Minkowski decomposition of Okounkov bodies on surfaces
MMinkowski decomposition of Okounkov bodies onsurfaces
Patrycja (cid:32)Luszcz-´Swidecka and David Schmitz
Abstract.
We prove that the Okounkov body of a big divisor with respect to a generalflag on a smooth projective surface whose pseudo-effective cone is rational polyhedraldecomposes as the Minkowski sum of finitely many simplices and line segments arisingas Okounkov bodies of nef divisors.
1. Introduction
The construction of Okounkov bodies associated to linear series on a projectivevariety, which was introduced by Okounkov and was given a theoretical frameworkin the seminal papers [6] and [8], recently attracted attention as it encodes plentyof information on geometric properties of line bundles. For example, the volumeof a big linear series essentially agrees with the euclidean volume of its associatedOkounkov body.Okounkov’s idea is to assign to a big divisor D on a smooth projective n -dimensional variety X a convex body ∆( D ) in n -dimensional euclidean space R n .The construction, which we sketch in section 2, depends on the choice of a flag ofsubvarieties Y • : X = Y ⊇ Y ⊇ . . . ⊇ Y n of codimensions i such that Y n is anon-singular point on each of the Y i .In ([8, Theorem B]), Lazarsfeld and Mustat¸ˇa prove the existence of a globalOkounkov body : for a smooth projective variety there is a closed convex cone ∆( X ) ⊆ R n × N ( X ) R such that the fiber over any big rational class ξ ∈ N ( X ) R of the map ϕ induced by the second projection is equal to ∆( ξ ). Additionally, in order to establishthe log-concavity relationvol X ( D + D ) /n (cid:62) vol X ( D ) /n + vol X ( D ) /n for any two big R -divisors, they deduce from the convexity of the global Okounkovbody the inclusion ∆( D ) + ∆( D ) ⊆ ∆( D + D ) . Here the left hand side denotes the Minkowski sum of ∆( D ) and ∆( D ), i.e., theset obtained by pointwise addition see [8, Corollary 4.12]).In general the above inclusion turns out to be strict (see Example 4.2). However,it would be desirable to know conditions for equality; in particular one would hope Mathematics Subject Classification:
Primary 14C20; Secondary 14J26.
Keywords.
Okounkov body, Minkowski decomposition, Zariski chamber.The second author was supported by DFG grant BA 1559/6-1. a r X i v : . [ m a t h . AG ] A p r to be able to decompose the Okounkov body of any big divisor as the Minkowskisum of “simple” bodies. Specifically, the following questions arise: is there a set Ωof big divisors such that the Okounkov body of any big divisor D with respect to anadmissible flag Y • decomposes as Minkowski sum of the bodies associated to divisorsin Ω? If so, can Ω be chosen to be finite?An affirmative answer to these questions was given in [9] in the case of the delPezzo surface X , the blow-up of the projective plane in three non-collinear points,equipped with a certain natural flag. We prove in this paper that the answers toboth questions are “yes” for a general admissible flag (see Proposition 2.1) on anysmooth projective surface whose pseudo-effective cone is rational polyhedral. Forexample, this is the case for all del Pezzo surfaces and, more generally, for surfaceswith big anticanonical class (see [4, Lemma 3.4]). We will see in the following sectionthat considering nef divisors is sufficient since the Okounkov body of any big divisoris a translate of the body associated to the positive part of its Zariski decomposition. Theorem.
Let X be a smooth projective surface such that Eff( X ) is rational poly-hedral, and let X = Y ⊇ Y ⊇ Y = { pt } be a general flag. Then there exists a finiteset Ω of nef Q - divisors such that for any nef Q -divisor D there exist non-negativerational numbers α P ( D ) such that D = (cid:88) P ∈ Ω α P ( D ) P and ∆ Y • ( D ) = (cid:88) P ∈ Ω α P ( D )∆ Y • ( P ) . (1.0.1) Definition.
A presentation D = (cid:80) α i D i as in (1.0.1) is called a Minkowski decom-position of D with respect to the Minkowski basis
Ω.The proof, which we present in section 3, includes the construction of the Minkowskibasis Ω as well as an effective method to determine a Minkowski decomposition ofany given nef Q -divisor. It depends on two features distinctive for surfaces, firstlya characterization of Okounkov bodies in terms of intersections with the positiveand negative part in the Zariski decomposition due to Lazarsfeld and Mustat¸ˇa, andsecondly on the Zariski chamber decomposition of the big cone introduced in [3].We sketch these results in section 2.Throughout this paper we work over the complex numbers. Acknowledgements.
We are grateful to Thomas Bauer and Tomasz Szemberg forhelpful comments and valuable discussions.
2. Okounkov bodies on surfaces
In this section we first give a quick review of Okounkov’s construction in arbitrarydimension (we refer to [8] for details), and then turn to additional features knownin the case of surfaces.As mentioned in the introduction, one assigns to a big divisor D on a smoothprojective n -dimensional variety X a convex body ∆( D ) in R n . The constructiondepends on the choice of a flag on X , i.e., a sequence Y • : X = Y ⊇ Y ⊇ . . . ⊇ Y n of subvarieties Y i of codimension i . A flag is admissible if Y n is a non-singular pointon each of the Y i . To an admissible flag, one assigns a function ν Y • : H ( X, O X ( D )) → Z n , by mapping a section s ∈ H ( X, O X ( D )) to the tuple ( ν ( s ) , . . . , ν n ( s )) where ν ( s ) := ord Y ( s ), ν ( s ) is given by the order of vanishing along Y of the section s ∈ H ( Y , O Y ( D − ν ( s ) Y )) determined by s , and so forth up to ν n ( s ). Re-peating this construction for integral multiples of D , we define the Okounkov body∆( D ) = ∆ Y • ( D ) to be the closed convex hull of the set S ( D ) := (cid:91) k (cid:62) (cid:8) k ν Y • ( s ) s ∈ H ( X, O X ( kD )) (cid:9) . Note that although the number of image vectors ( ν , . . . , ν n ) is equal to the dimensionof H ( X, O X ( kD )) for each k , the convex body ∆( D ) need not be polyhedral (see [8,Section 6.3]). By [8, Proposition 4.1], numerically equivalent divisors have identicalOkounkov bodies and for any positive integer p we have the scaling ∆( pD ) = p ∆( D ),so we can assign an Okounkov body to big rational classes in the N´eron-Severi vectorspace N ( X ) R . For non-rational classes this is not so straightforward. Instead, itfollows from the existence of global Okounkov bodies ([8, Theorem B]): There is aclosed convex cone ∆( X ) ⊆ R n × N ( X ) R such that the fiber over any big rationalclass ξ ∈ N ( X ) R of the map ϕ induced by the second projection is equal to ∆( ξ ).Consequently, the Okounkov body of a big real class is defined as its fiber under ϕ .Additionally, since the image of ∆( X ) under ϕ is the pseudo-effective cone Eff( X ),the construction can be extended to pseudo-effective real classes.From the existence of the global Okounkov body on X many interesting prop-erties of the volume function vol X : Big( X ) → R can quite easily be proved. Forexample, the log-concavity relationvol X ( D + D ) /n (cid:62) vol X ( D ) /n + vol X ( D ) /n for any two big R -divisors is a consequence of the Brunn-Minkowski theorem: fromthe convexity of the global Okounkov body we obtain the inclusion∆( D ) + ∆( D ) ⊆ ∆( D + D )with the Minkowski sum on the left hand side (see [8, Corollary 4.12]).For the remainder of this section, let X be a smooth projective surface with anadmissible flag X ⊇ C ⊇ { p } on it. Any pseudo-effective (rational) divisor D on X has a Zariski decomposition D = P D + N D , where P D is nef, and N D is effective, orthogonal to P D , and if it is not the zero-divisor, it has negative definite intersection matrix. Define µ C ( D ) := sup { t D − tC effective } and consider the functions α, β : [0 , µ C ( D )] → R , with α ( x ) = ord p ( N D − xC ), and β ( x ) = ord p ( N D − xC ) + ( C · ( P D − xC )) . Then by ([8, Theorem 6.4]), α and β are the upper and lower boundary functionsfor ∆( D ), respectively. Concretely,∆( D ) = (cid:8) ( x, y ) ∈ R (cid:54) x (cid:54) µ C ( D ) , α ( x ) (cid:54) y (cid:54) β ( x ) (cid:9) . The following proposition shows that in the situation of the theorem, in order todetermine the Okounkov body of a big divisor D it is sufficient to know the positivepart of the divisors D − tC for 0 (cid:54) t (cid:54) µ C ( D ). Proposition 2.1.
If the pseudo-effective cone
Eff( X ) is rational polyhedral and X ⊇ C ⊇ p is a general admissible flag, then C is big and nef as a divisor, and α ( x ) = 0 , β ( x ) = C · P D − xC for all (cid:54) x (cid:54) µ C ( D ) .Proof. If Eff( X ) on X is rational polyhedral, then in particular there are only finitelymany irreducible curves E on X with self-intersection E (cid:54)
0. Therefore, in ageneral flag X ⊇ C ⊇ p the irreducible curve C has positive self-intersection, so itis big and nef as a divisor. Furthermore, p is a non-singular point on C , which doesnot lie on any curve with negative self-intersection. Now by definition, the negativepart N D − xC in the Zariski decomposition of D − xC either is the zero-divisor, orhas negative definite intersection matrix. In the latter case, its support consists ofcurves with negative self-intersection, so in either case we have ord p ( N D − xC ) = 0for all x . (cid:3) Example 2.2.
For any 0 (cid:54) t (cid:54) C − tC is nef and effective, hence P C − tC = C − tC . So by the proposition, ∆( C ) is the simplex of height C andlength 1. Remark.
By [9, Corollary 2.2] the Okounkov body of a big divisor D with respectto a flag X ⊇ C ⊇ p such that C is not a component of N D is a translate bythe vector (0 , ord p ( N D )). In particular by the above proof, for a general flag on asurface with rational polyhedral pseudo-effective cone, the Okounkov bodies of anybig divisor and of its positive part coincide.Recall that by the main result of [3] on a smooth projective surface there exists alocally finite decomposition of Big( X ) into locally polyhedral subcones, the so called Zariski chambers , such that • the support of negative parts of divisors is constant on each chamber, • the volume function vol X ( · ) varies polynomially on the chambers, and • on the interior of each chamber the augmented base loci B + are constant.The basic idea of [3] is to consider for a big and nef divisor P the setΣ P := { D ∈ Big( X ) Neg( D ) = Null( P ) } , where Neg( D ) denotes the support of N D and Null( P ) is the set of irreduciblecurves orthogonal to P with respect to the intersection product. These sets givea decomposition of Big( X ) obviously satisfying the first property in the above list,while proving the remaining properties as well as local finiteness still requires quite aneffort. For an explicit description of chambers, passing to closures in [3, Proposition1.10] we obtain the identityΣ P = convex hull (cid:16) Nef( X ) ∩ Null( P ) ⊥ , Null( P ) (cid:17) , (2.2.1)from which we deduce the following useful statement about positive parts. Proposition 2.3.
Let P be a big and nef divisor on X with corresponding Zariskichamber Σ p . Then for all D , D ∈ Σ P we have P D + D = P D + P D , i.e., the positive parts of the Zariski decompositions vary linearly on the closure ofeach Zariski chamber.Proof. Let D = P + (cid:80) si =1 α i N i and D = P + (cid:80) si =1 β i N i be representationscorresponding to (2.2.1) with α i , β i (cid:62) N i ∈ Null( P ), and P , P nef. Clearly, P + P is nef and has intersection product zero with the N i . Furthermore, thedivisor (cid:80) si =1 ( α i + β i ) N i is effective and has negative definite intersection matrix.Thus D + D = ( P + P ) + s (cid:88) i =1 ( α i + β i ) N i is the Zariski decomposition. (cid:3)
3. Minkowski decomposition
In this section we prove the main theorem. Fix throughout a general admissible flag Y • : X ⊇ C ⊇ p on a smooth projective surface X .As stated in the introduction, the starting point for this investigation was theobservation from [8] that for any two pseudo-effective divisors D , D we have theinclusion ∆( D ) + ∆( D ) ⊆ ∆( D + D ) . This inclusion turns out to be strict in general. We refer to [9] for examples.On the other hand, one observes that the Okounkov body of a pseudo-effectivedivisor D with respect to Y • can always be decomposed as the Minkowski sum offinitely many simplices and line segments. (∆( D ) is the area of the upper rightquadrant bounded by the piecewise linear, concave function β .) The question thenis: do these elementary “building blocks” come up as Okounkov bodies themselves?As the theorem shows, the answer is “yes”.Before we prove the theorem, let us consider candidates for a Minkowski basis,i.e., nef divisors whose Okounkov bodies are of one of the elementary types mentionedabove. • For a nef divisor D with D = 0, for positive t none of the divisors D − tC iseffective since C by Proposition 2.1 is big and nef being the curve in a generaladmissible flag. Therefore, µ C ( D ) = 0, and ∆( D ) is the vertical line segmentof length C · D (see Figure 1). Figure 1: The Okounkov body ∆( D ) • If for a big and nef divisor D (cid:48) all the classes D (cid:48) − tC for 0 < t < µ C ( D (cid:48) ) lie inthe same Zariski chamber then by Proposition 2.3 the positive parts P D (cid:48) − tC vary linearly with t . Consequently, ∆( D (cid:48) ) is the simplex of height C · D (cid:48) andlength µ C ( D (cid:48) ) (see Figure 2). Figure 2: The Okounkov body ∆( D (cid:48) ) We now turn to the proof of the theorem. It consists of two parts: we firstconstruct the set Ω and then show how to find the presentation of any big and nefdivisor D in terms of elements of Ω which yields the Minkowski decomposition of D . Remark.
Effective representations of a nef divisor in terms of the Minkowskibasis are not unique. It is possible that such a representation is not a Minkowskidecomposition (see Example 4.2). This is why the second part of the proof isimportant as it shows how to pick the right decomposition.
Construction of a Minkowski basis
In the Zariski chamber decomposition of the big cone Big( X ) we assign to eachchamber an element of Ω as follows. Writing { N , . . . , N s } for the set of curves in thesupport of negative parts of divisors in a chamber Σ, we define the “correspondingMinkowski basis element” M as follows: Consider the linear subspace of N ( X ) R spanned by C together with the classes of the curves N i . Its intersection withthe subspace N ⊥ ∩ · · · ∩ N ⊥ s is a rational line, spanned by some integral divisor M = dC + (cid:80) α i N i . We will argue that d and the α i all have the same signs, andwe conclude that either M or − M is nef.The intersection matrix S of the divisor (cid:80) N i is negative definite with non-negative entries outside the diagonal. By the auxiliary result [3, Lemma 4.1] (seealso [1, Lemma A.1]), the inverse matrix S − has only negative entries. Therefore,and since CN i (cid:62)
0, the solution to the system of the equations S · ( α , . . . , α s ) t = − d ( CN , . . . , CN s ) t (3.0.1)for fixed d is a vector ( α , . . . , α s ) whose entries have the same sign as d . Fix apositive integral solution and set M = dC + (cid:80) α i N i . Note that since M lies in N ⊥ ∩ · · · ∩ N ⊥ s it is nef by the positivity of its coefficients and the nefness of C .Furthermore, it lies in the closure of Σ, or more concretely in the closure of the faceΣ ∩ Nef( X ). Remark.
Note that in the above construction different chambers can have the samecorresponding Minkowski basis element. For example, on the del Pezzo surface X with standard basis H, E , E and with a flag such that C has class H = π ∗ ( O P (1))the chambers Σ H , Σ H − E , and Σ H − E have M = H .Note also that the corresponding basis element to the nef cone is always C .We can now describe the Minkowski basis Ω: it consists of the divisors M Σ con-structed above together with one integral representative for each ray of the nef conenot contained in Big( X ).Note that since Eff( X ) is rational polyhedral the set Ω is finite. Note fur-thermore that the divisors in Ω have Okounkov bodies which cannot be decom-posed as Minkowski sums, i.e., in a sense the set Ω is minimal: By construc-tion, for all 0 < t < µ C ( M ) = d the class M Σ − tC lies in the cone spanned byNef( X ) ∩ N ⊥ ∩ · · · ∩ N ⊥ s and the N , . . . , N s , i.e., in the closure of the Zariski cham-ber Σ. Therefore, the positive part of M Σ − tC varies linearly, so ∆( M Σ ) is thesimplex of height C · M and length d , whereas the other basis elements D i lie in theboundary of Eff( X ), so µ C ( D i ) = 0 which means that the corresponding Okounkovbody is the vertical line segment of length C · D i . Algorithmic construction of Minkowski decompositions
To complete the proof we now describe how to find the Minkowski decomposition ofa given nef divisor D .If D is not big, then D = 0 and Ω contains some positive multiple D (cid:48) = βD .Thus ∆( D ) = 1 β ∆( D (cid:48) ) , and we are done.Otherwise, consider the Zariski chamber Σ corresponding to the big and nefdivisor D . Let M be the corresponding Minkowski basis element and set τ := sup { t D − tM nef } . Since nefness is defined by finitely many linear conditions, τ is rational. The nef Q -divisor D (cid:48) := D − τ M lies on the boundary of the face Nef( X ) ∩ Null( D ). If D (cid:48) = 0, we are done.Otherwise, we claim that∆( D ) = τ ∆( M ) + ∆( D (cid:48) ) , (3.0.2)so ∆( D ) decomposes into the elementary part τ ∆( M ) and the Okounkov body ofthe divisor D (cid:48) .For the proof we first note that by construction of the Minkowski basis, M lies,like D (cid:48) , on the boundary of the Zariski chamber Σ. Furthermore, as we have seenabove, the divisors M − tC lie in the closure of the chamber Σ for 0 < t < µ C ( M ).Thus by Proposition 2.3 we have P D − xC = P D (cid:48) + P τM − xC = D (cid:48) + P τM − xC for 0 (cid:54) x (cid:54) µ C ( τ M ).For the remaining µ C ( τ M ) (cid:54) x (cid:54) µ C ( D ) let (cid:102) M denote the divisor τ M − µ C ( τ M ) C (which is just τ ( (cid:80) α i N i ) in the above notation). We claim that forany t > (cid:102) M ) ⊆ Null( D ) ⊆ Null( D (cid:48) ) = B + ( D (cid:48) ) ⊆ B + ( D (cid:48) − tC ) = Null( P D (cid:48) − tC ) . The two equalities are given by [5, Example 1.10] and [5, Example 1.11] respectively.The first inclusion is clear since the N i are contained in Null( D ). The second onefollows from the fact that D (cid:48) is contained in the boundary of the face of the nefcone containing D , while the last inclusion is a direct consequence of the fact thatsubtracting a nef divisor can only augment the base locus.Note that in general for a big divisor E with Zariski decomposition E = P E + N E and an effective divisor F with support contained in Null( P E ) the decomposition E + F = P E + ( N E + F )is the Zariski decomposition: P E is nef, has trivial intersection with all componentsof ( N E + F ), and the latter divisor has negative definite intersection matrix. Inother words, adding an effective divisor F with support contained in Null( P D ) doesnot alter the positive part.Taking in the above consideration E and F to be D − xC and (cid:102) M respectively,we obtain the identity P D − xC = P D (cid:48) − ( x − µ C ( τM )) C for µ C ( τ M ) (cid:54) x (cid:54) µ C ( D ). Putting the two decompositions of positive parts to-gether, we get β D ( x ) = (cid:40) β τM ( x ) + C · D (cid:48) (cid:54) x (cid:54) µ C ( τ M ) β D (cid:48) ( x − µ C ( τ M )) µ C ( τ M ) (cid:54) x (cid:54) µ C ( D ) , which amounts to the claimed identity (3.0.2).Repeat the above procedure with the divisor D (cid:48) . This is possible because if D (cid:48) is big and nef, it defines a Zariski chamber Σ with M Σ (cid:54) = M , which can be seenas follows: if it were not the case, we would have Null( D (cid:48) ) ⊆ Null( M ), but thenit follows from D = M + D (cid:48) that Null( D (cid:48) ) ⊆ Null( D ), which is impossible. Thealgorithm terminates after at most ρ steps, since in every step the dimension of theface of the nef cone in which D lies decreases. Eventually, we end up with either 0or a divisor spanning an extremal ray of the nef cone. Such a divisor has a multiplein Ω, and we are done. (cid:3) Note that in order to determine the Minkowski decomposition of a given divisor D it is not necessary to know the whole Minkowski basis of X . Instead in every stepthe necessary basis element can be found based on knowledge of the intersectionmatrix of Null( D ) alone. In fact, the algorithm can be implemented for automatedcomputation, provided the intersection matrix of C together with the negative curveson X is known.
4. Del Pezzo surfaces
On a del Pezzo surface X the pseudo-effective cone is rational polyhedral by thecone theorem. Concretely, it is spanned by rational curves of self-intersection − X is either P , its blow-up X r in up to 8 general points, or P × P . Acomplete list of the ( − X r is well known [10, 8 Chapt IV] (cf. [2,Theorem 3.1] for an elementary proof): they are the exceptional curves E , . . . , E r together with the strict transforms of • lines through two of the p i , • irreducible conics through five of the p i , if r (cid:62) • irreducible cubics through six of the p i with a double point in one of them, if r (cid:62) • irreducible quartics through the eight points p i with a double point in threeof them, if r (cid:62) • irreducible quintics through the eight points p i with a double point in six ofthem, if r (cid:62) • irreducible sextics through the eight points p i with a double point in seven ofthem, and a triple point in one of them, if r (cid:62) X r consists of an irreducible curve C with a general point p onit where C is the strict transform of an irreducible member of the class O P ( k ) forsome k >
0. We consider the case k = 1 (the others work analogously) and write asusual H for the class of C . Let us construct a Minkowski basis for X r . Starting withany chamber Σ, we consider Neg(Σ) = { N , . . . , N s } , the support of negative partsof the divisors in Σ. Its intersection matrix, being negative definite with diagonalentries −
1, can have only zero entries outside the diagonal. In particular, we canimmediately read off the basis element M (Σ) from the system of equations (3.0.1):Setting d = 1, we obtain α i = N i · H for all i , hence we have M (Σ) = H + s (cid:88) i =1 ( N i · H ) N i . Let us determine the Okounkov bodies of this Minkowski basis element. It is clearthat µ H ( M (Σ)) = 1, since (cid:80) si =1 ( N i · H ) N i lies on the boundary of Eff( X r ). On theother hand, setting λ := H · ( H + s (cid:88) i =1 ( N i · H ) N i ) , by the argumentation in the proof of the theorem, ∆( M (Σ)) is the simplex of height λ and length 1, which we denote by ∆( λ, E with self-intersection E = 0. As we have seen above, their Okounkov body isthe vertical line segment of length H · E . The following statement thus is a directconsequence of the theorem. Proposition 4.1.
On a del Pezzo surface X r , for any big divisor D ⊆ X r thefunction β ( x ) bounding the Okounkov body is piecewise linear with integer slope oneach linear piece. X . Up to permutationof the E i , we have the possible supports for Zariski chambers with corresponding ba-sis elements displayed in Table 1 in the standard basis H, E , . . . , E r , with L i,j , C , C denoting the ( − Neg(Σ) M (Σ) E , . . . , E s HL , , . . . , L , s , E s +1 , . . . , E s + t ( s + 1) H − sE − E − · · · − E s L , , L , , L , , E , . . . , E t H − E − E − E C , L , , . . . , L , s , E s +1 , . . . , E s + t (5 + s ) H − (2 + s ) E − E − · · · − E s − E s +1 − · · · − E C , L , , L , , L , , E H − E − E − E − E − · · · − E C , C , L , , . . . , L , s (9 + s ) H − E − E − ( s + 4) E − E − · · · − E s − E s − . . . − E r C , C , L , , L , , L , H − E − E − E − E − E − E Table 1: Zariski chambers and corresponding Minkowski basis elements on X The additional Minkowski basis elements (corresponding to non-big nef classes)are the strict transforms of • lines through one of the p i , • irreducible conics through four of the p i .We thus get the following elementary bodies as building blocks for the Okounkovbody of any big divisor on X :∆(1 , , . . . , ∆(12 , , ∆(1 , , ∆(2 , . Example 4.2.
Consider the divisor D = 7 H − E − E − E − E − E on X . • For D = D = 7 H − E − E − E − E − E , we find Null( D ) = { E } ,so M ( D ) = H . With τ = 2 we get D = 5 H − E − E − E − E − E . • Now, Null( D ) = { C , L , , L , , L , , E } , so M ( D ) = 8 H − E − E − E − E − E . With τ = we get D = H − E − E − E − E . • then D = 0, so we are done.Cosequently, the Okounkov body of D is given as the Minkowski sum∆( D ) = ∆(2 ,
2) + ∆(8 ,
1) + ∆(1 , . Note on the other hand that we have the identity D = (3 H − E − E − E ) + (4 H − E − E − E )and both summands are Minkowski basis elements. Clearly, this representationcannot be a Minkowski decomposition (see Figures 4 and 5).1 Figure 3: The Okounkov body ∆( D ) as a Minkowski sum
5. Non-del-Pezzo examples
1. For a simple non-del-Pezzo example, let π : X → P be the blow-up of 3points on a line with exceptional divisors E , E , E . Choose C general inthe class H := π ∗ ( O P (1)) and p ∈ C a general point. This gives a flagas above. The pseudo-effective cone is spanned by the exceptional divisorstogether with the class D := H − E − E − E of the strict transform ofthe line joining the blown up points. We have 12 Zariski chambers: the nefchamber, the 7 chambers belonging to principal submatrices of the intersectionmatrix of E + E + E , the one corresponding to D ,and three chambers withsupport D together with one of the exceptional divisors. The correspondingMinkowski basis element is H for the first 8 chambers, 3 H − E − E − E forthe 9th, and 2 H − E i − E j for last three. The remaining elements of Ω are Figure 4: ∆(3 H − E − E − E ) Figure 5: ∆(4 H − E − E − E ) H − E , H − E , H − E . Let’s calculate the decomposition for the arbitrarilychosen divisor P = 15 H − E − E − E . • The divisor P is ample, so M = H ; with τ = 8 we get P = 7 H − E − E − E . • Now, Null( P ) = D , so M Σ = 3 H − E − E − E ; with τ = 1 we get P = 4 H − E − E . • In the next step, Null( P ) = { D, E } , so M Σ = 2 H − E − E ; with τ = 2,we get P = 0, and we are done.Thus we get the decomposition P = 8 · H + (3 H − E − E − E ) + 2 · (2 H − E − E )with corresponding Minkowski decomposition of the Okounkov body∆( P ) = 8∆( H ) + ∆(3 H − E − E − E ) + 2∆(2 H − E − E )= ∆(8 ,
8) + ∆(3 ,
1) + ∆(4 , .
2. (K3-surface)For an example of a surface which is not a blow-up of P let us consider a K3-surface. As Kov´acs proves in [7], for any 1 (cid:54) ρ (cid:54)
19 there exists a K3-surface X with Picard number ρ whose pseudo-effective cone is rational polyhedral,spanned by the classes of finitely many rational ( − X with Picard number 3 such that the pseudo-effectivecone is spanned by three ( − L , L , D forming a hyperplane section L + L + D such that L and L are lines and D is an irreducible conic. Thehyperplane section L + L + D has intersection matrix − − − . Therefore, the Zariski chamber decomposition consists of five chambers,namely the nef chamber, one chamber corresponding to each of the ( − D, L , L , and one chamber with support L + L . Pick C to be anirreducible curve with class L + L + D , i.e., a general hyperplane section,and p to be a point in C not on L , L , and D . Then the Minkowski basiselements corresponding to the above list of chambers are C , 3 L + 2 L + 2 D ,2 L + 3 L + 2 D , L + L + 2 D , and 2 L + 2 L + D . In addition, the Minkowskibasis Ω contains the curves L + D and L + D of self-intersection zero. Thus,by the theorem, the building blocks of Okounkov bodies of nef divisors on X are ∆(4 , , ∆(9 , , ∆(6 , , ∆(3 , . In particular, in contrast to the del Pezzo case, the slope of a linear piece ofthe bounding function β need not be integral for K3-surfaces.3 References [1] Bauer, Th., Funke, M.: Weyl and Zariski chambers on K3 surfaces. Forum Mathematicum 24,609-625 (2012)[2] Bauer, Th., Funke, M., Neumann, S.: Counting Zariski chambers on Del Pezzo surfaces.Journal of Algebra 324, 92-101 (2010)[3] Bauer, Th., K¨uronya, A., Szemberg, T.: Zariski chambers, volumes, and stable base loci. J.reine angew. Math. 576, 209-233 (2004)[4] Bauer, Th., Schmitz D.: Volumes of Zariski chambers. J. pure appl. Algebra 217, 153-164(2013)[5] Ein, L., Lazarsfeld, R., Mustat¸ˇa, M., Nakamaye, M., Popa, M.: Asymptotic invariants of baseloci. Ann. Inst. Fourier 56, No.6, 1701-1734 (2006)[6] Kaveh K., Khovanskii A.: Convex bodies and algebraic equations on affine varieties, Preprint2008, arXiv:0804.4095[7] Kov´acs, S.: The cone of curves of a K3 surface. Math. Ann. 300, 681-691 (1994)[8] Lazarsfeld, R., Mustat¸ˇa, M.: Convex bodies associated to linear series, Ann. Sci. Ec. Norm.Super. 42, 783-835 (2009)[9] (cid:32)Luszcz-´Swidecka, P.: On Minkowski Decompositions of Okounkov bodies on a Del Pezzosurface, Annales Universitatis Paedagogicae Cracoviensis, Studia Mathematica 10, 105-115(2011)[10] Manin, Y.: Cubic Forms. Algebra, Geometry, Arithmetic. North-Holland Mathematical Li-brary. Vol. 4. North-Holland, 1974.
Patrycja (cid:32)Luszcz-´Swidecka Institute of Mathematics, Jagiellonian University,ul. (cid:32)Lojasiewicza 6, 30-438 Krak´ow, Poland.
E-mail address: [email protected]
David Schmitz, Fachbereich Mathematik und Informatik, Philipps-Universit¨at Marburg,Hans-Meerwein-Straße, D-35032 Marburg, Germany.