# The minimal exponent of cones over smooth complete intersection projective
varieties

Qianyu Chen  Department of Mathematics, University of Michigan, 530 Church
Street, Ann Arbor, MI 48109, USA<EMAIL_ADDRESS>,  Bradley Dirks  Department
of Mathematics, Stony Brook University, Stony Brook, NY 11794, USA
<EMAIL_ADDRESS>and  Mircea Mustaţă  Department of Mathematics,
University of Michigan, 530 Church Street, Ann Arbor, MI 48109, USA
<EMAIL_ADDRESS>In memory of Lucian Bădescu

###### Abstract.

We compute the minimal exponent of the affine cone over a complete
intersection of smooth projective hypersurfaces intersecting transversely. The
upper bound for the minimal exponent is proved, more generally, in the
weighted homogeneous setting, while the lower bound is deduced from a general
lower bound in terms of a strong factorizing resolution in the sense of Bravo
and Villamayor.

###### 2020 Mathematics Subject Classification:

14F10, 14B05, 32S25

Q.C. was partially supported by NSF grant DMS-1952399, B.D. was partly
supported by NSF grant DMS-2001132 and NSF-MSPRF grant DMS-2303070, and M.M.
was partially supported by NSF grants DMS-2301463 and DMS-1952399.

##  1\. Introduction

Let $X$ be a smooth complex algebraic variety. If $Z$ is a nonempty
hypersurface in $X$, then the _minimal exponent_ $\widetilde{\alpha}(Z)$ was
defined by Saito in [Saito-B] using the Bernstein-Sato polynomial of a local
equation of $Z$, as follows. Recall that if $Z$ is defined in an open subset
$U$ of $X$ by $f\in\mathcal{O}_{X}(U)$, then the Bernstein-Sato polynomial of
$f$ is the monic polynomial $b_{f}(s)\in{\mathbf{C}}[s]$ of minimal degree
such that

$b_{f}(s)f^{s}\in{\mathcal{D}}_{U}[s]\cdot f^{s+1}.$

Here $f^{s}$ is a formal symbol on which the sheaf $\mathcal{D}_{U}$ of
differential operators on $U$ acts in the expected way. By a result of
Kashiwara [Kashiwara], all roots of $b_{f}$ are negative rational numbers. It
is easy to see, by specializing $s$ to $-1$, that if $Z|_{U}:=Z\cap U$ is
nonempty, then $b_{f}(-1)=0$. By definition,
$\widetilde{\alpha}(Z|_{U})=\widetilde{\alpha}(f)$ is the negative of the
largest root of $b_{f}(s)/(s+1)$ (with the convention that this is $\infty$ if
$b_{f}(s)=s+1$). In order to define $\widetilde{\alpha}(Z)$, one takes an open
cover $X=\bigcup_{i}U_{i}$ and
$\widetilde{\alpha}(Z)=\min_{i}\widetilde{\alpha}(Z|_{U_{i}})$, where the
minimum is over those $i$ such that $Z|_{U_{i}}$ is nonempty.

The minimal exponent of a hypersurface is an interesting invariant. A result
due to Lichtin and Kollár [Kollar] says that the minimal exponent refines an
important invariant of singularities in birational geometry, the _log
canonical threshold_ ${\rm lct}(X,Z)$; more precisely, we have

${\rm lct}(X,Z)=\min\big{\\{}\widetilde{\alpha}(Z),1\big{\\}}.$

It was shown by Saito [Saito-B] that $\widetilde{\alpha}(Z)>1$ if and only if
$Z$ has rational singularities. Moreover, we have
$\widetilde{\alpha}(Z)=\infty$ if and only if $Z$ is smooth. Recently, it was
shown that the minimal exponent characterizes the _higher Du Bois_ property of
the singularities of $Z$ (see [MOPW] and [Saito_et_al]) and the condition for
_higher rational singularities_ (see [FL] and [MP2]).

If $Z$ has isolated singularities, then the minimal exponent can be described
via asymptotic expansions of integrals along vanishing cycles, see [Malgrange]
and [Malgrange2]. In this incarnation, it has been extensively studied in
[AGZV] and is also known as the _Arnold exponent_ of $f$.

In [CDMO], the authors of the present article and Sebastián Olano introduced
and studied an extension of the minimal exponent $\widetilde{\alpha}(Z)$ to
the case when $Z$ is a complete intersection in $X$ of pure codimension $r$,
for any $r\geq 1$. The definition was in terms of the Kashiwara-Malgrange
filtration associated to $Z$ (the corresponding description in the
hypersurface case is a result due to Saito [Saito-MLCT]). One of the main
results in [CDMO] gave a description in terms of the minimal exponent of a
hypersurface, as follows. Suppose that $Z$ is defined in $X$ by
$f_{1},\ldots,f_{r}\in\mathcal{O}_{X}(X)$ and
$g=\sum_{j=1}^{r}f_{j}y_{j}\in\mathcal{O}_{Y}(Y)$, where
$Y=X\times{\mathbf{A}}^{r}$, with $y_{1},\ldots,y_{r}$ being the coordinates
on ${\mathbf{A}}^{r}$. If
$W=X\times\big{(}{\mathbf{A}}^{r}\smallsetminus\\{0\\}\big{)}$, then
$\widetilde{\alpha}(Z)=\widetilde{\alpha}(g|_{W})$. This description allows
deducing the main properties of the minimal exponent of local complete
intersections from the corresponding properties of the invariant in the case
of hypersurfaces. Results on the $V$-filtration from [BMS] allowed us to
relate again the minimal exponent to the log canonical threshold and to
rational singularities: we have

${\rm lct}(X,Z)=\min\big{\\{}\widetilde{\alpha}(Z),r\big{\\}}$

and $\widetilde{\alpha}(Z)>r$ if and only if $Z$ has rational singularities.
It was also shown in [CDMO] that one can use the minimal exponent to detect
how far the Hodge filtration on the local cohomology
$\mathcal{H}^{r}_{Z}(\mathcal{O}_{X})$ agrees with the pole order filtration,
extending the corresponding result for hypersurfaces from [Saito-MLCT] and
[MP]. In conjunction with results from [MP3], this implied that the minimal
exponent detects the higher Du Bois property of local complete intersections.
The fact that it also detects higher rational singularities in this setting
was subsequently shown in [CDM]. Finally, the minimal exponent can be
described in terms of the Bernstein-Sato polynomial $b_{\mathbf{f}}(s)$,
associated to ${\mathbf{f}}=(f_{1},\ldots,f_{r})$, that was introduced in
[BMS]: in this case we have $b_{\mathbf{f}}(-r)=0$ and it was shown in [Dirks]
that $\widetilde{\alpha}(Z)$ is the negative of the largest root of
$b_{\mathbf{f}}(s)/(s+r)$.

While many of the basic properties of the minimal exponent are by now
understood in the local complete intersection case, there are few known
explicit examples beyond codimension $1$. One example given in [CDMO] is that
of a complete intersection in ${\mathbf{A}}^{n}$, with an isolated singularity
at $0$, defined by homogeneous equations of the same degree $d$; in this case
we have $\widetilde{\alpha}(Z)=\tfrac{n}{d}$, extending a well-known formula
for hypersurfaces. Our main result in this note is the following extension to
the case when the homogeneous equations defining $Z$ have possibly different
degrees:

######  Theorem 1.1.

Let $f_{1},\ldots,f_{r}\in{\mathbf{C}}[x_{1},\ldots,x_{n}]$ be homogeneous
polyomials that form a regular sequence, with ${\rm deg}(f_{i})=d_{i}$ for
$1\leq i\leq r$, and such that $2\leq d_{1}\leq\ldots\leq d_{r}$. For every
$i$, we denote by $H_{i}$ the hypersurface defined by $f_{i}$ in
${\mathbf{A}}^{n}$ and by $Z$ the intersection $H_{1}\cap\ldots\cap H_{r}$. If
on ${\mathbf{A}}^{n}\smallsetminus\\{0\\}$ each $H_{i}$ is smooth and
$\sum_{i=1}^{r}H_{i}$ has simple normal crossings, then

(1) $\widetilde{\alpha}(Z)=\min\big{\\{}i+\tfrac{1}{d_{i}}(n-d_{1}-\ldots-
d_{i})\mid 1\leq i\leq r\big{\\}}=p+\tfrac{1}{d_{p}}(n-d_{1}-\ldots-d_{p}),$

where $p$ is the smallest $i\leq r$ that satisfies $d_{1}+\ldots+d_{i}>n$
(with the convention that $p=r$ if there is no such $i$).

We are interested, in particular, in the case when $\widetilde{\alpha}(Z)>{\rm
lct}(X,Z)$, that is, when $\widetilde{\alpha}(Z)>r$. The formula in the
theorem implies that this is the case if and only if $\sum_{i=1}^{r}d_{i}<n$.
We also recover the well-known facts that under the assumptions in the
theorem, the pair $(X,rZ)$ is log canonical if and only if
$\sum_{i=1}^{r}d_{i}\leq n$ and $Z$ has rational singularities if and only if
$\sum_{i=1}^{r}d_{i}<n$.

We also note that if $d_{1}=\ldots=d_{r}$ and we only assume that
$Z\cap\big{(}{\mathbf{A}}^{n}\smallsetminus\\{0\\}\big{)}$ is smooth, then
after replacing each $f_{i}$ by a general linear combination of
$f_{1},\ldots,f_{r}$, the Kleinman-Bertini theorem implies the condition that
on ${\mathbf{A}}^{n}\smallsetminus\\{0\\}$ each $H_{i}$ is smooth and
$\sum_{i=1}^{r}H_{i}$ has simple normal crossings. Therefore the above theorem
implies the formula for the minimal exponent in [CDMO, Example 4.23].

The upper bound for $\widetilde{\alpha}(Z)$ in Theorem 1.1 can be extended to
the weighted homogeneous case, even without assuming that the equations
themselves are homogeneous. The hypersurface case follows directly from a
well-known formula for the minimal exponent of an isolated singularity that is
nondegenerate with respect to its Newton polyhedron and the semicontinuity of
the minimal exponent in families. We then obtain the following result for
complete intersections: consider on $R={\mathbf{C}}[x_{1},\ldots,x_{n}]$ the
grading such that ${\rm deg}(x_{i})=w_{i}>0$ for $1\leq i\leq n$. For every
nonzero $f\in R$, we denote by ${\rm wt}(f)$ the smallest degree of a monomial
$x^{u}=x_{1}^{u_{1}}\cdots x_{n}^{u_{n}}$ that appears with a nonzero
coefficient in $f$.

######  Theorem 1.2.

With the above notation, suppose that
$f_{1},\ldots,f_{r}\in(x_{1},\ldots,x_{n})^{2}\subseteq R$ are such that ${\rm
wt}(f_{i})=d_{i}$, for $1\leq i\leq r$, with $d_{1}\leq d_{2}\leq\ldots\leq
d_{r}$. If $Z$ is a complete intersection of pure codimension $r$ in some
neighborhood of $0$, then

$\widetilde{\alpha}_{0}(Z)\leq\min\big{\\{}i+\tfrac{1}{d_{i}}(w_{1}+\ldots+w_{n}-d_{1}-\ldots-
d_{i})\mid 1\leq i\leq r\big{\\}}.$

For the precise definition of $\widetilde{\alpha}_{0}(Z)$ the local version of
the minimal exponent of $Z$, see Section 2. We expect that if, in addition,
$f_{1},\ldots,f_{r}$ are homogeneous with respect to the above grading and the
hypersurfaces $H_{i}$ defined by $f_{i}$ satisfy a suitable transversality
assumption on ${\mathbf{A}}^{n}\smallsetminus\\{0\\}$ (for example, each
$H_{i}$ is irreducible and $\sum_{i=1}^{r}H_{i}$ has simple normal crossings
in ${\mathbf{A}}^{n}\smallsetminus 0\\}$), then the inequality in Theorem 1.2
is an equality. When all $f_{i}$ have the same degree, this can be proved as
in [CDMO, Example 4.23]. When the degrees are different, however, we can only
prove the assertion in the usual homogeneous case.

The key ingredient in the proof of the lower bound for $\widetilde{\alpha}(Z)$
in Theorem 1.1 is a result of independent interest, giving a lower bound for
the minimal exponent of a local complete intersection $Z$ in $X$ in terms of a
suitable resolution of $(X,Z)$: a _strong factorizing resolution_ in the sense
of Bravo and Villamayor [BV]. Under the assumption that $Z$ is generically
reduced, this is a proper morphism $\pi\colon\widetilde{X}\to X$ which is an
isomorphism over the complement $X\smallsetminus Z_{\rm sing}$ of the singular
locus of $Z$, with $\widetilde{X}$ smooth, and such that the reduced
exceptional divisor $E$ and the strict transform $\widetilde{Z}$ of $Z$ have
simple normal crossings. Moreover, we have a factorization

(2)
$\mathcal{I}_{Z}\cdot\mathcal{O}_{\widetilde{X}}=\mathcal{I}_{\widetilde{Z}}\cdot\mathcal{O}_{\widetilde{X}}(-F),$

for an effective divisor $F$ supported on $E$, where $\mathcal{I}_{Z}$ and
$\mathcal{I}_{\widetilde{Z}}$ are the ideals of $Z$ and $\widetilde{Z}$ in $X$
and $\widetilde{X}$, respectively. Note that the usual Hironaka algorithm does
not guarantee the latter condition; the existence of strong factorizing
resolutions for all generically reduced $Z$ is the main result of [BV]. Given
such a resolution, we write $E=\sum_{j=1}^{N}E_{j}$ as the sum of prime
divisors and for every $j$, we denote by $a_{j}$ and $k_{j}$ the coefficients
of $E_{j}$ in the divisors $F$ and, respectively, the relative canonical
divisor $K_{\widetilde{X}/X}$.

######  Theorem 1.3.

Suppose that $X$ is a smooth complex algebraic variety and $Z$ is a reduced
subscheme of $X$ that is a local complete intersection, of pure codimension
$r$. If $\pi\colon\widetilde{X}\to X$ is a strong factorizing resolution of
$(X,Z)$ as above, then

$\widetilde{\alpha}(Z)\geq\min_{1\leq j\leq N}\frac{k_{j}+1}{a_{j}}.$

Note that if $Z$ is a hypersurface in $X$, then the condition (2) is
automatically satisfied, hence a strong factorizing resolution is simply a log
resolution of $(X,Z)$. In this case, the inequality in Theorem 1.3 was proved
in [MP, Corollary D] using the theory of Hodge ideals (see also [DM, Corollary
1.5] for a more elementary proof). We deduce the general case in Theorem 1.3
by reducing it to the case of hypersurfaces. In order to get the lower bound
for $\widetilde{\alpha}(Z)$ in Theorem 1.1, we construct an explicit strong
factorizing resolution of $({\mathbf{A}}^{n},Z)$.

##  2\. An upper-bound in the weighted homogeneous case

Our goal in this section is to prove Theorem 1.2. Let us begin by recalling
the local version of the minimal exponent discussed in the Introduction. If
$Z$ is a local complete intersection in the smooth variety $X$, of pure
codimension $r$, and $P\in Z$, then for every open neighborhood $U$ of $P$, we
have $\widetilde{\alpha}(Z\cap U)\geq\widetilde{\alpha}(Z)$ and
$\widetilde{\alpha}(Z\cap U)$ is constant if $U$ is small enough. This
constant value is denoted by $\widetilde{\alpha}_{P}(Z)$. It is then easy to
see that $\widetilde{\alpha}(Z)=\min_{P\in Z}\widetilde{\alpha}_{P}(Z)$. We
refer to [CDMO, Definition 4.16] and the discussion around it for details. Of
course, $\widetilde{\alpha}_{P}(Z)$ is defined if we only know that $Z$ is a
local complete intersection of codimension $r$ at $P$. If $Z$ is a
hypersurface defined by $f$, we also write $\widetilde{\alpha}_{P}(f)$ for
$\widetilde{\alpha}_{P}(Z)$.

We begin with the following result in the case of hypersurfaces. We let
$R={\mathbf{C}}[x_{1},\ldots,x_{n}]$ and use the notation in Theorem 1.2.

######  Proposition 2.1.

If $f\in R$ is nonzero and $0\in Z$ is a singular point, then

$\widetilde{\alpha}_{0}(f)\leq\frac{w_{1}+\ldots+w_{n}}{{\rm wt}(f)}.$

###### Proof.

We write $f=\sum_{u\in\Lambda}a_{u}x^{u}$, with $\Lambda$ finite and
$a_{u}\neq 0$ for all $u\in\Lambda$. Let $N\geq 2$ be such that $Nw_{i}>{\rm
wt}(f)$ for all $i$. We consider the family of hypersurfaces parametrized by
the open subset $U\subseteq{\mathbf{A}}^{|\Lambda|+n}$, with the hypersurface
corresponding to $v=\big{(}(c_{u})_{u\in\Lambda},b_{1},\ldots,b_{n}\big{)}$
being defined by
$h_{v}=\sum_{u\in\Lambda}c_{u}x^{u}+b_{1}x_{1}^{N}+\ldots+b_{n}x_{n}^{N}$
(here $U$ consists of those $v$ such that $h_{v}$ is nonzero). It is clear
that for $v\in U$ general, $h_{v}$ has an isolated singularity at $0$ and it
is nondegenerate with respect to its Newton polyhedron $P$ (recall that $P$ is
the convex hull of $\bigcup_{u}(u+{\mathbf{R}}_{\geq 0}^{n})$, where the union
over all monomials $x^{u}$ that appear with nonzero coefficient in the
equation $h$ of the hypersurface). In this case, it is known that the minimal
exponent at $0$ of such a hypersurface is $1/c$, where

$c=\min\big{\\{}t>0\mid(t,\ldots,t)\in P\big{\\}}$

(see [Varchenko], [EhlersLo], or [Saito-exponents]). Note that $P$ is the
convex hull of
$\big{(}\Lambda\cup\\{Ne_{1},\ldots,Ne_{n}\\}\big{)}+{\mathbf{R}}^{n}_{\geq
0}$, where $e_{1},\ldots,e_{n}$ is the standard basis of ${\mathbf{Z}}^{n}$.
Since $\sum_{i=1}^{n}u_{i}w_{i}\geq{\rm wt}(f)$ for all
$u\in\Lambda\cup\\{Ne_{1},\ldots,Ne_{n}\\}$, it follows that
$\sum_{i=1}^{n}u_{i}w_{i}\geq{\rm wt}(f)$ for all $u\in P$, and thus
$c\cdot\sum_{i=1}^{n}w_{i}\geq~{}{\rm wt}(f)$. On the other hand, it follows
from the semicontinuity of minimal exponents (see [MP, Theorem E(2)]) that for
every $v^{\prime}\in U$, we have
$\widetilde{\alpha}_{0}(h_{v^{\prime}})\leq\widetilde{\alpha}_{0}(h_{v})=1/c$,
when $v\in U$ general. In particular, this applies for $f$, and we get

$\widetilde{\alpha}_{0}(f)\leq\frac{1}{c}\leq\frac{w_{1}+\ldots+w_{n}}{{\rm
wt}(f)}.$

∎

Before giving the proof of Theorem 1.2, we give a lemma that describes the
infimum in this theorem.

######  Lemma 2.2.

Let $w\in{\mathbf{R}}$ and let $d_{1}\leq\ldots\leq d_{r}$ be positive
integers. If for $1\leq i\leq r$, we put

$\alpha_{i}:=\big{\\{}i+\tfrac{1}{d_{i}}(w-d_{1}-\ldots-d_{i})\mid 1\leq i\leq
r\big{\\}},$

then the following hold:

  1. i)

If $i\leq r-1$ is such that $d_{i}=d_{i+1}$, then $\alpha_{i}=\alpha_{i+1}$.

  2. ii)

If $i\leq r-1$ and $d_{i}<d_{i+1}$, then $\alpha_{i}\geq\alpha_{i+1}$ if and
only if $d_{1}+\ldots+d_{i}\leq w$.

  3. ii)

We have $\min_{i}\alpha_{i}=\alpha_{p}$, where $p$ is the smallest $i\leq r$
that satisfies $d_{1}+\ldots+d_{i}>w$ (with the convention that $p=r$ if there
is no such $i$).

###### Proof.

The first two assertions follow from the fact that for $i\leq r-1$, we have

$\alpha_{i}-\alpha_{i+1}=\frac{(w-d_{1}-\ldots-
d_{i})(d_{i+1}-d_{i})}{d_{i}d_{i+1}},$

and the third assertion is an easy consequence. ∎

We can now prove the upper bound for the minimal exponent of complete
intersections in terms of the weights of the defining equations.

###### Proof of Theorem 1.2.

For every $i$, with $1\leq i\leq r$, let

$\alpha_{i}=\big{\\{}i+\tfrac{1}{d_{i}}(w_{1}+\ldots+w_{n}-d_{1}-\ldots-
d_{i})\mid 1\leq i\leq r\big{\\}},$

and let $p$ be such that $\alpha_{p}=\min_{i}\alpha_{i}$. By Lemma 2.2i), we
may assume that if $p>1$, then $d_{p-1}<d_{p}$.

Let
$g=\sum_{j=1}^{r}f_{j}y_{j}\in\mathcal{O}({\mathbf{A}}^{n}\times{\mathbf{A}}^{r})$,
where $y_{1},\ldots,y_{r}$ are the coordinates on ${\mathbf{A}}^{r}$. It
follows from the description of the minimal exponent of $Z$ in terms of $g$
given in the Introduction that if
$U={\mathbf{A}}^{r}\smallsetminus\\{0\\}\subseteq U^{\prime}=(y_{p}\neq 0)$,
then

$\widetilde{\alpha}_{0}(Z)=\max_{V\ni 0}\widetilde{\alpha}(g|_{V\times
U})\leq\max_{V\ni 0}\widetilde{\alpha}(g|_{V\times U^{\prime}}),$

where $V$ runs over the open neighborhoods of $0$ in ${\mathbf{A}}^{n}$. We
put $z_{j}=y_{j}/y_{p}$ for $1\leq j\leq r$, $j\neq p$, so
$z_{1},\ldots,\widehat{z_{p}},\ldots,z_{r}$ can be viewed as coordinates on
${\mathbf{A}}^{r-1}$. Since $g$ is homogeneous of degree $1$ with respect to
$y_{1},\ldots,y_{r}$, it follows that if we put

$h=g/y_{p}=f_{1}z_{1}+\ldots+f_{p-1}z_{p-1}+f_{p}+f_{p+1}z_{p+1}+\ldots+f_{r}z_{r}\in\mathcal{O}({\mathbf{A}}^{n}\times{\mathbf{A}}^{r-1}),$

then

$\widetilde{\alpha}(g|_{V\times
U^{\prime}})=\widetilde{\alpha}(h|_{V\times{\mathbf{A}}^{r-1}})$

(we use here the fact that the minimal exponent does not change by pull-back
by a smooth surjective morphism, see for example [CDMO, Proposition 4.12]). We
thus conclude that

(3) $\widetilde{\alpha}_{0}(Z)\leq\widetilde{\alpha}_{(0,0)}(h).$

By assumption, we have $f_{p}\in(x_{1},\ldots,x_{n})^{2}$, and thus $h$ has a
singular point at $(0,0)$. If we consider the weight of $z_{j}$ to be
$d_{p}-d_{j}$ for $1\leq j\leq p-1$ and $\epsilon>0$ for $p+1\leq j\leq r$,
then we see that ${\rm wt}(h)=d_{p}$, hence it follows from Proposition 2.1
that

(4)
$\widetilde{\alpha}_{(0,0)}(h)\leq\frac{1}{d_{p}}\big{(}w_{1}+\ldots+w_{n}+(d_{p}-d_{1})+\ldots+(d_{p}-d_{p-1})+(r-p)\epsilon\big{)}=\alpha_{p}+\frac{r-p}{d_{p}}\epsilon.$

By combining (3) and (4), and letting $\epsilon$ go to $0$, we obtain the
inequality in the theorem. ∎

##  3\. A general lower bound via a strong factorizing resolution

In this section, we prove the lower bound on the minimal exponent in terms of
a strong factorizing resolution.

###### Proof of Theorem 1.3.

We may and will assume that $X$ is affine and $Z$ is defined by a regular
sequence $f_{1},\ldots,f_{r}\in\mathcal{O}_{X}(X)$. Let
$g=f_{1}y_{1}+\ldots+f_{r}y_{r}\in\mathcal{O}_{Y}(Y)$, where
$Y=X\times{\mathbf{A}}^{r}$, with $y_{1},\ldots,y_{r}$ being the coordinates
on ${\mathbf{A}}^{r}$. Let
$W=X\times\big{(}{\mathbf{A}}^{r}\smallsetminus\\{0\\}\big{)}$, so
$\widetilde{\alpha}(Z)=\widetilde{\alpha}(g|_{W})$.

Consider now the morphism

$\varphi=\pi\times{\rm
id}_{{\mathbf{A}}^{r}}\colon\widetilde{Y}=\widetilde{X}\times{\mathbf{A}}^{r}\to
Y.$

This is a projective morphism which is an isomorphism over the complement of
$Z_{\rm sing}\times{\mathbf{A}}^{r}$. The exceptional divisors of $\varphi$
are the $E_{i}\times{\mathbf{A}}^{r}$, with $1\leq i\leq N$. Moreover, it
follows from the definition of a strong factorizing resolution that we can
cover $\widetilde{X}$ by open subsets $V_{j}$, such that on each
$V_{j}\times{\mathbf{A}}^{r}$ we can write

$g\circ\varphi|_{V_{j}\times{\mathbf{A}}^{r}}=v_{j}\cdot\sum_{i=1}^{r}h_{i}y_{i},$

where the divisor ${\rm div}(v_{j})$ defined by $v_{j}$ is supported on
$G=E\times{\mathbf{A}}^{r}$ and $h_{1},\ldots,h_{r}$ generate the ideal of
$\widetilde{Z}$ in $V_{j}$. Moreover, the coefficient of
$E_{i}\times{\mathbf{A}}^{r}$ in ${\rm div}(v_{j})$ is $a_{i}$. Note that if
$V_{j}\cap\widetilde{Z}=\emptyset$, then $\sum_{i=1}^{r}h_{i}y_{i}$ defines a
smooth hypersurface in $V_{j}\times{\mathbf{A}}^{r}$, that has simple normal
crossings with $G$.

By assumption, $\widetilde{Z}$ is smooth, of codimension $r$ in
$\widetilde{X}$, and has simple normal crossings with $E$ (that is, both $E$
and $E|_{\widetilde{Z}}$ are reduced simple normal crossing divisors).
Therefore we may and will assume that for every $j$ such that
$V_{j}\cap\widetilde{Z}\neq\emptyset$, we have algebraic coordinates
$x_{1},\ldots,x_{n}$ on $V_{j}$ such that $h_{i}=x_{i}$ for $i\leq r$ and
$E|_{V_{j}}=\sum_{i=r+1}^{r+s}a_{i}\cdot{\rm div}(x_{i})$.

Let $\varphi_{W}\colon\varphi^{-1}(W)\to W$ be the restriction of $\varphi$
over $W$. Note that on $\varphi^{-1}(W)\cap(V_{j}\times{\mathbf{A}}^{r})$,
with $V_{j}\cap\widetilde{Z}\neq\emptyset$, the divisor defined by

$(x_{1}y_{1}+\ldots+x_{r}y_{r})\cdot\prod_{i=r+1}^{r+s}x_{i}^{a_{i}}$

has simple normal crossings. Since $g\circ\varphi$ clearly defines a simple
normal crossing divisor in $\varphi^{-1}(W)\cap(V_{j}\times{\mathbf{A}}^{r})$
when $V_{j}\cap\widetilde{Z}=\emptyset$, we conclude that $\varphi_{W}$ is a
log resolution of $(W,{\rm div}(g)|_{W})$ which is an isomorphism over
$W\smallsetminus V(g)$. The exceptional divisors of $\varphi_{W}$ are the
$E^{\prime}_{i}=E_{i}\times\big{(}{\mathbf{A}}^{r}\smallsetminus\\{0\\}\big{)}$
and the relative canonical divisor of $\varphi_{W}$ is
$\sum_{i=1}^{N}k_{i}E^{\prime}_{i}$. Moreover, the divisor ${\rm div}(g)|_{W}$
is reduced: its singular locus is contained in $Z_{\rm
sing}\times\big{(}{\mathbf{A}}^{r}\smallsetminus\\{0\\}\big{)}$ (see [CDMO,
Lemma 4.22]) and thus ${\rm div}(g)$ is generically reduced, hence reduced. In
addition, its strict transform on $\varphi^{-1}(W)$ is smooth: this is clear
on $V_{j}\times\big{(}{\mathbf{A}}^{r}\smallsetminus\\{0\\}\big{)}$ if
$V_{j}\cap\widetilde{Z}=\emptyset$, while if
$V_{j}\cap\widetilde{Z}\neq\emptyset$, it follows from the fact that it is
defined by $\sum_{i=1}^{r}x_{i}y_{i}$. We can thus apply the lower bound on
the minimal exponent of a hypersurface in terms of a log resolution (see [MP,
Corollary D] or [DM, Corollary 1.5]) to conclude that

$\widetilde{\alpha}(Z)=\widetilde{\alpha}(g|_{U})\geq\min_{1\leq i\leq
N}\frac{k_{i}+1}{a_{i}},$

which is the assertion in the theorem. ∎

##  4\. The formula in the homogeneous case

Our main goal in this section is to prove Theorem 1.1. In order to prove the
lower bound in the theorem, we will use Theorem 1.3. We thus proceed to
describe a strong factorizing resolution of $({\mathbf{A}}^{n},Z)$.

With the notation in Theorem 1.1, let $\pi_{1}\colon X_{1}\to{\mathbf{A}}^{n}$
be the blow-up of the origin, with exceptional divisor $E_{1}$. Suppose that
$k\geq 1$ and $1\leq p_{1},p_{2},\ldots,p_{k}$ are such that

$d_{1}=\ldots=d_{p_{1}}<d_{p_{1}+1}=\ldots=d_{p_{1}+p_{2}}<\ldots<d_{p_{1}+\ldots+p_{k-1}+1}=\ldots=d_{p_{1}+\ldots+p_{k}}.$

Note that $p_{1}+\ldots+p_{k}=r$. In order to simplify the notation, we put
$e_{j}=d_{p_{1}+\ldots+p_{j}}$ for $1\leq j\leq k$. We define a morphism
$\pi\colon Y\to{\mathbf{A}}^{n}$ to be the composition of $\pi_{1}$ with
$\sum_{i=1}^{k-1}(e_{i+1}-e_{i})$ smooth blow-ups, as follows. First, we
consider $(e_{2}-e_{1})$ blow-ups, each of these blowing up the intersection
of the previous exceptional divisor with the strict transforms of
$H_{1},\ldots,H_{p_{1}}$. We next consider $(e_{3}-e_{2})$ blow-ups, each of
these blowing up the intersection of the previous exceptional divisor with the
strict transforms of $H_{1},\ldots,H_{p_{1}+p_{2}}$, etc.

######  Proposition 4.1.

With the above notation, the composition $\pi\colon Y\to{\mathbf{A}}^{n}$ has
the following properties:

  1. i)

If $r\leq n-1$, then $\pi$ is a strong factorizing resolution of
$({\mathbf{A}}^{n},Z)$.

  2. ii)

If $r=n$, then $\pi$ is a log resolution of the pair $({\mathbf{A}}^{n},Z)$.

###### Proof.

We note that if $r\leq n-1$, then the assumption on $Z$ implies that it is
generically reduced, hence reduced, since it is a complete intersection and
thus Cohen-Macaulay. Therefore, in this case, it makes sense to say that $\pi$
is a strong factorizing resolution.

The blow-up $X_{1}$ is covered by affine open charts $U_{1},\ldots,U_{n}$,
where $U_{i}$ has coordinates

$x_{i},y_{1},\ldots,y_{i-1},y_{i+1},\ldots,y_{n}$

such that $x_{j}=x_{i}y_{j}$ for all $j\neq i$. Note that if
$I_{Z}=(f_{1},\ldots,f_{r})$, then

$I_{Z}\cdot\mathcal{O}_{U_{i}}=(x_{i}^{d_{1}}g_{1},\ldots,x_{i}^{d_{r}}g_{r}),$

where $g_{j}=f_{j}(y_{1},\ldots,y_{i-1},1,y_{i+1},\ldots,y_{n})$ for $1\leq
j\leq r$. Moreover, $E_{1}\cap U_{i}$ is defined by $x_{i}$ and if $r<n$, then
the strict transform $\widetilde{Z}$ of $Z$ on $X_{1}$ is defined in $U_{i}$
by $(g_{1},\ldots,g_{r})$, hence it is smooth. Note that if $k=1$ (that is, we
have $d_{1}=\ldots=d_{r}$), then $\pi_{1}$ is a strong factorizing resolution
when $r<n$ and is a log resolution of $(X,Z)$ for $r=n$. Therefore we are done
in this case.

Suppose now that $k>1$. We note that, by our assumption on
$f_{1},\ldots,f_{r}$, the hypersurfaces defined by $x_{i},g_{1},\ldots,g_{r}$
in $U_{i}$ are smooth and their sum has simple normal crossings. It follows
that for every point $P\in U_{i}$, we can find algebraic coordinates
$z_{0},\ldots,z_{n-1}$ in a neighborhood $W_{P}$ of $P$ such that the ideal
$I_{Z}\cdot\mathcal{O}_{W_{P}}$ is equal to:

Case 1. $(z_{1},\ldots,z_{r})$. This is the case when $P\not\in E_{1}$, when
we may assume that $W_{P}\cap E_{1}=\emptyset$. This case is clear: it follows
from the definition of $\pi$ that the morphism $Y\to X_{1}$ is an isomorphism
over $W_{P}$ and it is clear that above $W_{P}$ the condition for $\pi$ to be
a strong factorizing resolution (if $r<n$) or a log resolution of $(X,Z)$ (if
$r=n$) is satisfied.

Case 2. $(z_{0}^{d_{1}}z_{1},\ldots,z_{0}^{d_{r}}z_{r})$. This is the case
when $r<n$ and $P$ lies on the hypersurfaces defined by
$x_{i},g_{1},\ldots,g_{r}$. Note that $\widetilde{Z}$ is defined in $W_{P}$ by
$(z_{1},\ldots,z_{r})$.

Case 3. $(z_{0}^{d_{1}}z_{1},\ldots,z_{0}^{d_{q}}z_{q},z_{0}^{d_{q+1}})$, for
some $q<r$. This is the case when $P$ lies on $E_{1}$ and $g_{j}(P)=0$ for
$j\leq q$, but $g_{q+1}(P)\neq 0$. After getting rid of some redundant
generators, we may assume that $q=p_{1}+\ldots+p_{m}$. If $r<n$, then we see
that $\widetilde{Z}$ does not meet $W_{P}$ in this case.

We now consider the next blow-up $\pi_{2}\colon X_{2}\to X_{1}$ in our
sequence: we blow up along
$E_{1}\cap\widetilde{H_{1}}\cap\ldots\cap\widetilde{H_{p_{1}}}$, where
$\widetilde{H_{j}}$ denotes the strict transform of $H_{j}$ on $X_{1}$. Let’s
describe $\pi_{2}$ over the above open subset $W_{P}\subseteq U_{i}$ when we
are in Case 2 or Case 3. Note that we are blowing up along the zero locus of
$(z_{0},z_{1},\ldots,z_{p_{1}})$, which is smooth. Let $V_{j}$ be the chart in
$\pi_{2}^{-1}(W_{P})$ given by

$z_{\ell}=u_{\ell}\quad\text{for}\quad\ell\in\\{j,p_{1}+1,\ldots,n\\}\quad\text{and}\quad
z_{\ell}=u_{j}u_{\ell}\quad\text{for}\quad 0\leq\ell\leq p_{1},\ell\neq j$

for some $j$, with $1\leq j\leq p_{1}$. An easy computation shows that
$I_{Z}\cdot\mathcal{O}_{V_{j}}$ is generated by $u_{j}^{d_{1}+1}u_{0}^{d_{1}}$
in both Cases 2 and 3. Since $u_{j}$ defines the $\pi_{2}$-exceptional divisor
and $u_{0}$ defines the strict transform of $E_{1}$, we see that
$I_{Z}\cdot\mathcal{O}_{V_{j}}$ is the ideal of a divisor supported on the
exceptional locus. Therefore the condition for a strong factorizing resolution
(in the case $r<n$) or for a log resolution (in the case $r=n$) will be
trivially satisfied over $V_{j}$.

We next consider the chart $V_{0}$ in $\pi_{2}^{-1}(W_{P})$ given by

$z_{\ell}=u_{\ell}\quad\text{for}\quad\ell\in\\{0,p_{1}+1,\ldots,n\\}\quad\text{and}\quad
z_{\ell}=u_{0}u_{\ell}\quad\text{for}\quad 1\leq\ell\leq p_{1}.$

Note that the $\pi_{2}$-exceptional divisor is defined in this chart by
$u_{0}$. Again, an easy computation shows that $I_{Z}\cdot\mathcal{O}_{V_{0}}$
is equal to

$(u_{0}^{e_{1}+1}u_{1},\ldots
u_{0}^{e_{1}+1}u_{p_{1}},u_{0}^{e_{2}}u_{p_{1}+1},\ldots,u_{0}^{e_{k}}u_{r})$

in Case 2 and to

$(u_{0}^{e_{1}+1}u_{1},\ldots,u_{0}^{e_{1}+1}u_{p_{1}},\ldots,u_{0}^{e_{m+1}})$

in Case 3 (we recall that $m$ is such that $q=p_{1}+\ldots+p_{m}$). We thus
see that if we are in Case 2, after performing $(e_{2}-e_{1})$ such blow-ups,
we are in the situation where $k$ is replaced by $k-1$: in the only charts
that we need to consider, we have coordinates $v_{0},v_{1},\ldots,v_{n-1}$,
such that the pull-back of $I_{Z}$ is equal to

$(v_{0}^{e_{2}}v_{1},\ldots,v_{0}^{e_{2}}v_{p_{1}+p_{2}},v_{0}^{e_{3}}v_{p_{1}+p_{2}+1},\ldots,v_{0}^{e_{k}}v_{r}).$

If $k>2$, then the next blow-up is along the ideal
$(v_{0},v_{1},\ldots,v_{p_{1}+p_{2}})$, and the process continues as above. In
the end, we see that in the only charts that we need to consider, we have
coordinates $w_{0},\ldots,w_{n-1}$ such that the pull-back of $I_{Z}$ is
$w_{0}^{d_{r}}\cdot(w_{1},\ldots,w_{r})$. Therefore, in such a chart, the
condition for having a strong factorizing resolution is satisfied.

Similarly, if we are in Case 3, then after the first
$(e_{2}-e_{1})+\ldots+(e_{m+1}-e_{m})$ blow-ups, in the only charts that we
need to consider, we have coordinates $w_{0},\ldots,w_{n-1}$ such that the
pull-back of $I_{Z}$ is $(w_{0}^{e_{m+1}})$. Therefore, in this chart, we only
have the ideal of a divisor supported on the exceptional locus, so this
satisfies the condition for $\pi$ to be a strong factorizing resolution when
$r<n$ and to be a log resolution when $r=n$. This completes the proof of the
proposition. ∎

######  Remark 4.2.

With the notation in Proposition 4.1, it follows from the definition of $\pi$
that if $r<n$, then starting with $X_{1}$, at each step we blow up a smooth
center that is not contained in the strict transform of $Z$ on the respective
variety. In fact, with the notation in the proof, for every chart $U_{i}$ on
$X_{1}$ and for every exceptional divisor on $Y$ whose image in $X_{1}$
intersects $U_{i}$, we see that $g_{r}$ does not vanish along this image.

We can now prove the main result of this note.

###### Proof of Theorem 1.1.

For every $k$, with $1\leq k\leq n$, we put
$\alpha_{k}=k+\frac{n-d_{1}-\ldots-d_{k}}{d_{k}}$. By Lemma 2.2, we know that
$\min_{k}\alpha_{k}=\alpha_{p}$, where $p$ is as in the statement of the
theorem. Since the inequality

$\widetilde{\alpha}(Z)\leq\alpha_{p}$

follows from Theorem 1.2, we only need to prove the opposite inequality.

Suppose first that $\sum_{j=1}^{r}d_{j}>n$. Note that since
$\widetilde{\alpha}(Z)\leq\alpha_{n}<r$, we know that in this case we have
$\widetilde{\alpha}(Z)={\rm lct}(X,Z)$, and thus only need to show that ${\rm
lct}(X,Z)\geq\alpha_{p}$. For basic facts about log canonical thresholds
(including the definition), we refer to [Lazarsfeld, Chapter 9]. As in the
proof of Proposition 4.1, we consider the blow-up $\pi_{1}\colon
X_{1}\to{\mathbf{A}}^{n}$ of ${\mathbf{A}}^{n}$, with exceptional divisor
$E_{1}$. Note that $K_{X_{1}/{\mathbf{A}}^{n}}=(n-1)E_{1}$. We have seen in
the proof of Proposition 4.1 that we can cover $X_{1}$ by affine open charts
$U_{i}$, such that
$I_{Z}\cdot\mathcal{O}_{U_{i}}=(x_{i}^{d_{1}}g_{1},\ldots,x_{i}^{d_{r}}g_{r})$,
where $x_{i}$ defines $E_{1}$ in $U_{i}$, and the divisors defined by
$x_{i},g_{1},\ldots,g_{r}$ are smooth and their sum has simple normal
crossings. We need to show that if $G$ is a prime divisor on $W$, where
$\varphi\colon W\to X_{1}$ is such that $\pi_{1}\circ\varphi$ is a log
resolution of $(X,Z)$, with the valuation ${\rm ord}_{G}$ corresponding to
$G$, and if $a_{G}={\rm ord}_{G}(I_{Z})$ and $k_{G}$ is the coefficient of $G$
in $K_{W/X}$, then $\tfrac{k_{G}+1}{a_{G}}\geq\alpha_{p}$. Suppose that the
image of $G$ on $X_{1}$ intersects the chart $U_{i}$ and let $b_{0}={\rm
ord}_{G}(x_{i})$ and $b_{j}={\rm ord}_{G}(g_{j})$ for $1\leq j\leq r$. We may
and will assume that $b_{0}>0$: otherwise, since $Z\smallsetminus\\{0\\}$ is
smooth, of codimension $r$ in ${\mathbf{A}}^{n}\smallsetminus\\{0\\}$, we have
${\rm lct}({\mathbf{A}}^{n}\smallsetminus\\{0\\},Z\smallsetminus\\{0\\})=r$,
and thus $\tfrac{k_{G}+1}{a_{G}}\geq r>\alpha_{p}$. It is well-known that
since the divisor ${\rm div}(x_{i})+\sum_{j=1}^{r}{\rm div}(g_{j})$ has simple
normal crossings, if $k^{\prime}_{G}$ is the coefficient of $G$ in
$K_{W/X_{1}}$, then

$k^{\prime}_{G}+1\geq b_{0}+b_{1}+\ldots+b_{r}$

(see, for example, the proof of [Lazarsfeld, Lemma 9.2.19]). Since

$K_{W/X}=K_{W/X_{1}}+\varphi^{*}(K_{X_{1}/{\mathbf{A}}^{n}})=K_{W/X_{1}}+(n-1)\varphi^{*}(E_{1}),$

we have

$k_{G}+1=k^{\prime}_{G}+1+(n-1)b_{0}\geq nb_{0}+\sum_{j=1}^{r}b_{j}.$

Since

${\rm ord}_{G}(I_{Z})=\min\\{b_{0}d_{j}+b_{j}\mid 1\leq j\leq r\\},$

it follows that it is enough to show that

(5) $nb_{0}+\sum_{j=1}^{r}b_{j}\geq\alpha_{p}\cdot\min\\{b_{0}d_{j}+b_{j}\mid
1\leq j\leq r\\}.$

If we put $u_{j}=b_{j}/b_{0}$ for $1\leq j\leq r$ and
$M=\min\\{d_{j}+u_{j}\mid 1\leq j\leq r\\}$, then (5) becomes

(6) $n+\sum_{j=1}^{r}u_{j}\geq\alpha_{p}M.$

We define an increasing sequence $k_{1}<k_{2}<\ldots<k_{s}=r$ such that

$k_{1}=\max\\{j\mid 1\leq j\leq r,d_{j}+u_{j}=M\\},$

and if $k_{\ell}<r$, then

$k_{\ell+1}=\max\\{k>k_{\ell}\mid d_{k}+b_{k}=\min\\{d_{j}+u_{j}\mid
j>k_{\ell}\\}\big{\\}}.$

With this notation, the inequality (6) becomes

(7) $\frac{n+u_{1}+\ldots+u_{r}}{d_{k_{1}}+u_{k_{1}}}\geq\alpha_{p}.$

For $1\leq q\leq s$, let us put

(8)
$\beta_{k_{q}}:=\frac{n+k_{q}u_{k_{q}}+\sum_{j=1}^{k_{q}}(d_{k_{q}}-d_{j})+\sum_{j>k_{q}}u_{j}}{d_{k_{q}}+u_{k_{q}}}.$

For every $j<k_{1}$, we have $u_{j}\geq u_{k_{1}}+(d_{k_{1}}-d_{j})$, hence
the left-hand side of (7) is $\geq\beta_{k_{1}}$ and thus (7) follows if we
show

(9) $\beta_{k_{1}}\geq\alpha_{p}.$

The key step is to show that if $q<s$, then

(10) $\beta_{k_{q}}\geq\min\\{\alpha_{k_{q}},\beta_{k_{q+1}}\\}.$

Indeed, if we view $\beta_{k_{q}}$ as a function of $u_{k_{q}}$, since $0\leq
u_{k_{q}}\leq u_{k_{q+1}}+(d_{k_{q+1}}-d_{k_{q}})$, we see that
$\beta_{k_{q}}$ is bounded below by the minimum taken when $u_{k_{q}}=0$ and
when $u_{k_{q}}=u_{k_{q+1}}+(d_{k_{q+1}}-d_{k_{q}})$. In the former case, the
value is

$\frac{n+\sum_{j=1}^{k_{q}}(d_{k_{q}}-d_{j})+\sum_{j>k_{q}}u_{j}}{d_{k_{q}}}\geq\alpha_{k_{q}},$

while in the latter case, using the fact that $u_{j}\geq
u_{k_{q+1}}+d_{k_{q+1}}-d_{j}$ for $k_{q}<j\leq k_{q+1}$, the value is

$\frac{n+k_{q}(u_{k_{q+1}}+d_{k_{q+1}}-d_{k_{q}})+\sum_{j=1}^{k_{q}}(d_{k_{q}}-d_{j})+\sum_{j>k_{q}}u_{j}}{u_{k_{q+1}}+d_{k_{q+1}}}\geq\beta_{k_{q+1}}.$

We thus obtain the inequality in (10). Using the fact that
$\alpha_{k_{q}}\geq\alpha_{p}$ for all $q$ gives

$\beta_{k_{1}}\geq\min\\{\alpha_{p},\beta_{k_{s}}\\}.$

On the other hand, we have $k_{s}=r$, and thus

$\beta_{k_{s}}=\frac{n+ru_{r}+\sum_{j=1}^{r}(d_{r}-d_{j})}{d_{r}+u_{r}}.$

As above, if we view this as a function of $u_{r}$, we see that it is bounded
below by the minimum of its values when $u_{r}=0$ (which is $\alpha_{r}$) and
the value of the limit when $u_{r}$ goes to infinity (which is
$r>\alpha_{r}$). We thus conclude that $\beta_{k_{1}}\geq\alpha_{p}$,
completing the proof of (9), and thus the proof of the theorem when
$\sum_{j=1}^{r}d_{j}>n$.

Suppose now that $\sum_{j=1}^{r}d_{j}\leq n$. Note that since we assume
$d_{j}\geq 2$ for all $j$, we have $r<n$. By Theorem 1.3, in order to show
that $\widetilde{\alpha}(Z)\geq\alpha_{r}$, it is enough to show that if $G$
is a prime $\pi$-exceptional divisor on $Y$, then we have
$\tfrac{k_{G}+1}{a_{G}}\geq\alpha_{r}$ (note that we keep the notation in the
first part of the proof). We choose again a chart $U_{i}$ on $X_{1}$ that
intersects the image of $G$ and put $b_{0}={\rm ord}_{G}(x_{i})$ and
$b_{j}={\rm ord}_{G}(g_{j})$ for $1\leq j\leq r$. As before, it is enough to
show that

(11) $\tfrac{nb_{0}+b_{1}+\ldots+b_{r}}{a_{G}}\geq\alpha_{r},$

where $a_{G}=\min\\{b_{0}d_{j}+b_{j}\mid 1\leq j\leq r\\}$. A key point is
that, by construction, we have $b_{r}=0$ (see Remark 4.2). This implies that

(12) $a_{G}\leq b_{0}d_{r}.$

On the other hand, since $b_{j}\geq a_{G}-b_{0}d_{j}$ for $j\geq 1$, we have

$\tfrac{nb_{0}+b_{1}+\ldots+b_{r}}{a_{G}}\geq\frac{nb_{0}+\sum_{j=1}^{r}(a_{G}-b_{0}d_{j})}{a_{G}}=r+\tfrac{(n-d_{1}-\ldots-
d_{r})b_{0}}{a_{G}}\geq\alpha_{r},$

where the last inequality follows from (12), using the fact that
$n\geq\sum_{j=1}^{r}d_{j}$. This proves (11) and completes the proof of the
theorem. ∎

######  Remark 4.3.

In the statement of Theorem 1.1, we made the assumption that $d_{1}\geq 2$.
The general case can be easily reduced to this one: indeed, suppose that
$d_{q}=1<d_{q+1}$ for some $q\leq r-1$. In this case, $Z$ is isomorphic to a
closed subscheme $W$ of ${\mathbf{A}}^{n-q}$ defined by homogeneous equations
of degrees $d_{q+1}\leq\ldots\leq d_{r}$, and which satisfies the hypothesis
in Theorem 1.1. Moreover, by [CDMO, Proposition 4.14], we have
$\widetilde{\alpha}(Z)=\widetilde{\alpha}(W)+q$.

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