Cohomology Chambers on Complex Surfaces and Elliptically Fibered Calabi-Yau Three-folds

We determine several classes of smooth complex projective surfaces on which Zariski decomposition can be combined with vanishing theorems to yield cohomology formulae for all line bundles. The obtained formulae express cohomologies in terms of divisor class intersections, and are adapted to the decomposition of the effective cone into Zariski chambers. In particular, we show this occurs on generalised del Pezzo surfaces, toric surfaces, and K3 surfaces. In the second part we use these surface results to derive formulae for all line bundle cohomology on a simple class of elliptically fibered Calabi-Yau three-folds. Computing such quantities is a crucial step in deriving the massless spectrum in string compactifications.


Introduction and Summary
Vector bundle cohomology is an essential tool for string theory, being related to the degrees of freedom (particles) present in the low energy field theory limit. However, its computation is notoriously difficult and has been a major obstacle for progress in string phenomenology from its very beginning. In the last decade several computer implementations have been written to cope with this technical hurdle, automating laborious calculations that would otherwise be impossible to carry out in any practicable time [1][2][3]. These codes primarily deal with holomorphic line bundle cohomology on complex manifolds, since line bundles feature in many important contexts in string theory and moreover can be used as building blocks for higher rank vector bundles. Though extremely useful for practical purposes, such implementations remain limited in two respects. First, the algorithms become increasingly slow and eventually unworkable for manifolds with a large A novel approach to the problem has recently emerged through the observation that for many classes of complex manifolds of interest in string theory, line bundle cohomology is described by simple, often locally polynomial, functions [4][5][6]. To date, this observation has been checked to hold true for the zeroth as well as all higher cohomologies on several classes of two and three-dimensional complex manifolds which include certain complete intersections in products of projective spaces, toric varieties and hypersurfaces therein, all del Pezzo and all Hirzebruch surfaces [6][7][8][9][10][11]. The existence of simple closed-form expressions for cohomology is an interesting mathematical question in itself. For Physics, these provide an unexpected shortcut to incredibly hard computations needed for connecting String Theory to Particle Physics, making feasible the implementation of what is known in string model building as the 'bottom-up approach'. This involves working out the topology and geometry of the compactification space by starting from physical data, such as the number of quark and lepton families, and the number of vector-like matter states, which get encoded in the compactification data as dimensions of certain vector bundle cohomologies. The context in which cohomology formulae are, perhaps, the most relevant for attempting a bottom-up string model building approach is that of heterotic string compactifications on smooth Calabi-Yau three-folds with abelian internal fluxes described by sums of line bundles (see for instance Refs. [12][13][14][15][16][17][18][19]).
The existence of cohomology formulae has been discovered through a combination of direct observation [4-6, 8, 9] and machine learning [7,10] of line bundle cohomology dimensions computed algorithmically. A common feature of these formulae is that they involve a decomposition of the Picard group into disjoint regions, in each of which the cohomology function is polynomial or very close to polynomial. This pattern has been observed for the zeroth as well as all higher cohomologies, with a different region structure emerging for each type of cohomology. The number of regions often increases dramatically with the Picard number of the space. The origin of these formulae has been elucidated for certain complex surfaces in Refs. [9,11].

Simple example
A central aim in the present paper is to give a general understanding of the appearance of functions describing the zeroth cohomology of line bundles on certain classes of non-singular complex projective surfaces. In dimension two it suffices to understand the zeroth cohomology function since this implies the existence of formulae for the first and second cohomologies by Serre duality and the Hirzebruch-Riemann-Roch theorem.
We begin with a simple example.
Consider a del Pezzo surface of degree 7, obtained by blowing-up P 2 at two generic points, denoted as dP 2 in the Physics literature. Within the cone of effective line bundles (divisor classes), one finds [11] that the zeroth cohomology is given by the value of a piecewise polynomial function. Outside of the effective cone the zeroth cohomology is trivial. Figure 1 depicts the chambers, within each of which a single polynomial describes the zeroth cohomology. Region 0 corresponds to the nef cone, its interior being the Kähler cone. The Picard lattice of dP 2 is spanned by the hyperplane class H of P 2 and the two exceptional divisor classes E 1 and E 2 resulting from the two blow-ups. The effective cone (Mori cone) is generated by M 1 = E 1 , M 2 = E 2 , and M 3 = H − E 1 − E 2 . All three generators are rigid, satisfying M 2 i = −1. In the nef cone, a vanishing theorem due to Kawamata and Viehweg implies that all higher cohomologies are trivial and hence the zeroth cohomology is given by the index (the Euler characteristic), which is a polynomial function of degree 2. In the other regions, it turns out that the zeroth cohomology is given by the index of a shifted divisor. More explicitly, for an effective line bundle associated with a divisor class D, one has the following locally polynomial formula. Equivalently, one can capture this locally polynomial function in the single expression, where θ( · ) equals one for x ≥ 0 and zero otherwise.

Summary of results
The appearance of formulae as in Equation (1.1), which is a particularly simple example of a more general phenomenon, can be explained by combinining Zariski decomposition with vanishing theorems for cohomology, as we now briefly explain.
If D is an effective divisor, a theorem due to Zariski ensures that it can be uniquely decomposed as D = P + N , where P is nef and N effective, and P intersects no components in the curve decomposition of N . In general P and N are rational rather than integral divisors. In the case of an integral divisor D, When D is nef, N is trivial and D = P . When D is outside the nef cone, the positive part P always lies on the boundary of the nef cone. In the latter case, the prescription for Zariski decomposition implies that an effective divisor gets projected D → P to a face of the nef cone. Grouping divisors according to the face onto which they get projected gives rise to 'Zariski chambers', which are locally polyhedral subcones of the effective cone. Within a Zariski chamber, the support of N is fixed and the Zariski decomposition takes a fixed form.
The chamber structure induced on the interior of the effective cone (the big cone) by Zariski decomposition is a fairly recent result established in Ref. [20].
If the image of a Zariski chamber under the map [D] → [ P ] is covered by a vanishing theorem, then the index function can be 'pulled back' to give a single function for zeroth cohomology throughout the Zariski chamber. In this case the Zariski chamber becomes also a 'cohomology chamber'.
Promoting Zariski chambers to cohomology chambers requires a vanishing theorem that interacts well with the flooring. While the positive part P in a Zariski decomposition is nef, there is a round-down operation in the relation h 0 (S, O S (D)) = h 0 (S, O S ( P )), so that it is not sufficient for a vanishing theorem to apply to the nef cone. Additionally, most vanishing theorems involve a twist by the canonical bundle, which may push P even further away from the region covered by the vanishing theorems.
Cohomology formulae for complex surfaces While Zariski chambers exist for every smooth complex projective surface, whether these become cohomology chambers depends on the presence of appropriate vanishing theorems. Hence in this paper we consider several classes of surfaces on which there exist such vanishing theorems.
On all generalised del Pezzo surfaces and all projective toric surfaces, we prove that the zeroth line bundle cohomology is described throughout the Picard lattice by closed-form expressions. In the case of toric surfaces, it is possible to utilise the Demazure vanishing theorem.
Theorem. Let S be a smooth projective toric surface, and D an effective Z-divisor with Zariski decomposition D = P + N . Then Hence every Zariski chamber is upgraded to a cohomology chamber.
On generalised del Pezzo surfaces, we show one can use the Kawamata-Viehweg vanishing theorem. Here we find that one should instead use the round-up P of the positive part, rather than the round-down P . Hence every Zariski chamber, excluding its intersection with the boundary of the Mori cone, is upgraded to a cohomology chamber.
On the boundary, one can at least say for the subset of integral divisors D whose support has negative definite intersection matrix that the positive part is trivial P = 0 so that h 0 S, O S (D ) = h 0 S, O S = 1. In general this determines the cohomology on a number of faces of the Mori cone but not the entire boundary.
The expressions for P , and hence P and P , can be made very explicit, given knowledge of the Mori cone and the intersection form, and in particular are determined purely from intersection properties. Since the index is also computed from intersections, this means the calculation of any zeroth cohomology involves only intersection computations.
Concretely, note the prescription for constructing Zariski chambers is that every face F of the nef cone not contained in the boundary of the Mori cone gives rise to a Zariski chamber Σ F , by translating the face Note that the dual divisor M ∨ i k ,R is computed with respect to the set R = {M i1 , M i2 , . . . M in } and so can take different forms in different Zariski chambers. When D is integral as in the case of line bundles, the round-up and round-down are then given by the following simple expressions We also show that, alternatively, one can write a single expression for P throughout the effective cone.
Let I(S) be the set of rigid curves on the surface S, which is a subset of the set of Mori cone generators.
And let R(S) be the set of subsets of I(S) with negative definite intersection form. Every subset R ∈ R(S) corresponds to a set of generators of the Mori cone orthogonal to a face of the nef cone. In a given subset R ∈ R(S), for any element M i ∈ R one can define a unique effective dual divisor M ∨ i,R as above. Each element , one then has the following.
Proposition. Let D be an effective divisor on S with Zariski decomposition D = P + N Then At a practical level, to determine Zariski decompositions one requires knowledge of the subsets of the Mori cone generators on which the intersection form restricts to a negative definite matrix. These are the subsets of generators orthogonal to those faces of the nef cone that intersect the interior of the Mori cone, and hence directly determine the Zariski chambers. The subsets are straightforward to compute given knowledge of the Mori cone and the intersection form.
While for generalised del Pezzo surfaces and toric surfaces the Mori cone data can be computed algorithmically, in general this is not an easy matter. In the cases where the Mori cone data is not easily available, one can attempt to use the cohomology formulae described above 'backwards'. The proposal is that one would start with some partial knowledge of the zeroth cohomology, as determined from algorithmic methods, and then attempt to fit these results to the formulae in order to infer the Mori cone data.
We note again that, while the above framework applies only to the zeroth cohomology, formulae for the first and the second cohomology follow immediately via the index formula and Serre duality.

Cohomology formulae for elliptically fibered Calabi-Yau three-folds
Elliptically fibered Calabi-Yau three-folds are of particular significance in string theory, especially in the study of heterotic/F-theory duality (see Ref. [21,22] for some recent work on this duality involving line bundles).
Thus, in the second part of the paper we consider smooth elliptic Calabi-Yau three-folds realised as generic Weierstrass models with a single section over smooth compact two dimensional bases. The aim is to lift the cohomology formulae obtained for surfaces to the corresponding three-folds.
On such a three-fold X 3 , the cohomology of any line bundle L can be computed in terms of the cohomology of the pushforward bundle π * L and the higher direct image R 1 π * L under the projection map π : X 3 → B 2 to the base B 2 , by use of the Leray spectral sequence. We show that this sequence degenerates in our context, so that the lift of cohomology on the base to the three-fold is simply (1.9) The pushforward and higher direct image are simple sums of line bundles, written explicitly in Equation (5.17).
From these formulae, one can expect that the cohomology chambers of the base give rise on the three-fold to regions in which the cohomology function has a closed form. We study this phenomenon in detail for an elliptic fibration over the simplest base, P 2 . On the one hand, we show that it is indeed possible to determine regions and corresponding formulae describing all line bundle cohomologies on the Calabi-Yau three-fold. On the other hand, we make the point that this procedure is intricate, and not immediately transparent. Nevertheless, this provides the first proofs of cohomology formulae for three-folds of this kind.

Zariski decomposition
In this section we give a pedagogical introduction to Zariski decomposition. The reader familiar with the terminology and the basic ideas can safely skip to the following section.

Divisors
We start by reviewing some definitions involving divisors. Since we are dealing only with smooth projective surfaces, we will not distinguish between Weil and Cartier divisors. The group of divisors on a surface S is denoted by Div(S). A divisor D ∈ Div(S) is a Z-linear combination of irreducible codimension one subvarieties (irreducible curves), that is a finite sum D = i n i C i with n i ∈ Z, and the group operation is addition. The set of curves {C i } is called the support of D, which we denote by Supp(D). D is said to be effective if n i ≥ 0 for all i. A subdivisor P of D is a divisor such that D − P is effective.
Two divisors D and D are said to be linearly equivalent D ≡ D if they differ by the divisor of a mero- where ord i is the vanishing order (positive) or the pole order (negative) of f on the curve C i . Note that the divisor of a product of meromorphic functions is div(f g) = div(f ) + div(g). The class of a divisor D modulo linear equivalence is denoted by [D]. A linear equivalence class is said to be effective if it contains effective representatives.
There is also the related notion of a complete linear system of a divisor, denoted |D|, which is the set of all effective divisors linearly equivalent to D, which can of course be empty. If D is effective, one can think of |D| as the family of deformations of D, and its dimension dim |D| as the number of parameters of the family.
If D is effective but the only element in its complete linear system, then dim |D| = 0 and D is called 'rigid'.
The set of points common to every element of the complete linear system is called the base locus.
The group Div(S) can be extended to Div(S) ⊗ Q, whose elements are called Q-divisors. These are rational linear combinations of curves. Two Q-divisors D 1 and D 2 are said to be linearly equivalent if there exists an integer n such that nD 1 and nD 2 are integral and linearly equivalent. Elements of Div(S) will be referred to as integral or Z-divisors. An R-divisor is a Q-divisor multiplied by some real number.

Divisors and line bundles
A divisor D ∈ Div(S) determines a line bundle O S (D) such that D is a rational section of O S (D). If two divisors are linearly equivalent, their associated line bundles are isomorphic. Hence the group of divisors modulo linear equivalence is isomorphic to the group of line bundles up to bundle isomorphisms, which is called the Picard group Pic(S). Note in particular that the operation of adding divisors corresponds to taking the tensor product of the line bundles, . Below we will be interested only in line bundles up to isomorphism, so we will simply refer to a 'line bundle' when we mean a line bundle up to isomorphism, and we will write O S (D) rather than O S ([D]).
Particularly important for our purposes is the simple relationship between the zeroth cohomology of the line bundle O S (D) and the complete linear system of the divisor D, specifically where P( · ) denotes the projectivisation.

Intersections
If two curves C and C intersect transversely, there is a natural intersection product given by the number #{C ∩ C } of intersection points of the two subvarieties. More generally, if two curves do not share connected components, the geometric interpretation is still valid if the intersection points are weighted by the local intersection multiplicities (greater than 1 for non-transversal intersections). The product can be extended to include curves sharing connected components, by requesting that the following conditions are met: 1. Consistency with the natural case: C · C = #{C ∩ C } if C and C intersect transversely.

Symmetry:
These conditions give a unique intersection product C · C ∈ Z. In particular, the intersection of two curves C and C sharing a connected component is understood by replacing C by a linearly equivalent sum of curves that share no connected components with C. In this way, negative intersections naturally occur. Suppose a curve C is linearly equivalent to a distinct curve or an effective sum D of curves that shares no connected components with C. Then its self-intersection is nonnegative, C 2 = C · D ≥ 0. Hence, conversely, if C 2 < 0, then there must be no distinct effective divisor linearly equivalent to C. So one can conclude that there are no other elements in the complete linear system |C|, i.e. C is rigid. For this reason we also refer to a rigid curve as a 'negative' curve.
As divisors are linear combinations of curves, the above defines intersections between R-divisors. This gives rise to an important equivalence relation on Div(S) ⊗ R: two divisors D 1 and D 2 are called 'numerically equivalent' if D 1 · C = D 2 · C for every curve C in S. Note that by the third condition above, linearly equivalent divisors are also numerically equivalent, so numerical equivalence is in general a weaker condition.
In particular, note there is an intersection pairing Pic(S) × Pic(S) → Z.
On many common spaces, linear equivalence and numerical equivalence coincide. For instance, this is true on all compact toric varieties (see Proposition 6.3.15 in Ref. [23]), on all generalised del Pezzo surfaces, and on all projective Calabi-Yau manifolds of dimension greater than one (where X being Calabi-Yau is understood in the strict sense of having no holomorphic k-forms for 0 < k < dim(X)), and hence on all spaces we discuss explicitly below. Counter-examples to this include the elliptic curve, and products of curves of large genus.
An R-divisor D ∈ Div(S) ⊗ R is said to be nef if D · C ≥ 0 for every curve C in S. It follows that D is nef if and only if D · C j ≥ 0 for every C j ∈ Supp(D) since the intersection of distinct curves is non-negative. For a divisor D = x i C i , with C i irreducible components, its intersection matrix I(D) is defined as the symmetric

Cones
The natural arena for defining several important objects is the space of divisors modulo numerical equivalence.
This is called the Néron-Severi group NS(S), and we can define the corresponding real vector space NS(S) R = NS(S) ⊗ R. Within this vector space, the set of nef divisors naturally forms a cone Nef(S). To any cone one can associate a dual cone, which is the set of points having non-negative intersection with every element in the cone. The dual of the nef cone is the closure of the cone of effective divisors, and is called the Mori cone or the cone of pseudo-effective divisors, and is denoted by NE(S). The interior of the Mori cone is the big cone, whose elements are big divisors.
We note it is easy to see that a rigid curve must be a generator of the Mori cone as follows. Let C be a rigid curve and consider the hyperplane in the Néron-Severi group corresponding to zero intersection with C.
Any other Mori cone generator, being a distinct curve, must have non-negative intersection with C, and hence must lie on the hyperplane or be on the positive side of it. But C is on the negative side since C 2 < 0. Since C is effective, this is impossible unless C is also a generator.
Since linear equivalence and numerical equivalence coincide on the spaces we will discuss, the Néron-Severi group and the Picard group are isomorphic. Hence integral points in the above cones can be identified with line bundles up to isomorphism.

Detection of rigid divisors
An important idea in relation to Zariski decomposition is that of detecting via intersections rigid parts of a complete linear system. Suppose an effective divisor D has negative intersection D · C j < 0 with an irreducible curve C j . In the intersection the only possible negative contribution is from the self-intersection term x j (C j · C j ). Hence C j , which must be a rigid curve, must be in the divisor expansion of D, i.e. x j > 0. More strongly, there is clearly a lower bound on the coefficient x j of C j , Any linearly equivalent divisor has the same intersection with C j , and hence for any effective D ≡ D, i.e. any element of the linear complete system |D|, the same lower bound applies. In particular, removing this much of C j from every divisor in the complete linear system |D| leads to a linear system of equal size More generally, if C j is an irreducible negative divisor andD j an effective divisor that (1) intersects C j negatively and (2) has non-negative intersection with all other irreducible curves, thenD j can be used in order to detect the presence of C j in the expansion of D, provided that D ·D j < 0. As before, it follows that

Zariski decomposition
In Ref. [24], Zariski established the following result. Then D has a unique decomposition D = P + N , where P and N are Q-divisors such that 1. P is nef.
2. N is effective and if N = 0 then it has negative definite intersection matrix Zariski decomposition was extended to pseudo-effective divisors by Fujita in Ref. [25]. While N is effective, P is only pseudo-effective in general. Moreover, recalling that an R-divisor D R is a Q-divisor D Q multiplied by some real number a, the Zariski decomposition of D R can be defined by D R = a P D Q + a N D Q .

A pedagogical algorithm
In Section 3 we will present a way to implement Zariski decomposition with a simple formula, which will be the basis of the cohomology discussion in Section 4. In the present section, we present a pedagogical iterative algorithm, based on a classical proof for Zariski's theorem -see for example Theorem 14.14 in Ref. [26].
The algorithm begins with a naive guess of the support for the negative part N , as detected by negative intersections. A candidate Zariski decomposition is then constructed. However there then appear new negative intersections, so the process is iterated.
The following steps lead to the unique Zariski decomposition of an effective Q-divisor D. Let I(S) denote the set of all irreducible negative divisors on S. And set I = ∅.
1. Determine the set of curves {C ∈ I(S) | C · D < 0}. This set is non-empty, unless D is nef, in which case its Zariski decomposition is trivial. Incorporate these into the setĨ = I ∪ {C ∈ I(S) | C · D < 0}.
2. Construct the unique, effective Q-divisorÑ with supportĨ such thatÑ · C i = D · C i for all C i ∈Ĩ.
3. DefineP := D −Ñ . If this is nef, take P =P and N =Ñ . Otherwise, repeat the first two steps with The algorithm must terminate because each iteration increases the size of the setĨ, whileĨ is finite since . The uniqueness and effectiveness ofÑ at each stage follow respectively from Lemmas 14.12 and 14.9 of Ref. [26].

Example
Consider the Gorenstein Fano toric surface F 8 , whose ray diagram is depicted in Figure 2. In Appendix A.1 we provide a reminder on how to compute important properties of a toric surface from its ray diagram, which we apply here for the case of There are four linear relations between the six rays leading to the following weight system: The toric divisors D 1 , D 2 , D 3 , D 4 can be used as a basis for the Picard lattice. In terms of these, the expressions for D 5 and D 6 , read off from the weight system, are D 5 = D 1 + 2D 2 + D 3 and D 6 = D 2 + D 3 + D 4 . The self-intersections are given by: In this basis we will write divisors as D = ( · , · , · , · ). The intersection form (see Appendix A.1 for details about how to infer intersections from the toric data) is The Mori cone generators M i and the dual nef cone generators N j are given by (2.10) For applying the algorithm above, we need the set of rigid irreducible curves. In the present case these are simply the Mori cone generators.
With D = 2D 1 + D 2 + D 3 , let us apply the above algorithm to find its Zariski decomposition. Applying the steps of the algorithm presented above we have the following.
The Zariski decomposition of D is hence [P ] is in general not integral. In view of the cohomology result (2.14) below, we define the round-down version of the Q-divisor P as the Z-divisor P , obtained by rounding down each coefficient in the divisor expansion of P . The round-up P of a divisor is defined analogously. We prove the following result. This is summarised in the following theorem.
Theorem 2.5. Let S be a smooth projective surface, and let D be an effective Z-divisor with Zariski decomposition D = P + N . Then (2.14) Proof. See Proposition 2.3.21 in Ref. [27].
In fact, it is straightforward to see that the same result applies in the case of the round-up P .
Corollary 2.6. Let S be a smooth projective surface, and let D be an effective Z-divisor with Zariski decom- Proof. The ceiling P of the positive part is related to P by a fractional effective divisor, i.e.
where 0 ≤ f i < 1. Importantly, since D is integral, Supp(∆) ⊆ Supp(N ). From the properties of Zariski decomposition, recalled in Section 2.2, it is then trivial to verify that the above expression for P is in fact a Zariski decomposition, with positive part P and negative part ∆, so that in particular D and P

Iteration of Zariski decomposition and divisor rounding
While the positive part P in the Zariski decomposition D = P + N is nef, the same is not in general true of the round-down P or the round-up P . That is, the maps φ ↓ Z and φ ↑ Z , defined above, do not in general output in the nef cone. So it may be possible to perform a subsequent Zariski decomposition.
For simplicity we focus on the round-down P . Denoting P = D (1) , its Zariski decomposition can be written as This process can be iterated until for some n, P (n) is nef, which includes the possibility of being zero.
Equivalently, the iteration takes place as long as the Zariski decomposition gives a non-trivial negative part.
The fact that such an n must exist is clear, since at every iteration Zariski decomposition and flooring reduce at least one of the coefficients in the divisor expansion. It is useful to see a real example, which we choose from among the 16 Gorenstein Fano toric surfaces.

The F 8 example, once again
Consider the Gorenstein Fano toric surface F 8 , whose ray diagram is depicted in Figure 2 and whose properties we recalled in Section 2.2.
We also take again as our initial divisor D = 2D 1 + D 2 + D 3 , which being integral defines a line bundle.
In Section 2.2, we determined the Zariski decomposition of D to be Since P is not an integral divisor, P = P . In particular, P = D 2 . Noting the intersection properties we see that P is not nef. Hence we look for another Zariski decomposition, P = P (1) + N (1) . Applying again the algorithm of Section 2.2, we find straightforwardly P = 0 + D 2 ≡ P (1) + N (1) . In this particular case, P (1) = P (1) . Since P (1) = 0 is nef, the iteration process terminates here.
In terms of line bundles, the map i.e. the final bundle is the trivial line bundle, which is nef. For the preserved zeroth cohomology, we have

Zariski chambers
Let D = P + N be a Zariski decomposition. By varying the coefficients in N while holding the support fixed, yielding effective divisorsÑ , one obtains divisorsD whose Zariski decompositions areD = P +Ñ . By keeping P fixed and adding variousÑ with fixed support, one performs what might be called a 'Zariski composition'.
If Supp(N ) = ∅, then the positive part P lies on a boundary of the nef cone. To see this, note P · C = 0 for all curves C ∈ Supp(N ). As the C are rigid and hence generators of the Mori cone, C · (. . .) = 0 specifies a hyperplane which meets the nef cone along a boundary. But P is nef by definition, so P must lie on this boundary. Hence in a Zariski composition, one begins at a point on a boundary of the nef cone. One can then imagine varying the starting point across the entire boundary. The region reached by all such compositions will then be given by translating the entire boundary along the elements in Supp(N ).
This perspective was formalised in Ref. [20]. The authors showed that on any smooth projective surface the interior of the effective cone, which is the big cone, can be decomposed into rational locally polyhedral subcones called 'Zariski chambers' such that in each region the support of the negative part of the Zariski decomposition of the divisors is constant. Moreover, these subcones are in one-to-one correspondence with faces of the nef cone that intersect the big cone.
Definition 3.1. Let S be a smooth projective surface, and let F denote a face of the nef cone which intersects the big cone. The Zariski chamber Σ F associated with F is the subcone of the effective cone constructed by translating F along all negative curves that are orthogonal to F with respect to the intersection form, where the boundary between two chambers belongs to the chamber whose corresponding face has higher dimension.
Note that in general a Zariski chamber is a cone which is neither open nor closed.
To construct these chambers, one requires knowledge of at least two out of three of the Mori cone, nef cone, and intersection form, which may in general be non-trivial to determine. Moreover, note that since the possible supports for the negative part of a Zariski decomposition are in one-to-one correspondence with the collections {C A } of rigid curves which have negative definite intersection matrix, the same is true of the faces of the nef cone that intersect the big cone. The Zariski chambers are determined by knowledge of the set of such collections, which we denote R(S) on a surface S.
In Ref. [20] Zariski chambers were defined only in the interior of the effective cone, since the authors were interested in the volume properties of big line bundles. For our purposes we do not need to make this restriction. As such, we can extend Zariski chambers to the closure of the effective cone.

Map for fixed Supp(N )
As we now explain, within a Zariski chamber the form of the Zariski decomposition is fixed. Let D be an effective divisor with curve decomposition and Zariski decomposition where C A are rigid curves and the intersection matrix (C A · C B ) is negative definite. When the support of N is known, as throughout a Zariski chamber, the coefficients y A can be straightforwardly obtained as follows.
Lemma 3.2. Let D = P + N be the Zariski decomposition of an effective divisor. Then for every C A ∈ Supp(N ), the coefficient y A of C A in N is given by The divisor C ∨ A,Supp(N ) should be read 'the dual of C A with respect to the support of N '. We note it is a classic result in the context of Zariski decomposition that the divisor C ∨ A in Lemma (3.2) is effective (see for instance Lemma 14.9 of Ref. [26]). This lemma immediately gives the following formula. Note the rigid curves are a subset of the Mori cone generators, so we write M i for the elements in Supp(N ).

Example: Zariski chambers for the F 6 surface
The space F 6 is a toric surface that is not isomorphic to a Hirzebruch or del Pezzo surface, and in fact it is the lowest Picard number surface of this kind among the Gorenstein Fano toric surfaces. It is isomorphic to a blow-up of the Hirzebruch surface F 2 . We show the toric diagram with labelled toric divisors and the weight system: We can take as a divisor basis self-intersections are given by and the intersection form in the above basis is  Zariski chambers outside of the nef cone. We have labelled the rays of the Mori cone generators M i and the nef cone generators N j .
Following the prescription outlined above, we determine the set R(F 6 ) of collections of rigid curves with negative definite intersection matrix. In the present case, the rigid curves are precisely the Mori cone generators, so intersections between rigid curves are given by the matrix in Equation (3.4). Intersections between a subset of the rigid curves are given by restricting the matrix. For example, restricting to {M 1 , M 3 } gives In total there are five collections with negative definite intersection matrix, Correspondingly, one can note from Figure 4 that the nef cone has three codimension 1 faces and two codimension 2 faces that have a non-vanishing intersection with the big cone.
The Zariski chamber Σ R for R ∈ R(S) is given by translating along the elements in R the boundary of the nef cone spanned by generators orthogonal to all elements in R up to boundaries -which we recall belong to the subcone whose corresponding face has higher dimension. Hence, the five Zariski chambers of F 6 , which are sub-cones of the effective cone in addition to the nef cone, are which are depicted in Figure 4. For a Zariski chamber Σ i corresponding to translation by a single Mori cone generator, the duals are simply For the chambers with two Mori cone generators we have the following duals:

Alternative packaging
In some situations it is useful to repackage the information given by the Zariski chamber structure, instead writing a single formula that captures the behaviour of the decomposition throughout the effective cone.
Recall R(S) is the set of collections of rigid curves with negative intersection matrix. For a given R ∈ R(S), also recall that for any C A ∈ R, one can define a unique effective dual curve C ∨ A,R with respect to R, which has Supp(C ∨ A,R ) ⊆ R and which satisfies C ∨ A,R · C B = −δ AB for all C B ∈ R. Each element R ∈ R(S) with C A ∈ R hence determines a divisor C ∨ A,R , giving a set of possible duals for C A written as For example, in the F 6 case treated above, these are These sets S A provide an alternative way to determine a Zariski decomposition, as follows.
Lemma 3.4. Let D be an effective divisor with Zariski decomposition D = P + N , and let C A be a rigid curve. Then the following statements are true.
Proof. These are straightforward to prove from the fact that R(S) is precisely the set of possible supports for the negative parts in Zariski decomposition.
, both results are contained in the statement that the coefficient of C A in N is precisely the maximum of G A . Hence we have the following.
Proposition 3.5. Let D be an effective divisor. The negative part N of its Zariski decomposition is given by Proof. This follows immediately from Lemma (3.4).
The formula in Equation (3.9) has a natural interpretation in the context of detecting rigid curves by intersection, which was reviewed at the end of Section 2.1. For each rigid curve, the formula checks several candidate effective divisors to see which detects the maximal amount in D. The reason the number of candidates is small is because instead of checking every element in the cone of candidate divisors (see Section 2.1) it suffices to check the generators, as one can verify.
This perspective makes it clear that for a rigid curve C A the set of duals in S A can also be understood as the generators of the cone in the Néron-Severi group determined by the inequalitiesD · C A ≤ 0 andD · C i ≥ 0 for all C i ∈ I(S) where C i = C A , excluding generators lying on a nef cone boundary. We also note in passing that these cones are a subset of the (closures of the) simple Weyl chambers as defined in Ref. [28].

Cohomology chambers and formulae
For D an effective Z-divisor with Zariski decomposition D = P + N , Theorem (2.5) asserts the preservation of Proof. Within a Zariski chamber, the form of the map D → P is fixed, with N given by Equation (3.2). In the case of an integral divisor D, the round-up of P = D − N leaves D unchanged and so affects only the coefficients in N . Hence Combination with Theorem (2.5) then gives the stated result.
Since there is also the cohomology relation Proof. This is analogous to the proof of Theorem (4.1), using the cohomology relation of Corollary (2.6).
While the relations in Theorem 4.1 and Corollary 4.2 are valuable in themselves, unless something can be said about the cohomologies appearing on the right hand side, this is unlikely to be helpful in practice. On the classes of surfaces that we discuss below, something can indeed be said about these cohomologies, due to the existence of powerful vanishing theorems.

Vanishing theorems and 'pulling back' the index
A vanishing theorem asserts the triviality of a number of the cohomologies for a subclass of line bundles, given certain properties of the variety. Perhaps the most well-known vanishing theorem is that of Kodaira.
Theorem (Kodaira vanishing). On a smooth irreducible complex projective variety X, for ample divisor D, When a vanishing theorem ensures that all but one cohomology vanish, the remaining dimension can be computed from the index. For example, if Kodaira vanishing ensures that the higher cohomologies of a line bundle L vanish, then While individual cohomologies are generically difficult to compute, the index can be computed using only divisor intersection properties, due to the Hirzebruch-Riemann-Roch theorem. In the case of a surface S, where O S is the trivial bundle. Hence this gives a formula describing the sole non-trivial cohomology throughout the region of vanishing. Note in the surface case the formula is quadratic in the divisor D, or equivalently, quadratic in the integers specifying D with respect to a basis.
between integral divisor classes, defined in Equation (2.12). If a vanishing theorem applies across either of these images, then one can 'pull back' the index to give a formula for cohomology throughout the Zariski chamber Σ.
If instead the vanishing theorem applies throughout the image region φ ↑ Z Σ ∩ NS(S) , then In either situation, the Zariski chamber becomes a 'cohomology chamber', in which the zeroth cohomologies are given throughout by a single formula.
Proof. This is immediate given the cohomology preservation relations in Theorem (2.5) and Corollary (2.6).
Note that while the image of a Zariski chamber under the map [D] → [P ] lies on a boundary of the nef cone, due to the rounding operations involved in the maps φ ↓ Z and φ ↑ Z of integral divisor classes the images φ ↓ Z Σ∩NS(S) and φ ↑ Z Σ∩NS(S) will in general not lie entirely in the nef cone. Hence the vanishing theorems of interest are not simply those applying to the nef cone.
Remark 4.4. In the case that every Zariski chamber is a cohomology chamber, the zeroth cohomology is described throughout the effective cone by using the expressions in Proposition (4.3) within each Zariski chamber.
Though we do not know of cases where some Zariski chambers become cohomology chambers via the map φ ↓ Z while others become cohomology chambers via the map φ ↑ Z , this is a possibility. When all Zariski chambers are cohomology chambers via the same map, then using the packaging in Proposition (3.5) one can alternatively write everywhere where D is any effective Z-divisor, and where I(S) is the set of negative curves on S while G A (D) is defined above Proposition (3.5).
Note that since ind(D) is a quadratic polynomial in the divisor D (or equivalently in the integers specifying D with respect to a basis), the formula for the zeroth cohomology in a cohomology chamber is a polynomial in the divisor P or P . Since these involve rounding, the result is not a genuine polynomial in general. This is illustrated in the example in Section 4.2 below.

Iteration and cohomology chambers
While the integral divisors P and P are not in general nef, if one iterates the process of Zariski decomposition and rounding, eventually this will reach an integral nef divisor. Naively then it seems that a vanishing theorem throughout the nef cone is sufficient to upgrade each Zariski chamber to a cohomology chamber.
However, two integral divisors from the same Zariski chamber may pass through distinct chambers on their journey to the nef cone, so that the combined map by which an index expression in the nef cone is 'pulled back' would not be uniform throughout the original chamber, so that the Zariski chamber is not a cohomology chamber.
The following is an illustrative example. In Section 2.2 we considered the divisor D = 3D 1 + D 2 + 5D 3 on the Gorenstein Fano toric surface F 8 , and determined its Zariski decomposition to be In Section 2.3 we noted that P is not nef, so that a further Zariski decomposition is required. This decomposition is trivial, 11) and the process terminates here, since P (1) = P (1) = 0 is nef. Note one can check that P is not nef, so in either case it requires multiple steps to reach the nef cone.
Now consider instead the divisor D = 3D 1 + 2D 2 + 5D 3 , which one can check has a Zariski decomposition Since Supp(N ) = Supp(N ), D and D lie in the same Zariski chamber. However, in contrast to the case for D, the divisor P is nef, so that the process terminates after a single step.

Higher cohomologies
When all Zariski chambers are also cohomology chambers, the zeroth cohomology is described throughout the entire effective cone by a set of regions and corresponding formulae, and is by definition zero outside. The higher cohomologies can then be obtained throughout the Picard group by Serre duality and the Hirzebruch- (4.13) In particular, we see that the chambers for the second cohomology are given by simply reflecting through the origin and translating by K S the Zariski chambers, while intersections of chambers in these two sets give chambers for the first cohomology.

Toric surfaces
On toric varieties, there is a powerful vanishing theorem due to Demazure. See for example Chapters 9.2 and 9.3 of Ref. [23] for details and a proof.
Theorem (Demazure vanishing for Q-divisors). Let D be a nef Q-divisor on a toric variety X Σ whose fan Σ has convex support. Then Demazure's vanishing theorem is limited to toric varieties with convex support. However, this is not a restriction in the context of Zariski decomposition, because this condition holds when the toric variety is projective. To see this, note that a projective variety is compact. A toric variety is compact if and only if its fan is 'complete' (see for example Theorem 3.1.19 of Ref. [23]), which means its support is R n for some n.
But this support is clearly convex. So compact toric varieties, and in particular projective toric varieties, are covered by Demazure vanishing.
This implies that on any projective toric surface, every Zariski chamber is also a cohomology chamber.  Hence every Zariski chamber is upgraded to a cohomology chamber. Explicitly, if D lies in the Zariski chamber Moreover, on a projective toric surface the Zariski chamber decomposition is straightforward to implement, because the Mori cone, nef cone, and intersection form are all computed algorithmically from the toric data.

Cohomology chambers on Gorenstein Fano toric surfaces
A commonly used set of projective toric surfaces are the 16 Gorenstein Fano toric surfaces, whose fans are shown in Figure 10 in Appendix B. where we have included also the number |I(F i )| of rigid curves.
Proof. The intersection forms for the Gorenstein Fano toric surfaces are given in Appendix B. From these one can determine the subsets of the Mori cone generators on which the intersection form is negative definite.
These subsets count the Zariski chambers, together with the empty set which corresponds to the nef cone, and by the above discussion these are also cohomology chambers.
Example: cohomology chambers for the F 6 surface In Section 3 we have determined the Zariski chambers for the example of the Gorenstein Fano toric surface F 6 , and using the Demazure vanishing theorem these can be immediately upgraded to cohomology chambers.
From the intersection form, the subsets of Mori cone generators with negative definite intersection ma- Together with the nef cone, this gives six Zariski chambers, illustrated in Figure 4. In the upgrade to cohomology chambers, this gives the following formulae.
It is sometimes useful to express the cohomology formulae with respect to a basis. One obvious choice here is to write a general element D of the Néron-Severi group as a sum over the Mori cone generators, and across all Zariski chambers the results for P are The formulae describing cohomology follow from these by using the expression for the index in this basis, so that the zeroth cohomology in each Zariski chamber is given by the following table.

Kawamata-Viehweg vanishing and Zariski decomposition
On non-toric surfaces Demazure's vanishing theorem is unavailable. However there is the following generalisation of Kodaira vanishing (see for example Chapter 9.1.C of Ref. [29]).
Theorem (Kawamata-Viehweg vanishing for Q-divisors). Let X be a non-singular projective variety, and let B be a Z-divisor. Assume that where D is a nef and big Q-divisor, and ∆ = i a i D i is a Q-divisor with fractional coefficients 0 ≤ a i < 1 and with simple normal crossing support. Then We note the useful characterisation that on an irreducible projective variety X of dimension n a nef divisor Note that the round-up D = D + ∆ of a Q-divisor satisfies the requirements for B in the theorem, provided that ∆ = D − D has simple normal crossing support. Additionally, it is convenient to rewrite the theorem to state that higher cohomologies of O X (B) vanish when B is such that B − K X is nef and big. This gives the following corollary.
Corollary 4.7. Let S be a smooth projective surface and let P be a Q-divisor. If P − K S is nef and big, and P − P has simple normal crossing support, then Proof. This is immediate.
Here we have suggestively written P for the Q-divisor, as we are interested in applying this vanishing theorem to the positive part P of a Zariski decomposition, as in Proposition (4.3). While the positive part P of a Zariski decomposition is by definition nef, it is not in general true that P − K S is nef and big, nor is it necessarily true that the fractional part of P has simple normal crossing support. However, there is at least one obvious class of surfaces for which these conditions are satisfied for every Zariski decomposition of a Z-divisor. These are the generalised del Pezzo surfaces, as we now discuss.

Application to generalised del Pezzo surfaces
In order for Corollary (4.7) to apply to all Q-divisors P throughout the nef cone, it is necessary that P − K S be nef and big for every nef P . Clearly, nefness of all P − K S for all P requires that −K S be itself nef.
Additionally, recalling that a nef divisor D is big if and only if D 2 > 0, we check the self-intersection Since P is nef and effective, P 2 ≥ 0, while since P is nef and −K S is effective, P · (−K S ) ≥ 0. To guarantee that P − K S is always big, the final term must be positive, so that −K S must be big. A variety whose anticanonical divisor is nef and big is called 'weak Fano', or in two dimensions a 'generalised del Pezzo' surface.
All generalised del Pezzo surfaces except for the Hirzebruch surfaces

Classification of generalised del Pezzo surfaces
Up to isomorphism, a generalised del Pezzo surface is either P 1 ×P 1 , the Hirzebruch surface F 2 , or a blow-up of P 2 at up to 8 points in almost general position. The ordinary del Pezzo surfaces, on which the anti-canonical divisor is not just nef and big but ample, are P 1 × P 1 and the blow-ups of P 2 at points in general position. A useful invariant of a generalised del Pezzo surface S is the degree d = (−K S ) 2 . On a generalised del Pezzo surface given by the blow-up of P 2 in n points, the degree is d = 9 − n. In the remaining cases of P 1 × P 1 and F 2 the degree is 8. Note 1 ≤ d ≤ 9.
As already mentioned above, any curve on a generalised del Pezzo surface has self-intersection C 2 ≥ −2, and any curve on an ordinary del Pezzo surface has self-intersection C 2 ≥ −1. On a generalised del Pezzo surface, the number of curves with self-intersection C 2 = −2 is at most 9, while the number of curves with self-intersection C 2 = −1 is finite.
Generalised del Pezzo surfaces are classified in terms of their 'type', as defined below.
Definition (Definition 3 in Ref. [32]). Two generalised del Pezzo surfaces have the same type if there is an isomorphism of their Picard groups preserving the intersection form that gives a bijection between their sets of classes of negative curves.
This classification is particularly important for our purposes, since the decomposition of the Mori cone of a surface into Zariski chambers is determined by the Mori cone generators and the intersection form alone.
While in general the negative curves do not fully specify the Mori cone, there is the following theorem.
While this theorem does not cover the cases with degrees 9 or 8, there is up to isomorphism precisely one generalised del Pezzo surface with degree 9, P 2 , and three with degree 8, P 1 × P 1 , F 2 , and Bl 1 P 2 , which are all of distinct types. Hence the Mori cone and intersection form, and hence also the Zariski chambers, are fixed within a type.
Since the surfaces with degree d = 9 or d = 8 are toric and very simple, the classification of types of generalised del Pezzos can be restricted to d ≤ 7. With this restriction, the Picard group Pic(S) and its intersection form depend only on the degree of S. What then differs among generalised del Pezzo surfaces of the same degree are the classes in Pic(S) which are effective.
To cut a long story short, the type of a generalised del Pezzo surfaces S of degree d ≤ 7 is specified by three elements: its degree d, the incidence graph Γ of the (-2)-curves, which turns out to be always a disjoint union of Dynkin graphs of types A, D, E, and the number m of (-1)-curves, hence the notation S d,Γ,m .
For each degree d ≤ 7, the graphs describing the possible configurations of (-2)-curves correspond to the Dynkin diagrams of all the subsystems of the root systems R d (up to automorphisms of R d ) given in the following with the exception of the subsystems 7A 1 of R 2 and 7A 1 , 8A 1 and D 4 + 4A 1 of R 1 , which only occur in characteristic 2 (see [35,36]

Example: ordinary del Pezzo surfaces
Among the generalised del Pezzo surfaces are the ordinary del Pezzo surfaces. As well as the simple case of P 1 × P 1 , these are the blow-ups of P 2 at 0 ≤ n ≤ 8 points in general position, which we write as dP n . These surfaces are non-toric only for n > 3. The numbers of Zariski chambers z(dP n ) on dP n have been determined in Ref. [37]. By the above analysis, these are also the numbers of cohomology chambers. The numbers of chambers together with the numbers |I(dP n )| of negative curves (which must be (−1)-curves) are The formula can be alternatively written in the following form, which appeared in Refs. [9,11]: where the Picard number is ρ = 4, in addition to four toric types there are two non-toric types. These non-toric types correspond to root subsystems A 1 and A 2 . All six types are shown in Table 4 of Ref. [32]. We take the case of the subsystem A 1 as a simple example of a generalised del Pezzo surface S which is neither toric nor an ordinary del Pezzo surface.
The Picard group and intersection form depend only on the degree of the generalised del Pezzo surface.
For degree d = 6 the Picard lattice is spanned by l 0 and l i with i = 1, 2, 3, which we write collectively as l A .
We write a general element 4 A=0 k A l A in this basis as a vector (k 0 , k 1 , k 2 , k 3 ). In this basis the intersection form and the anti-canonical divisor class −K S are 1, 1, 1) .  The intersection form between the Mori cone generators is (4.33) The above data determines the structure of the Zariski chambers. From the intersection matrix (M A ·M B ), there are eleven subsets of the rigid curves {M A } which have a negative definite intersection form, which are Together with the nef cone, this gives eleven Zariski chambers. The zeroth cohomology is then given throughout the effective cone by the following formulae.

K3 surfaces
A complex K3 surface is a compact connected complex surface S with trivial canonical bundle and with H 1 (S, O S ) = 0. These are the Calabi-Yau surfaces, excluding, by the latter condition, a product of tori.
Among the smooth complex K3 surfaces, we restrict to the projective case, since this is the case in which Zariski decomposition can be applied. Below we will often say 'K3 surface' where we mean 'smooth projective complex K3 surface'.
On a K3 surface the Picard group and the Néron-Severi group coincide, so we will not need to make a distinction. Moreover, the only negative curves on a K3 surface are (−2)-curves. See Ref. [38] for more properties of K3 surfaces.

Vanishing in the big cone
The Kawamata-Viehweg vanishing theorem can be applied to K3 surfaces, bearing in mind that in the present case the canonical bundle is trivial, which leads to the following specialisation of Corollary (4.7).  Proof. Note both P and D are in the Mori cone, either in the interior or on the boundary. We prove the statement by showing that if D is on the boundary then so is P , and that if D is in the interior then so is P .
First suppose D is on the boundary. Then there exists a nef cone generator N 0 such that D · N 0 = 0. Since P = D − N , this means P · N 0 = −N · N 0 . But since P and N are both in the Mori cone, P · N 0 ≥ 0 and N · N 0 ≥ 0. This is consistent only if P · N 0 = 0, so that P is on the boundary of the Mori cone.
Next suppose D is in the interior. We have P 2 = P · (D − N ) = P · D, since P · N = 0 by definition. But since D is in the interior of the Mori cone, it follows that P · D > 0, unless P is numerically equivalent and hence linearly equivalent to 0. However, P is linearly equivalent to 0 only when D lies on the boundary of the Mori cone, which cannot happen since D is big. Therefore P 2 > 0, which implies P is big, i.e. in the interior of the Mori cone.
This immediately gives the following proposition.  For integral divisors lying on the boundary of the Mori cone, the zeroth cohomology is in general not determined by the current framework of combining Zariski decomposition with vanishing theorems, and these require a separate discussion, which we will not attempt here. However, in the special case of integral divisors on the boundary whose positive part is trivial, there is the following simple result. Proof. This is immediate, since in this case the negative part of D is D itself, so its positive part is trivial.
Example: quartic hypersurface in P 3 with Picard number 3 In Ref. [39] it was shown that there exist K3 surfaces S constructed as smooth quartic surfaces in P 3 with Hence, for any divisor D not on the boundary of the Mori cone, the zeroth cohomology is given by the formulae in the table below. We also include as an additional line the case of divisors on faces of the Mori cone which project to the origin under the map D → P : The zeroth cohomology is undetermined on the remaining parts of the Mori cone boundary, which are Using this expression, the zeroth cohomology formulae in Equation (4.37) become

Example: Weierstrass model
We now discuss a K3 surface S realised as a Weierstrass fibration of an elliptic curve over P 1 . The example we take can be realised as a hypersurface in a three-dimensional toric variety, whose fan is given by a triangulation of the surface of the polytope shown in Figure 6. The fibration π : S → P 1 has a single zero-section σ. The Mori cone is generated by M 1 = π * (H) and is the pullback of the hyperplane class (point) on the P 1 base. In this basis, the intersection form is In this basis the dual nef cone is generated by N 1 = (1, 0) and N 2 = (2, 1). There is only one subset of {M 1 , M 2 } on which the intersection form is negative definite, which is {M 2 }. As such, apart from the nef cone, there is only one Zariski chamber, Σ 2 , obtained by extending the face N 2 R ≥0 along M 2 . The other face of the nef cone, N 1 R ≥0 is on the boundary of the effective cone, and is not covered by our present cohomology discussion. We then obtain the following formula for the zeroth cohomology of effective line bundles.
In fact in this present simple case of a Weierstrass K3 surface, it is straightforward to find formulae describing cohomology by using the Leray spectral sequence to lift those on the base P 1 , analogously to the discussion in Section 5.2 below for three-folds. In particular, we can then find the formula for the zeroth cohomology on the remaining region, M 1 R ≥0 , which we include here for completeness to complement the above table.
It is not a surprising fact that the formula along the M 1 R ≥0 is linear, rather than quadratic in k 1 . This comes in agreement with the holomorphic Morse inequalities (see Remark 2.2.20 in Ref. [27]), according to which on a projective variety X of dimension n, if D is a nef divisor, then for every integer q ∈ [0, n] one has h q (X, O X (mD)) ≤ O(m n−1 ).

Cohomology chambers on elliptic Calabi-Yau three-folds
In the previous section we obtained formulae for line bundle cohomology on surfaces. One immediate application is to lift these formulae to higher-dimensional manifolds which use these surfaces as building blocks. An obvious construction of this kind is to consider fibrations over the surfaces studied above. In these constructions, the lift of cohomologies can be computed straightforwardly through the Leray spectral sequence. A class of fibrations which are both simple and have many applications in string theory are elliptically fibered Calabi-Yau three-folds, and we study these in the present section. We will consider the simplest setting, in which the generic fibration is smooth, since in this case the lift by the Leray spectral sequence is straightforward 1 .

Elliptically fibered Calabi-Yau three-folds
In this section we provide a brief summary on the construction and properties of elliptically fibered Calabi-Yau three-folds. This is provided as a reminder and can be skipped by a reader familiar with this material.

Weierstrass models
An elliptically fibered manifold X consists of a fibration π : X → B of an elliptic curve over a base manifold B, with a section σ : B − → X that embeds the base into the total space.
We focus on elliptic fibrations which can be constructed with a Weierstrass model. Note that every elliptic fibration is birationally equivalent to a Weierstrass one. In a Weierstrass model, the elliptic curve E(b) over a point b ∈ B is described as a hypersurface in an ambient space. A useful choice for the ambient space, for reasons that will become clear in a moment, is the weighted complex projective space P 231 [x : y : z]. The elliptic curve is defined by a degree six polynomial, which can by coordinate redefinitions be written in the form Here f (b) and g(b) are parameters that define the elliptic curve E(b).
A fibration of the elliptic curve is inherited if the space P 231 is fibered over the base. In order to fiber P 231 over the base, the homogeneous coordinates x, y, z are taken as sections of powers of a line bundle L on the base: . From the Weierstrass equation this means the parameters f and g must vary over the base as sections f ∈ Γ(L 4 ), g ∈ Γ(L 6 ).
Upon fibering, different choices of the ambient space for the elliptic curve are not equivalent, but rather determine the existence of sections of the fibration. In particular, the choice of P 231 ensures the existence of a single section, given by z = 0, as one can verify. Note that P 231 is the Gorenstein Fano toric surface F 10 whose ray diagram is shown in Figure 10. Making a different choice among these gives a different structure of sections. See for example Table 2 of Ref. [40], or the earlier Ref. [41].
One can check using adjunction that for the resulting manifold to be Calabi-Yau, the defining line bundle L must be chosen to be the anti-canonical bundle of the base, L = K −1 B . This leads to the requirement that the anti-canonical bundle K −1 B must have sections, which constrains the possible choices for the base spaces. This condition is true on for example toric surfaces and generalised del Pezzo surfaces. This is also true for K3 surfaces, but in this case the fibration is trivial, and the three-fold is a product K3 × T 2 .

Smoothness
The elliptic curve E(b) described in Weierstrass form in Equation (5.1) can be singular, depending on the values of f (b) and g(b). In particular, one can check that the elliptic curve E(b) is singular if the discriminant In the fibration of the elliptic curve, the discriminant varies over the base as a section ∆ ∈ Γ L 12 , and the elliptic curve is singular over the zero locus of this section. In the case of an elliptic fibration giving rise to a Calabi-Yau three-fold, ∆ ∈ Γ K −12

B
. Importantly however, the elliptic fibration is often smooth despite the singular elliptic fibers. In particular, only severe singularities of the fiber give rise to singularities of the fibration, with the severity of the fiber singularity essentially determined by the vanishing orders of f , g, and ∆. The singularities in the fibration can occur over loci in the base of various codimension.
In the case of singularities over codimension one loci in the base, the types of singularity in the fibration are summarised in Table 1, which is reproduced from Ref. [42]. Singularities over loci of higher codimension are more complicated. More details can be found for example in Ref. [43] and references contained therein.
ord(f ) ord(g) ord(∆) fiber-type singularity-type Table 1: Classification of singularities from fiber degeneracy over codimension one loci in the base.
In the case of a Weierstrass model with L = −K B2 , chosen to give rise to a Calabi-Yau three-fold X 3 , a necessary condition for a smooth fibration to exist is that the base B 2 does not contain curves C of self- To see this, first note that the self-intersection of a curve C on a surface B 2 is related to the genus g C of the curve by (K B2 + C) · C = 2g C − 2, so that Since g C ≥ 0, the right-hand side is negative if C · C < −2. But this negative intersection indicates the presence of C in every element of the complete linear system |−K B2 |, by the argument in Section 2.1 above.
The amount of C detected is even greater for multiples |−mK B2 |, and specifically any element takes the form One can check that if C · C < −2, then in the cases m = 4 and m = 6, the coefficient on the right is at least two. That is, any sections f ∈ Γ(K −4 B2 ) and g ∈ Γ(K −6 B2 ) must have vanishing orders of at least two over the curve C. Glancing at Table 1, we see that in this case the three-fold is singular over this locus.
This condition constrains the set of base spaces which can give rise to smooth fibrations. This condition holds on the generalised del Pezzo surfaces. Among the toric surfaces, the only cases satisfying this condition are the 16 cases which are also generalised del Pezzo -on any other toric surfaces the Weierstrass model will not be generically smooth.

Properties of the three-fold
When the Weierstrass model is smooth, the properties of the elliptically fibered manifold X follow straightforwardly from those of the base B, as we now discuss.
Associated to the projection map π : X → B is a pullback map π * , which lifts bundles on the base B to bundles on the total space X, and lifts divisors to divisors. This gives an injection π * : div(B) → div(X). In addition to the pullback divisors, there is also the section 2 σ. In particular, a basis of the Picard group on X is given by {σ , π * (D i )}, where {D 1 , D 2 , . . .} is a basis of the Picard group on the base. We write D 0 ≡ σ and D i ≡ π * (D i ). A line bundle L on X can hence be specified by an integer n along the section, and a pullback of a line bundle L on the base, The cone Eff(X) of effective divisors on X is trivially related to the Mori cone M(B) on the base, This is clear for example from the Leray spectral sequence below.
As well as the pullback there is the inclusion σ : B − → X. In the case of a three-fold X 3 over a complex surface B 2 , this sends divisors on the base to curves in the total space. These give rise to curve classes σ(D i ), which together with the fiber class F give a basis of curves {C 0 , C i } ≡ {F , σ(D i )}. If the anti-canonical bundle of the base is nef, as in our cases, then like the effective cone the Mori cone M(X 3 ) of the three-fold is trivially related to that of the base (see for example the argument in Ref. [44]), The intesections between the above curves and divisors are σ π * (D) The nef cone is the dual of the Mori cone with respect to these intersections. The triple intersection numbers are (5.10)

Lifting base cohomologies
The Leray spectral sequence On a fibration π : X → B, the cohomology of a bundle V on the total space X can be computed in terms of cohomologies on the base B. In particular, the relevant objects on the base are the cohomologies of the higher direct images R q π * V of the bundle, where q = 0, 1, . . ., which are in general sheaves. Note that R 0 π * V ≡ π * V is simply the pushforward. While we will not compute them explicitly, we note that R q π * V is equal to the sheaf associated to the presheaf where U are the open sets on the base, and we refer the reader to Chapter III.8 of Ref. [46] for details.
The relation between the cohomology of V and the cohomologies of the higher direct images is provided by the Leray spectral sequence. The definition of the Leray spectral sequence begins with the definition of the second page E p,q 2 as E p,q 2 = H p (B, R q π * V ) . .

(5.13)
When this iterative process stabilises, so that E p,q r = E p,q n ∀r ≥ n for some n, we write E p,q n ≡ E p,q ∞ , and the cohomologies of the bundle V on the total space X are given by (5.14)

Higher direct images
We recall from Section 5.1 above that on a generically smooth Weierstrass Calabi-Yau three-fold X 3 , a line bundle L can be written uniquely as where L is a line bundle on the base B 2 , and σ is the section of the fibration. For such a tensor product of bundles, the direct and higher direct images can be simplified by use of the projection formula (see for example Chapter III.8 in Ref. [46]), Hence, all higher direct images are determined by knowledge of those of O X3 (nσ). Recall also that the higher direct images vanish trivially unless i = 0, 1, so we require only π * O X3 (nσ) and R 1 π * O X3 (nσ), for all n. These have been worked out elsewhere (see for example Appendix C of Ref. [47]), and we collect the results below 3 .

(5.17)
Here 0 is the rank zero bundle, which gives 0 upon taking the tensor product with any other bundle, and for which all cohomologies are trivial.
For the below, it will be important that R i π * L is always a sum of line bundles. As the ith cohomology of a sum of line bundles is the sum of the ith cohomologies of the line bundles, it is straightforward to determine H i (B 2 , R j π * L) and hence the second page E p,q 2 given knowledge of line bundle cohomology on the base.

Degeneration of the Leray spectral sequence
In the present context, the Leray spectral sequence simplifies dramatically.
Proposition 5.1. On a generically smooth Weierstrass Calabi-Yau three-fold X 3 over a smooth complex projective base B 2 , the Leray spectral sequence for a line bundle L on X 3 degenerates at the second page, i.e.
Proof. First note from the dimension of the base the trivial vanishings H p B 2 , R q π * L = 0 unless p = 0, 1, 2.
Additionally, glancing at the expression in Equation (5.11), from the dimension of the fiber R q π * L = 0 unless q = 0, 1. Inserting these into Equation (5.13), one finds that every element in the third page is trivially related to the second page, E p,q 3 = E p,q 2 , except for two cases: E 0,1 3 and E 2,0 3 . The relations for these remaining terms are .
It is easy to check that if either one of E 0,1 2 and E 2,0 2 is zero, then these final relations too are trivial, i.e.  .17), we see immediately that either π * L = 0 or R 1 π * L = 0. So in these cases either E 0,1 2 = 0 or E 2,0 2 = 0. The only remaining case to check is n = 0. Here we have where in the final equality we used Serre duality. The first term is non-zero only when L ⊗ K B2 is in the effective cone, while the second is non-zero only when L * ⊗ K B2 is in the effective cone. For both to be non-zero, the Mori cone M must overlap its reflection −M through the origin after being shifted by −2K B2 .
But −K B2 is effective for the Weierstrass model to exist. If the Mori cone is strongly convex, an effective shift will separate M from −M without overlap. This holds on a projective surface, which proves the result.
More explicitly, the cohomology of L on the three-fold X 3 is given by the following relations

Explicit expressions
Given the relations (5.21), it only remains to plug in the expressions from Equation (5.17) for the higher direct images. Note it is sufficient to determine only the zeroth and first cohomologies, since the second and third are trivially related to these by Serre duality. Again writing L = O X3 (nσ) ⊗ π * L, one has h 0 (X 3 , L) = h 0 (B 2 , π * L) and h 1 (X 3 , L) = h 0 (B 2 , R 1 π * L) + h 1 (B 2 , π * L) , for n < 0 for n = 0, 1 for n > 0 for n < 0 for n = 0, 1 for n ≥ 2 When line bundle cohomologies on the base are described by simple formulae along the lines of Section 4, the above give expressions for line bundle cohomology on the three-fold. These expressions provide a simple and fast method to determine any line bundle cohomology. We note that, while these expressions can be guessed from raw data, for example by equation fitting or machine learning methods, as pursued in Refs. [7,8,10], the present approach has the advantages of giving a proof, and the knowledge there are no missed edge-cases.

Example
On a given base B 2 , we expect the expressions in Equation (5.22) to simplify substantially, to give compact regions and formulae describing all line bundle cohomology on the three-fold, analogous to the surface case.
Here we consider the simplest example, of a Weierstrass three-fold with base a projective plane, B 2 = P 2 .

Properties of the three-fold
We recall the discussion in Section 5. In the following discussion, we write L = O X3 nσ+kπ * (H) for a general line bundle L on the three-fold X 3 .
Additionally, we will use the shorthand notation O X3 (n, k) for such a line bundle.

Zeroth cohomology
On the projective plane P 2 , the zeroth cohomology of a line bundle is given by the Bott formula as the binomial and h 0 (X 3 , L) = 0 otherwise. Here θ( · ) is a step function, equal to one for x ≥ 0 and zero otherwise.
We plot the numerical values in Figure 8. From the expressions or the figure it is clear the effective cone is simply the positive quadrant, n, k ≥ 0. This cone further naturally splits into two regions (at least for n ≥ 2): when k ≥ 3n, all of the step functions in the sum are satisfied, while when k < 3n, the sum is cut off by the step functions. In the latter region, notably there will no dependence on n.
In the k ≥ 3n region of the effective cone, the above sum that appears for n ≥ 2 has unit coefficients,  Note that this expression happens to be zero when n = 1 (but not when n = 0), so we may include it in the n = 1 case as well. The full expression for h 0 (X 3 , L) in the n > 0 , k ≥ 3n region is then There remains only the n = 0 boundary. Here the cohomologies are given by k+2 2 = 1 2 (1 + k)(2 + k). When k < 3n, some step functions are not satisfied. In particular, the upper limit on the sum becomes k 3 . Since the only appearance of n in the sum is in the limit, we can simply make the replacement n → k 3 in Equation (5.29). From the lower limit, we should only include the sum when k 3 ≥ 2. However one can check the sum is anyway zero when n = 1, so the replacement is still correct when k 3 = 1, i.e. k = 3, 4, 5. Additionally, one can check that = 0 when k = 1, 2 (but not when k = 0). Hence the replacement n → k 3 in Equation (5.29) gives the correct expression for h 0 in the region k < 3n, n > 0, but not on the boundary k = 0, n ≥ 0. On the boundary h 0 (X 3 , L) is given simply by the non-sum term k+2 2 | k=0 = 1.
Glancing at the properties of the three-fold above, we see that the region n ≥ 0, k ≥ 3n precisely corresponds to the nef cone. Hence by Kodaira vanishing, we know that in the interior of this cone the zeroth cohomology must be given by the index. We see that this is indeed borne out by our results, from comparison of Equation (5.30) with the expression for the index in Equation (5.27).
We can then compactly write all the expressions in terms of the index as follows. We depict the regions where the formulae apply in Figure 8.

First cohomology
On the projective plane, all first cohomologies vanish. Hence the general expression for first cohomologies on the three-fold in Equation (5.21) simplifies, leaving only one term in the direct sum, Using the Bott formula for cohomologies on the projective plane, the expression in Equation (5.22) becomes and h 0 (X 3 , L) = 0 otherwise.
We plot the numerical values in Figure 9. The step functions in the sum are θ(k + 3) , θ(k + 6) , . . . , θ(k + 3(−n − 1)), which gives in addition to the line n = 0 the other boundary of the non-zero region as k = 3(n + 1). This cone then naturally splits into two regions (at least for n ≤ −2): when k ≥ −3, all step functions in the sum are satisfied, while when k < −3 the lower limit is raised by the step functions.
In the k ≥ −3 region, the sum that appears for n ≤ −2 has unit coefficients, giving This excludes the origin, on which one has h 1 O(0, 0) = 0.
So a more natural boundary is k ≥ 3n. We also note that this expression is manifestly invariant under the shift (k, n) → (k + 3, n + 1), which is obvious in the data in Figure 9. Since h 0 (X 3 , L) = h 2 (X 3 , L) = 0 in this region, the expresssion can also be written as where in the second equality we used Serre duality and the expression for h 0 (X 3 , L) from above. Finally, on the boundary n ≤ −1, k = 0, the expression is just given by Equation (5.34) in the case k = 0. We can also note that by Serre duality and the expression for h 0 (X 3 , L) above this is . Below we collect the above results using the index expressions, and depict the regions in Figure 9.

Insight into general structure
Above we have used the Leray spectral sequence to determine the regions and formulae describing all line bundle cohomology on a simple Weierstrass Calabi-Yau three-fold. However, it is clear that this route does not reflect the natural structure of the three-fold cohomology: the naive regions from the lift had to be redrawn, and several special cases had to be taken into account. Hence, while this method is viable for any given smooth Weierstrass model, it does not appear to provide insight into the structure of line bundle cohomology on Calabi-Yau three-folds more generally.

Conclusions
The The information defining a toric variety can be encoded in a fan, which is a collection of cones in R n whose generators correspond to integral points. The complex dimension of the corresponding toric variety equals the real dimension n of the fan. The one-dimensional cones are rays. For the example of P 2 , the fan is shown in the first diagram in Figure 10. There are three one-dimensional cones and three two-dimensional cones.
The fan determines the scaling equivalences and allowed coordinate vanishings as follows. Firstly, the generators v i of the rays are in one-to-one correspondence with the complex coordinates x i . A scaling identification between these coordinates corresponds to a linear dependency relation between the generators. Specifically, if then there is an identification (x 1 , . . . , x m ) ∼ (λ c1 x 1 , . . . , λ cm x m ) ∀λ ∈ C * . In the example of P 2 , one sees from Figure 10 that the three generators simply add to zero. One typically writes the weights c i in a table, called the 'weight system'.
The allowed simultaneous vanishings of the coordinates are determined by the full structure of the fan: if the generators corresponding to a number of coordinates share a common cone, then the coordinates are allowed to simultaneously vanish, otherwise they are not. In the example of P 2 , any two generators share a common cone, but all three do not. One typically writes the allowed vanishings in the 'Stanley-Reisner ideal'.

Divisors: intersections and linear equivalences
To each complex coordinate x i one can associate a toric divisor D i corresponding to the vanishing locus of x i .
Hence there are natural divisors corresponding to the rays in the fan. One particularly simple aspect of toric varieties is that the anti-canonical divisor class is given by the sum of the toric divisor classes, The toric divisors are not all linearly inequivalent. In particular, for any vector u in the space of the fan, taking the dot product with all generators v i in the toric diagram gives a linear equivalence relation, It is clear that there are as many independent such relations as the dimension of the fan. Hence on a toric variety, the dimension h 1,1 of the Picard group equals the number of rays minus the dimension of the fan.
The question of whether distinct toric divisors mutually intersect is the same as the question of whether the corresponding coordinates can simultaneously vanish. This is hence determined by whether the corresponding rays share a cone, as discussed above. Self-intersections are then determined by rewriting a toric divisor in terms of others using a linear equivalence relation, and using mutual intersections.
The fan of a toric surface is two-dimensional. In this case, the mutual intersections can be found straightforwardly: if two toric rays are neighbours, then the corresponding toric divisors have a mutual intersection of 1, otherwise it is 0. Self-intersections are determined as usual via the linear equivalence relations.

Surface case: Mori cone and irreducible negative self-intersection divisors
We are particularly interested in the case of toric surfaces, on which divisors and curves can be identified. In the main text we are interested in the Mori cone of these surfaces, and the irreducible, negative self-intersection divisors.
On a compact toric surface, it is straightforward to determine the Mori cone. The Mori cone generators are a subset of the toric divisor classes (see for example Theorem 6.3.20 of Ref. [23]). Since a Mori cone generator has non-positive self-intersection, these must be a subset of those with non-positive self-intersection. This makes it straightforward to determine the generators from the ray diagram: take the toric divisor classes with non-positive self-intersection, and throw away any that can be expressed as a non-negative sum of the others.
The nef cone follows as the dual of the Mori cone.
In Zariski decomposition, one is interested in the set of irreducible, negative self-intersection divisors. On a compact toric surface, these are precisely the toric divisor classes with negative self-intersection. To see this, note again that every irreducible, negative self-intersection divisor class is a generator of the Mori cone, and that the Mori cone is generated by toric divisor classes.

A.2 Toric description of Weierstrass models
Since in the simple Weierstrass models utilised in the main text the elliptic fiber is described by a polynomial in the coordinates of the weighted projective plane, which is toric, when the base is also toric it is easy to give a description of the three-fold as a hypersurface in a toric ambient space. In this section we provide a summary of this construction.
In a trivial product B 2 × P (2,3,1) , we would lift the rays of the base-space analogously, putting zeroes in the final two entries. The non-trivial fibering is achieved by using the non-zero entries for any ray (b 1 , b 2 ) of the base-space fan. The projection of the four-fold to the base corresponds to sending the last two coordinates to zero.
By taking the dot product of the above ray vectors with the vectors (0, 0, 1, 0) and (0, 0, 0, 1), we get the divisor equivalences where the sums run over all rays of the base fan. These equivalences correspond to the statement that the coordinates x and y are sections of K −2 B2 and K −3 B2 respectively. The coordinates of these rays determine the weight system, which tabulates the equivalences of coordinates under the various projective scalings. The rows correspond to linear combinations of the rays that sum to zero.  Here b i are the coordinates on the base, and for each a, (s a i ) is a row in the weight system of the base. We have also included on the right the sum of each row, which is the scaling of sections of the anti-canonical bundle of the toric four-fold.
In addition to the rays, one must specify the triangulation of the resulting polytope, i.e. the top-dimensional cones of the fan. To describe a fiber bundle, the triangulation should be taken to be simply the product triangulation, i.e. the same triangulation as for the fan of the product space B 2 × P (2,3,1) .
The elliptic Calabi-Yau is then described as a hypersurface inside this toric ambient space by a Weierstrass equation. One can check that the Weierstrass polynomial is a section of the anti-canonical bundle of the ambient space, so that the three-fold is indeed Calabi-Yau. This corresponds to each monomial having the scaling of the sum of all the columns in the weight system above.

Example
The simplest toric base space is P 2 . Writing u, v, and w for the homogeneous coordinates, the rays of the The top-dimensional cones have generators given by the product cones. Schematically, { c 1 , c 2 | c 1 ∈ Cones 2 (P 2 ) and c 2 ∈ Cones 2 (P (2,3,1) )} , where Cones 2 (P 2 ) = { u , v , u , w , v , w } and Cones 2 (P (2,3,1) ) = { x , y , x , z , y , z } . (A.9) The polynomials f and g in the Weierstrass equation are in this case homogeneous polynomials of degree 12 and 18 respectively in u, v, and w.

B The sixteen reflexive polytopes
A set of toric surfaces with many applications in string theory are the 16 Gorenstein Fano toric surfaces.
We also frequently take these as examples in the main text. In this appendix we collect for each of these surfaces the properties required to determine the Zariski chambers. These properties can be straightforwardly determined using the methods in described in Appendix A.1.
The ray diagrams of the 16 Gorenstein Fano toric surfaces are given by the 16 reflexive polytopes in Figure 10. The rank ρ(S) of the Picard group in each case is as follows.
Several of these spaces are isomorphic to Hirzebruch or ordinary del Pezzo surfaces. Specifically and, further, all of the others can be seen as blow-ups of Hirzebruch surfaces. Below we skip the spaces corresponding to the first two reflexive polytopes, which are isomorphic to P 2 and P 1 × P 1 and so are trivial. The ray diagram for F 3 has 4 rays, hence its Picard number is 2. We can use 2 of the toric divisors as a basis for the Picard lattice, and the remaining toric divisors can be expressed in this basis. The choice we use is In this basis the intersection form and the anti-canonical divisor are given by The Mori cone generators M i and the nef cone generators N j , which satisfy M i · N j = δ ij , are given by There is a single irreducible rigid divisor, namely M 2 .

Data for F 4
The ray diagram for F 4 has 4 rays, hence its Picard number is 2. We can use 2 of the toric divisors as a basis for the Picard lattice, and the remaining toric divisors can be expressed in this basis. The choice we use is In this basis the intersection form and the anti-canonical divisor are given by The Mori cone generators M i and the nef cone generators N j , which satisfy M i · N j = δ ij , are given by There is a single irreducible rigid divisor, namely M 1 .

Data for F 5
The ray diagram for F 5 has 5 rays, hence its Picard number is 3. We can use 3 of the toric divisors as a basis for the Picard lattice, and the remaining toric divisors can be expressed in this basis. The choice we use is In this basis the intersection form and the anti-canonical divisor are given by The Mori cone generators M i and the nef cone generators N j , which satisfy M i · N j = δ ij , are given by The irreducible rigid divisors D − k are just the Mori cone generators.

Data for F 6
The ray diagram for F 6 has 5 rays, hence its Picard number is 3. We can use 3 of the toric divisors as a basis for the Picard lattice, and the remaining toric divisors can be expressed in this basis. The choice we use is In this basis the intersection form and the anti-canonical divisor are given by The Mori cone generators M i and the nef cone generators N j , which satisfy M i · N j = δ ij , are given by The irreducible rigid divisors D − k are just the Mori cone generators.
Data for F 7 The ray diagram for F 7 has 6 rays, hence its Picard number is 4. We can use 4 of the toric divisors as a basis for the Picard lattice, and the remaining toric divisors can be expressed in this basis. The choice we use is The irreducible rigid divisors D − k are just the Mori cone generators.

Data for F 9
The ray diagram for F 9 has 6 rays, hence its Picard number is 4. We can use 4 of the toric divisors as a basis for the Picard lattice, and the remaining toric divisors can be expressed in this basis. The choice we use is The irreducible rigid divisors D − k are just the Mori cone generators.

Data for F 10
The ray diagram for F 10 has 6 rays, hence its Picard number is 4. We can use 4 of the toric divisors as a basis for the Picard lattice, and the remaining toric divisors can be expressed in this basis. The choice we use is The irreducible rigid divisors D − k are just the Mori cone generators.

Data for F 11
The ray diagram for F 11 has 7 rays, hence its Picard number is 5. We can use 5 of the toric divisors as a basis for the Picard lattice, and the remaining toric divisors can be expressed in this basis. The choice we use is The irreducible rigid divisors D − k are just the Mori cone generators.

Data for F 12
The ray diagram for F 12 has 7 rays, hence its Picard number is 5. We can use 5 of the toric divisors as a basis for the Picard lattice, and the remaining toric divisors can be expressed in this basis. The choice we use is The irreducible rigid divisors D − k are just the Mori cone generators.

Data for F 13
The ray diagram for F 13 has 8 rays, hence its Picard number is 6. We can use 6 of the toric divisors as a basis for the Picard lattice, and the remaining toric divisors can be expressed in this basis. The choice we use is The irreducible rigid divisors D − k are just the Mori cone generators.

Data for F 14
The ray diagram for F 14 has 8 rays, hence its Picard number is 6. We can use 6 of the toric divisors as a basis for the Picard lattice, and the remaining toric divisors can be expressed in this basis. The choice we use is The irreducible rigid divisors D − k are just the Mori cone generators.

Data for F 15
The ray diagram for F 15 has 8 rays, hence its Picard number is 6. We can use 6 of the toric divisors as a basis for the Picard lattice, and the remaining toric divisors can be expressed in this basis. The choice we use is The irreducible rigid divisors D − k are just the Mori cone generators.

Data for F 16
The ray diagram for F 16 has 9 rays, hence its Picard number is 7. We can use 7 of the toric divisors as a basis for the Picard lattice, and the remaining toric divisors can be expressed in this basis. The choice we use is The irreducible rigid divisors D − k are just the Mori cone generators.