Seshadri constants via Okounkov functions and the
Segre-Harbourne-Gimigliano-Hirschowitz Conjecture
M. Dumnicki, A. Küronya∗, C. Maclean, T. Szemberg†

arXiv:1304.0249v1 [math.AG] 31 Mar 2013

April 2, 2013

Abstract
In this paper we relate the SHGH Conjecture to the rationality of onepoint Seshadri constants on blow ups of the projective plane, and explain how
rationality of Seshadri constants can be tested with the help of functions on
Newton–Okounkov bodies.
Keywords Nagata Conjecture, SHGH Conjecture, Seshadri constants,
Okounkov bodies
Mathematics Subject Classification (2000) MSC 14C20

1

Introduction

Nagata’s conjecture and its generalizations have been a central problem in the theory
of surfaces for many years, and much work has been done towards verifying them
[19], [8], [13], [23], [9]. In this paper we open a new line of attack in which we
relate Nagata-type statements to the rationality of one-point Seshadri constants and
invariants of functions on Newton–Okounkov bodies. We obtain as a consequence
of our approach some evidence that certain Nagata-type questions might be false.
Seshadri constants were ﬁrst introduced by Demailly in the course of his work
on Fujita’s conjecture [10] in the late 80’s and have been the object of considerable
interest ever since. Recall that given a smooth projective variety X and a nef line
bundle L on X, the Seshadri constant of L at a point x ∈ X is the real number
ε(L; x) =def inf
C

L·C
,
multx C

(1)

where the inﬁmum is taken over all irreducible curves passing through x. An intriguing and notoriously diﬃcult problem about Seshadri constants on surfaces is
the question whether these invariants are rational numbers, see [17, Remark 5.1.13]
It follows quickly from
√ their deﬁnition that if a Seshadri constant is irrational then
it must be ε(L; x) = L2 , see e.g. [3, Theorem 2.1.5]. It is also known that Seshadri
constants of a ﬁxed line bundle L, take their maximal value on a subset in X which
is a complement of at most countably many Zariski closed proper subsets of X.
∗

During this project Alex Küronya was partially supported by DFG-Forschergruppe 790 “Classification of Algebraic Surfaces and Compact Complex Manifolds”, and the OTKA Grants 77476
and 81203 by the Hungarian Academy of Sciences.
†
Szemberg research was partially supported by NCN grant UMO-2011/01/B/ST1/04875

2
We denote this maximum by ε(L, 1). Similar notation is used for√multi-point
2
Seshadri constants, see [3, Deﬁnition 1.9]. In particular, if ε(L; x)
√ = L at some
point the same holds in a very general point on X and ε(L; 1) = L2 .
From a slightly diﬀerent point of view, Seshadri constants reveal information on
the structure of the nef cone on the blow-up of X at x, hence their study is closely
related to our attempts to understand Mori cones of surfaces.
An even older problem concerning linear series on algebraic surfaces is the conjecture formulated by Beniamino Segre in 1961 and rediscovered, made more precise
and reformulated by Harbourne 1986, Gimigliano 1987 and Hirschowitz 1988. (See
[8] for a very nice account on this development and related subjects.) In particular
it is known, [8, Remark 5.12] that the SHGH Conjecture implies the Nagata Conjecture. We now recall this conjecture, using by Gimigliano’s formuation, which will
be the most convenient form for us [12, Conjecture 3.3].
SHGH Conjecture. Let X be the blow up of the projective plane P 2 in s general
points with exceptional divisors E1 , . . . , Es . Let H denote the pullback to X of the
hyperplane bundle OP2 (1) on P2 . Let the integers d, m1 > . . . > ms > −1 with
d > m1 + m2 + m3 be given. Then the line bundle
dH −

s
X

mi Ei

i=1

is non-special.
The main result of this note is the following somewhat unexpected relation between
the SHGH Conjecture and the rationality problem for Seshadri constants.
Theorem 1.1. Let s > 9 be an integer for which the SHGH Conjecture holds true.
Let X be the blow up of the projective plane P2 in s general points. Then
a) either there exists on X an ample line bundle whose Seshadri constant at a
very general point is irrational;
b) or the SHGH Conjecture fails for s + 1 points.
Note that it is known that the SHGH conjecture holds true for s 6 9, [8, Theorem
5.1]. It is also known that Seshadri constants of ample line bundles on del Pezzo
surfaces (i.e. for s 6 8) are rational, see [22, Theorem 1.6]. In any case, the statement
of the Theorem is interesting (and non-empty) for s = 9. (See the challenge at the
end of the article.)
Corollary 1.2. If all one-point Seshadri constants on the blow-up of P2 in nine
general points are rational, then the SHGH conjecture fails for ten points.
An interesting feature of our proof is that the role played by the general position
of the points at which we blow up becomes clear.
In a diﬀerent direction, we study the connection between functions on Newton–
Okounkov bodies deﬁned by orders of vanishing, and Seshadri-type invariants. Our
main result along these lines is the following.
Theorem 1.3. Let X be a smooth projective surface, Y• an admissible ﬂag, L a
big line bundle on X, and let P ∈ X be an arbitrary point.
If

max

x∈∆Y• (L)

φordp (x) ∈ Q then ε(L, P ) ∈ Q .

3
Acknowledgements. We thank Cristiano Bocci, Sébastien Boucksom and Patrick
Graf for helpful discussions. Part of this work was done while the second author
was visiting the Uniwersytet Pedagogiczny in Cracow. We would like to thank the
Uniwersytet Pedagogiczny for the excellent working conditions.

2

Rationality of one point Seshadri constants and the SHGH
Conjecture

In this section we prove Theorem 1.1: we start with notation and preliminary
leamms. Let f : X → P2 be the blow up of P2 at s > 9 general points P1 , . . . , Ps
with exceptional divisors E1 , . . . , Es . We denote as usual by H = f ∗ (OP2 (1)) the
pull back of the hyperplane bundle and we let E = E1 + · · · + Es be the sum of exceptional divisors. We consider the blow up g : Y → X of X at P with exceptional
divisor F . Whilst The following result is well known, we include it for the lack of a
proper reference.
P
Lemma 2.1. If there exists a curve C ⊂ X in the linear system dH − si=1 mi Ei
computing Seshadri constant of a Q-line bundle L = H − αE, then there exists a
divisor Γ with multP1 Γ = . . . = multPs Γ = M computing Seshadri constant of L at
P , i.e.
L·C
L·Γ
=
= ε(L; P ).
multP Γ
multP C
Proof. Since the points P1 , . . . , Ps are general, there exist curves
Cσ = dH −

s
X

mσ(j) Ej

j=1

for all permutations σ ∈ Σs . Since the point P is general, we may take all these
curves to have the same multiplicity m at P . Summing over a cycle σ of length s in
Σs , we obtain a divisor
Γ=

s
X
i=1

Cσi = sdH −

s
s X
X
i=1 j=1

mσi (j) Ej = sdH − M E,

with M = m1 + . . . + ms . Note that the multiplicity of Γ at P equals sm. Taking
the Seshadri quotient for Γ we have
sd − αsM
d − αM
L·Γ
=
=
= ε(L; P )
sm
sm
m
hence Γ satisﬁes the assertions of the Lemma.
The following auxiliary Lemma will be used in the proof of Theorem 1.1. We
postpone its proof to the end of this section.
Lemma 2.2. Let s > 9 be an integer. The function
p
√
√
f (δ) = (2 s + 1 − s) 1 − sδ2 + s(1 − s + 1)δ + s − 2

(2)

is non-negative for δ satisfying

√

1
1
<δ< √ .
s
s+1

(3)

4
Proof of Theorem 1.1. Let δ be a rational number satisfying (3). Note that the
SHGH Conjecture implies the Nagata Conjecture [8, Remark 5.12] so that
1
ε(OP2 (1); s) = √
s
and hence the Q–divisor L = H − δE is ample. If ε(L; 1) is irrational, then we are
done.
√
So we proceed assuming that ε(L; 1) is rational and that it is not equal to L2
(this can be achieved changing δ a little bit if necessary). In particular, by Lemma
2.1 for a general point P ∈ X Seshadri constant ε(L; P ) there is a divisor Γ ⊂ P2
of degree γ with M = multP1 Γ = . . . = multPs γ and m = multP Γ whose proper
e on X computes the Seshadri constant
transform Γ
ε(L; P ) =

p
e
L·Γ
γ − δsM
=
< 1 − sδ2 .
m
m

This gives an upper bound on γ
γ<m

p

1 − sδ2 + δsM.

(4)

We need to prove that statement b) in Theorem 1.1 holds. Suppose not: the SHGH
Conjecture then holds for s + 1 points in P2 . The Nagata Conjecture then also holds
for s + 1 points and this gives a lower bound for γ, since for Γ we must have that
1
γ
.
>√
sM + m
s+1

(5)

γ > 2M + m.

(6)

γ < 2M + m.

(7)

We now claim that
Suppose not. We then have that

The real numbers
√
√
2 s+1−s
s − δs s + 1
a :=
and b :=
2 − δs
2 − δs
are positive. Multiplying (4) by a and (7) by b and adding we obtain
p
√
sM + m 6 γ s + 1 < sM + (b + a 1 − sδ2 )m,
where the ﬁrst inequality follows from (5). Subtracting sM in the left and in the
right term and dividing by m we obtain
p
1 < b + a 1 − sδ2 .
Plugging in the deﬁnition of a and b and rearranging terms we obtain that
p
√
√
(2 s + 1 − s) 1 − sδ2 + s − δs s + 1 < 2 − δs,
which contradicts Lemma 2.2. Hence (6) holds.

5
It follows now from the SHGH conjecture for s + 1 points (in the form stated in
the introduction) that the linear system
γH − M E − mF
on Y is non-special. Indeed the condition γ > 2M + m is (6) and the condition
γ
> √1s (because the Nagata Conjecture holds for s
γ > 3M is satisﬁed since sM
by hypothesis) and because we have assumed that s > 9. This system is also nonempty because the proper transform of Γ under g is its member. Thus by a standard
dimension count
0 6 γ(γ + 3) − sM (M + 1) − m(m + 1).
The upper bound on γ (4) together with the above inequality yields
p
p
0 6 (sδM + m 1 − sδ2 )(sδM + m 1 − sδ2 + 3) − m2 − m − sM − sM 2 .

(8)

Note that the quadratic term in (8) is a negative semi-deﬁnite form
p
(s2 δ2 − s)M 2 + 2sδ 1 − sδ2 M m − sδ2 m2 .

Indeed, the restrictions on δ made in (3) imply that the term at M 2 is negative.
The determinant of the associated symmetric matrix vanishes. These two conditions
imply together that the form is negative semi-deﬁnite. In particular this term of (8)
is non-positive. The linear part in turn is
p
(3sδ − s)M + (3 1 − sδ2 − 1)m,
which is easily seen to be negative. This provides the desired contradiction and
ﬁnishes the proof of the Theorem.
Remark 2.3. As it is well known, Nagata’s conjecture can be interpreted in terms of
the nef and Mori cones of the blow-up X of P2 at s general points. More precisely,
consider the following question: for what t > 0 does the ray H − tE meet the
boundary of the nef cone? The conjecture predicts that this ray should intersect the
boundaries of the nef cone and the eﬀective cone at the same time.
Considering the Zariski chamber structure of X (see [5]), we see that this is
equivalent to requiring that H − tE crosses exactly one Zariski chamber (the nef
cone itself). Surprisingly, it is easy to prove that H − tE cannot cross more than
two chambers.
Proposition 2.4. Let f : X → P2 be the blow up of P 2 in s general points with
exceptional divisors E1 , . . . , Es . Let H be the pull-back of the hyperplane bundle and
E = E1 + . . . + Es . The ray R = H − tE meets at most two Zariski chambers on X.
Proof. If ε = ε(OP2 (1); s) = √1s i.e. this multi-point Seshadri constant is maximal,
then the ray crosses only the nef cone.
P
If ε is submaximal, then there is a curve C = dH −
mi Ei computing this
d
P
Seshadri constant, i.e. ε = mi .
If this curve is homogeneous, i.e. m = m1 = · · · = ms , then we claim ﬁrst that
µ = µ(OP2 (1); E) = m/d. Indeed, this is an eﬀective divisor on the ray R and it is
not big (because big divisors on surfaces intersect all nef divisors positively, see [6,
Corollary 3.3]), so it must be the point where the ray leaves the big cone.

6
Now, suppose that for some ε < δ < µ the ray R crosses another Zariski chamber
wall. This means that there is a divisor eH −kE (obtained after possible symmetrization of a curve D with 0 intersection number with H − δE) with
(H − δE) · (eH − kE) = 0.
Hence e = δks < µks. On the other hand
(eH − kE) · (dH − µE) = e − kµs < 0,
implies that C is a component eH − kE which is not possible. Hence there are only
two Zariski chambers meeting the ray R in this case.
If the curve C is not homogeneous, then since the points are general there exist at
least (and also at most) s diﬀerent irreducible curves computing ε. All these curves
are in the support of the negative part of the Zariski decomposition of H − λE for
λ > ε. Hence their intersection matrix is negative deﬁnite and this is a matrix
of maximal dimension (namely s) with this property. This implies that R cannot
meet another Zariski chamber because the support of the negative part of Zariski
decompositions grows only when encountering new chambers, see [5].
It is interesting to compare this result with the following easy example, which
constructs rays meeting a maximal number of chambers.
Example 2.5. Keeping the notation from Proposition 2.4 let L = ( s(s+1)
+ 1)H −
2
E1 − 2E2 − . . .− sEs is an ample divisor on X and the ray R = L + λE crosses s + 1 =
ρ(X) Zariski chambers. Indeed, with λ growing, exceptional divisors E1 , E2 , . . . , Es
join the support of the Zariski decomposition of L − λE one by one. We leave the
details to the reader.
We conclude this section with the proof of Lemma 2.2.
√
s + 1) = 0 it is enough to show that f (δ) is inProof of Lemma
√ 2.2. Since f (1/
√
creasing for 1/ s + 1 6 δ 6 1/ s. Consider the derivative


√
√
δ
′
(9)
f (δ) = s 1 + √
(s − 2 s + 1) − s + 1 .
1 − sδ2
√
√
δ
The function h(δ) = √1−sδ
is increasing for 1/ s + 1 6 δ 6 1/ s since the nu2
merator is an increasing function of δ and the denominator is a decreasing function
1
) = 1 so that h(δ) > 1 holds for all δ. Since the coeﬃcient at
of δ. We have h( √s+1
h(δ) in (9) is positive we have
√
√
√

f ′ (δ) > s 1 + (s − 2 s + 1) − s + 1 = 1 + s − 3 s + 1 > 0,
which completes the proof.

3

Rationality of Seshadri constants and functions on Okounkov bodies

The theory of Newton–Okounkov bodies has emerged recently with work by Okounkov [21], Kaveh–Khovanskii [14], and Lazarsfeld–Mustaţă [18]. Shortly thereafter, Boucksom–Chen [7] and Witt-Nyström [20] have shown ways of constructing
geometrically signiﬁcant functions on Okounkov bodies, that were further studied

7
in [16]. In the context of this note the study of Okounkov functions was pursued
by the last three authors in [16]. We refer to [16] for construction and properties of
Okounkov functions.
In this section we consider an arbitrary smooth projective surface X and an
ample line bundle L on X. Let p ∈ X be an arbitrary point and let π : Y → X be
the blow up of p with exceptional divisor E. Recall that the Seshadri constant of L
at p can equivalently be deﬁned as
ε(L; p) = sup {t > 0 | π ∗ L − tE is nef } .
There is a related invariant
def

µ(L; p) = sup {t > 0 | π ∗ L − tE is pseudo-eﬀective} = sup {t > 0 | π ∗ L − tE is big} .
The invariant ε(L; p) is the value of the parameter λ where the ray π ∗ L − λE meets
the boundary of the nef cone of Y , and µ(L; p) is the value of λ where the ray meets
the boundary of the pseudo-eﬀective cone. The following relation between the two
invariants is important in our considerations.
Remark 3.1. If ǫ(L; p) is irrational, then
ǫ(L; p) = µ(L; p) .
In particular, if µ(L; p) is rational, then so is ǫ(L; p).
Rationality of µ(L; p) implies rationality of the associated Seshadri constants on
surfaces. This invariant appears in the study of the concave function ϕordp associated
to the geometric valuation on X deﬁned by the order of vanishing ordp at p. We
ﬁx some ﬂag Y• : X ⊇ C ⊇ {x0 } and consider the Okounkov body ∆Y• (L) deﬁned
with respect to that ﬂag. We deﬁne also a multiplicative ﬁltration determined by the
geometrical valuation ordP on the graded algebra V = ⊕k>0 Vk with Vk = H 0 (X, kL)
by
Ft (V ) = {s ∈ V : ordP (s) > t} ,
see [16, Example 3.7] for details. (All the above remains valid in the more general
context of graded linear series.) There is an induced ﬁltration F• (Vk ) on every
summand of V and one deﬁnes the maximal jumping numbers of both ﬁltrations as

emax (V, F• ) = sup {t ∈ R : ∃kFktVk 6= 0} and emax (Vk , F• ) = sup {t ∈ R : Ft Vk 6= 0}
respectively. Let ϕordP (x) = ϕF• (x) be the Okounkov function on ∆Y• (L) determined by ﬁltration F• , see [16, Deﬁnition 4.8]. It turns out that µ(L; p) is the
maximum of the Okounkov function ϕordP .
Proposition 3.2. With notation as above we have that

max ordp (s) | s ∈ H 0 (X, OX (mL))
µ(L; p) = lim sup
m
m→∞

=

max

x∈∆Y• (L)

φordp (x) .

Proof. Observe that
ordp (s) = ordE (π ∗ s) = max {m ∈ N | div(π ∗ s) − mE is eﬀective} .
Consequently,
µ(L; p) = sup {t ∈ R>0 | π ∗ L − tE is pseudo-eﬀective}

max ordp (s) | s ∈ H 0 (X, OX (mL)
,
= lim sup
m
m→∞

8
which gives the ﬁrst equality.
For the second equality, we observe ﬁrst that

max ordp (s) | s ∈ H 0 (X, OX (mL))

= emax (Vm , F• ) ,

and hence


max ordp (s) | s ∈ H 0 (X, OX (mL))
lim sup
m
m→∞

= emax (V, F• ) .

Since
emax (V, F• ) =

max

x∈∆Y• (L)

ϕordp (x)

by Theorem 3.4, we are done.

3.1

Independence of the maximum of an Okounkov function on the flag

In the course of this section the projective variety X can have arbitrary dimension.
Boucksom and Chen proved that though ϕF• and ∆(V• ) depend on the ﬂag Y• ,
the integral of ϕF• over ∆(V• ) is independent of Y• , [7, Remark 1.12 (ii)]. We prove
now that the maximum of the Okounkov function does not depend on the ﬂag. This
fact is valid in the general setting of arbitrary multiplicative ﬁltration F deﬁned on
a graded linear series V• .
Remark 3.3. Note that in general the functions ϕF• are only upper-semicontinuous
and concave, but not continuous on the whole Newton–Okounkov body as explained
in [16, Theorem 1.1]. They are however continuous provided the underlying body
∆(V• ) is a polytope (see again [16, Theorem 1.1]), which is the case for complete
linear series on surfaces [15].
Theorem 3.4 (Maximum of Okounkov functions). With the above notation, we
have that
max ϕF• (x) = emax (L, F• ).
x∈∆Y• (L)

In particular the left hand side does not depend on the flag Y• .
Proof. For any real t > 0, we consider the partial Okounkov body ∆t,Y• (L) associated
the graded linear series Vt,k ⊂ H 0 (kL) given by
def

Vt,k = Fkt (H 0 (kL)) .
Note that by deﬁnition
emax (L, F• ) = sup{t ∈ R| ∪k Vt,k 6= 0.}
In other words,
emax (L, F• ) = sup{t ∈ R|∆t,Y• (L) 6= ∅}.
Recall that by deﬁnition
ϕF• (x) = sup{t ∈ R|x ∈ ∆t,Y• (L)}.
and it is therefore immediate that ∀x
ϕF• (x) 6 emax (L, F• ).

9
from which it follows that
max

x∈∆Y• (L)

ϕF• (x) 6 emax (L, F• ).

Since the bodies ∆t,Y• (L) form a decreasing family of closed subsets of Rd , we have
that
∩t|∆t,Y• (L)6=∅ ∆t,Y• (L) 6= ∅.
Consider a point y ∈ ∩t|∆t,Y• (L)6=∅ ∆t,Y• (L) We then have that
y ∈ ∆t,Y• (L) ⇔ ∆t,Y• (L) 6= ∅
and hence
sup{t ∈ R|y ∈ ∆t,Y• (L)} = sup{t ∈ R|∆t,Y• (L) 6= ∅}
or in other words
ϕF• (y) = emax (L, F• )
from which it follows that
max

x∈∆Y• (L)

ϕF• (x) 6 emax (L, F• ).

This completes the proof of the theorem.

4

The effect of blowing up on Okounkov bodies and functions

We begin with an observation (valid in fact in arbitrary dimension, though we state
and prove it here only for surfaces.)
Proposition 4.1. Let S be an arbitrary surface with a fixed flag Y• and let f : X →
S be the blow up of S at a point P not contained in the divisorial part of the flag,
P ∈
/ Y1 . Let E be the exceptional divisor. And finally let D be a big divisor on S.
For any rational number λ such that 0 6 λ < µ(L; P ) we let Dλ be the Q–divisor
f ∗ D − λE. There is then a natural inclusion
∆Y• (Dλ ) ⊂ ∆Y• (D).
Moreover, a filtration F• on the graded algebra ⊕k>0 H 0 (S; kD) induces a filtration
F•λ on the graded (sub)algebra ⊕H 0 (X, kDλ ), where the sum is taken over all k
divisible enough. For associated Okounkov functions we have
ϕF•λ (x) 6 ϕF• (x)

(10)

for all x ∈ ∆Y• (Dλ ).
Remark 4.2. The best case scenario is that the functions φ are piecewise linear with
rational coeﬃcients over a rational polytope. Of these properties, some evidence for
the ﬁrst was given by Donaldson [11] in the toric situation. For the second condition,
it was proven in [1] that every line bundle on a surface has an Okounkov body which
is a rational polytope.

10
Proof. Note ﬁrst that since the blow up center is disjoint from all elements in the
ﬂag, one can take Y• to be an admissible ﬂag on X. (Strictly speaking one takes
f ∗ Y• as the ﬂag, but it should cause no confusion to identify ﬂag elements upstairs
and downstairs.)
Then, if k is suﬃciently divisible we have

H 0 (X, f ∗ kD − kλE) = s ∈ H 0 (S, kD) : ordP (s) > E ⊂ H 0 (S, kD).

The inclusion of the Okounkov bodies follows immediately under this identiﬁcation.
We can view the algebra associated to f ∗ D − E as a graded linear series on S.
The claim about the Okounkov functions follows from their deﬁnition, see [16,
Deﬁnition 4.8]. Indeed, the supremum arising in the deﬁnition of ϕF•λ is taken over
a smaller set of sections than it is for ϕF• .
The following examples illustrate various situations arising in the setting of
Proposition 4.1.
Example 4.3. Let ℓ be a line in X0 = P2 and let P0 ∈ ℓ be a point. We ﬁx the ﬂag
Y• : X0 ⊃ ℓ ⊃ {P0 } .

Let D = OP2 (1). Then ∆Y• (D) is simply the standard simplex in R2 .

1

∆Y• (D)
0

1

Let F• be the ﬁltration on the complete linear series of D imposed by the geometric
valuation ν = ordP0 and let ϕν be the associated Okounkov function. Then
ϕν (a, b) = a + b.
Indeed, given a point (a, b) with rational coordinates, we pass to the integral point
(ka, kb). This valuation vector can be realized geometrically by a global section in
H 0 (P2 , OP2 (k)) vanishing exactly with multiplicity ka along ℓ, exactly with multiplicity kb along a line passing through P0 diﬀerent from ℓ and along a curve of degree
k(1 − a − b) not passing through P0 .
The next example shows that even when the Okounkov body changes in the
course of blowing up, the Okounkov function may remain the same.
Example 4.4. Keeping the notation from the previous Example and from Proposition 4.1 let f : X1 = BlP1 P2 → X0 = P2 be the blow up of the projective plane in
a point P1 not contained in the ﬂag line ℓ with the exceptional divisor E1 . We work
now with a Q–divisor Dλ = f ∗ (OP2 (1)) − λE1 = H − λE1 , for some ﬁxed λ ∈ [0, 1].
A direct computation using [18, Theorem 6.2] gives that the Okounkov body has
the shape

11

1

∆Y• (Dλ )
0

1−λ

Thus we see that the Okounkov body of Dλ is obtained from that of D by intersecting
with a closed halfspace.
For the valuation ν = ordP0 , we get as above
ϕν (a, b) = a + b.
Let now k be an integer such that the point (ka, kb) is integral and kλ is also an
integer. Now we need to exhibit a section s in H 0 (P2 , OP2 (k)) satisfying the following
conditions:
a) s vanishes along ℓ exactly to order a;
b) s vanishes in the point P1 to order at least kλ;
c) s vanishes in the point P0 exactly to order b.
We let the divisor of s to consist of a copies of ℓ (there is no other choice here), of
b copies of the line through P0 and P1 , of kλ − b copies of any other line passing
through P1 (if this number is negative then this condition is empty) and of a curve
of degree k(1 − a − max {b, λ}) passing neither through P0 , nor through P1 .
Remark 4.5. Note that in the setting of Proposition 4.1 Okounkov bodies of divisors
Dλ always result from those of D by cutting with ﬁnitely many halfplanes. This is
an immediate consequence of [15, Theorem 5].
We conclude by showing that the inequality in (10) can be sharp, i.e. the blow
up process can inﬂuence the Okounkov function as well as the Okounkov body.
Example 4.6. Keeping the notation from the previous examples, let f : X6 → P2 be
the blow up of six general points P1 , . . . , P6 not contained in ℓ and chosen so that the
points P0 , P1 , . . . , P6 are also general. Let E1 , . . . , E6 denote the exceptional divisors
and set E = E1 +. . .+E6 . We consider the divisor D = H − 52 E. A direct computation
using [18, Theorem 6.2] (this requires computing Zariski decompositions this time,
see [2] for an eﬀective approach) yields the triangle with vertices at the origin and
in points (0, 1) and (1/25, 0) as ∆Y• (D). For the valuation ν = ordP0 we get now
ϕν (a, b) 6 4/15 < a + b

for (a, b) ∈ Ω = (x, y) ∈ R2 : x ∈ [0, 11/360) and b ∈ (4/15 − a, 1 − 25a] .

(11)

12

1

Ω
11 17
( 360
, 72 )

0

1
25

The reason for the above inequality is the following. Let (a, b) ∈ Ω be a valuation vector. Assume to the contrary that ϕ(a, b) > 4/15. It is well known that
ε(OP2 (1), P0 , . . . , P6 ) = 38 , see for instance [23, Example 2.4]. On the other hand a
section with the above valuation vector would have (after scaling to O(1)) multiplicities 2/5 at P1 , . . . , P6 and ϕν (a, b) > 4/15 at P0 . It would give Seshadri quotient
6·

2
5

3
1
< ,
8
+ ϕν (a, b)

a contradiction. This proves (11).
Since the SHGH Conjecture holds for 9 points, the ﬁrst challenge arising in the
view of our Theorem would be to compute the Okounkov body and the Okounkov
function associated to ordP0 as above for the system
22H − 7(E1 + · · · + E9 ).
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Seshadri
arXiv:0908.4502v4.

constants

on

rational

surfaces

with

anticanonical

pencils,

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Marcin Dumnicki, Jagiellonian University, Institute of Mathematics, Lojasiewicza 6,
PL-30-348 Kraków, Poland
E-mail address: Marcin.Dumnicki@im.uj.edu.pl
Alex Küronya, Budapest University of Technology and Economics, Mathematical Institute, Department of Algebra, Pf. 91, H-1521 Budapest, Hungary.
E-mail address: alex.kuronya@math.bme.hu
Current address: Alex Küronya, Albert-Ludwigs-Universität Freiburg, Mathematisches
Institut, Eckerstraße 1, D-79104 Freiburg, Germany.
Catriona Maclean, Institut Fourier, CNRS UMR 5582 Université de Grenoble, 100 rue
des Maths, F-38402 Saint-Martin d’Héres cedex, France
E-mail address: catriona@fourier.ujf-grenoble.fr
Tomasz Szemberg, Instytut Matematyki UP, Podchora̧żych 2, PL-30-084 Kraków, Poland.
E-mail address: szemberg@up.krakow.pl

