Alessandro Ruzzi

FANO SYMMETRIC VARIETIES WITH LOW RANK

Keywords: Symmetric varieties, Fano varieties. MSC 2010: 14M17, 14J45, 14L30

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Fano symmetric varieties with low rank

A Gorenstein (projective) normal algebraic variety X over C is called a Fano variety if the anticanonical divisor is ample. The Fano surfaces are classically called Del Pezzo surfaces. The importance of Fano varieties in the theory of higher dimensional varieties is similar to the importance of Del Pezzo surfaces in the theory of surfaces. Moreover Mori's program predicts that every uniruled variety is birational to a fiberspace whose general fiber is a Fano variety (with terminal singularities).

Let θ be an involution of a reductive group G (over C) and let H be a closed subgroup of G such that G θ ⊂ H ⊂ N G (G θ ). A symmetric variety is a normal G-variety with an open orbit isomorphic to G/H. The symmetric varieties are a generalization of the toric varieties. The toric smooth Fano varieties with rank at most four are been classified. By [START_REF] Alexeev | Boundedness of spherical Fano varieties[END_REF], Theorem 4.2 there is only a finite number of Fano smooth symmetric varieties with a fixed open orbit. In [START_REF] Ruzzi | Smooth projective symmetric varieties with Picard number equal to one[END_REF] we have classified the smooth compact symmetric varieties with Picard number one and G semisimple, while in [START_REF] Ruzzi | Geometrical description of smooth projective symmetric varieties with Picard number one[END_REF] we have given an explicitly geometrical description of such varieties; they are automatically Fano.

In this work, we want to classify the Fano symmetric varieties with low rank (and G semisimple). First, we consider a special case of arbitrary rank. We say that a variety X is quasi Q-Fano if -K X is a nef and big Q-divisor. Fixed an open orbit G/H with G semisimple, there is a unique maximal compactification between the ones which have only one closed orbit. Such variety is called the standard compactification. If it is also smooth, it is called the wonderful compactification; this is the case, for example, if H = N G (G θ ) (see [START_REF] Concini | Complete symmetric varieties, in Invariant Theory[END_REF], Theorem 3.1). We prove that the standard symmetric varieties are all quasi Q-Fano and we describe when they are Fano. We determine also the symmetric Fano varieties obtainable from a wonderful one by a sequence of blow-ups along closed orbits. In particular, we prove that such a variety must be either a wonderful one or the blow-up of a wonderful one along the unique closed orbit.

Next we consider the symmetric varieties of rank at most three. The rank of a symmetric variety X is defined as the rank of C(X) (B) /C * , where B is any fixed Borel subgroup of G. The symmetric varieties with rank one are all wonderful; moreover one can show that they are isomorphic, under the action of Aut 0 (X), either to a projective homogeneous variety G/P with P maximal, or to P n × P n (see [A83]). Thus they are all Fano.

We classify all the locally factorial (resp. smooth) Fano symmetric varieties of rank 2. When G is semisimple, we classify also the locally factorial (resp. smooth) symmetric varieties which are only quasi-Fano. In the proof of such result we obtain a classification of the toroidal Fano varieties of rank 2 with G semisimple (without assumption on the regularity).

Finally, we classify the smooth Fano symmetric varieties of rank three which are obtainable from a wonderful one by a sequence of blow-ups along G-subvarieties (in particular G is semisimple). This class of varieties is quite large; indeed any compact symmetric variety is dominated by a variety obtained from the wonderful one by a sequence of blow-ups along G-subvarieties of codimension two (see [START_REF] Concini | Complete symmetric varieties II, Intersection theory[END_REF], Theorem 2.4). This result on 3-rank varieties can be generalized to varieties obtainable from a generic wonderful varieties of rank 3 by a sequence of blow-ups along G-subvarieties (without suppose G/H symmetric).

Introduction and notations

In this section we introduce the necessary notations. The reader interested to the embedding theory of spherical varieties can see [START_REF] Knop | The Luna-Vust theory of spherical embeddings[END_REF], [START_REF] Brion | Variétés sphériques[END_REF] or [T06]. In [START_REF] Vust | Plongements d'espace symétriques algébriques: une classification[END_REF] is explained such theory in the particular case of symmetric varieties.

First definitions

Let G be a connected reductive algebraic group over C and let θ be an involution of G. Given a closed subgroup H such that G θ ⊂ H ⊂ N G (G θ ), we say that G/H is a symmetric space and that H is a symmetric subgroup. A normal G-variety is called a spherical variety if it contains a dense B-orbit (B is a chosen Borel subgroup of G). We say that a subtorus of G is split if θ(t) = t -1 for all its elements t; moreover it is a maximal split torus if it has maximal dimension. A maximal torus containing a maximal split torus is maximally split; any maximally split torus is θ stable (see [T06], Lemma 26.5). We fix arbitrarily a maximal split torus T 1 and a maximally split torus T containing T 1 . Let R G be the root system of G w.r.t. T . We can choose a Borel subgroup T ⊂ B such that, for any positive root α, either θ(α) = α or θ(α) is negative. Moreover, BH is dense in G (see [START_REF] Concini | Complete symmetric varieties, in Invariant Theory[END_REF], Lemma 1.2 and Proposition 1.3). In particular, every normal equivariant embedding of G/H is spherical; we call it a symmetric variety.

We can assume that G is the direct product of a simply connected, semisimple group with a central split torus.

Colored fans

Now, we introduce some details about the classification of the symmetric varieties by their colored fans (this classification is defined more generally for spherical varieties). Let D(G/H) be the set of B-stable prime divisors of G/H; its elements are called colors. We say that a spherical variety is simple if it contains one closed orbit. Let X be a simple symmetric variety with closed orbit Y . We define the set of colors of X as the subset D(X) of D(G/H) consisting of the colors whose closure in X contains Y . To each prime divisor D of X, we can associate the normalized discrete valuation

v D of C(G/H) whose ring is O X,D ; D is G-stable if and only if v D is G-invariant.
Let N be the set of all Ginvariant valuations of C(G/H) taking value in Z and let N (X) be the set of the valuations associated to the G-stable prime divisors of X. Observe that each irreducible component of X \ (G/H) has codimension one, because G/H is affine. Let S := T / T ∩ H ≃ T • (eH/H). One can show that the group C(G/H) (B) / C * is isomorphic to the character group χ(S) of S (see [START_REF] Vust | Plongements d'espace symétriques algébriques: une classification[END_REF], §2.3); in particular, it is a free abelian group. We define the rank l of G/H as the rank of χ(S). We can identify the dual group Hom Z (C(G/H) (B) / C * , Z) with the group χ * (S) of one-parameter subgroups of S; so we can identify χ * (S) R with Hom Z (χ(S), R). The restriction map to C(G/H) (B) /C * is injective over N (see [START_REF] Brion | Variétés sphériques[END_REF], §3.1 Corollaire 3), so we can identify N with a subset of χ * (S) R . We say that N is the valuation monoid of G/H. For each color D, we define ρ(D) as the restriction of v D to χ(S). In general, the map ρ : D(G/H) → χ * (S) R is not injective. Let C(X) be the cone in χ * (S) R generated by N (X) and ρ(D(X)). We say that the pair (C(X), D(X)) is the colored cone of X; it determines univocally X (see [START_REF] Brion | Variétés sphériques[END_REF], §3.3 Théorème).

Let Y be an orbit of a symmetric variety X.

The set {x ∈ X | G • x ⊃ Y } is an open simple G-
subvariety of X with closed orbit Y , because any spherical variety contains only a finite number of G-orbits. Let {X i } be the set of open simple subvarieties of X and define the set of colors of X, D(X), as i∈I D(X i ). The family F (X) := {(C(X i ), D(X i ))} i∈I is called the colored fan of X and determines completely X (see [START_REF] Brion | Variétés sphériques[END_REF], §3.4 Théorème 1). Moreover X is compact if and only if cone(N ) is contained in the support |F (X)| := i∈I C(X i ) of the colored fan (see [START_REF] Brion | Variétés sphériques[END_REF], §3.4 Théoremè 2).

Given a symmetric variety X we denote by ∆ (or by ∆ X ) the fan associated to the colored fan of X, by ∆(i) the set of i-dimensional cones in ∆ and by ∆[p] the set of primitive generators of the 1-dimensional cones of ∆. The fan ∆ is formed by all the faces of the cones C such that there is a colored cone (C, F ) ∈ F (X). The toric varieties are a special case of symmetric varieties. If X is a toric variety, then D(G/H) is empty and we need only to consider the fan ∆ X (actually the theory of colored fans is a generalization of the classification of toric varieties by fans).

Restricted root system

To describe the sets N and ρ(D(G/H)), we associate a root system to G/H. We can identify χ(T 1 ) R with χ(S) R because [χ(S) : χ(T 1 )] is finite. We call again θ the involution induced on χ(T ) R . The inclusion T 1 ⊂ T induces an isomorphism of χ(T 1 ) R with the (-1)-eigenspace of χ(T ) R under the action of θ (see [T06], §26). Denote by W G the Weyl group of G (w.r.t. T ). We can identify χ(T 1 ) R with its dual χ * (T 1 ) R by the restriction (

•, •) to χ(T 1 ) R of a fixed W G -invariant non-degenerate symmetric bilinear form on χ(T ) R . Let R 0 G be the set of roots fixed by θ and let R 1 G be R G \ R 0 G . Let R i,+ G := R i G ∩ R + G . The set R G,θ := {β -θ(β) | β ∈ R 1
G } is a root system in χ(S) R (see [START_REF] Vust | Plongements d'espace symétriques algébriques: une classification[END_REF], §2.3 Lemme), which we call the restricted root system of (G, θ); we call the non zero β -θ(β) the restricted roots. Usually we denote by β (resp. by α) a root of R G (resp. of R G,θ ); often we denote by ̟ (resp. by ω) a weight of R G (resp. of R G,θ ). We denote by R G = {β 1 , ..., β n } the basis of R G associated to B and by ̟ 1 , ..., ̟ n the fundamental weights of

R G . Let R i G be R G ∩ R i G . There is a permutation θ of R 1 G such that, ∀β ∈ R 1 G , θ(β) + θ(β) is a linear combination of roots in R 0 G . We denote by α 1 , ..., α s the elements of the basis R G,θ := {β -θ(β) | β ∈ R 1 G } of R G,θ . If R G,θ is irreducible we order R G,θ as in [Bo68]. Let b i be equal to 1 2 if 2α i belongs to R G,
:= -C + .
We say that a dominant weight ̟ ∈ χ(T ) is a spherical weight if V (̟) contains a non-zero vector fixed by G θ . In this case, V (̟) G θ is one-dimensional and θ(̟) = -̟, so ̟ belongs to χ(S) R . One can show that set of dominant weights of R G,θ is the set of spherical weights and that C + is the intersection of χ(S) R with the positive closed Weyl chamber of R G . Suppose β j -θ(β j ) = α i , then ω i is a positive multiple of ̟ j + ̟ θ(j) . More precisely, we have

ω i = ̟ j + ̟ θ(j) if θ(j) = j, ω i = 2̟ j if θ(j) = j and β j is orthogonal to R 0
G and ω i = ̟ j otherwise (see [START_REF] Chirivì | The ring of sections of a complete symmetric variety[END_REF], Theorem 2.3 or [T06], Proposition 26.4). We say that a spherical weight is regular if it is strictly dominant as weight of the restricted root system.

The sets N and D(G/H)

The set N is equal to C -∩ χ * (S); in particular, it consists of the lattice vectors of the rational, polyhedral, convex cone C -= cone(N ). The set ρ(D(G/H)) is equal to R ∨ G,θ and any fibre ρ -1 (α ∨ ) contains at most 2 colors. For any simple spherical variety X, N (X) is formed by the primitive generators of the 1-faces of C(X) which are contained in cone(N ). When X is symmetric, also ρ(X) can be recovered by C(X): its elements generate the 1-faces of C(X) which are not contained in C -. We say that (G, θ) indecomposable if the unique normal, connected, θ-stable subgroup of G is the trivial one. In this case the number of colors is at most equal to rank (G/H) + 1. If θ is indecomposable then there are three possibilities: i) G is simple; ii) G = Ġ × Ġ with Ġ simple and θ(x, y) = (y, x); iii) G = C * and θ(t) = t -1 . See [START_REF] Warner | Harmonic analysus on semi-simple Lie group[END_REF], §1.1 for a classification of the involution of a simple group. In [START_REF] Warner | Harmonic analysus on semi-simple Lie group[END_REF] (and in [START_REF] Ruzzi | Geometrical description of smooth projective symmetric varieties with Picard number one[END_REF], §1) are also indicated the Satake diagrams of the indecomposable involutions. The Satake diagram of any involution (G, θ) is obtained from the Dynkin diagram of G as follows: 1) the vertices corresponding to element of R 0 G (resp. of R 1 G ) are black (resp. white); 2) two simple roots β 1 , β 2 ∈ R 1 G such that θ(β 1 ) = β 2 are linked by a double-headed arrow.

If ♯D(G/H) > rank(G/H) and (G, θ) is indecomposable, we have two possibilities: 1)

G θ = H = N G (G θ ); 2) H = G θ and [G θ : N G (G θ )] = 2.
In the last case any element of N G (G θ )\ G θ exchanges two colors and R G,θ has type A 1 , B 2 or C n . We say that a simple restricted root α is exceptional if ♯ρ -1 (α ∨ ) = 2 and 2α is a restricted root. In this case the irreducible factor of R G,θ containing α is associated to an indecomposable factor of G/G θ as in 1). We say that also (G, θ) and any symmetric variety (with open orbit G/H) are exceptional. We denote by D α the sum of the colors in ρ -1 (α ∨ ) and by D ω the G-stable divisor corresponding to (R ≥0 ω, ∅) ∈ F(X).

Toroidal symmetric varieties

In this section we want to define a special class of varieties. We say that a spherical variety is toroidal if D(X) = ∅. Let (C, F ) be a colored cone of X, we say that the blow-up of X along the subvariety associated to (C, F ) is the blow-up of X along (C, F ). In the following of this section we suppose G semisimple. Then there is a special simple compactification of G/H because N G (H)/H is finite. This compactification, called the standard compactification X 0 , is associated to (cone(N ), ∅) and it is the maximal simple compactification of G/H in the dominant order. We define e i as the primitive positive multiple of -ω ∨ i (in χ * (S)), so ∆ X0 [p] = {e 1 , ..., e l }. The standard compactification is wonderful (i.e. it is also smooth) if and only if χ * (S) = Ze i . De Concini and Procesi have proved that X 0 is wonderful if

H = N G (G θ ), or equivalently χ * (S) =
Zω ∨ i (see [START_REF] Concini | Complete symmetric varieties, in Invariant Theory[END_REF] Theorem 3.1). The standard compactification X 0 contains an affine toric S-variety Z 0 , which is a quotient of an affine space by a finite group. The toroidal varieties are the symmetric varieties which dominates the standard compactification and are in one-to-one correspondence with the S-toric varieties which dominates Z 0 .

Let P be the stabilizer of the B-stable affine open set U := X 0 \ D(G/H) D. This open set is P -isomorphic to R u P × Z 0 , where R u P = β∈R 1,+ G U β is the unipotent radical of P and dim Z 0 = rank X 0 . To any toroidal variety X we associated the inverse image Z of Z 0 by the projection X → X 0 . Moreover, X \ D(G/H) D is P -isomorphic to R u P × Z. The toroidal varieties are also in one-to-one correspondence with a class of compact toric varieties in the following way. To a symmetric variety X, we associate the closure Z c of Z in X; Z c is also the inverse image of Z c 0 . The fan of Z is the fan ∆ X associated to the colored fan of X, while the fan of Z c consists of the translates of the cones of Z by the Weyl group

W G,θ ∼ = N G θ (T 1 )/C G θ (T 1 ) of R G,θ .

The Picard group

The class group of a symmetric variety is generated by the classes of the Bstable prime divisors modulo the relations div(f ) with f ∈ C(G/H) (B) . Indeed Cl(BH/H) = P ic(BH/H) is trivial. Given ω ∈ χ(S) we denote by f ω the element of C(G/H) (B) with weight ω and such that

f ω (H/H) = 1. A Weyl divisor D∈D(G/H) a D D + E∈N (X) b E E is a Cartier divisor if and only if, for any (C, F ) ∈ F (X), there is h C ∈ χ(S) such that h C (E) = a E ∀E ∈ C and h C (ρ(D)) = a D ∀D ∈ F . Let P L ′ (X)
be the set of functions on the support |F (X)| such that: 1) are linear on each colored cone; 2) are integer on χ * (S) ∩ |F (X)|. Let L(X) ⊂ P L ′ (X) be the subset composed by the restrictions of linear functions and let P L(X) := P L ′ (X)/L(X). The {h C }, corresponding to any Cartier divisor, defines an element of P L(X). If X is compact, there is an exact sequence (see [START_REF] Brion | Groupe de Picard et nombres caractéristiques des variétés sphérique[END_REF], Théorème 3.1):

0 → D∈D(G/H)\D(X) ZD → P ic(X) → P L(X) → 0.
A Cartier divisor is globally generated (resp. ample) if and only if the associated function is convex (resp. strictly convex) and h

C (ρ(D)) ≤ a D (resp. h C (ρ(D)) < a D ) ∀(C, F ) ∈ F (X)
and ∀D ∈ D(G/H) \ F . Given any linearized line bundle L, the space H 0 (X, L) is a multiplicity free G-module and, if L is globally generated, the highest weights of H 0 (X, L) are the elements of χ(S) ∩ hull({h C } dim C=l ), where hull({x 1 , ..., x m }) is the convex hull of x 1 , ..., x m (see [START_REF] Brion | Groupe de Picard et nombres caractéristiques des variétés sphérique[END_REF], §3). Thus, a Cartier divisor on a projective symmetric variety is nef if and only if it is globally generated. Moreover, a nef G-stable Cartier divisor on a projective symmetric variety is big if and only if the associated piecewise

linear function h is such that ( C∈∆(l) h C , R ∨ ) = 0 for each irreducible factor R ∨ of R ∨ G,θ (see [Ru09] Theorem 4.2).
In particular, when θ is indecomposable every non-zero nef G-stable divisor is big. When X is toroidal we have an exact split sequence

0 → P ic(X 0 ) → P ix(X) → P ic(Z) → 0. A normal variety X is locally factorial if the Picard group is isomorphic to the class group, while X is Q-factorial if P ic(X) Q ∼ = Cl(X) Q . A simple symmetric variety associated to a colored cone (C, F ) is locally factorial if: i)
C is generated by a subset of a basis of χ * (S) and ii) ρ is injective over F (see [START_REF] Brion | Curves and divisors in spherical varieties, in Algebraic groups and Lie Groups Austral[END_REF] for a general statement in the spherical case). When the variety is toroidal the locally factoriality is equivalent to the smoothness.

An

anticanonical divisor -K X of X is α∈R ∨ G,θ a α D α + E∈N (X) E with a α ω α = 2ρ-2ρ 0 . Here 2ρ := 2ρ RG = α∈R + G ω α is the sum of all the positive roots of R G , while 2ρ 0 := 2ρ R 0
G is the sum of the positive roots in R 0 G . Let k (or k X ) be the piecewise linear function associated to -K X . The anticanonical divisor -K X is linearly equivalent to a unique G-stable divisor -K X . The piecewise linear function k (or k X ) associated to -K X is equal to k -2ρ + 2ρ 0 over N (X) and to 0 over ρ(D(X)).

Indeed -K X is -K X + div( α∈R G,θ f aα α ), where f α ∈ C(G/H) (B) is an equation of D α (of weight ω α ). In particular k = k -2ρ + 2ρ 0 if X is toroidal.

Standard symmetric varieties

In the following, unless explicitly stated, we always suppose G semisimple (we will consider the general reductive case mainly in §5.3). Moreover, we often denote the normalizator N G (H) by N (H). In this section we show that all the standard symmetric varieties are quasi Q-Fano varieties; moreover we classify the Fano ones. When the rank of G/H is one, the standard compactification X 0 of G/H is the unique G-equivariant compactification. In such a case X 0 is an homogeneous projective variety w.r.t. Aut 0 (X) by [A83]; moreover it is wonderful and Fano, because either it is P n × P n or it has Picard number one.

First of all, we reduce ourselves to the indecomposable case. Write (G, θ) as a product (G j , θ) of indecomposable involutions and let X j be standard compactification of

G j /(G j ∩ H). G/H θ n, l wonderful Spin 2n+1 /(Spin l × Spin 2n+1-l ) BI n ≥ l ≥ 4 no Spin 7 /(Spin 3 × Spin 4 ) BI n = l = 3 no Sp 2n /N (GL n ) CI n = l ≥ 3 yes Spin 2l /N (Spin l × Spin l ) DI n = l ≥ 4 yes Spin 2n /H DI n = l ≥ 6 no (G θ ⊂ H N (G θ )) Spin 2n /(Spin l × Spin 2n-l ) DI n > l ≥ 4 no Spin 8 /(Spin 3 × Spin 5 ) DI n = 4, l = 3 no E 6 /N (C 4 ) EI yes E 7 /N (A 7 ) EV yes E 7 /A 7 EV no E 8 /D 8 EV III yes F 4 /(C 3 × A 1 ) F I yes G 2 /(A 1 × A 1 ) G yes Figure 1: Non Q-Fano standard indecomposable symmetric varieties Lemma 2.1 The variety X is (quasi) Q-Fano if and only if all the X i are (quasi) Q-Fano. Proof. The weight k X is equal to k Xi . . A standard symmetric variety is always Q-factorial; in particular, K X is a Q-Cartier divisor.
Moreover, if X is wonderful then also the X j are wonderful and X = X j (see [START_REF] Ruzzi | Effective and Big Divisors on a Projective Symmetric Variety[END_REF] Corollary 2.1). We have the following theorem: Theorem 2.1 Let X be a standard indecomposable symmetric variety. Let n be the rank of G and let l be the rank of X. Then:

• The anticanonical divisor of X is always a nef and big Q-divisor.

• Suppose X wonderful. Then it is not a Fano variety if and only:

i) if the involution induced on χ(S) R is -id; ii) R G,θ is different from A n and B n ; iii) H = N G (G θ ).
• The standard indecomposable varieties whose anticanonical divisor is not ample are compactifications of the symmetric spaces in Figure 1.

Proof. We have to determine when ( k C -, α i ) ≤ 0 for each i ∈ {1, ...l}. We can write -2ρ + 2ρ 0 as the sum of the spherical weights

-2 ρ = -2 βj∈R 1 G ̟ j and 2 ρ 0 = -2 βj ∈R 0 G ̟ j +2ρ 0 . Write β j -θ(β j ) = α i , so ( k C -, α i ) = 2(k C -, β j ) and (2 ρ 0 , α i ) = 4(ρ 0 , β j ) ≤ 0. Thus (-2ρ + 2ρ 0 )(α ∨ i ) = -1 if ω i = 2̟ j and (-2ρ + 2ρ 0 )(α ∨ i ) ≤ -2 otherwise. Suppose now H autonormalizing, i.e. H = N (G θ ); the case where H N (H) is very similar. We want to study k C -= - l i=1 α i . By the expression of the Cartan matrix of R G,θ , k C -(α ∨ i ) ≤ 1 for each i. Therefore k C -is always anti-dominant. If k C -is not regular, then there is (a unique) α i = 2β j ∈ R G,θ such that k X (α ∨ i ) = 1; in particular G is simple.
By the classification of the involutions by their Satake diagrams, k C - is not regular if and only if θ = -id over χ(S) and R G,θ is different by A l and B l .

Blows-ups along closed orbits

In this sections we want to prove a partial result in arbitrary rank. We restrict ourselves to the smooth toroidal case. For the toric varieties of rank 2 one can easily proves the following property (*):

Let Z be a smooth toric variety of rank 2 and let Z ′ be a smooth toric variety birationnaly proper over Z. If the anticanonical bundle of Z ′ is ample, then also the anticanonical bundle of Z is ample.

This allows to prove easily that a smooth toric variety proper over A 2 with ample anticanonical bundle is either A 2 or its blow-up in the origin. We would like to use a similar property to classify the smooth toroidal Fano symmetric varieties. Unfortunately the previous property (*) is false already in rank three. Indeed, let Z be the 3-dimensional toric variety whose fan ∆ has maximal cones cone(e 1 , e 2 , e 3 ) and cone(e 1 , e 2 , e 1 + e 2 -e 3 ), where {e 1 , e 2 , e 3 } is any basis of χ * (S). The function associated to its anticanonical bundle is linear, so -K Z is nef but non-ample. Furthermore, the blow-up Z ′ of Z along cone(e 1 , e 2 ) is Fano. We can take Z × A m as higher dimensional example.

In the previous example, we have considered a blow-up along a subvariety with strictly positive dimension. Now, we prove a property similar to (*) considering only blow-ups along compact orbits, i.e. S-fixed points. In the next section, we prove a much stronger statement when the rank is three.

Lemma 3.1 Let Z be a smooth l-dimensional toric variety whose fan contains two l-dimensional cones σ + and σ -such that: i) σ + ∩ σ -has dimension l -1 and ii) σ + ∪ σ -is strictly convex. Assume moreover that the piecewise linear function associated to the anticanonical bundle of Z is not strictly convex on σ + ∪ σ -. Then the anticanonical bundle of any toric variety obtained from Z by a sequence of blow-ups centred in S-fixed points is not ample.

We can reformulate the first hypothesis in a more combinatorial way. Indeed, we can write

σ + = cone(v 1 , ..., v l-1 , v + ) and σ -= cone(v 1 , ..., v l-1 , v -) with v 1 , ..., v l-1 , v + , v -primitive and v + + v -= a i v i ,
where the a i are positive integers, not all zero.

Proof of Lemma 3.1. The anticanonical bundle of a variety with satisfies the hypotheses of the lemma is not ample. Thus, it is sufficient to show that the blow-up Z ′ of Z centered in any σ ∈ ∆(l) satisfies again the hypotheses of the lemma. We can suppose σ = σ + by symmetry. Then the fan of Z ′ contains σ - and σ

′ := cone(v 1 , ..., v l-1 , v + + v i ). We have (v + + v i )+ v -= (a i + 1)v i , so Z ′ satisfies the hypotheses w.r.t. σ -and σ ′ .
Now we can classify the toric varieties with ample anticanonical bundle which are obtained from A l by a sequence of blow-ups centered in S-fixed points.

Proposition 3.1 Let Z be a smooth toric variety with ample anticanonical bundle which is obtained from A l by a sequence of blow-ups centered in S-fixed points, then it is either A l or the blow-up of A l in the S-fixed point.

Proof. One can easily see that the blow-up Z 1 of A l in the S-stable point has ample anticanonical bundle. The blow-up of Z 1 in the S-fixed point corresponding to cone(e 1 , ..., êj , ..., e l , e i ) satisfies the hypotheses of the previous lemma w.r.t. cone(e 1 , ..., êh , ..., e l , l i=1 e i ) and cone(e 1 , ..., êh , ..., êj , ..., e l , l i=1 e i , 2 l i=1 e i -e j ), where h = j. Thus, a symmetric variety obtained from a wonderful one by a sequence of blow-ups along closed orbits can be Fano only if it is either the wonderful variety or its blow-up along the closed orbit. We have already considered the wonderful case. Now, we prove that, when such blow-up is Fano, the rank of every indecomposable factor of R G,θ is at most 3. Lemma 3.2 Let X 1,...,l be the blow-up of the wonderful compactification of G/H and suppose that R G,θ contains an irreducible factor of rank at least three. If X 1,...,l is Fano then it is indecomposable, has rank 3 and H N (G θ ).

Proof. The weights { k C } associated to -K X 1,...,l are λ i = -2ρ + 2ρ 0 -(l - 2)e * i + j =i e * j with i = 1, ..., l. First, suppose G/H indecomposable and write e * i = -x i α i . If H N (G θ ), then R G,θ has type A 1 , B 2 or C l .
We consider two cases. First, suppose that there is

β h ∈ R 1 G orthogonal to R 0 G . Write β h -θ(β h ) = α j , so 0 > (λ j , α ∨ j ) = (-2ρ + 2ρ 0 , α ∨ j )) + ((l - 2)x j α ∨ j , α ∨ j ) -( i =j x i α ∨ i , α ∨ j ) ≥ -2 + 2(l -2)x j + 0. Observe that x -1 j ≤ 2, so l = 3 and H N (G θ ).
If there is not such a root, we have the following possibilities: 1) θ has type

AII and G/G θ is SL 2l+2 /Sp 2l+2 ; 2) θ has type CII and G/G θ is Sp 2n /(Sp 2l × Sp 2n-2l ); 3) θ has type DIII and G/G θ is SO 4l+2 /GL 2l+1 . Then there are β 3 , β 5 ∈ R 0 G orthogonal to R 0 G \{β 3 , β 5 } and β 4 ∈ R 1 G such that α 2 = β 3 + 2β 4 + β 5 . Moreover, (β 3 , β 3 ) = (β 4 , β 4 ) = (β 5 , β 5 ), α ∨ 2 = 1 (β2,β2) α 2 and x i = 1 if i < l. Thus 0 > (λ 2 , α ∨ 2 ) = (-2ρ + 2ρ 0 , α ∨ 2 )) + (l -2)(α 2 , α ∨ 2 ) -(α 1 , α ∨ 2 ) -x 3 (α 3 , α ∨ 2 ) ≥ -4 + 2(l -2) + 1 + x 3 , so we have again l = 3 and H N (G θ ).
Finally, suppose θ decomposable. Let (G, θ) = (G 1 , θ 1 ) × (G 2 , θ 2 ) with l ′ := rank (G 1 /G θ 1 ) ≥ 3 and define the weight λ ′ i for G 1 in an analogous way to the λ i . We have

λ i = λ ′ i -(l -l ′ )e * i + ω
where ω is orthogonal to R G1,θ . By the previous part of the proof there is always an i with

λ ′ i (α ∨ i ) ≥ -1, so λ i (α ∨ i ) ≥ λ ′ i (α ∨ i ) + 1 2 (α i , α ∨ i ) ≥ 0, a contradiction.
By an explicit analysis of the indecomposable involutions of rank at most three we obtain: Theorem 3.1 Let G/H be a symmetric space of rank l (> 1) associated to an involution θ and let X be a compact symmetric variety obtained from the wonderful compactification of G/H by a sequence of blow-ups along closed orbits.

1. If X is a Fano variety then either it is the wonderful variety X 0 or it is the blow-up X 1,...,l of X along the closed orbit.

2. If there is an indecomposable factor of (G, θ) of rank at least 3 then X 1,...,l is not Fano.

3. If (G, θ) has rank at least 6 and has an indecomposable factor of rank 2, then X 1,...,l is not Fano.

4. If X 1,...,l is Fano, the possibilities for the indecomposable factors of G/H are as in Figure 2 (we indicate also the eventual conditions on rank G/H so that such a factor can appear).

G ′ /H ′ θ|G ′ rank G/H rank G ′ /H ′ SL 3 A 2 l = 2 2 Spin 5 B 2 l = 2 2 SL 6 /N (Sp 6 ) AII l = 3 2 SL n+1 /S(GL 2 × GL n-1 ), n ≥ 4 AIII l = 2 2 Sp 2n /Sp 4 × Sp 2n-4 , n ≥ 5 CII l ≤ 3 2 Sp 8 /Sp 4 × Sp 4 CII l = 3 2 Sp 8 /N (Sp 4 × Sp 4 ) CII l = 2 2 SO 10 /GL 5 DIII l ≤ 3 2 E 6 /N (D 5 × C * ) EIII l ≤ 4 2 E 6 /N (F 4 ) EIV l ≤ 5 2 P SL 2 A 1 l = 2 1 SL 2 A 1 l ≤ 3 1 SL n+1 /N (SO n ) AI l = 2 1 SL n+1 /SO n AI l = 2 1 SL 6 /N (Sp 6 ) AII l ≤ 3 1 SL 6 /Sp 6 AII l ≤ 5 1 SL n+1 /N (S(GL 1 × GL n )) AIV l ≤ n + 1 1 SO 2n+1 /N (SO 1 × SO 2n ) BII l ≤ n + 1 1 SO 2n+1 /(SO 1 × SO 2n ) BII l ≤ 2n 1 Sp 2n /(Sp 2 × Sp 2n-2 ) CII l ≤ 2n 1 SO 2n /N (SO 1 × SO 2n-1 ) DII l ≤ n 1 SO 2n /(SO 1 × SO 2n-1 ) DII l ≤ 2n -1 1 F 4 /B 4 F II l ≤ 12 1 Figure 2: Fano X 1,...,l
4 Regular Fano varieties of rank 3

In this section we suppose X 0 wonderful; recall that {e 1 , e 2 , e 3 } is the basis of χ * (S) which generates C -. We classify all the Fano symmetric varieties obtainable by a wonderful symmetric variety of rank three from X 0 by a succession of blow-ups along G-subvarieties. This class of varieties contains many varieties; indeed each compact symmetric variety is dominated by a smooth toroidal variety obtained by a succession of blow-ups along G-subvarieties of codimension two. We begin proving a result similar to Lemma 3.1.

Let Z be the toric variety whose fan ∆ has maximal cones cone(v

1 , v 2 , v + ) and cone(v 1 , v 2 , v -= x 1 v 1 + x 2 v 2 -v + ), where {v 1 , v 2 , v + } is a basis of χ * (S),
x 1 +x 2 > 0 and x 1 ≥ x 2 ≥ 0. The anticanonical bundle of Z is ample if and only if x 1 = x 1 +x 2 = 1. In this case, Z is the blow up of A 3 along a stable subvariety of codimension 2. Moreover, the anticanonical bundle of Z is nef, but non-ample if and only if x 1 + x 2 = 2. We have two possibilities:

either v + + v -= v 1 + v 2 or v + + v -= 2v 1 .
In the first case we have a variety isomorphic to the variety Z of the previous section. This is the more problematic case, so we will study it in a second time.

Lemma 4.1 Let Z be a smooth 3-dimensional toric variety whose fan contains two maximal cones cone(v 1 , v 2 , v + ) and cone(v

1 , v 2 , v -) such that v + + v -= x 1 v 1 + x 2 v 2 ,
where x 1 and x 2 are integers with x 1 ≥ x 2 ≥ 0. Suppose moreover that x 1 ≥ 2. Then the anticanonical bundle of any toric variety Z ′ obtained from Z by a sequence of blow-ups along S-subvarieties is not ample.

Proof. Remark that the anticanonical bundle of Z is not ample. We say that a variety satisfies weakly the hypotheses of the lemma if x 1 + x 2 ≥ 2 (instead of x 1 ≥ 2). We use the following trivial observation:

x 1 + x 2 > 2 implies x 1 ≥ 2.
One can try to prove this lemma by induction as the Lemma 3.1. Unfortunately we can prove only the following weaker statement. Proof. We can suppose τ ⊂ cone(v 1 , v 2 , v -) by symmetry. If τ = cone(v 1 , v 2 ), we have three possibilities:

τ = cone(v 1 , v -), τ = cone(v 2 , v -) and τ = cone(v 1 , v 2 , v -). We always have ∆ Z ′ [p] = ∆ Z [p] ∪ {v ′ := v -+ b 1 v 1 + b 2 v 2 } with b 1 , b 2 ∈ {0, 1}. Moreover, ∆ Z ′ contains the cones cone(v 1 , v 2 , v + ) and cone(v 1 , v 2 , v ′ ) and we have v ′ + v + = (x 1 + b 1 )v 1 + (x 2 + b 2 )v 1 with (x 1 + b 1 ) + (x 2 + b 2 ) > 2, so Z 1 satisfies the hypotheses of the lemma. Finally let τ = cone(v 1 , v 2 ). The fan of Z ′ contains the cones cone(v 1 , v 1 + v 2 , v + ) and cone(v 1 , v 1 + v 2 , v -). We have v + + v -= (x 1 -x 2 )v 1 + x 2 (v 1 + v 2 ) with (x 1 -x 2 ) + x 2 = x 1 ≥ 2.
Now, we consider the general case. We have a sequence Z = Z 0 ← Z 1 ← ... ← Z i ← ... ← Z r = Z ′ where Z i+1 is the blow-up of Z i along the cone τ i . Let ∆ i = ∆ Zi and let j be the maximal index such that Z j satisfies the hypotheses (w.r.t. cone(w 1 , w 2 , w + ) and cone(w 1 , w 2 , w -)). By the previous lemma Z j+1 satisfies weakly the hypotheses, in particular its anticanonical bundle is not ample. By the maximality of j, Z j+1 does not satisfies the hypotheses, so Z j+1 contains a variety isomorphic to Z. Let ∆ be the fan of such variety. If ∆ r contains ∆ then -K Z ′ is not ample. Otherwise there is a minimal h such that ∆ is not contained in ∆ h+1 . We claim that Z h+1 satisfies the hypotheses of the lemma, a contradiction.

By the previous lemma τ j = cone(w 1 , w 2 ). We know that Z j+1 satisfies weakly the hypotheses w.r.t. σ + = cone(w 1 , w 1 + w 2 , w + ) and σ -= cone(w 1 , w 1 + w 2 , w -). Moreover, the open subvariety of Z j+1 with maximal cones σ + and σ -is isomorphic to Z. By the Lemma 4.2 2) we can suppose

τ h = cone(w 1 , w 1 + w 2 ).
The fan of Z h+1 contains two cones cone(w 1 +w 2 , w + , 2w 1 +w 2 ) and cone(w 1 + w 2 , w + , w ′ ), with w ′ = w 2 + x 1 (w 1 + w 2 ) + x 2 w + and x 1 , x 2 ≥ 0. Therefore Z h+1 satisfies the hypotheses of the lemma w.r.t. these cones. Indeed (2w 1 + w 2 ) + w ′ = (2 + x 1 )(w 1 + w 2 ) + x 2 w + . Now we want to study the varieties which contain an open subvariety isomorphic to Z. Observe that these varieties are never Fano varieties. Let Z be such a variety and let Z ′ be the blow-up of Z along the subvariety of Z associated to cone(e 1 , e 2 ). We prove that, if Z ′ satisfies the hypotheses of Lemma 4.1, then there are not Fano varieties obtainable from Z by a sequence of blow-ups. Lemma 4.3 Let Z be a smooth 3-dimensional toric variety whose fan contains cone(v 1 , v 2 , v 3 ) and cone(v 1 , v 2 , v 1 + v 2 -v 3 ) for suitable v 1 , v 2 , v 3 . Let Z ′ be the blow-up of Z along the stable subvariety corresponding to cone(v 1 , v 2 ) and let Z ′′ be a toric variety obtained from Z by a sequence of blow-ups along Ssubvarieties. If the anticanonical bundle of Z ′′ is ample, then Z ′′ is obtainable from Z ′ by a sequence of blow-ups along S-subvarieties.

Proof. We cannot proceed as in the previous lemma, because we do not know the other cones of ∆ Z . We have again a sequence

Z = Z 0 ← Z 1 ← ... ← Z i ← ... ← Z h = Z ′′ where π i+1 : Z i+1 → Z i is the blow-up along τ i . First of all, there is a (minimal) cone τ j contained in cone(v 1 , v 2 , v 3 , v 1 + v 2 -v 3 ), because
otherwise the anticanonical bundle of Z ′′ is not ample. By Lemma 4.1, τ j is cone(v 1 , v 2 ). We want to reorder the cones associated to the subvarieties along which we are blowing-up. Clearly this operation is not well defined in general.

We consider the following sequence of blow-ups:

Z = Z ′ 0 ← Z ′ 1 ← ... ← Z ′ i ← ... ← Z ′ j+1 , where π ′ 0 : Z ′ 1 → Z ′ 0 is the blow-up along τ j and π ′ i+1 : Z ′ i+1 → Z ′ i is the blow-up along τ i-1 for each i ≥ 1. Let ∆ ′ i = ∆ Z ′ i .
We show that these blow-ups are well defined and that

Z ′ j+1 = Z j+1 . The cone τ i-1 belongs to ∆ ′ i for each 1 ≤ i ≤ j because τ i is contained in |∆|\cone(v 1 , v 2 , v 3 , v 1 + v 2 -v 3 ) for each i ≤ j. Moreover, the elements of ∆ ′ 1 (3) not contained in cone(v 1 , v 2 , v 3 , v 1 + v 2 -v 3 ) are exactly the elements of ∆ 0 (3) \ {cone(v 1 , v 2 , v 3 ), cone(v 1 , v 2 , v 1 + v 2 -v 3 )}.
Z is the union of the following two open S-subvarieties: U 1 whose fan has maximal cones cone

(v 1 , v 2 , v 3 ) and cone(v 1 , v 2 , v 1 + v 2 -v 3 ); U 2 whose fan has maximal cones ∆(3) \ {cone(v 1 , v 2 , v 3 ), cone(v 1 , v 2 , v 1 + v 2 -v 3 )}.
The blow-up π ′ 0 induces an isomorphism between U 2 and its inverse image, because cone(v 1 , v 2 ) is not contained in any maximal cone of U 2 . In the same way π j induces an isomorphism between the inverse image of U 2 in Z j and its inverse image in Z j+1 . So the inverse image of U 2 in Z j+1 is isomorphic to the the inverse image of U 2 in Z ′ j+1 . Moreover π ′ j • ... • π ′ 2 induces an isomorphism between (π ′ 1 ) -1 (U 1 ) and its inverse image. In the same way π j-1 •...•π 1 induces an isomorphism between U 1 and its inverse image. So the inverse image of U 1 in Z j+1 is isomorphic to the the inverse image of U 1 in Z ′ j+1 . The lemmas follows because there is at most one morphism between two toric S-varieties extending the identity automorphism of S.

We now restrict the possible Fano symmetric varieties with rank three (and fixed G/H) which are obtainable as before to a finite explicit list.

Proposition 4.1 The toric varieties obtainable from A 3 by a sequence of blowups and with ample anticanonical bundle are, up to isomorphisms:

1. A 3 ; 2. a 2-blow-up of A 3 ; 3. the 3-blow-up of A 3 ; 4. the variety whose fan has maximal cones: cone(e 1 , e 1 + e 2 , e 1 + e 2 + e 3 ), cone(e 1 , e 3 , e 1 + e 2 + e 3 ), cone(e 2 , e 3 , e 1 + e 2 + e 3 ) and cone(e 2 , e 1 + e 2 , e 1 + e 2 + e 3 ). This variety is obtainable from A 3 by two consecutive blow-ups along subvarieties of codimension two; 5. the variety whose fan has maximal cones: cone(e 1 , e 3 , e 1 + e 2 + 2e 3 ), cone(e 1 , e 1 + e 2 + e 3 , e 1 + e 2 + 2e 3 ), cone(e 1 , e 2 , e 1 + e 2 + e 3 ), cone(e 2 , e 1 + e 2 + e 3 , e 1 + e 2 + 2e 3 ) and cone(e 2 , e 3 , e 1 + e 2 + 2e 3 ). This variety is obtainable from A 3 by a 3-blow up followed by a 2-blow up.

Proof. We proceed as follows: the anticanonical bundle of A 3 is ample, so we consider all the possible blow-ups of A 3 . Let Z be a blow-up of A 3 : 1) if Z satisfies the hypotheses of Lemma 4.1 we know that there are not toric varieties with ample anticanonical bundle and obtainable from Z by a sequence of blow-ups; 2) if Z satisfies the hypotheses of the Lemma 4.3 we study the variety Z ′ of such lemma; 3) finally, if the anticanonical bundle Z is ample, we reiterate the procedure. Observe that a priori it is possible that Z belongs to none of the previous cases. In the following, if two blow-ups of a given variety are isomorphic, we examine only one of them. Given a toric variety Z, let ∆ be its fan and let k be the piecewise linear function associated to -K Z . Suppose that all the maximal cones in ∆ are 3-dimensional. Remember that -K Z is ample if and only if, given any cone C ∈ ∆(3) and any

v ∈ ∆[p] with v / ∈ C, (k C )(v) < 1.
Let Z 0 = A 3 ; it has ample canonical bundle. Up to isomorphisms there are two blow-ups of A 3 : the blow-up Z 1 along cone(e 1 , e 2 ) and the blow-up Z 2 along cone(e 1 , e 2 , e 3 ).

One can show that the anticanonical bundle of Z 1 is ample because ∆ 1 (2) = {cone(e 1 , e 3 , e 1 + e 2 ), cone(e 2 , e 3 , e 1 + e 2 )} and ∆[p] = {e 1 , e 2 , e 3 , e 1 + e 2 }. The blow-ups of Z 1 are, up to isomorphisms: i) the blow-up Z 11 along cone(e 1 , e 3 ); ii) the blow-up Z 12 along cone(e 1 , e 1 + e 2 ); iii) the blow-up Z 13 along cone(e 1 + e 2 , e 2 , e 3 ); iv) the blow-up Z 14 along cone(e 1 + e 2 , e 3 ).

The variety Z 11 satisfies the hypotheses of the Lemma 4.3 w.r.t. cone(e 3 , e 1 + e 2 , e 1 + e 3 ) and cone(e 3 , e 1 + e 2 , e 2 ). Hence we have to study the blow-up Z 11b of Z 11 along cone(e 3 , e 1 + e 2 ). This variety satisfies the hypotheses of the Lemma 4.3 w.r.t. cone(e 1 + e 2 , e 1 + e 3 , e 1 ) and cone(e 1 + e 2 , e 1 + e 3 , e 1 + e 2 + e 3 ). Hence we have to study the the blow-up Z 11c of Z 11b along cone(e 1 + e 2 , e 1 + e 3 ). Z 11c satisfies the hypotheses of the Lemma 4.3 w.r.t. cone(e 1 +e 2 , e 1 +e 2 +e 3 , e 2 ) and cone(e 1 +e 2 , e 1 +e 2 +e 3 , 2e 1 +e 2 +e 3 ). Hence we have to consider the blow-up Z 11d of Z 11c along cone(e 1 + e 2 , e 1 + e 2 + e 3 ). Z 11d satisfies the hypotheses of the Lemma 4.1 w.r.t. cone(e 2 , e 1 +e 2 +e 3 , e 3 ) and cone(e 2 , e 1 +e 2 +e 3 , 2e 1 +2e 2 +e 3 ). Thus there are not toric varieties with ample anticanonical bundle and obtained from Z 11 by a sequence of blow-ups.

Z 12 satisfies the hypotheses of the Lemma 4.1 w.r.t. cone(e 3 , e 1 +e 2 , 2e 1 +e 2 ) and cone(e 3 , e 1 + e 2 , e 2 ). Z 13 satisfies the hypotheses of the Lemma 4.1 w.r.t. cone(e 3 , e 1 + e 2 , e 1 ) and cone(e 3 , e 1 + e 2 , e 1 + 2e 2 + e 3 ).

The anticanonical bundle of Z 14 is ample because ∆ 14 (2) = {cone(e 1 + e 2 , e 1 , e 1 + e 2 + e 3 ), cone(e 1 , e 3 , e 1 + e 2 + e 3 ), cone(e 2 , e 3 , e 1 + e 2 + e 3 ), cone(e 1 + e 2 , e 2 , e 1 + e 2 + e 3 )} and ∆[p] = {e 1 , e 2 , e 3 , e 1 + e 2 , e 1 + e 2 + e 3 }. The blow-ups of Z 14 are: i) the blow-up Z 141 of Z 14 along cone(e 1 , e 1 + e 2 ); ii) the blow-up Z 142 of Z 14 along cone(e 1 + e 2 , e 1 + e 2 + e 3 ); iii) the blow-up Z 143 of Z 14 along cone(e 3 , e 1 +e 2 +e 3 ); iv) the blow-up Z 144 of Z 14 along cone(e 2 , e 3 , e 1 +e 2 +e 3 ); v) the blow-up Z 145 of Z 14 along cone(e 2 , e 1 + e 2 , e 1 + e 2 + e 3 ); vi) the blow-up Z 146 of Z 14 along cone(e 1 , e 3 ); vii) the blow-up Z 147 of Z 14 along cone(e 1 , e 1 +e 2 +e 3 ).

The variety Z 141 satisfies the hypotheses of the Lemma 4.1 w.r.t. cone(e 1 + e 2 , e 1 +e 2 +e 3 , e 2 ) and cone(e 1 +e 2 , e 1 +e 2 +e 3 , 2e 1 +e 2 ). The variety Z 142 satisfies the hypotheses of the Lemma 4.1 w.r.t. cone(e 1 , e 1 + e 2 + e 3 , 2e 1 + 2e 2 + e 3 ) and cone(e 1 , e 1 + e 2 + e 3 , e 3 ). The variety Z 143 satisfies the hypotheses of the Lemma 4.1 w.r.t. cone(e 1 , e 1 + e 2 + e 3 , e 1 + e 2 ) and cone(e 1 , e 1 + e 2 + e 3 , e 1 + e 2 + 2e 3 ). The variety Z 144 satisfies the hypotheses of the Lemma 4.1 w.r.t. cone(e 3 , e 1 + e 2 + e 3 , e 1 ) and cone(e 3 , e 1 + e 2 + e 3 , e 1 + 2e 2 + 2e 3 ). The variety Z 145 satisfies the hypotheses of the Lemma 4.1 w.r.t. cone(e 2 , e 1 + e 2 + e 3 , e 3 ) and cone(e 2 , e 1 + e 2 + e 3 , 2e 1 + 3e 2 + e 3 ).

The variety Z 146 satisfies the hypotheses of the Lemma 4.3 w.r.t. cone(e 1 , e 1 + e 2 + e 3 , e 1 + e 3 ) and cone(e 1 , e 1 + e 2 + e 3 , e 1 + e 2 ). Hence we have to study the blow-up of Z 146 along cone(e 1 , e 1 + e 2 + e 3 ). This variety is Z 11c , so there are no toric varieties with ample anticanonical bundle which are obtained from Z 146 by a sequence of blow-ups.

The variety Z 147 satisfies the hypotheses of the Lemma 4.3 w.r.t. cone(e 1 + e 2 , e 1 + e 2 + e 3 , e 2 ) and cone(e 1 + e 2 , e 1 + e 2 + e 3 , 2e 1 + e 2 + e 3 ). Hence we have to study the blow-up Z 147b of Z 147 along cone(e 1 + e 2 , e 1 + e 2 + e 3 ). This variety satisfies the hypotheses of the Lemma 4.1 w.r.t. cone(e 2 , e 1 + e 2 + e 3 , e 3 ) and cone(e 2 , e 1 + e 2 + e 3 , 2e 1 + 2e 2 + e 3 ). Observe that we have classified the toric varieties with anticanonical bundle which are obtained from Z 1 by a sequence of blow-ups.

The anticanonical bundle of Z 2 is ample because ∆ 2 (2) = {cone(e 2 , e 3 , e 1 + e 2 +e 3 ), cone(e 1 , e 3 , e 1 +e 2 +e 3 ), cone(e 1 , e 2 , e 1 +e 2 +e 3 )} and ∆ 2 (p) = {e 1 , e 2 , e 3 , e 1 + e 2 + e 3 }. The blow-ups of Z 2 are, up to isomorphisms: i) the variety Z 14 ; ii) the blow-up Z 21 of Z 2 along cone(e 1 , e 2 , e 1 + e 2 + e 3 ); iii) the blow-up Z 22 of Z 2 along cone(e 3 , e 1 + e 2 + e 3 ).

Z 21 satisfies the hypotheses of the Lemma 4.1 w.r.t. cone(e 2 , e 1 + e 2 + e 3 , e 3 ) and cone(e 2 , e 1 + e 2 + e 3 , 2e 1 + 2e 2 + e 3 ).

The anticanonical bundle of Z 22 is ample because ∆ 22 (2) = {cone(e 1 , e 3 , e 1 + e 2 +2e 3 ), cone(e 1 , e 1 +e 2 +e 3 , e 1 +e 2 +2e 3 ), cone(e 1 , e 2 , e 1 +e 2 +e 3 ), cone(e 2 , e 1 + e 2 + e 3 , e 1 + e 2 + 2e 3 ), cone(e 2 , e 3 , e 1 + e 2 + 2e 3 )} and ∆ 22 [p] = {e 1 , e 2 , e 3 , e 1 + e 2 + e 3 , e 1 + e 2 + 2e 3 }.

The blow-ups of Z 22 are, up to isomorphisms: i) the variety Z 143 which satisfies the hypotheses of the Lemma 4.1; ii) the blow-up Z 221 of Z 22 along cone(e 3 , e 1 +e 2 +2e 3 ); iii) the blow-up Z 222 of Z 22 along cone(e 1 +e 2 +e 3 , e 1 +e 2 + 2e 3 ); iv) the blow-up Z 223 of Z 22 along cone(e 1 , e 1 +e 2 +2e 3 ); v) the blow-up Z 224 of Z 22 along cone(e 1 , e 3 ); vi) the blow-up Z 225 of Z 22 along cone(e 1 , e 1 +e 2 +e 3 ); vii) the blow-up Z 226 of Z 22 along cone(e 1 , e 3 , e 1 + e 2 + 2e 3 ); viii) the blow-up Z 227 of Z 22 along cone(e 1 , e 1 + e 2 + e 3 , e 1 + e 2 + 2e 3 ); ix) the blow-up Z 228 of Z 22 along cone(e 1 , e 2 , e 1 + e 2 + e 3 ).

Z 221 satisfies the hypotheses of the Lemma 4.1 w.r.t. cone(e 1 , e 1 + e 2 + 2e 3 , e 1 + e 2 + e 3 ) and cone(e 1 , e 1 + e 2 + 2e 3 , e 1 + e 2 + 3e 3 ). Z 222 satisfies the hypotheses of the Lemma 4.1 w.r.t. cone(e 1 , e 1 + e 2 + 2e 3 , e 3 ) and cone(e 1 , e 1 + e 2 + 2e 3 , 2e 1 + 2e 2 + 3e 3 ). Z 223 satisfies the hypotheses of the Lemma 4.1 w.r.t. cone(e 1 +e 2 +e 3 , e 1 +e 2 +2e 3 , e 2 ) and cone(e 1 +e 2 +e 3 , e 1 +e 2 +2e 3 , 2e 1 +e 2 +2e 3 ).

Z 224 satisfies the hypotheses of the Lemma 4.3 w.r.t. cone(e 1 , e 1 + e 2 + 2e 3 , e 1 + e 2 + e 3 ) and cone(e 1 , e 1 + e 2 + 2e 3 , e 1 + e 3 ). Hence we have to study the blow-up Z 224b of Z 224 along cone(e 1 , e 1 + e 2 + 2e 3 ). This variety satisfies the hypotheses of the Lemma 4.1 w.r.t. cone(e 1 , e 1 + e 2 + e 3 , e 2 ) and cone(e 1 , e 1 + e 2 + e 3 , 2e 1 + e 2 + 2e 3 ). Z 225 satisfies the hypotheses of the Lemma 4.3 w.r.t. cone(e 1 , e 1 +e 2 +2e 3 , 2e 1 +e 2 +e 3 ) and cone(e 1 , e 1 +e 2 +2e 3 , e 3 ). Hence we have to study the blow-up Z 225b of Z 225 along cone(e 1 , e 1 +e 2 +2e 3 ). Z 225b satisfies the

G/H θ m1 m 1 P SL 2 A 1 1 0 SL 2 A 1 2 1 SL 2 /N (SO 2 ) AI 0 0 h SL 2 /SO 2 AI 1 0 h SL 2 /N (Sp 2 ) AII 2 1 SL 2 /Sp 2 AII 4 3 SL n /S(GL 1 × GL n-1 ) AIV n n -1 e SO 2n+1 /N (SO 1 × SO 2n ) BII n -1 n -1 SO 2n+1 /(SO 1 × SO 2n ) BII 2n -1 2n -2 Sp 2n /(Sp 2 × Sp 2n-2 ) CII 2n -1 2n -2 SO 2n /S(O 1 × O 2n-1 ) DII n -1 n -2 SO 2n /(SO 1 × SO 2n-1 ) DII 2n -2 2n -3 F 4 /B 4 FII 11 10
Figure 3: Weights of rank 1 symmetric spaces hypotheses of the Lemma 4.3 w.r.t. cone(2e 1 + e 2 + e 3 , e 1 + e 2 + 2e 3 , e 1 + e 2 + e 3 ) and cone(2e 1 + e 2 + e 3 , e 1 + e 2 + 2e 3 , 2e 1 + e 2 + 2e 3 ). Hence we have to consider the blow-up Z 225c of Z 225b along cone(2e 1 + e 2 + e 3 , e 1 + e 2 + 2e 3 ). This variety satisfies the hypotheses of the Lemma 4.1 w.r.t. cone(e 1 +e 2 +e 3 , e 1 +e 2 +2e 3 , e 2 ) and cone(e 1 + e 2 + e 3 , e 1 + e 2 + 2e 3 , 3e 1 + 2e 2 + 3e 3 ). Z 226 satisfies the hypotheses of the Lemma 4.1 w.r.t. cone(e 1 , e 1 + e 2 + 2e 3 , e 1 + e 2 + e 3 ) and cone(e 1 , e 1 + e 2 + 2e 3 , 2e 1 + e 2 + 3e 3 ). Z 227 satisfies the hypotheses of the Lemma 4.1 w.r.t. cone(e 1 + e 2 + e 3 , e 1 + e 2 + 2e 3 , e 2 ) and cone(e 1 + e 2 + e 3 , e 1 + e 2 + 2e 3 , 3e 1 + 2e 2 + 3e 3 ). Z 228 satisfies the hypotheses of the Lemma 4.1 w.r.t. cone(e 1 , e 1 + e 2 + e 3 , e 1 + e 2 + 2e 3 ) and cone(e 1 , e 1 + e 2 + e 3 , 2e 1 + 2e 2 + e 3 ).

Let G/H be a symmetric space of rank three such that the standard compactification X 0 of G/H is wonderful. We introduce the following notations:

• we denote by X ij the blow-up of X 0 along (cone(e i , e j ), ∅);

• we denote by X 123 the blow-up of X 0 along the closed G-orbit;

• we denote by X 123,ij the blow-up of X 123 along (cone(e i , e j ), ∅);

• we denote by X 123,i the blow-up of X 123 along (cone(e i , e 1 + e 2 + e 3 ), ∅).

By [Ru09] Corollary 2.1, if (G, θ) = (G 1 , θ) × (G 2 , θ) and X 0 is wonderful, then H = H 1 × H 2 , where H i := H ∩ G i . Given an 1-rank symmetric space G i /H i , let ψ i (r) := -2ρ + 2ρ 0 -re * i , m i := max{r : ψ i (r)(α ∨ i ) < 0} and mi := max{r : ψ i (r)(α ∨ i ) ≤ 0}.
In Figure 3 are written the value of m i and mi for the various G i /H i . Moreover, we indicate by e (resp. by h) when G i /H i is exceptional (resp. hermitian non-exceptional). By an explicitly analysis we can prove the following theorems.

Theorem 4.1 Let G/H be an indecomposable symmetric space of rank 3 such that its standard compactification X 0 is wonderful. If X is a smooth Fano compactification of G/H obtained from X 0 by a sequence of blow-ups along Gsubvarieties, then it is X 0 , X 12 , X 13 , X 23 or X 123,13 . More precisely, the Fano ones are those which appears in Figure 4. Theorem 4.2 Let G/H be a symmetric space such that X 0 is wonderful and such that (G, θ) = (G 1 , θ)× (G 2 , θ) with rank G i /G θ i = i. If X is a smooth Fano compactification of G/H obtained from X 0 by a sequence of blow-ups along Gsubvarieties, then it is X 0 , X 12 , X 13 , X 23 , X 123 , X 123,12 , X 123,13 , X 123,23 , X 123,1 , X 123,12 or X 123,3 . More precisely, the classification of such varieties is as in Figure 5. In the second column, we indicate the conditions on m 1 so that X is Fano.

G/H θ X P SL 4 A 3 X 0 , X 13 SO 7 B 3 X 0 P Sp 6 C 3 X 0 , X 13 Sp 6 C 3 X 0 , X 13 SL 4 /N (SO 4 ) AI X 0 SL 8 /N (Sp 8 ) AII X 0 , X 12 , X 13 , X 23 , X 123,13 SL n+1 /S(L 3 × L n-2 ), n ≥ 6 AIII X 0 , X 13 SL 6 /N (S(L 3 × L 3 )) AIII X 0 SL 6 /S(L 3 × L 3 ) AIII X 0 SO 2n+1 /N (SO 3 × SO 2n-2 ) BI X 0 Sp 6 /N (GL 3 ) CI ∄ Sp 6 /GL 3 CI X 0 Sp 12 /N (Sp 6 × Sp 6 ) CII X 0 , X 12 , X 13 , X 23 Sp 12 /(Sp 6 × Sp 6 ) CII X 0 , X 12 , X 13 , X 23 , X 123,13 Sp 2n /(Sp 6 × Sp 2n-6 ), n > 6 CII X 0 , X 12 , X 13 , X 23 , X 123,13 SO 2n /N (SO 3 × SO 2n-3 ) DI X 0 SO 12 /N (GL 6 ) DIII X 0 , X 12 SO 12 /N (GL 6 ) DIII X 0 , X 12 SO 14 /GL 7 DIII X 0 , X 12 , X 13 , X 23 , X 123,13 E 7 /N (E 6 × C * ) EVII X 0 , X 12 E 7 /(E 6 × C * ) EVII X 0 , X 12
Theorem 4.3 Let G/H be a symmetric space such that X 0 is wonderful. Sup- pose that (G, θ) = (G 1 , θ) × (G 2 , θ) × (G 3 , θ) with rank G i /G θ i = 1
and let x r be the number of factors G i such that ψ i (r) is antidominant and regular. If X is a smooth Fano compactification of G/H obtained from X 0 by a sequence of blow-ups along G-subvarieties, then it is X 0 , X 12 , X 13 , X 23 , X 123 , X 123,12 , X 123,13 , X 123,23 , X 123,1 , X 123,2 or X 123,3 . More precisely, we have the following classification (depending on G/H):

• If x 1 ≤ 1, then the smooth Fano compactifications of G/H are X 0 , X 12 ,
X 13 and X 23 . In particular, there are four of them.

• If x 1 = 2, then there are five Fano varieties. Let i < j be the indices such that ψ i (1) and ψ j (1) are anti-dominant and regular. The smooth Fano compactifications of G/H are X 0 , X 12 , X 13 , X 23 and X 123,ij .

G 2 /H 2 m 1 X P SL 3 - X 0 , X 12 , X 13 , X 23 SO 5 - X 0 , X 12 Spin 5 - X 0 , X 12 , X 13 , X 23 G 2 - X 0 , X 13 SL 3 /N (SO 3 ) - X 0 SL 6 /N (Sp 6 ) - X 0 , X 12 , X 13 , X 23 , X 123,23 m 1 ≥ 1 X 123 , X 123,12 , X 123,13 SL n+1 /S(GL 2 × GL n-1 ) - X 0 , X 12 , X 13 , X 23 (n ≥ 5) m 1 ≥ 1 X 123,13 SL 5 /S(GL 2 × GL 3 ) - X 0 , X 12 , X 13 , X 23 SL 4 /N (S(GL 2 × GL 2 )) - X 0 SL 4 /S(GL 2 × GL 2 ) - X 0 , X 12 SO 5 /S(O 2 × O 3 ) - X 0 SO 5 /(SO 2 × SO 3 ) - X 0 SO 2n+1 /S(O 2 × O 2n-1 ) - X 0 , X 13 SO 2n+1 /(SO 2 × SO 2n-1 ) - X 0 , X 13 Sp 2n /Sp 4 × Sp 2n-4 - X 0 , X 12 , X 13 , X 23 , X 123,23 (n ≥ 5) m 1 ≥ 1 X 123 , X 123,12 , X 123,13 m 1 ≥ 2 X 123,2 Sp 8 /N (Sp 4 × Sp 4 ) - X 0 , X 12 , X 13 , X 23 Sp 8 /Sp 4 × Sp 4 - X 0 , X 12 , X 13 , X 23 , X 123,23 m 1 ≥ 1 X 123 , X 123,12 , X 123,13 SO 2n /S(O 2 × O 2n-2 ) - X 0 , X 13 SO 2n /SO 2 × SO 2n-2 - X 0 , X 13 SO 8 /N (GL 4 ) - X 0 , X 12 SO 8 /GL 4 - X 0 , X 12 SO 10 /GL 5 - X 0 , X 12 , X 13 , X 23 , X 123,23 m 1 ≥ 1 X 123 , X 123,12 , X 123,13 E 6 /D 5 × C * - X 0 , X 12 , X 13 , X 23 , X 123,23 m 1 ≥ 1 X 123 , X 123,12 , X 123,13 , X 123,1 m 1 ≥ 2 X 123,2 , X 123,3 E 6 /N (F 4 ) - X 0 , X 12 , X 13 , X 23 , X 123,23 m 1 ≥ 1 X 123 , X 123,12 , X 123,13 , X 123,1 m 1 ≥ 2 X 123,2 , X 123,3 G 2 /(A 1 × A 1 ) - ∄ Figure 5: Fano decomposable symmetric varieties of rank 3 6) Suppose R G,θ = BC 2 and cone(α ∨ 1 , v) ∈ ∆, with v := -xα ∨ 1 -yα ∨ 2 primitive. As before y = 1. Moreover, 0 < (-v, α 2 ) = -x + 1, so 0 < x < 1. 7) Suppose R G,θ = BC 2 and cone(α ∨ 2 , v) ∈ ∆, with v := -xα ∨ 1 -yα ∨ 2
primitive. As before x = 1 and 0 < (-v, α 1 ) = 2 -y, so v = -ω ∨ 1 is not regular. 8) Suppose R G,θ = G 2 and cone(α ∨ 1 , v) ∈ ∆, with v := -xα ∨ 1 -yα ∨ 2 primitive. As before y = 1 and 0 < (-v, α 2 ) ≤ -3x + 2, so 0 < x < 2 3 . 9) Suppose R G,θ = G 2 and cone(α ∨ 2 , v) ∈ ∆, with v := -xα ∨ 1 -yα ∨ 2 primitive. As before x = 1. Moreover, 0 < (-v, α 1 ) = 2 -y and 0 < (-v, α 2 ) = -3 + 2y; thus 3 < 2y < 4.

In the next two lemmas, we do not make any hypothesis on the regularity of X.

Lemma 5.2 Let X be a Fano symmetric variety with G semisimple. Then F (X) contains at most three colored 1-cones.

Proof. Suppose by contradiction that F (X) contains (σ i := cone(v i , v i+1 ), ∅), with i = 1, 2, 3 and v 1 , v 2 , v 3 , v 4 primitive. We can write v 2 as a positive linear combination xv 1 + yv 4 of v 1 and v 4 . Then

k σ2 (v 1 ) < k(v 1 ) = 1, k σ2 (v 4 ) < k(v 4 ) = 1 and 1 = k σ2 (v 2 ) = xk σ2 (v 1 ) + yk σ2 (v 4 ) ≤ 0.
Lemma 5.3 Let X be a Fano non-simple toroidal symmetric variety. Then X 0 is smooth and ∆[p] is either {e 1 , e 1 + re 2 , e 2 } or {e 1 , re 1 + e 2 , e 2 }. These varieties are smooth if and only if r = 1.

Proof. By the previous lemma, we have ∆[p] = {e 1 , e 2 , v} for an appropriate v. First suppose that X 0 is smooth, i.e. χ * (S) = Ze 1 ⊕ Ze 2 , and write v = x 1 e 1 + x 2 e 2 . For each i, let σ i = cone(e i , v) and {i, i c } = {1, 2}, so k σi is e * i + 1-xi x i c e * i c . If x 1 ≥ 2 and x 2 ≥ 2, then 1-x1 x2 and 1-x2 x1 are strictly negative integers, so x 1 ≥ x 2 + 1 ≥ x 1 + 2, a contradiction.

Suppose now that X 0 is singular. Then R G,θ is either A 2 or A 1 × A 1 . In the first case, e i = -3ω ∨ i and the strictly positive integer k

(-ω ∨ 1 -ω ∨ 2 ) is lesser than 1 3 k(-3ω ∨ 1 ) + 1 3 k(-3ω ∨ 2 ) = 2 3 , a contradiction. Finally suppose R G,θ = A 1 × A 1 and χ * (S) = Z2ω ∨ 2 ⊕ Z(ω ∨ 1 + ω ∨ 2 ). Then, there is i such that -ω ∨ 1 -ω ∨ 2 = av + b(-2ω ∨ i ) with a, b ≥ 0. The integer a + b = k(-ω 1 -ω 2 ) is strictly lesser than 1 2 k(-2ω 1 ) + 1 2 k(-2ω 2 ) = 1, a contradiction.
Remark that the previous two lemmas apply also to a toroidal symmetric variety with -K X ample, |F (X)| convex and generated by a basis of χ * (S) (without supposing X compact). Now, we state the main result of this section.

Theorem 5.1 Let G/H be a symmetric space of rank 2 (with G semisimple).

• If a (projective) symmetric variety X is Fano then ρ -1 (ρ(D(X))) = D(X).
If moreover X is locally factorial, then ♯ρ(D(X)) = ♯ρ -1 (ρ(D(X))).

• If R G,θ is irreducible and X is a Fano locally factorial equivariant compactification of G/H, we have exactly the following possibilities for ∆[p]:

1. ∆[p] = {e 1 , α ∨ 1 } if ♯ρ -1 (α ∨ 1 ) = 1 and R G,θ is not G 2 ; 2. ∆[p] = {e 2 , α ∨ 2 } if ♯ρ -1 (α ∨ 2 ) = 1 and R G,θ is not B 2 ; 3. ∆[p] = {e 2 , α ∨ 2 } if R G,θ = B 2 and H = G θ ; 2. ∆[p] = {α ∨ 2 , -α ∨ 2 , v 2 (r), v 2 (r + 1), ..., v 2 (r + s)} if i) s = 0, 1, ii) r + s ≤ m1 + 1, iii) either r + s > m 1 + 1 or ♯ρ -1 (α ∨ 2 ) = 2; 3. ∆[p] = {α ∨ 1 , -α ∨ 1 , v 1 (r), v 1 (r + 1), ..., v 1 (r + s)} if i) s ≥ 2 and ii) r + s ≤ m2 + 1; 4. ∆[p] = {α ∨ 2 , -α ∨ 2 , v 2 (r), v 2 (r + 1), ..., v 2 (r + s)} if i) s ≥ 2 and ii) r + s ≤ m1 + 1; 5. ∆[p] = {α ∨ 1 , -α ∨ 1 , -rα ∨ 1 + e 2 , w 1 (r, 1), ..., w 1 (r, s)} if r ≤ m2 and 2 ≤ s ≤ m1 + 1; 6. ∆[p] = {α ∨ 2 , -α ∨ 2 , -rα ∨ 2 + e 1 , w 2 (r, 1), ..., w 2 (r, s)} if r ≤ m1 and 2 ≤ s ≤ m2 + 1; 7. ∆[p] = {α ∨ 1 , α ∨ 2 , -α ∨ 1 -α ∨ 2 } if ♯D(G/H) > 2; 8. ∆[p] = {e 1 , e 1 + e 2 ,
∈ ρ(D(X)), ♯ρ -1 (α ∨ ) = 2 and 2α∈ / R G,θ . • If G/H is decomposable but χ * (S) = Z( 1 2 α ∨ 1 + 1 2 α ∨ 2 ) ⊕ Zα ∨ 2 , let v j (i) := -2i+1 2 α ∨ j -1 2 α ∨ j c and w j (x, y) := -2xy+y+2 2 α ∨ j -y 2 α ∨ j c .
We have the following locally factorial quasi Fano compactifications of G/H, which are not Fano:

1. ∆[p] = {α ∨ 1 , -α ∨ 1 , v 1 (r), v 1 (r + 1), ..., v 1 (r + s)} if i) r ≥ 0, ii) s ≥ 2 and iii) r + s ≤ m2+1 2 ; 2. ∆[p] = {α ∨ 2 , -α ∨ 2 , v 2 (r), v 2 (r + 1), ..., v 2 (r + s)} if i) r ≥ 0, ii) s ≥ 2 and iii) r + s ≤ m1+1 2 ; 3. ∆[p] = {α ∨ 1 , -α ∨ 1 , v 1 (r), v 1 (r + 1), ..., v 1 (r + s)} if i) r ≥ 0, ii) s = 0, 1 and iii) m2+1 2 < r + s ≤ m2+1 2 ; 4. ∆[p] = {α ∨ 2 , -α ∨ 2 , v 2 (r), v 2 (r + 1), ..., v 2 (r + s)} if i) r ≥ 0, ii) s = 0, 1 and iii) m1+1 2 < r + s ≤ m1+1 2 ; 5. ∆[p] = {α ∨ 1 , -α ∨ 1 , -2r+1 2 α ∨ 1 -1 2 α ∨ 2 , w 1 (r, 1), ..., w 1 (r, s)} if i) r ≤ m2-1 2 and ii) 2 ≤ s ≤ m1 + 1; 6. ∆[p] = {α ∨ 2 , -α ∨ 2 , -1 2 α ∨ 1 -2r+1 2 α ∨ 2 , w 2 (r, 1), ..., w 2 (r, s)} if i) r ≤ m1-1 2 and ii) 2 ≤ s ≤ m2 + 1; 7. ∆[p] = {-α ∨ 1 , -α ∨ 2 , -1 2 α ∨ 1 -1 2 α ∨ 2 }.
The last variety is smooth, while the other ones are smooth if and only if r = 0.

The idea of the proof is to utilize Theorem 5.1 thanks to the following lemma. Remark that, when θ is indecomposable, if -K X is nef it is also big (see §1.6).

Proof. The first property holds because the inequalities in the conditions for the nefness of a Cartier divisor are not strict. I) First suppose ρ non-injective. We have to consider the varieties whose fan is as in Theorem 5.1, but which are not Fano because ♯ρ -1 (ρ(D(X)) = ♯ρ(D(X)). In such cases ρ has to be injective over D(X), so that X is locally factorial. If θ is indecomposable we have the following possibilities:

i) R G,θ = BC 2 and ∆(2) = {cone(α ∨ 2 , -ω 2 )}; ii) R G,θ = B 2 , H = G θ and ∆(2) = {cone(α ∨ 1 , -2ω 1 )}.
If θ is decomposable, we have to consider all the possibilities listed in Theorem 5.1 which correspond to non-toroidal varieties with χ * (S) = Ze 1 ⊕ Ze 2 . We have also to substitute the conditions of type ψ i (m) < 0 with the corresponding conditions ψ i (m) ≤ 0.

Lemma 5.4 Let X be a projective symmetric variety with -K X nef, then there is a symmetric variety X ′ below X such that the piecewise linear function associated to -K X ′ is strictly convex (over the colored fan of X ′ ) and coincides with the function associated to -K X . If X is toroidal, we can choose X ′ toroidal.

Proof of Lemma 5.4. Let ∆ ′ be the fan whose maximal cones are the maximal cones over which k X is linear. Given any cone C ∈ ∆ ′ , define F C as {D ∈ D(G/H) : ρ(D) ∈ C}. We claim that {(C, F C )} C∈∆ ′ ,Int C∩C -=∅ is a colored fan associated to a symmetric variety which satisfies the conditions of the lemma. Remark that |∆ ′ | = |F (X)|. We have only to prove that the maximal cones of ∆ ′ are strictly convex.

Let C = cone(α ∨ 1 , ..., α ∨ r , -̟ ∨ 1 , ..., -̟ ∨ s ) ∈ ∆ ′ (l) (with ̟ ∨ 1 , ..., ̟ ∨ s dominant
) and suppose by contradiction that it contains the line generated by v = r i=1 a i α ∨ i + s j=1 b j (-̟ ∨ j ). Then C contains also Rv ′ , where v ′ = b j (-̟ ∨ j ); indeed -v ′ = -v + a i α ∨ i . Write -v ′ = α∈R G,θ c α α ∨ ; then C contains all the α ∨ such that c α = 0, because -v ′ ∈ C ∩ C + and any spherical weight is a positive rational combination of the simple restricted roots. Thus,

if v ′ = 0, k C (-v ′ ) = c α k C (α ∨ ) ≥ 0, while k C (v ′ ) = b j k C (-̟ ∨ j ) > 0.
Suppose now that v = r i=1 a i α ∨ i with a j = 0. Then C contains Rα ∨ j . Write -α ∨ j = r i=1 a ′ i α ∨ i + s i=1 b ′ i (-̟ ∨ i ) with positive coefficients, then there is j 0 such that b j0 = 0. So k C (-α ∨ j ) ≥ b j0 > 0 and k C (α ∨ j ) > 0, a contradiction. . II) Suppose X toroidal and let X ′ be as in Lemma 5.4. If X ′ = X and is simple, then G/H = G 2 /(SL 2 × SL 2 ). If X ′ = X and is simple, then it must be singular. Otherwise, given any w = x 1 e 1 + x 2 e 2 ∈ ∆ X [p] \ ∆ X ′ [p], we have x i ∈ Z >0 and x 1 + x 2 = k(w) = 1. If R G,θ = A 2 and H = G θ , then k X = -1 3 (ω 1 + ω 2 ) / ∈ χ(S), so X is not locally factorial. Thus R G,θ = A 1 × A 1 and χ * (S) =

Z2ω ∨ 1 ⊕ Z(ω ∨ 1 + ω ∨ 2 ). Let w = -x 1 α ∨ 1 -x 2 α ∨ 2 be in ∆ X [p] \ ∆ X ′ [p]; then 2x 1 , 2x 2 ∈ Z >0 and x 1 +x 2 = k(w) is 1. Thus ∆ X [p] = {-α ∨ 1 , -α ∨ 2 , -1 2 α ∨ 1 - 1 2 α ∨ 2 }.
In this case -K X is nef but non-ample. If X ′ is not simple, then the standard compactification of G/H has to be smooth because of Lemma 5.3. So it is sufficient to prove the following lemma.

Lemma 5.5 The smooth toric varieties birationally proper over A 2 with nef anticanonical divisor are, up to isomorphisms, A 2 and the varieties Z m , where Z m is the variety whose fan has maximal cones {cone(e 1 , e 1 + e 2 ), cone(e 1 + e 2 , e 1 + 2e 2 ), ..., cone(e 1 + (m -1)e 2 , e 1 + me 2 ), cone(e 1 + me 2 , e 2 )}.

Proof of Lemma 5.5. The piecewise linear k m function associated to the anticanonical bundle of Z m is linear on cone(e 1 , e 1 + me 2 ). It is easy to see that thus such function is convex. Now we show that, given any Z as in the hypotheses, it is isomorphic to Z m for an appropriate m. Notice, that any smooth toric variety birationally proper over A 2 is obtained by a sequence of blow-ups. Thus there is nothing e = -α ∨ , because there is an appropriate cone(α ∨ , v) in ∆(2). Suppose first that χ * (S) = Ze ⊕ Zf . We have to consider the following cases: 1) If ∆[p] = {f, -f, v}, then v = e+rf because {f, v} is a basis of χ * (S). We have to impose that k cone(f,e+rf ) (α ∨ ) = ψ(r-1)(α ∨ ) < 0 and that k cone(-f,e+rf ) (α ∨ ) = ψ(-r-1)(α ∨ ) < 0. 2) Suppose ∆[p] = {f, -f, v 1 , v 2 }. As before v i = e + x i f ; moreover x 1 = x 2 ± 1 because {v 1 , v 2 } is a basis of χ * (S). Suppose x 1 = x 2 + 1. We have to impose k cone(f,e+(x2+1)f ) (α ∨ ) = ψ(x 2 )(α ∨ ) < 0 and k cone(-f,e+x2f ) (α ∨ ) = ψ(-x 2 -1)(α ∨ ) < 0.

3) Suppose ∆[p] = {α ∨ , -f, v}. Then v = -α ∨ + rf = -mα ∨ ± f because X is smooth. But -f is not contained in cone(α ∨ , v), thus v = -α ∨ + f . 4) Suppose ∆[p] = {α ∨ , -f, v 1 , v 2 } and ∆(2) = {cone(α ∨ , v 1 ), σ := cone(v 1 , v 2 ), cone(-f, v 2 )}. Then v 1 = -mα ∨ +f and v 2 = -α ∨ +rf . Observe that R ≥0 v 1 = R ≥0 (-α ∨ + 1 m f ), so r ≤ 0. Furthermore v 2 -rv 1 = -(1 -rm)α ∨ = ±α ∨ , so mr is 0 or 2. Thus v 2 = -α ∨ . Moreover h σ (-f ) = m -1 < 1, so v 1 = -α ∨ + f . 5) Suppose ∆[p] = {α ∨ , v 1 , v 2 } and ∆(2) = {cone(α ∨ , v 1 ), cone(v 1 , v 2 ), cone(α ∨ , v 2 )}. Thus v 1 = -rα ∨ + f and v 2 = -mα ∨ -f with r, m > 0. Moreover, v 1 + v 2 = -(r + m)α ∨ = ±α ∨ , a contradiction. 6) Suppose ∆[p] = {α ∨ , v 1 , v 2 , v 3 } and ∆(2) = {cone(α ∨ , v 1 ), σ := cone(v 1 , v 2 ), cone(v 2 , v 3 ), cone (α ∨ , v 3 )}. Write v 2 = -xα ∨ + yf . By the smoothness of X, v 1 = -rα ∨ + f , v 3 = -mα ∨ -f , x = -my ± 1 and x = +ry ± 1. The last two conditions, plus x > 0 imply that v 2 = -α ∨ . Moreover k σ (v 3 ) = m + r -1 < 1, a contradiction. Now, suppose χ * (S) = Z f +α ∨ 2 ⊕ Zf . 1) The toroidal case can be studied as before. 2) Suppose that ∆(2) contains two cones σ ± := cone(v ± , α ∨ ). Let u = 1 2 α ∨ + 1 2 f . We have v + = -2m+1 2 α ∨ + 1 2 f = -(2m + 1)u + (m + 1)f and v -= -2r+1 2 α ∨ -1 2 f = -(2r + 1)u + rf with m, r ≥ 0. First, suppose that there is another v = -xu + yf in ∆[p]; here x > 0. By the smoothness, we have x(m + 1) = y(2m + 1) ± 1 and xr = y(2r + 1) ± 1. Thus x[(m + 1)(2r + 1) -r(2m + 1)] = ±1 ∓ 1, so x(m + r + 1) = 2, x(m + 1) = y(2m + 1) + 1 and xr = y(2r + 1) -1. If x = 1, then the previous three equations are not compatible. Instead, if x = 2 then ∆[p] = {α ∨ , -1 2 α ∨ + 1 2 f, -α ∨ , -1 2 α ∨ -1 2 f }. Next, suppose that ∆[p] = {α ∨ , v + , v -}; in particular {v + , v -} is a basis. Thus v + + v -= -(m + r + 1)α ∨ = ±α ∨ , so ∆[p] = {α ∨ , -1 2 α ∨ + 1 2 f, -1 2 α ∨ -1 2 f }. 3) Finally, suppose that ∆[p] contains α ∨ and -f . Then ∆ contains σ = cone(α ∨ , v 1 ) and cone(-f, v 2 ), with v 1 = -2m+1 2 α ∨ + 1 2 f , m ≥ 0 and v 2 = -1 2 α ∨ -2r+1 2 f . If v 1 = v 2 , then ∆[p] = {α ∨ , -f, -1 2 α ∨ + 1 2 f }. Otherwise, by Lemma 5.6, ∆ contains cone(v 1 , v 2 ). Furthermore, (2r + 1)v 1 + v 2 = ±α ∨ , hence (2r + 1)(2m + 1) is 1 or -3. But r ≥ 0 because v 2 / ∈ σ, so v 1 = -1 2 α ∨ + f and v 2 = -1 2 α ∨ -f .

Lemma 4. 2

 2 Let Z be a toric variety which satisfies weakly the hypotheses of Lemma 4.1 and let Z ′ be the blow-up of Z along a cone τ .1. If τ = cone(v 1 , v 2 ), then Z ′ satisfiesthe hypotheses of Lemma 4.1. 2. If Z satisfies the hypotheses of Lemma 4.1, then Z ′ satisfies weakly the hypotheses of Lemma 4.1.

Figure 4 :

 4 Figure 4: Fano indecomposable symmetric varieties of rank 3

  θ and equal to one otherwise; for each i we define α ∨ i as the coroot 2bi (αi,αi) α i . The set {α ∨ 1 , ..., α ∨ s } is a basis of the dual root system R ∨ G,θ . We call the elements of R ∨ G,θ the restricted coroots. Let ω 1 , ..., ω s be the fundamental weights of R G,θ w.r.t. {α 1 , ..., α s } and let ω ∨

1 , ..., ω ∨ s be the fundamental weights of R ∨ G,θ w.r.t. {α ∨ 1 , ..., α ∨ s }. Let C + be the positive closed Weyl chamber of χ(S) R and let C -

  e 1 + 2e 2 , ..., e 1 + (s -1)e 2 , e 1 + se 2 , e 2 )} if 2 ≤ s ≤ m1 + 1; 9. ∆[p] = {e 2 , e 1 + e 2 , 2e 1 + e 2 , ..., (s -1)e 1 + e 2 , se 1 + e 2 , e 1 )} if 2 ≤ s ≤ m2 + 1. These varieties are smooth if either they are toroidal or if, ∀α ∨

• If (x 1 , x 2 ) is equal to (3, 0) or to (3, 1), then there are eight Fano varieties: X 0 , X 12 , X 13 , X 23 , X 123 , X 123,12 , X 123,13 and X 123,23 .

• If (x 1 , x 2 ) = (3, 2), then there are nine Fano varieties. Suppose that the ψ j (2) with j = i are anti-dominant and regular, then the Fano varieties are: X 0 , X 12 , X 13 , X 23 , X 123 , X 123,12 , X 123,13 , X 123,23 and X 123,i .

• If (x 1 , x 2 ) = (3, 3), then there are eleven Fano varieties: X 0 , X 12 , X 13 , X 23 , X 123 , X 123,12 , X 123,13 , X 123,23 , X 123,1 , X 123,2 and X 123,3 .

Remark that the first part of the proof holds in a more general contest.

Corollary 4.1 Given any wonderful G-variety X 0 of rank 3 (even non symmetric) and any Fano variety X obtained from X 0 by a succession of blow-ups along G-stable subvarieties, then X is one of the following varieties: X 0 , X 12 , X 13 , X 23 , X 123 , X 123,12 , X 123,13 , X 123,23 , X 123,1 , X 123,2 or X 123,3 .

5 (Quasi) Fano symmetric varieties of rank 2

Fano symmetric varieties

In this section we consider the quasi-Fano locally factorial symmetric varieties with rank 2 (and G only reductive). Remark that to individuate univocally a projective 2-rank symmetric variety with ρ injective over ̺ -1 (̺(D(X)) is sufficient to give ∆[p]. We begin classifying the Fano varieties with G semisimple. First, we consider two special cases: i) G/H is indecomposable, while X is neither simple nor toroidal; ii) X is toroidal.

Lemma 5.1 Let X be a locally factorial projective symmetric variety. Suppose that G/H is indecomposable and that X is neither simple nor toroidal, then R G,θ = A 2 , H = G θ and F (X) contains (cone(-α ∨ 1 -α ∨ 2 ), ∅).

Proof. We do a case-to-case analysis. 1) Suppose R G,θ = A 2 and H = N (G θ ). Then there is cone(α ∨ 1 , v) in ∆, with v := -x 3 α ∨ 1 -y 3 α ∨ 2 ∈ int(C -) primitive. We can write -ω ∨ 2 as a positive integral combination of α ∨ 1 and v; thus y ∈ {1, 2}. Moreover, 0 < (-v, α 2 ) = -x 3 + 2y 3 , so 0 < x < 2y. If y = 1, then v = -1 3 (ω ∨ 1 +ω ∨ 2 ) / ∈ χ * (S). If y = 2, we have three possibilities: i

The previous varieties are singular in the following cases:

) and let m i be as in section §4. If χ * (S) = Ze 1 ⊕ Ze 2 , we have the following locally factorial Fano varieties:

Only the first two are smooth.

we have the following locally factorial Fano compactifications of G/H:

2 . The first is smooth, while the other ones are smooth if and only if r = 0.

Proof. The first property holds because -K X is linearly equivalent to a G-stable divisor. Let a i := (2ρ -2ρ 0 )(α ∨ i ) for each i; recall that a 1 , a 2 ≥ 1 (see §2). In the following we always suppose ♯ρ(D(X)) = ♯ρ -1 (ρ(D(X))).

I) Suppose X simple. If X is toroidal (i.e. X = X 0 ) have to exclude the following cases by

In the first case X 0 is not Fano, while in the last two cases

2 ) ≤ 0 < a 2 . Thus we have only to verify if X is locally factorial. We have to exclude two cases:

). II) Suppose now that X is not simple. In the toroidal case ∆[p] = {e 1 , e 2 , e 1 + e 2 } and X 0 is smooth by Lemma 5.3. This variety is Fano if and only if -2ρ + 2ρ 0 + e * i is antidominant and regular for each i. III) Assume moreover that X is not toroidal. Suppose first θ indecomposable. Then, by Lemma 5.1,

. By the Lemma 5.3 we have two possibilities for ∆

In the first case there are not other conditions because the weights of k are a 1 ω 1 -(a 1 +1)ω 2 and -ω 2 . In the last case we have to impose that a 1 , a 2 > 1. Indeed the weights of k are

2 ) < a 2 . In §2, we have seen that a 1 ≤ 1, a 2 ≤ 1 if and only if θ = -id.

IV) Suppose now θ decomposable. First suppose

The case with X 0 singular is very similar. We can apply the Lemma 5.3 to the maximal open toroidal subvariety X ′ of X (whose colored fan has support cone(-α ∨ 1 , -mα ∨ 1 + e 2 )). There are two possibilities for ∆ X ′ : its maximal cones are either

2 ) < 0. Now, we verify the smoothness of such varieties. First, we explain the conditions for a projective locally factorial symmetric variety X with rank two to be smooth (see [START_REF] Ruzzi | Smooth projective symmetric varieties with Picard number equal to one[END_REF], Theorems 2.1 and 2.2). Let Y be a open simple Gsubvariety of X whose closed orbit is compact; then the associated colored cone (C, F ) has dimension two. Write C = cone(v 1 , v 2 ) and let

2 ) be the dual cone; we take v 1 and v 2 primitive. If X is smooth then ρ is injective over F and ρ(F ) does not contain any exceptional root. Suppose such conditions verified and let R ′ the root subsystem of R G,θ generated by the simple roots α such that α ∨ ∈ ρ(F ) and ♯ρ -1

thus Y is smooth if and only if r = 0. The other cases are similar.

Using the Lemma 5.3, one can easily prove the following proposition:

Proposition 5.1 Let G/H be a symmetric space of rank 2 with X 0 smooth. If θ is indecomposable, the Fano toroidal compactifications of G/H are as in figure 6. If θ is decomposable, let m i be as in §4. Then, the Fano toroidal compactifications of G/H are the following ones:

Smooth quasi Fano varieties

Now, we want to classify the smooth (resp. locally factorial) quasi Fano symmetric varieties with rank two and G semisimple. A Gorenstein (projective) variety is called quasi Fano if its anticanonical divisor is big and nef.

Theorem 5.2 Let G/H be a symmetric space of rank 2 (with G semisimple).

• The nefness of the anticanonical bundle of a compactification of G/H depends only by the fan associated to the colored fan (and not by the whole colored fan).

• The fans of the locally factorial quasi Fano (but non-Fano) compactifications of an indecomposable symmetric space of rank 2 (with G semisimple) are whose in Figure 7 (we have also to request that ρ is injective over D(X)). Such a variety is singular if and only if ρ(D(X)) contains an exceptional root.

, let m i and mi as in §4. Supposing χ * (S) = Ze 1 ⊕ Ze 2 , let v j (i) := -iα ∨ j + e j c and w j (x, y) := -(xy + 1)α ∨ j + ye j c . We have the following locally factorial quasi Fano compactifications of G/H, which are not Fano (we always suppose ρ injective over D(X)): to prove if ♯∆[p] < 4. Suppose now that ♯∆[p] ≥ 4; we claim that up to isomorphisms, ∆ contains cone(e 1 , e 1 + e 2 ) and cone(e 1 + e 2 , e 1 + 2e 2 ). We know that τ = R ≥0 (e 1 + e 2 ) is contained in ∆.

First of all, we determine the restrictions of k to the cones containing τ and afterwards we will determine the cones themselves. Let σ = cone(e 1 + e 2 , b 1 e 1 + b 2 e 2 ) ∈ ∆(2) be a maximal cone containing τ and write k σ = x 1 e * 1 + x 2 e * 2 , so (k σ )(e 1 + e 2 ) = x 1 + x 2 = 1 and x i = (k σ )(e i ) ≤ 1 for each i. Hence k σ is e * i for an appropriate i and b i = (k σ )(b 1 e 1 + b 2 e 2 ) = 1. Because of the non-singularity of Z the only possibilities for σ are cone(e 1 + e 2 , e 1 ), cone(e 1 + e 2 , e 2 ), cone(e 1 + e 2 , e 1 + 2e 2 ) and cone(e 1 + e 2 , 2e 1 + e 2 ). The fan ∆ does not contain both cone(e 1 + e 2 , e 1 + 2e 2 ) and cone(e 1 + e 2 , 2e 1 + e 2 ); otherwise k cone(e1+e2,e1+2e2) (2e 1 + e 2 ) = 2 > 1 = k(2e 1 + e 2 ). Observe that if ∆ contains cone(e 1 + e 2 , 2e 1 + e 2 ), then Z is isomorphic to a variety whose fan contains cone(e 1 + e 2 , e 1 + 2e 2 ) by the isomorphism induced by the automorphism of χ * (S) that exchanges e 1 and e 2 . So the claim is proved.

Because of the non-singularity of Z, ∆ contains a cone σ = cone(e 1 +me 2 , e 2 ) for a suitable integer m; we want to show that Z is Z m . Let Z ′ be the open toric subvariety of Z whose fan ∆ ′ is ∆\{cone(e 1 + me 2 , e 2 ), cone(e 2 )}.

We claim that, for each integer r > 1, there is an unique variety Z ′ r with the two following properties: 1) the fan ∆ ′ r of Z ′ r has support cone(e 1 , e 1 + re 2 ); 2)

r is an open subvariety of a toric variety Z r with nef anticanonical bundle and birationally proper over A 2 . In particular, the anticanonical divisor of Z ′ r is nef. The open subvariety Z ′ r of Z r whose fan is ∆ r \{cone(e 1 + re 2 , e 2 ), cone(e 2 )}, satisfies these properties. So it is sufficient to prove the claim.

We show the claim for induction on r. We have already verified the basis of induction. Let Z ′ r be a variety that satisfies the hypotheses of the claim and let σ ′ be the unique cone in ∆ ′ r (2) which contains e 1 +re 2 . Because of the inductive hypothesis it is sufficient to show that σ ′ = cone(e 1 + re 2 , e 1 + (r -1)e 2 ).

Let k be the function associated to the anticanonical bundle of a fixed Z r . Let k σ ′ = x 1 e * 1 + x 2 e * 2 , then 1 = k σ ′ (e 1 + re 2 ) = x 1 + rx 2 and x i = k σ ′ (e i ) ≤ 1 for each i, so the unique possibilities for k σ ′ are e * 1 and -(r -1)e * 1 + e * 2 . Write

Because of the non-singularity of Z we have c 1 -1 = ±1, so there are two possibilities: either σ ′ = cone(e 1 + re 2 , e 2 ) or σ ′ = cone(e 1 + re 2 , 2e 1 + (2r -1)e 2 ). We exclude the first one because e 2 does not belong to | ∆ ′ r |. We exclude also the second one because k cone(e1,e1+e2

Suppose ∆ X [p] = {e 1 , e 1 + e 2 , e 1 + 2e 2 , ..., e 1 + re 2 , e 2 }. Then -K X is nef if and only if -2ρ + 2ρ 0 + e * 1 and -2ρ + 2ρ 0 -(r -1)e * 1 + e * 2 are antidominant. In such a case, if θ is decomposable, then the sum of these weights is regular, so -K X is big. If r = 1 and the previous weights are regular, then -K X is ample. Remark that if r = 1 and θ is decomposable, such weights are always regular. III) If X is not toroidal and θ is indecomposable, let X ′ be as in Lemma 5.4. By Lemma 5.1 and Theorem 5.

, up to reindexing. By Lemma 5.5 we have two

IV) Finally, suppose X non toroidal and θ decomposable. Suppose also X 0 smooth; the case X 0 singular is very similar. By the proof of Theorem 5.1, ρ(D(X)) cannot be ρ(D(G/H)). If ρ(D(X)) = α ∨ 1 , then, by the local factoriality of X, e 1 = -α ∨ 1 and there is r

1 + e 2 ) ∈ ∆. We apply Lemma 5.4 to the maximal open toroidal subvariety of X (whose colored fan has support cone(-α ∨ 1 , v ′ )). Then, by Lemma 5.5, ∆[p] has to be

). The non-trivial conditions are kσ1 (α ∨ 1 ) ≤ 0 and kσ2 (α ∨ 2 ) ≤ 0 (or equivalently s ≤ m1 + 1 and r ≤ m2 ). If s = 1, then r > m 2 because -K X is not ample. In the second case we can suppose s ≥ 2; k is linear over cone

2 ) ≤ 0, or, equivalently, r +s-1 ≤ m2 . In the first two cases kσ1 (α ∨ 2 ) < 0 and kσ2 (α ∨ 1 ) < 0, so -K X is big. In the last case, we proceed as in the proof of Theorem 5.1, obtaing m 2 + 1 < r ≤ m2 + 1. Moreover, kcone(α ∨ 1 ,v ′ ) (α ∨ 2 ) < 0 and kcone(-α ∨ 1 ,v ′ ) (α ∨ 1 ) < 0, so -K X is big. We can study the smoothness of all the previous varieties as in the proof of Theorem 5.1. Remark that if H = G θ , θ is decomposable, ♯ρ -1 (α ∨ 1 ) = 2 and (C(Y ), D(Y )) = (cone(α ∨ 1 , -rα ∨ 1 -α ∨ 2 ), F ), then Y is smooth if and only if ♯F = 1 and α 1 is not exceptional (but Y cannot be an open subvarieties of a Fano variety). The symmetric varieties in the statement are all projective. Indeed, if ∆ X [p] = {e 1 , e 1 + e 2 , e 1 + 2e 2 , ..., e 1 + (s -1)e 2 , e 1 + se 2 , e 2 )}, then the following divisor is ample: -pK X + s i=1 i 2 D e1+ie2 + m D(G/H)\D(X) D with p, m >> 0. Indeed the piecewise linear function associated to s i=1 i 2 D e1+ie2 is strictly convex on {C ∈ ∆ X : C ⊂ cone(e 1 , e 1 + s 2 )}. Moreover, k X is strictly convex on the fan with maximal cones cone(e 1 , e 1 + s 2 ) and cone(e 1 + s 2 , e 2 ). The other cases are similar to this one.

Symmetric Fano varieties with G reductive

In this section we consider the 2-rank locally factorial Fano symmetric varieties over which acts a group G which is only reductive. If G is a torus, then X is the projective space. So, we can suppose

In the last case H is generated by G θ and by n 1 n 2 , where and -r ≤ m1+3 2 ;

6.

). The only singular varieties are the ones in the cases 3) and 4).

Observe that X cannot be simple because the valuation cone is not strictly convex. We begin with a lemma similar to Lemma 5.2. Lemma 5.6 Let X be a Fano locally factorial symmetric variety with G as before. If X is toroidal, there are at most four colored 1-cones. Otherwise, there are at most three colored 1-cones.

Proof of Lemma 5.6. First suppose X toroidal. Then k(±f ) = 1. Let σ ∈ ∆(2) be a cone which does not contain neither f nor -f , then k σ (±f ) ≤ 0. So k σ is a multiple of e * . If there is another cone σ ′ ∈ ∆(2) with the same properties (and such that dim (σ ∩ σ ′ ) = 1), then k σ = k σ ′ . If X is not toroidal, we can study the maximal open toroidal subvariety X ′ of X as in Lemma 5.2 (because |F (X ′ )| is strictly convex).

Proof of Theorem 5.3. We have to request, as in Theorem 5.1, that ♯ρ(D(X)) = ♯ρ -1 (ρ(D(X))), but in this case ♯ρ(D(X)) ≤ 1. If X is not toroidal, then


A Wendell Rodrigues

Frédéric Guyomarc'h Lifl -Ustl

Jean-Luc Dekeyser
email: jean-luc.dekeyser@inria.fr

Programming Massively Parallel Architectures using MARTE: a Case Study

Nowadays, several industrial applications are being ported to parallel architectures. These applications take advantage of the potential parallelism provided by multiple core processors. Many-core processors, especially the GPUs(Graphics Processing Unit), have led the race of floating-point performance since 2003. While the performance improvement of generalpurpose microprocessors has slowed significantly, the GPUs have continued to improve relentlessly. As of 2009, the ratio between many-core GPUs and multicore CPUs for peak floating-point calculation throughput is about 10 times. However, as parallel programming requires a non-trivial distribution of tasks and data, developers find it hard to implement their applications effectively. Aiming to improve the use of many-core processors, this work presents an case-study using UML and MARTE profile to specify and generate OpenCL code for intensive signal processing applications. Benchmark results show us the viability of the use of MDE approaches to generate GPU applications.

I. INTRODUCTION

Advanced engineering and scientific communities have used parallel programming to solve their large scale complex problems for a long time. Despite the high level knowledge of the developers belonging to these communities, they find hard to implement their parallel applications effectively. Over recent years, using Graphics Processing Units (GPUs) has become increasingly popular and in fact important for seeking performance benefits in computationally intensive parallel applications. However, even for GPUs, there are not performance gains without challenges: first, the identification and exploitation of any parallelism in the application is responsibility of programmers. Often, this requires intimate understanding of the hardware and extensive re-factoring work rather than simple program transformations. Second, the high-level abstractions of a problem can hardly be expressed in the CUDA [START_REF]NVIDIA CUDA Compute Unified Device Architecture[END_REF] or OpenCL [START_REF]OpenCL -The open standard for parallel programming of heterogeneous systems[END_REF] programming model. Furthermore, subsequent manual optimisations distort any remaining abstractions in the application.

A way of addressing this abstraction is to provide a modelto-source transformation mechanism where the model is captured through a Model-Driven Environment (MDE) [START_REF] Rodrigues | An MDE Approach for Automatic Code Generation from MARTE to OpenCL[END_REF] and then the code generation is handled by templates. Gaspard2 [START_REF] Gamatié | A model driven design framework for massively parallel embedded systems[END_REF] is a framework that uses UML and the MARTE profile in order to implement its MDE approach. This paper shows a case-study using Gaspard2 to generate an application code for massively parallel architectures based on GPUs. Moreover, performance issues are applied to the model transformation and code generation steps in order to achieve optimization levels accomplished in manually written codes.

II. BACKGROUND

A. Massively Parallel Architectures

GPU is a manycore processor attached to a graphics card dedicated to calculating floating point operations. The GPU devotes more transistors to data processing rather data caching and flow control. This is the reason why the GPU is specialized for compute intensive. Nevertheless, even if GPUs are a manycore processors, their parallelism continues to scale with Moore's law. It is necessary to develop application software that transparently scales its parallelism. GPUs such as NVIDIA GeForce GTX 480 contain 15 Streaming Multiprocessors, each of which supports up to 1024 co-resident threads, so 30K threads can be created for a certain task. In addition, each multiprocessor executes groups, called warps, of 32 threads simultaneously. NVIDIA's actual CUDA architecture, codenamed Fermi, has features for general-purpose computing. Fundamentally, Fermi processors are still graphics processors, not general-purpose processors. The system still needs a host CPU to run the operating system, supervise the GPU, provide access to main memory, present a user interface, and perform everyday tasks that have little or no data-level parallelism.

B. OpenCL

OpenCL(Computing Language) is a standard for parallel computing consisting of a language, API, libraries and a runtime system. Originally, was proposed by Apple, and then turned over to the Khronos Group.

OpenCL also defines a programming language for writing kernels, which is an extension of C. Kernels are executed within their own memory domain and may not directly access host main memory. OpenCL usually defines a host where main programs are placed and one or more devices that executes kernels. Furthermore, device memory is divided into four distinct regions:

• Global memory, a kind of "device main memory", can be accessed by all work-items and the host in reads/writes.

• Constant memory is similar to global memory, except that work-items may only read from this memory.

• Local memory is read/write memory local to a workgroup, and is shared by all work-items of this group.

• Private memory is local to each work-item. The OpenCL programming language defines type qualifiers to specify in which memory region a variable is stored or a pointer points to. As a kernel can neither access host main memory nor dynamically allocate global and constant memory, all memory management must be done by the host. The OpenCL API provides functions to allocate linear memory blocks in global or constant memory, as well as to copy data to or from these blocks.

C. MARTE in Gaspard2 Context

MARTE (Modeling and Analysis of Real-Time and Embedded systems) [START_REF]Modeling and Analysis of Real-time and Embedded systems (MARTE)[END_REF] is a standard proposal of the Object Management Group (OMG). The primary aim of MARTE is to add capabilities to UML for model-driven engineering of real-time and embedded systems. UML provides the framework into which needed concepts are plugged. The MARTE profile enhances possibility to model software, hardware and relations between them. It also provides extensions to make performance and scheduling analysis and to take into account platform services. Gaspard2 [START_REF] Gamatié | A model driven design framework for massively parallel embedded systems[END_REF] is a framework based on MDE and MARTE profile. From a high-level abstraction model of application, architecture and allocation, Gaspard2 provides transformation chains and templates to code generation for several target platforms. One of these platforms is the hybrid (CPU+GPU) platform beneath the OpenCL API.

III. CASE STUDY: H.263 VIDEO DOWNSCALING

The case study concerned in this paper deals with specific aspect of H.263-based video compression standard, scaling. The scaling during video-compression is considerably important for previews or for streaming for small form factor devices, such as mobile phones. The application consists of a classical downscaler, which transforms a video signal, which, for instance, is expressed in Common Intermediate Format (CIF), into a smaller size video. In this situation, the downscaler can be composed of two components: a horizontal filter that reduces the number of pixels from 352-lines to 132lines and a vertical filter that reduces the number of pixels from 288-lines to 128-lines by interpolating packets of 8 pixels both row-and column-wise.

In a typical case of handling a 25-frames-per-second video signal lasting for 80-seconds, the downscaler may process up to 2000 frames in CIF format, with each input frame being represented by a two-dimensional array of size 352 × 288 and should emit 2000 output frames of size 132 × 128. Since each video pixels is encoded in 24-bit RGB colour model, the frame generation process is repeated for each frame and for each pixel of different colour space along two different directions. The final frame is produced by using these outputs from different colour space. Depending on the composing function, a broad-range of output colours are possible for each pixel and thus for each frame. The figure 1 illustrates this basic operation for a given frame in high-definition format.

HDTV Resolution 1920x1080 720x1080

DVD Resolution 720x480

Horizontal Filter Vertical Filter Fig. 1. Horizontal and vertical filter processes

As can be observed, the operations concerned with the scaling is highly parallel and repetitive. The interpolation is repeated for each frame, each pixel and for each colour channel.

A. Downscaler Model

The figure 2 gives us an overview of the downscaler application. The figure illustrates a model for a 300-frames video, even if we have 2000 frames in our testbed. For this example, we are going to analyse only the first repetitive task from the Horizontal Filter component. The other tasks have equivalent behaviour. The yhfk task has a multiplicity equals to [288,44]. It means this task is composed of 288x44 independent tasks (so-called Elementary Task), and thus, parallelizable. Each elementary task takes a pattern from the input array. A tiler stereotype1 do the tiling operation. It allows to split input data in patterns in accordance with tiler's array definitions of origin, paving and fitting. Besides features such as tiler specifications and task repetitions, MARTE profile is applied to OpenCL architecture definition (host and device) in order to grant task allocations. For this illustrated model, we add a specification to one host (CPU+Memory) and one device (GPU+Global Memory). Data and tasks are placed into memories and processors according to project interests. For instance, the six repetitive tasks in horizontal and vertical filters are allocated onto the GPU in order to generate kernels in the execution environment. Allocate stereotypes are used to map ports and tasks into HwRAMs and HwProcessors. These stereotypes will allow for creating all variables and relations between them. Additionally, in order to distinguish host from device, we modify the description attribute from HwResource stereotype.

B. MARTE to OpenCL Transformation Chain

Gaspard2 supplies a transformation engine that allows us chaining a set of model-to-model or model-to-text transformations. These transformations take into account model elements and properties and gather information to create cleaner models towards a target platform. A recent chain was added to Gaspard2 framework and it allows for automatic code generation from MARTE to OpenCL (see [START_REF] Rodrigues | An MDE Approach for Automatic Code Generation from MARTE to OpenCL[END_REF]). Subsequent paragraphs highlight some details of the designed model taken into account by these transformations. 2) Data Allocation: A critical problem in application modeling based on MDE is to manage the data allocation in the target platform. MARTE profile adds the flowPort stereotype to UML port element. The main attribute aggregated to a port element is the direction, which allows to define whether the port is input, output, or bidirectional. This information contributes to decide which elements are read-only variables.

By using UML links we can associate flowPorts to memories in architecture models. Each port has attributes and associations that allow us defining size and data type for example. Thus, developers can specify in their application models where the data will be stored and how much space will occupy the data. In the figure 4 we can see a simple example of allocation. Ports from different tasks are allocated into memory elements of their respective processors.

3) Performance Tuning: Usually connected ports allocated to different memory boxes, as seen in the data allocations illustration (figure 4), cause a data transfer between CPU and GPU. At a first sight, one can say that is a critical point to decrease the performance because subsequent kernels reuse

IV. RESULTS

Four versions of the Downscaler were tested. The first one is a sequential version using the same structure defined in the figure 2. The other two versions are OpenCL automatically generated and the last one is a manually written OpenCL version. We have used a transformation chain that transforms model to model using QVTO [START_REF] Omg | M2M/Operational QVT Language[END_REF] and model to text using Acceleo [START_REF] Obeo | Acceleo -Model to Text transformation[END_REF]. The first OpenCL code is a not optimized program without any further analysis on memory transfers. The second one regards the memory transfers between host and device. Minimize these transfers reduces notably the total GPU execution time. As seen in the figure 5, data transfers take a lot of time (more than 70%) in the Downscaler application. The communication takes more time than computing process by the work-items. Time analysis in figure 5 demonstrates the bigger spent time in y-component kernels due to their bigger topology and handled data. After the performance tuning, no time changes occurs in kernels (as it was expected). However, we verify about 30% and 70% faster transfer times in host to device and device to host respectively.

The figure 6 presents the total execution time of each implemented version. Both OpenCL codes give us good results with relation to CPU code. Using optimized transfers we can achieve speedups of 10x. In fact, structurally, the optimized version is really closer to manually written (considering the model designer is the code programmer). The decision of the topology and data transfers by the transformations (model compiler) were closely inspired by decisions taken if they would a human programmer. For the optimized version we achieve about 25% of speed-up. This is an enough expressive performance for two parallel implementations.

V. CONCLUSION

Even though we have used an application model not specially developed for GPU architectures, we have had good results at performance level. Parallel languages are hard to program and MDE approaches are well suitable to allow not specialized programmers creating parallel programs. The results presented in this work help us to certify the high potential of MARTE profile to create parallel applications for massively parallel processors. For the time being, we are implementing more optimization features in the transformation chain in order to ensure a stable and generic framework to create OpenCL applications exploiting, among other things, the memory hierarchy throughput.

Fig. 2 .Fig. 3 .

 23 Fig. 2. Downscaler Application Model

Fig. 5 .Fig. 6 .

 56 Fig. 5. Time Distribution on the GPU

defined in ArrayOL[START_REF] Boulet | Array-OL Revisited, Multidimensional Intensive Signal Processing Specification[END_REF] language and part of MARTE.