Fano Threefolds with 2-Torus Action

A picture book

Hendrik Süß

Received: December 16, 2013

Communicated by Thomas Peternell

Abstract. Following the work of Altmann and Hausen we give a combinatorial description for smooth Fano threefolds admitting a 2-torus action. We show that a whole variety of properties and invariants can be read off from this description. As an application we prove and disprove the existence of Kähler-Einstein metrics for some of these Fano threefolds, calculate their Cox rings and some of their toric canonical degenerations.

2010 Mathematics Subject Classification: Primary 14L30; Secondary 14J45, 32Q20 Keywords and Phrases: torus action, moment polytope, Fano variety, Kahler-Einstein metric, Cox ring

1 Introduction

Knowing the fan or the polytope of a projective toric variety does not only give a concrete construction of the variety but also enables us to explore its properties and to easily calculate a whole bunch of invariants. In low dimensions a close look on the corresponding pictures is enough to obtain detailed information about the variety. Loosely speaking we can even prove statements by drawing pictures. In [AH06] and [AHS08] the description of toric varieties by cones and fans was generalized to the case of an arbitrary T -variety, i.e. a variety X endowed with the effective action of an algebraic torus T. This general description consists of combinatorial data living on the Chow quotient X//T. A series of papers by different authors followed this approach to derive properties and calculate invariants out of the combinatorial data. This works especially well for the case of complexity-one torus action, i.e. when the torus has one dimension less then the variety and the Chow quotient is just a curve. The aim of this paper is to make these results available for Fano threefolds with an action of a two-dimensional torus. More precisely, we are going to state the needed combinatorial description for a list of those Fano threefolds and relate it to the classification of Mori-Mukai [MM86]. By the mentioned preliminary results we can immediately derive a lot of additional information about these Fano varieties. We want to highlight two concrete applications. First, by [HS10] we are able to calculate the Cox rings of these T-varieties. The knowledge of the Cox ring may give further inside in the geometry of the variety. For example the automorphism group can be studied by the methods from [AHHL12]. As a second application we consider the question of the existence of Kähler-Einstein metrics on Fano manifolds. In general this is a hard question. There are some known obstructions such as the vanishing of the Futaki character [Fut83]. There is also a sufficient criterion due to Tian [Tia87]. Recently it was shown that there is, indeed, a completely algebraic formulation of the Kähler-Einstein property by the means of so called K-stability [CDS14, Tia12]. Nevertheless, for a given Fano variety it’s still hard to check this stability condition directly. Again for toric Fano varieties the Kähler-Einstein property can be easily reformulated in terms of the defining polytope. By a result of Wang and Zhu a toric Fano variety is Kähler-Einstein if and only if the Futaki character vanishes [WZ04]. Moreover, the Futaki character can be easily calculated as the barycenter of the polytope corresponding to the toric Fano manifold [Mab87]. We are going to generalize the last result for our situation of complexity-one torus actions in order to show that the Futaki character does not vanish for some of the threefolds, and hence disprove the existence of Kähler-Einstein metrics for them. We are using an application of Tian’s criterion from [Süß13] to prove the existence of Kähler-Einstein metrics for three of the remaining ones. Our main conclusions drawn from the given combinatorial description are summarized in the following theorem.

Theorem 1.1. The Fano threefolds Q, 2.24^{∗}, 2.29–2.32, 3.8^{∗}, 3.10^{∗}, 3.18– 3.24, 4.4, 4.5, 4.7 and 4.8 from Mori’s and Mukai’s classification admit a 2-torus action. The moment polytopes together with their Duistermaat-Heckman measures are given in Section 5. The Cox rings, Futaki characters F (X) as well as the existence of a Kähler-Einstein metric can be found in the following table.

∗Only one variety of the family is known to admit a 2-torus action

Documenta Mathematica 19 (2014) 905–940

No. Cox ring  F(X) Kähler-Einstein
Q  k[\underline{T}]/(T_{1}T_{2} + T_{3}T_{4} + T_{5}^{2})  0  yes
2.24^{∗} k[\underline{T}]/(T_{1}T_{2}^{2} + T_{3}T_{4}^{2} + T_{5}T_{6}^{2})  0  yes
2.29 k[\underline{T}]/(T_{1}T_{2}^{2}T_{3} + T_{4}T_{5} + T_{6}^{2})  0  ?
2.30 k[\underline{T} , S_{1}]/(T_{1}T_{2} + T_{3}T_{4} + T_{5}^{2})  \binom{0}{−2}  no
2.31 k[\underline{T}]/(T_{1}T_{2} + T_{3}T_{4} + T_{5}T_{6}^{2})  \binom{−^{4}/3}{−^{4}/3}  no
2.32 k[\underline{T}]/(T_{1}T_{2} + T_{3}T_{4} + T_{5}T_{6})  0  yes
3.8^{∗} k[\underline{T} , S_{1}]/(T_{1}T_{2}^{2} + T_{3}T_{4}^{2} + T_{5}T_{6}^{2})  \binom{^{5}/3}{−^{5}/6}  no
3.10^{∗} k[\underline{T}]/(T_{1}T_{2}^{2}T_{3} + T_{4}T_{5}^{2}T_{6} + T_{7}^{2})  0  yes
3.18 k[\underline{T} , S_{1}]/(T_{1}T_{2}^{2}T_{3} + T_{4}T_{5} + T_{6}^{2})  \binom{0}{−^{7}/8}  no
3.19 k[\underline{T} , S_{1}, S_{2}]/(T_{1}T_{2} + T_{3}T_{4} + T_{5}^{2})  0  ?
3.20 k[\underline{T}]/(T_{1}T_{2} + T_{3}T_{4} + T_{5}T_{6}^{2}T_{7})  0  ?
3.21 k[\underline{T} , S_{1}]/(T_{1}T_{2}^{2} + T_{3}T_{4}^{2} + T_{5}T_{6})  \binom{^{7}/8}{^{7}/8}  no
3.22 k[\underline{T} , S_{1}, S_{2}]/(T_{1}T_{2} + T_{3}T_{4} + T_{5}^{2})  \binom{0}{−^{2}/3}  no
3.23 k[\underline{T} , S_{1}]/(T_{1}T_{2} + T_{3}T_{4} + T_{5}T_{6}^{2})  \binom{−^{13}/12}{−^{53}/24}  no
3.24 k[\underline{T} , S_{1}]/(T_{1}T_{2} + T_{3}T_{4} + T_{5}T_{6})  \binom{−^{2}/3}{−^{4}/3}  no
4.4  k[\underline{T} , S_{1}, S_{2}]/(T_{1}T_{2}^{2}T_{3} + T_{4}T_{5} + T_{6}^{2})  0  ?
4.5  k[\underline{T} , S_{1}, S_{2}]/(T_{1}T_{2}^{2} + T_{3}T_{4}^{2} + T_{5}T_{6}) \binom{^{5}/24}{^{5}/24}  no
4.7  k[\underline{T} , S_{1}, S_{2}]/(T_{1}T_{2} + T_{3}T_{4} + T_{5}T_{6})  0  ?
4.8  k[\underline{T} , S_{1}, S_{2}]/(T_{1}T_{2} + T_{3}T_{4} + T_{5}T_{6}) \binom{0}{−^{13}/12}  no

The paper is organized as follows. In Section 2 we recall the combinatorial description of complexity-one torus actions from [AIP^{+}11]. In Section 3 we describe equivariant polarizations and give a characterization of the Fano property. Moreover, we recall how to calculate the Cox ring from the combinatorial description and we state a simple formula for the Futaki character. In Section 4 we show how to find toric degenerations for our T-varieties. Finally in Section 5 we state the list with the combinatorial descriptions for all the considered Fano varieties, which in turn leads to the proof of Theorem 1.1. The appendix provides the proof for the formula for the Futaki character.

Acknowledgments

I would like to thank Ivan Arzhantsev, Nathan Ilten and Johan Martens for their helpful comments and conversations.

2 Combinatorial description of torus actions of complexity one

F-divisors

We consider a variety X/_{ℂ} with an effective action of an algebraic torus T ≅ (ℂ^{∗})^{r}. Then X is called a T -variety of complexity (dim X − dim T). In the following we restrict ourselves to the case of rational T -varieties of complexity one. Such objects where described and studied earlier in [KKMSD73, Tim08]. We are using the more general approach of [AH06] which describes T -varieties of arbitrary complexity, but restrict it to the case of rational complexity-one T -varieties. Our general references are [AH06, AHS08, AIP^{+}11]. We follow the standard terminology for toric varieties. The character lattice of the torus T is denoted by M and we set N := M^{∗}. For the associated vector spaces we write M_{ℚ} and N_{ℚ} or M_{ℝ} and N_{ℝ}, respectively. For a polyhedron Δ ⊂ N_{ℚ} we consider its tail (or recession) cone tail Δ defined by tail Δ := {v ∈ N_{ℚ} | Δ+ℚ_{≥0}·v = Δ}. For a complete polyhedral subdivision Ξ of N_{ℚ} the set of tail cones has the structure of a fan, which is called the tail fan of Ξ and will be denoted by tail Ξ. Now, consider a pair

S = ( ∑_{P ∈ℙ^{1}} S_{P} ⊗ P, deg S)

Here, ∑_{P} S_{P} ⊗ P is just a formal sum and S_{P} are complete polyhedral subdivisions of N_{ℚ} with some common tail fan denoted by tail S. For the moment the degree deg S is just a subset of N_{ℚ}. The subdivisions S_{P} are called slices of S. We assume that there are only finitely many non-trivial slices, i.e. those which differ from the tail fan. Note, that for every full-dimensional σ ∈ tail(S) there is a unique polyhedron Δ_{P}^{σ} in S_{P} with tail(Δ_{P}^{σ}) = σ.

Definition 2.1. A pair as above is called a (complete) f-divisor if for any full-dimensional σ ∈ tail(S) we have either deg S ∩ σ = ∅ or

∑_{P} Δ_{P}^{σ} = deg S ∩ σ ⊊ σ.

A (complete) f-divisor S as above corresponds to a rational complete T -variety X(S) of complexity one, see [AIP^{+}11, Section 5.3]. It comes with a rational quotient map π : X(S) −−→ ℙ^{1}.

Example 2.2 (Fano threefold 3.10). We consider the f-divisor S with three non-trivial slices and the degree given in Figure 1. It is straight forward to check, that the condition of Definition 2.1 is indeed fulfilled.

S_{0}  S_{∞}  S_{1}  deg S

Figure 1: f-divisor of the Fano threefold 3.10.

As we will see later, this f-divisor corresponds to a blow up of the quadric threefold in two disjoint conics, i.e. it is an element of the family 3.10 from the classification of Mori and Mukai.

By [LS10, Proposition 5.1, Theorem 5.3] we have the following criterion for the smoothness of X(S).

Theorem 2.3. The T-variety X(S) is smooth if and only if for every cone σ ∈ tail S one of the following two conditions is fulfilled

i. if σ ∩ deg S = ∅ then cone ℚ_{≥0} · (Δ × {1} + σ × {0}) is regular, for every P and every maximal polyhedron Δ ⊂ S_{P} with tail cone σ.

ii. if σ is maximal and σ ∩ deg S ≠ ∅ then for at most two points P = Q, R the polyhedron Δ_{P}^{σ} is not a lattice translate of σ and the cone

ℚ_{≥0} · ( ∑_{P ≠Q} Δ_{P}^{σ} × {1} ∪ σ × {0} ∪ Δ_{Q}^{σ} × {−1})

is regular.

Example 2.4 (Fano threefold 3.10 (continued)). We consider again the fdivisor from Example 2.2. There is no maximal polyhedron Δ with tail Δ ∩ deg S = ∅. Look at the shaded polyhedra in Figure 1. They have tail cone σ = ℚ_{≥0} · (−1, 0) + ℚ_{≥0} · (−1, 1). Only Δ_{1}^{σ} and Δ_{∞}^{σ} are not a lattice translate of σ and we obtain

ℚ_{≥0} · ((Δ_{1}^{σ} + (−1,0)) × {1} ∪ σ × {0} ∪ Δ_{∞}^{σ} × {−1})
= ℚ_{≥0} · (−1, 1, 2) + ℚ_{≥0} · (0, 0, −1) + ℚ_{≥0} · (0, −1, −2),

which is regular cone. Similarly for the other cones from the tail fan we get the same result. Hence, by Theorem 2.3 we obtain the smoothness of X(S).

Symmetries

In [AIP^{+}11, Section 5.3] there is also a characterization of f-divisors giving rise to isomorphic T-varieties. We consider isomorphisms φ : ℙ^{1} → ℙ^{1} and F : N → N′. Having this we define

F (φ^{∗}S) := (∑_{P}(F (S_{φ^{−1}(P)})) ⊗ P, F (deg S)) .

Definition 2.5. Two f-divisors S, S′ on ℙ^{1} are called equivalent if deg S = deg S′ and there are lattice points v_{P} ∈ N with ∑_{P} v_{P} = 0, such that S_{P} +v_{P} = S′_{P}. In this situation we write S ∼ S′.

Proposition 2.6. S and S′ lead to isomorphic T-varieties if and only if there are isomorphisms φ : ℙ^{1} → ℙ^{1} and F : N → N′, such that F (φ^{∗}S) ∼ S′.

By considering the case S = S′ we even obtain a description of the group of equivariant automorphisms Aut_{T} (X). Therefore, we consider the set Aut(S) of pairs (F, φ) as above, such that F (φ^{∗}S) ∼ S holds. This set becomes a group via (F, φ) ∘ (F′, φ′) := (F ∘ F′, φ′ ∘ φ). Note, that there is a natural action of Aut(S) on the characters M and on ℙ^{1}, by considering only the first or only the second component of the pairs, respectively. Now, we have the following proposition, due to [AIP^{+}11, Section 5.3].

Proposition 2.7. The group Aut(S) is isomorphic to Aut_{T} (X(S))/T.

Moreover, this isomorphism preserves the natural action of Aut_{T} (X(S))/T on M by conjugation. Hence, we will identify Aut_{T} (X(S))/T and Aut(S).

Definition 2.8. A T -variety of complexity one is called symmetric if there is a finite subgroup G ⊂ Aut_{T} (X(S)), with M^{G} = {0}.

Lemma 2.9. Consider a complete rational variety X with torus action of complexity ≤ 1. Let T be the maximal torus of the automorphisms. Then Aut_{T} (X)/T is finite and Aut_{T} (X) is reductive.

Proof. If X is a toric variety the statement is well known. Hence, we may consider the case of a complexity-one torus action. We consider the f-divisor S with X = X(S) with non-trivial slices S_{P_{1}}, …, S_{P_{ℓ}}. All other slices are assumed to be just lattice translates of the tail fan. Assume that ℓ = 2 then we may assume, that P_{1} = 0 and P_{2} = ∞. Now, every element φ ∈ ℂ^{∗} ⊂ Aut(ℙ^{1}) will give rise to a pair (id, φ) ∈ Aut(S). Hence, T was not a maximal torus and X is just a toric variety with an restricted torus action. Further on we may assume that ℓ ≥ 3. Consider an element (F, φ) of Aut(S). Since, F has to be an automorphism of the tail fan, there are only finitely many choices for F. Since, ℓ ≥ 3 the automorphism φ of ℙ^{1} is induced by a permutation of P_{1}, …, P_{ℓ}. Hence, there are only finitely many choices of for φ, as well. Since, X is complete Aut_{T} (X) is an algebraic group and we have a Levi decomposition Aut_{T} (X) = H ⋉R_{u}, where H is reductive and R_{u} is the unipotent radical. We may assume that T ⊂ H. Hence, we obtain Aut_{T} (X)/T ≅ H/T ×R_{u}. By the finiteness of Aut_{T} (X)/T we conclude that R_{u} is trivial and Aut_{T} (X) is reductive.

Proposition 2.10. Consider an f-divisor S on ℙ^{1} with at least three non-trivial slices. Then X = X(S) is symmetric if and only if M^{Aut(S)} = 0 holds.

Proof. One direction is obvious. Now, assume that M^{Aut(S)} = 0. By Lemma 2.9 we know that Aut(S) = Aut_{T} (X)/T is finite and Aut_{T} (X) is reductive. Now by the results of [Tit66] we can lift Aut(S) to a finite subgroup G ⊂ Aut_{T} (X), where G is not necessarily isomorphic to Aut(S). Now, we also have M^{G} = 0.

The notion of symmetry for toric varieties goes back to [BS99] it was generalized in [Süß13] to give a sufficient criterion for the existence of a Kähler-Einstein metric on X(S). For a vertex v in S_{P} let us denote by μ(v) the minimal natural number, such that μ(v) · v is a lattice point. We call μ(v) the multiplicity of v.

For P ∈ ℙ^{1} we set μ(P) = max{μ(v) | v ∈ S_{P}^{(0)}}, where S_{P}^{(0)} denotes the set of vertices in S_{P}.

Theorem 2.11. Let X = X(S) be a symmetric smooth Fano T -variety of complexity one, given by an f-divisor S. If one of the following conditions is fulfilled:

i. there are three points P_{1}, P_{2}, P_{3} ∈ ℙ^{1} such that μ(P_{i}) > 1 for i = 1, 2, 3,

ii. there are two points as in (i) which are swapped by an element of Aut(S),

iii. Aut(S) acts fixed-point-free on ℙ^{1},

then X is Kähler-Einstein.

Proof. This is only a reformulation of [Süß13, Theorem 1.1]. The condition that T acts on π^{−1}(P) with disconnected stabilizers is replaced by our condition μ(P) > 1. But this is equivalent by [HS10, Proposition 4.11].

Now, we make use of Proposition 2.10 to check the symmetry of a variety X(S).

Example 2.12 (Fano threefold 3.10 (continued)). We reconsider the f-divisor and variety from Example 2.2. By setting v_{0} = (−1, 0), v_{1} = (−1, 0), v_{∞} = (1, 1) and v_{P} = 0 for all other points P ∈ ℙ^{1} we see that S ∼ −S. Hence, the pair (− id_{N}, id_{ℙ^{1}}) is an element of Aut(S). Hence, X(S) is symmetric, since − id_{N} has only 0 as a fixed point. Moreover, in the slices S_{0}, S_{∞}, S_{1} there are vertices of multiplicity 2. By Theorem 2.11 (i) this shows, that this Fano threefolds is actually Kähler-Einstein.

3 Torus invariant divisors

In this section we recall the results of [PS11] concerning Weil and Carier divisors on T-varieties, see also [AIP^{+}11, IS10].

The class group

On a T-variety X(S) we find two different types of torus invariant prime divisors. Every vertex v in S_{P} correspond to a so called vertical prime divisors V_{v} := V_{P.v} which projects to the point P ∈ ℙ^{1}, via the quotient map π. We denote the set of all vertices in S_{P} by S_{P}^{(0)} and the disjoint union ∐_{P} S_{P}^{(0)} by S^{(0)}. As before, for a vertex v ∈ S_{P}^{(0)} we denote by μ(v) the smallest positive integer μ such that μ·v ∈ N. Horizontal prime divisors H_{ρ} project surjectively onto ℙ^{1} via the quotient map π. They correspond to the rays ρ in tail S which do not intersect deg S. We denote the set of all such rays by (tail S)^{×}.

Theorem 3.1 ([PS11, Cor. 3.15]). The divisor class group of X(S) is isomorphic to

⨁_{ρ∈(tail S)^{×}} ℤ · H_{ρ} ⊕ ⨁_{v∈S^{(0)}} ℤ · V_{v}

modulo the relations

∑_{v∈S_{P}^{(0)}} μ(v)V_{v} = ∑_{v∈S_{Q}^{(0)}} μ(v)V_{v},

0 = ∑_{ρ}⟨u, ρ⟩H_{ρ} + ∑_{P,v} μ(v)⟨u, v⟩V_{v} .

where P, Q ∈ ℙ^{1} and u ∈ M.

Corollary 3.2. The class group of X has rank

1 + ∑_{P}(#S_{P}^{(0)} − 1) + #(tail S)^{×} − dim N.

Example 3.3 (Fano threefold 3.10 (continued)). We are coming back to our T-variety from Example 2.2. Since every ray in tail S intersects the degree, by Theorem 3.1 the divisor class group of X is generated by the seven vertical divisors

D_{1} = V_{0,(0,0)}, D_{2} = V_{0,(−^{1}/2,0)}, D_{3} = V_{0,(−1,0)},
D_{4} = V_{∞,(0,0)}, D_{5} = V_{∞,(0,−^{1}/2)}, D_{6} = V_{∞,(0,−1)},
D_{7} = V_{1,(^{1}/2,^{1}/2)}.

The relations are given by the rows of the following matrix

M = ( \begin{matrix}1 & 2 & 1 & 0 & 0 & 0 & −2 \\ 0 & 0 & 0 & 1 & 2 & 1 & −2 \\ −1 & −1 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & −1 & −1 & 1 \end{matrix} )

Calculating the Smith normal form D = P · M · Q gives

D = ( \begin{matrix}1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 \end{matrix} ) , P = I_{4}, Q = ( \begin{matrix}0 & 0 & −1 & 1 & −1 & 1 & 1 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 1 & 0 & 1 & 1 & −1 & 1 & 1 \\ 0 & 1 & 0 & 2 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 1 & 0 & 1 & 1 \end{matrix} )

Hence, Cl(X) ≅ ℤ^{3}. Where the identification is given by the last three columns of Q, i.e. [D_{1}], [D_{3}] ↦ (−1, 1, 1), [D_{2}] ↦ (1, 0, 0), [D_{4}] ↦ (0, 0, 1), [D_{5}] ↦ (0, 1, 0), [D_{6}] ↦ (0, 0, 1), [D_{7}] ↦ (0, 1, 1).

Ample divisors

We consider a piecewise affine concave function Ψ : □ → Div_{ℝ} ℙ^{1}, defined on a lattice polytope □ ⊂ M_{ℝ}. It induces corresponding piecewise affine and concave functions Ψ_{P} : □ → ℝ via Ψ(u) = ∑_{P ∈ℙ^{1}} Ψ_{P} (u) · P. Using this notation we are going to define the crucial object of this section.

Definition 3.4. A divisorial polytope is a continuous piecewise affine concave function Ψ : □ → Div_{ℝ} ℙ^{1}, such that

i. for every u in the interior of □, deg Ψ(u) > 0,

ii. for every P the graph of Ψ_{P} is integral, i.e. has its vertices in M × ℤ.

We define the support function of Ψ to be the family Ψ^{∗} = {Ψ_{P}^{∗} }_{P ∈ℙ^{1}} of piecewise affine concave functions Ψ_{P}^{∗} : N_{ℝ} → ℝ with

Ψ_{P}^{∗} := min_{u∈□_{h}}(⟨u, v⟩ − Ψ_{P} (u)).

We call lin(Ψ^{∗}) := min⟨□, ·⟩ the linear part of Ψ^{∗}. It is a piecewise linear function on the normal fan of □.

Proposition 3.5 ([AIP^{+}11, Theorem 63], [IS10, Theorem 3.2]). There is a one-to-one correspondence between invariant ample divisors D on X(S) and divisorial polytopes Ψ : □ → Div_{ℝ} ℙ^{1}, with

i. Ψ_{P}^{∗} induces the subdivision S_{P}, i.e. the maximal polyhedra of S_{P} are the regions of affine linearity of Ψ_{P}^{∗},

ii. deg Ψ(u) = 0 holds for a point u ∈ □_{D} iff ⟨u, v⟩ = min⟨□, v⟩ for some v ∈ deg S.

Moreover, by [PS11, Corollary 3.19, Theorem 3.12] Ψ as in Proposition 3.5 corresponds to an anti-canonical divisor on X = X(S), if there exists an integral divisor K_{ℙ^{1}} = ∑ a_{P} · P of degree −2 on ℙ^{1}, such that

Ψ_{P}^{∗} (v) = −a_{P} − 1 +^{1}/μ(v), lin(Ψ^{∗})(v_{ρ}) = 1  (1)

for every vertex v ∈ S_{P} and every ray ρ of (tail S)^{×}. In particular, X will be Fano. In this case we say that Ψ is a divisorial polytope corresponding to the Fano T-variety X. Remark 3.6. By Proposition 3.5 (i) we have a one-to-one correspondence between facets of the graph of Ψ_{P} and vertices of S_{P}. A facet of the graph of Ψ_{P} projects to a region U ⊂ □ where Ψ_{P} is affine linear, i.e. Ψ_{P} |_{U} = ⟨·, v⟩ + c for some v ∈ N_{ℝ}. The corresponding vertex of S_{P} is exactly v and we have Ψ^{∗}(v) = c. Taking this into account we can reformulate condition (1) as follows.

i. For every facet F of the graph of Ψ_{P} + a_{P} + 1 there is a vertex v such that ⟨ · , μ(v)(v, 1)⟩ ≡ 1 on F. In particular, F has lattice distance 1 from the origin. Moreover, we have Ψ_{P} ≡ −a_{P} if S_{P} is trivial.

ii. the facets F of □ with deg Ψ|_{F} ≢ 0 have lattice distance 1 from the origin.

Definition 3.7. The volume of a divisorial polytope is defined, by

vol Ψ := ∫_{□}deg Ψ dμ.

Here, we are integrating with respect to the Euclidean measure μ induced by the inclusion M ⊂ M_{ℝ}.

Theorem 3.8 ([PS11, Proposition 3.31]). If D is ample on a T-variety X = X(S) of dimension n, then its top self intersection number is given by

(D)^{n} = n! vol Ψ_{D}.

Example 3.9 (Fano threefold 3.10 (continued)). In Figure 2 the divisorial polytope Ψ : □ → Div_{ℝ} ℙ^{1} for the canonical divisor is sketched. More precisely we give the non-trivial concave functions Ψ_{P} by drawing □ and the regions of affine linearity of Ψ_{P} and giving the values of Ψ at the vertices of these regions.

Ψ_{0} 0 0  −1−2 −1−2 0 0  ^{Ψ_{∞} 0 0  0 0 −2−1 −2 −1} ^{Ψ_{1} 0 0  1 2 3 3 2 1}  ^{deg Ψ 0 0  0 0 0 0 0 0 ^{1}/2}

Figure 2: A divisorial polytope for the threefold 3.10.

We have deg Ψ = 0 on the boundary of □ and deg S intersects all tail cones. Hence, condition (ii) of Proposition 3.5 is fulfilled. We will now check that the Ψ induces the subdivisions S_{P} and fulfills (1), for K_{ℙ^{1}} = −2 · [1]. Calculating Ψ^{∗} gives

Ψ_{0}^{∗} = min ( ⟨\binom{2}{1}, ·⟩ + 2, ⟨\binom{1}{2}, ·⟩ + 1, ⟨\binom{−1}{2}, ·⟩ + 0, ⟨\binom{−2}{1}, ·⟩ + 0,
⟨\binom{2}{−1}, ·⟩ + 2, ⟨\binom{1}{−2}, ·⟩ + 1, ⟨\binom{−1}{−2}, ·⟩ + 0, ⟨\binom{−2}{−1}, ·⟩ + 0).
Ψ_{∞}^{∗} = min ( ⟨\binom{2}{1}, ·⟩ + 0, ⟨\binom{1}{2}, ·⟩ + 0, ⟨\binom{−1}{2}, ·⟩ + 0, ⟨\binom{−2}{1}, ·⟩ + 0,
⟨\binom{2}{−1}, ·⟩ + 1, ⟨\binom{1}{−2}, ·⟩ + 2, ⟨\binom{−1}{−2}, ·⟩ + 2, ⟨\binom{−2}{−1}, ·⟩ + 1).
Ψ_{1}^{∗} = min ( ⟨\binom{2}{1}, ·⟩ − 2, ⟨\binom{1}{2}, ·⟩ − 1, ⟨\binom{−1}{2}, ·⟩ + 0, ⟨\binom{−2}{1}, ·⟩ + 0,
⟨\binom{2}{−1}, ·⟩ − 3, ⟨\binom{1}{−2}, ·⟩ − 3, ⟨\binom{−1}{−2}, ·⟩ − 2, ⟨\binom{−2}{−1}, ·⟩ − 1).

It’s not hard to check, that Ψ_{P}^{∗} is affine linear exactly on the maximal polyhedra of S_{P}. Hence, by Proposition 3.5 it corresponds to an ample divisor D on X(S). Moreover, we obtain

Ψ_{0}^{∗}(0, 0) = 0 h_{0}(−^{1}/2, 0) = ^{1}/2 h_{0}(−1, 0) = 0
Ψ_{∞}^{∗}(0, 0) = 0 h_{∞}(0, −^{1}/2) = ^{1}/2 h_{∞}(0, −1) = 0
Ψ_{1}^{∗}(^{1}/2,^{1}/2) = 1

Hence, (1) is fulfilled for our choice K_{ℙ^{1}} = −2 · [1] and D is an anti-canonical divisor. By elementary calculations for vol Ψ we get

vol Ψ = ∫_{□}deg Ψ = \frac{13}{3}.

Putting together the observations of Example 2.2, 3.3 and 3.9 we conclude that X is a smooth Fano threefold of Picard rank 3 and Fano degree 26. By [MM86] this is a blow up of the quadric threefold in two disjoint conics.

3.1 Moment map and Duistermaat-Heckman measure

In this section we give an interpretation of divisorial polytopes in terms of moment maps. The polarisation of a T-variety X by L = O(D), for an ample invariant divisor D, induces a symplectic structure on X by pulling back the Fubini-Study form of the corresponding torus equivariant embedding into projective space (at least for a suitable multiple of D). The action of the algebraic torus action on X induces a Hamiltonian action of the contained (maximal) compact torus T_{c} on the corresponding symplectic manifold. This action comes with a moment polytope P and a moment map ϕ : X → P. Now, the push forward of the canonical symplectic measure on X via the moment map gives a measure on P. This measure is called the Duistermaat-Heckman measure and the corresponding continuous density function f is called the Duistermaat-Heckman function. It is known to be piecewise polynomial and the degree corresponds to the complexity of the torus action, cf. [DH82]. The global sections L decompose in homogeneous components of weight u ∈ M. By [PS11, Proposition 3.23] we may express such a homogeneous component in terms of the corresponding divisorial polytope Ψ_{D}:

H^{0}(X, L)_{u} ≅ \begin{cases}H^{0}(ℙ^{1}, O(Ψ_{D}(u))) & , u ∈ □_{D} \\ 0 & , else \end{cases}  (2)

From this one can deduce that □ = □_{D} equals the moment polytope P, see [AIP^{+}11, Section 14.7]. More generally for powers if L we get by [IS10, Prop. 3.1]

H^{0}(X, L^{⊗k})_{k·u} = H^{0}(ℙ^{1}, O(k · Ψ_{D}(u)))  (3)

for u ∈ □ ∩ \frac{1}{k} · M. One obtains the value f(0) as the volume of the symplectic reduction of the moment fiber ϕ^{−1}(0). By the Kempf-Ness Theorem the symplectic reduction of ϕ^{−1}(0) by T_{c} is the same as the GIT quotient Y = X//^{L}T = Proj ⨁_{k} H^{0}(X, L^{⊗k})^{T}. The GIT quotient comes together with a polarization by a ℚ-line bundle O_{Y} (1), given by the invariant sections of L. This polarization is compatible with the symplectic structure on ϕ^{−1}(0)/T_{c}. Hence, we have vol ϕ^{−1}(0)/T_{c} = (O_{Y} (1))^{dim Y}. If 0 is in the interior of □ we get Y = X//^{L}T = ℙ^{1}. Moreover, by (2) we have O_{Y} (1) = O(Ψ(0)) and we get f(0) = deg(Ψ(0)). By shifting the linearization of L (and therefore the moment map) by u ∈ □ ∩ M we get f(u) = deg(Ψ(u)) for all lattice points in □. By (3) we get the same for rational points in □ and eventually by continuity for all points. Altogether we obtain Proposition 3.10. The Duistermaat-Heckman function corresponding to the polarization given by an invariant ample divisor D with divisorial polytope Ψ_{D} is given by deg Ψ : □_{D} → ℝ.

The other way around, we may interpret the divisorial polytope Ψ_{D} as a decomposition of the Duistermaat-Heckman function corresponding to the polarization of X by L. This shows the relation between our divisorial polytopes and the description of Hamiltonian torus actions given by Karshon and Tolman in [KT03, KT14].

The Cox ring

For a normal variety with a class group isomorphic to ℤ^{r} the Cox ring is defined as

Cox(X) = ⨁_{α∈ℤ^{r}} H^{0} (X, O(∑_{i} α_{i}D_{i})) ,

with D_{1}, …, D_{r} being a basis of Cl(X). For class groups which are not free the definition is little bit more involved. Since smooth Fano varieties always have free class group, we don’t have to discuss this here. In [HS10, Theorem 1.2] the Cox ring of a T-variety was calculated in terms of the combinatorial data. Due to [AIP^{+}11] we have the following reformulation of this theorem in our language.

Theorem 3.11 ([AIP^{+}11, Theorem 40]).

Cox(X) = \frac{ℂ [T_{v}, S_{ρ} | v ∈ S^{(0)}, ρ ∈ (tail S)^{×}]}{⟨T^{μ(0)} + cT^{μ(∞)} + T^{μ(c)} | c ∈ ℂ^{∗}⟩},

where T^{μ(P)} = ∏_{v∈S_{P}^{(0)}} T_{v}^{μ(v)}. The Cl(X)-grading is given by deg(T_{v}) = [V_{v}], deg(S_{ρ}) = [H_{ρ}].

Example 3.12 (Fano threefold 3.10 (continued)). For our f-divisor from Example 2.2 we get the Cox ring

ℂ[T_{1}, …, T_{7}]/⟨T_{1}T_{2}^{2}T_{3} + T_{4}T_{5}^{2}T_{6} + T_{7}⟩.

The grading is given by deg(T_{i}) = [D_{i}], with the notation of Example 3.3. Using the identification Cl X ≅ ℤ^{3} from Example 3.3 we obtain the weight matrix

T_{1} T_{2} T_{3} T_{4} T_{5} T_{6} T_{7}
( −1 1 −1 0 0 0 0)
1 0 1 0 1 0 1
1 0 1 1 0 1 1

The Futaki character

In [Fut83] Futaki introduced the invariant as an obstruction for the existence of Kähler-Einstein metrics on Fano manifolds. Hence, for a Kähler-Einstein Fano manifold the Futaki character has to vanish. Donaldson gave a an algebraic redefinition of this invariant in [Don02], which we are using in the following. Consider a ℂ^{∗}-action λ on a normal variety X of dimension n and an invariant Cartier divisor D. Now, let l_{k} = dim H^{0}(X, O(kD)) and w_{k} the total weight of the ℂ^{∗}-action on H^{0}(X, O(kD)), i.e. w_{k} = ∑_{i} i · dim H^{0}(X, O(kD))_{i}. Then l_{k} and w_{k} obtain expansions

l_{k} = a_{0}k^{n} + a_{1}k^{n−1} + O(k^{n−2}),
w_{k} = b_{0}k^{n+1} + a_{1}k^{n} + O(k^{n−1}).

Definition 3.13. The Futaki invariant of this ℂ^{∗}-action is defined as

F_{D}(λ) = 2 · \frac{a_{1}b_{0} − a_{0}b_{1}}{a_{0}} .

Remark 3.14. Note, that we choose a different scaling and sign as Donaldson for the invariant. More precisely his invariant equals −\frac{1}{2a_{0}} F_{D} in our notation. By plugging in one-parameter subgroups of an acting torus we obtain a linear map F_{D} : N → ℝ, hence an element of M_{ℝ}. If X is Fano we set F (X) := F_{(−K_{X})}. For a toric variety Donaldson in [Don02] gives an easy formula to actually calculate the Futaki character. Let Δ be the polytope corresponding to an ample divisor D on a toric variety. Then the Futaki character is given by

F_{D}(v) = ∫_{∂Δ} v dμ_{1} − \frac{vol_{1} ∂Δ}{vol Δ} · ∫_{Δ} v dμ.  (4)

Here, we are integrating over Δ with the Euclidean measure μ induced by the lattice M ⊂ M_{ℝ}. For integrating over the boundary we are using the facet measure μ_{1}, which is induced by (ℝ · δ) ∩ M for every facet δ ≺ Δ. The corresponding facet volume is denoted by vol_{1} and the facet barycenter by bc_{1}. Note, that up to scaling by (vol_{1} ∂Δ), for the Futaki character we obtain the difference of the barycenters of Δ and its boundary:

F_{D} = (vol_{1} ∂Δ) · (bc_{1}(∂Δ) − bc(Δ)).  (5)

It’s easy to see that for the standard polytope of the canonical divisor Δ = {u | ∀_{ρ∈Σ^{(1)}} : ⟨u, n_{ρ}⟩ ≥ −1} the formula simplifies to F (X) = vol Δ · bc Δ, i.e. the Futaki character is up to positive scaling given by the barycenter of the polytope. This was already observed by Mabuchi [Mab87]. For generalizing the last result to our case we define the barycenter \overline{bc}(Ψ) of a divisorial polytope Ψ : □ → Div_{ℝ} ℙ^{1} by

⟨\overline{bc}(Ψ), v⟩ = ∫_{□_{h}}v · deg Ψ dμ.

This gives the same as the projection to M_{ℝ} of the barycenter of

Δ := Δ(deg Ψ) := {(u, x) | u ∈ □, 0 ≤ x ≤ deg(u)}.

Theorem 3.15. For a Fano T-variety X and a corresponding divisorial polytope Ψ we have

F (X) = vol Ψ · \overline{bc}Ψ.

We postpone the proof to the appendix.

Remark 3.16. By Proposition 3.10 we may interpret the Futaki character as barycenter of the moment polytope with respect to the Duistermaat-Heckman measure (deg Ψ) · μ, see also [Mab87, Theorem 9.2.3] for closely related results.

Example 3.17. We would like to compute the Futaki invariant of the Fano threefold 3.23. Its divisorial polytope Ψ is given by Figure 3. We can read off Δ(deg Ψ) directly from the rightmost picture. The barycenter of this poly-

Ψ_{0} 0 0 0 0  0 −1 −3 −3 ^{Ψ_{∞} −1 0 0  −1 0 0} ^{Ψ_{1} 1 0 0  2 3 3} ^{deg Ψ^{1}/2 0 0 0  ^{3}/2^{3}/20 0 0}

Figure 3: Divisorial polytope of 3.23

tope is (0, −^{9}/40,^{37}/80)^{t} and the volume ^{20}/3. Note, that we have vol Ψ = vol Δ(deg Ψ). Hence, by applying Theorem 3.15 we get F (X) = \binom{0}{−^{3}/2} ≠ 0. In particular, there does not exists a Kähler-Einstein metric on X.

4 Toric degenerations

We are using the notation and results of [IV12] to describe degenerations to toric varieties via f-divisors. First note, that the torus action of a T-variety X(S) is not necessarily maximal. Consider an f-divisor S with two non-trivial slices S_{0} and S_{∞}. Remark 1.8 in [IV12] shows that the T -action on X(S) extends to the action of a torus T′ of full dimension. Hence, X is essentially toric with a fan Σ as follows: We embed the polyhedra of S_{0} in height 1 of N_{ℚ} × ℚ and consider the fan Σ_{0} consisting of the cones over these polyhedra. We are doing the same with S_{∞} embedded in height −1 and obtain Σ_{∞}. Now, we obtain Σ from Σ_{0} ∪ Σ_{∞} by joining cones which intersect in a common facet of the form σ ×{0}, such that σ ∩deg S ≠ ∅.

Example 4.1. Consider the f-divisor pictured in Figure 4 having two nontrivial slice S_{0} and S_{∞}. The fan Σ_{0} from above consists of five maximal cones:

S_{0}  S_{∞}  deg S

Figure 4: F-divisor of a downgraded toric variety.

σ_{0} = pos ((0, 0, 1), (−1, 0, 1), (−1, −1, 1), (0, −1, 1))
σ_{1} = pos ((0, 0, 1), (−1, 0, 1), (1, 1, 0), (−1, 1, 0))
σ_{2} = pos ((0, 0, 1), (0, −1, 1), (1, 1, 0), (1, −1, 0))
σ_{3} = pos ((−1, 0, 1), (−1, −1, 1), (−1, 1, 0), (−1, −1, 0))
σ_{4} = pos ((−1, −1, 1), (0, −1, 1), (1, −1, 0), (−1, −1, 0)).

The fan Σ_{∞} consists of the four cones

τ_{1} = pos ((1, 1, −2), (1, 1, 0), (−1, 1, 0))
τ_{2} = pos ((1, 1, −2), (1, 1, 0), (1, −1, 0))
τ_{3} = pos ((1, 1, −2), (−1, 1, 0), (−1, −1, 0))
τ_{4} = pos ((1, 1, −2), (1, −1, 0), (−1, −1, 0)).

We have to join four pairs of cones (σ_{i}, τ_{i}), i = 1, …, 4 and obtain the fan Σ consisting of the five maximal cones

δ_{0} = σ_{0}
δ_{1} = pos ((0, 0, 1), (−1, 0, 1), (1, 1, −2))
δ_{2} = pos ((0, 0, 1), (0, −1, 1), (1, 1, −2))
δ_{3} = pos ((−1, 0, 1), (−1, −1, 1), (1, 1, −2))
δ_{4} = pos ((−1, −1, 1), (0, −1, 1), (1, 1, −2)).

This is the fan of the projective cone over the quadric surface. Moreover, it is the face fan of the (reflexive) polytope

conv ((0, 0, 1), (−1, 0, 1), (−1, −1, 1), (0, −1, 1), (1, 1, −2))

which has ID 544395 in the classification of canonical toric Fano threefolds from [Kas10]^{†}.

Now, we consider a polyhedral subdivision Ξ. A Minkowski-decomposition of Ξ consist of polyhedral subdivisions Δ = Δ^{1} + … + Δ^{r} for every Δ ∈ Ξ such that i. (Δ ∩ ∇)^{i} = Δ^{i} ∩ ∇^{i}, for Δ, ∇ ∈ Ξ and Δ ∩ ∇ ≠ ∅,

†see http://grdb.lboro.ac.uk/search/toricf3c?ID=544395

Documenta Mathematica 19 (2014) 905–940

ii. For I ⊂ {1, …, r}, J ⊂ I ⊂ Ξ we have

∑_{i∈I} ⋂_{Δ∈I} Δ^{i} ≺ ∑_{i∈I} ⋂_{Δ∈J} Δ^{i}

We obtain polyhedral subdivisions Ξ^{i} := {Δ^{i} | Δ ∈ Ξ}. By abuse of notation we will also write Ξ = Ξ^{1} + … + Ξ^{r} for this situation. Such a decomposition is called admissible if for every vertex v of Ξ there is at most one of the corresponding vertices v_{i} ∈ Ξ_{i} with v = ∑_{i} v_{i} which is not a lattice point. We may start with an f-divisor S with non-trivial slices Ξ_{0} = S_{P_{0}}, …, Ξ_{ℓ} = S_{P_{ℓ}} and an admissible Minkowski decomposition Ξ_{0} = Ξ_{ℓ+1} + … + Ξ_{ℓ+r}. As described in [IV12, Sections 2 and 4] this data gives rise to a deformation of X(S). A general fiber of this deformation corresponds to an f-divisor S′ with non-trivial slices being exactly

S′_{P′_{1}} = Ξ_{1}, …, S′_{P′_{ℓ+r}} = Ξ_{ℓ+r}.

For some distinct points P′_{1}, … P′_{ℓ+r} ∈ ℙ^{1}. We now reverse the above procedure in order to obtain toric degenerations. Let’s start with an f-divisor and assume that all non-trivial slices are contained in {S_{P_{1}}, … S_{P_{r}}, S_{∞}} and the first r of them form an admissible Minkowski decomposition of some subdivision Ξ

Ξ = S_{P_{1}} + … + S_{P_{r}}.

Then X(S) is a deformation of the T-variety corresponding to the f-divisor

(Ξ ⊗ [0] + S_{∞} ⊗ [∞], deg S)

which describes a subtorus action on a toric variety as we have seen above.

Example 4.2. We consider the f-divisor with slices S_{0}, S_{∞}, S_{1} and degree sketched in Figure 5. The corresponding variety turns out to be the quadric threefold.

S_{0}  S_{∞}  S_{1}  deg S

Figure 5: F-divisor of the quadric threefold

Note, that S_{0} and S_{∞} form an admissible Minkowski decomposition, which is sketched in Figure 6.

=  +

Figure 6: An admissible Minkowski decomposition

Hence, X degenerates to the toric variety from Example 4.1. We find another toric degeneration to the canonical toric Fano variety with ID 547378 in [Kas10] by adding up Ξ = S_{0} + S_{1}. Adding up Ξ = S_{0} + S_{∞} + S_{1} will give a toric degeneration to a non-canonical toric Fano variety. Remark 4.3. Note, that we do not get all toric degeneration with this method, but only those which are equivariant with respect to the T -action on X.

5 Proof by pictures

Aim of this section is to proof Theorem 1.1. For every variety from the list in Theorem 1.1 we will state the f-divisor S and a divisorial polytope Ψ, which corresponds to a canonical divisor. To verify the proof for a particular threefold one has to check first, that the f-divisor and divisorial polytope actually correspond to the given threefold. This is done by carrying out the following procedure.

A.1 Applying Theorem 2.3: checking the smoothness of X(S) by checking the regularity of certain cones (Example 2.2).

A.2 Calculating the Picard rank of X(S) by Corollary 3.2.

A.3 Checking that Ψ corresponds to a canonical divisor (Example 3.9).

A.4 Calculating the Fano degree of X(S) by Theorem 3.8 (Example 3.9).

Now, by the classification in [MM86] we know that the given f-divisor describes a Fano threefold in the stated family. Hence, we constructed a Fano threefold with 2-torus action within this family. To check the other statement of Theorem 1.1 we have to apply additional steps

B.1 Applying Theorem 3.11 to calculate the Cox ring (Example 3.12).

B.2 Applying Theorem 3.15 to calculate the Futaki invariant (Example 3.17).

B.3 Checking symmetry as in Example 2.12.

B.4 Checking if F (X) = 0 and if possible apply Theorem 2.11 to check for the Kähler-Einstein property (Examples 2.12, 3.17).

B.5 “Adding up slices” to obtain toric degenerations (Example 4.2).

Beside the given examples we leave it to the reader to carry out the verification procedure for the given f-divisors. We give some hints how to interpret the pictures and information given below. As in the examples we first state the non-trivial slice S_{0}, S_{∞}, S_{1} and deg S of an f-divisor S. Next, we state the divisorial polytope of the canonical divisor by giving Ψ_{0}, Ψ_{∞} and Ψ_{1} (all other Ψ_{P} are assumed to be trivial). For convenience of the reader we state deg Ψ, as well. In the pictures • marks the origin and × the Futaki character (if it differs from the origin). If there exists equivariant degenerations to canonical toric varieties then we state the corresponding IDs for the Fano polytopes in the Graded Ring Database [BDK^{+}13].

Name :  Q Quadric threefold
Picard rank: 1  Fano Degree:  54  symm.: yes
Futaki-char.: 0  Degenerations: 544395, 547378
Cox ring: ℂ[T_{1}, …, T_{5}]/⟨T_{1}T_{2} + T_{3}T_{4} + T_{5}^{2}⟩  deg(\underline{T}) = (1, 1, 1, 1, 1)

0  −3 0  0 ^{0  0 0  −3} ^{0  3 0  3} ^{^{3}/2 0  0 0  0}
Name :  2.24 Divisor of bidegree (1, 2) in ℙ^{2} × ℙ^{2}
Picard rank:  2  Fano Degree:  54  symm.: yes
Futaki-char.:  0  Degenerations: —
Cox ring: ℂ[T_{1}, …, T_{6}]/⟨T_{1}T_{2}^{2} + T_{3}T_{4}^{2} + T_{5}T_{6}^{2}⟩ deg(\underline{T}) = ( \begin{matrix}1 & 0 & 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 & 0 & 1 \end{matrix} )

0 0 0 0  −2 −2  ^{0 0 0 0 −2 −2}  ^{2 2 0 0 2 2}  ^{0 0 0 0 0 0 ^{1}/2  ^{1}/2^{1}/2}

Name :  2.29 Blowup of Q in a conic
Picard rank:  2  Fano Degree:  40  symm.: yes
Futaki-char.:  0  Degenerations: 430083, 544339
Cox ring: ℂ[T_{1}, …, T_{6}]/⟨T_{1}T_{2}^{2}T_{3} + T_{4}T_{5} + T_{5}^{2}⟩  ( \begin{matrix}−1 & 1 & −1 & 0 & 0 & 0 \\ 1 & 0 & 1 & 1 & 1 & 1 \end{matrix} )

−1 0 0  −3 −1 0 ^{0 0 0  0 −2 −2} ^{1 0 0  3 3 2} ^{1 0 0 0  0 0 0}

Name :  2.30 Blow up of Q in a point
Picard rank:  2  Fano Degree:  46  symm.: no
Futaki-char.: \binom{0}{−2}  Degenerations: 520157, 544343
Cox ring: ℂ[T_{1}, …, T_{6}]/⟨T_{1}T_{2} + T_{3}T_{4} + T_{5}^{2}⟩  ( \begin{matrix}1 & 1 & 2 & 0 & 1 & −1 \\ 0 & 0 & −1 & 1 & 0 & 1 \end{matrix} )

0 0 0  −2−3 0 ^{0 0 −3 0 0} ^{0 0 3 2 3} ^{0 0  0 0 0 1^{3}/2}

Name :  2.31 Blow up of Q in a line
Picard rank:  2  Fano Degree:  46  symm.: no
Futaki-char.: −\binom{^{4}/3}{^{4}/3}  Degenerations: 520058, 520159
Cox ring: ℂ[T_{1}, …, T_{6}]/⟨T_{1}T_{2} + T_{3}T_{4} + T_{5}T_{6}^{2}⟩  ( \begin{matrix}0 & 1 & 1 & 0 & −1 & 1 \\ 1 & 0 & 0 & 1 & 1 & 0 \end{matrix} )

0 0 0  −2 −2 0  ^{0 0 0  0 −1 −3}  ^{0 0 0  2 3 3}  ^{0 0 0  0 0 0^{3}/2 1 1}

Name :  W or 2.32 Divisor of bidegree (1, 1) in ℙ^{2} × ℙ^{2}
Picard rank:  2  Fano Degree:  46  symm.: yes
Futaki-char.:  0  Degenerations: 520058, 520159
Cox ring: ℂ[T_{1}, …, T_{6}]/⟨T_{1}T_{2} + T_{3}T_{4} + T_{5}T_{6}⟩  deg(\underline{T}) = ( \begin{matrix}1 & 0 & 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 & 0 & 1 \end{matrix} )

−2 0 0 0 0 −2  ^{0 0 0 −2 −2 0}  ^{2 0 0 2 2 2}  ^{0 0 0 0 0 0 2}

Name :  3.8 Blowup of 2.24 in a line
Picard rank:  3  Fano Degree:  24  symm.:  no
Futaki-char.: \binom{^{5}/6}{−^{5}/3}  Degenerations: —
Cox ring: ℂ[T_{1}, …, T_{7}]/⟨T_{1}T_{2}^{2} + T_{3}T_{4}^{2} + T_{5}T_{6}^{2}⟩ deg(\underline{T}) = ( \begin{matrix}1 & 1 & 1 & 1 & 1 & 1 & 0 \\ 0 & 1 & 2 & 0 & 2 & 0 & 0 \\ 0 & 0 & −2 & 1 & 0 & 0 & 1 \end{matrix} )

0 0 0 0 −1 −2 −2  ^{0 0 0 0 −1 −1}  ^{2 2 0 0  2 2 0}  ^{0 0 0 0 0  0^{1}/2^{1}/2 ^{1}/2^{1}/2}

Name :  3.10 Blowup of Q in two disjoint invariant conics
Picard rank:  3  Fano Degree:  26  symm.:  yes
Futaki-char.:  0  Degenerations:  —
Cox ring: ℂ[T_{1}, …, T_{7}]/⟨T_{1}T_{2}^{2}T_{3} + T_{4}T_{5}^{2}T_{6} + T_{7}^{2}⟩ deg(\underline{T}) = ( \begin{matrix}0 & 1 & 0 & 0 & 1 & 0 & 1 \\ 1 & 0 & 1 & 0 & 1 & 0 & 1 \\ 1 & 0 & 1 & 1 & 0 & 1 & 1 \end{matrix} )

0 0  −1−2 −1−2 0 0  ^{0 0  0 0 −2−1 −2 −1}  ^{0 0  1 2 3 3 2 1}  ^{0 0  0 0 0 0 0 0 ^{1}/2}

Name :  3.18 Blowup of Q in a point and a conic
Picard rank:  3  Fano Degree:  36  symm.:  yes
Futaki-char.: \binom{0}{−^{7}/8}  Degenerations: 255687, 519918
Cox ring: ℂ[T_{1}, …, T_{7}]/⟨T_{1}T_{2}^{2}T_{3} + T_{4}T_{5} + T_{6}^{2}⟩ deg(\underline{T}) = ( \begin{matrix}0 & 1 & 0 & 2 & 0 & 1 & −1 \\ 1 & 0 & 1 & 2 & 0 & 1 & −1 \\ 1 & 0 & 1 & 1 & 1 & 1 & 0 \end{matrix} )

0 0  −3 −2 −1 −1 0 0 ^{0 0  0 0 −2 −2} ^{0 0 2 3 2 3} ^{0 0  0 0 0 0 1 1^{1}/2 ^{1}/2}

Name :  3.19 Blowup of Q in two points
Picard rank:  3  Fano Degree:  38  symm.:  yes
Futaki-char.:  0  Degenerations: 430426, 519917
Cox ring: ℂ[T_{1}, …, T_{7}]/⟨T_{1}T_{2} + T_{3}T_{4} + T_{5}^{2}⟩ deg(\underline{T}) = ( \begin{matrix}1 & 1 & 2 & 0 & 1 & −1 & 0 \\ 0 & 0 & −1 & 1 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 1 \end{matrix} )

0 0  −3 −2 0  −2 0 0 ^{0 0  0 0 −1  −1} ^{0 0  3 2 1  3} ^{0 0  0 0 0  0 1 1^{3}/2}

Name :  3.20 Blowup of Q in two disjoint lines
Picard rank:  3  Fano Degree:  38  symm.:  yes
Futaki-char.:  0  Degenerations:  255692, 430424
Cox ring: ℂ[T_{1}, …, T_{7}]/⟨T_{1}T_{2} + T_{3}T_{4} + T_{5}T_{6}^{2}T_{7}⟩ deg(\underline{T}) = ( \begin{matrix}0 & 1 & 1 & 0 & 1 & 0 & 0 \\ 1 & 1 & 1 & 1 & 0 & 1 & 0 \\ 1 & 0 & 0 & 1 & 0 & 0 & 1 \end{matrix} )

0 0 0 −2 −2 0 0 −1  ^{0 0 0 −1 0 0 −2−2}  ^{0 0 0 2 3 0 2 3}  ^{0 0 0 0 0 0 0 0^{3}/2 1 1 1 1}

Name :  3.21 Blowup of ℙ^{1}×ℙ^{2} in a curve of degree (2,1)
Picard rank:  3  Fano Degree:  38  symm.: no
Futaki-char.: \binom{^{7}/8}{^{7}/8}  Degenerations: 429943
Cox ring: ℂ[T_{1}, …, T_{7}]/⟨T_{1}T_{2}^{2} + T_{3}T_{4}^{2} + T_{5}T_{6}⟩ deg(\underline{T}) = ( \begin{matrix}1 & 0 & 1 & 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & 1 & 1 & 1 & 0 \\ 0 & 1 & 0 & 1 & 2 & 0 & 1 \end{matrix} )

0 0 0 0  −2 −2 −1 ^{0 0 0 0 −2 −2 −1} ^{2 2 1 2 2 1} ^{0 0 0 0 0 0 ^{1}/2^{1}/21 1}

Name :  3.22 Blowup of ℙ^{1} × ℙ^{2} in a conic in {0} × ℙ^{2}
Picard rank:  3  Fano Degree:  40  symm.:  no
Futaki-char.: \binom{0}{−^{2}/3}  Degenerations: 430417, 519928
Cox ring: ℂ[T_{1}, …, T_{7}]/⟨T_{1}T_{2} + T_{3}T_{4} + T_{5}^{2}⟩ deg(\underline{T}) = ( \begin{matrix}1 & 1 & 0 & 2 & 1 & 0 & 0 \\ 0 & 0 & 1 & −1 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 1 & 1 \end{matrix} )

0 0 0 0  0 −1 −3 −3 ^{−1 0 0  −1 0 0} ^{1 0 0  2 3 3} ^{^{1}/2 0 0 0  ^{3}/2^{3}/20 0 0}

Name :  3.23 Blowup of Q in a point and the strict

transform of a line passing through

Picard rank:  3  Fano Degree:  42  symm.:  no
Futaki-char.: \binom{−^{13}/12}{−^{53}/24}  Degenerations: 254876, 430425
Cox ring: ℂ[T_{1}, …, T_{7}]/⟨T_{1}T_{2} + T_{3}T_{4} + T_{5}T_{6}^{2}⟩ deg(\underline{T}) = ( \begin{matrix}1 & 0 & 1 & 0 & 1 & 0 & 0 \\ 1 & 1 & 1 & 1 & 0 & 1 & 0 \\ 1 & 1 & 2 & 0 & 0 & 1 & 1 \end{matrix} )

0 0  −2 −1 0 −2 0  ^{0 0  0 0 −1 −3}  ^{0 0  2 1 1 3 3}  ^{0 0  0 0 1 0 0^{3}/2 1}

Name :  3.24 Blowup of W in (0 : 0 : 1, ∗ : ∗ : 0)
Picard rank:  3  Fano Degree:  42  symm.:  no
Futaki-char.: \binom{^{2}/3}{−^{4}/3}  Degenerations: 429952, 430423
Cox ring: ℂ[T_{1}, …, T_{7}]/⟨T_{1}T_{2} + T_{3}T_{4} + T_{5}T_{6}⟩ deg(\underline{T}) = ( \begin{matrix}0 & 1 & 0 & 1 & 1 & 0 & 0 \\ 1 & 0 & 0 & 1 & 0 & 1 & −1 \\ 1 & 0 & 1 & 0 & 0 & 1 & 0 \end{matrix} )

−2 −2 0 0 0 0 0  ^{0 0  0 0 −2 −2}  ^{2 2  0 2  0 2 2}  ^{0 0 1 0 1  0 0 0 2}

Name :  4.4 Blow up of Q in 2 non-colinear points and the strict

transform of a conic passing through both of them

Picard rank:  4  Fano Degree:  32  symm.: yes
Futaki-char.: 0  Degenerations: 254352, 430357
Cox ring: ℂ[T_{1}, …, T_{8}]/⟨T_{1}T_{2}^{2}T_{3} + T_{4}T_{5} + T_{6}^{2}⟩ deg(\underline{T}) = ( \begin{matrix}0 & 1 & 0 & 2 & 0 & 1 & 0 & 1 \\ 1 & 0 & 1 & 2 & 0 & 1 & 0 & 1 \\ 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 \end{matrix} )

0 0  −3 −2 0  −2 −1 −1 0 0 ^{0 0  0 0 −1  −1 −1 0} ^{0 0  3 2 1  3} ^{0 0  0 0 0  0 1 1^{1}/2^{1}/2 ^{1}/2^{1}/2}

Name :  4.5 Blowup of ℙ^{1}×ℙ^{2} in curves of deg (2,1) and (1,0)
Picard rank:  3  Fano Degree:  32  symm.:  no
Futaki-char.: \binom{^{5}/24}{^{5}/24}  Degenerations: 255339
Cox ring: ℂ[T_{1}, …, T_{8}]/⟨T_{1}T_{2}^{2} + T_{3}T_{4}^{2} + T_{5}T_{6}⟩ deg(\underline{T}) = ( \begin{matrix}1 & 0 & 1 & 0 & 1 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 2 & 1 & 0 & 0 \\ 0 & 1 & 0 & 1 & 2 & 0 & 1 & 0 \end{matrix} )

0 0 0 0  −2 −2 −1 0^{0 0 0 0 −2−2 −1  0} ^{2 2 1 2 2 1} ^{^{1}/2  ^{1}/20 0 0 0 0 0 ^{1}/2^{1}/21 1}

Name :  4.7 Blowup of W in (0 : 0 : 1, ∗ : ∗ : 0)
and (∗ : ∗ : 0, 0 : 0 : 1)
Picard rank:  4  Fano Degree:  36  symm.: yes
Futaki-char.: 0  Degenerations: 254603, 255837
Cox ring: ℂ[T_{1}, …, T_{8}]/⟨T_{1}T_{2} + T_{3}T_{4} + T_{5}T_{6}⟩ deg(\underline{T}) = ( \begin{matrix}0 & 1 & 1 & 0 & 1 & 0 & 0 & 0 \\ 1 & 0 & 1 & 0 & 0 & 1 & 0 & 1 \\ 1 & 0 & 0 & 1 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 \end{matrix} )

−2 −2 0 0 0 −10 0  ^{0 0  0 0 −1 −1}  ^{2 2  0 0 0 2 2 2}  ^{0 0 1 0 1  0 1 1 0 0 2}

Name :  4.8 Blowup of (ℙ^{1})^{3} in a curve of degree (0, 1, 1)
Picard rank:  4  Fano Degree:  38  symm.:  no
Futaki-char.: \binom{0}{−^{13}/12}  Degenerations: 255836, 428555
Cox ring: ℂ[T_{1}, …, T_{8}]/⟨T_{1}T_{2} + T_{3}T_{4} + T_{5}T_{6}⟩ deg(\underline{T}) = ( \begin{matrix}0 & 1 & 1 & 0 & 1 & 0 & 0 & 0 \\ 1 & 0 & 1 & 0 & 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 1 & 0 & 1 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 \end{matrix} )

−2 −2 −10 0 0 0 0  ^{2 2 1  2 2 1}  ^{−2 −2 −1 0 0 0 0 0}  ^{0 0 0  0 0 0 2 1 2}

A Calculating the Futaki character

The Appendix is devoted to the, somewhat technical, proof of Theorem 3.15. For an invariant ample Cartier divisor D we have the divisorial polytope Ψ : □ → Div_{ℝ} ℙ^{1}, with □ ⊂ M_{ℝ}. We cannot directly use Donaldson’s result for the toric case, since the dimension of the global sections of O(kD) does not grow like the number of lattice points of a multiple of a polytope. But as it turns out it grows like the number of “lattice points” of a weighted sum of polytopes. More precisely, we consider the abelian group Π̃(V) generated by all polytopes in some ℝ-vector space V with the relations [Δ]+[∇] = [Δ∪∇]+[Δ∩∇], whenever Δ, ∇, Δ ∪ ∇ are polytopes. A subset of V which is covered by polyhedra ∇ = ⋃_{i∈I} Δ_{i} can be naturally identified with an element of this group

[∇] := ∑_{J⊂I}(−1)^{|J|+1} · [⋂_{j∈J} Δ_{j}] .  (6)

In particular this applies for the relative boundary dΔ of a polytope. We also introduce a boundary operator ∂ for the class of a polytope in Π̃.

∂[Δ] := \begin{cases}[dΔ] & , codim(Δ) = 0 \\ 2[Δ] − [dΔ] & , codim(Δ) = 1, \\ 0 & , else. \end{cases}  (7)

For an object Δ = ∑_{i} a_{i}[Δ_{i}] ⊂ Π̃ simply by linear continuation we have well defined notions of dilation, boundary, volume, numbers of lattice point, and barycenter:

k ∗ Δ := ∑_{i} a_{i} · [kΔ_{i}],
∂Δ := ∑_{i} a_{i} · ∂[Δ_{i}],
vol(Δ) := ∑_{i} a_{i} · vol(Δ_{i}),
N(Δ) := ∑_{i} a_{i} · N(Δ_{i}),

bc(Δ) := ∑_{i} a_{i} · \frac{vol(Δ_{i})}{vol(Δ)} · bc(Δ_{i}).

Similarly we can extend the facet volume vol_{1} and the facet barycenter bc_{1} to Π̃, where vol_{1} and bc_{1} are defined to be zero for polytopes of codimension ≠ 1. More generally, we may integrate linear forms on V over elements of Π̃.

Proposition A.1. For an element Δ ⊂ Π̃(ℝ^{m}) we have the following asymptotic formula for the number of lattice points

N(k ∗ Δ) = (vol Δ) · k^{m} + \frac{vol_{1} ∂Δ}{2} · k^{m−1} + O(k^{m−2}).

Proof. This is a well known fact for polytopes of maximal dimension (see also [Don02, Proposition 4.1.3] for a proof). The same fact follows for weighted sums of maximal dimensional polytopes just by linearity. Our definition (7) of a boundary ∂[P] makes it work for polytopes of lower dimension, as well.

We are now going to associate to a divisorial polytope Ψ an element [Ψ] of Π̃(M_{ℝ} × ℝ). We consider the natural projection M_{ℝ} × ℝ → M_{ℝ} and inclusion N_{ℝ} ↪ N_{ℝ} × ℝ. First, we need some notation. For a function f : ℝ^{n} ⊃ U → ℝ with f ≥ R let Δ^{R}(f) be the region enclosed by its graph and the R-level:

Δ^{R}(f) = {(u, a) ∈ M_{ℝ} × ℝ | u ∈ D, R ≤ a ≤ f(u)}.

If f is non-negative we set Δ(f) := Δ^{0}(f). Now, for a concave piecewise affine function f on □ we denote its graph by f̂ and associate an element [f] ⊂ Π̃ which correspond to the area enclosed by the graph of f and the 0-level. More precisely, we choose an integer R ≤ 0 such that f ≥ R and define

[f] = [Δ^{R}(f)] − [□×[R, 0]]  (8)

Note, that [f] does not depend on the particular choice of R. Now, to a divisorial polytope Ψ : □ → Div_{ℝ} ℙ^{1} we associate a class [Ψ] ∈ Π̃(M_{ℝ} × ℝ) by

[Ψ] := [˜□] + ∑_{P}[Ψ_{P}].  (9)

Here, we set ˜□ := □ × {0}. The following proposition justifies this definition.

Proposition A.2. For an ample divisor D on a T-variety X(S) of dimension n and the corresponding divisorial polytope Ψ = Ψ_{D} we have

i. (D)^{n} = vol Ψ = vol[Ψ],

ii. dim H^{0}(X, O(D)) = N([Ψ]).

iii. dim H^{0}(X, O(D))_{u} = N_{u}([Ψ]).

Here, N_{u}([Ψ]) is the number of lattice point that project to u ∈ M_{ℝ} if we consider the projection M_{ℝ} × ℝ → M_{ℝ} for every element in the sum (9).

Proof. By Theorem 3.8 we only have to prove vol Ψ = vol[Ψ] to obtain (i). The only summand of (9) that contributes to the volume is [Ψ_{P}], since all other polytopes have lower dimension. Now, by the basic rules of integration we have ∫_{□} Ψ_{P} = vol[Ψ_{P}]. Hence, we obtain

vol[Ψ] = ∑_{P} vol[Ψ_{P}] = ∑_{P} ∫_{□} Ψ_{P} = ∫_{□} deg Ψ = vol Ψ.

We prove (iii). Remember, that the global sections of weight u ∈ □ ∩ M are given by H^{0}(ℙ^{1}, O(Ψ(u))). Hence, we have to check that the number N_{u} of lattice points of [Ψ], that project to u equals dim H^{0}(ℙ^{1}, O(Ψ(u))). Looking at (9) and (8) gives

N_{u} = 1+∑_{P}(⌊Ψ_{P} (u)−R_{P} ⌋+1−(1−R_{P})) = 1+∑_{P}⌊Ψ_{P} (u)⌋ = 1+deg⌊Ψ(u)⌋,

since the intervals [R_{P} , Ψ_{P} (u)] and [R_{P} , 0] contain ⌊Ψ_{y}(u)−R_{P} ⌋+1 and 1−R_{P} integers, respectively. We have dim H^{0}(X, O(Ψ(u)) = 1 + deg⌊Ψ(u)⌋ and the claim follows. Obviously (ii) follows from (iii).

Theorem A.3. The Futaki character F_{D} : N_{ℝ} → ℝ of an divisor D with corresponding divisorial polytope Ψ = Ψ_{D} is given by

F_{D}(v) = ∫_{∂[Ψ]}v − \frac{vol_{1} ∂[Ψ]}{vol Ψ} ∫_{[Ψ]}v.  (10)

Proof. We are using exactly the same arguments as in the proof of Theorem 4.2.1 in [Don02]. For an element v ∈ N we are interested in the total weight of H^{0}(X, O(D)) with respect to this one-parameter subgroup. By Proposition A.2 it is given by

w(H^{0}(X, O(D))) = ∑_{u∈□∩M}⟨u, v⟩ · N_{u}([Ψ])

Note, that we have [Ψ_{kD}] = k ∗ [Ψ_{D}], by [IS10, Proposition 3.1]. We are now constructing an element Q of Π̃(M_{ℝ} × ℝ) fulfilling

N(k ∗ Q) − N(k ∗ [Ψ]) = ∑_{u∈□∩M}⟨u, v⟩ · N_{u}(k ∗ [Ψ]).  (11)

For every elementary summand [Δ_{i}] of (9) (which is the class of a polytope) we are using the construction from [Don02, Section 4.2], i.e. we fix an integer R, such that ⟨·, v⟩ ≤ R on Δ_{i} and define

Q_{i} := {(u, t) | u ∈ Δ_{i}, 0 ≤ t ≤ R − ⟨u, v⟩}.

Now, by Proposition A.2 and Proposition A.1 we have

l_{k} = N(k ∗ [Ψ]) = (vol[Ψ]) · k^{m} + \frac{vol_{1} ∂[Ψ]}{2} · k^{m−1} + O(k^{m−2}).

Moreover, from the proof of [Don02, Theorem 4.2.1] we have

N(kQ_{i}) = k^{m+1} ∫_{Δ_{i}}(R − v) + \frac{k^{m}}{2} ∫_{∂Δ_{i}}(R − v) + N(kΔ_{i}) + O(k^{m−1}).

By linearity and (11) we obtain

w_{k} = k^{m+1} ∫_{[Ψ]}(R − v) + \frac{k^{m}}{2} ∫_{∂[Ψ]}(R − v) + O(k^{m−1})

Now, plugging in the coefficient in Definition 3.13 gives the desired result.

Corollary A.4.

F_{D} = (vol_{1} ∂[Ψ]) · (\overline{bc_{1}}(∂[Ψ]) − \overline{bc}([Ψ])),

where \overline{bc} and \overline{bc_{1}}, respectively denotes the projection of the barycenters in M_{ℝ}× ℝ to M_{ℝ}.

Proof. If we plug into the right hand side of (10) the elements of a lattice basis e_{1}, …, e_{n} of N × ℤ, by the definition of a barycenter we obtain the coordinates of

L = (vol_{1} ∂[Ψ])(bc_{1}(∂[Ψ]) − bc([Ψ]))

with respect to the dual basis e_{1}^{∗}, …, e_{n}^{∗}. By definition L is an element of (N_{ℝ} × ℝ)^{∗} = M_{ℝ} × ℝ and F_{D} ∈ N_{ℝ}^{∗} is just the restriction L|_{N_{ℝ}}, i.e. the projection of L to M_{ℝ}.

To simplify our notation we set Δ := Δ(deg Ψ) and denote by Δ^{∂} the part of Δ → □ that projects to ∂□.

Lemma A.5. For the volume and barycenter of [Ψ] we get

\overline{bc}([Ψ]) = \overline{bc}(Δ)
vol([Ψ]) = vol(Δ).

Proof. In the proof of Proposition A.2 we have seen already that vol([Ψ]) = ∫_{□} deg Ψ and hence vol([Ψ]) = vol(Δ). For the projected barycenter we use the same argument. The only summands that contribute to \overline{bc}([Ψ]) are [Ψ_{P}]. We may interpret \overline{bc}([Ψ_{P}]) as a linear form on v ∈ N_{ℝ} and by definition of the barycenter and basics of integration we have

vol([Ψ_{P}]) · \overline{bc}([Ψ_{P}])(v) = ∫_{Δ^{R_{P}} (Ψ_{P})} v − ∫_{□×[R_{P} ,0]} v
= ∫_{□} (Ψ_{P} − R_{P}) · v + R_{P} · ∫_{□} v
= ∫_{□} Ψ_{P} · v.

Hence, we obtain

vol([Ψ]) · \overline{bc}[Ψ] = ∑_{P} vol([Ψ_{P}]) · \overline{bc}[Ψ_{P}]

= ∑_{P} ∫_{□} Ψ_{P} · v

= ∫_{□} deg Ψ · v

= vol(Δ)\overline{bc}(deg Ψ)

= vol([Ψ])\overline{bc}(deg Ψ).

Lemma A.6. If we assume that Ψ_{P} is constant for P ∉ {P_{1}, …, P_{r}}, then we can calculate volume and barycenter of ∂[Ψ] as follows.

\overline{bc_{1}}(∂[Ψ]) = \overline{bc_{1}} ([Δ^{∂}] + (2 − r)[˜□] + ∑_{i=1}^{r}[Ψ̂_{P_{i}}]) ,

vol_{1}(∂[Ψ]) = vol_{1} ([Δ^{∂}] + (2 − r)[˜□] + ∑_{i=1}^{r}[Ψ̂_{P_{i}}]) ,

where Ψ̂_{P_{i}} denotes the graph of Ψ_{P_{i}}.

Proof. When calculating vol_{1} and bc_{1} we have to consider only summands of ∂[Ψ] of codimension one. For every P we have

∂[Ψ_{P}] = [Ψ̂_{P}] − [□ × R_{P}] + [Δ(Ψ_{P} |_{∂□})] − [∂□ × [R_{P} , 0]] + …,

where … consists of lower dimensional summands. Now, we have \overline{bc_{1}}[□×R_{P}] = \overline{bc_{1}}[□] and vol_{1}[□ × R_{P}] = vol_{1}[□]. Applying Lemma A.5 to every facet of □ gives

\overline{bc_{1}} (∑_{P}[Δ^{R_{P}} (Ψ_{P} |_{∂□})] − [∂□ × [R_{P} , 0]]) = \overline{bc_{1}}[Δ^{∂}],

vol_{1} (∑_{P}[Δ^{R_{P}} (Ψ_{P} |_{∂□})] − [∂□ × [R_{P} , 0]]) = vol_{1}[Δ^{∂}].

Hence,

\overline{bc_{1}}∂ ∑_{i=1}^{r}[Ψ_{P_{i}}] = \overline{bc_{1}} ([Δ^{∂}] − r · [□] + ∑_{i=1}^{r}[Ψ̂_{P_{i}}])

vol_{1} ∂ ∑_{i=1}^{r}[Ψ_{P_{i}}] = vol_{1} ([Δ^{∂}] − r · [□] + ∑_{i=1}^{r}[Ψ̂_{P_{i}}])

Looking at (9) and taking into account our definition of the boundary for lower dimensional polytopes we see, that we still have to add \overline{bc_{1}}(∂ ˜□) = 2\overline{bc_{1}}(˜□) or vol_{1}(∂ ˜□) = 2 vol_{1}(˜□), respectively which gives the desired result.

For a polytope ∇ ⊂ M_{ℝ} × ℝ we define ∇^{+} to be the part of ∇ lying above the 0-level with respect to the last component. Similarly we define ∇^{−} to be the part below the 0-level. We define the pyramid pyr(∇) as the class pyr(∇) := [conv(∇^{+}, 0)] − [conv(∇^{−}, 0)] in Π̃. The pyramid operator extends by linearity to Π̃. Now we have the following equivalent to Lemma A.6 for the full-dimensional volume and barycenter.

Lemma A.7. If we assume that Ψ_{P} ≡ 0 for P ∉ {P_{1}, …, P_{ℓ}}, then we can calculate volume and barycenter of [Ψ] as follows.

\overline{bc}[Ψ] = \overline{bc} (pyr ([Δ^{∂}] + ∑_{i=1}^{ℓ}[Ψ̂_{P_{i}}])) ,

vol[Ψ] = vol (pyr ([Δ^{∂}] + ∑_{i=1}^{ℓ}[Ψ̂_{P_{i}}])) ,

Proof. For a polytope ∇ ⊂ M_{ℝ} × ℝ and a point u ∈ M_{ℝ} we denote by ∇_{u} the length of its fiber over u ∈ M_{ℝ}. Again we may extend this notion linearly to Π̃. For proving the claim, by Lemma A.5 it’s sufficient to show that

deg Ψ(u) = pyr(Δ^{∂})_{u} + ∑_{i=1}^{r} pyr(Ψ̂_{P_{i}})_{u}.

For a point u = w ∈ ∂□ this equality holds, since we have pyr(Ψ̂_{P_{i}})_{w} = 0 and pyr(Δ^{∂})_{w} = deg Ψ(w) by definition. Moreover, for the point u = 0 the equality is fulfilled, as well, since, pyr(Δ^{∂})_{0} = 0 and pyr(Ψ̂_{P_{i}})_{0} = Ψ_{P_{i}} holds. Now, the equality follows by linearity for all points on the line segment connecting w and 0, hence, for all points in □.

Proof of Theorem 3.15. Consider a Fano T-variety given by an f-divisor S with nontrivial slices S_{P_{1}}, …, S_{P_{r}}. First we choose a special representation for the canonical divisor K_{ℙ^{1}} = ∑_{j=1}^{r−2} P_{r+j} − ∑_{i=1}^{r} P_{i}. Where P_{1}, …, P_{2r−2} are distinct points on ℙ^{1}. Now, by Remark 3.6 the corresponding divisorial polytope fulfills

i. Ψ_{P} ≡ 0 for P ≠ P_{i}, i = 1, … 2r − 2,

ii. Ψ_{P} ≡ −1 for P = P_{i}, i = r + 1, … 2r − 2,

iii. The facets of the graph of Ψ_{P} have lattice distance 1 to the origin for P ≠ P_{i}, i = 1, … 2r − 2,

iv. the facets F of □ with deg Ψ|_{F} ≢ 0 have lattice distance 1 from the origin.

By Corollary A.4 we have

F (X) = vol_{1}([Ψ])(\overline{bc_{1}}∂[Ψ] − \overline{bc}[Ψ])

We set

C := [Δ^{∂}] + (2 − r)[□ × {−1}] + ∑_{i=1}^{ℓ}[Ψ̂_{P_{i}}].

Now, by Lemma A.6 we have vol_{1}([Ψ]) = vol_{1}(C) and \overline{bc_{1}}([Ψ]) = \overline{bc_{1}}(C). Moreover, by Lemma A.7 we obtain vol([Ψ]) = vol(pyr(C)) and \overline{bc}([Ψ]) = \overline{bc}(pyr(C)). Since, C is a sum over polytopes in lattice distance 1 from the origin we may apply Lemma A.8 and obtain

F (X) = vol_{1}(∂[Ψ])(\overline{bc_{1}}(∂[Ψ]) − \overline{bc}[Ψ])
= vol_{1}(C) · (\overline{bc_{1}}(C) − \overline{bc}(pyr(C)))
= vol(pyr(C)) · \overline{bc}(pyr(C))
= vol[Ψ] · \overline{bc}([Ψ])
= vol Δ · \overline{bc}(Δ).

Lemma A.8. Assume that Δ_{1}, …, Δ_{r} ⊂ M_{ℝ} × ℝ are polytopes of codimension 1 with lattice distance 1 from the origin. Then for C = ∑ a_{i} · [Δ_{i}] we have

vol_{1}(C) · (\overline{bc_{1}}(C) − \overline{bc}(pyr(C))) = vol(pyr(C)) · \overline{bc}(pyr(C)).

Proof. We set n = dim M_{ℝ} + 1. Now, by elementary geometry we have

vol_{1} Δ_{i} = n · vol(pyr(Δ_{i})), \overline{bc_{1}}(Δ_{i}) = \frac{n}{n − 1} · \overline{bc}(pyr(Δ_{i})).

It follows that vol_{1} C = n · vol pyr(C). Hence, we get

\overline{bc_{1}}(C) = ∑ a_{i} \frac{vol_{1}(Δ_{i})}{vol_{1}(C)}\overline{bc_{1}}(Δ_{i}) = ∑ a_{i} \frac{vol(pyr(Δ_{i}))}{vol(pyr(C))} · \frac{n}{n − 1} · \overline{bc}(pyr(Δ_{i}))

= \frac{n}{n − 1} · \overline{bc}(pyr(C)).

Now, plugging in \frac{n}{n−1}·\overline{bc}(pyr(C)) for \overline{bc_{1}}(C) into vol_{1}(C)·(\overline{bc_{1}}(C)−\overline{bc}(pyr(C))) gives the desired result.

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Hendrik Suess

School of Mathematics Maxwell Institute University of Edinburgh, J.C. Maxwell Building The King’s Buildings Mayfield Road Edinburgh EH9 3JZ United Kingdom