Seshadri constants via Okounkov functions and theSegre-Harbourne-Gimigliano-Hirschowitz Conjecture

M. Dumnicki, A. Kuronya∗, C. Maclean, T. Szemberg†

October 17, 2013

Abstract

In this paper we relate the SHGH Conjecture to the rationality of one-point Seshadri constants on blow ups of the projective plane, and explain howrationality of Seshadri constants can be tested with the help of functions onNewton–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 theoryof surfaces for many years, and much work has been done towards verifying them[20], [7], [14], [25], [8]. In this paper we open a new line of attack in which werelate Nagata-type statements to the rationality of one-point Seshadri constants andinvariants of functions on Newton–Okounkov bodies. We obtain as a consequenceof our approach some evidence that certain Nagata-type questions might be false.

Seshadri constants were first introduced by Demailly in the course of his workon Fujita’s conjecture [10] in the late 80’s and have been the object of considerableinterest ever since. Recall that given a smooth projective variety X and a nef linebundle L on X, the Seshadri constant of L at a point x ∈ X is the real number

ε(L;x) =def infC

L · CmultxC

, (1)

where the infimum is taken over all irreducible curves passing through x. An in-triguing and notoriously difficult problem about Seshadri constants on surfaces isthe question whether these invariants are rational numbers, see [18, Remark 5.1.13]It follows quickly from their definition that if a Seshadri constant is irrational thenit 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 whichis a complement of at most countably many Zariski closed proper subsets of X.

∗During this project Alex Kuronya was partially supported by DFG-Forschergruppe 790 “Clas-sification of Algebraic Surfaces and Compact Complex Manifolds”, and the OTKA Grants 77476and 81203 by the Hungarian Academy of Sciences.†The last named author was partially supported by NCN grant UMO-2011/01/B/ST1/04875

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We denote this maximum by ε(L; 1). Similar notation is used for multi-pointSeshadri constants, see [3, Definition 1.9]. In particular, if ε(L;x) =

√L2 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 informationon the structure of the nef cone on the blow-up of X at x, hence their study isclosely related to our attempts to understand Mori cones of surfaces, see also [9] fora somewhat different approach.

An even older problem concerning linear series on algebraic surfaces is the con-jecture formulated by Beniamino Segre in 1961 and rediscovered, made more preciseand reformulated by Harbourne 1986, Gimigliano 1987 and Hirschowitz 1988. (See[7] for a very nice account on this development and related subjects.) In particularit is known, [7, Remark 5.12] that the SHGH Conjecture implies the Nagata Conjec-ture. We now recall this conjecture, using by Gimigliano’s formulation, which willbe the most convenient form for us [13, Conjecture 3.3].

SHGH Conjecture. Let X be the blow up of the projective plane P 2 in s generalpoints with exceptional divisors E1, . . . , Es. Let H denote the pullback to X of thehyperplane bundle OP2(1) on P2. Let the integers d,m1 > . . . > ms > −1 withd > m1 +m2 +m3 be given. Then the line bundle

dH −s∑i=1

miEi

is non-special.

Remark 1.1. Note that there are various formulations of the SHGH Conjecturepresent in the literature. The version presented here addresses general points inP2, whereas e.g. the version discussed in [9, page 2] requires the points to be verygeneral. The results presented in [9] though certainly related are however of differentflavor to those discussed here.

The main result of this note is the following somewhat unexpected relation betweenthe SHGH Conjecture and the rationality problem for Seshadri constants.

Theorem 1.2. Let s > 9 be an integer for which the SHGH Conjecture holds true.Then

a) either there exist points P1, . . . , Ps ∈ P2, a line bundle L on BlP1,...,Ps P2 anda points P ∈ X such that

ε(L;P ) 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, [7, Theorem5.1]. It is also known that Seshadri constants of ample line bundles on del Pezzosurfaces (i.e. for s 6 8) are rational, see [24, Theorem 1.6]. In any case, the statementof the Theorem is interesting (and non-empty) for s = 9. (See the challenge at theend of the article.)

Corollary 1.3. If all one–point Seshadri constants on the blow-up of P2 in ninegeneral points are rational, then the SHGH conjecture fails for ten points.

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An interesting feature of our proof is that the role played by the general positionof 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 filtrations imposed by orders of vanishing, and Seshadri-type invariants. Our main result along these lines is the following.

Theorem 1.4. Let X be a smooth projective surface, Y• an admissible flag, L abig line bundle on X, and let P ∈ X be an arbitrary point.

If maxx∈∆Y• (L)

φordp(x) ∈ Q then ε(L,P ) ∈ Q .

Acknowledgements. We thank Cristiano Bocci, Sebastien Boucksom and PatrickGraf for helpful discussions. We would like also to thank David Schmitz for pointingout an inaccuracy in the original proof of Proposition 2.6. Finally, we thank thereferee for pointing out connections between our work and the de Fernex preprint[9] and extremely helpful remarks clarifying the statement of Theorem 1.2.

Part of this work was done while the second author was visiting the UniwersytetPedagogiczny in Cracow. We would like to thank the Uniwersytet Pedagogiczny forthe excellent working conditions.

2 Rationality of one point Seshadri constants and the SHGHConjecture

In this section we prove Theorem 1.2: we start with notation and preliminary lem-mas. To begin with, recall that the standard Cremona transformation of P 2 is thebirational mapping τ : P2 → P2 which is the composition of the blowing up of threenon-collinear points followed by blowing down the proper transforms of three linesjoining pairwise these points, see [11, Section 7] for beautiful elaborations on Cre-mona transformations. In suitable coordinates adapted to the three points blownup, τ can be written as

τ : P2 3 (x : y : z)→(

1

x:

1

y:

1

z

)∈ P2.

Now, we make it precise what we understand by general points in this note. Herewe follow Nagata, [21].

Definition 2.1. We say that the points P1, . . . , Ps ∈ P2 are in Cremona generalposition if no 3 of them are collinear and the same holds for all standard Cremonatransformations based at 3 of the given points.More generally, we say that the points are in m–Cremona general position for apositive integer m, if after an arbitrary sequence of m− 1 Cremona transformationsthe resulting points remain in Cremona general position.

Note that for any given m, the set of points in m–Cremona general position is aZariski open subset of (P2)×s.

Let f : X → P2 be the blow up of P2 at s > 9 general points P1, . . . , Ps withexceptional divisors E1, . . . , Es. We denote as usual by H = f∗(OP2(1)) the pullback of the hyperplane bundle. It is well-known that H,E1, . . . , Es form a basis ofthe Neron-Severi space N1(X)R of X. Let (d;m1, . . . ,ms) denote the class dH −

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m1E1− . . .−msEs ∈ N1(X)R. A standard Cremona transformation τ based on thefirst three points operates on the space N1(X)R as

(d;m1, . . . ,ms)→ (d+ k,m1 + k,m2 + k,m3 + k,m4, . . . ,ms),

where k = d−m1−m2−m3, see [21, Lemma 2.3]. We explain in the Appendix howthis action leads to the following Lemma.

Lemma 2.2. Let P1, . . . , Ps be a finite set of points in P2 in m–Cremona generalposition with m 0. Let C be an irreducible curve with mi = multPi C. Let σ ∈ Σs

be a permutation of s elements. Then there exists an irreducible curve Cσ withmultPi Cσ = mσ(i).

Keeping the notation introduced above we let E = E1 + · · · + Es be the sumof exceptional divisors of f . We consider the blow up g : Y → X of X at P withexceptional divisor F . The following result is a well-known consequence of Lemma2.2. We were not able to find a reference and we include a proof for the benefit of areader.

Lemma 2.3. If there exists a curve C ⊂ X in the linear system dH −∑s

i=1miEicomputing Seshadri constant of a Q-line bundle L = H − αE, then there exists adivisor Γ with multP1 Γ = . . . = multPs Γ = M computing Seshadri constant of L atP , i.e.

L · ΓmultP Γ

=L · C

multP C= ε(L;P ).

Proof. Since the points P1, . . . , Ps are general, there exist curves

Cσ = dH −s∑j=1

mσ(j)Ej

for all permutations σ ∈ Σs. Since the point P is general, we may take all thesecurves to have the same multiplicity m at P . Summing over a cycle σ of length s inthe symmetric group Σs, we obtain a divisor

Γ =

s∑i=1

Cσi = sdH −s∑i=1

s∑j=1

mσi(j)Ej = sdH −ME,

with M = m1 + . . . + ms. Note that the multiplicity of Γ at P equals sm. Takingthe Seshadri quotient for Γ we have

L · Γsm

=sd− αsM

sm=d− αMm

= ε(L;P )

hence Γ satisfies the assertions of the Lemma.

The following auxiliary Lemma will be used in the proof of Theorem 1.2. Wepostpone its proof to the end of this section.

Lemma 2.4. Let s > 9 be an integer. The function

f(δ) = (2√s+ 1− s)

√1− sδ2 + s(1−

√s+ 1)δ + s− 2 (2)

is non-negative for δ satisfying

1√s+ 1

< δ <1√s. (3)

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Proof of Theorem 1.2. If part a) in the Theorem holds, then we are done.Otherwise we are in the situation that for all points P1, . . . , Ps ∈ P2, for all line

bundles L on X = BlP1,...,Ps P2 and for all points P ∈ X

ε(L;P ) is a rational number. (4)

We assume also to contradiction that part b) is false, i.e. that the SHGH Conjectureholds for s+ 1 points.

Let U(j) denote an open set in (P2)j such that SHGH holds for all j-tuples fromU(j). Let

πj+1,j : (P2)j+1 → (P2)j

denote the projection onto the first j factors. Let

W (s) := πs+1,s(U(s+ 1)) ∩ U(s).

Since the projection is an open mapping W (s) is an open set. Let the points

P1, . . . , Ps ∈W (s) (5)

be fixed. Let δ be a rational number satisfying (3). Note that the SHGH Conjectureimplies the Nagata Conjecture [7, Remark 5.12] so that

ε(OP2(1); s) =1√s

and hence the Q–divisor L = H − δE is ample. Changing δ a little bit if necessary,we can assume that

√L2 is an irrational number.

By (4) ε(L;P ) is a rational number for all points P ∈ P2. From now on, we fixa point P in such a way that (P1, . . . , Ps, P ) ∈ U(s+ 1) (this is possible by (5)).

By Lemma 2.3 there is a divisor Γ ⊂ P2 of degree γ with M = multP1 Γ = . . . =multPs Γ and m = multP Γ whose proper transform Γ on X computes the Seshadriconstant

ε(L;P ) =L · Γm

=γ − δsM

m<√

1− sδ2.

This gives an upper bound on γ

γ < m√

1− sδ2 + δsM. (6)

Since we work under assumption that SHGH holds for s+ 1, the Nagata Conjecturealso holds for s + 1 points and this gives a lower bound for γ, since for Γ we musthave

γ

sM +m>

1√s+ 1

. (7)

We now claim thatγ > 2M +m. (8)

Suppose not. We then have that

γ < 2M +m. (9)

The real numbers

a :=2√s+ 1− s2− δs

and b :=s− δs

√s+ 1

2− δs

6

are positive. Multiplying (6) by a and (9) by b and adding we obtain

sM +m 6 γ√s+ 1 < sM + (b+ a

√1− sδ2)m,

where the first inequality follows from (7). Rearranging the right and left had sidesof this inequality we obtain that

(2√s+ 1− s)

√1− sδ2 + s− δs

√s+ 1 < 2− δs,

which contradicts Lemma 2.4. Hence (8) holds.It follows now from the SHGH conjecture for s+ 1 points (in the form stated in

the introduction) that the linear system

γH −ME−mF

on Y is non-special. Indeed the condition γ > 2M + m is (8) and the conditionγ > 3M is satisfied since γ

sM > 1√s

(because the Nagata Conjecture holds for s

points by hypothesis) and because we have assumed that s > 9. This system is alsonon-empty because the proper transform of Γ under g is its member. Thus by astandard dimension count

0 6 γ(γ + 3)− sM(M + 1)−m(m+ 1).

The upper bound on γ (6) together with the above inequality yields

0 6 (sδM +m√

1− sδ2)(sδM +m√

1− sδ2 + 3)−m2 −m− sM − sM2. (10)

Note that the quadratic term in (10) is a negative semi-definite form

(s2δ2 − s)M2 + 2sδ√

1− sδ2Mm− sδ2m2.

Indeed, the restrictions on δ made in (3) imply that the term at M2 is negative.The determinant of the associated symmetric matrix vanishes. These two conditionsimply together that the form is negative semi-definite. In particular this term of (10)is non-positive. The linear part in turn is

(3sδ − s)M + (3√

1− sδ2 − 1)m,

which is easily seen to be negative. This provides the desired contradiction andfinishes the proof of the Theorem.

Remark 2.5. As it is well known, Nagata’s conjecture can be interpreted in terms ofthe 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 theboundary of the nef cone? The conjecture predicts that this ray should intersect theboundaries 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 isequivalent to requiring that H − tE crosses exactly one Zariski chamber (the nefcone itself). Surprisingly, it is easy to prove that H − tE cannot cross more thantwo chambers.

Proposition 2.6. Let f : X → P2 be the blow up of P 2 in s general points withexceptional divisors E1, . . . , Es. Let H be the pull-back of the hyperplane bundle andE = E1 + . . .+Es. The ray R = H − tE meets at most two Zariski chambers on X.

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Proof. If ε = ε(OP2(1); s) = 1√s

i.e. this multi-point Seshadri constant is maximal,

then the ray crosses only the nef cone.If ε is submaximal, then there is a curve C = dH −

∑miEi computing this

Seshadri constant, i.e. ε = d∑mi

.If this curve is homogeneous, i.e. m = m1 = · · · = ms, then we put Γ := C.

Otherwise we define Γ = γH −M ·E as a symmetrization of C as in Lemma 2.3, i.e.we sum the curves Ck = dH −

∑mσk(i)Ei over a length s cycle σ ∈ Σs.

Let µ = Mγ . Note that µ = µ(L;E). Indeed, it is in any case µ 6 µ(L;E) by the

construction of Γ and a strict inequality would contradict the fact that Γ computesthe Seshadri constant.

Now, we claim that

H − tE =µ− tµ− ε

(H − εE) +t− ε

γ(µ− ε)Γ

is the Zariski decomposition of H − tE for ε 6 t 6 µ. Indeed, H − εE is nef bydefinition and it is orthogonal to all components of Γ, which together with the IndexTheorem implies that the intersection matrix of Γ is negative definite.

Thus the ray R after leaving the nef chamber remains in a single Zariski chamber.

It is interesting to compare this result with the following easy example, whichconstructs rays meeting a maximal number of chambers.

Example 2.7. Keeping the notation from Proposition 2.6 let L = ( s(s+1)2 + 1)H −

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, . . . , Esjoin the support of the Zariski decomposition of L − λE one by one. We leave thedetails to the reader.

We conclude this section with the proof of Lemma 2.4.

Proof of Lemma 2.4. Since f(1/√s+ 1) = 0 it is enough to show that f(δ) is in-

creasing for 1/√s+ 1 6 δ 6 1/

√s. Consider the derivative

f ′(δ) = s

(1 +

δ√1− sδ2

(s− 2√s+ 1)−

√s+ 1

). (11)

The function h(δ) = δ√1−sδ2 is increasing for 1/

√s+ 1 6 δ 6 1/

√s since the nu-

merator is an increasing function of δ and the denominator is a decreasing functionof δ. We have h( 1√

s+1) = 1 so that h(δ) > 1 holds for all δ. Since the coefficient at

h(δ) in (11) 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 Ok-ounkov [23], Kaveh–Khovanskii [15], and Lazarsfeld–Mustata [19]. Shortly there-after, Boucksom–Chen [6] and Witt-Nystrom [22] have shown ways of constructing

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geometrically significant functions on Okounkov bodies, that were further studiedin [17]. In the context of this note the study of Okounkov functions was pursuedby the last three authors in [17]. We refer to [17] for construction and properties ofOkounkov functions.

In this section we consider an arbitrary smooth projective surface X and anample line bundle L on X. Let p ∈ X be an arbitrary point and let π : Y → X bethe blow up of p with exceptional divisor E. Recall that the Seshadri constant of Lat p can equivalently be defined as

ε(L; p) = sup t > 0 |π∗L− tE is nef .

There is a related invariant

µ(L; p)def= 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 meetsthe boundary of the nef cone of Y , and µ(L; p) is the value of λ where the ray meetsthe boundary of the pseudo-effective cone. The following relation between the twoinvariants is important in our considerations.

Remark 3.1. If ε(L; p) is irrational, then

ε(L; p) = µ(L; p) .

In particular, if µ(L; p) is rational, then so is ε(L; p).

Rationality of µ(L; p) implies rationality of the associated Seshadri constants onsurfaces. This invariant appears in the study of the concave function ϕordp associatedto the geometric valuation on X defined by the order of vanishing ordp at p. Wefix some flag Y• : X ⊇ C ⊇ x0 and consider the Okounkov body ∆Y•(L) definedwith respect to that flag. We define also a multiplicative filtration determined by thegeometrical valuation ordP on the graded algebra V = ⊕k>0Vk with Vk = H0(X, kL)by

Ft(V ) = s ∈ V : ordP (s) > t ,

see [17, Example 3.7] for details. (All the above remains valid in the more generalcontext of graded linear series.) There is an induced filtration F•(Vk) on everysummand 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 : FtVk 6= 0

respectively. Let ϕordP (x) = ϕF•(x) be the Okounkov function on ∆Y•(L) deter-mined by filtration F•, see [17, Definition 4.8]. It turns out that µ(L; p) is themaximum of the Okounkov function ϕordP .

Proposition 3.2. With notation as above we have that

µ(L; p) = lim supm→∞

max

ordp(s) | s ∈ H0 (X,OX(mL))

m= max

x∈∆Y• (L)φordp(x) .

Proof. Observe that

ordp(s) = ordE(π∗s) = max m ∈ N | div(π∗s)−mE is effective .

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Consequently,

µ(L; p) = sup t ∈ R>0 |π∗L− tE is pseudo-effective

= lim supm→∞

max

ordp(s) | s ∈ H0 (X,OX(mL)

m,

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

max

ordp(s) | s ∈ H0 (X,OX(mL))

= emax(Vm,F•) ,

and hence

lim supm→∞

max

ordp(s) | s ∈ H0 (X,OX(mL))

m= emax(V,F•) .

Sinceemax(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•, [6, Remark 1.12 (ii)]. We provenow that the maximum of the Okounkov function does not depend on the flag. Thisfact is valid in the general setting of arbitrary multiplicative filtration F defined ona graded linear series V•.

Remark 3.3. Note that in general the functions ϕF• are only upper-semicontinuousand concave, but not continuous on the whole Newton–Okounkov body as explainedin [17, Theorem 1.1]. They are however continuous provided the underlying body∆(V•) is a polytope (see again [17, Theorem 1.1]), which is the case for completelinear series on surfaces [16].

Theorem 3.4 (Maximum of Okounkov functions). With the above notation, wehave that

maxx∈∆Y• (L)

ϕF•(x) = emax(L,F•).

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) associatedthe graded linear series Vt,k ⊂ H0(kL) given by

Vt,kdef= Fkt(H

0(kL)) .

Note that by definition

emax(L,F•) = supt ∈ R| ∪k Vt,k 6= 0.

In other words,emax(L,F•) = supt ∈ R|∆t,Y•(L) 6= ∅.

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Recall that by definition

ϕF•(x) = supt ∈ R|x ∈ ∆t,Y•(L).

and it is therefore immediate that ∀x

ϕF•(x) 6 emax(L,F•).

from which it follows that

maxx∈∆Y• (L)

ϕF•(x) 6 emax(L,F•).

Since the bodies ∆t,Y•(L) form a decreasing family of closed subsets of Rd, we havethat

∩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 hencesupt ∈ R|y ∈ ∆t,Y•(L) = supt ∈ R|∆t,Y•(L) 6= ∅

or in other wordsϕF•(y) = emax(L,F•)

from which it follows that

maxx∈∆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 stateand 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–divisorf∗D − λE. There is then a natural inclusion

∆Y•(Dλ) ⊂ ∆Y•(D).

Moreover, a filtration F• on the graded algebra ⊕k>0H0(S; kD) induces a filtration

Fλ• on the graded (sub)algebra ⊕H0(X, kDλ), where the sum is taken over all kdivisible enough. For associated Okounkov functions we have

ϕFλ•(x) 6 ϕF•(x) (12)

for all x ∈ ∆Y•(Dλ).

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Remark 4.2. The best case scenario is that the functions φ are piecewise linear withrational coefficients over a rational polytope. Of these properties, some evidence forthe first was given by Donaldson [12] in the toric situation. For the second condition,it was proven in [1] that every line bundle on a surface has an Okounkov body whichis a rational polytope.

Proof. Note first that since the blow up center is disjoint from all elements in theflag, one can take Y• to be an admissible flag on X. (Strictly speaking one takesf∗Y• as the flag, but it should cause no confusion to identify flag elements upstairsand downstairs.)

Then, if k is sufficiently divisible we have

H0(X, f∗kD − kλE) =s ∈ H0(S, kD) : ordP (s) > E

⊂ H0(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 [17,Definition 4.8]. Indeed, the supremum arising in the definition of ϕFλ•

is taken overa smaller set of sections than it is for ϕF• .

The following examples illustrate various situations arising in the setting ofProposition 4.1.

Example 4.3. Let ` be a line in X0 = P2 and let P0 ∈ ` be a point. We fix the flag

Y• : X0 ⊃ ` ⊃ P0 .

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

10

1

∆Y•(D)

Let F• be the filtration on the complete linear series of D imposed by the geometricvaluation 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 inH0(P2,OP2(k)) vanishing exactly with multiplicity ka along `, exactly with multi-plicity kb along a line passing through P0 different from ` and along a curve of degreek(1− a− b) not passing through P0.

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The next example shows that even when the Okounkov body changes in thecourse of blowing up, the Okounkov function may remain the same.

Example 4.4. Keeping the notation from the previous Example and from Propo-sition 4.1 let f : X1 = BlP1 P2 → X0 = P2 be the blow up of the projective plane ina point P1 not contained in the flag line ` with the exceptional divisor E1. We worknow with a Q–divisor Dλ = f∗(OP2(1))− λE1 = H − λE1, for some fixed λ ∈ [0, 1].A direct computation using [19, Theorem 6.2] gives that the Okounkov body hasthe shape

1− λ0

1

∆Y•(Dλ)

Thus we see that the Okounkov body of Dλ is obtained from that of D by intersectingwith 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 aninteger. Now we need to exhibit a section s in H0(P2,OP2(k)) satisfying the followingconditions:

a) s vanishes along ` 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 ` (there is no other choice here), ofb copies of the line through P0 and P1, of kλ − b copies of any other line passingthrough P1 (if this number is negative then this condition is empty) and of a curveof 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 divisorsDλ always result from those of D by cutting with finitely many halfplanes. This isan immediate consequence of [16, Theorem 5].

We conclude by showing that the inequality in (12) can be sharp, i.e. the blowup 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 bethe blow up of six general points P1, . . . , P6 not contained in ` and chosen so that thepoints P0, P1, . . . , P6 are also general. Let E1, . . . , E6 denote the exceptional divisors

13

and set E = E1+. . .+E6. We consider the divisorD = H− 25E. A direct computation

using [19, Theorem 6.2] (this requires computing Zariski decompositions this time,see [2] for an effective approach) yields the triangle with vertices at the origin andin 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 (13)

for (a, b) ∈ Ω =

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

.

125

( 11360 ,

1772)

0

1

Ω

The reason for the above inequality is the following. Let (a, b) ∈ Ω be a valua-tion vector. Assume to the contrary that ϕ(a, b) > 4/15. It is well known thatε(OP2(1), P0, . . . , P6) = 3

8 , see for instance [25, Example 2.4]. On the other hand asection with the above valuation vector would have (after scaling to O(1)) multiplic-ities 2/5 at P1, . . . , P6 and ϕν(a, b) > 4/15 at P0. It would give Seshadri quotient

1

6 · 25 + ϕν(a, b)

<3

8,

a contradiction. This proves (13).

Since the SHGH Conjecture holds for 9 points, the first challenge arising in theview of our Theorem would be to compute the Okounkov body and the Okounkovfunction associated to ordP0 as above for the system

22H − 7(E1 + · · ·+ E9).

Appendix: Cremona transformations and permutations.

Here we prove Lemma 2.2. It suffices to show the Lemma under the assumptionthat σ is a transposition. After renumbering the points, we can reduce the proof toshowing that one can exchange m1 and m2. If the number of points is very smalls 6 3, then there exists an automorphism of P2 exchanging P1 and P2 and keepingP3 fixed. Then Cσ is just the image of C under this isomorphism. For s > 4 weproceed as follows. We perform the standard Cremona transformation based onthe points P1, P3 and P4 first. Then we perform the transformation based on thepoints P2, P3, P4 and finally on P1, P3, P4. It is easy to check that this combinationtransforms the vector

(d;m1,m2,m3, . . . ,ms) to (d;m2,m1,m3, . . . ,ms).

14

The image of the curve C under this transformation remains irreducible, becauseCremona transformation induces an isomorphism on the blowing up.

Finally note that one can work in Lemma 2.2 with an explicit value of m, forexample with 3s. We don’t dwell on the question of finding an optimal m whichworks for the Lemma.

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Marcin Dumnicki, Jagiellonian University, Institute of Mathematics, Lojasiewicza 6,PL-30-348 Krakow, Poland

E-mail address: Marcin.Dumnicki@im.uj.edu.pl

Alex Kuronya, Budapest University of Technology and Economics, Mathematical Insti-tute, Department of Algebra, Pf. 91, H-1521 Budapest, Hungary.

E-mail address: alex.kuronya@math.bme.hu

Current address: Alex Kuronya, Albert-Ludwigs-Universitat Freiburg, MathematischesInstitut, Eckerstraße 1, D-79104 Freiburg, Germany.

Catriona Maclean, Institut Fourier, CNRS UMR 5582 Universite de Grenoble, 100 ruedes Maths, F-38402 Saint-Martin d’Heres cedex, France

E-mail address: catriona@fourier.ujf-grenoble.fr

Tomasz Szemberg, Instytut Matematyki UP, Podchorazych 2, PL-30-084 Krakow, Poland.E-mail address: szemberg@up.krakow.pl