On a combinatorial description of the Gorenstein index for varieties with torus action
Philipp Iber Eva Reinert Milena Wrobel



=1







              
              



 




		

		calc
		3d


patterns,matrix,arrows,shapes














C>c<L>l<R>r<theoremTheorem[section]
introthmTheoremlemma[theorem]Lemmaproposition[theorem]Propositioncorollary[theorem]Corollarycorollaryofproof[theorem]Corollary of proofconjecture[theorem]Conjecturedefinition*introintroconstrConstructiondefinition[theorem]Definitionconstruction[theorem]Construction*introdefDefinitionexample[theorem]Exampleremark[theorem]Remarksetting[theorem]Settingreminder[theorem]Reminderproblem[theorem]Problemnotation[theorem]NotationToDo[theorem]ToDointroexExampleequationtheoremℍ𝔾𝕃𝔽ℂ𝕂𝕋ℤℝ𝕊ℕℚℙ𝕏𝒜ℬ𝒞ℰℱ𝒢𝒦ℳ𝒪ℛ#1⟨ #1 ⟩1//reg




7045On a combinatorial description of the Gorenstein Index]On a combinatorial description of the Gorenstein Index for varieties with torus actionPhilipp Iber, Eva Reinert, Milena Wrobel]Philipp Iber, Eva Reinert, Milena WrobelInstitut für Mathematik, Universität Oldenburg,
26111 Oldenburg, Germanyphilipp.iber@uni-oldenburg.deInstitut für Mathematik, Universität Oldenburg,
26111 Oldenburg, Germanyeva.reinert1@uni-oldenburg.deInstitut für Mathematik, Universität Oldenburg,
26111 Oldenburg, Germanymilena.wrobel@uni-oldenburg.de[2010]14M25, 14J45, 52B20, 14L30abbrv§ ABSTRACT 
The anticanonical complex is a combinatorial tool that was invented to extend the features of the Fano polytope from toric geometry to wider classes of varieties.
In this note we show that the Gorenstein index of Fano varieties with torus action of complexity one (and even more general of the so-called general arrangement varieties) can be read off its anticanonical complex in terms of lattice distances in full analogy to the toric Fano polytope. As an application we give concrete bounds on the defining data
of almost homogeneous Fano threefolds of Picard number one having a reductive automorphism group with two-dimensional maximal torus depending on their Gorenstein index.
[
    [
    
=====



§ INTRODUCTION

The main objective of this article is to contribute to the development of combinatorial methods for the study of geometric properties of Fano varieties. 
The model case is toric geometry. Here we have the well-known one-to-one correspondence between toric Fano varieties X and the so-called Fano polytopesA_X. 
These polytopes allow to describe several algebraic and geometric invariants of the corresponding toric varieties in a purely combinatorial manner. One of these invariants is the Gorenstein index, that is the smallest positive integer ι_X such that ι_X-times the canonical divisor 𝒦_X of X is Cartier. In the toric case, this invariant is encoded in the lattice distances of the facets of the Fano polytope, see <cit.>. This fact has been used by several authors to contribute to the classification of toric Fano varieties of low Gorenstein index; see con, kasNil,BatBo,Ath,BrunsRoemer,BaNill.

The purpose of this note is to generalize this combinatorial Gorenstein criterion 
to Fano varieties X with an effective action of an algebraic torus  of higher complexity, where the latter means that the difference dim(X) - dim() is greater or equal to one. More precisely we consider Fano general arrangement varieties as introduced in <cit.>. These are varieties
X coming with a torus action of arbitrary complexity c that gives rise to a specific rational quotient X ^c,
the so called maximal orbit quotient, whose critical values form a general hyperplane arrangement.
Note that this class comprises i.a. all Fano varieties with torus action of complexity one as well as all toric varieties.

By realizing the Fano general arrangement varieties X as subvarieties of toric varieties Z, we
can replace the toric Fano polytope with a polyhedral complex, the so-called anticanonical complex𝒜_X, which is supported on the tropical variety of X ⊆ Z. 


[tdplot_main_coords,
			edge/.style=,
			leaf0/.style=,
			leaf1/.style=,
			leaf2/.style=,
			leaf0e/.style=,
			leaf1e/.style=,
			leaf2e/.style=,
			le/.style=,
			axis/.style=,
			scale=.45]
	 (o) at (0,0,0);
	 (leaf0m) at (-,-,);
	 (leaf0p) at (-,-,);
	 (leaf0vm) at (,,);
	 (leaf0vp) at (,,);
	 (linm) at (0,0,);
	 (linp) at (0,0,);
	 (leaf1m) at (,0,);
	 (leaf1p) at (,0,);
	 (leaf1vm) at (-,0,);
	 (leaf1vp) at (-,0,);
	 (leaf2m) at (0,,);
	 (leaf2p) at (0,,);
	 (leaf2vm) at (0,-,);
	 (leaf2vp) at (0,-,);
	
	 (v01) at (-1,-1,-2);
	 (v02) at (-2,-2,-1);
	 (v1) at (2,0,1);
	 (v2) at (0,3,2);
	 (e1) at (0,0,2);
	 (e2) at (0,0,-1);
	
	
	
	
	
	
	[edge,leaf2e] (o) – (v2);
	[edge,leaf2e] (v2) – (e1);
	[edge,leaf2e] (v2) – (e2);
	[leaf2e,opacity=.2] (e1) – (e2) – (v2) – cycle;
	


	
	[leaf2e] (v2) circle (2pt);
	
	
	
	
	[edge,leaf1e] (o) – (v1);
	[edge,leaf1e] (v1) – (e1);
	[edge,leaf1e] (v1) – (e2);
	[leaf1e,opacity=.2] (e1) – (e2) – (v1) – cycle;
	


	
	[] (v1) circle (2pt);  
	
	
	
	
	
	[edge,leaf0e] (o) – (v01);
	[edge,leaf0e] (o) – (v02);
	[edge,leaf0e] (v02) – (e1);
	[edge,leaf0e] (v01) – (e2);
	[edge,leaf0e] (v01) – (v02);
	[leaf0e,opacity=.2] (e1) – (o) – (v02) – cycle;
	[leaf0e,opacity=.2] (v01) – (o) – (v02) – cycle;
	[leaf0e,opacity=.2] (e2) – (o) – (v01) – cycle;
	



	
	[] (v01) circle (2pt);
	[] (v02) circle (2pt);
	
	
	
	[edge] (o) – (e1);
	[edge] (o) – (e2);
	
	
	
	[] (o) circle (2pt);
	[] (e1) circle (2pt);
	[] (e2) circle (2pt);
	
	[] (linm) node𝒜_X for X=V(T_0^2T_1 + T_2^2 + T_3^3)⊆_5,8,9,6;
	




The anticanonical complex was
introduced in <cit.> for varieties with torus action of complexity one and later generalized i.a. to the general arrangement case in hw19,hmw and has so far been successfully used 
for the classification of singular Fano varieties; see abhw,bhhn, bh, hmw, h_quadrics, hw19.
Our main result states that for Fano general arrangement varieties the Gorenstein index can be read off the anticanonical complex 
in full analogy to the toric Fano polytope case:

    Let X ⊆ Z be a Fano general arrangement variety with anticanonical complex 𝒜_X. Then the Gorenstein index ι_X of X equals the least common multiple of the lattice distances of the maximal cells in the boundary of 𝒜_X:ι_X := lcm(d(0, F);  F ∈∂𝒜_X).
    

As an application of our result
we consider the -factorial rational almost homogeneous Fano varieties of Picard number one with reductive automorphism group having a maximal torus of dimension two that were described in <cit.>. Recall that these varieties are uniquely determined up to isomorphy by their divisor class group graded Cox ring and their anticanonical class, see <ref>. For fixed Gorenstein index we give concrete bounds on these defining data of the varieties, see Propositions <ref>, <ref>, <ref>, <ref> and <ref>.

This allows in particular a computer aided search for varieties of small Gorenstein index.
For illustration we list all -factorial rational almost homogeneous Fano varieties of Picard number one with reductive automorphism group having a maximal torus of dimension two and Gorenstein index smaller or equal to three here:


Every three-dimensional -factorial Fano variety of Picard number one with reductive automorphism group having a maximal torus of dimension two and Gorenstein index ι_X smaller or equal to three is isomorphic to one of the following varieties X, specified by their Cl(X) graded Cox ring ℛ(X), a matrix [w_1…,w_r] of generator degrees and their anticanonical classes -𝒦_X as follows:


gorindvarcccccc
No.
   (X)   (X)   [w_1,…,w_r]   -_X   ι_X

   [T_1,T_2,T_3,T_4,T_5]/T_1^2+T_2 T_3+T_4 T_5      [[ 1 1 1 1 1 ]]   [[ 3 ]]   1

   [T_1,T_2,T_3,T_4,T_5]/T_1 T_2+T_3 T_4+T_5^2   ×_3   [[ 1 1 1 1 1; 1 2 2 1 0 ]]   [[ 3; 0 ]]   1

   [T_1,T_2,T_3,T_4,T_5]/T_1^2+T_2 T_3+T_4^3      [[ 3 3 3 2 1 ]]   [[ 6 ]]   1

   [T_1,T_2,T_3,T_4,T_5]/T_1^2+T_2 T_3+T_4^4   ×_2   [[ 2 2 2 1 1; 1 1 1 1 0 ]]   [[ 4; 0 ]]   1


   [T_1,T_2,T_3,T_4,T_5]/T_1 T_2+T_3 T_4+T_5^3   ×_4   [[ 1 2 2 1 1; 1 3 3 1 0 ]]   [[ 4; 0 ]]   2

   [T_1,T_2,T_3,T_4,T_5]/T_1^2+T_2 T_3+T_4^2 T_5^2   ×_2   [[ 2 2 2 1 1; 1 1 1 0 0 ]]   [[ 4; 1 ]]   2

   [T_1,T_2,T_3,T_4,T_5]/T_1^2+T_2 T_3+T_4^2 T_5^2   ×_4   [[ 2 2 2 1 1; 3 3 3 1 0 ]]   [[ 4; 0 ]]   2

   [T_1,T_2,T_3,T_4,T_5]/T_1^2+T_2 T_3+T_4^3 T_5^5      [[ 4 4 4 1 1 ]]   [[ 6 ]]   2

   [T_1,T_2,T_3,T_4,T_5]/T_1^2+T_2 T_3+T_4^3 T_5^7      [[ 8 8 8 3 1 ]]   [[ 12 ]]   2

   [T_1,T_2,T_3,T_4,T_5]/T_1^2+T_2 T_3+T_4 T_5^3      [[ 4 4 4 5 1 ]]   [[ 10 ]]   2

   [T_1,T_2,T_3,T_4,T_5]/T_1^2+T_2 T_3+T_4 T_5^3   ×_2   [[ 2 2 2 1 1; 1 1 1 0 0 ]]   [[ 4; 1 ]]   2

   [T_1,T_2,T_3,T_4,T_5]/T_1^2+T_2 T_3+T_4 T_5^5      [[ 8 8 8 1 3 ]]   [[ 12 ]]   2

   [T_1,T_2,T_3,T_4,T_5]/T_1^2+T_2 T_3+T_4 T_5^7   ×_3   [[ 4 4 4 1 1; 2 2 2 1 0 ]]   [[ 6; 0 ]]   2

   [T_1,T_2,T_3,T_4,T_5]/T_1^2+T_2 T_3+T_4^6   ×_2   [[ 3 3 3 1 2; 1 1 1 0 1 ]]   [[ 6; 0 ]]   2


   [T_1,T_2,T_3,T_4,T_5]/T_1 T_2+T_3 T_4+T_5^5   ×_3   [[ 2 3 3 2 1; 1 2 2 1 0 ]]   [[ 6; 0 ]]   3

   [T_1,T_2,T_3,T_4,T_5]/T_1 T_2+T_3 T_4+T_5^4   ×_5   [[ 1 3 3 1 1; 1 4 4 1 0 ]]   [[ 5; 0 ]]   3

   [T_1,T_2,T_3,T_4,T_5]/T_1 T_2+T_3 T_4+T_5^2   ×_3   [[ 1 3 3 1 2; 0 1 1 0 2 ]]   [[ 6; 0 ]]   3

   [T_1,T_2,T_3,T_4,T_5]/T_1 T_2+T_3 T_4+T_5^2   ×_3   [[ 3 1 1 3 2; 2 0 0 2 1 ]]   [[ 6; 0 ]]   3

   [T_1,T_2,T_3,T_4,T_5]/T_1 T_2+T_3 T_4+T_5^2   ×_9   [[ 1 1 1 1 1; 1 8 8 1 0 ]]   [[ 3; 0 ]]   3

   [T_1,T_2,T_3,T_4,T_5]/T_1^2+T_2 T_3+T_4^2 T_5^7      [[ 15 15 15  1  4 ]]   [[ 20 ]]   3

   [T_1,T_2,T_3,T_4,T_5]/T_1^2+T_2 T_3+T_4^2 T_5^8   ×_2   [[ 9 9 9 1 2; 1 1 1 0 1 ]]   [[ 12;  0 ]]   3

   [T_1,T_2,T_3,T_4,T_5]/T_1^2+T_2 T_3+T_4^2 T_5^10   ×_4   [[ 6 6 6 1 1; 3 3 3 1 0 ]]   [[ 8; 0 ]]   3

   [T_1,T_2,T_3,T_4,T_5]/T_1^2+T_2 T_3+T_4^3 T_5^9      [[ 6 6 6 1 1 ]]   [[ 8 ]]   3

   [T_1,T_2,T_3,T_4,T_5]/T_1^2+T_2 T_3+T_4^3 T_5^12      [[ 9 9 9 2 1 ]]   [[ 12 ]]   3

   [T_1,T_2,T_3,T_4,T_5]/T_1^2+T_2 T_3+T_4^3 T_5^18      [[ 15 15 15  4  1 ]]   [[ 20 ]]   3

   [T_1,T_2,T_3,T_4,T_5]/T_1^2+T_2 T_3+T_4^3 T_5^3      [[ 3 3 3 1 1 ]]   [[ 5 ]]   3

   [T_1,T_2,T_3,T_4,T_5]/T_1^2+T_2 T_3+T_4^3 T_5^3      [[ 9 9 9 1 5 ]]   [[ 15 ]]   3

   [T_1,T_2,T_3,T_4,T_5]/T_1^2+T_2 T_3+T_4^3 T_5^3   ×_5   [[ 3 3 3 1 1; 2 2 2 3 0 ]]   [[ 5; 0 ]]   3

   [T_1,T_2,T_3,T_4,T_5]/T_1^2+T_2 T_3+T_4^4 T_5^7      [[ 9 9 9 1 2 ]]   [[ 12 ]]   3

   [T_1,T_2,T_3,T_4,T_5]/T_1^2+T_2 T_3+T_4^4 T_5^8   ×_2   [[ 6 6 6 1 1; 1 1 1 1 0 ]]   [[ 8; 0 ]]   3

   [T_1,T_2,T_3,T_4,T_5]/T_1^2+T_2 T_3+T_4^4 T_5^10   ×_2   [[ 9 9 9 2 1; 1 1 1 1 0 ]]   [[ 12;  0 ]]   3

   [T_1,T_2,T_3,T_4,T_5]/T_1^2+T_2 T_3+T_4^5 T_5^7      [[ 6 6 6 1 1 ]]   [[ 8 ]]   3

   [T_1,T_2,T_3,T_4,T_5]/T_1^2+T_2 T_3+T_4^5 T_5^8      [[ 9 9 9 2 1 ]]   [[ 12 ]]   3

   [T_1,T_2,T_3,T_4,T_5]/T_1^2+T_2 T_3+T_4^5 T_5^10      [[ 15 15 15  4  1 ]]   [[ 20 ]]   3

   [T_1,T_2,T_3,T_4,T_5]/T_1^2+T_2 T_3+T_4 T_5^4   ×_3   [[ 3 3 3 2 1; 2 2 2 1 0 ]]   [[ 6; 0 ]]   3

   [T_1,T_2,T_3,T_4,T_5]/T_1^2+T_2 T_3+T_4 T_5^2      [[ 3 3 3 4 1 ]]   [[ 8 ]]   3

   [T_1,T_2,T_3,T_4,T_5]/T_1^2+T_2 T_3+T_4 T_5^5      [[ 6 6 6 7 1 ]]   [[ 14 ]]   3

   [T_1,T_2,T_3,T_4,T_5]/T_1^2+T_2 T_3+T_4^5      [[ 5 5 5 2 3 ]]   [[ 10 ]]   3

   [T_1,T_2,T_3,T_4,T_5]/T_1^2+T_2 T_3+T_4^8   ×_2   [[ 4 4 4 1 3; 1 1 1 0 1 ]]   [[ 8; 0 ]]   3

   [T_1,T_2,T_3,T_4,T_5]/T_1^2+T_2 T_3+T_4^9      [[ 9 9 9 2 1 ]]   [[ 12 ]]   3

   [T_1,T_2,T_3,T_4,T_5]/T_1^2+T_2 T_3+T_4^12   ×_2   [[ 6 6 6 1 1; 1 1 1 1 0 ]]   [[ 8; 0 ]]   3

   [T_1,T_2,T_3,T_4,T_5]/T_1^2+T_2 T_3+T_4^18   ×_2   [[ 9 9 9 1 2; 1 1 1 0 1 ]]   [[ 12;  0 ]]   3


§ BACKGROUND ON GENERAL ARRANGEMENT VARIETIES

We assume the reader to be familiar with the foundations of toric geometry; see cls, ful for introductory texts. In this section we recall the necessary facts and notions on general arrangement varieties. 
This class of varieties has been introduced in <cit.> and can be obtained by using the constructive description of varieties with torus action provided there: The construction is based on a result of <cit.> relating the Cox ringℛ(X) := ⊕_[D] ∈Cl(X)Γ(X, 𝒪_X(D))

of a variety X with torus action to that of a suitable rational quotient X  Y, the so-called maximal orbit quotient. It provides us with the Cox ring and an associated embedding of X into a toric variety Z_X. 
Specializing the procedure to the case that Y is the projective or the affine line, one retrieves the Cox ring based approach to rational varieties with torus action of complexity one developed in hh, hhs, hw_cpl1. The class of general arrangement varieties introduced in <cit.> can then be seen as a controlled step leaving the case of complexity one: These are varieties X with torus action and Y = ^n such that the critical values of the maximal orbit quotient X  Y form a general hyperplane arrangement.

We briefly recall the construction of graded rings R(A,P) that are defined by a pair of matrices and which turn out to be the Cox rings of general arrangement varieties; compare <cit.>:

Fix integers r ≥ c > 0
and n_0, …, n_r > 0 
as well as m ≥ 0. 
Set n := n_0 + … + n_r.
For every i = 0, …, r fix a tuple l_i ∈_>0^n_i and define a monomial 

T_i^l_i :=  
T_i1^l_i1⋯ T_in_i^l_in_i ∈ [T_ij,S_k;  0 ≤ i ≤ r,  1 ≤ j ≤ n_i,  1 ≤ k ≤ m].

We will also write [T_ij, S_k] for the above polynomial ring.
Let A :=(a_0, … a_r) be a a (c+1) × (r+1) matrix over  
such that any c+1 of its columns 
a_0, …, a_r are linearly independent.
For every t = 1, …, r-c, 
we obtain a polynomial
 
g_t 
 := [
[         a_0           …         a_c       a_c+t;     T_0^l_0           …     T_c^l_c T_c+t^l_c+t ]]
 ∈ [T_ij,S_k].

In the next step, we construct a grading on the factor ring
[T_ij, S_k]/g_1, …, g_r-c.

We build up 
an integral (r+s) × (n+m)-matrix P from an
r × (n+m)-matrix matrix P_0
built from the tuples of positive integers 
l_i, where i = 0, …, r and a s × (n+m)-matrix D
as follows

P:=
[
[ P_0;   D ]]
 := [
[ -l_0  l_1         0    0    …    0;    ⋮    ⋮    ⋱    ⋮    ⋮         ⋮; -l_0    0       l_r    0    …    0;          ;         D      ]],

whereby we require the columns of the matrix P to be pairwise different, primitive and generate ^r+s as a vector space.

Now, let e_ij∈^n 
and e_k ∈^m denote the 
canonical basis vectors
and consider the projection 

Q ^n+m → 
K := ^n+m / (P^*)
 
onto the factor group
by the row lattice of P.
Then the 
K-graded -algebra
associated with (A,P) 
is defined as
 
R(A,P)
 :=  [T_ij,S_k] / g_1,…,g_r-c,
(T_ij) :=  Q(e_ij),
    (S_k) :=  Q(e_k).

We note that the rings R(A,P) can be directly read off the matrices A and P. They are integral normal complete intersections.


From a ring R(A,P) as above, we obtain general arrangement varieties X together with an embedding X ⊆ Z into a toric variety Z via the following construction:


Let R(A,P) be as above. The generators T_ij,S_k of R(A,P) give rise to an embedding

X̅ := (R(A,P))[r, hook]
   Z̅:=^n+m. 


Fix any fan Σ in ^r+s having the columns of P as its primitive ray generators and denote by Z the toric variety with defining fan Σ.
Consider the linear map P^n+m→^r+s defined by P, set
Σ̂:={σ≼γ;  P(σ)∈Σ}, where γ⊆^n+m denotes the positive orthant, and denote by Ẑ the corresponding toric variety. Then we obtain a commutative diagram

X̅∩Ẑ[r, hook][d,"p"]
   Ẑ[d,"p"]


X(A,P,Σ)[r, hook]
   
Z


where p denotes the toric morphism corresponding to the linear map P
and X:=X(A,P,Σ)
is the closure of p(X̅∩^n+m)
inside Z.
By construction, the variety X is invariant under the subtorus action ^s⊆^r+s of the acting torus of Z.

  


The varieties X:=X(A,P,Σ)⊆ Z
are normal varieties with
dimension, invertible functions, divisor class group and Cox ring given in terms of their defining data by:
(X) = s + c,
    Γ(X,𝒪^*) = ^*,
    (X) = K,
    ℛ(X) = R(A,P). 

The torus action of ^s on X is effective and of complexity c, i.e. the general torus orbit is of codimension c.


Let X:=X(A,P,Σ) ⊆ Z arise from Construction <ref>. Then we call X ⊆ Z an explicit general arrangement variety. Moreover we call any -variety that is equivariantly isomorphic to an explicit general arrangement variety a general arrangement variety.

Let R(A,P) be as above and assume that the columns of P generate ^r+s as a cone. Let γ denote the positive orthant _≥0^n+m. We define a polyhedral cone 
Mov(R(A,P)) :=  ⋂_γ_0≼γ facet Q(γ_0) ⊆ K_.

Then any projective explicit general arrangement variety is of the form X(A, P, Σ(u)), where Σ(u) is constructed as follows: Let u ∈Mov(R(A,P))^∘ and set
Σ(u) := {P(γ_0^*);  γ_0 ≼γ,  u ∈ Q(γ_0)^∘},   where γ_0^* := cone(e_i;  e_i ∉γ_0).

In particular, up to isomorphy, a projective general arrangement variety X can be regained from its Cl(X)-graded Cox ring ℛ(X) and an ample class u ∈Cl(X)
in the above way. Moreover, if X is Fano, we may choose u = - 𝒦_X.
    Let X:=X(A,P,Σ) ⊆ Z be an explicit general arrangement variety. We note that in Construction <ref> we may successively remove all maximal cones σ∈Σ whose corresponding orbit does not intersect X, that is,
    X ∩^r+s· z_σ = ∅,
    where z_σ denotes the common limit point for t → 0 of all one-parameter subgroups t ↦ (t^v_1, …, t^v_r+s) of the acting torus ^r+s on Z with v ∈^r+s taken from the relative interior σ^∘⊆σ.
    We end up with a minimal fan Σ still defining the same general arrangement variety X.
    We call the toric variety corresponding to this minimal fan the minimal ambient toric variety of X and denote it with Z_X.


We end this chapter by a close investigation of the structure of the fan Σ of the minimal ambient toric variety Z_X of a general arrangement variety X. 

Let us briefly recall the basic notions on tropical varieties.
Let Z be a toric variety with acting torus .
For a closed subvariety X⊆ Z intersecting the torus  non trivially consider the vanishing ideal I(X ∩) in the Laurent polynomial ring (T). For every f ∈ I(X∩) let |Σ(f)| denote the support of the codimension one skeleton of the normal quasifan of its Newton polytope.
Then the tropical variety (X) ofX is defined as follows, 
see <cit.>:
(X) := ⋂_f ∈ I(X ∩) |Σ(f)| ⊆^(Z).

The following result of Tevelev then gives a criterion on which orbits of the toric variety Z are intersected by the embedded variety X in terms of the tropical variety trop(X), see <cit.>:

Let X⊆ Z be a closed embedding.
Then X intersects the torus orbit T · z_σ corresponding to the cone σ∈Σ non-trivially if and only if the relative interior σ^∘ intersects the tropical variety (X) non-trivially.


Using this criterion, we obtain that the cones occurring in the fan corresponding to the minimal ambient toric variety of an explicit general arrangement variety X ⊆ Z_X
are as follows: 



Let X(A, P, Σ)⊆ Z_X be an explicit general arrangement variety of complexity c and denote with Σ_^r the fan corresponding to the toric variety ^r. Then we have

|trop(X)| = |Σ^≤ c__r|×^s,
  where  Σ_^r^≤ c:={σ∈Σ_^r; (σ)≤ c}
We endow trop(X) with the following quasifan structure:
Denote by e_1,…,e_r+s the canonical basis of ^r+s and set e_0:=-∑_i=1^r e_i.
For any subset I⊆{0,…,r}  with 0 ≤ |I| ≤ c we set
λ_I:=(e_i; i∈ I) + (e_r+1,…,
e_r+s).
 
Then we have λ_I⊆(X) and these cones define quasifan structure on trop(X). More precisely we have
trop(X) = Σ^≤ c__r×^s = {σ×^s;  σ∈Σ^≤ c__r} = {λ_I;  I ⊆{0, …, r},  0 ≤ |I| ≤ c}.

The cones λ_I, where 1≤ |I| =: k are called the k-leaves of (X).
Moreover, we have the  lineality space of (X): 
λ_lin:= λ_∅ = ⋂λ_I = lin(e_r+1, …, e_r+s).
Using this quasifan structure on trop(X), we can distinguish between two types of cones that occur in the defining fan Σ of the minimal ambient toric variety Z_X of X:
A cone σ∈Σ is either a leaf cone, that means, σ⊆λ_I holds for a leaf λ_i ∈trop(X), or σ∈Σ is a big cone, that means
σ∩λ_i^∘≠∅ 
holds for all 1-leaves λ_i of (X).
Moreover, we call a big cone elementary big, if for every 0 ≤ i ≤ r there exists precisely one ray ϱ_i of σ with ϱ_i ⊆λ_i.



    Let X:=X(A,P, Σ) ⊆ Z_X be an explicit general arrangement variety.
    Let σ∈Σ be a cone with σ⊈λ_lin. Then the following statements are equivalent:
    
  * σ is a big cone.
        
  *  
        We have σ^∘∩λ_lin≠∅.
    As each cone in Σ_X is either a big cone or a leaf cone, we only need to show the implication (i) ⇒ (ii).
    So let σ be a big cone.
    For 0 ≤ i ≤ r we set
    
    J_i := {j ∈{1, …, n_i};  v_ij∈σ}  and  
    J:= {k ∈{1, …, m};  v_k ∈σ}.
    
    As σ is a big cone, none of the sets J_i is empty.
    We obtain α_i := ∑_j ∈ J_i l_ij > 0 for all 0 ≤ i ≤ r and conclude
    ∑_i=0^r α_0/α_i∑_j ∈ J_i v_ij + ∑_k ∈ J v_k ∈σ^∘∩λ_lin.    
    
    Let X:=X(A,P, Σ) ⊆ Z_X be an affine or complete explicit general arrangement variety and let σ∈Σ be a maximal big cone, where we mean maximal in Σ with respect to inclusion. Then we have
    dim(σ∩λ_lin) = dim(λ_lin).
    
    If X is affine, then Σ consists of precisely one maximal big cone. As by construction the columns of P generate ^r+s as a vector space, we conclude that σ intersects λ_lin in full dimension.

    So assume X is complete and let σ∈Σ_X be a maximal big cone. Then due to Lemma <ref>, there exists a point x ∈σ^∘∩λ_lin.
    Moreover, as X is complete, we have |trop(X)| ∩ |Σ_X|= |trop(X)|, and therefore
    
    λ _lin= ⋃_τ∈Σ
    (τ∩λ_lin)
        =(λ_lin) (τ∩λ_lin).

In particular there exists a cone τ∈Σ with (τ∩λ_lin)=(λ_lin) and x∈σ^∘∩τ.
As
σ∩τ≼σ 
and 
σ^∘∩τ≠∅ holds, we infer σ∩τ=σ and thus σ≼τ. Since σ is a maximal cone, we conclude σ =τ and hence (σ∩λ_lin)=(λ_lin).


§ THE GORENSTEIN INDEX VIA THE ANTICANONICAL COMPLEX

In this section we will describe how to read the Gorenstein index of an explicit general arrangement variety X ⊆ Z_X off its anticanonical complex. We start by shortly recalling the construction of the anticanonical complex for these varieties and some basic facts on lattice distances.


Let X(A, P, Σ) ⊆ Z_X be an explicit general arrangement variety. We consider the coarsest common refinement
Σ':= Σ⊓(X) := {σ∩τ;  σ∈Σ,  τ∈(X)},
where trop(X) is endowed with the quasifan structure defined in Remark <ref>.
Let φ Z' → Z be the toric morphism arising from the refinement of fans Σ' →Σ and let X' be the proper transform of X under φ. Then Z'→ Z is called a weakly tropical resolution of X and X'⊆ Z' fulfills the following conditions: 

  * X' ⊆ Z' is again a general arrangement variety.
    
  *  The fan Σ' consists of leaf cones.
    
  *  For any leaf cone σ∈Σ we have σ∈Σ'.

Using the results of <cit.> we obtain the following description of the anticanonical complex for general arrangement varieties:

Let X ⊆ Z_X be an explicit -Gorenstein general arrangement variety and let φ Z' → Z be its weakly tropical resolution.
For 0 ≤ i ≤ r we consider the following torus invariant divisors on Z_X:

D_Z^(i) := ∑_j=1^n_i (r-c)l_ijD_ϱ_ij - ∑_ϱ∈Σ^(1) D_ϱ.

Let σ' ∈Σ' be any cone. Then σ' is a leaf cone and there exists an index 0 ≤ i ≤ r with v_ij∉σ' for all 1 ≤ j ≤ n_i. Let u_σ'∈ M_ be any element with div(χ^u_σ') = D_Z^(i). Then the anticanonical complex of X ⊆ Z is given as 
𝒜_X:= ⋃_σ' ∈Σ' A_σ'      
A_σ' := σ' ∩{v ∈ N_;  u_σ', v≥ -1 }.

The relative interior𝒜_X^∘ of the anticanonical complex 𝒜_X is the interior of its support with respect to the tropical variety trop(X) 
and its boundary is 
∂𝒜_X := 𝒜_X ∖𝒜_X^∘,
which we will assume to be endowed with the polyhedral complex structure
inherited from 𝒜_X.
In particular, a cell of the anticanonical complex 𝒜_X lies in its boundary if and only if it does not contain 0.

We consider the Fano explicit general arrangement variety X := V(T_01^2T_02 + T_11^2 + T_21^3)⊆_5,8,9,6 =:Z with Cox ring R(A,P) defined by the matrices
A := [ -1  1  0; -1  0  1 ]  and  
P=[v_01,v_02,v_11,v_21] = [[ -2 -1  2  0; -2 -1  0  3; -1 -2  1  2 ]].

In particular, we have
ℛ(X) = R(A,P) = [T_01,T_02,T_11,T_21] / T_01^2T_02 + T_11^2 + T_21^3
with generator degrees [w_01,w_02,w_11,w_21] = [ 5 8 9 6 ]
and the fan Σ corresponding to the minimal ambient toric variety Z_X has the following three maximal cones: σ_1 := (v_01,v_11,v_21), σ_2 := (v_02,v_11,v_21) and σ_3 := (v_01,v_02).
The vertices of the anticanonical complex can then be calculated from these data using <cit.>. These are v_01, v_02, v_11, v_21 and the points u_1 = (0,0,2) and u_2=(0,0,-1) in the lineality space λ_lin of the tropical variety trop(X).

We draw the anticanonical complex 𝒜_X and its boundary ∂𝒜_X inside the tropical variety (X):

[tdplot_main_coords,
			edge/.style=thick,
			leaf0/.style=,
			leaf1/.style=,
			leaf2/.style=,
			leaf0e/.style=,
			leaf1e/.style=,
			leaf2e/.style=,
			le/.style=,
			axis/.style=,
			scale=.7]
	 (o) at (0,0,0);
	 (leaf0m) at (-,-,);
	 (leaf0p) at (-,-,);
	 (leaf0vm) at (,,);
	 (leaf0vp) at (,,);
	 (linm) at (0,0,);
	 (linp) at (0,0,);
	 (leaf1m) at (,0,);
	 (leaf1p) at (,0,);
	 (leaf1vm) at (-,0,);
	 (leaf1vp) at (-,0,);
	 (leaf2m) at (0,,);
	 (leaf2p) at (0,,);
	 (leaf2vm) at (0,-,);
	 (leaf2vp) at (0,-,);
	
	 (v01) at (-1,-1,-2);
	 (v02) at (-2,-2,-1);
	 (v1) at (2,0,1);
	 (v2) at (0,3,2);
	 (e1) at (0,0,2);
	 (e2) at (0,0,-1);
	
	
	
	
	
	[leaf2,opacity=.1] (linp)–(leaf2p)–(leaf2m)–(linm);
	[edge,leaf2e] (o) – (v2);
	[edge,leaf2e] (v2) – (e1);
	[edge,leaf2e] (v2) – (e2);
	[leaf2e,opacity=.2] (e1) – (e2) – (v2) – cycle;
	
	
	[leaf2e] (v2) circle (2pt) node[anchor=south west]v_21;
	
	
	
	[leaf1,opacity=.1] (linp)–(leaf1p)–(leaf1m)–(linm);
	[edge,leaf1e] (o) – (v1);
	[edge,leaf1e] (v1) – (e1);
	[edge,leaf1e] (v1) – (e2);
	[leaf1e,opacity=.2] (e1) – (e2) – (v1) – cycle;
	


	
	[] (v1) circle (2pt) node[anchor=west]v_11;  
	
	
	
	[leaf0,opacity=.1] (linp)–(leaf0p)–(leaf0m)–(linm);
	
	[edge,leaf0e] (o) – (v01);
	[edge,leaf0e] (o) – (v02);
	[edge,leaf0e] (v02) – (e1);
	[edge,leaf0e] (v01) – (e2);
	[edge,leaf0e] (v01) – (v02);
	[leaf0e,opacity=.2] (e1) – (o) – (v02) – cycle;
	[leaf0e,opacity=.2] (v01) – (o) – (v02) – cycle;
	[leaf0e,opacity=.2] (e2) – (o) – (v01) – cycle;
	

	
	[] (v01) circle (2pt) node[anchor=north east]v_02;
	[] (v02) circle (2pt) node[anchor=east]v_01;
	
	
	
	[edge] (o) – (e1);
	[edge] (o) – (e2);
	
	
	
	[] (o) circle (2pt);
	[] (e1) circle (2pt) node[anchor=south east]u_1;
	[] (e2) circle (2pt) node[anchor=north west]u_2;
	
	[] ((linm)+(0,0,-1)) node𝒜_X;

[xshift=10cm]


	 (o) at (0,0,0);
	 (leaf0m) at (-,-,);
	 (leaf0p) at (-,-,);
	 (leaf0vm) at (,,);
	 (leaf0vp) at (,,);
	 (linm) at (0,0,);
	 (linp) at (0,0,);
	 (leaf1m) at (,0,);
	 (leaf1p) at (,0,);
	 (leaf1vm) at (-,0,);
	 (leaf1vp) at (-,0,);
	 (leaf2m) at (0,,);
	 (leaf2p) at (0,,);
	 (leaf2vm) at (0,-,);
	 (leaf2vp) at (0,-,);
	
	 (v01) at (-1,-1,-2);
	 (v02) at (-2,-2,-1);
	 (v1) at (2,0,1);
	 (v2) at (0,3,2);
	 (e1) at (0,0,2);
	 (e2) at (0,0,-1);
	
	
	
	
	
	[leaf2,opacity=.1] (linp)–(leaf2p)–(leaf2m)–(linm);

	[edge,leaf2e] (v2) – (e1);
	[edge,leaf2e] (v2) – (e2);
	[leaf2e,opacity=.05] (e1) – (e2) – (v2) – cycle;
	
	[gray,anchor=south] (F6) at ((v2)!0.4!(e1)) F_6;
	[gray,anchor=west] (F7) at ((v2)!0.3!(e2)) F_7;
	
	[leaf2e] (v2) circle (2pt);
	
	
	
	[leaf1,opacity=.15] (linp)–(leaf1p)–(leaf1m)–(linm);

	[edge,leaf1e] (v1) – (e1);
	[edge,leaf1e] (v1) – (e2);
	[leaf1e,opacity=.05] (e1) – (e2) – (v1) – cycle;
	
	[leaf1e,anchor=west] (F4) at ((v1)!0.7!(e1)) F_4;
	[leaf1e,anchor=north west] (F5) at ((v1)!0.5!(e2)) F_5;
	
	[] (v1) circle (2pt);  
	
	
	
	[leaf0,opacity=.1] (linp)–(leaf0p)–(leaf0m)–(linm);
	

	[edge,leaf0e] (v02) – (e1);
	[edge,leaf0e] (v01) – (e2);
	[edge,leaf0e] (v01) – (v02);
	[leaf0e,opacity=.05] (e1) – (v02) – (v01) – (e2) – cycle;
	
	[leaf0e,anchor=south east] (F1) at ((v02)!0.5!(e1)) F_1;
	[leaf0e,anchor=north east] (F2) at ((v02)!0.5!(v01)) F_2;
	[leaf0e,anchor=north west] (F2) at ((v01)!0.5!(e2)) F_3;
	
	[] (v02) circle (2pt);
	[] (v01) circle (2pt); 
	
	
	
	[] (o) circle (2pt);
	[] (e1) circle (2pt);
	[] (e2) circle (2pt);
	
	[] ((linm)+(0,0,-1)) node∂𝒜_X;


The maximal cells of ∂𝒜_X are the line segments F_1 = (u_1,v_01), F_2 = (v_01,v_02), F_3 = (u_2,v_02), F_4 = (u_1,v_11), F_5 = (u_2,v_11), F_6 = (u_1,v_21) and F_7 = (u_2,v_21).



Now let us turn to lattices distances. A 
lattice subspace is an affine subspace A ⊆^n such that dim(A) = rk(A ∩^n). 
Note that any affine subspace A ⊆^n that contains an element of ^n is a lattice subspace. 
A lattice hyperplane is a lattice subspace of codimension 1. 

The lattice distanced(x,A) between a point x ∈^n and a lattice subspace A ⊆^n is the number of lattice hyperplanes H in the affine hull aff(A ∪{x}) lying between x and A, i.e.


d(x,A) := |{H ⊆aff(A ∪{x}); [ H  lattice hyperplane with  x ∉ H;           and  H ∩conv(A ∪{x}) ≠∅ ]}|.


It is well known that the lattice distance of a lattice hyperplane H ⊆^d and a point x ∈^d can be calculated as follows: We have

d(x, H) = |u_H,v - u_H,x|,

where u_H is a primitive normal of H and v is any point on H.
The lattice distance does not depend on unimodular transformations. For a convex set B ⊆^n with aff(B) a lattice subspace, we set
d(x,B) := d(x, aff(B)).


Theorem <ref> is a direct consequence of the following proposition:

    Let X ⊆ Z_X be an affine or complete explicit -Gorenstein general arrangement variety with anticanonical complex 𝒜_X. Then the Gorenstein index ι_X of X equals the least common multiple of the lattice distances of the maximal cells in the boundary of 𝒜_X:
    ι_X = lcm(d(0, F);  F ∈∂𝒜_X).
    [Example <ref> continued]
We calculate the Gorenstein index of the variety X = V(T_01^2T_02 + T_11^2 + T_21^3) ⊆_5,8,9,6 as described in Example <ref> 
using the above Proposition <ref>:
We have
[ d(0, F_1) = 4,  d(0,F_2) = 3, d(0, F_3) = 1, d(0, F_4) = 4,;  d(0,F_5) = 1, d(0, F_6) = 2,  d(0,F_7) = 1,                ]
and obtain
ι_X = lcm(d(0,F_i);   i = 1, …, 7) = 12.

    Let X ⊆ Z_X be a -Gorenstein general arrangement variety. Then
    
    c_σ:= min{m ∈_>0;  [ there exists  u ∈^r+s with  m· u ∈^r+s;       and div(χ^u)|_Z_σ = D_Z^(i)|_Z_σ ]}
    does not depend on the choice of i ∈{0, …, r} and 
    the Gorenstein index ι_X of X equals lcm(c_σ;  σ∈Σ).

We recall that the pullback homomorphism Cl(Z_X) →Cl(X) is an isomorphism on the level of divisor class groups as well as on the level of Picard groups, see <cit.>. In particular, as X is -Gorenstein, each of the (linear equivalent) divisors D_Z^(i) is -Cartier on Z_X and their Cartier index equals the Gorenstein index of X.
As Z_X is toric, for each σ∈Σ we have
D_Z^(i)|_Z_σ = div(χ^u)|_Z_σ
for some u ∈^r+s. Therefore, using Cl(Z_X) ≅Cl(Z_X)^, we conclude that the Cartier index of D_X^(i) on Z_σ equals c_σ. In particular, c_σ does not depend on the choice of i and the Cartier index of D_Z^(i) on Z equals lcm(c_σ;  σ∈Σ) as claimed.

Let H ⊆^r+s be a lattice hyperplane with 0 ∉ H.
Let e_1, …, e_r+s be the standard basis vectors and set e_0:=-∑ e_i and consider for 0 ≤ i ≤ r the lattice subspaces

H_i :=  H ∩λ_i,   with λ_i := cone(e_i) + lin(e_r+1, …, e_r+s).

If dim(H ∩lin(e_r+1, …, e_r+s)) = s - 1 and dim(H_i) = s holds for all 0 ≤ i ≤ r, then for any subset I ⊆{0, …, r} with |I| = r, we have

d(0,H)  = lcm(d(0,H_i);  i ∈ I).

We exemplarily prove the case I = {1, …,r}. 
Let b ∈^r+s with b,v = 1 for all v ∈ H and let m ∈_≥ 1 be the minimal element such that m · b ∈^r+s. Then m · b is a primitive normal of H and we have d(0,H) = m. Identifying lin(λ_i) with ^1+s via the projection
π_i ^r+s→^1+s,    (a_1, …, a_r+s) ↦ (a_i, a_r+1, …, a_r+s)

we can regard H_i as a lattice hyperplane in ^1+s and b^(i):=(b_i, b_r+1, …, b_r+s) fulfills b^(i),v =1 for all v∈ H_i. In particular, for the minimial m^(i)∈_≥ 1 with m^(i)· b^(i)∈^1+s we have d(0, H_i) = m^(i). Due to the structure of the b^(i),  we conclude

    d(0,H) = m = lcm(m^(i);  1 ≤ i ≤ r) = lcm(d(0,H_i);  1 ≤ i ≤ r).

We will make frequent use of the following straightforward statement about lattice distances of lattice subspaces:

    Let 0 ∉ A ⊆ M_ be a lattices subspace. Then for every lattice subspace 0 ∉ A' containing A we have d(0,A)| d(0,A') and in particular d(0,A) ≤ d(0,A'). Moreover, we have
    
    d(0,A) = min{d(0,H);  0∉ H ⊆ M_ lattice hyperplane with  A ⊆ H}.
     
    By replacing M_ with lin(A'), it suffices to show the first assertion for lattice hyperplanes.
    Applying a suitable unimodular transformation we may futhermore assume M = ^n+m and aff(A ∪{0})= ^n ⊆^n+m. 
    In particular, there exists a unique primitive normal u_A ∈^n of A with d(0,A) = u_A,v for any v ∈ A. Now let 0 ∉ H ⊆^n+m  be any hyperplane containing A. Then there is a primitive normal of H of the form u_H = (λ· u_A, u) for some u ∈^n-s and λ∈_> 0. We conclude
    u_A,v|u_H, v for any v ∈ A ⊆ H and thus d(0,A) | d(0,H). This shows the first assertion. 
    Moreover, the hyperplane 0 ∉ H with A⊆ H and primitive normal (u_A,0, …, 0) ∈^n fulfills d(0,A) = d(0,H) and we obtain the desired equality.    

Let σ∈Σ be any cone and let u ∈^r+s such that div(χ^u)|_Z_σ = D_Z^(i)|_Z_σ holds. In a first step we show that the lattice distance d(0, B_σ^(i)) with

B_σ^(i) :=  σ∩{v ∈ N_;  u, v = -1}
equals c_σ as defined in Lemma <ref>.
By construction, the hyperplanes H with normal u ∈^r+s fulfilling div(χ^u)|_Z_σ = D_Z^(i)|_Z_σ and u,v  = -1 for all v ∈ H are precisely the hyperplanes containing B_σ.
Moreover, for these hyperplanes H we have d(0,H) = m, where m ∈_>0 is the minimal integer with m · u ∈^r+s.
Using Lemma <ref> we conclude d(0,B_σ^(i)) = c_σ as claimed.

To complete the proof, we note that, in the notation of Construction <ref>, the cells of the anticanonical complex 𝒜_X that lie in its boundary are the polyhedra
C_σ' := σ' ∩{v ∈ N_;  u_σ', v = -1}.
In particular, we are left with showing
that 
d(B_σ^(i), 0) = lcm(d(C_σ', 0);  σ' ∈Σ'  with σ' ⊆σ)
holds for every σ' ∈Σ'.
As C_σ'⊆ B_σ^(i) for some 0 ≤ i ≤ r holds for all σ'⊆σ and d(0,B_σ^(i)) does not depend on the choice of i, we obtain ≥ using Lemma <ref>. For the inequality ≤ we distinguish between the two types of cones occurring in Σ. So, let σ∈Σ be a leaf cone. Then σ is not affected by the weakly tropical resolution, that means we have σ∈Σ'.
We conclude d(0,B_σ^(i)) = d(0,C_σ) = lcm(d(C_σ', 0);  σ' ∈Σ'  with σ' ⊆σ). 
Now let σ∈Σ be a big cone. As d(0,B_σ^(i)) does not depend on the choice of i, 
using Lemma <ref> it suffices to prove

d(0,B_σ^(i)) = lcm(d(0, C_σ(j));  σ(j) := λ_j ∩σ for  j ∈{0, …, r} with  j ≠ i).

for a maximal big cone σ.
In this situation, by construction of the anticanonical complex, see <ref>, we have C_σ(j) = B_σ^(i)∩λ_j. 
Using Proposition <ref>, we obtain dim(B_σ^(i)∩λ_lin) = s-1 and as σ is a big cone we have dim(C_σ(j)) = dim(B_σ^(i)∩λ_j) = s. In particular we can apply Lemma <ref> which proves the claim.


§ APPLICATIONS

In this section we apply our results to almost homogeneous Fano varieties X, where almost homogeneous means that the automorphism group of X has an open orbit in X. On the basis of the classification of all -factorial rational almost homogeneous Fano varieties with reductive automorphism group having a maximal torus of dimension two obtained in <cit.>, we give concrete bounds on the defining data depending on the Gorenstein index, see Propositions <ref>, <ref>, <ref>, <ref> and <ref>. This enables us to filter the varieties for those of small Picard number, see Corollary <ref> for the cases of Gorenstein index one, two and three.


Any almost homogeneous -factorial Fano threefold of Picard number one with reductive automorphism group having a maximal torus of dimension two is either the variety No. <ref> from Corollary <ref>, which is of Gorenstein index one, or arises up to isomorphy from one of the Settings <ref>, <ref>, <ref>, <ref> and <ref>, where we list the defining matrices A and P, the fan Σ of the minimal ambient toric variety Z_X and the vertices of the anticanonical complex: 


We have A := [ -1  1  0; -1  0  1 ] and 
P=[v_01,v_02, v_11, v_12, v_21] = 
[   -1   -1    1    1    0;   -1   -1    0    0 l_21;   -1    0    0    1    0;    0    0    0 d_12 d_21 ]
where l_21 > 1, d_12 > 2 and - d_21/d_12-1< l_21 < -d_21 and the maximal cones of the fan Σ corresponding to the minimal ambient toric variety are given as

    [ σ_1 := (v_01,v_02,v_11,v_21), σ_2 := (v_01,v_02,v_12,v_21),; σ_3 := (v_01,v_11,v_12,v_21), σ_4 := (v_02,v_11,v_12,v_21), ]

each of these is a big cone. Moreover, Σ contains the four elementary big cones, σ_1∩σ_3, σ_1∩σ_4, σ_2∩σ_3 and σ_2∩σ_4. 
The vertices of the anticanonical complex can then be calculated from these data using <cit.>. These are given as the columns of P together with the following points in the lineality space of the tropical variety trop(X):

    [          v_σ_1∩σ_3' =(0, 0, -l_21/1+l_21, d_21/1+l_21),                     v_σ_1∩σ_4' =(0, 0, 0, d_21/1+l_21),;           v_σ_2∩σ_3' =(0, 0, 0, d_12 l_21+d_21/1+l_21), v_σ_2∩σ_4' =(0, 0, 
    l_21/1+l_21, d_12 l_21+d_21/1+l_21) ]

Let X be a Fano variety arising from Setting <ref> and denote by ι_X its Gorenstein index. Then 
we have 2<d_12≤ 3ι_X and -ι_X ≤ k< 0 such that
(k d_12+ι_X ) l_21+ι_X |ι_X k^2 d_12  and   k d_21 = ι_X (l_21+1).   

In particular, for fixed Gorenstein index there are finitely many varieties arising via this setting.

Due to the structure of the defining fan Σ of Z_X we obtain that (v_21,v_σ_2∩σ_3',v_σ_2∩σ_4',0) is a cell in its anticanonical complex 𝒜_X, and using Proposition <ref> we obtain

    d(aff(v_21,   v_σ_2∩σ_3',v_σ_2∩σ_4'),0)
       = (d_12 l_21+d_21/(d_12 l_21+d_21,1+l_21), d_12 l_21+d_21/(d_12 l_21+d_21,d_12-d_21)) |ι_X.

In particular, this implies d_12 l_21+d_21|ι_X (1+l_21) and thus

    d_12 l_21+d_21≤ι_X (1+l_21).



Similarly, since (v_21,v_σ_1∩σ_3',v_σ_1∩σ_4',0)∈𝒜_X, we see that
 d(aff(v_21,v_σ_1∩σ_3',v_σ_1∩σ_4'),0) = d_21/(d_21,l_21+1)|ι_X.
In particular, there exists some k∈ such that 

    k d_21 = ι_X (l_21+1).

Note that because of l_21 < -d_21, we have -ι_X ≤ k < 0. Inserting this into (<ref>) yields

    d_12≤ι_X (1+l_21) - d_21/l_21≤2ι_X (1+l_21)/l_21≤ 3ι_X.



We notice that - d_21/d_12-1< l_21 and the identity (<ref>) ensure that -kd_12≠ι_X: If otherwise -kd_12 = ι_X, then -d_21/d_12-1 = -ι_X (l_21+1)/k (d_12-1) = d_12/d_12-1(l_21+1)>l_21.

Once again, we consider d_12 l_21+d_21|ι_X (1+l_21). Using (<ref>) we infer

    (k d_12+ι_X ) l_21+ι_X | k ι_X  l_21+k ι_X

Since (k d_12+ι_X )≠ 0 as seen above, we have 

    k ι_X  l_21+k ι_X    | (k ι_X  l_21+k ι_X)(k d_12+ι_X ).

Therefore,

    (k d_12+ι_X ) l_21+ι_X |    (k ι_X  l_21+k ι_X)(k d_12+ι_X ) - k ι_X ((k d_12+ι_X ) l_21+ι_X) 
       = ι_X k^2 d_12.

Thus, for fixed ι_X there are finitely many possibilities for d_12 and k. For each of these there are only finitely many possibilities for l_21 and thus also for d_21 by Equation (<ref>).

We have A := [ -1  1  0; -1  0  1 ] and 
P=[v_01, v_11, v_12, v_21, v_22] = 
[   -2    1    1    0    0;   -2    0    0 l_21 l_22;   -1    0    1    0    0; d_01    0    0 d_21 d_22 ]
where l_21, l_22 > 1, 2d_22 > -d_01l_22, -2d_21 > d_01l_21. and the maximal cones of the fan Σ corresponding to the minimal ambient toric variety are given as

    [ σ_1 := (v_01,v_11,v_12,v_21), σ_2 := (v_01,v_11,v_12,v_22),; σ_3 := (v_01,v_11,v_21,v_22), σ_4 := (v_01,v_12,v_21,v_22), ]

each of these is a big cone. Moreover, Σ contains the four elementary big cones, σ_1∩σ_3, σ_1∩σ_4, σ_2∩σ_3 and σ_2∩σ_4. 
The vertices of the anticanonical complex can then be calculated from these data using <cit.>. These are given as the columns of P together with the following points in the lineality space of the tropical variety trop(X):

    [  v_σ_1∩σ_3' = (0, 0, -l_21/2+l_21, 
    d_01 l_21+2 d_21/2+l_21),  v_σ_1∩σ_4' =(0, 0, 
    l_21/2+l_21, 
    d_01 l_21+2 d_21/2+l_21) ,; v_σ_2∩σ_3' =(0, 0, 
    -l_22/2+l_22, 
    d_01 l_22+2 d_22/2+l_22) ,    v_σ_2∩σ_4' =(0, 0, 
    l_22/2+l_22, 
    d_01 l_22+2 d_22/2+l_22) ]

Let X be a Fano variety arising from Setting <ref> and denote by ι_X its Gorenstein index. Then we end up in one of the following cases:

  * d_01=0, l_21=l_22|ι_X, 2|ι_X, d_21| (2+l_21)ι_X/2 and d_22| (2+l_22)ι_X/2.
    
  * d_01=0, l_21>l_22, 2|ι_X, 1<l_22<ι_X, 0<d_22<ι_X and 1≤ k <ι_X such that

    d_21/(d_21,d_22)|ι_X/2+k   and   k(l_21 d_22-d_21 l_22)=ι_X(d_22-d_21),

  * d_01=-1, 1<l_21<4ι_X, 0<s, s|ι_X(l_21+2), 0<k≤ι_X and 0<t such that

    t | 2ι_X^2s+2ksι_X   and   k(tl_21+sl_22)=2ι_X(t+s),

where d_21=l_21-s/2 and d_22=l_22+t/2; or the same with (s,l_21) and (t,l_22) interchanged.

In particular, for fixed Gorenstein index there are finitely many varieties arising via this setting.

By suitable subtracting the first row from the last one, we can reach d_01∈{0,-1}. We start with d_01=0. In this case the conditions change to l_21, l_22 > 1, d_22 > 0 and d_21 < 0. Let l_21≥ l_22 without loss of generality due to admissible operations. 

Due to the structure of the defining fan Σ of Z_X we obtain that (v_21, v_22,v_σ_1∩σ_4',v_σ_2∩σ_4',0) is a cell in 𝒜_X, and using Proposition <ref> we obtain 

    d(aff(v_21,   v_22,v_σ_1∩σ_4',v_σ_2∩σ_4'),0) = l_22 d_21-d_22 l_21/(l_22 d_21-d_22 l_21,d_21-d_22)=l_21=l_22|ι_X.

Hence, l_21 and l_22 are bounded. Using again the structure of the defining fan Σ of Z_X we obtain that  (v_01,v_σ_1∩σ_3',v_σ_1∩σ_4',0) and (v_01,v_σ_2∩σ_3',v_σ_2∩σ_4',0) are cells of 𝒜_X, and with Proposition <ref> we obtain 

    d(aff(v_01,v_σ_1∩σ_3',v_σ_1∩σ_4'),0)  = (2, 2d_21/(2d_21, 2+l_21)) |ι_X,
    
    d(aff(v_01,v_σ_2∩σ_3',v_σ_2∩σ_4'),0)  = (2, 2d_22/(2d_22, 2+l_22)) |ι_X.

In particular, using 2|ι_X, this leads to the desired d_21| (2+l_21)ι_X/2 and d_22| (2+l_22)ι_X/2.



Secondly, let l_21>l_22. As above, we obtain

    d(aff(v_21,   v_22,v_σ_1∩σ_4',v_σ_2∩σ_4'),0) 
       = (l_21 d_22-d_21 l_22/(l_21 d_22-d_21 l_22, l_21-l_22), l_21 d_22-d_21 l_22/(l_21 d_22-d_21 l_22,d_22-d_21)) |ι_X.

In particular, this implies l_21 d_22-d_21 l_22|ι_X(l_21-l_22) and l_21 d_22-d_21 l_22|ι_X(d_22-d_21). Thus we have

    ι_X(l_21-l_22)/l_21 d_22-d_21 l_22≥ 1   and  ι_X(d_22-d_21)/l_21 d_22-d_21 l_22≥ 1,

which yields

    (d_22-d_21)l_22≤ (ι_X-d_22)(l_21-l_22)   and    d_22 (l_21-l_22) ≤ (ι_X-l_22)(d_22-d_21).

 Since (d_22-d_21)l_22 >0 and d_22 (l_21-l_22)>0, we infer d_22<ι_X and l_22<ι_X by using l_21-l_22>0 and d_22-d_21>0. 

Once again, we consider l_21 d_22-d_21 l_22|ι_X(d_22-d_21). In particular, there exists some k∈ with k≥ 1 such that 

    k(l_21 d_22-d_21 l_22)=ι_X(d_22-d_21).

Note that because of l_22, l_21>1, we have l_21 d_22-d_21 l_22>d_22-d_21 and thus 1≤ k <ι_X. Using (<ref>) we obtain

    (ι_X-kl_22)d_21=(ι_X-kl_21)d_22

and thus d_21| (ι_X-kl_21)d_22. Due to the structure of the defining fan Σ of Z_X we obtain that  (v_01,v_σ_1∩σ_3',v_σ_1∩σ_4',0) is a cell in 𝒜_X, and using Proposition <ref> we obtain 

    d(aff(v_01,v_σ_1∩σ_3',v_σ_1∩σ_4'),0)  = (2, 2d_21/(2d_21, 2+l_21)) |ι_X.

In particular, this implies 2d_21|ι_X(2+l_21) and 2|ι_X. Thus, we have d_21|ι_X+ι_X/2l_21. For d̃_21:=d_21/(d_21,d_22) we infer d̃_21|ι_X+ι_X/2l_21 and d̃_21|ι_X -kl_21, and thus

    d̃_21|ι_X+ι_X/2l_21-(ι_X -kl_21)=(ι_X/2+k)l_21.

Since d_21 and l_21 are coprime, we obtain d̃_21|ι_X/2+k.
Thus, for fixed ι_X there are finitely many possibilities for d_22,l_22, d̃_21 and k. For each of these there are only finitely many possibilities for d_21 and thus also for l_21 by (<ref>).

We continue with d_01=-1. In this case the conditions change to l_21, l_22 > 1, -l_22+2d_22> 0 and l_21-2d_21 > 0. This yields l_21d_22-l_22d_21>0. We set 

    s:= l_21-2d_21∈_≥ 1   and    t:= -l_22+2d_22∈_≥ 1.

Then we have

    2(l_21d_22-l_22d_21)=l_21(l_22+t)-l_22(l_21-s)=tl_21+sl_22.

Due to the structure of the defining fan Σ of Z_X we obtain that (v_01,v_σ_1∩σ_3',v_σ_2∩σ_3',0) is a cell in 𝒜_X, and using Proposition <ref> we obtain 

    d(aff   (v_01,v_σ_1∩σ_3',v_σ_2∩σ_3'),0)  
       = (d_22-d_21/(l_21d_22-l_22d_21,d_22-d_21),l_21-l_22/(l_21d_22-l_22d_21,l_21-l_22)) |ι_X.

This implies l_21d_22-l_22d_21|ι_X(l_21-l_22) and l_21d_22-l_22d_21|ι_X(d_22-d_21), and thus

    l_21d_22-l_22d_21|ι_X((l_21-l_22)+2(d_22-d_21))=ι_X(s+t).

In particular, this yields

    tl_21+sl_22=2(l_21d_22-l_22d_21) | 2ι_X(t+s),

and because of l_21d_22-l_22d_21>0 and t+s>0 we obtain tl_21+sl_22/t+s≤ 2ι_X. 

We first consider t≥ s. Then we have 

    l_21/2< tl_21+sl_22/t+s≤ 2ι_X,

and this yields l_21<4ι_X. 
Due to the structure of the defining fan Σ of Z_X we obtain that 
(v_11,v_12, v_σ_1∩σ_3',v_σ_1∩σ_4',0) is a cell in 𝒜_X, and using Proposition <ref> we obtain 

    d(aff(v_11,v_12,v_σ_1∩σ_3',v_σ_1∩σ_4'),0)  = (s/(s,l_21+2)) |ι_X.

In particular, this implies s|ι_X(l_21+2), and l_21<4ι_X yields s< 4ι_X^2+2ι_X.
Similarly, since (v_11,v_12, v_σ_2∩σ_3',v_σ_2∩σ_4',0) ∈𝒜_X , we see that

    d(aff(v_11,v_12,v_σ_2∩σ_3',v_σ_2∩σ_4'),0)  = (t/(t,l_22+2)) |ι_X.

In particular, this implies t|ι_X(l_22+2). Furthermore, due to <ref> there exists some k∈ such that

    k(tl_21+sl_22)=2ι_X(t+s).

Note that because of l_21, l_22>1, we have tl_21+sl_22≥ 2(s+t), and thus 0<k≤ι_X. Using (<ref>) yields

    t(kl_21-2ι_X)=2ι_X s-ksl_22,

so we get t| 2ι_X s-ksl_22. Using t|ι_X(l_22+2) we obtain

    t|ι_X(2ι_X s-ksl_22) +ksι_X(l_22+2)=2ι_X^2s+2ksι_X.

Thus, for fixed ι_X there are finitely many possibilities for l_21, s,t and k. For each of these there are only finitely many possibilities for l_22 due to (<ref>).

For the case where s >t, one follows the same arguments as above with (s,l_21) and (t,l_22) interchanged.
In both cases, d_21 and d_22 are obtained by (<ref>).

We have A := [ -1  1  0; -1  0  1 ] and 
P=[v_01, v_11, v_12, v_21, v_22] = 
[   -2    1    1    0    0;   -2    0    0    1 l_22;   -1    0    1    0    0; d_01    0    0 d_21 d_22 ]
where l_22 > 1, d_22 > d_21l_22 + l_22, 2d_22 > -d_01l_22, -2d_21> d_01 and the maximal cones of the fan Σ corresponding to the minimal ambient toric variety are given as

    [ σ_1 := (v_01,v_11,v_12,v_21), σ_2 := (v_01,v_11,v_12,v_22),; σ_3 := (v_01,v_11,v_21,v_22), σ_4 := (v_01,v_12,v_21,v_22), ]

each of these is a big cone. Moreover, Σ contains the four elementary big cones, σ_1∩σ_3, σ_1∩σ_4, σ_2∩σ_3 and σ_2∩σ_4. 
The vertices of the anticanonical complex can then be calculated from these data using <cit.>. These are given as the columns of P together with the following points in the lineality space of the tropical variety trop(X):

    [                  v_σ_1∩σ_3' =(0, 0, -1/3, 
    d_01/3+2 d_21/3),                    v_σ_1∩σ_4' =(0, 0, 1/3, d_01/3+2 d_21/3),; v_σ_2∩σ_3' =(0, 0, 
    -l_22/2+l_22, 
    d_01 l_22+2 d_22/2+l_22),    v_σ_2∩σ_4' =(0, 0, l_22/2+l_22, 
    d_01 l_22+2 d_22/2+l_22) ]

Let X be a Fano variety arising from Setting <ref> and denote by ι_X its Gorenstein index. Then we have d_21=0, -3ι_X ≤ d_01<0, -3ι_X ≤ k_01<0 and 0 < k_22<ι_X such that

    ι_X( 3/k_01+2/k_22)l_22-2ι_X/k_22| 6 ι_X(k_22+k_01)   and   d_22 k_22 = ι_X (l_22-1).

In particular, for fixed Gorenstein index there are finitely many varieties arising via this setting.

By subtracting d_21 times the second row from the last one, we can reach d_21=0. The conditions change to l_22 > 1, d_22 > l_22, 2d_22 > -d_01l_22 and 0> d_01.
Due to the structure of the defining fan Σ of Z_X we obtain that (v_01,v_σ_1∩σ_3',v_σ_1∩σ_4',0) is a cell in 𝒜_X, and using Proposition <ref> we obtain

    d(aff(v_01,   v_σ_1∩σ_3',v_σ_1∩σ_4'),0)= d_01/(d_01,3)|ι_X.

In particular, this implies d_01| 3ι_X and thus there exists some k_01∈ with 

    d_01 k_01=3ι_X.

Note that because of -3ι_X ≤ d_01<0, we have -3ι_X ≤ k_01 <0. Similarly, since (v_21,v_22,v_σ_1∩σ_4',
v_σ_2∩σ_4',0)∈𝒜_X, we see that

    d(aff(v_21,v_22,   v_σ_1∩σ_4',v_σ_2∩σ_4'),0) = d_22/(l_22-1,d_22)|ι_X.

In particular, there exists some k_22∈ such that 

    d_22 k_22 = ι_X (l_22-1).

Note that because of 1<l_22 < d_22, we have 0 < k_22 < ι_X. Similarly, since (v_01,v_σ_2∩σ_3',
v_σ_2∩σ_4',0)∈𝒜_X, we see that

    d(aff(v_01,   v_σ_2∩σ_3',v_σ_2∩σ_4'),0) 
       = (d_01 l_22+2 d_22/(-d_01+d_22,d_01 l_22+2 d_22), d_01 l_22+2 d_22/(2+l_22,d_01 l_22+2 d_22)) |ι_X.

In particular, this implies  d_01 l_22+2 d_22|ι_X(2+l_22). Using (<ref>) and (<ref>) we obtain

    d_01 l_22+2 d_22= ι_X( 3/k_01+2/k_22)l_22-2ι_X/k_22_=: b|ι_X(2+l_22).

Assuming 3/k_01+2/k_22=0, then we get

    d_01 l_22+2 d_22= -2ι_X /k_22<0,

that contradicts d_01 l_22+2 d_22>0. Thus we have 

    0 ≠3/k_01+2/k_22=3k_22+2k_01/k_22 k_01.

Once again, we consider d_01 l_22+2 d_22|ι_X(2+l_22). By using (<ref>) we infer

    b | ι_X l_22+2ι_X | (3k_22+2k_01)(ι_X l_22+2ι_X).

This implies

    b | (3k_22+2k_01)(ι_X l_22+2ι_X)-k_22 k_01 b=6 ι_X(k_22+k_01).

Assuming k_22=-k_01, then we get

    d_01 l_22+2 d_22= ι_X ( 3/k_01-2/k_01)l_22+2ι_X/k_01=ι_X(l_22+2)/k_01<0,

that contradicts d_01 l_22+2 d_22>0.
Thus, for fixed ι_X there are finitely many possibilities for d_01,k_01 and k_22. For each of these there are only finitely many possibilities for l_22 and thus also for d_22 by (<ref>) and (<ref>).

We have A := [ -1  1  0; -1  0  1 ] and 
P=[v_01, v_11, v_12, v_21, v_22] = 
[   -2    1    1    0    0;   -2    0    0    1 l_22;   -1    0    1    0    0; d_01    0    0 d_21 d_22 ]
with l_22 > 1, 2d_22 > -d_01l_22, 1 - 2d_21 > d_01.
There are at least three maximal cones in the fan Σ corresponding to the minimal ambient toric variety, which are given as

    [ σ_1 := (v_01,v_11,v_12,v_22), σ_2 := (v_01,v_11,v_21,v_22),; σ_3 := (v_01,v_12,v_21,v_22). ]

Each of these is a big cone. If 2d_21+d_01≠ 0 holds, then there exists a fourth maximal cone of Σ that is a big cone, namely σ_4 := (v_01,v_11,v_12,v_21).
Moreover, Σ contains the four elementary big cones, σ_1∩σ_2, σ_1∩σ_3,τ_1 ≼σ_2, τ_2 ≼σ_3. 
The vertices of the anticanonical complex can then be calculated from these data using <cit.>. These are given as the columns of P together with the following points in the lineality space of the tropical variety trop(X):

    [                      v_τ_1' =(0, 0, -1/3, 
    d_01/3+2 d_21/3),                         v_τ_2' =(0, 0, 1/3, d_01/3+2 d_21/3); v_σ_1∩σ_2' =(0, 0, 
    -l_22/2+l_22, 
    d_01 l_22+2 d_22/2+l_22),    v_σ_1∩σ_3' =(0, 0, l_22/2+l_22, 
    d_01 l_22+2 d_22/2+l_22) ]

Let X be a Fano variety arising from Setting <ref> and denote by ι_X its Gorenstein index. Then we have
d_21=0, -2ι_X ≤ d_01≤ 0 and 0<k<2ι_X such that

    (d_01 k+2 ι_X )  l_22/k-2 ι_X/k| 2ι_X(d_01k+3ι_X)   and     d_22 k=ι_X (l_22-1).

In particular, for fixed Gorenstein index there are finitely many varieties arising via this setting.

By subtracting d_21 times the second row from the last one, we can reach d_21=0. The conditions change to l_22 > 1, 2d_22 > -d_01l_22 and 1  > d_01. Due to the structure of the defining fan Σ of Z_X we obtain that
(v_21,v_22,v_τ_1',v_σ_1 ∩σ_2',0) is a cell in 𝒜_X, and using Proposition <ref> we obtain

    d(aff(v_21,v_22,   v_τ_1',v_σ_1∩σ_2'),0) = d_22/(l_22-1,d_22)|ι_X.

In particular, this implies d_22|ι_X(l_22-1). Because of d_22>0 and l_22>1, we have

    d_22/l_22-1<ι_X.

Using 2d_22>-d_01l_22 we infer

    -d_01/2·l_22/l_22-1<ι_X,

that implies -2ι_X ≤ d_01≤ 0. If d_01=0, then the last coordinates of v_τ_1', v_τ_2', v_σ_1∩σ_2' and v_σ_1∩σ_3' would all be non-negative, which contradicts 0 ∈|𝒜_X| ^∘. Thus we have d_01<0 and d_22>l_22/2. 
Once again, we consider d_22|ι_X(l_22-1). Thus there exists k ∈ with 

    d_22 k=ι_X (l_22-1)   ⇔   d_22=ι_X(l_22-1)/k.

Note that because of d_22>l_22/2 we have 0<k<2ι_X. Since (v_11,v_12,v_σ_1 ∩σ_2',v_σ_1 ∩σ_3',0) is a cell in 𝒜_X, and using Proposition <ref> we obtain

    d(aff(v_11,v_12,   v_σ_1∩σ_2',v_σ_1∩σ_3'),0) = d_01 l_22+2 d_22/(2+l_22,d_01 l_22+2 d_22)|ι_X.

In particular, this implies d_01 l_22+2 d_22|ι_X (2+l_22). Inserting (<ref>) into this yields

    d_01 l_22+2 d_22=(d_01 k+2 ι_X )  l_22/k-2 ι_X/k_=:b|ι_X(2+l_22).

Assuming d_01k+2ι_X=0, then we get 2ι_X=-d_01k. With -2ι_X ≤ d_01<0 and 0<k<2ι_X we infer 2≤ -d_01, k<ι_X. This contradicts 2d_22 > -d_01l_22, thus we have d_01k+2ι_X≠ 0. Using this we get

    b | 2ι_X+ι_X l_22| (d_01k+2ι_X)(2ι_X+ι_X l_22).

This implies 

    b | (d_01k+2ι_X)(2ι_X+ι_X l_22)-kι_X b=2ι_X(d_01k+3ι_X).
 
Assuming d_01=-3ι_X/k, then we get

    d_01 l_22+2 d_22= (-3ι_X+2ι_X)l_22/k-2ι_X/k=-ι_X(l_22+2)/k<0,

that contradicts d_01 l_22+2 d_22>0.
Thus, for fixed ι_X there are finitely many possibilities for d_01 and k. For each of these there are only finitely many possibilities for l_22 and thus also for d_22 by (<ref>).

We have A := [ -1  1  0; -1  0  1 ] and 
P=[v_01,v_11,v_12,v_21,v_1]= 
[   -2    1    1    0    0;   -2    0    0 l_21    0;   -1    0    1    0    0;    1    0    0 d_21    1 ] with 1 < l_21 < -2d_21 < 2l_21 and the maximal cones of the Σ corresponding to the minimal ambient toric variety are given as

    [  σ_1 := (v_01,v_11,v_21,v_1),  σ_2 := (v_01,v_12,v_21,v_1),; σ_3 := (v_01,v_11,v_12,v_21),       σ_4 := (v_11,v_12,v_1), ]

where σ_1,σ_2,σ_3 are big cones and σ_4 is a leaf cone. Moreover, Σ contains two elementary big cones, σ_1∩σ_3 and σ_2∩σ_3. 
The vertices of the anticanonical complex can then be calculated from these data using <cit.>. These are given as the columns of P together with the following points in the lineality space of the tropical variety trop(X):

    v_σ_1∩σ_3' =(0, 0, -l_21/2+l_21, 
    l_21+2 d_21/2+l_21)
       and   v_σ_2∩σ_3' = (0, 0, 
     l_21/2+l_21, l_21+2 d_21/2+l_21).

Let X be a Fano variety arising from Setting <ref> and denote by ι_X its Gorenstein index. Then we have -ι_X ≤ k<-ι_X/2 such that

    l_21(2k+ι_X/ι_X) + 2|  4kι_X   and  k l_21/ι_X = d_21-1.

In particular, for fixed Gorenstein index there are finitely many varieties arising via this setting.

Due to the structure of the defining fan Σ of Z_X we obtain that (v_21,v_1,v_σ_1∩σ_3',0) is a cell in 𝒜_X, and using Proposition <ref> we obtain  d(aff(v_21,v_1,v_σ_1∩σ_3'),0) = l_21/(l_21,d_21-1)|ι_X.
In particular, this implies

    k l_21/ι_X = d_21-1

for some k∈. Because of l_21 < -2d_21 < 2l_21, we have -ι_X ≤ k <-ι_X/2.
Similarly, since (v_11,v_12,v_σ_1∩σ_3',v_σ_2∩σ_3',0)∈𝒜_X, we see that
 d(aff(v_11,v_12,v_σ_1∩σ_3',v_σ_2∩σ_3'),0) = l_21+2 d_21/(l_21+2 d_21,l_21+2)|ι_X.
In particular, we obtain (l_21+2 d_21)|ι_X(l_21+2). Using (<ref>), we infer

    (l_21 + 2k/ι_Xl_21+2)= (l_21(2k+ι_X/ι_X) + 2) |ι_X(l_21+2)

We notice that -ι_X≤ 2k+ι_X < 0. Therefore, ι_X(l_21+2)| (2k+ι_X)ι_X(l_21+2). Hence,

    (l_21(2k+ι_X/ι_X) + 2) | (2k+ι_X)ι_X(l_21+2) - ι_X^2(l_21(2k+ι_X/ι_X) + 2) = 4kι_X.

Thus, for fixed ι_X there are finitely many possibilities for k and thus finitely many possibilities for l_21.