Determinación de la velocidad mínima de fluidización

In this section we briefly discuss the geometrical properties of the almost Fano twofolds B=_Fk, dP_n and the toric surfaces. The discussion in this section is supplemented by the explicit computations of the K¨ahler cones ofdP_n in appendix 1.5 and the summary of the key geometric data of _Fk, dPn in Appendix 1.6, which is critical for the proof in Section 1.3.

8_{We allow here for rational coefficientsm}α

I,biin the expansion ofFα,π∗c1that are in the integral homol- ogyH(1,1)

(X,Z)in order to account for the possibility of K¨ahler generatorsDIthat only span a sublattice of

Hirzebruch Surfaces

The Hirzebruch surfaces_Fk areP1-bundles overP1 of the formFk=P(O ⊕ O(k)). There is an infinite family of such bundles for every positivek∈_Z≥0.

The isolated section of this bundle, S, and the fiber F are effective curves generating the Mori cone and spanning the entire second homology

H₂(_Fk,_Z) = ⟨S,F⟩. (1.14)

Their intersections read

S2= −k, S⋅F=1, F2=0. (1.15)

From this we deduce that the generatorsDi, i=1,2, of the K¨ahler cone, which are defined to be dual to the generators in (1.14), read

D₁=F, D₂=S+kF. (1.16)

The Chern classes on_Fk read

c₁(_Fk) =2S+ (2+k)F= (2−k)D₁+2D₂, c₂(_Fk) =4, (1.17)

which implies that the vectorbin (1.9) isb= (2−k,2)T.

Using (1.15), we compute the triple intersections in (1.7), in particular (1.8), as

C= ⎛ ⎜ ⎜ ⎝ 0 1 1 k ⎞ ⎟ ⎟ ⎠ , K₀₀₁=2, K₀₀₂=2+k, K₀₀₀=8, (1.18)

from which the curvature terms in (1.13) immediately follow as

T₀=92, T1=24, T2=24+12k (1.19)

We emphasize that_Fk by means of (1.17) is Fano fork<2 and almost Fano fork=2, since the coefficientb₁=2−k≥0. The general elliptic Calabi-Yau fibrationX overF_kwith k=0,1,2 is smooth and developsI₃-singularities fork=3 up toII∗-singularities fork=12, before terminal singularities occur for k>12 [136]. Thus, we focus on the Hirzebruch surfaces withk=0,1,2.

Del Pezzo Surfaces

The Fano del Pezzo surfacesdPnare the blow-up ofP2at up to eight generic points.9 Their second homology group is spanned by the pullback of the hyperplane on _P2_,

denoted byH, and the classes of the exceptional divisors, denoted asE_i,i=1, . . . ,n,

H₂(dPn,Z) = ⟨H,Ei=1,...,n⟩. (1.20)

The intersections of these classes read

H2=1, H⋅E_i=0, E_i⋅E_j= −δi j. (1.21)

The Chern classes ondP_nread

c₁(dP_n) =3H− n ∑ i=1

E_i, c₂(dP_n) =3+n. (1.22)

9_{See [94, 57] for recent computations of refined BPS invariants on del Pezzo surfaces as well as their} interpretation in M-/F-theory.

The Mori cone ofdP_nforn>1 is spanned by the curvesΣobeying [58, 59]

Σ2= −1, Σ⋅ [K_dP−1

n] =1, (1.23)

where[K_dP−1

n]is the anti-canonical divisor indPn, which is dual toc1(dPn). By adjunction,

we see that the curves obeying (1.23) obey the necessary condition for being _P1’s. By solving the conditions (1.23) with the ansatz a₀H+ ∑n_i=1aiEi for a0,ai∈_Z, we obtain a cone that is simplicial, i.e. generated by h(1,1,)(B) =1+n generators, for n=0,1,2 and non-simplicial forn>2. The number of generators, beginning withdP2, furnish irreducible

representations ofA₁, A₁×A₂,A₄, D₅, E_n, forn=6,7,8, which concretely are3, 2⊗3, 10, 16,27,56,248.10 _{For the simplicial cases the Mori cone reads}

P2 ∶ ⟨H⟩, dP1 ∶ ⟨E1,H−E1⟩, dP2 ∶ ⟨E1,E2,H−E1−E2⟩ (1.24)

and we refer to appendix 1.5 for more details on the non-simplicial cases.

Consequently, also the K¨ahler cones of the dP_n, which are the dual of the Mori cones defined by (1.23), are non-simplicial for n>2. The K¨ahler cone is spanned by rational curvesΣobeying

Σ2=0, Σ⋅ [K_dP−1_n] =2 or Σ2=1, Σ⋅ [K_dP−1_n] =3, (1.25)

which again implies by adjunction that Σ=_P1. The solutions over the integers of these conditions yield the generators of the K¨ahler cone ofdP_nwhich again follow the represen- tation theory of the above mentioned Lie algebras. The number of generators, starting with dP0, is 1, 2, 3, 5, 10, 26, 99, 702 and 19440, see appendix 1.5. In the simplicial cases, the

10_{The genuine roots inH}

2(dP_n)are the−2-curves orthogonal to[K_dP−1

n], i.e.αi=Ei−Ei+1,i=1, . . . ,n−1,

K¨ahler cone generators read

P2∶D1=H, dP1∶D1=H−E1,D2=H, dP2∶D1=H−E1,D2=H−E2,D3=H (1.26)

Generically, forn≥2 the vectorc₁(dP_n)is the center both of the K¨ahler and Mori cone. This implies that for all del Pezzo surfaces, the coefficients b_i are positive. For the sim- plicial K¨ahler cones, this can be computed explicitly. For the non-simplicial cases we will argue in appendix 1.5, that a covering of the K¨ahler cone by simplicial subcones, i.e. sub- cones with h(1,1) _{generators, with all b}

i≥0 always exists. We note that for all dPn, the defining property of the K¨ahler cone (1.25), together with (1.7), implies the intersections

K_00i=2,3, K₀₀₀=9−n. (1.27)

In addition, by explicit computations we check in general that allC_{i j} ≥0 for all pairs of K¨ahler cone generators. The intersections (1.27) together with (1.21), (1.22) further imply that the curvature terms in (1.13) read

T0=102−10n, Ti=24,36 (1.28)

For the three simplicial cases of_P2_,dP

1anddP2, we compute the matrices (1.8) in the basis

(1.26) as C_P2 =1, C_dP1= ⎛ ⎜ ⎜ ⎝ 0 1 1 1 ⎞ ⎟ ⎟ ⎠ , C_dP2= ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 0 1 1 1 0 1 1 1 1 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ . (1.29)

We emphasize that the del PezzosdPnby means of (1.27) are Fano forn<9 and almost Fano forn=9, sincec2₁=0. The surface dP9 is the rational elliptic surface. Its Mori cone

cone is infinite dimensional. We will only consider the Fano del Pezzo surfacesdP_n,n<9.

Toric Surfaces from Reflexive Polytopes

Toric surfaces obtained from fine star triangulations of reflexive polytopes are smooth al- most Fano twofolds.11 There are 16 such polytopes in two dimensions, which are displayed in Figure 1.1.

A number of these twofolds are simply toric descriptions of previously described sur- faces. Specifically, these are_P2_,dP

1,dP2,dP3,F0andF2which are described by polytopes

1, 3, 5, 7, 2 and 4, respectively. From the form of some of the other polytopes it is clear that they can be obtained from_P2_,dP

1,dP2,or dP3 via toric blow-up. For example, reflecting

polytope 7 through the vertical axis going through its center and performing a toric blow- up associated to the point(−1,1), one obtains polytope 12. Thus, the smooth Fano surface associated to polytope 12 is a toric realization ofdP₄at a non-generic point in its complex structure moduli space.

The toric varieties associated to all these 16 reflexive polytopes can be constructed explicitly using the software package Sage [61]. The intersections (1.7), (1.8) are readily constructed in a given fine star triangulation and the K¨ahler cone can be obtained. We summarize the geometric data necessary for the computation of the bounds derived below in the proof in Appendix 1.6.

11_{See the recent [60] for a systematic study of the quantum geometry of the elliptically fibered Calabi-Yau} manifolds over these bases.

13 14 15 16

9 10 11 12

5 6 7 8

1 2 3 4

Figure 1.1: The sixteen two-dimensional reflexive polytopes which define the almost Fano toric surfaces via their fine star triangulations.