OKOUNKOV BODIES ASSOCIATED TO PSEUDOEFFECTIVE DIVISORS
SUNG RAK CHOI, YOONSUK HYUN, JINHYUNG PARK, AND JOONYEONG WON
Abstract. The Okounkov body is a convex body associated to a big divisor on a smooth
projective variety with respect to an admissible flag. We introduce two different ways to associate
the Okounkov bodies with pseudoeffective divisors, and show that these convex bodies reflect
asymptotic invariants of given pseudoeffective divisors.
1. Introduction
For a divisor D on a smooth projective variety X of dimension n, one can associate a convex
body ∆(D) in Rn called the Okounkov body. After the pioneering works by Lazarsfeld-Mustaţă
([LM]) and Kaveh-Khovanskii ([KK]), motivated by earlier works by Okounkov ([O1], [O2]), the
Okounkov bodies ∆(D) have received a considerable amount of attention in algebraic geometry
in a variety of flavors. It is believed that this convex body carries rich information of the
asymptotic invariants of D. For example, if D is big, then the Euclidean volume of ∆(D) in Rn
is equal to the volume vol(D) of D up to the constant n! ([LM, Theorem A]). However, little is
known about the Okounkov bodies for pseudoeffective divisors. One of the annoying phenomena
is that for a pseudoeffective divisor D which is not big, the associated convex body ∆(D) may
not be full dimensional in Rn so that its Euclidean volume in Rn is zero. Nevertheless, it is still
tempting to study what kind of positivity of the divisor is encoded in such convex bodies.
The purpose of this paper is to introduce and study two different convex bodies ∆val (D) and
lim
∆ (D) associated to a pseudoeffective divisor D. The main idea for the construction of these
convex bodies is to require the admissible flags to satisfy certain conditions so that ∆val (D) and
∆lim (D) encode the asymptotic invariants of the divisor D. If D is big, then the admissible flag
can be chosen arbitrarily, and we have
∆val (D) = ∆lim (D) = ∆(D).
Thus our constructions give generalizations of the usual Okounkov bodies ∆(D) in the case
where D is big.
Turning to the details, we first recall the construction of the Okounkov body which is equivalent to the ones given by Lazarsfeld-Mustaţă ([LM]) and Kaveh-Khovanskii ([KK]). Let D be
a big divisor on a smooth projective variety X of dimension n. Fix an admissible flag on X
Y• : X = Y0 ⊃ Y1 ⊃ Y2 ⊃ · · · ⊃ Yn−1 ⊃ Yn = {x}
where each Yi is an (n − i)-dimensional irreducible subvariety of X. For D0 ∈ |D|R := {D0 |
D ∼R D0 ≥ 0}, define a valuation-like function
ν(D0 ) = νY• (D0 ) := ν1 (D0 ), ν2 (D0 ), · · · , νn (D0 ) ∈ Rn≥0
as follows. Set ν1 (D0 ) := ordY1 (D0 ), ν2 (D0 ) := ordY2 ((D0 − ν1 (D0 )Y1 )|Y1 ), and so on. The
Okounkov body ∆Y• (D) = ∆(D) of D with respect to Y• is defined as the closure of the convex
hull of νY• (|D|R ) in Rn≥0 . See Subsection 3.1 for more details.
Date: August 17, 2015.
Key words and phrases. Okounkov body, pseudoeffective divisor, Kodaira dimension, restricted volume,
Nakayama locus.
1
2
SUNG RAK CHOI, YOONSUK HYUN, JINHYUNG PARK, AND JOONYEONG WON
Note that if a given divisor D is effective, i.e., the Iitaka dimension κ(D) is nonnegative,
then the above construction can be carried out without any changes. We consider an admissible
flag Y• which contains a subvariety U ⊂ X such that κ(D) = dim U and the natural map
H 0 (X, OX (bmDc)) → H 0 (U, OU (bmD|U c)) is injective for every integer m ≥ 0. We call such U
a Nakayama locus of D (see Subsection 2.3). If D is big, then X itself is the unique Nakayama
locus of D.
Definition 1.1. Let D be a divisor with κ(D) ≥ 0 on a smooth projective variety X, and Y•
an admissible flag on X which contains a Nakayama locus Yn−κ(D) = U . The Okounkov body
∆Y• (D) of D with respect to such Y• is called the valuative Okounkov body of D and we denote
val
it by ∆val
Y• (D) = ∆ (D).
We will prove that the intersection of general ample divisors of dimension κ(D) is a Nakayama
locus of D. Thus in Definition 1.1 we may take a general admissible flag Y• constructed with
successive intersections of general ample divisors.
Theorem A (=Theorem 3.12). Let D be a divisor with κ(D) ≥ 0 on a smooth projective variety
X. Fix an admissible flag Y• containing a Nakayama locus U of D such that Yn = {x} is a
general point. Then we have
1
volX|U (D).
κ(D)!
val
dim ∆val
Y• (D) = κ(D) and volRκ(D) (∆Y• (D)) =
For convenience, we define dim(point) := 0 and volR0 (point) := 1.
We remark that ∆val
Y• (D) is not a numerical invariant (see Remark 3.13). We may have
val
∆ (D) = ∅ even for a pseudoeffective divisor D. See Subsection 3.2 for further properties of
valuative Okounkov bodies.
We associate another convex body ∆lim
Y• (D) to a divisor D which reflect more numerical
properties of D. Let D be a pseudoeffective divisor on a smooth projective variety X, i.e.,
the numerical Iitaka dimension κν (D) ≥ 0. We call a smooth subvariety V ⊆ X a positive
volume locus of D if κν (D) = dim V , V 6⊆ B− (D) and vol+
X|V (D) > 0 (see Subsection 2.4). See
Subsection 2.2 for the definition of the augmented restricted volume vol+
X|V (D). If D is big, then
X itself is the unique positive volume locus of D.
Definition 1.2. Let D be a pseudoeffective divisor on a smooth projective variety X, and Y•
an admissible flag which contains a positive volume locus Yn−κν (D) = V of D. We define the
limiting Okounkov body of a pseudoeffective divisor D with respect to such Y• as
\
lim
∆lim
(D) := lim ∆Y• (D + εA) =
∆Y• (D + εA)
Y• (D) = ∆
ε→0+
ε>0
where A is an ample divisor.
We will see that the intersection of general ample divisors of dimension κν (D) is a positive
volume locus of D. Thus as with ∆val (D), in Definition 1.2 we can take a general admissible
flag.
Theorem B (=Theorem 3.20). Let D be a pseudoeffective divisor on a smooth projective variety
X. Fix an admissible flag Y• containing a positive volume locus V of D. Then we have
lim
dim ∆lim
Y• (D) = κν (D) and volRκν (D) (∆Y• (D)) =
1
vol+
X|V (D).
κν (D)!
By construction, it is easy to see that ∆lim
Y• (D) is a numerical invariant. Actually, even more
is true.
OKOUNKOV BODIES ASSOCIATED TO PSEUDOEFFECTIVE DIVISORS
3
Theorem C (=Theorem 3.23). Let D, D0 be pseudoeffective divisors on a smooth projective
lim
0
variety X. Then D ≡ D0 if and only if ∆lim
Y• (D) = ∆Y• (D ) for all admissible flags Y• containing
0
a positive volume locus V of both D and D .
This theorem generalizes [LM, Proposition 4.1] and [J, Theorem A]. See Subsection 3.3 for
further properties of limiting Okounkov bodies.
Organization. We start in Section 2 by collecting relevant basic facts on asymptotic base locus,
augmented restricted volume, Nakayama locus, and positive volume locus. Section 3 is the main
part of this paper. We recall the construction of the Okounkov body ∆(D) of a big divisor
D in Subsection 3.1. We also introduce and study the valuative Okounkov body ∆val (D) and
the limiting Okounkov body ∆lim (D) of a pseudoeffective divisor D in Subsections 3.2 and 3.3,
respectively. Finally, in Section 4, we exhibit various examples.
2. Preliminaries
Throughout this paper, we work over the field C of complex numbers. A variety in this paper
is assumed to be smooth, reduced and irreducible. By a divisor on a variety X, we always
mean an R-divisor unless otherwise stated. A divisor D is pseudoeffective if its numerical class
[D] ∈ N1 (X)R := N1 (X) ⊗ R lies in Eff(X), the closure of the cone N1 (X)R spanned by the
classes of effective divisors. A divisor is big if its numerical class lies in Big(X), the interior of
Eff(X).
In this section, we collect basic facts used throughout the paper.
2.1. Asymptotic base locus.
Let D be a pseudoeffective divisor on a variety X. When D is a Q-divisor, we define the
stable base locus of D as
\
SB(D) :=
Bs(mD)
m
where the intersection is taken over all positive integers m such that mD are Z-divisors, and
Bs(mD) denotes the base locus of the linear system |mD|. The augmented base locus B+ (D) is
defined as
\
B+ (D) :=
SB(D − A)
A
where the intersection is taken over all ample divisors A such that D − A are Q-divisors. The
restricted base locus B− (D) of D is defined as
[
B− (D) :=
SB(D + A).
A
where the union is taken over all ample divisors A such that D + A are Q-divisors. We have
B− (D) ⊆ SB(D) ⊆ B+ (D) for a Q-divisor D. One can check that a divisor D is ample
(respectively nef) if and only if B+ (D) = ∅ ( B− (D) = ∅). The asymptotic base loci B+ (D) and
B− (D) depend only on the numerical class of D while SB(D) does not ([La, Example 10.3.3]).
Let V ⊆ X be an irreducible subvariety of X. When D is big, define the asymptotic valuation
of V at D as
ordV (||D||) := inf{ordV (D0 ) | D ≡ D0 ≥ 0}.
When D is only pseudoeffective, define ordV (||D||) := limε→0+ ordV (||D + εA||) for some ample divisor A. This definition is independent of the choice of A, and the number σnum (D)
depends only on the numerical class [D] ∈ N1 (X)R . It is a birational invariant: for a birational morphism f : Y → X with the exceptional divisor E such that f (E) = V , we have
ordV (||D||) = ordE (||f ∗ (D)||) (see [BBP, Lemma 1.4]). The restricted base locus B− (D) can be
characterized as follows.
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SUNG RAK CHOI, YOONSUK HYUN, JINHYUNG PARK, AND JOONYEONG WON
Theorem 2.1 ([ELMNP1, Proposition 2.8],[N, Lemma V.1.9]). Let D be a pseudoeffective
divisor on a variety X, and V ⊆ X an irreducible subvariety. Then V ⊆ B− (D) if and only if
ordV (||D||) > 0.
2.2. Augmented restricted volume.
To introduce the notion of augmented restricted volume of a divisor along a subvariety, we
first recall the classical volume function and restricted volume function. Let D be a Q-divisor
on a variety X of dimension n. Recall that the volume of D is defined as
volX (D) := lim sup
m→∞
h0 (X, OX (mD))
.
mn /n!
The volume volX (D) depends only on the numerical class of D. Furthermore, this function
uniquely extends to a continuous function
volX : Big(X) → R.
Let V be a subvariety of X such that V 6⊆ B+ (D). For the natural restriction map ϕ :
H 0 (X, OX (D)) → H 0 (V, OV (D)), we denote the image Im(ϕ) by H 0 (X|V, D) and its dimension
by h0 (X|V, D). The restricted volume of D along V is defined as
volX|V (D) := lim sup
m→∞
h0 (X|V, mD)
mdim V / dim V !
([ELMNP2, Definition 2.1]). The restricted volume volX|V (D) depends only on the numerical
class of D. Furthermore, this function uniquely extends to a continuous function
volX|V : BigV (X) → R
where BigV (X) is the set of all R-divisor classes ξ such that V is not properly contained in any
irreducible component of B+ (ξ). Note that by letting V = X, we recover the usual volume function volX|X (D) = volX (D). For more details on the functions volX and volX|V , see [ELMNP2],
[La], [Le], etc.
By [ELMNP2, Theorem 5.2], volX|V (D) = 0 if V is an irreducible component of B+ (D).
Similarly, volX (D) = 0 if D is not big (i.e., X = B+ (D)). Thus the functions volX|V and volX
do not capture the subtle asymptotic properties of pseudoeffective divisors that are not big on
V or X. In such situations, the following function seems useful.
Definition 2.2. Let D be a pseudoeffective divisor on X, and V ⊆ X be a subvariety such that
V 6⊆ B− (D). For an ample divisor A on X, we define the augmented restricted volume of D
along V as
vol+
X|V (D) := lim volX|V (D + εA).
ε→0+
We write
vol+
X|X (D)
=
vol+
X (D).
The definition is independent of the choice of A. As with volX and volX|V , one can check
that the augmented restricted volume vol+
X|V (D) depends only on the numerical class of D.
By the continuity of the function volX|V , we see that vol+
X|V (D) coincides with volX|V (D) for
D ∈ BigV (X). In particular, volX (D) = vol+
X (D) for any big divisor D.
For D ∈ BigV (X), the following inequalities hold by definition:
volX|V (D) ≤ vol+
X|V (D) ≤ volV (D|V ).
Example 2.3. Let S be a relatively minimal rational elliptic surface, and H an ample divisor
on S. Take a general element V ∈ |kH| for a sufficiently large k > 0. Then volS|V (−KS ) =
lim supm→∞
h0 (S,OS (−mKS ))
m
which is independent of k and H.
However, one can see that
OKOUNKOV BODIES ASSOCIATED TO PSEUDOEFFECTIVE DIVISORS
5
vol+
S|V (−KS ) = k(−KS )·H which can be arbitrarily large depending on k. Thus volS|V (−KS ) <
vol+
S|V (−KS ).
2.3. Nakayama locus.
The aim of this subsection is to introduce and study Nakayama loci of divisors, which are
closely related to the Iitaka dimension.
Definition 2.4. For a divisor D on a variety X, let N(D) = {m ∈ Z>0 | |bmDc| =
6 ∅}. For
dim
|bmDc|
m ∈ N(D), let ΦmD : X 99K P
be the rational map defined by the linear system
|bmDc|. We define the Iitaka dimension of D as the following value
max{dim Im(ΦmD ) | m ∈ N(D)} if N(D) 6= ∅
κ(D) :=
−∞
if N(D) = ∅.
Remark 2.5. If D is a divisor such that κ(D) ≥ 0, then there exists an integer m0 ≥ 1 such
that h0 (X, OX (bmm0 Dc)) ∼ mκ(D) for m 0 ([N, Theorem II.3.7]). For this m0 , we have
κ(D) = κ(m0 D). Thus in many situations below, we may assume that m0 = 1.
Remark 2.6. If D ∼R D0 and D, D0 are effective, then κ(D) = κ(D0 ). However, the Iitaka
dimension κ(D) is neither a ∼R nor numerical invariant in general ([Le, Example 6.1]).
Definition 2.7. Let D be a divisor on a variety X such that κ(D) ≥ 0. A smooth subvariety
U ⊆ X is called a Nakayama locus of D if κ(D) = dim U and the natural map
H 0 (X, OX (bmDc)) → H 0 (U, OU (bmD|U c))
is injective (or equivalently, H 0 (X, IU ⊗ OX (bmDc)) = 0 where IU is an ideal sheaf of U in X)
for every integer m ≥ 0.
By definition, U = X is the (unique) Nakayama locus of D if and only if D is big. More
generally, Nakayama loci have the following property.
Lemma 2.8. Let D be a divisor such that κ(D) ≥ 0, and U ⊆ X its Nakayama locus. Then
D|U is big.
Proof. By the definition of Nakayama locus, we have
h0 (U, OU (bmD|U c)) ≥ h0 (X, OX (bmDc))
for every integer m ≥ 0. Since h0 (X, OX (bmDc)) ∼ mκ(D) for m 0, the assertion follows. The following lemma guarantees the existence of Nakayama loci.
Nakayama locus is not unique in general.
It also shows that a
Proposition 2.9 (cf. [N, Lemma V.2.11]). Let D be a divisor on X such that κ(D) ≥ 0. Any
general subvariety U ⊆ X of dimension κ(D) is a Nakayama locus of D.
Proof. Consider a dominant rational map ϕm : X 99K Zm induced by a complete linear series
|bmDc| for any integer m > 0 such that |bmDc| =
6 ∅. We have dim Zm ≤ κ(D) = dim U . Let
f : Y → X be the blow-up at U with the exceptional divisor E. Since U ⊆ X is general,
f ∗ (bmDc) − kE is not effective for any k > 0. Thus H 0 (X, IU ⊗ OX (bmDc)) = 0, and hence,
the assertion follows.
Corollary 2.10. If the effective divisors D, D0 on a variety X satisfy D ∼R D0 , then D, D0
have the same Nakayama loci.
Proof. The given condition implies that κ(D) = κ(D0 ) ≥ 0. By proposition 2.9, any general
subvariety U ⊆ X of dimension κ(D) is a Nakayama locus of D.
6
SUNG RAK CHOI, YOONSUK HYUN, JINHYUNG PARK, AND JOONYEONG WON
2.4. Positive volume locus.
The aim of this subsection is to introduce and study positive volume loci of divisors, which
are closely related to the numerical Iitaka dimension and the restricted volume. First, we review
the numerical Iitaka dimension.
Definition 2.11. Let D be a divisor on a variety X. We define the numerical Iitaka dimension
of D as the nonnegative integer
h0 (X, OX (bmDc + A))
>0
κν (D) := max k ∈ Z≥0 lim sup
mk
m→∞
for some fixed ample Cartier divisor A if h0 (X, OX (bmDc + A)) 6= ∅ for infinitely many m > 0
and we let κν (D) := −∞ otherwise.
The numerical Iitaka dimension κν (D) depends only on the numerical class [D] ∈ N1 (X)R .
One can easily check that κ(D) ≤ κν (D) holds and the inequality is strict in general (see [Le,
Example 6.1]). However, if κν (D) = dim X, then κ(D) = dim X. See [Le] and [N] for detailed
properties of κ and κν .
By the following theorem, the numerical Iitaka dimension κν (D) and vol+
X|V (D) are closely
related.
Theorem 2.12 ([Le]). Let D be a pseudoeffective divisor on a variety X. Then we have
κν (D) = max{dim W | vol+
X|W (D) > 0 and W 6⊆ B− (D)}.
The numerical Iitaka dimension κν (D) actually coincides with many other invariants defined
with D. For details, see [Le].
Definition 2.13. Let D be a pseudoeffective divisor on a variety X. A subvariety V ⊆ X of
dimension κν (D) such that vol+
X|V (D) > 0 and V 6⊆ B− (D) is called a positive volume locus
of D. (In other words, the maximum in Theorem 2.12 is attained at a positive volume locus
V ⊆ X.)
Theorem 2.12 guarantees the existence of a positive volume locus.
Example 2.14. Let S be a relatively minimal rational surface with a reducible singular fiber,
and E a (−2)-curve in a singular fiber. Then E 6⊆ B− (−KS ) but vol+
S|E (−KS ) = 0. In this
example, we see that V 6⊆ B− (D) and dim V = κν (D) does not imply vol+
X|V (D) > 0.
By definition, if D is a big divisor on X, then V = X is the unique positive volume locus of
D. Thus we have the following obvious bigness criterion.
Proposition 2.15. A pseudoeffective divisor D is big if and only if vol+
X (D) > 0.
We prove some notable properties of positive volume locus.
Lemma 2.16. If D and D0 are pseudoeffective divisors on a variety X such that D ≡ D0 , then
they have the same positive volume loci.
Proof. Note first that B− (D) = B− (D0 ). The statement then follows from the fact that
0
vol+
X|V (D) depends only on the numerical class [D] for any V 6⊆ B− (D) = B− (D ).
Let V be a smooth subvariety of a variety X. A birational morphism f : Y → X is said to
be V -birational if V is not contained in the center of the f -exceptional locus. We denote by Ve
the proper transform of V on Y . Then the pair (Y, Ve ) is called a V -birational model of X (see
[Le, Definition 2.10]).
OKOUNKOV BODIES ASSOCIATED TO PSEUDOEFFECTIVE DIVISORS
7
Proposition 2.17. Let V ⊆ X be a smooth positive volume locus of a pseudoeffective divisor D
on a projective variety X. If f : Y → X is a V -birational morphism, then Ve is also a positive
volume locus of f ∗ D.
Proof. The birational transform Ve ⊆ Y of V has dimension dim Ve = dim V = κν (D) = κν (f ∗ D).
Note that Ve 6⊆ B− (f ∗ (D)). Indeed, the inclusion Ve ⊆ B− (f ∗ (D)) would imply V = f (Ve ) ⊆
f (B− (f ∗ (D)) = B− (D). Since vol+
X|V (D) > 0 by definition, it is enough to show the inequality
∗
vol+
X|V (D) ≤ volY |Ve (f D).
First of all, since V 6⊆ B− (D) by definition, for any ample divisor A on X such that D + A is a
Q-divisor, we have V 6⊆ B+ (D + A). We can also check that Ve 6⊆ B+ (f ∗ D + f ∗ A) using [BBP,
Proposition 2.3]. The following computation yields the desired inequality:
vol+
X|V (D) = lim volX|V (D + A)
||A||→0
= lim volY |Ve (f ∗ D + f ∗ A) (by [ELMNP2, Lemma 2.4])
||A||→0
≤ vol+ e (f ∗ D).
Y |V
(Here,
P|| · || is any suitable norm on divisors. For example, we can let ||D|| = max{|ai |} if
D = ai Di .)
Proposition 2.18. Let D be a pseudoeffective divisor on a variety X of dimension n. If
κν (D) < n, then for any ample Cartier divisor A with a sufficiently large integer m > 0, the
locus V := A1 ∩ · · · ∩ An−κν (D) satisfies vol+
X|V (D) > 0, where Ai ∈ |mA| are general members.
In particular, V is a smooth positive volume locus of D.
Proof. By Theorem 2.12, there exists a positive volume locus W of D. In particular, we have
vol+
X|W (D) > 0. We can take a sequence {Hi }i∈Z≥0 of ample divisors on X such that each D +Hi
is a Q-divisor and Hi → 0 as i → ∞. For a large and sufficiently divisible integer k, choose
1 , · · · , E κν (D) ∈ |k(D + H )|. By taking a sufficiently large integer
κν (D) general divisors Ek,i
i
k,i
m > 0, we can make mA sufficiently positive so that we have
κ (D)
1
∩ · · · ∩ Ek,iν
#(V ∩ Ek,i
κ (D)
1
∩ · · · ∩ Ek,iν
\ SB(D + Hi )) ≥ #(W ∩ Ek,i
\ SB(D + Hi )).
Thus by applying [ELMNP2, Theorem B], we obtain
κ (D)
volX|V (D + Hi ) = lim
1 ∩···∩E ν
#(V ∩Ek,i
k,i
k→∞
≥ lim
\SB(D+Hi ))
kκν (D)
κ (D)
1 ∩···∩E ν
#(W ∩Ek,i
k,i
k→∞
vol+
X|V (D)
This immediately implies that
we can assume that V 6⊆ B− (D).
\SB(D+Hi ))
kκν (D)
= volX|W (D + Hi ).
≥ vol+
X|W (D) > 0. Since A1 , · · · , An−κν (D) are general,
It is known that D|V is pseudoeffective for any subvariety V ⊆ X such that V 6⊆ B− (D).
Thus κ(D|V ) ≥ 0.
Theorem 2.19. Let V be a positive volume locus of a pseudoeffective divisor D on a projective
variety X. Then vol+
V (D|V ) > 0. In particular, D|V is big on V and κν (D) = κν (D|V ) = dim V .
Proof. We prove the statement for a pseudoeffective Cartier divisor D. The R-divisor case then
follows from the log-concavity property of the restricted volume function ([ELMNP2, Theorem
A]). Fix a sufficiently ample Cartier divisor A on X. Since V ⊆ X is a positive volume locus of
D, there exists a constant C0 > 0 such that for any large integer k > 0 we have
lim sup
m→∞
h0 (X|V, m(D + k1 A))
> C0
mκν (D)
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SUNG RAK CHOI, YOONSUK HYUN, JINHYUNG PARK, AND JOONYEONG WON
where m is taken over all positive integers such that m(D + k1 A) is a Z-divisor. Note that
h0 (X|V, m(D + k1 A)) ≤ h0 (V, m(D|V + k1 A|V )). Thus for any large integer k > 0, we have
h0 (V, m(D|V + k1 A|V ))
lim sup
> C0
mκν (D)
m→∞
which implies vol+
V (D|V ) > 0. By Lemma 2.15, D|V is big on V .
3. Okounkov body of a pseudoeffective divisor
In this section, we introduce and study two Okounkov bodies ∆val (D) and ∆lim (D) for pseudoeffective divisors D.
3.1. Okounkov body ∆(D).
We first recall the construction of the Okounkov body of a divisor D or a graded linear series
W• associated to D from [LM]. We remark that the construction of ∆(D) in the introduction
is equivalent to the one given below ([KL]).
Throughout this subsection, we fix an admissible flag on X
Y• : X = Y0 ⊃ Y1 ⊃ Y2 ⊃ · · · ⊃ Yn−1 ⊃ Yn = {x}
where each Yi is an (n − i)-dimensional smooth subvariety of X. We first assume that D is a
big Cartier divisor on X. For a section s ∈ H 0 (X, OX (D)) \ {0}, we define the function
ν(s) = νY• (s) := (ν1 (s), ν2 (s), · · · , νn (s)) ∈ Zn≥0
as follows. Let ν1 = ν1 (s) := ordY1 (s). Using a local equation f1 for Y1 in X, define a section
s01 = s ⊗ f1−ν1 ∈ H 0 (X, OX (D − ν1 Y1 )).
Since s01 does not vanish identically along Y1 , its restriction s01 |Y1 defines a nonzero section
s1 := s01 |Y1 ∈ H 0 (Y1 , OY1 (D − ν1 Y1 )) \ {0}.
Now take ν2 = ν2 (s) := ordY2 (s1 ). Continuing this manner, we obtain the nonnegative integers
νi (s) for all 1 ≤ i ≤ n so that ν(s) ∈ Zn≥0 . Each integer νi = νi (s) does not depend on the choice
of the local equation fi chosen to define s0i and si .
Recall that a graded linear series W• (D) = {Wm := Wm (D)}m≥0 associated to D consists of
subspaces Wm ⊆ H 0 (X, OX (mD)) with W0 = C satisfying Wm · Wl ⊆ Wm+l for all m, l ≥ 0.
The graded semigroup of W• is
Γ(W• ) := {(ν(s), m) | Wm 6= {0} and s ∈ Wm \ {0} } ⊆ Zn≥0 × Z≥0 .
Definition 3.1. Under the same notations as above, we associate the convex body ∆(W• ) called
the Okounkov body of a graded linear series W• with respect to an admissible flag Y• on X as
follows:
∆(W• ) = ∆Y• (W• ) := Σ(Γ(W• )) ∩ (Rn≥0 × {1})
where Σ(Γ(W• )) denotes the closure of the convex cone in Rn≥0 × R≥0 spanned by Γ(W• ). If
W• is a complete graded linear series (i.e., Wm = H 0 (X, O(mD)) for each m), then we denote
∆(D) = ∆(W• ).
We naturally treat ∆(W• ) as an object in Rn , i.e., ∆(W• ) ⊆ Rn . By construction, the
Okounkov body ∆(D) depends on the choice of the admissible flag Y• . By the homogeneity of
∆(D) ([LM, Proposition 4.1]), we can extend the construction of ∆(D) to Q-divisors D and
even to R-divisors using the continuity of ∆(D).
Remark 3.2. Note that the above constructions of ∆(D) and ∆(W• ) work as long as H 0 (X, OX (mD)) 6=
0 or Wm 6= 0 for some m. Thus we do not need to assume that a given divisor D is big to define
the Okounkov body ∆(D).
OKOUNKOV BODIES ASSOCIATED TO PSEUDOEFFECTIVE DIVISORS
9
Remark 3.3. Let Y• be an admissible flag on a variety X of dimension n. For each integer k
such that 0 ≤ k ≤ n, we can define the k-th partial flag Yk• of Y• as
Yk• : Yk ⊃ Yk+1 ⊃ · · · ⊃ Yn = {x}.
Then Yk• is an admissible flag on a smooth projective variety Yk of dimension n − k. Thus for
a Cartier divisor D on X, we can define the function
νYk• : H 0 (X|Yk , OX (D)) \ {0} → Zn−k
≥0
as we did above. Let
ΓYk (D) := {(νYk• (s), m)|s ∈ H 0 (X|Yk , OX (mD)) \ {0}, m ≥ 0}.
The k-th partial Okounkov body of a divisor D on X with respect to the k-th partial flag Yk• is
defined as
n−k
× {1})
∆Yk• (D) := Σ(ΓYk (D)) ∩ (R≥0
n−k
× R≥0 spanned by ΓYk (D). These
where Σ(ΓYk (D)) is the closure of the convex cone in R≥0
partial flags and Okounkov bodies will be useful in the next subsections.
Under the following conditions, the Okounkov body ∆(W• ) behaves well.
Definition 3.4 ([LM, Definitions 2.5 and 2.9]).
(1) We say that a graded linear series W• satisfies Condition (B) if Wm 6= 0 for all m 0, and
for all sufficiently large m, the rational map ϕm : X 99K P(Wm ) defined by |Wm | is birational
onto its image.
(2) We say that a graded linear series W• satisfies Condition (C) if
i. for every m 0, there exists an effective divisor Fm such that the divisor Am := mD−Fm
is ample, and
ii. for all sufficiently large p, we have
H 0 (X, OX (pAm )) ⊆ Wpm ⊆ H 0 (X, OX (pmD)).
If W• is complete, i.e., Wm = H 0 (X, OX (mD)) for all m ≥ 0 and D is big, then it automatically satisfies Condition (C).
Theorem 3.5 ([LM, Theorem 2.13]). Suppose that a graded linear series W• satisfies Condition
(B) or Condition (C). Then for any admissible flag Y• (with a general choice of the point
Yn = {x} when W• satisfies Condition (B)), we have
dim ∆(W• ) = dim X(= n) and volRn (∆(W• )) =
1
· volX (W• )
n!
where
volX (W• ) := lim
m→∞
dim Wm
.
mn /n!
Note that [LM, Theorem 2.13] also requires Condition (A) ([LM, Definition 2.4]), but it is
automatically satisfied since we always assume our variety X is projective.
Remark 3.6. It is well known by [LM, Proposition 4.1] that for a fixed admissible flag Y• on
X, if D is a big divisor on X, then ∆Y• (D) depends only on the numerical class of D. If D is
only pseudoeffective, it is no longer true. See Remark 3.13.
Remark 3.7. The k-th partial Okounkov body ∆Yk • (D) depends on the numerical class of D
if D|Yk is big.
10
SUNG RAK CHOI, YOONSUK HYUN, JINHYUNG PARK, AND JOONYEONG WON
3.2. Valuative Okounkov body ∆val (D).
Now we define the valuative Okounkov body ∆val (D) of a divisor D with κ(D) ≥ 0 on a
smooth projective variety of dimension n.
Definition 3.8. Let D be a divisor on a smooth projective variety X of dimension n such that
κ(D) ≥ 0 and U ⊆ X is a smooth Nakayama locus of D. Let Y• be an admissible flag containing
U with a general choice of Yn = {x}:
Y• : X = Y0 ⊃ Y1 ⊃ Y2 ⊃ · · · ⊃ Yn−κ(D) = U ⊃ · · · ⊃ Yn = {x}.
The valuative Okounkov body ∆val
Y• (D) associated to D is defined as
val
n
∆val
Y• (D) = ∆ (D) := ∆Y• (D) ⊆ R .
For a divisor D with κ(D) = −∞, we define ∆val (D) := ∅.
Obviously, the definition of ∆val (D) depends on the choice of the Nakayama locus U and the
admissible flag Y• on X containing U . We will see below that in fact ∆val (D) depends only on
the choice of a Nakayama locus of D and the (n − κ(D))-th partial flag on it.
Remark 3.9. If D is big, then X is the unique Nakayama locus of D. Thus ∆val (D) coincides
with ∆(D).
Consider the restricted complete graded linear series W• (D|U ) of a Cartier divisor D along U
which is given by Wm (D|U ) := H 0 (X|U, mD) for each m ≥ 0. Recall that H 0 (X|U, mD) is the
image of the restriction map ϕU : H 0 (X, OX (mD)) → H 0 (U, OU (mD|U )). Note that W• (D|U )
is a graded linear subseries of W• (D|U ) on U . Thus we can consider the function
νYn−κ(D)• : Wm (D|U ) \ {0} → Zκ(D) .
Since the natural restriction map ϕU is injective for each m, we may treat the function νYn−κ(D)•
as defined on the sections s ∈ H 0 (X, OX (mD)) \ {0} by letting
νYn−κ(D)• (s) := νYn−κ(D)• (ϕU (s)) ∈ Zκ(D) .
Proposition 3.10. Let s ∈ H 0 (X, OX (mD)) \ {0}. Then under the settings as above, we have
νY• (s) = (0, . . . , 0, νYn−κ(D)• (s)).
| {z }
n−κ(D)
In particular, ∆val (D) ⊆ {0}n−κ(D) × Rκ(D) ⊆ Rn and we can regard ∆val (D) as a subset of
Rκ(D) .
Proof. By the definition of Nakayama locus, we have for all m ≥ 0
H 0 (X, IU ⊗ OX (mD)) = 0.
Let k be any integer such that 1 ≤ k ≤ n−κ(D). Then since Yk ⊇ U , it follows that H 0 (X, IYk ⊗
OX (mD)) = 0. Thus we obtain νk (s) = 0. This implies that
νn−κ(D)+1 (s) = νYn−κ(D)•,1 (s|Yn−κ(D) ) = νYn−κ(D)•,1 (ϕU (s)).
This shows the required equality.
Remark 3.11. The valuative Okounkov body ∆val (D) can be constructed alternatively as
follows. Consider the (n−κ(D))-th partial flag Yn−κ(D)• of an admissible flag Y• on X containing
a Nakayama locus U = Yn−κ(D) fo D. Since W• (D|U ) is a graded linear subseries associate to
D|U on U , we can define the Okounkov body ∆Yn−κ(D)• (W• (D|U )) ⊆ Rκ(D) as in Definition 3.1.
By regarding ∆val (D) as a subset of Rκ(D) (Proposition 3.10), we have
∆val (D) = ∆Yn−κ(D)• (W• (D|U )).
OKOUNKOV BODIES ASSOCIATED TO PSEUDOEFFECTIVE DIVISORS
11
Thus ∆Yn−κ(D)• (W• (D|U )) gives an alternative construction of ∆val (D).
The following is the main property of ∆val (D).
Theorem 3.12. Let D be a divisor such that κ(D) ≥ 0 on a smooth projective variety X, and
fix an admissible flag Y• containing a Nakayama locus U of D such that Yn = {x} is a general
point. Then we have
1
dim ∆val (D) = κ(D) and volRκ(D) (∆val (D)) =
volX|U (D).
κ(D)!
Proof. We first consider the case where D is a Cartier divisor. By Remark 3.11, we have
∆val (D) = ∆Yn−κ(D)• (W• (D|U )). Since Yn = {x} is assumed to be general, by Theorem 3.5
it suffices to show that W• (D|U ) satisfies Condition (B). The required equalities follow from
the properties of a Nakayama locus: h0 (X, OX (mD)) = dim Wm ∼ mκ(D) for m 0 and
dim U = κ(D). By the similar argument as in [LM], we can easily extend the results to Q- or
R-divisors. We leave the details to the interested readers.
Remark 3.13. It is natural to ask whether ∆val (D) is a numerical invariant of D or not. If
D is a big divisor on X, then by Remark 3.9, ∆val (D) = ∆(D), thus it depends only on the
numerical class of D. However, in general, we have κ(D) < κ(D0 ) for effective divisors D, D0
even if D ≡ D0 . Thus for such divisors D, D0 , we have ∆val (D) 6= ∆val (D0 ) since their dimensions
are different.
On the other hand, we have the following.
Proposition 3.14. If two effective divisors D, D0 satisfy D ∼R D0 , then ∆val (D) = ∆val (D0 )
with respect to any admissible flag Y• containing a Nakayama locus U with a general choice of
Yn = {x}.
Proof. It is trivial.
Remark 3.15. The converse of Proposition 3.14 is false in general. Let D be a nef divisor such
that κ(D) = 0 and D 6∼ 0 (e.g., if X is the blow-up of P2 at 9 general points, then −KX satisfies
such properties). Then ∆val (D) = ∆val (0) = {(0, 0, · · · , 0) ∈ Rn }.
3.3. Limiting Okounkov body ∆lim (D).
Here, we define and study the limiting Okounkov body ∆lim (D) of a pseudoeffective divisor
D on a smooth projective variety X of dimension n.
Definition 3.16. Let D be a pseudoeffective divisor D on a variety X of dimension n with a
positive volume locus V ⊆ X. Fix an admissible flag Y• containing V . The limiting Okounkov
body ∆lim
Y• (D) of D with respect to Y• as
lim
∆lim
(D) := lim ∆Y• (D + εA) ⊆ Rn
Y• (D) = ∆
ε→0+
where A is any ample divisor. If D is not a pseudoeffective divisor, then ∆lim (D) := ∅.
Remark 3.17. Here we do not assume the generality of Yn = {x} as we did for ∆val
Y• (D) in
Definition 3.8.
We also introduce the following alternative construction (Proposition 3.19). This construction
is often more convenient to study the asymptotic invariants of pseudoeffective divisors. To begin
with, we first consider a pseudoeffective Cartier divisor D on a variety X of dimension n. Let
V ⊆ X be a positive volume locus of D. Fix an admissible flag V• on V :
V• : V = V0 ⊃ V1 ⊃ · · · ⊃ Vκν (D) = {x}.
12
SUNG RAK CHOI, YOONSUK HYUN, JINHYUNG PARK, AND JOONYEONG WON
Let A be an ample Cartier divisor on X. For each integer k ≥ 1, consider the restricted
graded linear series W•k := W• (kD + A|V ) of kD + A along V given by Wm (kD + A|V ) =
h0 (X|V, m(kD + A)) for m ≥ 0. We define the restricted limiting Okounkov body of a Cartier
divisor D with respect to a positive volume locus V of D as
1
∆V• (W•k ) ⊆ Rκν (D) .
∆lim
V• (D) := lim
k→∞ k
By the continuity, we can extend this definition for any pseudoeffective R-divisor D.
Definition 3.18. Let D be a pseudoeffective divisor D on a variety X with a positive volume
locus V ⊆ X. The restricted limiting Okounkov body ∆lim
V• (D) of D with respect to V• as
κν (D)
∆lim
V• (D) := lim ∆V• (D + εA) ⊆ R
ε→0+
for some ample divisor A on X. If D is not pseudoeffective, then we define ∆lim
V• (D) := ∅.
This definition depends on the choice of V and the admissible flag V• on it, but it is indeκν (D) . By the
pendent of the choice of A. By definition, ∆lim
V• (D) is a closed convex subset of R
κ
(D)
n−κ
(D)
κ
(D)
n
lim
ν
inclusion R ν
= {0}
×R ν
,→ R , we will often treat ∆V• (D) as a subset of Rn .
If D is big, then V = X and by the continuity of the Okounkov bodies, ∆lim (D) coincides with
the original ∆(D).
The following Proposition shows that if an admissible flag Y• contains a positive volume locus
Yn−κν (D) = V of D, then the limiting Okounkov body ∆lim
Y• (D) in Definition 3.16 depends only
on the partial flag Yn−κ(D)• of Y• .
Proposition 3.19. Let D be a pseudoeffective divisor on a smooth projective variety X of
dimension n, and fix an admissible flag Y• containing a positive volume locus V = Yn−κν (D) of
D. Then
lim
∆lim
Y• (D) = ∆V• (D).
In particular, ∆lim
Y• (D) depends on the (n − κν (D))-th partial flag V• and the numerical class of
D.
Proof. If κν (D) = n, then there is nothing to prove. Therefore, we assume that κν (D) < n. Fix
an ample divisor A on X. By definition, the positive volume locus Yn−κν (D) = V is not contained
in B− (D). Since Yi ⊇ Yn−κν (D) for 0 ≤ i ≤ n − κν (D), we have Yi 6⊆ B+ (D + εA) for any ε > 0.
Thus [LM, Theorem 4.26] implies that if 0 ≤ i ≤ n − κν (D), then ∆Y• (D + εA) ∩ ({0}i × Rn−i ) =
∆Yi• (D + εA). By taking the limit ε → 0, we obtain for 0 ≤ i ≤ n − κ(D)
i
n−i
∆lim
) = ∆lim
Y• (D) ∩ ({0} × R
Yi• (D).
lim
lim
n−κ(D) × Rκ(D) . Suppose
Thus to prove ∆lim
Y• (D) = ∆V• (D), it is enough to check ∆Y• (D) ⊆ {0}
that this does not hold. Then the integer
i
n−i
l := max{i | ∆lim
}
Y• (D) ⊆ {0} × R
lim
satisfies l < n − κν (D). Furthermore, we also have dim ∆lim
Y• (D) = dim ∆Yl• (D) = n − l. By
[LM, Lemma 2.16 and (2.7) in p.804], we have
1
volX|Yl (D + εA).
volRn−l (∆Yl• (D + εA)) =
(n − l)!
If we take the limit ε → 0, we obtain
1
volRn−l (∆lim
vol+
Yl• (D)) =
X|Yl (D).
(n − l)!
lim
Since dim ∆lim
Yl• (D) = n − l, we have volRn−l (∆Yl• (D)) > 0. On the other hand, since Yl )
+
Yn−κν (D) = V , volX|Yl (D) = 0 holds by Theorem 2.12. Thus we have a contradiction.
OKOUNKOV BODIES ASSOCIATED TO PSEUDOEFFECTIVE DIVISORS
13
Note that by Proposition 2.18, if each Yi is general, then Y• always contains a positive volume
locus. Thus for such general admissible flag Y• , ∆lim
Y• (D) also satisfies the same properties of
lim
∆V• (D) in Definition 3.18.
The following is the main property of the limiting Okounkov body ∆lim
V• (D).
Theorem 3.20. Let D be a pseudoeffective divisor on a smooth projective variety X. Fix a
positive volume locus V ⊆ X of D and consider an admissible flag Y• on X containing V . Then
we have
1
lim
vol+
dim ∆lim
Y• (D) = κν (D) and volRκν (D) (∆Y• (D)) =
X|V (D).
κν (D)!
Proof. By Proposition 3.19, we will prove the assertion for the V• instead of Y• . It is enough
to prove the case where D is a Cartier divisor since the R-divisor case follows by the standard
argument. Since V 6⊆ B− (D) by definition, it follows that V 6⊆ B+ (kD + A) for any k ≥ 1. Let
W•k := W• (kD + A|V ) be the restricted graded series of kD + A along V for k ≥ 1. Then W•k
satisfies Condition (C) by [LM, Lemma 2.16]. It follows from Theorem 3.5 that for each k ≥ 1,
dim ∆V• (W•k ) = dim V = κν (D)
and
1
volX|V (kD + A).
κν (D)!
The first equality implies dim ∆lim (D) = κν (D). The second equality yields the following
computation:
volRκν (D) (∆V• (W•k )) =
volRκν (D) (∆lim (D)) = volRκν (D)
lim 1 ∆(W•k )
k→∞ k
1
k
κ (D) volRκν (D) ∆(W• )
k→∞ k ν
1
lim κν (D)!
volX|V D + k1 A
k→∞
+
1
κν (D)! volX|V (D).
= lim
=
=
If D is a big divisor, then by [LM] and [J] it is known that ∆Y• (D) = ∆Y• (D0 ) for all flags
Y• on X if and only if D ≡ D0 . Remark 3.13 implies that the statement is false in general
in the pseudoeffective case for ∆(D) (as well as ∆val (D)). We prove below that ∆lim (D) is a
appropriate generalization of ∆(D) which makes the statement true.
Proposition 3.21. Let D be a pseudoeffective divisor on a smooth projective variety X. Fix a
positive volume locus V ⊆ X of D and consider an admissible flag Y• on X containing V . Then
∆lim
Y• (D) depends on the numerical class of D.
Proof. By Proposition 3.19, we only have to prove the assertion for the V• instead of Y• . Suppose
that D, D0 are pseudoeffective divisors such that D ≡ D0 . Then Theorem 2.19 implies that
(D + εA)|V , (D0 + εA)|V are big on V for a fixed ample divisor A and any ε > 0. By Remark
3.7, we have ∆V• (D + εA) = ∆V• (D0 + εA) for any admissible flags V• on V . Thus
0
lim
0
∆lim
V• (D) = lim ∆V• (D + εA) = lim ∆V• (D + εA) = ∆V• (D ).
ε→0+
ε→0+
For a pseudoeffective divisor D, define the following set of divisors
Y• is an admissible flag with Y1 = E such that
lim
n(∆ (D)) := E lim
.
∆Y• (D)x1 =0 = ∅
Lemma 3.22. The divisors in n(∆lim (D)) are precisely the divisorial components of B− (D)
and the set n(∆lim (D)) is finite.
14
SUNG RAK CHOI, YOONSUK HYUN, JINHYUNG PARK, AND JOONYEONG WON
Proof. Let Y• be an admissible flag on X such that Y1 = E. By [LM], it is known that for any
ample divisor A and ε > 0,
ordE (||D + εA||) = inf {x1 |(x1 , · · · , xn ) ∈ ∆Y• (D + εA) } .
If we let ε → 0, then we have
o
n (D)
.
ordE (||D||) = inf x1 (x1 , · · · , xn ) ∈ ∆lim
Y•
By [ELMNP2], ordE (||D||) > 0 if and only if E ⊆ B− (D). Thus we obtain the desired statement.
The set n(∆(D)) is finite by [N].
We prove the converse of Proposition 3.21.
Theorem 3.23. Let D and D0 be pseudoeffective divisors on X. Then the following are equivalent:
(1) D ≡ D0
lim
0
(2) ∆lim
Y• (D) = ∆Y• (D ) with respect to any admissible flags Y• on X.
Proof. The equivalence (1) ⇔ (2) is known for big divisors by [LM, Proposition 4.1] and [J,
Theorem A]. Thus we assume that D, D0 are not big. The implication (1)⇒(2) follows by
Proposition 3.21. To prove (1)⇐(2), we use the argument used to prove the big case by Jow in
[J].
Let E1 , · · · , El be the divisorial components of B− (D). For any sufficiently general admissible
flag Y• on X such that Yn−1 6⊆ B+ (D), we have
lim
volR (∆
Y• (D)x1 =···=xn−1 =0)
= volR
=
=
lim ∆Y• (D + εA)
ε→0+
x1 =···=xn−1 =0
lim volR (∆Y• (D + εA)x1 =···=xn−1 =0 )
ε→0+
lim volX|Yn−1 (D + εA)
P
= Yn−1 · D − ordEi (||D||)
ε→0+
(by [J, Theorem 3.4 (b)])
(by Lemma 3.24)
where the summation in the last line is taken over all Ei (i = 1, · · · , l) such that Yn−1 ∩ Ei 6= ∅.
As can be seen in the proof of Lemma 3.22, the summation depends only on ∆lim
Y• (D). Therefore,
lim (D 0 ), then we have Y
0 . As in [J], if consider sufficiently
if ∆lim
(D)
=
∆
·
D
=
Y
·
D
n−1
n−1
Y•
Y•
1 ,··· ,Y ρ
general flags Y•1 , · · · , Y•ρ (where ρ = dim N1 (X)R ) such that Yn−1
n−1 form a basis of
0
N1 (X)R , then we can conclude that D ≡ D .
Now we are left to prove the following lemma which is used in the proof.
Lemma 3.24 (cf. [J, Corollary 3.3]). Let Y• be a sufficiently general admissible flag on X.
For a pseudoeffective divisor D on X, let E1 , E2 , · · · , El be the divisorial components of B− (D),
then
X
vol+
ordEi (||D||)
X|Yn−1 (D) = lim volX|Yn−1 (D + εA) = Yn−1 · D −
ε→0+
where the last summation is taken over all i such that Yn−1 ∩ Ei 6= ∅.
Proof. Suppose first that D is big. Since Yn−1 is very general, we have D ∈ BigYn−1 (X). Thus
vol+
X|Yn−1 (D) = volX|Yn−1 (D) and the statement is nothing but [J, Corollary 3.3].
Now let D be a pseudoeffective divisor. Applying the statement for the case of big divisors
to volX|Yn−1 (D + εA), we obtain
vol+
X|Yn−1 (D) =
lim volX|Yn−1 (D + εA)
ε→0+
P
lim (Yn−1 · (D + εA) − ordEi (||D + εA||))
ε→0+
P
= Yn−1 · D − ordEi (||D||).
=
OKOUNKOV BODIES ASSOCIATED TO PSEUDOEFFECTIVE DIVISORS
15
Remark 3.25. Let D be a pseudoeffective Cartier divisor on X and V ⊆ X be its positive
volume locus of dimension κν (D). Let Y• be an admissible flag on X whose (n − κν (D))-th
partial flag Y(n−κν (D))• induces an admissible flag V• on V . To construct the limiting Okounkov
body ∆lim
V• (D), we used the the restricted complete linear series W• (X|V, D) which is a subseries
of the complete linear series W•0 = {H 0 (V, OV (mD|V ))}m≥0 . Using the linear series W•0 instead,
we can construct a convex body ∆YV• (D|V ) associated to the restricted divisor D|V on V . We
can easily verify the following:
dim ∆V• (D|V ) = κν (D) and
1
volV (D|V ).
κν (D)!
The proofs are left to the readers. Let Y• be an admissible flag on X which is general enough
so that it contains both Nakayama locus U and positive volume locus V of a pseudoeffective
divisor D. Then we have the following inclusions:
volRκν (D) ∆V• (D|V ) =
(#)
lim
lim
∆val
Y• (D) ⊆ ∆Y• (D) = ∆V• (D) ⊆ ∆V• (D|V ).
This confirms the inclusions volX|V (D) ≤ vol+
X|V (D) ≤ volV (D|V ) which we saw in Subsection
2.3. If D is big, then all the inclusions are equalities. However, we will see in the next section
that if D is not big, then they are strict in general.
4. Examples
In this section, we exhibit various examples and counterexamples related to our results.
First, recall that in defining ∆val (D) and ∆lim (D), we need to fix a Nakayama locus and a
positive volume locus of D and the flags on them, respectively. The following examples show
that for some badly chosen flags, Theorems 3.12 and 3.20 do not hold.
Example 4.1. Let π : S → P2 be a blow-up of P2 at two distinct points with exceptional
divisors E1 and E2 and L := π ∗ OP2 (1). Consider a non-big effective divisor D := L − E1 + E2 ,
and fix an admissible flag
{x} ⊆ E1 ⊆ S.
Here we note that E1 is neither a Nakayama locus nor a positive volume locus of D. However,
we can still follow the same construction for ∆val (D) and ∆lim (D) with respect to any admissible
flag. We have
∆val (D) = ∆lim (D) = {(x1 , x2 ) ∈ R2 | 1 ≤ x1 ≤ 2 and x2 = 0}
whose Euclidean volume in the x1 -axis is 1. However, note that B− (D) = B+ (D + εA) = E2
for an ample divisor A and a small ε > 0. Thus we see that volX|E1 (D) = vol+
X|E1 (D) = 0. We
also remark that ∆val (D) = ∆lim (D) is not in the x2 -axis as usual. This is due to the fact that
E1 is not a positive volume locus of D.
Examples 4.2 and 4.3 show that the inequalities in (#) of Remark 3.25 can be strict.
Example 4.2. In this example, we see that ∆val (D) ( ∆lim (D) even if κ(D) = κν (D). Let
S be a relatively minimal rational surface, and H a sufficiently positive ample divisor. Take a
general element V ∈ |H|. Fix an admissible flag
{x} ⊆ V ⊆ S.
Note that κ(−KS ) = κν (−KS ) = 1, and V is both a Nakayama locus and a positive volume locus of D. Thus we have volR1 (∆val (−KS )) = volS|V (−KS ) and volR1 (∆lim (−KS )) =
16
SUNG RAK CHOI, YOONSUK HYUN, JINHYUNG PARK, AND JOONYEONG WON
+
vol+
S|V (−KS ). However, we saw in Example 2.3 that volS|V (−KS ) < volS|V (−KS ). Thus
∆val (−KS ) ( ∆lim (−KS ).
Example 4.3. Let S := P(E) where E is a rank two vector bundle on an elliptic curve C such
that it is a nontrivial extension of OC by OC , and let H be the tautological divisor of P(E).
Then we can easily check that H is nef and κ(H) = 0 but κν (H) = 1. Let F be a fiber of
the natural ruling π : S → C. Note that any point in S is a Nakayama locus of H and F is a
positive volume locus of H. Thus take an admissible flag
{x} ⊆ F ⊆ S.
Then we can easily compute the following.
(1) ∆val (H) = {(0, 0)} and volS|{x} (H) = 0.
(2) ∆lim (H) = {(x1 , x2 ) | x1 = 0 and 0 ≤ x2 ≤ 1} and vol+
S|F (H) = 1.
(3) ∆(H|F ) = ∆lim (H).
For ∆lim (H), we see the convergence of limε→0+ ∆(H + εA) for any ample divisor A in the
following picture.
x2
1+ε
∆(H + εA)
1
∆lim (H)
x1
In fact, we use the following for computing the limiting Okounkov body ∆lim (D) of a pseudoeffective divisor D (cf. [LM, Theorem 6.4])
Theorem 4.4. Let S be a smooth projective surface, and let D be a pseudoeffective divisor
on S. Fix an admissible flag {x} ⊆ C ⊆ S. Let a := multC N where D = P + N is the
Zariski decomposition, and let µ := sup{s ≥ 0 | D − sC is pseudoeffective}. Consider a divisor
Dt := D − tC for a ≤ t ≤ µ. Denote by Dt = Pt + Nt the Zariski decomposition. Let
α(t) := ordx (Nt |C ) and β(t) := α(t) + C.Pt . Then the limiting Okounkov body of D is given by
∆lim (D) = {(x1 , x2 ) ∈ R2 | a ≤ x1 ≤ µ and α(x1 ) ≤ x2 ≤ β(x2 )}.
Proof. Let Dε := D + εA for some ample divisor A, and let Dtε := Dε − tC. Denote by
Dtε = Ptε + Ntε the Zariski decomposition. We see that the Zariski decomposition Dt = Pt + Nt
is given by Pt = limε→0+ Ptε and Nt = limε→0+ Ntε . Since ∆lim (D) = limε→0+ ∆(D + εA), the
assertion now follows from [LM, Theorem 6.4].
Remark 4.5. In Theorem 4.4, we do not need to assume that an admissible flag contains a
positive volume locus.
Lastly, we give an example for which the inequalities (#) of Remark 3.25 are equalities.
Example 4.6. Let X be a smooth projective toric variety with a torus T . For definitions and
notations for toric varieties, we refer to [F]. When D is an irreducible big T -divisor, for any fixed
OKOUNKOV BODIES ASSOCIATED TO PSEUDOEFFECTIVE DIVISORS
17
admissible flag Y• , ∆lim (D) = ∆Y• (D) is the same as PD up to translation. For pseudoeffective
divisor D, we have
\
\
PD+εA = PD .
∆(D + εA) =
∆lim (D) = lim ∆(D + εA) =
ε→0+
ε>0
ε>0
For any projective toric variety, we have κ(D) = κν (D), which implies dim ∆val (D) = dim ∆lim (D).
In fact, we can show that ∆val (D) = ∆lim (D). To see this, it is enough to show that volX|V (D) =
vol+
X|V (D) = limε→0+ volX|V (D + εA) for a fixed positive volume locus V . Let V (τ ) be a T invariant irreducible closed subvariety of X, and π the projection of MR to (Mτ )R = MR ∩ τ ⊥ .
Then we observe that
(1) dim H 0 (X, OX (D)) = the number of lattice points in PD , and
(2) dim Im(H 0 (X, OX (D)) → H 0 (V, OV (D|V ))) = the number of lattice points in π(PD ).
T
It follows that π(PD ) = ε>0 π (PD+εA ). Thus volX|V (D) = vol+
X|V (D).
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Center for Geometry and Physics, Institute for Basic Science, Pohang, Gyeongbuk, Korea
E-mail address: [email protected]
School of Mathematics, Korea Institute for Advanced Study, Seoul, Korea
E-mail address: [email protected]
School of Mathematics, Korea Institute for Advanced Study, Seoul, Korea
E-mail address: [email protected]
Department of Mathematical Sciences, KAIST, Daejeon, Korea
E-mail address: [email protected]