# Multiplicities of jumping numbers

Swaraj Pande  Department of Mathematics
University of Michigan
Ann Arbor, MI 48109-1043
USA<EMAIL_ADDRESS>
###### Abstract.

We study multiplicities of jumping numbers of multiplier ideals in a smooth
variety of arbitrary dimension. We prove that the multiplicity function is a
quasi-polynomial, hence proving that the Poincaré series is a rational
function. We further study when the various components of the quasi-polynomial
have the highest possible degree and relate it to jumping numbers contributed
by Rees valuations. Finally, we study the special case of monomial ideals.

###### Key words and phrases:

multiplier ideals, jumping numbers, Poincaré series, Rees valuations, monomial
ideals

###### 2020 Mathematics Subject Classification:

Primary 14F18; Secondary 13D40

This project was partially supported by NSF Grant DMS 1801697.

##  1\. Introduction

Let $X$ be a smooth variety over an algebraically closed field of
characteristic $0$. Associated to each closed subscheme $Z$ of $X$ is a family
of ideals, called the multiplier ideals, that quantify the singularities of
$Z$. The multiplier ideals are indexed by a positive real parameter $c$
(denoted by $\mathscr{I}(c\cdot Z)$), and form a decreasing family of ideals
of $\mathcal{O}_{X}$.

As $c$ varies over the positive real numbers, the stalks of these ideals
$\mathscr{I}(c\cdot Z)_{x}$ at any point $x$ change exactly at a discrete set
of rational numbers $c_{i}$ called the _jumping numbers_ of $Z$ at $x$. So we
get a descending chain of ideals

$\mathcal{O}_{X,x}\supsetneqq\mathscr{I}(c_{1}\cdot
Z)_{x}\supsetneqq\mathscr{I}(c_{2}\cdot Z)_{x}\supsetneqq\dots$ (1.1)

where $c_{1},c_{2},\dots$ are the jumping numbers of $Z$ at $x$. The jumping
numbers were defined and studied in [Ein+04]. They are important invariants of
the singularity of $Z$ at a point.

Now fix a closed point $x\in X$ and the local ring $(A,\mathfrak{m})$ at $x$.
If $Z$ is supported only at $x$, then all the non-trivial multiplier ideals at
$x$ will be $\mathfrak{m}$-primary ideals. In this situation, the jumping
numbers come with a natural multiplicity denoted by $m(c)$ that measures the
difference between successive multiplier ideals in the chain (1.1). Some of
these multiplicities appear in the Hodge-Spectrum of a general member of
$\mathfrak{a}$ as proved by Budur in [Bud03].

The purpose of this paper is to study multiplicities of jumping numbers in
detail for arbitrary dimensional smooth varieties. Fixing a jumping number
$c$, we obtain another jumping number if we add any positive integer, so it is
natural to consider the sequence of multiplicities of the jumping numbers
$c+n$, as $n$ ranges through the natural numbers. We prove

######  Theorem (3.2).

Fix any jumping number $c$ of a point scheme on a smooth variety. Then the
sequence of multiplicities $(m(c+n))_{n\in\mathbb{N}}$ is polynomial in $n$ of
degree less than the dimension of $X$.

This generalizes the work in dimension two of Alberich-Carramiñana, Àlvarez
Montaner, Dachs-Cadefau and González-Alonso. They proved the dimension two
case in [AADG17] by computing the multiplicities explicitly in terms of the
intersection matrix of a log resolution. We use numerical intersection theory
of divisors as developed by Kleiman in [Kle66].

The coefficient of $n^{d-1}$ in the above polynomial turns out to have
interesting features. For example, it predicts the relevance of Rees
valuations for the computation of jumping numbers. For this reason, we call
this coefficient the _Rees Coefficient_ of $c$ and prove

######  Theorem (4.5).

The Rees Coefficient of a jumping number $c$ is positive if and only if
$c+d-1$ is a jumping number contributed by a Rees valuation of $\mathfrak{a}$,
in the sense of [ST07] ($d$ is the dimension of $X$).

We note that the Rees coefficient is not necessarily the “leading coefficient”
of the polynomial $m(c+n)$ because it can be zero; see Example 4.7. This
theorem has some interesting consequences, for example producing easy examples
of jumping numbers of any $\mathfrak{m}$-primary ideal contributed by its Rees
valuations (Corollary 4.10). Another interesting feature is that the Rees
Coefficients of jumping numbers modulo $\mathbb{Z}$ sum to the Hilbert-Samuel
multiplicity of the ideal $\mathfrak{a}$; see Theorem 4.14.

We also give a formula for calculating multiplicities and Rees coefficients
for monomial ideals (Theorem 5.1). In particular, this proves that the Rees
coefficient of every jumping number of a monomial ideal is positive (Corollary
5.3).

Finally, we examine a generating function for these multiplicities, proving in
Theorem 6.1 that the _Poincaré Series_ of $\mathfrak{a}$ is a rational
function in a suitable sense in higher dimensions, generalizing the previous
results from [GM10] and [AADG17] in dimension two. This theorem was
independently proved by Àlvarez Montaner and Núñez-Betancourt in [AN21] using
algebraic methods.

The paper is organised as follows: In the second section, we review the basic
theory of multiplier ideals and numerical intersection theory that will be
used in the rest of the paper. The third section is dedicated to proving
Theorem 3.2 and computing the Rees coefficient. We analyze the properties of
the Rees coefficient in the fourth section. The fifth section is dedicated to
the special case of monomial ideals. In the final section, we prove the
rationality of the Poincaré Series.

Notation: $k$ will always denote an algebraically closed field of
characteristic zero. The set of natural numbers denoted by $\mathbb{N}$ will
always denote the set of non-negative integers.

### Acknowledgements

I would first like to thank my PhD thesis advisor Prof. Karen Smith for
generously sharing her time and ideas, and closely guiding me throughout this
project. I would like to thank Josep Àlvarez Montaner and Luis Núñez-
Betancourt for communicating their results and kindly sharing their preprint
[AN21] with me. I also thank them for suggesting a useful modification to the
statement of Theorem 6.1. Finally I would like to thank Shelby Cox, Andrew
Gordon, Sayantan Khan and Yinan Nancy Wang for helpful conversations.

##  2\. Review of Multiplier Ideals and Intersection Theory

###  2.1. Multiplier Ideals

Let $X$ be a smooth variety over an algebraically closed field $k$ of
characteristic $0$. Say the dimension of $X$ is $d$. Fix a coherent ideal
sheaf $\mathfrak{a}$ of $\mathcal{O}_{X}$ and let $Z$ be the subscheme defined
by $\mathfrak{a}$. We will now recall the definition of the multiplier ideals
$\mathscr{I}(c\cdot Z)$ interchangeably denoted by
$\mathscr{I}(\mathfrak{a}^{c})$, referring the reader to [Laz04a] for details.

Let $\mu:Y\to X$ be a log resolution of the ideal $\mathfrak{a}$. This means
that $\mu$ is a proper, birational map, $Y$ is smooth and
$\mathfrak{a}\cdot\mathcal{O}_{Y}=\mathcal{O}_{Y}(-F)$ where $F$ is an
effective divisor and $F+K_{Y|X}$ has simple normal crossing support, where
$K_{Y|X}$ denotes the relative canonical divisor of $\mu$. In this paper, for
simplicity, we in addition require that log resolutions be projective maps.
Such log resolutions exist by Hironaka’s theorem on resolution of
singularities.

Then, for any $c>0$ the _multiplier ideal_ is defined as

$\mathscr{I}(\mathfrak{a}^{c})=\mu_{*}(\mathcal{O}_{Y}(K_{Y|X}-\lfloor
cF\rfloor))$

The multiplier ideal is independent of the choice of a log resolution $\mu$.
As $c$ varies over positive real numbers, the stalks of the multiplier ideal
$\mathscr{I}(\mathfrak{a}^{c})_{x}$ at any point $x$ decrease and change
exactly at a discrete set of rational numbers $c_{i}$ called the _jumping
numbers_ of the ideal $\mathfrak{a}$ at $x$.

The jumping numbers are important invariants of the singularity of $Z$ at $x$.
For example, the smallest of the jumping numbers is the well-known log
canonical threshold ($lct$) of the subscheme. When $Z$ is defined by a
polynomial $f$, the jumping numbers in the interval $(0,1]$ are all negative
roots of the _Bernstein-Sato polynomial_ of $f$, as proved in [Ein+04],
generalizing the result for the log canonical threshold in [Kol97]. This
result was further generalized to arbitrary subschemes in [BMS06], after
defining a suitable Bernstein-Sato polynomial. The jumping numbers were also
connected with uniform Artin-Rees numbers in [Ein+04].

We fix the notation from the previous paragraph throughout the paper for the
log resolution, the relative canonical divisor and the effective divisor $F$.

####  2.1.1. Candidate Jumping Numbers:

Let $K_{Y|X}=\sum_{i}k_{i}E_{i}$ and $F=\sum_{i}a_{i}D_{i}+\sum_{i}b_{i}E_{i}$
for irreducible divisors $D_{i}$ and exceptional irreducible divisors $E_{i}$.
Then, we call the numbers of the form $\frac{n}{b_{i}}$ and $\frac{n}{a_{i}}$
the _candidate jumping numbers_ of $\mathfrak{a}$. These are precisely the
values of $c$ where the divisor $K_{Y|X}-\lfloor cF\rfloor$ changes, hence the
set of jumping numbers is certainly contained in the set of candidate jumping
numbers. This is slightly different from the candidate jumping numbers as
defined in [ST07] which is the set of rational numbers where the divisor
$K_{Y|X}-\lfloor cF\rfloor$ changes and is not effective. The main reason for
this deviation is that now for every candidate jumping number $c$, its
fractional part $\\{c\\}$ is also a candidate jumping number. Hence, every
jumping number $c$ can be written as $\\{c\\}+\lfloor c\rfloor$ i.e. an
integer translate of a candidate jumping number that lies in the interval
$(0,1]$. The candidate jumping numbers make it clear that the set of jumping
numbers is rational and discrete. Moreover, there is a uniform $\varepsilon$
such that $c_{i+1}-c_{i}>\varepsilon$ for any two consecutive jumping numbers
$c_{i}$ and $c_{i+1}$.

We now recall two of the main results from the theory of multiplier ideals in
the form that we will use them. The reference for these results is Chapter 9
of [Laz04a].

######  Theorem 2.1.

(Local Vanishing Theorem)

Let $\mathfrak{a}\subset\mathcal{O}_{X}$ be an ideal, and $\mu:Y\to X$ be a
log resolution of $\mathfrak{a}$ and let
$\mathfrak{a}\cdot\mathcal{O}_{Y}=\mathcal{O}_{Y}(-F)$ for an effective
divisor $F$. Then for any $c>0$,

$R^{j}\mu_{*}\mathcal{O}_{Y}(K_{Y|X}-\lfloor
cF\rfloor)=0\quad\mathrm{for}\quad j>0$

######  Theorem 2.2.

(Skoda’s Theorem)

Let $\mathfrak{a}\subset\mathcal{O}_{X}$ be an ideal and $c>0$ be a rational
number and $m\geq d$ be an integer. Then we have,

$\mathscr{I}(\mathfrak{a}^{c+m})=\mathfrak{a}\cdot\mathscr{I}(\mathfrak{a}^{c+m-1})$

###  2.2. Numerical Intersection Theory

Now we review Kleiman’s numerical intersection theory of divisors as developed
in [Kle66]. We only state the main facts that we need here.

Let $Y$ be a proper scheme over $k$ of dimension $d$. Let
$\mathscr{L}_{1},\dots,\mathscr{L}_{r}$ be line bundles on $Y$, $D$ an
effective Cartier divisor and $\mathscr{F}$ a coherent sheaf on $Y$. Then we
have

######  Theorem 2.3.

([Kle66], Snapper) Consider the function
$f(m_{1},\dots,m_{r})=\chi(\mathscr{F}\otimes\mathscr{L}_{1}^{m_{1}}\otimes\dots\otimes\mathscr{L}_{r}^{m_{r}})$
for $m_{1},\dots,m_{r}\in\mathbb{Z}$, where $\chi$ denotes the Euler
characteristic. Then there is a polynomial $P(x_{1},\dots,x_{r})$ with
coefficients in $\mathbb{Q}$ and of total degree $\leq
s=\dim\mathrm{Supp}(\mathscr{F})$ such that
$P(m_{1},\dots,m_{r})=f(m_{1},\dots,m_{r})$ for all
$m_{1},\dots,m_{r}\in\mathbb{Z}$.

######  Definition 2.4.

([Kle66], Section 2, Definition 1) Suppose that
$\dim\mathrm{Supp}(\mathscr{F})\leq r$, then define the intersection number
$(\mathscr{F};\mathscr{L}_{1}\cdot\ldots\cdot\mathscr{L}_{r})$ to be the
coefficient of the term $x_{1}\dots x_{r}$ in the polynomial
$P(x_{1},\dots,x_{r})$ as in Theorem 2.3.

For any coherent sheaf $\mathscr{F}$ with
$\dim(\mathrm{Supp}(\mathscr{F}))\leq r$, this defines an integer valued
multi-linear form on Pic$(Y)^{r}$. If a line bundle $\mathscr{L}$ is defined
by a Cartier divisor $D$, then we sometimes write $D$ instead of $\mathscr{L}$
in the intersection form. We will use the following properties of the
intersection numbers:

######  Proposition 2.5.

  1. (1)

([Kle66], Section 2, Proposition 4) If $\mathscr{F}$ is a locally free sheaf,
then

$(\mathscr{F};\mathscr{L}_{1}\cdot\ldots\cdot\mathscr{L}_{d-1}\cdot\mathcal{O}_{Y}(D))=(\mathscr{F}|_{D};\mathscr{L}_{1}|_{D}\cdot\ldots\cdot\mathscr{L}_{d-1}|_{D})$

  2. (2)

([Kle66], Section 2, Proposition 3) If $\mathscr{F}^{\prime}$ and
$\mathscr{F}^{\prime\prime}$ are two other coherent sheaves and there is an
exact sequence

$0\to\mathscr{F}^{\prime}\to\mathscr{F}\to\mathscr{F}^{\prime\prime}\to 0$

then
$(\mathscr{F};\mathscr{L}_{1}\cdot\ldots\cdot\mathscr{L}_{d})=(\mathscr{F}^{\prime};\mathscr{L}_{1}\cdot\ldots\cdot\mathscr{L}_{d})+(\mathscr{F}^{\prime\prime};\mathscr{L}_{1}\cdot\ldots\cdot\mathscr{L}_{d})$

  3. (3)

If $\mathscr{L}$ is any line bundle on $Y$ and $P(x)$ is the polynomial such
that $P(n)=\chi(\mathscr{F}\otimes\mathscr{L}^{n})$ for $n\in\mathbb{Z}$
(which exists by the theorem above), then if $a$ is the coefficient of $x^{d}$
in $P(x)$, then $d!\times
a=(\mathscr{F};\mathscr{L}\cdot\ldots\cdot\mathscr{L})$ ($d$ copies of
$\mathscr{L}$).

  4. (4)

([Kle66], Section 2, Proposition 5, Corollary 1,2) Let
$V=\mathrm{Supp}(\mathscr{F})$ and $V_{1},\dots,V_{s}$ its irreducible
components. Let $l_{i}=\mathrm{length}(\mathscr{F}\otimes\mathcal{O}_{V_{i}})$
where $\mathcal{O}_{V_{i}}$ is the stalk of $\mathcal{O}_{Y}$ at the generic
point of $V_{i}$, then, if $r\geq\dim(V)$

$(\mathscr{F};\mathscr{L}_{1}\cdot\ldots\cdot\mathscr{L}_{r})=\sum_{i=1}^{s}l_{i}\times(\mathscr{L}_{1}|_{V_{i}}\cdot\ldots\cdot\mathscr{L}_{r}|_{V_{i}})$

In particular, if $\mathscr{F}$ is locally free, then
$(\mathscr{F};\mathscr{L}_{1}\cdot\ldots\cdot\mathscr{L}_{d})=(\mathscr{L}_{1}\cdot\ldots\cdot\mathscr{L}_{d})$.

  5. (5)

([Kle66], Section 2, Proposition 6) Let $\pi:Y^{\prime}\to Y$ be a map of
finite type of integral projective varieties of dimension $d$, then
$(\pi^{*}\mathscr{L}_{1}\cdot\ldots\cdot\pi^{*}\mathscr{L}_{d})_{V^{\prime}}=deg(\pi)\times(\mathscr{L}_{1}\cdot\ldots\cdot\mathscr{L}_{d})_{V}$

###### Proof.

(3): We note that if $Q(x_{1},\dots,x_{d})$ is the polynomial such that
$Q(m_{1},\dots,m_{d})=\chi(\mathscr{F}\otimes\mathscr{L}^{m_{1}}\otimes\ldots\otimes\mathscr{L}^{m_{d}})$,
then $Q(x_{1},\dots,x_{d})=P(x_{1}+\dots+x_{d})$. Since, $P(x)$ has a degree
at most $d$, the coefficient of $x_{1}\dots x_{d}$ in $P(x_{1}+\dots+x_{d})$
is $d!\times a$ which by definition is the required intersection number.

∎

######  Notation.

Since, all our computations of multiplier ideals will be local around a point,
we sometimes assume that $X$ is a smooth affine variety. For convenience, we
also assume to have fixed a projective compactification $\overline{X}$ of $X$.
Then, whenever we have a proper birational map $\mu:Y\to X$, we automatically
have a compactification $\overline{Y}$ of $Y$. Suppose $E_{i}\subset Y$ is a
collection of exceptional divisors such that $\mu(E_{i})$ is a point (in which
case $E_{i}$ is projective over $k$). If $\mathscr{F}$ is a locally free sheaf
on $Y$, then the intersection number $(\mathscr{F};E_{1}\cdot\ldots\cdot
E_{d})$ denotes $(\overline{\mathscr{F}};E_{1}\cdot\ldots\cdot E_{d})$ where
$\overline{\mathscr{F}}$ is any extension of $\mathscr{F}$ to a coherent sheaf
on $\overline{Y}$. This number is independent of the choice of
compactifications and the choice of extension of $\mathscr{F}$ since it is
equal to the intersection number
$(\mathscr{F}|_{F_{d}};\mathcal{O}_{F_{d}}(E_{1})\cdot\ldots\cdot\mathcal{O}_{F_{d}}(E_{d-1}))-(\mathscr{F}|_{G_{d}};\mathcal{O}_{G_{d}}(E_{1})\cdot\ldots\cdot\mathcal{O}_{G_{d}}(E_{d-1}))$
where $E_{d}=F_{d}-G_{d}$ for $F_{d},G_{d}$ complete (over $k$), effective
divisors on $Y$.

##  3\. The Polynomial of a Jumping Number

Let $X$ be a smooth variety of dimension $d$ over an algebraically closed
field $k$ of characteristic $0$. We fix $(A,\mathfrak{m})$ to be the local
ring of a closed point $x\in X$. Let $\mathfrak{a}$ be an
$\mathfrak{m}-$primary ideal, corresponding to some closed subscheme of $X$
supported at $x$. We recall the notation from section 2: $\mu:Y\to X$ is a log
resolution of the ideal $\mathfrak{a}$ so that
$\mathfrak{a}\cdot\mathcal{O}_{Y}=\mathcal{O}_{Y}(-F)$ where $F$ is an
effective divisor and $F+K_{Y|X}$ has simple normal crossing support, where
$K_{Y|X}$ denotes the relative canonical divisor of $\mu$. Abusing notation,
we write $\mathscr{I}(\mathfrak{a}^{c})$ for the stalk of the multiplier ideal
at $x$ and think of it as an ideal in $A$.

######  Definition 3.1.

[Ein+04] For any jumping number $c>0$ of $\mathfrak{a}$, the multiplicity of
$c$ is

$m(c):=\dim_{k}\mathscr{I}(\mathfrak{a}^{c-\varepsilon})/\mathscr{I}(\mathfrak{a}^{c})$
(3.1)

for $\varepsilon>0$ small enough.

The multiplicity is well-defined because $\mathscr{I}(\mathfrak{a}^{c})$ is
either $A$ or an $\mathfrak{m}-$primary ideal for all $c$ (since
$\mathscr{I}(\mathfrak{a}^{c})\supset\mathscr{I}(\mathfrak{a}^{\lceil
c\rceil})\supset\mathfrak{a}^{\lceil c\rceil}$). Of course, we can define
$m(c)$ for any positive real number $c$ by expression (3.1), but it will be
non-zero if and only if $c$ is a jumping number, by the definition of jumping
number.

Our focus in this section is on the function $m(c+n)$ where $c>0$ is any fixed
real number and $n$ varies over the natural numbers $\mathbb{Z}_{\geq 0}$. So,
we define the function

$f_{c}(n):=m(c+n)$

The main theorem about $f_{c}(n)$ is the following:

######  Theorem 3.2.

For each $c>0$, the function $f_{c}(n)$ is a polynomial function of $n$ of
degree less than the dimension $d$ of $X$.

Before we prove the theorem, it will be convenient to introduce some notation
and the notion of a _jumping divisor_.

######  Definition 3.3.

Let $\mathfrak{a}\cdot\mathcal{O}_{Y}=\mathcal{O}_{Y}(-F)$, where $\mu:Y\to X$
is a log resolution. For $\varepsilon>0$ small enough, the divisor $\lfloor
cF\rfloor-\lfloor(c-\varepsilon)F\rfloor$ is reduced. Further, for even
smaller $\varepsilon$, there are no candidate jumping numbers between
$c-\varepsilon$ and $c$. So, in this case $\lfloor
cF\rfloor-\lfloor(c-\varepsilon)F\rfloor$ is a unique reduced effective
divisor (i.e. it doesn’t change as $\varepsilon\to 0$). So we set

$E^{c}=\lfloor cF\rfloor-\lfloor(c-\varepsilon)F\rfloor,$

where $\varepsilon$ is a sufficiently small positive number. We call $E^{c}$
the _jumping divisor_ of $c$. If $c$ is not a candidate jumping number of
$\mathfrak{a}$, then $E^{c}$ is zero and its support is empty.

The notion of a jumping divisor was introduced as the _maximal jumping
divisor_ in [AAD16] along with its _minimal_ variant in the two dimensional
case. For simplicity, we drop the adjective ‘maximal’ from the name and do not
discuss the minimal jumping divisor here.

######  Remark 3.4.

Let $\mu:Y\to X$ be any log resolution of $\mathfrak{a}$ (an
$\mathfrak{m}$-primary ideal) as above. Then
$\mu_{*}(\mathcal{O}_{Y}(-F))=\overline{\mathfrak{a}}$, the integral closure
of $\mathfrak{a}$. Hence, if $E\subset Y$ is an irreducible component of $F$
then $\mu(E)=x$. Hence, each component of $F$ (and of $E^{c}$) is a projective
variety over $k$ (since we are already assuming our log resolutions are
projective maps) and is contracted by $\mu$ to a point. Moreover, all
exceptional divisors for $\mu$ that are contracted to $x$ occur as irreducible
components of $F$. These will be the only kind of exceptional divisors we will
need to consider throughout this paper.

We can now prove Theorem 3.2.

##### Proof of Theorem 3.2

Since the multiplicities $m(c)$ depend only on the stalk of the multiplier
ideal at $x$, we may shrink $X$ to an open neighbourhood of $x$ and assume
that $X$ is affine. We now claim that for any $c>0$, the multiplicity $m(c)$
can be calculated as an Euler-Characteristic on a log resolution (with
notation as fixed in the beginning here):

$m(c):=\dim_{k}\mathscr{I}(\mathfrak{a}^{c-\varepsilon})/\mathscr{I}(\mathfrak{a}^{c})=\chi_{E^{c}}(\mathscr{L}|_{E^{c}})=\dim_{k}(\Gamma(E^{c},\mathscr{L}|_{E^{c}}))$
(3.2)

where $\mathscr{L}$ is the line bundle on $Y$ corresponding to the divisor
$K_{Y|X}+E^{c}-\lfloor cF\rfloor=K_{Y|X}-\lfloor(c-\varepsilon)F\rfloor$.

To see this, we start with the exact sequence

${0}$${\mathcal{O}_{Y}(-E^{c})}$${\mathcal{O}_{Y}}$${\mathcal{O}_{E^{c}}}$${0}$

Tensoring with the invertible sheaf associated to $\mathscr{L}$, we get

${0}$${\mathcal{O}_{Y}(\mathscr{L})\otimes\mathcal{O}_{Y}(-E^{c})}$${\mathcal{O}_{Y}(\mathscr{L})}$${\mathcal{O}_{E^{c}}(\mathscr{L}|_{E^{c}})}$${0}$
(3.3)

Now, observing that
$\mathcal{O}_{Y}(\mathscr{L})\otimes\mathcal{O}_{Y}(-E^{c})\cong\mathcal{O}_{Y}(K_{Y|X}-\lfloor
cF\rfloor)$, the Local Vanishing Theorem 2.1 tells us that the sequence
remains exact after applying $\mu_{*}$. Thus we have the exact sequence

${0}$${\mathscr{I}(\mathfrak{a}^{c})}$${\mathscr{I}(\mathfrak{a}^{c-\varepsilon})}$${\Gamma(E^{c},\mathscr{L}|_{E^{c}})}$${0}$

where the first map is just the inclusion of the ideal
$\mathscr{I}(\mathfrak{a}^{c})$ inside
$\mathscr{I}(\mathfrak{a}^{c-\varepsilon})$. Hence, the multiplicity
$m(c)=\dim_{k}(\Gamma(E^{c},\mathscr{L}|_{E^{c}}))$.

Since $X$ is affine, applying the Local Vanishing Theorem 2.1 again, we have
that the first two sheaves in the short exact sequence (3.3) have vanishing
higher cohomology. Using the long exact sequence for (3.3) of sheaf cohomology
groups, we get:

$H^{p}(E^{c},\mathscr{L}|_{E^{c}})=0\quad\text{for all $p>0$.}$

Hence, the Euler-Characteristic of $\mathscr{L}|_{E^{c}}$ is equal to
$\dim_{k}(\Gamma(E^{c},\mathscr{L}|_{E^{c}}))=m(c)$, proving our claim.

By the observation (3.2) above, we have

$f_{c}(n)=\chi\Big{(}\mathcal{O}_{Y}\big{(}K_{Y|X}-\lfloor(c-\varepsilon)F\rfloor\big{)}|_{E^{c}}\otimes\mathcal{O}_{Y}\big{(}-nF\big{)}|_{E^{c}}\Big{)}$
(3.4)

Since we are fixing $c$ and varying $n$, we may apply Snapper’s Theorem 2.3 on
the complete scheme $E^{c}$ over $k$ with $r=1$,
$\mathscr{F}=\mathcal{O}_{Y}\big{(}K_{Y|X}-\lfloor(c-\varepsilon)F\rfloor\big{)}|_{E^{c}}$
and $\mathscr{L}_{1}=\mathcal{O}_{Y}\big{(}-F\big{)}|_{E^{c}}$, there is a
polynomial $Q_{c}(x)$ of degree at most the dimension of $E^{c}$ (= $d-1$)
such that
$Q_{c}(n)=\chi\Big{(}\mathcal{O}_{Y}\big{(}K_{Y|X}-\lfloor(c-\varepsilon)F\rfloor\big{)}|_{E^{c}}\otimes\mathcal{O}_{Y}\big{(}-nF\big{)}|_{E^{c}}\Big{)}$
for all $n\geq 0$. ∎

We can also explicitly determine the coefficient of $n^{d-1}$ in the
polynomial $m(c+n)$:

######  Theorem 3.5.

Fix $c>0$. With notation as before, the coefficient $\rho_{c}$, of the term
$n^{d-1}$ in the polynomial $f_{c}(n)$ describing the multiplicities $m(c+n)$
can be computed on the log resolution $Y$ using the formula:

$\rho_{c}=(-1)^{d-1}\frac{(F\cdot\ldots\cdot F\cdot E^{c})}{(d-1)!}.$ (3.5)

###### Proof.

Let $\mathscr{F}_{c}=\mathcal{O}_{Y}(K_{Y|X}-\lfloor(c-\varepsilon)F\rfloor)$
and $\mathscr{L}=\mathcal{O}_{Y}(-F)$. Then
$f_{c}(n)=\chi(\mathscr{F}\otimes\mathscr{L}^{n})$ by (3.4). Using part (3) of
Proposition 2.5, we have,
$\rho_{c}=\frac{(\mathscr{F}_{c}|_{E^{c}};\mathscr{L}|_{E^{c}}\cdot\ldots\cdot\mathscr{L}|_{E^{c}})}{(d-1)!}$.
By part (1) of Proposition 2.5, this can be computed on any projective closure
of $Y$ as:

$\frac{(\mathscr{F}_{c}|_{E^{c}};\mathscr{L}|_{E^{c}}\cdot\ldots\cdot\mathscr{L}|_{E^{c}})}{(d-1)!}=\frac{(\mathscr{F}_{c};\mathscr{L}\cdot\ldots\cdot\mathscr{L}\cdot
E^{c})}{(d-1)!}=(-1)^{d-1}\frac{(\mathscr{F}_{c};F\cdot\ldots\cdot F\cdot
E^{c})}{(d-1)!}$

Since $\mathscr{F}_{c}$ is a line bundle on $Y$, using part (4) of Proposition
2.5, we have:

$\rho_{c}=(-1)^{d-1}\frac{(F\cdot\ldots\cdot F\cdot E^{c})}{(d-1)!}$

∎

######  Definition 3.6.

Given an $\mathfrak{m}-$primary ideal $\mathfrak{a}\subset\mathcal{O}_{X,x}$,
we define the _Rees Coefficient_ of $c>0$ (denoted by $\rho_{c}$) to be the
coefficient of the term $n^{d-1}$ in the polynomial $f_{c}(n)$. Equivalently,
in light of Theorem 3.2, $\rho_{c}$ is the unique number such that

$m(c+n)=\rho_{c}n^{d-1}+o(n^{d-2})\quad\text{as $n\to\infty$}$

The Rees Coefficient of a jumping number will be studied in more detail in the
next section.

######  Remark 3.7.

When $d=2$, the polynomial $f_{c}(n)$ has degree at most 1. In this case, we
have,

$m(c+n)=m(c)+\rho_{c}n,$

where $\rho_{c}=-F\cdot E^{c}$. This recovers the _linear_ formula for
$m(c+n)$ proved in [AADG17].

In higher dimensions, the polynomials have other coefficients that can be
computed as follows:
If $f_{c}(n)=m(c)+\alpha_{1}^{c}n+\dots+\alpha_{d-1}^{c}n^{d-1}$, then

$\alpha_{j}^{c}=\frac{1}{j!}\int\limits_{E^{c}}c_{1}(\mathscr{L}|_{E^{c}})^{j}\cap\tau_{E^{c},j}(\mathscr{F}_{c}|_{E^{c}})$
(3.6)

where $\mathscr{L}$ and $\mathscr{F}_{c}$ are as above, $c_{1}$ denotes the
first Chern class of the line bundle and $\tau_{E^{c},j}$ denotes the degree
$j$ component of the Todd class of the sheaf. This formula comes from equation
3.4 and the Riemann-Roch Theorem for singular varieties. See [Ful98], Example
18.3.6 for the details.

The interpretation of the multiplicity $m(c)$ as the dimension of global
sections as in (3.2) also implies that the function under consideration,
$f_{c}(n)$ is a non-decreasing function which we now note:

######  Proposition 3.8.

Fix any $c>0$, then $m(c+1)\geq m(c)$.

###### Proof.

As before, we assume $X$ is affine. It is sufficient to deal with the case
when $c$ is a jumping number since otherwise $m(c)=0$ and $m(c+1)\geq 0$ by
definition. So we assume that $c$ is a jumping number, hence $m(c)>0$. By
(3.2), we need to show that

$\dim_{k}\Gamma\Big{(}E^{c},\mathcal{O}_{Y}(K_{Y|X}-\lfloor(c+1-\varepsilon)F\rfloor)|_{E^{c}}\Big{)}\geq\dim_{k}\Gamma\Big{(}E^{c},\mathcal{O}_{Y}(K_{Y|X}-\lfloor(c-\varepsilon)F\rfloor)|_{E^{c}}\Big{)}$

Denoting by $\mathscr{F}$ and $\mathscr{L}$ the invertible sheaves
$\mathcal{O}_{Y}(K_{Y|X}-\lfloor(c-\varepsilon)F\rfloor)$ and
$\mathcal{O}_{Y}(-F)$ on $Y$ respectively, we need to prove:

$\dim_{k}\Gamma\Big{(}E^{c},\mathscr{F}|_{E^{c}}\otimes\mathscr{L}|_{E^{c}}\Big{)}\geq\dim_{k}\Gamma\Big{(}E^{c},\mathscr{F}|_{E^{c}}\Big{)}$

But, since $\mathfrak{a}\cdot\mathcal{O}_{Y}=\mathcal{O}_{Y}(-F)$,
$\mathscr{L}$ is generated by its global sections (by the generators of
$\mathfrak{a}$) on $Y$, the same is true for $\mathscr{L}|_{E^{c}}$ on
$E^{c}$. In particular, $\mathscr{L}|_{E^{c}}$ has a non-zero global section.
Hence, choosing a non-zero global section of $\mathscr{L}|_{E^{c}}$, we have
an injective map:

$\mathcal{O}_{E^{c}}\hookrightarrow\mathcal{O}_{E^{c}}(\mathscr{L})$

Tensoring with $\mathscr{F}|_{E^{c}}$ preserves injectivity since
$\mathscr{F}$ was invertible. Hence, we have

$\mathcal{O}_{E^{c}}(\mathscr{F})\hookrightarrow\mathcal{O}_{E^{c}}(\mathscr{L}\otimes\mathscr{F})$

Since taking global sections also preserves injectivity, our claim is proved.
∎

######  Remark 3.9.

This proposition implies that if $c$ is a jumping number of $\mathfrak{a}$
then so is $c+1$. This also follows directly from the definition of multiplier
ideals and jumping numbers. Though this proposition says that if we fix a
‘$c$’ then $m(c+n)$ is non-decreasing with $n$, this does not imply that the
multiplicity $m(c)$ increases with $c$ (i.e. as $c$ varies over the jumping
numbers). In fact, this is not true. As we’ll see in the next section,
$m(c+n)$ increases at different rates for different jumping numbers $c$.

######  Remark 3.10.

Theorem 3.2 can be thought of as a finiteness statement for multiplier ideals
(of $\mathfrak{m}-$primary ideals) analogous to Skoda’s Theorem 2.2 and coming
from the Local Vanishing Theorem 2.1. A simple illustration of this finiteness
idea is the result that the jumping numbers of $\mathfrak{a}$ are periodic
after $d-1$ where $d$ is the dimension of $X$. This result is a well-known
consequence of Skoda’s Theorem. But, we’ll give a different proof of this
using Theorem 3.2 and the previous Proposition 3.8:

######  Proposition 3.11.

If $c>d-1$, then $c$ is a jumping number of $\mathfrak{a}$ if and only if
$c+1$ is a jumping number of $\mathfrak{a}$.

###### Proof.

By Proposition 3.8, if $c$ is a jumping number, then $c+1$ is also a jumping
number (without any assumptions on $c$). So its enough to prove that if $c+1$
is a jumping number of $\mathfrak{a}$ then so is $c$. But if not, then again
by Proposition 3.8, $m(c-n)=0$ for all $n$ such that $0\leq n\leq d-1$. But,
$m(c+n)$ is a polynomial of degree at most $d-1$. So, the $d$ zeroes of
$m(c+n)$ implies that $m(c+n)$ is identically zero. This means that $c+1$ can
not be a jumping number either. ∎

##  4\. Properties of the Rees coefficient

Fix a smooth variety $X$ of dimension $d$ and an $\mathfrak{m}-$primary ideal
$\mathfrak{a}$ in the local ring $(A,\mathfrak{m})$ of a closed point $x$.
Fixing a candidate jumping number $c$ for $\mathfrak{a}$ (see section 2.1 for
the definition), Theorem 3.2 says that the function $m(c+n)$ describing the
multiplicity of the jumping number $c+n$ is a polynomial of degree at most
$d-1$. We now investigate properties of the Rees coefficient $\rho_{c}$ of
$c$–that is, the coefficient of the term $n^{d-1}$ in $m(c+n)$. As proved in
Theorem 3.5, $\rho_{c}$ can also be computed as:

$\rho_{c}=(-1)^{d-1}\frac{(F\cdot\ldots\cdot F\cdot E^{c})}{(d-1)!}$

on a log resolution $\mu:Y\to X$ of $\mathfrak{a}$ where
$\mathfrak{a}\cdot\mathcal{O}_{Y}=\mathcal{O}_{Y}(-F)$ and $E^{c}=\lfloor
cF\rfloor-\lfloor(c-\varepsilon)F\rfloor$. The following observations about
$\rho_{c}$ are straight forward given that $m(c+n)$ is an integer polynomial
(in $n$) of degree at most $d-1$ and $\rho_{c}$ is the coefficient of
$n^{d-1}$ :

  1. (1)

$\rho_{c}\geq 0$ for all $c>0$

  2. (2)

$\rho_{c}\in\frac{1}{(d-1)!}\mathbb{N}_{\geq 0}$

  3. (3)

$\rho_{c}=0$ if $c$ is not a candidate jumping number of $\mathfrak{a}$

  4. (4)

$\rho_{c}=\rho_{c+n}$ for any positive integer $n$

  5. (5)

$\rho_{c}>0\implies c+n$ is a jumping number for some $n$.

Since $\rho_{c}$ is the as same $\rho_{c+n}$ for any natural number $n$, we
think of $\rho_{c}$ as a $\mathbb{Q}$-valued function on
$\mathbb{Q}/\mathbb{Z}$. This function is an interesting invariant of the
ideal $\mathfrak{a}$ and describes interesting properties of jumping numbers
(or rather their classes in $\mathbb{Q}/\mathbb{Z}$) of $\mathfrak{a}$. We
turn to explaining these properties.

###  4.1. Criteria for Positivity

We first discuss the issue of when the Rees coefficient $\rho_{c}$ is
positive. More precisely, we ask for which classes in $\mathbb{Q}/\mathbb{Z}$
of candidate jumping numbers $c$ is the Rees coefficient $\rho_{c}$ positive?
Equivalently, for which jumping numbers $c$, does the multiplicity $m(c+n)$
grow like $n^{d-1}$ as $n\to\infty$? In particular, we ask if this holds for
the classes of all jumping numbers? The answer to this question in general is
no. In this subsection, we prove several criteria for when the Rees
coefficient is positive. The following Theorem 4.1 is a summary of the results
of this section:

######  Theorem 4.1.

Let $\mathfrak{a}\subset\mathcal{O}_{X}$ be an ideal of a subscheme supported
at a closed point $x$ and fix $c>0$. Then the following are equivalent:

  1. (i)

$\rho_{c}>0$.

  2. (ii)

The polynomial $m(c+n)$ has degree $d-1$ in $n$.

  3. (iii)

The jumping divisor $E^{c}$ contains a _Rees valuation_ on some (equivalently
any) log-resolution of $\mathfrak{a}$.

  4. (iv)

$c+d-1$ is a jumping number _contributed by a Rees valuation_.

  5. (v)

The class of $c$ in $\mathbb{Q}/\mathbb{Z}$ is _contributed by a Rees
valuation_ (where $d=\dim X$).

We first explain the statement of the theorem and the various terms appearing
in it before giving the proof.

##### Jumping numbers contributed by divisors

The notion of a jumping number _contributed by a divisor_ on a log resolution
was defined in [ST07] by Smith and Thompson. This notion was studied further
in [Tuc10] and [Bau+18]. This definition naturally generalizes to classes of
jumping numbers in $\mathbb{Q}/\mathbb{Z}$. In this terminology, Theorem 4.1
characterizes the classes _contributed by_ the Rees valuations on any log-
resolution as exactly the classes $[c]$ such that the Rees coefficient
$\rho_{c}$ is positive (hence the choice of name). We now review the
definition of contribution by a divisor here. Recall that $E^{c}$ is the
jumping divisor defined as $E^{c}=\lfloor
cF\rfloor-\lfloor(c-\varepsilon)F\rfloor$.

######  Definition 4.2.

Let $\mu:Y\to X$ be a log resolution of an ideal
$\mathfrak{a}\subset\mathcal{O}_{X}$ and $E\subset Y$ be a prime exceptional
divisor. Then a jumping number $c$ of $\mathfrak{a}$ is said to be
_contributed by $E$_ if $E\subset E^{c}$ and

$\mathscr{I}(\mathfrak{a}^{c})=\mu_{*}(\mathcal{O}_{Y}(K_{Y|X}-\lfloor
cF\rfloor))\subsetneqq\mu_{*}(\mathcal{O}_{Y}(K_{Y|X}-\lfloor cF\rfloor+E))$

We say that a class $[c]\in\mathbb{Q}/\mathbb{Z}$ is _contributed by $E$_ if
$c+n$ is contributed by $E$ for some natural number $n$.

The condition that $E\subset E^{c}$ is just saying that $c$ is a candidate
jumping number for $E$. We also note that contribution by a prime divisor
depends only on the valuation defined by $E$ and not on the chosen log
resolution. So we say that a jumping number is “contributed by a valuation
$\nu$” if it is contributed by a divisor on a log resolution whose associated
valuation is $\nu$.

##### Rees valuations:

Let $\mathfrak{a}\subset\mathcal{O}_{X}$ be an ideal. If $\nu:\tilde{X}\to X$
denotes the normalization of the blow-up of $X$ along $\mathfrak{a}$, then
$\mathfrak{a}\cdot\mathcal{O}_{\tilde{X}}=\mathcal{O}_{\tilde{X}}(-E)$, for an
effective exceptional divisor $E$ on $\tilde{X}$. Then the valuations
corresponding to the prime components of $E$ are called the _Rees valuations_
of $\mathfrak{a}$.

Let $(A,\mathfrak{m})$ be a local ring and $\mathfrak{a}$ be an
$\mathfrak{m}-$primary ideal. The set of Rees valuations of $\mathfrak{a}$ is
a minimal set of discrete, rank one valuations $\nu_{1},\dots,\nu_{r}$ of $R$
satisfying the property that:

For all $n\in\mathbb{N}$, the integral closure
$\overline{\mathfrak{a}^{n}}=\bigcap_{i}\big{(}\mathfrak{a}^{n}V_{i}\cap
A\big{)}$ where $V_{i}$ is the valuation ring of $\nu_{i}$

In other words, the Rees valuations are a minimal set of valuations that
determine the integral closure of all the powers of the ideal $\mathfrak{a}$.
We refer to Chapter 10 of [HS06] for details about Rees valuations. This is
also explained in Section 9.6 of [Laz04a].

Given the normalized blowup $\tilde{X}$, or more generally any normal variety
$Y$ mapping properly and birationally to $X$, we abuse terminology by saying
“$E$ is Rees valuation of $\mathfrak{a}$” to refer to a prime exceptional
divisor $E$ on $Y$ corresponding to a Rees valuation of $\mathfrak{a}$.

We now turn to the proof of Theorem 4.1.

##### Proof of Theorem 4.1:

We first note the following easy implications in the theorem:

The equivalence (i) $\iff$ (ii) is clear from the definition 3.6 of the Rees
coefficient. The implication (iv) $\implies$ (iii) follows from the definition
4.2 of jumping numbers contributed by divisors (so does (v) $\implies$ (iii)).
(iv) $\implies$ (v) follows immediately from the definition 4.2 of classes of
jumping numbers contributed by a divisor. So to complete the proof we will
prove that (i) $\iff$ (iii) and (iii) $\implies$ (iv). These will be proved as
the two main theorems of this subsection.

We first prove (i) $\iff$ (iii) of Theorem 4.1:

######  Theorem 4.3.

Fix $c>0$. The Rees coefficient $\rho_{c}$ is positive if and only if on some
(equivalently every) log resolution, the jumping divisor $E^{c}$ contains a
divisor corresponding to a Rees valuation.

Let $\mu:Y\to X$ be a log resolution of the $\mathfrak{m}$-primary ideal
$\mathfrak{a}$. Let $\mathfrak{a}\cdot\mathcal{O}_{Y}=\mathcal{O}_{Y}(-F)$.
When $X$ is a surface, it follows from the results of Lipman, from the 1969
paper [Lip69] that, for any irreducible exceptional curve $E\subset Y$,
$(-F\cdot E)$ is non-negative and is positive exactly when $E$ corresponds to
a Rees valuation of $\mathfrak{a}$. It follows from the formula 3.5 that the
Rees coefficient $\rho_{c}$ is positive if and only if a Rees valuation
appears in the jumping divisor $E^{c}$. Theorem 4.3 can be thought of as a
generalization of Lipman’s result in this context to higher dimensions. To
prove this theorem, we need a key lemma which is a generalization to higher
dimensions of a version of Lemma 21.2 from [Lip69].

######  Lemma 4.4.

If $E\subset Y$ is an irreducible component of $F$, then the intersection
number

$(\underbrace{-F\cdot\ldots\cdot-F}_{d-1\text{ times}}\cdot E)\geq 0,$

where $d$ is the dimension of $X$. Moreover, $(-F\cdot\ldots\cdot-F\cdot E)$
is positive if and only if the divisor $E$ is a Rees valuation of
$\mathfrak{a}$.

###### Proof.

Since the relevant intersection numbers depend only on a neighbourhood of $x$
in $X$, we assume $X$ is affine. For convenience, let $\mathscr{L}$ denote the
invertible sheaf $\mathcal{O}_{Y}(-F)$. Then since
$\mathfrak{a}\cdot\mathcal{O}_{Y}\cong\mathscr{L}$, it is generated on $Y$ by
its global sections (by the pull-back of the generators of $\mathfrak{a}$).
Hence, so is the restriction of $\mathscr{L}$ to $E$. Hence,
$\mathscr{L}|_{E}$ is a nef line bundle on $E$ (since further restriction to
any curve is also globally generated). Since $E$ is a projective variety of
dimension $d-1$ over $k$, it follows from a theorem of Kleiman (See Theorem
1.4.9 in [Laz04]), that we have

$\underbrace{\mathscr{L}|_{E}\cdot\ldots\cdot\mathscr{L}|_{E}}_{d-1\text{
times}}\geq 0$

This proves the first claim.

Now, let $\tilde{\mu}:\tilde{X}\to X$ be the normalized blow-up of
$\mathfrak{a}$. Since $Y$ is normal (it is smooth) and
$\mathfrak{a}\cdot\mathcal{O}_{Y}$ is locally principal, the universal
property of normalization and of blowing-up provides a factorization

$\mu:Y\xrightarrow{\pi}\tilde{X}\xrightarrow{\tilde{\mu}}X$

i.e. $\mu=\tilde{\mu}\circ\pi$. Let $\tilde{\mathscr{L}}$ be the invertible
sheaf $\mathfrak{a}\cdot\mathcal{O}_{\tilde{X}}$ on $\tilde{X}.$ Then, since
$\tilde{\mathscr{L}}$ is relatively ample for $\tilde{\mu}$, if
$E\subset\tilde{X}$ is any exceptional divisor, then
$\tilde{\mathscr{L}}|_{E}$ is ample. We also have that
$\mathscr{L}=\pi^{*}(\tilde{\mathscr{L}})$.

Let $B=\\{G_{j}\\}$ denote the finite set of prime exceptional divisors on
$\tilde{X}$ and $C=\\{E_{i}\\}$ denote the finite set of irreducible
components of $F$ on $Y$. Then $\pi^{*}(G_{j})=\sum_{j}\lambda_{ij}E_{i}$ for
some $\lambda_{ij}\geq 0$ and every $E\in C$ appears with a positive
coefficient for some $j$—that is, for each $i$, $\lambda_{ij}>0$ for some $j$.
Now, Proposition 2.5 (5), applied on the compactifications of $Y$ and
$\tilde{X}$, implies:

$(\tilde{\mathscr{L}}\cdot\ldots\cdot\tilde{\mathscr{L}}\cdot
G_{j})=(\mathscr{L}\cdot\ldots\cdot\mathscr{L}\cdot\pi^{*}{G_{j}})=\sum_{i}\lambda_{ij}(\mathscr{L}\cdot\ldots\cdot\mathscr{L}\cdot
E_{i})$ (4.1)

But, from Proposition 2.5 (1), we have
$(\mathscr{L}\cdot\ldots\cdot\mathscr{L}\cdot
E_{i})=(\mathscr{L}|_{E_{i}}\cdot\ldots\cdot\mathscr{L}|_{E_{i}})$ and
$(\tilde{\mathscr{L}}\cdot\ldots\cdot\tilde{\mathscr{L}}\cdot
G_{j})=(\tilde{\mathscr{L}}|_{G_{j}}\cdot\ldots\cdot\tilde{\mathscr{L}}|_{G_{j}})$.
Since each $E_{i}$ is irreducible, we have that $\lambda_{ij}>0\iff
E_{i}\subset\pi^{-1}(G_{j})$ and in that case, $\pi$ restricts to a map from
$E_{i}$ to $G_{j}$:

${E_{i}}$${G_{j}}$${Y}$${\tilde{X}}$$\scriptstyle{\pi|_{E_{i}}}$$\scriptstyle{\pi}$

So, $\mathscr{L}|_{E_{i}}\cong\pi|_{E_{i}}^{*}\tilde{\mathscr{L}}|_{G_{j}}$.

Consider the restriction $\pi|_{E_{i}}:E_{i}\to G_{j}$. Since $\tilde{X}$ is
normal, this restriction map is birational exactly when $E_{i}$ is the strict
transform of $G_{j}$, that is, when $E_{i}$ is a Rees valuation of
$\mathfrak{a}$. In this case, the degree of $\pi|_{E_{i}}$ is one. Otherwise,
the $\pi(E_{i})\subsetneqq G_{j}$ has dimension strictly less than the
dimension of $G_{j}$ and hence degree of $\pi|_{E_{i}}$ is zero. Using
Proposition 2.5(5) again, we have:

$(\mathscr{L}\cdot\ldots\cdot\mathscr{L}\cdot
E_{i})=(\mathscr{L}|_{E_{i}}\cdot\ldots\cdot\mathscr{L}|_{E_{i}})=\begin{cases}(\tilde{\mathscr{L}}|_{G_{j}}\cdot\ldots\cdot\tilde{\mathscr{L}}|_{G_{j}})&\quad\text{if
}E_{i}\text{ is a Rees valuation}\\\ 0&\quad\text{ otherwise}\end{cases}$

Since $\tilde{\mathscr{L}}|_{G_{j}}$ is ample,
$(\tilde{\mathscr{L}}|_{G_{j}}\cdot\ldots\cdot\tilde{\mathscr{L}}|_{G_{j}})>0$.
This completes the proof that: $(-F\cdot\ldots\cdot-F\cdot E)$ is positive
exactly when $E$ corresponds to a Rees valuation of $\mathfrak{a}$. ∎

##### Proof of Theorem 4.3:

The theorem now follows from the Lemma and Theorem 3.5:

$\rho_{c}=\frac{(-F\cdot\ldots\cdot-F\cdot E^{c})}{(d-1)!}=\sum_{E\subset
E^{c}}\frac{(-F\cdot\ldots\cdot-F\cdot E)}{(d-1)!}$

and the characterization in the above lemma of when the intersection numbers
on the right-hand side is positive. ∎

Finally, we conclude the proof of Theorem 4.1, by proving (iii) $\implies$
(iv).

######  Theorem 4.5.

Let $\mu:Y\to X$ be a log-resolution of $\mathfrak{a}$. For any $c>0$, suppose
the jumping divisor $E^{c}$ contains a Rees valuation $E$, then $c+d-1$ is a
jumping number contributed by $E$ (where $d=\dim X$).

##### Proof of Theorem 4.5:

Since the subscheme corresponding to $\mathfrak{a}$ is supported at the point
$x$, we may replace $X$ by any affine open containing $x$ and assume $X$ is
affine. Further, since by assumption $E^{c}$ is non-zero, $c$ must be a
candidate jumping number (defined in section 2.1). Let $q>0$ be a candidate
jumping number and $G$ be a prime exceptional divisor with $G\subset E^{q}$.
Then $q$ is a jumping number contributed by $G$ if and only if
$H^{0}(G,\mathcal{O}_{G}(K_{Y|X}-\lfloor qF\rfloor+G)|_{G})\neq 0$. This
follows from the following exact sequences:

${0}$${\mathcal{O}_{Y}(-G)}$${\mathcal{O}_{Y}}$${\mathcal{O}_{G}}$${0}$

Tensoring with $\mathcal{O}_{Y}(K_{Y|X}-\lfloor qF\rfloor+G)$, we get

$0\to\mathcal{O}_{Y}(K_{Y|X}-\lfloor
qF\rfloor)\to\mathcal{O}_{Y}(K_{Y|X}-\lfloor
qF\rfloor+G)\to\mathcal{O}_{G}((K_{Y|X}-\lfloor qF\rfloor+G)|_{G})\to 0$

Applying $\mu_{*}$, the Local Vanishing Theorem 2.1 and since $X$ is affine by
assumption, we see that

$\mu_{*}(\mathcal{O}_{Y}(K_{Y|X}-\lfloor
qF\rfloor+G))/\mu_{*}(\mathcal{O}_{Y}(K_{Y|X}-\lfloor qF\rfloor))\cong
H^{0}(G,\mathcal{O}_{G}(K_{Y|X}-\lfloor qF\rfloor+G)|_{G})$

Further, since $G$ is a smooth divisor, we have

$\mathcal{O}_{G}(K_{Y|X}-\lfloor
qF\rfloor+G)|_{G}\cong\mathcal{O}_{G}(K_{G}-\lfloor qF\rfloor|_{G})$

where we have used the adjunction formula for $G\subset Y$ and the fact that
$G$ maps to a point on $X$ (since $G\subset E^{q}$), hence the canonical
bundle $\omega_{X}$ pulls back to the trivial bundle on $G$. So, in fact, we
have that

$\text{$q$ is a jumping number contributed by $G$ if and only if
}H^{0}(G,\mathcal{O}_{G}(K_{G}-\lfloor qF\rfloor|_{G}))\neq 0$ (4.2)

Hence, the statement that $q+n$ is a jumping number contributed by $G$ for a
natural number $n$, is equivalent to the statement that

the line bundle $\mathcal{O}_{G}(K_{G}-(\lfloor
qF\rfloor)|_{G})\otimes\mathcal{O}_{G}(-nF|_{G})$ has a non-zero section.
(4.3)

So, to prove the theorem, we need to use that the divisor $E\subset E^{c}$ is
a Rees valuation to conclude that the sheaf $\mathcal{O}_{E}(K_{E}-(\lfloor
cF\rfloor)|_{E})\otimes\mathcal{O}_{E}(-(d-1)F|_{E})$ has a non-zero global
section. The main idea of the proof is to push this sheaf forward to the
normalized blow-up of $\mathfrak{a}$ and use vanishing theorems to bound the
Castelnuovo-Mumford regularity of the push-forward. The technique is very
similar to the one used to bound the regularity of the Grauert-Riemenschneider
sheaf. For this, we need the following version of the Kawamata-Viehweg
vanishing theorem:

######  Theorem ([Laz04a], Theorem 9.1.18).

Let $X$ be a smooth projective variety of dimension $n$ and let $N$ be an
integral divisor on $X$. Assume that

$N\equiv_{num}B+\Delta$

where $B$ is a big and nef $\mathbb{Q}-$divisor and $\Delta=\sum
a_{i}\Delta_{i}$ is a $\mathbb{Q}-$divisor with simple normal crossings
support and such that $0\leq a_{i}<1$ for each $i$. Then,

$H^{i}(X,\mathcal{O}_{X}(K_{X}+N))=0\quad\text{for all }i>0.$

Continuing with the proof of Theorem 4.5, like in the proof of Theorem 4.4, we
consider the normalized blow-up $\tilde{X}$ of the ideal $\mathfrak{a}$ and by
the universal property of blowing up and normalization, the map $\mu$ factors
through $\tilde{X}$. The Rees valuation $E$ on $Y$ is the strict transform of
an exceptional divisor $\tilde{E}$ on $\tilde{X}$ and $E$ maps birationally
onto $\tilde{E}$ (since $\tilde{X}$ is normal and $\tilde{E}$ is a divisor).
This is summarized in the picture below:

${E}$${\tilde{E}}$${\\{x\\}}$${Y}$${\tilde{X}}$${X}$$\scriptstyle{f|_{E}}$$\scriptstyle{f}$

Now, let $\mathscr{L}=\mathcal{O}_{Y}(-F)=\mathfrak{a}\cdot\mathcal{O}_{Y}$
and $\tilde{\mathscr{L}}=\mathfrak{a}\cdot\mathcal{O}_{\tilde{X}}$. Then,
$\tilde{\mathscr{L}}$ is very-ample on $\tilde{X}$ and
$\mathscr{L}=f^{*}(\tilde{\mathscr{L}})$. So, using the projection formula
along $f|_{E}$, we have

$H^{0}(E,\mathcal{O}_{E}(K_{E}-\lfloor
cF\rfloor|_{E})\otimes\mathscr{L}^{n})\cong
H^{0}(\tilde{E},f|_{E*}(\mathcal{O}_{E}(K_{E}-\lfloor
cF\rfloor|_{E}))\otimes\tilde{\mathscr{L}}^{n})$

Now, $\mathscr{L}$ is a globally generated line bundle on $Y$. Hence, by the
Kleiman-Bertini theorem, we may choose a smooth divisor (possibly
disconnected) $D$ linearly equivalent to $-F$. Further, we may also assume
that $D$ intersects $E$ transversally (in particular, no component of $D$ is
$E$). Having chosen such a $D$, we note that $-\lfloor cF\rfloor=\lceil-
cF\rceil\sim_{\mathbb{Q}}cD+\sum a_{i}E_{i}$ as $\mathbb{Q}-$divisors with
$0\leq a_{i}<1$ and where $E_{i}$ are the irreducible components of $F$ in
some order. Let $\varDelta=\sum a_{i}E_{i}$, then since $E\subset E^{c}$, the
coefficient of $E$ in $cF$ is already an integer, hence the coefficient of $E$
in $\varDelta$ is $0$. So, we have $-\lfloor
cF\rfloor|_{E}\sim_{\mathbb{Q}}cD|_{E}+\varDelta|_{E}$. Since the support of
$\varDelta$ is the union of divisors in the support of $F$ and does not
contain $E$, it follows that $\varDelta|_{E}$ has simple normal crossings
support (since $F$ has snc support). Further, all the coefficients of
$\varDelta|_{E}$ are in the interval $[0,1)$. Next, $\mathscr{L}|_{E}$ is the
pullback of $\tilde{\mathscr{L}}|_{\tilde{E}}$ along $f|_{E}$; since
$f|_{E}:E\to\tilde{E}$ is a projective birational map, and
$\tilde{\mathscr{L}}$ is very ample on $\tilde{E}$, $\mathscr{L}|_{E}$ is big
and nef on $E$. So $cD|_{E}$ is a big and nef $\mathbb{Q}$-divisor. Since $E$
is smooth and projective, we can apply the Kawamata-Viehweg vanishing theorem
on $E$ with $N=-\lfloor cF\rfloor|_{E}\equiv_{num}cD|_{E}+\varDelta|_{E}$ to
conclude that

$H^{i}(E,\mathcal{O}_{E}(K_{E}-\lfloor cF\rfloor|_{E}))=0\quad\text{for all
$i>0$.}$

By the same argument, for all natural numbers $n\in\mathbb{N}_{\geq 0}$, we
have

$H^{i}(E,\mathcal{O}_{E}(K_{E}-\lfloor(c+n)F\rfloor|_{E}))=0\quad\text{for all
$i>0$ and all $n\geq 0$.}$ (4.4)

Now, Lemma 4.3.10 of [Laz04] implies that
$R^{j}f|_{E*}\mathcal{O}_{E}(K_{E}-\lfloor(c+n)F\rfloor|_{E}))=0$ for all
$j>0$ and all $n\geq 0$. By the Leray spectral sequence, we then have
isomorphisms

$H^{i}(E,\mathcal{O}_{E}(K_{E}-\lfloor(c+n)F\rfloor|_{E}))\cong
H^{i}(\tilde{E},f|_{E*}\mathcal{O}_{E}(K_{E}-\lfloor(c+n)F\rfloor|_{E}))$
(4.5)

for all $i\geq 0$ and all $n\geq 0$. So, if $\mathscr{F}$ denotes the
(coherent, since $f|_{E}$ is proper) sheaf
$f|_{E*}\mathcal{O}_{E}(K_{E}-\lfloor cF\rfloor|_{E})$ on $\tilde{E}$, then
putting together (10) and (11), we get

$H^{i}(\tilde{E},\mathscr{F}\otimes\tilde{\mathscr{L}}^{n}|_{\tilde{E}})=0\quad\text{for
all $i>0$ and $n\geq 0$}$ (4.6)

Finally, (13) implies that
$\mathscr{F}\otimes\tilde{\mathscr{L}}^{d-1}|_{\tilde{E}}$ is $0-$regular with
respect to the very ample line bundle $\tilde{\mathscr{L}}|_{\tilde{E}}$ i.e.

$H^{i}(\tilde{E},\mathscr{F}\otimes\tilde{\mathscr{L}}^{d-1}|_{\tilde{E}}\otimes\tilde{\mathscr{L}}^{-i}|_{\tilde{E}})=0\quad\text{for
all $i>0$}$

This is because since the dimension of $\tilde{E}$ is $d-1$, we only need to
check the vanishing of cohomology groups till $i=d-1$. But if $i\leq d-1$,
then (13) gives the required vanishing. So, by Mumford’s Theorem (Theorem
1.8.5, [Laz04]), we get that
$\mathscr{F}\otimes\tilde{\mathscr{L}}^{d-1}|_{\tilde{E}}$ is globally
generated, in particular,

$H^{0}(\tilde{E},\mathscr{F}\otimes\tilde{\mathscr{L}}^{d-1}|_{E})\neq 0$

So, using (10), we are done. This completes the proof of Theorem 4.5 and hence
of Theorem 4.1. ∎

######  Remark 4.6.

The Rees Coefficient is not always positive. We now give an example of an
ideal in the polynomial ring $k[x,y]$ for which the coefficient $\rho_{c}$ is
zero for some of the jumping numbers (including the log-canonical threshold).
We first recall the properties of $\rho_{c}$ when specialized to the two
dimensional case. These properties were observed in [AADG17] and are
specializations of the theory developed so far:

  1. (1)

$\rho_{c}=-F\cdot E^{c}$ where $E^{c}$ is the jumping divisor of $c$

  2. (2)

If $c>1$ is a jumping number and $c-1$ is not a jumping number, then
$\rho_{c}>0$

  3. (3)

If $c\in(0,1]$ is a jumping number such that $m(c)=m(c+1)$, then $\rho_{c}=0$

######  Example 4.7.

Let $X=\mathbb{A}^{2}$. Consider the ideal $\mathfrak{a}=(X^{5}+Y^{3},Y^{4})$
in the polynomial ring $k[X,Y]$. Then, a log resolution $Y$ of $\mathfrak{a}$
can be computed by hand and requires $9$ successive blow-ups. Let
$E_{1},\dots,E_{9}$ denote the exceptional divisors obtained in the order they
are labelled. Then, the intersection matrix for the resolution is:

$\begin{pmatrix}-3&0&1&0&0&0&0&0&0\\\ 0&-3&0&1&0&0&0&0&0\\\
1&0&-2&1&0&0&0&0&0\\\ 0&1&1&-2&1&0&0&0&0\\\ 0&0&0&1&-2&1&0&0&0\\\
0&0&0&0&1&-2&1&0&0\\\ 0&0&0&0&0&1&-2&1&0\\\ 0&0&0&0&0&0&1&-2&1\\\
0&0&0&0&0&0&0&1&-1\\\ \end{pmatrix}$

The relative canonical divisor is
$K_{Y|X}=E_{1}+2E_{2}+4E_{3}+7E_{4}+8E_{5}+9E_{6}+10E_{7}+11E_{8}+12E_{9}$. If
$\mathfrak{a}\cdot\mathcal{O}_{Y}=\mathcal{O}_{Y}(-F)$, then
$F=3E_{1}+5E_{2}+9E_{3}+15E_{4}+16E_{5}+17E_{6}+18E_{7}+19E_{8}+20E_{9}$. With
this information, we can compute the jumping numbers of $\mathfrak{a}$ along
with the multiplicities using the formula from [AADG17]. The formula has been
implemented in the package [DC] on _Macaulay2_ , [GS]. This gives us that the
log-canonical threshold of $\mathfrak{a}$ is $\frac{8}{15}$ and the other
jumping numbers less than $1$ are $\frac{11}{15},\frac{13}{15}$ and
$\frac{14}{15}$. The jumping divisor for each of these is $E_{4}$ and
$\rho_{8/15}=-F\cdot E_{4}=5+9-2\times 15+16=0$.

######  Remark 4.8.

These theorems suggest that the classes of the jumping numbers in
$\mathbb{Q}/\mathbb{Z}$ naturally come in various types depending on the type
of growth of the multiplicities of its translates. The type of jumping numbers
for which the multiplicity grows fastest i.e. for which $\rho_{c}>0$ is
described by Theorem 4.1. The next easy Corollary states that the class of
integers in $\mathbb{Q}/\mathbb{Z}$ is always of this type. This recovers the
well-known fact that the dimension $d$ of $X$ is always a jumping number for
any $\mathfrak{m}-$primary ideal $\mathfrak{a}$. Moreover, $d$ is always
contributed by any of the Rees valuations of $\mathfrak{a}$. In another
direction, we also get an upper bound on the number of Rees valuations of
$\mathfrak{a}$ in terms of growth of the multiplicities of the jumping numbers
$d+n$:

######  Corollary 4.9.

For any $\mathfrak{m}-$primary ideal $\mathfrak{a}$, the Rees coefficient
$\rho_{1}$ is positive. In fact, the number $(d-1)!\times\rho_{1}$ is at least
the number of Rees valuations of the ideal $\mathfrak{a}$. In particular, $d$
is always a jumping number of $\mathfrak{a}$ contributed by each of the Rees
valuations of $\mathfrak{a}$ (where $d$ is the dimension of $X$).

###### Proof.

Since the jumping divisor $E^{1}=F_{red}$, it contains all the exceptional
divisors corresponding to the Rees valuations. Moreover, since the
intersection numbers $(-F\cdot\ldots\cdot-F\cdot E)$ are non-negative integers
(by Lemma 4.4), each Rees valuation adds at least $1/(d-1)!$ to $\rho_{1}$.
The last claim now follows immediately from Theorem 4.5 since $E^{1}$ contains
each of the Rees valuations. ∎

Further, Theorem 4.1 and the polynomial nature of the multiplicities $m(c+n)$
give us more jumping numbers in the interval $(d-1,d]$ (where $d$ is the
dimension of $X$). The following observations generalize similar ones when
$d=2$ made in [AADG17].

######  Corollary 4.10.

With notation as before, let
$\mathfrak{a}\cdot\mathcal{O}_{Y}=\mathcal{O}_{Y}(-F)$ and for any prime
exceptional divisor $E$ in the support of $F$ on $Y$, let $ord(E)$ denote the
coefficient of $E$ in the divisor $F$. Let $E_{1},\dots,E_{r}$ be all the
prime divisors corresponding to the Rees valuations of $\mathfrak{a}$. Then,

  1. (1)

For each $1\leq i\leq r$, the number $\frac{\ell}{ord(E_{i})}$ is a jumping
number contributed by $E_{i}$ whenever $\ell$ is an integer such that
$\frac{\ell}{ord(E_{i})}$ lies in the interval $(d-1,d]$.

  2. (2)

If $c$ is a jumping number in the interval $(d-1,d]$ such that $c$ is not an
integer translate of a smaller jumping number, then $c$ is of the form in part
(1).

  3. (3)

More generally, if $c<d$ is a jumping number such that $c$ is not an integer
translate of a smaller jumping number, then $m(c+n)$ is polynomial (in $n$) of
degree at least $\lfloor c\rfloor$.

###### Proof.

Clearly (3) $\implies$ (2). So, we just prove (1) and (3):

  1. (1)

We know that for any number of the form $\frac{\ell}{ord(E_{i})}$, $E_{i}$ is
contained in the jumping divisor and hence, $\rho_{\frac{\ell}{ord(E_{i})}}>0$
by Theorem 4.3. Hence, the claim now follows immediately from Theorem 4.5.

  2. (3)

If $c$ is a jumping number such that $c-n$ is not a jumping number for any
$n$, $m(\\{c\\}+n)$ is a polynomial which is zero for $0\leq n\leq\lfloor
c\rfloor-1$. This means the degree must be at least $\lfloor c\rfloor$.

∎

######  Remark 4.11.

To summarize what we now know about the jumping numbers of $\mathfrak{a}$, we
have proved that among all jumping numbers of $\mathfrak{a}$, there are
numbers of the form $\frac{\ell}{ord(E)}$ in the interval $(d-1,d]$ for $E$
ranging over the Rees valuations that are very easy to understand. Indeed, the
number $ord(E)$ can be calculated directly from the normalized blow-up of the
ideal $\mathfrak{a}$ (instead of a log-resolution of $\mathfrak{a}$).
Moreover, every jumping number in the interval $(d-1,d]$ is either of this
form or an integer translate of a jumping number in the interval $(0,d-1]$. In
this context, we recall the result of Budur from [Bud03] mentioned in the
introduction which implies that the jumping numbers of $\mathfrak{a}$ in the
interval $(0,1)$ can be interpreted as coming from the cohomology of the
Milnor fiber of a general element of $\mathfrak{a}$. Even though we are far
from understanding all jumping numbers, we raise the following two questions
towards understanding all the jumping numbers.

######  Question 4.12.

For each integer $j$ in the interval $[0,d-2]$, how do we characterize the
classes of jumping numbers $[c]$ such that $m(c+n)$ grows like $n^{j}$? In
particular, can we characterise such classes of jumping numbers in terms of
contribution by divisors?

######  Question 4.13.

Given a class of jumping numbers $[c]\in\mathbb{Q}/\mathbb{Z}$ contributed by
a Rees valuation, how do we find the smallest jumping number in $[c]$? More
generally, how do we find the smallest jumping number in any given class of
jumping numbers in $\mathbb{Q}/\mathbb{Z}$? Equivalently, how do we
characterize the jumping numbers $c\leq d-1$ of $\mathfrak{a}$ such that $c$
is not a translate of a smaller jumping number of $\mathfrak{a}$.

###  4.2. Relation to Hilbert-Samuel Multiplicity

In this section, we relate the Hilbert-Samuel multiplicity of the ring $A$
with respect to the ideal $\mathfrak{a}$ to the Rees Coefficients $\rho_{c}$
associated to the jumping numbers of $\mathfrak{a}$. Recall that the Hilbert-
Samuel multiplicity of the ring $A$ with respect to $\mathfrak{a}$,
$e_{\mathfrak{a}}(A)$ is defined to be the number

$e_{\mathfrak{a}}(A)=\lim\limits_{n\to\infty}\frac{d!\,\dim_{k}(A/\mathfrak{a}^{n})}{n^{d}}$

The number $e_{\mathfrak{a}}(A)$ is always a positive integer. We recall that
for any $c>0$, the number $\rho_{c}$ was defined in 3.6 such that
$m(c+n)=\rho_{c}n^{d-1}+o(n^{d-2})$.

######  Theorem 4.14.

If $d$ is the Krull dimension of $A$, the following equality holds:

$\sum\limits_{c\in(0,1]}\rho_{c}=\frac{e_{\mathfrak{a}}(A)}{(d-1)!}$

The main idea relating these two numbers is that the Hilbert-Samuel
multiplicity can be computed using multiplier ideals, which is a consequence
of Skoda’s Theorem:

######  Lemma 4.15.

$e_{\mathfrak{a}}(A)=\lim\limits_{n\to\infty}\frac{d!\,\dim_{k}(A/\mathscr{I}(\mathfrak{a}^{n}))}{n^{d}}$

###### Proof.

For $n\geq d$, we have

$\mathfrak{a}^{n}\subset\mathscr{I}(\mathfrak{a}^{n})\subset\mathfrak{a}^{n-d+1}$

where the second containment holds because of Skoda’s Theorem 2.2 which says
that
$\mathscr{I}(\mathfrak{a}^{n})\subset\mathfrak{a}^{n-d+1}\cdot\mathscr{I}(\mathfrak{a}^{d-1})$
for $n\geq d$. This gives us that

$l(A/\mathfrak{a}^{n-d+1})\leq l(A/(\mathscr{I}(\mathfrak{a}^{n})))\leq
l(A/\mathfrak{a}^{n})$

When we divide by $n^{d}$ and take limit as $n\to\infty$, since both the first
and third terms approach $e_{\mathfrak{a}}(A)/d!$, so does the middle term. ∎

##### Proof of Theorem 4.14

We first note that $\rho_{c}$ can be non-zero only for the candidate jumping
numbers (defined in 2.1.1). Hence, the sum is finite. Now to prove the
proposition, we use Lemma 4.15 to compute $e_{\mathfrak{a}}(A)$ as:

$\frac{e_{\mathfrak{a}}(A)}{d!}=\lim\limits_{n\to\infty}\frac{\dim_{k}(A/\mathscr{I}(\mathfrak{a}^{n}))}{n^{d}}=\lim\limits_{n\to\infty}\frac{\sum\limits_{c\leq
n}m(c)}{n^{d}}=\lim\limits_{n\to\infty}\frac{\sum\limits_{c\in(0,1]}\sum\limits_{j=0}^{n-1}m(c+j)}{n^{d}}$

Since $m(c+j)=\rho_{c}j^{d-1}+o(j^{d-1})$ as $n\to\infty$, by using the fact
that $\sum\limits_{j=0}^{n}j^{i}=\frac{n^{i+1}}{i+1}+o({n^{i}})$ as
$n\to\infty$, we have
$\sum\limits_{j=0}^{n-1}m(c+j)=\frac{\rho_{c}}{d}n^{d}+o(n^{d-1})$ as
$n\to\infty$. Using this, we have

$\frac{e_{\mathfrak{a}}(A)}{d!}=\lim\limits_{n\to\infty}\frac{\sum\limits_{c\in(0,1]}\frac{\rho_{c}}{d}n^{d}+o(n^{d-1})}{n^{d}}=\frac{\sum\limits_{c\in(0,1]}\rho_{c}}{d}$

∎

##  5\. Special Case: Monomial Ideals

In this section, we derive an explicit formula for the multiplicities and Rees
coefficients (Theorem 5.1) of an arbitrary cofinite monomial ideal (See
Definitions 3.1 and 3.6). In particular, we see that the Rees valuations are
always positive in this case (Corollary 5.3).

We introduce the relevant notation. Let $X=\mathbb{A}^{d}$,
$R=k[x_{1},\dots,x_{d}]$ and $\mathfrak{a}$ be an $\mathfrak{m}-$primary
monomial ideal where $\mathfrak{m}=(x_{1},\dots,x_{d})$. Fix
$M=\mathbb{N}^{d}$ inside $M_{\mathbb{R}}=\mathbb{R}^{d}$. For any
$v=(v_{1},\dots,v_{d})\in M$, we write $x^{v}=x_{1}^{v_{1}}\dots
x_{d}^{v_{d}}$ for the corresponding monomial. For any ideal generated by
monomials $\mathfrak{a}$ in $R$, the _Newton Polyhedron_
$P(\mathfrak{a})\subset\mathbb{R}^{d}$ of $\mathfrak{a}$ is the convex hull of
all vectors $v\in M$ such that $x^{v}$ belongs to $\mathfrak{a}$. Note that
$P(\mathfrak{a})$ is an unbounded polyhedral (i.e. bounded by polygonal faces)
region in the first orthant of $\mathbb{R}^{d}$. For any positive real number
$c$, let $P(c\cdot\mathfrak{a})$ denote the polyhedron obtained by scaling the
vectors in $P(\mathfrak{a})$ by a factor of $c$. A theorem of Howald provides
a formula for the multiplier ideals of $\mathfrak{a}$:

######  Theorem.

(Howald) Fix a positive real number $c$. The multiplier ideal
$\mathscr{I}(\mathfrak{a}^{c})$ is the monomial ideal generated by all
monomials $x^{v}$ such that

$v+\mathbf{1}\in Int(P(c\cdot\mathfrak{a})),$

where $\mathbf{1}$ is the vector $(1,\dots,1)$ and
$Int(P(c\cdot\mathfrak{a}))$ denotes the interior of the scaled Newton
polyhedron $P(c\cdot\mathfrak{a})$.

We can now state our formula for the multiplicity and Rees coefficients of a
jumping number:

######  Theorem 5.1.

Fix a jumping number $c$ for $\mathfrak{a}$. The multiplicity of $c$ is

$m(c)=\\#\\{\bigcup_{i}cP_{i}^{\circ}\cap M\\},$

where the $P_{i}$ are the bounded faces of the Newton polyhedron
$P(\mathfrak{a})$ and

$cP_{i}^{\circ}:=cP_{i}-(\bigcup_{j}j^{th}\text{ coordinate
hyperplane})=cP_{i}\cap(\mathbf{1}+M).$

Moreover, the Rees coefficient $\rho_{c}$ is given by

$\rho_{c}=\sum\limits_{c\mathcal{H}_{i}\cap
M\neq\emptyset}\mathrm{vol}(P_{i}),$

where $\mathcal{H}_{i}$ is the unique hyperplane containing the face $P_{i}$.

To prove Theorem 5.1, we need the following Lemma:

######  Lemma 5.2.

For each jumping number $c$ of the monomial ideal $\mathfrak{a}$, the
$k$-vector space
$\mathscr{I}(\mathfrak{a}^{c-\epsilon})/\mathscr{I}(\mathfrak{a}^{c})$ is
isomorphic to the vector space generated by the monomials in $\partial
P(c\cdot\mathfrak{a})\cap(\mathbf{1}+M)$, where $\partial
P(c\cdot\mathfrak{a})$ is the boundary of the dilated (by $c$) Newton
polyhedron $P(c\cdot\mathfrak{a})$.

###### Proof.

Howald’s theorem implies that a real number $c>0$ is a jumping number of
$\mathfrak{a}$ if and only if $cP_{i}^{\circ}\cap M$ is non-empty for some
$i$. Moreover, the monomials that are contained in
$\mathscr{I}(\mathfrak{a}^{c-\epsilon})$ but not in
$\mathscr{I}(\mathfrak{a}^{c})$ are exactly the monomials $x^{v}$ such that
$v+\mathbf{1}$ is on the boundary of $P(c\cdot\mathfrak{a})$. Therefore, the
$k-$vector space
$\mathscr{I}(\mathfrak{a}^{c-\epsilon})/\mathscr{I}(\mathfrak{a}^{c})$ is
isomorphic to the vector space generated by the monomials in $\partial
P(c\cdot\mathfrak{a})\cap(\mathbf{1}+M)$. ∎

##### Proof of Theorem 5.1

The condition that $\mathfrak{a}$ is an $\mathfrak{m}-$primary monomial ideal
is equivalent to the condition that the region $P(\mathfrak{a})$ intersects
each coordinate axis. In this case, the boundary of $P(\mathfrak{a})$ is the
union of the bounded faces $P_{1},\dots,P_{r}$ and the unbounded faces exactly
along the coordinate hyperplanes. The faces $P_{1},\dots,P_{r}$ are polytopes
of dimension $d-1$ with vertices in $M$. Each polytope $P_{i}$ is contained in
a unique hyperplane $\mathcal{H}_{i}\subset\mathbb{R}^{d}$.

For the first claim, we note that addition by 1 gives a one-to-one
correspondence between vectors in $M$ and the vectors in $M$ that do not lie
on any of the coordinate hyperplanes. Hence, the vectors $v$ in $M$ such that
$v+\textbf{1}\in\partial P(c\cdot\mathfrak{a})$ are in one-to-one
correspondence with the vectors in $\partial P(c\cdot\mathfrak{a})$ that do
not lie on any of the coordinate hyperplanes. But, since the components of the
boundary that do not lie on the coordinate hyperplanes are exactly the bounded
components $P_{1},\dots,P_{r}$, this proves the first claim. So the formula
for the multiplicity follows from this and Lemma 5.2 above.

For $c>0$, let $cP_{i}$ denote the polytope obtained by scaling each vector in
$P_{i}$ by $c$ and let $c\mathcal{H}_{i}$ be the corresponding scaled
hyperplane.

For the last claim, we use the following fact: Let $Q$ be a convex polyhedron
of dimension $\ell$ in $\mathbb{R}^{\ell}$. Then,

$\mathrm{vol}(Q)=\lim\limits_{t\to\infty}\frac{\\#(tQ\cap\mathbb{Z}^{\ell})}{t^{\ell}}.$

We actually need a slightly more general version of this formula: Suppose
$\\{Q_{n}\\}_{n\geq 1}$ is a sequence of polyhedra of dimension $\ell$ in
$\mathbb{R}^{\ell}$ such that $Q_{n}$ is some (possibly non-integer) translate
of $c_{n}Q_{1}$ where $\\{c_{n}\\}$ is a sequence of real numbers such that
$c_{n}\to\infty$ as $n\to\infty$. Then

$\mathrm{vol}(Q_{1})=\lim\limits_{n\to\infty}\frac{\\#(Q_{n}\cap
M)}{c_{n}^{\ell}}.$

This statement actually follows immediately from the previous statement with
the additional observation that when we translate a polyhedron by any vector,
then the difference in the number of lattice points is bounded above by a
fixed multiple of volume of the boundary of the polyhedron. Hence, the
difference doesn’t really matter for the limit.

Suppose $c\mathcal{H}_{i}\cap M\neq\emptyset$, then the number points of
$(c+n)\mathcal{H}_{i}\cap M$ in $P((c+n)\cdot\mathfrak{a})$ is the same as the
number of lattice points in a translate of $\frac{c+n}{c}(cP_{i})$ in
$c\mathcal{H}_{i}$. This is because, any lattice point on $P_{i}$ gives us a
natural bijection between the lattice points in $c\mathcal{H}_{i}$ and
$(c+n)\mathcal{H}_{i}$ (by translating). Under this translation, we can
identify $(c+n)P_{i}$ in $(c+n)\mathcal{H}_{i}$ with a translate of
$\frac{c+n}{c}(cP_{i})$ in $c\mathcal{H}_{i}$. Hence, using the formula above
with $\ell=d-1$, $\mathbb{R}^{d-1}=c\mathcal{H}_{i}$, $Q_{n}=(c+n)P_{i}$ and
$c_{n}=\frac{c+n}{c}$, we have

$\mathrm{vol}(P_{i})=\frac{1}{c^{d-1}}\mathrm{vol}(cP_{i})=\lim\limits_{n\to\infty}\frac{\\#((\frac{c+n}{c}P_{i})\cap
M)}{(\frac{c+n}{c})^{d-1}}.$

Since the number of lattice points in the intersection of $(c+n)P_{i}$ and the
coordinate hyperplanes or the intersection of $(c+n)P_{i}$ and $(c+n)P_{j}$
for $j\neq i$ grows at a rate of at most $n^{d-2}$, the formula for $\rho_{c}$
follows. ∎

It follows immediately from the theorem that:

######  Corollary 5.3.

If $c$ is a jumping number of an $\mathfrak{m}-$primary monomial ideal
$\mathfrak{a}\subset k[x_{1},\dots,x_{d}]$, then the Rees coefficient
$\rho_{c}$ is positive.

##  6\. Poincaré Series

For any closed subscheme $Z$ supported at a closed point of $X$, we can
assemble the multiplicities of jumping numbers (Definition 3.1) into a
generating function called the Poincaré Series of $Z$. More precisely, this is
the generating function defined by:

$\phi_{Z}(T)=\sum\limits_{c\in(0,\infty)}m(c)T^{c}=\sum\limits_{c\in(0,1]}\sum\limits_{n=0}^{\infty}m(c+n)T^{c+n}$

This series is clearly only a countable sum, because the jumping numbers of
$Z$ are a discrete subset of rational numbers. Moreover, there is an
$\ell\in\mathbb{N}$ such that every jumping number has a denominator a factor
of $\ell$ i.e. every jumping number is of the form $\frac{n}{\ell}$ for
$n\in\mathbb{N}$. Then, having chosen such an $\ell$ and setting
$z=T^{1/\ell}$, we see that $\phi_{Z}(T)$ is actually a power series in $z$.
We use the polynomial nature of $m(c+n)$ proved in Theorem 3.2 to prove that
this power series is actually a rational function:

######  Theorem 6.1.

$\phi_{Z}(z)$ is a rational function of $z$ where $z=T^{1/\ell}$ and $\ell$ is
as above. In fact, we have the following formula:

$\phi_{Z}(T)=\sum\limits_{c\in(0,1]}\Big{(}\frac{m(c)}{(1-T)}+\frac{\gamma_{c,1}T}{(1-T)^{2}}+\dots+\frac{\gamma_{c,d-2}T}{(1-T)^{d-1}}+\frac{\rho_{c}(d-1)!T}{(1-T)^{d}}\Big{)}T^{c}$
(6.1)

for some rational numbers $\gamma_{c,i}$.

###### Proof.

For any $c\in(0,1]$, by Theorem 3.2 we have

$m(c+n)=\alpha_{0,c}+\alpha_{1,c}n+\ldots+\alpha_{d-1,c}n^{d-1}$

(Of course, $\alpha_{d-1,c}=\rho_{c}$ and $\alpha_{0,c}=m(c)$). Now, the list
of polynomials
$p_{0}(n)=1,p_{1}(n)=n,p_{2}(n)=\dbinom{n+1}{2},\dots,p_{d-1}(n)=\dbinom{n+d-2}{d-1}$
has exactly one polynomial of each degree between $0$ and $d-1$. So, these
polynomials form a basis of $\mathbb{Q}$ of the space of rational polynomials
of degree less than $d$. So, we can write the polynomial $m(c+n)$ in $n$ in
terms of the $p(i)$’s:

$m(c+n)=\sum\limits_{i=0}^{d-1}\gamma_{c,i}p_{i}(n)\quad\text{for some
rational numbers }\gamma_{c,i}$ (6.2)

Inspecting the constant and the leading coefficient, we get
$\gamma_{c,0}=m(c)$ and $\gamma_{c,d-1}=(d-1)!\times\rho_{c}$.

$\displaystyle\phi_{Z}(T)$
$\displaystyle=\sum\limits_{c\in(0,\infty)}m(c)T^{c}=\sum\limits_{c\in(0,1]}\sum\limits_{n=0}^{\infty}m(c+n)T^{c+n}$
$\displaystyle=\sum\limits_{c\in(0,1]}T^{c}\sum\limits_{n=0}^{\infty}(\gamma_{c,0}p_{o}(n)+\gamma_{c,1}p_{1}(n)+\ldots+\gamma_{c,d-1}p_{d-1}(n))T^{n}$
$\displaystyle=\sum\limits_{c\in(0,1]}T^{c}\Big{(}\gamma_{c,0}\sum\limits_{n=0}^{\infty}p_{o}(n)T^{n}+\gamma_{c,1}\sum\limits_{n=0}^{\infty}p_{1}(n)T^{n}+\ldots+\gamma_{c,d-1}\sum\limits_{n=0}^{\infty}p_{d-1}(n)T^{n}\Big{)}.$

Now the Theorem follows from the following elementary observation:

$\sum_{n=0}^{\infty}p_{i}T^{n}=\begin{cases}\frac{1}{1-T}\quad\text{if }i=0\\\
\frac{T}{(1-T)^{i+1}}\quad\text{if }i\geq 1\end{cases}$

When $i=0$ the formula is clear. For $i\geq 1$, the formula is equivalent to

$\frac{1}{(1-T)^{i+1}}=\sum_{j=0}^{\infty}\dbinom{j+i}{i}T^{j}$

which is a well-known combinatorial identity counting degree $j$ monomials in
$i+1$ variables. ∎

######  Remark 6.2.

This theorem was proved independently by Àlvarez Montaner and Núñez-Betancourt
in [AN21] using algebraic methods. They also prove an analogous theorem for
Test Ideals, which are positive characteristic counterparts of multiplier
ideals. We are also grateful to them for conveying the explicit form for the
Poincaré Series as in equation 6.1.

######  Remark 6.3.

The case $d=2$ of Theorem 6.1 was proved by Alberich-Carramiñana et al. in
[AADG17], generalizing the work of Galindo and Monserrat in the case of simple
complete ideals in [GM10].

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