Minkowski bases on algebraic surfaces with rational polyhedral pseudo-effective cone

The purpose of this note is to study the number of elements in Minkowski bases on algebraic surfaces with rational polyhedral pseudo-effective cone.


Introduction
Lazarsfeld and Mustaţȃ initiated in [4] a systematic study of Okounkov bodies. These are convex bodies ∆(D) in R n attached to big divisors D on a smooth projective variety X of dimension n. They depend on the choice of a flag of subvarieties (Y n , Y n−1 , . . . , Y 1 ) of codimensions n, n − 1, . . . , 1 in X respectively, such that Y n is a non-singular point of each of the Y i 's. We refer to [4] for details of the construction and a very enjoyable introduction to this circle of ideas.
Okounkov bodies are subject of intensive ongoing research. Luszcz-Świdecka observed in [5] that for a del Pezzo surface there are finitely many basic bodies, called the Minkowski basis, such that all other bodies are obtained as their Minkowski sums (therefor the name of the basis). Building upon these ideas, Luszcz-Świdecka and Schmitz introduced in [6] an algorithmic construction of Minkowski bases for algebraic surfaces with rational polyhedral pseudo-effective cone Eff(X).
In the present note we consider a natural question of how many elements there are in a Minkowski bases in the set-up of [6]. The answer is closely related to the partition of the big cone of arbitrary smooth projective surfaces introduced in [2].

Preliminaries
In this section we introduce the notation and collect some basic ideas underlying the present note. By a curve we mean here an irreducible and reduced complete subscheme of dimension 1. For a divisor D on a smooth projective surface X we denote by D ⊥ the set of curves intersecting D with multiplicity zero, i.e. D ⊥ := {C ⊂ X : D.C = 0}.
We begin with a tool fundamental for understanding linear series on algebraic surfaces.
Definition 2.1 (Zariski decomposition). Let X be a smooth projective surface and let D be a pseudoeffective Q-divisor on X. Then there exist Q-divisors P D and N D such that a) D = P D + N D ; b) P D is a nef divisor and N D is either zero or it is supported on a union of curves N 1 , . . . , N r with negative definite intersection matrix; c) N i ∈ (P D ) ⊥ for each i = 1, . . . , r.
Let (x, C) be a flag on a surface X. Let D be a big divisor on X with Zariski decomposition D = P D +N D . Lazarsfeld and Mustaţȃ give in [4] the description of ∆(D) as the area enclosed between the graphs of functions α(t) and β(t) defined for real numbers t between 0 and sup {s ∈ R : D − sC is effective} as follows.
Recently, the authors of [6] presented a different approach to describing Okounkov bodies for a certain class of smooth complex projective surfaces.
, where x is a general point and C is a big and nef curve on X there exists a finite set of nef divisors MB(x , C ) = {P 1 , ..., P r } such that for a big and nef R-divisor D there exist uniquely determined non-negative real numbers a i 0 with where the first sum indicates the numerical equivalence of divisors and the second sum is the Minkowski sum of convex bodies.
The Theorem above justifies the following definition. The proof of Theorem 2.2 in [6] gives in particular a simple way to construct Minkowski basis elements based on the Bauer -Küronya -Szemberg decomposition of the big cone Big(X) [2]. . Let X be a smooth complex projective surface. Then there is a locally finite decomposition of the big cone of X into rational locally polyhedral subcones Σ such that in the interior of each subcone Σ the support Neg(Σ) of the negative part of the Zariski decomposition of the divisors in the subcone is constant. Now, the idea of Luszcz-Świdecka and Schmitz is to assign to a chamber Σ an element in the Minkowski basis M Σ . Specifically, let C be a big and nef curve in the interior of Σ.
It is convenient to work in the sequel with a compact slice Nef H (X) of the nef cone Nef(X) defined as Finally, we write NnB(X) for the number of numerical equivalence classes of nef and non-big divisors in Nef H (X) and we write Zar(X) for the number of Zariski chambers in the BKS-decomposition of Big(X).

The cardinality of Minkowski bases
In the view of Remark 2.4 it is natural to ask how many elements there are in the Minkowski basis. We will show here that the answer depends on the choice of the flag and that the number is a sharp upper bound for the number of elements in the Minkowski basis. The number of negative curves on surfaces with Eff(X) rational polyhedral is finite, hence the number of Zariski chambers on such surfaces is finite as well. This number can be large. For example for del Pezzo surfaces X i obtained as the blow ups of P 2 in i ∈ {1, ..., 8} general points we have see [1]. Now, we explain that the second summand in (1) is also finite.

Lemma 3.1 (Nef, non-big divisors). Let X be a surface with Eff(X) rational polyhedral. Then there is only a finite number of nef and non-big divisors in Nef H (X).
Proof. Assume to the contrary that there are two divisors N 1 , N 2 , which are nef and not big, such that for all t ∈ [0, 1] the divisors tN 1 + (1 − t)N 2 lie on the common face (here the rational polyhedrality assumption comes into the play). Thus (tN 1 + (1 − t)N 2 ) 2 = 0 for every t ∈ [0, 1], what implies that N 1 .N 2 = 0. It means that the intersection matrix of N 1 , N 2 is the zero matrix of size 2 × 2, which contradicts the index theorem.
Now we relate the number in (1) to the geometry of the solid Nef H (X).
Proof. Let G be a face of Nef H (X). If G = Nef H (X) then this corresponds to f ρ−1 = 1 and is accounted for by 1 on the right hand side in the formula (3). Otherwise we distinguish two cases: either G is a vertex of Nef H (X) which is not big, hence G 2 = 0 or G is a big vertex or a face of dimension 1. The first case occurs (f 0 ) nb times and is accounted for by the second summand on the right in (3). The second case corresponds to the third summand in (3). Indeed, given a nef and big divisor D there exists a Zariski chamber Σ D with Neg(Σ D ) = D ⊥ . This follows from Nakamaye's result [7,Theorem 1.1]. Thus the inequality in (3) is established.
For the reverse inequality it suffices to show that distinct Zariski chambers determine distinct faces of Nef H (X). To this end let Σ be a Zariski chamber. By [2] there is a face of Nef H (X) orthogonal to the support of Neg(Σ). The injectivity of this assigment Σ → Neg(Σ) ⊥ follows again from the aforementioned result of Nakamaye. Now we are in a position to prove our main result. Proof. Given Zariski chambers Σ 1 , Σ 2 with Neg(Σ i ) = N i 1 , . . . , N i r i for i = 1, 2, we associate to them elements of the Minkowski basis. Since A is ample we have A.N i j > 0 for j = 1, . . . , r i and i = 1, 2. This fact implies that to distinct chambers Σ 1 , Σ 2 there belong distinct basis elements. This can be seen as follows. We order negative curves in such a way that N 1 j = N 2 j for j = 1, . . . , s and all other curves in both sets Neg(Σ i ) are mutually distinct. Assume that M Σ 1 = M Σ 2 . Then Intersecting both sides of (4) with the nef divisor M Σ 1 we obtain Since N 2 ℓ .M Σ 1 > 0 for ℓ s + 1, we conclude that r 2 = s. Similarly, intersecting with M Σ 2 we get r 1 = s. All this implies that Σ 1 = Σ 2 , see also [1].
Example 3.4 (Del Pezzo surfaces). Using the above theorem we can compute the cardinality of Minkowski basis for del Pezzo surfaces X i with respect to a fixed ample flag (x, A). To this end we need to compute the number of nef non-big curves on X i . Let C = aH − b j E j be such a curve, where as usually π i : X i → P 2 is the blow up at i general points with exceptional divisors E 1 , . . . , E i and H = π * i (O P 2 (1)). First we observe that C is a rational curve. This follows from adjunction since On the other hand 0 = C 2 = a 2 − b 2 j .
It is elementary to check that (5) and (6)  Note that all solutions can be obtained from C(1) applying standard Cremona transformations. This verifies again that an irreducible nef non-big curve on a del Pezzo surface is rational.
Counting all curves C(j) on the appropriate surface X i and taking (2)