\operatorname{codim}c_{1}^{-1}(z_{1})=l$ in $C(Z_{1})$, then the multiplicity
at $z_{1}$ of polar varieties of $(Z_{1},z_{1})$ of codimension $d-l+1$ is
zero. But then, by Teissier’s result, so is the multiplicity at $z_{2}$ of the
polar varieties of $(Z_{2},z_{2})$ of dimension $d-l+1$. Hence
$\operatorname{codim}c_{2}^{-1}(z_{2})=l$ in $C_{2}(Z_{2})$. ∎

Next we show that the dc property is stable under infinitesimal deformations.

######  Proposition 8.5.

Suppose $X\rightarrow Y$ is family with equidimensional fibers and
equidimensional reduced total space $X$. Assume that $Y$ is smooth of
dimension one and $X_{y_{0}}$ has an isolated dc singularity at a point
$x_{0}$. Then $X_{y}$ is a dc singularity for each $y$ closed enough to
$y_{0}$.

###### Proof.

Let $J_{\mathrm{rel}}(X)$ be the relative Jacobian module of $X\rightarrow Y$,
and let $\mathcal{F}$ be a free module that contains it. Set
$\mathcal{F}(y):=\mathcal{F}\otimes_{\mathcal{O}_{Y}}k(y)$ for each point
$y\in Y$. The image of $J_{\mathrm{rel}}(X)$ in $\mathcal{F}(y)$ is the
Jacobian module $J(X_{y})$ of $X_{y}$. Let $S$ be the singular locus of $X$.
By replacing $X$ with a small enough neighborhood of $x_{0}$ we can assume
that $S$ is finite over $Y$. Then

$H_{S}(\mathcal{F}^{n}/J_{\mathrm{rel}}(X)^{n})\otimes_{\mathcal{O}_{Y}}k(y_{0})\hookrightarrow
H_{S_{y_{0}}}(\mathcal{F}(y_{0})^{n}/J(X_{y_{0}})^{n}).$

Thus $\varepsilon(J_{\mathrm{rel}}(X))(y_{0})\leq\varepsilon(J(X_{y_{0}}))$.
Because $X_{y_{0}}$ is dc at each point of $S_{y_{0}}$, then
$\varepsilon(J(X_{y_{0}}))=0$ and so
$\varepsilon(J_{\mathrm{rel}}(X))(y_{0})=0$. Applying the LVF and noting that
the intersection number on its right-hand side is nonnegative, we obtain
$\varepsilon(J_{\mathrm{rel}}(X))(y)=0$. By Prp. 4.7
$\varepsilon(J_{\mathrm{rel}}(X))(y)=\varepsilon(J(X_{y}))$ for $y$ close
enough to $y_{0}$. Thus $\varepsilon(J(X_{y}))=0$ locally at each singular
point of $X_{y}$. So by Prp. 8.3 $X_{y}$ is a dc singularity. ∎

The next three results are the key ingredients in the proof of the main
theorem of this section. The first one shows that the codimension of the
fibers of conormal spaces can only increase under transverse maps. The second
result due to Trivedi allows us to deform holomorphic maps in such a way so
that the deformations become transverse to a predetermined collection of
submanifolds in the target space. The third result due to Buchweitz allows us
to obtain deformations of singularities obtained from pullbacks of holomorphic
maps.

######  Definition 8.6.

Let $M$ and $N$ be complex manifolds. Let $g\colon M\rightarrow N$ be a smooth
map. We say that $g$ is transvserse to a submanifold $S$ of $N$ at a point
$m\in M$ and we denote it by $g\pitchfork_{m}S$, if either $g(m)\not\in S$ or
$g(m)\in S$ and $Dg_{m}(T_{m}M)+T_{g(m)}S=T_{g(m)}N$. If
$V=\bigsqcup_{i=1}^{q}V_{i}$ is a stratification of $V\subset N$ then by
$g\pitchfork_{K}V$ we mean that $g$ is transverse to each stratum $V_{i}$ at
all points in $K\subset M$.

Suppose $(V,0)\subset(\mathbb{C}^{m},0)$ is a complex analytic subvariety. We
say that $V=\bigsqcup_{i=1}^{q}V_{i}$ is Whitney A (respectively Whitney B)
stratification if pairs of nearby strata satisfy Whitney condition A
(respectively Whitney B). The existence of these stratifications was
established by Whitney [Wh64] as discussed in Sct. 6. Recall that a Whitney B
stratification is a Whitney A stratification.

Let $g\colon(\mathbb{C}^{n},0)\rightarrow(\mathbb{C}^{m},0)$ be a holomorphic
map. Consider the diagram where the right square is cartesian:

$\begin{CD}T^{*}\mathbb{C}^{n}@<{}<{(dg)^{*}}<g^{*}T^{*}\mathbb{C}^{m}@>{g_{\pi}}>{}>T^{*}\mathbb{C}^{m}\\\
@V{}V{\pi_{n}}V@V{}V{\pi}V@V{}V{\pi_{m}}V\\\
\mathbb{C}^{n}<EMAIL_ADDRESS>
We say that $A$ is a conic subset of $T^{*}\mathbb{C}^{m}$ if its fibers are
invariant in the fibers of $T^{*}\mathbb{C}^{m}$ under a multiplication with a
nonzero complex number. For a closed conic subset $A$ of $T^{*}\mathbb{C}^{m}$
we say that $g$ is noncharacteristic with respect to $g$ if
$g_{\pi}^{-1}(A)\cap\mathrm{Ker}((dg)^{*})$ is the zero-section of
$g^{*}T^{*}\mathbb{C}^{m}$. Denote by $T_{V}^{*}\mathbb{C}^{m}$ the closure in
$\mathbb{C}^{m}\times\mathbb{C}^{m*}$ of the pairs of $(v,\eta_{v})$ where $v$
is a smooth point of $V$ and $\eta_{v}$ is a conormal vector at $v$. Define
$T_{X}^{*}\mathbb{C}^{n}$ analogously. Note that $T_{V}^{*}\mathbb{C}^{m}$ and
$T_{X}^{*}\mathbb{C}^{n}$ are conical. By projectivizing with respect to
vertical homotheties in $T^{*}\mathbb{C}^{m}$ and $T^{*}\mathbb{C}^{n}$ we
obtain $\mathbb{P}(T_{V}^{*}\mathbb{C}^{m})=C(V)$ and
$\mathbb{P}(T_{X}^{*}\mathbb{C}^{n})=C(X)$.

######  Theorem 8.7.

Assume $m\geq n$. Let $(X,0)\subset(\mathbb{C}^{n},0)$ and
$(V,0)\subset(\mathbb{C}^{m},0)$ be equidimensional reduced complex analytic
varieties with $g^{-1}V=X$ and $\operatorname{codim}X=\operatorname{codim}V$.
Assume $g^{-1}V^{\prime}$ is an irreducible component of $X$ for each
irreducible component $V^{\prime}$ of $V$. Assume that $g\pitchfork_{0}V$
where $V$ is given a Whitney B stratification. Then set-theoretically

$C(X)_{x}=(dg)^{*}(g_{\pi}^{-1}(C(V)_{0}))$

for each $x\in g^{-1}(0)$. Furthermore,
$\operatorname{codim}(C(X)_{x},C(X))\geq\operatorname{codim}(C(V)_{0},C(V))$.
In particular, if $V$ is dc singularity at $0$, then so is $X$ at $x$.

###### Proof.

Let $(V,0)=\bigsqcup_{i=1}^{q}V_{i}$ be a Whitney $B$ stratification of
$(V,0)$. Suppose $V_{1}$ is the smooth locus of $V$ and $0\in V_{2}$. First we
show that there exists a neighborhood $U$ of $0$ in $(\mathbb{C}^{m},0)$ such
that $g$ is transverse to each point in $U\cap V_{i}$ for $i=1,\ldots q$. Note
that $f$ is transverse at $U\cap V_{2}$ for sufficiently small $U$ by openess
of transversality. We will show that $g$ is transverse to $U\cap V_{1}$ for
$U$ sufficiently small (the proof for the rest of the strata is the same).
Suppose $x_{1},x_{2},\ldots$ is a sequence of points in $X$ converging to $x$
with $x\in g^{-1}(0)$ such that $g$ fails to be transverse at
$g(x_{1}),g(x_{2}),\ldots$ Then
$Dg(T_{x_{j}}\mathbb{C}^{n})+T_{g(x_{j})}V_{1}\neq T_{g(x_{j})}\mathbb{C}^{m}$
for $j=1,2,\ldots$. Assume that as $x_{j}\rightarrow x$ the linear spaces
$T_{g(x_{j})}V_{1}\rightarrow T$. Because $(V_{1},V_{2})$ satisfies Whitney
condition A at $0$ it follows that $T_{0}V_{2}\subset T$. This would imply
that $Dg(T_{x}\mathbb{C}^{n})+T_{0}V_{2}\neq T_{0}\mathbb{C}^{m}$ which
contradicts our assumption $g\pitchfork_{0}V_{1}$. Replace $V_{i}$ by $U\cap
V_{i}$ for each $i$ where $U$ is a sufficiently small neighborhood of the
origin in $\mathbb{C}^{m}$. Because $g$ is transverse to $V_{i}$, then
$\bigsqcup_{i=1}^{q}g^{-1}(V_{i})$ is a Whitney B stratification of $V$ (see
[Sc03, pg. 257]).

Set $X_{1}:=g^{-1}V_{1}$ and $X_{2}:=g^{-1}V_{2}$. Let $T_{X_{2}}$ be a
transversal of $X_{2}$ through $x$. Because $g$ is transverse at $V_{2}$, it
follows from the local diffeomorphism theorem, after $V_{2}$ and $X_{2}$ are
replaced by small enough neighborhood around $0$ and $x$, that
$(T_{V_{2}},0):=g((T_{X_{2}},x))$ is transversal of $V_{2}$ through $0$ and it
is diffeomorphic to $T_{X_{2}}$. Let $X_{x}$ the fiber over $x$ of the
transverse retraction from $X$ to $X_{2}$ induced by $T_{X_{2}}$. Similarly,
denote by $V_{0}$ the fiber over $0$ of a transverse retraction from $V$ to
$V_{2}$ induced by $T_{V_{2}}$.

Because $(X_{2},x)$ is smooth, then its defining equations are part of regular
system of parameters in $\mathcal{O}_{X,x}$. Thus the family
$(X,x)\rightarrow(X_{2},x)$ obtained from the transversal retraction induced
by $T_{X_{2}}$ has equidimensional fibers. Because $(X_{1},X_{2})$ satisfies
Whitney B at $x$, by [GR19, Thm. 3.1] set-theoretically,
$C(X)_{x}=C(X_{x})_{x}$ where the conormal spaces are taken in
$\mathbb{C}^{n}$ and $C(X)_{x}$ is the fiber of $C(X)$ over $x$.

By assumption the inverse image of each irreducible component of $(V,0)$ is an
irreducible component of $(X,x)$. Thus the family $(V,0)\rightarrow(V_{2},0)$
obtained from the transversal retraction induced by $T_{V_{2}}$ has
equidimensional fibers. Hence by [GR19, Thm. 3.1] we have a set-theoretic
equality $C(V)_{0}=C(V_{0})_{0}$. Therefore, it’s enough to show that
$C(X_{x})_{x}=(dg)^{*}(C(V_{0})_{0})$.

By hypothesis we can arrange $T_{V_{2}}$ such that the set in $V_{0}$ where
the preimage of $g$ is empty does not contain a component of $V_{0}$. Thus
$g((X_{0})_{\mathrm{sm}})$ is dense in $V_{0}$. Because $g$ is a
diffeomorphism between $T(X_{2})$ and $T(V_{2})$ it follows that $g$ is a
bijection between $(X_{0})_{\mathrm{sm}}$ and $g((X_{0})_{\mathrm{sm}})$, and
the last set is open. We will show that $C(X_{x})_{x}$ and $C(V_{0})_{0}$ are
isomorphic.

Set $\Lambda:=g_{\pi}^{-1}(T_{V_{0}}^{*}\mathbb{C}^{m})$ where
$T_{V_{0}}^{*}\mathbb{C}^{m}$ is the closure of $(v,\eta_{v})$ in
$\mathbb{C}^{m}\times\mathbb{C}^{m*}$ with $v$ a smooth point of $V_{0}$ and
$\eta_{v}$ a conormal vector at $v$. Set $\Lambda^{\prime}:=dg^{*}(\Lambda)$.
Because the restriction of $g$ is a diffeomorphism between $T_{X_{2}}$ and
$T_{V_{2}}$, then $g_{\pi}^{-1}(A)\cap\mathrm{Ker}((dg)^{*})$ is the zero
section for every closed conic $A\subset T_{T_{V_{2}}}^{*}\mathbb{C}^{m}$.
Thus $g$ is noncharacteristic with respect to $\Lambda$. In particular, by
Lem. 4.3.1 in [Sc03, pg. 255]
$(dg)^{*}\colon\Lambda\rightarrow\Lambda^{\prime}$ is proper.

Let $\omega_{n}$ and $\omega_{m}$ be the Liouville $1$-forms on
$T^{*}\mathbb{C}^{n}$ and $T^{*}\mathbb{C}^{m}$ respectively. We have

$\omega_{m}|T_{V_{0}}^{*}\mathbb{C}^{m}\equiv 0\implies
g_{\pi}^{*}\omega_{m}|\Lambda\equiv 0\implies(dg)^{*}\omega_{n}|\Lambda\equiv
0\implies\omega_{n}|\Lambda^{\prime}\equiv 0$

where the last implication follows from the properness of the map $(dg)^{*}$
and Prp. 8.3.11 in [KS90, pg. 332]. Because $\Lambda^{\prime}$ is closed and
$\omega_{n}$ vanishes at its smooth points, by [TF18, Prp. 2.2]
$\Lambda^{\prime}$ equals the conormal of its image under $\pi_{n}$, i.e.
$\Lambda^{\prime}=T_{X_{x}}^{*}\mathbb{C}^{n}$. Note that
$\mathbb{P}(T_{V_{0}}^{*}\mathbb{C}^{m})=C(V_{0})$ and
$\mathbb{P}(T_{X_{x}}^{*}\mathbb{C}^{n})=C(X_{x})$. Thus, set-theoretically,
$C(X)_{x}=(dg)^{*}(g_{\pi}^{-1}(C(V)_{0}))$ which proves the first claim.

Let $f_{1}(y_{1},\ldots,y_{m}),\ldots,f_{s}(y_{1},\ldots,y_{m})$ be equations
for $V_{0}$ in $(\mathbb{C}^{m},0)$. Observe that $\mathbb{P}(\Lambda)$ is the
$\mathrm{Projan}$ of the Rees algebra of the coherent
$\mathcal{O}_{X_{x},x}$-module generated by the columns of the matrix
$(\frac{\partial f_{i}}{\partial y_{j}}\circ g)$. In particular,
$\mathbb{P}(\Lambda)$ is analytic. Because
$\mathbb{P}(\Lambda)\rightarrow\mathbb{P}(\Lambda^{\prime})$ is a proper map
between analytic spaces whose generic fiber is of dimension $m-n$, by upper
semi-continuity of fiber dimension and the dimension formula we get
$\dim(\Lambda_{x})\geq\dim\Lambda^{\prime}_{x}+m-n$. Because
$\Lambda_{x}=C(V)_{0}$ and $\Lambda^{\prime}_{x}=C(X)_{x}$, then

$\operatorname{codim}(C(X)_{x},C(X))\geq\operatorname{codim}(C(V)_{0},C(V)).$

In particular, if $V$ is dc singularity at $0$, then so is $X$ at $x$. ∎

Denote by $\mathcal{H}(M,N)$ the complete metric space of holomorphic maps
between two complex manifold $M$ and $N$ with the weak topology induced by the
weak topology of $C^{\infty}(M,N)$. The following result due to Trivedi is a
generalization of Thom’s classical transversality result [Tm69] to the complex
analytic setting.

######  Theorem 8.8 (Trivedi [Tr13], Thm. 2.1 and Thm. 3.1).

Let $M$ be a Stein manifold and $N$ be an Oka manifold. Let $V$ be a complex
analytic variety in $N$ and let $V=\bigsqcup_{i=1}^{u}V_{i}$ be a Whitney A
stratification. Then for any compact subset $K$ in $M$, the set of maps
$\\{g\in\mathcal{H}(M,N):g\pitchfork_{K}V_{i}\\}$ is open and dense in
$\mathcal{H}(M,N)$.

Next we record a key algebraic result due to Buchweitz which allows to obtain
flat deformations of varieties obtained from pullbacks of holomorphic maps
between complex affine spaces by deforming the components of these maps.

######  Theorem 8.9 (Buchweitz, 4.3.4 in [Bu81]).

Let $F\colon(\mathbb{C}^{n},0)\rightarrow(\mathbb{C}^{m},0)$ be a holomorphic
map. Let $(X,0)\subset(\mathbb{C}^{n},0)$ and $(V,0)\subset(\mathbb{C}^{m},0)$
be complex analytic varieties with
$\operatorname{codim}X=\operatorname{codim}V$ and $g^{-1}V=X$. Assume
$g^{-1}V^{\prime}$ is an irreducible component of $X$ for each irreducible
component $V^{\prime}$ of $V$ and assume that $(V,0)$ is Cohen–Macaulay. Then
for every unfolding
$\tilde{F}\colon(\mathbb{C}^{n}\times\mathbb{C}^{k},0)\rightarrow(\mathbb{C}^{m},0)$
the family $\tilde{F}^{-1}(V)\rightarrow(\mathbb{C}^{k},0)$ is a flat
deformation of $X$.

Consider the map
$F\colon(\mathbb{C}^{n},0)\rightarrow\mathrm{Hom}(\mathbb{C}^{l},\mathbb{C}^{l+s})$
given by $l+s$ by $l$ matrix $M_{X}$ with entries complex analytic functions
on a neighborhood of the origin in $\mathbb{C}^{n}$. We say $X$ is
determinantal of type $(l+s,l,u)$ if $X$ consists of the points for which
$\mathrm{rk}(M_{X})<u$ and $X$ is of codimension $(l+s-u+1)(l-u+1)$ in
$\mathbb{C}^{n}$. Denote by $\Sigma^{u}$ the closed subset of
$\mathrm{Hom}(\mathbb{C}^{l},\mathbb{C}^{l+s})$ consisting of linear maps of
rank less than $i$. Then $X=F^{-1}\Sigma^{u}$.

An isolated determinantal singularity $X$ of type $(l+s,l,u)$ is smoothable if
$\dim X\leq 2l+s-2u+3$ [Wah81, Thm. 6.2]. By the Hilbert–Burch theorem [E95,
Sct. 20.4] Cohen–Macaulay codimension $2$ varieties are determinantal of type
$(l,l+1,l)$. In particular, they are smoothable up to dimension $3$ ([Sc77],
cf. pg. 19–20 in [Art76]) and nonsmoothable in dimension $4$ and higher unless
they are complete intersections (M. Zach, priv. comm., 2019).

We say $X$ is Pfaffian if it’s defined by the $2u\times 2u$ Pfaffians of a
skew-symmetric matrix. Isolated Pfaffian singularities of a
$(2l+1)\times(2l+1)$ skew-symmetric matrix are smoothable if $\dim X\leq
4(l-u)+6$ [Wah81, Thm. 6.3]. Gorenstein codimension $3$ singularities are
Pfaffian with $u=l$ by the structure theorem of Buchsbaum and Eisenbud [BE77,
Thm. 2.1].

The following theorem, which combines theorems 8.7, 8.8 and 8.9 is the main
result of this section.

######  Theorem 8.10.

Let $F\colon(\mathbb{C}^{n},0)\rightarrow(\mathbb{C}^{m},0)$ be a holomorphic
map. Let $(X,0)\subset(\mathbb{C}^{n},0)$ and $(V,0)\subset(\mathbb{C}^{m},0)$
be reduced complex analytic varieties with
$\operatorname{codim}X=\operatorname{codim}V$ and $g^{-1}V=X$. Assume $(V,0)$
is Cohen–Macaulay dc singularity. Then there exists an embedded deformation
$\mathcal{X}\rightarrow W$ of $X$ such that $\mathcal{X}_{w}$ is dc for
generic $w$. In particular, determinantal, with the exception of the case
$u=2$ and $s=0$, and Pfaffian singularities deform to dc singularities.

###### Proof.

Let $V=\bigsqcup_{i=1}^{u}V_{i}$ be a Whitney B stratification. Deforming the
entries of $F$ by generic linear forms on $(\mathbb{C}^{m},0)$ by Thm. 8.9 we
obtain an unfolding $\tilde{F}$ of $F$ such that
$\tilde{F}^{-1}V\rightarrow(\mathbb{C}^{m},0)$ is a flat deformation of $X$.
Set $\mathcal{X}:=\tilde{F}^{-1}V$ and $W:=(\mathbb{C}^{m},0)$. By Thm. 8.8
$\tilde{F}_{w}\pitchfork_{0}V$ for generic $w$ as complex affine spaces are
Oka and Stein. By Thm. 8.7 $\mathcal{X}_{w}$ is dc singularity.

Each determinantal or Pfaffian variety $X$ is obtained as $F^{-1}V$ where $V$
is the generic determinantal or Pfaffian singularity of appropriate sizes.
Note that $V$ is irreducible and Cohen–Macaulay by a result of Eagon and
Hochster [EH71] in the determinantal case, and by a result of Kleppe and
Laksov [KL80] in the Pfaffian case. In the determinantal case
$\operatorname{codim}(C(V)_{0},C(V))$ is $(u-1)(s+u-1)$ by [GR16, Prp. 2.6]
which is greater or equal to $2$ unless $u=2$ and $s=0$.

Suppose $V$ is the generic Pfaffian variety of $(2l+1)\times(2l+1)$ skew-
symmetric matrices given by the $2u\times 2u$ Pfaffians (the case of $2l\times
2l$ skew-symmetric matrices is analogous). Then
$\operatorname{codim}(C(V)_{0},C(V))=(2l+1)(2l)/2-(u-1)(2(2l+1)-2u+1)$ by
[LS17, Thm. 5.9 ]. Because $u$ is at most $l$, then
$2l+1>(\sqrt{17}+1)/2+2(u-1)$. Thus $\operatorname{codim}(C(V)_{0},C(V))\geq
2$. ∎

Suppose $X$ is isolated determinantal singularity with $u=2$ and $s=0$. Then
$n\leq l^{2}$. If $n<l^{2}$, then $\tilde{F}_{w}$ will miss the singular locus
of $V$ which is the origin. Thus by the implicit function theorem
$\mathcal{X}_{w}:=\tilde{F}_{w}^{-1}V$ is smooth. Suppose $n=l^{2}$ and $X=V$.
Then $X$ is the affine cone over the Segre embedding of
$\mathbb{P}^{l-1}\times\mathbb{P}^{l-1}$ in $\mathbb{P}^{l^{2}-1}$. By [GR16,
Prp. 2.6] $C(X)_{0}$ is the hypersurface in $\mathbb{P}^{l^{2}-1}$ cut out by
the determinant of the generic $l\times l$ matrix. All determinantal
infinitesimal deformations of $X$ are isomorphic to $X$; hence not dc.

An immediate consequence of the structure theorems of Hilbert–Buch and
Buchsbaum–Eisenbud is that their versal deformation spaces are smooth (see pg.
67–68 and pg. 76 in [Har10]). Thus we have following corollary.

######  Corollary 8.11.

Suppose $X$ is Cohen–Macaulay codimension $2$ or Gorenstein codimension $3$
singularity. Then the generic deformation of $X$ is dc singularity.

Next we give an example of isolated singularities that do not deform to dc
singularities.

######  Example 8.12.

Let $Z$ be abelian variety and $\mathcal{L}$ an ample line bundle on $Z$. Then
$\mathcal{L}^{3}$ gives a projectively normal embedding
$Z\hookrightarrow\mathbb{P}^{N-1}=\mathbb{P}H^{0}(\mathcal{L})$ where $N=\dim
H^{0}(Z,\mathcal{L})$(see [Koi76]). Denote by $\Theta$ the tangent sheaf of
$Z$. Then $H^{1}(\Theta(i))=0$ for each $i$ because the tangent bundle of any
group varieity is trivial. Also, $H^{1}(\mathcal{O}_{Z}(i))=0$ for each $i$ by
the vanishing theorem in [Mum70, pg.150]. Thus $Z$ is strongly rigid [Sch73,
pg.155], which by [Sch73, Thm. 2, pg.159] implies that the versal deformation
space of $C_{Z}$ is isomorphic to the versal deformation space of $Z$ in
$\mathbb{P}^{N-1}$; every deformation of $C_{Z}$ is conical. In particular,
$C_{Z}$ is not smoothable (this was proved without the hypothesis of
projective normality by Kollár and Kovács in [KK18]).

Let $C_{\mathcal{Z}}\rightarrow Y$ be a deformation of $C_{Z}$ with
$(C_{\mathcal{Z}})_{y_{0}}=C_{Z}$ induced by a deformation
$\mathcal{Z}\rightarrow Y$ of $Z$ with $\mathcal{Z}\subset
Y\times\mathbb{P}^{N-1}$ and $\mathcal{Z}_{y_{0}}=Z$ for some closed point
$y_{0}\in Y$. To show that the fiber of the conormal over the vertex of the
cone $(C_{\mathcal{Z}})_{y}=C_{\mathcal{Z}_{y}}$ is of maximal dimension for
each $y$, by the correspondence established in Ex. 8.2, it is enough to show
that the dual of $\mathcal{Z}_{y}$ is a hypersurface in the dual of $P^{N-1}$,
i.e. $\mathrm{def}(\mathcal{Z}_{y})=0$. Suppose $\mathrm{def}(Z)=r\geq 1$.
Then by [Tev05, Thm. 1.8 (i)] $Z$ is ruled by projective subspaces of
dimension $r$. But that’s impossible because any morphism from
$\mathbb{P}^{1}$ to a group variety is constant. Thus $\mathrm{def}(Z)=0$. We
claim that $\mathrm{def}(\mathcal{Z}_{y})=0$ for each $y$ close enough to
$y_{0}$.

Let $e$ be the identity element in $Z$. Because $Z$ is smooth, after $Y$ is
replaced by sufficiently small neighborhood of $y_{0}$, then there exists
$Y^{\prime}\subset Z$ passing through $e$ such that $r\colon
Y^{\prime}\rightarrow Y$ is étale. Consider the family
$Z\times_{Y}Y^{\prime}\rightarrow Y^{\prime}$ with the section
$y^{\prime}\rightarrow(y^{\prime},y^{\prime})$ for $y^{\prime}\in Y^{\prime}$.
Then for each $y^{\prime}$ the fiber over $y^{\prime}$ is isomorphic
$Z_{r(y^{\prime})}$. By [MF82, Thm. 6.14] $Z\times_{Y}Y^{\prime}$ is abelian;
hence, there are no projective spaces contained in it and thus
$\mathcal{Z}_{y}$ is not ruled. Therefore, $C_{\mathcal{Z}_{y}}$ is not a dc
singularity.

As mentioned before all Cohen–Macaulay codimension $2$ singularities of
dimension at most $3$ are smoothable. The result is sharp because the cone in
$\mathbb{C}^{6}$ over the Segre embedding of
$\mathbb{P}^{1}\times\mathbb{P}^{2}$ in $\mathbb{P}^{5}$, which is a
Cohen–Macaulay codimension $2$ singularity, is rigid and hence not smoothable
(this is the first example of a nonsmoothable singularity due to Thom). In
fact, in dimension $4$ and higher Cohen–Macaulay codimension $2$ singularities
are nonsmoothable unless they are complete intersections. Next we consider a
class of $4$-dimensional isolated nonsmoothable Cohen–Macaulay codimension $2$
singularities suggested to me by T. Gaffney. The generic deformations of this
isolated singularity are singular but dc by Cor. 8.11. For this class of
singularities the restricted local volume associated with the the relative
conormal spaces takes particularly nice form - it’s a sum of a Buchsbaum–Rim
multiplcity and a polar multiplicity.

######  Example 8.13.

To each polynomial $h(w,x,y)$ which defines an isolated singularity in
$(\mathbb{C}^{3},0)$ associate the Cohen–Macaulay codimension $2$ singularity
$X_{h}$ in $(\mathbb{C}^{6},0)$ defined by the vanishing of the $2$ by $2$
minors of the following matrix

(40) $F_{h}:=\begin{pmatrix}u&x\\\ v&y\\\ h(w,x,y)&z\end{pmatrix}.$

where we view the presentation matrix above as a map
$F_{h}\colon(\mathbb{C}^{6},0)\rightarrow\mathrm{Hom}(\mathbb{C}^{2},\mathbb{C}^{3})$.
Observe that $X_{h}$ is not smoothable (see [Art76, pg. 19–20]). Note that
$X_{h}=F_{h}^{-1}(\Sigma^{2})$. A one-parameter deformation $\mathcal{X}_{h}$
of $X_{h}$ with fibers $\mathcal{X}_{h}(t)$ is obtained by perturbing the
entries of the presentation matrix (40). Then by Cor. 8.11 the generic fiber
$\mathcal{X}_{h}(t)$ is a dc singularity.

Apply Cor. 2.7 to the pair of modules: the relative Jacobian module
$J_{\mathrm{rel}}(\mathcal{X}_{h})$, and the normal module of
$\mathcal{X}_{h}$ which in this case is $F_{h}^{*}(J(\Sigma^{2}))$, the
pullback of the Jacobian module of $\Sigma^{2}$ (see [GRu16, Prp. 2.11] and
the discussion preceeding [GR16, Prp. 2.4]). Observe that
$J_{\mathrm{rel}}(\mathcal{X}_{h})$ specializes to the Jacobian module
$J(\mathcal{X}_{h}(t))$ of each fiber $\mathcal{X}_{h}(t)$. Because
$\mathcal{X}_{h}(t)$ is a dc singularity for generic $t$, the module
$F_{h}^{*}(J(\Sigma^{2}))$ is integrally dependent on $J(\mathcal{X}_{h}(t))$.
So, for generic $t$ the Buchsbaum–Rim multiplicity vanishes:

$e(J(\mathcal{X}_{h}(t),F_{h}^{*}(J(\Sigma^{2})))=0.$

For dimension reasons the codimension $4$ polar variety
$\Gamma_{4}(\Sigma^{2})$ is empty (see the discussion that follows [GRu16,
Prp. 2.14]). Hence, by [GR16, Thm. 2.5] $\Gamma_{4}(F_{h}^{*}(J(\Sigma^{2})))$
is empty. Thus the MPT yields
$e(J(X_{h}),F_{h}^{*}(J(\Sigma^{2}))=m_{4}(X_{h})$ where $m_{4}(X_{h})$ is the
degree of the polar curve of $\mathcal{X}_{h}$. Applying the LVF with the
observation that the generic term on the left-hand side vanishes by Prp. 8.3,
we get

$\varepsilon(J_{\mathrm{rel}}(\mathcal{X}_{h}))(0)=e(J(X_{h}),F_{h}^{*}(J(\Sigma^{2}))).$

Thus we reduced the problem of computing the restricted local volume to
computing a relative Buchsbaum–Rim multiplicity, which as defined in Sct. 2
can be computed as a sum of intersection numbers of the blowup of
$\mathrm{Proj}(\mathcal{R}(F_{h}^{*}(J(\Sigma^{2})))$ with center the ideal
generated by $J(X_{h})$. First we need to find
$\mathrm{Proj}(\mathcal{R}(F_{h}^{*}(J(\Sigma^{2})))$. Recall that

$B_{X_{h}}:=\overline{\\{(x,l_{1},l_{2})|x\in
X_{h},l_{1}\in\mathbb{P}(\mathrm{Ker}(M_{X_{h}}(x))),l_{2}\in\mathbb{P}(\mathrm{Ker}(M_{X_{h}}^{t}(x))\\}}$

which sits inside $X_{h}\times\mathbb{P}^{1}\times\mathbb{P}^{2}$ is set-
theoretically equal to $\mathrm{Proj}(\mathcal{R}(F_{h}^{*}(J(\Sigma^{2})))$
by [GR16, Thm. 3.7]. The morphism between
$X_{h}\times\mathbb{P}^{1}\times\mathbb{P}^{2}$ and
$X_{h}\times\mathbb{P}(\mathrm{Hom}(\mathbb{C}^{2},\mathbb{C}^{3}))$ is given
by

$(x,[S_{1},S_{2}],[T_{1},T_{2},T_{3}])\rightarrow\left(x,\begin{bmatrix}S_{1}T_{1}&S_{2}T_{1}\\\
S_{1}t_{2}&S_{2}T_{2}\\\ S_{1}T_{3}&S_{2}T_{3}\end{bmatrix}\right).$

An easy computation shows that $B_{X_{h}}$ is cut out locally at the chart
$[1,s],[1,t_{1},t_{2}]$ from $\mathbb{C}[x,y,w,s,t_{1},t_{2}]$ by
$u+sx=0,v+sy=0,f+sz=0,t_{1}x+t_{2}y+z=0$. Hence $B_{X_{h}}$ is a complete
intersection. Then $\mathrm{Proj}(\mathcal{R}(F_{h}^{*}(J(\Sigma^{2})))$ set-
theoretically is a complete intersection. But it is generically reduced
because $X_{h}$ is reduced. So,
$\mathrm{Proj}(\mathcal{R}(F_{h}^{*}(J(\Sigma^{2})))$ is reduced. Thus

(41)
$\mathrm{Proj}(\mathcal{R}(F_{h}^{*}(J(\Sigma^{2})))\simeq\mathbb{C}(x,y,w)[s,t_{1},t_{2}]/(f-s(t_{1}x+t_{2}y)).$

where $\mathbb{C}(x,y,w)$ is localized at the origin. Our next task is to
compute the ideal induced by $J(X_{h})$ in
$\mathcal{R}(F_{h}^{*}(J(\Sigma^{2}))$. Set

$G\colon\mathrm{Hom}(\mathbb{C}^{2},\mathbb{C}^{3})\rightarrow\mathbb{C}$

such that $G^{-1}(0)=\Sigma^{2}$. From the chain rule $D(G\circ F)=(D(G)\circ
F)\circ D(F)$ it follows that the ideal $\mathcal{I}_{J}$ generated by
$J(\mathcal{X}_{h}(0))$ in $\mathcal{R}(F_{h}^{*}(J(\Sigma^{2}))$ is generated
by $D(F)$. An easy computation shows that the generators for $\mathcal{I}_{J}$
are $t_{1},t_{2},st_{1}+f_{x},st_{2}+f_{y},s,f_{w}$. Thus by (41)
$\mathbb{V}(\mathcal{I}_{J})=\mathrm{Spec}(\mathbb{C}[x,y,w]/\langle
f,J(f)\rangle)$ where $J(f)$ is the Jacobian ideal of $f$. Therefore, the
computation of the Buchsbaum–Rim multiplicity reduces to computing the
Hilbert–Samuel multiplicity of $\mathcal{I}_{J}$ in
$\mathbb{C}(x,y,w)[s,t_{1},t_{2}]/(f-s(t_{1}x+t_{2}y))$. The latter ring is
Cohen–Macaulay of dimension $5$. Therefore, if $\mathcal{I}_{J}^{\prime}$ is a
reduction of $\mathcal{I}_{J}$, i.e. an ideal generated by $5$ generic
$\mathbb{C}$-linear combinations of the generators, then the Hilbert–Samuel
multiplicity $e(\mathcal{I}_{J})$ of $\mathcal{I}_{J}$ is equal to
$\dim_{\mathbb{C}}\mathbb{C}(x,y,w)[s,t_{1},t_{2}]/(\mathcal{I}_{J}^{\prime},f-s(t_{1}x+t_{2}y))$.
For generators of $\mathcal{I}_{J}^{\prime}$ we can choose $t_{1},t_{2},s$ and
$2$ generic linear combinations of $f_{x},f_{y}$ and $f_{w}$. Thus,

$e(\mathcal{I}_{J})=\dim_{\mathbb{C}}\mathbb{C}[x,y,w]/(f,\alpha_{1}f_{x}+\alpha_{2}f_{y}+\alpha_{3}f_{w},\beta_{1}f_{x}+\beta_{2}f_{y}+\beta_{3}f_{w})$

for generic $\alpha_{i}$ and $\beta_{i}$. Hence $e(\mathcal{I}_{J})$ is equal
to the Hilbert–Samuel multiplicity $e(J(f))$ of $J(f)$ in $\mathbb{C}[x,y,w]$.
Finally,

$\varepsilon(J_{\mathrm{rel}}(\mathcal{X}_{h}))(0)=\mu(f)+\mu(f\cap H)$

where $\mu(f)$ is the Milnor number of $f$ and $\mu(f\cap H)$ is the Milnor
number of $\mathbb{V}(f)\cap H$ for a generic hyperplane section $H$ in
$(\mathbb{C}^{3},0)$.

## References

  * [AK12] Altman, A., and Kleiman, S., ”A term in commutative algebra,” Wordwide Center of Mathematics, 2012.
  * [Art76] Artin, M., “ Lectures on Deformations of Singularities,” Tata Inst. Fund. Res. Bombay (1976) (notes by C. S. Seshardi and Alan Tannenbaum).
  * [BGG80] Briancon, J., Galligo, A., Granger, M., “Déformations équisingulieres des germes de courbes gauches réduites”, Mémoires de la S. M. F. 2e série, tome 1 (1980), 1–69.
  * [BE77] Buchsbaum, D., and Eisenbud, D., Algebraic structures for finite free resolutions and some structure theorems for ideals of codimension $3$, Amer. J. Math. 99, 447–485.
  * [BR64] Buchsbaum, D.A., and Rim, D.S., A generalized Koszul complex. II. Depth and multiplicity, Trans. Amer. Math. Soc., 111 (1964), 197–224.
  * [Bu81] Buchweitz, R., Thése, Paris, 1981.
  * [BG80] Buchweitz, R., Greul, G., The Milnor number of complex curve singularities, Invent. Math. 58, (1980), 241–281.
  * [Cut15] Cutkosky, S. D., Asymptotic multiplicities, Journal of Algebra, 442 (2015), 260–298.
  * [CHS10] Cutkosky,S.D., Herzog, J., and Srinivasan, H., Asymptotic growth of algebras associated to powers of ideals, Math. Proc. Camb. Philos. Soc. 148 (2010), 55–72.
  * [Das21] Das, S., Epsilon multiplicity for graded algebras, Journal of Pure and Applied Algebra, 225 Issue 10, 2021.
  * [EH71] Eagon, J., and Hochster, M., Cohen-Macaulay rings, invariant theory and the generic perfection of determinantalloci, Amer. J. Math. 93 (1971), 1020–1058.
  * [ELMNP] Ein, L., Lazarsfeld, R., Mustata, M., Nakamaye, M., Popa, M., Restricted volumes and base loci of linear series, Amer. J. Math. 131 (2009), no.3, 607–651.
  * [Ein86] Ein, L., Varieties with small dual varieties. $\rm{I}$, Invent. Math. 86 (1986), no. 1, 63–74.
  * [Ein85] Ein, L., Varieties with small dual varieties. $\rm{II}$, Duke Math. J. 52 (1985), no. 4, 895–907.
  * [EHU03] Eisenbud, D., Huneke, C., and Ulrich, B., What is the Rees algebra of a module, Proc. Amer. Math. Soc., 131 (2003), no. 3, 701–708.
  * [E95] Eisenbud, D., ”Commutative Algebra with a view toward Algebraic Geometry,” GTM, 150. Springer-Verlag, New York, 1995.
  * [Fle77] Flenner, H., Die Sätze von Bertini für lokale Ringe,(German), Math. Ann. 229 (1977), no. 2, 97–111.
  * [Ful13] Fulger, M., Local volumes of Cartier divisors over normal algebraic varieties, Ann. Inst. Fourier (Grenoble), 63 (2013), 1793–1847.
  * [Ful84] Fulton, W., “Intersection Theory.” Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 2. Springer-Verlag, Berlin, 1984.
  * [GRu16] Gaffney, T., Ruas, M., Equisingularity and EIDS, http://arxiv.org/abs/1602.00362 [math.CV].
  * [GR19] Gaffney, T., Rangachev, A., The $A_{f}$ condition and relative conormal spaces for functions with non-vanishing derivative, Int. J. Math. 30, no. 10, (2019).
  * [GR16] Gaffney, R., Rangachev, A., Pairs of modules and determinantal isolated singularities, arXiv:1501.00201v2 [math.CV].
  * [Gaf08] Gaffney, T., The Multiplicity Polar Theorem and isolated singularities, J. Alg. Geom., 18 (2009), 547–574.
  * [Gaf04] Gaffney, T., The Multiplicity polar theorem, arXiv:math/0703650v1 [math.CV]
  * [Gaff04] Gaffney, T., Polar methods, invariants of pairs of modules and equisingularity, in Real and complex singularities, eds T. Gaffney and M.A.S. Ruas, Contemp. Math., 354, Amer. Math. Soc., Providence, RI, 2004, 113–135.
  * [GaM99] Gaffney, T., and Massey, D., Trends in equisingularity theory, in the proceedings of the Liverpool conference in honor of CTC Wall, edited by W. Bruce and D. Mond, London Math. Soc. Lecture Notes 263 (Cambridge University Press, 1999), 207–248.
  * [GK99] Gaffney, T., Kleiman, S., Specialization of integral dependence for modules, Invent. Math., 137 (1999), 541–574.
  * [Gaf96] Gaffney, T., Multiplicities and equsingularity of ICIS germs, Invent. Math., 123 (1996), 209–-220.
  * [Gaf93] Gaffney, T., Polar multiplicities and equisingularity of map germs, Topology, 31 (1993), 185–223.
  * [Gaf92] Gaffney, T., Integral closure of modules and Whitney equisingularity, Invent. Math., 107 (1992), 301–-322.
  * [GM80] Goresky, M., MacPherson, R., Intersection Homology Theory, Topology, 19 (1980), 135–162.
  * [Gra72] Grauert, H., Über die Deformationen isolierter Singularitäten analytischer Mengen, Invent. Math. 15 (1972), 171–198.
  * [Gr67] Grothendieck, A., “Local Cohomology” (notes by R. Hartshorne), Lecture Notes in Math. 41, Springer-Verlag, Heidelberg (1967), 106 pp.
  * [Gr68] Grothendieck, A., Cohomologie locale des faisceaux cohérents et théorèms de Lefschetz locaux et globaux (SGA 2), Séminaire de Géométrie Algébrique du Bois-Marie 1962 (North-Holland, Amsterdam, 1968).
  * [HMX18] Hacon, C., McKernan, J., Xu, C., Boundedness of moduli of varieties of general type, J. Eur. Math. Soc. 20, (2018), 865–901.
  * [Har10] Hartshorne, R., ”Deformation theory,” Springer-Verlag, GTM 257, 2010.
  * [Har77] Hartshorne, R., “Algebraic Geometry, ” Springer, GTM 52, 1977.
  * [HMS84] Henry, J. P. G., Merle, M., Sabbah, C., Sur la condition de Thom stricte pour un morphisme analytique complexe, Ann. Scient. Éc. Norm. Sup. 17 (1984), 227–268.
  * [HIO88] Hermann, M., Ikeda, S., and Orbanz, U., ”Equimultiplicity and blowing up. An algebraic study,” with an appendix by B. Moonen, Springer-Verlag, 1988.
  * [Hir76] Hironaka, H., Stratification and Flatness, in ”Real and Complex singularities, Nordic Summer School, Oslo, 1976,” Sijthoff and Noordhoff, 1977, 199–265.
  * [Hir70] Hironaka, H., Normal cones in analytic Whitney Stratifications, Publ. Math. I.H.E.S., 36 (1970), 127–138.
  * [Hir64] Hironaka, H., Resolution of singularities of an algebraic variety over a field of characteristic zero I, II, Ann. of Math. (2) 79 (1964), 109–203; ibid. (2) 79 (1964), 205–326.
  * [Hir58] Hironaka, H., A note on algebraic geometry over ground rings. The invariance of Hilbert characteristic functions under the specialization process, Ill. Math. 2 (1958), 355–366.
  * [HS15] Huneke, C., Smirnov, I., Prime filtrations of the powers of an ideal, Bull. London Math. Soc. 47 (2015), 585–592.
  * [KS90] Kashiwara, M., Schapira, P., “Sheaves on Manifolds,” Springer-Verlag, Berlin, 1990.
  * [KK12] Kaveh, K., Khovanskii, A. G., Newton-Okounkov bodies, semigroups of integral points, graded algebras and intersection theory, Annals of Mathematics 176 (2012), 1–54.
  * [KUV] Kleiman, S., Ulrich, B., and Validashti, J., Integral Dependence and Equisingularity, in preparation.
  * [KT01] Kleiman, S., Thorup, A.,The exceptional fiber of a generalized conormal space, an appendix to Gaffney, T., The theory of integral closure of ideals and modules: applications and new developments. NATO Sci. Ser. II Math. Phys. Chem., 21, New developments in singularity theory (Cambridge, 2000), 379–404, Kluwer Acad. Publ., Dordrecht, 2001.
  * [KT00] Kleiman, S., and Thorup, A., Conormal geometry of maximal minors, J. Algebra 230 (2000), 204 – 221.
  * [Kle99] Kleiman, S., Equisingularity, Multiplicity, and Dependence, in Commutative algebra and algebraic geometry in honor of M. Fiorentini (edited by F. van Oystayen), Lecture Notes in Pure and Applied Math. 206 (1999), Marcel Dekker, Inc.
  * [KT94] Kleiman, S., Thorup, A., A geometric theory of the Buchsbaum-Rim multiplicity, J. Algebra 167 (1994), 168–-231.
  * [Kle74] Kleiman, S., _The transversality of a general translate,_ Compositio Math. 28 (1974), 287–297.
  * [KL80] Kleppe, H., Laksov, D., The algebraic structure and deformation of Pfaffian schemes, J. Algebra 64 (1980), 167–189.
  * [Koi76] Koizumi, S., Theta relations and projective normality of Abelian varieties, Amer. J. Math. 98 (1976), no. 4, 865–889.
  * [KK18] Kollár, J., and Kovács, S., _Deformations of log canonical and F-pure singularities,_ Algebr. Geom. 7, no. 6 (2020), 758–780.
  * [Kol18] Kollár, J., “Families of varieties of general type,” https://web.math.princeton.edu/~kollar/book/modbook20170720-hyper.pdf
  * [LS17] Lakshmibai,V., and Singh, R., Conormal Varieties on the Cominuscule Grassmannian arxiv.org/pdf/1712.06737.pdf [math.CV]
  * [Laz04] Lazarsfeld, R., “Positivity in algebraic geometry”I, II Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge 48,49. Springer-Verlag, Berlin, 2004.
  * [LT88] Lê, D. T., Teissier, B., Limites de’espaces tangent en géométrie analytique, Comment. Math. Helvetici 63 (1988), 540–578.
  * [LM83] Lê, D. T., Mebkhout, Z., Introduction to linear differential systems, in “Singularities,” Proc. Sympos. Pure Math., vol. 40, Part 2, Amer. Math. Soc., Providence, 1983, 31–-63.
  * [LT81] Lê Dung Tráng and B. Teissier, Variétés polaires locales et classes de Chern des variétés singulières. (French) [Local polar varieties and Chern classes of singular varieties], Ann. of Math. (2) 114 (1981), no. 3, 457–491.
  * [Lip82] Lipman, J., Equimultiplicity, reduction and blowing-up, in “Commutative algebra: analytic methods,” Dekker Lecture Notes in Pure and Applied Math. 68 (1982), 111-–48.
  * [Mas03] Massey, D., “Numerical control over complex analytic singularities.” Memoirs of the AMS, vol. 163, no. 778, 2003.
  * [M73] Mather, J.N., Stratifications and mappings, Dynamical Systems, Academic Press (1973), 195–232.
  * [M76] Mather, J.N., How to stratify manifolds and jet spaces, Springer Lecture Notes in Mathematics, 535 (1976), 128–176.
  * [Mat86] Matsumura, H., “Commutative ring theory.” Cambridge University Press, 1986.
  * [Mat70] Matsumura, H., “Commutative algebra.” W. A. Benjamin Co., New York (1970), xii + 262 pp.
  * [MF82] Mumford, D., and Fogarty, J., “Geometric Invariant Theory.” Springer-Verlag, New York, 1982.
  * [Mum70] Mumford, D., “Abelian varieties.” With appendices by C. P. Ramanujam and Yuri Manin, Oxford University Press India, 1970.
  * [Oko03] Okounkov, A., Why would multiplicities be log-concave? in “The orbit method in geometry and physics”, Progr. Math. 213, 2003, 329–347.
  * [Ram73] Ramanujam, C. P., On a geometric interpretation of multiplicity, Invent. Math. 22 (1973), 63–-67.
  * [Ran21b] Rangachev, A., Deficient Conormal Singularities, (in preparation)
  * [Ran21a] Rangachev, A., Asymptotics of intersection numbers and the Local Volume Formula, (in preparation).
  * [Ran20] Rangachev, A., A valuation theorem for Noetherian rings, (submitted), Accessible at https://arxiv.org/abs/2011.14749
  * [Ran19] Rangachev, A., Associated primes and integral closure of Noetherian rings, Journal of Pure and Applied Algebra, 225 Issue 5, 2021.
  * [Ran18] Rangachev, A., Associated points and integral closure of modules, Journal of Algebra, 508, (2018), 301–338.
  * [Sc77] Schaps, M., Deformations of Cohen–Macaulay schemes of codimension 2 and nonsingular deformations of space curves, Amer. J. Math. 99 (1977), 669–-685.
  * [Sch73] Schlessinger, M., On Rigid Singularities. Rice Institute Pamphlet - Rice University Studies, 59, no. 1 (1973), Rice University.
  * [Sc03] Schürmann, J., “Topology of Singular Spaces and Constructible Sheaves.” Mongrafie Matematyczne (IMPAN), Birkhäuser, 2003.
  * [SH06] Swanson, I., and Huneke, C., “Integral closure of ideals, rings, and modules.” London Mathematical Society Lecture Note Series, vol. 336, Cambridge University Press, Cambridge, 2006.
  * [ST71] Siu, Y.-T., and Trautmann, G., “Gap-sheaves and extension of coherent analytic subsheaves.” Lecture Notes in Math., vol. 172, Springer, 1971.
  * [Stks] The Stacks Project Authors, Stacks Project, http://stacks.math.columbia.edu, 2016.
  * [SUV01] Simis, A., Ulrich, B., Vasconcelos, W., Codimension, multiplicy and integral extensions, Math. Proc. Camb. Phil. Soc., 130 (2001), 237–257.
  * [TF18] Teissier, B., Flores, A.G., Local polar varieties in the geometric study of singularities, Annales de la Faculté des Sciences de Toulouse, 27, no. 4 (2018), 679–775.
  * [T81] Teissier, B., Variétés polaires II: Multiplicité polaires, sections planes, et conditions de Whitney, Actes de la conférence de géometrie algébrique á la Rábida, Springer Lecture Notes, 961 (1981), 314–491.
  * [T75] Teissier, B., Introduction to equisingularity problems, Proc. Symposia pure math. vol., 29, Amer. Math. Soc. (1975), 575–84.
  * [T73] B. Teissier, Cycles évanescents, sections planes et conditions de Whitney. (French) Singularités à Cargése (Rencontre Singularités en Géométrie Analytique, Institut d’études Scientifiques de Cargése, 1972), pp. 285–362. Asterisque, Nos. 7 et 8, Soc. Math. France, Paris, 1973.
  * [Tev05] Tevelev, E., “Projective Duality and Homogeneous Spaces.” Encyclopedia of Mathematical Sciences, vol. 133, Springer, 2005.
  * [Tm69] Thom, R., Ensembles et morphismes stratifies, Bull. Amer. Math. Soc., 75 (1969), 240–284.
  * [Tr13] Trivedi, S., Stratified transversality of holomorphic maps, Internat. J. Math. 24 (2013), no. 13, 12 pp.
  * [Z71] Zariski, O., Presidential Address, Bulletin of the American Mathematical Society, 77 No. 4 (1971), 481–491.
  * [UV11] Ulrich, B., Validashti, J., Numerical criteria for integral dependence, Math. Proc. Camb. Phil. Soc., 151, (2011), 95–102.
  * [V76] Verdier, J.L., Stratifications de Whitney et Théoréme de Bertini–Sard, Invent. Math., 36 (1976), 295–312.
  * [Wah81] Wahl, J., Smoothings of normal surface singularities, Topology, 20 (1981), 219–246.
  * [Wh64] Whitney, H., Tangents to analytic variety, Ann. Math., 81 (1964), 496–549.


CERN NA63

# Precision Measurement of Trident Production in Strong Electromagnetic Fields

Christian F. Nielsen  Department of Physics and Astronomy, Aarhus University,
8000 Aarhus, Denmark  Robert Holtzapple  Department of Physics, California
Polytechnic State University, San Luis Obispo, California 93407, USA  Mads M.
Lund  Department of Physics and Astronomy, Aarhus University, 8000 Aarhus,
Denmark  Jeppe H. Surrow  Department of Physics and Astronomy, Aarhus
University, 8000 Aarhus, Denmark  Allan H. Sørensen  Department of Physics and
Astronomy, Aarhus University, 8000 Aarhus, Denmark  Marc B. Sørensen
Department of Physics and Astronomy, Aarhus University, 8000 Aarhus, Denmark
Ulrik I. Uggerhøj  Department of Physics and Astronomy, Aarhus University,
8000 Aarhus, Denmark

(September 3, 2024)

###### Abstract

We demonstrate experimentally that the trident process $e^{-}\rightarrow
e^{-}e^{+}e^{-}$ in a strong external field, with a spatial extension
comparable to the effective radiation length, is well understood
theoretically. The experiment, conducted at CERN, probes values for the strong
field parameter $\chi$ up to 2.4. Experimental data and theoretical
expectations using the Local Constant Field Approximation show remarkable
agreement over almost 3 orders of magnitude in yield.

##  I Introduction

When an electron impinges on an electrostatic potential barrier, it may
penetrate or be reflected. Classically, for electron energies less than the
barrier height, the electron is always reflected. In non-relativistic quantum
mechanics, an exponentially damped tunneling into the barrier is predicted,
with no transmission when the potential remains higher than the electron
energy beyond the classical turning point. In relativistic quantum theory,
however, an undamped electron-current is present beyond the classical turning
point provided the barrier rises sufficiently abruptly and high, even if it is
infinitely high. This was shown in 1929 by Oscar Klein Klein (1929) for a step
barrier in one of the first applications of the Dirac equation. It became
known as the ’Klein paradox’. As later shown by Fritz Sauter Sauter (1931a,
b), inspired by a supposition by Niels Bohr, the potential has to rise with
the rest energy of the electron, $mc^{2}$, over its Compton wavelength,
$\hbar/mc$, for transmission to occur with substantial probability. The
corresponding field strength,

$\mathcal{E}_{0}=m^{2}c^{3}/e\hbar\simeq 1.32\times
10^{16}\leavevmode\nobreak\ \mathrm{V/cm},$ (1)

later became known as the critical or Schwinger field.

Previous studies of the Klein paradox have been limited to theory Greiner  _et
al._ (1985); Krekora  _et al._ (2004); Giachetti and Sorace (2008), e.g. with
phenomena analogous to the Klein paradox, possibly observable in graphene
Katsnelson  _et al._ (2006); Calogeracos (2006); Buchanan (2006); Bøggild  _et
al._ (2017); Nguyen and Charlier (2018). Some studies have been partly
motivated by heuristic arguments linking the Klein paradox, strong field pair
production and Hawking radiation from black holes Müller  _et al._ (1977);
Holstein (1998, 1999). Today, the Klein paradox is explained by the creation
of electron-positron pairs at the boundary.

In the following, we demonstrate experimentally that the trident process
$e^{-}\rightarrow e^{-}e^{+}e^{-}$ becomes significant when a strong external
field is ’turned on’. In our experiment this happens when a crystal is
rotated, relative to the beam direction, from a ’random’ orientation, where
the field experienced by a penetrating particle is the sum of essentially
randomly placed screened Coulomb fields, to an axial orientation, where the
atoms along the axis act coherently in deflecting the particle. This
effectively creates a continuous field which is macroscopic along the main
direction of motion and of critical magnitude in the particle rest frame.
Regardless of the orientation of the crystal, there are two contributions to
the trident process, the direct and a ”two-step” or ”cascade” contribution,
where the incoming electron first emits a real photon which then converts to
an electron-positron pair. In our case, the direct trident production, the
emission of real photons, and conversion of photons into pairs, all proceed in
essentially constant fields.

The trident process for electrons penetrating amorphous material where they
interact with screened target nuclei, has been investigated in detail since
the thirties, see e.g. Crane and Halpern (1939); Powell (1949); Occhialini
(1949); Hooper  _et al._ (1951); Barkas  _et al._ (1952); Hooper  _et al._
(1952); Block  _et al._ (1954); Heitler (1954); Kelner (1967); Grossetête  _et
al._ (1968); Böhm  _et al._ (1973); Baier and Katkov (2008). As a result, the
amorphous yield of trident events is considered well known and the
contributions to the total trident yield for the direct and two-step processes
are similar for material thicknesses of a few percent of the radiation length
$X_{0}$ Kelner (1967); Baier and Katkov (2008).

For extended electromagnetic fields, the trident process depends on the
strong-field parameter $\chi$ defined through Ritus (1985); Baier  _et al._
(1998); Berestetskii  _et al._ (1989)

$\chi^{2}=(F_{\mu\nu}{p}^{\nu})^{2}/{m^{2}c^{2}}\mathcal{E}_{0}^{2},$ (2)

where $F_{\mu\nu}$ is the electromagnetic field strength tensor and
${p}^{\nu}$ is the four-momentum. For values of $\chi$ smaller than unity,
pair production by photons is exponentially suppressed Ritus (1985); Baier
_et al._ (1998), therefore the trident process is also suppressed.
Consequently, the scale of the electric fields required for the trident
process to occur are of the order $\mathcal{E}_{0}$. Such fields are not yet
possible to produce in the laboratory. Nevertheless, a relativistic particle,
with a relatively large Lorentz factor e.g. $\gamma\simeq 10^{5}-10^{6}$, will
experience an electric field in its own rest frame that, with a suitable field
configuration, may be Lorentz boosted and can reach $\chi>1$, which enables
the trident process. The trident process in plane wave background fields has
been, and is being investigated in great detail, both experimentally
(SLAC-E144 experiment Bula  _et al._ (1996)) and theoretically (see e.g. the
comprehensive review Fedotov  _et al._ (2022) and references therein). Most
theoretical investigations are driven by several upcoming and ongoing
experiments colliding GeV electron beams with tightly focused laser pulses to
probe the trident process Abramowicz  _et al._ (2021); E-320 collaboration
(2018).

Aligned crystals have been a known source of strong electric fields for
decades, see e.g. Uggerhøj (2005); Baier  _et al._ (1998) for details on
strong field effects in crystals. Depending on the element and orientation of
the crystal, fields of the order $10^{9}-10^{11}$ V/cm are readily available
in the laboratory frame. Due to the coherent action of the screened nuclei
along crystallographic directions, this field is of truly macroscopic extent
for Ge and Si crystals, where single crystals can be grown in principle to
meters in length. For a charged particle moving in a field that is purely
electric in the laboratory, and essentially transverse to the direction of
motion as in the case of an aligned crystal, the strong field parameter
reduces to $\chi\simeq\gamma\mathcal{E}/\mathcal{E}_{0}$. At CERN, electrons
and positrons with up to $\gamma\simeq 10^{6}$ are available making it
possible to achieve values of $\chi>1$ in the rest frame of the electron in an
aligned crystal.

In 2007 an attempt to measure the production of electron-positron pairs
through the trident process $e^{-}\rightarrow e^{-}e^{+}e^{-}$ in strong
electromagnetic fields was made with participation from one of the present
authors (UIU) Esberg  _et al._ (2010), that resulted in significant
discrepancies theory. In this paper we report on a recent precision
measurement on the trident process, impinging 200 GeV electrons on the same
400 $\mu$m germanium crystal target as in 2007, oriented along the $\langle
110\rangle$ axis, showing an extraordinary agreement between theory and
experiment. Thus, the discrepancies observed in the 2007 experiment are yet to
be explained. It is our conjecture that the 2007 experimental values are
incorrect possibly due to improper alignment of the crystal.

Figure 1: Experimental setup. A schematic representation of the experimental
setup in the H4 beam line in the SPS North Area at CERN. The symbols “Mi”,
with $i=0,\ldots,7$, denote ‘Mimosa-26’ position sensitive detectors.

##  II Experiment

The experiment was performed by the NA63 collaboration at the H4 beamline of
the CERN SPS that provided a 200 GeV electron beam having a
$\sigma_{x}\simeq\sigma_{y}\simeq 105$ $\mu$rad divergence impinging on the
400 $\mu$m thick $\langle 110\rangle$ oriented germanium single crystal.
Figure 1 is a schematic of the setup where M0-M7 are MIMOSA-26 position
sensitive CMOS-based pixel detectors Baudot  _et al._ (2009). The detectors
have a resolution of a few $\mu$m and an active area of $1.1\times 2.1$ cm2
containing $576\times 1152$ pixels. The crystal target is mounted on a
goniometer that allows us to set the crystal orientation with $\mu$rad
precision. The MDX27 magnet provides an integrated magnetic field of $0.072$
Tm and, together with the detectors in Arm 1 and Arm 2, forms a magnetic
spectrometer allowing us to measure the energy of each charged particle from
the deflection angle in the magnet. The crystal is situated inside a vacuum
chamber at $\simeq 300$ K. To reduce scattering and background, all mimosas
are placed in closed compartments that are continuously flushed with helium.
The total material contributing to the background before the MDX27 magnet, in
units of the radiation length, amounts to $\simeq 1.1\%$.

The incoming electron rate was set to ensure that only a single primary
electron was present in the setup per event. We therefore have essentially
complete information about the particle and its secondaries.

Three measurement series were performed; a background measurement with no
target, a ’random’ measurement where the crystal is rotated far away from any
major crystallographic orientation thus acting as an amorphous target, and a
measurement with the aligned crystal. In the latter, the crystal was aligned
with the beam centroid along the $\langle 110\rangle$ axis through a careful
procedure where enhancement of radiation upon the passage of crystalline
planes creates a stereogram from which the location of the axis can be found
with a precision of about $20\leavevmode\nobreak\ \mu$rad.

Figure 2: Positron spectrum of reconstructed trident events. Solid lines are
simulations while squares and triangles are experimental data points. Blue and
Red show the trident spectrum in aligned orientation, magenta and cyan show
the amorphous yield while black and green is the background. The right figure
is the same as the left but with a linear vertical axis. The inset shows a
zoom of the low energy part, which largely overlaps with the energy range
measured in 2007 Esberg  _et al._ (2010).

##  III Data Analysis and Theoretical Comparison

The data analysis routine begins by finding complete single particle tracks
after the target by matching tracks found in Arm 1 and Arm 2 in the center of
the magnet. A track in Arm 1 is only accepted if it simultaneously overlaps
with a hit in M0 and M1. To find possible trident events, all complete single
particle tracks are combined and matched through strict matching criteria. The
energy resolution of the magnetic spectrometer for a single particle track was
measured to be $\sigma_{E}/E\simeq 6.7\%$ at 200 GeV (including $dp/p\simeq
1\%$ from the beamline) and it remains relatively uniform down to particle
energies of a few GeV.

To find possible trident events, all complete single particle tracks are
combined and matched through strict matching criteria. The energy resolution
of the magnetic spectrometer for a single particle track was measured to be
$\sigma_{E}/E\simeq 6.7\%$ at 200 GeV (including $dp/p\simeq 1\%$ from the
beamline) and it remains relatively uniform down to particle energies of a few
GeV.

Figure 2 shows the experimentally measured positron spectrum of all three
measurement series together with theoretical predictions. The theoretical
curves labeled ”Sim” result from a full scale Monte-Carlo simulation of the
experimental setup, starting with a beam of electrons identical to what was
provided by the SPS. The simulation propagates primary electrons individually
through each element of the setup, where they are subject to multiple Coulomb
scattering and emission of photons, that can later pair produce, and direct
trident production. When a particle penetrates a detector, its position is
saved in a data file identical to the experimental data file containing
$(x,y,z)$ positions from each detector from particle hits. The simulated data
files are then analyzed with the experimental analysis routine. As in the
experiment, we simulate all three measurement conditions and the background is
subtracted from the amorphous and aligned curves after the analysis. To
account for detector efficiencies, we fit a linear energy dependent
efficiency, $f(E)=aE+b$, to the ratio between the experimental and simulated
curves for the amorphous case after background subtraction. All simulated
curves shown in fig. 2 are then the result of the experimental analysis
routine multiplied by the linear energy efficiency factor with the fitting
parameters $a=-0.0012\pm 0.001\text{GeV}^{-1}\quad\text{and}\quad b=0.55\pm
0.08$. The value for $b$, found by this procedure, agrees well with
expectations based on the efficiency of the Mimosa detectors that depend on
set threshold values, and the value of $a$ is small.

When particles penetrate amorphous material, the probability of the two-step
trident process is modeled by Bethe-Heitler photon emission followed by Bethe-
Heitler pair production as described in e.g, Workman and Others (2022). The
scattering centers are completely decoupled and non-linear effects, e.g.
photons emitted in the background that pair produce in the crystal or vice
versa, are then taken into account by the simulation of the experiment.
Therefore, it is important that all elements are included correctly in the
setup and that the background simulation matches the measurement. The direct
trident process is modeled using the theory described in Kelner (1967); Baier
and Katkov (2008). Amorphous material in the background and the crystal
oriented in ’random’ are modeled identically, the processes depend on the
nuclear charge $Ze$ and the radiation length $X_{0}$. Figure 2 shows that the
theoretical curves and experimental data are in good agreement for the
amorphous case, as in the 2007 experiment Esberg  _et al._ (2010). Using the
detector-efficiency correction on the background, we also see good agreement,
which strongly indicates that the background contribution is well understood.

When a particle hits the aligned crystal, a separate simulation code is
employed which is described in detail in Nielsen (2019, 2022). The simulation
solves the equation of motion for each time step based on the Lorentz force in
the electric fields modeled by the Doyle-Turner continuum string potentials
Doyle and Turner (1968); Andersen (2018). For every time step the local value
of $\chi$ is evaluated and the probability of photon emission, based on the
Local Constant Field Approximation (LCFA), is evaluated. This photon is then
propagated forward through the crystal where the probability of pair
production, using the LCFA, is again evaluated and the produced pair is
propagated through the setup (details on the implementation of these processes
can be found in Nielsen (2022)).

The applicability of the LCFA requires that the angular excursions of the
projectile over a formation length are wider than the light cone
$1/\gamma=2.6$ $\mu$rad. In the channeling regime, the angular excursions of a
charged particle are on the order of the critical Lindhard angle Lindhard
(1965); Andersen (2018), which in our case is 57 $\mu$rad. For entry angles
larger than the critical Lindhard angle, but smaller than the Baier angle
$U_{0}/mc^{2}=0.4$ mrad, the angular deflections remain larger than $1/\gamma$
Sørensen (1996). Here $U_{0}=215$ eV is the continuum string potential depth,
in this case for a single $<110>$ row of Ge atoms. As a result, with a beam
divergence of $\simeq 105$ $\mu$rad, the LCFA is appropriate for nearly all
particles within the beam. As a measure of the applicability of the constant-
field approximation under channeling conditions, the authors of Baier  _et
al._ (1998) introduced the parameter $\rho_{c}=\xi^{2}=2U_{0}\gamma/mc^{2}$
(where $\xi$ is known as the classical non-linearity parameter Di Piazza  _et
al._ (2012); Fedotov  _et al._ (2022) in the strong-field laser community),
which under channeling conditions has a value of $\simeq 330$ for 200 GeV
electrons. The large value of $\rho_{c}$ verifies that treating the local
field as constant is a good approximation.

For the aligned crystal, the direct trident process is modeled through the
Weizsäcker-Williams method of virtual quanta Jackson (1975); Baier  _et al._
(1998), together with the LCFA for the pair production vertex. For constant
electric fields, the Weizsäcker-Williams method has been investigated and
deviates around $10\%$ from methods evaluating the actual direct two-vertex
Feynman diagrams King and Ruhl (2013). This difference has marginal influence
in our case due to the dominance of the two-step process.

Figure 2 shows that the theoretical prediction for the aligned case agrees
remarkably well with the experimental data points throughout the entire
spectrum which spans several orders of magnitude.

The relative importance of the direct process and the two-step process can be
estimated by comparing the virtual Weizsäcker-Williams photon intensity
(radiated energy per photon-energy interval) with the real photon intensity.
The virtual photon intensity is given by the fine-structure constant up to a
logarithmic factor. The real photons have a fairly flat intensity spectrum
given by $L/X$, where $L$ is the target thickness and $X$ the effective
radiation length defined as $X=E/(dE/dx)$ where $dE/dx$ is the energy-loss
rate per unit length due to radiation. Therefore, the two processes are
comparable in strength for a target thickness of order a percent of the
effective radiation length (see also King and Ruhl (2013); King and Fedotov
(2018)). This is the case for the amorphous setting in our experiment
(’random’ setting), where $X=X_{0}=2.30$ cm and $L/X_{0}=1.7$ %. Accordingly,
the simulations show that $\sim 50\%$ of all tridents come from the direct
process in the amorphous setting. For the aligned case, the effective
radiation length $X$ is much shorter than $X_{0}$ due to the strong-field
effects. The stronger radiation causes the direct process to only contribute a
few percent to the total pair rate.

In the two-step process, the theoretical model applied averages over the
photon polarization. The polarization has been predicted to have significant
effects in strong constant fields King  _et al._ (2013); Seipt and King
(2020). For axially aligned crystals, the experimentally measured photon
spectrum is polarization averaged due to each projectile having a unique
trajectory through the crystal. If a real photon is emitted with a specific
polarization, this photon will also follow a unique trajectory due to the
uniqueness of the emitting particle. This means the pair production process
also becomes polarization averaged, and modeling the real photon as
unpolarized, is a good approximation.

##  IV Conclusion

By impinging 200 GeV electrons on a $\langle 110\rangle$ oriented 400 $\mu$m
thick germanium crystal, we investigated the trident process in strong
electric fields. Our experimental results are in remarkably good agreement
with theory based on LCFA, the locally constant field approximation.

The inset in fig. 2 shows the low-energy tail (which is subject to a major
non-linear drop in detection efficiency) corresponding to the energy range
studied in the 2007 experiment Esberg  _et al._ (2010). Here too, there is
good agreement between theory and experiment. Accordingly, the discrepancy in
2007 most likely was due to an unknown experimental error.

In our case the experimental spectrum is dominated by the two-step process. To
quantify the direct process using an aligned crystal, the crystal will have to
be much thinner than what was used here. An ultra thin crystal greatly reduces
the production rate, making it a challenging measurement, even if the crystal
is cooled.

##  V Acknowledgments

The numerical results presented in this work were partly obtained at the
Centre for Scientific Computing Aarhus (CSCAA) and with support from Nvidia’s
GPU grant program. This work was partially supported by the U.S. National
Science Foundation (Grant No. PHY-1535696, and PHY-2012549) and from the
Danish National Instrument Center for CERN Experiments (NICE).

## References

  * Klein (1929) O. Klein, Z. Phys. 53, 157 (1929).
  * Sauter (1931a) F. Sauter, Z. Phys. 69, 742 (1931a).
  * Sauter (1931b) F. Sauter, Z. Phys. 73, 547 (1931b).
  * Greiner  _et al._ (1985) W. Greiner, B. Müller, and J. Rafelski,  _Quantum Electrodynamics of Strong Fields_ (Springer, 1985).
  * Krekora  _et al._ (2004) P. Krekora, Q. Su, and R. Grobe, Phys. Rev. Lett. 92, 040406 (2004).
  * Giachetti and Sorace (2008) R. Giachetti and E. Sorace, Phys. Rev. Lett. 101, 190401 (2008).
  * Katsnelson  _et al._ (2006) M. Katsnelson, K. Novoselov, and A. Geim, Nature Physics 2, 620 (2006).
  * Calogeracos (2006) A. Calogeracos, Nature Physics 2, 579 (2006).
  * Buchanan (2006) M. Buchanan, Nature Physics 2, 721 (2006).
  * Bøggild  _et al._ (2017) P. Bøggild, J. M. Caridad, C. Stampfer, G. Calogero, N. R. Papior, and M. Brandbyge, Nature Communications 8, 15783 (2017).
  * Nguyen and Charlier (2018) V. H. Nguyen and J.-C. Charlier, Phys. Rev. B 97, 235113 (2018).
  * Müller  _et al._ (1977) B. Müller, W. Greiner, and J. Rafelski, Phys. Lett. A 63, 181 (1977).
  * Holstein (1998) B. R. Holstein, American Journal of Physics 66, 507 (1998).
  * Holstein (1999) B. R. Holstein, American Journal of Physics 67, 499 (1999).
  * Crane and Halpern (1939) H. R. Crane and J. Halpern, Phys. Rev. 55, 838 (1939).
  * Powell (1949) C. F. Powell, Il Nuovo Cimento (1943-1954) 6, 379 (1949).
  * Occhialini (1949) G. P. S. Occhialini, Il Nuovo Cimento (1943-1954) 6, 413 (1949).
  * Hooper  _et al._ (1951) J. Hooper, D. King, and A. Morrish, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 42, 304 (1951), https://doi.org/10.1080/14786445108561268 .
  * Barkas  _et al._ (1952) W. H. Barkas, R. W. Deutsch, F. C. Gilbert, and C. E. Violet, Phys. Rev. 86, 59 (1952).
  * Hooper  _et al._ (1952) J. Hooper, D. King, and A. Morrish, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 43, 853 (1952), https://doi.org/10.1080/14786440808520233 .
  * Block  _et al._ (1954) M. M. Block, D. T. King, and W. W. Wada, Phys. Rev. 96, 1627 (1954).
  * Heitler (1954) W. Heitler,  _The quantum theory of radiation_ , 3rd ed. (Oxford University Press, London, 1954).
  * Kelner (1967) S. R. Kelner, Sov. J. Nucl. Phys. 5, 778 (1967).
  * Grossetête  _et al._ (1968) B. Grossetête, R. Tchapoutian, D. J. Drickey, and D. Yount, Phys. Rev. 168, 1475 (1968).
  * Böhm  _et al._ (1973) J. Böhm, A. Mátlová, J. Trková, and Z. Trka, Nuclear Physics B 57, 355 (1973).
  * Baier and Katkov (2008) V. N. Baier and V. M. Katkov, JETP Letters 88, 80 (2008).
  * Ritus (1985) V. I. Ritus, J. Sov. Laser Res. 6, 497 (1985).
  * Baier  _et al._ (1998) V. N. Baier, V. M. Katkov, and V. M. Strakhovenko,  _Electromagnetic processes at high energies in oriented single crystals_ (Singapore, Singapore: World Scientific (1998) 554 p, 1998).
  * Berestetskii  _et al._ (1989) V. B. Berestetskii, E. M. Lifshitz, and L. P. Pitaevskii,  _Quantum Electrodynamics_ (Pergamon, New York, 1989).
  * Bula  _et al._ (1996) C. Bula, K. T. McDonald, E. J. Prebys, C. Bamber, S. Boege, T. Kotseroglou, A. C. Melissinos, D. D. Meyerhofer, W. Ragg, D. L. Burke, R. C. Field, G. Horton-Smith, A. C. Odian, J. E. Spencer, D. Walz, S. C. Berridge, W. M. Bugg, K. Shmakov, and A. W. Weidemann, Phys. Rev. Lett. 76, 3116 (1996).
  * Fedotov  _et al._ (2022) A. Fedotov, A. Ilderton, F. Karbstein, B. King, D. Seipt, H. Taya, and G. Torgrimsson, “Advances in QED with intense background fields,”  (2022).
  * Abramowicz  _et al._ (2021) H. Abramowicz _et al._ , Eur. Phys. J. ST 230, 2445 (2021).
  * E-320 collaboration (2018) E-320 collaboration, “Probing Strong-Field QED at FACET-II,” (unpublished experimental proposal, 2018).
  * Uggerhøj (2005) U. I. Uggerhøj, Rev. Mod. Phys. 77, 1131 (2005).
  * Esberg  _et al._ (2010) J. Esberg, K. Kirsebom, H. Knudsen, H. D. Thomsen, E. Uggerhøj, U. I. Uggerhøj, P. Sona, A. Mangiarotti, T. J. Ketel, A. Dizdar, M. M. Dalton, S. Ballestrero, and S. H. Connell (CERN NA63), Phys. Rev. D 82, 072002 (2010).
  * Baudot  _et al._ (2009) J. Baudot, G. Bertolone, A. Brogna, G. Claus, C. Colledani, Y. Degerli, R. De Masi, A. Dorokhov, G. Dozière, W. Dulinski, M. Gelin, M. Goffe, A. Himmi, F. Guilloux, C. Hu-Guo, K. Jaaskelainen, M. Koziel, F. Morel, F. Orsini, M. Specht, I. Valin, G. Voutsinas, and M. Winter, in  _2009 IEEE Nuclear Science Symposium Conference Record (NSS/MIC)_ (2009) pp. 1169–1173.
  * Workman and Others (2022) R. L. Workman and Others (Particle Data Group), PTEP 2022, 083C01 (2022).
  * Nielsen (2019) C. F. Nielsen, Computer Physics Communications , 107128 (2019).
  * Nielsen (2022) C. Nielsen, Computer Physics Communications 278, 108425 (2022).
  * Doyle and Turner (1968) P. A. Doyle and P. S. Turner, Acta Crystallographica Section A 24, 390 (1968).
  * Andersen (2018) J. U. Andersen, “Notes on channeling,”  (2018), lecture notes, Aarhus University, https://phys.au.dk/publikationer/lecture-notes/.
  * Lindhard (1965) J. Lindhard, Kong. Danske Vidensk. Selsk, Mat.-Fys. Medd 34, no. 14, 1 (1965).
  * Sørensen (1996) A. H. Sørensen, Nuclear Instruments and Methods in Physics Research Section B: Beam Interactions with Materials and Atoms 119, 2 (1996).
  * Di Piazza  _et al._ (2012) A. Di Piazza, C. Müller, K. Z. Hatsagortsyan, and C. H. Keitel, Rev. Mod. Phys. 84, 1177 (2012).
  * Jackson (1975) J. D. Jackson,  _Classical Electrodynamics_ (Wiley, New York, 1975).
  * King and Ruhl (2013) B. King and H. Ruhl, Phys. Rev. D 88, 013005 (2013).
  * King and Fedotov (2018) B. King and A. M. Fedotov, Phys. Rev. D 98, 016005 (2018).
  * King  _et al._ (2013) B. King, N. Elkina, and H. Ruhl, Phys. Rev. A 87, 042117 (2013).
  * Seipt and King (2020) D. Seipt and B. King, Phys. Rev. A 102, 052805 (2020).