Smooth projective horospherical varieties of Picard group $\mathbb{Z}^2$

We classify all smooth projective horospherical varieties of Picard group $\mathbb{Z}^2$ and we give a first description of their geometry via the Log Minimal Model Program.


Introduction
In this paper, varieties are irreducible algebraic varieties over C and groups are linear algebraic groups over C. And we study varieties that belong to the family of horospherical varieties. Let us first introduce this family.

Results of the paper
We classify and give a first study of the geometry of smooth projective horospherical varieties of Picard group Z 2 . For toric varieties, these are only decomposable projective bundles over projective spaces [Kle88]. But for horospherical varieties, there are many other cases.
Indeed, in addition to homogeneous spaces, products of two varieties and decomposable projective bundles over projective spaces, we distinguish several other types of such horospherical varieties. We classify them in this paper, in particular by studying their Log MMP.
To state as nicely as possible the classification of smooth projective horospherical varieties of Picard group Z 2 , we extend the notion of simple roots to the groups C * and {1}. We first briefly recall the case of simple groups (in this paper, a simple group has positive semi-simple rank).
If G is a simply connected simple group, we fix a maximal torus contained in a Borel subgroup B of G, then it defines a root system and in particular a set of simple roots. To each simple root α are associated a fundamental weight denoted by α and a fundamental G-module denoted by V ( α ). More generally, if χ is a dominant weight (a non-negative sum of fundamental weights) we denote by V (χ) the G-module associated to χ: it is the unique irreducible G-module that contains a unique B-stable line where B acts with weight χ. A non-zero element of the B-stable line of V (χ) is called a highest weight vector (of weight χ) and the stabilizer of the B-stable line of V (χ) is denoted by P (χ) (it is a parabolic subgroup of G containing B).
In this paper, if G = C * , we call the identity automorphism of C * the simple root of G; we denote it by α, and we set α = α. Then the natural C * -module C is denoted by V ( α ) where α is the simple root of C * . And for any n ∈ Z, V (n α ) is the C * -module C where C * acts with weight n α ; in particular, any character of C * is dominant. Moreover, if G = {1}, we call the trivial morphism from G to C * the simple root of G; we denote it by α, and we set α = 0. In these two cases a highest weight vector is any non-zero vector.
Suppose now that G is a product G 0 × · · · × G t of simply connected simple groups, C * and {1}. A simple root of G is a simple root of some G i and it is said to be trivial if G i is equal to C * or {1}. Moreover if χ 0 , . . . , χ t are respectively dominant weights of G 0 , . . . , G t , the G-module associated to χ = χ 0 + · · · + χ t is the tensor product V (χ 0 ) ⊗ · · · ⊗ V (χ t ) and a highest weight vector of this G-module is a decomposable tensor product of highest weight vectors.
In Definition 3.9, we define two types of projective horospherical varieties X 1 and X 2 with Picard group Z 2 . We describe them explicitly as the closure of some G-orbit of a sum of highest weight vectors in the projectivization of a G-module, with the convention above. These varieties depend on the group G, on a simple root β, on a tuple α of, possibly trivial, simple roots of G and on a tuple a of positive integers.
We can now state the two main results of this paper.

Some known results on horospherical varieties 2.1. First definitions, first properties of divisors, and smoothness criterion
In this section, we present the classification of horospherical varieties in terms of colored fans Then we give the criteria for divisors to be Cartier, globally generated, and ample. And we state the smoothness criterion. All are generalizations of the theory of toric varieties (without colors).
Let G be a connected reductive group. Fix a maximal torus T and a Borel subgroup B containing T . Denote by U the unipotent radical of B, by S the set of simple roots of (G, B, T ), by X(T ) the lattice of characters of T (or B) and by X(T ) + ⊂ X(T ) the monoid of dominant characters.
For any lattice L we denote by L Q the Q-vector space L ⊗ Z Q.

Definition 2.1. A horospherical variety X is a normal G-variety with an open orbit isomorphic to G/H
where H is a subgroup of G containing U . Then G/H is a torus fibration over the flag variety G/P where P is the parabolic subgroup of G containing B defined as the normalizer of H in G. The dimension of the torus is called the rank of G/H or the rank of X and is denoted by n.
We denote by M the sublattice of X(T ) consisting of characters of P whose restrictions to H are trivial. Its dual lattice is denoted by N . (The lattices M and N are of rank n.) Let R be the subset of S consisting of simple roots that are not simple roots of P (i.e., simple roots associated to fundamental weights some multiples of which are characters of P ).
For any simple root α ∈ R, the restriction of the coroot α ∨ to M is a point of N , which we denote by α ∨ M . We denote by σ the map α −→ α ∨ M from R to N . Definition 2.2.
(1) A colored cone of N Q is a pair (C, F ) where C is a convex cone of N Q and F is a subset of R (called the set of colors of the colored cone), such that (i) C is generated by finitely many elements of N and contains {α ∨ M | α ∈ F }, (ii) C does not contain any line and F does not contain any α such that α ∨ M is zero. (2) A colored face of a colored cone (C, F ) is a pair (C , F ) such that C is a face of C and F is the set of α ∈ F satisfying α ∨ M ∈ C . (3) A colored fan is a finite set F of colored cones such that (i) any colored face of a colored cone of F is in F , and (ii) any element of N Q is in the relative interior of at most one colored cone of F .
The main result of Luna-Vust Theory of spherical embeddings is the following classification result (see for example [Kno91]).

Theorem 2.3 (D. Luna-T. Vust). There is an explicit one-to-one correspondence between G-isomorphism classes of horospherical G-varieties with open orbit G/H and colored fans.
Complete G/H-embeddings correspond to complete fans, i.e., to fans such that N Q is the union of the first components of their colored cones.
If G = (C * ) n and H = {1}, we recover the well-known classification of toric varieties in terms of fans. If X is a G/H-embedding, we denote by F X the colored fan of X in N Q and we denote by F X the subset ∪ (C,F )∈F X F of R, called the set of colors of X.
From now on, X is a complete horospherical variety as above. Let us recall now the characterization of Cartier, Q-Cartier, globally generated and ample divisors of horospherical varieties, due to M. Brion in the more general case of spherical varieties ( [Bri89]).
First, we describe the B-stable prime divisors of X. We denote by X 1 , . . . , X m the G-stable prime divisors of X. The valuations of C(X) defined by the order of zeros and poles along these divisors define primitive elements of N , denoted by x 1 , . . . , x m respectively.
And the B-stable but not G-stable prime divisors of X are the closures in X of B-stable prime divisors of G/H, which are the inverse images by the torus fibration G/H −→ G/P of the Schubert divisors of the flag variety G/P . The Schubert divisors of G/P can be naturally indexed by the subset of simple roots R. Hence, we denote the B-stable but not G-stable prime divisors of X by D α with α ∈ R (note that σ (α) is the element of N defined by the valuation of C(X) defined by the zeros and poles along the divisor D α ).
Theorem 2.4 (cf. Section 3.3 in [Bri89]). Any divisor of X is linearly equivalent to a linear combination of X 1 , . . . , X m and D α with α ∈ R. Now, let D = m i=1 a i X i + α∈R a α D α be a Q-divisor of X. (2) Suppose that D is Q-Cartier. Then D is globally generated (resp. ample) if and only if the piecewise linear function h D is convex (resp. strictly convex) and for any α ∈ R\F X , we have h D (α ∨ M ) ≤ a α (resp. h D (α ∨ M ) < a α ). Suppose that D is a Q-Cartier Q-divisor. We define the pseudo-moment polytope of (X, D) to be the polytopeQ D in M Q given by the following inequalities, where χ ∈ M Q : (h D ) + χ ≥ 0 and for any α ∈ R\F X , a α + χ(α ∨ M ) ≥ 0. Let v 0 := α∈R a α α , we define the moment polytope of (X, D) to be the polytope Q D := v 0 +Q D .
(3) Suppose that D is a Cartier divisor. Note that the weight of the canonical section of D is v 0 . Then the G-module H 0 (X, D) is the direct sum (with multiplicities one) of the irreducible G-modules of highest weights χ + v 0 with χ inQ D ∩ M.
From now on, a divisor of a horospherical variety is always supposed to be B-stable, i.e., of the form m i=1 a i X i + α∈R a α D α .
Note that the characterizations of Cartier, Q-Cartier, globally generated and ample divisors can be also applied without the completeness assumption. In particular, Corollary 2.7 also does not need the completeness assumption.
To formulate the smoothness criterion we need to give the following definition.
Definition 2.8. ([Pas06,Def. 2.4]) Let R 1 and R 2 be two disjoint subsets of S. Let Γ R 1 ∪R 2 be the maximal subgraph of the Dynkin diagram of G whose vertices are in R 1 ∪ R 2 . The pair (R 1 , R 2 ) is said to be smooth if, for any connected component Γ of Γ R 1 ∪R 2 , (1) there is at most one vertex of Γ in R 2 and, (2) if α ∈ R 2 is a vertex of Γ , then Γ is of type A or C and α is a short extremal simple root of Γ .
Theorem 2.9 (cf. Theorem 2.6 of [Pas06]). Let X be a locally factorial horospherical variety. Then X is smooth if and only if for any colored cone (C, F ) of F X , the pair (S\R, F ) is smooth.
Corollary 2.10 (cf. Proposition 2.17 of [Pas06]). Let X be a smooth horospherical variety. Any G-stable subvariety of X is a smooth horospherical variety.
Remark 2.11. If X is a toric variety, Theorem 2.9 is trivial because locally factorial toric varieties are smooth, or because for any colored cone (C, F ) of F X , the pair (S\R, F ) is necessarily (∅, ∅) (indeed the root system or the Dynkin diagram of a torus is empty).

Log MMP via moment polytopes
The MMP [Pas15] and Log MMP [Pas17] of horospherical varieties can be completely computed and described by studying one-parameter families of polytopes. In this subsection, we recall the main results of this theory, as briefly as we can, in order to use them in Section 5.
From the previous section, to any horospherical variety X, are associated a parabolic subgroup P and a sublattice M of X(P ); and moreover, any ample B-stable Q-Cartier Q-divisor D defines a pseudo-moment polytopeQ and a moment polytope Q. In fact, the map (X, D) −→ (P , M, Q,Q) classifies polarized projective horospherical varieties in terms of quadruples (P , M, Q,Q). Definition 2.12. A quadruple (P , M, Q,Q) is called admissible if it satisfies the following: • P is a parabolic subgroup of G containing B, M is a sublattice of X(P ), Q is a polytope of X(P ) Q included in X(P ) + Q andQ is a polytope of M Q ; • there exists (a unique) v 0 ∈ X(P ) Q such that Q = v 0 +Q; • the polytopeQ is of maximal dimension in M Q (i.e., its interior in M Q is not empty); • the polytope Q intersects the interior of X(P ) + Q . Example 2.13. Suppose that X(P ) = Z 1 ⊕ Z 2 and M = Z 2 , then Q andQ are vertical segments of the same length,Q is in Q 2 and Q is in Q ≥0 1 ⊕ Q ≥0 2 (but not in Q 2 ). In Figure 1, we draw three possible pairs (Q,Q) to get three admissible quadruples (P , M, Q,Q) respectively corresponding to polarized varieties (X, D 1 ), (X, D 2 ) and (X , D ), with D 1 D 2 and X X . Proposition 2.14 (Corollary 2.10 of [Pas17] together with Propositions 2.10 and 2.11 [Pas15]).
(1) The map (X, D) −→ (P , M, Q,Q) is a bijection from the set of isomorphism classes of polarized projective horospherical varieties to the set of admissible quadruples. (2) It induces a bijection between the set of G-orbits in X and the set of non-empty faces of Q (orQ), preserving the natural orders of both sets. Also, the G-orbit in X associated to a non-empty face F = v 0 +F of Q is isomorphic to a horospherical homogeneous space corresponding to (P F , M F ) where P F is the minimal parabolic subgroup of G containing P and M F is the maximal sublattice of M such that (P F , M F , F,F) is an admissible quadruple. Moreover (P F , M F , F,F) is the quadruple associated to the (horospherical) closure in X of the G-orbit associated to F (polarized by some D F we do not need to explicit here).
Example 2.15. Consider the moment polytopes of Example 2.13. And suppose that D 1 , D 2 and D are very ample (otherwise it would be enough to consider multiples of the divisors and of the polytopes).
From Proposition 2.14, we easily get the following result.
Corollary 2.16. Let (X, D) be a polarized projective horospherical variety and (P , M, Q,Q) be the corresponding admissible quadruple. Let F be a non-empty face of Q (orQ) and Ω be the corresponding G-orbit in X. Then We can also describe G-equivariant morphisms between horospherical G-varieties, in terms of moment polytopes [Pas15,2.4]. We summarize, very briefly, this description here.
Without loss of generality, we can reduce to dominant G-equivariant morphisms, i.e. G-equivariant morphisms from a G/H-embbedding to a G/H -embbedding where H ⊂ H , i.e., G-equivariant morphisms that extend the projection G/H −→ G/H . In that case, we have P ⊂ P and M ⊂ M. We keep the same notations as above for the data associated to G/H and we use the same notations with prime for the data associated to G/H . Let X be a projective G/H-embedding corresponding to an admissible quadruple (P , M, Q,Q) and let X be a projective G/H -embedding corresponding to an admissible quadruple (P , M , Q ,Q ). Then the projection G/H −→ G/H extends to a G-equivariant morphism from X to X if and only if for any nonempty face F of Q, the set of facets (or the corresponding halfspaces in M Q ) and the set of walls of X(P ) + Q that contain F define naturally a non-empty face F of Q . Moreover in that case the G-orbit of X corresponding to F is sent to the G-orbit of X corresponding to F . Example 2.17. Consider the varieties X and X of Example 2.13. Each vertex of Q, which is a facet, naturally correspond to a vertex of Q . But, the vertex 2 1 of Q is contained in a wall of X(P ) + Q and will correspond to the empty face of Q. Then, here, there exists a G-equivariant morphism φ from X to X Figure 2. Moment polytopes and G-equivariant morphisms but there is no such morphism from X to X. Moreover, φ is an isomorphism outside one closed G-orbit where φ is the projection G/(P ( 1 ) ∩ P ( 2 )) −→ G/P ( 2 ).
To complete this example, consider some G/H of rank 2 such that P has a unique fundamental weight . In Figure 2 we draw 3 moments polytopes of G/H and another moment polytope of a horospherical homogeneous space G/H of rank 1 with P = G (in fact G/H C * and the segment corresponds to the variety P 1 ). We also draw all G-equivariant morphisms between the corresponding varieties. Note that this picture is similar to Figure 8 with moment polytopes instead of pseudo-moment polytopes.
We also emphasis some vertices and some edges to illustrate images of G-orbits. More precisely, if we focus at the G-orbits distinguished by a •, φ 0 restricts to the projection G/P ( ) −→ pt. If we focus at the G-orbits distinguished by a non-dashed rectangle, φ + 0 restricts to the fibration P 1 −→ pt and φ 1 restricts to the identity morphism P 1 −→ P 1 . If we focus at the G-orbits distinguished by a dashed rectangle, φ 0 and φ + 0 restrict to identity morphisms and φ 1 restricts to a fibration to a point.
where B, C and v = v 0 + v 1 are such that, for any ≥ 0 small enough,Q and Q are respectively the pseudo-moment and moment polytope of (X, D + (K X + ∆)).
Note that the matrices A, B and C can be easily computed. Indeed, A is given by the primitive elements of the rays of the colored fan of X and the images of the colors of G/H; the coefficients of B are the opposites of the coefficients of D; and the coefficients of C are the opposites of the coefficients of K X + ∆. Also, the coefficients of v 0 and v 1 correspond to the coefficients of the D α 's for D and K X + ∆ respectively.
We can rewrite the conclusion of Theorem 2.18 more precisely as the existence of rational numbers 0 := 0,0 < · · · < 0,k 0 < 0,k 0 +1 = 1,0 < · · · · · · < 1,k 1 < 1,k 1 +1 = 2,0 < · · · < p,k p < p,k p +1 = max (with p ≥ 1, and for any i ∈ {0, . . . , p}, k i ≥ 0) such that, (P , M, Q ,Q ) is an admissible quadruple if and only if ∈ [0, max [, and for , η ∈ [0, max [ the following three assertions are equivalent: • X is isomorphic to X η (where X and X η are the varieties associated to the admissible quadruples (P , M, Q ,Q ) and (P , M, Q η ,Q η ) respectively); • the faces of Q (orQ ) and Q η (orQ η ) are "the same", in the following sense: up to deleting inequalities corresponding to some x j with j ∈ {1, . . . , m} but without changingQ andQ η , we have that for any set I of rows, the face ofQ corresponding to I (defined by replacing inequalities by equalities for the rows in I) is non empty if and only the face ofQ η corresponding to I is non empty; • there exists i ∈ {0, . . . , p} such that and η are both in Moreover, for any i ∈ {0, . . . , p} and k ∈ {1, . . . , k i } there are morphisms from X to X i,k with < i,k big enough and > i,k small enough, defining flips. For any i ∈ {1, . . . , p}, there are morphisms from X to X i,0 with < i,0 big enough, defining divisorial contractions. Actually, divisorial contractions appear exactly when an inequality corresponding to some x j with j ∈ {1, . . . , m} becomes superfluous to defineQ .
Also, there exists P and M such that (P , M , Q max ,Q max ) is an admissible quadruple associated to a variety X max and such that there is a fibration from X to X max with < max big enough. Moreover, the general fiber of this fibration is a horospherical variety and can be described.
In fact all fibers could be described with the following strategy: consider a G-orbit G/H of X max and list all G-orbits of X with < max big enough that are sent to G/H by the fibration, then if there is a unique biggest such G-orbit Ω, the fibers over G/H are isomorphic to the closure of L · v where L is a Levi subgroup of H and v is the projectivization of a sum of highest weight vectors in Ω. Note that in this paper, there will always be such a biggest G-orbit.
All morphisms above are G-equivariant and the image of any G-orbit can be described as follows. To a face of Q (orQ ) we can associate the maximal set of rows for which equality holds for any element x of the face (in the inequalities Ax ≥ B + C). And similarly to a set of rows we can also naturally associate a face of Q (may be empty). For any and i,k as above, for any face F ofQ , we construct a face ofQ i,k by taking the maximal set of rows associated to F and then the face F i,k associated to these rows. Then, since there is a morphism φ from X to X i,k , the non-empty face F i,k corresponds to the G-orbit image by φ of the G-orbit corresponding to F . Several examples illustrating Theorem 2.18, in rank 2, are given in Sections 5.2 and 5.4.

Reduction to three cases
In this section, we only use Luna-Vust theory and Corollary 2.7 to reduce to the three main cases of Theorem 1.1.
Remark 3.2. If X is a toric variety, R = ∅ then we are necessarily in Case (2), and the lemma is already known [Kle88, Theorem 1], and X is the decomposable projective bundle P(O ⊕ O(a 1 ) ⊕ · · · ⊕ O(a r ) over P s .
We now detail each case.
Case (0): In the case where n = 0, X is the complete homogeneous variety G/P (and F X = ∅). And then |R| = 2. Case (1): Consider the fanF := {C | (C, F ) ∈ F X } associated to the colored fan F X (in fact it is the fan of the toric fiber Y of the toroidal varietyX := G × P Y obtained from X by erasing all colors of X). Since X is locally factorial, the fanF is the fan of a smooth toric variety of Picard number 1 (because |F X (1)| = n + 1). Then it is well-known that such a fan is the fan of the projective space P n . In particular, there exists a basis (e 1 , . . . , e n ) of N such thatF = {C I | I {0, . . . , n}} where e 0 := −e 1 − · · · − e n and C I is the cone generated by the e i with i ∈ I. Denote by β the unique element of R\F X . Then, up to reordering the e i 's (for i ∈ {0, . . . , n}), we can suppose that σ (β) is in C {1,...,n} and equals a 1 e 1 + · · · + a n e n with 0 ≤ a 1 ≤ · · · ≤ a n . Case (2): As above, consider the fanF . Since X is locally factorial, it is the fan of a smooth toric variety of Picard number 2 (because |F X (1)| = n + 2). Then, by [Kle88, Theorem 1], there exist integers r ≥ 1, s ≥ 1, 0 ≤ a 1 ≤ · · · ≤ a r and a basis (u 1 , . . . , u r , v 1 , . . . , v s ) of N such thatF = {C I,J | I {0, . . . , r} and J {1, . . . , s + 1}}, where u 0 := −u 1 − · · · − u r , v s+1 := a 1 u 1 + · · · + a r u r − v 1 − · · · − v s and C I,J is the cone generated by the u i 's with i ∈ I and the v j 's with j ∈ J. We conclude by using the following facts: for any α ∈ F X and for any (C, F ) ∈ F X , we have α ∈ F if and only if σ (α) ∈ C; and for any α ∈ F X , σ (α) is the primitive element of an edge of F X (using again Corollary 2.7). Remark 3.3. In section 5, we will use the MMP or the Log MMP to study and compare geometrically all these varieties X. We can already describe some Mori fibrations from these varieties, by using the following description of G-equivariant morphisms between horospherical varieties in terms of colored fans ( [Kno91]). Let G/H and G/H be two horospherical homogeneous spaces with H ⊂ H , and denote by π : G/H −→ G/H the projection. We keep the same notations as before for the data associated to G/H and we use the same notations with prime for the data associated to G/H . In particular, we have M ⊂ M, P ⊂ P and R ⊂ R. By duality, we also have a projection π * : N Q −→ N Q . Let X be a G/H-embedding with colored fan F X and let X be a G/H -embedding with colored fan F X . Then the morphism π extends to a G-equivariant morphism from X to X if and only if for any colored cone (C, F ) ∈ F X , there exists a colored cone (C , F ) ∈ F X such that π * (C) ⊂ C and F ∩ R ⊂ F . Case (0): If X is a complete homogeneous variety G/P of Picard group Z 2 , then the MMP gives two Mori fibrations from X to the complete homogeneous varieties G/P 1 and G/P 2 of Picard group Z, where P 1 and P 2 are the maximal proper parabolic subgroups of G containing B such that P = P 1 ∩ P 2 . Note moreover that G/P is a product if and only if Aut 0 (G/P ) is not simple.
Then we can easily check the condition above to prove that there exists a G-equivariant morphism from X to G/P ( β ). Note that the general fiber of this fibration is smooth horospherical of Picard group Z (in particular, it is homogeneous or one of the two-orbit varieties described in [Pas09]). Case (2): Let P be the parabolic subgroup containing B (and P ) such that R :=σ −1 ({v j | j ∈ {1, . . . , s + 1}}).
Let M be the sublattice of M orthogonal to Zu 1 ⊕ · · · ⊕ Zu r ⊂ N . The pair (P , M ) corresponds to a horospherical homogeneous space G/H with H containing H. Also the dual lattice N of M is the image of the projection from N to Zu 1 ⊕ · · · ⊕ Zu r . We denote by v 1 , . . . , v s+1 the images of v 1 , . . . , v s+1 in N , in particular v s+1 = −v 1 − · · · − v s . And finally we denote by F X the colored fan The colored fan F X corresponds to a G/H -embedding X . Then we can check the condition above to prove that there exists a G-equivariant morphism from X to X , which is a Mori fibration. Note that X and the general fiber of this fibration are smooth horospherical varieties of Picard group Z (in particular, they are homogeneous or one of the two-orbit varieties described in [Pas09]).
In the rest of the paper, in cases (1) and (2), we will denote this fibration by ψ : X −→ Z.

Description via polytopes
We now describe X embedded in the projectivization of a G-module, by choosing the smallest ample Cartier divisor of X and by applying Corollary 2.6. We first study the nef cone of X, which is 2-dimensional.
Recall that any Cartier divisor of X is linearly equivalent to a B-stable divisor, and any prime G-stable divisor corresponds to an edge of F X that is not generated by some σ (α) with α ∈ F X , and any other B-stable prime divisor is the closure of a color of G/H. Then in Cases (1) and (2), we have n + 2 prime B-stable divisors that we can denote naturally as follows: Case (1): D n+1 = D β ; for any i ∈ {0, . . . , n}, D i is the B-stable divisor corresponding to the edge generated by e i (which equals D α with α ∈ F X = R\{β} if and only if the edge is generated by σ (α), and which is G-stable otherwise). Case (2): for any i ∈ {0, . . . , r}, D i is the B-stable divisor corresponding to the edge generated by u i ; and for any j ∈ {1, . . . , s + 1}, D j+r is the B-stable divisor corresponding to the edge generated by v i (which equals D α with α ∈ F X = R if and only if the edge is generated by σ (α), and which is G-stable otherwise).
By Theorem 2.4, one checks that D 0 and D n+1 are globally generated but not ample. We also check that for any a and b in Q, aD 0 + bD n+1 is Cartier if and only if a and b are integers.
Before applying Corollary 2.6, we reduce to the case where G is the product of simply connected simple groups and a torus, with the following lemma.
Lemma 3.5 (cf. proof of Proposition 3.10 in [Pas06]). Let G := [G, G] and let T be the torus P /H. Then X is also a horospherical G × T -variety. Moreover, ifĜ is the universal cover ofĜ , X is also a horospherical G × T-variety.
Without loss of generality by the lemma, we now assume that G is the product G of simply connected simple groups and a torus T . In particular, P is the product of a parabolic subgroup of G with T , and the characters of P are sums of weights of the maximal torus of G and characters of T . Hence a basis of With these assumptions, we get the following result.
Lemma 3.6. The embedding of X given by the ample Cartier divisor D 0 + D n+1 is: where χ 0 = 0, χ 1 , . . . , χ n are characters of T , and for any i ∈ {0, . . . , n}, i is either α if e i = σ (α) with α ∈ F X or 0 otherwise. Case (2): Proof. In each case, we describe the pseudo-moment polytope of (X, D 0 + D n+1 ) in a particular basis of M and then the moment polytope of (X, D 0 + D n+1 ). Then we use Corollary 2.6 to conclude.
Case (1): By the previous lemma and the description of the images of colors, for any i ∈ {1, . . . , n}, the element e * i is of the form χ i + i − 0 + a i β , where χ 1 , . . . , χ n are characters of T and for any i ∈ {0, . . . , n}, i is either α if e i = σ (α) with α ∈ F X or 0 otherwise. The pseudo-moment polytope of (X, D 0 + D n+1 ) is the simplex with vertices 0, e * 1 , . . . , e * n . The weight of the canonical section of D 0 + D n+1 is 0 + β , where 0 is either α if e 0 = σ (α) with α ∈ F X or 0 otherwise. Hence, the moment polytope of (X, D 0 + D n+1 ) is the simplex with vertices 0 + 0 + β = χ 0 + 0 + (1 + a 0 ) β and (χ i Case (2): By the previous lemma and the description of the images of colors, for any i ∈ {1, . . . , r} the element u * i is of the form χ i + i − 0 + a i n+1 and for any j ∈ {1, . . . , s} the element v * j is of the form χ r+j + r+j − n+1 , where χ 1 , . . . , χ n are characters of T , and for any i ∈ {0, . . . , n + 1}, i is either The pseudo-moment polytope of (X, D 0 + D n+1 ) is the polytope with the following vertices: 0, u * 1 , . . . , u * r , v * 1 , . . . , v * s and u * i + (a i + 1)v * j for any 1 ≤ i ≤ r and for any 1 ≤ j ≤ s. Note that the lattice points of this polytope are exactly 0, v * 1 , . . . , v * s and for any 1 ≤ i ≤ r all the points of the form In particular, the lattice points of the pseudo-moment polytope translated by 0 + n+1 are exactly the where the sum is taken over all s+2-tuples of non-negative integers Recall that, by Lemma 3.5, (χ 1 , . . . , χ n ) is a basis of X(T ). Hence, there exists a subtorus S of T such that: (χ i|S ) i∈{1,...,n}, i =0 is a basis of X(S), and for any i ∈ {1, . . . , n} such that i 0, we have χ i|S = 0.

Lemma 3.7. In both cases (1) and (2), X is also a horospherical G × S-variety.
Proof. Consider Case (1). For any i ∈ {1, . . . , n} such that i 0, the G-orbit and the G × S-orbit of the highest weight vector v We can replace χ i + i with α i such that * if χ i|S = 0 and i 0, α i is a simple root of G (that is supposed to be a product of simply connected simple groups); * S is a product of C * 's whose trivial simple roots are the α i 's with i such that χ i|S 0 and i = 0; * if i = 0 or n + 1, χ i|S = 0, and i = 0, we have that α i is the trivial root of {1}.
This finally gives the following proposition.
Proposition 3.8. Let X be a smooth projective horospherical variety of Picard group Z 2 as in Case (1) or (2). Then X is isomorphic to a smooth closure of a G-orbit of a sum of highest weight vectors as follows where G is the product G 0 × · · · × G t of simply connected simple groups, C * and {1}: where * n ≥ 1 and β is a (non-trivial) simple root of G 0 ; * α 0 , . . . , α n are distinct simple roots (may be trivial) of G distinct from β; * for any k ∈ {1, . . . , t}, G k = {1} if and only if k = 1 and α 0 is the trivial root of G 1 ; * and 0 = a 0 ≤ a 1 ≤ · · · ≤ a n are integers.
These two cases of Proposition 3.8 justify the definition of two types of varieties. In Case (2), we already only consider the case where s = 1 to simplify the definition ; we will prove in Section 4.3 that we can reduce to this case. Definition 3.9. Let G = G 0 × · · · × G t be a product of simply connected simple groups, C * and {1} (with t ≥ 0).

Reduction to the cases of Theorem 1.1
This section we define the restricted conditions mentioned in Theorem 1.1, and we prove that we can reduce the cases of Proposition 3.8 to the varieties X 1 (G, β, α, a) and X 2 (G, α, a) with these restricted conditions.

Smooth horospherical varieties and G-modules
To prove Theorem 1.1 from Proposition 3.8, we replace sums of irreducible G-modules with irreducible G-modules with G ⊂ G as soon as we can. Then we enlarge the group G and we reduce to "smaller" cases (for example to horospherical varieties with smaller rank). For this, we first need to apply the smoothness criterion to X (Theorem 2.9), which comes from the fact that horospherical G-modules (i.e. G-modules that are horospherical as varieties) are the C * -modules C, the SL d -modules V ( 1 ) = C d and Sp d -modules (with d even) V ( 1 ) = C d . And then we use easy facts as "the As in [Pas09, Theorem 1.7], the smoothness criterion reveals 8 configurations including the 5 configurations that give the five families of horospherical two-orbit varieties corresponding to non-homogeneous smooth projective horospherical varieties of Picard group Z. We recall these 8 configurations in the following definition.
Definition 4.1. Let K be a simple algebraic group over C and let γ, δ be two simple roots of K. The triple (K, γ, δ) is said to be smooth if (type of K, γ, δ) is one of the following 8 cases, up to exchanging γ and δ (with the notation of Bourbaki [Bou75]).
We say that the triple (type of K, γ, δ) is smooth of two-orbit type if it is one of the cases 3, 4, 5, 7 or 8 above.
Remark 4.2. The smooth triples of two-orbit type correspond bijectively to the isomorphism classes of non-homogeneous projective smooth horospherical varieties with Picard group Z. These varieties have two orbits under the action of their automorphism groups, which are given in [Pas09, Theorem 1.11] and justify that all these varieties are distinct.
Here we also need to introduce another "smooth object" (only used in Case (1)). Definition 4.3. Let K be a simple algebraic group over C and let β be a simple root of K and let R be a subset of simple roots of K, all distinct from β. Let n be a non-negative integer. Denote by L the Levi subgroup of the maximal parabolic subgroup P ( β ) of K, then the semi-simple part of L is a quotient by a finite central group of a product of simple groups L 1 , . . . , L q (with q ≥ 0).
The quadruple (K, β, R, n) is said to be smooth if (1) n = 1, R = {γ, δ} such that γ and δ are simple roots of the same L k such that the triple (L k , γ, δ) is smooth; (2) or for any k ∈ {1, . . . , q}, at most one simple root of L k is in R, and if γ ∈ R is a simple root of L k , then L k is of type A or C and γ is a short extremal simple root of L k .
We can list all smooth quadruples (K, β, R, n) (see the appendix). We remark, in particular, that R is at most of cardinality 3.
We can now define the restricted conditions that allow us to state Theorems 1.1 and 1.3.
Definition 4.4. Let X = X 1 (G, β, α, a) as in Definition 3.9. Recall that R 0 is the maximal subset of {α 0 , . . . , α n } consisting of simple roots of G 0 . We say that X satisfies the restricted condition (a), (b) or (c) respectively if it satisfies all the following properties including (a), (b) or (c) respectively.
(2) If R 0 is empty, then G 0 is the universal cover of the automorphism group of G/P ( β ).
(3) If i < j and a i = a j then α j ∈ R 0 . Moreover, if α i and α j are in R 0 , we suppose them to be ordered with Bourbaki's notation as simple roots of G 0 . (4) One of the three following cases occurs.
In the two next cases, the map {α 0 , . . . , α n }\R 0 −→ {1, . . . , t} is surjective and strictly increasing, and for any k ∈ {1, . . . , t}, either G k is isomorphic to some SL d k and α i k is the first simple root of G k , or G k is isomorphic to C * or {1} and α i k is the trivial simple root of G k . (b) The simple root α n is not trivial (in particular if a n−1 = a n ). (c) The simple root α n is trivial (and then a n−1 < a n ).
Definition 4.5. Let X = X 2 (G, α, a) as in Definition 3.9. We say that X satisfies the restricted condition (a), (b) or (c) respectively if it satisfies all the following properties including (a), (b) or (c) respectively.
(3) One of the three following cases occurs.
In the two next cases: t = n, the map {α 0 , . . . , α n−1 } −→ {0, . . . , t − 1} is surjective and strictly increasing; and for any i ∈ {1, . . . , t}, either G i is isomorphic to some SL d i and α i is the first simple root of G i , or G i is isomorphic to C * or {1} and α i is the trivial simple root of G i . (b) The simple root α n−1 is not trivial. (c) The simple root α n−1 is trivial.
Remark 4.6. In Theorem 1.1, the decomposable projective bundles over projective spaces are the horospherical varieties X in Case (1) with restricted condition (b) or (c), and such that R 0 = ∅ and β is the first simple root of G 0 = SL d 0 for some d 0 ≥ 2 (and 0 < a 1 < · · · < a n ).
Example 4.7. The three varieties given in Examples 3.11 do not satisfy the restricted condition. Indeed, for W 1 we have a 2 = a 3 but α 3 is not a simple root of G 0 . For W 2 , we have a 0 = a 1 and G 2 = Sp 8 . And for W 3 , (G t , α n , α n+1 ) is not smooth of two-orbit type.
The colored fan F 1 is the complete colored fan whose maximal colored cones are generated by all u 0 , . . . , u n except one and with all possible colors except β, where (u 1 , . . . , u n ) is a basis of N and u 0 = −u 1 − · · · − u n . Recall also that the map σ is injective from the set R\{β} of colors of the horospherical variety to {u 0 , . . . , u n } and σ (β) = β ∨ M is a 1 u 1 + · · · + a n u n . The colored fan F 2 is the complete colored fan whose maximal colored cones are generated by all u 0 , . . . , u r , v 1 , . . . , v s+1 except one u i and one v j , and with all possible colors, where (u 1 , . . . , Recall also that the map σ is injective from the set R of colors of the horospherical variety to {u 0 , . . . , u r , v 1 , . . . , v s }.

Lemma 4.8.
Case (1): The quadruple (G 0 , β, R 0 , n) is smooth. If there exist 0 ≤ i < j ≤ n such that α i and α j are simple roots of the same simple group G k with k ∈ {1, . . . , t} then n = 1, i = 0 and j = 1 (also t = k = 1). Moreover in that case, the triple (G k , α i , α j ) is smooth. otherwise, for any i ∈ {0, . . . , n}, the simple root α i is either trivial or in G 0 or the short extremal simple root of some simple group G k with k ∈ {1, . . . , t} that is of type A or C. Case (2): If there exist 0 ≤ i < j ≤ n + 1 such that α i and α j are simple roots of the same simple group G k with k ∈ {0, . . . , t} then either r = 1, i = 0 and j = 1, or s = 1, i = n and j = n + 1. Moreover in that case, the triple (G k , α i , α j ) is smooth. For any i ∈ {0, . . . , n}, such that the simple root α i is the unique α j of a simple group G k with k ∈ {0, . . . , t}, the root α i is either trivial or the short extremal simple root of G k that is of type A or C.

Proof.
Case (1): With notation of Definition 4.3 (with K = G 0 ), suppose γ and δ are two simple roots of the same L j . If n > 1, then there exists a maximal colored cone of F X that contains γ ∨ M and δ ∨ M . By applying Theorem 2.9, we get a contradiction. Then n = 1 and applying Theorem 2.9 to the two one-dimensional colored cones of F X , we have that the pairs (R 0 \{β, δ}, γ) and (R 0 \{β, γ}, δ) are smooth, so that (L j , γ, δ) is smooth (from a case by case study done in [Pas09, Proof of Theorem 1.7]).
Suppose that α is the unique simple root of L j in R 0 . By applying Theorem 2.9 to the colored cone (Q ≥0 α ∨ M , {α}) we get that L j is of type A or C and α is a short extremal simple root of L j . This finishes the proof of the smoothness of (G 0 , β, R 0 , n).
If there exist 0 ≤ i < j ≤ n such that α i and α j are simple roots of the same simple group G k with k ∈ {1, . . . , t} then as above Theorem 2.9 implies that n = 1 and (G k , α i , α j ) is smooth. The fact that i = 0, j = 1 and t = k = 1 is obvious. Now, let i ∈ {0, . . . , n} such that the simple root α i is the unique α j of a simple group G k with k ∈ {1, . . . , t} and suppose that α i is not trivial. Apply again Theorem 2.9 to the colored cone (Q ≥0 α ∨ M , {α}) to get that α i is the short extremal simple root G k with k ∈ {1, . . . , t} that is of type A or C. This finishes the proof of the lemma in Case (1). Case (2): Suppose there exist 0 ≤ i < j ≤ n + 1 such that α i and α j are simple roots of the same simple group G k with k ∈ {0, . . . , t}. Then Theorem 2.9 implies that (G k , α i , α j ) is smooth (still from the case by case study done in [Pas09, Proof of Theorem 1.7]). But this also gives a contradiction if there exists a maximal colored cone of F X that contains α ∨ i,M and α ∨ j,M . This contradiction occurs if and only if 0 ≤ i ≤ r and r + 1 ≤ j ≤ n + 1, or 0 ≤ i, j ≤ r and r ≥ 2, or r + 1 ≤ i, j ≤ n + 1 and s ≥ 2.
We conclude the proof of the lemma in Case (2) as in Case (1).
With notation of Bourbaki [Bou75] (we put primes to write differently fundamental weights of G from those of G).
Moreover in each case, the projectivizations of the G-orbit and the G-orbit have the same dimension, in particular the two projective varieties defined as the closure of these two orbits in the corresponding projective spaces are the same.
(2) Cases (c), (d) and (e) correspond to the triples of Definition 4.1 that are not of two-orbit type.
Proof. The first two items are easy and left to the reader. The last three items are given in [Pas09, Propositions 1.8, 1.9 and 1.10].
In Case (2), we need the following generalization of Lemma 4.9.
where the sum is taken over all (τ + 1)-tuples of non-negative integers where the sum is taken over all (τ + 1)-tuples of non-negative integers With notation of Bourbaki [Bou75] (we put primes to write differently fundamental weights of G from those of G).
(e) Let G = Spin 2d (with d ≥ 4) and G = Spin 2d+1 . Then Moreover in each case, the projectivizations the G-orbit and the G-orbit have the same dimension, in particular the two projective varieties defined as the closure of these two orbits in the corresponding projective spaces are the same.
Proof. Remark that for a = 1 the lemma is Lemma 4.9. For any a ≥ 1, we denote by V a the G-module that we consider in each case.
Consider the horospherical G-variety X defined as the closure of the G-orbit of a sum x 1 of highest weight vectors in P(V 1 ): it is a smooth projective variety with Picard group Z (it is isomorphic to P d−1 , P d−1 , the quadric Q 2d−2 , the Grassmannian Gr(i + 1, d + 1), Spin(2d + 1)/P ( d ) respectively). Moreover V * 1 is the G-module of global sections of O X (1). And, for any a ≥ 1, the G-module V * a is the set of global sections of O X (a). But, in each case, X is also a homogeneous projective G-variety G/P ( ) (with = 1 , 1 , 1 , i+1 and d respectively) by Lemma 4.9, then V a is also the irreducible G-module V G (a ). Also, the image of x 1 in P(V a ) is the projectivization of a highest weight vector in V G (a ) for a good choice of a Borel subgroup of G (because G · x 1 is the homogeneous projective G-variety G/P ( )).

Proof of Theorem 1.1 in Case (1)
The first part is already proved by Proposition 3.8 and Lemma 4.8, in particular X is embedded as the closure of the G-orbit of a sum of highest weight vectors in It remains to prove that we can suppose that * G 0 is the universal cover of the automorphism group of G 0 /P ( β ) if R 0 is empty; * if i < j and a i = a j then α j ∈ R 0 ; * and some groups G k of type C can be replaced by groups of type A.
• If R 0 is empty and G 0 is not the universal cover of the automorphism group of G 0 /P ( β ), then where G 0 is the universal cover of Aut(G 0 /P ( β )) and (G 0 , β, G 0 , β ) is one of the following: (Sp 2m , 1 , SL 2m , 1 ), (G 2 , 1 , Spin 7 , 1 ), (Spin 2m+1 , m , Spin 2m+2 , m ) or (Spin 2m+1 , m , Spin 2m+2 , m+1 ). In any case, and X is isomorphic to the closure of the G-orbit of a sum of highest weight vectors in • Suppose that there is 0 ≤ i < j ≤ n such that α i and α j are simple roots of the same simple group among G 1 , . . . , G t . Then by Lemma 4.8, we have n = 1, i = 0, j = 1 (also t = 1) and the triple (G 1 , α i , α j ) is smooth. In particular, X is embedded as the closure of the G-orbit of a sum of highest weight vectors in If a 1 = 0, the G-module V ( α 0 + β ) ⊕ V ( α 1 + (1 + a 1 ) β ) is isomorphic to the tensor product of the G 0 -module V ( β ) by the G 1 -module V ( α 0 )⊕V ( α 1 ), so that X is the product of G/P ( β ) by the smooth projective horospherical variety of Picard group Z defined as the closure of the G 1 -orbit of a sum of highest weight vectors in P(V ( α 0 ) ⊕ V ( α 1 )).
We conclude that if X is not a product, X is as in Case (1a) (with a 1 > 0).
From now on, we suppose that there is no 0 ≤ i < j ≤ n such that α i and α j are simple roots of the same simple group among G 1 , . . . , G t .
• Suppose that there exists 0 ≤ i < j ≤ n such that a i = a j and both α i and α j are not simple roots of G 0 .
Up to reordering, assume that α i and α j are simple roots of G 1 and G 2 (t ≥ 2). Note that if i = 0 and α 0 is trivial, G 1 = {1}. By Lemma 4.8, G 1 and G 2 are {1}, C * (d k = 1 in these two cases), SL d k (with d k ≥ 2) or Sp d k (with d k ≥ 2 even) and α i , respectively α j , is either a trivial root or a short extremal root of G 1 , respectively G 2 .
Let G = G 0 × G 3 × · · · × G t × SL d 1 +d 2 . By Lemma 4.9 ((a) if i > 0 or α 0 is not trivial and (b) otherwise), the And X is a subvariety of the closure X of the G-orbit of a sum of highest weight vectors in P under the action of G.
Then we can replace, without changing X, the product of the two simple groups corresponding to two simple roots α i and α j with a i = a j , with a unique simple group of type A. Note that n decreases by this change. (Also note that, if i = 0 and α 0 is trivial then the new α 0 is not trivial any more.) With similar arguments, we can also replace any group G 1 , . . . , G t , of type C and that contains a unique simple root α i , by a group of type A.
• What we did just above also works in the cases where n = 1, a 1 = 0, α 0 and α 1 are simple roots of G 1 and G 2 (and t = 2). In that case, this proves that X is the closure of the SL d ×G 0 -orbit of a highest weight vector in P C d V ( β ) . Hence, in that case, X is isomorphic to P d−1 × G 0 /P ( β ). Hence, we conclude the proof by iteration.

Proof of Theorem 1.1 in Case (2)
The first part is already proved by Proposition 3.8 and Lemma 4.8, in particular X is embedded as the closure of the G-orbit of a sum of highest weight vectors in where the sum is taken over all s + 2-tuples of non-negative integers (i, b 1 , . . . , b s+1 ) such that 0 ≤ i ≤ r and s+1 j=1 b j = 1 + a i . It remains to prove that we can suppose that * s = 1, α n , α n+1 are both simple roots of G t and (G t , α n , α n+1 ) is smooth of two-orbit type; * 0 < a 1 < · · · < a r ; * and some groups G k of type C can be replaced by groups of type A.
• Suppose first that s > 1, or s = 1 and α n , α n+1 are not simple roots of the same simple group G k . Up to reordering and applying Lemma 4.8, for any j ∈ {1, . . . , s}, α r+j is either a trivial root of G t−s+j that is C * or {1}, or a short extremal simple root of G t−s+j that is of type A or C. Moreover, the simple groups G t−s+1 , . . . , G t contain no other α i with i ∈ {0, . . . , r}. Also, G t−s+j = {1} if and only if j = s and α r+s is trivial.
We now apply Lemma 4.11 ((a) if α r+s is not trivial and (b) otherwise). Hence, there exists d ≤ 2 such that, with G ⊂ G := G 0 × · · · × G t−s × SL d , we have X is a subvariety of the closure X of the G-orbit Ω X of a sum of highest weight vectors in P, and dim((G t+1−s × · · · × G t )/P ∩ (G t+1−s × · · · × G t ) = d − s − 1. In particular the dimension of Ω X (which is horospherical of rank r) equals the dimension of G/H. Hence, X = X. Now remark that X is a horospherical variety as in Case (1).
• From now on, we suppose that s = 1 (and n = r + 1), and that α n , α n+1 are both simple roots of G t (up to reordering). In particular, X is of type X 2 (G, α, a) and then is embedded as the closure of the G-orbit of a sum of highest weight vectors in Note now that for any k ∈ {0, . . . , t}, G k = {1} if and only if k = 0 and α 0 is trivial.
Recall that, by Lemma 4.8, α 0 , . . . , α r are not simple roots of G t and the triple (G t , α n , α n+1 ) is smooth. Then X is embedded as the closure of the G-orbit of a sum of highest weight vectors in If (G t , α n , α n+1 ) is not of two-orbit type, we can apply Lemma 4.11 ((c), (d) or (e)): we get G ⊂ G with ((1 + a i ) ) , X is a subvariety of the closure X of the G-orbit Ω X of a sum of highest weight vectors in P, and dim(G t /P ∩ G t ) + 1 = dim(G t /P ( )).
In particular the dimension of Ω X (which is horospherical of rank r) equals the dimension of G/H. Hence, X = X. And remark that X is a horospherical variety as in Case (1).
• Now suppose that r > 1, or r = 1 and α 0 , α 1 are not simple roots of the same simple group. Let i i in {0, . . . , r} such that a i = a i . Up to reordering and applying Lemma. 4.8, α i and α i are, trivial or short extremal, simple roots respectively of G 0 and G 1 that are C * , {1} or simple groups of type A or C. Moreover G 0 and G 1 contain no other α k 's.
We can apply Lemma 4.9 ((a) if i > 0 or α 0 is trivial and (b) otherwise) to get G ⊂ G := SL d ×G 2 · · · × G t such that X is a subvariety of the closure X of the G-orbit Ω X of a sum of highest weight vectors in P, and dim((G 0 × G 1 )/P ∩ (G 0 × G 1 )) + 1 = d − 1. In particular the dimension of Ω X (which is horospherical of rank (r − 1) + 1) equals the dimension of G/H. Hence, X = X. Now remark that X is either a horospherical variety as in Case (2) of rank one less than X, or a horospherical variety as in Case (1) if r = 1.
With similar arguments, we can also replace any group G 0 , . . . , G t−1 , of type C and that contains a unique simple root α i , by a group of type A.
• By iteration of the above process, we can now assume that 0 < a 1 < · · · < a r , or that r = 1 (and t = 1) and α 0 , α 1 are two simple roots of G 0 . In the second case, note that by Lemma. 4.8, the triple (G 0 , α 0 , α 1 ) is smooth.
Suppose r = 1, α 0 , α 1 are two simple roots of G 0 and that a 1 = a 0 = 0. Then, X is the closure of the G 0 × G 1 -orbit of a sum of highest weight vectors in Hence in that case, X is the product of two varieties: the closure of the G 0 -orbit of a sum of highest weight vectors in P (V G 0 ( α 0 ) ⊕ V G 0 ( α 1 )) and the closure of the G 1 -orbit of a sum of highest weight vectors in Hence, in any case we can assume that 0 < a 1 < · · · < a r . This finishes the proof of Theorem 1.1.

The MMP and Log MMP for smooth projective horospherical varieties of Picard group Z 2
The main goal of this section is to prove Theorem 1.3. For this we apply the Log MMP from the horospherical varieties X 1 and X 2 .
The principle of the Log MMP is the following. We begin with a pair (X, ∆) where X is a not too singular projective variety and ∆ is a Q-divisor such that K X + ∆ is Q-Cartier. We want to contract curves having negative intersection with K X + ∆ in order to get a new variety with smaller Picard number. In general, we can do this by choosing an extremal ray (whose curves have negative intersection with K X + ∆) in the cone of effective curves up to numerical equivalence.
In our context, note that this cone is two dimensional and then has two extremal rays; this explains why we have two ways to do the Log MMP.
After contracting a curve it may happen that the new variety is too singular, so that we have to partially desingularize it in a natural and unique way; we call this a flip.
To continue the program, we have to choose again an extremal ray in the cone of effective curves of the new variety, until we finish with a minimal model (when there is no curve with negative intersection with K X + ∆) or a fibration (when the dimension decreases).
For horospherical varieties, we can compute a Log MMP to the end just by choosing an ample divisor at the beginning (and not an extremal ray at each step), and by considering a one-parameter family of polytopes (Theorem 2.18).

Generalities
Let X be a smooth projective horospherical variety with Picard group Z 2 . Here, we suppose that X is as in Case (1) or (2) of Lemma 3.1 (or Theorem 1.1).
By Proposition 3.4, up to linear equivalence, the ample Cartier divisors of X are of the form D = d 0 D 0 + d n+1 D n+1 with positive integers d 0 and d n+1 .
We can apply [Pas15] to the polarized variety (X, D) and obtain a description of the MMP from X, via moment polytopes (if X is Fano, we obtain two different paths of the program depending on the choice of d 0 and d n+1 ; if X is not Fano, we obtain a unique path of the program).
Moreover, we can also choose a B-stable Q-divisor ∆ of X and apply [Pas17] to the polarized pair ((X, D), ∆) and obtain a description of the Log MMP from (X, ∆), via moment polytopes as described in Section 2.2. To get a uniform Log MMP for any smooth projective horospherical variety with Picard group Z 2 , we choose D = D 0 + D n+1 and ∆ = −D i − K X for i ∈ {0, n + 1}.
Remark 5.1. In Case (1), an anticanonical divisor of X is (see for example [Pas08, Proposition 3.1]) In particular, X is Fano (i.e., −K X ample) if and only if b β > n i=1 a i b i . To describe the MMP from X we could choose the ample divisor D = ( n i=0 b i )D 1 + (b β + 1)D β , so that D + K X is ample for any ∈ [0, 1[ and D + K X ∼ ( n i=0 a i b i + 1)D β is not ample but globally generated. Then, for that choice of D, the MMP from X consists of the Mori fibration to G/P ( β ) described in Remark 3.3. Moreover, this Mori fibration is also the unique contraction of the Log MMP obtained with the choices D = D 0 + D n+1 and ∆ = −D 0 − K X in Theorem 2.18 (in that case, Q 1 is a multiple of β ).
In Case (2), an anticanonical divisor of X is is the color D α i (respectively D r+j is the color D α r+j ). In particular, X is Fano if and only if the inequality s+1 j=1 b r+j > r i=0 a i b i is s. To describe the MMP from X we could choose the ample divisor D = ( r i=0 b i )D 0 + (1 + s+1 j=1 b r+j )D n+1 , so that D + K X is ample for any ∈ [0, 1[ and D + K X ∼ (1 + r i=0 a i b i )D n+1 is not ample but globally generated. Then, for that choice of D, the MMP from X consists of the Mori fibration ψ from X to Z described in Remark 3.3. Moreover, this Mori fibration is also the unique contraction of the Log MMP obtained with the choices D = D 0 + D n+1 and ∆ = −D 0 − K X in Theorem 2.18 (in that case, Q 1 is a simplex of dimension s).
Hence, in both cases, we will describe the Log MMP obtained with the choices D = D 0 + D n+1 and ∆ = −D n+1 − K X .
In the next four subsections, X is one the varieties of Theorem 1.1 in Case (1) or (2). We begin by constructing the families of polytopes for the log pairs (X, ∆ = −D n+1 − K X ) with the choice of ample divisor D = D 0 + D n+1 , and then we describe in detail the Log MMP's obtained with these families.
We draw, in Figure 3, these polytopes for = 0 in different cases with the hyperplane H 0 := {x ∈ M Q | a 1 x 1 + a 2 x 2 = −1}. Note that there is no such hyperplane if a 2 = a 1 = 0.
• If a n = 0,Q =Q 0 for any ∈ [0, 1] and it is empty if > 1. Moreover, for any ∈ [0, 1], Q intersects the interior of X(P ) + Q if and only if < 1. In that case, the Log MMP described by the family (Q ) ∈Q ≥0 consists of a fibration φ 0 : Figure 3. The polytopesQ 0 in the cases where a 1 = 1 and a 2 = 2, a 1 = 0 and a 2 = 1 and a 1 = a 2 = 1 respectively The fibers of this fibration can be easily computed (by the strategy given in Section 2.2) because the faces of Q 0 are "the same" as the faces of Q 1 and then the fibration induces a bijection between the sets of G-orbits of X and Y 0 . Then the fibers of φ 0 are isomorphic to the homogeneous projective spaces ( i∈I P ( α i ))/(P ( β ) ∩ i∈I P ( α i )) (of Picard group Z), with ∅ I ⊂ {0, . . . , n}. Here, we use the following notation: if α i is trivial, P ( α i ) = G (and otherwise, it is the proper maximal parabolic subgroup of G associated to α i ).
In the next proposition, we give a description of the non-empty faces ofQ by distinguishing whether a face is in the hyperplane H or not.
Note Proposition 5.3 (recall that a 0 = 0 and that a n 0 here).
The polytopeQ is of dimension n if and only if < max n i=0 (1 + a i ) = 1 + a n . Suppose now that < 1 + a n . The non-empty faces ofQ are the distinct F I and F I,β (with I {0, . . . , n}) defined as follows: In particular, the facets ofQ are: F i with i ∈ {0, . . . , n − 1} (for any < 1 + a n ), F n if < 1 + a n−1 , F ∅,β if > 1, and F n,β if = 1 and a n−1 = 0. Moreover, we can write any face ofQ as the intersection of all the facets that contain it, as follows. For any I {0, . . . , n} such that < max i I (1 + a i ), F I = i∈I F i . For any I {0, . . . , n} such that min i I (1 + a i ) < < max i I (1 + a i ), F I,β = F ∅,β ∩ i∈I F i . For any I {0, . . . , n} such that = min i I (1 + a i ) = max i I (1 + a i ), F I,β = F n,β ∩ i∈I F i if = 1, n ∈ I and a n−1 = 0 or F I,β = i∈I F i if 1, n I or a n−1 0. . , n} such that a i > − 1 if and only if a n > − 1 (because 0 = a 0 ≤ · · · ≤ a n ). This proves the first statement of the proposition. Suppose now that < 1 + a n . For any non-empty face F ofQ , either F H and F is the intersection of a non-empty face ofQ with H + , or F ⊂ H and F is the intersection of a non-empty face ofQ with H .
Let  (1 + a i ). Also, in that latter case, the dimension of F I is the same as the dimension of F I ; in particular the non-empty F I that are not included in H are all distinct.
Similarly, F I,β is not empty if and only if there exist i and j not in I (may be equal) such that e * i ∈ H + and e * j H ++ (i.e., a i ≥ −1 and a j ≤ −1). Then To describe the facets, it is sufficient to find the F i with < max j i (1 + a j ), the F i,β with equal to both min j i (1+a j ) and max j i (1+a j ), and F ∅,β with 1 = min n i=0 (1+a i ) < < max n i=0 (1+a i ) = 1+a n . We easily find the F i with i ∈ {0, . . . , n − 1} for any < 1 + a n , and F n for any < 1 + a n−1 . We conclude by noticing that, for any i ∈ {0, . . . , n}, we have = min j i (1 + a j ) = max j i (1 + a j ) < 1 + a n if and only if i = n and 0 = a 0 = · · · = a n−1 (and in particular, = 1).
To get the last statement, apply the fact that any face of a polytope is the intersection of the facets containing it.
, if a n−1 = a n ) or the simple root α n is not trivial (i.e., when X is as in Case (1b) of Theorem 1.1); , if a n−1 < a n ) and the simple root α n is trivial (i.e., when X is as in Case (1c) of Theorem 1.1).
Proof. We apply the theory described in Section 2.2, in particular the fact that the isomorphism classes of the varieties X are obtained by looking at the 's for which "the faces of Q change". Note first that, by Proposition 5.3, (P , M, Q ,Q ) is an admissible quadruple if and only if < 1 + a n . Also, the facets ofQ are: F i with i ∈ {0, . . . , n − 1}, F n if < 1 + a n−1 , F ∅,β if > 1, and F n,β (orthogonal to α ∨ n,M ) if = 1 and a n−1 = 0. In particular, for any , η ∈ [0, 1 + a n [, if a n−1 0, the facets of Q and Q η are "the same" if and only if and η are both in [0, 1] or ]1, 1 + a n−1 [ or [1 + a n−1 , 1 + a n [ (which may be empty). And if a n−1 = 0, the facets of Q and Q η are "the same" for any , η ∈ [0, 1 + a n [ (indeed, in that case, the facets F n if < 1, F ∅,β if > 1, and F n,β if = 1 are "the same", in particular all orthogonal to β ∨ M = a n α ∨ n,M ). Figure 4. The Log MMP described by the polytopesQ in the case where n = 2, a 1 = 1, a 2 = 2 and α 2 is not trivial.
We now use a consequence of the proof of Proposition 5.3: for any I {0, . . . , n}, i∈I F i is not empty if and only if ≤ max i I (1 + a i ), F ∅,β ∩ i∈I F i is not empty if and only if min i I (1 + a i ) ≤ ≤ max i I (1 + a i ) and F n,β ∩ i∈I F i is not empty if and only if min i I (1 + a i ) = = max i I (1 + a i ). In particular for any l ∈ {0, . . . , k − 2}, suppose that for I = {i l+1 , . . . , n} and that i∈I F i is not empty; suppose also that for I = {0, . . . , i l − 1} and that F ∅,β ∩ i∈I F i is not empty; then = 1 + a i l . Similarly for any l ∈ {0, . . . , k − 2}, suppose that for I = {i l+1 − 1, . . . , n} and i∈I F i is not empty; suppose also that for I = {0, . . . , i l − 1} and that F ∅,β ∩ i∈I F i is not empty; then ∈ [1 + a i l , 1 + a i l+1 ]. If i k n, F n is still a facet of Q and what we did above with l ∈ {0, . . . , k − 2} can be done as well with l = k − 1.
Hence, this proves that if the two varieties X and X η are isomorphic then and η are in one of the subsets described in the corollary.
To conclude, we have to prove that the two varieties X and X η are isomorphic when and η are in one of these subsets. It is obvious from Proposition 5.3 except in the case where i k = n and the simple root α n is trivial. But in that case, all polytopes Q with ∈ [1 + a n−1 , 1 + a n [= [1 + a i k−1 , 1 + a i k [ are simplexes with facets F i for i ∈ {0, . . . , n − 1} and F ∅,β or F n,β if = 1 + a n−1 = 1, i.e., they could be defined even deleting the row corresponding to the simple root α n that is trivial, so that their faces are "the same".
We can reformulate this corollary as follows, and get the first statement of Theorem 1.3 in Case (1). We denote X 0 = X and for any l ∈ {1, · · · , k}, X l := X with ∈]1 + a i l−1 , 1 + a i l [, and for any l ∈ {0, · · · , k}, Y l := X 1+a i l .
Example 5.6. In the five different cases with n = 2 and a 2 0, we illustrate this corollary in terms of polytopes in Figures 4, 5, 6, 7 and 8.

Proof of the last statement of Theorem 1.3 in Case (1)
The previous section proves that a i 1 , . . . , a i k are invariants of X. To finish the proof of Theorem 1.3 in Case (1), we have to prove that G 0 , . . . , G t , α 0 , . . . , α n , β and i 1 , . . . , i k are also invariants of X. For this, we have to describe some exceptional loci and some fibers of the different morphisms of the Log MMP. Figure 5. The Log MMP described by the polytopesQ in the case where n = 2, a 1 = 1, a 2 = 2 and α 2 is trivial. Figure 6. The Log MMP described by the polytopesQ in the case where n = 2, a 1 = 0, a 2 = 1 and α 2 is not trivial.
We first distinguish two cases by the following result.
Proposition 5.7. Define the simple subgroups of P ( β ) as in Definition 4.3. * Suppose that n = 1 and that α 0 and α 1 are two simple roots of the same simple subgroup of P ( β ). Then, the fiber of ψ : X −→ G/P ( β ) is either a homogeneous variety different from a projective space (a quadric Q 2m with m ≥ 2, a Grassmannian Gr(i, m) with p ≥ 5 and 2 ≤ i ≤ m − 2, or a spinor variety Spin(2m + 1)/P ( m ) with m ≥ 4), or a two-orbit variety as in [Pas09]. * Suppose that n > 1 or that α 0 and α 1 are not two simple roots of the same simple subgroup of P ( β ).
Then, the fiber of ψ : X −→ G/P ( β ) is a projective space.
Proof. The fiber of ψ : X −→ G/P ( β ) is the smooth projective P ( β )-variety of Picard group Z isomorphic to the closure of the P ( β )-orbit of a sum of highest weight vectors in P := P(V ( α 0 )⊕· · ·⊕V ( α n )). Hence, the proposition is a consequence of [Pas09, Section 1]. • In the case where n = 1 and that α 0 and α 1 are two simple roots of the same simple subgroup of P ( β ), G = G 0 , the Log MMP described by Corollary 5.5 consists of a fibration if a 1 = 0, or a flip and a fibration if a 1 > 0.
-Suppose first that a 1 = 0. There are two cases to deal with.
In particular, the pair (G/P ( α 0 ), G/P ( β )) is an invariant of X. Then if G 0 is not the universal cover of the automorphism group of G/P ( β ) it must be the universal cover of the automorphism group of G/P ( α 0 ), so that G 0 is an invariant of X. Also, φ −1 0 (G/P ( α 0 )) = G/(P ( α 0 ) ∩ P ( β )), then the pair (α 0 , β) is an invariant of X up to symmetries of the Dynkin diagram of G 0 .
Moreover, if β is fixed, the possible symmetries are the ones (which fixed β) in type A m with m ≥ 5 odd, β = m+1 2 and any α 0 , type E 6 with β = 4 and α 0 = 1 , 3 , 5 or 6 , and type D m with m ≥ 4, β = i for any i ∈ {1, . . . , m − 2} and α 0 = m−1 or m . The description of the fiber of ψ : X −→ G/P ( β ), together with Remark 4.2, implies that α 0 and α 1 are also invariants of X up to symmetries of the Dynkin diagram of G 0 .
Otherwise (this occurs only in types D and E), G 0 is the universal cover of the automorphism group of G/P ( β ), and then G 0 and β are invariants of X up to symmetries of the Dynkin diagram of G 0 .
In particular, the pair (G/P ( α 0 ), G/P ( α 1 )) is an invariant of X and then the pair (α 0 , α 1 ) is also an invariant of X up to symmetries of the Dynkin diagram of G 0 .
In particular X, Y 0 and X 1 have two closed G-orbits and one open G-orbit so that we easily compute exceptional locus and fibers as follows. For example, the exceptional locus of φ 0 : X −→ Y 0 is the G-orbit of X isomorphic to G/(P ( α 0 ) ∩ P ( β )). Then the universal cover of its automorphism group G 0 is an invariant of X. And then β is also an invariant of X up to symmetries of the Dynkin diagram of G 0 .
Note now that the exceptional locus of φ 0 is sent to the G-orbit of Y 0 isomorphic to G/P ( α 0 ) so that the triple (G/P ( α 0 ), G/P ( α 1 ), G/P ( β )) is an invariant of G. Also the (same) description of the fiber of ψ : X −→ G/P ( β ) implies that the subgroup or P ( β ) and the pair (α 0 , α 1 ) are invariants of X (up to symmetries in type A, D and E as in the case where a 1 = 0). Hence, the triple (β, α 0 , α 1 ) is an invariant of X up to symmetries of the Dynkin diagram of G 0 .
• Now we suppose that n > 1 or that α 0 and α 1 are not two simple roots of the same simple subgroup of P ( β ).
We define various exceptional loci in X as follows. Let l ∈ {0, . . . , k − 1}, define E l to be the closure in X of the set of points x ∈ X such that x is in the open isomorphism set of the first l contractions and x is in the exceptional locus of φ l .
Proposition 5.8. For any l ∈ {0, . . . , k} the exceptional locus E l is the closure in X of the G-orbit associated to the non-empty face F I l of Q with I l := {i l+1 , . . . , n}. In particular E l is isomorphic to the closure of the G-orbit of a sum of highest weight vectors in and E l is a smooth projective horospherical of Picard group Z 2 as in Case (1), unless l = 0, i 1 = 1 so that E l is homogeneous (projective of Picard group Z or Z 2 ).
Note that for l = k, I k = ∅ and E k = X.
We denote by Ω l I and Ω l I,β the G-orbits of X l associated to the non-empty faces F l I and F l I,β of the polytopeQ l . We denote by ω l I and ω l I,β the G-orbits of Y l = X 1+a i l associated to the non-empty faces F In particular, we have φ l (Ω l I l ) = ω l I l ∪{0,...i l −1},β (which is also φ l (Ω l I l ,β ) if l ≥ 1). But Ω l I l and ω l I l ∪{0,...i l −1},β are non isomorphic horospherical homogeneous spaces by Proposition 2.14, so that Ω l I l is in the exceptional locus of φ l . Moreover, if Ω is a G-orbit of X l not contained in Ω l I l , it is of the form Ω l I or Ω l I,β where I l I. Hence, in that case φ l (Ω) = Ω. And then the exceptional locus of φ l is Ω l I l . Note that Ω 0 I l , . . . , Ω l−1 I l are not in the exceptional locus of φ 0 , . . . , φ l−1 respectively, to conclude that E l = Ω 0 I l . We use again Proposition 2.14 to see that E l = Ω 0 I l corresponds to the admissible quadruple (P F , M F , F,F) with F = F 0 I l (and with some ample divisor of E l ). Then we conclude by Corollaries 2.6 and 2.10.
Definition 5.9. We say that the fibers ofφ l are locally maximal over ω ⊂ E l if the dimensions of the fibers ofφ l over any point of ω are the same and bigger than the dimension of the fibers ofφ l over any point of a neighborhood of ω that is not in ω.
We say that the fibers ofφ l are locally almost maximal over ω ⊂ E l if there exists ω ω such that the fibers ofφ l are locally maximal over ω and the fibers ofφ l |φ l −1 (E l \ω ) are locally maximal over ω\ω ⊂ E l \ω .
We now prove the following result, which implies in particular that i 1 , . . . , i k are invariant of X.
The mapφ l is surjective and we distinguish four distinct cases.
(1) we have i l+1 − i l = 1 and α i l is not a simple root of G 0 . The fibers ofφ l are locally maximal over E l and dim E l − dim E l−1 = 1 + dim E l (here we set dim E −1 := dim G/P ( β ) − 1 so that it still holds for l = 0). Moreover, E l is homogeneous and isomorphic to G/P ( α i l ) (which is a point if α i l is trivial).
(2) we have i l+1 − i l = 1 and α i l is a simple root of G 0 . The fibers ofφ l are locally maximal over E l and dim E l − dim E l−1 1 + dim E l (also here dim E −1 := dim G/P ( β ) − 1 so that it still holds for l = 0). Moreover, E l is homogeneous and isomorphic to G/P ( α i l ).
(3) we have i l+1 − i l > 1 and α i l is not a simple root of G 0 . The fibers ofφ l are locally maximal over a unique proper subset of E l , which is a closed G-orbit ω of E l isomorphic to G/P ( α i l ). Also the fibers of φ l are locally almost maximal over exactly i l+1 − i l − 1(> 0) subvarieties of E l containing ω , respectively of dimensions dim G/P ( α i l ) + dim G/P ( α j ) + 1 with j ∈ {i l + 1, . . . , i l+1 − 1}. (4) we have i l+1 − i l > 1 and α i l is a simple root of G 0 . The fibers ofφ l are locally maximal over i l+1 − i l closed G-orbits, which are respectively isomorphic to G/P ( α j ) with j ∈ {i l , . . . , i l+1 − 1}.
Moreover, in the four cases, the dimension of the fibers over all pointed subsets of E l are as follows.
(2) The dimension of the fibers ofφ l is (3) The dimension of the locally maximal fibers ofφ l is 1 + dim E l−1 (in particular dim G/P ( β ) if l = 0). And for any j ∈ {i l + 1, . . . , i l+1 − 1}, the dimension of locally almost maximal fibers ofφ l over the subset of E l of dimension dim G/P ( α i l ) + dim G/P ( α j ) + 1 is (4) For any j ∈ {i l , . . . , i l+1 −1}, the dimension of locally maximal fibers ofφ l over the closed G-orbit isomorphic to G/P ( α j ) is Proof. We so that the dimension δ l,ω of a fiber ofφ l over ω is The dimension δ l,ω is the biggest when I is as big as possible (it would be I = {0, . . . , n} if it was allowed to define ω). Moreover, if we remove from I some i, the dimension changes if and only if j is such that α i is in G 0 (i.e., α i is not trivial and not the only simple root α j in a simple group of G different from G 0 , by hypothesis). From this, we will deduce the different following cases. We easily deduce the following.
Corollary 5.11. With the notation of Proposition 5.10: for any j ∈ {0, . . . , n}, we have In particular, for any l ∈ {0, . . . , k}, the sets And then we conclude the proof of Case (1) of Theorem 1.3 (i.e., that G 0 , β, α 0 , . . . , α n are invariants of X) by the following lemma (still in the case where n > 1 or that α 0 and α 1 are not two simple roots of the same simple subgroup of P ( β )).
Lemma 5.12. Let G, G be two products of simply connected simple groups and C * 's. Let β, β be two simple roots of two of the simple factors G 0 and G 0 of G and G respectively. And let α 0 , . . . , α n , respectively α 0 , . . . , α n be simple roots of G, G both as in Case (1) of Theroem 1.1 (with the same integers k and i 1 , . . . , i k ).
Proof. We proceed in several steps.
Step 1. For any l ∈ {0, . . . , k}, α i l R 0 if and only if α i l R 0 , and in that case, α i l and α i l are both extremal simple roots of SL m+1 with m = dim P ( β )/(P ( β ) ∩ P ( α j )) = dim P ( β )/(P ( β ) ∩ P ( α j )). Indeed, we have that α i l R 0 if and only if and this is equivalent to saying that α i l R 0 . The second statement is obvious from the hypothesis on the α i 's and α i 's. Note that α i l +1 , . . . , α i l+1 −1 are in R 0 by hypothesis. Step
In particular,Q is the intersection of the closed half-space H + := {x ∈ M Q | a 1 x 1 +· · ·+a r x r −x r+1 ≥ −1} withQ 0 . We denote by H ++ the interior of H + and by H the hyperplane H + \H ++ .
Example 5.13. If n = 2 (i.e., r = s = 1) we have a 1 > 0, and either α 1 is trivial or not. We draw, in Figure 9, such a polytope for = 0 with the hyperplane H 0 :  Remark that, if = min i I (1 + a i ) = max i I (1 + a i ), then I = {0, . . . , r}\{i} where i is such that = 1 + a i . Note also thatQ 1+a r is the point u * r so that Q 1+a r is the point α r .  (1 + a i ). Hence, the dimension of F I,2 is the dimension of F I minus 1 if < max i I (1 + a i ) and it equals the dimension of F I if = max i I (1 + a i ). In the first case, the F I,2 are all distinct and yield all non-empty faces ofQ included in H but not in F ∅,1 . In the second case, To get the last statements, use again the fact that a facet is a face of codimension 1 and that any face of a polytope is the intersection of the facets containing it.
From Proposition 5.14, we deduce the following result. Proof. We apply the theory described in Section 2.2, in particular the fact that the isomorphism classes of the varieties X are obtained by looking at the 's for which "the faces of Q change". Note first that, by Proposition 5.14, (P , M, Q ,Q ) is an admissible quadruple if and only if < 1 + a r . Also, the facets ofQ are: F i (orthogonal to α ∨ i,M ) with i ∈ {0, . . . , r − 1}, F r (orthogonal to α ∨ r,M ) if < 1 + a r−1 , F ∅,1 (orthogonal to α ∨ r+1,M ) and F ∅,2 (orthogonal to α ∨ r+2,M ). In particular, for any , η ∈ [0, 1 + a r [, the facets of Q and Q η are "the same" if and only if and η are both in [0, 1 + a r−1 [ or [1 + a r−1 , 1 + a r [.
We now use a consequence of the proof of Proposition 5.14: for any I {0, . . . , r}, i∈I F i is not empty if and only if ≤ max i I (1 + a i ), F ∅,1 ∩ i∈I F i is not empty if and only if ≤ max i I (1 + a i ), F ∅,2 ∩ i∈I F i is not empty if and only if ≤ max i I (1 + a i ), and F ∅,1,2 ∩ i∈I F i is not empty if and only if we have min i I (1 + a i ) ≤ ≤ max i I (1 + a i ). In particular, for any i ∈ {0, . . . , r − 2}, suppose that for I = {i + 1, . . . , r} and that i∈I F i is not empty; suppose also that for I = {0, . . . , i − 1} and that F ∅,1,2 ∩ i∈I F i is not empty; then = 1 + a i . Similarly for any i ∈ {0, . . . , r − 2}, suppose that for I = {i + 2, . . . , n} and that i∈I F i is not empty; suppose also that for I = {0, . . . , i −1} and that F ∅,1,2 ∩ i∈I F i is not empty; then ∈ [1+a i , 1+a i+1 ].
Hence, this proves that if two varieties X and X η are isomorphic then and η are a one of the subsets described in the corollary.
To conclude, we have to prove that the two varieties X and X η are isomorphic when and η are in one of these subsets. It is obvious from Proposition 5.14 except in the case where the simple root α n is trivial. But in that case, all polytopes Q with ∈ [1 + a r−1 , 1 + a r [ could be defined even deleting the row corresponding to the simple root α r that is trivial, so that their faces are "the same" (they are simplexes with facets F i for i ∈ {0, . . . , r − 1}, F ∅,1 and F ∅,2 ).
We can reformulate this corollary as follows, and get the first statement of Theorem 1.3 in Case (2). We denote X 0 = X and for any i ∈ {1, · · · , r}, X i := X with ∈]1 + a i−1 , 1 + a i [ and for any i ∈ {0, · · · , r}, Y i := X 1+a i .
Corollary 5.16. The family (Q ) ∈Q ≥0 describes a Log MMP from X as follows: * r flips φ i : X i −→ Y i ←− X i+1 : φ + i for any i ∈ {0, · · · , r − 1} and a fibration φ r : X r −→ Y r , if the simple root α r is not trivial; , · · · , r − 2}, followed by a divisorial contraction φ r−1 : X r−1 −→ Y r−1 X r and a fibration X r −→ Y r pt, if the simple root α r is trivial.
Example 5.17. In the two different cases with n = 2 and a 1 = 2, we illustrate this corollary in terms of polytopes in Figures 10 and 11.

Proof of the last statement of Theorem 1.3 in Case (2)
The previous section proves that a 1 , . . . , a r are invariants of X. To finish the proof of Theorem 1.3 in Case (2), we have to prove that G 0 , . . . , G t and α 0 , . . . , α r+2 are also invariants. Since the "first" Log MMP consists of a fibration ψ : X −→ Z where Z is a two-orbit variety embedded in P(V ( α r+1 ) ⊕ V ( α r+2 )) as in [Pas09], G t , α r+1 and α r+2 are invariants of X. As in Case (1), we will describe some exceptional loci and some fibers of different morphisms of the Log MMP, but we first distinguish two cases by the following result. Proposition 5.18. Suppose that r = 1 and that α 0 and α 1 are two simple roots of G 0 (and then t = 1). Then, the general fiber of ψ : X −→ Z is either a homogeneous variety different from a projective space (a quadric Q 2m with m ≥ 2, a Grassmannian Gr(i, m) with m ≥ 5 and 2 ≤ i ≤ m − 2, or a spinor variety Spin(2m + 1)/P ( m ) with m ≥ 4), or a two-orbit variety as in [Pas09]. Suppose that r > 1 or that α 0 and α 1 are simple roots of G 0 and G 1 respectively. Then, the general fiber of ψ : X −→ Z is a projective space.
• In the case where r = 1 and that α 0 and α 1 are two simple roots of G 0 , G = G 0 ×G 1 and the description of the general fiber of ψ : X −→ G/P ( β ), with Remark 4.2, implies that G 0 , α 0 and α 1 are invariants of X.
• Now we suppose that r > 1 or that α 0 and α 1 are not two simple roots of the same simple subgroup of P ( β ).
We define various exceptional loci in X as follows. Let i ∈ {0, . . . , r}, define E i to be the closure in X of the set of points x ∈ X such that x is in the open isomorphism set of the first i contractions and x is in the exceptional locus of φ i .
Proposition 5.19. For any i ∈ {0, . . . , r} the exceptional locus E i is the closure in X of the G-orbit associated to the non-empty face F I i with I i := {i + 1, . . . , r}. In particular E i is isomorphic to the closure of the G-orbit of a sum of highest weight vectors in hence for i ∈ {1, . . . , r}, E i is a smooth projective horospherical variety of Picard group Z 2 as in Case (2), and E 0 is the product a two-orbit variety with a homogeneous variety (projective of Picard group Z).
Note that E r = X and that in any case, the rank of the horospherical G-variety E i is i + 1.
We denote by Ω i I , Ω i I,1 , Ω i I,2 and Ω i I,1,2 the G-orbits of X i associated to the non empty faces F i I , F i I,1 , F i I,2 and F i I,1,2 of the polytopeQ i . We denote by ω i I , ω i I,1 , ω i I,2 and ω i I,1,2 the G-orbits of Y i = X 1+a i associated to the non-empty faces F In particular, we have φ i (Ω i I i ) = ω i I\{i},1,2 . But Ω i I i and ω i {0,...,r}\{i},1,2 are non-isomorphic horospherical homogeneous spaces by Proposition 2.14, so that Ω i I i is in the exceptional locus of φ l . Moreover, if Ω is a G-orbit of X i not contained in Ω i I i , it is of the form Ω i I , Ω i I,1 , Ω i I,2 or Ω i I,1,2 where I i I. Hence, in that case φ i (Ω) = Ω. And then the exceptional locus of φ i is Ω i I i . Note that Ω 0 I i , . . . , Ω l−1 I i are not in the exceptional locus of φ 0 , . . . , φ i−1 respectively, to conclude that E i = Ω 0 I i . We use again Proposition 2.14 to see that E i = Ω 0 I i corresponds to the admissible quadruple (P F , M F , F,F) with F = F 0 I i (and with some ample divisor of E i ). Then we conclude by Corollaries 2.6 and 2.10.
Corollary 5.21. The dimension of the fibers ofφ i is In particular the dimensions d j of the G/P ( α j )'s, which are projective space under G i = SL d j +1 , are invariants of X.
Proof. Since r > 1, or r = 1 and α 0 , α 1 are not two simple roots of the same simple subgroup of G, the simple roots α 0 , . . . , α r are respectively the first simple roots of G 0 , . . . , G r that are of type A. (And α r+1 , α r+2 are simple roots of G r+1 .) Then the corollary can be easily deduced from the proposition.

Appendix
Proposition 6.1. Let (K, β, R, n) be a smooth quadruple. Then we are in one of the following cases, up to symmetries.