DISCRETENESS OF F -JUMPING NUMBERS AT ISOLATED
NON-Q-GORENSTEIN POINTS

arXiv:1605.03825v2 [math.AG] 11 Apr 2017

PATRICK GRAF AND KARL SCHWEDE
Abstract. We show that the F -jumping numbers of a pair (X, a) in positive characteristic have no limit points whenever the symbolic Rees algebra of −KX is finitely generated
outside an isolated collection of points. We also give a characteristic zero version of this result, as well as a generalization of the Hartshorne–Speiser–Lyubeznik–Gabber stabilization
theorem describing the non-F -pure locus of a variety.

1. Introduction
By now it is well understood that there is an interesting connection between multiplier
ideals in characteristic zero, defined via resolution of singularities, and test ideals in positive
characteristic, defined via the behavior of the Frobenius map. Recall that for any complex
pair (X, a), the multiplier ideal J (X, at ) gets smaller as t increases, but it does not change
if we increase t just slightly: J (X, at ) = J (X, at+ε ) for 0 < ε ≪ 1. Hence it makes sense
to define the jumping numbers of (X, a) as those real numbers ti such that J (X, ati ) (
J (X, ati −ε ) for ε > 0. By analogy, the F -jumping numbers are the real numbers ti where
the test ideal jumps or changes: τ (X, ati ) ( τ (X, ati −ε ) for ε > 0.
The discreteness and rationality of (F -)jumping numbers has been studied by many authors, e.g. [ELSV04, BdFFU15, Har06, BMS08, BMS09, KLZ09, BSTZ10, KZ14, ST14,
KSSZ14]. In characteristic zero, discreteness and rationality of jumping numbers is elementary if X is Q-Gorenstein, but rationality fails in general [Urb12, Theorem 3.6]. Discreteness
remains an open problem, with several special cases known, e.g. if the non-Q-Gorenstein
locus of X is zero-dimensional [Gra16, Theorem 1.4] (and [Urb12, Theorem 5.2] for an earlier, weaker version). For test ideals, discreteness and rationality are known whenever the
algebra of local sections
M


R X, −(KX + ∆) :=
OX ⌊−m(KX + ∆)⌋
m≥0

(also known as the symbolic Rees algebra) is finitely generated [BSTZ10, Sch11b, CEMS14].
In this paper, we prove the following result.
Theorem A (Discreteness of F -jumping numbers, Theorem 4.2). Suppose that X is a
normal variety over an F -finite field
 k of positive characteristic and that ∆ ≥ 0 is a Qdivisor such that R X, −(KX + ∆) is finitely generated except at an isolated collection of
points. Suppose a ⊆ OX is a nonzero coherent ideal sheaf. Then the F -jumping numbers of
(X, ∆, a) have no limit points.
Date: April 11, 2017.
2010 Mathematics Subject Classification. 13A35, 14F18.
Key words and phrases. Test ideals, F -jumping numbers, Q-Gorenstein, multiplier ideals.
The first-named author was supported in part by the DFG grant “Zur Positivität in der komplexen
Geometrie”. The second named author was supported in part by the NSF grant DMS #1064485, NSF FRG
Grant DMS #1501115, NSF CAREER Grant DMS #1501102.
1

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P. GRAF AND K. SCHWEDE

The corresponding statement in characteristic zero is also new.
Theorem B (Discreteness of jumping numbers, Theorem 4.3). Suppose that X is a normal variety over a field k of characteristic zero and that ∆ ≥ 0 is a Q-divisor such that
R X, −(KX + ∆) is finitely generated except at an isolated collection of points. Suppose
a ⊆ OX is a nonzero coherent ideal sheaf. Then the jumping numbers of (X, ∆, a) have no
limit points.
We would like to point out that the question whether F -jumping numbers are always
rational is still open. However the characteristic zero counterexample mentioned above
suggests that maybe one should expect a negative answer.
The method used to prove these results builds upon [Urb12] and [Gra16]. In particular,
we prove global generation of (Frobenius pushforwards of) sheaves used to compute test
ideals after twisting by a sufficiently ample divisor H. If X is projective, discreteness of
F -jumping numbers follows quickly, since the twisted test ideals are globally generated by
vector subspaces within the finite-dimensional vector space H 0 (X, OX (H)). The general
case is easily reduced to the projective case by a compactification argument.
Using these same methods, we also obtain a generalization of the Hartshorne–Speiser–
Lyubeznik–Gabber stabilization theorem. Let us motivate the result briefly. Notice that
if R is a ring, we have canonical maps HomR (F∗e R, R) −
→ R obtained by evaluation at 1.
These images yield a descending chain of ideals Je . If KR is Cartier, it follows from [HS77,
Lyu97, Gab04] that these images stabilize, giving a canonical scheme structure to the nonF -pure locus of X = Spec R. Blickle and Böckle also proved a related stabilization result
for arbitrary rings (and even more) [BB11, Bli09] but their result does not seem to imply
that Je = Je+1 for e ≫ 0 (Blickle obtained another result which implies stabilization of a
different set of smaller ideals). However, as a corollary of our work, we obtain the following
generalization, also see [CEMS14, Proposition 3.7].
Theorem C (HSLG-type stabilization, Theorem 4.4). Suppose that X is a normal variety
over an F -finite field k of characteristic p > 0. Set



eval@1
Je := Image F∗e OX (1 − pe )KX ∼
= Hom OX (F∗e OX , OX ) −−−−→ OX ⊂ OX .
If R(X, −KX ) is finitely generated except at an isolated collection of points, then Je = Je+1
for all e ≫ 0.

Remark 1.1. There should be a more general version of Theorem 4.4 with R X, −(KX +∆)
in place of R(X, −KX ), but for the proof one would probably need to generalize Theorem 3.1
further.
We end the introduction by pointing out some geometric and cohomological conditions
on the singularities of X which ensure that our assumptions on the anticanonical algebra
of X are satisfied.
Proposition 1.2 (Klt or rational singularities and finite generation). Let X be a normal
variety over a field k. Assume any of the following:
(1.2.1) char(k) = 0 and there is a Q-divisor D ≥ 0 such that the pair (X, D) is klt except
at an isolated collection of points.
(1.2.2) dim X ≤ 3 and X has pseudorational singularities except at an isolated collection
of points.
Then for any Q-Weil divisor B on X, the algebra of local sections R(X, B) is finitely generated except at an isolated collection of points. Hence the assumptions of Theorems 4.2, 4.3
and 4.4 are satisfied in this case.

F -JUMPING NUMBERS AT ISOLATED NON-Q-GORENSTEIN POINTS

3

For the definition of pseudorational singularities, see [LT81, Section 2, p. 102]. An equivalent condition, emphasizing the point of view of extendability of differential forms, is given
in [LT81, Section 4, Corollary on p. 107]. From the latter condition, it is easy to see that
for a klt threefold pair (X, D), the space X has pseudorational singularities except at an
isolated collection of points since X is Cohen–Macaulay in codimension two anyways.
Acknowledgements. The authors first began working on this project at the 2015 Summer
Research Institute on Algebraic Geometry held at the University of Utah. Furthermore they
would like to thank the referee for helpful suggestions.
2. Preliminaries
Convention 2.1. Throughout this paper, all schemes are Noetherian and separated and of
finite type over a field which in characteristic p > 0 is always assumed to be F -finite. In
characteristic p > 0, F : X −
→ X denotes the absolute Frobenius map, acting on an affine
scheme U = Spec R by r 7→ r p .
The material in this section is mostly well-known to experts, and collected for convenience
of the reader.
2.1. Grothendieck duality. We will use the following special case of Grothendieck dual→ Y be a finite map and F , G coherent sheaves
ity [KM98, Proposition 5.67]. Let f : X −
on X and on Y , respectively. Then there is a natural f∗ OX -linear isomorphism
(2.1.1)

Hom OY (f∗ F , G ) = f∗ Hom OX (F , f ! G ),

where f ! G := Hom OY (f∗ OX , G ). Furthermore we will use the fact that if X is essentially of
finite type over an F -finite field, then F ! ωX ∼
= ωX where ωX is the canonical sheaf of X.
Suppose now that X is a normal integral scheme of finite type over an F -finite field of
characteristic p > 0. Then we have a canonical map (called the trace map)
F∗e ωX −
→ ωX
which under (2.1.1) corresponds to id ∈ F∗e Hom OX (ωX , ωX ). Let KX be a canonical divisor
on X. Twisting by OX (−KX ) and reflexifying yields

→ OX
F∗e OX (1 − pe )KX −

and then for any effective Weil divisor D ≥ 0 by restriction we obtain a map

→ OX .
(2.1.2)
tr : F∗e OX (1 − pe )KX − D −


Using (2.1.1) again, the left-hand side sheaf is identified with Hom OX F∗e OX (D), OX . It
is straightforward to check that under this identification, (2.1.2) becomes the “evaluation
at 1” map
 eval@1
Hom OX F∗e OX (D), OX −−−−→ OX .
2.2. Test ideals. We recall the following definition of test ideals from the literature.
Definition 2.2 ([HT04, Lemma 2.1], [Sch11c, Proof of 3.18]). If X is a normal F -finite
scheme, ∆ ≥ 0 is a Q-divisor on X, a is an ideal sheaf and t ≥ 0 is a real number then for
any sufficiently large effective Weil divisor C, we define the sheaf
X X 

e
τ (X, ∆, at ) :=
φ F∗e a⌈t(p −1)⌉ · OX (−C)
e
e≥0 φ∈C∆



where C∆e = Hom OX F∗e OX ⌈(pe − 1)∆⌉ , OX .

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P. GRAF AND K. SCHWEDE

Remark 2.3. The choice of C is philosophically the same as the choice of a test element
usually included in the (local) literature mentioned above. Note that for any affine chart
U = Spec R ⊆ X, if c ∈ R is an appropriate test element and C is a Weil divisor on X
such that OX (−C) U ⊆ c · OU , then one can always find another test element d ∈ R with
d · OU ⊆ OX (−C) U . It follows that our definition of the test ideal is indeed independent
of the choice of C.
Lemma 2.4. Notation as above. Then for any e0 ≥ 0, we have
X X 

e
τ (X, ∆, at ) =
φ F∗e a⌈t(p −1)⌉ · OX (−C) .
e
e≥e0 φ∈C∆

Proof. The inclusion “⊇” is clear. For “⊆”, choose C1 a sufficiently large Cartier divisor
such that OX (−C1 ) is contained in the right-hand side, and put C ′ = C + pe0 −1 C1 . Then
we see that
X X 
 X X  e ⌈t(pe −1)⌉

e
φ F∗e a⌈t(p −1)⌉ · OX (−C ′ ) ⊆
φ F∗ a
· OX (−C)
e
e≥0 φ∈C∆

e
e≥e0 φ∈C∆

Pe0 −1
P
since we can split up the sum on the left-hand side as e=0
( · · · ) + e≥e0 ( · · · ). But the
left-hand side is just τ (X, ∆, at ), hence the lemma is proved.

If C is in fact Cartier, an easy direct computation yields

X

e ⌈t(pe −1)⌉
φ
F
a
·
O
(−C)
X
∗
e
φ∈C∆
i
h
 eval@1


e
= Image F∗e a⌈t(p −1)⌉ · Hom OX F∗e OX ⌈(pe − 1)∆⌉ + C , OX −−−−→ OX ,
hence
τ (X, ∆, at )
h
i
X


 eval@1
e
=
Image F∗e a⌈t(p −1)⌉ · Hom OX F∗e OX ⌈(pe − 1)∆⌉ + C , OX −−−−→ OX
e≥e0
i
h 
X
 tr
e
=
Image F∗e a⌈t(p −1)⌉ · OX (1 − pe )KX − ⌈(pe − 1)∆⌉ − C −−→ OX
e≥e0
i
h 
X
 tr
e
=
Image F∗e a⌈t(p −1)⌉ · OX ⌊(1 − pe )(KX + ∆)⌋ − C −−→ OX .
e≥e0

However, for our purposes this definition of the test ideal is not quite optimal. Fortunately,
this is easy after adjusting C.
Lemma 2.5. With notation as above, assume that C is Cartier. Then for any e0 ≥ 0 we
have
h 
i
X
 tr
e
Image F∗e a⌈tp ⌉ · OX ⌊(1 − pe )(KX + ∆)⌋ − C −−→ OX .
τ (X, ∆, at ) =
e≥e0

Proof. Without loss of generality, and in view of Remark 2.3, we may assume that X =
Spec R is affine and that OX (−C) = c · OX for some c ∈ R. Choose b ∈ a⌈t⌉ nonzero, so that
e
e
e
ba⌈t(p −1)⌉ ⊆ a⌈t⌉ a⌈t(p −1)⌉ ⊆ a⌈tp ⌉ . Replacing C = div(c) by div(bc) we obtain our desired
formula.


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5

2.3. Multiplier ideals. In this section we work over a field of characteristic zero. The
theory of non-Q-Gorenstein multiplier ideals was developed by [DH09]. The starting point
is a notion of pullback for Weil divisors.
Definition 2.6 ([DH09, Def. 2.6]). Let f : Y −
→ X be a proper birational morphism
between normal varieties, and let D be an integral Weil divisor on X. Then the natural
pullback f ♮ D of D along f is defined by
∗∗
OY (−f ♮D) = OX (−D) · OY ,
where we consider OX (−D) ⊂ KX as a fractional ideal sheaf on X.

We will consider triples (X, ∆, at ) consisting of a normal variety X, an effective Q-divisor
∆ ≥ 0, a nonzero coherent ideal sheaf a ⊂ OX and a real number t ≥ 0. In the case ∆ = 0,
the following definition was made in [DH09, Definition 4.8].
Definition 2.7 ([CEMS14, Def. 2.19]). Let (X, ∆, at ) be a triple, and let m ∈ N be a
positive integer such that m∆
→ X be a log resolution of the pair
 is integral. Let f : Y −
X, OX (−m(KX + ∆)) + a in the sense of [DH09, Definition 4.1]. Let Z be the Cartier
divisor on Y such that a · OY = OY (−Z). Then we define



1 ♮
Jm (X, ∆, at ) := f∗ OY KY − m
f m(KX + ∆) − tZ .
One shows that this is a coherent ideal sheaf on X, independent of the choice of the
resolution f . Furthermore, Jm (X, ∆, at ) ⊂ Jkm(X, ∆, at ) for any integer k > 0. Thus by
the Noetherian property of X, the following definition makes sense.
Definition 2.8. The multiplier ideal J (X, ∆, at ) of a triple as above is defined to be the
unique maximal element of the family

Jm (X, ∆, at ) | m ≥ 1 and m∆ is integral ,
i.e. it is equal to Jm (X, ∆, at ) for m sufficiently divisible.

We will need the following notion of compatible boundaries, which is a straightforward
generalization of the ∆ = 0 case in [DH09, Definition 5.1].
Definition 2.9. Let (X, ∆, at ) be a triple, and fix an integer m ≥ 2 such that m∆ is
integral. Given a log resolution f : Y −
→ X of X, OX (−m(KX + ∆)) + a , a Q-Weil divisor
∆′ on X is called m-compatible for (X, ∆, at ) with respect to f if the following hold:
(i) KX + ∆ + ∆′ is Q-Cartier,
(ii) m∆′ is integral and ⌊∆′ ⌋ = 0,
(iii) no component of ∆′ is contained in supp(∆) ∪ supp(OX /a),


(iv) f is a log resolution of (X, ∆ + ∆′ ), OX (−m(KX + ∆)) + a ,

1 ♮
(v) KY + f∗−1 ∆′ − f ∗ (KX + ∆ + ∆′ ) = KY − m
f m(KX + ∆) .

Proposition 2.10. Let (X, ∆, at ) be a triple, and fix an integer m ≥ 2 such that m∆ is
integral and J (X, ∆, at ) = Jm (X, ∆, at ). Then for any m-compatible boundary ∆′ we have

J (X, ∆, at ) = J (X, ∆ + ∆′ ); at ,

where the right-hand side is a multiplier ideal in the usual Q-Gorenstein sense [Laz04b,
Definition 9.3.60].
Proof. The proof is analogous to [DH09, Proposition 5.2], and thus it is omitted.



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P. GRAF AND K. SCHWEDE

The existence of compatible boundaries is ensured by the following theorem, cf. [DH09,
Theorem 5.4] and [Gra16, Theorem 4.4]. The Weil index of a triple (X, ∆, at ) is defined to
be the smallest positive integer m0 such that m0 (KX + ∆) is integral.
Theorem 2.11. Let (X, ∆, at ) be a triple of Weil index m0 , and let k ≥ 2 be an integer.
Choose an effective Weil divisor D on X such that m0 (KX + ∆) − D is Cartier, and let
L ∈ Pic X be a line bundle such that L (−kD) := L ⊗
 OX (−kD) is globally generated.
Pick a finite-dimensional subspace V ⊂ H 0 X, L (−kD) that generates L (−kD), and let
M be the divisor of a general element of V . Then
1
M
∆′ :=
km0
is a (km0 )-compatible boundary for (X, ∆, at ).


Proof. Let f : Y −
→ X be a log resolution of (X, ∆), OX (−km0 (KX +∆))+OX (−kD)+a ,
and set E := f ♮ (kD). Then



f ♮ km0 (KX + ∆) = f ♮ km0 (KX + ∆) − kD + kD = km0 · f ∗ KX + ∆ − m10 D + E

and so we have
KY −

1
♮
km0 f


km0 (KX + ∆) = KY − f ∗ KX + ∆ −

1
m0 D



−

1
km0 E.

Let M be the divisor of a general element of V . Then since L (−kD) is generated by V ,
we see that M is reduced with no component contained in supp(∆) ∪ supp(OX /a). Put
G = M + kD, a Cartier divisor. Since also f ∗ L ⊗ OY (−E) is generated by the (pullbacks
of the) sections in V , we have that f ∗ G = f∗−1 M + E.
1
M . Then by the above, conditions (ii)–(iv) of Definition 2.9 are
Now set ∆′ = km
0
satisfied. Also it is clear that
KX + ∆ + ∆′ = KX + ∆ −

1
m0 D

+

1
km0 G

is Q-Cartier, so (i) is fulfilled. To check (v), note that
KY + f∗−1 ∆′ − f ∗ (KX + ∆ + ∆′ )


1
= KY + f∗−1 ∆′ − f ∗ KX + ∆ + ∆′ − km
G
−
0

1
E
= KY − f ∗ KX + ∆ − m10 D − km
0

1
= KY − km
f ♮ km0 (KX + ∆) .
0

1
∗
km0 f G

This proves the theorem.



3. Global generation at isolated non-finitely generated points
The following theorem is a positive characteristic version of [Gra16, Theorem 7.1]. But
even in characteristic zero (cf. Remark 3.2), the present result is stronger than that theorem.
Most notably, we remove the divisibility condition in [Gra16, Theorem 7.1] and replacethe
“KX + ∆ is Q-Cartier” condition with the weaker requirement that R X, −(KX + ∆) be
finitely generated.
Theorem 3.1. Let X be a normal projective d-dimensional variety over an F -finite field
k of characteristic p > 0. Further let D be a Weil Q-divisor L
and C a Cartier divisor on
X. Suppose that W ⊆ X is a closed set such that R(X, D) := m≥0 OX (⌊mD⌋) is finitely
generated on X \ W , let W0 be the set of isolated points of W and put
U = (X \ W ) ∪ W0 .

F -JUMPING NUMBERS AT ISOLATED NON-Q-GORENSTEIN POINTS

7

Then there exists an ample Cartier divisor H on X such that for all e ≥ 0, m ≥ 0,
ℓ ≥ max{m, pe } and any nef Cartier divisor N , the sheaf

F∗e OX ⌊mD + ℓH⌋ − C + N

is globally generated on U (as an OX -module). Furthermore, for any ample Cartier divisor
H ′ on X fixed in advance, H can be taken to be a sufficiently high multiple of H ′ .

Remark 3.2. Theorem 3.1 continues to hold over an arbitrary field k of characteristic zero
if one interprets F = idX and pe = 1. The proof does not require any changes.
Proof of Theorem 3.1. The strategy is similar to [Gra16, Theorem
 7.1]. We will find a
→ F∗e OX ⌊mD + ℓH⌋ − C + N so that the cokernel is
globally generated sheaf F∗e Fm,ℓ ֒−
supported on W . The proof is divided into three steps.
Step 1: Blowing up. Let N0 be a positive integer such that N0 D is an integral Weil divisor.
It follows that the Veronese subring R(X \ W, N0 D) is also Noetherian and R(X \ W, D)
is a finite R(X \ W, N0 D)-module [GHNV90, Lemma 2.4]. By [GHNV90, Theorem 3.2(3)],
making N0 more divisible if necessary we may assume that R(X \ W, N0 D) is generated in
degree 1 as a graded ring.
Let f : Y −
→ X be the normalized blowup of the fractional ideal sheaf OX (N0 D). Then
we have f −1 (X \ W ) = Proj R(X \ W, N0 D). In particular, f is a small morphism over
X \ W by [KM98, Lemma 6.2]. Furthermore if we write
.
∗
OY (B) = f OX (N0 D) torsion = OX (N0 D) · OY ,
then B is Cartier and f -ample by [Gra16, Theorem 6.2]. Thus by [Har77, II, Proposition
7.10] or [KM98, Prop. 1.45] there exists a very ample Cartier divisor A on X so that B +f ∗ A
is globally ample on Y .
Step 2: Vanishing. Now for any integer m ∈ N0 , write uniquely
m = q m N0 + r m

with 0 ≤ rm ≤ N0 − 1,

i.e. qm = ⌊m/N0 ⌋. Fix a nef Cartier divisor N on X and form the sheaf


Gm := OY (qm B + f ∗ N ) ⊗ OX ⌊rm D − C⌋ · OY .
|
{z
}
=:Hm

Claim 3.3. There exists an m0 ≥ 1 such that for all ℓ ≥ m ≥ m0 , i ≥ 1, and any nef
Cartier divisor P on Y we have

(3.3.1)
H i Y, Gm ⊗ OY (ℓf ∗ A + P ) = 0 and
(3.3.2)

f∗ Gm ≃ R f∗ Gm .

Proof of Claim 3.3. Let us make two easy observations: Firstly, the sheaf Hm can take on
only finitely many values. Secondly, qm −
→ ∞ as m −
→ ∞. Hence (3.3.1) follows from Fujita
vanishing [Fuj83, Theorem 1] applied to the ample divisor B + f ∗ A on Y upon writing

H i Y, Gm ⊗ OY (ℓf ∗ A + P )

= H i Y, Hm ⊗ OY (qm B + ℓf ∗ A + f ∗ N + P )

= H i Y, Hm ⊗ OY qm (B + f ∗ A) + (ℓ − qm ) f ∗ A + f ∗ N + P .
| {z }
≥0

Similarly, making m0 even larger if necessary, (3.3.2) follows from relative Serre vanish
ing [Laz04a, Theorem 1.7.6] for the f -ample divisor B.

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P. GRAF AND K. SCHWEDE

Claim 3.4. Put H = b · A where b > d = dim X. Then for every e ≥ 0, m ≥ m0 ,
and ℓ ≥ max{m, pe }, the sheaf F∗e f∗ Gm ⊗ OX (ℓH) is 0-regular with respect to A, and
hence globally generated as an
 OX -module. Furthermore its first cohomology group vanishes,
H 1 X, F∗e f∗ Gm ⊗ OX (ℓH) = 0.
Proof of Claim 3.4. We need to show that for every 1 ≤ j ≤ d, we have


H j X, F∗e f∗ Gm ⊗ OX (ℓH) ⊗ OX (−jA) = 0.
The left-hand side is


e
e
j
= H X, F∗ f∗ Gm ⊗ OX (ℓb − p j)A

= H j X, f∗ Gm ⊗ OX (ℓb − pe j)A

= Hj X, R f∗ Gm ⊗ OX (ℓb − pe j)A

= H j Y, Gm ⊗ OY (ℓb − pe j)f ∗ A
=0

projection formula
F e is finite(a)
(3.3.2)
composition of derived functors
(3.3.1) and ℓb − pe j ≥ m.

The assertion ℓb − pe j ≥ m in the last line is justified since
ℓb ≥ ℓ + ℓd ≥ m + pe d ≥ m + pe j.
Any coherent sheaf 0-regular with respect to A is globally generated by Castelnuovo–
Mumford regularity [Laz04a, Theorem 1.8.5]. The desired vanishing

H 1 X, F∗e f∗ Gm ⊗ OX (ℓH) = 0
follows by the same argument as above, leaving off the OX (−jA).



Step 3: Reflexification and global generation.

Claim 3.5. For any e, m, ℓ ≥ 0, the reflexive hull of F∗e f∗ Gm ⊗ OX (ℓH) is equal to

F∗e OX ⌊mD + ℓH⌋ − C + N .

Furthermore the cotorsion of F∗e f∗ Gm ⊗ OX (ℓH) is supported on W , i.e. in the natural
short exact sequence


0 −→ F∗e f∗ Gm ⊗ OX (ℓH) −→ F∗e OX ⌊mD + ℓH⌋ − C + N −→ Qe,m,ℓ −→ 0
the support of Qe,m,ℓ is contained in W .
Proof of Claim 3.5. Let V ⊂ X be the maximal open subset over which f is an isomorphism.
Note that codimX (X \ V ) ≥ 2. We see that


(f∗ Gm ) V = OV (qm N0 D + N ) ⊗ OV ⌊rm D − C⌋ = OV ⌊mD⌋ − C + N ,
the first equality holding by definition and the second one because N0 D is Cartier on V .
Pushing this forward by the inclusion V ֒−
→ X, we get

∗∗
(3.5.1)
(f∗ Gm ) = OX ⌊mD⌋ − C + N .
Observe that Frobenius pushforward commutes with taking the reflexive hull. Hence twisting (3.5.1) by OX (ℓH) and applying F∗e ( · ) proves the first part of the claim. For the second
part, use the fact that f is small over X \ W and that the pushforward of a reflexive sheaf
under a small birational map is again reflexive.

(a) One may also argue by noting that F e ( · ) leaves the abelian sheaf structure, and hence sheaf coho∗

mology, unchanged.

F -JUMPING NUMBERS AT ISOLATED NON-Q-GORENSTEIN POINTS

9

Returning to the proof of Theorem 3.1, choose a point x ∈ U = (X \ W ) ∪ W0 , and
assume first that m ≥ m0 . We see that Q = Qe,m,ℓ is globally generated at x since either
◦ x ∈ W0 and then by Claim 3.5, x is an isolated point of supp Q or Q is even zero at x, or
◦ x ∈ X \ W and then Q definitely is zero at x.

Hence F∗e OX ⌊mD + ℓH⌋ − C + N is globally generated at x by Claim 3.4 and [Gra16,
Lemma 7.3].
To finish the proof, we still need to take care of the sheaves

for e ≥ 0, 0 ≤ m < m0 , and ℓ ≥ max{m, pe }.
(3.5.2)
F∗e OX ⌊mD + ℓH⌋ − C + N

To this end, notice that arguing as in the proof of Claim 3.4, for 1 ≤ j ≤ d we have


H j X, F∗e OX ⌊mD + ℓH⌋ − C + N ⊗ OX (−jA)


= H j X, OX ⌊mD⌋ − C ⊗OX (ℓb − pe j) A + N .
| {z }
|
{z
}
finitely many values

≥pe (b−d)

Hence by Fujita vanishing, taking b sufficiently large in Claim 3.4, we may assume that the
sheaves (3.5.2) are 0-regular with respect to A. In particular they are globally generated
on U .
To justify the last claim of Theorem 3.1, simply note that for any ample Cartier divisor
H ′ on X given in advance, we may pick A to be a sufficiently high multiple of H ′ and then
also H will be a multiple of H ′ .

4. Proof of main results
In this section we prove the results announced in the introduction. But first we state a
weak result on global generation of test ideals. Compare with [Mus11, Sch11a, Kee08].

Proposition 4.1. Suppose X is a normal projective variety over an F -finite field k of
characteristic p > 0, ∆ ≥ 0 is a Q-divisor, a isa nonzero coherent ideal sheaf and t0 > 0 is
a real number. Suppose that R X, −(KX + ∆) is finitely generated away from a closed set
W ⊆ X and that W0 ⊆ W is the set of isolated points of W . Set U = (X \ W ) ∪ W0 . Then
there exists an ample divisor H such that
τ (X, ∆, at ) ⊗ OX (H)
is globally generated on U for all t ∈ [0, t0 ].
Proof. Choose an effective Cartier divisor C ≥ 0 on X so that OX (−C) ⊆ τ (X, ∆, at0 ) ⊆
τ (X, ∆, at ). By Lemma 2.5, for any t ∈ [0, t0 ] we have
i
h 
X
 tr
e
(4.1.1)
τ (X, ∆, at ) =
Image F∗e a⌈tp ⌉ OX ⌊(1 − pe )(KX + ∆)⌋ − C −−→ OX .
e≥0

Now fix an ample Cartier divisor A on X so that
a⌈t⌉ ⊗ OX (A)
is globally generated for all t ∈ [0, t0 ]. We then observe that for all m > 0 and t ∈ [0, t0 ],
(4.1.2)

a⌈mt⌉ ⊗ OX (mA)

is also globally generated. The reason is that since ⌈mt⌉ ≤ m⌈t⌉, the ideal a⌈mt⌉ can be
written as the product of m ideals of the form a⌈s⌉ for various values of s ∈ [0, t0 ]. For
t for the k-vector space of global sections of the sheaf (4.1.2). It
ease of notation, write Wm
follows that for every m > 0, the map
t
Wm
⊗k OX (−mA) −
→ a⌈mt⌉

10

P. GRAF AND K. SCHWEDE

is surjective. Combining with (4.1.1), we get that
i
h 
X

e
→ OX
τ (X, ∆, at ) =
Im F∗e a⌈tp ⌉ OX ⌊(1 − pe )(KX + ∆)⌋ − C −
e≥0

=

X

i
h 

→ OX
Im F∗e Wpte ⊗k OX (−pe A) ⊗ OX ⌊(1 − pe )(KX + ∆)⌋ − C −

=

X

i
h 

→ OX .
Im F∗e Wpte ⊗k OX ⌊(pe − 1)(−KX − ∆ − A)⌋ − C − A −

e≥0

e≥0

Now choose an ample divisor H that satisfies the conclusion of Theorem 3.1 relative to the
divisor D = −(KX + ∆ + A), where C + A takes the role of C. Then for m = pe − 1, ℓ = pe
and N = 0 we get that


(4.1.3)
F∗e OX ⌊(pe − 1)(−KX − ∆ − A)⌋ − C − A ⊗ OX (H)
is globally generated (as an OX -module) over U for all e ≥ 0. Hence also
τ (X, ∆, at ) ⊗ OX (H)
is globally generated over U , being a quotient of a direct sum of sheaves of the form (4.1.3).
This completes the proof.

Theorem 4.2. Suppose that X is a normal variety over an F -finite
 field k of positive
characteristic and that ∆ ≥ 0 is a Q-divisor such that R X, −(KX +∆) is finitely generated
except at an isolated collection of points. Suppose a ⊆ OX is a nonzero coherent ideal sheaf.
Then the F -jumping numbers of (X, ∆, a) have no limit points.
Proof. By [BSTZ10, Proposition 3.28], we may assume that X is affine. Let X denote the
closure of X in some projective space. By normalizing, we may also assume that X is
normal. There exists a Q-divisor ∆ ≥ 0 on X and a coherent ideal sheaf a ⊂ OX which
restrict to ∆ and a, respectively.
Pick an arbitrary real number t0 > 0. By Proposition 4.1, we know that there exists an
ample Cartier divisor H on X such that

τ X, ∆, at ⊗ OX (H) is globally generated on X ⊂ X for all t ∈ [0, t0 ].

Note that τ X, ∆, at X = τ (X, ∆, at ) since X ⊂ X is open. Now it follows from [Gra16,
Lemma 8.2] that for any strictly increasing sequence of numbers 0 ≤ s0 < s1 < · · · < t0 ,
the corresponding sequence of test ideals
τ (X, ∆, as0 ) ⊃ τ (X, ∆, as1 ) ⊃ · · ·
stabilizes. Hence the set of F -jumping numbers of (X, ∆, a) does not have a limit point in
the interval [0, t0 ]. As t0 > 0 was chosen arbitrarily, this proves the theorem.

Theorem 4.3. Suppose that X is a normal variety overa field k of characteristic zero and
that ∆ ≥ 0 is a Q-divisor such that R X, −(KX + ∆) is finitely generated except at an
isolated collection of points. Suppose a ⊆ OX is a nonzero coherent ideal sheaf. Then the
jumping numbers of (X, ∆, a) have no limit points.
Proof. The proof follows quite closely along the lines of [Gra16, Theorem 8.1]. For the
reader’s convenience, we give a sketch of the argument here.
Arguing by contradiction, assume that there is a strictly increasing and bounded above
sequence 0 ≤ s0 < s1 < · · · of jumping numbers of (X, ∆, a). As above, we may assume that
there is a triple (X, ∆, a) containing (X, ∆, a) as an open subset and such that X is normal
and projective. Let m0 be the Weil index of (X, ∆, a). By Theorem 3.1 in combination

F -JUMPING NUMBERS AT ISOLATED NON-Q-GORENSTEIN POINTS

11

with Remark 3.2, using Theorem 2.11 we can construct for each k ≥ 2 a Q-Weil divisor ∆k
on X such that ∆k := ∆k X is a (km0 )-compatible boundary for (X, ∆, a) and furthermore
the Q-linear equivalence class of ∆k does not depend on k. The last property is crucial, as
it enables us to find an ample Cartier divisor H on X such that
J (X, ∆ + ∆k , asℓ ) ⊗ OX (H)

is globally generated for all k ≥ 2, ℓ ≥ 0,

using [Gra16, Proposition 8.3]. Since ∆k is (km0 )-compatible, it follows that J (X, ∆, asℓ )⊗
OX (H) is globally generated on X for all ℓ ≥ 0. By [Gra16, Lemma 8.2], this implies that
the sequence of ideals
J (X, ∆, as0 ) ⊃ J (X, ∆, as1 ) ⊃ · · ·
stabilizes when restricted to X. Since J (X, ∆, asℓ ) X = J (X, ∆, asℓ ), this contradicts the
assumption that each si is a jumping number of (X, ∆, a).

Theorem 4.4. Suppose that X is a normal variety over an F -finite field k of characteristic
p > 0. Set



eval@1
Je := JeX := Image F∗e OX (1 − pe )KX ∼
= Hom OX (F∗e OX , OX ) −−−−→ OX ⊂ OX .
If R(X, −KX ) is finitely generated except at an isolated collection of points, then Je = Je+1
for all e ≫ 0.
Proof. First notice that Je ⊇ Je+1 since F∗e OX ֒→ F∗e+1 OX and Hom OX (−, OX ) is contravariant. As in the proof of Theorem 4.2, we may assume that X is an open subset of
some normal projective variety X. By Theorem 3.1, we know there exists an ample divisor
H on X such that


F∗e OX (1 − pe )KX + pe H = F∗e OX (1 − pe )KX ⊗ OX (H)
is globally generated on X, as an OX -module, for all e ≥ 0. Hence its image JeX ⊗ OX (H)
is also globally generated on X. But JeX X = JeX since X ⊂ X is open. Hence we see that

Je = Je+1 for e ≫ 0 by [Gra16, Lemma 8.2]. This completes the proof.
Finally we prove the final statement from the introduction.
Proof of Proposition 1.2. For (1.2.1), we need to prove that for every characteristic zero
klt pair (X, D) and for every Q-divisor B on X, the algebra R(X, B) is finitely generated.
This is well-known to experts (see e.g. [Kol08, Theorem 92]), but for completeness’ sake we
provide a proof.
The question is local, so we may assume that B is effective and that KX + D ∼Q 0. Let
π: Y −
→ X be a small Q-factorial modification, which exists
by [BCHM10, Corollary 1.4.3].

For some rational 0 < ε ≪ 1, the pair Y, π∗−1 (D + εB) is klt. The map π being small, we
have


π∗ R Y, KY + π∗−1 (D + εB) = R(X, KX + D + εB).
By [BCHM10, Theorem 1.2(3)], the left-hand side is finitely generated. Hence so is the
right-hand side. Since KX + D ∼Q 0 and ε ∈ Q, we see that R(X, KX + D + εB) and
R(X, B) have isomorphic Veronese subalgebras. We conclude by [GHNV90, Lemma 2.4
and Theorem 3.2].
Concerning (1.2.2), after shrinking X we may assume that X = Spec R is affine and has
pseudorational singularities. If the singular locus of X is zero-dimensional, we are clearly
done. So let p ∈ Spec R be the generic point of a one-dimensional component of Sing(X).
Localizing at p, we obtain a two-dimensional pseudorational germ U := Spec Rp −
→ X with
e
closed point m := pRp . Let π : U −
→ U be a desingularization of U , with exceptional divisor

12

P. GRAF AND K. SCHWEDE

e . The Grothendieck spectral sequence associated to the composition of functors
E ⊂ U
Γm ◦ π∗ = ΓE yields an exact sequence
e, Oe) −
e , O e ),
HE1 (U
→ Hm0 (U, R1 π∗ OUe ) −
→ Hm2 (U, OU ) −
→ HE2 (U
U
|
|
{z U }
{z
}
|
{z
}
=0

e ,O e )
=H 1 (U
U

injective by pseudorationality

e , O e ) = 0, so
where the first term is zero due to [Lip78, Theorem 2.4]. It follows that H 1 (U
U
U has rational singularities in the sense of Lipman [Lip69, Definition 1.1].
Now consider a Q-Weil divisor B on X. By [Lip69, Proposition 17.1], the restriction of B
to U is Q-Cartier and then B itself is Q-Cartier in a neighborhood of p ∈ X. Applying this
argument to every one-dimensional component of Sing(X), we see that except at an isolated
collection of points, B is Q-Cartier and in particular R(X, B) is finitely generated.

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PG: Lehrstuhl für Mathematik I, Universität Bayreuth, 95440 Bayreuth, Germany
URL: www.pgraf.uni-bayreuth.de/en/
E-mail address: patrick.graf@uni-bayreuth.de
KS: Department of Mathematics, The University of Utah, 155 S 1400 E Room 233, Salt Lake
City, UT 84112, USA
URL: www.math.utah.edu/~schwede/
E-mail address: schwede@math.utah.edu

