D ec 2 01 8 Smooth projective horospherical varieties of Picard group Z 2

We classify all smooth projective horospherical varieties of Picard group Z2 and we give a first description of their geometry via the Log Minimal Model Program Mathematics Subject Classification: 14E30, 14J45, 14M17, 52B20


Introduction
In this paper, varieties are algebraic varieties over C and groups are algebraic groups over C.
Smooth projective horospherical varieties of Picard group Z are known since 2009 [Pas09] and give useful examples in various theories. For toric varieties there are only projective spaces. But for horospherical varieties, in addition to homogeneous spaces, there are 5 families of two-orbit varieties (two of them are infinite families).
Here we classify and give a first study of the geometry of smooth projective horospherical varieties of Picard group Z 2 . For toric varieties, there 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 types of other such horospherical varieties. We classify them in this paper, in particular by studying their Log MMP.
To write as nice 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.
If G is a 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 is 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 * , the simple root α of G denotes the identity endomorphism of C * 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 (n̟ α ) is the C * -module C where C * acts with weight n̟ α ; in particular, any character of C * is dominant. Moreover, if G = {1}, the simple root α of G denotes the trivial morphism from G to C * 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 simple groups, C * and {1}. A simple root of G is a simple root of some G i and it is said to be imaginary if it 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.
We can now write the two main results of this paper. Theorem 1.1. Let X be a smooth projective horospherical variety with Picard group Z 2 . Suppose that X is not the product of two varieties. Then X is isomorphic to one of the following horospherical variety (which we still denote by X).
Let G = G 0 × G 1 × · · · × G t be a product of simply connected simple groups, C * and {1}.
Case (0): t = 0, G 0 is simple and X is an homogeneous variety G 0 /P where P is the intersection of two maximal (proper) parabolic subgroup of G 0 .
Case (1): t ≥ 0 and X is the closure of the G-orbit of a sum of highest weight vectors in where n ≥ max{1, t}, β is a simple root of G 0 , α 0 , . . . , α n are distinct simple roots of G distinct from β and 0 = a 0 ≤ a 1 ≤ · · · ≤ a n are integers, satisfying the following properties.
If R 0 is empty, then G 0 is the universal cover of the automorphism group of G/P (̟ β ).
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 .
And one of the three following cases occurs.
In the two next cases, t = t ′ , 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 imaginary simple root of G k . Moreover, G k is isomorphic to {1} if and only if k = 1, i 1 = 0 and α i 1 is imaginary. (b) The simple root α n is not imaginary or a n−1 = a n .
(c) The simple root α n is imaginary and a n−1 < a n .
Case (2): t ≥ 2 and X is the closure of the G-orbit of a sum of highest weight vectors in where n ≥ 2, 0 = a 0 < a 1 < · · · < a n−1 are integers, and α 0 , . . . , α n+1 are distinct simple roots of G satisfying the following properties.
And one of the three following cases occurs.
In the two next cases: t = n and for any i ∈ {1, . . . , t}, either G i is isomorphic to some SL d i and α i the first simple root of G i , or G i is isomorphic to C * or {1} and α i is the imaginary simple root of G i .
(b) The simple root α n−1 is not imaginary.
(c) The simple root α n−1 is imaginary. Remark 1.2. In Theorem 1.1, the decomposable projective bundles over projective spaces are the horospherical varieties X in Cases (1b) and (1c) with 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 ).
The horospherical varieties described in Theorem 1.1 are all distinct. This is a consequence of the following result. Theorem 1.3. Let X be one of the varieties described in Theorem 1.1. Then "the" Log MMP from X gives the following in each case.
Case (0): There are two Mori fibrations from X, respectively into Y and Z, with (general) fibers respectively not isomorphic to Z and Y .
Case (1): (a) A "first" Log MMP consists of a Mori fibration from X to G/P (̟ β ) with general fibers not isomorphic to a projective space (but isomorphic to another homogeneous variety or to a two-orbit variety) and a "second" one consists of a flip from X followed by a fibration.
(b) A "first" Log MMP consists of a Mori fibration from X to G/P (̟ β ) with general fibers isomorphic to a projective space and a "second" one consists of a finite sequence (may be empty) of flips from X followed by a fibration.
(c) A "first" Log MMP consists of a Mori fibration from X to G/P (̟ β ) with general fibers isomorphic to a projective space and a "second" one consists of a finite sequence (may be empty) of flips from X followed by a divisorial contraction.
Case (2): A "first" Log MMP consists of a fibration ψ to a two-orbit variety, the general fiber F ψ of ψ and a "second" Log MMP are described as follows.
(a) F ψ is not isomorphic to a projective space (but isomorphic to another homogeneous variety or to a two-orbit varity) and a "second" Log MMP consists of a flip from X followed by a fibration.
(b) F ψ is isomorphic to a projective space and a "second" Log MMP consists of a finite sequence (not empty) of flips from X followed by a fibration.
(c) F ψ is isomorphic to a projective space and a "second" Log MMP consists of a finite sequence (may be empty) of flips from X followed by a divisorial contraction.

Remark 1.4.
• In the paper (Proposition 3.3), we prove that for any smooth projective horospherical variety X with Picard group Z 2 , the nef cone of X is generated by the two elements of a basis of Pic(X), then this gives us two canonical ways to choose the log pair to compute Log MMP from X (see Section 5 for more details). Also, in Cases (1) and (2), one of the "two canonical" Log MMP is "naturally" defined (see Remark 3.2) and only consists of a fibration.
• In Case (1b), if the sequence of flips is empty, we get two fibrations from X. They could be both into homogeneous varieties. But one and only one of these fibrations has all its fibers isomorphic to each others. (On the contrary, in Case (0), each fibration has all their fibers isomorphic to each others.) The paper is organized as follows. We first recall in Section 2 the results on horospherical varieties that we use in the paper. Then, in Section 3, we easily describe a first (but not optimal) combinatorial classification, containing many repetitions. In Section 4, we give a first geometric description of all these latter cases in order to reduce the number of cases and prove Theorem 1.1. Then, in Section 5, we prove Theorem 1.3, by studying the Log MMP of all varieties of Theorem 1.1.
2 Some known results on horospherical varieties 2.1 First definitions, first properties of divisors, and smooth criterion 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 cone of dominant characters.
For any lattice L we denote by L Q the Q-vector space L ⊗ Z Q.
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 it 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 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 (ie, simple roots associated to fundamental weights that 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 .
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 couple (C ′ , F ′ ) such that C ′ is a face of C and F ′ is the set of α ∈ F satisfying α ∨ M ∈ C ′ . 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, (ii) and any element of N Q is in the 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]). Complete G/H-embeddings correspond to complete fans, ie, to fans such that N Q is the union of their colored cones.
If G = (C * ) n and H = {1}, we recover the well-known classification of toric varieties. 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.
We now recall 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 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 ). Moreover, if D is a divisor, D is Cartier if and only if it is Q-Cartier and the linear functions defines as above can be identified to elements of M .
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 ). 3. Suppose that D is a Q-Cartier Q-divisor. LetQ D be the polytope in M Q (called pseudo-moment polytope) defined 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 , then the polytope v 0 +Q D is called the moment polytope of D (or (X, D)).

4.
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, ie, of the form m i=1 a i X i + α∈R a α D α . Theorem 2.5. ([Pas06, Theorem 0.3]) Let X be a projective horospherical variety and let D be an ample Cartier divisor of X. Suppose that X is smooth.
Then D is very ample.
Since H ⊃ U and the unique U -stable lines of irreducible G-modules are the lines generated by highest weight vectors, we deduce from Theorems 2.4 and 2.5 the following result. (See also [Pas15,Remark 2.13] to explain why we can ignore duals.) Corollary 2.6. Let X be a smooth projective horospherical variety and let D be an ample Cartier divisor of X. Then X is isomorphic to the closure of the G-orbit of a sum of highest weight vectors in P(⊕ χ∈Q D ∩M V (χ + v 0 )).
From Theorem 2.4, we can also deduce a locally factorial criterion.
Corollary 2.7. A horospherical variety X is locally factorial if and only if for any colored cone (C, F) of F X , C is generated by a basis of N and the map σ : α −→ α ∨ M induces an injective map from F to this basis.
In particular if X is locally factorial, the Picard number of X is given by the following formula where F X (1) is the set of edges (one-dimensional colored cones) of F X .
To write the smooth criterion we need to give the following definition.
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 • there is at most one vertex of Γ in R 2 and, • 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.

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, is 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 2.11. 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 (ie, its interior in M Q is not empty); • the polytope Q intersects the interior of X(P ) + Q .
1. The map (X, D) −→ (P, M, Q,Q) is a bijection from the set of isomorphic 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 homogenous 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).
In particular, we easily get the following consequence.
Corollary 2.13. 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 fix a basis of M (and consider the dual basis for N ). Also we choose an order in {x 1 , . . . , x m } ∪ {α ∨ M | α ∈ R}. Then we define A ∈ M m+|R|,n (Q) whose rows are the coordinates of the vectors of {x 1 , . . . , x m } ∪ {α ∨ M | α ∈ R} in the chosen basis.
Theorem 2.14. Let X be a Q-factorial projective horospherical variety and let ∆ be a B-stable Q-divisor of X. Then for any (general) choice of an ample B-stable Q-Cartier Q-divisor D of X, a Log MMP from the pair (X, ∆) is described by the following oneparameter families of polytopes where B, C and v ǫ = v 0 + ǫv 1 are such that, for any ǫ ≥ 0 small enough,Q ǫ and Q ǫ are respectively the pseudo-moment and moment polytope of (X, D + ǫ(K X + ∆)).
Also, there exists P ′ and M ′ such that (P ′ , M ′ , Q ǫmax ,Q ǫmax ) is an admissible quadruple associated to a variety X ǫmax and such that there is a fibration from X ǫ to X ǫmax with ǫ < ǫ max big enough. Moreover, the general fibers of this fibration is a horospherical variety and can be described.
All morphisms above are G-equivariant and images of any G-orbit can be described as follows. To a face of Q ǫ (orQ ǫ ) we can associate the maximal set of rows for which equality holds for any element x of the face (in the inequalities Ax ≥ B + ǫC). And similarly to a set of rows we can also naturally associate a face of Q ǫ (may be empty). For any ǫ and ǫ i,k as above, for any face F ǫ ofQ ǫ , we construct a face ofQ ǫ i,k by taking the maximal set of rows associated to F ǫ and then the face F ǫ i,k associated to these rows. Then, since there is a morphism φ from X ǫ to X ǫ i,k , the non-empty face F ǫ i,k corresponds to the G-orbit image by φ of the G-orbit corresponding to F ǫ .
3 First combinatorial classification and first geometric description

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.
Lemma 3.1. Let X be a smooth projective horospherical variety with Picard group Z 2 . Then one the three following cases occurs (with notation of Section 2).
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.
We conclude by 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 3.2. 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.
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(G/P ) is not simple.
Case (1): There exists a G-equivariant morphism ψ from X to G/P (̟ β ). Note that the general fiber of ψ is smooth of Picard group Z (in particular, it is homogeneous or one of the two-orbit varieties described in [Pas09]).
Case (2): Let P Z be the parabolic subgroup containing B (and P ) such that R Z := R P Z = σ −1 ({v j | j ∈ {1, . . . , s + 1}}). Let M Z be the sublattice of M orthogonal to Zu 1 ⊕· · ·⊕Zu r ⊂ N . The pair (P Z , M Z ) corresponds to a horospherical homogeneous space G/H Z with H Z containing H. Also the dual lattice N Z of M Z is the image of the projection from N to Zu 1 ⊕ · · · ⊕ Zu r . We denote by v 1,Z , .
. The colored fan F Z corresponds to a G/H Z -embedding Z. Moreover, we have a G-equivariant morphism ψ from X to Z, it is a Mori fibration. Note that Z and the general fiber of ψ are smooth horospherical varieties of Picard group Z (in particular, they are homogeneous or one of the two-orbit varieties described in [Pas09]).

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 prime B-stable divisor is 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 if not).
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 if not).
Proposition 3.3. In both cases (1) and (2), the nef cone of X is generated by D 0 and D n+1 . In particular, Proof. By Theorem 2.4, we prove 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 to apply 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 and remark.
Lemma 3.4. [Pas06, proof of Proposition 3.10] We can suppose that G is the product of a semi-simple group with a torus by replacing G by the product of its semi-simple part G ′ := [G, G] and the torus T = P/H. 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 M ≃ X(T) is of the form (χ i + θ i ) i∈{1,...,n} such that (χ i ) i∈{1,...,n} form a basis of M = X(T), and the θ i 's are weights of the maximal torus of G ′ .
Remark 3.5. We can moreover assume G ′ to be the product of simply connected simple groups by taking the universal cover of G ′ .
With these assumptions, we get the following result.
Case (1): Consider the basis (e * 1 , . . . , e * n ) of M that is dual to the basis (e 1 , . . . , e n ) of N . 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 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 if not.
Lemma 3.7. In both cases, X is also a horospherical G ′ × S-variety.
We can replace • 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 imaginary 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, α i is the imaginary root of {1}.
It 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}: Case (1): where n ≥ 1; β is a simple root of G 0 ; α 0 , . . . , α n are distinct simple roots (may be imaginary) of G distinct from β; for any k ∈ {1, . . . , t}, G k = {1} if and only if k = 1 and α 0 is imaginary; and 0 = a 0 ≤ a 1 ≤ · · · ≤ a n are integers.
Note that the two cases in Proposition 3.8, with s = 1 in Case (2), are similar to the ones of Theorem 1.1.
4 Reduction to the cases of Theorem 1.1

Smooth horospherical varieties and G-modules
To prove Theorem 1.1 from Proposition 3.8, we glue together G-modules as soon as we can, in order to enlarge the group G and reduce to "smaller" cases. For this, we first need to apply the smooth criterion to X (Theorem 2.9), which comes from the fact that smooth horospherical G-modules are the C * -modules C, the SL d -modules V (̟ 1 ) = C d and Sp d - As in [Pas09, Theorem 1.7], the smooth 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]). Definition 4.2. 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 distinct from β. Let n be a non-negative integer. Denote by L a Levi subgroup of the maximal parabolic subgroup P (̟ β ), 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 root of the same L k so 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 quadruple (K, β, R, n) (see the appendix). We remark, in particular, that R is at most of cardinality 3.
We obtain the following result by applying the smooth criterion to the smooth projective horospherical variety X with Picard group Z 2 in both cases (1) and (2). Here, we suppose that X is as in Proposition 3.8, and in Case (1) 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.
If not, for any i ∈ {0, . . . , n}, the simple root α i is either imaginary or in G 0 or the short extremal simple root of one of a 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}. Then α i is either imaginary or the short extremal simple root of one of G k that is of type A or C.

Proof.
Case (1): With notation of Definition 4.2, 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 prove that the couples (R 0 \{β, δ}, γ) and (R 0 \{β, γ}, δ) are smooth, so that (L j , γ, δ) is smooth.
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 . It 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 imaginary. 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. It 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. But it also gives a contradiction if there exists a maximal colored cone of F X that contains α ∨ iM and α ∨ jM . 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.
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 4.4.
where the sum is taken over all (τ + 1)-uplets of non-negative integers where the sum is taken over all (τ + 1)-uplets 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 4.4. 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 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 4.4, then V a is also the irreducible G-module V G (a̟). Also, the image of x 1 in P(V a ) is the projectivization of a highest weight vector of weight in V G (a̟) 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)
A first part is already proved by Proposition 3.8 and Lemma 4.3, 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 G 0 /P (̟ β ) is isomorphic to G ′ 0 /P (̟ β ′ ) 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 ), or (Spin 2m+1 , ̟ m , Spin 2m+2 , ̟ m or ̟ 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 G 1 , . . . , G t . Then by Lemma 4.3, 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 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 imaginary, G 1 = {1}. By Lemma 4.3, 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 an imaginary root or a short extremal root of G 1 , respectively G 2 . Let 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.
We can now compare the dimension of the open G-orbit Ω X of X with the dimension of the open G-orbit of X. Indeed Ω X is isomorphic to a horospherical homogeneous space of rank n − 1 over ( Hence Ω X and G/H have the same dimension, so that X = X. 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 decrease by this change. (Also note that, if i = 0 and α 0 is imaginary then the new α 0 is not imaginary 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, it 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)
A first part is already proved by Proposition 3.8 and Lemma 4.3, 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-uplets 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 4.3, for any j ∈ {1, . . . , s}, α r+j is either an imaginary 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 imaginary.
We now apply Lemma 4.6 ((a) if α r+s is not imaginary and (b) if not). Hence, there exists d ≤ 2 such that, with G ⊂ G : 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) (case that we previously deal with).
• 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 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 imaginary.
Recall that, by Lemma 4.3, 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 4.6 ((c), (d) or (e)) to get G ⊂ G : ), 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. 4.3, α i and α i ′ are, imaginary 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 4.4 ((a) if i > 0 or α 0 is imaginary and (b) if not) 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 argument, 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. 4.3, 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 (̟ 0 ) ⊕ V G 0 (̟ 1 ))) and the closure of the G 1 -orbit of a sum of highest weight vectors in P ((V G 1 (̟ 2 ) ⊕ V G 1 (̟ 3 ))).
Hence, in any case we can assume that 0 < a 1 < · · · < a r . This finish 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.

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 3.3, up to linear equivalence, the ample Cartier divisors of X are the 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]) 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 + ǫK X is ample for any ǫ ∈ [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 3.2. 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 Case (2), an anticanonical divisor of X is In particular, X is Fano if and only if s+1 j=1 b r+j > r i=0 a i b i . 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 + ǫK X is ample for any ǫ ∈ [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 3.2. 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 .
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 1, these polytopes for ǫ = 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.
The fibers of this fibration can be easily computed 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 . More precisely, we deduce the fibers of φ 0 from the description of Gorbits of X and Y 0 given in Section 2.2: they 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 imaginary, P (̟ α i ) = G (and if not, it is the (proper) maximal parabolic subgroup of G associated to α i ).
• Suppose now that a n = 0, thenQ ǫ is the intersection of the simplexQ = Conv(e * 0 , e * 1 , . . . , e * n ) with the closed half-space H ǫ + := {x ∈ M Q | a 1 x 1 + · · · + a n x n ≥ ǫ − 1}, where e * 0 := 0. We denote by H ǫ ++ the interior of H ǫ + and by H ǫ the hyperplane H ǫ + \H ǫ ++ . In the next proposition, we give a description of the non-empty faces ofQ ǫ by distinguishing whether a face is in the hyperplane H ǫ or not.
Note first that the non-empty faces of the simplexQ are the F I := Conv(e * i | i ∈ {0, . . . , n}\I), with I {0, . . . , n}. In particular, the facets ofQ are the F i := F {i} and for any I {0, . . . , n}, F I = i∈I F i .
(Recall that a 0 = 0 and that a n = 0 here.) Proposition 5.3. The polytopeQ ǫ is of dimension n if and only if ǫ < max n i=0 (1 + a i ) = 1 + a n .
Proof. The polytopeQ ǫ is of dimension n if and only ifQ intersects H ǫ ++ if and only if there exists i ∈ {0, . . . , n} such that e * i ∈ H ǫ ++ if and only if there exists i ∈ {0, . . . , n} such that a i > ǫ − 1 if and only if a n > ǫ − 1 (because 0 = a 0 ≤ · · · ≤ a n ). This proves the first statement of the proposition.
Suppose now that ǫ < 1 + a n . For any non-empty face F ofQ ǫ , either F ⊂ H ǫ and F is the intersection of a non-empty face ofQ with H ǫ + , or F ⊂ H ǫ and F is the intersection of a non-empty face ofQ with H ǫ .
Let  (1 + a i ). Also, in that latter case, the dimension of F ǫ I is the same as the dimension of F I ; in particular the non-empty F ǫ I that are not included in H ǫ are all distinct.
Similarly, F ǫ 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 ǫ + and e * j ∈ H ǫ ++ (ie, a i ≥ ǫ − 1 and a j ≤ ǫ − 1). Then F ǫ I,β is not empty if and only min i ∈I (1 + a i ) ≤ ǫ ≤ max i ∈I (1 + a i ). Moreover, F ǫ I,β is not empty and included in no proper face of F I (ie, H ǫ intersects the relative interior of F I ) if and only if there exist i = j not in I such that e * i ∈ H ǫ ++ and e * j ∈ H ǫ + (ie, a i > ǫ − 1 and a j < ǫ − 1) or for any i ∈ I we have e * i ∈ H ǫ (ie, a i = ǫ − 1). Then F ǫ I,β is not empty and included in no proper face of F I if and only min i ∈I (1 + a i ) < ǫ < max i ∈I (1 + a i ) or ǫ = min i ∈I (1 + a i ) = max i ∈I (1 + a i ). Note also that the non-empty F ǫ I,β that are not included in a proper face of F I are all distinct and describe all non-empty faces ofQ ǫ included in H ǫ . This finishes the proof of the second statement of the proposition.
To get the last statement, apply the fact that any face of a polytope is the intersection of the facets containing it.
Proof. We apply the theory described in Section 2.2, in particular the fact that the isomorphic classes of the varieties X ǫ are obtained with looking at the ǫ's for which "the faces of Q ǫ change". Note first that, by Proposition 5.3, (P, M, Q ǫ ,Q ǫ ) is an admissible quadruple if and only if ǫ < 1 + a n .
Hence, it proves that if the two varieties X ǫ and X η are isomorphic then ǫ and η are a one of the subsets described in the corollary.
To conclude, we have to prove that the two varieties X ǫ and X η are isomorphic when ǫ and η are in one of these subsets. It is obvious with Proposition 5.3 except in the case where i k = n and the simple root α n is imaginary. But in that case, all polytopes Q ǫ with ǫ ∈ [1 + a n−1 , 1 + a n [= [1 + a i k−1 , 1 + a i k [ are simplexes with facets F ǫ i for i ∈ {0, . . . , n − 1} and F ǫ ∅,β or F ǫ n,β if ǫ = 1 + a n−1 = 1, ie, they could be defined with deleting the row corresponding to the simple root α n that is imaginary, so that their faces are "the same".
Corollary 5.5. The family (Q ǫ ) ǫ∈Q ≥0 describes a Log MMP from X as follows: • k flips φ l : X l −→ Y l ←− X l+1 : φ + l for any l ∈ {0, · · · , k − 1} and a fibration φ k : X k −→ Y k , if i k = n or the simple root α n is not imaginary; and the simple root α n is imaginary.
Example 5.6. In the fives different cases with n = 2 and a 2 = 0, we illustrate this corollary in terms of polytopes in Figures 2, 3, 4, 5 and 6 5.3 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 0 Figure 3: The Log MMP described by the polytopesQ ǫ in the case where n = 2, a 1 = 1, a 2 = 2 and α 2 is imaginary.  Figure 6: The Log MMP described by the polytopesQ ǫ in the case where n = 2 and a 1 = a 2 = 1.
of the different morphisms of the Log MMP.
We first distinguish two cases by the following result.
Proposition 5.7. Define the simple subgroups of P (̟ β ) as in Definition 4.2.
• Suppose that n = 1 and that α 0 and α 1 are two simple roots of the same simple subgroup of P (̟ β ).
• Suppose that n > 1 or that α 0 and α 1 are not two simple roots of the same simple subgroup of P (̟ β ).
• 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.
In particular, G/P (̟ α 0 ) (as 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. And then β is also an invariant of X up to symmetries of the Dynkin diagram of G 0 . The description of the fiber of ψ : X −→ G/P (̟ β ) implies that α 0 and α 1 are also invariants of X unless may be if two simple subgroups of P (̟ β ) have the same type (and rank ≥ 2). This could happens if and only if: G 0 is of type A m with m ≥ 5 odd and ̟ β = ̟ m+1 2 , or G 0 is of type E 6 and ̟ β = ̟ 3 . In these two cases α 0 and α 1 are invariants of X up to symmetries.
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 . As for the case where a 1 = 0, the (same) description of the fiber of ψ : X −→ G/P (̟ β ) implies that the pair (α 0 , α 1 ) is an invariant of X (up to symmetries). 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 couple (α 0 , α 1 ) is an invariant of X (still up to symmetries).
• 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 different 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 isomorphic 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 ǫ l I and F ǫ l I,β of the polytopeQ ǫ 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 1+a i l I and F 1+a i l I,β of the polytopeQ 1+a i l . Recall that, for any ǫ ∈ Q ≥0 , we have an order on the G-orbits of X ǫ compatible with the order on the non-empty faces ofQ ǫ : in particular Ω l I ⊂ Ω l I ′ and Ω l I,β ⊂ Ω l I ′ ,β respectively if and only if I ′ ⊂ I, and Ω l I,β ⊂ Ω l I (as soon as these orbits are defined, ie, as soon as the corresponding faces are non-empty).
For any I {0, . . . , n} such that there exists i ≥ i l not in I (ie, such that Ω l I is defined), φ l (Ω l I ) = ω l I if there exists i ≥ i l+1 not in I, and φ l (Ω l I ) = ω l I∪{0,...i l −1},β if for any i ≥ i l+1 , i ∈ I. Indeed I ∪ {0, . . . i l − 1} is the minimal subset of {0, . . . , n} containing I such that ω l I∪{0,...i l −1},β is defined and there is no I ′ containing I such that ω l I ′ is defined. And for any I {0, . . . , n} such that there exist i ≥ i l and i ′ < i l not in I (ie, such that Ω l I,β is defined), φ l (Ω l I,β ) = ω l I,β if there exists i ≥ i l+1 not in I, and φ l (Ω l I,β ) = ω l 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 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 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.
Proposition 5.10. (We still are 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 ).) Let l ∈ {0, . . . , k}.
The mapφ l is surjective and we distinguish four distinct cases.
1. 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 stays true for l = 0). Moreover, E ′ l is homogeneous isomorphic to G/P (̟ α i l ) (which is a point if α i l is imaginary).
2. 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 3. 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 ofφ l are locally almost maximal over exactly i l+1 − i l − 1(> 0) subsets 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. 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, we can compute with Corollary 2.13 the dimensions of the fibers over all pointed subsets of E ′ l .

The dimension of 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 of 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 of the closed G-orbit isomorphic to G/P (̟ α j ) is Proof. We keep the notation of the proof of Proposition 5.8.
Let ω be a G-orbit of Y l in ω l I l ∪{0,...i l −1},β . Then there exists I {0, . . . , n} con- where the union is taken over all J such that J ∩ I l−1 = I ∩ I l−1 . In particular,φ l is surjective and φ l −1 (ω) = Ω l I∩I l−1 . We then compute dim(ω) = dim(F l I,β ) + dim(G/ i ∈I P (̟ α i )), and dim(Ω l I∩I l−1 ) = dim(F I∩I l−1 ) + dim(G/P (̟ β ) ∩ i ∈I∩I l−1 P (̟ α i )), so that the dimension of a fiber ofφ l over ω is These dimensions are the biggest when I is the biggest (in particular when I = {0, . . . , n}, which is not allowed to define ω). Moreover, if we remove to I some i, the dimension changes if and only if j is such that α i is in G 0 (ie, α i is not imaginary 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.
And then we conclude the proof of Case (1) of Theorem 1.3 (ie, 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 simple subgroups 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 ).
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 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.
Note also thatQ 1+ar is the point u * r so that Q 1+ar is the point ̟ αr .
Proof is not empty and included in no proper face of F I,1 if and only if there exist i and j not in I such that u * i ∈ H ǫ ++ and u * j ∈ H ǫ + if and only if there exist i and j not in I such that a i > ǫ − 1 and a j < ǫ − 1 (ie, a i < ǫ − 1 and a j > ǫ − 1) or for any i ∈ I we have u * i ∈ H ǫ (ie, a i = ǫ − 1). Then F ǫ I,1,2 is not empty and included in no proper face of F I,1 if and only if min i ∈I (1 + a i ) < ǫ < max i ∈I (1 + a i ) or ǫ = min i ∈I (1 + a i ) = max i ∈I (1 + a i ). In particular, the dimension of F ǫ I,1,2 is the dimension of F I,1 minus 1 if min i ∈I (1+a i ) < ǫ < max i ∈I (1+a i ) and it equals the dimension of F I,1 if ǫ = min i ∈I (1 + a i ) = max i ∈I (1 + a i ). Note also that the non-empty F ǫ I,1,2 that are not included in a proper face of F I,1 are all distinct and describe all non-empty faces ofQ ǫ included in H ǫ ∩ F ∅,1 . This finishes the proof of the second statement of the proposition.
To get the last statements, apply 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.
Corollary 5.15. The isomorphic classes of the horospherical varieties X ǫ associated to the polytopes in the family (Q ǫ ) ǫ∈Q ≥0 are given by the following subsets of Q ≥0 : • ]1 + a r−1 , 1 + a r [ and {1 + a r−1 } if the simple root α r is not imaginary (ie, when X is as in Case (2b) of Theorem 1.1); • [1 + a r−1 , 1 + a r [ if the simple root α r is imaginary (ie, when X is as in Case (2c) of Theorem 1.1).
Proof. We apply the theory described in Section 2.2, in particular the fact that the isomorphic classes of the varieties X ǫ are obtained with looking at the ǫ's for which "the faces of Q ǫ change". Note first that, by Proposition 5.14, (P, M, Q ǫ ,Q ǫ ) is an admissible quadruple if and only if ǫ < 1 + a r .
Also, the facets ofQ ǫ are: Hence, it proves that if two varieties X ǫ and X η are isomorphic then ǫ and η are a one of the subsets described in the corollary.
To conclude, we have to prove that the two varieties X ǫ and X η are isomorphic when ǫ and η are in one of these subsets. It is obvious with Proposition 5.14 except in the case where the simple root α n is imaginary. But in that case, all polytopes Q ǫ with ǫ ∈ [1 + a r−1 , 1 + a r [ could be defined with deleting the row corresponding to the simple root α r that is imaginary, so that their faces are "the same" (they are simplexes with facets F ǫ i for i ∈ {0, . . . , r − 1}, F ǫ ∅,1 and F ǫ ∅,2 ).
Corollary 5.16. The family (Q ǫ ) ǫ∈Q ≥0 describes a Log MMP from X as follows: i for any i ∈ {0, · · · , r − 1} and a fibration φ r : X r −→ Y r , if the simple root α r is not imaginary; i for any i ∈ {0, · · · , k − 2}, 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 n is imaginary.
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 8 and 9.

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 0 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.
• 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 ) 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 different 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 isomorphic 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 loci 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 of Picard group Z 2 as in Case (2), and E 0 is the product a two-orbit variety with a homogeneous (projective of Picard group Z) variety.
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 ǫ i I , F ǫ i I,1 , F ǫ i I,2 and F ǫ i I,1,2 of the polytopeQ ǫ 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 1+a i I , F 1+a i I,1 , F 1+a i I,2 and F 1+a i I,1,2 of the polytopeQ 1+a i . Recall that, for any ǫ ∈ Q ≥0 , we have an order on the G-orbits of X ǫ compatible with the order on the non-empty faces ofQ ǫ : in particular Ω i I ⊂ Ω i I ′ , Ω i I,1 ⊂ Ω i I ′ ,1 , Ω i I,2 ⊂ Ω i I ′ ,2 and Ω i I,1,2 ⊂ Ω i I ′ ,1,2 respectively if and only if I ′ ⊂ I, and Ω i I,1 ⊂ Ω i I , Ω i I,2 ⊂ Ω i I , Ω i I,1,2 ⊂ Ω i I, 1 and Ω i I,1,2 ⊂ Ω i I,2 (as soon as these orbits are defined, ie, as soon as the corresponding faces are non-empty).
For any I {0, . . . , r} such that there exists j ≥ i not in I (ie, such that Ω i I is defined), φ i (Ω i I ) = ω i I if there exists j ≥ i + 1 not in I, and φ i (Ω i I ) = ω i I\{i},1,2 if for any j ≥ i + 1, j ∈ I. Indeed I ∪ {0, . . . i − 1} = I\{i} is the minimal subset of {0, . . . , r} containing I such that ω i I\{i},1,2 is defined and there is no I ′ containing I such that ω i I ′ , ω i I ′ ,1 or ω i I ′ ,2 is defined. Similarly, with k = 1 or 2, for any I {0, . . . , r} such that there exists j ≥ i not in I (ie, such that Ω i I,k is defined), φ i (Ω i I,k ) = ω i I,k if there exists j ≥ i + 1 not in I, and But Ω i I i and ω i {0,...,r}\{i},1,2 are not isomorphic horospherical homogeneous spaces by Proposition 2.12, 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 2.12 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.
Proposition 5.20. For any i ∈ {0, . . . , r}, E ′ i is a closed G-orbit of Y i isomorphic to G/P (̟ α i ) (which is a point if α i is imaginary). In particular, the mapφ i is surjective.

2.
R is empty or one of the following case occurs.