# An estimate for $F$-jumping numbers via the roots of the Bernstein-Sato
polynomial

Mircea Mustaţă  Department of Mathematics, University of Michigan, 530 Church
Street, Ann Arbor, MI 48109, USA<EMAIL_ADDRESS>
###### Abstract.

Given a smooth, irreducible complex algebraic variety $X$ and a nonzero
regular function $f$ on $X$, we give an effective estimate for the difference
between the jumping numbers of $f$ and the $F$-jumping numbers of a reduction
$f_{p}$ of $f$ to characteristic $p\gg 0$, in terms of the roots of the
Bernstein-Sato polynomial $b_{f}$ of $f$. In particular, we get uniform
estimates only depending on the dimension of $X$. As an application, we show
that if $b_{f}$ has no roots of the form $-\operatorname{lct}(f)-n$, with $n$
a positive integer, then the $F$-pure threshold of $f_{p}$ is equal to the log
canonical threshold of $f$ for $p\gg 0$ with
$(p-1)\operatorname{lct}(f)\in{\mathbf{Z}}$.

###### 2020 Mathematics Subject Classification:

13A35, 14F18, 14F10

The author was partially supported by NSF grants DMS-2001132 and DMS-1952399.

##  1\. Introduction

Let $X$ be a smooth, irreducible complex algebraic variety and let
$f\in\mathcal{O}_{X}(X)$ be a nonzero regular function defining the
hypersurface $H\subset X$. One associates to $f$ a sequence of coherent ideals
$\mathcal{J}(f^{\lambda})\subseteq\mathcal{O}_{X}$, the _multiplier ideals_ of
$f$, depending on the parameter $\lambda\in{\mathbf{R}}_{\geq 0}$. They can be
defined either in terms of a log resolution of the pair $(X,H)$, or in terms
of integrability conditions. These ideals give interesting invariants of the
singularities of the hypersurface $H$ and they play an important role in
vanishing theorems. They satisfy
$\mathcal{J}(f^{\lambda})\subseteq\mathcal{J}(f^{\mu})$ if $\mu\leq\lambda$
and there is an increasing sequence of positive rational numbers
$(\lambda_{m})_{m\geq 1}$, with $\lim_{m\to\infty}\lambda_{m}=\infty$, such
that $\mathcal{J}(f^{\lambda})$ is constant for
$\lambda\in[\lambda_{m-1},\lambda_{m})$ for all $m\geq 1$ (with the convention
that $\lambda_{0}=0$). These are the _jumping numbers_ of $f$ and the smallest
jumping number $\lambda_{1}$ is the _log canonical threshold_
$\operatorname{lct}(f)$. For an introduction to multiplier ideals and jumping
numbers, see [Lazarsfeld, Chapter 9].

It has long been understood that many classes and invariants of singularities
that appear in birational geometry in characteristic 0 have analogues in
positive characteristic, defined via the Frobenius morphism. Suppose now that
$Y$ is a regular scheme of characteristic $p>0$, which we assume to be
$F$-finite (this means that the Frobenius morphism $F\colon Y\to Y$ is a
finite morphism). If $g\in\mathcal{O}_{Y}(Y)$ is everywhere nonzero, then Hara
and Yoshida [HY] defined a sequence of ideals
$\tau(g^{\lambda})\subseteq\mathcal{O}_{Y}$, the (generalized) _test ideals_
of $g$, depending on the parameter $\lambda\in{\mathbf{R}}_{\geq 0}$. These
ideals turn out to satisfy many of the formal properties that multiplier
ideals satisfy in characteristic $0$. In particular,
$\tau(g^{\lambda})\subseteq\tau(g^{\mu})$ if $\mu\leq\lambda$ and it was shown
in [BMS2] that there is an increasing sequence of positive rational numbers
$(\alpha_{m})_{m\geq 1}$, with $\lim_{m\to\infty}\alpha_{m}=\infty$, such that
$\tau(g^{\lambda})$ is constant for $\lambda\in[\alpha_{m-1},\alpha_{m})$ for
all $m\geq 1$ (with the convention that $\alpha_{0}=0$). These are the _$F$
-jumping numbers_ of $g$ and the smallest such invariant $\alpha_{1}$ is the
_$F$ -pure threshold_ $\operatorname{fpt}(g)$, introduced by Takagi and
Watanabe in [TW].

The most interesting results and open problems in this area are related to the
comparison between multiplier ideals and test ideals via reduction mod $p$.
Suppose that $X$ is a smooth, irreducible complex algebraic variety and
$f\in\mathcal{O}_{X}(X)$ is nonzero. After choosing a model $(X_{A},f_{A})$ of
$(X,f)$ over a finitely generated ${\mathbf{Z}}$-subalgebra
$A\subseteq{\mathbf{C}}$, for every closed point $t\in{\rm Spec}(A)$ we obtain
a pair $(X_{t},f_{t})$ in positive characteristic (note that the field $k(t)$
is a finite field). As it is typical in this setting, we always allow
replacing $A$ by a localization $A_{a}$, in order to preserve certain
properties from characteristic $0$ (for example, in this way we may assume
that $X_{A}$ is smooth over ${\rm Spec}(A)$ and that $f_{A}$ defines a
relative Cartier divisor over ${\rm Spec}(A)$). The following is a key result
from [HY]:

######  Theorem 1.1.

With the above notation, after possibly replacing $A$ by a localization
$A_{a}$, the following hold:

  1. i)

For every closed point $t\in{\rm Spec}(A)$, we have
$\tau(f_{t}^{\lambda})\subseteq\mathcal{J}(f^{\lambda})_{t}$ for all
$\lambda\in{\mathbf{R}}_{\geq 0}$.

  2. ii)

For every $\lambda\in{\mathbf{R}}_{\geq 0}$, if $t\in{\rm Spec}(A)$ is a
closed point such that ${\rm char}\,k(t)\gg 0$, then
$\tau(f_{t}^{\lambda})=\mathcal{J}(f^{\lambda})_{t}$.

The assertion in ii) is the deepest one. We note that the condition on ${\rm
char}\,k(t)$ here can’t be made independent of $\lambda$, in general. However,
we have the following open problem concerning the relation between multiplier
ideals and test ideals:

######  Conjecture 1.2.

With the above notation, there is a dense subset $T$ of closed points in ${\rm
Spec}(A)$ such that

$\tau(f_{t}^{\lambda})=\mathcal{J}(f^{\lambda})_{t}\quad\text{for all}\quad
t\in T,\lambda\in{\mathbf{R}}_{\geq 0}.$

It is known that this conjecture is equivalent to a deep conjecture in
arithmetic geometry, the Weak Ordinarity conjecture (see [Mustata] and [MS],
as well as [BST] for the case of a singular ambient variety). It is
instructive to see what the above theorem and conjecture say about the
relation between $\operatorname{lct}(f)$ and $\operatorname{fpt}(f_{t})$.
First, the assertion in Theorem 1.1i) says that after possibly replacing $A$
by a localization $A_{a}$, we may assume that
$\operatorname{fpt}(f_{t})\leq\operatorname{lct}(f)$ for all closed points in
$t\in{\rm Spec}(A)$. On the other hand, the result in Theorem 1.1ii) implies
that

(1) $\lim_{{\rm
char}\,k(t)\to\infty}\operatorname{fpt}(f_{t})=\operatorname{lct}(f).$

Finally, Conjecture 1.2 predicts that there is a dense subset $T$ of closed
points in ${\rm Spec}(A)$ such that
$\operatorname{fpt}(f_{t})=\operatorname{lct}(f)$ for all $t\in T$.

Our main goal in this note is to give an effective estimate of the limit in
(1) and a generalization of this to arbitrary jumping numbers, in terms of the
roots of the _Bernstein-Sato polynomial_ $b_{f}(s)$ of $f$. Recall that
$b_{f}(s)$ is the monic polynomial of smallest degree such that

(2) $b_{f}(s)f^{s}\in\mathcal{D}_{X}[s]\bullet f^{s+1},$

where $\mathcal{D}_{X}$ is the sheaf of differential operators on $X$. Here
$f^{s}$ has to be interpreted as a formal symbol on which differential
operators act in the expected way. The existence of a polynomial $b_{f}(s)$ as
in (2) was proved by Bernstein in [Bernstein] when $X={\mathbf{A}}^{n}$ and by
Kashiwara [Kashiwara] in general (in fact, for arbitrary holomorphic functions
on complex manifolds). Furthermore, it was shown in [Kashiwara] that all roots
of $b_{f}$ are negative rational numbers.

The following is our estimate for the $F$-pure thresholds of the reductions of
$f$.

######  Theorem 1.3.

Let $X$ be a smooth complex algebraic variety and $f\in\mathcal{O}_{X}(X)$
nonzero. If $(X_{A},f_{A})$ gives a model of $(X,f)$ over $A$, then after
possibly replacing $A$ by a localization $A_{a}$, for every closed point
$t\in{\rm Spec}(A)$ with ${\rm char}\,k(t)=p_{t}$, the following holds: if $i$
is a positive integer such that
$(p_{t}-i)\cdot\operatorname{lct}(f)\in{\mathbf{Z}}$, then

$\operatorname{fpt}(f_{t})>\operatorname{lct}(f)-\frac{m_{i}}{p_{t}},$

where $m_{i}=\max\big{\\{}\beta\mid
b_{f}(-\beta)=0,\beta-(i\cdot\operatorname{lct}(f)+1-\lceil
i\cdot\operatorname{lct}(f)\rceil)\in{\mathbf{Z}}_{\geq 0}\big{\\}}$.

Note that if we write $\operatorname{lct}(f)=\tfrac{a}{b}$, with $a$ and $b$
relatively prime, then it is enough to consider in the theorem only those $i$
with $1\leq i\leq b$ and relatively prime to $b$. In particular, we get an
explicit lower bound for $\operatorname{fpt}(f_{t})$, in terms of the roots of
$b_{f}$, that works for all $t$. Using a lower bound for the roots of the
Bernstein-Sato polynomial from [Saito_microlocal], we will also see that we
may replace $m_{i}$ in the theorem by $\dim(X)$ (see Remark 3.2).

In fact, we prove a more general statement (see Theorem 3.1), in which
$\operatorname{lct}(f)$ is replaced by any jumping number of $f$. The main
ingredient in the proof of Theorem 1.3 is a result from [MZ] which says that
there is such a lower bound, but for which $m_{i}$ is only determined, in a
complicated way, by a log resolution (this, in turn, relied on the techniques
in [HY] that give (1)). The other ingredient is the elementary observation,
which goes back to [MTW], that for every closed point $t\in{\rm Spec}(A)$, the
integer $\lceil\operatorname{fpt}(f_{t})p_{t}\rceil-1$ gives a root of $b_{f}$
mod $p_{t}$.

We note that the assertion in Theorem 1.3 when $\operatorname{lct}(f)=1$ has
been recently obtained by Dodd in [Dodd], using completely different methods.
In fact, his result provided us with the motivation for revisiting this circle
of ideas.

It is well-known that $-\operatorname{lct}(f)$ is a root of $b_{f}$ (in fact,
this is the largest root of the Bernstein-Sato polynomial, see [Kollar,
Section 10]). When $b_{f}$ has no roots of the form
$-\operatorname{lct}(f)-n$, with $n\in{\mathbf{Z}}_{>0}$, we obtain the
following result in the direction of the conjectural existence of prime
reductions with $F$-pure threshold equal to $\operatorname{lct}(f)$.

######  Corollary 1.4.

Let $X$ be a smooth complex algebraic variety and $f\in\mathcal{O}_{X}(X)$
nonzero such that $b_{f}\big{(}-\operatorname{lct}(f)-n\big{)}\neq 0$ for
every $n\in{\mathbf{Z}}_{>0}$. If $(X_{A},f_{A})$ gives a model of $(X,f)$,
then after possibly replacing $A$ by a localization $A_{a}$, for every closed
point $t\in{\rm Spec}(A)$ with ${\rm char}\,k(t)=p_{t}$, we have
$\operatorname{fpt}(f_{p})=\operatorname{lct}(f)$ as long as
$(p_{t}-1)\cdot\operatorname{lct}(f)\in{\mathbf{Z}}$.

Acknowledgment. I am grateful to Christopher Dodd for sharing with me a
preliminary version of [Dodd] and for several discussions on this topic. I
would also like to thank Shunsuke Takagi for some good questions on the first
version of this note. Last but not least, I am indebted to the anonymous
referee for several useful suggestions, especially for Example 3.5.

##  2\. A brief review of $F$-jumping numbers

Let $Y$ be a regular $F$-finite scheme of characteristic $p>0$ and let
$g\in\mathcal{O}_{Y}(Y)$ be everywhere nonzero (in other words, $g$ is nonzero
on every connected component of $Y$). Note that since $Y$ is regular, the
Frobenius morphism $F\colon X\to X$ is also flat by a famous result of Kunz
[Kunz]. We do not recall the definition of the test ideals $\tau(g^{\lambda})$
since we will not need it. We refer the reader to [HY] for the original
definition and to [BMS1] for a simpler one in our setting (when the ambient
scheme is assumed to be regular and $F$-finite).

We only review here the description of the $F$-jumping numbers of $g$ as
$F$-thresholds, following [MTW] and [BMS1]. We note that if
$Y=U_{1}\cup\ldots\cup U_{N}$ is an open cover, then the set of $F$-jumping
numbers of $g$ is the union of the sets of the jumping numbers of
$g|_{U_{i}}$, for $1\leq i\leq N$. By taking a suitable affine cover of $Y$,
we may thus reduce the study of $F$-jumping numbers to the case of an affine
scheme. In order to conform to the presentation in [MTW] and [BMS1], we assume
that $Y$ is affine and let $R=\mathcal{O}_{Y}(Y)$.

Given a proper ideal $J\subseteq R$ such that $g\in{\rm rad}(J)$, for every
$e\geq 1$ we put

$\nu^{J}_{g}(p^{e})=\max\\{r\in{\mathbf{Z}}_{\geq 0}\mid g^{r}\not\in
J^{[p^{e}]}\\}.$

Recall that $J^{[p^{e}]}=(h^{p^{e}}\mid h\in J)$. The flatness of the
Frobenius morphism gives

$\big{(}J^{[p^{e+1}]}:g^{pr}\big{)}=\big{(}J^{[p^{e}]}:g^{r}\big{)}^{[p]},$

hence $g^{r}\not\in J^{[p^{e}]}$ implies $g^{pr}\not\in J^{[p^{e+1}]}$. We
thus have

$\frac{\nu^{J}_{g}(p^{e})}{p^{e}}\leq\frac{\nu^{J}_{g}(p^{e+1})}{p^{e+1}}\quad\text{for
all}\quad e\geq 1$

and the _$F$ -threshold of $g$ with respect to $J$_ is

$c^{J}(g):=\sup_{e\geq
1}\frac{\nu^{J}_{g}(p^{e})}{p^{e}}=\lim_{e\to\infty}\frac{\nu^{J}_{g}(p^{e})}{p^{e}}.$

One can show that $c^{J}(g)<\infty$ and we have

(3) $\frac{\nu^{J}_{g}(p^{e})}{p^{e}}<c^{J}(g)\quad\text{for all}\quad e\geq
1$

(see [MTW, Remark 1.2] and [MTW, Proposition 1.7(5)]). What’s special when
working with a principal ideal such as $(g)$, is that if $g^{r+1}\in
J^{[p^{e}]}$, then $g^{(r+1)p}\in J^{[p^{e+1}]}$. We thus have

$\frac{\nu^{J}_{g}(p^{e})+1}{p^{e}}\geq\frac{\nu^{J}_{g}(p^{e+1})}{p^{e+1}}\quad\text{for
all}\quad e\geq 1,$

which immediately implies

(4) $c^{J}(g)\leq\frac{\nu^{J}_{g}(p^{e})+1}{p^{e}}\quad\text{for all}\quad
e\geq 1.$

By combining the inequalities (3) and (4), we obtain

(5) $\nu^{J}_{g}(p^{e})+1=\lceil c^{J}(g)\cdot p^{e}\rceil\quad\text{for
all}\quad e\geq 1$

(see [MTW, Proposition 1.9]).

The $F$-thresholds are relevant for us since the set of $F$-jumping numbers of
$g$ coincides with the set of all $F$-thresholds of $g$, when the ideal $J$
varies (see [BMS1, Corollary 2.30]). We note that unlike in _loc_. _cit_. we
do not consider $0$ as an $F$-jumping number, which corresponds to the fact
that we require $J$ to be a proper ideal of $\mathcal{O}_{X}$.

As we have mentioned in the Introduction, a basic result about the $F$-jumping
numbers of $g$ is that they form a discrete set of rational numbers (see
[BMS1, Theorem 3.1] for the case when $R$ is essentially of finite type over a
field and [BMS2, Theorem 1.1] for the general case). We note that the
corresponding result about jumping numbers in characteristic 0 is an immediate
consequence of their description in terms of a log resolution, see
[Lazarsfeld, Lemma 9.3.21].

As it is the case for jumping numbers in characteristic $0$, some $\lambda>1$
is an $F$-jumping number of $g$ if and only if $\lambda-1$ has the same
property. This follows, for example, from the fact that
$\tau(g^{\alpha})=g\cdot\tau(g^{\alpha-1})$ for every $\alpha>1$ (see for
example [BMS1, Proposition 2.25]). A more interesting fact, that is peculiar
to positive characteristic, is that if $\lambda$ is an $F$-jumping number of
$g$, then so is $p\lambda$. This follows from the fact that $p\cdot
c^{J}(g)=c^{J^{[p]}}(g)$, see [BMS1, Proposition 3.4(1)]. By combining these
two facts one can show that the $F$-pure threshold $\operatorname{fpt}(g)$
can’t lie in certain intervals: more precisely, for every $e\geq 1$ and every
integer $a$, with $1\leq a\leq p^{e}-1$, we have

(6)
$\operatorname{fpt}(g)\not\in\left(\tfrac{a}{p^{e}},\tfrac{a}{p^{e}-1}\right).$

This is due to the fact that if
$\operatorname{fpt}(g)\in\big{(}\tfrac{a}{p^{e}},\tfrac{a}{p^{e}-1}\big{)}$,
then $p^{e}\cdot\operatorname{fpt}(g)-a$ is an $F$-jumping number of $g$ that
is $<\operatorname{fpt}(g)$, a contradiction; see [BMS2, Proposition 4.3] for
details.

We end this discussion of $F$-jumping numbers with the following result that
we will need:

######  Proposition 2.1.

Suppose that $Y={\rm Spec}(R)$ is a smooth affine scheme of finite type over
the finite field $k$ and $k\subseteq k^{\prime}$ is a finite field extension.
If $g\in R$ is nowhere zero and $g^{\prime}=g\otimes 1\in
R^{\prime}=R\otimes_{k}k^{\prime}$, then

$\tau({g^{\prime}}^{\lambda})=\tau(g^{\lambda})\cdot R^{\prime}\quad\text{for
all}\quad\lambda\in{\mathbf{R}}_{\geq 0}.$

In particular, $g$ and $g^{\prime}$ have the same $F$-jumping numbers.

The first assertion is a very special case of [HT, Theorem 3.3], which applies
since the homomorphism $R\to R^{\prime}$ is finite and étale. The second
assertion is an immediate consequence.

##  3\. The proof of the main result

Our goal is to prove the following more general version of Theorem 1.3.

######  Theorem 3.1.

Let $X$ be a smooth, irreducible complex algebraic variety,
$f\in\mathcal{O}_{X}(X)$ nonzero, $\lambda>0$ a jumping number of $f$, and
$\lambda^{\prime}<\lambda$ such that $\mathcal{J}(f^{\alpha})$ takes the same
value for all $\alpha\in[\lambda^{\prime},\lambda)$. If $(X_{A},f_{A})$ gives
a model of $(X,f)$ over $A$, then after possibly replacing $A$ by a
localization $A_{a}$, for every closed point $t\in{\rm Spec}(A)$ with ${\rm
char}\,k(t)=p_{t}$, there is an $F$-jumping number $\mu$ for $f_{t}$ in the
interval $(\lambda^{\prime},\lambda]$, and for every such $\mu$, if
$(p_{t}-i)\lambda\in{\mathbf{Z}}$, then

$\mu>\lambda-\frac{m_{i}}{p_{t}},$

where $m_{i}=\max\big{\\{}\beta\mid b_{f}(-\beta)=0,\beta-(\lambda
i+1-\lceil\lambda i\rceil)\in{\mathbf{Z}}_{\geq 0}\\}$.

######  Remark 3.2.

In the setting of Theorem 3.1, if $\lambda=\tfrac{a}{b}$, with $a$, $b$
relatively prime, and

$I=\\{i\mid 1\leq i\leq b,\,{\rm gcd}(i,b)=1\\},$

then for every $t$ there is $i\in I$ such that
$(p_{t}-i)\lambda\in{\mathbf{Z}}$. The assertion in the theorem thus implies
that if $m=\max_{i\in I}m_{i}$, then $\mu>\lambda-\frac{m}{p_{t}}$.

In fact, we can get a uniform bound just in terms of $n=\dim(X)$. More
precisely, it follows from [Saito_microlocal] that all roots of $b_{f}(s)$ are
$>-n$; therefore, in the situation in the theorem, we have
$\mu>\lambda-\tfrac{n}{p_{t}}$ for all $t$.

Note that by taking $\lambda=\operatorname{lct}(f)$ in Theorem 3.1, we deduce
the assertion in Theorem 1.3. Hence from now on we focus on Theorem 3.1.

######  Remark 3.3.

If $X$, $f$, and $A$ are as in the theorem, and $\lambda\in{\mathbf{R}}_{>0}$
is not a jumping number of $f$, then there is $\lambda^{\prime}<\lambda$ such
that $\mathcal{J}(f^{\lambda})=\mathcal{J}(f^{\lambda^{\prime}})$. Applying
Theorem 1.1 for both $\lambda$ and $\lambda^{\prime}$, we see that if
$t\in{\rm Spec}(A)$ is a closed point with ${\rm char}\,k(t)\gg 0$, then there
is no $F$-jumping number of $f_{t}$ in the interval
$(\lambda^{\prime},\lambda]$.

Before giving the proof of Theorem 3.1, we make a few preliminary remarks
concerning models for $(X,f)$ over $A$. This is standard material, for more
details we refer to [MS, Section 2.2].

The assumption in Theorem 3.1 is that $A\subseteq{\mathbf{C}}$ is a finite
type algebra over ${\mathbf{Z}}$ and $X_{A}$ is a scheme of finite type over
$A$ and $f_{A}\in\mathcal{O}_{X_{A}}(X_{A})$ are such that we have an
isomorphism $X_{A}\times_{A}{\mathbf{C}}\simeq X$ such that the pull-back of
$f_{A}$ is mapped to $f$. By generic smoothness, after possibly replacing $A$
by a localization $A_{a}$, we may and always assume that $X_{A}$ is smooth
over ${\rm Spec}(A)$ and $f_{A}$ defines a relative Cartier divisor in $X_{A}$
over ${\rm Spec}(A)$.

If $A\subseteq A^{\prime}\subseteq{\mathbf{C}}$, where $A^{\prime}$ is another
finite type algebra over ${\mathbf{Z}}$, then we have
$X_{A^{\prime}}=X\times_{A}A^{\prime}$ and the image $f_{A^{\prime}}$ of
$f_{A}$ in $\mathcal{O}_{X_{A^{\prime}}}(X_{A^{\prime}})$, which give a model
for $(X,f)$ over $A^{\prime}$. By general properties of finite type morphisms,
there is a nonzero $a\in A$ such that ${\rm Spec}(A_{a})$ is contained in the
image of ${\rm Spec}(A^{\prime})$. Moreover, if $t\in{\rm Spec}(A_{a})$ is a
closed point, then there is a closed point $t^{\prime}\in{\rm
Spec}(A^{\prime})$ whose image is $t$. In particular, the field extension
$k(t)\hookrightarrow k(t^{\prime})$ is an extension of finite fields, hence we
may apply Proposition 2.1 to the elements of an affine open cover of $X_{t}$
to conclude that $f_{t}$ and $f_{t^{\prime}}$ have the same $F$-jumping
numbers. Therefore it is enough to prove the theorem for $X_{A^{\prime}}$ and
$f_{A^{\prime}}$, hence we are free to replace $A$ by a larger algebra with
the same properties.

We can now prove our main result.

###### Proof of Theorem 3.1.

If $X=U_{1}\cup\ldots\cup U_{N}$ is an affine open cover, then $b_{f}={\rm
lcm}(b_{f}|_{U_{i}})$. Since the set of jumping numbers of $f$ is the union of
the sets of jumping numbers of $f|_{U_{i}}$, for $1\leq i\leq N$, it is
straightforward to see that it is enough to prove the assertion in the theorem
for each $U_{i}$. Hence from now on we may and will assume that $X={\rm
Spec}(R)$ is affine.

Suppose that $X_{A}={\rm Spec}(R_{A})$. For every closed point $t\in{\rm
Spec}(A)$, let $R_{t}=R_{A}\otimes_{A}k(t)$ be the $k(t)$-algebra
corresponding to $X_{t}$. By definition of the Bernstein-Sato polynomial,
there is $P(s)\in D_{R}[s]$ such that

$b_{f}(s)f^{s}=P(s)\bullet f^{s+1},$

where $D_{R}$ is the ring of differential operators of $R$. After possibly
replacing $A$ by a larger finitely generated ${\mathbf{Z}}$-subalgebra of
${\mathbf{C}}$, we may assume that the denominators of the roots of $b_{f}$
are invertible in $A$ and that we have $P_{A}(s)\in D^{(0)}_{R_{A}/A}[s]$ such
that

$b_{f}(s)f_{A}^{s}=P_{A}(s)\bullet f_{A}^{s+1},$

where $D_{R_{A}/A}^{(0)}\subseteq{\rm End}_{A}(R_{A})$ is the subring
generated by $R_{A}$ and ${\rm Der}_{A}(R_{A})$. In this case, for every
closed point $t\in{\rm Spec}(A)$ with ${\rm char}\,k(t)=p_{t}$, we get a
corresponding relation

$\overline{b_{f}}(s)f_{t}^{s}=P_{t}(s)\bullet f_{t}^{s+1},$

where $\overline{b_{f}}\in{\mathbf{F}}_{p_{t}}[s]$ is the image of $b_{f}$ and
$P_{t}(s)$ has coefficients in the subring $D_{R_{t}/k(t)}^{(0)}\subseteq{\rm
End}_{k(t)}(R_{t})$ generated by $R_{t}$ and ${\rm Der}_{k(t)}(R_{t})$. In
particular, for every $m\in{\mathbf{Z}}_{\geq 0}$, we have

(7) $\overline{b_{f}}(m)f_{t}^{m}\in D_{R_{t}/k(t)}^{(0)}\bullet
f_{t}^{m+1}\subseteq R_{t}.$

A key observation, going back to [MTW, Proposition 3.11] is that for every
proper ideal $J$ in $R_{t}$ and every $e\geq 1$, we have

(8)
$\overline{b_{f}}\big{(}\nu_{f_{t}}^{J}(p_{t}^{e})\big{)}=0\,\,\text{in}\,\,{\mathbf{F}}_{p_{t}}.$

Indeed, if we take $m=\nu^{J}_{f_{t}}(p_{t}^{e})$, then by definition we have
$f^{m}\not\in J^{[p_{t}^{e}]}$, while $f^{m+1}\in J^{[p_{t}^{e}]}$. Since the
ideal $J^{[p_{t}^{e}]}$ is preserved by the action of $D_{R_{t}/k(t)}^{(0)}$,
the assertion in (8) is a consequence of the formula in (7).

The other main ingredient in our proof is [MZ, Theorem B(i)], which says that
there is $C>0$ such that after possibly replacing $A$ by some localization
$A_{a}$, we may assume that for every closed point $t\in{\rm Spec}(A)$ with
${\rm char}\,k(t)=p_{t}$, there is always an $F$-jumping number
$\mu\in(\lambda^{\prime},\lambda]$ for $f_{t}$ and every such $F$-jumping
number satisfies

(9) $\lambda-\tfrac{C}{p_{t}}\leq\mu\leq\lambda.$

Without any loss of generality, we may and will assume that
$C\in{\mathbf{Z}}$, in which case it follows from (9) that

(10) $0\leq\lceil\lambda p_{t}\rceil-\lceil\mu p_{t}\rceil\leq C.$

Let us pick one such $t\in{\rm Spec}(A)$ and $\mu$ as above. Using the
description of $F$-jumping numbers as $F$-thresholds discussed in the previous
section, we see that there is a proper ideal $J$ in $R_{t}$ such that
$\mu=c^{J}(f_{t})$. In this case, it follows from (5) that $\lceil\mu
p_{t}\rceil=\nu^{J}_{f_{t}}(p_{t})+1$. Let us choose a positive integer $d$
such that the coefficients of $b_{f}$ lie in $\tfrac{1}{d}{\mathbf{Z}}$ and
$d\lambda^{i}\in{\mathbf{Z}}$ for $i\leq{\rm deg}(b_{f})$. We may and will
assume that $p_{t}$ does not divide $d$. We deduce from (8) that

(11) $b_{f}\big{(}\lceil\mu p_{t}\rceil-1\big{)}\equiv 0\,\,({\rm
mod}\,\,p_{t})\,\,\text{in}\,\,\tfrac{1}{d}{\mathbf{Z}}.$

Let $k=\lceil\lambda p_{t}\rceil-\lceil\mu p_{t}\rceil$, so $0\leq k\leq C$ by
(10). Since we can write $\lambda p_{t}=\lambda(p_{t}-i)+\lambda i$ and
$\lambda(p_{t}-i)\in{\mathbf{Z}}$, it follows that

$\lceil\lambda p_{t}\rceil=\lambda(p_{t}-i)+\lceil\lambda i\rceil.$

The condition (11) thus implies

$b_{f}\big{(}\lambda(p_{t}-i)+\lceil\lambda i\rceil-k-1\big{)}\equiv
0\,\,({\rm mod}\,\,p_{t})\,\,\text{in}\,\,\tfrac{1}{d}{\mathbf{Z}},$

and thus

(12) $b_{f}\big{(}\lceil\lambda i\rceil-\lambda i-k-1)\equiv 0\,\,({\rm
mod}\,\,p_{t})\,\,\text{in}\,\,\tfrac{1}{d}{\mathbf{Z}}.$

Since $k$ is an integer bounded independently of $t$ and $\mu$, it can take
only finitely many values. After possibly inverting finitely many prime
integers, we may assume that every $k$ that appears when we vary $t\in{\rm
Spec}(A)$, appears for infinitely many primes $p_{t}$. In this case, condition
(12) implies that for every such $k$, we have

(13) $b_{f}\big{(}\lceil\lambda i\rceil-\lambda i-k-1)=0.$

By definition, we thus have

(14) $k+1+\lambda i-\lceil\lambda i\rceil\leq m_{i}.$

Note now that we have

$\mu>\frac{\lceil\mu p_{t}\rceil-1}{p_{t}}=\frac{\lceil\lambda
p_{t}\rceil-k-1}{p_{t}}=\frac{\lambda(p_{t}-i)+\lceil\lambda
i\rceil-k-1}{p_{t}}\geq\lambda-\frac{m_{i}}{p_{t}},$

where the last inequality follows from (14). This completes the proof of the
theorem. ∎

We next deduce our result on the equality between $\operatorname{lct}(f)$ and
$\operatorname{fpt}(f_{t})$.

###### Proof of Corollary 1.4.

It follows from Theorem 1.3 that under our assumption, we may assume that for
every closed point $t\in{\rm Spec}(A)$ such that
$(p_{t}-1)\cdot\operatorname{lct}(f)\in{\mathbf{Z}}$, we have

$\operatorname{fpt}(f_{t})>\operatorname{lct}(f)-\frac{\operatorname{lct}(f)}{p_{t}},$

where $p_{t}={\rm char}\,k(t)$. Moreover, we also have

$\operatorname{fpt}(f_{t})\leq\operatorname{lct}(f)$

by Theorem 1.1i). In order to prove that
$\operatorname{fpt}(f_{t})=\operatorname{lct}(f)$, it is enough to show that
we can’t have
$\operatorname{fpt}(f_{t})\in\left(\operatorname{lct}(f)-\tfrac{\operatorname{lct}(f)}{p_{t}},\operatorname{lct}(f)\right)$.
This follows from (6). Indeed, let us write
$\operatorname{lct}(f)=\tfrac{a}{b}$, with ${\rm gcd}(a,b)=1$. Since
$(p_{t}-1)\cdot\operatorname{lct}(f)\in{\mathbf{Z}}$, we can write
$p_{t}-1=bc$ for some positive integer $c$. We then have

$\operatorname{lct}(f)=\frac{ac}{p_{t}-1}\quad\text{and}\quad\operatorname{lct}(f)-\frac{\operatorname{lct}(f)}{p_{t}}=\frac{ac}{p_{t}},$

hence (6) gives the assertion we need. ∎

######  Example 3.4.

Let $f=x^{2}+y^{3}\in{\mathbf{C}}[x,y]$. It is well-known that
$\operatorname{lct}(f)=\tfrac{5}{6}$ (see [Lazarsfeld, Example 9.2.15]) and in
fact

$b_{f}(s)=\left(s+\tfrac{5}{6}\right)(s+1)\left(s+\tfrac{7}{6}\right)$

(see for example [Kashiwara2, Example 6.19]). In this case we can take
$A={\mathbf{Z}}$ and the model given by $({\mathbf{A}}_{{\mathbf{Z}}}^{2},f)$,
where we view $f$ as an element of ${\mathbf{Z}}[x,y]$. If we denote by
$f_{p}$ the image of $f$ in ${\mathbf{F}}_{p}[x,y]$, with $p\gg 0$, then it
follows from Corollary 1.4 that if $p\equiv 1$ (mod $6$), then
$\operatorname{fpt}(f_{p})=\operatorname{lct}(f)=\tfrac{5}{6}$, and it follows
from Theorem 1.3 that if $p\equiv 5$ (mod $6$), then

$\operatorname{fpt}(f_{p})\geq\tfrac{5}{6}-\tfrac{7}{6p}.$

In fact, it is known that in this case
$\operatorname{fpt}(f_{p})=\tfrac{5}{6}-\tfrac{1}{6p}$, see [MTW, Example
4.3].

The following example was pointed out by the anonymous referee.

######  Example 3.5.

Consider the polynomials $f,g\in{\mathbf{C}}[x,y]$, where

$f=x^{5}+y^{4}\quad\text{and}\quad g=x^{5}+x^{3}y^{2}+y^{4}.$

The roots of $b_{f}(s)$ are the negatives of

$\tfrac{9}{20},\tfrac{13}{20},\tfrac{7}{10},\tfrac{17}{20},\tfrac{9}{10},\tfrac{19}{20},1,\tfrac{21}{20},\tfrac{11}{10},\tfrac{23}{20},\tfrac{13}{10},\tfrac{27}{20},{\bf\tfrac{31}{20}}$

and the roots of $b_{g}(s)$ are the negatives of

$\tfrac{9}{20},{\bf\tfrac{11}{20}},\tfrac{13}{20},\tfrac{7}{10},\tfrac{17}{20},\tfrac{9}{10},\tfrac{19}{20},1,\tfrac{21}{20},\tfrac{11}{10},\tfrac{23}{20},\tfrac{13}{10},\tfrac{27}{20}$

(see [Yano, Sections 11 and 18]). This is a well-known example in which $f$
and $g$ are part of a family of isolated singularities, with constant Milnor
number, but such that the Bernstein-Sato polynomials are different. In both
cases we may take $A={\mathbf{Z}}$ and the models given by
$({\mathbf{A}}^{2}_{\mathbf{Z}},f)$ and $({\mathbf{A}}^{2}_{\mathbf{Z}},g)$,
respectively, where we view $f$ and $g$ as elements of ${\mathbf{Z}}[x,y]$.
Note that we have ${\rm lct}(f)=\tfrac{9}{20}={\rm lct}(g)$ and

$\operatorname{fpt}(f_{p})=\tfrac{9}{20}\,\,\text{if}\,\,p\equiv
1\,(\text{mod}\,20)\quad\text{and}\quad\operatorname{fpt}(f_{p})=\tfrac{9p-11}{20p}\,\,\text{if}\,\,p\equiv
19\,(\text{mod}\,20)$

(this follows, for example, from [Hernandez, Theorem 3.1]), while

$\operatorname{fpt}(g_{p})=\tfrac{9}{20}\,\,\text{if}\,\,p\equiv
1\,(\text{mod}\,20)\quad\text{and}\quad\operatorname{fpt}(g_{p})=\tfrac{9p-11}{20(p-1)}\,\,\text{if}\,\,p\equiv
19\,(\text{mod}\,20)$

(see [MTW, Example 4.5]). Note that this matches the predictions provided by
Corollary 1.4, namely that

$\operatorname{fpt}(f_{p})=\operatorname{lct}(f)\quad\text{and}\quad\operatorname{fpt}(g_{p})=\operatorname{lct}(g)\quad\text{if}\quad
p\equiv 1\,(\text{mod}\,20)$

and those provided by Theorem 1.3, namely that

$\operatorname{fpt}(f_{p})>\operatorname{lct}(f)-\tfrac{31}{20p}\quad\text{and}\quad\operatorname{fpt}(g_{p})>\operatorname{lct}(g)-\tfrac{11}{20p}\quad\text{if}\quad
p\equiv 19\,(\text{mod}\,20).$

Note that in this case the bound satisfied by $g$ when $p\equiv 19$ (mod $20$)
is _not_ satisfied by $f$ (in fact, for $f$ this becomes equality), but this
is allowed due to the presence of the root $-\tfrac{31}{20}$ of $b_{f}(s)$.

######  Remark 3.6.

Suppose now that $X$ is a smooth, irreducible, $n$-dimensional complex
algebraic variety and $\mathfrak{a}$ is an arbitrary nonzero coherent ideal
sheaf on $X$. Recall that we can associate multiplier ideals and test ideals
to non-principal ideal as well (see [Lazarsfeld, Chapter 9] for the case of
multiplier ideals and [BMS1] for the case of test ideals) Let $\lambda>0$ be a
jumping number of $\mathfrak{a}$, and let $\lambda^{\prime}<\lambda$ be such
that $\mathcal{J}(\mathfrak{a}^{\alpha})$ takes the same value for all
$\alpha\in[\lambda^{\prime},\lambda)$. We fix an integer $d>\lambda$. We claim
that if $(X_{A},\mathfrak{a}_{A})$ gives a model of $(X,\mathfrak{a})$, then
after possibly replacing $A$ by a localization $A_{a}$, for every closed point
$t\in{\rm Spec}(A)$ with ${\rm char}\,k(t)=p_{t}$, there is an $F$-jumping
number $\mu$ for $\mathfrak{a}_{t}$ in the interval
$(\lambda^{\prime},\lambda]$, and for every such $\mu$, we have

(15) $\mu>\lambda-\frac{dn}{p_{t}}.$

Indeed, in order to see this, we may again assume that $X={\rm Spec}(R)$ is
affine and let $f_{1},\ldots,f_{r}\in R$ be generators of $\mathfrak{a}$. For
$1\leq i\leq d$, let $h_{i}$ be a general linear combination of
$f_{1},\ldots,f_{r}$, with ${\mathbf{C}}$-coefficients, and let
$h=\prod_{i=1}^{d}h_{i}$. In this case it follows from [Lazarsfeld,
Proposition 9.2.28] that

(16)
$\mathcal{J}(\mathfrak{a}^{\alpha})=\mathcal{J}(h^{\alpha/d})\quad\text{for
all}\quad\alpha\leq\lambda.$

After possibly enlarging $A$, we may assume that we have a model $h_{A}\in
R_{A}$ for $h$, such that $h_{A}\in\mathfrak{a}_{A}^{d}$.

We may assume the existence of an $F$-jumping number for $\mathfrak{a}_{t}$ in
the interval $(\lambda^{\prime},\lambda]$ by Theorem 1.1 (more precisely, by
its version for arbitrary ideals), hence we only need to prove that

(17)
$\tau(\mathfrak{a}_{t}^{\lambda^{\prime}})=\tau(\mathfrak{a}_{t}^{\lambda-\tfrac{dn}{p_{t}}}).$

Note that we may assume that we have the equalities

(18)
$\tau(\mathfrak{a}_{t}^{\lambda^{\prime}})=\mathcal{J}(\mathfrak{a}^{\lambda^{\prime}})_{t}=\mathcal{J}(h^{\tfrac{\lambda^{\prime}}{d}})_{t}=\tau(h_{t}^{\tfrac{\lambda^{\prime}}{d}})=\tau(h_{t}^{\tfrac{\lambda}{d}-\tfrac{n}{p_{t}}}),$

where the first and third equalities follow from (1.1), the second one follows
from (16), and the fourth one follows from Remark 3.2. On the other hand, we
have the inclusions

(19)
$\tau(h_{t}^{\tfrac{\lambda}{d}-\tfrac{n}{p_{t}}})\subseteq\tau(\mathfrak{a}_{t}^{\lambda-\tfrac{dn}{p_{t}}})\subseteq\tau(\mathfrak{a}_{t}^{\lambda^{\prime}}),$

where the first inclusion follows from $h_{t}\in\mathfrak{a}_{t}^{d}$ and the
second one follows from $\lambda^{\prime}\leq\lambda-\tfrac{dn}{p_{t}}$. By
combining (18) and (19), we obtain, in particular, the equality (17). This
completes the proof of our assertion.

We finally note that for an arbitrary ideal $\mathfrak{a}$ there is a notion
of Bernstein-Sato polynomial, whose roots are negative rational numbers, see
[BMS]. It is an interesting question whether the bound in (15) can be refined
by taking into account these roots as in Theorem 3.1. In fact, everything in
the proof of the theorem carries through in this setting, with the exception
of one point: it is not clear that having $\lambda p_{t}-C\leq\mu p_{t}$
implies that $\lambda p_{t}-\nu^{J}_{\mathfrak{a}}(p_{t})$ stays bounded.

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