Multigraded Factorial Rings

and Fano Varieties with Torus Action

Jürgen Hausen, Elaine Herppich, Hendrik Süß

Received: November 15, 2009 Revised: December 12, 2010

Communicated by Thomas Peternell

Abstract. In a first result, we describe all finitely generated factorial algebras over an algebraically closed field of characteristic zero that come with an effective multigrading of complexity one by means of generators and relations. This enables us to construct systematically varieties with free divisor class group and a complexity one torus action via their Cox rings. For the Fano varieties of this type that have a free divisor class group of rank one, we provide explicit bounds for the number of possible deformation types depending on the dimension and the index of the Picard group in the divisor class group. As a consequence, one can produce classification lists for fixed dimension and Picard index. We carry this out expemplarily in the following cases. There are 15 non-toric surfaces with Picard index at most six. Moreover, there are 116 non-toric threefolds with Picard index at most two; nine of them are locally factorial, i.e. of Picard index one, and among these one is smooth, six have canonical singularities and two have non-canonical singularities. Finally, there are 67 nontoric locally factorial fourfolds and two one-dimensional families of non-toric locally factorial fourfolds. In all cases, we list the Cox rings explicitly.

2010 Mathematics Subject Classification: 13A02, 13F15, 14J45

Keywords and Phrases: Multigraded factorial ring, Fano variety, torus action Introduction

Let 𝕂 be an algebraically closed field of characteristic zero. A first aim of this paper is to determine all finitely generated factorial 𝕂-algebras R with an effective complexity one multigrading R = ⊕_{u∈M}R_{u} satisfying R_{0} = 𝕂; here effective complexity one multigrading means that with d := dim R we have M ≅ ℤ^{d−1} and the u ∈ M with R_{u} ≠ 0 generate M as a ℤ-module. Our result extends work by Mori [23] and Ishida [17], who settled the cases d = 2 and d = 3. An obvious class of multigraded factorial algebras as above is given by polynomial rings. A much larger class is obtained as follows. Take a sequence A = (a_{0}, …, a_{r}) of vectors a_{i} ∈ 𝕂^{2} such that (a_{i}, a_{k}) is linearly independent whenever k ≠ i, a sequence 𝔫 = (n_{0}, …, n_{r}) of positive integers and a family L = (l_{ij}) of positive integers, where 0 ≤ i ≤ r and 1 ≤ j ≤ n_{i}. For every 0 ≤ i ≤ r, we define a monomial

f_{i} := T_{i1}^{l_{i1}} ⋯ T_{in_{i}}^{l_{in_{i}}} ∈ 𝕂[T_{ij}; 0 ≤ i ≤ r, 1 ≤ j ≤ n_{i}],

for any two indices 0 ≤ i, j ≤ r, we set α_{ij} := det(a_{i}, a_{j}), and for any three indices 0 ≤ i < j < k ≤ r, we define a trinomial

g_{i,j,k} := α_{jk}f_{i} + α_{ki}f_{j} + α_{ij}f_{k} ∈ 𝕂[T_{ij}; 0 ≤ i ≤ r, 1 ≤ j ≤ n_{i}].

Note that the coefficients of g_{i,j,k} are all nonzero. The triple (A, 𝔫, L) then defines a 𝕂-algebra

R(A, 𝔫, L) := 𝕂[T_{ij}; 0 ≤ i ≤ r, 1 ≤ j ≤ n_{i}] / ⟨g_{i,i+1,i+2}; 0 ≤ i ≤ r − 2⟩.

It turns out that R(A, 𝔫, L) is a normal complete intersection, see Proposition 1.2. In particular, it is of dimension

dim R(A, 𝔫, L) = n_{0} + … + n_{r} − r + 1.

If the triple (A, 𝔫, L) is admissible, i.e., the numbers gcd(l_{i1}, …, l_{in_{i}}), where 0 ≤ i ≤ r, are pairwise coprime, then R(A, 𝔫, L) admits a canonical effective complexity one grading by a lattice K, see Construction 1.7. Our first result is the following.

Theorem 1.9. Up to isomorphy, the finitely generated factorial 𝕂-algebras with an effective complexity one grading R = ⊕_{M}R_{u} and R_{0} = 𝕂 are

(i) the polynomial algebras 𝕂[T_{1}, …, T_{d}] with a grading deg(T_{i}) = u_{i} ∈ ℤ^{d−1} such that u_{1}, …, u_{d} generate ℤ^{d−1} as a lattice and the convex cone on ℚ^{d−1} generated by u_{1}, …, u_{d} is pointed,

(ii) the (K × ℤ^{m})-graded algebras R(A, 𝔫, L)[S_{1}, …, S_{m}], where R(A, 𝔫, L) is the K-graded algebra defined by an admissible triple (A, 𝔫, L) and deg S_{j} ∈ ℤ^{m} is the j-th canonical base vector.

The further paper is devoted to normal (possibly singular) d-dimensional Fano varieties X with an effective action of an algebraic torus T. In the case dim T = d, we have the meanwhile extensively studied class of toric Fano varieties, see [3], [27] and [4] for the initiating work. Our aim is to show that the above Theorem provides an approach to classification results for the case dim T = d − 1, that means Fano varieties with a complexity one torus action. Here, we treat the case of divisor class group Cl(X) ≅ ℤ; note that in the toric setting this gives precisely the weighted projective spaces. The idea is to consider the Cox ring

R(X) = ⨁_{D∈Cl(X)} Γ(X, O_{X}(D)).

The ring R(X) is factorial, finitely generated as a 𝕂-algebra and the T -action on X gives rise to an effective complexity one multigrading of R(X) refining the Cl(X)-grading, see [5] and [15]. Consequently, R(X) is one of the rings listed in the first Theorem. Moreover, X can be easily reconstructed from R(X); it is the homogeneous spectrum with respect to the Cl(X)-grading of R(X). Thus, in order to construct Fano varieties, we firstly have to figure out the Cox rings among the rings occuring in the first Theorem and then find those, which belong to a Fano variety; this is done in Propositions 1.11 and 2.5. In order to produce classification results via this approach, we need explicit bounds on the number of deformation types of Fano varieties with prescribed discrete invariants. Besides the dimension, in our setting, a suitable invariant is the Picard index [Cl(X) : Pic(X)]. Denoting by ξ(μ) the number of primes less or equal to μ, we obtain the following bound, see Corollary 2.2: for any pair (d, μ) ∈ ℤ_{>0}^{2}, the number δ(d, μ) of different deformation types of d-dimensional Fano varieties with a complexity one torus action such that Cl(X) ≅ ℤ and μ = [Cl(X) : Pic(X)] hold is bounded by

δ(d, μ) ≤ (6dμ)^{2ξ(3dμ)+d−2}μ^{ξ(μ)^{2}+2ξ((d+2)μ)+2d+2}.

In particular, we conclude that for fixed μ ∈ ℤ_{>0}, the number δ(d) of different deformation types of d-dimensional Fano varieties with a complexity one torus action Cl(X) ≅ ℤ and Picard index μ is asymptotically bounded by d^{Ad} with a constant A depending only on μ, see Corollary 2.4. In fact, in Theorem 2.1 we even obtain explicit bounds for the discrete input data of the rings R(A, 𝔫, L)[S_{1}, …, S_{m}]. This allows us to construct all Fano varieties X with prescribed dimension and Picard index that come with an effective complexity one torus action and have divisor class group ℤ. Note that, by the approach, we get the Cox rings of the resulting Fano varieties X for free. In Section 3, we give some explicit classifications. We list all non-toric surfaces X with Picard index at most six and the non-toric threefolds X with Picard index up at most two. They all have a Cox ring defined by a single relation; in fact, for surfaces the first Cox ring with more than one relation occurs for Picard index 29, and for the threefolds this happens with Picard index 3, see Proposition 3.5 as well as Examples 3.4 and 3.7. Moreover, we determine all locally factorial fourfolds X, i.e. those of Picard index one: 67 of them occur sporadic and there are two one-dimensional families. Here comes the result on the locally factorial threefolds; in the table, we denote by w_{i} the Cl(X)-degree of the variable T_{i}.

Theorem 3.2. The following table lists the Cox rings R(X) of the threedimensional locally factorial non-toric Fano varieties X with an effective two torus action and Cl(X) = ℤ.

No. R(X)  (w_{1}, …, w_{5})  (−K_{X})^{3}
1  𝕂[T_{1}, …, T_{5}] / ⟨T_{1}T_{2}^{5} + T_{3}^{3} + T_{4}^{2}⟩  (1, 1, 2, 3, 1)  8
2  𝕂[T_{1}, …, T_{5}] / ⟨T_{1}T_{2}T_{3}^{4} + T_{4}^{3} + T_{5}^{2}⟩ (1, 1, 1, 2, 3)  8
3  𝕂[T_{1}, …, T_{5}] / ⟨T_{1}T_{2}^{2}T_{3}^{3} + T_{4}^{3} + T_{5}^{2}⟩ (1, 1, 1, 2, 3)  8
4  𝕂[T_{1}, …, T_{5}] / ⟨T_{1}T_{2} + T_{3}T_{4} + T_{5}^{2}⟩  (1, 1, 1, 1, 1)  54
5  𝕂[T_{1}, …, T_{5}] / ⟨T_{1}T_{2}^{2} + T_{3}T_{4}^{2} + T_{5}^{3}⟩ (1, 1, 1, 1, 1)  24
6  𝕂[T_{1}, …, T_{5}] / ⟨T_{1}T_{2}^{3} + T_{3}T_{4}^{3} + T_{5}^{4}⟩ (1, 1, 1, 1, 1)  4
7  𝕂[T_{1}, …, T_{5}] / ⟨T_{1}T_{2}^{3} + T_{3}T_{4}^{3} + T_{5}^{2}⟩ (1, 1, 1, 1, 2)  16
8  𝕂[T_{1}, …, T_{5}] / ⟨T_{1}T_{2}^{5} + T_{3}T_{4}^{5} + T_{5}^{2}⟩ (1, 1, 1, 1, 3)  2
9  𝕂[T_{1}, …, T_{5}] / ⟨T_{1}T_{2}^{5} + T_{3}^{3}T_{4}^{3} + T_{5}^{2}⟩ (1, 1, 1, 1, 3)  2

Note that each of these varieties X is a hypersurface in the respective weighted projective space ℙ(w_{1}, …, w_{5}). Except number 4, none of them is quasismooth in the sense that Spec R(X) is singular at most in the origin; quasismooth hypersurfaces of weighted projective spaces were studied in [21] and [7]. In Section 4, we take a closer look at the singularities of the threefolds listed above. It turns out that number 1,3,5,7 and 9 are singular with only canonical singularities and all of them admit a crepant resolution. Number 6 and 8 are singular with non-canonical singularities but admit a smooth relative minimal model. Number two is singular with only canonical singularities, one of them of type cA_{1}, and it admits only a singular relative minimal model. Moreover, in all cases, we determine the Cox rings of the resolutions.

The authors would like to thank Ivan Arzhantsev for helpful comments and discussions and also the referee for valuable remarks and many references.

1 UFDs with complexity one multigrading

As mentioned before, we work over an algebraically closed field 𝕂 of characteristic zero. In Theorem 1.9, we describe all factorial finitely generated 𝕂algebras R with an effective complexity one grading and R_{0} = 𝕂. Moreover, we characterize the possible Cox rings among these algebras, see Proposition 1.11. First we recall the construction sketched in the introduction.

Construction 1.1. Consider a sequence A = (a_{0}, …, a_{r}) of vectors a_{i} = (b_{i}, c_{i}) in 𝕂^{2} such that any pair (a_{i}, a_{k}) with k ≠ i is linearly independent, a sequence 𝔫 = (n_{0}, …, n_{r}) of positive integers and a family L = (l_{ij}) of positive integers, where 0 ≤ i ≤ r and 1 ≤ j ≤ n_{i}. For every 0 ≤ i ≤ r, define a monomial

f_{i} := T_{i1}^{l_{i1}} ⋯ T_{in_{i}}^{l_{in_{i}}} ∈ 𝕂[T_{ij}; 0 ≤ i ≤ r, 1 ≤ j ≤ n_{i}],

for any two indices 0 ≤ i, j ≤ r, set α_{ij} := det(a_{i}, a_{j}) = b_{i}c_{j} − b_{j}c_{i} and for any three indices 0 ≤ i < j < k ≤ r define a trinomial

g_{i,j,k} := α_{jk}f_{i} + α_{ki}f_{j} + α_{ij}f_{k} ∈ 𝕂[T_{ij}; 0 ≤ i ≤ r, 1 ≤ j ≤ n_{i}].

Note that the coefficients of this trinomial are all nonzero. The triple (A, 𝔫, L) then defines a ring

R(A, 𝔫, L) := 𝕂[T_{ij}; 0 ≤ i ≤ r, 1 ≤ j ≤ n_{i}] / ⟨g_{i,i+1,i+2}; 0 ≤ i ≤ r − 2⟩.

Proposition 1.2. For every triple (A, 𝔫, L) as in 1.1, the ring R(A, 𝔫, L) is a normal complete intersection of dimension

dim R(A, 𝔫, L) = n − r + 1, n := n_{0} + … + n_{r}.

Lemma 1.3. In the setting of 1.1, one has for any 0 ≤ i < j < k < l ≤ r the identities

g_{i,k,l} = α_{kl} · g_{i,j,k} + α_{ik} · g_{j,k,l}, g_{i,j,l} = α_{jl} · g_{i,j,k} + α_{ij} · g_{j,k,l}.

In particular, every trinomial g_{i,j,k}, where 0 ≤ i < j < k ≤ r is contained in the ideal ⟨g_{i,i+1,i+2}; 0 ≤ i ≤ r − 2⟩.

Proof. The identities are easily obtained by direct computation; note that for this one may assume a_{j} = (1, 0) and a_{k} = (0, 1). The supplement then follows by repeated application of the identities.

Lemma 1.4. In the notation of 1.1 and 1.2, set X := V (𝕂^{n}, g_{0}, …, g_{r−2}), and let z ∈ X. If we have f_{i}(z) = f_{j}(z) = 0 for two 0 ≤ i < j ≤ r, then f_{k}(z) = 0 holds for all 0 ≤ k ≤ r.

Proof. If i < k < j holds, then, according to Lemma 1.3, we have g_{i,k,j}(z) = 0, which implies f_{k}(z) = 0. The cases k < i and j < k are obtained similarly.

Proof of Proposition 1.2. Set X := V (𝕂^{n}; g_{0}, …, g_{r−2}), where g_{i} := g_{i,i+1,i+2}. Then we have to show that X is a connected complete intersection with at most normal singularities. In order to see that X is connected, set ℓ := ∏ n_{i} ∏ l_{ij} and ζ_{ij} := ℓn_{i}^{−1}l_{ij}^{−1}. Then X ⊆ 𝕂^{n} is invariant under the 𝕂^{∗}-action given by

t · z := (t^{ζ_{ij}}z_{ij})

and the point 0 ∈ 𝕂^{n} lies in the closure of any orbit 𝕂^{∗}·x ⊆ X, which implies connectedness. To proceed, consider the Jacobian J_{g} of g := (g_{0}, …, g_{r−2}). According to Serre’s criterion, we have to show that the set of points of z ∈ X with J_{g}(z) not of full rank is of codimension at least two in X. Note that the Jacobian J_{g} is of the shape

J_{g} = ( δ_{0 0} δ_{0 1} δ_{0 2}  0  0 ; 0 δ_{1 1} δ_{1 2} δ_{1 3} 0 ; ⋮ ; 0 δ_{r−3 r−3} δ_{r−3 r−2} δ_{r−3 r−1}  0 ; 0  0 δ_{r−2 r−2} δ_{r−2 r−1} δ_{r−2 r} )

where δ_{ti} is a nonzero multiple of the gradient δ_{i} := grad f_{i}. Consider z ∈ X with J_{g}(z) not of full rank. Then δ_{i}(z) = 0 = δ_{k}(z) holds with some 0 ≤ i < k ≤ r. This implies z_{ij} = 0 = z_{kl} for some 1 ≤ j ≤ n_{i} and 1 ≤ l ≤ n_{k}. Thus, we have f_{i}(z) = 0 = f_{k}(z). Lemma 1.4 gives f_{s}(z) = 0, for all 0 ≤ s ≤ r. Thus, some coordinate z_{st} must vanish for every 0 ≤ s ≤ r. This shows that z belongs to a closed subset of X having codimension at least two in X.

Lemma 1.5. Notation as in 1.1. Then the variable T_{ij} defines a prime ideal in R(A, 𝔫, L) if and only if the numbers gcd(l_{k1}, …, l_{kn_{k}}), where k ≠ i, are pairwise coprime.

Proof. We treat exemplarily T_{01}. Using Lemma 1.3, we see that the ideal of relations of R(A, 𝔫, L) can be presented as follows

⟨g_{s,s+1,s+2}; 0 ≤ s ≤ r − 2⟩ = ⟨g_{0,s,s+1}; 1 ≤ s ≤ r − 1⟩.

Thus, the ideal ⟨T_{01}⟩ ⊆ R(A, 𝔫, L) is prime if and only if the following binomial ideal is prime

𝔞 := ⟨α_{s+1 0}f_{s} + α_{0s}f_{s+1}; 1 ≤ s ≤ r − 1⟩ ⊆ 𝕂[T_{ij}; (i, j) ≠ (0, 1)].

Set l_{i} := (l_{i1}, …, l_{in_{i}}). Then the ideal 𝔞 is prime if and only if the following family can be complemented to a lattice basis

(l_{1}, −l_{2}, 0, …, 0), …, (0, …, 0, l_{r−1}, −l_{r}).

This in turn is equivalent to the statement that the numbers gcd(l_{k1}, …, l_{kn_{k}}), where 1 ≤ k ≤ r, are pairwise coprime.

Definition 1.6. We say that a triple (A, 𝔫, L) as in 1.1 is admissible if the numbers gcd(l_{i1}, …, l_{in_{i}}), where 0 ≤ i ≤ r, are pairwise coprime.

Construction 1.7. Let (A, 𝔫, L) be an admissible triple and consider the following free abelian groups

E := ⨁_{i=0}^{r} ⨁_{j=1}^{n_{i}} ℤ·e_{ij}, K := ⨁_{j=1}^{n_{0}} ℤ·u_{0j} ⊕ ⨁_{i=1}^{r} ⨁_{j=1}^{n_{i}−1} ℤ·u_{ij}

and define vectors u_{in_{i}} := u_{01} + … + u_{0r} − u_{i1} − … − u_{in_{i}−1} ∈ K. Then there is an epimorphism λ: E → K fitting into a commutative diagram with exact rows

[figure: 0; E; α; e_{ij}↦l_{ij}e_{ij}; η; e_{ij}↦u_{ij}; E; e_{ij}↦\overline{e}_{ij}; λ; ⨁_{i,j} ℤ/l_{ij}ℤ; ≅; 0; 0; K; β; K; ⨁_{i,j} ℤ/l_{ij}ℤ; 0]

Define a K-grading of 𝕂[T_{ij}; 0 ≤ i ≤ r, 1 ≤ j ≤ n_{i}] by setting deg T_{ij} := λ(e_{ij}). Then every f_{i} = T_{i1}^{l_{i1}} ⋯ T_{in_{i}}^{l_{in_{i}}} is K-homogeneous of degree

deg f_{i} = l_{i1}λ(e_{i1}) + … + l_{in_{i}}λ(e_{in_{i}}) = l_{01}λ(e_{01}) + … + l_{0n_{0}}λ(e_{0n_{0}}) ∈ K.

Thus, the polynomials g_{i,j,k} of 1.1 are all K-homogeneous of the same degree and we obtain an effective K-grading of complexity one of R(A, 𝔫, L).

Proof. Only for the existence of the commutative diagram there is something to show. Write for short l_{i} := (l_{i1}, …, l_{in_{i}}). By the admissibility condition, the vectors v_{i} := (0, …, 0, l_{i}, −l_{i+1}, 0, …, 0), where 0 ≤ i ≤ r−1, can be completed to a lattice basis for E. Consequently, we find an epimorphism λ: E → K having precisely lin(v_{0}, …, v_{r−1}) as its kernel. By construction, ker(λ) equals α(ker(η)). Using this, we obtain the induced morphism β: K → K and the desired properties.

Lemma 1.8. Notation as in 1.7. Then R(A, 𝔫, L)_{0} = 𝕂 and R(A, 𝔫, L)^{∗} = 𝕂^{∗} hold. Moreover, the T_{ij} define pairwise nonassociated prime elements in R(A, 𝔫, L).

Proof. The fact that all elements of degree zero are constant is due to the fact that all degrees deg T_{ij} = u_{ij} ∈ K are non-zero and generate a pointed convex cone in K_{ℚ}. As a consequence, we obtain that all units in R(A, 𝔫, L) are constant. The T_{ij} are prime by the admissibility condition and Lemma 1.5, and they are pairwise nonassociated because they have pairwise different degrees and all units are constant.

Theorem 1.9. Up to isomorphy, the finitely generated factorial 𝕂-algebras with an effective complexity one grading R = ⊕_{M}R_{u} and R_{0} = 𝕂 are

(i) the polynomial algebras 𝕂[T_{1}, …, T_{d}] with a grading deg(T_{i}) = u_{i} ∈ ℤ^{d−1} such that u_{1}, …, u_{d} generate ℤ^{d−1} as a lattice and the convex cone on ℚ^{d−1} generated by u_{1}, …, u_{d} is pointed,

(ii) the (K × ℤ^{m})-graded algebras R(A, 𝔫, L)[S_{1}, …, S_{m}], where R(A, 𝔫, L) is the K-graded algebra defined by an admissible triple (A, 𝔫, L) as in 1.1 and 1.7 and deg S_{j} ∈ ℤ^{m} is the j-th canonical base vector.

Proof. We first show that for any admissible triple (A, 𝔫, L) the ring R(A, 𝔫, L) is a unique factorization domain. If l_{ij} = 1 holds for any two i, j, then, by [15, Prop. 2.4], the ring R(A, 𝔫, L) is the Cox ring of a space ℙ_{1}(A, 𝔫) and hence is a unique factorization domain.

Now, let (A, 𝔫, L) be arbitrary admissible data and let λ: E → K be an epimorphism as in 1.7. Set n := n_{0} + … + n_{r} and consider the diagonalizable groups

𝕋^{n} := Spec 𝕂[E],  H := Spec 𝕂[K],  H_{0} := Spec 𝕂[⊕_{i,j}ℤ/l_{ij}ℤ].

Then 𝕋^{n} = (𝕂^{∗})^{n} is the standard n-torus and H_{0} is the direct product of the cyclic subgroups H_{ij} := Spec 𝕂[ℤ/l_{ij}ℤ]. Moreover, the diagram in 1.7 gives rise to a commutative diagram with exact rows

[figure: 0; 𝕋^{n}; 𝕋^{n}; (t_{ij}^{l_{ij}})↤(t_{ij}); H_{0}; 0; 0; H; ı; H; ȷ; H_{0}; ≅; 0]

where t_{ij} = χ^{e_{ij}} are the coordinates of 𝕋^{n} corresponding to the characters e_{ij} ∈ E and the maps ı, ȷ are the closed embeddings corresponding to the epimorphisms η, λ respectively.

Setting deg T_{ij} := e_{ij} defines an action of 𝕋^{n} on 𝕂^{n} = Spec 𝕂[T_{ij}]; in terms of the coordinates z_{ij} corresponding to T_{ij} this action is given by t·z = (t_{ij}z_{ij}). The torus H acts effectively on 𝕂^{n} via the embedding ȷ: H → 𝕋^{n}. The generic isotropy group of H along V (𝕂^{n}, T_{ij}) is the subgroup H_{ij} ⊆ H corresponding to K → K/λ(E_{ij}), where E_{ij} ⊆ E denotes the sublattice generated by all e_{kl} with (k, l) ≠ (i, j); recall that we have K/λ(E_{ij}) ≅ ℤ/l_{ij}ℤ.

Now, set l′_{ij} := 1 for any two i, j and consider the spectra X := Spec R(A, 𝔫, L) and X′ := Spec R(A, 𝔫, L′). Then the canonical surjections 𝕂[T_{ij}] → R(A, 𝔫, L) and 𝕂[T_{ij}] → R(A, 𝔫, L′) define embeddings X → 𝕂^{n} and X′ → 𝕂^{n}. These embeddings fit into the following commutative diagram

[figure: 𝕂^{n}; π; (z_{ij}^{l_{ij}})↤(z_{ij}); 𝕂^{n}; X′; X]

The action of H leaves X invariant and the induced H-action on X is the one given by the K-grading of R(A, 𝔫, L). Moreover, π: 𝕂^{n} → 𝕂^{n} is the quotient map for the induced action of H_{0} ⊆ H on 𝕂^{n}, we have X = π^{−1}(X′), and hence the restriction π: X → X′ is a quotient map for the induced action of H_{0} on X. Removing all subsets V (X; T_{ij}, T_{kl}), where (i, j) ≠ (k, l) from X, we obtain an open subset U ⊆ X. By Lemma 1.8, the complement X \U is of codimension at least two and each V (U, T_{ij}) is irreducible. By construction, the only isotropy groups of the H-action on U are the groups H_{ij} of the points of V (U, T_{ij}). The image U′ := π(U) is open in X′, the complement X′ \ U′ is as well of codimension at least two and H/H_{0} acts freely on U′. According to [22, Cor. 5.3], we have two exact sequences fitting into the following diagram

\xymatrix{ &  & 1 \ar[d] \\  &  & Pic(U′) \ar[d]^{π^{∗}} \\ 1 \ar[r] & 𝕏(H_{0}) \ar[r]^{α} & Pic_{H_{0}}(U) \ar[r]^{β} \ar[d]^{δ} & Pic(U) \\  &  & _{∏i,j}𝕏(H_{ij})}

Since X′ is factorial, the Picard group Pic(U′) is trivial and we obtain that δ is injective. Since H_{0} is the direct product of the isotropy groups H_{ij} of the Luna strata V (U, T_{ij}), we see that δ ∘ α is an isomorphism. It follows that δ is surjective and hence an isomorphism. This in turn shows that α is an isomorphism. Now, every bundle on U is H-linearizable. Since H_{0} acts as a subgroup of H, we obtain that every bundle is H_{0}-linearizable. It follows that β is surjective and hence Pic(U) is trivial. We conclude Cl(X) = Pic(U) = 0, which means that R(A, 𝔫, L) admits unique factorization. The second thing we have to show is that any finitely generated factorial 𝕂algebra R with an effective complexity one multigrading satisfying R_{0} = 𝕂 is as claimed. Consider the action of the torus G on X = Spec R defined by the multigrading, and let X_{0} ⊆ X be the set of points having finite isotropy G_{x}.

Then [15, Prop 3.3] provides a graded splitting

R ≅ R′[S_{1}, …, S_{m}],

where the variables S_{j} are identified with the homogeneous functions defining the prime divisors E_{j} inside the boundary X \X_{0} and R′ is the ring of functions of X_{0}, which are invariant under the subtorus G_{0} ⊆ G generated by the generic isotropy groups G_{j} of E_{j}. Since R′_{0} = R_{0} = 𝕂 holds, the orbit space X_{0}/G has only constant functions and thus is a space ℙ_{1}(A, 𝔫) as constructed in [15, Section 2]. This allows us to proceed exactly as in the proof of Theorem [15, Thm 1.3] and gives R′ = R(A, 𝔫, L). The admissibility condition follows from Lemma 1.5 and the fact that each T_{ij} defines a prime element in R′.

Remark 1.10. Let (A, 𝔫, L) be an admissible triple with 𝔫 = (1, …, 1). Then K = ℤ holds, the admissibility condition just means that the numbers l_{ij} are pairwise coprime and we have

dim R(A, 𝔫, L) = n_{0} + … + n_{r} − r + 1 = 2.

Consequently, for two-dimensional rings, Theorem 1.9 specializes to Mori’s description of almost geometrically graded two-dimensional unique factorization domains provided in [23].

Proposition 1.11. Let (A, 𝔫, L) be an admissible triple, consider the associated (K × ℤ^{m})-graded ring R(A, 𝔫, L)[S_{1}, …, S_{m}] as in Theorem 1.9 and let μ: K × ℤ^{m} → K′ be a surjection onto an abelian group K′. Then the following statements are equivalent.

(i) The K′-graded ring R(A, 𝔫, L)[S_{1}, …, S_{m}] is the Cox ring of a projective variety X′ with Cl(X′) ≅ K′.

(ii) For every pair i, j with 0 ≤ i ≤ r and 1 ≤ j ≤ n_{i}, the group K′ is generated by the elements μ(λ(e_{kl})) and μ(e_{s}), where (i, j) ≠ (k, l) and 1 ≤ s ≤ m, for every 1 ≤ t ≤ m, the group K′ is generated by the elements μ(λ(e_{ij})) and μ(e_{s}), where 0 ≤ i ≤ r, 1 ≤ j ≤ n_{i} and s ≠ t, and, finally the following cone is of full dimension in K′_{ℚ}:

⋂_{(k,l)} cone(μ(λ(e_{ij})), μ(e_{s}); (i, j) ≠ (k, l)) ∩ ⋂_{t} cone(μ(λ(e_{ij})), μ(e_{s}); s ≠ t).

Proof. Suppose that (i) holds, let p: X̂′ → X′ denote the universal torsor and let X′′ ⊆ X′ be the set of smooth points. According to [14, Prop. 2.2], the group H′ = Spec 𝕂[K′] acts freely on p^{−1}(X′′), which is a big open subset of the total coordinate space Spec R(A, 𝔫, L)[S_{1}, …, S_{m}]. This implies the first condition of (ii). Moreover, by [14, Prop. 4.1], the displayed cone is the moving cone of X′ and hence of full dimension. Conversely, if (ii) holds, then the K′- graded ring R(A, 𝔫, L)[S_{1}, …, S_{m}] can be made into a bunched ring and hence is the Cox ring of a projective variety, use [14, Thm. 3.6].

2 Bounds for Fano varieties

We consider d-dimensional Fano varieties X that come with a complexity one torus action and have divisor class group Cl(X) ≅ ℤ. Then the Cox ring R(X) of X is factorial [5, Prop. 8.4] and has an effective complexity one grading, which refines the Cl(X)-grading, see [15, Prop. 2.6]. Thus, according to Theorem 1.9, it is of the form

R(X) ≅ 𝕂[T_{ij}; 0 ≤ i ≤ r, 1 ≤ j ≤ n_{i}][S_{1}, …, S_{m}] / ⟨g_{i,i+1,i+2}; 0 ≤ i ≤ r − 2⟩,
g_{i,j,k} := α_{jk}T_{i1}^{l_{i1}} ⋯ T_{in_{i}}^{l_{in_{i}}} + α_{ki}T_{j1}^{l_{j1}} ⋯ T_{jn_{j}}^{l_{jn_{j}}} + α_{ij}T_{k1}^{l_{k1}} ⋯ T_{kn_{k}}^{l_{kn_{k}}} .

Here, we may (and will) assume n_{0} ≥ … ≥ n_{r} ≥ 1. With n := n_{0} + … + n_{r}, we have n + m = d + r. For the degrees of the variables in Cl(X) ≅ ℤ, we write w_{ij} := deg T_{ij} for 0 ≤ i ≤ r, 1 ≤ j ≤ n_{i} and u_{k} = deg S_{k} for 1 ≤ k ≤ m. Moreover, for μ ∈ ℤ_{>0}, we denote by ξ(μ) the number of primes in {2, …, μ}. The following result provides bounds for the discrete data of the Cox ring.

Theorem 2.1. In the above situation, fix the dimension d = dim(X) and the Picard index μ = [Cl(X) : Pic(X)]. Then we have

u_{k} ≤ μ for 1 ≤ k ≤ m.

Moreover, for the degree γ of the relations, the weights w_{ij} and the exponents l_{ij}, where 0 ≤ i ≤ r and 1 ≤ j ≤ n_{i} one obtains the following.

(i) Suppose that r = 0, 1 holds. Then n + m ≤ d + 1 holds and one has the

bounds

w_{ij} ≤ μ for 0 ≤ i ≤ r and 1 ≤ j ≤ n_{i},

and the Picard index is given by

μ = lcm(w_{ij}, u_{k}; 0 ≤ i ≤ r, 1 ≤ j ≤ n_{i}, 1 ≤ k ≤ m).

(ii) Suppose that r ≥ 2 and n_{0} = 1 hold. Then r ≤ ξ(μ) − 1 and n = r + 1 and m = d − 1 hold and one has

w_{i1} ≤ μ^{r} for 0 ≤ i ≤ r,  l_{01} ⋯ l_{r1} | μ,  l_{01} ⋯ l_{r1} | γ ≤ μ^{r+1},

and the Picard index is given by

μ = lcm(gcd(w_{j1}; j ≠ i), u_{k}; 0 ≤ i ≤ r, 1 ≤ k ≤ m).

(iii) Suppose that r ≥ 2 and n_{0} > n_{1} = 1 hold. Then we may assume l_{11} > … > l_{r1} ≥ 2, we have r ≤ ξ(3dμ) − 1 and n_{0} + m = d and the bounds

w_{01}, …, w_{0n_{0}} ≤ μ, l_{01}, …, l_{0n_{0}} < 6dμ,

w_{11}, l_{21} < 2dμ, w_{21}, l_{11} < 3dμ,

w_{i1} < 6dμ, l_{i1} < 2dμ for 2 ≤ i ≤ r,
l_{11} ⋯ l_{r1} | γ < 6dμ,

and the Picard index is given by

μ = lcm(w_{0j}, gcd(w_{11}, …, w_{r1}), u_{k}; 1 ≤ j ≤ n_{0}, 1 ≤ k ≤ m).

(iv) Suppose that n_{1} > n_{2} = 1 holds. Then we may assume l_{21} > … > l_{r1} ≥ 2, we have r ≤ ξ(2(d + 1)μ) − 1 and n_{0} + n_{1} + m = d + 1 and the bounds

w_{ij} ≤ μ for i = 0, 1 and 1 ≤ j ≤ n_{i}, w_{21} < (d + 1)μ,
w_{ij}, l_{ij} < 2(d + 1)μ for 0 ≤ i ≤ r and 1 ≤ j ≤ n_{i},
l_{21} ⋯ l_{r1} | γ < 2(d + 1)μ,

and the Picard index is given by

μ = lcm(w_{ij}, u_{k}; 0 ≤ i ≤ 1, 1 ≤ j ≤ n_{i}, 1 ≤ k ≤ m).

(v) Suppose that n_{2} > 1 holds and let s be the maximal number with n_{s} > 1. Then one may assume l_{s+1,1} > … > l_{r1} ≥ 2, we have r ≤ ξ((d + 2)μ) − 1 and n_{0} + … + n_{s} + m = d + s and the bounds

w_{ij} ≤ μ, for 0 ≤ i ≤ s,
w_{ij}, l_{ij} < (d + 2)μ for 0 ≤ i ≤ r and 1 ≤ j ≤ n_{i},
l_{s+1,1} ⋯ l_{r1} | γ < (d + 2)μ,

and the Picard index is given by

μ = lcm(w_{ij}, u_{k}; 0 ≤ i ≤ s, 1 ≤ j ≤ n_{i}, 1 ≤ k ≤ m).

Putting all the bounds of the theorem together, we obtain the following (raw) bound for the number of deformation types.

Corollary 2.2. For any pair (d, μ) ∈ ℤ_{>0}^{2}, the number δ(d, μ) of different deformation types of d-dimensional Fano varieties with a complexity one torus action such that Cl(X) ≅ ℤ and [Cl(X) : Pic(X)] = μ hold is bounded by

δ(d, μ) ≤ (6dμ)^{2ξ(3dμ)+d−2}μ^{ξ(μ)^{2}+2ξ((d+2)μ)+2d+2}.

Proof. By Theorem 2.1 the discrete data r, 𝔫, L and m occuring in R(X) are bounded as in the assertion. The continuous data in R(X) are the coefficients α_{ij}; they stem from the family A = (a_{0}, …, a_{r}) of points a_{i} ∈ 𝕂^{2}. Varying the a_{i} provides flat families of Cox rings and hence, by passing to the homogeneous spectra, flat families of the resulting Fano varieties X.

Corollary 2.3. Fix d ∈ ℤ_{>0}. Then the number δ(μ) of different deformation types of d-dimensional Fano varieties with a complexity one torus action, Cl(X) ≅ ℤ and Picard index μ := [Cl(X) : Pic(X)] is asymptotically bounded by μ^{Aμ^{2}/ log^{2} μ} with a constant A depending only on d.

Corollary 2.4. Fix μ ∈ ℤ_{>0}. Then the number δ(d) of different deformation types of d-dimensional Fano varieties with a complexity one torus action, Cl(X) ≅ ℤ and Picard index μ := [Cl(X) : Pic(X)] is asymptotically bounded by d^{Ad} with a constant A depending only on μ.

We first recall the necessary facts on Cox rings, for details, we refer to [14]. Let X be a complete d-dimensional variety with divisor class group Cl(X) ≅ ℤ. Then the Cox ring R(X) is finitely generated and the total coordinate space \overline{X} := Spec R(X) is a factorial affine variety coming with an action of 𝕂^{∗} defined by the Cl(X)-grading of R(X). Choose a system f_{1}, …, f_{ν} of homogeneous pairwise nonassociated prime generators for R(X). This provides an 𝕂^{∗}-equivariant embedding

\overline{X} → 𝕂^{ν}, \overline{x} ↦ (f_{1}(\overline{x}), …, f_{ν}(\overline{x})).

where 𝕂^{∗} acts diagonally with the weights w_{i} = deg(f_{i}) ∈ Cl(X) ≅ ℤ on 𝕂^{ν}. Moreover, X is the geometric 𝕂^{∗}-quotient of X̂ := \overline{X} \ \{0\}, and the quotient map p: X̂ → X is a universal torsor. By the local divisor class group Cl(X, x) of a point x ∈ X, we mean the group of Weil divisors WDiv(X) modulo those that are principal near x.

Proposition 2.5. For any \overline{x} = (\overline{x}_{1}, …, \overline{x}_{ν}) ∈ X̂ the local divisor class group Cl(X, x) of x := p(\overline{x}) is finite of order gcd(w_{i}; \overline{x}_{i} ≠ 0). The index of the Picard group Pic(X) in Cl(X) is given by

[Cl(X) : Pic(X)] = lcm_{x∈X}(| Cl(X, x)|).

Suppose that the ideal of \overline{X} ⊆ 𝕂^{ν} is generated by Cl(X)-homogeneous polynomials g_{1}, …, g_{ν−d−1} of degree γ_{j} := deg(g_{j}). Then one obtains

−K_{X} = ∑_{i=1}^{ν} w_{i} − ∑_{j=1}^{ν−d−1} γ_{j}, (−K_{X})^{d} = (∑_{i=1}^{ν} w_{i} − ∑_{j=1}^{ν−d−1} γ_{j})^{d} \frac{γ_{1} ⋯ γ_{ν−d−1}}{w_{1} ⋯ w_{ν}}

for the anticanonical class −K_{X} ∈ Cl(X) ≅ ℤ. In particular, X is a Fano variety if and only if the following inequality holds

∑_{j=1}^{ν−d−1} γ_{j} < ∑_{i=1}^{ν} w_{i}.

Proof. Using [14, Prop. 2.2, Thm. 4.19], we observe that X arises from the bunched ring (R, 𝔉, Φ), where R = R(X), 𝔉 = (f_{1}, …, f_{ν}) and Φ = {ℚ_{≥0}}.

The descriptions of local class groups, the Picard index and the anticanonical class are then special cases of [14, Prop. 4.7, Cor. 4.9 and Cor. 4.16]. The anticanonical self-intersection number is easily computed in the ambient weighted projective space ℙ(w_{1}, …, w_{ν}), use [14, Constr. 3.13, Cor. 4.13].

Remark 2.6. If the ideal of \overline{X} ⊆ 𝕂^{ν} is generated by Cl(X)-homogeneous polynomials g_{1}, …, g_{ν−d−1}, then [14, Constr. 3.13, Cor. 4.13] show that X is a well formed complete intersection in the weighted projective space ℙ(w_{1}, …, w_{ν}) in the sense of [16, Def. 6.9].

We turn back to the case that X comes with a complexity one torus action as at the beginning of this section. We consider the case n_{0} = … = n_{r} = 1, that means that each relation g_{i,j,k} of the Cox ring R(X) depends only on three variables. Then we may write T_{i} instead of T_{i1} and w_{i} instead of w_{i1}, etc.. In this setting, we obtain the following bounds for the numbers of possible varieties X (Fano or not).

Proposition 2.7. For any pair (d, μ) ∈ ℤ_{>0}^{2} there is, up to deformation, only a finite number of complete d-dimensional varieties with divisor class group ℤ, Picard index [Cl(X) : Pic(X)] = μ and Cox ring

𝕂[T_{0}, …, T_{r}, S_{1}, …, S_{m}] / ⟨α_{i+1,i+2}T_{i}^{l_{i}} + α_{i+2,i}T_{i+1}^{l_{i+1}} + α_{i,i+1}T_{i+2}^{l_{i+2}}; 0 ≤ i ≤ r − 2⟩.

In this situation we have r ≤ ξ(μ)−1. Moreover, for the weights w_{i} := deg T_{i}, where 0 ≤ i ≤ r and u_{k} := deg S_{k}, where 1 ≤ k ≤ m, the exponents l_{i} and the degree γ := l_{0}w_{0} of the relation one has

l_{0} ⋯ l_{r} | γ,  l_{0} ⋯ l_{r} | μ,  w_{i} ≤ μ^{ξ(μ)−1},  u_{k} ≤ μ.

Proof. Consider the total coordinate space \overline{X} ⊆ 𝕂^{r+1+n} and the universal torsor p: X̂ → X as discussed before. For each 0 ≤ i ≤ r fix a point \overline{x}(i) = (\overline{x}_{0}, …, \overline{x}_{r}, 0, …, 0) in X̂ such that \overline{x}_{i} = 0 and \overline{x}_{j} ≠ 0 for j ≠ i hold. Then, denoting x(i) := p(\overline{x}(i)), we obtain

gcd(w_{j}; j ≠ i) = | Cl(X, x(i))| | μ.

Consider i, j with j ≠ i. Since all relations are homogeneous of the same degree, we have l_{i}w_{i} = l_{j}w_{j}. Moreover, by the admissibility condition, l_{i} and l_{j} are coprime. We conclude l_{i}|w_{j} for all j ≠ i and hence l_{i}| gcd(w_{j}; j ≠ i). This implies

l_{0} ⋯ l_{r} | l_{0}w_{0} = γ, l_{0} ⋯ l_{r} | μ.

We turn to the bounds for the w_{i}, and first verify w_{0} ≤ μ^{r}. Using the relation l_{i}w_{i} = l_{0}w_{0}, we obtain for every l_{i} a presentation

l_{i} = l_{0} · \frac{w_{0} ⋯ w_{i−1}}{w_{1} ⋯ w_{i}} = η_{i} · \frac{gcd(w_{0}, …, w_{i−1})}{gcd(w_{0}, …, w_{i})}

Documenta Mathematica 16 (2011) 71–109

with suitable integers 1 ≤ η_{i} ≤ μ. In particular, the very last fraction is bounded by μ. This gives the desired estimate:

w_{0} = \frac{w_{0}}{gcd(w_{0}, w_{1})} · \frac{gcd(w_{0}, w_{1})}{gcd(w_{0}, w_{1}, w_{2})} ⋯ \frac{gcd(w_{0}, …, w_{r−2})}{gcd(w_{0}, …, w_{r−1})} · gcd(w_{0}, …, w_{r−1}) ≤ μ^{r}.

Similarly, we obtain w_{i} ≤ μ^{r} for 1 ≤ i ≤ r. Then we only have to show that r + 1 is bounded by ξ(μ), but this follows immediately from the fact that l_{0}, …, l_{r} are pairwise coprime. Finally, to estimate the u_{k}, consider the points \overline{x}(k) ∈ X̂ having the (r + k)-th coordinate one and all others zero. Set x(k) := p(\overline{x}(k)). Then Cl(X, x(k)) is of order u_{k}, which implies u_{k} ≤ μ.

Lemma 2.8. Consider the ring 𝕂[T_{ij}; 0 ≤ i ≤ 2, 1 ≤ j ≤ n_{i}][S_{1}, …, S_{k}]/⟨g⟩ where n_{0} ≥ n_{1} ≥ n_{2} ≥ 1 holds. Suppose that g is homogeneous with respect to a ℤ-grading of 𝕂[T_{ij}, S_{k}] given by deg T_{ij} = w_{ij} ∈ ℤ_{>0} and deg S_{k} = u_{k} ∈ ℤ_{>0}, and assume

deg g < ∑_{i=0}^{2} ∑_{j=1}^{n_{i}} w_{ij} + ∑_{i=1}^{m} u_{i}.

Let μ ∈ ℤ_{>1}, assume w_{ij} ≤ μ whenever n_{i} > 1, 1 ≤ j ≤ n_{i} and u_{k} ≤ μ for 1 ≤ k ≤ m and set d := n_{0} + n_{1} + n_{2} + m − 2. Depending on the shape of g, one obtains the following bounds.

(i) Suppose that g = η_{0}T_{01}^{l_{01}} ⋯ T_{0n_{0}}^{l_{0n_{0}}} + η_{1}T_{11}^{l_{11}} + η_{2}T_{21}^{l_{21}} with n_{0} > 1 and coefficients η_{i} ∈ 𝕂^{∗} holds, we have l_{11} ≥ l_{21} ≥ 2 and l_{11}, l_{21} are coprime. Then, one has

w_{11}, l_{21} < 2dμ,  w_{21}, l_{11} < 3dμ,  deg g < 6dμ.

(ii) Suppose that g = η_{0}T_{01}^{l_{01}} ⋯ T_{0n_{0}}^{l_{0n_{0}}} + η_{1}T_{11}^{l_{11}} ⋯ T_{1n_{1}}^{l_{1n_{1}}} + η_{2}T_{21}^{l_{21}} with n_{1} > 1 and coefficients η_{i} ∈ 𝕂^{∗} holds and we have l_{21} ≥ 2. Then one has

w_{21} < (d + 1)μ, deg g < 2(d + 1)μ.

Proof. We prove (i). Set for short c := (n_{0} + m)μ = dμ. Then, using homogeneity of g and the assumed inequality, we obtain

l_{11}w_{11} = l_{21}w_{21} = deg g < ∑_{i=0}^{2} ∑_{j=1}^{n_{i}} w_{ij} + ∑_{i=1}^{m} u_{i} ≤ c + w_{11} + w_{21}.

Since l_{11} and l_{21} are coprime, we have l_{11} > l_{21} ≥ 2. Plugging this into the above inequalities, we arrive at 2w_{11} < c+w_{21} and w_{21} < c+w_{11}. We conclude w_{11} < 2c and w_{21} < 3c. Moreover, l_{11}w_{11} = l_{21}w_{21} and gcd(l_{11}, l_{21}) = 1 imply l_{11}|w_{21} and l_{21}|w_{11}. This shows l_{11} < 3c and l_{21} < 2c. Finally, we obtain

deg g < c + w_{11} + w_{21} < 6c.

We prove (ii). Here we set c := (n_{0} + n_{1} + m)μ = (d + 1)μ. Then the assumed inequality gives

l_{21}w_{21} = deg g < ∑_{i=0}^{1} ∑_{j=1}^{n_{i}} w_{ij} + ∑_{i=1}^{m} u_{i} + w_{21} ≤ c + w_{21}.

Since we assumed l_{21} ≥ 2, we can conclude w_{21} < c. This in turn gives us deg g < 2c for the degree of the relation.

Proof of Theorem 2.1. As before, we denote by \overline{X} ⊆ 𝕂^{n+m} the total coordinate space and by p: X̂ → X the universal torsor.

We first consider the case that X is a toric variety. Then the Cox ring is a polynomial ring, R(X) = 𝕂[S_{1}, …, S_{m}]. For each 1 ≤ k ≤ m, consider the point \overline{x}(k) ∈ X̂ having the k-th coordinate one and all others zero and set x(k) := p(\overline{x}(k)). Then, by Proposition 2.5, the local class group Cl(X, x(k)) is of order u_{k} where u_{k} := deg S_{k}. This implies u_{k} ≤ μ for 1 ≤ k ≤ m and settles Assertion (i).

Now we treat the non-toric case, which means r ≥ 2. Note that we have n ≥ 3. The case n_{0} = 1 is done in Proposition 2.7. So, we are left with n_{0} > 1. For every i with n_{i} > 1 and every 1 ≤ j ≤ n_{i}, there is the point \overline{x}(i, j) ∈ X̂ with ij-coordinate T_{ij} equal to one and all others equal to zero, and thus we have the point x(i, j) := p(\overline{x}(i, j)) ∈ X. Moreover, for every 1 ≤ k ≤ m, we have the point \overline{x}(k) ∈ X̂ having the k-coordinate S_{k} equal to one and all others zero; we set x(k) := p(\overline{x}(k)). Proposition 2.5 provides the bounds

w_{ij} = deg T_{ij} = | Cl(X, x(i, j))| ≤ μ for n_{i} > 1, 1 ≤ j ≤ n_{i},

u_{k} = deg S_{k} = | Cl(X, x(k))| ≤ μ for 1 ≤ k ≤ m.

Let 0 ≤ s ≤ r be the maximal number with n_{s} > 1. Then g_{s−2,s−1,s} is the last polynomial such that each of its three monomials depends on more than one variable. For any t ≥ s, we have the “cut ring”

R_{t} := 𝕂[T_{ij}; 0 ≤ i ≤ t, 1 ≤ j ≤ n_{i}][S_{1}, …, S_{m}] / ⟨g_{i,i+1,i+2}; 0 ≤ i ≤ t − 2⟩

where the relations g_{i,i+1,i+2} depend on only three variables as soon as i > s holds. For the degree γ of the relations we have

(r − 1)γ = (t − 1)γ + (r − t)γ
= (t − 1)γ + l_{t+1,1}w_{t+1,1} + … + l_{r1}w_{r1}
< ∑_{i=0}^{r} ∑_{j=1}^{n_{i}} w_{ij} + ∑_{i=1}^{m} u_{i}

= ∑_{i=0}^{t} ∑_{j=1}^{n_{i}} w_{ij} + w_{t+1,1} + … + w_{r1} + ∑_{i=1}^{m} u_{i}.

Since l_{i1}w_{i1} > w_{i1} holds in particular for t + 1 ≤ i ≤ r, we derive from this the inequality

γ < \frac{1}{t − 1} (∑_{i=0}^{t} ∑_{j=1}^{n_{i}} w_{ij} + ∑_{i=1}^{m} u_{i}) .

To obtain the bounds in Assertions (iii) and (iv), we consider the cut ring R_{t} with t = 2 and apply Lemma 2.8; note that we have d = n_{0} + n_{1} + n_{2} + m − 2 for the dimension d = dim(X) and that l_{22} ≥ 0 is due to the fact that X is nontoric. The bounds w_{ij}, l_{0j} < 6dμ in Assertion (iii) follow from l_{ij}w_{ij} = γ < 6dμ and l_{i1} < 2dμ follows from l_{i1} | w_{21} for 3 ≤ i ≤ r. Moreover, l_{i1} | w_{11} for 2 ≤ i ≤ r implies l_{11} ⋯ l_{r1} | γ = l_{11}w_{11}. Similarly w_{ij}, l_{ij} < 2(d + 1)μ in Assertion (iv) follow from l_{ij}w_{ij} = γ < 2(d + 1)dμ and l_{21} ⋯ l_{r1} | γ = l_{21}w_{21} follows from l_{i1} | w_{21} for 3 ≤ i ≤ r. The bounds on r in (iii) in (iv) are as well consequences of the admissibility condition. To obtain the bounds in Assertion (v), we consider the cut ring R_{t} with t = s. Using n_{i} = 1 for i ≥ t+1, we can estimate the degree of the relation as follows:

γ ≤ \frac{(n_{0} + … + n_{t} + m)μ}{t − 1} = \frac{(d + t)μ}{t − 1} ≤ (d + 2)μ.

Since we have w_{ij}l_{ij} ≤ deg g_{0} for any 0 ≤ i ≤ r and any 1 ≤ j ≤ n_{i}, we see that all w_{ij} and l_{ij} are bounded by (d + 2)μ. As before, l_{s+1,1} ⋯ l_{r1} | γ is a consequence of l_{i1} | γ for i = s + 2, …, r and also the bound on r follows from the admissibility condition. Finally, we have to express the Picard index μ in terms of the weights w_{ij} and u_{k} as claimed in the Assertions. This is a direct application of the formula of Proposition 2.5. Observe that it suffices to work with the p-images of the following points: For every 0 ≤ i ≤ r with n_{i} > 1 take a point \overline{x}(i, j) ∈ X̂ with ij-coordinate T_{ij} equal to one and all others equal to zero, for every 0 ≤ i ≤ r with n_{i} = 1 whenever n_{i} = 1 take \overline{x}(i, j) ∈ X̂ with ij-coordinate T_{ij} equal to zero, all other T_{st} equal to one and coordinates S_{k} equal to zero, and, for every 1 ≤ k ≤ m, take a point \overline{x}(k) ∈ X̂ having the k-coordinate S_{k} equal to one and all others zero.

We conclude the section with discussing some aspects of the not necessarily Fano varieties of Proposition 2.7. Recall that we considered admissible triples (A, 𝔫, L) with n_{0} = … = n_{r} = 1 and thus rings R of the form

𝕂[T_{0}, …, T_{r}, S_{1}, …, S_{m}] / ⟨α_{i+1,i+2}T_{i}^{l_{i}} + α_{i+2,i}T_{i+1}^{l_{i+1}} + α_{i,i+1}T_{i+2}^{l_{i+2}}; 0 ≤ i ≤ r − 2⟩.

Proposition 2.9. Suppose that the ring R as above is the Cox ring of a nontoric variety X with Cl(X) = ℤ. Then we have m ≥ 1 and μ := [Cl(X) : Pic(X)] ≥ 30. Moreover, if X is a surface, then we have m = 1 and w_{i} = l_{i}^{−1}l_{0} ⋯ l_{r}.

Proof. The homogeneity condition l_{i}w_{i} = l_{j}w_{j} together with the admissibility condition gcd(l_{i}, l_{j}) = 1 for 0 ≤ i ≠ j ≤ r gives us l_{i} | gcd(w_{j}; j ≠ i). Moreover, by Proposition 1.11, every set of m + r weights w_{i} has to generate the class group ℤ, so they must have greatest common divisor one. Since X is nontoric, l_{i} ≥ 2 holds and we obtain m ≥ 1. To proceed, we infer l_{0} ⋯ l_{r} | μ and l_{0} ⋯ l_{r} | deg g_{ijk} from Proposition 2.5. As a consequence, the minimal value for μ and deg g_{ijk} is obviously 2 · 3 · 5 = 30. what really can be received as the following example shows. Note that if X is a surface we have m = 1 and gcd(w_{i}; 0 ≤ i ≤ r) = 1. Thus, l_{i}w_{i} = l_{j}w_{j} gives us deg g_{ijk} = l_{0} ⋯ l_{r} and w_{i} = l_{i}^{−1}l_{0} ⋯ l_{r}.

The bound [Cl(X) : Pic(X)] ≥ 30 given in the above proposition is even sharp; the surface discussed below realizes it.

Example 2.10. Consider X with R(X) = 𝕂[T_{0}, T_{1}, T_{2}, T_{3}]/⟨g⟩ with g = T_{0}^{2} + T_{1}^{3} + T_{2}^{5} and the grading

deg T_{0} = 15, deg T_{1} = 10, deg T_{2} = 6, deg T_{3} = 1.

Then we have gcd(15, 10) = 5, gcd(15, 6) = 3 and gcd(10, 6) = 2 and therefore [Cl(X) : Pic(X)] = 30. Further X is Fano because of

deg g = 30 < 32 = deg T_{0} + … + deg T_{3}.

Let us have a look at the geometric meaning of the condition n_{0} = … = n_{r} = 1. For a variety X with an action of a torus T, we denote by X_{0} ⊆ X the union of all orbits with at most finite isotropy. Then there is a possibly non-separated orbit space X_{0}/T; we call it the maximal orbit space. From [15], we infer that n_{0} = … = n_{r} = 1 holds if and only if X_{0}/T is separated. Combining this with Propositions 2.7 and 2.9 gives the following.

Corollary 2.11. For any pair (d, μ) ∈ ℤ_{>0}^{2} there is, up to deformation, only a finite number of d-dimensional complete varieties X with a complexity one torus action having divisor class group ℤ, Picard index [Cl(X) : Pic(X)] = μ and maximal orbit space ℙ_{1} and for each of these varieties the complement X \ X_{0} contains divisors.

Finally, we present a couple of examples showing that there are also non-Fano varieties with a complexity one torus action having divisor class group ℤ and maximal orbit space ℙ_{1}.

Example 2.12. Consider X with R(X) = 𝕂[T_{0}, T_{1}, T_{2}, T_{3}]/⟨g⟩ with g = T_{0}^{2} + T_{1}^{3} + T_{2}^{7} and the grading

deg T_{0} = 21, deg T_{1} = 14, deg T_{2} = 6, deg T_{3} = 1.

Then we have gcd(21, 14) = 7, gcd(21, 6) = 3 and gcd(14, 6) = 2 and therefore [Cl(X) : Pic(X)] = 42. Moreover, X is not Fano, because its canonical class K_{X} is trivial

K_{X} = deg g − deg T_{0} − … − deg T_{3} = 0.

Example 2.13. Consider X with R(X) = 𝕂[T_{0}, T_{1}, T_{2}, T_{3}]/⟨g⟩ with g = T_{0}^{2} + T_{1}^{3} + T_{2}^{11} and the grading

deg T_{0} = 33, deg T_{1} = 22, deg T_{2} = 6, deg T_{3} = 1.

Then we have gcd(22, 33) = 11, gcd(33, 6) = 3 and gcd(22, 6) = 2 and therefore [Cl(X) : Pic(X)] = 66. The canonical class K_{X} of X is even ample:

K_{X} = deg g − deg T_{0} − … − deg T_{3} = 4.

The following example shows that the Fano assumption is essential for the finiteness results in Theorem 2.1. Remark 2.14. For any pair p, q of coprime positive integers, we obtain a locally factorial 𝕂^{∗}-surface X(p, q) with Cl(X) = ℤ and Cox ring

R(X(p, q)) = 𝕂[T_{01}, T_{02}, T_{11}, T_{21}] / ⟨g⟩, g = T_{01}T_{02}^{pq−1} + T_{11}^{q} + T_{21}^{p};

the Cl(X)-grading is given by deg T_{01} = deg T_{02} = 1, deg T_{11} = p and deg T_{21} = q. Note that deg g = pq holds and for p, q ≥ 3, the canonical class K_{X} satisfies

K_{X} = deg g − deg T_{01} − deg T_{02} − deg T_{11} − deg T_{21} = pq − 2 − p − q ≥ 0.

3 Classification results

In this section, we give classification results for Fano varieties X with Cl(X) ≅ ℤ that come with a complexity one torus action; note that they are necessarily rational. The procedure to obtain classification lists for prescribed dimension d = dim X and Picard index μ = [Cl(X) : Pic(X)] is always the following. By Theorem 1.9, we know that their Cox rings are of the form R(X) ≅ R(A, 𝔫, L)[S_{1}, …, S_{m}] with admissible triples (A, 𝔫, L). Note that for the family A = (a_{0}, …, a_{r}) of points a_{i} ∈ 𝕂^{2}, we may assume

a_{0} = (1, 0),  a_{1} = (1, 1),  a_{2} = (0, 1).

The bounds on the input data of (A, 𝔫, L) provided by Theorem 2.1 as well as the criteria of Propositions 1.11 and 2.5 allow us to generate all the possible Cox rings R(X) of the Fano varieties X in question for fixed dimension d and Picard index μ. Note that X can be reconstructed from R(X) = R(A, 𝔫, L)[S_{1}, …, S_{n}] as the homogeneous spectrum with respect to the Cl(X)-grading. Thus X is classified by its Cox ring R(X). In the following tables, we present the Cox rings as 𝕂[T_{1}, …, T_{s}] modulo relations and fix the ℤ-gradings by giving the weight vector (w_{1}, …, w_{s}), where w_{i} := deg T_{i}. The first classification result concerns surfaces.

Theorem 3.1. Let X be a non-toric Fano surface with an effective 𝕂^{∗}-action such that Cl(X) = ℤ and [Cl(X) : Pic(X)] ≤ 6 hold. Then its Cox ring is precisely one of the following.

[Cl(X) : Pic(X)] = 1
No. R(X)  (w_{1}, …, w_{4}) (−K_{X})^{2}
1  𝕂[T_{1}, …, T_{4}]/⟨T_{1}T_{2}^{5} + T_{3}^{3} + T_{4}^{2}⟩  (1, 1, 2, 3)  1

[Cl(X) : Pic(X)] = 2
No. R(X)  (w_{1}, …, w_{4}) (−K_{X})^{2}
2  𝕂[T_{1}, …, T_{4}]/⟨T_{1}^{4}T_{2} + T_{3}^{3} + T_{4}^{2}⟩  (1, 2, 2, 3)  2

[Cl(X) : Pic(X)] = 3
No. R(X)  (w_{1}, …, w_{4}) (−K_{X})^{2}
3  𝕂[T_{1}, …, T_{4}]/⟨T_{1}^{3}T_{2} + T_{3}^{3} + T_{4}^{2}⟩  (1, 3, 2, 3)  3
4  𝕂[T_{1}, …, T_{4}]/⟨T_{1}T_{2}^{3} + T_{3}^{5} + T_{4}^{2}⟩  (1, 3, 2, 5)  1/3
5  𝕂[T_{1}, …, T_{4}]/⟨T_{1}^{7}T_{2} + T_{3}^{5} + T_{4}^{2}⟩  (1, 3, 2, 5)  1/3

[Cl(X) : Pic(X)] = 4
No. R(X)  (w_{1}, …, w_{4}) (−K_{X})^{2}
6  𝕂[T_{1}, …, T_{4}]/⟨T_{1}^{2}T_{2} + T_{3}^{3} + T_{4}^{2}⟩  (1, 4, 2, 3)  4
7  𝕂[T_{1}, …, T_{4}]/⟨T_{1}^{6}T_{2} + T_{3}^{5} + T_{4}^{2}⟩  (1, 4, 2, 5)  1

[Cl(X) : Pic(X)] = 5
No. R(X)  (w_{1}, …, w_{4}) (−K_{X})^{2}

8  𝕂[T_{1}, …, T_{4}]/⟨T_{1}T_{2} + T_{3}^{3} + T_{4}^{2}⟩  (1, 5, 2, 3)  5
9  𝕂[T_{1}, …, T_{4}]/⟨T_{1}^{5}T_{2} + T_{3}^{5} + T_{4}^{2}⟩  (1, 5, 2, 5)  9/5
10  𝕂[T_{1}, …, T_{4}]/⟨T_{1}^{9}T_{2} + T_{3}^{7} + T_{4}^{2}⟩  (1, 5, 2, 7)  1/5
11  𝕂[T_{1}, …, T_{4}]/⟨T_{1}^{7}T_{2} + T_{3}^{4} + T_{4}^{3}⟩  (1, 5, 3, 4)  1/5

[Cl(X) : Pic(X)] = 6
No. R(X)  (w_{1}, …, w_{4}) (−K_{X})^{2}
12  𝕂[T_{1}, …, T_{4}]/⟨T_{1}^{4}T_{2} + T_{3}^{5} + T_{4}^{2}⟩  (1, 6, 2, 5)  8/3
13  𝕂[T_{1}, …, T_{4}]/⟨T_{1}^{8}T_{2} + T_{3}^{7} + T_{4}^{2}⟩  (1, 6, 2, 7)  2/3
14  𝕂[T_{1}, …, T_{4}]/⟨T_{1}^{6}T_{2} + T_{3}^{4} + T_{4}^{3}⟩  (1, 6, 3, 4)  2/3
15  𝕂[T_{1}, …, T_{4}]/⟨T_{1}^{9}T_{2} + T_{3}^{3} + T_{4}^{2}⟩  (1, 3, 4, 6)  2/3

Proof. As mentioned, Theorems 1.9, 2.1 and Propositions 1.11, 2.5 produce a list of all Cox rings of surfaces with the prescribed data. Doing this computation, we obtain the list of the assertion. Note that none of the Cox rings listed is a polynomial ring and hence none of the resulting surfaces X is a toric variety. To show that different members of the list are not isomorphic to each other, we use the following two facts. Firstly, observe that any two minimal systems of homogeneous generators of the Cox ring have (up to reordering) the same list of degrees, and thus the list of generator degrees is invariant under isomorphism (up to reordering). Secondly, by Construction 1.7, the exponents l_{ij} > 1 are precisely the orders of the non-trivial isotropy groups of one-codimensional orbits of the action of the torus T on X. Using both principles and going through the list, we see that different members X cannot be T -equivariantly isomorphic to each other. Since all listed X are non-toric, the effective complexity one torus action on each X corresponds to a maximal torus in the linear algebraic group Aut(X). Any two maximal tori in the automorphism group are conjugate, and thus we can conclude that two members are isomorphic if and only if they are T -equivariantly isomorphic.

We remark that in [28, Section 4], log del Pezzo surfaces with an effective 𝕂^{∗}- action and Picard number 1 and Gorenstein index less than 4 were classified. The above list contains six such surfaces, namely no. 1-4, 6 and 8; these are exactly the ones where the maximal exponents of the monomials form a platonic triple, i.e., are of the form (1, k, l), (2, 2, k), (2, 3, 3), (2, 3, 4) or (2, 3, 5). The remaining ones, i.e., no. 5, 7, and 9-15 have non-log-terminal and thus non-rational singularities; to check this one may compute the resolutions via resolution of the ambient weighted projective space as in [14, Ex. 7.5]. With the same scheme of proof as in the surface case, one establishes the following classification results on Fano threefolds.

Theorem 3.2. Let X be a three-dimensional locally factorial non-toric Fano variety with an effective two torus action such that Cl(X) = ℤ holds. Then its Cox ring is precisely one of the following.

No. R(X)  (w_{1}, …, w_{5})  (−K_{X})^{3}
1  𝕂[T_{1}, …, T_{5}] / ⟨T_{1}T_{2}^{5} + T_{3}^{3} + T_{4}^{2}⟩  (1, 1, 2, 3, 1)  8
2  𝕂[T_{1}, …, T_{5}] / ⟨T_{1}T_{2}T_{3}^{4} + T_{4}^{3} + T_{5}^{2}⟩ (1, 1, 1, 2, 3)  8
3  𝕂[T_{1}, …, T_{5}] / ⟨T_{1}T_{2}^{2}T_{3}^{3} + T_{4}^{3} + T_{5}^{2}⟩ (1, 1, 1, 2, 3)  8
4  𝕂[T_{1}, …, T_{5}] / ⟨T_{1}T_{2} + T_{3}T_{4} + T_{5}^{2}⟩  (1, 1, 1, 1, 1)  54
5  𝕂[T_{1}, …, T_{5}] / ⟨T_{1}T_{2}^{2} + T_{3}T_{4}^{2} + T_{5}^{3}⟩ (1, 1, 1, 1, 1)  24
6  𝕂[T_{1}, …, T_{5}] / ⟨T_{1}T_{2}^{3} + T_{3}T_{4}^{3} + T_{5}^{4}⟩ (1, 1, 1, 1, 1)  4
7  𝕂[T_{1}, …, T_{5}] / ⟨T_{1}T_{2}^{3} + T_{3}T_{4}^{3} + T_{5}^{2}⟩ (1, 1, 1, 1, 2)  16
8  𝕂[T_{1}, …, T_{5}] / ⟨T_{1}T_{2}^{5} + T_{3}T_{4}^{5} + T_{5}^{2}⟩ (1, 1, 1, 1, 3)  2
9  𝕂[T_{1}, …, T_{5}] / ⟨T_{1}T_{2}^{5} + T_{3}^{3}T_{4}^{3} + T_{5}^{2}⟩ (1, 1, 1, 1, 3)  2

The singular threefolds listed in this theorem are rational degenerations of smooth Fano threefolds from [18]. The (smooth) general Fano threefolds of the corresponding families are non-rational see [12] for no. 1-3, [8] for no. 5, [20] for no. 6, [30, 29] for no. 7 and [19] for no. 8-9. Even if one allows certain mild singularities, one still has non-rationality in some cases, see [13], [9, 25], [10], [6].

Theorem 3.3. Let X be a three-dimensional non-toric Fano variety with an effective two torus action such that Cl(X) = ℤ and [Cl(X) : Pic(X)] = 2 hold. Then its Cox ring is precisely one of the following.

No.  R(X)  (w_{1}, …, w_{5})  (−K_{X})^{3}
1  𝕂[T_{1}, …, T_{5}]/⟨T_{1}^{4}T_{2} + T_{3}^{3} + T_{4}^{2}⟩  (1, 2, 2, 3, 1)  27/2
2  𝕂[T_{1}, …, T_{5}]/⟨T_{1}^{4}T_{2}^{3} + T_{3}^{5} + T_{4}^{2}⟩  (1, 2, 2, 5, 1)  1/2
3  𝕂[T_{1}, …, T_{5}]/⟨T_{1}^{8}T_{2} + T_{3}^{5} + T_{4}^{2}⟩  (1, 2, 2, 5, 1)  1/2
4  𝕂[T_{1}, …, T_{5}]/⟨T_{1}^{4}T_{2} + T_{3}^{3} + T_{4}^{2}⟩  (1, 2, 2, 3, 2)  16

5  𝕂[T_{1}, …, T_{5}]/⟨T_{1}^{4}T_{2}^{3} + T_{3}^{5} + T_{4}^{2}⟩  (1, 2, 2, 5, 2)  2^{6  𝕂[T_{1}, …, T_{5}]/⟨T_{1}^{8}T_{2} + T_{3}^{5} + T_{4}^{2}⟩  (1, 2, 2, 5, 2)  2}^{7  𝕂[T_{1}, …, T_{5}]/⟨T_{1}T_{2}^{5} + T_{3}^{3} + T_{4}^{2}⟩  (1, 1, 2, 3, 2)  27/2}^{8  𝕂[T_{1}, …, T_{5}]/⟨T_{1}T_{2}^{9} + T_{3}^{5} + T_{4}^{2}⟩  (1, 1, 2, 5, 2)  1/2}^{9  𝕂[T_{1}, …, T_{5}]/⟨T_{1}^{3}T_{2}^{7} + T_{3}^{5} + T_{4}^{2}⟩  (1, 1, 2, 5, 2)  1/2}^{10  𝕂[T_{1}, …, T_{5}]/⟨T_{1}T_{2}^{11} + T_{3}^{3} + T_{4}^{2}⟩  (1, 1, 4, 6, 1)  1/2}^{11  𝕂[T_{1}, …, T_{5}]/⟨T_{1}^{5}T_{2}^{7} + T_{3}^{3} + T_{4}^{2}⟩  (1, 1, 4, 6, 1)  1/2}^{12  𝕂[T_{1}, …, T_{5}]/⟨T_{1}T_{2}^{11} + T_{3}^{3} + T_{4}^{2}⟩  (1, 1, 4, 6, 2)  2}^{13  𝕂[T_{1}, …, T_{5}]/⟨T_{1}^{5}T_{2}^{7} + T_{3}^{3} + T_{4}^{2}⟩  (1, 1, 4, 6, 2)  2}^{14  𝕂[T_{1}, …, T_{5}]/⟨T_{1}^{2}T_{2}^{5} + T_{3}^{3} + T_{4}^{2}⟩  (1, 2, 4, 6, 1)  2}^{15  𝕂[T_{1}, …, T_{5}]/⟨T_{1}^{10}T_{2} + T_{3}^{3} + T_{4}^{2}⟩  (1, 2, 4, 6, 1)  2}^{16  𝕂[T_{1}, …, T_{5}]/⟨T_{1}T_{2}^{2} + T_{3}^{3} + T_{4}^{2}⟩  (2, 2, 2, 3, 1)  16}^{17  𝕂[T_{1}, …, T_{5}]/⟨T_{1}T_{2}^{4} + T_{3}^{5} + T_{4}^{2}⟩  (2, 2, 2, 5, 1)  2}^{18  𝕂[T_{1}, …, T_{5}]/⟨T_{1}^{2}T_{2}^{3} + T_{3}^{5} + T_{4}^{2}⟩  (2, 2, 2, 5, 1)  2}^{19  𝕂[T_{1}, …, T_{5}]/⟨T_{1}T_{2}^{2} + T_{3}T_{4} + T_{5}^{3}⟩ (1, 1, 1, 2, 1)  81/2}^{20  𝕂[T_{1}, …, T_{5}]/⟨T_{1}T_{2}^{4} + T_{3}T_{4}^{2} + T_{5}^{5}⟩ (1, 1, 1, 2, 1)  5/2}^{21  𝕂[T_{1}, …, T_{5}]/⟨T_{1}^{2}T_{2}^{3} + T_{3}T_{4}^{2} + T_{5}^{5}⟩ (1, 1, 1, 2, 1)  5/2}^{22  𝕂[T_{1}, …, T_{5}]/⟨T_{1}T_{2}^{3} + T_{3}^{2}T_{4} + T_{5}^{4}⟩ (1, 1, 1, 2, 1)  16}^{23  𝕂[T_{1}, …, T_{5}]/⟨T_{1}T_{2}^{4} + T_{3}^{3}T_{4} + T_{5}^{5}⟩ (1, 1, 1, 2, 1)  5/2}^{24  𝕂[T_{1}, …, T_{5}]/⟨T_{1}^{2}T_{2}^{3} + T_{3}^{3}T_{4} + T_{5}^{5}⟩ (1, 1, 1, 2, 1)  5/2}^{25  𝕂[T_{1}, …, T_{5}]/⟨T_{1}T_{2}^{3} + T_{3}^{2}T_{4} + T_{5}^{2}⟩ (1, 1, 1, 2, 2)  27}^{26  𝕂[T_{1}, …, T_{5}]/⟨T_{1}T_{2}^{5} + T_{3}^{2}T_{4}^{2} + T_{5}^{3}⟩ (1, 1, 1, 2, 2)  3/2}^{27  𝕂[T_{1}, …, T_{5}]/⟨T_{1}T_{2}^{5} + T_{3}^{4}T_{4} + T_{5}^{3}⟩ (1, 1, 1, 2, 2)  3/2}^{28  𝕂[T_{1}, …, T_{5}]/⟨T_{1}^{2}T_{2}^{4} + T_{3}^{4}T_{4} + T_{5}^{3}⟩ (1, 1, 1, 2, 2)  3/2}^{29  𝕂[T_{1}, …, T_{5}]/⟨T_{1}T_{2}^{5} + T_{3}^{4}T_{4} + T_{5}^{2}⟩ (1, 1, 1, 2, 3)  8}^{30  𝕂[T_{1}, …, T_{5}]/⟨T_{1}^{3}T_{2}^{3} + T_{3}^{4}T_{4} + T_{5}^{2}⟩ (1, 1, 1, 2, 3)  8}^{31  𝕂[T_{1}, …, T_{5}]/⟨T_{1}T_{2}^{7} + T_{3}^{2}T_{4}^{3} + T_{5}^{2}⟩ (1, 1, 1, 2, 4)  1}^{32  𝕂[T_{1}, …, T_{5}]/⟨T_{1}^{3}T_{2}^{5} + T_{3}^{2}T_{4}^{3} + T_{5}^{2}⟩ (1, 1, 1, 2, 4)  1}^{33  𝕂[T_{1}, …, T_{5}]/⟨T_{1}T_{2}^{7} + T_{3}^{6}T_{4} + T_{5}^{2}⟩ (1, 1, 1, 2, 4)  1}^{34  𝕂[T_{1}, …, T_{5}]/⟨T_{1}^{3}T_{2}^{5} + T_{3}^{6}T_{4} + T_{5}^{2}⟩ (1, 1, 1, 2, 4)  1}^{35  𝕂[T_{1}, …, T_{5}]/⟨T_{1}T_{2}^{3} + T_{3}T_{4} + T_{5}^{4}⟩ (1, 1, 2, 2, 1)  27}

36  𝕂[T_{1}, …, T_{5}]/⟨T_{1}T_{2}^{5} + T_{3}T_{4}^{2} + T_{5}^{6}⟩ (1, 1, 2, 2, 1)  3/2^{37  𝕂[T_{1}, …, T_{5}]/⟨T_{1}T_{2}^{3} + T_{3}T_{4} + T_{5}^{2}⟩ (1, 1, 2, 2, 2)  16}^{38  𝕂[T_{1}, …, T_{5}]/⟨T_{1}T_{2}^{5} + T_{3}T_{4}^{2} + T_{5}^{3}⟩ (1, 1, 2, 2, 2)  6}^{39  𝕂[T_{1}, …, T_{5}]/⟨T_{1}^{2}T_{2}^{4} + T_{3}T_{4}^{2} + T_{5}^{3}⟩ (1, 1, 2, 2, 2)  6}^{40  𝕂[T_{1}, …, T_{5}]/⟨T_{1}^{3}T_{2}^{3} + T_{3}T_{4}^{2} + T_{5}^{2}⟩ (1, 1, 2, 2, 2)  27/2}^{41  𝕂[T_{1}, …, T_{5}]/⟨T_{1}^{3}T_{2}^{5} + T_{3}T_{4}^{3} + T_{5}^{2}⟩ (1, 1, 2, 2, 2)  32}^{42  𝕂[T_{1}, …, T_{5}]/⟨T_{1}T_{2}^{5} + T_{3}T_{4}^{2} + T_{5}^{2}⟩ (1, 1, 2, 2, 3)  4}^{43  𝕂[T_{1}, …, T_{5}]/⟨T_{1}T_{2}^{7} + T_{3}T_{4}^{3} + T_{5}^{2}⟩ (1, 1, 2, 2, 4)  32}^{44  𝕂[T_{1}, …, T_{5}]/⟨T_{1}T_{2}^{9} + T_{3}T_{4}^{4} + T_{5}^{2}⟩ (1, 1, 2, 2, 5)  1/2}^{45  𝕂[T_{1}, …, T_{5}]/⟨T_{1}T_{2}^{9} + T_{3}^{2}T_{4}^{3} + T_{5}^{2}⟩ (1, 1, 2, 2, 5)  1/2}^{46  𝕂[T_{1}, …, T_{5}]/⟨T_{1}^{3}T_{2}^{7} + T_{3}T_{4}^{4} + T_{5}^{2}⟩ (1, 1, 2, 2, 5)  1/2}^{47  𝕂[T_{1}, …, T_{5}]/⟨T_{1}^{3}T_{2}^{7} + T_{3}^{2}T_{4}^{3} + T_{5}^{2}⟩ (1, 1, 2, 2, 5)  1/2}^{48  𝕂[T_{1}, …, T_{5}]/⟨T_{1}^{5}T_{2}^{5} + T_{3}T_{4}^{4} + T_{5}^{2}⟩ (1, 1, 2, 2, 5)  1/2}^{49  𝕂[T_{1}, …, T_{5}]/⟨T_{1}^{5}T_{2}^{5} + T_{3}^{2}T_{4}^{3} + T_{5}^{2}⟩ (1, 1, 2, 2, 5)  1/2}^{50  𝕂[T_{1}, …, T_{5}]/⟨T_{1}T_{2} + T_{3}T_{4} + T_{5}^{3}⟩ (1, 2, 1, 2, 1)  48}^{51  𝕂[T_{1}, …, T_{5}]/⟨T_{1}^{2}T_{2} + T_{3}^{2}T_{4} + T_{5}^{4}⟩ (1, 2, 1, 2, 1)  27}^{52  𝕂[T_{1}, …, T_{5}]/⟨T_{1}T_{2}^{2} + T_{3}T_{4}^{2} + T_{5}^{5}⟩ (1, 2, 1, 2, 1)  10}^{53  𝕂[T_{1}, …, T_{5}]/⟨T_{1}T_{2}^{2} + T_{3}^{3}T_{4} + T_{5}^{5}⟩ (1, 2, 1, 2, 1)  10}^{54  𝕂[T_{1}, …, T_{5}]/⟨T_{1}^{3}T_{2} + T_{3}^{3}T_{4} + T_{5}^{5}⟩ (1, 2, 1, 2, 1)  10}^{55  𝕂[T_{1}, …, T_{5}]/⟨T_{1}^{4}T_{2} + T_{3}^{4}T_{4} + T_{5}^{6}⟩ (1, 2, 1, 2, 1)  3/2}^{56  𝕂[T_{1}, …, T_{5}]/⟨T_{1}^{2}T_{2} + T_{3}^{2}T_{4} + T_{5}^{2}⟩ (1, 2, 1, 2, 2)  32}^{57  𝕂[T_{1}, …, T_{5}]/⟨T_{1}^{2}T_{2}^{2} + T_{3}^{4}T_{4} + T_{5}^{3}⟩ (1, 2, 1, 2, 2)  6}^{58  𝕂[T_{1}, …, T_{5}]/⟨T_{1}^{4}T_{2} + T_{3}^{4}T_{4} + T_{5}^{3}⟩ (1, 2, 1, 2, 2)  6}^{59  𝕂[T_{1}, …, T_{5}]/⟨T_{1}^{4}T_{2} + T_{3}^{4}T_{4} + T_{5}^{2}⟩ (1, 2, 1, 2, 3)  27/2}^{60  𝕂[T_{1}, …, T_{5}]/⟨T_{1}^{2}T_{2}^{3} + T_{3}^{2}T_{4}^{3} + T_{5}^{2}⟩ (1, 2, 1, 2, 4)  4}^{61  𝕂[T_{1}, …, T_{5}]/⟨T_{1}^{2}T_{2}^{3} + T_{3}^{6}T_{4} + T_{5}^{2}⟩ (1, 2, 1, 2, 4)  4}^{62  𝕂[T_{1}, …, T_{5}]/⟨T_{1}^{6}T_{2} + T_{3}^{6}T_{4} + T_{5}^{2}⟩ (1, 2, 1, 2, 4)  4}^{63  𝕂[T_{1}, …, T_{5}]/⟨T_{1}^{4}T_{2}^{3} + T_{3}^{4}T_{4}^{3} + T_{5}^{2}⟩ (1, 2, 1, 2, 5)  1/2}^{64  𝕂[T_{1}, …, T_{5}]/⟨T_{1}^{8}T_{2} + T_{3}^{4}T_{4}^{3} + T_{5}^{2}⟩ (1, 2, 1, 2, 5)  1/2}^{65  𝕂[T_{1}, …, T_{5}]/⟨T_{1}^{8}T_{2} + T_{3}^{8}T_{4} + T_{5}^{2}⟩ (1, 2, 1, 2, 5)  1/2}^{66  𝕂[T_{1}, …, T_{5}]/⟨T_{1}^{2}T_{2} + T_{3}T_{4} + T_{5}^{4}⟩ (1, 2, 2, 2, 1)  32} 67  𝕂[T^{1}, …, T^{5}]/⟨T 4 1T^{2} + T^{3}T 2 4 + T 6 5⟩ (1, 2, 2, 2, 1)  6

68  𝕂[T_{1}, …, T_{5}]/⟨T_{1}^{4}T_{2} + T_{3}T_{4}^{2} + T_{5}^{2}⟩ (1, 2, 2, 2, 3)  16^{69  𝕂[T_{1}, …, T_{5}]/⟨T_{1}^{4}T_{2}^{3} + T_{3}T_{4}^{4} + T_{5}^{2}⟩ (1, 2, 2, 2, 5)  2}^{70  𝕂[T_{1}, …, T_{5}]/⟨T_{1}^{4}T_{2}^{3} + T_{3}^{2}T_{4}^{3} + T_{5}^{2}⟩ (1, 2, 2, 2, 5)  2}^{71  𝕂[T_{1}, …, T_{5}]/⟨T_{1}^{8}T_{2} + T_{3}T_{4}^{4} + T_{5}^{2}⟩ (1, 2, 2, 2, 5)  2}^{72  𝕂[T_{1}, …, T_{5}]/⟨T_{1}^{8}T_{2} + T_{3}^{2}T_{4}^{3} + T_{5}^{2}⟩ (1, 2, 2, 2, 5)  2}^{73  𝕂[T_{1}, …, T_{5}]/⟨T_{1}T_{2}T_{3}^{10} + T_{4}^{3} + T_{5}^{2}⟩ (1, 1, 1, 4, 6)  1/2}^{74  𝕂[T_{1}, …, T_{5}]/⟨T_{1}T_{2}^{2}T_{3}^{9} + T_{4}^{3} + T_{5}^{2}⟩ (1, 1, 1, 4, 6)  1/2}^{75  𝕂[T_{1}, …, T_{5}]/⟨T_{1}T_{2}^{3}T_{3}^{8} + T_{4}^{3} + T_{5}^{2}⟩ (1, 1, 1, 4, 6)  1/2}^{76  𝕂[T_{1}, …, T_{5}]/⟨T_{1}T_{2}^{4}T_{3}^{7} + T_{4}^{3} + T_{5}^{2}⟩ (1, 1, 1, 4, 6)  1/2}^{77  𝕂[T_{1}, …, T_{5}]/⟨T_{1}T_{2}^{5}T_{3}^{6} + T_{4}^{3} + T_{5}^{2}⟩ (1, 1, 1, 4, 6)  1/2}^{78  𝕂[T_{1}, …, T_{5}]/⟨T_{1}^{2}T_{2}^{3}T_{3}^{7} + T_{4}^{3} + T_{5}^{2}⟩ (1, 1, 1, 4, 6)  1/2}^{79  𝕂[T_{1}, …, T_{5}]/⟨T_{1}^{2}T_{2}^{5}T_{3}^{5} + T_{4}^{3} + T_{5}^{2}⟩ (1, 1, 1, 4, 6)  1/2}^{80  𝕂[T_{1}, …, T_{5}]/⟨T_{1}^{3}T_{2}^{4}T_{3}^{5} + T_{4}^{3} + T_{5}^{2}⟩ (1, 1, 1, 4, 6)  1/2}^{81  𝕂[T_{1}, …, T_{5}]/⟨T_{1}T_{2}T_{3}^{2} + T_{4}^{3} + T_{5}^{2}⟩ (1, 1, 2, 2, 3)  27/2}^{82  𝕂[T_{1}, …, T_{5}]/⟨T_{1}T_{2}^{3}T_{3} + T_{4}^{3} + T_{5}^{2}⟩ (1, 1, 2, 2, 3)  27/2}^{83  𝕂[T_{1}, …, T_{5}]/⟨T_{1}^{2}T_{2}^{2}T_{3} + T_{4}^{3} + T_{5}^{2}⟩ (1, 1, 2, 2, 3)  27/2}^{84  𝕂[T_{1}, …, T_{5}]/⟨T_{1}T_{2}T_{3}^{4} + T_{4}^{5} + T_{5}^{2}⟩ (1, 1, 2, 2, 5)  1/2}^{85  𝕂[T_{1}, …, T_{5}]/⟨T_{1}T_{2}^{3}T_{3}^{3} + T_{4}^{5} + T_{5}^{2}⟩ (1, 1, 2, 2, 5)  1/2}^{86  𝕂[T_{1}, …, T_{5}]/⟨T_{1}T_{2}^{5}T_{3}^{2} + T_{4}^{5} + T_{5}^{2}⟩ (1, 1, 2, 2, 5)  1/2}^{87  𝕂[T_{1}, …, T_{5}]/⟨T_{1}T_{2}^{7}T_{3} + T_{4}^{5} + T_{5}^{2}⟩ (1, 1, 2, 2, 5)  1/2}^{88  𝕂[T_{1}, …, T_{5}]/⟨T_{1}^{2}T_{2}^{2}T_{3}^{3} + T_{4}^{5} + T_{5}^{2}⟩ (1, 1, 2, 2, 5)  1/2}^{89  𝕂[T_{1}, …, T_{5}]/⟨T_{1}^{2}T_{2}^{6}T_{3} + T_{4}^{5} + T_{5}^{2}⟩ (1, 1, 2, 2, 5)  1/2}^{90  𝕂[T_{1}, …, T_{5}]/⟨T_{1}^{3}T_{2}^{3}T_{3}^{2} + T_{4}^{5} + T_{5}^{2}⟩ (1, 1, 2, 2, 5)  1/2}^{91  𝕂[T_{1}, …, T_{5}]/⟨T_{1}^{3}T_{2}^{5}T_{3} + T_{4}^{5} + T_{5}^{2}⟩ (1, 1, 2, 2, 5)  1/2}^{92  𝕂[T_{1}, …, T_{5}]/⟨T_{1}^{4}T_{2}^{4}T_{3} + T_{4}^{5} + T_{5}^{2}⟩ (1, 1, 2, 2, 5)  1/2}^{93  𝕂[T_{1}, …, T_{5}]/⟨T_{1}T_{2}T_{3}^{5} + T_{4}^{3} + T_{5}^{2}⟩ (1, 1, 2, 4, 6)  2}^{94  𝕂[T_{1}, …, T_{5}]/⟨T_{1}T_{2}^{3}T_{3}^{4} + T_{4}^{3} + T_{5}^{2}⟩ (1, 1, 2, 4, 6)  2}^{95  𝕂[T_{1}, …, T_{5}]/⟨T_{1}T_{2}^{5}T_{3}^{3} + T_{4}^{3} + T_{5}^{2}⟩ (1, 1, 2, 4, 6)  2}^{96  𝕂[T_{1}, …, T_{5}]/⟨T_{1}T_{2}^{7}T_{3}^{2} + T_{4}^{3} + T_{5}^{2}⟩ (1, 1, 2, 4, 6)  2}^{97  𝕂[T_{1}, …, T_{5}]/⟨T_{1}T_{2}^{9}T_{3} + T_{4}^{3} + T_{5}^{2}⟩ (1, 1, 2, 4, 6)  2}^{98  𝕂[T_{1}, …, T_{5}]/⟨T_{1}^{2}T_{2}^{4}T_{3}^{3} + T_{4}^{3} + T_{5}^{2}⟩ (1, 1, 2, 4, 6)  2}

99  𝕂[T_{1}, …, T_{5}]/⟨T_{1}^{2}T_{2}^{8}T_{3} + T_{4}^{3} + T_{5}^{2}⟩ (1, 1, 2, 4, 6)  2^{100  𝕂[T_{1}, …, T_{5}]/⟨T_{1}^{3}T_{2}^{5}T_{3}^{2} + T_{4}^{3} + T_{5}^{2}⟩ (1, 1, 2, 4, 6)  2}^{101  𝕂[T_{1}, …, T_{5}]/⟨T_{1}^{3}T_{2}^{7}T_{3} + T_{4}^{3} + T_{5}^{2}⟩ (1, 1, 2, 4, 6)  2}^{102  𝕂[T_{1}, …, T_{5}]/⟨T_{1}^{4}T_{2}^{6}T_{3} + T_{4}^{3} + T_{5}^{2}⟩ (1, 1, 2, 4, 6)  2}^{103  𝕂[T_{1}, …, T_{5}]/⟨T_{1}^{5}T_{2}^{5}T_{3} + T_{4}^{3} + T_{5}^{2}⟩ (1, 1, 2, 4, 6)  2}^{104  𝕂[T_{1}, …, T_{5}]/⟨T_{1}^{2}T_{2}T_{3} + T_{4}^{3} + T_{5}^{2}⟩ (1, 2, 2, 2, 3)  16}^{105  𝕂[T_{1}, …, T_{5}]/⟨T_{1}^{2}T_{2}T_{3}^{3} + T_{4}^{5} + T_{5}^{2}⟩ (1, 2, 2, 2, 5)  2}^{106  𝕂[T_{1}, …, T_{5}]/⟨T_{1}^{4}T_{2}T_{3}^{2} + T_{4}^{5} + T_{5}^{2}⟩ (1, 2, 2, 2, 5)  2}^{107  𝕂[T_{1}, …, T_{5}]/⟨T_{1}^{6}T_{2}T_{3} + T_{4}^{5} + T_{5}^{2}⟩ (1, 2, 2, 2, 5)  2}

The varieties no. 2,3 and 25, 26 are rational degenerations of quasismooth varieties from the list in [16]. In [11] the non-rationality of a general (quasismooth) element of the corresponding family was proved. The varieties listed so far might suggest that we always obtain only one relation in the Cox ring. We discuss now some examples, showing that for a Picard index big enough, we need in general more than one relation, where this refers always to a presentation as in Theorem 1.9 (ii).

Example 3.4. A Fano 𝕂^{∗}-surface X with Cl(X) = ℤ such that the Cox ring R(X) needs two relations. Consider the ℤ-graded ring

R = 𝕂[T_{01}, T_{02}, T_{11}, T_{21}, T_{31}]/⟨g_{0}, g_{1}⟩,

where the degrees of T_{01}, T_{02}, T_{11}, T_{21}, T_{31} are 29, 1, 6, 10, 15, respectively, and the relations g_{0}, g_{1} are given by

g_{0} := T_{01}T_{02} + T_{11}^{5} + T_{21}^{3}, g_{1} := α_{23}T_{11}^{5} + α_{31}T_{21}^{3} + α_{12}T_{31}^{2}

Then R is the Cox ring of a Fano 𝕂^{∗}-surface. Note that the Picard index is given by [Cl(X) : Pic(X)] = lcm(29, 1) = 29.

Proposition 3.5. Let X be a non-toric Fano surface with an effective 𝕂^{∗}- action such that Cl(X) ≅ ℤ and [Cl(X) : Pic(X)] < 29 hold. Then the Cox ring of X is of the form

R(X) ≅ 𝕂[T_{1}, …, T_{4}]/⟨T_{1}^{l_{1}}T_{2}^{l_{2}} + T_{3}^{l_{3}} + T_{4}^{l_{4}}⟩.

Proof. The Cox ring R(X) is as in Theorem 1.9, and, in the notation used there, we have n_{0} + … + n_{r} + m = 2 + r. This leaves us with the possibilities n_{0} = m = 1 and n_{0} = 2, m = 0. In the first case, Proposition 2.9 tells us that the Picard index of X is at least 30.

So, consider the case n_{0} = 2 and m = 0. Then, according to Theorem 1.9, the Cox ring R(X) is 𝕂[T_{01}, T_{02}, T_{1} …, T_{r}] divided by relations

g_{0,1,2} = T_{01}^{l_{01}}T_{02}^{l_{02}} +T_{1}^{l_{1}} +T_{2}^{l_{2}}, g_{i,i+1,i+2} = α_{i+1,i+2}T_{i}^{l_{i}} +α_{i+2,i}T_{i+1}^{l_{i+1}} +α_{i,i+1}T_{i+2}^{l_{i+2}},

where 1 ≤ i ≤ r − 2. We have to show that r = 2 holds. Set μ := [Cl(X) : Pic(X)] and let γ ∈ ℤ denote the degree of the relations. Then we have γ = w_{i}l_{i} for 1 ≤ i ≤ r, where w_{i} := deg T_{i}. With w_{0i} := deg T_{0i}, Proposition 2.5 gives us

(r − 1)γ < w_{01} + w_{02} + w_{1} + … + w_{r}.

We claim that w_{01} and w_{02} are coprime. Otherwise they had a common prime divisor p. This p divides γ = l_{i}w_{i}. Since l_{1}, …, l_{r} are pairwise coprime, p divides at least r − 1 of the weights w_{1}, …, w_{r}. This contradicts the Cox ring condition that any r + 1 of the r + 2 weights generate the class group ℤ. Thus, w_{01} and w_{02} are coprime and we obtain

μ ≥ lcm(w_{01}, w_{02}) = w_{01} · w_{02} ≥ w_{01} + w_{02} − 1.

Now assume that r ≥ 3 holds. Then we can conclude

2γ < w_{01} + w_{02} + w_{1} + w_{2} + w_{3} ≤ μ + 1 + γ (\frac{1}{l_{1}} + \frac{1}{l_{2}} + \frac{1}{l_{3}} )

Since the numbers l_{i} are pairwise coprime, we obtain l_{1} ≥ 5, l_{2} ≥ 3 and l_{3} ≥ 2. Moreover, l_{i}w_{i} = l_{j}w_{j} implies l_{i} | w_{j} and hence l_{1}l_{2}l_{3} | γ. Thus, we have γ ≥ 30. Plugging this in the above inequality gives

μ ≥ γ (2 − \frac{1}{l_{1}} − \frac{1}{l_{2}} − \frac{1}{l_{3}} ) − 1 = 29.

The Fano assumption is essential in this result; if we omit it, then we may even construct locally factorial surfaces with a Cox ring that needs more then one relation.

Example 3.6. A locally factorial 𝕂^{∗}-surface X with Cl(X) = ℤ such that the Cox ring R(X) needs two relations. Consider the ℤ-graded ring

R = 𝕂[T_{01}, T_{02}, T_{11}, T_{21}, T_{31}]/⟨g_{0}, g_{1}⟩,

where the degrees of T_{01}, T_{02}, T_{11}, T_{21}, T_{31} are 1, 1, 6, 10, 15, respectively, and the relations g_{0}, g_{1} are given by

g_{0} := T_{01}^{7}T_{02}^{23} + T_{11}^{5} + T_{21}^{3}, g_{1} := α_{23}T_{11}^{5} + α_{31}T_{21}^{3} + α_{12}T_{31}^{2}

Then R is the Cox ring of a non Fano 𝕂^{∗}-surface X of Picard index one, i.e, X is locally factorial.

For non-toric Fano threefolds X with an effective 2-torus action Cl(X) ≅ ℤ, the classifications 3.2 and 3.3 show that for Picard indices one and two we only obtain hypersurfaces as Cox rings. The following example shows that this stops at Picard index three.

Example 3.7. A Fano threefold X with Cl(X) = ℤ and a 2-torus action such that the Cox ring R(X) needs two relations. Consider

R = 𝕂[T_{01}, T_{02}, T_{11}, T_{12}, T_{21}, T_{31}]/⟨g_{0}, g_{1}⟩

where the degrees of T_{01}, T_{02}, T_{11}, T_{12}, T_{21}, T_{31} are 1, 1, 3, 3, 2, 3, respectively, and the relations are given by

g_{0} = T_{01}^{5}T_{02} + T_{11}T_{12} + T_{21}^{3}, g_{1} = α_{23}T_{11}T_{12} + α_{31}T_{21}^{3} + α_{12}T_{31}^{2}.

Then R is the Cox ring of a Fano threefold with a 2-torus action. Note that the Picard index is given by

[Cl(X) : Pic(X)] = lcm(1, 1, 3, 3) = 3.

Finally, we turn to locally factorial Fano fourfolds. Here we observe more than one relation in the Cox ring even in the locally factorial case.

Theorem 3.8. Let X be a four-dimensional locally factorial non-toric Fano variety with an effective three torus action such that Cl(X) = ℤ holds. Then its Cox ring is precisely one of the following.

No. R(X)  (w_{1}, …, w_{6})  (−K_{X})^{4}
1  𝕂[T_{1}, …, T_{6}]/⟨T_{1}T_{2}^{5} + T_{3}^{3} + T_{4}^{2}⟩  (1, 1, 2, 3, 1, 1)  81
2  𝕂[T_{1}, …, T_{6}]/⟨T_{1}T_{2}^{9} + T_{3}^{2} + T_{4}^{5}⟩  (1, 1, 2, 5, 1, 1)  1
3  𝕂[T_{1}, …, T_{6}]/⟨T_{1}^{3}T_{2}^{7} + T_{3}^{2} + T_{4}^{5}⟩  (1, 1, 2, 5, 1, 1)  1
4  𝕂[T_{1}, …, T_{6}]/⟨T_{1}T_{2}T_{3}^{4} + T_{4}^{3} + T_{5}^{2}⟩  (1, 1, 1, 2, 3, 1)  81
5  𝕂[T_{1}, …, T_{6}]/⟨T_{1}T_{2}^{2}T_{3}^{3} + T_{4}^{3} + T_{5}^{2}⟩  (1, 1, 1, 2, 3, 1)  81
6  𝕂[T_{1}, …, T_{6}]/⟨T_{1}T_{2}T_{3}^{8} + T_{4}^{5} + T_{5}^{2}⟩  (1, 1, 1, 2, 5, 1)  1
7  𝕂[T_{1}, …, T_{6}]/⟨T_{1}T_{2}^{2}T_{3}^{7} + T_{4}^{5} + T_{5}^{2}⟩  (1, 1, 1, 2, 5, 1)  1
8  𝕂[T_{1}, …, T_{6}]/⟨T_{1}T_{2}^{3}T_{3}^{6} + T_{4}^{5} + T_{5}^{2}⟩  (1, 1, 1, 2, 5, 1)  1
9  𝕂[T_{1}, …, T_{6}]/⟨T_{1}T_{2}^{4}T_{3}^{5} + T_{4}^{5} + T_{5}^{2}⟩  (1, 1, 1, 2, 5, 1)  1
10 𝕂[T_{1}, …, T_{6}]/⟨T_{1}^{2}T_{2}^{3}T_{3}^{5} + T_{4}^{5} + T_{5}^{2}⟩  (1, 1, 1, 2, 5, 1)  1
11 𝕂[T_{1}, …, T_{6}]/⟨T_{1}^{3}T_{2}^{3}T_{3}^{4} + T_{4}^{5} + T_{5}^{2}⟩  (1, 1, 1, 2, 5, 1)  1
12 𝕂[T_{1}, …, T_{6}]/⟨T_{1}T_{2} + T_{3}T_{4} + T_{5}^{2}⟩  (1, 1, 1, 1, 1, 1)  512
13 𝕂[T_{1}, …, T_{6}]/⟨T_{1}T_{2}^{2} + T_{3}T_{4}^{2} + T_{5}^{3}⟩  (1, 1, 1, 1, 1, 1)  243

Documenta Mathematica 16 (2011) 71–109

14 𝕂[T_{1}, …, T_{6}]/⟨T_{1}T_{2}^{3} + T_{3}T_{4}^{3} + T_{5}^{4}⟩  (1, 1, 1, 1, 1, 1)  64^{15 𝕂[T_{1}, …, T_{6}]/⟨T_{1}T_{2}^{4} + T_{3}T_{4}^{4} + T_{5}^{5}⟩  (1, 1, 1, 1, 1, 1)  5}^{16 𝕂[T_{1}, …, T_{6}]/⟨T_{1}T_{2}^{4} + T_{3}^{2}T_{4}^{3} + T_{5}^{5}⟩  (1, 1, 1, 1, 1, 1)  5}^{17 𝕂[T_{1}, …, T_{6}]/⟨T_{1}^{2}T_{2}^{3} + T_{3}^{2}T_{4}^{3} + T_{5}^{5}⟩  (1, 1, 1, 1, 1, 1)  5}^{18 𝕂[T_{1}, …, T_{6}]/⟨T_{1}T_{2}^{3} + T_{3}T_{4}^{3} + T_{5}^{2}⟩  (1, 1, 1, 1, 2, 1)  162}^{19 𝕂[T_{1}, …, T_{6}]/⟨T_{1}T_{2}^{5} + T_{3}T_{4}^{5} + T_{5}^{3}⟩  (1, 1, 1, 1, 2, 1)  3}^{20 𝕂[T_{1}, …, T_{6}]/⟨T_{1}T_{2}^{5} + T_{3}^{2}T_{4}^{4} + T_{5}^{3}⟩  (1, 1, 1, 1, 2, 1)  3}^{21 𝕂[T_{1}, …, T_{6}]/⟨T_{1}T_{2}^{5} + T_{3}T_{4}^{5} + T_{5}^{2}⟩  (1, 1, 1, 1, 3, 1)  32}^{22 𝕂[T_{1}, …, T_{6}]/⟨T_{1}T_{2}^{5} + T_{3}^{3}T_{4}^{3} + T_{5}^{2}⟩  (1, 1, 1, 1, 3, 1)  32}^{23 𝕂[T_{1}, …, T_{6}]/⟨T_{1}T_{2}^{7} + T_{3}T_{4}^{7} + T_{5}^{2}⟩  (1, 1, 1, 1, 4, 1)  2}^{24 𝕂[T_{1}, …, T_{6}]/⟨T_{1}T_{2}^{7} + T_{3}^{3}T_{4}^{5} + T_{5}^{2}⟩  (1, 1, 1, 1, 4, 1)  2}^{25 𝕂[T_{1}, …, T_{6}]/⟨T_{1}^{3}T_{2}^{5} + T_{3}^{3}T_{4}^{5} + T_{5}^{2}⟩  (1, 1, 1, 1, 4, 1)  2}^{26 𝕂[T_{1}, …, T_{6}]/⟨T_{1}T_{2}T_{3}T_{4}^{3} + T_{5}^{3} + T_{6}^{2}⟩ (1, 1, 1, 1, 2, 3)  81}^{27 𝕂[T_{1}, …, T_{6}]/⟨T_{1}T_{2}T_{3}^{2}T_{4}^{2} + T_{5}^{3} + T_{6}^{2}⟩ (1, 1, 1, 1, 2, 3)  81}^{28 𝕂[T_{1}, …, T_{6}]/⟨T_{1}T_{2}T_{3}T_{4}^{7} + T_{5}^{5} + T_{6}^{2}⟩ (1, 1, 1, 1, 2, 5)  1}^{29 𝕂[T_{1}, …, T_{6}]/⟨T_{1}T_{2}T_{3}^{2}T_{4}^{6} + T_{5}^{5} + T_{6}^{2}⟩ (1, 1, 1, 1, 2, 5)  1}^{30 𝕂[T_{1}, …, T_{6}]/⟨T_{1}T_{2}T_{3}^{3}T_{4}^{5} + T_{5}^{5} + T_{6}^{2}⟩ (1, 1, 1, 1, 2, 5)  1}^{31 𝕂[T_{1}, …, T_{6}]/⟨T_{1}T_{2}T_{3}^{4}T_{4}^{4} + T_{5}^{5} + T_{6}^{2}⟩ (1, 1, 1, 1, 2, 5)  1}^{32 𝕂[T_{1}, …, T_{6}]/⟨T_{1}T_{2}^{2}T_{3}^{2}T_{4}^{5} + T_{5}^{5} + T_{6}^{2}⟩ (1, 1, 1, 1, 2, 5)  1}^{33 𝕂[T_{1}, …, T_{6}]/⟨T_{1}T_{2}^{2}T_{3}^{3}T_{4}^{4} + T_{5}^{5} + T_{6}^{2}⟩ (1, 1, 1, 1, 2, 5)  1}^{34 𝕂[T_{1}, …, T_{6}]/⟨T_{1}T_{2}^{3}T_{3}^{3}T_{4}^{3} + T_{5}^{5} + T_{6}^{2}⟩ (1, 1, 1, 1, 2, 5)  1}^{35 𝕂[T_{1}, …, T_{6}]/⟨T_{1}^{2}T_{2}^{2}T_{3}^{3}T_{4}^{3} + T_{5}^{5} + T_{6}^{2}⟩ (1, 1, 1, 1, 2, 5)  1}^{36 𝕂[T_{1}, …, T_{6}]/⟨T_{1}T_{2}T_{3} + T_{4}T_{5}^{2} + T_{6}^{3}⟩ (1, 1, 1, 1, 1, 1)  243}^{37 𝕂[T_{1}, …, T_{6}]/⟨T_{1}T_{2}T_{3}^{2} + T_{4}T_{5}^{3} + T_{6}^{4}⟩ (1, 1, 1, 1, 1, 1)  64}^{38 𝕂[T_{1}, …, T_{6}]/⟨T_{1}T_{2}T_{3}^{3} + T_{4}T_{5}^{4} + T_{6}^{5}⟩ (1, 1, 1, 1, 1, 1)  5}^{39 𝕂[T_{1}, …, T_{6}]/⟨T_{1}T_{2}T_{3}^{3} + T_{4}^{2}T_{5}^{3} + T_{6}^{5}⟩ (1, 1, 1, 1, 1, 1)  5}^{40 𝕂[T_{1}, …, T_{6}]/⟨T_{1}T_{2}^{2}T_{3}^{2} + T_{4}T_{5}^{4} + T_{6}^{5}⟩ (1, 1, 1, 1, 1, 1)  5}^{41 𝕂[T_{1}, …, T_{6}]/⟨T_{1}T_{2}^{2}T_{3}^{2} + T_{4}^{2}T_{5}^{3} + T_{6}^{5}⟩ (1, 1, 1, 1, 1, 1)  5}^{42 𝕂[T_{1}, …, T_{6}]/⟨T_{1}T_{2}T_{3}^{2} + T_{4}T_{5}^{3} + T_{6}^{2}⟩ (1, 1, 1, 1, 1, 2)  162}^{43 𝕂[T_{1}, …, T_{6}]/⟨T_{1}T_{2}T_{3}^{4} + T_{4}T_{5}^{5} + T_{6}^{3}⟩ (1, 1, 1, 1, 1, 2)  3}^{44 𝕂[T_{1}, …, T_{6}]/⟨T_{1}T_{2}T_{3}^{4} + T_{4}^{2}T_{5}^{4} + T_{6}^{3}⟩ (1, 1, 1, 1, 1, 2)  3}

45 𝕂[T_{1}, …, T_{6}]/⟨T_{1}T_{2}^{2}T_{3}^{3} + T_{4}T_{5}^{5} + T_{6}^{3}⟩ (1, 1, 1, 1, 1, 2)  3^{46 𝕂[T_{1}, …, T_{6}]/⟨T_{1}T_{2}^{2}T_{3}^{3} + T_{4}^{2}T_{5}^{4} + T_{6}^{3}⟩ (1, 1, 1, 1, 1, 2)  3}^{47 𝕂[T_{1}, …, T_{6}]/⟨T_{1}^{2}T_{2}^{2}T_{3}^{2} + T_{4}T_{5}^{5} + T_{6}^{3}⟩ (1, 1, 1, 1, 1, 2)  3}^{48 𝕂[T_{1}, …, T_{6}]/⟨T_{1}T_{2}^{2}T_{3}^{3} + T_{4}^{3}T_{5}^{3} + T_{6}^{2}⟩ (1, 1, 1, 1, 1, 3)  32}^{49 𝕂[T_{1}, …, T_{6}]/⟨T_{1}T_{2}^{2}T_{3}^{3} + T_{4}T_{5}^{5} + T_{6}^{2}⟩ (1, 1, 1, 1, 1, 3)  32}^{50 𝕂[T_{1}, …, T_{6}]/⟨T_{1}T_{2}T_{3}^{4} + T_{4}^{3}T_{5}^{3} + T_{6}^{2}⟩ (1, 1, 1, 1, 1, 3)  32}^{51 𝕂[T_{1}, …, T_{6}]/⟨T_{1}T_{2}T_{3}^{4} + T_{4}T_{5}^{5} + T_{6}^{2}⟩ (1, 1, 1, 1, 1, 3)  32}^{52 𝕂[T_{1}, …, T_{6}]/⟨T_{1}T_{2}T_{3}^{6} + T_{4}T_{5}^{7} + T_{6}^{2}⟩ (1, 1, 1, 1, 1, 4)  2}^{53 𝕂[T_{1}, …, T_{6}]/⟨T_{1}T_{2}T_{3}^{6} + T_{4}^{3}T_{5}^{5} + T_{6}^{2}⟩ (1, 1, 1, 1, 1, 4)  2}^{54 𝕂[T_{1}, …, T_{6}]/⟨T_{1}T_{2}^{2}T_{3}^{5} + T_{4}T_{5}^{7} + T_{6}^{2}⟩ (1, 1, 1, 1, 1, 4)  2}^{55 𝕂[T_{1}, …, T_{6}]/⟨T_{1}T_{2}^{2}T_{3}^{5} + T_{4}^{3}T_{5}^{5} + T_{6}^{2}⟩ (1, 1, 1, 1, 1, 4)  2}^{56 𝕂[T_{1}, …, T_{6}]/⟨T_{1}T_{2}^{3}T_{3}^{4} + T_{4}T_{5}^{7} + T_{6}^{2}⟩ (1, 1, 1, 1, 1, 4)  2}^{57 𝕂[T_{1}, …, T_{6}]/⟨T_{1}T_{2}^{3}T_{3}^{4} + T_{4}^{3}T_{5}^{5} + T_{6}^{2}⟩ (1, 1, 1, 1, 1, 4)  2}^{58 𝕂[T_{1}, …, T_{6}]/⟨T_{1}^{2}T_{2}^{3}T_{3}^{3} + T_{4}T_{5}^{7} + T_{6}^{2}⟩ (1, 1, 1, 1, 1, 4)  2}^{59 𝕂[T_{1}, …, T_{6}]/⟨T_{1}^{2}T_{2}^{3}T_{3}^{3} + T_{4}^{3}T_{5}^{5} + T_{6}^{2}⟩ (1, 1, 1, 1, 1, 4)  2}^{60 𝕂[T_{1}, …, T_{6}]/⟨T_{1}T_{2} + T_{3}T_{4} + T_{5}T_{6}⟩  (1, 1, 1, 1, 1, 1)  512}^{61 𝕂[T_{1}, …, T_{6}]/⟨T_{1}T_{2}^{2} + T_{3}T_{4}^{2} + T_{5}T_{6}^{2}⟩ (1, 1, 1, 1, 1, 1)  243}^{62 𝕂[T_{1}, …, T_{6}]/⟨T_{1}T_{2}^{3} + T_{3}T_{4}^{3} + T_{5}T_{6}^{3}⟩ (1, 1, 1, 1, 1, 1)  64}^{63 𝕂[T_{1}, …, T_{6}]/⟨T_{1}T_{2}^{3} + T_{3}T_{4}^{3} + T_{5}^{2}T_{6}^{2}⟩ (1, 1, 1, 1, 1, 1)  64}^{64 𝕂[T_{1}, …, T_{6}]/⟨T_{1}T_{2}^{4} + T_{3}T_{4}^{4} + T_{5}T_{6}^{4}⟩ (1, 1, 1, 1, 1, 1)  5}^{65 𝕂[T_{1}, …, T_{6}]/⟨T_{1}T_{2}^{4} + T_{3}T_{4}^{4} + T_{5}^{2}T_{6}^{3}⟩ (1, 1, 1, 1, 1, 1)  5}^{66 𝕂[T_{1}, …, T_{6}]/⟨T_{1}T_{2}^{4} + T_{3}^{2}T_{4}^{3} + T_{5}^{2}T_{6}^{3}⟩ (1, 1, 1, 1, 1, 1)  5}^{67 𝕂[T_{1}, …, T_{6}]/⟨T_{1}^{2}T_{2}^{3} + T_{3}^{2}T_{4}^{3} + T_{5}^{2}T_{6}^{3}⟩ (1, 1, 1, 1, 1, 1)  5}^{68 𝕂[T_{1}, …, T_{7}]/ ⟨ \begin{matrix}T_{1}T_{2}+T_{3}T_{4}+T_{5}T_{6}, \\ αT_{3}T_{4}+T_{5}T_{6}+T_{7}^{2} \end{matrix} ⟩  (1, 1, 1, 1, 1, 1, 1) 324}^{69 𝕂[T_{1}, …, T_{7}]/ ⟨ \begin{matrix}T_{1}T_{2}^{2}+T_{3}T_{4}^{2}+T_{5}T_{6}^{2}, \\ αT_{3}T_{4}^{2}+T_{5}T_{6}^{2}+T_{7}^{3} \end{matrix} ⟩  (1, 1, 1, 1, 1, 1, 1) 9}

where in the last two rows of the table the parameter α can be any element from 𝕂^{∗} \ {1}.

By the result of [26], the singular quintics of this list are rational degenerations of smooth non-rational Fano fourfolds.

4 Geometry of the locally factorial threefolds

In this section, we take a closer look at the (factorial) singularities of the Fano varieties X listed in Theorem 3.2. Recall that the discrepancies of a resolution φ: X̃ → X of a singularity are the coefficients of KX̃ − φ^{∗}K_{X}, where K_{X} and KX̃ are canonical divisors such that KX̃ − φ^{∗}K_{X} is supported on the exceptional locus of φ. A resolution is called crepant, if its discrepancies vanish and a singularity is called canonical (terminal), if it admits a resolution with nonnegative (positive) discrepancies. By a relative minimal model we mean a projective morphism X̃ → X such that X̃ has at most terminal singularities and its relative canonical divisor is relatively nef.

Theorem 4.1. For the nine 3-dimensional Fano varieties listed in Theorem 3.2, we have the following statements.

(i) No. 4 is a smooth quadric in ℙ^{4}.

(ii) Nos. 1,3,5,7 and 9 are singular with only canonical singularities and all admit a crepant resolution.

(iii) Nos. 6 and 8 are singular with non-canonical singularities but admit a smooth relative minimal model.

(iv) No. 2 is singular with only canonical singularities, one of them of type cA_{1}, and admits only a singular relative minimal model.

The Cox ring of the relative minimal model X̃ as well as the the Fano degree of X itself are given in the following table.

No.  R(X̃)  (−K_{X})^{3}
1  𝕂[T_{1}, …, T_{14}]/(T_{1}T_{2}T_{3}^{2}T_{4}^{3}T_{5}^{4}T_{6}^{5} + T_{7}^{3}T_{8}^{2}T_{9} + T_{10}^{2}T_{11}⟩  8
2  𝕂[T_{1}, …, T_{9}]/⟨T_{1}T_{2}T_{3}^{2}T_{4}^{4} + T_{5}T_{6}^{2}T_{7}^{3} + T_{8}^{2}⟩  8
3  𝕂[T_{1}, …, T_{8}]/⟨T_{1}T_{2}^{2}T_{3}^{3} + T_{4}T_{5}^{3} + T_{6}T_{7}^{2}⟩  8
4  𝕂[T_{1}, …, T_{5}]/⟨T_{1}T_{2} + T_{3}T_{4} + T_{5}^{2}⟩  54
5  𝕂[T_{1}, …, T_{6}]/⟨T_{1}T_{2}^{2} + T_{3}T_{4}^{2} + T_{5}^{3}T_{6}⟩  24
6  𝕂[T_{1}, …, T_{6}]/⟨T_{1}T_{2}^{3} + T_{3}T_{4}^{3} + T_{5}^{4}T_{6}⟩  4
7  𝕂[T_{1}, …, T_{7}]/⟨T_{1}T_{2}^{3} + T_{3}T_{4}^{3} + T_{5}^{2}T_{6}⟩  16
8  𝕂[T_{1}, …, T_{7}]/⟨T_{1}T_{2}^{5} + T_{3}T_{4}^{5} + T_{5}^{2}T_{6}⟩  2
9  𝕂[T_{1}, …, T_{46}]/ ⟨ \begin{matrix}T_{1}T_{2}T_{3}T_{4}^{2}T_{5}^{2}T_{6}^{3}T_{7}^{3}T_{8}^{4}T_{9}^{4}T_{10}^{5} + \\ + T_{11}···T_{18}T_{19}^{2}···T_{24}^{2}T_{25}^{3}T_{26}^{3} + T_{27}···T_{32}T_{33}^{2} \end{matrix} ⟩  2

For the proof, it is convenient to work in the language of polyhedral divisors introduced in [1] and [2]. As we are interested in rational varieties with a complexity one torus action, we only have to consider polyhedral divisors on the projective line Y = ℙ^{1}. This considerably simplifies the general definitions and allows us to give a short summary. In the sequel, N ≅ ℤ^{n} denotes a lattice and M = Hom(N, ℤ) its dual. For the associated rational vector spaces we write N_{ℚ} and M_{ℚ}. A polyhedral divisor on the projective line Y := ℙ^{1} is a formal sum

D = ∑_{y∈Y} D_{y} · y,

where the coefficients D_{y} ⊆ N_{ℚ} are (possibly empty) convex polyhedra all sharing the same tail (i.e. recession) cone D_{Y} = σ ⊆ N_{ℚ}, and only finitely many D_{y} differ from σ. The locus of D is the open subset Y (D) ⊆ Y obtained by removing all points y ⊆ Y with D_{y} = ∅. For every u ∈ σ^{∨} ∩ M we have the evaluation

D(u) := ∑_{y∈Y} min_{v∈D_{y}}⟨u, v⟩·y,

which is a usual rational divisor on Y (D). We call the polyhedral divisor D on Y proper if deg D ⊊ σ holds, where the polyhedral degree is defined by

deg D := ∑_{y∈Y} D_{y}.

Every proper polyhedral divisor D on Y defines a normal affine variety X(D) of dimension rk (N) + 1 coming with an effective action of the torus T = Spec 𝕂[M]: set X(D) := Spec A(D), where

A(D) := ⨁_{u∈σ^{∨}∩M} Γ(Y (D), O(D(u))) ⊆ ⨁_{u∈M} 𝕂(Y) · χ^{u}.

A divisorial fan, is a finite set Ξ of polyhedral divisors D on Y, all having their polyhedral coefficients D_{y} in the same N_{ℚ} and fulfilling certain compatibility conditions, see [2]. In particular, for every point y ∈ Y, the slice

Ξ_{y} := {D_{y}; D ∈ Ξ}

must be a polyhedral subdivision. The tail fan is the set Ξ_{Y} of the tail cones D_{Y} of the D ∈ Ξ; it is a fan in the usual sense. Given a divisorial fan Ξ, the affine varieties X(D), where D ∈ Ξ, glue equivariantly together to a normal variety X(Ξ), and we obtain every rational normal variety with a complexity one torus action this way. Smoothness of X = X(Ξ) is checked locally. For a proper polyhedral divisor D on Y, we infer the following from [28, Theorem 3.3]. If Y (D) is affine, then X(D) is smooth if and only if cone({1} × D_{y}) ⊆ ℚ × N_{ℚ}, the convex, polyhedral cone generated by {1} × D_{y}, is regular for every y ∈ Y (D). If Y (D) = Y holds, then X(D) is smooth if and only if there are y, z ∈ Y such that D = D_{y}y + D_{z}z holds and cone({1} × D_{y}) + cone({−1} × D_{z}) is a regular cone in ℚ × N_{ℚ}. Similarly to toric geometry, singularities of X(D) are resolved by means of subdividing D. This means to consider divisorial fans Ξ such that for any y ∈ Y, the slice Ξ_{y} is a subdivision of D_{y}. Such a Ξ defines a dominant morphism X(Ξ) → X(D) and a slight generalization of [2, Thm. 7.5.] yields that this morphism is proper.

Proposition 4.2. The 3-dimensional Fano varieties No. 1-8 listed in Theorem 3.2 and their relative minimal models arise from divisorial fans having the following slices and tail cones.

1_{^{(−1,1) −_{\frac{4}{5}} −_{\frac{3}{4}} −_{\frac{2}{3}} −_{\frac{1}{2}}}  \frac{1}{3} \frac{1}{2} 1  1^{\frac{1}{2}}  ^{(0,1) (0,-1) (-1,6) (1,0)}}

2_{\begin{matrix}(− \frac{1}{4} , \frac{1}{2} ) \\ (0, \frac{1}{2} ) \\ (−1,1) \end{matrix}  \begin{matrix}(− \frac{2}{3} ,0) \\ (− \frac{1}{2} ,0) \end{matrix}  (1,− \frac{1}{2} )  ^{(-4,3) (1,0) (2,-2.6) (2,-1) (1,-1) (0,1) (-1,1)}}

3_{\begin{matrix}( \frac{1}{2} , \frac{1}{2} ) \\ ( \frac{2}{3} , \frac{1}{3} ) \end{matrix}  (0,− \frac{1}{3} )  (− \frac{1}{2} ,0)  ^{(-3,-2) (1,0) (0,1) (-1,0) (1,1) (0,-1)}}

4_{(−1,0)  (0,−1)  ( \frac{1}{2} , \frac{1}{2} )  ^{(-1,1) (1,1) (1,-1) (-1,-1)}}

5_{(− \frac{1}{2} ,0)  (0,− \frac{1}{2} )  ( \frac{1}{3} , \frac{1}{3} )  ^{(-1,2) (1,1) (2,-1) (-1,-1) (0,-1) (-1,0)}}

6_{(− \frac{1}{3} ,0)  (0,− \frac{1}{3} )  ( \frac{1}{4} , \frac{1}{4} )  ^{(-1,3) (1,1) (3,-1) (-1,-1) (0,-1) (-1,0)}}

7_{( \frac{2}{3} , \frac{1}{3} )  (0,− \frac{1}{3} )  (− \frac{1}{2} ,0)  ^{(1,2) (1,0) (-3,-2) (-1,0) (2,1) (0,-1)}}

8_{( \frac{3}{5} , \frac{1}{5} )  (0,− \frac{1}{5} )  (− \frac{1}{2} ,0)  ^{(1,2) (1,0) (-5,-2) (-1,0) (3,1) (0,-1)}}

The above table should be interpreted as follows. The first three pictures in each row are the slices at 0, 1 and ∞ and the last one is the tail fan. The divisorial fan of the fano variety itself is given by the solid polyhedra in the pictures. Here, all polyhedra of the same gray scale belong to the same polyhedral divisor. The subdivisions for the relative minimal models are sketched with dashed lines. In general, polyhedra with the same tail cone belong all to a unique polyhedral divisor with complete locus. For the white cones inside the tail fan we have another rule: for every polyhedron Δ ∈ Ξ_{y} with the given white cone as its tail there is a polyhedral divisor Δ · y + ∅ · z ∈ Ξ, with z ∈ {0, 1, ∞} \ {y}. Here, different choices of z lead to isomorphic varieties, only the affine covering given by the X(D) changes. In order to prove Theorem 4.1, we also have to understand invariant divisors on X = X(Ξ) in terms of Ξ, see [15, Prop. 4.11 and 4.12] for details. A first type of invariant prime divisors, is in bijection D_{y,v} ↔ (y, v) with the vertices (y, v), where y ∈ Y and v ∈ Ξ_{y} is of dimension zero. The order of the generic isotropy group along D_{y,v} equals the minimal positive integer μ(v) with μ(v)v ∈ N. A second type of invariant prime divisors, is in D_{ϱ} ↔ ϱ with the extremal rays ϱ ∈ Ξ_{Y}, where a ray ϱ ∈ Ξ_{Y} is called extremal if there is a D ∈ Ξ such that ϱ ⊆ D_{Y} and deg D ∩ ϱ = ∅ holds. The set of extremal rays is denoted by Ξ_{Y}^{×}. The divisor of a semi-invariant function f · χ^{u} ∈ 𝕂(X) is then given by

div(f · χ^{u}) = − ∑_{y∈Y} ∑_{v∈Ξ_{y}^{(0)}} μ(v) · (⟨v, u⟩ + ord_{y}f) · D_{y,v} − ∑_{ϱ∈Ξ_{Y}^{×}}⟨n_{ϱ}, u⟩ · D_{ϱ}.

Next we describe the canonical divisor. Choose a point y_{0} ∈ Y such that Ξ_{y_{0}} = Ξ_{Y} holds. Then a canonical divisor on X = X(Ξ) is given by

K_{X} = (s − 2) · y_{0} − ∑_{Ξ_{y}≠Ξ_{Y}} ∑_{v∈Ξ_{i}^{(0)}} D_{y,v} − ∑_{ϱ∈Ξ_{Y}^{×}} E_{ϱ}.

Proposition 4.3. Let D be a proper polyhedral divisor with Y (D) = ℙ_{1}, let Ξ be a refinement of D and denote by y_{1}, …, y_{s} ∈ Y the points with Ξ_{y_{i}} ≠ Ξ_{Y}. Then the associated morphism φ: X(Ξ) → X(D) satisfies the following.

(i) The prime divisors in the exceptional locus of φ are the divisors D_{y_{i},v} and D_{ϱ} corresponding to v ∈ Ξ_{y_{i}}^{(0)} \ D_{y_{i}}^{(0)} and ϱ ∈ Ξ_{Y}^{×} \ D^{×} respectively.

(ii) Then the discrepancies along the prime divisors D_{y_{i},v} and D_{ϱ} of (i) are computed as

d_{y_{i},v} = −μ(v) · (⟨v, u′⟩ + α_{y}) − 1, d_{ϱ} = −⟨v_{ϱ}, u′⟩ − 1,

where the numbers α_{i} are determined by

(\begin{matrix}−1 & −1 & … & −1 & 0 \\ μ(v_{1}^{1}) & 0 & … & 0 & μ(v_{1}^{1})v_{1}^{1} \\ ⋮ & ⋮ & & ⋮ & ⋮ \\ μ(v_{1}^{r_{1}}) & 0 & … & 0 & μ(v_{1}^{r_{1}})v_{1}^{r_{1}} \\ . & . & . \\ 0 & 0 & … & μ(v_{s}^{1}) & μ(v_{s}^{1})v_{s}^{1} \\ ⋮ & ⋮ & & ⋮ & ⋮ \\ 0 & 0 & … & μ(v_{s}^{r_{s}}) & μ(v_{s}^{r_{s}})v_{s}^{r_{s}} \\ 0 & 0 & … & 0 & n_{ϱ_{1}} \\ ⋮ & ⋮ & & ⋮ & ⋮ \\ 0 & 0 & … & 0 & n_{ϱ_{r}} \end{matrix} ) · (α_{y_{1}} ; ⋮ ; α_{y_{s}} ; u) = (2 − s ; 1 ; ⋮ ; 1 ; 1 ; ⋮ ; 1)

Proof. The first claim is obvious by the characterization of invariant prime divisors. For the second claim note that by [24, Theorem 3.1] every Cartier divisor on X(D) is principal. Hence, we may assume

ℓ · K_{X} = div(f · χ^{u}), div(f) = ∑_{y} α_{y} · y.

Then our formulæ for div(f · χ^{u}) and K_{X} provide a row for every vertex v_{i}^{j} ∈ Ξ_{y_{i}}, i = 0, …, s, and for every extremal ray ϱ_{i} ∈ Ξ^{×}, and ℓ^{−1}(α, u) is the (unique) solution of the above system.

Note, that in the above Proposition, the variety X(D) is ℚ-Gorenstein if and only if the linear system of equations has a solution.

Proof of Theorem 4.1 and Proposition 4.2. We exemplarily discuss variety number eight. Recall that its Cox ring is given as

R(X) = 𝕂[T_{1}, …, T_{5}]/(T_{1}T_{2}^{5} + T_{3}T_{4}^{5} + T_{5}^{2})

with the degrees 1, 1, 1, 1, 3. In particular, X is a hypersurface of degree 6 in ℙ(1, 1, 1, 1, 3), and the self-intersection of the anti-canonical divisor can be calculated as

(−K_{X}^{3}) = 6 · \frac{(1 + 1 + 1 + 1 + 3 − 6)^{3}}{1 · 1 · 1 · 1 · 3} = 2.

The embedding X ⊆ ℙ(1, 1, 1, 1, 3) is equivariant, and thus we can use the technique described in [1, Sec. 11] to calculate a divisorial fan Ξ for X. The result is the following divisorial fan; we draw its slices and indicate the polyhedral divisors with affine locus by colouring their tail cones D_{Y} ∈ Ξ_{Y} white:

Ξ_{0}^{( \frac{3}{5} , \frac{1}{5} )}  Ξ_{1}^{(0,− \frac{1}{5} )}  Ξ_{∞}^{(− \frac{1}{2} ,0)}  ^{Ξ_{Y}_{(1,2)}_{(1,0)}_{(-5,-2)}_{(-1,0)}}

One may also use [15, Cor. 4.9.] to verify that Ξ is the right divisorial fan: it computes the Cox ring in terms of Ξ, and, indeed, we obtain again R(X). Now we subdivide and obtain a divisorial fan having the refined slices as indicated in the following picture.

( \frac{3}{5} , \frac{1}{5} )  (0,− \frac{1}{5} )  (− \frac{1}{2} ,0)  ^{(1,2) (1,0) (-5,-2) (-1,0) (3,1) (0,-1)}

Here, the white ray ℚ_{≥0} · (1, 0) indicates that the polyhedral divisors with that tail have affine loci. According to [15, Cor. 4.9.], the corresponding Cox ring is given by

R(X̃) = 𝕂[T_{1}, …, T_{7}]/⟨T_{1}T_{2}^{5} + T_{3}T_{4}^{5} + T_{5}^{2}T_{6}⟩.

We have to check that X̃ is smooth. Let us do this explicitly for the affine chart defined by the polyhedral divisor D with tail cone D_{Y} = cone((1, 2), (3, 1)). Then D is given by

D = ((\frac{3}{5}, \frac{1}{5}) + σ) · \{0\} + ([−\frac{1}{2}, 0] × 0 + σ) · \{∞\}.

Thus, cone({1} × D_{0}) + cone({−1} × D_{∞}) is generated by (5, 3, 1), (−2, −1, 0) and (−1, 0, 0); in particular, it is a regular cone. This implies smoothness of the affine chart X(D). Furthermore, we look at the affine charts defined by the polyhedral divisors D with tail cone D_{Y} = cone(1, 0). Since they have affine locus, we have to check cone({1} × D_{y}), where y ∈ Y. For y ≠ 0, 1, we have D_{y} = D_{Y}. In this case, cone({1} × D_{y}) is generated by (1, 1, 0), (0, 1, 0) and thus is regular. For y = 0, we obtain that cone({1} × D_{y}) is generated by (5, 3, 1), (1, 0, 0), (0, 1, 0) and this is regular. For y = 1 we get the same result. Hence, the polyhedral divisors with tail cone D_{y} = cone(1, 0) give rise to smooth affine charts. Now we compute the discrepancies according to Proposition 4.3. The resolution has two exceptional divisors D_{∞,0} and E_{(1,0)}. We work in the chart defined by the divisor D ∈ Ξ with tail cone D_{Y} = cone((1, 2), (1, 0)). The resulting system of linear equations and its unique solution are given by

( \begin{matrix}−1 & −1 & −1 & 0 & 0 & −1 \\ 5 & 0 & 0 & 3 & 1 & 1 \\ 0 & 1 & 0 & 0 & 0 & 1 \\ 0 & 5 & 0 & 0 & −1 & 1 \\ 0 & 0 & 2 & −1 & 0 & 1 \end{matrix} ) ,  (\begin{matrix}α_{0} \\ α_{1} \\ α_{∞} \\ u \end{matrix}) = (\begin{matrix}0 \\ 1 \\ 0 \\ −1 \\ 4 \end{matrix}) .

The formula for the discrepancies yields d_{∞,0} = −1 and d_{(1,0)} = −2. In particular, X has non-canonical singularities. By a criterion from [24, Sec. 3.4.], we know that D_{∞,0} + 2 · E_{(1,0)} is a nef divisor. It follows that X̃ is a minimal model over X.

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Jürgen Hausen Mathematisches Institut Universität Tübingen Auf der Morgenstelle 10 72076 Tübingen Germany Elaine Herppich Mathematisches Institut Universität Tübingen Auf der Morgenstelle 10 72076 Tübingen Germany

Hendrik Süß

Institut für Mathematik LS Algebra und Geometrie BTU Cottbus PF 10 13 44 03013 Cottbus Germany 110

Documenta Mathematica 16 (2011)